Question: In a category \( \mathcal{C} \), let \( A \) and \( B \) be objects. If there exists a natural isomorphism \( \mathcal{C}(-,A) \cong \mathcal{C}(-,B) \), does it follow that \( A \cong B \)? Provide a justification for your answer.

Answer: A \cong B


Question: Find the interval of convergence for the Maclaurin series of the function \( \ln(\sin x / x) \).

Answer: (-\pi, \pi)


Question:
Find the number of solutions in natural numbers to the equation \(x_{1} + x_{2} + x_{3} + x_{4} + y_{1} + y_{2} = 6\), where \(0 \leq x_{i} \leq 2\) and \(y_{i}\) is divisible by 3, by calculating the coefficient of \(x^{6}\) in the generating function:
\[f(x) = (1 + x + x^{2})^{4} \left(\frac{1}{1 - x^{3}}\right)^{2}.\]

Answer: 45


Question: Evaluate the limit \( \lim_{z \to \infty} \left( \sqrt{z-2i} - \sqrt{z-i} \right) \), where \( z \) is a complex number.

Answer: 0


Question:
If the polynomial \( f(x) = ax^3 + bx - c \) is divisible by the polynomial \( g(x) = x^2 + bx + c \), determine the type of progression that the coefficients \( a, b, c \) are in. Choose from the following options:
1. Arithmetic Progression
2. Geometric Progression
3. Harmonic Progression
4. Arithmetic and Geometric Progression

Answer: 2


Question: Let $\mathbb{C}$ be the field of complex numbers and $\mathbb{Q}$ the field of rational numbers. Let $G$ be an additive subgroup of $\mathbb{Q}$. Consider a commutative algebra $R$ over $\mathbb{C}$ that is a domain and $G$-graded, with each $G$-graded space being 1-dimensional. Is it true that $R \cong \mathbb{C}[G]$, the group algebra over $G$?

Answer: \mathbb{C}[G]


Question:
Let \(a_{n}\) be a sequence with positive terms such that \( \lim_{n \to \infty} \frac{a_{n}}{n} = 0 \) and \( \limsup\limits_{n \to \infty}\frac{a_1 + a_2 + \cdots + a_n }{n} \in \mathbb{R}.\) Find the value of:
\[ \lim_{n \to \infty} \frac{a_1^2 + a_2^2 + \cdots + a_n^2}{n^2}. \]

Answer: 0


Question: A sphere with a radius of 1 is placed inside an inverted pyramid. The top of the pyramid is a horizontal square, and the other faces are equilateral triangles. Determine the distance from the center of the sphere to the apex of the pyramid.

Answer: \sqrt{3}


Question: Let $U_1, U_2, U_3, U_4, U_5$ be independent random variables, each uniformly distributed on the interval (0, 1). Let $R$ be the distance between the minimum and maximum of these variables. Find the expected value $E(R)$. Express your answer as a single number.

Answer: \dfrac{2}{3}


Question: Find the curvature of the curve defined by the parametric equation \( r(t) = (\cos(-3t), \sin(-3t), 5t) \) at \( t = \frac{\pi}{6} \).

Answer: \dfrac{9}{34}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_{0}^{n} \left(1-\frac{3x}{n}\right)^n e^{\frac{x}{2}} \, dx \]

Answer: \dfrac{2}{5}


Question: Let $A$ be an $n \times n$ matrix with rank $k$. Define the subspaces $L = \{B \in \mathcal{M}_{n \times n} : BA = 0 \}$ and $R = \{C \in \mathcal{M}_{n \times n} : AC = 0 \}$. Determine the dimensions of $L$ and $R$.

Answer: n(n - k)


Question: Evaluate the limit: $$\lim_{x\to 1} \frac{(1-x)(1-x^2)\cdots(1-x^{2n})}{[(1-x)(1-x^2)\cdots(1-x^n)]^2}$$

Answer: \dbinom{2n}{n}


Question: Calculate the limit: $$\lim_{x\to0}\frac{1}{x}\left(\sqrt[3]{\frac{1-\sqrt{1-x}}{\sqrt{1+x}-1}}-1\right)$$

Answer: \dfrac{1}{6}


Question: If $a, b, c > 0$ are integers such that $(a, b) = (b, c) = (a, c) = 1$, is it necessary that $(c^2, ab) = (ab, a^n - b^n) = (c^2, a^n - b^n) = 1$ for $n \geq 3$? Answer 'yes' or 'no' and justify your answer.

Answer: No


Question:
Determine whether the first-order predicate formula \((A \land B) \Rightarrow C\) is valid, where:

- \(A = \forall a (\neg P(a,a))\)
- \(B = \forall a \forall b \forall c (P(a,b)\wedge P(b,c)\Rightarrow P(a,c))\)
- \(C = \forall a \forall b (P(a,b)\Rightarrow \neg P(b,a))\)

A formula is valid if there is no truth assignment that makes it false.

Answer: Valid


Question: Given the equation \( \theta+\tan^{-1}\left(\frac{6}{4-x}\right)+\tan^{-1}\left(\frac{3}{x}\right)=\pi \), find \( \frac{d\theta}{dt} \) when \( x=4 \) and \( \frac{dx}{dt}=2 \text{ cm/sec} \).

Answer: -\dfrac{7}{75}


Question: Given the function \(f(x) = \lim_{n\to \infty}\dfrac{\left(1- \cos \left(1- \tan \left(\dfrac {\pi}{4}-x\right)\right)\right)(x+1)^n+ \lambda\sin((n - \sqrt{n^2 -8n})x)}{x^2(x+1)^n+x}\), where \(x \neq 0\), determine the value of \(f(0) + 2\lambda\) if \(f(x)\) is continuous at \(x=0\).

Answer: 3


Question: Given a least-squares problem \(||Ax-b||=0\) solved using a backward stable algorithm (QR decomposition with Householder projectors), and a condition number \(\kappa(A)=10^5\), determine how many digits of accuracy the solution \(x\) should have when using double-precision floating-point arithmetic (\(10^{-16}\)).

Answer: 11


Question: Reduce the equation \(6xy + 8y^2 - 12x - 26y + 11 = 0\) to its canonical form of a second-order curve. Identify the type of conic section and determine the necessary rotation angle and translation of the origin to achieve this form. Verify that the canonical form is \(\frac{X^2}{1} - \frac{Y^2}{9} = 1\), with a rotation angle of \(\arctan{3}\) and a translation to the point \((-1, 2)\).

Answer: \frac{X^2}{1} - \frac{Y^2}{9} = 1


Question:
Given the sequences \(y_t\) and \(z_t\) defined by the equations:
\[ y_t = 6y_{t-1} - 20z_{t-1} \]
\[ z_t = -17y_{t-1} - 12z_{t-1} \]
Derive a second-order difference equation of the form:
\[ y_{t+1} + ny_t + ry_{t-1} = 0 \]
where \(n\) and \(r\) are constants. The auxiliary equation is given by:
\[ m^2 + nm + r = 0 \]
with solutions \(m = h\) and \(m = k\), where \(h > k\). Calculate the value of \(h\) to two decimal places.

Answer: 17.52


Question: Given an irrational number \( r \in \mathbb{R} \), determine whether the set \( r\mathbb{Z} \) is dense in the topological space \( \mathbb{R}/\mathbb{Z} \).

Answer: Yes


Question: Evaluate the integral \( \int_0^{\sqrt{\pi / 2}} \left( \int_x^{\sqrt{\pi / 2}} \sin(y^2) \, dy \right) dx \).

Answer: \dfrac{1}{2}


Question:
Determine the Galois group of the polynomial \(x^4 + 8x + 12\) over \(\mathbb{Q}\) using its cubic resolvent and discriminant. The cubic resolvent is given by \(R_3(f) = x^3 - 12x - 64\). Calculate the discriminant of \(R_3(f)\) and use the following criteria to determine the Galois group:

- \(\text{Gal}(f) \cong A_4\) if \(R_3(f)\) is irreducible and \(\text{disc}(R_3)\) is a square.
- \(\text{Gal}(f) \cong S_4\) if \(R_3(f)\) is irreducible and \(\text{disc}(R_3)\) is not a square.
- \(\text{Gal}(f) \cong V_4\) if \(R_3(f)\) is reducible and \(\text{disc}(R_3)\) is a square.
- \(\text{Gal}(f) \cong D_4\) or \(\mathbb{Z}/4\mathbb{Z}\) if \(R_3(f)\) is reducible and \(\text{disc}(R_3)\) is not a square.

Given that \(\text{disc}(R_3) = -103700\), determine the irreducibility of \(R_3(f)\) and conclude the Galois group of \(x^4 + 8x + 12\).

Answer: S_4


Question: Evaluate the limit: \[ \lim_{n\to+\infty} \prod_{k=1}^n \left(1+\frac{k}{n^2}\right). \]

Answer: \sqrt{e}


Question: Find the coefficient of \(x^4\) in the expansion of \((1 + 3x + 2x^3)^{12}\).

Answer: 40887


Question: Find the limit of the sequence \( n!\left(\frac{e}{n}\right)^n \) as \( n \to \infty \).

Answer: \infty


Question: Determine the number of real zeros of the function \( f(x) = \sec(x) - e^{-x^2} \).

Answer: 1


Question: Determine the number of distinct real solutions to the equation \(f(f(x))=0\), where \(f(x) = x^3 - 3x + 1\).

Answer: 7


Question: Evaluate the limit: $$\lim _{x\to \,1} \frac{\left(1-x^{1/2}\right)\left(1-x^{1/3}\right)\cdots \left(1-x^{1/n}\right)}{(1-x)^{n-1}}$$

Answer: \dfrac{1}{n!}


Question: Determine the number of orders on the rational numbers \( \mathbb{Q} \) that are compatible with the standard field operations. Is this number finite, infinite, or uncountably infinite?

Answer: 1


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{1-\cos 5x}{\cos 6x-1} \] without using L'Hopital's Rule.

Answer: -\dfrac{25}{36}


Question: Calculate \( \lim_{n \to \infty} I_n \), where \( I_n = \int_{0}^{n} \sqrt[n]{x} \cdot e^{-x} \, dx \). Use the dominated convergence theorem to justify the interchange of limit and integration.

Answer: 1


Question: Find \( \lim_{n \to \infty} \left( \sqrt[3]{n^3+1} - \sqrt{n^2+1} \right) \).

Answer: 0


Question: Find the tightest upper bound for the expression \(\prod_{i = 1}^k A_i!\) given that \(\sum_{i=1}^k A_i \leq C\). Express your answer in terms of \(C\) and \(k\).

Answer: C!


Question: Determine the cardinality of any 3-Sylow subgroup in the group of all invertible $4 \times 4$ matrices with entries in the field of 3 elements.

Answer: 729


Question: Evaluate the limit: $$ \lim_{m \to 0}\left(\frac{1}{m} \int_{\sqrt{\frac{\pi}{2}}}^{\sqrt{\frac{\pi}{2}}+m}\sin(x^2)\,dx\right) $$

Answer: 1


Question: Let \( R = \mathbb{Z}[\sqrt{-5}] \) and \( I = (2, 1+\sqrt{-5}) \). Prove that \( I^2 = (2) \) in \( R \).

Answer: I^2 = (2)


Question: Given the differential equation \( \frac{dP}{dt} = 0.8P \left(1 - \frac{P}{350}\right) \), where \( P \) is the number of people infected per week after the outbreak of a disease, and knowing that at the outbreak, 7 people were infected, determine the initial condition for \( P \).

Answer: P(0) = 7


Question: A fair coin is flipped 100 times in a row. For each flip, if it comes up heads, you win $2; if it comes up tails, you lose $1. You start with $50, and if you run out of money, you must stop prematurely. If you don't run out of money, you stop after 100 flips. What is the expected value of your total money at the end of the game?

Answer: 100


Question: Determine whether the following statement is true or false: Every sequence of sets has a converging subsequence, where a sequence of sets \(A_1, A_2, A_3, \ldots\) is said to converge if \(\limsup_{n \to \infty} A_n = \liminf_{n \to \infty} A_n\).

Answer: B


Question: Determine the value of \( g(z) = \int_C \frac{2s^2-s-2}{s-z} \, ds \) when \( |z| > 3 \), where \( C \) is the circle \( |s| = 3 \). Use the Cauchy Integral Theorem and Cauchy-Goursat Theorem to justify your answer.

Answer: 0


Question: Find the maximum value of distinct integers \(a\) and \(b\) such that both \(x^a + 1\) and \(x^b + 1\) are factors of the polynomial \(1 + x + x^2 + \dots + x^{255}\). Determine the maximum value of \(a + b\).

Answer: 192


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{1}{x} \int_{0}^{x} (1+ \sin(t))^{1/t} \, dt. \] Justify the use of L'Hospital's Rule in this context, considering the behavior of \((1 + \sin(t))\) at \(t = 0\).

Answer: e


Question: Calculate \( \lim_{n\to\infty} n^{\gamma} \int_{-1/n}^{1/n} |f(x)| \, dx \). Assume \( p > 1 \), \( \gamma \leq \frac{p-1}{p} \), and \( f \in L^p((-1/2, 1/2)) \).

Answer: 0


Question: Evaluate the complex integral \( \int_{\gamma} {\frac{z}{\overline z}}dz \) where \( \gamma \) is the edge of the region \( \{1 < |z| < 2 \text{ and } \Im (z) > 0\} \).

Answer: \dfrac{4}{3}


Question: Calculate \( f^{(2001)}(0) \) for the function \( f(x) = e^{-x}\sin(x) \).

Answer: 2^{1000}


Question: Determine the number of conjugacy classes in the set $A = \{\mu \in S_8: |\mu| = 6\}$, where $S_8$ is the symmetric group on 8 elements.

Answer: 5


Question: Determine whether the following statement is true: If all eigenvalues of the Jacobian matrix \(\frac{\partial f}{\partial x}(x_0)\) have negative real parts, then the equilibrium point \(x_0\) of the autonomous system \(\dot x = f(x)\) is exponentially stable.

Answer: True


Question: Two particles are positioned at the vertices of a cube. If at a given time $t$ the particles are on the same edge, they remain in the same position up to time $t + 1$. Otherwise, one of them is chosen at random with equal probability, and at time $t + 1$ it is moved to one of the 3 neighboring vertices with equal probability. If the particles start at opposite vertices of the same face, what is the expected value of the least time they are on the same edge?

Answer: 2


Question: Find all positive integer solutions \((x, y)\) to the equation \(7^x - 3^y = 4\).

Answer: (1, 1)


Question: Evaluate the line integral \( \oint_C \left(3y^2 + e^{\cos x}\,dx\right) + \left(\sin y + 5x^2\,dy\right) \), where \( C \) is the boundary of the region in the upper half-plane \( (y \ge 0) \) between the circles \( x^2 + y^2 = a^2 \) and \( x^2 + y^2 = b^2 \), with \( 0 < a < b \). Use Green's Theorem to determine the value of the integral.

Answer: -4(b^3 - a^3)


Question: Derive the conditional density of $X_1$ given $X_1 + X_2 = y$, where $X_1$ and $X_2$ are independent and identically distributed random variables with densities $f_{X_1}(x_1) = e^{-x_1}$ and $f_{X_2}(x_2) = e^{-x_2}$, respectively.

Answer: \dfrac{1}{y}


Question: Determine the isomorphism type of the quotient group \( \mathbb{Z} \times \mathbb{Z} / \langle(1,1)\rangle \) using the Fundamental Theorem of Finitely Generated Abelian Groups.

Answer: \mathbb{Z}


Question: Evaluate \( \lim_{n \rightarrow \infty} \int_0^n \left(1 - \frac{x}{n}\right)^n \cos\left(\frac{x}{\sqrt{n}}\right) e^{x/2} \, dx \).

Answer: 2


Question: Let $f(x)$ be a quadratic polynomial with leading coefficient 1, satisfying the equation \( f(f(x) + x) = f(x)(x^2 + 786x + 439) \) for all real numbers \( x \). Determine the value of \( f(3) \).

Answer: 2015


Question: Given the inequality $4a^2 + 3ab + b^2 \leq 2016$, find the maximum value of $a + b$ where $a$ and $b$ are real numbers.

Answer: 48


Question:
Let $a > 1$ be a real number that is not an integer, and let $k$ be the smallest positive integer such that $[a^k] > [a]^k$, where $[r]$ denotes the largest integer less than or equal to $r$. Determine which of the following statements is always true:

A) $k \leq 2([a]+1)^2$
B) $k \leq ([a]+1)^4$
C) $k \leq 2^{([a]+1)}$
D) $k \leq \frac{1}{(a-[a])}+1$

Answer: D


Question: Determine the number of roots of the equation \( 2\cos^{2} \theta = 1 + 3^{\sin \theta} + 3^{\csc \theta} \) in the interval \([0, 2\pi]\).

Answer: 2


Question:
Determine whether the sequence defined by the function \( f(n) \) reaches zero for any finite number of iterations when starting with \( n = 53 \). The function is defined as follows:
\[
f(n) = \begin{cases}
(n-1)^2, & \text{if } n \equiv 1 \pmod{4} \\
\lfloor n/4 \rfloor, & \text{otherwise}
\end{cases}
\]
Does \( f^k(53) = 0 \) for any finite \( k \)?

Answer: No


Question: Calculate the contour integral \( \oint_C f(z) \, dz \) using residue theory, where \( f(z) = \exp(-1/z) \sin(1/z) \) and \( C \) is the contour defined by \( |z| = 1 \).

Answer: 2\pi i


Question: Suppose you sample 12 observations randomly from a uniform distribution on the interval (0,2). Calculate the approximate probability that the average of your sample will be less than 0.5.

Answer: 0.0013


Question: Find the minimum value of the function \( f(x,y,z) = z \) subject to the constraints \( x + y + z = 1 \) and \( x^2 + y^2 = 1 \).

Answer: 1 - \sqrt{2}


Question: Estimate the combinatorial sum \( \sum _{i=0}^{cn} {cn\choose {i}}{(1-c)n \choose t-i} 2^{t-i} \) as \( n \to \infty \). Can it be bounded by \( 2^{c'n} \) for some \( 0 < c' < \log_2(3) \)?

Answer: Yes


Question: What is the minimum number of evaluations needed to determine all coefficients of a polynomial \( p(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where all coefficients \( a_i \) are positive integers, by evaluating the polynomial at chosen points? Provide a method or reasoning for your answer.

Answer: 2


Question: Let $X$ and $Y$ be metric spaces, where $Y$ is complete. Suppose there exists a homeomorphism $f: X \to Y$ that is uniformly continuous. Is it true that $X$ must also be complete? Provide a justification for your answer.

Answer: Yes


Question: Find a lower bound for the expression \( \operatorname{Tr} \bigl( \bigl( (A+I )^{-1} \bigr)^2\bigr) \), where \( A \in \mathbb{R}^{n \times n} \) is a symmetric positive semi-definite matrix with \( \operatorname{Tr}(A) \le n \) and \( I \) is the identity matrix. Is it true that \( \operatorname{Tr} \bigl( \bigl( (A+I )^{-1} \bigr)^2\bigr) \ge \frac{1}{4} \)? Justify your answer.

Answer: \dfrac{1}{4}


Question:
Given a graph $G=(V, E)$, the probability of the event that there exists a subgraph $A$ with at most $k$ vertices and Euclidean diameter at least $Ck$ is at least $p$. Determine the complement probability of this event. Is it given by:
\[P\left(\begin{array}{l}
\text{for any subgraphs } A \text{ with at} \\
\text{most $k$ vertices and Euclidean} \\
\text{diameter at least } Ck \end{array} \right) \le 1-p\text{?}\]

Answer: Yes


Question: Let $H$ be a Hilbert space and $H' \le H$ a subspace such that the inner product in $H'$ is the same as the inner product of $H$ restricted to $H'$. Consider an automorphism $f: H \to H$ of Hilbert spaces that fixes $H'$ pointwise. If $a, b \in H$ satisfy $\|a\| = \|b\|$ and $\text{dist}(a, H') = \text{dist}(b, H')$, is there an automorphism $f: H \to H$ which fixes $H'$ pointwise and maps $a$ to $b$?

Answer: No


Question:
Let $V$ be a real inner product space with an orthogonal basis $B = \{v_1, v_2, v_3\}$ where $\|v_i\| = 1$ for all $i$, $\langle v_1, v_2 \rangle = \langle v_1, v_3 \rangle = 0$, and $\langle v_2, v_3 \rangle = \frac{1}{2}$. Consider the operator $T$ on $V$ with the matrix representation $_B(T)_B = \begin{pmatrix} \alpha & 0 & 0 \\ 0 & \beta & 0 \\ 0 & 0 & \gamma \end{pmatrix}$. Determine which of the following statements is correct:

1. $T$ is self-adjoint for all $\alpha, \beta, \gamma \in \mathbb{R}$.
2. If $\alpha = 0$, there exist $\beta, \gamma$ such that $T$ is orthogonal.
3. If $\alpha = 1$, then $T$ is orthogonal if and only if $|\beta| = |\gamma| = 1$.
4. $T$ is self-adjoint if and only if $\beta = \gamma$. (correct answer)
5. Given $\alpha \in \mathbb{R}$, there exist $\beta, \gamma \in \mathbb{R}$, $\beta \neq \gamma$, such that $T$ is self-adjoint.

Choose the correct option.

Answer: 4


Question:
Let $G$ be a simple group of order 60. Determine which of the following statements is correct:

1. $G$ has six Sylow 5-subgroups.
2. $G$ has four Sylow 3-subgroups.
3. $G$ has a cyclic subgroup of order 6.
4. $G$ has a unique element of order 2.

Note: $60 = 2^2 \cdot 3 \cdot 5$. Consider the divisibility conditions for the number of Sylow subgroups and the properties of simple groups.

Answer: 1


Question: A sphere with a radius of 17 has a cylindrical hole with a radius of 8 drilled through its center. Calculate the volume of the resulting solid.

Answer: 4500\pi


Question: Evaluate the infinite series: \[ \sum_{n=1}^{\infty} \frac{1}{n!(2^n)}. \]

Answer: \sqrt{e} - 1


Question: Given that $X$ and $Y$ are jointly distributed random variables such that $Y|X=x \sim \text{Bin}(n,x)$ and $X \sim U(0,1)$, and knowing that $E[Y]= \frac{n}{2}$ and $\text{Var}(Y)=\frac{n^2}{12}+\frac{n}{6}$, find the covariance $\text{cov}(X,Y)$ without using techniques from the Bayesian section of the chapter.

Answer: \dfrac{n}{12}


Question: Find the multiplicative inverse of 95 modulo 101.

Answer: 84


Question: Find the function \( f(x) \) that satisfies the functional equation \[ f(x) = \frac{1}{2}\left[f(xy)+f\left(\frac{x}{y}\right)\right] \] for all \( x \in \mathbb{R^{+}} \), given that \( f(1)=0 \) and \( f'(1) = 2. \)

Answer: 2 \ln x


Question: Determine whether the following statement is true or false: If \( f_n(x) \) is a bounded sequence in \((C[0,1], || \cdot ||_\infty)\) for all \( n \), then there exists a subsequence \( f_{n_k}(x) \) such that \( f_{n_k}(x) \) converges uniformly in \((C[0,1], || \cdot ||_\infty)\).

Answer: B


Question: Let \(f(x)\) be a real-valued twice differentiable function on the interval \([1,5]\) such that \(f(1) = f(5) = 0\) and \(f(x) = f'(x) + f''(x)\) for all \(x \in [1,5]\). Find the value of \(f(2) + f(4) - f'(3)\).

Answer: 0


Question: Determine whether there exist non-zero integers $a$ and $b$ such that $a^2 = 2b^2$. Provide a justification for your answer.

Answer: No


Question: Evaluate the value of the integral: \[ \int_0^1 e^{x^2} \, dx + \int_1^e \sqrt{\ln x} \, dx \]

Answer: e


Question: Use Green's Theorem to evaluate the line integral \( \int_C (e^x+\cos(x)+2y)\,dx+(2x-\frac{y^2}{3})\,dy \), where \( C \) is the boundary of the ellipse given by \( \frac{(x-2)^2}{49}+\frac{(y-3)^2}{4}=1 \).

Answer: 0


Question:
Find the limit of the function as \( x \to 0 \):
\[ \lim_{x\rightarrow 0} \frac{a^{x} b^{x} - b^{x} - a^{x} + 1}{x^{2}} \]

Answer: (\ln a)(\ln b)


Question: Calculate the line integral \( \oint_C y^2\,dx + 3xy\,dy \) using Green's Theorem, where \( C \) is the semicircular region defined by \( x^2 + y^2 \leq 1 \) and \( y \geq 0 \).

Answer: \dfrac{2}{3}


Question: Let $f$ be a holomorphic function in the open ball $B(0,R)$ such that $|f(z)| \leq e^{-\frac{1}{|z|}}$ for all $0<|z|<R$. Prove that $f(z) = 0$ for all $z$ in $B(0,R)$.

Answer: 0


Question: Let $V$ be a finite-dimensional complex vector space and $G \subset GL(V)$ be a finite group. Consider the quotient variety $V/G$. Determine the codimension of the singular locus in $V/G$.

Answer: 2


Question: Determine the number of roots (counting multiplicity) of the polynomial \( p(z) = z^5 + 11z^2 - 4z - 2 \) inside the open ball \( B_2(0) \) in the complex plane.

Answer: 2


Question: Determine the number of elements of order 35 in the group \( \mathbb{Z}_{15} \times \mathbb{Z}_{35} \).

Answer: 144


Question: Can a countable infinite vector space over the field \( \mathbb{C} \) have a finite basis? Justify your answer.

Answer: No


Question: Find the integer solutions to the equation: \[ 3\sqrt{x + y} + 2\sqrt{8 - x} + \sqrt{6 - y} = 14 \]

Answer: (4, 5)


Question: Given a stationary ARMA(1,1) process \(x_t = 0.7 x_{t-1} + u_t + 0.2 u_{t-1}\), where \(u_t\) is white noise with standard deviation \(\sigma(u_t) = 4\), find \( \Bbb Var(x_{t+1}|x_t, u_t)\).

Answer: 16


Question: Find the positive value of $k$ such that the equation $ke^x - x = 0$ has exactly one real solution.

Answer: \dfrac{1}{e}


Question: Let $M$ be a smooth manifold and $Z$ a nowhere zero vector field on $M$, meaning $Z(p) \neq 0$ for all $p \in M$. Does there exist a one-form $\gamma \in \Omega^1(M)$ such that $\gamma(Z) = 1$?

Answer: Yes


Question: Given a function $f(x)$ such that $f(x+y) = f(x)f(y)$, $f(5) = 2$, and $f'(0) = 3$, find the value of $f'(5)$. Provide your answer as a single number.

Answer: 6


Question: Determine the number of solutions for the equation \( e^z - z = a \) where \( a \) is a complex number with \( \text{Re}(a) > 1 \), and \( \text{Re}(z) < 0 \).

Answer: 1


Question: Let \( f(z) = f(x+iy) = u(x,y) + iv(x,y) \) be an entire function. Determine the values of \( a \) and \( b \) such that the function \( g(z) = au^2(x,y) - bv^2(x,y) \) is harmonic.

Answer: a = b


Question: Determine the number of solutions to the equation \( \underbrace{\cos(\cos(\ldots(\cos x)\ldots))}_{n\text{ times }\cos}=x \) in the interval \((0, \frac{\pi}{2})\), where \(n\) is a positive integer.

Answer: 1


Question: Determine the number of group homomorphisms from \(\Bbb Z_3\) to \(\text{Aut}(\Bbb Z_7)\).

Answer: 3


Question: Find the value of the expression \(mn + 11m + 11n - m^2 - n^2\) that allows for the unique identification of a pair of integers \((m, n)\).

Answer: 121


Question: Determine the number of solutions to the equation \((\arcsin x)^3 + (\arccos x)^3 = a \pi^3\) given that \(a < \frac{1}{32}\).

Answer: 0


Question: Prove that every differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $(2h)f'(x) = f(x+h) - f(x-h)$ for all $x \in \mathbb{R}$ and all $h$ has the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Answer: f(x) = ax^2 + bx + c


Question: Determine whether the vector field $F(x,y,z) = (yx, x+z, yz)$ is conservative. Consider the surface $S = \{(x,y,z): y = x^2 + z^2, y \in [0,1] \}$ and compute the line integral \( \oint_{\partial S} F \cdot dr \). Is $F$ conservative?

Answer: 0


Question:
Consider a function \( f: \mathbb{R} \to \mathbb{R} \). Which of the following subsets of \( \mathbb{R} \) is impossible to be the set of points where \( f \) is discontinuous?

A. Empty set
B. Rational numbers
C. Irrational numbers
D. Positive real numbers
E. \( \mathbb{R} \)

Answer: C


Question: Determine the spacing $h$ in a table of equally spaced values of the function \( f(x) = \sqrt{x} \) on the interval \([1,2]\), such that interpolation with a quadratic polynomial yields an accuracy of \(5 \times 10^{-8}\).

Answer: \dfrac{1}{79}


Question: Does there exist an infinite \(\sigma\)-algebra that contains only countably many members? Justify your answer.

Answer: No


Question: Determine whether the following statement is true: \[ \lim_{\varepsilon\to0}\frac{f(g(x+\varepsilon),x+\varepsilon)-f(g(x+\varepsilon),x)}{\varepsilon}=(\partial_2f)_{(g(x),x)} \] where $f:\mathbb{R}^2\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ are continuously differentiable functions.

Answer: True


Question: Determine the zero divisors of the ring of Gaussian integers $\mathbb{Z}[i]$. Provide a set of elements that satisfy the zero divisor condition.

Answer: \emptyset


Question: Is the space $C_c^\infty(\Omega)$ dense in $W^{1,p}(\Omega) \cap C_0(\Omega)$ for a bounded open set $\Omega \subset \mathbb{R}^N$ and $1 \leq p < \infty$? Here, $C_0(\Omega)$ denotes the space of continuous functions on $\overline{\Omega}$ that are zero on $\partial\Omega$, and $W^{1,p}(\Omega) \cap C_0(\Omega)$ is equipped with the norm $\|\cdot\|_{W^{1,p}} + \|\cdot\|_{L^\infty}$. Provide a justification for your answer.

Answer: Yes


Question: Determine the number of distinct ways to arrange 6 orange marbles and 6 red marbles in a circle, considering rotations of the same arrangement as identical. Assume marbles of the same color are indistinguishable.

Answer: 80


Question: Determine whether there exist uncountable subsets of the real line that can be covered by arbitrarily small intervals, according to the definition of $S$-small sets.

Answer: Yes


Question: Find a closed form expression for the determinant of the Hankel matrix formed by the sequence \( \{ 1, a, a^2, \dots, a^{2n+2} \} \).

Answer: 0


Question: Consider a function \( f: X \to \mathbb{R} \) where \( X \) is a subset of \( \mathbb{R} \), and let \( x_0 \) be a limit point of \( X \). If \( f \) is monotone increasing and differentiable at \( x_0 \), prove that \( f'(x_0) \geq 0 \).

Answer: f'(x_0) \geq 0


Question: Find all real functions $F(x)$ such that for any real numbers $x_1$ and $x_2$, the inequality \( F(x_1) - F(x_2) \le (x_1 - x_2)^2 \) holds.

Answer: F(x) = C


Question: Determine the number of points \((x, y)\) over the finite field \(\mathbb{F}_p\), where \(p\) is a prime number such that \(p \equiv 2 \pmod{3}\), that satisfy the equation \(y^2 = x^3 + 1\).

Answer: p


Question: Compute the limit \( \lim_{x \to \infty} \frac{\log x}{x^{1/k}} \) for an arbitrary positive constant \( k \).

Answer: 0


Question: Find the Fréchet derivative of the functional $\|Au-f\|^2$, where $A$ is a bounded linear operator on an infinite-dimensional Hilbert space. Express your answer in terms of $A$, $u$, and $f$.

Answer: 2A^*(Au - f)


Question: Consider a continuous and surjective map \( \gamma: [0,T] \rightarrow [0,1] \times [0,1] \). Compute the winding number of \( \gamma \) around a point \( a \) where \( a \notin \text{Im}(\gamma) \).

Answer: 0


Question: Let $m$ and $n$ be positive integers. Suppose we have $m$ homogeneous polynomial equations with complex coefficients, where each of the $n$ variables appears at least once in all the polynomials, and no other variables are present. If $m < n$, do these $m$ equations always have at least one non-trivial common root in the field of complex numbers, where "non-trivial" means not all variables are zero?

Answer: Yes


Question: Let \( f \in C^{\infty}(\mathbb{R}^2) \) be a smooth function with compact support. Suppose \( \text{supp}(f) \subseteq [a,b] \times [c,d] \) and \( \varepsilon > 0 \). Can \( f(x,y) \) be approximated by a sum of the form \( \sum_{i=1}^m g_i(x) h_i(y) \), where \( g_i, h_i \in C^{\infty}(\mathbb{R}) \) are smooth with compact support, such that \( \text{supp}(g_i) \subseteq [a,b] \) and \( \text{supp}(h_i) \subseteq [c,d] \), and \( \| f(x,y) - \sum_{i=1}^m g_i(x) h_i(y) \|_\infty < \varepsilon \)?

Answer: Yes


Question: Determine whether the subset \( \mathcal{E} = \{V \in GL_n(\mathbb{R}): V = (v_1, \dots, v_r, *, \dots, *)\} \), where \( \{v_1, \dots, v_r\} \) is a fixed linearly independent set in \( \mathbb{R}^n \) with \(1 \le r < n\) and \(*\) represents any vector not in the span of \( \{v_1, \dots, v_r\} \), has two or more connected components.

Answer: Yes


Question: Evaluate the integral \( \int_0^{\pi/2}\frac{\sin^3t}{\sin^3t+\cos^3t}dt \).

Answer: \dfrac{\pi}{4}


Question: Find the minimum value of \( \cot^2 A + \cot^2 B + \cot^2 C \) in a triangle \( \triangle ABC \), where \( A + B + C = \pi \).

Answer: 1


Question: Evaluate the integral \( \int_0^\pi \frac{\sin\frac{21x}{2}}{\sin \frac x2} \, dx \).

Answer: \pi


Question: Evaluate the integral \( \int_0^\infty \frac{dx}{(1+x^2)^2} \).

Answer: \dfrac{\pi}{4}


Question: Find the residue of the function \( z^4 e^{\frac{1}{z-1}} \) at \( z = 1 \).

Answer: \dfrac{167}{40}


Question: Let \( F \) be a subfield of the reals, and let \( A \) and \( B \) be matrices with entries in \( F \) such that \( \det A < 0 \) and \( \det B > 0 \). Must there exist an element \( t \in F \) such that \( \det (tA + (1-t)B) = 0 \)?

Answer: No


Question: Evaluate the limit: \[ \lim_{n\to\infty} \frac{\cot{\frac{2}{n}}+n\csc{\frac{3}{n^3}}}{\csc{\frac{3}{n}} + n\cot{\frac{2}{n^2}}} \]

Answer: \infty


Question: In a 5vs5 game, what is the probability that Player 1 will achieve an ACE (all 5 kills)? Assume that Player 1 wins each gunfight with a probability of 0.5. Calculate the probability that Player 1 will get all 5 kills in this game.

Answer: \dfrac{1}{32}


Question:
Maximize the sum $x + y + z$ subject to the following constraints:
\[
\begin{align*}
2x + y + z & \leq 66, \\
x + 2y + z & \leq 60, \\
x + y + 2z & \leq 70, \\
x + 2y + 3z & \leq 110, \\
3x + y + 2z & \leq 98, \\
2x + 3y + z & \leq 89.
\end{align*}
\]
What is the maximum possible value of $x + y + z$?

Answer: 48


Question: If $T : B \rightarrow B$ is a linear continuous operator and $B$ is a Banach space, is $T$ a closed operator?

Answer: Yes


Question: Evaluate the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \) for the portion of the cylinder \( x^2 + y^2 = 1 \) between the planes \( z = 1 \) and \( z = 4 \).

Answer: 6\pi


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{1}{x^2} \int_0^x \frac{t + t^2}{1 + \sin(t)} \, dt \]

Answer: \dfrac{1}{2}


Question: Suppose 20 real numbers are chosen independently from the interval [0, 20] with uniform probability. Find a lower bound for the probability that their average lies between 8 and 12.

Answer: \dfrac{22}{27}


Question: Determine the values of $n \geq 2$ for which there exists a permutation $\sigma$ in the alternating group $A_n$ such that the order of $\sigma$ is greater than $n$.

Answer: n \geq 8


Question: Solve the equation \(x = e^{-1/x}\) for \(x\) analytically.

Answer: -\frac{1}{W(-1)}


Question: Evaluate the integral \( \int_{0}^{\infty} \frac{1}{\cosh(x)} \, dx \) using contour integration.

Answer: \dfrac{\pi}{2}


Question: Let \( \gcd(p, q) = 1 \) and consider the set \( Y = \{(t^p, t^q) \in \mathbb{C}^2 \} \). Determine the ideal \( I(Y) \) in \( \mathbb{C}[x,y] \).

Answer: (x^q - y^p)


Question: Given that \( \int^{\infty}_{-\infty}e^{tx}f(x)dx = \sin^{-1}\left(t-\sqrt{\frac{1}{2}}\right) \), find the value of \( \left(\int^{\infty}_{-\infty}xf(x)dx\right)^2 \).

Answer: 2


Question: Determine the radius of convergence of the power series \( \sum_{k=0}^\infty \frac{(-1)^k}{k!(n+k)!}\frac{z^{n+2k}}{2^{n+2k}} \), where \( z \) is a complex number.

Answer: \infty


Question: Find the minimum value of \( \int_0^1 (1+x^2)f^2(x) \, dx \) where \( f \) is a continuous function on \([0,1]\) such that \( \int_0^1 f(x) \, dx = 1 \).

Answer: \dfrac{4}{\pi}


Question: Find a 3-regular graph with the smallest number of vertices that contains a path of 6 vertices (P6) as an induced subgraph.

Answer: 10


Question: Determine the singularities of the function \( f(z) = \sin\left(\frac{3}{2-z}\right) \) and find the radius of convergence of its Maclaurin series.

Answer: 2


Question: Consider a machine with two parts, X and Y. The time until part X fails follows an Exponential distribution with rate \( \lambda = 1 \), and the time until part Y fails follows an Exponential distribution with rate \( \lambda = 0.05 \). When part X fails, it is replaced in 1 time unit, during which the machine is not operational and part Y does not age. When part Y fails, the machine is considered dead. Calculate the expected time until the machine is dead.

Answer: 40


Question: Find a graph where the number of edges \(|E| = 2|V|-3\) and the degree of each vertex \(\deg(v) = 3\).

Answer: K_{3,3}


Question: Evaluate the limit: \[ \lim_{(x,y)\rightarrow(0,0)}(1+xy)^{\frac{1}{|x|+|y|}} \]

Answer: 1


Question: What is the units digit of the smallest integer $n$ such that $7n$ has 1984 digits?

Answer: 3


Question: Evaluate the limit: \[ \lim_{n\to\infty} \frac{f\left(\frac{3}{2}(3+\sqrt{7})^n\right)}{g\left(\frac{1}{2}(2+\sqrt{2})^n\right)} \] where \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R} \to \mathbb{R} \) are periodic functions with periods \( \frac{3}{2} \) and \( \frac{1}{2} \) respectively, such that \( \lim_{x\to 0}\frac{f(x)}{x}=1 \) and \( \lim_{x\to 0}\frac{g(x)}{x}=2 \).

Answer: 0


Question: Determine the number of ways to distribute \( n \) distinct balls into \( n+1 \) boxes, where \( n \) of the boxes are identical and one box is distinct. Express your answer in terms of Bell numbers.

Answer: B_{n+1}


Question: Calculate the volume above the plane \( z = 0 \), under the surface \( z = x^2 + y^2 \), and inside the cylinder \( x^2 + y^2 = 2x \) using a double integral. Express the integral in polar coordinates and evaluate it to find the volume.

Answer: \dfrac{3\pi}{2}


Question: Calculate the spectral radius of the Volterra operator $A$ acting on $L^2[0,1]$, defined by $$A(f)(x)=\int_0^x f(t) dt,$$ using the spectral radius formula for bounded linear operators: $$\rho(A)=\lim_{n\rightarrow \infty} \|A^n\|^{1/n}.$$

Answer: 0


Question: Evaluate the double integral \( \int_{0}^{1}\int_{\sqrt{x}}^{1} \sin(\pi y^3)\, dy\, dx \).

Answer: \dfrac{2}{3\pi}


Question: Determine the limit of the sequence \( \left(\frac{\ln(n+p)}{\ln(n)}\right)^{n\ln(n)} \) as \( n \to \infty \).

Answer: e^p


Question: A group \( G \) of order 15 acts on a set of order 22 with no fixed points. Determine the number of orbits in this group action.

Answer: 6


Question: Let \( \vec{a} \) and \( \vec{b} \) be unit vectors. If \( \vec{c} \) is a vector such that \( \vec{c} + (\vec{c} \times \vec{a}) = \vec{b} \), find the maximum value of \(|(\vec{a} \times \vec{b}) \cdot \vec{c}|\) and express it as \( \frac{A}{10} \). Determine the value of \( A \).

Answer: 5


Question: Given the normed linear space \(\mathbb{R}^2\) with the norm \(\|(x,y)\| = |x| + |y|\), let \(f\) be a bounded linear functional on \(X = \{(x,y) : x = y\}\) defined by \(f(x,y) = 3x\). If \(g(x,y) = \alpha x + \beta y\) is a Hahn-Banach extension of \(f\) on \(\mathbb{R}^2\), find the value of \(\alpha - \beta\).

Answer: 0


Question: Calculate the integral \( \int_B \frac{(x^2+y^2)z}{(x^2+y^2+z^2)^2} \, d\lambda_3(x,y,z) \), where \( B = \{(x,y,z) \in \mathbb{R}^3 : x^2+y^2+z^2 \le 4, z \ge 0\} \).

Answer: \pi


Question: Let \( a_1 = a \) and \( a_{n+1} = \cos(a_n) \) for all \( n \in \mathbb{N} \). Determine the value of \( \lim\limits_{n\rightarrow \infty}(a_{n+2}-a_{n}) \).

Answer: 0


Question: Provide an example of a differentiable function $f$ such that $f'(0) > 0$, but there exists no $\delta > 0$ such that $f$ is strictly increasing in the interval $(-\delta, \delta)$. Verify that the function \[ g(x) = \begin{cases} x + 2x^2 \sin(1/x) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] satisfies these conditions.

Answer: g(x)


Question:
Let \( G \) be the group of \( 2 \times 2 \) non-singular matrices under multiplication over the field \( \mathbb{Z}_3 \). Define a map \( f: G \to \mathbb{R}^* \) such that \( f(A) = |A| \), where \( \mathbb{R}^* \) is the group of non-zero real numbers under multiplication. Determine the order of the kernel of \( f \), \( o(\ker f) \). Choose the correct answer:

(a) 12
(b) 24
(c) 48
(d) none of the above

Answer: B


Question: Evaluate the limit: \[ \lim_{x\to-\infty}\frac{2x^5+x^2}{\sqrt{2x^{10}+x^2}} \]

Answer: -\sqrt{2}


Question: Find the sum of all natural factors of \(10!\).

Answer: 15334088


Question: For a number field $k$ with characteristic 0, determine whether algebraic $k$-groups of multiplicative type are always linear.

Answer: Yes


Question: Determine if there exists a positive integer \( n \) such that \( n! - (n+1)^9 \) is a prime number.

Answer: No


Question: Let $a$ and $b$ be natural numbers with $a > 1$, and let $p$ be a prime number. If the quadratic equation $ax^2 + bx + c = p$ has two distinct integral solutions for $x$, determine the number of integral roots of the equation $ax^2 + bx + c = 2p$. Provide your answer as a single integer.

Answer: 0


Question: Solve the recurrence relation \( T(n) = nT(n - 1) + 1 \) with the initial condition \( T(n) = \Theta(1) \) for \( n \leq 1 \) using iterative substitution to find the asymptotic behavior of \( T(n) \).

Answer: \Theta(n!)


Question: Suppose for two operators $X$ and $Y$, it is given that $e^X e^Y = e^{X + Y}$. Does this imply that $X$ and $Y$ commute, i.e., $[X,Y] = 0$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty}\int_{-\infty}^{\infty}e^{-x^{2n}}\,dx \]

Answer: 2


Question: Evaluate the limit: \[ \lim_{x \to 0}\frac{x-\sin(x)\cos(x)}{\sin(x)-\sin(x)\cos(x)}. \]

Answer: \dfrac{4}{3}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{(2n)!}{2^n (n!)^2} \]

Answer: \infty


Question: Let $M$ be a manifold with boundary $\partial M$. Form the manifold $M'$ by attaching a half-infinite cylinder $\partial M\times[0,\infty)$ to $M$ along its boundary. In other words, \[M'=M\cup_{\partial M}\partial M\times[0,\infty),\] where we identify $\partial M\sim\partial M\times\{0\}$. Let $g$ be a Riemannian metric on $M$. Does there always exist a Riemannian metric $g'$ on $M'$ such that the restriction of $g'$ to $M$ is equal to $g$?

Answer: Yes


Question: Determine the smallest integer $n$ such that the group $(\mathbb{Z}_6,+)$ is isomorphic to a subgroup of the symmetric group $S_n$. Provide your answer as a single integer.

Answer: 5


Question: If the 26 letters of the alphabet are arranged in a circle such that no two vowels are adjacent, what is the probability that the letters 'a' and 'b' are next to each other?

Answer: \dfrac{2}{21}


Question: Evaluate the limit: \[ \lim_{x\rightarrow0}\frac{6x\cdot\sin x-6\cdot\sin\left(x^{2}\right)+x^{4}}{x^{5}\left(e^{x}-1\right)} \]

Answer: \dfrac{21}{20}


Question: Does there exist a countable proper subset $S$ of $\mathbb{Q}$ such that $\mathbb{Q} \setminus S$ is infinite but not dense in $\mathbb{R}$?

Answer: Yes


Question: If $L$ is a bounded linear functional on a Hilbert space $H$, such that $Lx=(x,y)$ for all $x \in H$ and some $y \in H$, is it true that $\|L\|=\|y\|$? Justify your answer.

Answer: \|L\| = \|y\|


Question: Find the value of \( n \) such that \( \sum_{r=0}^{n-1}\log _2\left(\frac{r+2}{r+1}\right) = \prod_{r = 10}^{99}\log _r(r+1) \).

Answer: 3


Question: Evaluate the double integral \( \iint_\mathcal{D}\sqrt{x^2+y^2}\,dx\,dy \), where \( \mathcal{D} = \{(x,y) \mid x^2+y^2 \leq 2x\} \).

Answer: \dfrac{32}{9}


Question: Evaluate the sum \( \sum_{k=0}^n (-1)^k \binom{2n+1}{n+k+1} \).

Answer: \dbinom{2n}{n}


Question: Let \( M \) be an infinite dimensional vector space over a field \( K \), and let \( f_1, f_2, \ldots, f_r \in M^* \) be a set of linearly independent vectors in the dual space \( M^* = \text{Hom}_K(M, K) \), with \( r \geq 2 \). Does there exist a vector \( m \in M \) such that \( f_1(m) \neq 0 \) and \( f_i(m) = 0 \) for all \( i \geq 2 \)?

Answer: Yes


Question: Determine whether the set of sinusoidal functions with period $2\pi$, defined as $B = \{ w: \mathbb{R} \to \mathbb{R} \mid w(t) = A \sin(t + \phi), A \geq 0, \phi \in [0, 2\pi) \}$, is closed under addition. Specifically, if $w_1(t) = A_1 \sin(t + \phi_1)$ and $w_2(t) = A_2 \sin(t + \phi_2)$ are elements of $B$, is $w_1(t) + w_2(t)$ also an element of $B$?

Answer: Yes


Question:
Is it sufficient to prove that a statement $P(\beta)$ is true for all ordinals $\beta$ if the following conditions are met?

1. $P(\emptyset)$ is true.
2. $P(\beta) \implies P(\beta + 1)$ for $\beta$ a successor ordinal.
3. $P(\lambda)$ is true for $\lambda$ a limit ordinal.

In other words, can transfinite induction be completed without verifying the case for successors of limit ordinals?

Answer: No


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a continuous function such that the limits \( \lim_{x \to \infty} f(x) \) and \( \lim_{x \to -\infty} f(x) \) exist. Is the function \( f \) uniformly continuous on \( \mathbb{R} \)?

Answer: Yes


Question: Find the probability limit of \( \frac{\frac{1}{n}{\sum}_{i = 1}^{n} X_i}{\frac{1}{n}{\sum}_{i = 1}^{n} Y_i} \) where \(X_1, \ldots, X_n\) and \(Y_1, \ldots, Y_n\) are independent and identically distributed random variables with mean \(\mu\) and variance \(\sigma^2\).

Answer: 1


Question: Does there exist a locally compact Hausdorff topological group that is not a sequential space?

Answer: Yes


Question: Determine whether a singular algebraic variety $A$ can be birationally equivalent to a smooth algebraic variety $B$ over the complex numbers $\mathbb{C}$. Provide a justification for your answer.

Answer: Yes


Question: Find the correlation between \( \max(X,Y) \) and \( \min(X,Y) \), where \( X \) and \( Y \) are independent and identically distributed random variables from a uniform distribution on the interval [0,1].

Answer: \dfrac{1}{2}


Question: Let $H$ be a Hilbert space and $L(H)$ be the set of linear operators on $H$. Suppose $S, T \in L(H)$ with $S \geq 0$ and $\|Sx\| = \|Tx\|$ for every $x \in H$. Can you conclude that $S = \sqrt{T^*T}$?

Answer: S = \sqrt{T^*T}


Question: Is it possible for random variables $X$, $Y$, and $Z$ to have the same distribution and satisfy the equation $X = U(Y + Z)$, where $U \sim U(0, 1)$, and $Y$ and $Z$ are independent of $U$ and each other? Justify your answer using moment generating functions.

Answer: Yes


Question: Determine the order of the automorphism group of the abelian group \( \mathbb{Z}_{8} \times \mathbb{Z}_{4} \), where \( \mathbb{Z}_{8} \) and \( \mathbb{Z}_{4} \) are cyclic groups of order 8 and 4, respectively.

Answer: 128


Question: Determine whether \(36^{36} + 41^{41}\) is divisible by 77.

Answer: Yes


Question:
Based on the reasoning provided, determine whether $GL(n;\mathbb{C})$ is algebraic or not. Choose the correct conclusion:
1) $GL(n;\mathbb{C})$ is algebraic
2) $GL(n;\mathbb{C})$ is not algebraic
3) Both 1) and 2) can be correct, as some subsets can be algebraic in one affine space and not in another of different dimension.

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \rightarrow \infty} \left( x^{2/3} -(x^2+1)^{1/3} \right) \]

Answer: 0


Question: Determine the number of triplets \((m, n, p)\) where \(p\) is a prime number that satisfy the equation \(2^m p^2 + 1 = n^5\).

Answer: 1


Question: Evaluate the limit \( \lim_{n \rightarrow \infty} \frac{\frac{(\sqrt{n}+n-1)!}{(\sqrt{n})!(n-1)!}}{2^{\sqrt{n}}} \). Show that this limit is equal to infinity.

Answer: \infty


Question: Find the intersection of the plane $z = 4 + x + y$ with the cylinder $x^2 + y^2 = 4$. Then, calculate the surface integral \( \int\int_{S}(x^2z+y^2z)\,dS \), where \( S \) is the surface defined by this intersection.

Answer: 32\sqrt{3}\pi


Question: Evaluate the limit as $x \to 0$ of the expression: $$\left(\sum_{r=1}^{n} r^{1/\sin^2 x}\right)^{\sin^2 x}$$

Answer: n


Question: Evaluate the integral \( \int_C \frac {3z^5 - 10z^3} {z^6-5z^4+10}\ dz \) where \( C = \{z \in \mathbb{C} : |z| < 2 \} \).

Answer: 4\pi i


Question:
Determine whether the following claim is true or false:

Let $f,g :[a,b] \to \mathbb{R}$ be bounded functions with $g$ continuous and $f>g$. Then there exists a continuous function $h: [a,b] \to \mathbb{R}$, such that $f>h>g$. Provide your answer as either "True" or "False."

Answer: False


Question: In a game involving two players, A and B, two standard six-sided dice are rolled repeatedly. Player A wins if a total of 2 is rolled first, while player B wins if two consecutive totals of 7 are rolled first. What is the probability that player A wins the game?

Answer: \dfrac{7}{13}


Question: Determine the interval of convergence for the series \( \sum_{n=1}^{\infty} n \exp(-x \sqrt{n}) \).

Answer: (0, \infty)


Question: Evaluate the limit \( \lim_{n \rightarrow \infty } \frac {[(n+1)(n+2)\cdots(n+n)]^{1/n}}{n} \) using the Cesáro-Stolz theorem.

Answer: \dfrac{4}{e}


Question: Evaluate the limit: $$\lim_{x \to 0}\lim_{n \to \infty} \frac{n}{\left(1^{\left(\cot^2x \right)}+2^{\left(\cot^2x \right)}+\ldots+n^{\left(\cot^2x \right)}\right)^{\left(\tan^2x \right)}}$$

Answer: 0


Question: Find the Laplace Transform of the function $u(t-1)\delta(t)$. Provide your answer and justify your reasoning.

Answer: 0


Question:
Given the equations:
\[ y^2 + z^2 + yz = a^2 \]
\[ z^2 + x^2 + zx = b^2 \]
\[ x^2 + y^2 + xy = c^2 \]
and the condition:
\[ yz + zx + xy = 0, \]
find the combination of signs such that \( a \pm b \pm c = 0 \).

Answer: a + b - c = 0


Question: Let $X$ be a set such that for every metric space $Y$ containing $X$, the set $X$ is open in $Y$. Must $X$ be the empty set?

Answer: \emptyset


Question: Let $I$ be an ideal in a regular ring $R$ that can be generated by $n$ elements. Let $P$ be an associated prime of $I$. Is the height of $P$ bounded above by $n$? Justify your answer.

Answer: Yes


Question: Provide an example of a set \( A \) with measure zero such that the closure of \( A \), denoted \( \bar{A} \), is equal to the entire set of real numbers \( \mathbb{R} \).

Answer: \mathbb{Q}


Question: Let $X$ and $Y$ be independent and identically distributed random variables, each following an exponential distribution with rate 2. Define $T = X + Y$. Determine the conditional density function $f_X(x | T = t)$. Provide your answer in terms of $x$ and $t$.

Answer: \dfrac{1}{t}


Question: Determine the fundamental group \( \pi_1(S^2, (0,0,1)) \) of the unit sphere with base point \((0,0,1)\).

Answer: 0


Question: Determine whether the function \( f(z) = \frac{1}{e^{1/z} + 1} \) has a removable singularity at \( z = \infty \).

Answer: Yes


Question: What linear algebra operation transforms a matrix $V$ such that $V'^T M V' = I$, where $I$ is the identity matrix, given that $V^T M V = E$?

Answer: V' = V E^{-1/2}


Question: Find an upper bound for \( \mathbb{E}\left[\frac{1}{1+M}\right] \), where \( M \) follows a Binomial distribution \( B(n,p) \). Express the upper bound in terms of \( n^{-1} \), and consider the case where \( n \) is sufficiently large.

Answer: \dfrac{1}{pn}


Question: Evaluate the limit: $$\lim_{x \rightarrow +\infty} \frac{4x^{100}+x^9+\sqrt{x}+e^x}{x^{100}-\sin x+4e^x} $$

Answer: \dfrac{1}{4}


Question: Compute the limit \( \lim_{t \to + \infty} \int_0^{+ \infty} \frac{ \mathrm{d} x}{e^x+ \sin (tx)} \).

Answer: \dfrac{\pi}{2}


Question: Evaluate the limit: $$\lim_{x \to 0}\frac{2\mu(e^x(x^2-x+1)-1)-e^x(x^2-2x+2)+2}{x(2\mu(e^x(x-1)+1)-e^x(x-2)-2)}=\mu$$

Answer: \mu


Question: Determine the limit of the sequence \((a_n)\) as \(n\) approaches infinity, where \(a_n = \frac{1}{(n+1)^2} + \frac{1}{(n+2)^2} + \cdots + \frac{1}{(n+n)^2}\).

Answer: 0


Question:
Calculate the work done by the vector field \( \vec{F}(x,y,z) = (2xe^y + 2x)\vec{i} + (x^2e^y)\vec{j} + (3z^2)\vec{k} \) along the curve \( C: \vec{r}(t) = \frac{2t}{1+t^2}\vec{i} + (t-t^3)\vec{j} + \frac{2}{1+t^2}\vec{k} \) from \( t = 0 \) to \( t = 1 \). Use the formula for work done by a vector field along a curve:
\[ W = \int_0^1 \vec{F}(f(t), g(t), h(t)) \cdot \vec{r}\;'(t) \; \operatorname d\!t \]

Answer: -5


Question:
Calculate the limit \( \lim_{n \to \infty} x_n \) for the sequence defined by:
\[ x_n = \frac{n}{2^{n+1}} \sum_{i=1}^{n} \frac{2^i}{i} \]

Answer: 1


Question: Evaluate the integral \( \int_{0}^{\frac{\pi}{2}} \frac{1}{1+\tan(x)^{\sqrt{2}}} \, dx \).

Answer: \dfrac{\pi}{4}


Question: Calculate the line integral of the vector field \( F(x,y) = (\sin(x) + 3x^2, 2x - e^{-y^2}) \) along the path \( \gamma_1(t) = (\cos(t), \sin(t)) \) for \( 0 \leq t \leq \pi \). The path is the upper half of the unit circle from \((1,0)\) to \((-1,0)\).

Answer: \pi - 2


Question: Let $F$ be the free group on two elements $x$ and $y$. Define $x^{-1}$ as $X$ and $y^{-1}$ as $Y$. Consider the elements $g_1 = xyxY$, $g_2 = XYXyxYxyXyxYXYXyxYxyxyXYY$, $g_3 = XYXyx$, $g_4 = YxyXyxYXYXyyx$, $g_5 = YxyXYxyxyX$, and $g_6 = xxYXXy$. Let $G$ be the subgroup of $F$ generated by $g_1, g_2, g_3, g_4, g_5, g_6$. Prove that $G$ is a free group of rank 6.

Answer: 6


Question: Evaluate the triple integral \( \iiint (x-1)(y-1) \,dx\,dy\,dz \) over the region \( D:=\left \{ (x,y,z) \in\mathbb{R}^3 :x^2+y^2 \leq z \leq 2x+2y+2 \right \} \).

Answer: 0


Question: Find the vector \( w \in W \) that is closest to \((0,2,1,0)\), where \( W = \text{span}\left(\{(1,-1,0,0),(1,2,0,-1),(1,0,0,1)\}\right) \subseteq \mathbb{R}^4 \).

Answer: (0, 2, 0, 0)


Question: Let $Y_1, \ldots, Y_{10}$ be a random sample from a normal distribution with mean $0$ and variance $\sigma^2$. Find the distribution of the statistic $\frac{10 \bar{Y}^2}{S^2}$, where $\bar{Y}$ is the sample mean and $S^2$ is the sample variance.

Answer: F_{1,9}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_{[0,1]^n} \frac{x_1^5 + \ldots + x_n^5}{x_1^4 + \ldots + x_n^4} \, dx_1 \cdots dx_n. \]

Answer: \dfrac{5}{6}


Question: Determine the cardinality of the set \( A = \bigcap_{k=1}^{\infty} A_k \), where \( A_k = \{ a \in \mathbb{N}^{\mathbb{N}} \mid \sum_{i=100k}^{100k+99} a_i = 200 \} \).

Answer: 2^{\aleph_0}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left(1 - \frac{2x}{n^2}\right)^{\frac{n(n-1)}{2}} \] where \( x \) is a constant of order 1.

Answer: e^{-x}


Question: Let $G$ be a finite solvable non-$p$-group, and let $A$ be a maximal subgroup of $G$. Suppose $A$ is of primary index $p^n$, meaning $|G : A| = p^n$ where $p$ is a prime and $n \geq 1$. If $|G| = p_1^{n_1}p_2^{n_2}\cdots p_l^{n_l}p^{m+n}$ with $n_i \geq 1$ and $m \geq 0$, and $P_i$ is a Sylow $p_i$-subgroup and $P$ is a Sylow $p$-subgroup of $G$, can we obtain a Sylow basis $\{P_1, P_2, \cdots, P_l, P\}$ with $l \geq 1$ such that $P_i \subseteq A$ for $1 \leq i \leq l$?

Answer: Yes


Question:
Given constants \( C \geq 1 \) and \( \epsilon > 0 \), determine if there exists a number \( N = N(C, \epsilon) \) such that for every set \( S \subseteq S^1 \) with cardinality \( C \), there is a Laurent polynomial \( f(z) = \sum_{n = -N}^N a_n z^n \) satisfying the following conditions:

1. \( f \geq 0 \) on \( S^1 \).
2. \( f(z) \geq 1 \) for each \( z \in S \).
3. \( \|f\|_{L^1} < \epsilon \).

Answer: Yes


Question: Determine whether the following statement is true or false: If a point is a source for a one-dimensional smooth map $f(x)$, meaning $x$ satisfies $|f'(x)| > 1$, then this point is sensitive to initial conditions. Specifically, for every $\epsilon > 0$, there exists an $\bar{x}$ and a $d > 0$ such that $|f^k(\bar{x}) - f^k(x)| \geq d$ for some integer $k$. Justify your answer.

Answer: True


Question: Evaluate the integral \( \int_0^{+\infty} \left( \frac{\arctan t}{t} \right)^2 dt \) without using complex analysis.

Answer: \pi \ln 2


Question: Let $q \in \mathbb{C}$ be a primitive $n$th root of unity, for some $n > 1$. Consider the $n^2 \times n^2$ matrix \( M = \left( q^{ki+lj} \right)_{(k,l),(i,j)} \) indexed by all pairs \((k,l), (i,j) \in \{0,\cdots, n-1\}^2\) using lexicographic order. Are these matrices $M$ always invertible?

Answer: Yes


Question: Evaluate the limit \( \lim_{r \to 0^+} \frac{\int_{B_r} (r^2 - \| x \|^2) \Delta (u^2(x)) \, dx}{\int_{\partial B_r} u^2 \, dS} \), where \( B_r = \{ x = (x_1, \ldots, x_n) \in \mathbb{R}^n : \| x \| \leq r \} \) and \( u : \mathbb{R}^n \to \mathbb{R} \) is a smooth function with \( u(0) > 0 \). The norm is the usual Euclidean norm, and \( \Delta \) denotes the Laplace operator.

Answer: 0


Question: Evaluate the limit \( \lim_{n \to \infty} \left(x_n - \ln(n+1)\right) \) for the sequence defined by \( x_{n+1} = x_n + e^{-x_n} \) with an initial value \( x_0 \) being a real number.

Answer: 0


Question: Let $G$ be a subgroup of $GL(n,F)$, where $F$ is a field of characteristic $p$. Suppose that for every element $g \in G$, $g^m = 1$ for some integer $m$ such that $p$ does not divide $m$. Can the group $G$ be infinite? Justify your answer.

Answer: Yes


Question: Let $f$ be an analytic function on a region $D$ that includes the point $z=0$. For every natural number $n$ such that $\frac{1}{n}$ is in $D$, the function satisfies the condition $|f(\frac{1}{n})| \leq \dfrac{1}{2^n}$. Must the function $f$ be constant?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to \infty} \frac{(\ln(x))^x}{x^{\ln(x)}} \]

Answer: \infty


Question: Calculate the limit using the mean value theorem: $$\lim_{x \rightarrow 0} \frac{\arctan(x^2+x-1)+\frac{\pi}{4}}{x^2+3x}$$

Answer: \dfrac{1}{6}


Question:
Evaluate the limit: \[ \lim_{n \to \infty}\left(\frac{1}{2^{\frac{n}{2}}\Gamma{\frac{n}{2}}}\int_{n+\sqrt{2n}}^{\infty}e^{-\frac{t}{2}}t^{\frac{n}{2}-1}dt\right) \]

Choose the correct answer:
(A) 0.5
(B) 0
(C) 0.0228
(D) 0.1587

Answer: D


Question: Is the space $S^2 \setminus \{p\}$, where $p$ is a point on the 2-sphere $S^2$, homeomorphic to the Euclidean plane $\mathbb{R}^2$? Provide a justification for your answer.

Answer: Yes


Question: Compute the integral of the function \( f(z) = \frac{e^{-z}}{z^2} \) over the positively oriented circle \( |z| = 3 \) using the residue theorem.

Answer: -2\pi i


Question: Solve the equation \(m^4 - n^4 = 5(m^3 + n^3)\) in the set of natural numbers.

Answer: (6, 3)


Question: A polyhedron consists of only triangular and quadrilateral faces. Each vertex of the polyhedron has a degree of four, and the polyhedron contains exactly four quadrilateral faces. Determine the number of triangular faces in the polyhedron.

Answer: 8


Question: What is the tensor product of two real numbers? Specifically, calculate \( 3 \otimes 4 \) and determine if it is equivalent to classical multiplication.

Answer: 12


Question: Two players each throw a dart at the set of the first uncountable ordinal. The first dart hits ordinal $n_1$, and the second dart hits ordinal $n_2$. The player who hits the higher ordinal number wins. What is the probability that the second player wins?

Answer: \dfrac{1}{2}


Question: Let \( n \geq 1 \) and let \( A \) be an \( n \times n \) matrix with real entries such that \( A^k = O \) for some \( k \geq 1 \). Let \( I \) be the \( n \times n \) identity matrix. Find the value of \( \det(I+A) \).

Answer: 1


Question: Find the minimum value of \( n \) such that \( \frac{(2i)^n}{(1+i)^{n-2}} \) is a positive integer. Note that \( n \) does not have to be positive.

Answer: -2


Question: What is the infimum of the Lebesgue measure \( \lambda(\bigcup_{k=1}^{\infty}J_k) \), where \( J_k=(a_k-\frac{1}{2^{k}}, a_k+\frac{1}{2^k}) \) and \( (a_k) \) is an enumeration of the rationals? Is this infimum equal to 1?

Answer: 1


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{\sqrt{n^2+3n+1} - \sqrt{n^2+3n-1}}{\ln(1+n) - \ln(2+n)} \]

Answer: -1


Question: Evaluate the integral \( \int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx \).

Answer: \pi \ln 2


Question: Find the limit of the sequence \( \left(\frac{3n+4}{3n-2}\right)^{2n+1} \) as \( n \to \infty \).

Answer: e^4


Question: Is there a family \( \mathcal{A} \subset \mathcal{P}(\mathbb{R}) \) such that \( |\mathcal{A}| \le 2^{\aleph_0} \) and \( \mathcal{A} \) is not contained in any countably generated \( \sigma \)-algebra?

Answer: Yes


Question: Let $R$ be a commutative Noetherian ring where all associated primes are minimal primes. Suppose $r \in R$ is a zero-divisor such that $r/1 = 0$ in $R_P$ for every minimal prime $P$ of $R$. Is it true that $r = 0$ in $R$? Justify your answer.

Answer: Yes


Question: Let $S$ be the disk $|z|<3$ in the complex plane, and let $f: S \to \mathbb{C}$ be an analytic function such that \( f\left(1+\frac{\sqrt{2}}{n}i\right) = -\frac{2}{n^2} \) for each natural number \( n \). Find the value of \( f(\sqrt{2}) \).

Answer: 3 - 2\sqrt{2}


Question: Find the norm of the operator \( A: L^1(0,3) \rightarrow L^1(0,3) \) defined by \((Af)(x) = (x+5)f(x)\).

Answer: 8


Question: Compute the convolution \((f*g)(t)\) where \(f(t) = \delta(t-c)\) and \(c\) is a positive constant. Express your answer in terms of \(g(t)\).

Answer: g(t - c)


Question: Given the function \( f(x) = 2x^4 + 3x^3 - 3x^2 - 6x + a \), determine the range of values for \( a \) such that \( f(1) \cdot f(2) < 0 \).

Answer: (-32, 4)


Question:
Find an explicit expression for the determinant of the matrix $A_n \in \mathbb{R}^{n \times n}$, where the matrix is defined by:
\[\left[a_{i,j}\right] = \begin{cases}
1 & i=j \\
-j & i = j+1\\
i & i = j-1 \\
0 & \text{other cases}
\end{cases}\]

Answer: n!


Question: Let \(X_1, \ldots, X_n\) be iid Bernoulli\((p)\) random variables with \(0 < p < 1\). Given a sample where \(x_1 = \ldots = x_n = 1\), and assuming the prior distribution of \(p\) is Uniform\((0,1)\), consider a Bayesian test for \(H_0: p \geq 0.9\) versus \(H_1: p < 0.9\). Find the necessary and sufficient condition on \(n\) such that \(H_0\) is rejected.

Answer: n \leq 5


Question: Can every partition-free family in $2^{[n]}$ be enlarged to one of size $2^{n-1}$? A family $\mathcal{F} \subset 2^{[n]}$ is partition-free if it does not contain any partition of $[n]$. Determine if it is always possible to enlarge any partition-free family to reach the size $2^{n-1}$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether it is possible for a function \( f(x) \) to satisfy both of the following limits simultaneously: \[ \lim_{x\to+\infty}x\left[f(x)\right]^5=0 \] and \[ \lim_{x\to+\infty}x\left[f(x)\right]^3=+\infty. \]

Answer: Yes


Question: Let $f:[-1,1]\rightarrow \mathbb{R}$ be a continuous function such that $\int_{-1}^{1}f(t)\, dt = 2$. Evaluate the limit: $$\lim_{n\to\infty} \int_{-1}^{1}f(t)\sin^2(nt)\,dt.$$ Choose the correct answer from the following options: \(a)\ 0, \(b)\ 1, \(c)\ f(1) - f(-1), \(d)\ \text{Does not exist}.

Answer: B


Question: Let $X$ be a locally compact Hausdorff space. Does the diagonal $\Delta X \subset X \times X$ have a closed neighborhood $N$ such that the canonical projection maps $N \to X$ are proper?

Answer: Yes


Question:
Given the polynomial \(x^3 - 2x^2 - 3x - 4 = 0\), let \(\alpha, \beta, \gamma\) be its three distinct roots. Calculate the value of the expression:
\[
\frac{\alpha^6 - \beta^6}{\alpha - \beta} + \frac{\beta^6 - \gamma^6}{\beta - \gamma} + \frac{\gamma^6 - \alpha^6}{\gamma - \alpha}.
\]

Answer: 608


Question: Calculate the probability $P(H > 2N)$ using the joint density function $f_{NH}(n,h) = 0.1e^{-0.5n-0.2h}$, where $N$ and $H$ are continuous random variables following exponential distributions with rates $\lambda = 0.5$ and $\lambda = 0.2$, respectively.

Answer: \dfrac{5}{9}


Question: Let $E_1$ and $E_2$ be two extensions of a field $F$ such that $[E_2:F]=2$ and $E_1 \cap E_2 = F$. If $E_2 = F(\alpha)$, find the degree $[E_1(\alpha):E_1]$.

Answer: 2


Question: Find the largest possible value of \(\delta\) such that if \(|x-y|<\delta\), then \(|\cos x - \cos y|<\sqrt{2}\).

Answer: \dfrac{\pi}{2}


Question: Is there a rational map with rational coefficients that has order 5?

Answer: No


Question: Given the sequences \( a_n = \int_0^1 \max(x, b_{n-1}) \, dx \) and \( b_n = \int_0^1 \min(x, a_{n-1}) \, dx \), determine the limit of the sequence \( \{a_n + b_n\} \).

Answer: 1


Question: Consider an absolutely summable sequence \( \sum_{n=1}^{\infty}|a_{n}|<\infty \). Is it true that for every \( \epsilon>0 \), there exists a \( q >0 \) such that \( \sum_{n=1}^{\infty}|a_{n+q}|<\epsilon \)?

Answer: Yes


Question:
Consider the cubic equation \(x^3 - ax + b = 0\), where \(a\) and \(b\) are positive real numbers, and all roots are real. Let \(\alpha\) be the root with the smallest absolute value. Determine the range of \(\alpha\) from the following options:

(A) \(\frac{2b}{3a} < \alpha \leq \frac{b}{a}\)

(B) \(\frac{b}{a} < \alpha \leq \frac{3b}{2a}\)

(C) \(-\frac{b}{a} < \alpha \leq \frac{b}{a}\)

(D) \(-\frac{b}{a} < \alpha \leq \frac{2b}{3a}\)

Answer: B


Question:
Determine where the function \( f(x,y) \) is differentiable, where:
\[
f(x,y)=
\begin{cases}
\frac{x^3 y}{x^2 + y^4} & (x,y) \ne (0,0) \\
0 & (x,y)=(0,0)
\end{cases}
\]

Answer: \mathbb{R}^2


Question: If a polynomial \( f(x) \in \mathbb{Q}[x] \) has a splitting field of degree 16 over \( \mathbb{Q} \), are the roots of \( f(x) \) solvable by radicals?

Answer: Yes


Question: Suppose $f$ and $g$ are both increasing functions from $[0,1]$ to $[0,1]$ and are equal almost everywhere. Do $f$ and $g$ have the same points of discontinuity? Justify your answer.

Answer: Yes


Question: Given the limit \( \lim_{n \to \infty} \frac{\sqrt{16n^2+3}}{(1+a_n)n+5\cos n} = \frac{7}{6} \), calculate \( \lim_{n \to \infty} a_n \).

Answer: \dfrac{17}{7}


Question: Evaluate the limit: \[ \lim_{n\to\infty} \frac{\left(\sum_{r=1}^n\sqrt{r}\right)\left(\sum_{r=1}^n\frac1{\sqrt{r}}\right)}{\sum_{r=1}^n r} \]

Answer: \dfrac{8}{3}


Question: Let \(X\) be a normed space and \(\alpha \in X^*\) be a non-zero functional. Suppose \(c \notin \text{Ker}(\alpha)\). For each \(a \in X\), can \(a\) be uniquely written as \(a = b + kc\) where \(b \in \text{Ker}(\alpha)\) and \(k\) is a scalar?

Answer: Yes


Question: Let \( T: \mathbb{R}^4 \to \mathbb{R}^4 \) be a linear map such that the null space of \( T \) is \( \{(x,y,w,z) \in \mathbb{R}^4 : x+y+w+z=0\} \) and the rank of \( (T-4I_4) \) is 3. If the minimal polynomial of \( T \) is \( x(x-4)^{\alpha} \), find the value of \( \alpha \).

Answer: 1


Question: A fair die is rolled repeatedly. Let $X$ be the number of rolls needed to obtain a 5, and $Y$ the number of rolls needed to obtain a 6. Calculate the expected value $E[X \mid Y=2]$, where $Y=2$ indicates that a 6 is rolled on the second roll.

Answer: \dfrac{33}{5}


Question: John and Peter are solving a math problem independently. The probability that John will solve it correctly is \( \frac{1}{8} \), and the probability that Peter will solve it correctly is \( \frac{1}{12} \). If both solve incorrectly, they receive the same result with a probability of \( \frac{1}{1001} \). Given that both have received the same result, what is the probability that their result is correct?

Answer: \dfrac{13}{14}


Question: In triangle \(ABC\), the cevian \(BD\) and the median \(CM\) are drawn. The angle bisector of \(\angle BCM\) bisects \(BD\) at \(N\). Given that \(AD = 2\), \(DC = 1\), and \(BC = \sqrt{3}\), calculate the measure of \(\angle ABD\).

Answer: 30^\circ


Question: Consider the distance function defined for $z, z' \in \{ -1,1 \}^m$ as $d(z,z') = \min_{\delta \in \{ -1,1\}}\sum_{i=1}^m \mathbb{1}_{z_i \neq \delta z'_i}$. Determine whether this distance function satisfies the triangle inequality.

Answer: Yes


Question: Given the functional equation $f(f(x)) = x^4$ for all real $x$ and the condition $f(1) = 1$, find the value of $f(0)$. Assume $f$ is a function from the reals to the reals.

Answer: 0


Question: Evaluate the limit \( \lim_{x \to 0} \int^{3x}_x \frac{\sin t}{t^2}dt \). Determine if the limit exists and find its value.

Answer: \ln 3


Question: Evaluate the improper integral \( \int_{1}^{\infty} \frac{\ln{(2x-1)}}{x^2} \, dx \).

Answer: 2 \ln 2


Question: Let $D$ be a non-empty connected open subset of $\mathbb{C}^n$. Let $f$ be a complex-valued continuous function on $D$. Let $Z$ be the set of zeros of $f$. Suppose $f$ is holomorphic on $D - Z$. Is $f$ holomorphic on $D$?

Answer: Yes


Question: Let \( A \in M_{nn}(\mathbb{K}) \) be a matrix such that \( AB = BA \) for all \( B \in M_{nn}(\mathbb{K}) \). Prove that \( A = aI_n \) for some \( a \in \mathbb{K} \).

Answer: A = aI_n


Question: Is there a series of rational numbers \( \frac{p(n)}{q(n)} \) such that \( \lim_{n\to\infty} \frac{p(n)}{q(n)} = \pi \), where \( p(n) \) and \( q(n) \) are whole numbers and known functions of \( n = 1, 2, 3, \ldots \)?

Answer: Yes


Question: Find the value of \[ \lim_{n\to\infty}\sum_{k=1}^n \frac{1\cdot 3\cdot 5\cdots (2k-1)}{3\cdot 6\cdot 9\cdots 3k}. \]

Answer: \sqrt{3} - 1


Question: Determine the number of consecutive composite integers that follow \( k! + 1 \).

Answer: k - 1


Question: Evaluate the limit: \[ L = \lim_{n \to \infty} \left(\frac{1}{n^2+1} + \frac{2}{n^2+2} + \cdots + \frac{n}{n^2+n} \right) \]

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{x\to 0^+}\left(\frac{\ln x}{(1+x)^2}-\ln\left(\frac{x}{1+x}\right)\right) \]

Answer: 0


Question: Compute the value of the right derivative of the function \( f(x) = \sin(x)^{\cos(x)} + \cos(x)^{\sin(x)} \) at \( x = 0 \).

Answer: 1


Question: Determine the number of zeros of the polynomial \( \frac{1}{4} z^{6} + z^{4} - z^{3} + 1 \) inside the unit circle.

Answer: 0


Question: Evaluate the integral: \[ \int_{-\infty}^{\infty} \frac{1}{\cosh(kx)} \, dx \]

Answer: \dfrac{\pi}{k}


Question: Find the value of \(\sigma\) for a Gaussian distribution centered at 0 such that 90% of the distribution's integral is within the interval \([-10, 10]\).

Answer: 6.08


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{f(2+\sin x)-f(2+x\cos x)}{x-\sin x} \] where the function \( f(x) \) is differentiable and \( f'(x) \) is continuous on \((-\infty, \infty)\) with \( f'(2) = 14 \).

Answer: 28


Question: Consider the matrix \( A = I_9 - 2u^Tu \) where \( u = \frac{1}{3}[1, 1, 1, 1, 1, 1, 1, 1, 1] \). If \( p \) and \( q \) are two distinct eigenvalues of \( A \), find \( |p-q| \).

Answer: 2


Question: Determine the entire function $f(z)$ such that $|f(z) + e^z| > |e^z|$ for all $z \in \mathbb{C}$.

Answer: f(z) = c e^z


Question: Determine the order of the zeros of the function \( f(z) = \frac{\sin z}{z} \).

Answer: 1


Question: Consider a complete graph with $n$ vertices. Each edge has a value of $1$ or $-1$, each with a probability of 50%. For any partition of the vertices into groups, the score of a vertex is the sum of the values of its edges to the other vertices in its group. Let $p_n$ be the probability that, for some partition, no vertex can switch to another group or form a singleton group to obtain a better score. Determine whether $p_n \rightarrow 1$ as $n \rightarrow \infty$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit as \( n \to \infty \) of \( \frac{\log(n!)}{n \log(n)} \).

Answer: 1


Question: Two cars, A and B, start at different positions on a road. Car A drops its first marker 5 feet from its starting position, and car B drops its first marker 9 feet from its starting position. Both cars then drop subsequent markers at increasing distances, with each subsequent marker being 2 feet further than the previous one. Determine the distance from the starting position where both cars drop a marker at the same location for the first time.

Answer: 9


Question: Let $A \in \operatorname{GL}_n(\mathbb{R})$ be a symmetric positive definite matrix with off-diagonal entries $\leq 0$. Must all the entries of $A^{-1}$ be positive?

Answer: No


Question: Evaluate the limit: \[ \lim\limits_{n\rightarrow\infty}\sum\limits_{k=2}^n\frac{1-e^{-(x_k-x_{k-1})}}{1+e^{-(x_k-x_{k-1})}} \] where the points \(\{x_1,..,x_n\} \in [0,1]\) are ordered \(x_1 < x_2 < ... < x_n\) and form a partition of \([0,1]\). The maximum difference \(x_j - x_{j-1}\) tends to zero. Determine if the limit is \(\frac{1}{2}\).

Answer: \dfrac{1}{2}


Question: Estimate \(\sqrt{24}\) to three decimal places using a power series expansion. Use the function \(f(x) = \sqrt{25-x}\) and expand it using the binomial series. Calculate the series up to the 7th term and ensure the error is within three decimal places using the Lagrange form of the remainder. What is the estimated value of \(\sqrt{24}\)?

Answer: 4.899


Question: Evaluate the definite integral \( \int_{-\pi}^\pi \frac{\cos^2x}{1+a^x}dx \) where \( a > 0 \).

Answer: \dfrac{\pi}{2}


Question:
Solve for \( x \) in the equation:
\[
26280 = 2 \times \sqrt{\frac{149,598,000,000 - x}{1.4}} + \frac{x}{10,217,588,910,387,300,000}
\]

Answer: 149356276560


Question: Two points \((x_1, y_1)\) and \((x_2, y_2)\) on the curve \(y = x^4 - 2x^2 - x\) have a common tangent line. Find the value of \(|x_1| + |x_2| + |y_1| + |y_2|\).

Answer: 4


Question: Compute the complex number \(\left(\frac{\sqrt{5} - 1}{4} + \frac{\sqrt{2\sqrt{5} + 10}}{4}i\right)^5\) and express your answer in the form \(a + bi\).

Answer: 1


Question: Calculate the limit: $$\lim_{n\to +\infty} \frac{\tan\left(\frac{\pi n}{2n+1}\right)}{\sqrt[3]{n^3+2n-1}}$$

Answer: \dfrac{4}{\pi}


Question: Given complex numbers $a$, $b$, and $c$ such that $|a| = |b| = |c| = 2$, find the maximum value of the expression $|a-b||b-c| + |b-c||c-a| + |c-a||a-b|$. Express your answer in terms of trigonometric functions if necessary.

Answer: 36


Question: Is it true that for any integer $n$, there exists a threshold $p_0$ such that for any finite group $G$ of Lie type with rank at most $n$ and characteristic $p \geq p_0$, and for any irreducible $\mathbb{F}_p$ representation $V$ of $G$ with dimension at most $n$, the first cohomology group $H^1(G,V)$ vanishes?

Answer: Yes


Question: Find the residue of the function \( \frac{z+2}{z(z+1)} \) at \( z = -1 \) by expanding it into a Laurent series centered around \( z = -1 \).

Answer: -1


Question: Find the value of \( \frac{d}{dx}(f^{-1}(x)) \) at \( x = \pi \) for the function \( f(x) = (2x-\pi)^3 + 2x - \cos x \).

Answer: \dfrac{1}{3}


Question: For the open unit disk $D$ in the complex plane, let $f: D \rightarrow \mathbb{C}$ be a holomorphic and one-to-one function. Does there exist a holomorphic function $g: D \rightarrow \mathbb{C}$ such that $d^2 = f'$?

Answer: Yes


Question: Given $n$ flips of a coin with success probability $p$, calculate the expected number of $k$-length win streaks in $n$ flips.

Answer: (n - k + 1)p^k


Question: Determine whether the field extension $K = \mathbb{Q}(\sqrt{2}, \sqrt{3})$ is a normal extension over $\mathbb{Q}$. Specifically, is it true that for every irreducible polynomial $f \in \mathbb{Q}[X]$ which has at least one root in $K$, $f$ splits completely in $K$?

Answer: Yes


Question: Determine the highest degree of polynomial for which the following Gaussian quadrature formula is exact: $$\int_{-1}^1 f(x) \, dx = \frac{5}{9}f\left(-\sqrt{\frac{3}{5}}\right) + \frac{8}{9}f(0) + \frac{5}{9}f\left(\sqrt{\frac{3}{5}}\right).$$

Answer: 5


Question: Evaluate the line integral \( \oint_C \vec{F} \cdot d\vec{R} \) using Stokes' theorem, where the vector field is \( \vec{F} = (ye^x, x^2+e^x, z^2e^z) \) and the curve \( C \) is given by \( \vec{r}(t) = (1+\cos t, 1+\sin t, 1-\cos t-\sin t) \) for \( 0 \leq t \leq 2\pi \).

Answer: 2\pi


Question: Let $X$ be a metric space, and $\mathscr{B}$ the $\sigma$-algebra generated by open sets of $X$. Consider the metric space $(\mathscr{B},d)$ with the metric $d(A,B)=m(A\Delta B)$, where $m$ is any Borel probability measure on the measurable space $(X,\mathscr{B})$. Can a countable dense subset be found in this metric space?

Answer: Yes


Question: Let $G$ be a topological group acting on a topological space $X$. For an element $g \in G$, define the map $f: X \rightarrow X$ by $f(x) = g * x$. Determine whether the map $f$ is continuous.

Answer: Yes


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function such that \( \lim_{x\to\infty} f(x) \) exists and is finite. If \( \lim_{x\to\infty} xf'(x) \) exists, find this limit.

Answer: 0


Question: Let $b$ be an integer greater than 2, and let $N_b = 1_b + 2_b + \cdots + 100_b$ (the sum contains all valid base $b$ numbers up to $100_b$). Compute the number of values of $b$ for which the sum of the squares of the base $b$ digits of $N_b$ is at most 512.

Answer: 30


Question:
Consider the continuous-time state-space system given by \( \dot{x}=Ax+Bu, \quad y=Cx. \) The system is sampled at times \( kh, k \in \mathbb{N} \), with a sampling time \( h = 3 \). The input function \( u(t) \) is constant between two subsequent sampling times, i.e., \( u(t)=u_{k} \) for \( t \in [kh, (k+1)h] \). The exact discretization results in:
\[ x_{k+1}=A_dx_k+B_dx_k \]
\[ y_k=C_dx_k \]
with matrices
\[ A_d = \begin{bmatrix} e^3 & 3e^3 \\ 0 & e^3 \end{bmatrix}, \quad B_d = \begin{bmatrix} e^3-1 \\ 0 \end{bmatrix}, \quad C_d = \begin{bmatrix} e^3 & e^3 \end{bmatrix} \]
Which of the following continuous-time state-space systems yields the discrete-time state-space system after exact discretization is applied?

A) \( \left[ \begin{array}{c|c}
A & B \\
\hline
C &  \\ \end{array} \right]
=\left[ \begin{array}{cc|c}
1 & 1 & 0 \\
0 & 1 & 1 \\
\hline
1 & 1 \\ \end{array} \right] \)

B) \( \left[ \begin{array}{c|c}
A & B \\
\hline
C &  \\ \end{array} \right]
=\left[ \begin{array}{cc|c}
1 & 1 & 1 \\
0 & 1 & 0 \\
\hline
1 & 1 \\ \end{array} \right] \)

C) \( \left[ \begin{array}{c|c}
A & B \\
\hline
C &  \\ \end{array} \right]
=\left[ \begin{array}{cc|c}
1 & 1 & 1 \\
0 & 1 & 0 \\
\hline
e^3 & e^3 \\ \end{array} \right] \)

D) \( \left[ \begin{array}{c|c}
A & B \\
\hline
C &  \\ \end{array} \right]
=\left[ \begin{array}{cc|c}
1 & 3 & 1 \\
0 & 1 & 0 \\
\hline
1 & 1 \\ \end{array} \right] \)

E) \( \left[ \begin{array}{c|c}
A & B \\
\hline
C &  \\ \end{array} \right]
=\left[ \begin{array}{cc|c}
1 & 1 & e^3-1 \\
0 & 1 & 0 \\
\hline
e^3 & e^3 \\ \end{array} \right] \)

Answer: C


Question: Evaluate the infinite series: \[ \sum_{n=1}^{\infty} \frac{n}{n^{4} + n^{2} + 1} \]

Answer: \dfrac{1}{2}


Question: Calculate the exact value of \( \tan^2{\left(\frac{\pi}{5}\right)}+\tan^2{\left(\frac{2\pi}{5}\right)} \) without using a calculator.

Answer: 10


Question: For every real number $x$, the equation \( 6 + f(x) = 2f(-x) + 3x^2 \int_{-1}^{1}f(t) \, dt \) holds true. Determine the value of \( \int_{-1}^{1} f(x) \, dx \).

Answer: 4


Question: Given that $f(f(x)) = x^2 - x + 1$, find the value of $f(0)$. Assume $f: \mathbb{R} \to \mathbb{R}$ without any further restrictions.

Answer: 1


Question: Find the limit of the sequence \( \frac{2 \sqrt[n]{n!}}{n} \) as \( n \to \infty \).

Answer: \dfrac{2}{e}


Question: Evaluate the limit \( \lim_{x \to 0}{\frac{\sqrt{1+x\sin{x}}-\sqrt{\cos{2x}}}{\tan^{2}{\frac{x}{2}}}} \) without using l'Hopital's rule.

Answer: 6


Question: Find the limit: \[ \lim_{x \to +\infty} \tan \left( \left( \frac{\pi x}{2x+1} \right)^{1/x} \right) \]

Answer: \tan 1


Question:
Determine the radius of convergence \( R \) for the power series:
\[ \sum_{n=1}^{\infty} \left(n \sin{\frac{1}{n}}\right)^{n} x^n. \]

Answer: 1


Question: Evaluate the surface integral \( \int \int_S 3z \, d\sigma \), where the surface \( S \) is the portion of the plane \( 2x+y+z=2 \) lying in the first octant.

Answer: 2\sqrt{6}


Question: Determine whether the supremum \( \sup_{x \in \mathbb{R}} \sum_{k \in \mathbb{Z}} \frac{|k|}{(1+|x-k|)^{100}} \) is finite.

Answer: \infty


Question: Evaluate the limit: \[ \lim_{R \to \infty} \int_{\gamma_R} \frac{e^{iz}}{z^2}dz = 0 \] where \( \gamma_R \) is the arc of the circle with center at 0 and radius \( R \), from \( R \) to \( -R \), contained in the upper half-plane.

Answer: 0


Question: Determine the Morley rank of the type $x = x$ in the theory $ACF_{0}$, where $ACF_{0}$ denotes the theory of algebraically closed fields of characteristic 0.

Answer: 1


Question: Find the function $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfies the equation $f(x) f(y) - f(xy) = x + y$ for all real numbers $x$ and $y$.

Answer: x + 1


Question: Find the most general entire functions of the form \( f(z) = u(x) + iv(y) \), where \( u \) and \( v \) depend only on one real variable.

Answer: f(z) = az + b


Question:
Find the function \( f(x) \) that satisfies the functional equation:
\[ x^2 f(x) + f(1-x) = 2x - x^4 \] for all real \( x \).

Answer: 1 - x^2


Question: Find the differentiable function \( f: [1, \infty) \rightarrow \mathbb{R} \) such that \( f(1) = 0 \) and \( y(f(xy) - f(x)) = F(y) + y \), where \( F \) is a primitive of \( f \).

Answer: \ln x


Question: Find the multiplicative inverse of 7 in \( \mathbb{Z}_3 \).

Answer: 1


Question: Is every linear orthogonal operator on a subspace of $\mathbb{R}^n$ surjective? An orthogonal operator $T$ is defined as a linear operator that preserves the dot product, i.e., $T(\vec{x}) \cdot T(\vec{y}) = \vec{x} \cdot \vec{y}$. Provide a justification for your answer.

Answer: Yes


Question: Determine the distance between the function $f(x) = x^2$ and the subspace $K$, where $K$ is the kernel of the linear functional $I: f \mapsto \int_{-1}^{1} f(x) \, dx$ in the normed linear space $G = C[-1,1]$ with the supremum norm. Express your answer as $d(x^2, K) = \inf_{g \in K} \|x^2 - g(x)\|$.

Answer: \dfrac{1}{3}


Question: Find the residue of the function \( \frac{\sin z}{z^3(z-1)(z-3)} \) at \( z = 0 \).

Answer: \dfrac{4}{9}


Question: Evaluate the limit \( \lim_{n\to\infty}\frac{1}{\ln(\ln n)}\sum_{k=2}^n\frac{1}{k\ln k} \).

Answer: 1


Question:
Let \( f \) be an odd function such that \( \lim_{x \rightarrow \infty} [f(x) + x - 1] = 0 \). Determine the asymptote of the graph of the function \( g \), which is the restriction of \( f \) to \( \mathbb{R}^- \). Choose from the following options:

A) \( y = x - 1 \)
B) \( y = x + 1 \)
C) \( y = -x - 1 \)
D) \( y = -x + 1 \)

Answer: C


Question:
Consider the group \( G = \mathbb{Q}/\mathbb{Z} \), where \( \mathbb{Q} \) is the group of rational numbers and \( \mathbb{Z} \) is the group of integers. Let \( n \) be a positive integer. Is there a cyclic subgroup of order \( n \) in \( G \)?

Choose one of the following options:

- Not necessarily
- Yes, a unique one
- Yes, but not necessarily a unique one
- Never

Answer: B


Question: Find the volume of the region \( F = \{(x,y,z) \in \mathbb{R}^{3} \mid 0 \leq x \leq 1, y^{2} + z^{2} \leq x^{2}(1-x)^{2} \} \).

Answer: \dfrac{\pi}{30}


Question: Evaluate the infinite product: \[ \prod_{x \geq 2} \frac{x^2+1}{x^2-1} \] for \( x \in \mathbb{Z}^+ \).

Answer: \dfrac{\sinh(\pi)}{\pi}


Question: Find the limit, if it exists: \[ \lim_{x \to -\infty} \frac{\sqrt{4x^6 - x}}{x^3 + 2} \]

Answer: -2


Question: Use Stokes' Theorem to evaluate the integral \( \iint_\Sigma y \, \mathrm{d}y \, \mathrm{d}z + z \, \mathrm{d}z \, \mathrm{d}x + x \, \mathrm{d}x \, \mathrm{d}y \), where \( \Sigma \) is the upper half-sphere oriented outwardly. The hint provided is that \( (y,z,x) = \text{Curl}(-xy, -yz, -zx) \).

Answer: 0


Question: Evaluate the contour integral \( \frac{1}{2\pi i}\int_{\gamma} \frac{\cos(z)}{z^7-2z^6}dz \), where \(\gamma\) is the path that traverses the circle centered at \(1/4\) with radius \(1/2\) counter-clockwise.

Answer: \dfrac{1}{192}


Question: Find the last two digits of $7^{100} - 3^{100}$. Use Euler's theorem to simplify the computation.

Answer: 00


Question: Determine whether the limit \( \lim_{n \to \infty} a_n \) exists for the sequence \( a_n = \left(1-\frac{1}{\sqrt{2}}\right)\left(1-\frac{1}{\sqrt{3}}\right)\cdots\left(1-\frac{1}{\sqrt{n+1}}\right) \), where \( n \ge 1 \). If it exists, find its value.

Answer: 0


Question: Can a subset-enumerating function, defined as a function \( f: \mathbb{N}^4 \to \mathbb{N} \) such that for all integers \( N \geq n \geq 1 \), the set \( \{ (f(N,n,k,1), f(N,n,k,2), \ldots, f(N,n,k,N)) \mid 1 \leq k \leq \binom{N}{n} \} = S_{N,n} \), be primitive recursive?

Answer: Yes


Question: Determine if the area of a parallelogram projected onto the xy-plane is given by \( Q \cos(\alpha) \), where \( Q \) is the area of the original parallelogram, and \( \alpha \) is the angle between the plane of the parallelogram and the xy-plane. Justify your answer geometrically.

Answer: Q \cos(\alpha)


Question: Determine all complex numbers \(z\) such that \(\left| \frac{z-a}{1-\bar{a} z} \right| = 1\), where \(a\) is a complex number with \(|a| < 1\).

Answer: |z| = 1


Question: Calculate the limit of the function \( \frac{(ax+1)^{-(1+\frac{1}{ax})}\ln(ax+1)}{x} \) as \( x \rightarrow 0 \), where \( a \) is a positive constant.

Answer: \dfrac{a}{e}


Question: Determine the value of \( a \) for which the function \( h(x, y) = (x - 1)^2 + 2a(x - 1)y + y^2 + y^4 \) has an extreme point at \((1, 0)\) that is either a maximum or a minimum, but not a saddle point.

Answer: [-1, 1]


Question: Find the probability density function (PDF) of the random vector \((\min(X,Y), \max(X,Y))\), where \(X\) and \(Y\) are independent random variables with cumulative distribution functions (CDFs) given by \(F_X(x) = x\) and \(F_Y(x) = x\) for \(0 \leq |x| \leq 1\).

Answer: 2


Question: Find the value of \(a\) in the interval \((0, \infty)\) such that the inequality \(a^x - x^a \geq 0\) holds for all \(x > 0\).

Answer: e


Question: In an experiment involving two urns, Urn 1 contains 3 blue balls and 1 red ball, while Urn 2 contains 1 blue ball and 3 red balls. It is equally likely to choose either urn. Players draw a ball at random, replace it, and guess if the urn is majority blue or red based on the ball's color. Each player sees the previous player's guess but not the ball. A cascade occurs if the first two players draw the same color, influencing the third player's guess regardless of their draw. What is the probability that a wrong cascade occurs after the first two players guess?

Answer: \dfrac{1}{16}


Question: Consider a vector $\mathbf{X} \in \mathbb{R}^n$ where each component is subGaussian with parameter $s$. Determine if the vector $\mathbf{X}$ is subGaussian, and if so, find its subGaussian parameter.

Answer: s\sqrt{n}


Question: Let $A \subset I_0$ be an arbitrary set such that $\text{Int}(A) \neq \emptyset$. Is the outer measure $\mu^*(A)$ always positive?

Answer: Yes


Question: Find the generator of the kernel of the homomorphism \(\phi : \mathbb{R}[x] \to \mathbb{C}\) defined by \(f(x) \mapsto f(2+i)\).

Answer: x^2 - 4x + 5


Question:
Which of the following definitions of smoothness on an arbitrary subset is considered standard? Let $M$ be a smooth manifold, $A \subseteq M$ be an arbitrary subset of $M$, and $f: A \rightarrow \mathbb{R}$.

1. $f$ is smooth if and only if there is an open set $U$, $A \subseteq U$, and a smooth function $\bar{f}: U \rightarrow \mathbb{R}$ such that $f = \bar{f}$ on $A$.

2. $f$ is smooth if and only if for every $p \in A$ there is an open neighborhood $U$ of $p$ and a smooth function $\bar{f}: U \rightarrow \mathbb{R}$ such that $\bar{f} = f$ on $U \cap A$.

Select the standard definition.

Answer: 2


Question:
Suppose $a$ is a real number for which all the roots of the equation $x^{4} - 2ax^{2}+x+a^{2}-a=0$ are real. Determine the correct bound on $a$ from the following options:

(a) $a<-rac{2}{3}$
(b) $a=0$
(c) $0<a<\frac{3}{4}$
(d) $a\ge \frac{3}{4}$

Answer: D


Question: Determine the maximum number of points that can be placed inside a semicircle of radius $r$ such that no two points are closer than $r$ apart. The points can be placed anywhere within the semicircle, including on the straight line, inside the area, or on the circumference.

Answer: 5


Question: Consider a seller who must sell a single private value good in a second-price sealed-bid auction. There are two potential buyers, each with a valuation that can take on one of three values, \(\theta_i \in \{0, 1, 2\}\), each value occurring with an equal probability of \(\frac{1}{3}\). The players’ values are independently drawn. The seller can set a reserve price \(r \geq 0\) that modifies the auction rules as follows: if both bids are below \(r\), the good is not sold; if both bids are at or above \(r\), the regular auction rules apply; if only one bid is at or above \(r\), that bidder obtains the good and pays \(r\) to the seller. Determine the optimal reserve price \(r\) for the seller.

Answer: 2


Question: Let $G$ be a finite abelian group and let $m = \max \{o(x) : x \in G\}$. Is it true that $o(x) \mid m$ for all $x \in G$?

Answer: Yes


Question: Consider a sphere of radius $r$ centered at the origin. Two great circles of the sphere lie on the xy-plane and xz-plane. A part of the sphere is removed such that the cross-section of the remaining part, perpendicular to the x-axis, is a square with vertices on the great circles. Compute the volume of the remaining part of the sphere.

Answer: \dfrac{8}{3} r^3


Question: Determine the $w^{*}$-closure of the finite rank operators in $B(H)$, where $B(H)$ denotes the bounded operators on a Hilbert space $H$. Is this closure the entire $B(H)$?

Answer: B(H)


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}n^2\left[\left(1+\frac{1}{1+n}\right)^{n+1}-\left(1+\frac{1}{n}\right)^{n}\right] \]

Answer: \dfrac{e}{2}


Question: Evaluate the complex integral \( I = \oint_C f(z)\,dz \), where \( f(z) = \sqrt{z^2-1} \) is a holomorphic function for \( 1 < |z| < \infty \) and \( C \) is a circle of radius \( R \) centered at the origin in the complex plane.

Answer: -\pi i


Question: Find all analytic functions \( f: \mathbb{C} \rightarrow \mathbb{C} \) such that \(|f'(z)|\) is constant on curves where the real part of \( f \) is constant.

Answer: f(z) = az + b


Question: Determine the values of $x \in \mathbb{R}$ for which the series \( \sum_{n=1}^{\infty} \frac{1}{n} 10^{l(n)}(2-x)^n \) is convergent, where \( l(n) \) denotes the number of digits of \( n \).

Answer: (1, 3)


Question: Evaluate the limit: \[ \lim_{n\to \infty} \sum_{r=0}^{n} \frac{\binom{n}{r}}{(r+4)n^r} \]

Answer: 6 - 2e


Question: Find the sum of the series: \[ \sum_{n=0}^\infty \frac{1+n}{3^n} \]

Answer: \dfrac{9}{4}


Question: If \(p\) is a prime number and \(G\) is a non-abelian group of order \(p^3\), determine the number of elements in the center of the group \(G\), denoted as \(Z(G)\).

Answer: p


Question: Given a three-dimensional Lie algebra $\mathfrak{g}$ over $\mathbb{R}$ spanned by $X, Y, Z$ with the relations $[X,Y]=Y$ and $[X,Z]=Y+Z$, determine the Lie bracket $[Y, Z]$. Provide your answer as an expression in terms of $X, Y, Z$.

Answer: 0


Question: Determine whether the following statement is true or false: If \( \liminf_{x\to\infty}f(x)>c \), where \( c>0 \), then for any \( \epsilon >0 \), there exists \( x_{\epsilon}>0 \) such that \( f(x)>c+\epsilon \) for all \( x\geq x_{\epsilon} \).

Answer: B


Question:
Find the limit of the sequence \( \{a_n\} \) defined by the recurrence relations:

\[ a_1 = \frac{1}{2} \]
\[ a_2 = 1 \]
\[ a_n = \frac{1}{2}a_{n-1} + \sqrt{a_{n-2}} \quad \text{for every} \; n \geq 3 \]

Answer: 4


Question: Solve the congruence \(x^7 \equiv 5 \pmod{11}\) using primitive roots.

Answer: 4


Question: Find all pairs of prime numbers \(p\) and \(q\) such that \(p^{q+1} + q^{p+1}\) is a perfect square.

Answer: (2, 2)


Question: A bag contains sticks of integer lengths, and the longest stick is 120 units long. If no three sticks can be selected to form a triangle, what is the maximum number of sticks that can be in the bag?

Answer: 11


Question: Compute the integral \( I = \int_C z \frac{f'(z)}{f(z)} dz \), where \( C = \{z : z = e^{i\theta}, 0 \leq \theta \leq 2\pi \} \) and \( f \) is analytic with a simple unique root at \( z_0 \) inside the circle \( C \).

Answer: 2\pi i z_0


Question: Determine the fundamental group of the space $E_\tau = S^1/\tau^{\mathbb{Z}}$, where $\tau \in \mathbb{S}^1$ is not a root of unity, with the basepoint being the equivalence class of $1$. This space is path-connected.

Answer: 0


Question: Find the sum of the series: \[ 1 + \frac{1^2 + 2^2}{2!} + \frac{1^2 + 2^2 + 3^2}{3!} + \cdots \]

Answer: \dfrac{17e}{6}


Question: Evaluate \( \lim_{n \to \infty} \int_{n}^{n+1} \frac{1}{\sqrt{x^{3}+x+1}} \, dx \).

Answer: 0


Question: Determine the values of $a$ for which the integral \( \int_0^{\infty} x^a e^{\text{sgn}(a)x} \, dx \) converges.

Answer: (-1, 0)


Question: Find all pairs of positive integers \((a, b)\) such that \(2^a + 5^b\) is a perfect square.

Answer: (2, 1)


Question: Evaluate the triple integral \( \iiint_D xydzdydx \) where the region \( D \) is bounded by \( z = 4 - x^2 - y^2 \), \( x^2 + y^2 = 1 \), and \( z = 0 \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to a} \frac{\log(x-a)}{\log(e^x-e^a)} \] without using L'Hôpital's Rule.

Answer: 1


Question: Determine all primitive Pythagorean triples \((a, b, c)\) that form an arithmetic progression, where \(a^2 + b^2 = c^2\) and \(\gcd(a, b, c) = 1\).

Answer: (3, 4, 5)


Question: Compute the surface integral \( \iint_S \mathbf{F} \cdot \mathbf{N} \, dS \), where \( \mathbf{F}(x,y,z) = 2yz\mathbf{i} + (\tan^{-1}(xz))\mathbf{j} + e^{xy}\mathbf{k} \) and \( \mathbf{N} \) is the outward normal vector to the surface \( S \), which is the sphere defined by \( x^2 + y^2 + z^2 = 1 \).

Answer: 0


Question: Find the value of the 20th derivative of the function \( \arcsin(x) \) at \( x = 0 \).

Answer: 0


Question: Let $f$ be a holomorphic function on the unit disc $D$ with zeros at $\frac{1}{2}$ and $-\frac{1}{2}$, such that $|f(z)| < 1$ for all $z \in D$. Prove that $|f(i/2)| \leq \frac{8}{17}$.

Answer: \dfrac{8}{17}


Question: Is it necessary for the basis $B$ to be orthonormal for the matrix representation of a linear operator $T$ to be a normal matrix?

Answer: No


Question: Evaluate the limit \( \lim_{n\rightarrow\infty} \sum^{n}_{k=0}\left|\frac{2\pi\cos(k\pi(3-\sqrt{5}))}{n}\right| \).

Answer: 4


Question: Find all natural numbers \( n \) such that \( \phi(n^4+1) = 8n \), where \( \phi(n) \) is Euler's totient function.

Answer: 2


Question: Find all integer values of \( n \) such that \( \sqrt{\frac{4n-2}{n+5}} \) is a rational number.

Answer: 13


Question: Determine whether the set $X = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 = 1\}$ is homeomorphic to the set $Y = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 = 1\} \cup \{(x,y) \in \mathbb{R}^2 : (x-2)^2 + y^2 = 1\}$. Answer with 'Yes' or 'No' and justify your answer.

Answer: No


Question:
Given that all roots of the equation \(z^4+az^3+bz^2+cz+d=0\) (where \(a, b, c, d \in \mathbb{R}\)) are of unit modulus, determine which of the following statements are correct:

(A) \(|a| \leq 4\)

(B) \(|b| \leq 4\)

(C) \(|b| \leq 6\)

(D) \(|c| \leq 4\)

Answer: ACD


Question: Find a function $f$ such that $f \notin L^{p}(\mathbb{R})$ for all $p > 0$, but there exists a constant $c > 0$ such that for all $t > 0$, the measure of the set $\{x \in \mathbb{R} : |f(x)| > t\}$ is bounded by $\frac{c}{t}$. Provide an example of such a function.

Answer: f(x) = \dfrac{1}{|x|}


Question: Let \((R, \mathfrak{m}_R)\) be a local domain where \(R\) is integral and \(\mathfrak{m}_R\) is the unique maximal ideal of \(R\). Suppose that \(\mathfrak{m}_R\) is finitely generated. Is the Krull dimension of \(R\) necessarily finite?

Answer: Yes


Question: In an acute-angled triangle \( \triangle ABC \), let \( H \) be the orthocenter. The altitudes \( AD, BE, \) and \( CF \) are drawn from vertices \( A, B, \) and \( C \) to the opposite sides. Find the value of \( \frac{AH}{AD} + \frac{BH}{BE} + \frac{CH}{CF} \).

Answer: 2


Question: Provide an example of a continuous map from $[0, ∞)$ onto the unit circle that is not a homeomorphism.

Answer: f(t) = (\cos t, \sin t)


Question: Let $X$ and $Y$ be independent random variables, each uniformly distributed on the interval $(0,1)$. Define $Z = X + Y$. Determine whether the conditional densities $f_{X \mid Z=z}(s)$ and $f_{Y \mid Z=z}(s)$ are equal for all $s$ and $z$. Justify your answer.

Answer: Yes


Question: Let $f, g \in k[x, y, z]$, where $k$ is an algebraically closed field. If the two polynomials do not share a single zero, do there exist polynomials $m, n \in k[x, y, z]$ such that $mf + ng = 1$? Provide a justification for your answer.

Answer: Yes


Question: Suppose $X_1, X_2, \ldots, X_n$ are independent random variables, each uniformly distributed over the interval $[0, 1]$. Let $X_{(n)}$ be the largest and $X_{(n-1)}$ be the second largest of these variables. Calculate the expected value of the ratio $\frac{X_{(n-1)}}{X_{(n)}}$. In other words, find $\mathbb{E}\left[\frac{X_{(n-1)}}{X_{(n)}}\right]$.

Answer: \dfrac{n-1}{n}


Question: Evaluate the limit \( \lim_{x \to 1} \frac{P(x^n)}{P(x^m)} \) where \( P \) is a polynomial such that \( P(1) = 0 \), and \( n, m \in \mathbb{N}^* \).

Answer: \dfrac{n}{m}


Question: Let \( y = g(x) \) be the inverse of the bijective mapping \( f: \mathbb{R} \to \mathbb{R} \) where \( f(x) = 3x^3 + 2x \). Find the area bounded by the graph of \( g(x) \), the x-axis, and the ordinate at \( x = 5 \).

Answer: \dfrac{13}{4}


Question: Let $a > 0$ and $u:[0,a] \to [0,+\infty)$ be a continuous function. Show that if there exists $L \geq 0$ such that for all $t \in [0,a]$, $u(t) \le \int_0^t Lu(s)\,ds$, then for all $t \in [0,a]$, $u(t) = 0$. Prove this statement.

Answer: 0


Question: Let \( u \in C(\overline{\mathbb{R}_+^2}) \) be a bounded harmonic function in the upper half-plane \( \mathbb{R}_+^2 \), satisfying \( u(x,0) \to \pi \) as \( x \to \infty \) and \( u(x,0) \to 0 \) as \( x \to -\infty \). Compute the limit of \( u(r\cos\theta, r\sin\theta) \) as \( r \to \infty \) for each \( 0 < \theta < \pi \).

Answer: \pi - \theta


Question: Is the induced map on the spectrum of rings surjective if $A$ is contained in $B$ and $B$ is not necessarily an integral extension of $A$?

Answer: No


Question:
Let $a_n = \sin (\pi/n)$. For the sequence $a_1, a_2, \ldots$, what is the supremum? Choose the correct option:

(a) $0$ and it is attained
(b) $0$ and it is not attained
(c) $1$ and it is attained
(d) $1$ and it is not attained

Answer: C


Question: Let $A, B, C$ be random variables with $\text{cov}(A, B) = 0$ and $\text{corr}(B, C) = 1$. Determine if $\text{cov}(A, C) = 0$ is true.

Answer: 0


Question: Evaluate the contour integral: $$\oint_{|z|=2} \tan{z}\,dz$$ where $|z|=2$ represents the circle of radius 2 centered at the origin in the complex plane.

Answer: -4\pi i


Question: Is the following statement a theorem of ZF set theory? If $M$ is a countable transitive model of ZF, then for each subset $X$ of an element of $M$, there exists a natural number $n$ such that $\bigcup^n X \in M$, where $\bigcup^n$ denotes the $n$-iterative union operator defined by $\bigcup^0 X = X$ and $\bigcup^{n+1} X = \bigcup (\bigcup^n X)$.

Answer: Yes


Question: If a function $f(x)$ approaches 0 faster than $x$ approaches infinity, does it imply that its derivative $f'(x)$ also approaches 0 faster than $x$ approaches infinity?

Answer: No


Question: Evaluate the surface integral \( \int_S z^2 \, dS \), where \( S \) is the surface of the cube defined by \( \{-1 < x < 1, -1 < y < 1, -1 < z < 1\} \).

Answer: \dfrac{40}{3}


Question:
Prove that the following inequality holds:
\[ \int_1^4 \int_0^1 (x^2+\sqrt{y})\cos(x^2y^2) \, dx \, dy \leq 9. \]

Answer: 9


Question:
Consider a two-place position predicate \( \varphi \). Determine which of the following statements is correct:

1) All models of \( \varphi \) are infinite.
2) All models of \( \varphi \) are finite.
3) \( \varphi \) has both infinite and finite models.
4) \( \varphi \) has no models.

Select the correct statement.

Answer: 3


Question: Does there exist a function $f$ that is $n$-times differentiable on an interval $I$ but is not $(n+1)$-times differentiable anywhere on $I$?

Answer: Yes


Question: Find the value of \( x \) such that \( 13^{11^7} \equiv x \pmod{10} \), where \( x \in [0,9] \cap \mathbb{Z} \).

Answer: 7


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{(n^5+2)^{1/4} - (n^2+1)^{1/3}}{(n^4+2)^{1/5} - (n^3+1)^{1/2}} \]

Answer: 0


Question: Evaluate the sum \( \sum\limits_{a=0}^{\infty}\sum\limits_{b=0}^{\infty}\sum\limits_{c=0}^{\infty}\frac{1}{(a+b+c)!} \).

Answer: \dfrac{7e}{2}


Question: Are there infinitely many primes $p$ such that $p^2 + 1$ is divisible by a prime greater than $p$?

Answer: Yes


Question: Does there exist an infinite group that contains all finite groups as subgroups?

Answer: Yes


Question: Consider a vector bundle $E$ of rank $k$ over a smooth manifold $M$ of dimension $m$. Can $E$ be immersed in $\mathbb{R}^{2m + k}$?

Answer: Yes


Question: Determine all values of the parameter \( a \) such that the sum of the solutions to the equation \( \sin(\sqrt{ax-x^2})=0 \) is 100.

Answer: 25


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_0^1 \sqrt{\frac{1}{x} + n^2 x^{2n}} \, dx. \]

Answer: 3


Question:
Given the joint probability density function:
\[
f_{X,Y}(x,y) = \begin{cases}
ye^{-xy-y} & \text{if } x > 0, y > 0 \\
0 & \text{otherwise}
\end{cases}
\]
Calculate the distribution of the product $XY$. Assume $X$ and $Y$ are dependent random variables.

Answer: e^{-z}


Question: Given positive numbers $a_1, a_2, \ldots, a_n$, let $A$ and $H$ denote their arithmetic mean and harmonic mean, respectively. Determine if there is a known lower bound for the ratio $\frac{H}{A}$.

Answer: 0


Question:
Determine whether the following interpolation inequality holds for all functions \( u \in C^\infty(\mathbb{R}) \):
\[ \int_{\mathbb{R}} u^3 \, dx \lesssim \int_{\mathbb{R}} (u')^2 \, dx + \int_{\mathbb{R}} u^2 \, dx. \]

Answer: Yes


Question: Identify the set operation that generates the group \(\mathbb{Z}_4\).

Answer: +_4


Question: Let $G$ be a group. Suppose there exist a finite sequence of elements $a_1, \cdots, a_n$ and a finite sequence of subgroups $H_1, \cdots, H_n$ such that $G = \bigcup_{i=1}^n a_iH_i$ is a disjoint union. If $(G: H_1) = \cdots = (G : H_n) < \infty$, is it true that $H_1 = \cdots = H_n$?

Answer: No


Question: Evaluate the integral \( \oint_C \frac{\sin z}{z^2 + \pi^2} \, dz \) along the closed contour \( |z - 2i| = 2 \). Use Cauchy's integral formula to find the value of the integral.

Answer: i \sinh \pi


Question: Evaluate the triple integral \( \iiint_E xyz \, dxdydz \) where the region \( E \) is defined by \( 4 \leq x^2 + y^2 + z^2 \leq 9 \), \( x \leq 0 \), and \( z \leq 0 \).

Answer: 0


Question: If $g$ is the inverse of a function $f$ and $f'(x) = \frac{1}{1 + x^5}$, find the expression for $g'(x)$. Assume $g(x)$ is differentiable where applicable.

Answer: 1 + [g(x)]^5