Question: A triangle in space has vertices at \((8,0,3)\), \((0,8,3)\), and \((8,8,a^2+3)\), where \(a\) is a real number. Express the area of the triangle as a function of \(a\), denoted as \(f(a)\), and find the minimum value of this function.

Answer: 32


Question: Let $S$ be a set with infinite cardinality $\kappa_1$. Determine the cardinality $k$ of the set of subsets of $S$ with cardinality $\kappa \le \kappa_0 < \kappa_1$. Assume $\kappa_0$ can be either finite or infinite. What is $k$?

Answer: \kappa_1^{\kappa_0}


Question: Let $f(x)$ be a periodic function such that for all $x$, \( f(x+346) = \frac{1+f(x)}{1-f(x)} \). Determine the minimum period of $f(x)$. \( \boxed{} \)

Answer: 1384


Question: Find a positive number \( \delta \) such that if \( 0 < |x-1| < \delta \), then \( |f(x)g(x) - h(x) - 1| < \frac{1}{1000} \), given that \( |f(x) - 2| < |x-1|^2 \), \( |g(x)-3| < 2|x-1| \), and \( |h(x)-5| < |x-1| \).

Answer: \dfrac{1}{10000}


Question: Let $X_1, X_2, X_3$ be independent random variables each uniformly distributed on the interval $(0, 1)$. Find the expected value of the expression $E\left[\frac{X_1 + X_2}{X_1 + X_2 + X_3}\right]$. Provide your answer as a single value.

Answer: \dfrac{2}{3}


Question: Calculate the line integral \( \int_C \mathbf{F} \cdot \mathrm{d}\mathbf{r} \), where \( \mathbf{F} = \langle y, x+2y \rangle \) and \( C \) is the upper semicircle from \((0,1)\) to \((2,1)\).

Answer: 2


Question: Which of the following statements about biholomorphic functions \( f \colon D(0, 1) \to D(0, 1) \) is true?\n\n(a) \( f \) must be constant.\n(b) \( f \) must have a fixed point.\n(c) \( f \) must be a rotation.\n(d) \( f \) must fix the origin.\n\nSelect the correct option.

Answer: B


Question:
Given the associative and commutative binary operation \(x * y = 3xy - 3x - 3y + 4\) with a neutral element \(e = \frac{4}{3}\), compute the result of the operation:
\[ \frac{1}{1017} * \frac{2}{1017} * \cdots * \frac{2014}{1017}. \]

Answer: 1


Question: Consider a minimal smooth conic bundle $S$ of dimension two with curves $C$ and $F$ such that $C^2 < 0$ and $F^2 = 0$. Let $D$ be a pseudoeffective divisor on $S$ such that $D \sim_{\mathbb{Q}} aC + bF$ where $a, b \in \mathbb{Q}$. Is it true that $mD$ must be an integral effective divisor for some $m \geq 1$?

Answer: Yes


Question: Given the cubic equation \(x^3 - 2kx^2 - 4kx + k^2 = 0\), one root is less than 1, another root is in the interval (1, 4), and the third root is greater than 4. The value of \(k\) lies in the interval \((a+\sqrt{b}, b(a+\sqrt{6}))\) where \(a, b \in \mathbb{N}\). Find the value of \((a+b)^3 + (ab+2)^2\).

Answer: 2007


Question: Given a graph G with a $K_t$ minor, let H be the smallest subgraph of G that contains $K_t$ as a minor. Suppose H can be partitioned into sets $V_1, V_2, \ldots, V_t$ such that each $H[V_i]$ is connected and $\\chi(H[V_i]) < t$ for all $i \in \{1, 2, \ldots, t\}$. If the clique number of H is less than t, can we conclude that $\chi(H) \leq t$?

Answer: Yes


Question: Find the number of non-negative integer solutions to the equation \( x + y^2 + z = x^2z + y \).

Answer: \infty


Question: Find the root of the transcendental equation: \[ e^{2x}(2x-1) + 1 = 0 \]

Answer: 0


Question: Suppose \((M,d)\) is a separable and complete metric space. Let \(C_b(M)\) be the space of all bounded continuous functions \(f: M \to \mathbb{R}\), and let \(\mathcal{P}(M)\) be the space of all Borel probability measures on \(M\). Endow \(\mathcal{P}(M)\) with the weak topology, which is the coarsest topology that makes \(\mu \mapsto \int_M f \, d\mu\) continuous for all \(f \in C_b(M)\). Suppose \(K \subset \mathcal{P}(M)\) is relatively weakly compact, and \(L \subset M\) satisfies \(\delta_x \in K\) for all \(x \in L\), where \(\delta_x\) denotes the Dirac delta. Is \(L\) relatively compact in \((M,d)\)?

Answer: Yes


Question: Determine the values of $z \in \mathbb{C}$ for which the series \( \sum_{n=0}^{\infty} z^n A^n \) converges absolutely, where \( A \) is an operator in the Hilbert space \( L^2(0,2\pi) \) defined by \( (Af)(x) = \frac{1}{\pi} \int_0^{2\pi} [\cos(x)\cos(y) + \sin(x)\sin(y)] f(y) \, dy \).

Answer: |z| < 1


Question: Consider the sequences \( \{a_n\}_{n=1}^{\infty} \) and \( \{b_n\}_{n=1}^{\infty} \) defined by \( a_n=(2^n+3^n)^{\frac{1}{n}} \) and \( b_n=\dfrac{n}{\displaystyle\sum_{i=1}^{n}\frac{1}{a_i}} \). Determine the limit of the sequence \( \{b_n\}_{n=1}^{\infty} \) as \( n \to \infty \).

Answer: 3


Question: If two congruent $n \times n$ matrices $A$ and $B$ are both diagonalized with all non-zero eigenvalues, and the eigenvalues are ordered in descending order down the diagonal, must the number of positive eigenvalues of $A$ be equal to the number of positive eigenvalues of $B$? Justify your answer by considering the properties of congruence.

Answer: Yes


Question: Suppose we have a matrix $A$ with its $LU$-decomposition such that $A=LU$, and $U$ is ill-conditioned (i.e., $\| U \| \| U^{-1} \|$ is large). Does this imply that $A$ is ill-conditioned?

Answer: Yes


Question: Determine whether the cardinality of the superstructure \(\mathbb{V}\) over \(\mathbb{R}\) and the set of finite subsets \(\mathcal{P}_{fin}(\mathbb{V})\) is \(\beth_\omega\).

Answer: \beth_\omega


Question: Evaluate the integral \( \int_{C}\frac{2dz}{(2i+1)z^2+6iz+2i-1} \) where \( C \) is the unit circle, by finding the residues of the function inside the unit circle.

Answer: \pi


Question: How many integer points (x, y, z) satisfy the equation \(35x + 55y + 77z = 1\) and are contained within a cube of side length 30 centered at the origin (0, 0, 0)?

Answer: 10


Question: Let \( \{f_{n}\}_{n\geq 1} \) be an orthonormal basis for a separable Hilbert space \( H \). For any \( f \in H \), it is known that \( \sum_{n=1}^{\infty}|\langle f, f_{n}\rangle|^{2}=\|f\|^{2} \). Consider two disjoint subsets \( I \) and \( J \) of the integers \( \{1,2,3,...\} \) such that \( I\cup J=\{1,2,3,...\} \). Prove or disprove that the following holds: \[ \sum_{n\in I}|\langle f, f_{n}\rangle|^{2}+\sum_{n\in J}|\langle f, f_{n}\rangle|^{2}=\|f\|^{2}. \]

Answer: \sum_{n\in I}|\langle f, f_{n}\rangle|^{2}+\sum_{n\in J}|\langle f, f_{n}\rangle|^{2}=\|f\|^{2}


Question: Suppose $f_n \in L_{loc}^1(\mathbb{R}^n)$ are convex functions that converge to a function $f$ in $L_{loc}^1(\mathbb{R}^n)$. Is the function $f$ necessarily convex?

Answer: Yes


Question: Let $G = S_5$ and $g = (1 2 3)$. Determine the number of elements in the subgroup $H = \{x \in G \ :\ xg = gx\}$. What is $|H|$?

Answer: 6


Question: Determine whether there exists a homeomorphism \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(2x) = 3f(x) \) for all \( x \in \mathbb{R} \).

Answer: Yes


Question:
Let \( D = \text{diag}(d_1, \dots, d_n) \) be a real diagonal matrix, where \( 0 \le d_1 \le d_2 \le \dots \le d_n \). Let \( a_1 < a_2 < \dots < a_m \) be its distinct eigenvalues (counted without multiplicities). Suppose \( A \) is a real symmetric \( n \times n \) matrix satisfying \( A^2 = D \). Must \( A \) be of the form:
\[
A = \begin{pmatrix}
\sqrt{a_1} B_1 & & & & 0 \\
& \sqrt{a_2} B_2 & & & \\
& & \sqrt{a_3} B_3 & & \\
& & & \ddots & \\
0 & & & & \sqrt{a_m} B_m
\end{pmatrix}
\]
where \( B_i \) are symmetric and \( B_i^2 = I \), and the size of \( B_i \) corresponds to the multiplicity of \( a_i \) as an eigenvalue of \( D \)?

Answer: Yes


Question: A fair coin is tossed repeatedly, and the outcomes are recorded as a sequence of 'H' (heads) and 'T' (tails). Let $X$ be the number of tosses required for the sequence 'HTH' to appear for the first time. Calculate the expected value $\mathbb{E}(X)$. Provide your answer as a single number.

Answer: 10


Question: Determine the values of \(m\) and \(k\) for which the series \(\sum_{n=1}^{\infty} \frac{\sqrt[m]{n!}}{\sqrt[k]{(2n!)}}\) converges.

Answer: k \leq 2m


Question: Evaluate the double integral \( \iint_{Q} (x+y)^{2013} \, dx \, dy \), where \( Q \) is the square \([-1,1]^2 \subset \mathbb{R}^2\).

Answer: 0


Question: Find the smallest positive integer \( a \), greater than 1000, such that the equation \( \sqrt{a - \sqrt{a + x}} = x \) has a rational root.

Answer: 1057


Question: Prove that the Legendre symbol \( \left(\frac{4}{p}\right) \) is congruent to 1 for any odd prime \( p \).

Answer: 1


Question: Find all integer values of $x$ such that $54x^3 + 1$ is a perfect cube.

Answer: 0


Question: Find the limit of the sequence given by \( \frac{1}{\sqrt{n}} \left( 1 + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}} \right) \) as \( n \to \infty \).

Answer: 2


Question: Find all continuous functions \( f \) such that \( f(x) - 1999f\left(\frac{2x}{1-x^2}\right) = 18 \) for \(|x| \neq 1\).

Answer: -\dfrac{1}{111}


Question: Given two matrices \(A, B \in \mathrm{SL}_2(\mathbb{Z})\) that are conjugate, find a matrix \(C \in \mathrm{SL}_2(\mathbb{Z})\) such that \(A C = C B\).

Answer: C


Question: Evaluate the limit \( \lim\limits_{x \to 0}\frac {1}{x^3}\int_0^x \frac{t\ln (1+t)}{t^4+4}\mathrm{d}t \) without using L'Hôpital's rule.

Answer: \dfrac{1}{12}


Question: Prove that the set \( A = \{ f \in C([0,1], \mathbb{R}) \mid \max f(x) \le 1 \} \) is closed in the space of continuous functions on \([0,1]\).

Answer: A


Question: Find the constant \( c \) such that the random variable \( Y = \frac{c(X_1 + X_2)}{\sqrt{(X_3)^2 + (X_4)^2}} \) has a t-distribution.

Answer: 1


Question: Evaluate the limit \( \lim_{x\to 0}\frac{\int_0^{x^2}e^{-t^2}dt}{x^2} \).

Answer: 1


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

Let \( \{f_n\} \) be a sequence of integrable functions defined on an interval \([a,b]\). If \( f_n(x) \to 0 \) almost everywhere, then \( \int_{a}^{b} f_n(x) \, dx \to 0 \).

Answer: B


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

Answer: \dfrac{4}{e}


Question: The equation of the ellipse is \( \frac{(x-4)^2}{16} + \frac{(y-3)^2}{9} = 1 \). Find the equation of the curve that represents the reflection of this ellipse about the line \( x-y-2=0 \). The equation of the reflected curve is given by \( 16x^2 + 9y^2 + k_1 x - 36y + k_2 = 0 \). Calculate \( \frac{k_1 + k_2}{33} \).

Answer: 4


Question: Determine the value of \(c\) such that \(\lim_{x\to\infty} \left(\frac{x+c}{x-c}\right)^x = e\).

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{x \to \infty} x^2 \int_0^x e^{t^3-x^3} \, dt \]

Answer: \dfrac{1}{3}


Question: Evaluate the value of \( \lim_{R \to \infty} \int_{C_R} \frac {e^{iz}}{z}dz \), where \( C_R \) is a curve in the complex plane parameterized as \( \gamma(t)= Re^{it}, 0\leq t\leq\pi \).

Answer: 0


Question: Determine if there exists a real number \( \lambda \) such that \( e^{\lambda t}(\lambda u(t,x) + g(u(t,x))) \ge 0 \) for all \((t,x) \in (0,T) \times \Omega\), where \( T > 0 \), \( d \in \mathbb{N} \), \( \Omega \subseteq \mathbb{R}^d \) is bounded and open, \( u \in C([0,T] \times \overline{\Omega}) \cap C^{1,2}((0,T) \times \overline{\Omega}) \) is nonnegative, and \( g \in C^1(\mathbb{R}) \) with \( g(0) = 0 \).

Answer: Yes


Question: Let $f,g\colon\mathbb{D}^2\to\mathbb{C}$ be analytic functions in the bi-disc $\mathbb{D}^2$ with no common zeros. Determine whether there exist analytic functions $u,v\colon \mathbb{D}^2\to\mathbb{C}$ such that $uf+vg=1$.

Answer: Yes


Question:
Evaluate the limit using Maclaurin series:
\[
\lim_{x\to0}\frac{xe^{2x}+xe^{-2x}-4x^{3}-2x}{\sin^{2}x-x^{2}}
\]

Answer: 0


Question: Determine whether the following statement is true or false: If $q$ is a prime number and $x \nmid (q-1)$, then $2^x \not\equiv 1 \pmod{q}$. Justify your answer.

Answer: False


Question: Find a polynomial \( p(x) \) of degree \( \le 2 \) that approximates \( f(x) = x^{1/3} \) and minimizes the error \[ E = (f(-1)-p(-1))^2 + \int_{-1}^1 (f(x)-p(x))^2 \, dx + (f(1)-p(1))^2. \]

Answer: \dfrac{15}{14}x


Question: Evaluate the limit: \[ \lim_{x\to 3^+} \left\lfloor\frac{5^x-x^5}{2^x-x^2}\right\rfloor \] where \([x]\) denotes the floor function.

Answer: 118


Question: Find the uniformly most powerful (UMP) test at significance level \( \alpha = 0.05 \) for the hypothesis test concerning a geometric distribution with probability mass function \( P(X=k) = p^k(1-p) \), where \( 0 < p < 1 \). The null hypothesis is \( H_0: p \leq 0.95 \) and the alternative hypothesis is \( H_1: p > 0.95 \). Determine the rejection region for this test.

Answer: X \geq 59


Question: Let \( A, B, \) and \( C \) be the roots of the polynomial \( x^3 - 4x - 8 = 0 \). Find the numerical value of the expression \( \frac{A+2}{A-2} + \frac{B+2}{B-2} + \frac{C+2}{C-2} \).

Answer: 7


Question: Find the expectation of the Ito integral: \( \int_0^4 B_t^2 \, dB_t \).

Answer: 0


Question: Given the differential equation \( xy'' + y' + k^2xy = 0 \), transform it by setting \( x = \alpha t \) such that it becomes a Bessel equation. Determine the value of \( \alpha \) that achieves this transformation and verify the correct application of the chain rule in the process.

Answer: \dfrac{1}{k}


Question:
Let $x_1, x_2, \dots, x_k \ge 0$ be non-negative real numbers. Determine whether the following inequality holds:
\[ k \left( \sum_{i=1}^k x_i^3 \right)^2 \ge \left( \sum_{i=1}^k x_i^2 \right)^3. \]

Answer: Yes


Question: Evaluate the integral \( I = \oint_{|z|=1} \frac{\cos(z)}{z(e^{z}-1)} \, dz \) along the unit circle.

Answer: -\pi i


Question: Evaluate the integral: $$\int_{-\infty}^{\infty} x^2 \mathrm{e}^{-\frac{1}{2}x^2} \; \mathrm{d}x$$

Answer: \sqrt{2\pi}


Question: Evaluate \[ \lim_{n \to \infty}\dfrac{\dfrac{\ln2}{2}+\dfrac{\ln3}{3}+\cdots+\dfrac{\ln n}{n}}{\ln^2 n}. \]

Answer: \dfrac{1}{2}


Question: Evaluate the limit: $$\lim_{x \to \infty} \frac{\arctan(x+1) - \arctan(x)}{\sin\left(\frac{1}{x+1}\right) - \sin\left( \frac 1x\right)}$$

Answer: -1


Question: Assume that $P(z)$ and $Q(z)$ are complex polynomials such that $P(z) = Q(z)$ for all $z$ on the unit circle $S = \{z : |z| = 1\}$. Does it follow that there exist a polynomial $f(z)$, positive integers $m$ and $n$, and a complex number $w$ on the unit circle such that $P(z) = f(z^n)$ and $Q(z) = f(wz^m)$?

Answer: Yes


Question: Determine the number of solutions to the congruence equation \(x^2 \equiv -1211 \pmod{2020}\).

Answer: 8


Question: Let $A$ be a separable, simple, non-unital C*-algebra, and let $\varphi$ be an approximately inner automorphism on $A\otimes\mathcal{K}$. This means there exists a sequence of unitaries $v_n$ in the multiplier algebra $\mathcal{M}(A\otimes\mathcal{K})$ such that $v_n x v_n^* \to \varphi(x)$ for all $x \in A\otimes\mathcal{K}$. Consider the induced automorphism $\varphi$ on the multiplier algebra. Let $e_{11} \in \mathcal{K}$ be the rank one projection. Determine if $1\otimes e_{11}$ is Murray-von-Neumann equivalent to $\varphi(1\otimes e_{11})$ in the multiplier algebra $\mathcal{M}(A\otimes\mathcal{K})$. Assume $A$ is simple and non-unital.

Answer: Yes


Question: Let \( S = \{ x \in \mathbb{R} : 2x^2 \cos\frac{1}{x} = 1 \} \) and \( T = \{ x \in \mathbb{R} : 2x^2 \cos\frac{1}{x} \leq 1 \} \cup \{0\} \). Under the usual metric on \( \mathbb{R} \), which of the sets \( S \) or \( T \) is complete?

Answer: T


Question:
Let \( f(z) = z + \frac{1}{z} \) for \( z \in \mathbb{C} \) with \( z \neq 0 \). Which of the following is always true?

(a) \( f \) is analytic on \( \mathbb{C} \setminus \{0\} \)
(b) \( f \) is a conformal mapping on \( \mathbb{C} \setminus \{0\} \)
(c) None of the above

Answer: a


Question: Let $(B_{t})_{t \geq 0}$ be a one-dimensional Brownian motion and $(\phi(t,0))_{t \geq 0}$ be its local time at $0$. Determine whether $\lim_{t \uparrow \infty}\phi(t,0)=\infty$ almost surely.

Answer: \infty


Question: Find all functions \( f : \mathbb{Z} \to \mathbb{Q} \) such that \[ f \left( \frac{x+y}{3} \right) = \frac{f(x) + f(y)}{2} \] for all \( x, y \in \mathbb{Z} \) where \( \frac{x+y}{3} \in \mathbb{Z} \).

Answer: f(x) = c


Question: Find the limit \( \lim_{n\to\infty} \int_0^n(1+ \frac xn)^{-{n}} \log(2+\cos(x/n))dx \) and justify your reasoning using appropriate convergence theorems.

Answer: \log 3


Question: Given that \(\sin(\pi x) = a_0 + \sum\limits_{n=1}^{\infty} a_n \cos(n\pi x)\) for \(0 < x < 1\), find the value of \((a_0 + a_1)\pi\).

Answer: 2


Question:
Consider the initial value problem given by the partial differential equation:

\[ (x-y) u_{x} + (y-x-u) u_{y} = u \]

with the initial condition \( u(x,0) = 1 \). Determine which of the following expressions the solution satisfies:

1. \( u^2(x-y+u) + (y-x-u) = 0 \)
2. \( u^2(x+y+u) + (y-x-u) = 0 \)
3. \( u^2(x-y+u) - (x+y+u) = 0 \)
4. \( u^2(y-x+u) + (x+y-u) = 0 \)

Answer: 2


Question: Calculate \( \lim_{n\rightarrow \infty}\int_{1}^{\infty}{\frac{\sqrt{x}\log{nx}\sin{nx}}{1+nx^{3}}} \, dx \).

Answer: 0


Question: Let \( f(x) \in C[0,1] \) be a decreasing function and \( \lambda \in (0,1) \). Define \( I_1 = \int_{0}^{\lambda} f(x) \, dx \) and \( I_2 = \lambda \int_{0}^{1} f(x) \, dx \). Find the numerical relationship between \( I_1 \) and \( I_2 \).

Answer: I_1 \geq I_2


Question: Evaluate the integral \( \int\limits_0^{\pi}\log (1-\cos x)\, dx \).

Answer: -\pi \ln 2


Question: If \( \frac{\partial u}{\partial v} = a \), is it true that \( \frac{\partial v}{\partial u} = \frac{1}{a} \)?

Answer: \dfrac{1}{a}


Question: Evaluate the integral \( \int_{|z|=1} ze^{1/z^2} \, dz \) using the Cauchy Residue Theorem.

Answer: 2\pi i


Question:
Let $X$ be a Polish space and $(\mu_i)_{i=1}^{\infty}$ be a sequence of probability measures in the Wasserstein space $\mathcal{P}(X)$ on $X$. Let $(\beta_i)_{i=1}^{\infty}$ be a summable sequence in $(0,\infty)$. For every positive integer $k$, define the probability measures
$$
\nu_k = (\sum_{1\leq i\leq k}\beta_i)^{-1} \sum_{1\leq i\leq k} \beta_i \mu_i
$$
and define the probability measure
$$
\nu_{\infty} = (\sum_{i=1}^{\infty}\beta_i)^{-1} \sum_{i=1}^{\infty} \beta_i \mu_i.
$$
Determine if $\lim\limits_{k\to\infty} \mathcal{W}(\nu_k,\nu_{\infty}) = 0$, where $\mathcal{W}$ denotes the Wasserstein distance on $\mathcal{P}(X)$. Provide a quantitative justification for your answer.

Answer: 0


Question: Determine whether the function \( g : \ell_2 \to \mathbb{R} \) defined by \[ g(x) = \sum_{n=1}^{\infty} \frac{x_n}{n} \] is continuous.

Answer: Yes


Question: Let \( T_1: \mathbb{R}^n \rightarrow \mathbb{R}^m \) and \( T_2: \mathbb{R}^m \rightarrow \mathbb{R}^n \) be linear transformations such that the composition \( T_1 T_2 \) is bijective. Determine the ranks of \( T_1 \) and \( T_2 \).

Answer: m


Question: Find the derivative of the complex function \( f(z) = \ln z := \ln |z| + \arg (z)i \). Express the derivative in terms of \( z \).

Answer: \dfrac{1}{z}


Question: Determine whether the following statement is true or false: If $X^*AX$ is real for all $X$ in $\Bbb{C^{n}}$, then the matrix $A$ must be Hermitian.

Answer: True


Question: Find the highest lower bound for the probability of getting exactly one head when tossing $d$ pairwise independent coins, each with a probability of $1/d$ of landing heads.

Answer: \dfrac{1}{d}


Question: Evaluate the limit: \[ \lim_{x \to 0}\frac{\ln \cos 3x}{\ln \cos (-x)} \] without using L'Hospital's rule.

Answer: 9


Question: Given two real square matrices $A$ and $B$ such that $A^TA = B^TB$, determine if there exists an orthogonal matrix $Q$ such that $A = QB$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int\limits_0^{\infty} \frac{t^2dt}{(t^2+1)^2} \).

Answer: \dfrac{\pi}{4}


Question: Find the maximum value of $x^2y$ subject to the constraint $x^2 + 2y^2 = 6$.

Answer: 4


Question: Do two isomorphic finite field extensions have the same dimension over their base field? Specifically, if \(E = F(u_1, \cdots, u_n) \cong \bar{E} = F(v_1, \cdots, v_m)\), do the two extensions necessarily have the same dimension over \(F\)?

Answer: Yes


Question: Let $G$ be a $p$-solvable group, and let $M_{1}$ and $M_{2}$ be finite-dimensional projective irreducible $kG$-modules, where the characteristic of $k$ is $p$. Determine whether the tensor product $M_{1} \otimes_{k} M_{2}$ is a semisimple $kG$-module.

Answer: Yes


Question: What is the simply connected cover of the Lie group $SO(2,\mathbb{R})$?

Answer: \mathbb{R}


Question: Determine the Lebesgue measure of the set \(A = \{ (x,0) : x \in [0,1] \} \subset \mathbb{R}^2\).

Answer: 0


Question: In tetrahedron $DABC$, the volume is $9$ cubic units. Given that $\angle ACB = \frac{\pi}{6}$ and $2AD + AC + BC = 18$, find the length of $AD$. Express your answer as a single number.

Answer: 3


Question: Find the last four digits of the number \(2^{10^{2018}}\).

Answer: 9376


Question:
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for any real numbers \( x \) and \( y \), the following equation holds:
\[ f(f(x) + x f(y)) = f(x) + xy. \]

Answer: f(x) = x


Question: Determine the values of \(x\) for which the series \(\sum_{n=0}^{\infty} \frac{n!}{(1+n)^n}x^n\) converges.

Answer: (-e, e)


Question: Find the maximum value of the sequence \( \{a_i\}_{i=1}^{\infty} \) where \( a_i = \left\lfloor \frac{10^{i+1}}{7} \right\rfloor - 100 \left\lfloor \frac{10^{i-1}}{7} \right\rfloor \).

Answer: 85


Question: Evaluate the limit: $$ \lim_{x\to0}\left(\sqrt{\frac{1}{x} + \sqrt{\frac{1}{x} + \sqrt{\frac{1}{x}}}} - \sqrt{\frac{1}{x} - \sqrt{\frac{1}{x} + \sqrt{\frac{1}{x}}}}\,\right) $$

Answer: 1


Question: Find the value of $a + b$ if the equations $x^2 + y^2 - 12x - 6y - 4 = 0$ and $x^2 + y^2 - 4x - 12y - k = 0$ have simultaneous real solutions $(x, y)$ if and only if $a \leq k \leq b$.

Answer: 68


Question: Find the orthogonal complement $U^\perp$ of the subspace $U$ in the vector space $V$ of all continuous functions on $[0,1]$, where $U$ is defined as the set of functions $f$ such that $f(0) = 0$. The inner product on $V$ is given by $\langle f,g \rangle = \int_{0}^{1}f(x)g(x)dx$. Determine the set of functions in $U^\perp$.

Answer: \{0\}


Question: Compute the line integral \( \int_C \omega \), where \( \omega = -y\sqrt{x^2+y^2}dx + x\sqrt{x^2+y^2}dy \) and \( C \) is the circle \( x^2+y^2=2x \), using Green's Theorem.

Answer: \dfrac{32}{3}


Question: If $A$ is a negative definite symmetric matrix, can $A^2$ be negative definite? Justify your answer using the spectral decomposition theorem.

Answer: No


Question:
Consider the curve given by the parametric equations \( x^i = x^i(s) \) for \( i = 1, 2, \ldots, N \), where \( x^i \) are coordinates. Determine if the following expression is zero:
\[ \frac{\partial}{\partial x^i}\left(\frac{dx^j}{ds}\right) = 0 \]

Answer: 0


Question: Find the 4x4 matrix $A$ with entries $a_{ij} \in \{1, -1\}$ that maximizes the determinant $\det(A)$. What is the maximum value of $\det(A)$?

Answer: 16


Question:
Given the integral representation of the arccosine function:
\[ \arccos(x) = \int_{1}^{x} -\frac{1}{\sqrt{1 - t^2}} \, \mathrm{d}t \]
If \(\theta\) is a known angle such that:
\[ \theta = \int_{1}^{x} -\frac{1}{\sqrt{1 - t^2}} \, \mathrm{d}t \]
Determine if it is possible to solve for the upper bound \(x\) in the integral.

Answer: x = \cos(\theta)


Question: Evaluate the expression: \[ \sec \frac{2\pi}{7} + \sec \frac{4\pi}{7} + \sec \frac{6\pi}{7} \]

Answer: -4


Question:
Evaluate the integral \( \int_0^1 f^{2022}(x) \, dx \) where \( f(x) = x^3 - \frac{3}{2}x^2 + x + \frac{1}{4} \) and \( f^n(x) = f \circ f \circ \ldots \circ f(x) \) (\( n \) times). Choose the correct answer from the options below:

\( A) \ 2022^{3/2} \)

\( B) \ 0 \)

\( C) \ 2022 \)

\( D) \) None of the above.

Answer: D


Question: Given that the variables $a$, $b$, and $c$ are independent and identically distributed random variables with an exponential distribution with a common parameter $1$, calculate the probability that the polynomial $ax^2 + bx + c$ has real roots.

Answer: \dfrac{1}{3}


Question: Find the volume of the solid obtained by rotating the region bounded by the curves \( x = y^2 \) and \( x = 1 - y^2 \) about the line \( y = 3 \).

Answer: 4\sqrt{2}\pi


Question:
Solve the following system of equations to find the maximum value of the function:

Maximize \( f(x, y) = xy \) subject to:
\[ \begin{cases} x^2 + y^2 + z^2 = 1 \\ x + y + z = 0 \end{cases} \]

Using the Lagrangian method, solve the system:
\[ \begin{cases} y + 2\lambda_1 x + \lambda_2 = 0 \\ x + 2\lambda_1 y + \lambda_2 = 0 \\ x^2 + y^2 + z^2 = 1 \\ x + y + z = 0 \end{cases} \]

Answer: \dfrac{1}{6}


Question: Determine the number of distinct roots of the polynomial \( p(x) = x^{12} + 2x^6 + 1 \) over the field \( \mathbb{F}_3 \).

Answer: 0


Question: Find the natural number \(k\) that maximizes the function \(f(k) = \frac{k^2}{1.001^k}\). No calculus or calculators are allowed.

Answer: 2001


Question: Evaluate the integral \( \int_{[0,1]^2} \frac{1}{x+y} \, d\lambda_2(x,y) \).

Answer: 2 \ln 2


Question: Is there a lattice $L$ of cardinality $\kappa$, where $\kappa$ is an infinite cardinal, such that $L$ contains no prime ideals?

Answer: Yes


Question: Find the maximum absolute interpolation error over the interval \([-1, 1]\) when the polynomial \(x^4\) is approximated by a polynomial of degree less than or equal to 2, which interpolates \(x^4\) at \(x = -1, 0, 1\).

Answer: \dfrac{1}{4}


Question: Does convergence in probability of a sequence of random variables $X_n$ to a random variable $X$, with $X \in L^2$, imply convergence in $L^2$? Specifically, if $X_n \stackrel{Prob}{\longrightarrow} X$ and $X \in L^2$, does it follow that $X_n \stackrel{L^2}{\longrightarrow} X$?

Answer: No


Question: Suppose that \( \sum_{j=1}^{\infty} a_j \) is a convergent series with \( a_j \geq 0 \). Determine whether it is true that \( \lim_{n \to \infty} S_n = 0 \), where \( S_n = \sum_{j=1}^{n-1} \frac{a_j}{n-j} \).

Answer: 0


Question: Determine whether there exists a unique function \( f \in C([0, 1]) \) such that for all \( x \in [0, 1] \), the equation \[ f(x) = \int_0^x K(x, y)\,f(y)\,dy + g(x) \] holds, given that \( K \in C([0, 1] \times [0, 1]) \) and \( g \in C([0, 1]) \).

Answer: Yes


Question: Calculate the expected sojourn time for cars that decide to take gas at a station where cars arrive according to a Poisson process with rate \( \frac{1}{3} \). An arriving car finding \( n \) cars at the station leaves with probability \( q_n = \frac{n}{4} \) and joins the queue with probability \( 1 - q_n \), for \( n = 0, 1, 2, 3, 4 \). Cars are served in order of arrival, and the service time is \( \text{Exponential}(\frac{1}{3}) \).

Answer: \dfrac{384}{71}


Question: Does there exist an equivalent renorming of the space $C[0,1]$ of real-valued continuous functions on $[0,1]$ such that the dual norm is strictly convex?

Answer: Yes


Question: Let $X_1, \ldots, X_n$ be independent and identically distributed observations from a $N(0,1)$ distribution. Define $\overline{X} = \frac{1}{n}\sum_{i=1}^n X_i$ and $S^2 = \frac{1}{n}\sum_{i=1}^n (X_i - \overline{X})^2$. Determine if $\sqrt{n}\sup_x |\Phi(\frac{x-\overline{X}}{S}) - \Phi(x)| \stackrel{p}{\to} 0$, where $\Phi(\cdot)$ is the standard normal cumulative distribution function.

Answer: No


Question:
The army needs to test \( n \) soldiers for a disease. There is a blood test that accurately determines when a blood sample contains blood from a diseased soldier. The army presumes, based on experience, that the fraction of soldiers with the disease is approximately equal to some small number \( p \).

Approach (1) is to test blood from each soldier individually; this requires \( n \) tests. Approach (2) is to randomly group the soldiers into \( g \) groups of \( k \) soldiers, where \( n = gk \). For each group, blend the \( k \) blood samples of the people in the group, and test the blended sample. If the group-blend is free of the disease, we are done with that group after one test. If the group-blend tests positive for the disease, then someone in the group has the disease, and we need to test all the people in the group for a total of \( k+1 \) tests on that group. Since the groups are chosen randomly, each soldier in the group has the disease with probability \( p \), and it is safe to assume that whether one soldier has the disease is independent of whether the others do.

What is the expected number of tests in Approach (2) as a function of the number of soldiers \( n \), the disease fraction \( p \), and the group size \( k \)?

Additionally, show how to choose \( k \) so that the expected number of tests using Approach (2) is approximately \( n\sqrt{p} \). Hint: Since \( p \) is small, you may assume that \( (1 - p)^k \approx 1 \) and \( \ln(1-p) \approx -p \).

Answer: n\sqrt{p}


Question:
Given that \( f(t) \) is a differentiable function satisfying \( \int_{t}^{f(t)}\sqrt{1+ 9x^4} \, dx = 1 \) for \( 0 < t < f(t) \), find the limit:
\[ \lim _{t \to \infty}t^3 \left(1-\{f'(t)\}^2 \right) \]

Answer: \dfrac{4}{3}


Question: Evaluate the integral \( I = \iiint_V (1-x) \,dV \) where \( V \) is the region defined by \( x, y, z \ge 0 \) and below the plane \( 3x + 2y + z = 6 \).

Answer: 3


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} \int_0^{+\infty} \frac{e^{-n^2x}}{\sqrt{|x-n^2|}} \, dx \] Use the hint: for \( n > 2 \), consider the integral as \[ \int_0^{+\infty} = \int_0^1 + \int_1^{n^2-n} + \int_{n^2-n}^{n^2+n} + \int_{n^2+n}^{+\infty} \]

Answer: 0


Question: Find the minimal value of the expression: \[ \log_{\frac{3}{2}x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\log_{\frac{3}{2}x_{2}}\left(\frac{1}{2}-\frac{1}{36x_{3}^{2}}\right)+\cdots+ \log_{\frac{3}{2}x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right) \] where \(x_{1}, x_{2}, \ldots, x_{n}\) are real numbers in the interval \(\left(\frac{1}{4},\frac{2}{3}\right)\).

Answer: 2n


Question: Evaluate the limit: \[ \lim_{(x,y)\to(0,0)}\frac{1-\cos(x^2+y^2)}{(x^2+y^2)x^2y^2} \]

Answer: \infty


Question: Integrate the complex function \( f(z) = xy \) over the circle \( C = [0, r] \).

Answer: 0


Question:
Evaluate whether the following limit holds true:
\[ \lim\limits_{n\to\infty} \int_{-\infty}^{+\infty}f(x)\sin(nx)dx = 0 \]
when \( \int_{-\infty}^{+\infty}f(x)dx \) is convergent but not absolutely convergent.

Answer: 0


Question: Determine if there exists an odd prime integer \( q \) such that \( p = \frac{q^3+1}{2} \) is also a prime number.

Answer: No


Question: Consider the logistic sequence defined by \( x_{n+1} = r x_n (1 - x_n) \) with \( r = 1 \) and initial condition \( x_1 = \frac{1}{2} \). Determine the rate at which this sequence converges to zero. Specifically, prove or disprove that \( \lim_{n \to \infty} n x_n = 1 \).

Answer: 1


Question: Six points \( A, B, C, D, E, F \) are placed in order on a circle, and the line segments \( AD, BE, CF \) intersect at a single point. Given that the lengths of the segments are \( AB = 1, BC = 2, CD = 3, DE = 4, EF = 5, \) and \( FA = \frac{k}{8} \), find the value of \( k \).

Answer: 15


Question: Evaluate the double integral: \[ \int_{R=0}^{\infty}\int_{\theta=0}^{2\pi}R\sin\theta \cos(gR\sin\theta)\,dR\,d\theta \]

Answer: 0


Question: Solve for $x$ in the equation: \( \sin^{-1}\left(\frac{2x}{1+x^2}\right) = 2\tan^{-1}(x) \).

Answer: [-1, 1]


Question: Calculate the expected value of \( \frac{S_1}{S_1+S_2} \), where \( S_1 = X_1 + X_2 \) and \( S_2 = X_2 + X_3 \), and \( X_k \) are Laplace-distributed random variables defined on \( \Omega = \{1, \ldots, 6\}^3 \) with \( X_k(\omega_1, \omega_2, \omega_3) = \omega_k \) for \( k \in \{1, 2, 3\} \).

Answer: \dfrac{1}{2}


Question: Let \( A \) be an \( n \times n \) symmetric invertible matrix with real positive elements. Determine the maximum number of zero elements in \( A^{-1} \).

Answer: (n-1)(n-2)


Question: Find the value of \(a\) such that \[ \lim_{n\to\infty} \frac{1^a+2^a+3^a+\cdots+n^a}{(n+1)^{a-1}[(na+1)+(na+2)+(na+3)+\cdots+(na+n)]} = \frac{1}{60}. \]

Answer: 7


Question: Find the differential of the integral \( \int_0^t B_s \, ds \) with respect to time \( t \), where \( B_s \) is the standard Brownian motion at time \( s \).

Answer: B_t \, dt


Question: Calculate the flow of the vector field \( \overrightarrow{V}=x \overrightarrow{i}+y\overrightarrow{j}+z\overrightarrow{k} \) through the lateral surface of the cylinder defined by \( x^2+y^2=a^2 \) and \( 0 \le z \le H \), where \( a, H > 0 \).

Answer: 2\pi a^2 H


Question:
Consider the function \( f(z) = e^{\frac{1}{z - \sin z}} \). Determine the nature of the singularity of \( f \) at \( z = 0 \). Choose one of the following options:

(a) \( f \) has a pole at \( z = 0 \).
(b) \( f \) has an essential singularity at \( z = 0 \).
(c) \( f \) has a removable singularity at \( z = 0 \).
(d) \( f \) is analytic at \( z = 0 \).

Answer: B


Question: Find the parameter $a$ such that the sequence defined by $x_0 = a$ and $x_{n+1} = x_n^2 - 4x_n + 6$ converges. What is the condition for convergence?

Answer: [1, 3]


Question: Find the limit: \[ \lim_{n \to \infty} \sqrt{\frac{2^n+20^n-7^{-n}}{(-3)^n+5^n}} \]

Answer: \infty


Question: Given the differential equation \( y'' + 2ay' + b^2 y = \cos(x) \) with parameters satisfying \( 0 \leq a < b < 1 \), find the unique $2\pi$-periodic solution. Determine the value of \( a \) for which the amplitude of this solution is maximized.

Answer: 0


Question: Calculate the arc length of the curve defined by \( \sqrt[3]{x^2} + \sqrt[3]{y^2} = \sqrt[3]{9} \).

Answer: 18


Question: Evaluate the integral \( \int_3^{11} \frac{dx}{(x+5)\sqrt{1+x}} \).

Answer: \dfrac{\pi}{12}


Question: If \( \lim\limits_{x\to \infty} \frac{f(x)}{g(x)}=1 \), does it follow that \( \lim\limits_{x\to\infty} \frac{\log f(x)}{\log g(x)}=1 \)? Justify your answer.

Answer: No


Question: Find the maximum value of the expression \(8 \cdot 27^{\log_{6}x} + 27 \cdot 8^{\log_{6}x} - x^3\) for \(x > 0\).

Answer: 216


Question:
Find the 3rd derivative of \( \cos(x) \) using the formula:
\[ \frac{d^n}{dx^n}f(x)=\lim_{\epsilon\to0}\frac{1}{\epsilon^n}\sum_{j=0}^n{((-1)^j\frac{\Gamma(n+1)}{j!\Gamma{(n+1-j)}}f(x-j\epsilon))} \]
where \( \Gamma(x) \) is the Gamma function.

Answer: \sin(x)


Question:
Show that the determinant of the following matrix is zero:
\[
\begin{vmatrix}
1 & \wp(v) & \wp'(v) \\
1 & \wp(w) & \wp'(w) \\
1 & \wp(v+w) & -\wp'(v+w)
\end{vmatrix} = 0
\]
where \(\wp\) denotes the Weierstrass elliptic function.

Answer: 0


Question: Let \( \tau \) be a fixed transcendental number. Can any transcendental number \( \sigma \) be expressed in the form \( \tau^n + m \), where \( n \in \mathbb{N} \) and \( m \in \mathbb{Z} \)?

Answer: No


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

Let \(n > 1\) be an integer, and let \(H = (V, E)\) be a hypergraph such that for all finite \(E_0 \subseteq E\), the hypergraph \((V, E_0)\) can be colored with \(n\) colors. Then \(H\) can be colored with \(n\) colors.

Answer: True


Question: Determine the number of real roots of the equation \( e^x + e^{-x} = 2\sin(x^3) \).

Answer: 0


Question: Does $X_n = O_P(a_n)$ and $a_n \to 0$ imply $X_n \stackrel{a.s.}{\to} 0$? Provide a justification for your answer.

Answer: No


Question: Let $R$ be a principal ideal domain (PID). Consider the sets $X_0=\{v_0,v_1,v_2\}$ and $X_1=\{e_1,e_2,e_3\}$, and let $C_i$ be the free $R$-module on $X_i$ for $i=0,1$. Define the $R$-module homomorphism $f: C_1 \to C_0$ by $f(e_1)=v_1-v_0$, $f(e_2)=v_2-v_1$, and $f(e_3)=v_0-v_2$. Prove that the quotient module $C_0/\operatorname{Im}(f)$ is isomorphic to $R$.

Answer: R


Question: Let \( G = \{z : 0 < |z| < 1\} \) and \( f:G\to\mathbb{C} \) be an analytic function. Suppose \( \gamma \) is a closed rectifiable curve in \( G \) such that the winding number \( n(\gamma,a)=0 \) for all \( a \) in \( \mathbb{C}\setminus G \). What is the value of \( \int_{\gamma}f \, dz \)?

Answer: 0


Question: If $XYZ=ZXY$, does $e^Xe^Ye^Z=e^Ze^Xe^Y$ hold for matrices $X$, $Y$, and $Z$?

Answer: No


Question: Find an analytical function $f(z)$ such that $f(z + \sin i) = f(z)$ and $f(z + \cos i) = f(z)$. Provide the function that satisfies these conditions.

Answer: f(z) = C


Question: Maximize the function \( f(x_1, x_2, x_3) = x_2 + x_3 - (x_2^2 + x_3^2) \) subject to the constraints \( x_1 + x_2 + x_3 = 1 \) and \( x_i > 0 \) for \( i = 1, 2, 3 \).

Answer: \dfrac{1}{2}


Question: Find the maximum value of \(|f(z)|\) for the function \(f(z) = z^2 + iz + 3 - i\) within the domain \(|z| \leq 1\).

Answer: 2\sqrt{5}


Question: Determine whether the following statement is true or false: If random vectors $X_{n} \xrightarrow{d} X$, then the coordinates $X_{n,i} \xrightarrow{d} X_{i}$. Provide a justification for your answer.

Answer: True


Question: For any positive definite matrix $A$, does there exist some $m > 0$ such that $A - mI$ is positive semi-definite?

Answer: Yes


Question: Compute the degree of the zero cycle in the Fano variety of lines on a smooth quadric $Q \subset \mathbb{P}_k^3$, where $F_{osc} = \{[l] \in F(Q), \exists P\text{ plane such that } 2l \subseteq P \cap Q\}$.

Answer: 0


Question: Let \( \sigma_k(n) = \sum_{d \vert n} d^k \) be the sum of the \( k^{th} \)-powers of the divisors of \( n \). Let \( \nu_2(x) \) denote the 2-adic valuation of the integer \( x \). Fix \( k, \ell \in \mathbb{N} \). If \( k \) and \( \ell \) have the same parity, is it true that \( \nu_2(\sigma_k(n)) = \nu_2(\sigma_{\ell}(n)) \)?

Answer: Yes


Question: Let $k$ be an infinite field and $f(X,Y) \in k[X,Y]$ be a polynomial with two variables. If $f(X,Y)$ vanishes on all points $(x,y) \in k^2$, i.e., \(f(x,y) = 0, \forall (x,y) \in k^2\), can we conclude that $f(X,Y) = 0$? In other words, is $k[X,Y]$ isomorphic to the ring of polynomial functions $k[X,Y]^* := \{f : k^2 \longrightarrow k, (x,y) \mapsto f(x,y) \mid f(X,Y) \in k[X,Y]\}$?

Answer: Yes


Question: Determine the number of natural numbers \( n \) such that there is no natural number \( M \) for which \( n^2 \) can be expressed as a sum of distinct squares of natural numbers \( a_i \).

Answer: 6


Question: Determine if there exists a number \( n \) such that when \( n \) is concatenated with itself twice, the resulting number is a perfect square. Specifically, find if there exists \( n \) such that \( n = a_0 a_1 \ldots a_0 a_1 \ldots a_0 a_1 \ldots (10) = a \cdot 10^{2m} + a \cdot 10^m + a = k^2 \) where \( 10^{m - 1} \le a < 10^m \).

Answer: No


Question: Evaluate the limit: \[ \lim_{n\to\infty}\sum_{i=0}^{n} \left(\frac{i}{n}\right)^{1.5} \]

Answer: \infty


Question:
Can the expression \( \sum_i{d(m_i,Pn_i)^2} \) be minimized over \( P \) using linear least squares? Suppose \( P \) is a \( 2 \times 2 \) matrix and both \( m_i \) and \( n_i \) are given 2-dimensional vectors in Cartesian coordinates, where \( d \) is the Euclidean distance. Is the following formulation correct for minimizing the expression?

Rewrite the problem as \( b - Ax \) and minimize over \( x \):

\[ i \leq N \]
\[ P = \begin{pmatrix}p_1 & p_2 \\ p_3 & p_4\end{pmatrix} \mapsto x = \begin{pmatrix}p_1 \\ p_2 \\ p_3 \\ p_4\end{pmatrix} \]
\[ b \in \mathbb{R}^{2N}; \quad b = \begin{pmatrix}m_{1,1} \\ m_{1,2} \\ m_{2,1} \\ m_{2,2} \\ \vdots \\ m_{N,2}\end{pmatrix} \]
\[ A \in \mathbb{R}^{2N \times 4} \]
\[ A = \begin{pmatrix}n_{1,1} & n_{1,2} & 0 & 0 \\ 0 & 0 & n_{1,1} & n_{1,2} \\ n_{2,1} & n_{2,2} & 0 & 0 \\ 0 & 0 & n_{2,1} & n_{2,2} \\ \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & n_{N,1} & n_{N,2}\end{pmatrix} \]

Answer: Yes


Question: Calculate the variance of the expression \(12(X_1 + X_2 + X_3)\), where \(X_1\) is a random variable uniformly distributed on \((0,1)\), \(X_2\) is uniformly distributed on \((0,X_1)\), and \(X_3\) is uniformly distributed on \((0,X_2)\).

Answer: \dfrac{565}{12}


Question:
Let \( f \) be a homogeneous polynomial with integral coefficients in four variables \( a, b, c, \) and \( d \). Suppose \( f \) is invariant under simultaneous rotation of \((a,b)\) and \((c,d)\) by the same angle. Define \( V = f^{-1}(0) \). For \( R > 0 \), let \( B_R \) be the ball of radius \( R \) in \( \mathbb{R}^4 \). Determine the smallest exponent \( n \) such that:
\[ \#(V \cap B_R \cap \mathbb{Z}^4) \le C_{f,\epsilon}R^{n+\epsilon}, \]
where \( C_{f,\epsilon} \) is a constant depending only on \( f \) and \( \epsilon \). Specifically, is it possible to take \( n = 2 \)?

Answer: 2


Question: Calculate the double integral \( \int\int_D (2x^2+y)\,dx\,dy \) where \(D\) is the region bounded by the curves: \(x = 0\), \(x= 1\), \(y= 0\), \(y=\frac{1}{x}\), and \(y = x^2 + 1\). Use the transformation \(\begin{cases} u = xy \\ v = y -x^2 \end{cases}\) to evaluate the integral over the new region \(E\) defined by \(\begin{cases} u-1 \leq v \leq 1 \\ 0 \leq u \leq 1 \end{cases}\).

Answer: \dfrac{3}{2}


Question:
A point is randomly selected on each side of a unit square with vertices at $(0,0)$, $(0,1)$, $(1,1)$, and $(1,0)$. Let $x_1$ and $x_2$ be the abscissae of the points on the lower and upper sides, respectively, and $y_1$ and $y_2$ be the ordinates of the points on the right and left sides, respectively. The area of the quadrilateral formed by these points is given by:
\[ 1 - \frac{1}{2}(x_1y_2 + (1-x_1)y_1 + (1-y_1)(1-x_2) + x_2(1-y_2)). \]
Find the mathematical expectation of this area.

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\sum_{k=1}^n \arcsin\frac{k}{n^2} \]

Answer: \dfrac{1}{2}


Question: Two natural numbers $x$ and $y$ are chosen at random. What is the probability that $x^2 + y^2$ is divisible by 10?

Answer: \dfrac{9}{50}


Question: Evaluate the limit: \[ \lim \limits_{x \to 0} \frac {e^{-1/x^2}}{x} \] without using L'Hôpital's rule.

Answer: 0


Question: Determine the smallest number of same-suited cards, n, out of 9 cards (2 cards for each of 2 players and 5 community cards) such that there is a 50% probability that at least one player has a flush.

Answer: 6


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

Answer: \dfrac{1}{e}


Question: Evaluate the integral \( \int_{-\infty}^{\infty} u(uu')'\,dx \) given that \( u = u(x) \), \( u' = \frac{du}{dx} \), and \( u(x) \rightarrow 0 \) and \( u'(x) \rightarrow 0 \) as \( x \rightarrow \pm \infty \).

Answer: 0


Question: Find the solution to the equation \( \cos x = x \cos(x - \cos x) \). Express your answer as Dottie's number, \( D \).

Answer: D


Question: Determine the convergence order of the sequence \(\left(1+\frac{1}{n}\right)^{\frac{1}{2}}\).

Answer: 1


Question: Determine for which \( \alpha > 0 \) the series \( \sum_{n=1}^{\infty} \frac{ne^n - \log(1+n)}{n^{\alpha+1}} \) converges.

Answer: \emptyset


Question: Is the argument that the cardinality of \( V_\omega \) is \( \aleph_0 \) valid, given that each \( V_n \) in the von Neumann Hierarchy has finitely many elements and \( V_\omega = \bigcup_{n < \omega} V_n \)?

Answer: \aleph_0


Question: Evaluate the limit: $$\lim_{x\rightarrow0} \frac{\tan^3(3x)-\sin^3(3x)}{x^5}$$

Answer: \dfrac{729}{2}


Question:
Consider a function \( f \) that is discontinuous on a subset of \( \mathbb{R} \). Which of the following subsets makes it impossible for \( f \) to be discontinuous?

A. Empty set
B. Rational numbers
C. Irrational numbers
D. Positive real numbers
E. Real numbers

Select the correct answer.

Answer: C


Question: Given an infinite set $X$, does there exist two distinct sets that are not elements of $X$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \rightarrow 0} \frac{1}{x^4} \int_{\sin{x}}^{x} \arctan{t} \, dt \]

Answer: \dfrac{1}{6}


Question: Let a complex-valued function \( f \) be analytic on \( D=\{z\in\mathbb{C}:R_1<|z-z_0|<R_2\} \) where \( 0<R_1,R_2<\infty. \) Moreover, in \( D, \) \( f \) can be represented as \[ f(z)=\sum_{n=-\infty}^\infty a_n(z-z_0)^n \quad (z\in D). \] Is this the Laurent series expansion of \( f \) in \( D? \)

Answer: Yes


Question: Given a sequence \((x_n)\) defined by \(x_1 > 0\) and \(x_{n+1} = -\ln(x_1 + x_2 + \cdots + x_n)\), find the sum of the series \(\sum_{n=1}^\infty x_n\).

Answer: 1


Question: Let \( f:[-1,1] \to \mathbb{R} \) be a continuously differentiable function such that \( \int_{-1}^{1} f(x)\,dx = \pi \) and \( \int_{-1}^{1} f'(x) x^n\,dx = 0 \) for all integers \( n \geq 2014! \). Determine the value of \( f(0) \).

Answer: \dfrac{\pi}{2}


Question: Determine the number of automorphisms of the ordered field of real numbers \((\mathbb{R}, 0, 1, +, \cdot, \leq)\).

Answer: 1


Question:
Compute the limit of the sequence defined by \( x_0 \) as a real number and \( x_{n+1} = x_n + e^{-x_n} \):
\[ \lim_{n\to\infty} \frac{x_n}{\ln {n}} \]

Answer: 1


Question: If \( u = u(x) > 0 \) is a smooth non-constant function that is subharmonic in \( \mathbb{R}^n \), can we conclude that \( u \) is unbounded in \( \mathbb{R}^n \)? Provide a justification for your answer.

Answer: Yes


Question: Let \( f : \mathbb{R} \to \mathbb{R} \) be a function such that \( \frac{f(x) f(y) - f(xy)}{3} = x + y + 2 \) for all \( x, y \in \mathbb{R} \). Find the function \( f(x) \).

Answer: x + 3


Question:
Determine the value of \( n = m \) such that the width of the 95% confidence interval for \( p_1 - p_2 \) is at most 0.1, regardless of the sampling results. The confidence interval is given by:
\[ \hat{p}_1 - \hat{p}_2 \pm z_{0.025} \sqrt{\frac{\hat{p}_1\hat{q}_1}{m} + \frac{\hat{p}_2\hat{q}_2}{n} } \]
where \( \hat{p}_1 = \frac{X}{m} \) and \( \hat{p}_2 = \frac{Y}{n} \), with \( X \sim \text{Binomial}(m, p_1) \) and \( Y \sim \text{Binomial}(n, p_2) \).

Answer: 769


Question: Evaluate the double integral of the floor function over the rectangular region: \[ \int_{1}^{3} \int_{2}^{5} \lfloor x+y \rfloor \, dy \, dx \]

Answer: 30


Question: Calculate the expected value \( \mathbb{E}\left[\max\{X_1, X_2, X_3\} \min\{X_1, X_2, X_3\}\right] \), where \( X_1, X_2, X_3 \) are independent and identically distributed random variables uniformly distributed over the interval \([0, 1]\).

Answer: \dfrac{1}{5}


Question: The first quadrant region enclosed by the x-axis and the graph of \( y = ax - x^2 \) traces out a solid of the same volume whether it is rotated about the x-axis or the y-axis. Determine the value of \( a \).

Answer: 5


Question: Determine the domain in the complex plane where the function \( f(z) = \int\limits_{-1}^{1} \frac{e^{tz}}{1+t^2} \, dt \) is defined and holomorphic.

Answer: \mathbb{C}


Question: Given a function $f :[0,1]\rightarrow \mathbb{R}$ satisfying \(\int_0^1|f(t)|dt<\infty\), and for a given \(\epsilon >0\), there exists a continuous function \(g\) on \([0,1]\) such that \(\|f-g\|_{L^1([0,1])}<\epsilon\). Can we find such a continuous function \(g\) such that \(Z(f)\subseteq Z(g)\), where \(Z(f)\) is the zero set of \(f\)?

Answer: No


Question: Find all values of \( \alpha \in \mathbb{R} \) such that for every continuous function \( f: [0,1] \to [0,1] \), there exists a \( c \in [0,1] \) satisfying \( f(c) = \alpha c \).

Answer: [1, \infty)


Question:
Find the radius of convergence of the series:
\[ \sum \frac{(K/e)^{3K}}{3K!}(7x)^K. \]

Answer: \dfrac{27}{7}


Question: Determine the smallest integer $m_n$ such that if $m_n$ points are chosen inside an equilateral triangle with side length 1, at least two of them are at a distance of $\frac{1}{n}$ or less.

Answer: n^2 + 1


Question: Find the value of \(x\) such that \(P_k(m,n) = P(x, n)\), where \(P_k(m,n)\) is the number of partitions of \(m\) with exactly \(n\) parts, each part at least \(k\). Express \(x\) in terms of \(m\), \(k\), and \(n\).

Answer: m - n(k - 1)


Question: Find the polynomial $p \in P_2$ that minimizes the $L^2$ norm $||x^3 - p||$. Use the orthogonal basis $\{1, x, x^2 - \frac{2}{6}\}$ and express $p$ as $\alpha_0 + \alpha_1 x + \alpha_2 (x^2 - \frac{2}{6})$, where $\alpha_i = \frac{\langle x^3, \psi_i \rangle}{\langle \psi_i, \psi_i \rangle}$. Compute the coefficients $\alpha_i$ and determine $p$. \(\langle \cdot, \cdot \rangle\) denotes the inner product.

Answer: \dfrac{3}{5}x


Question: Determine whether \(3^{1/2}\) is an element of the field \(\mathbb{Q}(3^{1/3}+3^{1/2})\).

Answer: Yes


Question: Given three periodic trigonometric functions $f_1(x)$, $f_2(x)$, and $f_3(x)$, where $x > 0$, the function $F(x) = f_1(x) \cdot f_2(x) \cdot f_3(x)$ is defined. It is known that $f_1(x)$ is positive for $\frac{2}{3}$ of its domain, $f_2(x)$ is positive for $\frac{1}{2}$ of its domain, and $f_3(x)$ is positive for $\frac{1}{2}$ of its domain. Determine the portion of the domain for which $F(x)$ is positive.

Answer: \dfrac{1}{2}


Question: Let $A$ and $B$ be $3 \times 3$ matrices such that $\det(A) = 0$, and every element $b_{ij}$ of matrix $B$ is 1. It is also given that $\det(A + B) = 1$. Find $\det(A + 2014B)$. \( \text{Express your answer as a single number.} \)

Answer: 2014


Question: Suppose we have a finite-dimensional vector space and a linear map $T$ such that for any vector $v$, the norm of $T(v)$ is not greater than the norm of $v$. Determine whether the linear map $T - \sqrt{2} I$ is surjective.

Answer: Yes


Question: Given the ciphertexts \((c_1, c_2, c_3) = (13, 19, 3)\) and the corresponding plaintexts \((m_1, m_2, m_3) = (5, 10, 23)\), determine the modulus \( p \) used in the affine cipher.

Answer: 79


Question: Find the limit of the sequence \( \prod_{k=1}^{n} \frac{2k-1}{2k} \) as \( n \to \infty \).

Answer: 0


Question: Find the principal value of \( \left[\frac{e}{2}(-1-\sqrt{3}i)\right]^{3\pi i} \).

Answer: -e^{2\pi^2}


Question: Given a natural number \( N \), determine if there exists a self-complementary graph \( G \) such that the maximum degree \( \Delta(G) \) is at least \( N \) times the minimum degree \( \delta(G) \).

Answer: Yes


Question: Compute the limit \( \lim_{n \to \infty} \frac{n!}{n^n} \).

Answer: 0


Question: If $a_n$ is a strictly increasing unbounded sequence, does the series $\sum_n \frac{a_{n+1} - a_n}{a_n}$ necessarily diverge?

Answer: Yes


Question: Determine the number of subfields of the splitting field of the polynomial $x^5 - 5$ over $\mathbb{Q}$. Assume that the splitting field is a Galois extension of $\mathbb{Q}$ and that the Galois group has order 20.

Answer: 14


Question: Evaluate the limit: \[ \lim_{(x,y) \to (0,0)} \frac{e^{-1/(x^2+y^2)}}{x^4+y^4} \]

Answer: 0


Question: Determine the number of tangents to the curve \( y = f(x) \) that are parallel to the line \( x + y = 0 \) for \( x > 0 \), where \( f(x) = \int_{1}^{x} \left( \sqrt{\log t} - \frac{1}{2}\log\sqrt{t} \right) dt \).

Answer: 1


Question:
Given a circle with center $C$ and radius $a$, points $P$ and $Q$ lie on the circle such that the angle $\widehat{PCQ}$ is $2\theta$. Determine the value of $\sin\theta$ that maximizes the inradius of triangle $CPQ$. Choose from the following options:

- $\sin\theta=\frac{\sqrt3-1}{2\sqrt2}$
- $\sin\theta=\frac{\sqrt5-1}{2}$
- $\sin\theta=\frac{\sqrt5+1}{2}$
- $\sin\theta=\frac{\sqrt5-1}{4}$

Answer: B


Question: Evaluate the integral \( \int_{0}^{2\pi}|2e^{it}-1|^2 2ie^{it} \, dt \).

Answer: -8\pi i


Question: Evaluate \( \lim_{x\to\infty}\left(\frac{1}{x}+2^{\frac{1}{x}}\right)^x \).

Answer: 2e


Question: Determine whether the limit \( \lim\limits_{(x,y)\to(1, 1)} \frac{\tan(y-1)\sin^2(x-y)}{(x-1)^2+(y-1)^2} \) exists.

Answer: 0


Question: Determine the number of conjugacy classes in a group of order 625, given that its center has an order of 25.

Answer: 145


Question:
Evaluate the limit if it exists:
\[ \lim\limits_{h \to 0} \frac{f(h)-f(2h)+f(3h)-f(4h)+\ldots+f(2013h)}{h} \]
Given that \( f(0) = 0 \) and \( f'(0) = -1 \).

Answer: -1007


Question: Given the equation \(x^2 + xy + y^3 = 1\), find the value of \(y'''(1)\).

Answer: 42


Question: Determine the number of points $z$ in the unit disk such that $e^z = 3z^4$.

Answer: 4


Question: Evaluate the limit: \[ \lim_{x \to 1} \frac{2^{\sin(\pi x)} - 2^{\tan(\pi x)}}{x-1} \] without using L'Hospital's rule or derivatives.

Answer: -2\pi \ln 2


Question: Let \( \vec{v} = \vec{i} \times (\vec{j} \times (\vec{k} \times (\vec{i} \times (\vec{j} \times (\vec{k} \times (\vec{i} \times (\vec{j} \times (\vec{j} \times \vec{k})))))))) \). Find the value of \( ||\vec{v}|| \).

Answer: 1


Question: Two teams, each with seven players numbered 1 to 7, compete in a series of games. In each game, the player with the same number from each team plays against each other. The loser is eliminated and replaced by the next player from the same team. This continues until all players from one team are eliminated. Let \( N \) be the number of possible sequences of games. Find the remainder when \( N \) is divided by 1000.

Answer: 432


Question: Consider a surjective and non-decreasing function \( f: \{1, 2, 3, \ldots, 13\} \rightarrow \{1, 2, 3, \ldots, 9\} \). What is the probability that \( f(7) = 4 \)?

Answer: \dfrac{20}{99}


Question: Find the volume of the solid obtained by revolving the region $D = \{(x, 0, z) \mid (x-1)^2 + z^2 \leq 1\}$ about the $z$-axis.

Answer: 2\pi^2


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{\sqrt[n]{e} + 2\sqrt[n]{e^2} + \cdots + n\sqrt[n]{e^n}}{n^2} \]

Answer: 1


Question: Given a positive definite matrix \(A\) such that \(x^T A x \geq C_A |x|^2\) for all vectors \(x\), find the constant \(C_B\) such that \(y^T A^{-1} y \geq C_B |y|^2\) for all vectors \(y\), in terms of \(C_A\).

Answer: \dfrac{1}{C_A}


Question: Find the smallest integer \( n \) such that \( 100^n > 100! \).

Answer: 79


Question: Evaluate the double integral \( \iint_{D} \frac{x^2-y^2}{1+x^4+y^4} \, \mathrm{d}x \, \mathrm{d}y \), where \( D \) is a bounded and closed domain in \( \mathbb{R}^2 \) that is symmetric with respect to the line \( y = x \).

Answer: 0


Question: Determine the radius of convergence for the power series \( \sum_{n=1}^\infty \frac{(2n)!}{(n!)^2} z^n \).

Answer: \dfrac{1}{4}


Question: Is the monadic fragment of inclusive first order logic decidable?

Answer: Yes


Question: Let \( G \) be a plane graph of order \( n \) and size \( m \) such that every interior region of \( G \) is bounded by a triangle, and the exterior region is bounded by a \( k \)-cycle, where \( k > 2 \). Express \( m \) in terms of \( n \) and \( k \). Use Euler's formula \( n - m + r = 2 \), where \( r \) is the number of regions, to find the relationship.

Answer: 3n - k - 3


Question: Calculate an upper bound for the norm of the linear functional \( \phi_n: (C[0,1], \Vert{\cdot}\Vert_\infty) \to \mathbb{R} \) defined by \( \phi_n(x) = \int_0^1 t^n x(t) \, dt \), which is independent of \( n \).

Answer: 1


Question: Determine whether the equation \(3x^3+5y^5+z^3+4z=13\) defines \(z\) as a function of \(x\) and \(y\) at the point \((1,1,1)\).

Answer: Yes


Question: Given a list of distinct elements, calculate the total number of permutations of all possible subsets of the list, including the empty set. For example, for the list [1, 2], the subsets are [], [1], [2], and [1, 2], with permutations counted as 1, 1, 1, and 2 respectively, totaling 5 permutations. Determine this total number for the list [1, 2, 3].

Answer: 16


Question:
Let \((X_{n})_{n \in \mathbb{N}}\) be a sequence of independent random variables, each with a Bernoulli distribution with parameter \(\frac{1}{2}\). Define the random variable \(Z_{n}\) for each \(n \in \mathbb{N}\) as follows:
\[ Z_{n} = \sum_{k=1}^{n} \frac{X_{k}}{2^k} \]
Find the limit in probability of the sequence \((Z_{n})_{n \in \mathbb{N}}\).

Answer: U(0,1)


Question: Determine the least natural number \(k\) such that \(a(k) > 1\), where \(a(n)\) is a sequence defined by \(a(0) = \frac{1}{2}\) and \(a(n+1) = a(n) + \frac{a(n)^2}{2013}\) for natural numbers \(n\).

Answer: 2014


Question:
Consider the ring $S = K[[S_1,\ldots,S_n]]$ and the elements $f_1,\ldots,f_d \in S[[X_1,\ldots,X_d]]$. Let ${\frak P} = (f_1,\ldots,f_d)$ be the prime ideal generated by these elements. Suppose ${\frak P}$ satisfies the following conditions:

1. ${\frak P} \cap S = 0$.
2. ${\mathrm{ht}}({\frak P}) = d$.
3. None of $\overline{f_1},\ldots,\overline{f_d} \in K[[X_1,\ldots,X_d]]$ is zero.

Is the ring $S[[X_1,\ldots,X_d]]/{\frak P}$ necessarily finite over $S$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to \pi/8 } \frac {\cot 4x - \cos 4x}{(\pi-8x)^3} \]

Answer: \dfrac{1}{16}


Question: Calculate the residue of the function \( f(z) = \frac{z(e^z-1)\cos(z)}{\sinh^3(z)} \) at \( z = 0 \).

Answer: 1


Question: Find all pairs of positive integers \((x, y)\) such that \(12^x - 5^y = 19\).

Answer: (2, 3)


Question: Determine the largest value of \( t \) such that for all positive \( x, y, z \), the inequality \((xy + xz + yz) \left(\frac{1}{x+y} + \frac{1}{x+z} + \frac{1}{y+z}\right)^2 \geq t\) holds.

Answer: \dfrac{25}{4}


Question: Construct a bipartite graph and determine an ordering of its vertices such that a greedy coloring algorithm uses at least 2014 colors.

Answer: 2014


Question: Evaluate the limit: $$\lim_{x\to 0} \frac{\sin(x)-\arctan(x)}{x^2\ln(1+2x+x^2)}$$

Answer: \dfrac{1}{12}


Question: Find the Bayes estimate of \( \theta \) based on a single observation of 5 from a distribution that is uniform on the interval \(0\) to \(\theta\). Use square-error loss and a prior distribution of \(\theta\) which has probability density function \(p(\theta) = \theta \cdot \mathrm{e}^{-\theta}\) where \(0<\theta <\infty\).

Answer: 6


Question: Let \( f(x) = 30 - 2x - x^3 \). Determine the number of positive integral values of \( x \) that satisfy the inequality \( f(f(f(x))) > f(f(-x)) \).

Answer: 2


Question: Let $h(x,y)$ be a function such that $h(x,y) = a \frac{\partial h(x,y)}{\partial x} + b \frac{\partial h(x,y)}{\partial y}$ for some constants $a, b \in \mathbb{R}$. If $|h(x,y)| \le M$ for every $(x,y) \in \mathbb{R}^2$, prove that $h(x,y)$ is identically zero.

Answer: 0


Question: Evaluate the limit: \[ \lim _{x\to 0}\left(\frac{\cos \left(x\right)-1+\frac{x}{2}\sin \left(x\right)}{\ln ^4\left(1+x\right)}\right) \]

Answer: -\dfrac{1}{24}


Question: Find the distance between the two points where the line is tangent to the curve \( y = x(1 + 2x - x^3) \) at exactly two points.

Answer: 2\sqrt{2}


Question: Aiming for the center, you throw 100 darts at a dartboard, and 25 of those darts hit within 15 mm from the center. Approximate the standard deviation of your dart throws using this information.

Answer: 20


Question: Calculate the expected value of the sum of two standard normally distributed random variables, \( x_1 \) and \( x_2 \), given that their difference is zero. Specifically, find \( \mathop{\mathbb{E}}[x_1 + x_2 \mid x_1 - x_2 = 0] \).

Answer: 0


Question: Let $f(x)$ be a positive, continuous, monotone, and integrable function on the interval $(0,1]$. Determine whether $\lim_{x \rightarrow 0} xf(x) = 0$. Justify your answer.

Answer: 0


Question: Find the Galois group of the splitting field of the polynomial $x^4 - 6x^2 + 7$ over $\mathbb{Q}$. The roots of this polynomial are $\pm \sqrt{3 \pm \sqrt{2}}$. Determine the structure of the Galois group.

Answer: D_4


Question: Determine whether the polar dual of a semi-algebraic convex body is also semi-algebraic. A convex body $C \subset \mathbb{R}^n$ is semi-algebraic if it can be expressed as \( C = \bigcap_{i \in I} \{ x \in \mathbb{R}^n \mid p_i(x) \le 0 \} \) for polynomials $p_i \in \mathbb{R}[X_1, \ldots, X_n]$ and a finite index set $I$. The polar dual is defined as \( C^\circ = \{ y \in \mathbb{R}^n \mid \langle x, y \rangle \le 1 \text{ for all } x \in C \} \).

Answer: Yes


Question: Evaluate the integral \( \int_{-\pi/2}^{\pi/2} \frac{\log(1+b\sin x)}{\sin x}\,\mathrm{d}x \) given that \(|b|<1\).

Answer: \pi \arcsin b


Question: Let $T_1: \mathbb{R}^n \to \mathbb{R}^m$ and $T_2: \mathbb{R}^m \to \mathbb{R}^n$ be linear transformations such that the composition $T_1T_2$ is bijective. Determine the ranks of $T_1$ and $T_2$.

Answer: m


Question: Given a subgroup $G \leq S_p$, where $p$ is a prime number and $|G| < p$, does the action of $G$ on the set $\{1, 2, \ldots, p\}$ necessarily have a fixed point?

Answer: Yes


Question: Consider two independent Gaussian random variables $X_1$ and $X_2$, each with mean zero and variance $1/2$. Let $E[|X_1|^2] = \delta_1$ and $E[|X_2|^2] = \delta_2$, where $\delta_2 > \delta_1$. Does the inequality $|X_2|^2 > |X_1|^2$ always hold? Provide a justification for your answer.

Answer: No


Question: Evaluate the integral: \[ \int_{-\infty}^\infty e^{-\pi x^2} \cos(2\pi x w) \, dx, \quad w \in \mathbb{R}. \]

Answer: e^{-\pi w^2}


Question: Evaluate the contour integral \( \int_{C_1(1)} \frac{1}{1+e^z} \, \text{d}z \), where \( C_1(1) \) is the circle of radius 1 centered at 1 in the complex plane.

Answer: 0


Question: Let $A$ be the set of all algebraic integers, and let $A'$ be the set of all $\alpha \in A$ such that $\frac{\alpha}{p} \not\in A$ for all primes $p$. If $\alpha \in A'$, does it follow that $\alpha^n \in A'$ for all positive integers $n$?

Answer: No


Question: Does the ring $k[x,y,z,w]/(wx-yz)$ contain any nonconstant invertible polynomials? Here, $k$ is an algebraically closed field, and $k[x,y,z,w]$ is the polynomial ring. Determine if there exists a nonconstant polynomial $\bar{h}$ in $A(X) = k[x,y,z,w]/(wx-yz)$ such that $\bar{h}$ is invertible.

Answer: No


Question: Consider a function $h$ and its derivative $h'$. Is it possible for there to be a point $x_0$ such that $h'(x_0) = r$ for some $r \in \mathbb{R}$, but $\lim_{x \to x_0} h'(x) \neq r$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\rightarrow \frac{\pi}{4}}(\tan x)^ {\tan 2x} \]

Answer: \dfrac{1}{e}


Question: Consider a finite field extension \( L/K \). Define the norm of an element in \( L \) as the product of each \( K \)-embedding evaluated at that element. Can the norm of a non-algebraic integer be an integer?

Answer: Yes


Question: Consider a series of i.i.d. random variables \((X_n)_{n\in\mathbb{N}}\). Is it possible that \( \limsup_{n\to\infty} \frac{X_n}{n} = 0 \quad a.s.\) does not hold? Provide a justification for your answer.

Answer: Yes


Question: Given that $a^{-1}ba = b^2$ and $|a| = 3$ in a group, find the order of the element $b$, where $b \neq e$ and $b$ is not of order 2.

Answer: 7


Question: Calculate the limit: \( \lim_{n\to\infty} \frac{n!}{n^{\log(\log(n))}} \)

Answer: \infty


Question: Is the quotient of a Hausdorff topological group by a closed subgroup also Hausdorff? Provide a simple yes or no answer.

Answer: Yes


Question: Evaluate the double summation \( \sum_{j=0}^{n} \sum_{i=j}^{n} \binom{n}{i} \binom{i}{j} \).

Answer: 3^n


Question: Evaluate the infinite product \( \prod _{n=2}^\infty \left(1+ \frac{1}{n^2}+\frac{1}{n^4}+\frac{1}{n^6}+\cdots \right) \).

Answer: 2


Question: Given a Doppler shift channel where the input signal is $x(t)$ with Fourier transform $X(f)$, and the output is $Y(f) = X(f-f_1)$, find the channel response $H(f) = \frac{Y(f)}{X(f)} = \frac{X(f-f_1)}{X(f)}$. Determine the inverse Fourier transform of $H(f)$, denoted as $h(t)$. Provide your answer as $h(t)$. \( \boxed{} \)

Answer: h(t) = e^{j2\pi f_1 t}


Question: Calculate the area of the region defined by the inequality \( \sin^2(\pi x) + \sin^2(\pi y) \le 1 \) for \( x, y \in [-1, 1] \).

Answer: 2


Question:
Find the power series solution to the Laguerre differential equation:
\[ xy'' + (1-x)y' - y = 0 \]
with the initial condition \( P(0) = 1 \).

Answer: e^x


Question: Given the initial value problem \( x'(t) = \sin(x(t)) \) with \( x(0) = x_0 \) where \( x_0 \in (0, \pi) \), find \( \lim_{t \to +\infty} x(t) \).

Answer: \pi


Question: Evaluate the limit: $$\lim_{x \to \infty}\frac{1}{x}\int_0^x|\sin(t)|dt$$

Answer: \dfrac{2}{\pi}


Question: Determine if it is possible to count the number of pairs \(\{(p,\ell)\in P\times L: p\; \text{lies above}\; \ell\}\) in time \(\tilde{O}(N)\), where \(P\) is a set of \(N\) points in the unit square and \(L\) is a set of \(N\) non-vertical lines.

Answer: Yes


Question: Verify the equation: \[ \sum_{k=0}^n k^{2}\binom{n}{k} = n(n+1)2^{n-2} \] for non-negative integers \( n \).

Answer: n(n+1)2^{n-2}


Question: Consider a bit string $x = x(1), \\dots , x(n)$ containing exactly $m$ occurrences of 1. We perform the following operation on $x$: choose a random pair of indices $(i,j)$ and swap $x(i)$ and $x(j)$, so that $x'(i) = x(j)$, $x'(j) = x(i)$, while $x'(k) = x(k)$ if $k \neq i$ and $k \neq j$. Let $X_1 = x$, and let $X_2, \\dots, X_N$ be obtained by such a sequence of operations. Show that for each $i$, $P(X_N (i) = 1) \rightarrow \frac{m}{n}$ as $N \rightarrow \infty$. Hint: Consider the last time $i$ was swapped.

Answer: \dfrac{m}{n}


Question: Given that \( \sum_{k=0}^{\infty} a_{k}^{2}9^{k} \le 5 \), prove that \( \sum_{k=0}^{\infty} |a_{k}|2^{k} \le 3 \).

Answer: 3


Question: Five men crash-land on a deserted island and gather coconuts into one big pile. During the night, each man, while on watch, divides the coconuts into five equal piles, finds one extra coconut, gives it to a monkey, takes one pile for himself, and recombines the remaining coconuts. This process is repeated by each of the five men. What is the smallest number of coconuts that could have been in the original pile?

Answer: 3121


Question: Determine whether the following statement is true: \( \lim_{n \to \infty} \mathbb{P}\left\{\frac{\sum_{i=1}^nX_i}{\sqrt{n}} < x\right\} = \Phi(x) \), where \( \{X_n\}_{n=1}^\infty \) is an i.i.d sequence with mean 0 and variance 1, and \( \Phi(x) \) is the cumulative distribution function of a standard normal variable.

Answer: True


Question: Is the unit ball in the dual space $D[0,1]^*$ separable in the weak* topology?

Answer: No


Question: Find the colimit of the diagram \( D: \mathfrak{F} \rightarrow \mathfrak{F}\text{unc}(\mathfrak{C}^{opp}, \mathfrak{Set}) \) where \( D \) maps an object \( x \) (an element of \( F(X) \)) to the presheaf \( \operatorname{Hom}_\mathfrak{C}(-, X) \) and an arrow \( \phi \) from \( \operatorname{Hom}_\mathfrak{F}(x, y) \) to the natural transformation \( \phi_* \) of left multiplication on \( \phi \).

Answer: F


Question:
Determine which of the following recurrent sequences converges to any solution of the equation \(x \cdot \sin(x) = 1\):

1. \(x_{n+1} = \arcsin(1/x_{n}), x_0 = 0\)
2. \(x_{n+1} = 1/\sin(x_n), x_0 = 1\)
3. \(x_{n+1} = 1/\sin(x_n), x_0 = 0\)
4. \(x_{n+1} = \cos(x_n), x_0 = 1\)

Answer: 2


Question: Let $S_1$ be the focus of the parabola $y^2 = 8x$ and $PQ$ be the common chord of the circle $x^2 + y^2 - 2x - 4y = 0$ and the given parabola. Find the area of the triangle formed by the points $P$, $Q$, and $S_1$. Provide your answer as a single numerical value.

Answer: 4


Question: Verify whether the following statement is correct: For regular distributions $T_f$ and $T_g$, where $f$ is defined on $\mathbb{R}^n$ and $g$ on $\mathbb{R}^m$, the tensor product $T_f \otimes T_g$ satisfies $\langle T_f \otimes T_g, \varphi \rangle = \int_{\mathbb{R}^n \times \mathbb{R}^m} f(x) g(y) \varphi(x, y) \, dx \, dy$. Thus, $T_f \otimes T_g$ is a regular distribution $T_h$, where $h(x,y)=f(x)g(y)$. Is this statement correct?

Answer: Yes


Question: Is it sufficient to verify that a linear map $f: A \to B$ between two algebras $A$ and $B$ over a field $k$ satisfies $f(ab) = f(a)f(b)$ for elements $a, b$ of a basis of $A$ in order to conclude that $f$ is a morphism of algebras?

Answer: Yes


Question: In the right triangle $ABC$ with $\angle B = 90^\circ$, a perpendicular is drawn from vertex $C$ to $BC$ at point $D$, such that $BD$ intersects $AC$ at $P$. Given that $AB = 4$, $BP = n$, $PC = m$, and $CD = 6$, calculate the circumradius of triangle $BPC$. Express your answer in terms of $m$ and $n$.

Answer: \dfrac{5mn}{24}


Question: Calculate the result of adding the point \((0, 4)\) to itself on the elliptic curve defined by \(y^2 = x^3 + 1\) over the finite field \(F_5\).

Answer: (0, 1)


Question: Suppose a sample of size 5 is taken from a continuous distribution. What is the probability that the population median lies between the smallest and largest numbers in this sample?

Answer: \dfrac{15}{16}


Question: For the function \(f(x)=\frac{k\cos{x}+1}{\sin^2{x}}\), where \(x\) ranges over all real numbers except integer multiples of \(\pi\), and \(k\) is a constant, it is given that \(f(x)>0\) for all values where \(f(x)\) is defined. Determine the positive integer difference between the largest and smallest possible values of \(k\).

Answer: 2


Question: Let \(x \in [2009, 2010]\) and \(y \in [2008, 2009]\). Find the minimum value of \(a > 0\) such that the inequality \((x+y)\left(\frac{1}{x} + \frac{a}{y}\right) \ge 9\) holds for all \(x\) and \(y\) in their respective intervals.

Answer: \dfrac{7}{2}


Question: Let $T$ be the smallest positive real number such that the tangent to the helix \( \cos t \mathbf{i} + \sin t \mathbf{j} + \frac{t}{\sqrt{2}} \mathbf{k} \) at \( t = T \) is orthogonal to the tangent at \( t = 0 \). Find the line integral of the vector field \( \mathbf{F} = x \mathbf{j} - y \mathbf{i} \) along the section of the helix from \( t = 0 \) to \( t = T \).

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


Question: Find all positive integer solutions \((a, b, c)\) to the equation \(3^c = 2^a + 2^b + 1\) given that \(b > a\).

Answer: (4, 6, 4)


Question: Given a sequence $(f_n)$ in $\Sigma^+$, where each $f_n$ is measurable and nonnegative, and $f_{n+1} \geq f_n$ almost everywhere for each $n$, let $f = \limsup_n f_n$. Can it be deduced that $f$ is measurable?

Answer: Yes


Question: A particle follows the path defined by \(r(t) = (2t^{3/2},2t+1, \sqrt{5} t )\) and starts at \(t=0\). At what time will the particle have traveled a total of 14 units?

Answer: 3


Question: Let $X \in M_n(\mathbb{C})$ be a positive matrix, meaning it has positive eigenvalues and is self-adjoint. Determine if there exists a vector $\xi \in \mathbb{C}^n$ such that $X = [\xi_i \overline{\xi}_j]_{i,j}$. Answer with "Yes" or "No."

Answer: No


Question: What is the probability of correctly guessing a specific number from the set of natural numbers, \( \mathbb{N} \)? Assume that \( \mathbb{N} \) contains infinitely many numbers.

Answer: 0


Question: Let $X$ be a Poisson random variable with $E[X] = \ln(2)$. Calculate $E[\cos(\pi X)]$. Provide your answer as a single value.

Answer: \dfrac{1}{4}


Question: Let \( \phi : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) be a linear mapping with eigenvalues 2, 1, and -1. Determine the trace of the mapping \( \phi^{\wedge 2} : \Lambda^{2}\mathbb{R}^3 \rightarrow \Lambda^{2}\mathbb{R}^3 \).

Answer: -1


Question: Transform the complex number \(\frac{(1+i)^{29}}{1-i}\) into the form \(a + bi\).

Answer: -16384i


Question: Are the definitions of nontangential limits in the disc, as given by Def 1 and Def 2, equivalent for bounded analytic functions?

Answer: Yes


Question: Let $(W_t)_{t \geq 0}$ be a Brownian motion. Define $\tau = \inf\{t \ge 0 : W_t = a + bt\}$, where $a < 0$. Compute $E[\tau]$. Assume $E(\tau) < \infty$. What is the expected value of $\tau$?

Answer: -\dfrac{a}{b}


Question: Find the inverse Laplace transform of the function \( \frac{4s}{(s^2+4)^2} \).

Answer: t \sin(2t)


Question: Let $X$ and $Y$ be two independent exponential random variables with means $1$ and $3$, respectively. Calculate $P(X > Y)$. Express your answer as a single probability value.

Answer: \dfrac{1}{4}


Question: Evaluate the integral \( \int_{C(1)} (z-a)^{-n} (z-b)^{-n} \, dz \) where \(|a|<|b|<1\).

Answer: 0


Question: Let \( f(x) = x^2 + 2x - t^2 \) and consider the roots \( \alpha(t) \) and \( \beta(t) \) of the equation \( f(x) = 0 \), where \( \alpha < \beta \) and \( t \) is a real parameter. Define \( I(t) = \int_{\alpha}^{\beta} f(x) \, dx \). Determine the maximum value of \( I(t) \), denoted as \( \lambda \). Find \( \lambda \).

Answer: -\dfrac{4}{3}


Question: Find the minimum value of \( \vert 2^m - 11^n \vert \) for positive integers \( m \) and \( n \).

Answer: 3


Question: Calculate \( \mathbb{E}[W(s)\mathrm{e}^{W(s)}] \) where \( W(s) \) is a standard Brownian motion at time \( s \).

Answer: s e^{s/2}


Question:
Find the function \( f(x) \) that satisfies the integral equation:
\[ f(x) = \int_{0}^{x} te^t f(x-t) \, \mathrm{d}t \]

Answer: 0


Question: Evaluate the limit: \[ \lim_{r \to \infty} \frac{\int_0^{\pi/2}\left(\frac{x}{\pi/2}\right)^{r-1}\cos x\,dx}{\int_0^{\pi/2}\left(\frac{x}{\pi/2}\right)^r\cos{x}\,dx} \]

Answer: 1


Question: Evaluate the summation \( S = \sum_{r=0}^{1010} \binom{1010}{r} \sum_{k=2r+1}^{2021}\binom{2021}{k} \).

Answer: 2^{3030}


Question: Evaluate the surface integral \( \int_S y \, dS \), where \( S \) is the part of the plane \( z = 1 + y \) that lies inside the cone \( z = \sqrt{2(x^2 + y^2)} \).

Answer: 2\pi


Question: Evaluate the limit: \[ \lim_{n \to +\infty} \frac{\sqrt[n]{n!}}{n} \]

Answer: \dfrac{1}{e}


Question: Consider the curve given by \( y = x^3 \). A tangent is drawn at a point \( P_1 \) on this curve, other than the origin \((0,0)\), and it intersects the curve again at point \( P_2 \). The tangent at \( P_2 \) intersects the curve again at \( P_3 \), and this process continues. If the ratio of the area of triangle \( \Delta P_1P_2P_3 \) to the area of triangle \( \Delta P_2P_3P_4 \) is \( \lambda \), find the value of \( \lambda \).

Answer: \dfrac{1}{16}


Question: If $R = \mathbb{Z}[\sqrt{c}]$ is a unique factorization domain (UFD), where $c$ is an integer that is not a square, does it necessarily follow that $R$ is also a principal ideal domain (PID)?

Answer: Yes


Question: Let \( f:[0,1]\to[0,1] \) be a continuous, strictly increasing function such that \( f(1)=1 \) and \( f(x)>x \) for all \( x\in[0,1) \). For any \( x_0\in[0,1) \), define the sequence \( x_k \) inductively by \( x_k = f(x_{k-1}) \). Prove that \( \lim_{k\to\infty}x_k=1 \).

Answer: 1


Question: Evaluate the double integral \( \iint \frac{x^2}{y^3} \, dA \) using a change of variables over the region defined by the curves \( y=2x, \; y=x, \; y=x^2, \; y=2x^2 \).

Answer: \dfrac{1}{4}


Question: Determine the structure of the quotient group $\mathbb{Z}^2/((m,n)\mathbb{Z})$, where $m$ and $n$ are integers greater than one and coprime.

Answer: \mathbb{Z}


Question: Let $(R,B(R),m)$ be a Lebesgue measurable space on $R$. For each $n \in \mathbb{N}$, define $f_n(x) = \arctan(n|x|)$. Show that \( \lim_{n \to \infty} \int_{[-1,1]} f_n \, dm = \pi. \)

Answer: \pi


Question: Calculate the mass of a wire bent in the shape of the curve \( r(t) = (3t,3t^2,2t^3) \) for \( 0 < t < 1 \), given that the density at \( r(t) \) is \( 1+t \) grams per unit length.

Answer: 8


Question: Determine the values of \(a\) such that the inequality \(3x^2 + y^2 \ge -ax(x+y)\) holds for all real numbers \(x\) and \(y\).

Answer: [-2, 6]


Question: Determine the convergence radius of the power series \( P(z) = \sum_{j=1}^{\infty}\left(\frac{j-3}{j}\right)^{j^2}z^j \) using the root test and the limit \( e^x = \lim_{j\to\infty} (1+x/j)^j \).

Answer: e^3


Question:
Determine the condition under which the exact sequence
\[ 0 \to \mathbb{Z}/n\mathbb{Z} \to \mathbb{Z}/nm\mathbb{Z} \to \mathbb{Z}/m\mathbb{Z} \to 0 \]
is split, where the first map sends \( k+n\mathbb{Z} \to mk+nm\mathbb{Z} \) and the second map sends \( k+nm\mathbb{Z} \to k+m\mathbb{Z} \).

Answer: \gcd(n, m) = 1


Question: Find the area of the region that lies inside both curves $r = 1 - \cos \theta$ and $r = \frac{1}{2}$. Express the area in the form $\frac{a \pi}{b} - \frac{c \sqrt{d}}{e}$, where $a$, $b$, $c$, $d$, and $e$ are positive integers. Calculate $a + b + c + d + e$. Provide your answer as a single integer.

Answer: 23


Question: Given a finite poset $S = \{x_1, x_2, \ldots, x_n\}$ with $k$ ordering relations that generate all the ordering of the poset, determine the number of ways to construct a totally ordered set $(S, \geq)$ that respects these $k$ ordering relations. For example, if $S = \{a, b, c, d\}$ with relations $a > b$ and $b > d$, find the number of ways to totally order $S$.

Answer: 4


Question:
For all planar graphs on $n$ vertices with a minimum vertex degree of at least 3, determine which of the following statements is true:

(i) In any planar embedding, the number of faces is at least $\frac{n}{2} + 2$.
(ii) In any planar embedding, the number of faces is less than $\frac{n}{2} + 2$.
(iii) There is a planar embedding in which the number of faces is less than $\frac{n}{2} + 2$.
(iv) There is a planar embedding in which the number of faces is at most $\frac{n}{\delta + 1}$.

Use the inequality $2e \geq 3f$ for planar graphs and Euler's formula $n - e + f = 2$ to determine the correct statement.

Answer: (i)


Question: Consider a sequence of random variables \( f_n \) defined as follows: for each \( n \), flip a fair coin. If the result is Heads, set \( f_n = 2 \); if Tails, set \( f_n = \frac{1}{2} \). Is there a nonzero probability that the product \( \prod_{n=1}^m f_n \) can become arbitrarily large as \( m \) increases? Justify your answer.

Answer: Yes


Question:
Evaluate the volume of the solid below the plane \(z = 1+y\), bounded by the coordinate planes and the vertical plane \(2x + y = 2\). Use the triple integral:
\[ V=\int_{x=0}^1\int_{y=0}^{2-2x}\int_{z=0}^{1+y}\,\mathrm dz\mathrm dy\mathrm dx \]
Verify the result by changing the order of integration and ensuring the limits are correctly set.

Answer: \dfrac{5}{3}


Question: Determine the cardinality of the set of all infinite sequences composed of the digits $0, 1, -1$, such that each sequence contains each digit at least once.

Answer: \mathfrak{c}


Question: Evaluate the limit: \[ \lim_{n\to \infty} \frac{1 + \sqrt[3]{2} + \sqrt[3]{3} + \cdots + \sqrt[3]{n}}{n^{4/3}}. \]

Answer: \dfrac{3}{4}


Question:
Given a sequence of $P$ positive integers, determine the number of good sets. A good set is defined as a sequence where:

1. If an integer $L$ appears in the sequence, then $L-1$ must also appear in the sequence.
2. The first occurrence of $L-1$ must come before the last occurrence of $L$.

For example, when $P=3$, there are 6 good sets. Find the number of good sets for a given $P$. Provide your answer as a single integer.

Answer: P!


Question: Determine the kernel of the linear map \(d: V \to W\), where \(V\) is a two-dimensional vector space with basis \(\{1,x\}\) and \(W = \bigoplus_{i = 1}^{\infty} V^{\otimes 2}\). The map \(d\) is defined as \(d = \bigoplus_{i =1}^{\infty} d_i\), with \(d_i : V \to V \otimes V\) given by \(d_i (1) = 1\otimes x + x \otimes 1\) and \(d_i (x) = x \otimes x\) for each \(i \in \mathbb{N}\).

Answer: \{0\}


Question: A particle is moving on the $x$-axis such that \(\frac {dx} {dt} = (x - 1) (x + 2) (x - 3)\). Here, $x$ denotes the $x$-coordinate of the particle and $t$ denotes time. The particle is initially positioned such that it does not wander off to infinity. Determine the point of equilibrium the particle will be close to after a sufficiently long time.

Answer: 1


Question: Let $p$ be a prime number. Determine the order of the group $G$, which consists of all $2 \times 2$ matrices over $\mathbb{Z}_p$ with determinant $1$, under matrix multiplication.

Answer: p(p^2 - 1)


Question: Find the least ordinal \( \beta > 0 \) such that \( \alpha \beta = \beta \) for a given \( \alpha > 1 \).

Answer: \alpha^{\omega}


Question: Let $A$ be a local ring with maximal ideal $m$. Suppose there exists a positive integer $k$ such that $m^k = m^{k+1}$. Must it be true that $m^k = 0$?

Answer: No


Question: Find the remainder when \(x^{100} + 2x + 10\) is divided by \(x - 11\) in \(\mathbb{Z}_{17}[x]\).

Answer: 2


Question: Find the power series representation and the radius of convergence for the function \( f(z) = \frac{1}{1+z^2} \) centered at \( z = 1 \).

Answer: \sqrt{2}


Question: Solve the first order homogeneous differential equation \( \frac{dy}{dx}=(x+y)^2 \) with the initial condition \( y(0)=0 \).

Answer: y = \tan(x) - x


Question: Evaluate the integral \( \int_{\lvert z\rvert = 2} \frac{\lvert dz\rvert}{\lvert z-1\rvert^2} \).

Answer: \dfrac{4\pi}{3}


Question: Find the remainder when $51!$ is divided by $61$. Use Wilson's theorem or any other method to solve the problem.

Answer: 27


Question: Given that $f\left(\frac{x}{y}\right) = \frac{f(x)}{f(y)}$ for all $x, y \in \mathbb{R}$ with $y \neq 0$, and $f'(x)$ exists for all $x$, and $f(2) = 4$, find the value of $f(5)$.

Answer: 25


Question: In a Hilbert space $X$ with an inner product $\langle x_1, x_2 \rangle$, consider a convex cone $X_c \subseteq X$. Define the projection $P$ of $x \in X$ onto the cone $X_c$ as \( y = Px = \underset{z \in X_c}{\text{argmin}} \langle x-z, x-z \rangle. \) Is the projection $y$ unique?

Answer: Yes


Question: Given the power series \( f(z)=\sum^{\infty}_{n=0}a_n(z-i)^n \) and \( g(z)=\sum^{\infty}_{n=0}b_n(z+i)^n \), both with a radius of convergence of \( \frac{3}{2} \), determine the radius of convergence for the series \( h(z)=f(z)+g(z) \) centered at 0.

Answer: \dfrac{1}{2}


Question: Let $f$ be an entire function such that $f(0) = 1$ and $|f(z) - e^z \cdot \sin 2z| < 4$ for all $z \in \mathbb{C}$. Determine the value of $f(1)$. \( \boxed{} \)

Answer: 1 + e \sin 2


Question: Compute the limit: \[ \lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left( \frac{k}{n^2} \right)^{\frac{k}{n^2} +1} \]

Answer: \dfrac{1}{2}


Question: In the Poincaré disk model of the hyperbolic plane, consider a line formed by two random ideal points on the boundary of the disk. What is the expected value of the distance from this line to the center of the disk, assuming the curvature is \(-1\)? The distance is defined as the length of the line segment perpendicular to the line that connects the line to the center of the disk.

Answer: \dfrac{4G}{\pi}


Question: Do there exist two recursively enumerable sets $A$ and $B$ such that none of the sets $A \cap B$, $A \cup B$, $A - B$, and $B - A$ is recursive?

Answer: Yes


Question: Find the least nonnegative integer \( a \) such that \( a \equiv 3^{340} \pmod{341} \).

Answer: 56


Question: Let $f$ be a surjective entire function of the complex plane. Determine whether there exists a real number $r_0 > 0$ such that the set $f^{-1}(D(0,r))$ is connected for any $r \geq r_0$. Provide a justification for your answer.

Answer: Yes


Question: Given that \( \sum_{n=0}^{\infty} a_n \) is a conditionally convergent series with \( a_n \neq 0 \) for all \( n \), determine \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \), assuming the limit exists. Justify your answer.

Answer: 1


Question:
Find the constant \( g \) such that the size of the test is \( \beta \) for the following scenario: You have a random sample \( X_1, X_2, \dots, X_n \) drawn from a uniform distribution on \([\varphi, \varphi + 1]\). To test the null hypothesis \( H_0 : \varphi = 0 \) against the alternative hypothesis \( H_1 : \varphi > 0 \), the test is defined as:
\[\text{Reject} \ H_0 \ \ \ \text{if} \ \ \ X_{(n)} \ge 1 \ \text{or} \ X_{(1)} \ge g,\]
where \( g \) is a constant, \( X_{(1)} = \min\{X_1, X_2, \dots, X_n\}\), and \( X_{(n)} = \max\{X_1, X_2, \dots, X_n\}\). Determine \( g \) such that the test has size \( \beta \).

Answer: 1 - \beta^{1/n}


Question: Find the partial derivative of the function \( f = \text{tr}((ABC)(ABC)^T) \) with respect to the matrix \( B \).

Answer: 2 A^\top A B C C^\top


Question:
Evaluate the integral \( \int\limits_{0}^{1} \left( \prod\limits_{r=1}^{n}(x+r) \right) \left( \sum\limits_{k=1}^{n}\frac{1}{x+k} \right) dx \). The value of the integral is:

(A) \( n \)  \hspace{1cm} (B) \( n! \)  \hspace{1cm} (C) \( (n+1)! \)  \hspace{1cm} (D) \( n \cdot n! \)

Answer: D


Question:
Consider a space $R_{M\times N}$ of size $M\times N$. Define the 2D Discrete Fourier Transform of $f \in R_{M\times N}$ as:
\[
\tilde{f}[m,n]=\sum_{p=0}^{M-1}\sum_{q=0}^{N-1}f[p,q]e^{\frac{-i 2\pi p m}{M} +\frac{-i2\pi q n}{N}},\ \ \  0\leq m < M,\ \ 0 \leq n < N
\]
If $f[m,n]$ and $g[m,n]$ are related by a translation, i.e., $f[m, n] = g[m + u_0 \mod M, n + v_0 \mod N]$, determine the rank of the matrix:
\[
H[m,n]= \frac{\tilde{f}[m,n]\tilde{g}[m,n]^*}{|\tilde{f}[m,n]\tilde{g}[m,n]^*|}
\]
where $H[m,n]=e^{\frac{i 2\pi mu_0 \mod M}{M}}e^{\frac{i 2\pi nv_0 \mod N}{N}}$. What is the rank of this matrix?

Answer: 1


Question: Prove using an epsilon-delta argument that \( \lim_{(x,y)\to(0,0)}\frac{x^3y^2}{x^4+y^4} = 0 \).

Answer: 0


Question: Let $x_1, x_2, \ldots, x_n$ be real numbers such that $x_i = -x_{n-i+1}$ for $1 \leq i \leq n$ and $n > 2$. Calculate the sum $S = \sum_{\substack{i,j,k=1 \\ i \neq j \neq k}}^n x_i x_j x_k$. What is the value of $S$?

Answer: 0


Question: Compute the integral \( \int_0^\pi \lfloor\cot (x)\rfloor \, dx \), where \( \lfloor \cdot \rfloor \) denotes the floor function.

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


Question: Given $m$ orthonormal vectors $u_1, ..., u_m \in \mathbb{R}^n$ where $m < n$, find a linear transformation $W: \mathbb{R}^n \rightarrow \mathbb{R}^m$ such that the vectors $Wu_1, ..., Wu_m \in \mathbb{R}^m$ are orthonormal.

Answer: W = U^\top


Question: Given a graph $G = (V,E)$ where each vertex $v \in V$ has degree 4, determine if it is always possible to orient the edges such that each vertex has exactly two incoming edges and two outgoing edges.

Answer: Yes


Question: Let $F$ be a subfield of the complex numbers. Define $n$ linear functionals on $F^n$ ($n \geq 2$) by $f_k(x_1, \dotsc, x_n) = \sum_{j=1}^n (k-j) x_j$, for $1 \leq k \leq n$. Determine the dimension of the subspace of $F^n$ that is annihilated by the functionals $f_1, \dotsc, f_n$. Provide your answer as a single integer.

Answer: n-2


Question: Let \( \{N(t), t \geq 0\} \) be a Poisson process with rate \( \lambda \). Define \( T_1 \) as the time of the first event, \( T_n \) as the elapsed time between the \((n-1)\)-th and the \(n\)-th event, and \( S_n \) as the time of the \(n\)-th event, for \( n = 1, 2, \ldots \). Find \( P[T_1 < s \mid N(t) = 1] \) for \( s \leq t \).

Answer: \dfrac{s}{t}


Question: Let \( N \sim \mathcal{P}(\lambda) \) and \( (X_i)_{i \geq 1} \) be iid random variables with \( X_i \sim Be(p) \). If \( N \) and \( (X_i)_{i \geq 1} \) are independent for all \( i \), calculate \( P(\mathbb{E}(X_1+\ldots+X_N|N)=0) \).

Answer: e^{-\lambda}


Question: Suppose we have matrices $A_1, A_2, \ldots, A_m$ and $B_1, B_2, \ldots, B_m$ such that $A_i, B_i \in \mathbb{R}^{p_i \times p_{i+1}}$ and $\|A_i - B_i\|_F \leq \varepsilon$ for $1 \leq i \leq m$, where $\|\cdot\|_F$ denotes the Frobenius norm. Determine the upper bound, in terms of $\varepsilon$, for the quantity $\|A_1 \cdot A_2 \cdot \ldots \cdot A_m - B_1 \cdot B_2 \cdot \ldots \cdot B_m\|_\infty$, where $\|\cdot\|_\infty$ is the maximum absolute value of the entries of a matrix.

Answer: m \varepsilon


Question: Let $f(x) = \frac{x}{1+x}$ and $g(x) = \frac{rx}{1-x}$. Determine the number of real numbers $r$ such that $f(g(x)) = g(f(x))$ for infinitely many real numbers $x$. Find the number of elements in the set $S$ of such $r$.

Answer: 2


Question:
Determine the oscillation of the function \( f \) at the point \((0,0)\), where \( f \) is defined as:
\[
f(x,y)=
\begin{cases}
\frac{xy}{x^2+y^2} & (x,y)\neq(0,0) \\
0 & (x,y)=(0,0)
\end{cases}
\]
Use the definition of oscillation:
\[
O(f,c) = \inf_{c\in U}\sup_{x_1,x_2\in U}|f(x_1)-f(x_2)|
\]
where \( U \) is an open subset containing \( c \).

Answer: 1


Question: Evaluate whether the integral \( \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{d}{dy}(uv) \, dy \, dx \) equals zero, where \( u(x,y), v(x,y) \in L^2 \). Justify your answer.

Answer: 0


Question: Consider the extended real line \( \bar{\mathbb{R}} \) and let \( 0 < q < \infty \). Suppose \( x_n \) is a sequence in \( \bar{\mathbb{R}} \) with \( x_n \geq 0 \) for all \( n \), and \( x_n \to +\infty \). Determine the limit \( \lim_{n \to +\infty} (x_n)^{-1/q} \).

Answer: 0


Question: Evaluate the contour integral \( \int\limits_{|z| = 2} \frac{z^4 \, dz}{z^4 + 1} \).

Answer: 0


Question: Let \(p\) be a prime number and \(k \geq 1\). Determine the number of pairs \((x, y)\) in \(\mathcal{F}_p^k\) such that \(\text{Tr}(y) = N(x)\), where \(\text{Tr}\) denotes the trace and \(N\) denotes the norm.

Answer: p^{2k - 1}


Question:
Let \( w_1, w_2, \ldots, w_{n-1} \) be the complex roots of the equation \( z^n - 1 = 0 \) that are not equal to 1, where \( n \) is an odd integer. Given the identity \( \frac{1-\bar{w}}{1+w} + \frac{1-w}{1+\bar{w}} = 2 - w - \bar{w} \), prove that:
\[ \sum_{k=1}^{n-1} \frac{1-\bar{w_k}}{1+w_k} = n \]

Answer: n


Question:
Given three circles with centers X, Y, and Z, each having a radius of \( \sqrt{\frac{1}{\pi}} \), and the area of intersection between circles A and B is 0.23, calculate the distance between the centers of these circles. Use the formula for the area of a lens:
\[
A_{\text{lens}} = r^2 \cos^{-1}\left(\frac{d^2 + r^2 - R^2}{2dr}\right) + R^2 \cos^{-1}\left(\frac{d^2 + R^2 - r^2}{2dr}\right) - 2\Delta
\]
where
\[
\Delta = \frac{1}{4}\sqrt{(-d + r + R)(d - r + R)(d + r - R)(d + r + R)}
\]
and \( r = R = \sqrt{\frac{1}{\pi}} \).

Answer: 0.74


Question: Minimize \( (3+2a^2)(3+2b^2)(3+2c^2) \) given that \( a+b+c=3 \) and \( a, b, c > 0 \).

Answer: 125


Question: Find the limit \( \lim_{k\rightarrow\infty}k\int_{[1-\frac{1}{k}, 1]}f(x) \, \text{d}x \) if it exists, where \( f \) is a continuous function with compact support in \([0, 2]\).

Answer: f(1)


Question: Evaluate the line integral \( \int_C F \cdot dr \) where \( F = (5xy^3)i + (3x^2y^2)j \) along the curve \( C \) consisting of the x-axis from \( x=0 \) to \( x=4 \), the parabola \( y=16-x^2 \) up to the y-axis, and the y-axis down to the origin.

Answer: 24576


Question: Determine whether the sequence \( \left(1 + \frac{1}{n+2}\right)^{3n\log(n)} \) converges or diverges. If it converges, find its limit.

Answer: \infty


Question: Consider an integral Dedekind scheme $S$ and a finite flat surjective morphism $f: X \longrightarrow \mathbf{P}^1_{S}$, where $X$ is an integral normal scheme. Let $\eta$ be the generic point of $S$. The morphism $f_\eta: X_\eta \longrightarrow \mathbf{P}^1_{K(S)}$ is a finite morphism of curves over $K(S)$. Determine whether $f$ is the normalization of $\mathbf{P}^1_S$ in the function field of $X_\eta$. Provide a justification for your answer.

Answer: Yes


Question: Consider the non-linear differential equation \( y'(t) = y^2 + t \). Determine the maximum number of times the solution \( y(t) \) is differentiable for \( t \in \mathbb{R} \).

Answer: \infty


Question: Determine whether there are any non-identity automorphisms in the group of automorphisms of the smallest subfield of $\mathbb{C}$ that contains the solutions to $x^4-2=0$. Justify your answer.

Answer: Yes


Question: Let \( A \) be a \( 2 \times 2 \) real matrix such that \( \det(A + A^T) = 8 \) and \( \det(A + 2A^T) = 27 \). Find \( \det(A) \).

Answer: 11


Question: Find the limit of the function \( f(x) = x\left(\left(1+\frac{1}{x}\right)^{x}-e\right) \) as \( x \to \infty \).

Answer: -\dfrac{e}{2}


Question: Determine the number of automorphisms of the structure \( \langle \omega, < \rangle \).

Answer: 1


Question: Determine whether there exists a compact orientable Riemannian manifold that does not have a compact totally geodesic submanifold of codimension 1.

Answer: Yes


Question: Evaluate the limit: \[ L = \lim_{x \to 0}\left(\sum_{k=1}^n k^{\csc^2 x}\right)^{\sin^2 x} \] where \( n \) is a natural number.

Answer: n


Question: Evaluate the limit: $$\lim_{n\to \infty}\Bigg\lvert\left(\frac{3^n+\left(-4\right)^n}{5}\right)^{\frac{1}{n}}\Bigg\rvert$$

Answer: 4


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

Answer: 0


Question: Determine whether the Sorgenfrey Line is monotonically monolithic.

Answer: No


Question: Find the smallest positive integer \( x \) such that \( a_{1834} \equiv x \pmod{100} \), where \( a_1 = 13 \) and for \( n \geq 2 \), \( a_n = 13^{a_{n-1}} \).

Answer: 53


Question: Find the remainder when the infinite power tower \(7^{7^{7^{7^{.^{.^{.^{\infty}}}}}}}\) is divided by 13.

Answer: 6


Question: Determine whether the following statement is true: For a random vector ${\bf x} \in \mathbb{C}^{N \times 1}$ with a uniform distribution, is it true that $\mathbb{E}\left[{\bf x} {\bf x}^H \right] = \mathbb{E}\left[{\bf x}\right] \mathbb{E}\left[{\bf x} \right]^H$?

Answer: False


Question: Evaluate the integral \( \int _{-3}^{3} u(x-2) \delta(x-3) \, dx \), where \( u(x) \) is the unit step function and \( \delta(x) \) is the Dirac delta function.

Answer: 1


Question: Evaluate the integral \( \int_{-2}^2 \int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}} \int_{\sqrt{x^2+y^2}}^2 xz \, dz \, dx \, dy \) by changing to cylindrical coordinates.

Answer: 0


Question: Find the value of \( \lim_{n\rightarrow \infty}\sqrt[n]{\frac{(27)^n(n!)^3}{(3n)!}} \).

Answer: 1


Question: How many 13-card bridge hands, drawn from a standard 52-card deck, contain one or more quads (four cards of the same rank)?

Answer: 21717689136


Question: Show that the integral \( \int_{0}^{\infty} \sin^2{\left[\pi\left(x+\frac{1}{x}\right)\right]}\,\mathrm{d}x \) diverges.

Answer: \infty


Question: Evaluate the integral \( \int_{0}^{\infty } e^{-t}\cdot t^{3}\cdot \sin(t) \, dt \) using the Laplace transform method.

Answer: 0


Question: The line \( \frac{x\cos \theta }{2} + y\sin \theta = 1 \), where \( \theta \in \left( \frac{\pi}{18}, \frac{\pi}{15} \right) \), intersects the ellipse \( x^2 + 2y^2 = 6 \) at points P and Q. Determine the value of \( K \) such that the angle between the tangents to the ellipse at points P and Q is \( \frac{\pi}{K} \).

Answer: 2


Question: Compute the Lebesgue integral \( \int_0^2 f(t)\,dt \), where the function \( f: [0, 2]\to \mathbb{R}_{+} \) is defined by \( f(t)=m(\{x\in [0, \pi]: t\leq 1+\cos (3x)\leq 3t\}) \).

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


Question: Provide an example of a finitely generated $R$-module that is torsion-free but not free.

Answer: (x, y)


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

Answer: 0


Question: Suppose $d \ge 2$ and $S$ is a finite simplicial complex of dimension $2d$ that is simply connected, has all its homology in middle degree, and has torsion-free homology in degree $d$. Is $S$ necessarily homotopy equivalent to a wedge of spheres of dimension $d$?

Answer: Yes


Question: Evaluate the limit: \(\lim_{x\to 0} \frac{e^{\cos^2x}-e}{ \tan^2x}\)

Answer: -e


Question: Consider a sequence $(a_i)_{i=1}^\infty$ defined by $a_0 = 0$ and $a_{k+1} = a_k + \frac{1}{d+a_k}$ for each $k \geq 0$, where $d > 0$ is a fixed real number. Determine whether this sequence diverges to infinity as $k \to \infty$. Provide a justification for your answer.

Answer: \infty


Question:
Let \( x_n \) be a sequence of positive non-zero numbers such that for all integers \( n \), the following holds:
\[ \sum_{k=1}^n kx_k = \prod_{k=1}^n x_k^k. \]
Find \( \lim_{n \to \infty} x_n. \)

Answer: 1


Question: Simplify the integral \( I = \int_{\Bbb{R}^2}\Delta\varphi(x)\log|x|^2\ dx \) using Green's identity, where \( \varphi \in C_c^\infty(\Bbb{R^2}) \).

Answer: 4\pi \varphi(0)


Question: Determine whether the function \( d: \mathcal{P}(X) \times \mathcal{P}(X) \to [0,1] \), defined by \( d(p,q) := \sup_{A \in \mathcal{B}(X)} |p(A) - q(A)| \), is a measurable function, where \( X \) is a standard Borel space and \( \mathcal{P}(X) \) is the space of Borel probability measures on \( X \) endowed with the topology of weak convergence.

Answer: Yes


Question: Let $f$ be a function such that for every real number $t$, \( \int_t^{f(t)} \frac{x^4}{1+x^2} \, dx = 2 \). Calculate \( \lim_{x \to \infty} f'(x) \).

Answer: 1


Question: Let $a, b, c$ be the roots of the polynomial $x^3 - 9x^2 + 11x - 1 = 0$. Define $s = \sqrt{a} + \sqrt{b} + \sqrt{c}$. Calculate the value of $s^4 - 18s^2 - 8s$. \[ \boxed{} \]

Answer: -37


Question: If \(\alpha\) is a root of the equation \(x^{x-\sqrt{x}} = \sqrt{x}+1\), find the value of \(\alpha + \frac{1}{\alpha}\).

Answer: 3


Question: Find the limit of the sequence \( \lim_{n\to\infty}((n^3+n^2)^{1/3}-(n^3+1)^{1/3}) \) without using the identity \( a^3-b^3=(a-b)(a^2+ab+b^2) \).

Answer: \dfrac{1}{3}


Question: In a unitary ring $R$, if $a, b \in R$ such that $ab = 1$, is $a$ necessarily invertible? Justify your answer.

Answer: No


Question: Determine all values of $c \in \mathbb{R}$ such that the initial value problem \( y'(t) = e^{-y(t)^2}, \ y(0) = c \) has exactly one solution.

Answer: \mathbb{R}


Question: Determine the value of \(a\) within the interval \(2.55 < a < 2.85\) such that the functions \(e^x\) and \(x^a\) intersect exactly once for \(x > 0\).

Answer: e


Question: Solve the heat equation \( u_{t} = u_{xx} \) for \( 0 \le x \le 1 \) with the boundary conditions \( u(0,t) = 0 \) and \( u(1,t) = 1 \), and the initial condition \( u(x,0) = x \), using the method of separation of variables.

Answer: u(x, t) = x


Question: Find the z-coordinate of the center of mass for the cone defined by \( C = \{ (x,y,z) \in \mathbb{R}^3 : \sqrt{x^2+y^2} \leq z \leq 1 \} \), assuming the cone has a constant mass density.

Answer: \dfrac{3}{4}


Question: Let $S$ be an $n$-surface in $\mathbb{R}^{n+k}$, defined by an open set $U \subseteq \mathbb{R}^{n+k}$ and a smooth map $f : U \to \mathbb{R}^k$ such that $S = f^{-1}(\{0\})$ and $\text{rank} \, Df(a) = k$ for all $a \in S$. Is it true that any path connected component of $S$ is again an $n$-surface?

Answer: Yes


Question:
In a two-dimensional Cartesian coordinate system, there are two points $A(2, 0)$ and $B(2, 2)$ and a circle $c$ with radius $1$ centered at the origin $O(0, 0)$. If $P$ is a point on the circle $c$, determine the minimum value of the expression:
\[ f = 2\sqrt{2}\lvert{PA}\rvert + \lvert{PB}\rvert. \]

Answer: 5


Question: Determine the number of non-negative integers \( n \) that satisfy the equations \( n^2 = p + q \) and \( n^3 = p^2 + q^2 \), where \( p \) and \( q \) are integers.

Answer: 3


Question: Find the coefficient of the basis form \( dx^{1} \wedge dx^{2} \wedge \cdots \wedge dx^{2n-1} \wedge dx^{2n} \) when the 2-form \( \omega = dx^{1} \wedge dx^{2} + dx^{3} \wedge dx^{4} + \cdots + dx^{2n-1} \wedge dx^{2n} \) is wedged with itself \( n \) times.

Answer: n!


Question: Does every Steinhausen set in $[0,1]$ have positive Lebesgue measure? A Steinhausen set is defined as a set $A \subseteq [0,1]$ such that $0 \in \operatorname{int}(A-A)$ and $\mu(A) \geq 0$. Provide a justification for your answer.

Answer: No


Question: Assume $A$ and $B$ are infinite abelian groups, with $B$ as a subgroup of $A$. If every homomorphism from $B$ to $\mathbb{Z}$ can be extended to a homomorphism from $A$ to $\mathbb{Z}$, is it true that for all $a \in B$ and all natural numbers $n$, if there exists $c \in A$ such that $nc = a$ in $A$, then there exists $x \in B$ such that $nx = a$ in $B$?

Answer: Yes