Question: Consider the infinite strip \(S= \{x \in \mathbb{R}^3 : a < x_1 < b \} \) in three-dimensional Euclidean space. Determine whether the only \(L^2\) harmonic function in this strip is the zero function.

Answer: 0


Question: Given a monotone increasing concave function \( f(x):\mathcal{R}_{\geq 0} \to \mathcal{R}_{\geq 0} \), determine if the following inequality holds: \[ f(d_1)+f(d_2)-f(d_1+d_2) \leq f(d_3)+f(d_4)-f(d_3+d_4) \] given that \( d_1<d_3 \) and \( d_2<d_4 \).

Answer: Yes


Question: Compute the value of the surface integral \( \iint_S{F \cdot n} \, dS \), where \( S \) is the hemisphere defined by \( x^2+y^2+z^2=1 \) with \( z \ge 0 \), and \( F(x,y,z) = x\mathbf{i} + y\mathbf{j} \). Here, \( n \) is the unit outward normal vector of \( S \). Use the explicit representation \( z=\sqrt{1-x^2-y^2} \) to perform the computation.

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


Question: If $f$ is a bounded function and $f^2$ is Riemann integrable, is $f$ necessarily Riemann integrable? Answer yes or no and justify your answer.

Answer: No


Question: Determine whether the set \( \left\{ (x,y) \in \mathbb{R}^2 : \sin\left(\frac{x^3}{x^2+y^4}\right) > 0 \right\} \) is open in \( \mathbb{R}^2 \) with the Euclidean topology.

Answer: Yes


Question: Given a set of numbers \([1, 2, 3, 4, 5, 6, 7, 8, 9, 10]\), calculate the expected value of the largest number when selecting 3 numbers at random from the set. Assume each number is selected with equal probability and selections are independent. What is the expected value of the largest number in such a selection?

Answer: \dfrac{319}{40}


Question: Given the differential equation \( r''(t) = \frac{-1}{r(t)^2} \) with initial conditions \( r(0) = 1 \) and \( r'(0) = 0 \), find the value of \( r'(t) \) when \( r(t) = 0 \).

Answer: -\infty


Question: Evaluate the limit \( \lim\limits_{x \to 0} \left(\frac{\sin x}{x}\right)^{\frac{1}{x^{2}}} \).

Answer: e^{-\frac{1}{6}}


Question: Determine whether it is possible for two non-isomorphic graphs to have the same spectrum of their adjacency matrices.

Answer: Yes


Question: Determine the number of functions \( f: \mathbb{N} \to \mathbb{N} \) such that \( \text{LCM}(f(n), n) - \text{HCF}(f(n), n) < 5 \) for all natural numbers \( n \).

Answer: 2880


Question: Calculate the volume of the region in space that lies below the plane $z = 3 - 2y$ and above the paraboloid $z = x^2 + y^2$. Use polar coordinates to set up the integral.

Answer: 8\pi


Question:
Let \( A \in \mathbb{R}^{n \times n} \) be the matrix given by:
\[
A = \begin{pmatrix} 1 & 0 & \cdots & 0 & 1 \\
-1 & 1 & \ddots & \vdots & \vdots \\
\vdots & \ddots & \ddots & 0 & \vdots \\
\vdots & & \ddots & 1 & 1 \\
-1 & \dots & \dots & -1 & 1 \end{pmatrix}
\]
Determine the value of \( \|A\|_\infty \|A^{-1}\|_\infty \). Given that \( \|A\|_\infty = n \), find \( A^{-1} \) and calculate the product.

Answer: n


Question: Determine the number of roots, counting multiplicities, of the equation \(2z^{5}-6z^{2}+z+1=0\) in the annulus \(1\leq |z|<2\).

Answer: 3


Question: Verify that for all positive integers \( n \), the following equation holds: \[ \sum_{1 \leq j \leq n} \mu(j) \left\lfloor \frac{n}{j} \right\rfloor = 1. \] Here, \( \mu(j) \) is the Möbius function and \( \left\lfloor x \right\rfloor \) denotes the greatest integer less than or equal to \( x \).

Answer: 1


Question: Calculate the sum of the infinite series: \[ \sum_{n=0}^{\infty} \frac{(n+1)(2n+1)}{3^n} \]

Answer: \dfrac{27}{4}


Question: Evaluate the integral \( \int_C \bar{z} \, dz \) where the curve \( C \) is the unit circle described by \( |z| = 1 \). Determine if this integral equals zero.

Answer: 2\pi i


Question: Evaluate the cumulative distribution function (CDF) of the standard normal distribution at 0.5365, i.e., find \( N(0.5365) \), where \( N(x) \) denotes the CDF of the standard normal random variable.

Answer: 0.7042


Question: Let \( f \) be analytic in the open unit disk and continuous on the boundary. Is there a sequence of polynomials that converges uniformly to \( f \) on the closed unit disk?

Answer: Yes


Question: Find the smallest integer \( n \) such that the symmetric group \( S_n \) contains an even permutation of order 312.

Answer: 26


Question: Let \((M, \omega)\) be a closed symplectic manifold, and let \(f, g \in C^\infty(M)\) be two smooth functions. Consider the Hamiltonian flows \(\phi_t\) and \(\psi_t\) in \(Ham(M, \omega)\) generated by the Hamiltonian vector fields \(X_f\) and \(X_g\), respectively. Prove that the flows commute for all \(t \in [0,1]\) if and only if the Poisson bracket \(\{f, g\} = \omega(X_f, X_g)\) is identically zero.

Answer: \{f, g\} = 0


Question: Compute the limit using the squeeze theorem and basic limit properties: \[ \lim_{n \rightarrow \infty} \dfrac{\tan\left(\dfrac{\pi}{n}\right)}{n\sin^2\left(\dfrac{2}{n}\right)} \]

Answer: \dfrac{\pi}{4}


Question: Determine the rank of an $n \times n$ matrix $M$, where the elements are defined as $M_{i,i} = 1 - 1/n$ and $M_{i,j} = -1/n$ for $i \neq j$. Prove that the rank of this matrix is $(n-1)$. Provide a detailed explanation of your solution.

Answer: n-1


Question: Calculate the surface area of the conical surface defined by the equation \(3z^2 = x^2 + y^2\) for \(0 \leq z \leq 2\).

Answer: 8\sqrt{3}\pi


Question: Consider a graph with 2n vertices and n^2 edges that contains no triangles. Determine whether this graph is necessarily bipartite.

Answer: Yes


Question: Let $A$ be a $4 \times 4$ real matrix, and consider the complex linear operator $T: \mathbb{C}^4 \to \mathbb{C}^4$ with matrix $A$ relative to the canonical basis of $\mathbb{C}^4$. Given that $i$ is an eigenvalue of $T$ with associated eigenvectors $(-i, 1-i, 1, 0)$ and $(0, 1+i, 0, 2)$, evaluate $A^{15}$. Assume that the complex conjugate of $i$ is also an eigenvalue with its corresponding eigenvectors.

Answer: -A


Question: Find the inverse of 4 in the 5-adic integers, \( \mathbb{Z}_5 \). Using this inverse, determine an integer \( m \in \mathbb{Z}_{>0} \) such that \( 4m \equiv 1 \pmod{5^4} \).

Answer: 469


Question: Evaluate the limit: $$\lim_{n \rightarrow \infty} \int^{n}_{0} \left(1+\frac{x}{n}\right)^{-n} \log\left(2+ \cos\left(\frac{x}{n}\right)\right)\ \mathrm{d}x.$$

Answer: \log 3


Question: Solve the differential equation: \[ f'(x) = 6x^2 - 4x - f(x+1) \]

Answer: 6x^2 - 28x + 50


Question: Find all analytic functions \( f: \mathbb{C} \longrightarrow \mathbb{C} \) such that \( f(3z) - f(2z) = 5f(z) \) for all \( z \in \mathbb{C} \) and \( f(1) = 3 \).

Answer: 3z^2


Question: Find the solution to the equation $u = e^{-u}$. Is there a single solution?

Answer: W(1)


Question: Evaluate the limit: \[ \lim _{x\to 0}\frac{\int _0^{x^2}\frac{t^3}{1+t^6}dt}{x^8} \]

Answer: \dfrac{1}{4}


Question: Evaluate the expression: $$\frac{29\int_0^1 (1-x^4)^7\,dx}{4\int_0^1 (1-x^4)^6\,dx}$$ without expanding.

Answer: 7


Question: Find the coordinates of all points P from which two tangent lines to the ellipse \( \frac{x^2}{17} + \frac{y^2}{8} = 1 \) are perpendicular to each other.

Answer: x^2 + y^2 = 25


Question: Let \( \{X_t \mid t\in\mathbb{N}\} \) be an iid sequence of normal random variables on a probability space \((\Omega,\mathcal{F},P)\). Define \(A(\omega) = \{X_t(\omega) \mid t\in\mathbb{N}\}\) for all \(\omega\in\Omega\) and let \(\overline{A(\omega)}\) be its closure. Let \(\ell\) be the Lebesgue measure. Determine the probability \(P(\{\omega \mid \ell(\overline{A(\omega)}) = 0\})\).

Answer: 0


Question: Let $A \in \mathbb{R}^{n \times n}$ be an orthogonal matrix and let $\varepsilon > 0$. Does there exist a rational orthogonal matrix $B \in \mathbb{R}^{n \times n}$ such that $\|A-B\| < \varepsilon$? An orthogonal matrix $A$ satisfies $A^T = A^{-1}$, and a rational matrix has all entries as rational numbers.

Answer: Yes


Question: Determine the values of the genus $g$ for which there exists a rank 2, degree $2g-2$, holomorphic vector bundle $E \to C$ over a curve of genus $g$ such that $H^0(E) = 0$.

Answer: g \geq 0


Question: Let \((r_k)_{k \in \mathbb{N}}\) be an enumeration of \(\mathbb{Q}\). Define \(U_{1/p} = \bigcup_{k \in \mathbb{N}} \left(r_k - \frac{1/p}{2^k}, r_k + \frac{1/p}{2^k}\right) \subset \mathbb{R}\). Determine the set \(\bigcap_{p \in \mathbb{N}^*} U_{1/p}\).

Answer: \mathbb{Q}


Question:
Given a random variable $X$ that follows a Student's $t$ distribution with $k$ degrees of freedom, determine the distribution of $Y = X^2$. The probability density function of $X$ is given by:
$$f_X(x;k) = \frac{\Gamma\left(\frac{k+1}{2}\right)}{\sqrt{k\pi} \ \Gamma \left(\frac{k}{2}\right)}{\left(1+\frac{x^2}{k}\right)}^{-\frac{k+1}{2}}$$

Answer: F(1, k)


Question: Let \( g: \mathbb{R} \to \mathbb{R} \) be a strictly increasing, continuous function such that \( g(x) > x \) and \( \lim\limits_{x \rightarrow -\infty} g(x) = -\infty \). Is there a continuous function \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(f(x)) = g(x) \)?

Answer: Yes


Question: Determine the asymptotic decay rate of the sequence \( \{b_n\}_{n=1}^{\infty} \), where \( b_n \) is the \( n \)-th largest number in the sequence \( a_{i,j} = \frac{1}{i^2 j^2} \) for \( i,j=1,2,\dots \). Specifically, find if \( b_n = O(n^{-k}) \) for some \( k \).

Answer: 2


Question: Calculate the integral \( \int_0^{2016} f(x) \, dx \) where the function \( f(x) \) satisfies the following properties: \( f(x) + f(-x) = 2 \) and \( f(1-x) = f(1+x) \).

Answer: 2016


Question: Find the limit: \[ \lim_{x\rightarrow +\infty}\left(x\arctan x-x\dfrac{\pi}{2}\right) \]

Answer: -1


Question: Determine the radius of convergence of the power series \( \sum_{n = 0}^{\infty} \frac{x^n}{a^{n+1} -1} \), where \( a \) is a constant greater than 1.

Answer: a


Question: For how many non-negative integers \( n \) from 0 to 300 (inclusive) is the binomial coefficient \( \binom{300}{n} \) not divisible by 5?

Answer: 9


Question: In a 2-state continuous-time Markov chain, the process stays in state A for an exponentially distributed amount of time with a mean of 2 hours before moving to state B. The process stays in state B for an exponentially distributed amount of time with a mean of 3 hours. Calculate the probability that at least one state change occurs within one hour, given that the process starts in state B.

Answer: 1 - e^{-1/3}


Question:
Given that $1, x,$ and $x^2$ are solutions of a second-order linear non-homogeneous differential equation on the interval $-1 < x < 1$, determine the general solution involving arbitrary constants. Choose the correct option:

(a) $c_1(1-x) + c_2(x - x^2) +1$

(b) $c_1(x) + c_2 ( x^2) +1$

(c) $c_1(1+x) + c_2(1 + x^2) +1$

(d) $c_1 + c_2 x + x^2$

Answer: a


Question: Show that the average value of the function \( f(x, y) = \sqrt{s^2 - x^2 - y^2} \) over the region \( D = \{(x, y) : x^2 + y^2 \leq s^2\} \) is \( \bar{z} = \frac{2s}{3} \).

Answer: \dfrac{2s}{3}


Question: Apply Green's theorem to evaluate the line integral of the vector field \( F(x,y) = (x+y, x-y) \) over the positively oriented region bounded by the circle \( x^2 + y^2 = 9 \) and the curve \( |x| + |y| = 1 \).

Answer: 0


Question: Determine the values of \(a\) for which the series \(\sum_{n=1}^{\infty} \frac{n^a}{1+n^3}\) converges.

Answer: a < 2


Question:
Find the smallest value of $b$ such that for all $0 \leq a < b$, the series \( \sum\limits_{n=1}^{\infty}\dfrac{(n!)^2a^n}{(2n)!} \) converges. Choose from the following options:

A) $1$
B) $2\log 2$
C) $2$
D) $\sqrt{2}$
E) $4$

Answer: E


Question:
Consider the curve given by \( y^2 = x^3 \). Let \( P(x_1, y_1) \) be a point on the curve, and the tangent at \( P \) intersects the curve again at \( Q(x_2, y_2) \). If the lines \( OP \) and \( OQ \) make angles \( \alpha \) and \( \beta \) with the x-axis, respectively, find the value of \( \frac{\tan\alpha}{\tan\beta} \).

(A) \(-1\)
(B) \(-2\)
(C) 2
(D) \sqrt{2}

Answer: B


Question: Evaluate whether the integral of the function \(f(x_1, \ldots, x_n) = \prod_{i=1}^n x_i\) over the closed unit disk \(\overline{D}_1(0) = \{x \in \mathbb{R}^n : \sum_{i=1}^n x_i^2 \leq 1\}\) is zero, where \(n\) is a positive integer and \(\lambda\) is the Lebesgue measure.

Answer: 0


Question:
Let \(\alpha, \beta, \gamma, \delta\) be fixed real numbers, and \(x\) a variable in \([0,\pi)\). Consider the expression:
\[
(\alpha^2+\beta^2)\cos^2(x) + 2(\alpha\gamma+\beta\delta)\cos(x)\sin(x) + (\gamma^2+\delta^2)\sin^2(x)
\]
Suppose this expression has a maximum at \(x=M\) and a minimum at \(x=m\). Prove that \(|M-m|=\pi/2\).

Answer: \dfrac{\pi}{2}


Question:
Evaluate the integral \( \int_{0}^{1} x^x \, dx \) and express the answer in terms of the constants:

\[ A = \frac{1}{1^1} + \frac{1}{2^2} + \frac{1}{3^3} + \cdots \]
\[ B = \frac{1}{2^2} + \frac{1}{4^4} + \frac{1}{6^6} + \cdots \]

Answer: A - 2B


Question:
Let $f(x)$ be a differentiable function, and suppose there exists some $a$ where $f'(a) \ne 0$. Calculate the limit:
\[ \lim_{h\rightarrow0}\frac{f(a+3h)-f(a-2h)}{f(a-5h)-f(a-h)}. \]

Answer: -\dfrac{5}{4}


Question: Let $F$ be an absolutely irreducible faithful representation of a group $G$ with degree $n$. Suppose $G'$ is a subgroup of the center $Z(G)$ of $G$. Show that the index $|G:Z(G)| = n^2$. Is the condition $G' \subseteq Z(G)$ necessary for this result?

Answer: n^2


Question: Compute the Lebesgue integral \( \int_{[1, \infty)} \frac{1}{x^2} \, d\mu \).

Answer: 1


Question: Evaluate the limit of the sequence as \( n \to \infty \): \[ \lim_{n \to \infty} \frac{n! - n^3}{n^5 - e^n + 3} \]

Answer: -\infty


Question: What is the probability that the series \( S_\infty = \sum\limits_{n=1}^\infty \frac{t_n}{n} \) converges, where \( t_n \) is a random variable taking values in \( \{-1, 1\} \) with \( \text{P}(t_n=1) = \text{P}(t_n=-1) = \frac{1}{2} \)?

Answer: 1


Question: Calculate \((x+1)^4\) in the field \(GF(3)[x]^*_{x^2+1}\).

Answer: 2


Question: Let $X$ be a compact metric space, $u: X \to [0, 1]$ an upper semi-continuous function, and $l: X \to [0, 1]$ a lower semi-continuous function such that $u(x) < l(x)$ for each $x \in X$. Determine whether there exists a continuous function $f: X \to [0, 1]$ such that $u(x) < f(x) < l(x)$ for each $x \in X$. Provide a justification for your answer.

Answer: Yes


Question: Let $K$ be a field and $f: K - \{0\} \rightarrow K - \{0\}$ be a function such that $f(f(x)) = x^{-1}$ for all $x \in K - \{0\}$ and $f(1) \neq 1$. Given that the equation $f^2(x) - f(x) + 1 = 0$ has a unique solution in $K - \{0\}$, determine the value of $f(2)$. \( \text{Provide your answer in the form } \boxed{\text{answer}}. \)

Answer: 1


Question: Find the limit of the sequence \(a_n = \left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right)\cdots\left(1-\frac{1}{n^2}\right)\).

Answer: \dfrac{1}{2}


Question:
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $\lim _{n \rightarrow \infty} f^{n}(x)$ exists for every $x \in \mathbb{R}$, where $f^{n}(x)=f \circ f^{n-1}(x)$ for $n \geq 2$. Define the sets:
\[ S = \left\{\lim _{n \rightarrow \infty} f^{n}(x): x \in \mathbb{R}\right\} \quad \text{and} \quad T=\{x \in \mathbb{R}: f(x)=x\} \]
Which of the following is necessarily true?
A. $S \subset T$
B. $T \subset S$
C. $S = T$
D. None of the above

Answer: C


Question:
Given the integral \( I(a) = \int_0^{\infty}\frac{e^{-ax}}{1+x^2}dx \), it is known that the solution can be expressed as:
\[ I(a) = (-Si(a) + c_1)\cos(a) + (Ci(a) + c_2)\sin(a) \]
where \( Si(a) \) and \( Ci(a) \) are the sine and cosine integral functions, respectively. It is given that \( c_1 = \frac{\pi}{2} \) using the condition \( I(0) = \frac{\pi}{2} \). Determine the value of the constant \( c_2 \).

Answer: 0


Question: Evaluate the double integral \( \iint_B (x^2 + y^2) \, dx \, dy \) over the region \( B \) in the first quadrant bounded by the curves \( xy = 1 \), \( xy = 3 \), \( x^2 - y^2 = 1 \), and \( x^2 - y^2 = 4 \), using the change of variables \( u = x^2 - y^2 \) and \( v = xy \).

Answer: 3


Question:
Which of the following definitions correctly describes a confidence interval of level \(1 - \alpha\)?

A) An interval \(\mathcal{I}_n\) whose boundaries do not depend on \(\bar{\theta}\) (but are possibly a function of the sample) such that \(\Bbb P_{\theta}(\mathcal{I}_n \ni \bar{\theta}) \geq 1 - \alpha \;\;\forall \;\theta \in \Theta.\)

B) An interval \(\mathcal{I}_n\) whose boundaries do not depend on \(\bar{\theta}\) (but are possibly a function of the sample) such that \(\Bbb P_{\theta}(\mathcal{I}_n \ni \theta) \geq 1 - \alpha \;\;\forall \;\theta \in \Theta.\)

C) An interval \(\mathcal{I}_n\) whose boundaries do not depend on \(\bar{\theta}\) (but are possibly a function of the sample) such that \(\Bbb P_{\bar{\theta}}(\mathcal{I}_n \ni \bar{\theta}) \geq 1 - \alpha.\)

Answer: B


Question:
Consider the function \( f(z) = z + 2z^2 + 3z^3 + \cdots = \sum_{n \geq 0} nz^n \) defined on the open disk \( \{z \mid |z| < 1\} \). Choose the correct option:

(a) \( f \) is not injective but attains every complex value at least once.
(b) \( f \) is injective but does not attain every complex value.
(c) \( f \) is injective and attains every complex value.
(d) None of the above.

Answer: B


Question: Determine the minimal positive integer \( r \) such that \( \int_{-1}^{1} \frac{35}{32}(1-t^2)^3 t^r \, dt \neq 0 \). This integer \( r \) is known as the order of the kernel function \( K(t) = \frac{35}{32}(1-t^2)^3 \mathbf{1}(|t| \leq 1) \).

Answer: 2


Question: Find the limit: \[ g(x) = \lim_{J \to +\infty} \frac{1}{2J} \coth\left(\frac{x}{2J}\right) \]

Answer: \dfrac{1}{x}


Question:
Calculate the limit of the function involving the Gamma function for a given integer value of \( n \):

\[ f(n) = \lim_{c \rightarrow 0} \frac{\Gamma(-n + c) + \Gamma(-n - c)}{2} \]

Find \( f(n) \) for \( n = 0 \).

Answer: -\gamma


Question: Does the reconstruction error in PCA decrease as more principal components are used? Consider a dataset with $N$ dimensions, where $N$ is a large positive integer. If PCA is applied to this dataset, will the reconstruction error (measured in the $\ell^2$-norm) decrease monotonically to $0$ as the number of components $d$ increases from $1$ to $N$?

Answer: Yes


Question: Find the value of \(c\) such that the substitution \(x = u + c\) transforms the equation \(x^3 - 12x^2 + 45x - 54 = 0\) into the equation \(u^3 - 3u^2 = 0\).

Answer: 3


Question:
If \(z\) is a complex number satisfying \(|z^3+z^{-3}| \le 2\), what is the maximum possible value of \(|z+z^{-1}|\)?

(A) \(2\)
(B) \(2^{1/3}\)
(C) \(2\sqrt{2}\)
(D) \(1\)

Answer: A


Question:
Find the moment generating function $M_Y(t)$ of the random variable $Y = |X-2|$, where $X$ has the probability density function:
\[ f_X(x) = \begin{cases} \frac{1}{2}, & 0 \le x \le 1 \\ \frac{1}{4}, & 3 \le x \le 5 \end{cases} \]
Then, use the moment generating function to calculate the expected value $E(Y)$. Note that $E[Y] = \frac{d(M_Y(t))}{dt}\bigg|_{t=0}$.

Answer: \dfrac{7}{4}


Question: Determine all positive integers $k$ such that $10^k + 1$ is divisible by 49.

Answer: k \equiv 21 \pmod{42}


Question: Solve the equation \(x^2 + 4(\sqrt{1 - x} + \sqrt{1 + x}) - 8 = 0\).

Answer: 0


Question: Find the direct limit of the sequence of groups \( \frac{1}{n}\mathbb{Z} \).

Answer: \mathbb{Q}


Question: Evaluate the limit: \[ \lim_{x\to0^{+}}\left[\ln\frac{f(2+3x)\sin(x)}{f(2-5x)(x+1)}\right] \] given that $f(x)$ is a continuous, differentiable function with $f'$ continuous, $f(2)=0$, $f(x)>0$ for all $x\neq2$, and $f'(2)=7$. Use the given conditions to determine the limit.

Answer: -\infty


Question: Evaluate the limit: \[ \lim_{n\to\infty} \sum_{r=0}^n \frac{\binom{n}{r}}{(n^r)(r+3)} \]

Answer: e - 2


Question: Calculate the monoid of natural transformations from the identity functor to itself in the category of abelian groups, \( \text{Id}_{\mathfrak{Ab}} \rightarrow \text{Id}_{\mathfrak{Ab}} \).

Answer: \mathbb{Z}


Question: Given \( \Delta = \left|\begin{array}{ccc} a & b & c \\ c & a & b \\ b & c & a \end{array}\right| \), find the value of the determinant \( \left|\begin{array}{ccc} a^2 - bc & b^2 - ca & c^2 - ab \\ c^2 - ab & a^2 - bc & b^2 - ca \\ b^2 - ca & c^2 - ab & a^2 - bc \end{array}\right| \) in terms of \( \Delta \).

Answer: \Delta^2


Question: A bag contains an infinite number of coins, each with a probability of landing heads on any given flip uniformly distributed between 0 and 1. A coin is drawn at random from this bag. Given that the first flip results in a head, what is the probability that the next flip will also be a head?

Answer: \dfrac{2}{3}


Question: Consider $f \in \mathbb{C}[X,Y]$ as an irreducible, non-singular curve. Let $A = \mathbb{C}[X,Y] / (f)$ be the coordinate ring of $f$. Choose a curve $g \in \mathbb{C}[X,Y]$ with no component in common with $f$. Let $(x_i,y_i)$ be a common zero of $f$ and $g$ with multiplicity $r_i$, where $1 \leq i \leq n = | Z(f,g)|$. Is it true that the ideal $(\overline{g})$ of $A$ can be factorized as $\prod_{i \leq n} (\overline{X}-x_i,\overline{Y} - y_i)^{r_i}$?

Answer: Yes


Question: Solve the initial value problem for the differential equation \((\sqrt{xy}-x)\,dy+y\, dx=0\) with the initial condition \(y(1)=0\).

Answer: 0


Question: Let U be a random variable uniformly distributed on the interval [-1,1]. Define V = U^2. Calculate the correlation coefficient, corr(U, V).

Answer: 0


Question: Let \( f:[a, b]\to\mathbb{R} \) be a twice differentiable function satisfying \( f(x)=f'(x)+f''(x) \) for each \( x\in[a,b] \) and \( f(a)=f(b)=0 \). Find all such functions \( f \).

Answer: 0


Question: Given two complex numbers \(a\) and \(b\), maximize the expression \(a\overline{b} + b\overline{a}\) subject to the constraint \(|a|^2 + |b|^2 = 1\).

Answer: 1


Question: Two line segments $OA$ and $OB$ are inclined to each other at $120^\circ$ at point $O$. $P$ is a fixed point on the internal angle bisector of $\angle AOB$. Given that $OA = OB = 20\text{ cm}$ and $OP = 10\text{ cm}$, two points are chosen on the line segments $OA$ and $OB$ such that the two points and $P$ form an equilateral triangle. The two points may be on the same line segment or on different line segments. If the number of such non-congruent equilateral triangles that can be made is $k$, find the value of $\left\lfloor \frac{200}{k} \right\rfloor$. \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to $x$. \( \boxed{} \)

Answer: 100


Question: Given a sequence of real numbers $a_1, a_2, \ldots, a_{2013}$ such that $a_1 > 3$, $a_{2013} = 2013$, and for $n = 1, 2, \ldots, 2012$, $a_{n + 1} = \frac{-a_n + a_n\sqrt{4a_n - 3}}{2}$, determine the value of \[ a_1 + \sum_{i = 1}^{2012}\frac{a_{i + 1}^3}{a_i^2 + a_ia_{i + 1} + a_{i + 1}^2}. \]

Answer: 4025


Question: Find the residue of the function \( \frac{\cot z}{z^4} \) at \( z = 0 \).

Answer: -\dfrac{1}{45}


Question: Find the number of ways to make change for 260 cents using coins of 1 cent, 2 cents, and 8 cents. Use binomial coefficients in your solution.

Answer: 2211


Question: Given that $X$ is uniformly distributed on $[0,1]$ and the conditional distribution of $Y$ given $X=x$ is uniform on $[1-x,1]$, find the probability $P(X+Y > \frac{3}{2})$. Provide your answer to four decimal places.

Answer: 0.1534


Question: Determine how many roots the polynomial \( z^4 + 3z^2 + z + 1 \) has in the right-half complex plane, where \( \text{Re}(z) > 0 \).

Answer: 2


Question: Determine the irreducible representations of the tensor product $\varphi_2 \otimes \varphi_3$ for the $SU(2)$ group, where $\varphi_n$ denotes the standard irreducible representation with highest weight $n$.

Answer: \varphi_1 \oplus \varphi_3 \oplus \varphi_5


Question: Evaluate the improper integral: $$\int_{0}^{\infty}\int_{y}^{\infty} \sin\left(\frac{\pi y}{2x}\right) \frac{e^{-x}}{x} \, dx \, dy.$$

Answer: \dfrac{2}{\pi}


Question: Does there exist a sequence of smooth functions $(f_n) \subset C^\infty([0,1])$ such that $\|f_n\|_{L^2} = 1$ for all $n$, but $\|f'_n\|_{L^2} \to \infty$?

Answer: Yes


Question:
Find the sum of all possible values of $xyz$ given that $x, y, z \in \mathbb{Z}$ satisfy the equation:
\[ x^2 + 2y^2 + 2z^2 - 2x - 6y - 10z + 2xy + 2yz + 14 = 0. \]

Answer: -6


Question: Let $X$ be a smooth projective algebraic curve over $\mathbb{C}$. Let $d$ be a positive integer. Determine whether there exists a smooth projective curve $Y$ and a finite morphism $f : Y \rightarrow X$ such that $f$ has degree $d$. Provide a justification for your answer.

Answer: Yes


Question: Given that $a^4 + a^3 + a^2 + a + 1 = 0$, find the value of $a^{2000} + a^{2010} + 1$. Provide your answer as a single number.

Answer: 3


Question:
Calculate the expectation \( \mathbb{E}(Y) \) where \( Y = X^6 \) and \( X \sim \mathcal{N}(0,1) \). Use the Law of the Unconscious Statistician (LOTUS) and integration by parts to evaluate the integral:
\[ \mathbb{E}(Y) = \int_{-\infty}^{\infty} x^6 \cdot \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2} \, dx \]

Answer: 15


Question: Determine whether there exists a positive integer \( k \) such that for every finite, simple, undirected graph \( G = (V, E) \) and any two maximum matchings \( M_1 \) and \( M_2 \) in \( G \), the inequality \( |M_1| - |M_2| \leq k \) holds. If such a \( k \) exists, find its value.

Answer: 0


Question: Suppose $f$ is a continuous function such that $f(0) = 0$, $f(1) = 1$, and $f'(x) > 0$ for all $x$ in the interval $[0, 1]$. Given that $\int_{0}^{1} f(x) \, dx = \frac{1}{3}$, find the value of the integral $\int_{0}^{1} f^{-1}(y) \, dy$. \( \text{Note: } f^{-1} \text{ denotes the inverse function of } f. \)

Answer: \dfrac{2}{3}


Question: Consider a $10 \times 10$ nilpotent matrix $A$ such that $A^3 = 0$ but $A^2 \neq 0$. Determine whether there exists a non-zero vector that lies in both the column space and the null space of $A$. In other words, is $\text{col}(A) \cap \text{nul}(A) \neq \{0\}$?

Answer: Yes


Question: Let $V = W \oplus W^{\bot}$ and $T: V \to W$ be the corresponding projection. Show that the adjoint of $T$, denoted $T^{*}$, is equal to $T$.

Answer: T^{*} = T


Question: Consider a finite simple graph $H$ with $v_H \ge 3$ vertices and $e_H \ge 3$ edges. A graph $H$ is defined as strict 2-balanced if \( \frac{e_H-1}{v_H-2} > \frac{e_K-1}{v_K-2} \) for all proper subgraphs $K$ with $v_K \ge 3$. It is defined as strict balanced if \( \frac{e_H}{v_H} > \frac{e_K}{v_K} \) for all proper subgraphs $K$ with $v_K \ge 1$. Determine whether a strict 2-balanced graph must also be strict balanced. Assume the density of the graph is greater than 2.

Answer: Yes


Question: Given that the moment generating function of a random variable $X$ is $M_X(u) = e^{5u + 2u^2}$, determine the probability that the stock price $S = e^X$ is between $70$ and $90$, given that $S > 50$. Assume $X$ follows a normal distribution with mean $\mu = 5$ and variance $\sigma^2 = 4$. Provide your answer as a probability.

Answer: 0.0675


Question:
Evaluate the limit:

If \(f(x) = \sum_{k=1}^{n} \frac{1}{\sin 2^k x}\) and \(g(x) = f(x) + \frac{1}{\tan 2^n x}\), find the value of
\[ \lim_{x \to 0} \left( (\cos x)^{g(x)} + \left(\frac{1}{\cos x}\right)^{\frac{1}{\sin x}} \right). \]

Answer: 2


Question: Let \( f(x) = x^3 - 6x^2 + x + 4 \) and \( g(x) = x^5 - 6x + 1 \). Use the Euclidean algorithm to find and justify that the greatest common divisor (GCD) of \( f(x) \) and \( g(x) \) is \( 1 \).

Answer: 1


Question: Let $A$ be an $n \times n$ real matrix, $X$ be an $a \times n$ real matrix, and $Y$ be an $n \times b$ real matrix. Define $M_{n-1} = XA^{n-1}Y$ and $M_n = XA^nY$. Is it possible to choose $X$, $A$, and $Y$ such that there exist indices $i$ and $j$ for which $M_{n-1}(i,j) = 0$ but $M_n(i,j) \neq 0$? Consider whether Cayley-Hamilton's theorem can assist in answering this question.

Answer: Yes


Question: Calculate the line integral \( \oint_\Gamma x \, dx + (x+y) \, dy + (x+y+z) \, dz \) using Stokes' theorem, where the curve \( \Gamma \) is parameterized by \( x = \sin{t}, y = \cos{t}, z = \sin{t} + \cos{t} \) for \( t \in [0, 2\pi] \).

Answer: -\pi


Question:
Find the function \( f(x) \) such that for all \( a > 1 \) and \( b > 1 \), the following equation holds:
\[ \int_{3}^{ab+2} f(x) \, dx = \int_{3}^{a+2} f(x) \, dx + \int_{3}^{b+2} f(x) \, dx. \]

Answer: \dfrac{1}{x - 2}


Question: Determine the radius of convergence for the Maclaurin series of \(\tan(z)\) without explicitly finding the series.

Answer: \dfrac{\pi}{2}


Question: Determine the cardinality of the set of all polynomials with complex coefficients, \(\mathbb{C}[x]\), using an expression built from cardinal operations, 2, and \(\omega\).

Answer: 2^{\omega}


Question: Given a \(3 \times 3\) matrix \(A\) with \(\det(A) = 6\), and the equation \(A(A^{17} - cI) = 0\), find the value of \(c\).

Answer: 6^{17/3}


Question:
Let \( f \) be a non-negative differentiable function on \([0,1]\) such that \( \int_{0}^{x} \sqrt{1-(f'(t))^2} \ dt = \int_{0}^{x} f(t) \ dt \) for \( 0 \le x \le 1 \) and \( f(0)=0 \). Determine which of the following is true:

1. \( f\left(\frac{1}{2}\right) < \frac{1}{2} \) and \( f\left(\frac{1}{3}\right) > \frac{1}{3} \).
2. \( f\left(\frac{1}{2}\right) > \frac{1}{2} \) and \( f\left(\frac{1}{3}\right) > \frac{1}{3} \).
3. \( f\left(\frac{1}{2}\right) < \frac{1}{2} \) and \( f\left(\frac{1}{3}\right) < \frac{1}{3} \).
4. \( f\left(\frac{1}{2}\right) > \frac{1}{2} \) and \( f\left(\frac{1}{3}\right) < \frac{1}{3} \).

Answer: 3


Question: Find the particular solution of the differential equation \( t^2y'' - t(t+2)y' + (t+2)y = 6t^3 \) for \( t > 0 \) using the method of variation of parameters. Given that \( y_1(t) = t \) and \( y_2(t) = te^t \) are solutions of the corresponding homogeneous equation, determine the particular solution.

Answer: -6t^2 - 6t


Question: Evaluate the line integral \( \oint_C \mathbf{F} \cdot \mathbf{T} \, ds \) for the vector field \( \mathbf{F}(x,y) = xy \, \mathbf{i} + x^2 \, \mathbf{j} \), where \( C \) is the rectangle with vertices \((0,0), (3,0), (3,1), (0,1)\), and \( \mathbf{T} \) is the unit tangent vector to \( C \) directed anticlockwise.

Answer: \dfrac{9}{2}


Question: Evaluate the integral \( \int_0^{2\pi} \tan \frac{\theta}{8} \, d\theta \) using complex analysis techniques.

Answer: 4 \ln 2


Question: Evaluate the limit: \[ \lim_{x \to 0^+} \frac{e^{-\frac{1}{x}}}{x^{2}} \]

Answer: 0


Question: If \( \sigma \) is a nontrivial automorphism of a field \( K \), is the induced automorphism \( \phi \) of \( GL(n,K) \), obtained by applying \( \sigma \) to the entries of matrices in \( GL(n,K) \), necessarily an outer automorphism of \( GL(n,K) \)?

Answer: Yes


Question: Simplify the ordinal expression \((\omega + 1) + \omega\).

Answer: \omega \cdot 2


Question: Given a function $f(x)$ for $x>0$ that is positive, with a derivative $f'(x)$ having only one root at $x=a$ where $f''(a)>0$, and both $f(x)$ and $f'(x)$ are continuous and differentiable, determine if $x=a$ is the global minimum of the function.

Answer: Yes


Question: A chemist uses two instruments, A and B, to measure the amount of a substance \(\mu\) in food. The measurements from instrument A are \(X_1, X_2, \ldots, X_n\) and from instrument B are \(Y_1, Y_2, \ldots, Y_m\), with \(\sigma_X = \sigma_Y = 1.6\). The chemist uses the estimator \(\hat{\mu} = a\bar{X} + b\bar{Y}\), where \(\bar{X}\) and \(\bar{Y}\) are the sample means. Given \(n = 12\) and \(m = 18\), find the value of \(a\) such that \(\hat{\mu}\) is an unbiased estimator with minimum variance.

Answer: \dfrac{2}{5}


Question: Evaluate the limit: $$\lim_{z \to \infty}\int_{\mathbb{C}}|w|e^{-|z-w|^2}dA(w)$$ where $dA = rdrd\theta$ is the area measure in polar coordinates.

Answer: \infty


Question: Find the residue of the function \( \frac{\cot(\pi z)}{z^2} \) at the pole \( z = 0 \).

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


Question: Is the space of connections modulo gauge equivalence on a compact four-manifold paracompact in the Sobolev topology?

Answer: Yes


Question: Let $A \in \mathbb{R}^{n \times n}$ be a matrix with $\|A\| < 1$, and let $\vec{b} \in \mathbb{R}^{n}$ be an eigenvector of $A$ corresponding to the largest eigenvalue $0 < \lambda < 1$. Is it possible for $\vec{b}$ to also be an eigenvector of $(I-A)$ corresponding to the largest eigenvalue?

Answer: Yes


Question: Find the $2n$-th derivative of the function $(x^2-1)^n$ and verify that it equals $(2n)!$. Express your answer as $\frac{d^{2n}}{dx^{2n}}(x^2-1)^n = (2n)!$. Show your work.

Answer: (2n)!


Question: Solve for $x$ in the equation: \[ \cos^\pi(x) + \sin^\pi(x) = 2^x \]

Answer: 0


Question: Given a function $f \in L^1(\mathbb{T})$ and $\epsilon > 0$, determine whether there exists a continuous function $g \in L^1(\mathbb{T})$ such that $\|f - g\|_1 < \epsilon$. Here, $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ is the unit circle, and the continuity of $g$ refers to the continuity of its periodic extension to $\mathbb{R}$.

Answer: Yes


Question: Determine whether the polynomial \( f(x) = x^4 - x^3 + 14x^2 + 5x + 16 \) can be expressed as the product of two polynomials of degree two with integer coefficients. Answer true or false.

Answer: False


Question: Evaluate the limit \( \lim_{x \to \infty} \int_0^\infty \frac{t e^{-xt}}{1-e^t} \, dt = 0 \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to 0}\frac{xe^{2x}+xe^{x}-2e^{2x}+2e^{x}}{(e^{x}-1)^{3}} \]

Answer: \dfrac{1}{6}


Question: For integer solutions greater than 1 to the equation \(x^2 - 3y^2 = 1\), can \(\frac{y+1}{2}\) be a square number?

Answer: No


Question: Does the apparent contradiction in the sum of densities of deficient odd numbers, abundant integers, and perfect numbers imply an error in one of the cited papers?

Answer: Yes


Question: Evaluate the limit \( \lim_{n \to \infty} \left( \frac{f(a + \frac{1}{n})}{f(a)} \right)^n \) given that the function \( f \) is differentiable at \( x = a \), with \( f'(a) = 2 \) and \( f(a) = 5 \).

Answer: e^{\frac{2}{5}}


Question: Let $V$ be a finite-dimensional vector space over a field $F$. Suppose $B$ is a basis of $V$ such that \( \left\{\begin{bmatrix} w \\ w \end{bmatrix} \mid w \in B \right\} \) is a basis for $V^2$. Determine the dimension of $V$. \( \boxed{} \)

Answer: 0


Question: Given a homomorphism \( \phi : (\mathbb{Z} \oplus \mathbb{Z}) \rightarrow (G, +) \) defined by \( (3,2) \mapsto x \) and \( (2,1) \mapsto y \), find \( \phi((4,4)) \).

Answer: 4x - 4y


Question: Determine the number of permutations of the set \( \{1, 2, \, \ldots, 9\} \) in which at least one odd integer is in its natural position.

Answer: 157824


Question: A cylinder has a circumference of 4 and a height of 3. Point A is directly above point B, and the distance from B to C along the circumference is 1. The shortest distance through the cylinder from point A to C is given by \( \sqrt{\frac{M+N\pi^2}{P\pi^2}} \), where M, N, and P are positive integers. Find the smallest value of the sum of M, N, and P.

Answer: 18


Question:
Evaluate the limit:
\[ \lim_{L \to 0} \int_0^L \int_0^L \left|\frac{L - x - y}{L} \right| f(x,t) f(y,t) \, \mathrm{d}x \, \mathrm{d}y, \]
where \( f(x,t): \mathbb{R}\times\mathbb{R}_+ \rightarrow [0,M] \) is continuously differentiable in \( x \) and \( M < \infty \). Determine if this limit converges to \( f(0,t)^2 \).

Answer: 0


Question: Let $A_1 A_2 \dotsb A_{11}$ be a regular 11-gon inscribed in a circle of radius 2. Let $P$ be a point such that the distance from $P$ to the center of the circle is 3. Calculate the value of $PA_1^2 + PA_2^2 + \dots + PA_{11}^2$. \(\boxed{\text{Answer}}\)

Answer: 143


Question: Determine the limit of the series \( S = 1 + r + \frac{1}{2}r^2 + \frac{1}{3}r^3 + \cdots \) if it exists, given that the series converges for \( r < 1 \).

Answer: 1 - \ln(1 - r)


Question: Evaluate the ratio \( \frac{A}{B} \) where \( A = \int_{\sin^2 t}^{1+\cos^2 t} x f[x(2-x)] \, dx \) and \( B = \int_{\sin^2 t}^{1+\cos^2 t} f[x(2-x)] \, dx \).

Answer: 1


Question: Find a real number \( \lambda \) such that \((A+B)^{-1} \leq (A+\lambda I)^{-1}\), where \(A\) and \(B\) are real symmetric non-singular matrices and \(A+B\) is non-singular.

Answer: \lambda_{\min}(B)


Question: Let $X$ be a connected CW complex with a finite fundamental group $\pi_1(X)$. Let $Y$ be a CW complex, and consider a continuous map $f: X \to Y \times S^1$. Prove that there exists a map $g: X \to Y$ such that $f$ is homotopic to the map $(g,1): X \to Y \times S^1$.

Answer: g


Question: Find a rational number $a \in \mathbb{Q}$ such that $a$ is not the norm of any element in the degree 3 Galois extension $E = \mathbb{Q}[x]/(x^3 + x^2 - 2x - 1)$ over $\mathbb{Q}$. The norm map is given by \( N_{E/\mathbb{Q}}(a+bx+cx^2) = a^3-a^2b-2ab^2+b^3+5ac^2-abc-b^2c+6ac^2-2bc^2+c^3 \).

Answer: 2


Question: Determine the values of $s$ for which the integral \[ \int_0^\infty \frac{\mathrm{d} x}{x \cdot \ln x \cdot \ln^{(2)} x \cdot \ln^{(3)} x \cdots (\ln^{(k)} x)^s } \] converges, where $\ln^{(k)} x$ denotes the $k$-fold iterated logarithm of $x$. Provide a condition or range for $s$ that ensures convergence.

Answer: s > 1


Question: Determine the number of positive integers $x$ for which the expression $x^4 + x^3 + x^2 + x + 1$ is a perfect square.

Answer: 1


Question: Assume that $X_j$ and $Y_j$ are independent Poisson distributed random variables with the same rate $\lambda > 0$ for all $j = 0, 1, 2, \ldots$. Define $U_j$ such that $U_j(\omega) = Y_{X_j(\omega)}(\omega)$ for all $\omega \in \Omega$. Find $\mathbb{E}[U_j]$. Provide your answer in terms of $\lambda$.

Answer: \lambda


Question:
Evaluate the limit of the multivariable function as \((x, y)\) approaches \((1, -1)\):
\[ \lim_{x\to1, y\to-1}{\frac{\arcsin{(x^5+y^5)}}{x^5+\tan{(x^4y+xy^4)}+\log{(1+x^3y^2+x^2y^3)}+y^5}} \]

Answer: \dfrac{5}{3}


Question:
Let $T$ be a linear map defined on a subspace $S$ of a Euclidean vector space $V$, satisfying the contraction condition:
\[ d(T(v_1),T(v_2)) \leq d(v_1,v_2) \quad \text{for } v_1,v_2\in S \]
where $d(v_1,v_2)$ is the Euclidean distance on $V$. Does there always exist a linear map $T'$ defined on $V$ such that:
\[ d(T'(v_1),T'(v_2)) \leq d(v_1,v_2) \quad \text{for all } v_1, v_2\in V \]
and $T'(v) = T(v)$ for all $v \in S$?

Answer: Yes


Question:
For discrete probability distributions \( P \) and \( Q \) defined on the same probability space, \( \mathcal{X} \), the relative entropy from \( Q \) to \( P \) is defined as
\[
D_{\mathrm{KL}}(P \| Q)=\sum_{x \in \mathcal{X}} P(x) \log \left(\frac{P(x)}{Q(x)}\right)
\]
Assume the two distributions are \( P=[p, 0, \ldots,0, 1-p] \) and \( Q=\text{binomial}(n,q)=\binom{n}{k} q^{k}(1-q)^{n-k}, \forall k \in \{0,1,2,...n\} \).
Both distributions have \( n+1 \) mass points. Find the parameter \( q \) that minimizes \( D_{\mathrm{KL}}(P \| Q) \).

Answer: 1 - p


Question: Find a sufficient statistic \(T(X)\) that serves as an unbiased estimator for \(1/p\) in a geometric distribution with probability mass function \(f(x;p) = p(1-p)^x\), where \(x = 0, 1, 2, \ldots\) and \(0 < p < 1\).

Answer: X + 1


Question: Determine whether the limit \( \lim_{(x,y)\rightarrow (0,0)} \frac{x^3-x^2y+xy^2-y^3}{x^2+y^2} \) exists.

Answer: 0


Question: Find the remainder when \(3^{2015} + 7^{2015}\) is divided by 50.

Answer: 0


Question: Given the equation \( px^2 + y^2 + qz^2 + 2yz + zx + 3xy = 0 \) represents a pair of perpendicular planes, find the value of \( p - q \).

Answer: 5


Question: Consider the language \(\{a^ib^jc^k\mid i+j=k\geq1  \}\). Convert the given context-free grammar to Greibach Normal Form (GNF) and determine the number of productions in the GNF grammar.

Answer: 7


Question: Find the radius of convergence for the complex power series \( \sum n^p z^n \).

Answer: 1


Question: For what value of $a > 1$ is the integral $\int_a^{a^2} \frac{1}{x} \ln\left(\frac{x-1}{32}\right) \, dx$ minimized?

Answer: 3


Question: Calculate the value of $19^{33} \pmod{20413}$. Provide your answer as a single integer.

Answer: 6338


Question: Given a binomial distribution $X$ with $n=2$ and unknown parameter $p$, the probabilities are as follows: $P(X=0)=0.175$, $P(X=1)=0.45$, and $P(X=2)=0.375$. Find the maximum likelihood estimate of the parameter $p$. Provide your answer as a decimal rounded to two decimal places.

Answer: 0.60


Question: Let $Y \sim \text{Exp}(3)$. The random variable $X$ is defined such that if $Y = y$, then $X$ is uniformly distributed in the interval $[y, 3y]$. Compute the covariance of $X$ and $Y$. Use the formula $\text{Cov}(X,Y) = E[XY] - E[X] \cdot E[Y]$. Given that $E[Y] = \frac{1}{3}$, find $E[XY]$ and $E[X]$ to determine the covariance.

Answer: \dfrac{2}{9}


Question: Does there exist a Riemann integrable function \( f: [0,1]^n \to \mathbb{R} \) such that for every non-empty open subset \( U \subset (0,1)^n \), the set \( \{x \in U \mid f \text{ is discontinuous at } x \} \) is uncountable?

Answer: Yes


Question: Evaluate the limit \( \lim_{n \to \infty} \left( \frac{1}{n+1} + \frac{1}{n+2} + \ldots + \frac{1}{2n} \right) \) using sequential methods.

Answer: \ln 2


Question: In a group \(G\), it is given that \(a^5 = e\), where \(e\) is the identity element of \(G\). If \(a, b \in G\) and \(aba^{-1} = b^2\), determine the order of the element \(b\).

Answer: 31


Question: Find the expected value of the maximum of three independent exponential random variables X, Y, and Z, with rate parameters \( \lambda = 1, 2, 3 \) respectively.

Answer: \dfrac{73}{60}


Question: Compute the value of the infinite series \( \sum_{n=0}^{\infty}{\frac{n(n+2)}{3^n}} \).

Answer: 3


Question: Let \( \alpha \) be a root of the irreducible polynomial \( f(X) = X^3 + X + 3 \). Express \( \alpha^3 \) in terms of the basis \( \{1, \alpha, \alpha^2\} \) for the field extension \( \mathbb{Q}(\alpha) \).

Answer: -3 - \alpha


Question: Evaluate the double integral: \[ \int_{v=0}^{\infty}\int_{u=0}^{\infty}\frac{\sqrt{uv}}{(u+v)^2}e^{-\frac{(u+v)}{2}}dudv. \]

Answer: \dfrac{\pi}{4}


Question: Determine \( \lim_{n\to\infty} \mathbb{P}\left(\sum_{i=1}^n X_i \leq \frac{n}{2}\right) \), where \(X_1, X_2, \ldots, X_n\) are independent and uniformly distributed random variables on \([0,1]\).

Answer: \dfrac{1}{2}


Question: Calculate the value of \(2^{2^{23}} \mod 100\).

Answer: 56


Question: Determine the intersection \(\bigcap_{\epsilon > 0} S_\epsilon\), where \(S_\epsilon = \{ n \in \mathbb{N} : \sin(n) > 1 - \epsilon \}\).

Answer: \emptyset


Question: Is there a polyhedral graph where every vertex has an even degree greater than 4?

Answer: No


Question: Determine the periodicity of the triples of numbers \( n \) for which \( ec(n) \equiv 1666667 \pmod{666667} \), where \( ec(n) \) is defined as the concatenation of two consecutive Mersenne numbers. The known values of \( n \) include 18, 19, 20, 17094, 34188, 51282, 68376, 85196, 85197, 85198, 205128, 222222, 239316, 255551, 255552, and 255553. What is the periodicity of these triples?

Answer: 17094


Question: Find all right-angled triangles with a hypotenuse of length \(2^{2015.5}\) and integer lengths for the other two sides.

Answer: 2^{2015}


Question: Let \( T = \{v_1, \dots, v_{2n}\} \subseteq \mathbb{C}^n \) be a set of \( \mathbb{R} \)-linearly independent vectors. Consider the \( 2^n \) subsets \( S \subseteq T \) of size \( n \) that contain exactly one of \( v_i \) and \( v_{n+i} \) for each \( i = 1, \dots, n \). Is one of these subsets necessarily a \( \mathbb{C} \)-basis for \( \mathbb{C}^n \)?

Answer: Yes


Question: Given that \( \frac{1+\alpha}{1-\alpha}, \frac{1+\beta}{1-\beta}, \frac{1+\gamma}{1-\gamma} \) are the roots of the cubic equation \( f(x) = 0 \), where \( \alpha, \beta, \gamma \) are the real roots of the cubic equation \( 3x^3 - 2x + 5 = 0 \), find the number of negative real roots of the equation \( f(x) = 0 \).

Answer: 1


Question: Given the functions \( f(x) = \int_3^{x^2} \frac{\sqrt{1+t^6}}{t} \, dt \) and \( F(x) = \int_3^x f(t) \, dt \), find the value of \( F''(2) \).

Answer: \sqrt{4097}


Question: Evaluate the triple integral \( \iiint_{K} (x^{2} - z^{2}) \, dx \, dy \, dz \), where the region \( K \) is defined by the inequalities \( x \geq 0 \), \( y \geq 0 \), \( z \geq 0 \), and \( x+y+z \leq 1 \).

Answer: 0


Question: Consider a prime number $p$ such that $p \nmid n$. Let $\mathscr{P}$ be a prime ideal lying above $p$ in the field extension $\mathbb{Q}(\zeta_n)$. Is it always true that $\mathscr{P} \nmid n$? Answer true or false.

Answer: A


Question: Evaluate the integral \( \oint\limits_{|z|=2018}\dfrac{\pi^{\frac{1}{z}}}{(1+3z^2)^{2018}}\,\mathrm dz \).

Answer: 0


Question: How many reduced fractions \(\frac{a}{b}\) exist such that \(ab = 20!\) and \(0 < \frac{a}{b} < 1\), where \(a\) and \(b\) are positive integers?

Answer: 128


Question: Consider the Hopf bundle \( \mathbb{S}^3 \to \mathbb{S}^2 \) and the usual irreducible representations \( \rho_n: U(1) \to \operatorname{Gl}(\mathbb{C}) \) for \( n \in \mathbb{Z} \). Given that the first Chern class of the Hopf bundle is \( c_1(\operatorname{Hopf}) = -1 \), determine the first Chern class \( c_1\left(\operatorname{Hopf} \times_{\rho_n} \mathbb{C}\right) \).

Answer: -n


Question: Evaluate the definite integral \( \int_{0}^{\infty} \frac{\log x}{1+x^2} \, dx \).

Answer: 0


Question: Let $f, g \in L^{2}(\mathbb{R})$ and $\{x_{n}\} \subset (0, \infty)$ such that $x_{n} \to \infty$ as $n \to \infty$. Assume that \[ \int_{\mathbb{R}} \left|\frac{1}{\sqrt{x_{n}}}f(y/x_{n})-g(y)\right|^2 dy \to 0 \] as $n \to \infty$. Can we conclude that $f = 0$?

Answer: f = 0


Question: Let $S$ be a $C^*$-embedded subset in a topological space $X$. Determine whether every cozero set of $S$ can be expressed as $U \cap S$, where $U$ is a cozero set of $X$. Provide a justification for your answer.

Answer: Yes


Question: Find the sum of the infinite series \( \sum_{n=1}^{\infty} \frac{1}{x_n} \), where the sequence \( \{x_n\} \) is defined recursively by \( x_1 = 2 \) and \( x_{n+1} = x_n^2 - x_n + 1 \) for \( n \geq 1 \).

Answer: 1


Question: Let $(X, \mu)$ be a finite measure space and let $A$ be a non-negative self-adjoint operator that generates a contraction semigroup $e^{tA}$ on $L^2(X, \mu)$. If $e^{tA}$ is also bounded on $L^\infty(X, \mu)$, is $e^{tA}$ bounded on $L^p(X, \mu)$ for $1 \leq p \leq \infty$? Provide a justification for your answer.

Answer: Yes


Question: Minimize the function \( f(x, y, z) = x^2 + y^2 + z^2 \) subject to the constraint \( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 0 \).

Answer: 0


Question: Find the multiplicative inverse of the polynomial $2x+1$ in the quotient ring $F/(m(x))$, where $m(x) = x^2 + x + 2$ is irreducible over $F = \mathbb{Z}/3\mathbb{Z}$.

Answer: x + 2


Question: Determine if the infinite product \( \prod_{m=1}^\infty \frac{1}{m^2} \) has a closed form.

Answer: 0


Question: Find the remainder when \(2^{7!}\) is divided by 2987.

Answer: 755


Question: Find \( \lim_{n\to\infty} \sqrt{n} \left(\log\left(\frac{x}{e\sqrt{n}} + 1\right)\right) \).

Answer: \dfrac{x}{e}


Question: Compare the numbers $2^{317}$ and $81^{50}$ without using a calculator. Determine which number is larger.

Answer: 2^{317}


Question: Does there exist a group $G$ such that for a subgroup $H$ of $G$, the conjugate $gHg^{-1}$ is a proper subgroup of $H$ for some $g \in G$?

Answer: Yes


Question: Given the function \( f(x) = 3 + x^2 + \tan\frac{\pi x}{2} \) for \(-1 < x < 1\), find the value of \((f^{-1})'(3)\).

Answer: \dfrac{2}{\pi}


Question: Find the value of \( f^{-1}(12) \) for the inverse function of \( f(x) = 5 + x + \ln(x-6) \).

Answer: 7


Question: Evaluate the expression \( s_1 t_k \delta_{ii} \delta_{k1} \delta_{nn} \) where all indices run from 1 to 3. What is the result?

Answer: 9 s_1 t_1


Question: Consider a matrix $A_{3\times3} = (a_{ij})$, where $a_{ij}\in \mathbb{Z}$ and $|a_{ij}|\le 9$ for $1\le i,j\le 3$. Determine the maximum possible value of $\det(A)$. Provide a solution without using brute-force methods.

Answer: 2916


Question: Find the function $f(x)$ that minimizes the integral $\int_{0}^{1} f^2(x) \, dx$ subject to the constraints $\int_{0}^{1} f(x) \, dx = 1$ and $\int_{0}^{1} x f(x) \, dx = 1$.

Answer: 6x - 2


Question: Let \( f \in \mathcal{C}^1(\mathbb{R} \to \mathbb{R}) \) such that both \( \lim_{x \to \infty} f(x) \) and \( \lim_{x \to \infty} f'(x) \) exist. Prove that \( \lim_{x \to \infty} f'(x) = 0. \)

Answer: 0


Question: Determine the largest integer $k \in \mathbb{N}$ such that there exist non-trivial events $A_1, A_2, \ldots, A_k \subseteq \Omega$ that are independent, where $\mathbb{P}$ is the uniform distribution on $\Omega = \{1, 2, \ldots, 10\}$. An event $A \subseteq \Omega$ is non-trivial if $0 < \mathbb{P}(A) < 1$. What is the largest such $k$?

Answer: 2


Question: Calculate the length of the line segment given by the equation \( y = x \) within the unit disk using the Poincaré metric. The set is defined as \( A = \{x^2 + y^2 < 1\} \) with the metric \( ds^2 = \frac{4dx^2 + 4dy^2}{(1-x^2-y^2)^2} \).

Answer: \infty


Question: Evaluate the line integral \( \oint_C \mathbf{F} \cdot \mathbf{T} \, ds \) for the vector field \( \mathbf{F}(x,y) = xy \, \mathbf{i} + x^2 \, \mathbf{j} \), where \( C \) is the rectangle with vertices \((0,0), (3,0), (3,1), (0,1)\), and \( \mathbf{T} \) is the unit tangent vector to \( C \) directed anticlockwise. Calculate the integral directly without using Green's Theorem.

Answer: \dfrac{9}{2}


Question: In an M/M/60 queueing model with an arrival rate of \( \lambda = 150 \) customers per hour and a service rate of \( \mu = 2 \) customers per hour per server, the system starts with 100 customers, occupying all 60 servers and placing 40 customers in the queue. Calculate the probability that a customer arriving immediately after this initial rush (the 41st in the queue) will have a queue wait time of less than \( \frac{1}{3} \) of an hour (20 minutes).

Answer: 0.468


Question: Determine the number of integers between 1 and 1,000,000 whose digits sum to 30.

Answer: 50877


Question: Let $X$ be a Banach space. Suppose $x_n \overbrace{\rightarrow}^w x$ (weak convergence) and $x_n \overbrace{\rightarrow}^s z$ (strong convergence). Prove or disprove that $x = z$.

Answer: x = z


Question: Evaluate the contour integral \( \oint_{|z| = 1} \! dz \log\left(\frac{z - a}{z - b}\right) \) given \(|a| < 1\) and \(|b| < 1\).

Answer: 2\pi i (b - a)


Question: For which values of $a \in \mathbb{N}$ does $a^{n!} - 1$ become divisible by $n$, given that $n \neq 1$ and $n \in \mathbb{N}$?

Answer: 1


Question: Determine the norm of the functional \(f\) in the dual space \((\ell^{\infty})^{*}\), where for \(x \in \ell^{\infty}\), \(f(x) = \sum_{n \in \mathbb{N}} x_{n} 2^{-n}\).

Answer: 1


Question: Suppose $f$ is a real-valued continuous non-constant function defined on all of $\mathbb{R}$. Let $A = \text{image} f$. Suppose there exists $L > 0$ such that for every half-open interval $I \subseteq \mathbb{R}$ with $|I| = L$, $\text{image} f|_{I} = A$. Must $f$ be periodic? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \( \lim\limits_{n \to \infty} \frac{\ln(n!)}{n} \)

Answer: \infty


Question: Determine whether there exists a function that is continuous at every irrational number, discontinuous at every rational number, and of bounded variation on the interval $[0,1]$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following proposition is true: Let $Y$ be a proper closed subspace of a normed space $X$. Let $z \in X \setminus Y$. Then $\inf \{\| z + y \| \mid y \in Y \} > 0$. Is this proposition true?

Answer: True


Question: Given that $P(\mu - t\sigma < X < \mu + t\sigma) \ge 1 - \frac{1}{t^2} = 0.98$, find the smallest integer $k$ such that $38k = \mu + t\sigma = (0.75)(48k) + \sqrt{50} \cdot \sqrt{48k \cdot 0.75 \cdot 0.25}$. Round up if necessary to ensure the probability is at least 0.98.

Answer: 113


Question: Determine the rate of divergence of the series \( \sum_{m=1}^{\infty}J_{0}(mx) \) as \( x \to 0^{+} \).

Answer: \dfrac{1}{x}


Question: Determine if there exists a positive integer \(k\) such that the number \(1987^k - 1\) ends with 1987 zeros.

Answer: Yes


Question: Suppose \( f \) is a non-negative function in \( L^{1}[0,\infty) \) such that \( \int_{0}^{\infty}f(x)dx=1 \). Determine whether the following limit is true: \[ \lim_{n\to \infty} \dfrac{\int_{\frac{1}{n}}^{1}\frac{f(x)}{x}dx}{n}=0. \]

Answer: 0


Question:
Find all continuous functions \( f \) on the interval \([0,1]\) such that for every \( x \in (0,1) \), the following equation holds:
\[ \int_0^x f(t) \, dt = \int_x^1 f(t) \, dt. \]

Answer: 0


Question: Calculate the area of the finite region bounded by the x-axis, the curve \( y = \frac{1}{\sqrt{1-x^2}} \), and the vertical asymptotes at \( x = -1 \) and \( x = 1 \).

Answer: \pi


Question: An ant is sitting at the center of a circle with a radius of 3 meters. Every minute, the ant randomly chooses a direction and moves 1 meter in that direction. On average, how many minutes does it take for the ant to leave the circle? Assume that if the ant reaches the edge of the circle within a minute, it is considered to have left the circle at the end of that minute.

Answer: 9


Question:
Calculate the expected value \( \mathbb{E}(X^2Y^2) \) for a bivariate normal distribution where \( (X,Y) \) is a random vector with \( \mathbb{E}[X] = \mathbb{E}[Y] = 0 \), \( \text{Var}(X) = \text{Var}(Y) = 1 \), and \( \text{Cov}(X,Y) = \theta \) where \( \theta \in (-1,1) \). The joint probability density function of \( (X,Y) \) is given by:

\[ p(x,y) = \frac{1}{2\pi\sqrt{1-\theta^2}} \exp\left\{-\frac{1}{2(1-\theta^2)} \left[ x^2 - 2\theta xy + y^2 \right]\right\}. \]

Use the transformation \( u = \frac{x-y}{\sqrt{2}} \) and \( v = \frac{x+y}{\sqrt{2}} \) to evaluate the integral:

\[ \mathbb{E}(X^2Y^2) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{(u^2-v^2)^2}{2} \cdot p\left(\frac{u+v}{\sqrt{2}}, \frac{u-v}{\sqrt{2}}\right) \mathrm{d}u \mathrm{d}v. \]

Answer: 1 + 2\theta^2


Question:
Given the differential equation \( \frac{dy}{dx}= \frac{(x+y)^2}{(x+2)(y-2)} \), determine which of the following expressions represents its solution:

(a) \((x+2)^4 \left(1+\frac{2y}{x}\right)= ke^{\frac{2y}{x}}\)

(b) \((x+2)^4 \left(1+ 2\frac{(y-2)}{x+2}\right)= ke^{\frac{2(y-2)}{x+2}}\)

(c) \((x+2)^3 \left(1+ 2\frac{(y-2)}{x+2}\right)= ke^{\frac{2(y-2)}{x+2}}\)

(d) None of these

Select the correct option.

Answer: B


Question: Calculate \( \lim_{x \to 0^{-}}(1-2^x)^{\sin(x)} \) using Taylor expansion.

Answer: 1


Question:
Consider the function \( f:[0,1] \to \mathbb{R} \) defined by
\[
f(x) =
\begin{cases}
x^2 & \text{if } x > 0, \\
1 & \text{if } x=0.
\end{cases}
\]
This function is convex. Determine the subdifferential of \( f \) at \( x = 0 \).

Answer: \emptyset


Question: In how many ways can 9 black and 9 white rooks be placed on a 6 × 6 chessboard such that no white rook can capture a black rook? A rook can capture another piece if it is in the same row or column as the other piece, with no other pieces between them.

Answer: 400


Question: Let $f$ and $g$ be rotations of the plane about distinct points, with arbitrary nonzero angles of rotation $\theta$ and $\phi$. Does the group generated by $f$ and $g$ contain a translation? Answer yes or no.

Answer: Yes


Question: Given the functional equation \( f\left(x-\frac{1}{x}\right) = x^3-\frac{1}{x^3} \), find the expression for \( f(-x) \).

Answer: -x^3 - 3x


Question: Let $R$ be a discrete valuation ring (DVR) and let $K$ be its fraction field. Consider an automorphism $\phi$ of $K$. Can $R$ be a proper subset of $\phi(R)$? Provide a justification for your answer.

Answer: No


Question: Evaluate the limit: \[ \lim_{x\to 0} \left(\frac{1}{1-\cos(x)} - \frac{2}{x^2}\right) \] without using L'Hôpital's rule.

Answer: \dfrac{1}{6}


Question: Evaluate the limit: \[ \lim_{x \to \infty} \frac{1}{x} \int_0^x \frac{dt}{1+x^2 \cos^2 t} \]

Answer: 0


Question: Given that \( \sin A + \sin^2 A = 1 \) and \( a \cos^{12} A + b \cos^8 A + c \cos^6 A - 1 = 0 \), find the value of \( b + \frac{c}{a} + b \).

Answer: 4


Question: Mary and her brothers play a game called "all or one" to decide who will do the dishes. The game is played by at least three people, and in each round, each player chooses a natural number simultaneously. A player wins if they choose a certain number and all others choose the same different number. Mary chooses the number of available numbers, \( k > 1 \), such that the probability of a fixed person winning in the first round is an integer percentage. Given that \( k \) and the total number of players \( n \) (including Mary) are relatively prime, determine how many brothers Mary has.

Answer: 2


Question:
Find the value of the infinite sum:
\[ S = 1 + \frac{1}{3} + \frac{1\cdot 3}{3\cdot 6} + \frac{1\cdot 3\cdot 5}{3\cdot 6\cdot 9} + \cdots \]

Answer: \sqrt{3}


Question:
Solve the equation for real and positive parameters $a$, $b$, and $c$:
\[ \sqrt{a+bx}+\sqrt{b+cx}+\sqrt{c+ax}=\sqrt{b-ax}+\sqrt{c-bx}+\sqrt{a-cx} \]

Answer: 0


Question:
Determine the value of \( \alpha \) such that the Runge-Kutta method given by
\[ y_{j+1}=y_j+\frac{h}{2 \alpha}f(t_j, y_j) + h\left(1- \frac{1}{2 \alpha}\right)f(t_j + \alpha h, y_j + \alpha h f(t_j, y_j)) \]
has an order of consistency of 2.

Answer: 1


Question: Let $B$ be a C*-algebra and $\{b_{i}\}_{i=1}^{n}\subset B$ be a linearly independent set. If $\{f_{k}\}\subset B$ is an approximate unit, can we find a sufficiently large $k$ such that $\{b_{1}f_{k}, b_{2}f_{k},...,b_{n}f_{k}\}$ is a linearly independent set?

Answer: Yes


Question: Suppose $Z_1, Z_2, \ldots$ are iid $\operatorname{Bernoulli}\left(\frac{1}{2}\right)$ random variables and let $S_n = Z_1 + \ldots + Z_n$. Let $T$ denote the smallest $n$ such that $S_n = 3$. Calculate $\operatorname{Var}(T)$. Provide your answer as a single number.

Answer: 6


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{k+n} \]

Answer: \ln 2


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

Given an isosceles triangle with equal sides of length $b$ and base angle $A$ $(<\pi/4)$, let $O$ and $I$ be the circumcenter and incenter of the triangle, respectively. Then, the distance between $O$ and $I$ is given by:
$$OI= \left|\frac{b \cos(3A/2)}{2\sin(A) \cos (A/2)}\right|.$$

Answer: True


Question: Let $A$ be an Artin algebra. Suppose $S_1$ and $S_2$ are simple modules in $	ext{mod}(A)$, and $P(S_1)$ is the projective cover of $S_1$. If there exists a non-zero module homomorphism $f: P(S_1) \rightarrow S_2$, prove that $S_1 \cong S_2$.

Answer: S_1 \cong S_2


Question: Let $E$ be any spectrum such that $E \wedge I_{\mathbf{Q/Z}}$ is not contractible. Is it true that $\langle E\rangle \geq \langle X\rangle$ for some nontrivial finite spectrum $X$?

Answer: Yes


Question: Find the hypervolume of the region in 4-dimensional space \( \mathbb{R}^4 \) defined by the inequalities \( w \ge 0, x \ge 0, y \ge 0, z \ge 0 \) and \( w + 2x + 3y + 4z \le 6 \).

Answer: \dfrac{9}{4}


Question: Determine the range of values for $a$ such that the series \( \sum_{n=1}^\infty \frac{(\log_e n)^{2012}}{n^a} \) is convergent.

Answer: (1, \infty)


Question: Determine the area of the region in the complex plane defined by \(-1 < x \le 2\) and \(-1 \le y < 3\) under the linear transformation \(T(z) = (1 + i \sqrt{3})z + 2 - i\).

Answer: 48


Question: Given a matrix $A_{n\times n}$ such that $I + A + A^2 + A^3 + A^4 = 0$, determine the dimension of $\operatorname{span}(I, A, A^2, A^3, \ldots)$. What can be said about this dimension?

Answer: 4


Question:
Let \( p \) be a prime and \( p \nmid b \). Prove that the following sum of Legendre symbols holds:
\[ \sum_{a=1}^{p-1}\left(\frac{a(a+b)}{p}\right) = -1. \]

Answer: -1


Question: How many polynomials of degree 3 are contained in the ideal \( I = (x^5 + x^2 + x + 1, x^7 + x^5 + x^3 + x) \) in \( \mathbb{F}_2[x] \)?

Answer: 2


Question: Evaluate the limit: \[ \lim_{y \to 1^-} \left(1-y + \ln(y)\right)\int_0^y \frac{dx}{(x-1) \ln(x)} \] without using Taylor expansion.

Answer: 0


Question: Calculate the flux of the vector field \( F(x,y,z) = [x^2y+z^2, x\cos(z)-xy^2, x^3+3z] \) across the lateral surface of the elliptic cylinder defined by \( x^2+4y^2=1 \) for \( 0 < z < 8 \). The unit normal \( \hat{N} \) points away from the \( z \)-axis. Use the divergence theorem to find the flux across the lateral surface, excluding the top and bottom surfaces at \( z=0 \) and \( z=8 \).

Answer: 0


Question: In a first-order language L with only the predicate = and no constants or function symbols, determine the smallest number of variables required to write a sentence A that is valid in a structure M if and only if M has at least n elements in its domain.

Answer: n


Question: Calculate \( \int_\gamma (y^{2018} + y^2e^{xy^2})\,dx + (x^{2018} + 2xye^{xy^2})\,dy \) where \( \gamma \) is the positively oriented unit circle \( \{x^2+y^2 = 1\} \). Use Green's theorem to evaluate the integral.

Answer: 0


Question: Find all integer pairs \((m, n)\) that satisfy the equation \(m^3 - 5m + 10 = 2^n\).

Answer: (2, 3)


Question: Evaluate \( \int_{\delta S} w \) where \( w = (x + y)dz + (y + z)dx + (x + z)dy \) and \( S \) is the upper part of the unit sphere. Use both direct evaluation and Stokes' theorem to find the integral.

Answer: 0


Question: Let \( f(x) = \prod\limits_{n=1}^{100} (x-n)^{n(101 - n)} \). Find the value of \( \frac{f(101)}{f'(101)} \).

Answer: \dfrac{1}{5050}


Question: Solve the equation: \[ \sin\left(\frac{x}{x-1}\right) + 2 \tan^{-1}\left(\frac{1}{x+1}\right) = \frac{\pi}{2} \]

Answer: 0


Question: Calculate the volume of the region defined by the inequalities \(0 \le z \le x^2 + 4y^2\) and \(x^2 + y^2 \le 1\) using the triple integral \(\iiint_K 1 \, dx \, dy \, dz\).

Answer: \dfrac{5\pi}{4}


Question: Evaluate the limit \( \lim_{x \to -1^+} \tan(\pi x) \cdot \ln(1+x) \) without using calculus.

Answer: 0


Question: Let $S_4$ be the symmetric group on 4 symbols. Consider a homomorphism $f: S_4 \to \mathbb{R}^*$, where $\mathbb{R}^*$ denotes the non-zero real numbers. Determine the number of elements in the set $\{ x \in S_4 : f(x) = 1 \}$. \( \text{Provide your answer as a single number.} \)

Answer: 12


Question: Consider the vectors in \( \mathbb{R}^n \) given by \( a_1=(1,1,\ldots,1) \), \( a_2=(-1,1,\ldots,1) \), \( a_{3}=(-1,-1,1,\ldots,1) \), ..., \( a_n=(-1,-1,\ldots,-1,1) \). Given \( 2n \) positive numbers \( \lambda_i \) for \( 1 \leq i \leq 2n \), define the regions \( P_i \) as follows: for \( 1 \leq i \leq n \), \( P_i=\{x:~ \langle x,a_i\rangle\leq \lambda_i\} \), and for \( n+1 \leq i \leq 2n \), \( P_i=\{x:~ \langle x,-a_{i-n}\rangle\leq \lambda_i\} \). Is the intersection \( \cap_{i=1}^{2n}P_i \) a compact region?

Answer: Yes


Question: Given the vectors \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2\hat{i}+2\hat{j}+\hat{k}, \vec{c}=5\hat{i}+\hat{j}-\hat{k}\), find the area of the region formed by the position vector \(\vec{r}\) that satisfies the equation \(\hat{r}\cdot \vec{a}=5\) and \(|\vec{r}-\vec{b}|+|\vec{r}-\vec{c}|=4\). Determine the area closest to an integer.

Answer: 4


Question: Compute \( \lim_{x\to 0^{+}}x^{x^{x}} \).

Answer: 0


Question: Calculate \( \lim_{x\to\infty}\frac{\ln(x^2-x+1)}{\ln(x^{10}+x+1)} \).

Answer: \dfrac{1}{5}


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

Answer: -\dfrac{1}{2}


Question: Find the value of \(x\) such that \(9^{89} \equiv x \pmod{1000}\) and \(0 \leq x \leq 999\).

Answer: 289


Question: Find the value of \( f^{(97)}(0) \) for the function \( f(x) = x^2 \cos{9x} \) using its Maclaurin series expansion.

Answer: 0


Question: Find the discriminant of the number field \( \mathbb{Q}(\sqrt{3},\sqrt{5}) \).

Answer: 3600


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

Answer: \infty


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

Answer: -\dfrac{1}{4}


Question: Evaluate the limit: \[ \lim_{x \rightarrow 0}\frac{3}{x}\left(\frac{1}{\tanh(x)}-\frac{1}{x}\right) \] using L'Hôpital's rule.

Answer: 1


Question: Find the limit: \[ \lim_{n\to\infty} n(\sqrt[n]{e}-1) \] without using Heine's theorem or L'Hôpital's rule.

Answer: 1


Question: Let $P$ be a poset with all binary joins and all chain-shaped joins (of arbitrary cardinality). Is $P$ necessarily a complete lattice?

Answer: Yes


Question: Suppose you have a unitary matrix $U$ such that $\overline{U}U=D$ for some diagonal unitary matrix $D$ over $\mathbb{C}$. Given that $U^2$ is symmetric, determine if it is true that $D^2=I$, which is equivalent to $D$ commuting with $U$. Provide a justification for your answer.

Answer: D^2 = I


Question: Determine whether the inequality \((|z|-|s|)^2 \leq \left| \frac{s-z}{1-\overline{s}z} \right|^2\) holds for all complex numbers \(z, s\) in the unit disc \(\mathbb{D}\).

Answer: Yes


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

Answer: -\dfrac{2}{3}


Question: Calculate the $L_\infty$ distance between the points $(a,\infty)$ and $(b,\infty)$ in a persistence diagram, where $a, b < \infty$. Explain why the infinity component can be ignored in this calculation.

Answer: |a - b|


Question: Evaluate the integral \( \int_{0}^{3} (x^2+1) \, d[x] \), where \([x]\) is the greatest integer function.

Answer: 17


Question: Calculate the work done by the force field \( \vec{F} (x,y) = y \ \vec{i} - x \ \vec{j} \) on an object moving along the parabola \( y = x^2 - 1 \) from the point \((1,0)\) to \((-2,3)\). The work done is given by the line integral \( \int_{C} y \, dx - x \, dy \).

Answer: 6


Question: Determine the radius of convergence of the series \( \sum_{n\ge 0}{\log(n!)x^n} \). You may use Stirling's approximation: \( n!\sim_\infty \left(\frac{n}{e}\right)^n\sqrt{2 \pi n} \).

Answer: 1


Question: For an integer $n \geq 3$, determine whether there exists a finite non-abelian group with exactly $n$ conjugacy classes.

Answer: Yes


Question: Determine the total number of zeros in the expression \((10^{10} + 10^{11} + 10^{12} + \ldots + 10^{2009})^{2010}\).

Answer: 20100


Question: Let \( f(x) = 5x^2 + Ax^{-5} \) for \( x > 0 \), where \( A \) is a positive constant. Find the smallest \( A \) such that \( f(x) \geq 28 \) for all \( x > 0 \).

Answer: 256


Question: Consider two probability density functions on $[0,1]$: $f_0(x) = 1$, and $f_1(x) = 2x$. Among all tests of the null hypothesis $H_0: X \sim f_0(x)$ versus the alternative $X \sim f_1(x)$, with significance level $\alpha = 0.1$, determine the maximum possible power of the test.

Answer: 0.19


Question: Calculate the 47th root of 11 in the multiplicative group of integers modulo 91, \( \mathbb{Z}_{91}^* \), using Euler's Theorem.

Answer: 58


Question: Find the multiplicative order of 15 modulo 257.

Answer: 32


Question: Find the maximum value of \(|f(w)|\) for \(f(w) = \frac{w}{4w^2 - 1}\) when \(|w| \geq 1\).

Answer: \dfrac{1}{3}


Question: Determine whether the function \( f: [0,1] \rightarrow \mathbb{R} \) defined by \( f(x) = x \) if \( x = 1/n \) for some \( n \in \mathbb{N} \) and \( f(x) = 0 \) otherwise, is Riemann-integrable. If it is, find the value of the integral.

Answer: 0


Question:
Solve the linear programming problem using the simplex method, starting with the basic feasible solution where $x_1=4$ and $x_2=0$. Maximize $z = -x_1 + 2x_2$ subject to the constraints:

\[ 3x_1 + 4x_2 = 12 \]
\[ 2x_1 - x_2 \leq 12 \]
\[ x_1 \geq 0, \quad x_2 \geq 0 \]

Answer: 6


Question: Let $X \sim U(0, 1)$ and $Y \sim U(0, 2)$ be independent random variables. Calculate the expected value of the absolute difference between $X$ and $Y$, denoted as $\mathbb{E}[|X - Y|]$. Provide your answer as a single numerical value.

Answer: \dfrac{2}{3}


Question: Compute the line integral \( \oint_C \mathbf{H} \; d\mathbf{r} \) for the vector field \( \mathbf{H}(x,y,z) = (-x, y, e^{z^2}) \), where \( C \) is the closed curve that lies on the intersection of the region \( V \subset \mathbb{R}^3 \) inside the cylinder \( x^2 + y^2 \le 1 \) and between the surfaces \( z = -2 \) and \( z = xy \).

Answer: 0


Question: Calculate the limit: \[ \lim_{n \to \infty} \left( \frac{1}{2\log(2)} + \frac{1}{3\log(3)} + \dots + \frac{1}{n\log n} \right) \]

Answer: \infty


Question: Let $x_5$ be the remainder when the positive integer $x$ is divided by 5. Compute the sum of all elements in the solution set of the equation: $$x^5(x_5)^5 - x^6 - (x_5)^6 + x(x_5) = 0.$$

Answer: 1300


Question: Find the number of positive integer solutions to the equation \( x^4 + 4y^4 + 16z^4 + 64 = 32xyz \).

Answer: 0


Question:
Determine the value(s) of the parameter \( a \) for which it is possible to find explicit formulas (without integrals) for the solution to the differential equation:
\[ \frac{dy}{dt} = aty + e^{-t^2} \]

Answer: -2


Question: Let \( \Omega \subset \mathbb{R}^N \) be a bounded open smooth domain, and let \( C_0(\Omega) \) denote the set of bounded continuous functions with compact support. Suppose \( \mu_n \to \mu \) in the weak star sense, meaning \( \int_\Omega f d\mu_n \to \int_\Omega f d\mu \) for all \( f \in C_0(\Omega) \), where \( \mu_n, \mu \geq 0 \). Is it true that \( \mu_n(\Omega) \to \mu(\Omega) \)?

Answer: No


Question: Given that \( \sin(\theta_1) + \sin(\theta_2) + \sin(\theta_3) = 0 \) and \( \cos(\theta_1) + \cos(\theta_2) + \cos(\theta_3) = 0 \), find the value of \( \sin^2(\theta_1) + \sin^2(\theta_2) + \sin^2(\theta_3) \).

Answer: \dfrac{3}{2}


Question: Consider a measure space $(\Omega, \mathcal{F}, P)$ and a partition $\mathcal{D} = \{D_1, D_2, D_3, \ldots \}$, which is a countable collection of disjoint subsets of $\Omega$ such that $\Omega = \bigcup_{i \in \mathbb{N}} D_i$. Let $\sigma(\mathcal{D})$ be the sigma algebra generated by $\mathcal{D}$. Verify whether the following statement is true: \( \forall A \in \sigma(\mathcal{D}), A = \bigcup_{k \in N} D_k \) where $N \subseteq \mathbb{N}$. Provide a justification for your answer.

Answer: True


Question: Let $X$ be a compact metric space and $\mu$ be a positive, finite, atomless, regular measure on the $\sigma$-algebra of Borel subsets of $X$. Does there exist a set $C \subset X$ such that $C$ is uncountable and has $\mu$-measure zero?

Answer: Yes


Question:
Let $n = p_1 p_2 \cdots p_k$ be the product of pairwise distinct odd primes. Define $X$ as the set of elements in $\operatorname{Aut}(\mathbb{Z}/n\mathbb{Z})$ of order 1 or 2. For each $\psi \in X$, consider the homomorphism $\varphi_\psi : \mathbb{Z}/2\mathbb{Z} \to \operatorname{Aut}(\mathbb{Z}/n\mathbb{Z})$ given by:
\[ x \mapsto \begin{cases}\operatorname{id} & \text{if } x = 0 \\ \psi & \text{if } x = 1 \end{cases} \]
This defines a semidirect product $G_\psi = \mathbb{Z}/n\mathbb{Z} \rtimes_{\varphi_\psi} \mathbb{Z}/2\mathbb{Z}$. Are the groups $G_\psi$ pairwise non-isomorphic for all $\psi \in X$?

Answer: Yes


Question: Let \((X_1,\cdots,X_m)\) and \((Y_1,\cdots,Y_n)\) be bases for \(\mathbb{R}^m\) and \(\mathbb{R}^n\), respectively. Do the \(mn\) matrices \(X_iY_j^T\) form a basis for the vector space \(\mathbb{R}^{m\times n}\) of all \(m\times n\) matrices? Justify your answer.

Answer: Yes


Question: Determine the radius of convergence of the Taylor series of the function \( f(z) = \frac{z+e^z}{(z-1+i)(z^2-2)(z-3i)} \) at the point \( a = 0 \).

Answer: \sqrt{2}


Question:
Find the minimal solution to the following system of linear congruences:
\[ 3x \equiv 1 \pmod{8} \\
   x \equiv 7 \pmod{12} \\
   x \equiv 4 \pmod{15} \]

Answer: 19


Question: Find the minimum value of the function \( f(x,y) = 3(x+1) + 2(y-1) \) subject to the constraint \( x^2 + y^2 = 4 \) using Lagrange Multipliers.

Answer: 1 - 2\sqrt{13}


Question: Find all integer solutions for \(n\) such that \(n^n = 2n\).

Answer: 2


Question: Compute the integral \( \int_{\partial D(0,1)} \frac{dz}{8z^3-1} \) using the Cauchy integral formula.

Answer: 0


Question: Let A be an n x n real symmetric matrix. If there exists an orthogonal matrix P such that \( P^{-1} A P \) is diagonal, can we conclude that \( P \in \text{SO}_n(\mathbb{R}) \), meaning \( P \in \text{O}_n(\mathbb{R}) \) with \( \det(P) = 1 \)? Provide a justification for your answer.

Answer: No


Question: Is there a finite index torsion-free subgroup $G$ of $GL_n(\mathbb{Z})$, where $n \ge 3$, such that the coinvariants group $\mathbb{Z}^n_G$ is finite? Here, $G$ acts on $\mathbb{Z}^n$ in the standard way, and $\mathbb{Z}^n_G$ is defined as the quotient of $\mathbb{Z}^n$ by the subgroup generated by the set $\{gz - z : g \in G, z \in \mathbb{Z}^n\}$.

Answer: Yes


Question: Let $A$ be a finite abelian group of order $360$ which does not contain any elements of order $12$ or $18$. How many elements of order $6$ does $A$ contain?

Answer: 56


Question: Evaluate the limit \( \lim _{n \to \infty} \int_{-\infty}^{\infty} \frac{e^{-x^{2} / n}}{1+x^{2}} d x \).

Answer: \pi


Question: Find the degree of the field extension \([\mathbb{Q}(\sqrt{5}, \sqrt{3}, \sqrt{2}): \mathbb{Q}(\sqrt{3}, \sqrt{2})]\).

Answer: 2


Question: How many roots of the polynomial \( p(z) = z^8 + 3z^7 + 6z^2 + 1 \) are contained in the annulus \( \{1 < |z| < 2\} \)?

Answer: 5


Question: Given two vectors $u \in \mathbb{R}^n$ and $v \in \mathbb{R}^n$, and a symmetric positive definite matrix $X$, consider the function $f(X) = u^T X v$. Calculate the gradient of $f$ with respect to $X$. Determine whether $\nabla f(X) = uv^T$ or $\nabla f(X) = vu^T$ is the correct expression for the gradient.

Answer: uv^T


Question: Evaluate the limit: \( \lim_{x\to0^-}{(1+\tan(9x))^{\frac{1}{\arcsin(5x)}}} \)

Answer: e^{\frac{9}{5}}


Question: Find the remainder when \( 9876^{3456789} (9^{99})^{5555} - 6789^{3414259} \) is divided by 14.

Answer: 7


Question: Determine the value of \( \lim\sup_{n \to \infty} \frac{E_n}{n!} \), where \( E_n \) is the number of permutations in \( S_n \) that can be obtained by using each transposition exactly once.

Answer: \dfrac{1}{2}


Question:
Compute the Fourier series for the function \( f(t) \) defined as follows:
\[
f(t) = \begin{cases} 50, & \text{for } t \in (0,2) ;\\
100 , & \text{for } t=2.\end{cases}
\]

Answer: 50


Question: Find the value of \(a\) such that the line \(y = a\) intersects the graph of the function \(f(x) = 2^{x+1} + 2^{4-x}\) at two points \(P\) and \(Q\), and the length of the line segment \(PQ\) is 4.

Answer: 17\sqrt{2}


Question: Determine the center of the group of quaternions, defined as \( Q = \langle i, j, k \mid i^2 = j^2 = k^2 = ijk = -1 \rangle \). The center \( Z(Q) \) is the set of elements in \( Q \) that commute with every element in \( Q \). Identify \( Z(Q) \).

Answer: \{\pm 1\}


Question: Find the number of automorphisms of the splitting field $L=\mathbb{Q}(\sqrt[4]{3},i)$ of the polynomial $X^4-3$ over the rationals. Express your answer as an integer.

Answer: 8


Question: Let \(A\subseteq \mathbb{R}\) be a Lebesgue-measurable set. We say that \(A\) is locally \(\varepsilon\)-dense if for any  \(\varepsilon > 0\), there exist \(x < y\in\mathbb{R}\) such that \(\frac{\mu(A\cap[x,y])}{y-x} \geq 1-\varepsilon\), where \(\mu\) denotes the Lebesgue measure on \(\mathbb{R}\). If \(A\) is locally \(\varepsilon\)-dense, does it necessarily have positive measure?

Answer: Yes


Question: Determine whether the function $f(z) = 2z$ can be expressed in the form $f(z) = \frac{az+b}{cz+d}$, where $a, b, c, d \in \mathbb{R}$ and $ad-bc=1$. If it cannot, explain why.

Answer: Yes


Question:
Determine the condition on $m$ such that the equation \(x^4+2x^3+mx^2+2x+1=0\) has four different real roots. Choose the correct option:

(a) \(m<3\);
(b) \(m<2\);
(c) \(m<-6\);
(d) \(1<m<3\);
(e) \(-6<m<2\)

Answer: C


Question: Let $X$ be a paracompact Hausdorff space, and let $\\{X_i\\}_{i \ge 0}$ be a collection of open subsets of $X$ such that $X_i \subset X_{i+1}$ and $\bigcup_{i \ge 0} X_i = X$. Determine whether there exists a continuous function $f: X \to \mathbb{R}$ such that $f(x) \ge i + 1$ for all $x \notin X_i$.

Answer: Yes


Question: Find the greatest common divisor (GCD) of the polynomials \(a^3 - 3ab^2\) and \(b^3 - 3ba^2\), where \(a, b \in \mathbb{Z}\).

Answer: 1


Question: Calculate the expected value of $a^b$, where $a$ and $b$ are independent random variables uniformly distributed over the interval (0, 1).

Answer: \ln 2


Question: Determine the condition under which the directional derivative of the function \( f=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} \) in the direction \( \textbf{r}=(x,y,z) \) is equal to the magnitude of the gradient \( |\nabla f| \).

Answer: a = b = c


Question: Find and classify the stationary point of the function \( f(x, y) = 2x^4 - 3x^2y + y^2 \).

Answer: (0, 0)


Question: The base of a sand pile covers the region in the xy-plane bounded by the parabola \(x^2 + y = 6\) and the line \(y = x\). The height of the sand above the point \((x, y)\) is given by \(x^2\). (i) Express the volume of the sand as a double integral. (ii) Express the volume as a triple integral. (iii) Calculate the volume of the sand pile.

Answer: \dfrac{125}{4}


Question: Evaluate the integral \( \int_0^{\pi/8} \cos(3x) \cos(4x) \cos(5x) \, dx \) and express it in the form \( \frac{k}{24} \). Find the constant \( k \).

Answer: 1 + 2\sqrt{2}


Question: Find the minimum value of \[ P = \frac{1}{1+2x} + \frac{1}{1+2y} + \frac{3-2xy}{5-x^2-y^2} \] given that \( x, y \in (-\sqrt{2}, \sqrt{2}) \) and \( x^4 + y^4 + 4 = \frac{6}{xy} \).

Answer: -\dfrac{5}{3}


Question: Are polynomials dense in the space of continuous functions from the unit circle to the complex numbers, $C(S^1)$, when equipped with the supremum norm?

Answer: No


Question: Determine the range of values of \(\lambda\) for which the transformation \(T\) defined by \((Tf)(x) = x + \lambda \int_0^x (x-t)f(t)\) is a contraction with respect to the supremum norm on \(C([0,1])\).

Answer: (-2, 2)


Question: Find the last two digits of \(6^{513}\).

Answer: 16


Question: Let $f: \mathbb{C}^5 \rightarrow \mathbb{C}^7$ be a linear function such that $f(2i e_1 + e_3) = f(e_2)$ and $\mathbb{C}^7 = X \oplus \text{Im}(f)$. Determine the dimension of $X$. \( \text{Im}(f) \) denotes the image of $f$. \( e_1, e_2, e_3 \) are standard basis vectors in \( \mathbb{C}^5 \).

Answer: 3


Question:
Find the best approximation of the function $y(t) = t^3$ in the normed linear space $C[0,1]$ with respect to the orthogonal complement of the subspace $X_0 = \{x \in C[0,1] : \int_{0}^{1} t^2 x(t) \, dt = 0\}$. The inner product is defined as $\langle x, y \rangle = \int_{0}^{1} x(t) \overline{y(t)} \, dt$. Determine $x_0(t)$, the best approximation of $y(t)$, from the following options:

A. $\frac{4}{5}t^{2}$
B. $\frac{5}{6}t^{2}$
C. $\frac{6}{7}t^{2}$
D. $\frac{7}{8}t^{2}$

Answer: B


Question: Given a polynomial $Q(x)$ with real coefficients such that $Q(1) = 1$ and \( \frac{Q(2x)}{Q(x+1)} = 8 - \frac{56}{x+7} \) for all \( x \neq -7 \) and \( Q(x+1) \neq 0 \), find the value of $Q(-1)$. \( \boxed{} \)

Answer: -\dfrac{5}{21}


Question: Evaluate the contour integral \( \int_{|z|=3}\frac{2z^2-z+1}{(z-1)^2(z-2)}dz \).

Answer: 4\pi i


Question: Evaluate the expression: \[ n^{n-\ell} \cdot \sum\limits_{k=1}^\ell \binom{n}{k} k! \begin{Bmatrix} \ell \\ k \end{Bmatrix} \] and determine if it simplifies to $n^n$.

Answer: n^n


Question:
Determine whether there exists a set of 4042 distinct positive integers that can be partitioned into two subsets of 2021 elements each, such that both the sums and the sums of squares of the elements in each subset are equal. Specifically, find if there exist integers such that:
\[a_1+a_2+\cdots+a_{2021}=b_1+b_2+\cdots+b_{2021}\] and
\[a_1^2+a_2^2+\cdots+a_{2021}^2=b_1^2+b_2^2+\cdots+b_{2021}^2.\]

Answer: Yes


Question: Evaluate the integral \( \int_0^{2\pi} \frac{\cos(3\theta)}{5-4\cos(\theta)}\mathrm{d}\theta \) using complex analysis techniques.

Answer: \dfrac{\pi}{12}


Question: Find all integer solutions \((x, y, z)\) to the equation \(x^2 + y^2 + z^2 = 2xyz\).

Answer: (0, 0, 0)


Question:
Consider the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) defined by:
\[
f(x)=\begin{cases}
0 & \text{if } x=0 \text{ or } x \notin \mathbb{Q} \\
\frac{1}{q} & \text{if } x=\frac{p}{q} \text{ with } p, q \neq 0 \text{ and } p, q \text{ relatively prime}
\end{cases}
\]
Determine whether the statement "For any \( x_0 \in \mathbb{R} \), \( \lim_{x \rightarrow x_0} f(x) = 0 \)" is true.

Answer: True


Question: Evaluate the limit: \[ \lim_{x\to \infty} \frac{\int_1^x (t^2(e^{1/t}-1)-t)\,dt}{x^2\ln\left(1+\frac{1}{x}\right)} \]

Answer: \dfrac{1}{2}


Question: Let $X$ be the number of tosses of a fair coin required to get the first head. If $Y | X = n$ is distributed as $Binomial(n, 1/2)$, find $P(Y = 1)$. Express your answer as a probability.

Answer: \dfrac{4}{9}


Question: Calculate the value of the integral \( \int_0^1 \int_y^1 (x^2 + y^2) \, dx \, dy \) using the change of variables \( x = u + v \) and \( y = u - v \). Given that the Jacobian determinant is \(|J| = 2\) and the transformed function is \(2(u^2 + v^2)\), set up and evaluate the new integral in terms of \(u\) and \(v\).

Answer: \dfrac{1}{3}


Question: For which values of \(a\) does the differential equation \(y'' + 2y' + ay = 0\) with boundary conditions \(y(0) = y(\pi) = 0\) have non-trivial solutions?

Answer: a = n^2 + 1


Question:
Let $\mathcal H$ be a Hilbert space, and let $\{x_i : i \in I \}$ be an orthonormal basis in $\mathcal H$. Consider a linear operator $T \in L(\mathcal H)$. Does the following equality hold for any set of scalars $\{\lambda_i\}$:
\[ T\left( \sum_{i \in I} \lambda_i x_i \right) = \sum_{i \in I} \lambda_i T(x_i) \?\]

Answer: Yes


Question: Find the least positive integer \( n \) such that \( A^n = I \), where \( A = \begin{bmatrix} \sin x & -\sin y \\ \sin y & \sin x \end{bmatrix} \) and \( x = \frac{\pi}{18} \), \( y = \frac{4\pi}{9} \).

Answer: 9


Question: In how many ways can you collect a total of $20 from 4 different children and 3 different adults, if each child can contribute up to $6 and each adult can give up to $10, with each individual contributing a nonnegative whole number of dollars?

Answer: 112567


Question: Find the integer between 2000 and 3000 that divides the number $85^9 - 21^9 + 6^9$. Compute that integer.

Answer: 2240


Question: Given that $X \sim U(0,1)$ and $Y \sim U(X,1)$, where $X$ and $Y$ are continuous uniformly distributed random variables, find the expected value $E(Y^2)$. Provide the method used to find $E(Y^2)$.

Answer: \dfrac{11}{18}


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

Answer: \dfrac{1}{2}


Question: Find the value of \( \lim_{n\to\infty}\int_{1}^{e}(\ln x)^n\ dx \).

Answer: 0


Question: Evaluate the line integral \( \int_{c} (x^2y^2 + z) \, ds \), where \( c \) is the curve formed by the intersection of the surfaces \( x^2 + y^2 = 4 \) and \( x^2 + y^2 + (z-2)^2 = 4 \).

Answer: 16\pi


Question: Evaluate the limit \( L = \lim_{x \to 0^+}\frac{3^{\ln x}}{x} \).

Answer: 0


Question: Given a complex number in rectangular form, convert it to polar form. If the tangent of the argument is given by \( \tan\theta = 2-\sqrt{3} \), find the exact value of \( \theta \).

Answer: \dfrac{\pi}{12}


Question: Determine the value of \( f(k) = \lim_{n \rightarrow \infty} \frac{|S_n(k)|}{\pi(n)} \), where \( S_n(k) = \{1 \leq m \leq n : m \text{ has } k \text{ ones in its binary representation and } m \text{ is prime}\} \) and \( \pi(x) \) is the prime counting function.

Answer: 0


Question: Determine the splitting field \( \mathbb{F}_q \) of the polynomial \( x^4 + x + 1 \) over \( \mathbb{F}_2 \).

Answer: \mathbb{F}_{16}


Question: Find \( \omega_0 \) in the set of complex numbers such that the substitution \( z = \omega - \omega_0 \) transforms the cubic equation \( z^3 + Az^2 + Bz + C = 0 \) into \( \omega^3 - m\omega - n = 0 \). Assume the constants are real numbers.

Answer: \dfrac{A}{3}


Question: Determine the number of zeros of the function \( f(z) = \frac{1}{3}e^z - z \) within the unit disc \( |z| < 1 \).

Answer: 1


Question: Let \(n \in \mathbb{Q}\) be a number such that \(\sqrt{n} + \sqrt[3]{n} \in \mathbb{N}\). Determine the form of \(n\).

Answer: m^6


Question: Find the value of \( k \) such that \( \binom{404}{4} - \binom{4}{1} \cdot \binom{303}{4} + \binom{4}{2} \cdot \binom{202}{4} - \binom{4}{3} \cdot \binom{101}{4} = (101)^k \).

Answer: 4


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

Answer: \ln 2


Question: Determine whether there exists a function $f$ such that at a point $c$, the second derivative $f''(c) = 1$, the first derivative $f'(c) = 0$, and the second derivative $f''$ is not continuous on any open interval containing $c$. Provide a justification for your answer.

Answer: Yes


Question: Given a ring $R$ and submodules $M_1, M_2$ of a module $M$ with $K \le M$, if $M_1 \cap K = M_2 \cap K$ and $M_1 + K = M_2 + K$, does it follow that $M_1 = M_2$? Justify your answer.

Answer: No


Question: Calculate the limit: \[ \lim_{n \to \infty}(n^2+1)\cdot(\ln(n^2-4)-2\ln(n)) \]

Answer: -4


Question:
Given the group $G$ with the presentation $G=\langle x,y \mid x^5=y^2=e, x^2y=yx \rangle$, determine which of the following groups $G$ is isomorphic to:

(a) $\mathbb{Z}_5$
(b) $\mathbb{Z}_{10}$
(c) $\mathbb{Z}_2$
(d) $\mathbb{Z}_{30}$

Select the correct option.

Answer: B


Question: Compute the dimension of the Zariski Tangent space of the affine scheme \(\operatorname{Spec} A\), where \(A = \mathbb{Z}[2i] \cong \mathbb{Z}[x]/(x^2 + 4)\), at the point corresponding to the maximal ideal \((2, x)\).

Answer: 2


Question: Let \( T: \mathbb{C}^{11} \rightarrow \mathbb{C}^{11} \) be a linear transformation such that \( \dim \ker T = 4 \), \( \dim \ker T^3 = 9 \), and \( \dim \ker T^4 = 11 \). Find \( \dim \ker T^2 \).

Answer: 7


Question: True or false: Every harmonic function on \( \Bbb C \setminus \{0\} \) has a harmonic conjugate.

Answer: B


Question: Let $G$ be a connected self-centered graph with radius $r(G)$ and diameter $d(G)$ both equal to $m < \infty$. Does $G$ always contain a cycle of length $C_{2m}$ or $C_{2m+1}$ as a subgraph?

Answer: Yes


Question: Find the irreducible polynomial over \( \mathbb{Q} \) for the element \( a = \sqrt{2} + \sqrt{3} \).

Answer: x^4 - 10x^2 + 1


Question: Is there a sequence in the interval $(0,1)$ such that the product of all its terms is $\frac{1}{2}$?

Answer: Yes


Question:
Given the joint probability density function of the bivariate random variables \((X, Y)\):
\[ f_{(X,Y)}(x,y) = \begin{cases} 8xy, & 0 \le y \le x \le 1 \\ 0, & \text{elsewhere} \end{cases} \]
Calculate the conditional probability \( P\left( X \le \frac{1}{2} \mid Y \ge \frac{1}{4} \right) \).

Answer: \dfrac{1}{25}


Question:
Given a Markov chain with states $S = \{1, 2, 3\}$ and the transition matrix:
\[ P = \begin{pmatrix} 0.65 & 0.28 & 0.07 \\ 0.15 & 0.67 & 0.18 \\ 0.12 & 0.36 & 0.52 \end{pmatrix} \]
If the initial state is $2$, calculate the probability that after 3 transitions, the chain is in state $3$ without visiting state $1$ in any of these transitions. Express this probability as $P(X_3=3, X_2 \neq 1, X_1 \neq 1 \mid X_0 = 2)$. Provide your answer as a decimal rounded to four decimal places.

Answer: 0.2039


Question:
Prove by induction using the recurrence relation of Legendre polynomials that \(P_n(1) = 1\) for all \(n \geq 0\). The recurrence relation is given by:

\[(n+1)P_{n+1}(x) - (2n+1)xP_n(x) + nP_{n-1}(x) = 0\] for \(n \geq 1\), with initial conditions \(P_0(x) = 1\) and \(P_1(x) = x\).

Answer: 1


Question: Calculate the probability \( P(X_n = 1) \) for the random variable \( X_n = \lfloor(2^n w)\rfloor \mod 2 \) on the probability space \(([0,1], B([0,1]), \lambda)\), where \( w \in [0,1] \) and \( n \in \mathbb{N} \).

Answer: \dfrac{1}{2}


Question: Evaluate the infinite series \( \sum_{n=0}^\infty \frac{n^2}{n!} \).

Answer: 2e


Question: Is it true that every Banach space has at least one extreme point of its dual unit ball that is normed by some point in the space? Specifically, given a Banach space $X$, does there exist an extreme point $x^*$ of the unit ball $B_{X^*}$ in the dual space $X^*$ such that $x^*(x) = 1$ for some $x \in X$ with $\|x\| \leq 1$?

Answer: Yes


Question: Calculate the value of \( 111^{4733} \mod 9467 \). Given that 9467 is a prime number and 111 is coprime to 9467, use Euler's theorem to find the result.

Answer: 9466


Question: Evaluate the contour integral \( \oint_{C}\frac{e^{\frac{1}{z}}}{z-1}dz \) where \( C: |z-1|=\frac{3}{2} \).

Answer: 2\pi i


Question: Find the degree of the splitting field of \((x^5-3)(x^5-7)\) over \(\mathbb{Q}\).

Answer: 100


Question: Determine whether the following statement is true or false: If $G$ is a simple finite group, then its fusion ring is different from the fusion rings of any other groups.

Answer: True


Question: Let $f:X \to Y$ be a smooth surjective morphism of irreducible Noetherian schemes over $\mathbb{C}$. Assume $X$ and $Y$ are not necessarily smooth over $\mathbb{C}$. Prove or disprove that the induced morphism of tangent spaces from $T_xX$ to $T_{f(x)} Y$ is surjective for all $x \in X$. \(\text{True or False?}\)

Answer: A


Question: Evaluate the definite integral: $$\int_{\pi/2}^{5\pi/2} \frac{e^{\tan^{-1} \sin x}}{e^{\tan^{-1} \sin x}+e^{\tan^{-1} \cos x}}\,dx$$

Answer: \pi


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty}\left(n^2\int_n^{3n}\frac{x+1}{x^4+1} \, dx\right) \] without directly calculating the integral \( \int_n^{3n}\frac{x+1}{x^4+1} \, dx \).

Answer: \dfrac{4}{9}


Question: Find the spectral radius of the matrix \( A = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & c & 0 \\ 0 & -c & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \), where \( c \in \mathbb{R} \).

Answer: \max\{1, |c|\}


Question: Let $a$ and $b$ be $p$-dimensional non-zero column vectors. Determine the number of distinct eigenvalues of the matrix $ab^t + ba^t$. Choose the correct option: (a) 1, (b) 2, (c) 1 or 2, (d) 1 or 2 or 3.

Answer: D


Question: Given a non-unit fraction \( \frac{a}{b} \) in lowest terms, where \( 0 < \frac{a}{b} < 1 \), and \( \frac{1}{n} \) is the largest unit fraction less than \( \frac{a}{b} \), let \( \frac{a'}{b'} = \frac{a}{b} - \frac{1}{n} \). Is it true that \( 0 < a' < a \)? Assume \( a, b, n \) are natural numbers.

Answer: Yes


Question: Find the arc length of the hypocycloid given by the equation \(x^{2/3} + y^{2/3} = 1\) oriented clockwise.

Answer: 6


Question: Let $f$ be a continuously differentiable function such that $L=\lim_{x\to\infty}(f(x)+f'(x))$ is finite. Prove or disprove: If $\lim_{x\to\infty} f'(x)$ exists, then it must be equal to $0$.

Answer: 0


Question: Consider a symmetric multisublinear continuous functional \( S: V\times\cdots\times V \rightarrow \mathbb{R} \) that is positively homogeneous and subadditive in every component, satisfying certain additional assumptions. Let \( T: U\times\cdots\times U \rightarrow \mathbb{R} \) be a symmetric multilinear continuous functional dominated by \( S \) on \( U\times\cdots\times U \). Does there exist an extension of \( T \) to \( V\times\cdots\times V \) that is dominated by \( S \) on \( V\times\cdots\times V \)?

Answer: Yes


Question:
Determine whether there exists a function \( f: [-1, 1] \to \mathbb{R}_+ \) such that:
\[
\int_{-1}^1 f(x) (1 - |x|) \, \mathrm{d}x = r, \quad \text{and} \quad \int_{-1}^1 (f(x))^k (1 - |x|) \, \mathrm{d}x = r^{k+1},
\]
where \( r > 1 \) and \( k > 1 \) is a fixed integer.

Answer: Yes


Question:
Let \( p \) be a prime number with \( p \ge 3 \). Consider the sum:
\[ \sum_{k=1}^{p-1} \frac{(p-1)!}{k} \]
Is this sum congruent to 0 modulo \( p \)? In other words, does \( p \) divide the sum \( q \) where:
\[ q = \frac{(p-1)!}{1} + \frac{(p-1)!}{2} + \cdots + \frac{(p-1)!}{p-1} \]
Provide a justification for your answer.

Answer: Yes


Question:
Given that the equation \(x^4 + 4rx + 3s = 0\) has no real roots, determine the correct relationship between \(r\) and \(s\):

(a) \(r^2 < s^3\)
(b) \(r^2 > s^3\)
(c) \(r^4 < s^3\)
(d) \(r^4 > s^3\)

Answer: C


Question: Assume that $A\in\mathbb{C}^{n\times n}$ is a Hermitian positive semi-definite matrix and $z\in\mathbb{C}^{n}$ such that $z\neq 0$. If $z^*Az=0$, is $z$ an eigenvector of $A$ associated with eigenvalue $0$?

Answer: Yes


Question:
Consider the equation $e^x - 4x^2 = 0$ which has a root between $4$ and $5$. Using the fixed point iteration method with the iteration function $g(x) = \frac{1}{2}e^{\frac{x}{2}}$, determine the behavior of the iteration. Does it:
1. Diverge
2. Converge
3. Oscillate
4. Converge monotonically
Choose the correct option.

Answer: 1


Question: Evaluate the integral \( \int_0^{+\infty} \frac{1}{(1+x^2)(1+x^{\phi})}\, dx \) where \( \phi > 0 \).

Answer: \dfrac{\pi}{4}


Question: Let $k$ be a field of characteristic zero, $\mathfrak{g}$ a finite-dimensional Lie algebra over $k$, and let $A,B$ be associative $k$-algebras. Suppose that $\mathfrak{g}$ acts on $A$ and $B$, and that each of these actions admits a quantum moment map. Then the associative algebra $A\otimes_kB$ also has a $\mathfrak{g}$ action and admits a quantum moment map. Is it true that $$(A\otimes_kB)//\mathfrak{g}\cong (A//\mathfrak{g})\otimes_k(B//\mathfrak{g})$$ where "$//\mathfrak{g}$" denotes the operation of quantum Hamiltonian reduction?

Answer: No


Question: Three points $A$, $B$, and $C$ form an equilateral triangle with side length 1. Each point moves towards the next point in the sequence ($A$ to $B$, $B$ to $C$, and $C$ to $A$) at a constant speed of $v = 50$. Determine the time $T$ at which all points meet at the center of the triangle.

Answer: \dfrac{1}{75}


Question: Let $f \in C^\infty(\mathbb{T}^d)$ be a smooth function on the $d$-dimensional torus, satisfying $f \ge 0$ and $\hat{f}(0) = 1$. Is it true that $|\hat{f}(k)| < 1$ for all $k \neq 0$?

Answer: Yes


Question: Let $a, b \in M_n(\mathbb{R})$. Consider the function $f: \mathbb{R}^2 \to M_n(\mathbb{R})$, defined by $(t, s) \mapsto e^{t \cdot a} e^{s \cdot b} e^{-t \cdot a} e^{-s \cdot b}$. Prove that $$\left( \frac{\partial^2 f}{\partial t \, \partial s} \right)\bigg|_{t = s = 0} = ab - ba.$$

Answer: ab - ba


Question: Calculate the flux of the vector field \( \vec{F} = 4x \hat{i} - 2y^2 \hat{j} + z^2 \hat{k} \) through the surface \( S = \{(x,y,z) : x^2 + y^2 = 4, 0 \leq z \leq 3\} \).

Answer: 48\pi


Question:
Given the expression \( A = \sqrt{x + \frac{2}{x}} - \sqrt{x - \frac{2}{x}} \) and the condition \( x >> 1 \), determine which of the following options provides the most accurate primary formula for calculating \( A \):

1. \( \frac{1}{x\sqrt{x}} \) for \( x > \sqrt{\frac{2}{u}} \)
2. \( \frac{2}{x\sqrt{x}} \) for \( x > \sqrt{\frac{2}{u}} \)
3. \( \frac{1}{x\sqrt{x}} \) for \( x > \frac{2}{u} \)
4. \( \frac{2}{x\sqrt{x}} \) for \( x > \frac{1}{u} \)

Note: The variable \( u \) is not defined. Choose the option that is true.

Answer: 2


Question: Determine the range of values for the parameter \(a\) such that the equation \(2x^3 - 3x^2 - 12x + a = 0\) has all roots real and distinct.

Answer: (-7, 20)


Question: Determine whether the sequence \((x_n)_n\) is convergent, given that \(\frac{2018}{x_n} = \left(\frac{x_{n+1}}{x_n}\right)^{(n+1)/2}\) for all \(n \geq 1\).

Answer: 2018


Question: Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. Given $g \in G$ and $X \in \mathfrak{g}$ such that \( \frac{d}{dt} \bigg|_{t=0} ge^{tX}g^{-1} = X \), determine the value of \( \frac{d}{dt} \bigg|_{t=0} e^{-tX} g e^{tX} \).

Answer: 0


Question: Let \( w = \frac{\sqrt{3} + i}{2} \) and define the set \( P = \{ w^n : n = 1, 2, 3, \ldots \} \). Consider the half-planes \( H_1 = \{ z \in \mathbb{C} : \text{Re}(z) > \frac{1}{2} \} \) and \( H_2 = \{ z \in \mathbb{C} : \text{Re}(z) < -\frac{1}{2} \} \), where \( \mathbb{C} \) is the set of all complex numbers. If \( z_1 \in P \cap H_1 \) and \( z_2 \in P \cap H_2 \), find the angle \( \angle z_1 O z_2 \), where \( O \) represents the origin.

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


Question: For a given constant \(0 < a < 1\), find the positive value of \(x\) such that \(x^{a^x} = a^{x^a}\).

Answer: a


Question: Determine whether the function \( \frac{\sqrt{x}}{\ln(x)-1} \) is strictly increasing for \( x \ge 21 \).

Answer: Yes


Question: Find the function \( f(x) \) such that \( f(f(x)) = 6x - f(x) \), given that \( f: [0,\infty) \rightarrow [0,\infty) \) and \( f(x) > 0 \) for all \( x \in (0,\infty) \).

Answer: 2x


Question: What is the derivative of the fractional part of $x$, denoted as $\{x\}$?

Answer: 1


Question: What is the limit, as \( n \to \infty \), of the expected distance between two points chosen uniformly at random within a unit-radius hypersphere in \( \mathbb{R}^n \)?

Answer: \sqrt{2}


Question: In how many ways can the set \( \{a_{1}, a_{2}, \ldots, a_{9}, b_{1}, b_{2}, \ldots, b_{9}, c_{1}, c_{2}, \ldots, c_{9}\} \) be partitioned into 9 subsets of the form \( \{ a_{i}, b_{j}, c_{k} \} \)?

Answer: (9!)^2


Question: Evaluate the Lebesgue integral \(\int_0^1 f(x)\, dx\), where the function \(f:[0, 1]\to \mathbb{R}\) is defined as follows: for each \(x\in (0, 1]\), consider the decimal representation \(x=0.d_1d_2\cdots d_n\cdots\). Define \(f(x)=0\) if \(x\) is rational and \(f(x)=18n\) if \(x\) is irrational, where \(n\) is the number of zeros in the decimal representation of \(x\) immediately after the decimal point up to the first non-zero digit of \(x\).

Answer: 2


Question: Given a sequence of real numbers \( \{x_a\} \), if \( x_{a+1} - x_a \to 0 \) and \( x_{2a} - 2x_a \to 0 \), does it follow that \( x_a \to 0 \)? Justify your answer.

Answer: Yes


Question: Find the smallest integer \( n > 4 \) such that the expression \( \binom{3n-1}{11} + \binom{3n-1}{12} + \binom{3n}{13} + \binom{3n+1}{14} \) is divisible by 101.

Answer: 33


Question: Evaluate the integral \( \int_{-\pi/3}^{\pi/3} (1 - \tan^4(x)) \, dx \) without using the secant function in your solution.

Answer: 0


Question: Given that \(\lim_{x\to\infty} (x-f(x))=2\), find the value of \[\lim_{x\to\infty} \frac{\sqrt{x+1}-\sqrt{f(x)}}{\sqrt{x}-\sqrt{f(x)}}.\]

Answer: \dfrac{3}{2}


Question: In a chess match against a skilled opponent, there is a 75% chance that any given game will end in a draw. For games that do not end in a draw, there is a 2/3 chance that you will win and a 1/3 chance that you will lose. The first player to win 2 games in a row is declared the winner of the match. What is the probability that you will win the match?

Answer: \dfrac{26}{33}