Question: Use Stokes' Theorem to evaluate \( \iint_{S} (\nabla \times \vec{F}) \cdot \vec{n} \, dS \), where \( S \) is the hemisphere defined by \( x^2 + y^2 + z^2 = 4 \) for \( y \geq 0 \) and oriented in the positive \( y \) direction. The vector field is given by \( \vec{F} = \langle ze^y, x\cos(y), xz\sin(y) \rangle \).

Answer: -4\pi


Question: Determine the fundamental group of the space obtained by removing a point from a double torus. Use the van Kampen theorem to find the fundamental group of the space $P \backslash \{ x \}$, where $P$ is a double torus.

Answer: F_4


Question: Is the set of natural numbers \(\mathbb{N}\) a Banach space when equipped with the norm \(|x-y|\) from \(\mathbb{R}\)?

Answer: No


Question: Evaluate the line integral \( \int_C (y-z)\,dx + (z-x)\,dy + (x-y)\,dz \), where the curve \( C \) is the intersection of the cylinder \( x^2+y^2=1 \) and the plane \( x-z=1 \).

Answer: 0


Question: Evaluate the integral \( \int_0^{2\pi}\frac{4 \cdot \cos(x)}{5-4 \cdot \cos(x)}dx \) using the method of residues.

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


Question: Determine whether there exists a non-trivial group homomorphism from the symmetric group $S_3$ to the cyclic group $\mathbb{Z}/3\mathbb{Z}$. Answer with True or False.

Answer: False


Question: For a linear operator $T_k:H \to H$ defined by \[ T_ku=\sum^\infty_{n=1}\frac{1}{n^3}\langle u,e_n\rangle e_n+k\langle u,z\rangle z, \] where $z=\sqrt{6/\pi^2}\sum^\infty_{n=1}e_n/n$, show that for negative $k$, the operator $T_k$ has at most one negative eigenvalue.

Answer: 1


Question: Evaluate the complex integral \( \int_{\gamma} \frac{-\cos(1/z)}{\sin(1/z)z^2} \, dz \), where \( \gamma \) is the positively oriented circle \( \partial B(0,2) \).

Answer: -2\pi i


Question: Evaluate the definite integral \( \int_{0}^{\pi/2} \sin 2x \ln(\tan x) \, dx \).

Answer: 0


Question:
Consider the sequence \((x_n)\) defined by \(x_0 = 1\), \(x_n = 2x_{n−1} + 1\) if \(n\) is odd, and \(x_n = 3x_{n−1} + 2\) if \(n\) is even. Determine the value of \(\lim_{n\to \infty} \frac{\log(x_n)}{n}\). Choose the correct option:

(a) The limit does not exist.
(b) \(\lim_{n\to \infty} \frac{\log(x_n)}{n} = \frac{\log6}{2}\)
(c) \(\lim_{n\to \infty} \frac{\log(x_n)}{n} = \log5\)
(d) \(\lim_{n\to \infty} \frac{\log(x_n)}{n} = \frac{5}{2}\)

Answer: B


Question: Determine the value of \( \alpha \) that makes the metric \( g = (1-x_1^2-x_2^2-\cdots-x_n^2)^{-\alpha}(dx_1\otimes dx_1+dx_2\otimes dx_2+\cdots+dx_n\otimes dx_n) \) geodesically complete on the set \( \{\vec{x}\in \mathbb{R}^n : x_1^2+x_2^2+\cdots+x_n^2<1\} \) in \( \mathbb{R}^n \).

Answer: 2


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{(1-x)(1-2x)(1-3x)\cdots(1-85x) - 1}{x} \]

Answer: -3655


Question: Every day, you put either 1 or 2 dollars into a piggy bank, each with a probability of \( \frac{1}{2} \). What is the expected number of 2-dollar bills in the piggy bank when the total amount reaches at least 100 dollars for the first time?

Answer: \dfrac{301}{9}


Question: Let \(X\) be a Banach space, and let \(T : X \to X\) be an \(\alpha\)-Lipschitz mapping with \(\alpha < 1\). Let \(z_*\) be the fixed point of \(T\). Suppose \(\{\gamma_t\}_{t=1}^\infty\) is a sequence such that \(0 < \gamma_t < 1\) for all \(t\) and \(\sum_t \gamma_t = \infty\). Let \(x_0 \in X\) be arbitrary, and for \(t \geq 1\), define \(x_t = (1 - \gamma_t) x_{t-1} + \gamma_t T x_{t-1}\). Prove or disprove that \(x_t \to z_*\).

Answer: x_t \to z_*


Question: Find the number of permutations of the letters AABBCCDD such that no two adjacent letters are the same.

Answer: 864


Question: Find the closure of \(l^1(\mathbb{N})\) in \(l^2(\mathbb{N})\).

Answer: l^2(\mathbb{N})


Question:
Determine the final polynomial for the recurrence relation given by:
\[ ax_n - bx_{n-1} - 2x = (xa-b)x_{n-1} - ax_{n-2} - 2x \]
where \( a = x^3(x^2-3) - (x^2-1) \) and \( b = x(x^2-3) \). What is the final polynomial?

Answer: x^5 - 3x^3 - x^2 + 1


Question: Determine the values of $m$ and $n$, where $n$ is the number of necklaces that can be formed using 17 identical pearls and two identical diamonds, and $m$ is the number of necklaces that can be formed using 17 identical pearls and 2 different diamonds.

Answer: 9


Question: Calculate the flux of the vector field \( F(x,y,z) = (y, x, z) \) downwards through the surface \( z = 16 - x^2 - y^2 \).

Answer: -128\pi


Question: Find the limit: \[ \lim_{n\to\infty}\frac{n^{\sqrt{n}}}{2^n} \]

Answer: 0


Question: Determine the semiclassical principal symbol $\sigma_h$ of the operator $h^2\Delta-1$, where $\Delta=-\sum_j\partial^{2}_{x_j}$. Explain why the principal symbol is $|\xi|^2-1$ instead of just $|\xi|^2$. What is the correct expression for $\sigma_h(h^2\Delta-1)$?

Answer: |\xi|^2 - 1


Question: Let $X$ be a non-compact weakly pseudoconvex manifold. Does there exist a smooth plurisubharmonic exhaustion function $\varphi$ on $X$ such that $d\varphi$ is never zero outside some compact set $K\subset X$?

Answer: Yes


Question: Given a sample $X$ of one hundred observations $X_i$ such that $\forall i: X_i \in \mathbb{N}[-101, 101]$, with unique minimum $A$ and unique maximum $B$, the mean of the sample is $M = \bar{X}$. After negating $A$ and $B$, the mean of the modified sample becomes $-M$. What is the maximum possible value of $M$ in the original sample?

Answer: 1


Question: Calculate the work done by the force \( F(x,y,z) = e^{y+2z}(i+xj+2xk) \) from the point \((0,0,0)\) to \((1,1,1)\) along the curve defined by \(x=t\), \(y=t^2\), and \(z=t^4\).

Answer: e^3


Question: Let \( A \subset \{(x,y) \in \mathbb{R}^2 : x = y\} \) be a set such that the one-dimensional Lebesgue measure of \( A \) is positive. Is it true that the projection of \( A \) onto the x-axis and y-axis is also a set of positive one-dimensional Lebesgue measure?

Answer: Yes


Question: Solve the congruence $M^{49} \equiv 2 \pmod{19}$. Find the integer $M$ that satisfies this equation.

Answer: 14


Question: For a Hecke L-function, determine if it is an Artin L-function when all of the local eigenvalues are roots of unity.

Answer: Yes


Question: For a prime number \( p \), determine the positive integer \( k \) such that the set \( \{1, 2, \ldots, k\} \) can be partitioned into \( p \) subsets with equal sums of elements.

Answer: 2p - 1


Question:
Let \(x\) and \(y\) be positive numbers. Which of the following conditions always implies \(x^y \geq y^x\)?

(a) \(x \leq e \leq y\)
(b) \(y \leq e \leq x\)
(c) \(x \leq y \leq e\) or \(e \leq y \leq x\)
(d) \(y \leq x \leq e\) or \(e \leq x \leq y\)

Answer: D


Question: Determine the region \( \Omega \) in the complex plane such that the series \( \sum_{n=1}^\infty \frac{1}{n^2} \exp\left(\frac{nz}{z-2}\right) \) converges for any \( z \in \Omega \).

Answer: \{ z \in \mathbb{C} \mid |z - 1| \leq 1 \} \setminus \{2\}


Question: Solve the equation: \[ 4^x - 4^{\sqrt{x}+1} = 3 \times 2^{x+\sqrt{x}} \]

Answer: 4


Question: Use the Divergence Theorem to evaluate the surface integral \( \iint_S F \cdot dS \), where \( F = (x^3y + z)i + xze^yj + (xy − 3z^4)k \) and \( S \) consists of the five lower faces of the cube \([0, 1]\times[0, 1]\times[0, 1]\), excluding the face with \( z = 1 \). Hint: Close the surface to apply the theorem.

Answer: \dfrac{e}{4}


Question: Determine whether there exists a dense open subset of \( \mathbb{R} \) with an uncountable complement.

Answer: Yes


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

Answer: e


Question: Let \(u_1, u_2, u_3, u_4\) be four linearly independent vectors in \(\mathbb{R}^n\). Determine the value of \(a\) for which the set \(\{u_1+4u_2, u_2+3u_3, u_3+3u_4, u_4+au_1\}\) is linearly dependent.

Answer: \dfrac{1}{36}


Question: Let $U$ be the open unit disk, and let $f: U \rightarrow U$ be a holomorphic function such that $f(1/2) = 1/3$. Determine the maximum value of $|f'(1/2)|$.

Answer: \dfrac{32}{27}


Question: Evaluate the integral \( \int \int_{\mathbb{R}^2} x^2 e^{-\sqrt{x^2+y^2}} \, dx \, dy \).

Answer: 6\pi


Question: Suppose $I$ is an ideal of a ring $R$ such that $|I| = 33$. Can we conclude that $I$ is a principal ideal?

Answer: Yes


Question: Compute the integral \( \int_R (x^2-y^2)\,dx\,dy \) over the region \( R \) in the first quadrant defined by the inequalities \( 3 \geq y-x \geq 0 \) and \( 5 \geq xy \geq 2 \). Use the change of variables \( u = y-x \) and \( v = xy \) to evaluate the integral.

Answer: -\dfrac{27}{2}


Question: Find the sum \( \sum_{i = 0}^{\infty}\frac{F_i}{7^i} \), where \( F_i \) is the \( i \)-th Fibonacci number.

Answer: \dfrac{7}{41}


Question:
Determine the cardinality of the group $\mathrm{SL}_2(\mathbb{Z}_p)$, where $p$ is a prime number. Choose the correct answer from the following options:

- $p(p-1)(p+1)$
- $p^2(p-1)$
- $p^3$
- $p^2(p-1)+p$

Answer: A


Question: In \(\triangle ABC\), which is an equilateral triangle, point \(O\) is located inside the triangle such that \(\angle OAB = x\), \(\angle OBC = 42^\circ\), and \(\angle OCB = 54^\circ\). Find the value of \(x\) without using trigonometry.

Answer: 48


Question: Compare the smallest positive roots of the equations \( \cos(\sin x) = x \) and \( \sin(\cos x) = x \) within the interval \([0, \frac{\pi}{2}]\). Let \(x_1\) be the root of \(\sin(\cos x) = x\) and \(x_2\) be the root of \(\cos(\sin x) = x\). Determine which is greater: \(x_1\) or \(x_2\).

Answer: x_2


Question: Evaluate the limit: \[ \lim_{n\to\infty}{n^2 \int_{\frac{1}{n+1}}^{\frac{1}{n}} \frac{\arctan(nx)}{\arcsin(nx)} \, dx} \]

Answer: \dfrac{1}{2}


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

Answer: x^3 - 3x


Question: Is the following statement true for all $0 < T \leq 1$? For $X_t \sim N(0, 1)$ independently defined for each $t \in (0, 1]$, does the equality \( P\left(\sup_{t \leq T} X_t \leq \alpha\right) = P(X_1 \leq \alpha)^T \) hold?

Answer: No


Question: Compute the integral \( \int_{\sigma} \omega \) for the differential form \( \omega = 2\; d\psi \) over the semicircle defined by the differentiable singular 1-simplex \( \sigma(t) = (-\sin(\pi t), \cos(\pi t)) \) on the manifold \( M = \mathbb{S}^{1}-\{(1, 0) \} \). Use the coordinate system \( (M, \psi) \) where \( M \ni (x, y)  \overset{\psi}{\mapsto} \frac{y}{1-x} \in \mathbb{R} \).

Answer: -4


Question: Consider a $10 \times 10$ matrix $A$. Each element $a_{ij}$ of the matrix is multiplied by $\frac{i}{j}$, where $1 \leq i,j \leq 10$. Determine how the determinant of the matrix changes after this transformation.

Answer: 1


Question: The axis of a light in a lighthouse is tilted. When the light points east, it is inclined upward at 1 degree. When it points north, it is inclined upward at 2 degrees. What is the maximum angle of elevation of the light beam above the horizontal plane?

Answer: \sqrt{5}


Question: Let \( G \) be a finite group and \( \chi \) its character. If the degree of \( \chi \) is greater than 1, is it always true that \( \langle \chi, \chi \rangle \ge 1 \)? Justify your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: If \( \{u_n\} \) is a sequence in a Hilbert space \( H \) that converges weakly to its limit \( u \in H \) and the sequence satisfies \( \limsup_{n \rightarrow \infty} \|u_n \| \leq \|u\| \), then \( \lim_{n \rightarrow \infty} \|u_n \| = \|u\|.\)

Answer: True


Question: Does every primitive digraph have a directed cycle? A digraph is a directed graph, and a directed cycle is a directed circuit where the only repeated vertices are the first and last vertices. A digraph is primitive if its adjacency matrix is primitive, meaning there exists a positive integer \( k \) such that \( A^k > 0 \) (all entries of \( A^k \) are positive).

Answer: Yes


Question:
Determine whether there exist four real numbers $x, y, a, b$ such that the expression \( z = (x-a)^2 + (y-b)^2 \) holds true for the quantity:
\[ z = (2^{p}-1)\beta^{2^{p-2}}k - \beta^{2^{p-1}} > 0 \]
where \( p \) is a prime number, \( k \) is a natural number, and \( \beta = \sqrt{3} + 2 \).

Answer: Yes


Question: Let $p$ be a prime number and $\mathbb{F}_{p}$ be a finite field of order $p$. Consider the general linear group $G=GL_{n}(\mathbb{F}_{p})$ and the unitriangular group $U_{n}$, consisting of $n \times n$ upper triangular matrices with ones on the diagonal over $\mathbb{F}_{p}$. Let $T$ be the diagonal subgroup of $G$. A diagonal automorphism $\varphi_{D}$ of $U_{n}$ is defined by $\varphi_{D}(M)=DMD^{-1}$ for $D \in T$. A subgroup $U$ of $U_{n}$ is invariant under the diagonal automorphisms if $\varphi_{D}(M) \in U$ for all $M \in U$. Determine whether all subgroups of order $p^{2}$ of $U_{n}$ are invariant under the diagonal automorphisms of $U_{n}$. Provide a justification for your answer.

Answer: No


Question: Find the point at which the density function of the $\chi^2(n)$ distribution reaches its maximum.

Answer: \max(n - 2, 0)


Question: Determine the structure of the quotient group $S_n/S_{n-1}$, where $S_{n-1}$ is the subgroup of $S_n$ consisting of permutations that fix the element $n$. Specifically, identify the group structure of $S_3/S_2$.

Answer: \mathbb{Z}_3


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sum_{j=0}^{n} \sum_{i=0}^j \frac{i^2+j^2}{n^4+ijn^2} \]

Answer: \dfrac{1}{4}


Question: Is there a left-invariant Riemannian metric on \(\text{SL}_3(\mathbb{R})\) such that the geodesics through the identity are precisely the integral curves of the invariant, smooth vector fields?

Answer: No


Question: Let $f: \mathbb{R}^3 \to \mathbb{R}$ be a polynomial function and $T: \mathbb{R}^3 \to \mathbb{R}^3$ be an invertible linear map. If $\nabla f(P) \neq 0$ for all $P \in \mathbb{R}^3 - \{0\}$, does it follow that $\nabla (f \circ T)(P) \neq 0$ for all $P \in \mathbb{R}^3 - \{0\}$? Justify your answer.

Answer: Yes


Question: Consider a holomorphic homeomorphism $f: U \rightarrow U$ of the unit disc $U$. Must $f$ extend to a continuous map $f: \overline{U} \rightarrow \overline{U}$? Provide a justification for your answer.

Answer: Yes


Question: Determine the number of solutions to the equation \( e^{\cot^2\theta} + \sin^2\theta - 2\cos^2 2\theta + 4 = 4\sin\theta \) in the interval \([0, 10\pi]\).

Answer: 5


Question:
Find the solutions of the equation $F'(x)=0$, where $F: [0,\pi] \rightarrow \mathbb{R}$ is defined by:
\[ F(x) = x + \int_{0}^{\cos x} \sqrt{1-x^2} \, dx \]

Answer: \dfrac{\pi}{2}


Question: Given the polynomial \(x^4 - 3x^3 - 27x^2 - 13x + 42\), find the value of \(\frac{1}{r_1 + 1} + \frac{1}{r_2 + 1} + \frac{1}{r_3 + 1} + \frac{1}{r_4 + 1}\), where \(r_1, r_2, r_3, r_4\) are the roots of the polynomial.

Answer: -\dfrac{7}{8}


Question:
Consider the function \( f(x) = e^{-x} \sin(x) g(x) \), where \( g: \mathbb{R^+} \to (-M, M) \) for some positive real \( M \). Determine the nature of the number of solutions to the equation \( |f(n_1)| + |f(n_2)| + \cdots + |f(n_k)| = 1 \), where all \( n_i \) are natural numbers. Choose the correct option:

A) The number of solutions is infinite for some values of \( k \) only.
B) The number of solutions is finite for all values of \( k \).
C) The number of solutions can be infinite for all values of \( k > 1 \).
D) None of these.

Answer: B


Question: Evaluate the sum \( \sum_{i=0}^{100}\sum_{j=0}^{100}\frac{\binom{100}{i}\cdot \binom{100}{j}}{\binom{200}{i+j}} \).

Answer: 201


Question: Consider the permutations $a = (13)(24)$ and $b = (234)$ in the symmetric group $S_4$. Determine the number of elements in the group generated by $a$ and $b$. Provide a systematic approach to find the size of this group.

Answer: 12


Question: Given that \(f(x) = \sum_{n=0}^{\infty} a_n x^n\) converges on \((-R, R)\) for some \(R > 0\), and that \(f(0) = 1\) and \(f'(x) = f(x)\) for all \(x \in (-R, R)\), find the coefficients \(a_n\) in the power series representation of \(f(x)\).

Answer: \dfrac{1}{n!}


Question: Evaluate the integral: $$\int_{0}^{\infty} \dfrac{1-\cos x}{x^{2}} dx$$

Answer: \dfrac{\pi}{2}


Question: Calculate the sum \( \sum_{n=1}^{\infty} 2^{-n} \left( \frac{1}{n} - \frac{1}{n+1} \right) \).

Answer: 1 - \ln 2


Question: Find the last four digits of \(2^{2020}\) using Euler's theorem.

Answer: 8576


Question: Evaluate the integral \( \int_{|z-i| = 10} \left(z+\frac{1}{z}\right)^4 \, dz \) using the Residue Theorem.

Answer: 0


Question: Compute \(75^{75^{75}} \mod 32\).

Answer: 19


Question: A cyclic hexagon has side lengths of 2, 2, 7, 7, 11, 11, in that order. Find the length of its circumradius.

Answer: 7


Question: Find the number of non-negative integer solutions to the equation \(a + b + c + d + e = 30\) given the constraints \(10 \leq e\) and \(4 \leq d \leq 7\).

Answer: 3025


Question: Let $z$ and $w$ be complex numbers such that $|2z - w| = 25$, $|z + 2w| = 5$, and $|z + w| = 2$. Find $|z|$. Express your answer as a single number.

Answer: 9


Question: Let \( X \) and \( Y \) be independent random variables, each following a geometric distribution with parameter \( p \). Compute \( E(X \mid X+Y=k) \) for all \( k = 2, 3, \ldots \).

Answer: \dfrac{k}{2}


Question: Find the residue at $z = \pi$ for the function \[ \frac{z^2 + \sin(z)}{(z - \pi)^4}. \]

Answer: \dfrac{1}{6}


Question: Consider a function \( f: \Bbb R^k \supset A \rightarrow \Bbb R^m \) such that for all \( x, y \in A \), the condition \( |f(x) - f(y)| \le |x - y| \) holds. Is it true that such a function can always be extended to a function \( f': \Bbb R^k \rightarrow \Bbb R^m \) that satisfies \( f'_{|A} = f \) and preserves the condition \( |f'(x) - f'(y)| \le |x - y| \) for all \( x, y \in \Bbb R^k \)? Assume the Euclidean metric is used in \( \Bbb R^n \).

Answer: Yes


Question: Evaluate the infinite series: \( \sum_{n=1}^\infty \frac{n^2}{3^n} \).

Answer: \dfrac{3}{2}


Question: Find the values of \(k\) such that the function \(f(x) = x^k |x|\) is three times differentiable at the origin.

Answer: k \geq 3


Question: Calculate the double integral of the function \( f(x, y) = 3x \) over the region bounded above by \( y = x(2-x) \) and below by \( x = y(2-y) \).

Answer: \dfrac{11}{20}


Question:
Determine the number of possible values of positive integers \((m, n, p, q)\) such that the following equations hold:
\[ x = 1^p + 2^p + 3^p + \ldots + m^p \]
\[ x = 1^q + 2^q + 3^q + \ldots + n^q \]
where \(p, q, m, n > 1\) and \(p \neq q\).

Answer: 0


Question: Let $X_k$, $k \geq 0$, be iid random variables with pmf $p_j = P(X_1 = j)$ for $1 \leq j \leq m$, where $\sum_{j = 1}^{m}p_j = 1$. Define $T = \min\{n > 0 : X_n = X_0\}$ as the first time we observe the initial value again. Compute $E[T]$. \(E[T]\) is the expected value of the first time the random variable $X_n$ equals $X_0$ again.

Answer: m


Question: Let $M$ be an oriented, compact, connected $n$-dimensional smooth manifold with boundary. Determine whether the $n$-th singular homology group $H_n(M)$ is zero.

Answer: 0


Question: Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that \( f' \) exists and satisfies the functional equation: \[ f(x) = f\left(\frac{x}{2}\right) + \frac{x}{2} \cdot f'(x), \forall x \in \mathbb{R}. \]

Answer: f(x) = ax + b


Question: Compute the expectation \( \mathbb{E}[\log|X|] \) where \( X \) is uniformly distributed on the unit ball \( \{u \in \mathbb{R}^{2} : |u| \leq 1\} \).

Answer: -\dfrac{1}{2}


Question: Evaluate the limit without using derivatives: \[ \lim\limits_{x\to 0} \left( \frac{a^{\sin x}+b^{\sin x} }{2} \right)^{\frac{1}{x}} \] where \(a > 0\) and \(b > 0\) are constants.

Answer: \sqrt{ab}


Question: Find the limit \( \lim_{n \to \infty}\sqrt[n]{a_n} \) for the sequence defined by \( a_1 = 1 \) and \( a_{n+1} = n \cdot (1 + \ln a_n) \).

Answer: 1


Question: Evaluate the integral \( I = \int_{x=0}^{1} \int_{y=x}^{1} (x^2 + y^2) \, dy \, dx \) using polar coordinates.

Answer: \dfrac{1}{3}


Question: Evaluate the limit using the Riemann integral: \[ \lim_{n \to +\infty} \frac{1}{n}\sqrt[n]{\frac{(2n)!}{n!}} \]

Answer: \dfrac{4}{e}


Question: Given a finite measure \( \mu \) on the measurable space \( (X, \mathscr{F}) \) and a measurable function \( f : X \to [0, +\infty) \), does there exist a continuous function \( g : [0, +\infty) \to [0, +\infty) \) such that \( g(x) \) is increasing and tends to \(+\infty\) as \( x \to +\infty \), and \( \int_X g(f(x)) \, \mu(dx) < +\infty \)? Assume \( \mu(X) = 1 \).

Answer: Yes


Question: Let $f: [0,1] \rightarrow \mathbb{R}$ be a differentiable function with $f(0) = 0$, $f(1) = 1$, and $f'(0) = \frac{1}{2}$. Determine if there exists a point $a \in (0,1)$ such that $f'(a) = \frac{2}{\pi}$. Justify your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left( \frac{(n+1)^{100} + (n+2)^{100} + \cdots + (n+101)^{100}}{n^{99}} - 101n \right) \]

Answer: 515100


Question: Sixty points are marked on a circle, with thirty points colored red, twenty points colored blue, and ten points colored green. These points divide the circle into sixty arcs. Each arc is assigned a number based on the colors of its endpoints: an arc between a red and a green point is assigned a number 1, an arc between a red and a blue point is assigned a number 2, and an arc between a blue and a green point is assigned a number 3. Arcs between two points of the same color are assigned a number 0. What is the greatest possible sum of all the numbers assigned to the arcs?

Answer: 100


Question: Determine the minimum value of \( \frac{a^2+b^2+c^2}{d^2} \) where \(a, b, c, d\) are the sides of a quadrilateral.

Answer: \dfrac{1}{3}


Question: Let \( X \) and \( Y \) be independent standard Gaussian random variables, i.e., \( X, Y \sim N(0,1) \). Define \( Z = \frac{X}{Y} \), which follows a Cauchy distribution. Determine whether the conditional expectation \( E[Z|Y] \) exists and if it equals zero.

Answer: 0


Question:
Given the joint probability density function of $X$ and $Y$:
\[ f_{X,Y}(x,y) = \begin{cases} 2 & \text{if}\ 0 \le y \le x \le 1 \\
 0 & \text{otherwise} \end{cases} \]
find the probability density function of the random variable $Z = \frac{Y}{X}$.

Answer: 1


Question: Let \( f: [0,1] \to \mathbb{R} \) be a continuous function that is differentiable on \((0,1)\) and satisfies \( f(0) = f(1) = 0 \). Determine the number of solutions to the equation \( f(x) = f'(x) \) within the interval \((0,1)\).

Answer: 1


Question: Prove that the following equation holds for all real numbers x: \[ \tan\left(\operatorname{mod}\left(x,\pi\right)-\left(\frac{\pi}{2}\right)\right)=-\cot\left(x\right) \] where \( \operatorname{mod}(x,\pi) \) denotes \( x \) modulo \( \pi \).

Answer: \tan\left(\operatorname{mod}\left(x,\pi\right)-\frac{\pi}{2}\right) = -\cot(x)


Question: Find the general solution of the differential equation \( xy'' + y' = -1 \) with the boundary conditions \( y(0) = 1 \) and \( y(1) = 0 \).

Answer: -x + 1


Question: Given the boundary value problem in the open unit ball $B = \{(x,y) \in \mathbb{R}^{2} : x^2 + y^2 < 1\}$, where $\Delta u(x,y) = 0$ in $B$ and $u(x,y) = \sin(x)$ on $\partial B$, use the mean value theorem to compute $u(0,0)$. Provide your answer as a single value.

Answer: 0


Question:
Given the permutations $A$, $B$, and $C$, where $AXB = C$, find the number of inversions in the permutation $X$. The permutations are defined as follows:

\[
A = \begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 \\
2 & 3 & 1 & 5 & 7 & 6 & 4
\end{pmatrix}, \\
B = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
7 & 6 & 5 & 4 & 3 & 2 & 1
\end{pmatrix}, \\
C = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
4 & 5 & 7 & 6 & 2 & 3 & 1
\end{pmatrix}
\]

Permutations are applied from right to left, meaning $AB$ implies $B$ is applied first. Calculate the number of inversions in $X$. \( \text{Note: An inversion is a pair } (i, j) \text{ such that } i < j \text{ and } X(i) > X(j). \)

Answer: 6


Question: Evaluate the triple integral \( \iiint _{W} zy\, dz\,dy\,dx \), where \( W \) is the volume inside the sphere \( x^{2} +y^{2} +z^{2} =1 \) and the cone \( z^{2} =x^{2} +y^{2} \).

Answer: 0


Question: Determine the computational cost of computing the power of a matrix, $A^k$, where $A \in \mathbb{R}^{n \times n}$ and $k$ is a positive integer.

Answer: O(n^3 \log k)


Question:
A clock has a minute hand of 10 cm long and an hour hand of 7 cm long. The hands move at a constant rate. At what time is the distance between the tips of the hands changing at the greatest rate?

A. 12:00
B. 12:05
C. 12:10
D. 12:15

Answer: C


Question: Determine the multiplicity of the 0 eigenvalue of the Laplacian matrix for a weakly connected directed graph.

Answer: 1


Question: Is it true that for any positive integer $n$, there exists an $n \times n$ real orthogonal matrix with all entries bounded in absolute value by $C/\sqrt{n}$, where $C$ is an absolute constant?

Answer: Yes


Question: Find the remainder when \( m \) is divided by 17, given that \( \frac{1}{1}+\frac{1}{2}+\frac{1}{3} +\dots+\frac{1}{33}=\frac{m}{33!} \).

Answer: 1


Question: Find the scalar function \( f(x, y) \) such that its gradient is given by \( \nabla f = \langle 4y \sin(xy), 4x \sin(xy) \rangle \).

Answer: -4\cos(xy)


Question: Using only a compass, construct the point D such that it is the intersection of line AB and a line perpendicular to AB passing through point C, where C is not on line AB.

Answer: D


Question: Let $k$ be a field, and consider an integral domain over $k$ that is a localization of a finitely generated $k$-algebra at a multiplicative set. Is this integral domain also a localization of an integral domain of finite type over $k$ at a multiplicative set?

Answer: Yes


Question: Does the space $\mathbb{N}^\mathbb{N}$, with $\mathbb{N}$ endowed with the discrete topology, contain a discrete subspace of size $2^{\aleph_0}$?

Answer: No


Question:
Evaluate the following limit:
\[
\lim_{x\to\infty}\left(\lim_{n\to\infty}\frac{\lfloor 1^2(\sin x)^x\rfloor+\lfloor 2^2(\sin x)^x\rfloor+\dots+\lfloor n^2(\sin x)^x\rfloor}{n^3}\right)
\]

Answer: 0


Question: Let $X_n$ be a Gaussian random variable with mean 0 and variance $1/n$. Define $Z_n = \frac{X_n^2}{\log(1+X_n^2)}$. Find $\lim_{n \to \infty} \mathbb{E}[Z_n]$. \( \lim_{n \to \infty} \mathbb{E} \frac{X_{n}^{2}}{\log(1+X_{n}^{2})} \)

Answer: 1


Question: Determine whether every infinite group has a subgroup.

Answer: Yes


Question: In a round-robin tennis tournament with eight players, each player pays an entry fee of $17.50. A player who wins $N$ sets earns $N^2$ dollars. Prove that the total entry fees collected are sufficient to cover the maximum possible prize money distributed.

Answer: 140


Question: Let $X$ be a discrete random variable with the distribution function $P(X=(-1)^k k)=\frac{c}{k^2\log k}$ for integer $k\geq 2$, where $c$ is a normalization constant. Determine whether $xP(|X|>x) \rightarrow 0$ as $x\rightarrow \infty$. Justify your answer.

Answer: 0


Question: Find the remainder when the number formed by concatenating all natural numbers from 1 to 2018 is divided by 180.

Answer: 138


Question: Let $a, b, c$ be non-negative real numbers satisfying $a \le b + c$. Determine the positive real values of $n$ for which the inequality $a^n \le b^n + c^n$ holds.

Answer: (0, 1]


Question:
Let $A$ and $B$ be two $n \times p$ data matrices, where $n$ is the number of samples and $p$ is the number of features. Both $A$ and $B$ are centered, meaning each feature has zero mean. Consider the eigenvalue decompositions of their covariance matrices:
\[ \frac{1}{n-1} A^T A = V_A L_A V^T_A \]
\[ \frac{1}{n-1} B^T B = V_B L_B V^T_B \]
Suppose $A$ and $B$ have the same principal components, i.e.,
\[ A V_A = B V_B \]
Does it follow that $B$ is a rotation of $A$, i.e.,
\[ B = AQ \] for some orthogonal matrix $Q$?

Answer: Yes


Question: Find the volume of the solid in the first octant that is bounded by the cylinder $x^2 + y^2 = 4$, the plane $z = y$, the $xy$ plane, and the $yz$ plane.

Answer: \dfrac{8}{3}


Question: Determine the smallest integer \( n \) such that the function \( g(x) = \ln \ln \ldots \ln (|x| + 3) \), where the natural logarithm is applied \( n \) times, is surjective onto \((-\infty, \infty)\).

Answer: 4


Question: Evaluate the limit: \[ \lim_{m \rightarrow \infty} \frac{Q(m)}{\log(m)} \] where \( Q(m) = \bigl| \{ n : m\leq n \leq m + \log(m) \text{ and } n = p \cdot q\text{, where }p,q\text{ are prime} \} \bigr| \).

Answer: 0


Question: Consider the endomorphism \( \phi: C(\mathbb{R}) \to C(\mathbb{R}) \) defined by \( \phi(f)(x) = x^2 f(x) \). Determine whether this endomorphism has an infinite norm, where the norm of a function \( f \) is defined as \( \|f\| = \sup_{x \in \mathbb{R}} |f(x)| \).

Answer: \infty


Question: Find the minimum value of the function \( a = \frac{{(1+t^2)}^3}{t^4} \) using basic inequalities, without employing calculus.

Answer: \dfrac{27}{4}


Question: Evaluate \( \lim_{n \to \infty} I_n \), where \( I_n = \int_{0}^{1} \frac{f(x)}{1+nx} \, dx \) and \( f(x) \) is an integrable function on \([0,1]\).

Answer: 0


Question: Consider $k$ rooms, each containing $n$ lights. Each light is independently on with probability $p$. As $n$ approaches infinity while $k$ remains fixed, determine the limit of the probability that Room 1 has the maximum number of lights on, possibly sharing this maximum with other rooms.

Answer: \dfrac{1}{k}


Question: A spherical planet has its center at the origin $(0, 0, 0)$ and a radius of $20$. The temperature at any point on the surface of this planet is given by the function $T(x, y, z) = (x + y)^2 + (y - z)^2$ degrees. Calculate the average temperature on the surface of this planet.

Answer: \dfrac{1600}{3}


Question:
Calculate the determinant of the following $n \times n$ matrix, where $n$ is a multiple of $3$:
\[\begin{pmatrix}
0 & 0 & 1 & & & & & & &\\
& & & 0 & 0 & 1 & & & &\\
& & & & & &\ddots\\
& & & & & & & 0 & 0 & 1\\
0 & 1 & 0 & & & & & & &\\
& & & 0 & 1 & 0 & & & &\\
& & & & & &\ddots\\
& & & & & & & 0 & 1 & 0\\
1 & 0 & 0 & & & & & & &\\
& & & 1 & 0 & 0 & & & &\\
& & & & & &\ddots\\
& & & & & & & 1 & 0 & 0
\end{pmatrix}\]

Answer: (-1)^{n/3}


Question: Find the value of a + b + c + d for the equations of the two lines tangent to the parabola \( f(x) = x^2 + 4x + 2 \) that pass through the point (2, -12). The equations of the tangent lines are \( y = ax + b \) and \( y = cx + d \).

Answer: -40


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

Answer: 0


Question: Find the smallest positive integer $k$ such that for any $k$ vectors $v_1, v_2, \ldots, v_k \in \mathbb{R}^{2016}$, there exist real numbers $a_1, a_2, \ldots, a_k$, not all zero, satisfying $a_1v_1 + a_2v_2 + \ldots + a_kv_k = 0$ and $a_1 + a_2 + \ldots + a_k = 0$. Determine $k$. \( \boxed{2017} \)

Answer: 2017


Question: Let \( G \) be a finite \( p \)-group with \( |G| = p^n \) and center \( Z(G) = \langle w \rangle \). Suppose there exists a maximal subgroup of \( G \) containing \( Z(G) \). Does there exist an automorphism \( f \in \operatorname{Aut}(G) \) such that \( f(w) = w^k \) for some integer \( k \not\equiv 1 \pmod{p} \)?

Answer: No


Question: Let $X_1, X_2, \ldots, X_n$ be independent random variables, each with the same distribution, mean $\mu$, and variance $\sigma^2$. Define $\bar{X} = \frac{1}{n}(X_1 + X_2 + \ldots + X_n)$. Show that \[ \mathbb{E}\left(\sum_{i=1}^n (X_i - \bar{X})^2\right) = (n-1)\sigma^2. \]

Answer: (n-1)\sigma^2


Question: Compute the double integral \( \iint_D \frac{1}{(1+(x+2y)^2)^2} \,dx\,dy \) where \( D \) is the region defined by \( x \geq 0 , \, y \geq 0, \, 1 \leq x+2y \leq 2 \). The region \( D \) forms a trapezoid with vertices at \((1,0), \, (2,0), \, (0, 1/2), \, (0,1)\). Use contour lines to transform the double integral into a single integral and evaluate it.

Answer: \dfrac{3}{40}


Question: Given that \( X_1, X_2, \ldots, X_n \) are i.i.d. random variables from a population with the density function \( f(x; \theta) = \frac{2\theta^2}{x^3} \) for \( x \geq \theta \) and 0 elsewhere, find the Maximum Likelihood Estimator (MLE) of \( \theta \). Determine if this estimator is unbiased.

Answer: X_{(1)}


Question: Find the number of positive integer pairs \((p, n)\), where \(p\) is a prime number, such that \[ p^2 = \frac{(2^n + 1)^5 - 1}{2^n} \].

Answer: 1


Question: Find all points on the portion of the plane \(x+y+z=5\) in the first octant where the function \(f(x,y,z)=xy^2z^2\) attains its maximum value.

Answer: (1, 2, 2)


Question:
Let \( f \) be an analytic function on the closed unit disk \( \{z \in \mathbb{C} \mid |z| \leq 1\} \) such that \( |f(z)| \leq 1 \) for all \( z \) in the disk. Determine which of the following is not a possible value of \( (e^{f})^{''}(0) \):

(a) \( 2 \)
(b) \( 6 \)
(c) \( \frac{7}{9}e^{1/9} \)
(d) \( \sqrt{2} + i\sqrt{2} \)

Answer: B


Question: Evaluate the limit: \[ \lim_{n\to\infty} \frac{n}{\ln(n)}\left(n^{1/n}-1\right) \]

Answer: 1


Question: Evaluate the limit: \[ \lim_{n\to\infty} \left(\frac{1}{n+1} + \frac{1}{n+1} + \dots +\frac{1}{2n}\right) \] Choose the correct answer from the following options: \(0\), \(1\), \(2\), \(\log 2\).

Answer: \log 2


Question: Can every finite graph be embedded in \(\mathbb{R}^3\)?

Answer: Yes


Question:
Find the values of \( \lambda \) and \( \mu \) in \( \mathbb{Q} \) such that the greatest common divisor of the polynomials \( 2X^3-7X^2+\lambda X+3 \) and \( X^3-3X^2+ \mu X +3 \) is a polynomial of degree 2. Choose the correct pair from the following options:

(a) \( \lambda = -1, \mu = 2 \)
(b) \( \lambda = 0, \mu = 0 \)
(c) \( \lambda = 2, \mu = 0 \)
(d) \( \lambda = 2, \mu = -1 \)
(e) \( \lambda = -1, \mu = -1 \)
(f) \( \lambda = 0, \mu = 2 \)

Answer: D


Question: Use Stokes' Theorem to transform the surface integral \( \int_S \text{rot}\vec{F}\cdot \vec{dS} \) into a line integral and calculate it for \( \vec{F}(x,y,z)=(y,z,x) \), where \( S \) is the part of the paraboloid \( z=1-x^2-y^2 \) with \( z\ge 0 \) and the normal vector has a non-negative \( z \) component. Assume \( \vec{F}(x,y,z) \) is conservative.

Answer: -\pi


Question: Given the matrix \( A = \begin{bmatrix} a & 1 & -1 \\ 1 & a & -1 \\ 1 & -1 & a \end{bmatrix} \), where \( a \in \mathbb{R} \), if \( A \) is invertible, calculate the value of \( (A)_{11} \times (A^{-1})_{11} \). Assume \( A \) is invertible for \( a \neq 0, 1, 2 \).

Answer: 1


Question: If $H$ is a subgroup of a group $G$ and $N$ is a normal subgroup of $G$, must $N$ be normal in the subgroup $HN$?

Answer: Yes


Question: Compute the line integral \( \int_{C}(x^2+y)dx + (z+x)dy + (x+2y)dz \) where \( C \) is the intersection of the cylinder \( x^2+y^2=4 \) and the plane \( x+y=z \).

Answer: 0


Question: Determine the number of solutions to the quadratic congruence \(2x^2 + 7x - 5 \equiv 0 \pmod{1099}\).

Answer: 0


Question: If a sequence of continuous functions \( \{f_n\} \) converges pointwise to a discontinuous function \( g \), does it imply that \( \{f_n\} \) does not converge uniformly to \( g \)? Answer yes or no.

Answer: Yes


Question: Prove that the following identity holds for all non-negative integers \( n \): \[ \sum_{k=0}^n (-1)^{k-1} k \binom{n}{k} = 0 \]

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to0^+} (\sin x)^{2/\ln x}$$

Answer: e^2


Question: If $D$ is an open, connected set in $\mathbb{C}$, is it true that any uncountable subset of $D$ has a limit point in $D$? Provide a justification for your answer.

Answer: Yes


Question: Calculate the triple integral \( \iiint_D z \, dx \, dy \, dz \) where the region \( D \) is defined by \( x^2 + y^2 = 1 \) and \( 0 \le z \le \sqrt{x^2 + y^2} \). Use cylindrical coordinates to evaluate the integral.

Answer: \dfrac{\pi}{4}


Question: Calculate the flow rate of the vector field \( f(x,y,z) = \left( \frac{-x}{(x^2+y^2+z^2)^{\frac{3}{2}}}, \frac{-y}{(x^2+y^2+z^2)^{\frac{3}{2}}}, \frac{-z}{(x^2+y^2+z^2)^{\frac{3}{2}}} \right) \) out of the surface \( S \), where \( S = \{(x,y,z) \; | \; x^2+y^2+z^2=1 \} \).

Answer: -4\pi


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a continuous function and define \( \Phi(x) = \int_0^x (x-t)f(t) \, dt \). Prove that \( \Phi(x) \) is twice differentiable and find \( \Phi''(x) \).

Answer: f(x)


Question: Determine the interval of absolute convergence for the series \(\sum_{n=1}^\infty \frac{n}{2^n(2n-1)}(x-1)^n\).

Answer: (-1, 3)


Question: Calculate the value of the double summation \( \sum\sum_{1\leq i<j\leq 100} (\alpha_i\alpha_j)^5 \), where \( \alpha_1, \alpha_2, \dots, \alpha_{100} \) are the 100th roots of unity.

Answer: 0


Question: Determine all values of $x$ for which the series \( \sum^\infty_{n=1}\Big(1+\frac{1}{2}+\ldots+\frac{1}{n}\big)\frac{\sin nx}{n} \) converges.

Answer: \mathbb{R}


Question: Find the fundamental group of the orbit space \(\mathbb{C}^*/\Gamma\), where \(\mathbb{C}^* = \mathbb{C} \setminus \{0\}\), and \(\Gamma = \{\varphi^n : \varphi(z) = 4^n z, n \in \mathbb{Z}\}\) acts on \(\mathbb{C}^*\) in the natural way.

Answer: \mathbb{Z} \times \mathbb{Z}


Question: Evaluate the contour integral \( \int_{|z|=2} \frac{|z|e^z}{z^2} \, dz \).

Answer: 4\pi i


Question: Given that $f(z+1) = c f(z)$ for a constant $c$ and a complex function $f$, determine the expression for $f(3(z+1))$ in terms of $f(3z)$. Verify if $f(3(z+1)) = c^3 f(3z)$ is correct.

Answer: f(3(z+1)) = c^3 f(3z)


Question: Evaluate the sum of the series: \[ \sum_{n=0}^\infty \frac{(-1)^{n+1}n}{(2n+1)!} \cdot \pi^{2n} \]

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \[ \lim _{p\to \infty }\left(p\ln \left(e\left(1+\frac{1}{p}\right)^{1-p}\right)\right) \]

Answer: \dfrac{3}{2}


Question: Find the limit \( \lim_{N \to \infty}\frac{\sum\limits_{i=1}^{N}\frac{\log i}{i}}{(\log N)^2}. \)

Answer: \dfrac{1}{2}


Question: Determine the number of subgroups of order 3 in a non-abelian group of order 39.

Answer: 13


Question:
Given a continuous function \( f(x) \) defined on \((0, \infty)\) such that \( \lim_{x \to \infty} f(x) = 1 \), find the limit:
\[ L = \lim_{x \to \infty} \mathrm{e}^{-x} \int_{0}^{x} f(y) \mathrm{e}^{y} \, \mathrm{d}y \]

Answer: 1


Question: Maximize \((a-3)(b-2)(c-1)\) for positive real numbers \(a, b, c\) given that \(a + b + c = 18\).

Answer: 64


Question: Compute the intersection multiplicity of the elliptic curve given by \(x^3 + xz^2 + z^3 + y^2z + yz^2 = 0\) and the curve \(x = 0\) at the point \((0, 1, 1)\). Note: \(f(x, y, z) = x^3 + xz^2 + z^3 + y^2z + yz^2 \in \mathbb{F}_2[x, y, z]\).

Answer: 0


Question: Evaluate the surface integral \( \int_Y xyz \, dS \), where \( Y \) is the part of the sphere \( x^2 + y^2 + z^2 = 1 \) that lies above the region \( \{ y \leq x, x \leq 0, 0 \leq x^2 + y^2 \leq 1 \} \).

Answer: \dfrac{1}{16}


Question: You have 4 unit vectors in 3D space, and you know all 6 angles between them. If you add another unit vector to this set, how many additional angles are needed to uniquely identify this new vector?

Answer: 3


Question: If a subset of \(\mathbb{R}\) is closed and bounded with respect to a metric equivalent to the Euclidean metric, must it be compact? Justify your answer.

Answer: No


Question: Let \(G\) be a torsion-free group with a subgroup \(H\) of finite index, where \(H \cong \mathbb{Z}\). Is \(G\) isomorphic to \(\mathbb{Z}\)?

Answer: Yes


Question: Find the number of 10-digit numbers such that each digit that appears in the number appears exactly twice. Note that the number cannot start with the digit 0.

Answer: 25719120


Question: Given a free abelian group of rank $n$ with two bases $\{x_1, \ldots, x_n\}$ and $\{y_1, \ldots, y_n\}$, show that if the change of basis matrices $A = (a_{ij})$ and $B = (b_{ij})$ satisfy $y_i = \sum_j a_{ij} x_j$ and $x_i = \sum_j b_{ij} y_j$, then the product $AB$ is the identity matrix $I_n$. Prove that $AB = I_n$.

Answer: AB = I_n


Question: Determine whether the vortex vector field \( F(x,y)=\left(\frac{2xy}{(x^2+y^2)^2},\frac{y^2-x^2}{(x^2+y^2)^2}\right) \) is conservative. Consider the application of the extended Green's Theorem for an arbitrary simple closed curve \( C \) that does not pass through the origin and includes a circular hole \( C' \) with radius \( a \) centered at the origin. Given that \( \frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}=0 \) and \( \int_{C'}\vec{F}\cdot d\vec{r}=0 \), does this imply that \( \int_C\vec{F}\cdot d\vec{r}=0 \) and that the vector field is conservative?

Answer: Yes


Question: Calculate the variance of the expression \((X_1 - X_2)^2\), where \(X_1\) and \(X_2\) are independent random variables from a normal distribution \(\mathrm{N}(\mu, \sigma^2)\).

Answer: 8\sigma^4


Question: Find the minimum value of the expression \(1000a + \frac{1}{1000b(a-b)}\) given that \(a > b > 0\).

Answer: 30


Question:
Given a positive sequence \((x_n)\) such that \(\lim_{n\to\infty}\frac{x_n}{n}=0\) and \(\limsup_{n\to\infty} \frac{x_1+x_2+\ldots+x_n}{n}\in\mathbb{R}\), find the limit:
\[ \lim_{n\to\infty} \frac{x^2_1+x^2_2+\ldots+x^2_n}{n^2} \]

Answer: 0


Question: Evaluate the line integral \( \int_\gamma \sin(y-x)\, dx + (2xy + \sin(x-y))\, dy \) along the path \( \gamma: y = \sqrt{x} \) for \( 0 \leq x \leq 1 \).

Answer: \dfrac{1}{2}


Question: In triangle $ABC$, $BC=115$, $AC=127$, and $AB=89$. Let $P$ be a point varying on the circumcircle of triangle $ABC$. Let $M$ and $N$ be the feet of the perpendiculars from $P$ to $AB$ and $AC$, respectively. Find the maximum value of the length of $MN$. Can this problem be solved using pure geometry?

Answer: 115


Question: Consider the $p$-adic Haar measure integration. Let $p$ be a prime number, $V$ be a non-empty subset of $\mathbb{Z}_{p}$, and $d\mu$ be the Haar measure on $\mathbb{Z}_{p}$ with normalization $\int_{\mathbb{Z}_{p}}d\mu=1$. Given $a,b\in\mathbb{Z}_{p}$ with $a\neq0$, and a $d\mu$-integrable function $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}$, define $aV+b=\{ ax+b:x\in V\}$. Is the following change of variables formula correct? Answer yes or no. $$\int_{V}f\left(ax+b\right)d\mu\left(x\right)=\frac{1}{\left|a\right|_{p}}\int_{aV+b}f\left(x\right)d\mu\left(x\right)$$ If it is not correct, provide the correct formula.

Answer: Yes


Question:
Compute the infinity norm \(||B^{-1}||_{\infty}\) for the square real matrix \(B\) defined by:
\[ b_{ii}=1, \quad b_{ij}=-1 \text{ for } i<j, \quad b_{ij}=0 \text{ for } i>j. \]

Given that \(B = I + N\), where \(N\) is a strictly lower triangular matrix, find \(||B^{-1}||_{\infty}\).

Answer: 2^{n-1}


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

Answer: 2


Question:
Given a directed graph $G=(V,E)$ with vertex set $V=\{1,2,...,n\}$, where the weight of each edge $(i,j)$ is denoted by $w(i, j)$, and $w(i,j)= + \infty$ if the edge $(i,j)$ is not present, and $w(i,i)=0$ for each vertex $i$. Using dynamic programming, determine which of the following recursive relations $d[i,j,k]$ correctly computes the shortest path between any two vertices $i$ and $j$ in the graph:

$I)$
$$d[i,j,k]=\begin{cases}w(i,j) & k=1\\ \min_{1 \leq r \leq n} \left\{ d[i,r,k-1] +w(r,j) \right\}& k>1 \end{cases}$$

$II)$
$$d[i,j,k]=\begin{cases}w(i,j) & k=0\\ \min \left\{ {d[i,j,k-1],d[i,k,k-1]+d[k,j,k-1]}\right\}  & k>0 \end{cases}$$

$III)$
$$d[i,j,k]=\begin{cases}w(i,j) & k=1\\ \min_{1 \leq r \leq n} \left\{ {d[i,r,\lfloor k/2\rfloor ]}+d[r,j, \lceil k/2\lceil ] \right\} & k>1 \end{cases}$$

Select the correct recursive relation.

Answer: II


Question: Solve the exponential equation: \[ \frac{e^x}{x} = \frac{2e^{\frac{3}{2}}}{3} \]

Answer: \dfrac{3}{2}


Question: Determine whether the following assertion is true or false: If $K \subset \mathbb{R}$ is a compact set with positive Lebesgue measure, then the interior of $K$ is non-empty.

Answer: B


Question: Let $f: [0,1] \to \mathbb{R}$ be a continuous function and $A$ a finite subset of $[0,1]$. Given $\epsilon > 0$, is there a polynomial $p$ such that \(|f(x) - p(x)| \le \epsilon\) for all $x \in [0,1]$ and $p(a) = f(a)$ for all $a \in A$?

Answer: Yes


Question: Calculate the expression \( f''_{xx} + f''_{yy} \) for the function \( f(x,y) = \ln \sqrt{(x-1)^2 + (y+1)^2} \).

Answer: 0


Question: Does there exist a $4 \times 4$ matrix with entries of 0's and 1's that is row equivalent to the identity matrix modulo 2, but not row equivalent to the identity matrix over the integers?

Answer: Yes


Question: Determine whether the following statement is true or false: A group is commutative if and only if every arc from vertex $a$ to vertex $b$ in its digraph is two-way for all $a, b \in G$. Provide a justification for your answer.

Answer: False


Question: If a prime number $p$ divides the number of elements of order $k$ (where $k \neq p$) in a finite group $G$, does it follow that $p$ divides the order of $G$? Answer 'yes' or 'no' and justify your answer.

Answer: No


Question: Find the volume of the tetrahedron formed by the intersection of the planes given by the equations $x+y=0$, $y+z=0$, $z+x=0$, and $x+y+z-1=0$. Express your answer as a fraction.

Answer: \dfrac{2}{3}


Question: Evaluate the integral \( \int_0^1 \sqrt{\frac{\log(1/t)}{t}} \,\mathrm{d}t \) and verify that its value is \( \sqrt{2\pi} \).

Answer: \sqrt{2\pi}


Question: Evaluate the limit: \[ \lim_{n \to \infty} n \int_0^\pi \{ -x \}^n \, dx \] where \( \{ x \} \) denotes the fractional part of \( x \).

Answer: 4


Question: Given a sheaf $F$ on a topological space $X$ and an open subset $U \subseteq X$, let $F|_U$ denote the restricted sheaf of $F$ to $U$. For any point $y \in U$, is it true that the stalk $F_y$ is equal to the stalk $(F|_U)_y$?

Answer: Yes


Question: Evaluate the complex integral \( \oint_C \frac{\sin(z)}{z(z-\pi/4)} dz \), where \( C \) is a circle of radius \( \pi \) centered at the origin.

Answer: 4\sqrt{2}i


Question: Evaluate the double integral \( \iint_D |x^3 y^3|\, \mathrm{d}x \mathrm{d}y \), where the region \( D \) is defined by \( \{(x,y)\mid x^2+y^2\leq y \} \).

Answer: \dfrac{1}{336}


Question: Given that the moment-generating function (MGF) of a random variable $X$ is $\beta^t e^{t^2}$, where $\beta > 0$, find the probability $\Bbb P (X > \log_e(\beta))$. Assume that $X$ is a normally distributed random variable.

Answer: \dfrac{1}{2}


Question: Find the splitting field $F$ of the polynomial $x^5 + x + 1$ over $\mathbb{Z}_2$, and determine the degree $[F: \mathbb{Z}_2]$. Given that $x^5 + x + 1$ factors as $(x^2 + x + 1)(x^3 + x^2 + 1)$ over $\mathbb{Z}_2$, and both factors are irreducible, calculate the degree of the field extension.

Answer: 6


Question:
Determine the constant \( c \) such that:
\[ \int_{0}^{x} f(t) \, dt = \int_{x}^{1} t^2 f(t) \, dt + \frac{x^8}{8} + \frac{x^6}{6} - \frac{c}{24} \]
where \( f: [0,1] \to \mathbb{R} \) is a continuous function.

Answer: 3


Question: Given three random variables $A$, $B$, and $C$, if $\operatorname{Cov}(A,B) \geq 0$ and $\operatorname{Cov}(B,C) \geq 0$, is it necessarily true that $\operatorname{Cov}(A,C) \geq 0$? Answer true or false.

Answer: False


Question: Let \( E \subset \mathbb{R} \) be a set of positive Lebesgue measure. Determine if there exists \( l > 0 \) such that the intersection \( \bigcap_{-l \leq t \leq l} (t+E) \) is a set of positive Lebesgue measure.

Answer: No


Question:
Given that $Y_1, Y_2, Y_3, Y_4, Y_5, Y_6$ are independent normal random variables with mean $0$ and variance $\sigma^2$, find the distribution of the following ratio:
\[ T = \frac{Y_1^2 + Y_2^2 + Y_3^2}{Y_4^2 + Y_5^2 + Y_6^2} \]

Answer: F(3,3)


Question: Determine whether the following statement is true or false: A matrix with entries in an Euclidean Domain is invertible if and only if the matrix can be row reduced to the identity matrix.

Answer: A


Question: Let $X$ be a non-empty set. Consider the power set $\mathcal{P}(X)$. Define an edge between $a, b \in \mathcal{P}(X)$ if their symmetric difference is a singleton, i.e., $\textrm{card}((a\setminus b) \cup (b\setminus a)) = 1$. For a finite set $X$, the graph has chromatic number 2. Determine if the graph still has chromatic number 2 when $X$ is infinite.

Answer: 2


Question: Suppose $A$ and $B$ are two elements of $M_n(\mathbb{R})$ such that their characteristic polynomials are equal. If $A=C^2$ for some $C\in M_n(\mathbb{R})$, then $B=D^2$ for some $D\in M_n(\mathbb{R})$. Is this statement true or false?

Answer: False


Question: Solve for the permutation $x$ in the symmetric group $S_4$ such that $$(1\ 3\ 4)x(3\ 2)=(1\ 2\ 3\ 4).$$

Answer: (1\ 2\ 3)


Question: Evaluate the limit \( \lim_{n \to \infty} \int_{[0,n]} \left(\frac{n+x}{n+2x}\right)^n \, d\lambda \).

Answer: 1


Question: If a topological space $X$ satisfies $X \times X \simeq \mathbb{R}^{2}$, must $X$ be homeomorphic to $\mathbb{R}$? Assume no prior conditions on $X$ being a manifold.

Answer: Yes


Question: Evaluate the one-sided limit: \[ \lim_{x\rightarrow \pi/2^+} \frac{\sqrt{1+\cos 2x}}{\sqrt{\pi}-\sqrt{2x}} \] without using L'Hôpital's rule.

Answer: -\sqrt{2\pi}


Question: Compute the limit: \[ \lim_{n \to \infty} \frac{1}{n^{2013}} \sum_{k=1}^n k^{2012} \]

Answer: \dfrac{1}{2013}


Question: Find the number of matrices \( A \) in \( F_4^{5 \times 3} \) such that \( \text{rank}_{F_4}(A) = 3 \).

Answer: 1051807680


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

Answer: \ln 2


Question: Find the limit: \[ \lim_{(x, y)\to (0, 0)} \frac{x^5 - y^7}{x^4 + y^4}. \]

Answer: 0


Question: Is it true that for every positive integer $k$, there exist distinct natural numbers $x_1, x_2, \ldots, x_k$ such that $\phi(x_1) = \phi(x_2) = \cdots = \phi(x_k)$, where $\phi$ is Euler's totient function?

Answer: Yes


Question: Find the function $f(x)$ that satisfies the equation \( \int_0^x f(t)\,dt = \int_x^1 t^2f(t)dt + \frac{x^{16}}{8} + \frac{x^{18}}{9} + c \), where $f$ is defined and continuous for all real $x$.

Answer: 2x^{15}


Question: Determine the number of group homomorphisms from the symmetric group $S_3$ to the alternating group $A_3$. $S_3$ has order 6, and $A_3$ has order 3.

Answer: 1


Question: Find the function \( g(x) \) such that the differential equation \((x+y)\text{d}x + g(x)\text{d}y = 0\) has an integrating factor of \(x\).

Answer: \dfrac{x}{2}


Question: Find the value of the expression \( \cot^{-1}(\tan 2x) + \cot^{-1}(-\tan 3x) \).

Answer: \pi + x


Question: Find the limit \( \lim _{ x \to 0} \frac{\sqrt[3]{1+6x+3x^2+3x^3+3x^4}-\sqrt[4]{1+8x+4x^2+4x^3-2x^4}}{6x^2} \).

Answer: \dfrac{1}{3}


Question: Given a prime number $p$, if $p^m = p^n + 2 \cdot p^k$, prove that $p = 3$.

Answer: 3


Question: Let $M$ be a smooth compact manifold, and let $p \in M$. Given a fixed Riemannian metric $g_0$ on $M$, determine whether there exists a constant $C > 0$ such that for any Riemannian metric $g \ge g_0$, the volume of the geodesic ball with respect to $g$ satisfies $\text{Vol}_g(B_g(1,p)) \ge C$. Provide a justification for your answer.

Answer: Yes


Question: Determine the largest possible cardinality of a set of mutually disjoint perfect matchings in the complete graph $K_\omega$, where $K_\omega$ is the complete graph on an infinite set of vertices.

Answer: \omega


Question: Calculate the value of the infinite sum: \[ 1 + \frac{4}{7} + \frac{9}{7^{2}} + \frac{16}{7^{3}} + \frac{25}{7^{4}} + \ldots \]

Answer: \dfrac{49}{27}


Question:
Find the index of the subgroup \( H \) in \( \mathrm{GL}_2(\mathbb{Z}) \), where \( H \) is defined as:
\[
H = \left\{\begin{bmatrix} a & b \\\ c & d \end{bmatrix} \mid a\equiv 1 \pmod{3},\ b\equiv 0 \pmod{3},\ c\equiv 0 \pmod{3},\ d\equiv 1 \pmod{3} \right\}.
\]

Answer: 48


Question:
A biquadratic function \( y = ax^4 + bx^3 + cx^2 + dx + e \) (where \( a \neq 0 \)) touches the line \( y = px + q \) at points \( \alpha \) and \( \beta \) (with \( \alpha < \beta \)). The area of the region bounded by the graphs of the biquadratic function and the line is given by \( \frac{a(\alpha - \beta)^5}{\lambda} \). Find the value of \( \lambda \).

At the points \( \alpha \) and \( \beta \), the slope of the line \( p \) is equal to the derivative of the biquadratic function, i.e., \( 4a\alpha^3 + 3b\alpha^2 + 2c\alpha + d = 4a\beta^3 + 3b\beta^2 + 2c\beta + d \).

Answer: 30


Question: Let \( S \) be the surface defined by \( \{(x,y,z) : x^{2} + y^{2} + 2z = 2, z \geq 0 \} \). Given the vector field \( \mathbf{F} = (y, xz, x^{2} + y^{2}) \), find the net flux of \( \mathbf{F} \) through the surface \( S \). Assume \( \mathbf{n} \) is the outward normal vector.

Answer: 2\pi


Question: Evaluate the lambda calculus expression: \[ (\;(\lambda f.\lambda x. f(f(f(x))))\;(\lambda g.\lambda y.g(g(y))) \;)(\lambda x.x+1)(0) \]

Answer: 8


Question:
Let \( \sum_{n=1}^\infty a_n, \ a_n > 0 \) be a convergent series. Consider the following statements:

- **P:** The series \( \sum_{n=1}^\infty \sqrt{a_n} \) is always convergent.
- **Q:** The series \( \sum_{n=1}^\infty \frac{a_1 + a_2 + \ldots + a_n}{n} \) is always divergent.

Determine which of the following is true:

- Both P and Q are true.
- P is true but Q is false.
- Both P and Q are false.
- P is false and Q is true.

Answer: D


Question: Let $E/\mathbb{Q}$ be a finite dimensional field extension of $\mathbb{Q}$ with $\dim E/ \mathbb{Q} = 6$. Suppose $\mathbb{Q} \subset F_1 \subset E$ and $\mathbb{Q} \subset F_2 \subset E$ are subextensions with $\dim F_1 / E = 2$ and $\dim F_2 / E = 3$. Let $z_1 \in F_1$ and $z_2 \in F_2$ with $z_1, z_2 \not\in \mathbb{Q}$. Must there exist $c_1, c_2 \in \mathbb{Q}$ such that $\mathbb{Q}(c_1 z_1 + c_2 z_2) = E$?

Answer: Yes


Question: Does the forgetful functor from the category of topological groups, \( \mathbf{TopGrp} \), to the category of topological spaces, \( \mathbf{Top} \), admit a left adjoint? Specifically, for an arbitrary topological space \( X \), can we find a topological group \( \Gamma X \) and a continuous map \( \iota:X\to\Gamma X \) such that for any topological group \( G \) and any continuous map \( f:X\to G \), there exists a unique continuous group homomorphism \( f^\prime:\Gamma X\to G \) satisfying \( f=f^\prime\circ\iota \)?

Answer: Yes


Question:
Let \( p \) and \( q \) be distinct prime numbers, and let \( a \) and \( b \) be integers such that \( a^e \equiv a \pmod{p} \) and \( b^e \equiv b \pmod{q} \). By the Chinese Remainder Theorem, there exists a unique integer \( x \) modulo \( pq \) such that:
\[ x \equiv a \pmod{p} \]
\[ x \equiv b \pmod{q} \]
Calculate \( x^e \mod pq \).

Answer: x


Question: Determine the values of $x$ for which the series \( \sum_{n=2}^{\infty} \frac{x^{n^2}}{n \log(n)} \) converges.

Answer: [-1, 1)


Question: Does the sandwich theorem apply to divergent sequences? Specifically, given a divergent sequence \( \{v_n\} \), can we find two sequences \( \{u_n\} \) and \( \{w_n\} \) such that \( \{u_n\} < \{v_n\} < \{w_n\} \), and if both \( \{u_n\} \) and \( \{w_n\} \) diverge to \( L = \pm \infty \), can we conclude that \( \{v_n\} \) also diverges to \( L = \pm \infty \)?

Answer: Yes


Question: Determine the number of points within the interval \([0, 6\pi]\) where the function \( f(x) \) is continuous, given that \( f(x) = \tan x \) for rational \( x \) and \( f(x) = x^2 + 1 \) for irrational \( x \).

Answer: 6


Question: Evaluate the integral \( \int_0^\infty \int_0^t e^{-t}\frac{\sin(\tau)}{\tau}d\tau \, dt \) using the Laplace transform.

Answer: \dfrac{\pi}{4}


Question: Determine the values of \(a \in \mathbb{R}\) for which the function \(f_a(x,y) = (x^2 + y^2)^a\) is Lebesgue-integrable over the set \(\mathbb{R}^2 \setminus E\), where \(E = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 \le 1, y \le \sqrt{|x|} \}\).

Answer: (-\frac{3}{2}, -1)


Question: Determine if there exists a function \( f(x) \) such that \( f(xy) = f(x) + f(y) \) for all positive integers \( x \) and \( y \), \( f(x) \geq 0 \) for any positive integer \( x \), \( f(x) = 0 \) if \( x \) has the digit '6' in the one's place, and \( f(1920) = 420 \).

Answer: Yes


Question:
Find all pairs of prime integers \((p, q)\) that satisfy the Diophantine equation:
\[ p^2 + pq + 275p + 10q = 2008. \]

Answer: (7, 2)


Question: Find the directional derivative of the function \( f(x,y) = \begin{cases}\frac{x^3+y^3}{x^2-y^2},\ x^2-y^2\neq 0 \\\ \,\ \,\ 0 \,\ \,\ ,x^2-y^2=0\end{cases} \) at the point \((0,0)\) in the direction of the vector \(\langle\frac{4}{5},\frac{3}{5} \rangle\).

Answer: \dfrac{13}{5}


Question:
Let $X$ be a random variable with the moment generating function (MGF) given by:
\[ M_{X}(t) = \frac{6}{\pi^{2}}\sum_{n\ge1}\frac{e^{\frac{t^2}{2n}}}{n^2}, \; t \in \mathbb{R} \]
Determine the probability that $X$ belongs to the set of rational numbers, $Q$. Choose the correct option:

A) 0

B) 1/4

C) 1/2

D) 3/4

Provide the correct answer.

Answer: A


Question:
Which of the following sets is not a null set as defined by the given criteria? A set $X \subseteq \mathbb{R}$ is a null set if for every $\epsilon > 0$, there exists a countable collection $\{(a_k, b_k)\}_{k=1}^\infty$ of open intervals such that $X \subseteq \cup_{k=1}^\infty (a_k, b_k)$ and $\sum_{k=1}^\infty (b_k - a_k) \le \epsilon$. Choose the set that is not a null set:

- Every finite set
- $\mathbb{Q}^c$, the set of irrational numbers
- $\mathbb{Q}$, the set of rational numbers
- $\mathbb{N}$, the set of natural numbers

Answer: B


Question: Compute the integral \( \int_\gamma \frac{e^{2z} + \sin z}{z - \pi} \,\text{d}z \) where \(\gamma\) is the circle \(|z-2|=2\) traversed counter-clockwise, using Cauchy's integral formula.

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


Question: Consider a circular arc with radius 1 that subtends an angle of \( x \) radians. The center of the circle is \( O \), and point \( C \) is the intersection of two tangent lines at points \( A \) and \( B \). Let \( T(x) \) be the area of triangle \( ABC \). Find \( \lim_{x \to 0} \frac{T(x)}{x^3} \).

Answer: \dfrac{1}{8}


Question: Determine the values of \( p \) for which the limit \( \lim \limits_{(x,y) \to (0,0)} \frac{|x||y|^{p}}{x^2+y^2} \) exists.

Answer: p > 1


Question: Let $E$ be a Lebesgue measurable set on $\mathbb{R}^n$. Does there exist a sequence of open sets $G_m$ and compact sets $K_m$ such that $K_m \subset E \subset G_m$ and $\mu((\cap G_m) \setminus (\cup K_m)) = 0$?

Answer: Yes


Question: Determine whether the sequence \( \{S_n\} \) is convergent, where \[ S_n = e^{-n} \sum_{k=0}^n \frac{n^k}{k!}. \]

Answer: \dfrac{1}{2}


Question: Derive the variance of the Wiener increment \( \Delta W(t) = W(t + \Delta t) - W(t) \) for a Wiener process \( W(t) \), given that \( \langle W(t) \rangle = 0 \) and \( \langle W(t)^2 \rangle = t \). Verify that the variance \( \langle (\Delta W)^2 \rangle \) is \( \Delta t \).

Answer: \Delta t


Question: Evaluate the limit: \(\lim_{x\rightarrow\pm\infty}\frac{e^{2/x}-1}{\ln\frac{2x+1}{2x}}\).

Answer: 4


Question: Find the function \( f(x) \) that satisfies the functional equation \( f(x+y) = f(x)f(y) \) for all real numbers \( x \) and \( y \), given that \( f'(0) = 2 \).

Answer: e^{2x}


Question: Given the numbers 1 through 9, determine the number of permutations of a five-number sequence where each number can be used at most twice. For example, sequences like 12345 and 15233 are valid, but sequences like 14333 and 10101 are not. What is the total number of such permutations?

Answer: 52920


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

Answer: \ln 2


Question: Calculate the integral \( \int_\gamma f(z) \, dz \) where \( f(z) = \frac{z}{z^2 -1} \) and \( \gamma(t) = i + e^{it} \) for \( t \in [0, 2\pi] \), using the Cauchy integral formula.

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\to2}\frac{\sqrt[3]{x^2-x-1}-\sqrt{x^2-3x+3}}{x^3-8} \] without using L'Hôpital's rule.

Answer: \dfrac{1}{24}


Question:
Find the eccentricity of the conic given by the equation:
\[ \left(x\tan 10^\circ+y\tan 20^\circ+\tan 30^\circ\right)\left(x\tan 120^\circ+y\tan 220^\circ+\tan 320^\circ\right)+2018=0 \]

Answer: \sqrt{2}


Question: Find the Fisher Information of a random variable $X$ that follows a Poisson distribution with parameter $\mu$, i.e., $X \sim \text{Poisson}(\mu)$. The probability mass function is given by $P(X = i) = \exp(-\mu) \mu^i/i!$ for $i \in \{0,1,2,\ldots\}$. Calculate the Fisher Information of $X$ with respect to $\mu$. \[ I(\mu) = \mathbb{E}\left[ \left( \frac{\partial}{\partial \mu} \ln P(X = i) \right)^2 \right] \]

Answer: \dfrac{1}{\mu}


Question:
For which real number $z$ are there positive integers $a$, $b$, and $c$ that satisfy the following system of equations?

1. $a + b + c = 57$
2. $a^2 + b^2 - c^2 = z$
3. $z \cdot c = 2017$

Answer: 2017


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{x - \ln(x+1)}{x \sin(2x)} \] without using L'Hopital's rule.

Answer: \dfrac{1}{4}


Question: In a game played inside the unit disc $\mathcal{D}$, player $\mathcal{A}$ chooses a point $p_0 \in \mathcal{D}$. At each step $n$, player $\mathcal{B}$ draws a line $\ell_n$ through $p_{n-1}$, and $\mathcal{A}$ chooses a point $p_n \in \mathcal{D} \cap \ell_n$. Player $\mathcal{B}$ wins if they can force the sequence $(p_n)$ to converge; otherwise, $\mathcal{A}$ wins. If both players play randomly, what is the probability that $\mathcal{B}$ wins?

Answer: 0


Question: Identify the generator of the n-th homology group of $\mathbb{R}P^n$ for odd $n$, given that $H_n(\mathbb{R}P^n) = \mathbb{Z}$.

Answer: [ \mathbb{R}P^n ]


Question: Evaluate the value of the following double summation: \[ \sum_{r=0}^{2019} \sum_{k=0}^{r} (-1)^k (k+1)(k+2) \binom{2021}{r-k} \]

Answer: 2^{2019}


Question: Consider the limit \( \lim_{x\to 0} \frac{f_1(x)-f_2(x)}{f_3(x)} \), where \( f_1(x) \) and \( f_2(x) \) are transcendental functions such that \( \lim_{x\to 0} f_1(x)-f_2(x)=0 \), and \( f_3(x) \) is a polynomial function with \( f_3(0)=0 \). Can this limit be a rational number?

Answer: Yes


Question: Evaluate the integral \( \int_C \nabla(r^4) \cdot \hat{n} \, ds \) in terms of the moments of inertia \( a \) and \( b \) about the \( x \) and \( y \) axes, respectively. Here, \( C \) is a closed plane curve, \( \hat{n} \) is the unit outward normal vector, and \( r = \sqrt{x^2 + y^2} \). Express your answer in the form \( 16(a + b) \).

Answer: 16(a + b)


Question: Does there exist a simple graph, called "prime," which has no non-trivial automorphisms, meaning it only has the identity automorphism?

Answer: Yes


Question: Find the limit \( \mathcal{T} = \lim\limits_{n\to\infty} nt_n \), where \( t_n = \frac{1}{2n+1} - \frac{1}{2n+2} + \frac{1}{2n+3} - \frac{1}{2n+4} + \ldots + \frac{1}{4n-1} - \frac{1}{4n} \). Relate this limit to the Riemann sums for the function \((1+x)^{-2}\).

Answer: \dfrac{1}{8}


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

Answer: -\dfrac{1}{3}


Question:
Determine the correct option(s) for the values of $a$ and $L$ that satisfy the equation:
\[
\frac{\int_{0}^{4\pi}e^t(\sin^6at+\cos^4at)\,dt}{\int_{0}^{\pi}e^t(\sin^6at+\cos^4at)\,dt}=L.
\]
Options:
\[\begin{align*}
\text{A}. & \quad a = 2, \quad L=\frac{e^{4\pi}-1}{e^\pi-1} \\
\text{B}. & \quad a = 2, \quad L=\frac{e^{4\pi}+1}{e^\pi+1} \\
\text{C}. & \quad a = 4, \quad L=\frac{e^{4\pi}-1}{e^\pi-1} \\
\text{D}. & \quad a = 4, \quad L=\frac{e^{4\pi}+1}{e^\pi+1}
\end{align*}\]

Answer: AC


Question: Determine whether there exist $2^{\aleph_0}$ pairwise non-isomorphic torsion-free countable groups.

Answer: Yes


Question: Evaluate the limit \( \lim_{n \to \infty} \frac{[x] + [2x] + [3x] + \ldots + [nx]}{n^2} \), where \([\cdot]\) denotes the greatest integer function.

Answer: \dfrac{x}{2}


Question: Let \( p(x) = x^2 + bx + c \), where \( b \) and \( c \) are integers. If \( p(x) \) is a factor of both \( x^4 + 6x^2 + 25 \) and \( 3x^4 + 4x^2 + 28x + 5 \), find \( p(1) \).

Answer: 4


Question: Evaluate the double integral \( \iint \frac{(x^2+y^2)^2}{x^2y^2} \, dx \, dy \) over the region common to the circles \( x^2+y^2=7x \) and \( x^2+y^2=11y \).

Answer: 77


Question: Suppose all solutions \( u(x,t) \) to the partial differential equation \( au_x + bu_t = 0 \) satisfy the condition \( u(1,2) = u(3,6) \). Determine the value of \( \frac{b}{a} \).

Answer: 2


Question: Evaluate the limit: \[ \lim_{x \to -\infty} \frac{x}{(\sqrt{x^2} + \sqrt{x^4 + 1}) + \sqrt{2} (\sqrt{x^4 + 1} + x^2)} \]

Answer: 0


Question: Can the Banach space \( C([0,1]) \), consisting of continuous real-valued functions on the interval \([0,1]\) with the supremum norm, be expressed as a countable union of compact sets \( K_1 \subset K_2 \subset K_3 \subset \dots \)?

Answer: No


Question:
Let \( f: \mathbb{R} \to \mathbb{R} \) be a function such that:
\[ \lim_{x \to 1} \frac{f^2(x) + f(x) - 6}{x-1} = 5 \]
If \[ \lim_{x \to 1} \frac{f(x) - 2x^2}{x-1} = a \] where \( a \in \mathbb{R} \), find the value of \( a \) without using derivatives.

Answer: -3


Question: Let $G$ be a finite group and $H$ a subgroup of $G$ with index $n$. Prove that there exists a normal subgroup $K$ of $G$ such that $K \subset H$ and the order of $K$ divides $n!$.

Answer: K


Question: Evaluate the limit: $$\lim_{x\to0}\frac{x\sin{(\sin{x})}-\sin^2{x}}{x^6}$$

Answer: \dfrac{1}{18}


Question:
Find the limit of the sequence as \( n \to \infty \):
\[ \lim_{n \to \infty} \left(1^2 + 2^2 + 3^2 + \ldots + n^2\right)^{\frac{1}{n}} \]

Answer: 1


Question: Determine the limit \( \lim_{\varepsilon \searrow 0} \varepsilon \mathbf{E}(|X|^{1-\varepsilon}) \), where \( X \) is a Cauchy random variable with density \( f(x) = \frac{1}{\pi} \frac{1}{1+x^2} \).

Answer: \dfrac{2}{\pi}


Question: Let \(X_1, X_2,\) and \(X_3\) be a random sample from a \(N(3,12)\) distribution. Define the sample mean \(\bar{X} = \frac{1}{3} \sum_{i=1}^{3} X_i\) and the sample variance \(S^2 = \frac{1}{2} \sum_{i=1}^{3} (X_i - \bar{X})^2\). Calculate the probability \(P(1.65 < \bar{X} \leq 4.35, 0.12 < S^2 \leq 55.26)\). Choose the correct answer from the following options: \((A) = 0.49\), \((B) = 0.50\), \((C) = 0.98\), \((D) = \text{None of the above}\).

Answer: A


Question: Consider the semi-infinite chain complexes \( \cdots \rightarrow A_i \rightarrow A_{i+1} \rightarrow \cdots \) and \( \cdots \rightarrow B_i \rightarrow B_{i+1} \rightarrow \cdots \) over an additive category, where \( A_i = B_i = 0 \) for \( i > 0 \). Let \( f: A \rightarrow B \) be a chain map such that, for every \( k \geq 0 \), \( f \) is chain homotopic to a chain map which is zero in degrees \( -k, \ldots, -1, 0 \). Is such a map \( f \) always null-homotopic?

Answer: Yes


Question: Let $X$ and $Y$ be independent random variables, each following a standard normal distribution, $N(0,1)$. Calculate the probability $P(|Y| \ge \frac{1}{\sqrt{3}}|X|)$. Provide your answer as a single probability value.

Answer: \dfrac{2}{3}


Question: Given that \( f(x), f'(x), xf(x) \in L^{2}(\mathbb{R}) \), prove that \( \lim_{|x|\rightarrow\infty}x|f(x)|^{2}=0 \).

Answer: 0


Question: Suppose \(f:\mathbb{R} \to [-8,8]\) is an onto function defined by \(f(x) = \dfrac{bx}{(a-3)x^3 + x^2 + 4}\) where \(a,b \in \mathbb{R}^+\). Determine the value of \(a+b+p+q\) if the set of all values of \(m\) for which the equation \(f(x) = mx\) has three distinct real solutions in the open interval \((p,q)\).

Answer: 43


Question: Find the least possible value of \( m \) such that the function \( f(x) = 2x^2 + 3x - m\log x \) is monotonically decreasing on the interval \((0,1)\).

Answer: 7


Question: Let $H$ be a Hilbert space, and let $y$ and $z$ be fixed vectors in $H$. Define the operator $T: H \to H$ by $T(x) = \langle x, z \rangle y$ for all $x \in H$. Calculate the operator norm $\|T\|$. Provide your answer as a single value.

Answer: \|y\| \|z\|


Question: Find the sum of the fifth powers of the roots of the polynomial equation \(x^3 + qx + r = 0\).

Answer: 5qr


Question: In a C*-algebra $A$, consider an arbitrary cyclic representation $\pi: A \rightarrow B(H)$. If $\pi(a) \geq 0$ for all $a \in A$, can we conclude that $a \geq 0$ in $A$?

Answer: No


Question: Calculate the expected value of the reciprocal of a random number selected uniformly from the interval (0, 1).

Answer: \infty


Question:
Find the coefficient of $x^{15}$ in the expansion of the product $(1-x)(1-2x)(1-2^2x)(1-2^3x)\cdots (1-2^{15}x)$. Choose the correct answer from the following options:

(a) $2^{105}-2^{121}$

(b) $2^{121}-2^{105}$

(c) $2^{120}-2^{104}$

(d) $2^{110}-2^{108}$

Answer: A


Question: Let $A$, $B$, and $C$ be strictly upper triangular matrices of order $n$ over a finite field $\mathbb{F}_q$. Determine if it is possible for $AC = CB$ while $AC \neq 0$ and $CB \neq 0$. Provide a justification for your answer.

Answer: Yes


Question:
Given the velocity components for a flow field:
\[ u = 16x^2+y, \quad v = 10, \quad w = yz^2 \]
and a rectangular region \( R \) in the \( xy \)-plane formed by the points \((0,0)\), \((10,0)\), \((10,5)\), \((0,5)\), determine the circulation for this flow field around the border of \( R \) (traveling counter-clockwise).

Answer: -50


Question: Determine the minimum number of terms required in the Maclaurin series to approximate the square root of $e$ with an error less than $10^{-3}$. Use the inequality \( \left| \frac{e^c}{(n+1)!} \cdot 0.5^{(n+1)} \right| \leq 10^{-3} \), where $0 < c < 0.5$, to find your answer.

Answer: 5


Question: Let \( k \) be a real closed field. Determine if the group of square classes \( k^*/k^{*2} \) is isomorphic to \( \mu_2 \).

Answer: \mu_2


Question: Find a basis for the vector space of $\mathbb{C}G$-homomorphisms $\operatorname{Hom}_{\mathbb{C}G}(V_1,V_2)$, where $G=C_4=\langle g\mid g^4=1\rangle$, and $V_1,V_2$ are $\mathbb{C}G$-modules with bases $\mathcal{B}_1=\{v_1,v_2\}$ and $\mathcal{B}_2=\{w_1,w_2\}$, respectively. The action of $G$ on $V_1$ and $V_2$ is given by $[g]_{V_1}=\begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}$ and $[g]_{V_2}=\begin{pmatrix}-3 & 8\\ -1 & 3\end{pmatrix}$. Determine if the set of homomorphisms is trivial or find a non-zero basis.


Question: Calculate the limit \( \lim\limits_{n \to \infty} {(\sin n)^{\frac{1}{n}}} \).

Answer: 1


Question:
Consider the polynomial \(P(x) = (x^2 + x + 1)^{2015} + x + 1\) with roots \(x_k\) for \(1 \le k \le 4030\). Evaluate the sum:
\[ \sum_{k=1}^{4030} \frac{1}{x_k} \]

Answer: -1008


Question: Let $X$ have the uniform distribution $\mathcal{U}(0,1)$, and let the conditional distribution of $Y$, given $X=x$, be $\mathcal{U}(0,x)$. Find the probability $P(X+Y \geq 1)$. Provide your answer as a single probability value.

Answer: 1 - \ln 2


Question: Find the remainder when \((n^3 - n)(5^{8n+4} + 3^{4n+2}) - 2017\) is divided by 3804 for all integers \(n \geq 2\). What is the value of the remainder \(R\)?

Answer: 1787


Question: Let $G$ be a connected graph with 100 vertices, where each vertex has a minimum degree of 10. Prove that $G$ contains a path with at least 21 vertices.

Answer: 21


Question: Evaluate the limit: \[ \lim_{x\to0^+}\dfrac { -1+\sqrt { \tan(x)-\sin(x)+\sqrt { \tan(x)-\sin(x)+\sqrt { \tan(x)-\sin(x) } +\cdots } } }{ -1+\sqrt { { x }^{ 3 }+\sqrt { { x }^{ 3 }+\sqrt { x^{ 3 } } +\cdots } } } \]

Answer: \dfrac{1}{2}


Question: Verify the Divergence Theorem for the vector field \( \mathbf{F}(x,y,z) = x^2\hat{\imath} + z\hat{\jmath} \) over the region \( D \), where \( D \) is the upper half-ball defined by \( x^2 + y^2 + z^2 \leq 1, z \geq 0 \). Calculate both the surface integral \( \iint \mathbf{F} \cdot \mathbf{n} \, dS \) and the volume integral \( \iiint \nabla \cdot \mathbf{F} \, dV \) and verify that they are equal.

Answer: 0


Question: Determine the group generated by the union of the subgroups $W = \{z \in \mathbb{C} \mid |z|=1\}$ and $\mathbb{R}_0^+$ within the multiplicative group of non-zero complex numbers $(\mathbb{C}_0, \cdot)$. Is this group equal to $\mathbb{C}_0$?

Answer: \mathbb{C}^\times


Question: Determine the radius of convergence of the power series \( \sum_{n=1}^{\infty} \frac{n^{n!}}{(n^n)!} x^n \). Use the hint: \( \forall n \in \mathbb{N}, 3^n (n!) > n^n \).

Answer: \infty


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

Answer: \dfrac{1}{2}


Question: In the ring $\mathbb{Z}[\sqrt{-2}]$, determine whether the ideal $\langle 14, \sqrt{-2} \rangle$ is the whole ring or a proper principal ideal. If it is a proper principal ideal, find the element that generates it.

Answer: \sqrt{-2}


Question: Using the Chinese Remainder Theorem, determine the number of solutions to the congruence $b^{1104} \equiv 1 \pmod{5 \times 13 \times 17}$, given that $\gcd(b, 1105) = 1$.

Answer: 768


Question: Let $M$ be an oriented closed manifold and $G$ a group. Let $x$ be a class in $H^*(M;\mathbb{Q})$. Suppose that for every class $y \in H^*(BG,\mathbb{Q})$ and every continuous map $f\colon M\to BG$, we have that $\langle x\cup f^*y,[M]\rangle=0$, where $[M]$ is the fundamental class in $H_*(M,\mathbb{Q})$. Does it follow that $x=0$?

Answer: No


Question: Let $p$ be an odd prime, and let $E/\mathbb{Q}_p$ be a finite extension that contains a primitive $p$-th root of unity $\zeta_p$ but not a primitive $p^2$-th root of unity $\zeta_{p^2}$. Let $a,b \in E^*$ such that $[E : \mathbb{Q}_p(a,b,\zeta_p)] = p^m$ for some $m \geq 1$. Determine if the Hilbert symbol $(a,b)_p$ calculated over $E$ is necessarily equal to $1$. Provide a justification for your answer.

Answer: 1


Question: Determine the number of integer solutions for the equation \( \sum_{i=1}^n i a_i = n \), where \( a_i \in \{0, \dotsc, n\} \).

Answer: p(n)


Question: Given that \(x^{x^4} = 4\), find the value of \(x^{x^8} + x^{x^2}\).

Answer: 258


Question: Determine whether the maximum rotation effect of multiplying a vector by a positive definite matrix is less than 90 degrees.

Answer: Yes


Question: Determine the number of unique Tsuro tiles that can exist. Each tile has 8 entry points, and each point is connected to exactly one other point. Consider the uniqueness of tiles with respect to rotations.

Answer: 35


Question: Let $V$ be a finite-dimensional unitary vector space, and let $T: V \to V$ be a positive semidefinite linear transformation. Suppose there exists an integer $k \geq 1$ such that $T^k = I$. Must $T$ be the identity operator? Justify your answer.

Answer: Yes


Question: Find the differentiable function \( f : \mathbb{R} \to \mathbb{R} \) that satisfies the equation \( f\left(\frac{x+y}{3}\right) = \frac{2 + f(x) + f(y)}{3} \) for all real numbers \( x \) and \( y \), given that \( f'(0) = 2 \).

Answer: 2x + 2


Question: Compute the limit: \( \lim_{n\to \infty} n\sin(2\pi n! e) \).

Answer: 2\pi


Question: Let $A, B, C \in k[x]^{n \times n}$ be polynomial square matrices with $\deg(A), \deg(C) \leq d$, where the degree of a matrix is defined as the maximum of the degrees of its entries. Assume all matrices have full rank $n$ over $k(x)$ and that $B$ is unimodular over $k[x]$, meaning $\deg \det B = 0$. Does $AB = C$ imply $\deg(B) \leq d$?

Answer: Yes


Question: Determine the number of tuples \((k_1, k_2, \ldots, k_{26})\) such that each \(k_i\) is either 0, 1, or 3, and the sum \(k_1 + k_2 + \ldots + k_{26} = 15\).

Answer: 853423740


Question: Given that $f'(x) = g(x)$ and $g'(x) = -f(x)$ for all real $x$, and $f(5) = 2 = f'(5)$, find the value of $f^2(10) + g^2(10)$. Provide your answer as a single number.

Answer: 8


Question: Find the area of the spherical triangle formed by the points \((0, 0, 1)\), \((0, 1, 0)\), and \(\left(\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}\)\) on the unit sphere.

Answer: \dfrac{\pi}{4}


Question: Consider an immersion of a sphere \( f: S^2 \to \mathbb{R}^3 \) where the equator is an immersed circle in the \( xy \)-plane that contracts to a point in the \( \pm z \) directions by running along two incompatible immersed disks. Does \( f(S^2) \) bound an immersed 3-ball, i.e., does \( f \) extend to an immersion of a 3-ball?

Answer: Yes


Question: Given two distinct integers $a$ and $b$ such that $\operatorname{gcd}(a,b)=1$ and $a < b$, are there infinitely many primes of the form $a + bm$, where $m$ is a positive integer?

Answer: Yes


Question: Find the smallest positive integer $x$ such that the last digit of $x$ is 6, and if this 6 is moved to the front of the number, the resulting number is four times $x$. What is $x$?

Answer: 153846


Question: Find all positive integer pairs \((a, b)\) such that \(a^{a^a} = b^b\).

Answer: (1, 1)


Question: Determine whether the following statement is true or false: If $f^2 \in \mathcal{L}^1(\mathbb{R})$, does it imply that $f \in \mathcal{L}^1(\mathbb{R})$? Justify your answer.

Answer: False


Question: The number of accidents follows a Poisson distribution with a mean of 12. Each accident generates 1, 2, or 3 claimants with probabilities \(\frac{1}{2}, \frac{1}{3}, \frac{1}{6}\) respectively. Calculate the variance in the total number of claimants.

Answer: 40


Question:
Determine the maximum value of \(a\) in the interval \((0,1)\) such that the sum of the middle binomial coefficients
\[ 2^{-n} \sum_{k=(n/2)-\lfloor n^a \rfloor}^{(n/2)+\lfloor n^a \rfloor} \binom{n}{k} \]
goes to zero as \(n\) approaches infinity.

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{(x,y)\to(0,0)} \frac{e^{-\frac{1}{x^2+y^2}}}{x^6+y^6} \]

Answer: 0


Question: Let $\mathcal{E}'(\mathbb{R})$ be the space of all compactly supported distributions on $\mathbb{R}$. For $f \in \mathcal{E}'(\mathbb{R})$, let $\widehat{f}$ denote the entire extension of the Fourier transform of $f$. If $f_n \to f$ in $\mathcal{E}'(\mathbb{R})$ as $n \to \infty$, does $\widehat{f}_n$ converge to $\widehat{f}$ uniformly on compact subsets of $\mathbb{C}$ as $n \to \infty$?

Answer: Yes


Question: Compute the integral \( \int_0^\infty \mathrm{si}(x)\mathrm{d}x \), where \( \mathrm{si}(x) = \int_x^\infty \frac{\sin(t)}{t}\mathrm{d}t \) for all \( x > 0 \).

Answer: 1


Question:
Let $f$, $g$, and $h$ be bounded functions on the closed interval $[a,b]$ such that $f(x) \le g(x) \le h(x)$ for all $x \in [a,b]$. Let $P=\{a=a_0<a_1<a_2<\cdots <a_n=b\}$ be a partition of $[a,b]$. We denote by $U(P,f)$ and $L(P,f)$ the upper sum and lower Riemann sum of $f$ with respect to the partition $P$, and similarly for $g$ and $h$. Which of the following statements is necessarily true?

(A) If $U(P,h)-U(P,f)<1$ then $U(P,g)-L(P,g)<1$
(B) If $L(P,h)-L(P,f)<1$ then $U(P,g)-L(P,g)<1$
(C) If $U(P,h)-L(P,f)<1$ then $U(P,g)-L(P,g)<1$
(D) If $L(P,h)-U(P,f)<1$ then $U(P,g)-L(P,g)<1$

Determine the correct statement.

Answer: C


Question: Given the quadratic equations \(x^2 - 10ax - 11b = 0\) with roots \(c\) and \(d\), and \(x^2 - 10cx - 11d = 0\) with roots \(a\) and \(b\), find the value of \(a + b + c + d\).

Answer: 1210


Question: Let $Q$ be a binomial random variable with parameters $N$ and $p$. Determine whether $\mathbb{E}[N/Q \mid Q>0] \rightarrow 1/p$ as $N \rightarrow \infty$. Justify your answer.

Answer: \dfrac{1}{p}


Question: Find the remainder when the sum \( 10^{10} + 10^{10^2} + 10^{10^3} + \ldots + 10^{10^{100}} \) is divided by 7.

Answer: 1


Question: Let \( f(x) = \begin{cases} e^{-1/x} , & x > 0 \\ 0, &x \leq 0 \end{cases} \). Find \( f^{(n)} (0) \) for all \( n \in \mathbb{N} \). Justify your answer.

Answer: 0


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

Answer: 2\pi


Question: Determine the number of real roots of the equation \(x^8 - x^5 + x^2 - x + 1 = 0\).

Answer: 0


Question: Determine the conditions under which the series \( \sum_{n=1}^{\infty} \frac{x^n}{\sin(n\theta)} \) converges for \( \theta = t\pi \), where \( t \) is an irrational number, such as \( \theta = \sqrt{3}\pi \).

Answer: |x| < 1


Question: Evaluate the limit \( \lim_{(x,y)\rightarrow 0}\frac{xy^2}{x^2+y^4} \) along the path \( x = y^2 \).

Answer: \dfrac{1}{2}


Question: Compute the winding number $n(\gamma, 0)$ for the curve $\gamma(t) = \cos t + 3i \sin t$ where $0 \leq t \leq 4\pi$. What is the value of $n(\gamma, 0)$?

Answer: 2


Question: Suppose $x_1, x_2,$ and $x_3$ are roots of the equation $(11 - x)^3 + (13 - x)^3 - (24 - 2x)^3 = 0$. What is the sum $x_1 + x_2 + x_3$?

Answer: 36


Question: Is the set of all valid equalities over the two-sorted structure \((\mathbb{R}, \mathbb{R}^3)\), using operations such as addition, subtraction, multiplication, vector addition, vector subtraction, dot product, cross product, scalar multiplication, and constants 0, 1, and the zero-vector \(\bf0\), finitely generated? Specifically, can this set be determined by a finite set of equations?

Answer: Yes


Question: Given a graph $G$ composed of two cycles $C_x$ and $C_y$ that share an edge, where $x, y \ge 3$, find a formula for the number of spanning trees of $G$. The graph $G$ has $x + y - 1$ edges. Provide the formula in terms of $x$ and $y$.

Answer: xy - 1


Question: Is every continuous function \( f:[a,b]\to \mathcal{L}^p(H) \) Riemann integrable, where \( \mathcal{L}^p(H) \) is the Schatten \(*\)-ideal on a separable Hilbert space \( H \) for \( p \in (0,1) \)?

Answer: Yes


Question: Evaluate the integral \( \int\limits_{-1/2}^{1/2}\frac{dx}{\sqrt{1-x^2}\arccos^2x} \).

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


Question: Find the value of \(x\) such that \(\int_{0}^{x}\{t\}dt = \int_{0}^{\{x\}}tdt\), where \(x > 0\), \(x \notin \mathbb{Z}\), and \(\{x\}\) is the fractional part of \(x\).

Answer: (0,1)


Question: Let $S \subset \mathbb{R}^n$ be a compact set, and let $\mathfrak{C} = \{B_1, B_2, \cdots, B_m \}$ be a finite covering of $S$ with $m$ open balls in $\mathbb{R}^n$. Define $R = \min(r(B_1), \cdots, r(B_m))$, where $r(B_i)$ denotes the radius of the open ball $B_i$. Determine whether there exists an absolute constant $c_n$, depending only on $n$ and not on $m$, such that for any two points $Q_1, Q_2 \in S$ with $d(Q_1, Q_2) \leq R$, there exist points $P_1, P_2, \cdots, P_{c_n}$ and balls $B_{i_1}, B_{i_2}, \cdots, B_{i_{c_n}}$ such that $P_j, P_{j+1} \in B_{i_{j+1}}$ for all $j < c_n$, $P_1, Q_1 \in B_{i_1}$, and $P_{c_n}, Q_2 \in B_{i_{c_n}}$. If true, this would provide another proof for the fact that if $f: K \rightarrow \mathbb{R}^n$ is continuous with $K$ compact, then $f$ is uniformly continuous.

Answer: Yes


Question: Given two independent random variables $X$ and $Y$, each with a continuous uniform distribution on the interval $[0, 1]$, find the expected value of the random variable $Z = \max(X, Y)$. Express your answer as a single number.

Answer: \dfrac{2}{3}


Question: Let G be a nontrivial finite group. Does there exist an irreducible representation of G over an arbitrary field with a dimension greater than or equal to the cardinality of G?

Answer: No


Question: Let $A$ be an associative $\mathbb{C}$-algebra with three generators $E$, $H$, $F$, and three defining relations: $$HE - EH = 2E, \ HF - FH = -2F, \ EF - FE = H.$$ Let $M$ be a nonzero finite dimensional representation of $A$. Does there exist an element $v_0 \in M$, $v_0 \neq 0$, and $\lambda \in \mathbb{C}$ such that $E(v_0) = 0$ and $H(v_0) = \lambda \cdot v_0$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{1}{n+\sqrt{(k^2-k+1)}}. \]

Answer: \ln 2


Question: Evaluate the limit \( \lim_{x\to \infty} \frac{1}{x}\int_{2}^{x}\left(1-\frac{\cos^2{t}}{t}\right)\, dt \).

Answer: 1


Question: Let $M$ be an oriented $m$-manifold without boundary, $X \in \mathfrak{X}(M)$ be a vector field, and $\omega \in \Omega^m_c(M)$ be a compactly supported top-degree differential form. Prove or disprove that \( \int_M \mathcal{L}_X \omega = 0 \), where $\mathcal{L}_X$ is the Lie derivative.

Answer: 0


Question: Consider a compact oriented 4-manifold $X$ with nonempty boundary and a second Betti number $b_2(X) \geq 1$. Can the intersection form of $X$ be trivial, meaning it is represented by the zero matrix?

Answer: Yes


Question: Calculate the limit: $$\lim_{x \to 0}\int_{\sin x}^{x}\frac{dt}{t^3(1+\sqrt{t})}$$

Answer: \dfrac{1}{6}


Question: Given a diagonalizable square matrix $A$ with eigendecomposition $A = S_A D_A S_A^{-1}$, express the matrix $S_{A \oplus A}$ that diagonalizes the Kronecker sum $A \oplus A = A \otimes I + I \otimes A$ in terms of $S_A$. Here, $I$ is the identity matrix of the same size as $A$. Assume that $A \otimes I$ and $I \otimes A$ are simultaneously diagonalizable.

Answer: S_A \otimes S_A


Question: An urn contains 6 red balls and 4 green balls. Balls are drawn one by one without replacement until the urn is empty, and the order of colors is noted. Let $X$ be the number of red balls drawn in the first five draws, and $Y$ be the number of red balls drawn in the last five draws. Compute the covariance $\text{Cov}(X, Y)$. \( \text{Cov}(X, Y) = ? \)

Answer: -\dfrac{2}{3}


Question: Evaluate the limit: \[ \lim_{x\rightarrow \frac{\pi}{2}}\frac{\sin x-(\sin x)^{\sin x}}{1-\sin x+\ln (\sin x)} \] without using L'Hôpital's Rule or series expansion.

Answer: 2


Question: Given that \( \frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 1 \), find the value of the expression \( \frac{a^2}{b+c} + \frac{b^2}{a+c} + \frac{c^2}{a+b} \).

Answer: 0


Question: Find the number of orbits in the set \( \{ 1, 2, 3, 4, 5, 6, 7, 8 \} \) under the action of the cyclic subgroup \( \langle (1356) \rangle \) of \( S_8 \).

Answer: 5


Question: Verify whether the following expression is an identity for the exponential function: \[ e^x = \lim_{n \to \infty}\left(\frac{1+0.5\frac{x}{n}}{1-0.5\frac{x}{n}}\right)^n. \]

Answer: e^x


Question: Find the limit: \[ \lim_{n \to \infty} \frac{1}{(n!)^{1/n}} \]

Answer: 0


Question: Compute the expected value of the ratio \( Y = \frac{X_i}{\sum_{j=1, j \neq i}^{n} X_j} \), where \( X_i \) are i.i.d. exponential random variables with mean 1. Is the expected value \( \mathbb{E}[Y] = \frac{1}{n-1} \)?

Answer: \frac{1}{n-2}


Question:
Determine whether the inverse Fourier transform is a linear transform. Consider the inverse Fourier Transform and the Fourier Transform:
\[f(x) = \int_{-\infty}^\infty F(k)e^{2\pi i k x}dk \]
\[F(k) = \int_{-\infty}^\infty f(x)e^{-2\pi i k x}dx\]
The Fourier transform is linear. Is the inverse Fourier transform also a linear transform?

Answer: Yes


Question:
Find the minimum value of the expression
\[ \frac{\sin x + \cos x}{\sin x + \tan x} + \frac{\sin x + \cos x}{\cos x + \cot x} + \frac{\tan x + \cot x}{\cos x + \tan x} + \frac{\tan x + \cot x}{\sin x + \cot x} \]
for \(0 < x < \frac{\pi}{2}\). Use Engel's inequality as a hint.

Answer: 4


Question: Calculate the area of the sphere defined by \(x^2 + y^2 + z^2 = a^2\) that lies outside the regions defined by the cylinders \(x^2 + y^2 - ax = 0\) and \(x^2 + y^2 + ax = 0\).

Answer: 8a^2


Question: Let $S = \{z : |z| = 1\}$ be the multiplicative group of complex numbers with modulus 1. Suppose $G$ and $H$ are subgroups of $S$ with orders 8 and 10, respectively. Determine the order of the intersection of $G$ and $H$. What is the order?

Answer: 2


Question: Let $X$, $Y$, and $Z$ be three independent uniform random variables on $[0, 1]$. Compute the probability $P(XY < Z^2)$. Provide your answer as a single probability value.

Answer: \dfrac{5}{9}


Question: Determine the radius of convergence for the series \( \sum_{n \ge 0} c_n x^{2n} \), where \( c_n \) is a periodic sequence with period 2, i.e., \( c_{n+2} = c_n \) for all \( n \ge 0 \). Use the root test to justify your answer.

Answer: 1


Question: Find the Cauchy Principal Value of the integral \( \int_0^\infty \frac{x^{-p}}{x-1} \, dx \) for \(|p|<1\).

Answer: \pi \cot(\pi p)


Question:
Consider the sequences defined by the following summations:
\[ S_n = \sum_{k=1}^n \frac{\tan^{-1} \frac{k}{n}}{n} \quad \text{and} \quad T_n = \sum_{k=0}^{n-1} \frac{\tan^{-1} \frac{k}{n}}{n} \quad \text{for}~n \in \mathbb{N} \]
Determine which of the following statements is false:

(A) \( S_n > \frac{\pi - \ln 4}{4} \)

(B) \( T_n < \frac{\pi - \ln 4}{4} \)

(C) \( \lim_{n \to \infty} S_n > \lim_{n \to \infty} T_n \)

(D) None of these

Answer: C


Question: Evaluate the limit: \[ \lim_{x\to\infty}\left({\left(\frac{x^2+5}{x+5}\right)}^{1/2}\sin{\left({\left(\frac{x^3+5}{x+5}\right)}^{1/2}\right)}-(x-5)^{1/2}\sin{\left({\left(x^2-5x+25\right)}^{1/2}\right)}\right)=0 \]

Answer: 0


Question: Determine whether a Householder matrix commutes with a unitary matrix. Specifically, if $A$ is a unitary matrix and $P = I - 2\alpha\alpha^H$ where $0 \neq \alpha \in \mathbb{F}^{n \times 1}$ and $\lVert \alpha \rVert = 1$, does $PA = AP$ hold true?

Answer: No


Question: Calculate the expected return from a game where you flip a fair 6-sided die. You win the amount of money equivalent to the number that shows up. You can play a maximum of three times but must decide whether to stop or continue after each throw. How much should you be willing to pay to play this game?

Answer: \dfrac{14}{3}


Question: Given a lattice \((L; \leq)\), if every non-bottom element of \(L\) is join irreducible, must \(L\) be linearly ordered? Answer yes or no.

Answer: Yes


Question: Let $V$ be a finite-dimensional vector space over $\mathbb{C}$, and let $T: V \to V$ be a linear transformation with minimal polynomial $x^{13}(x-1)$. Determine the minimal polynomial of the transformation $T^4$. \( \text{Provide your answer as a polynomial.} \)

Answer: x^4(x - 1)


Question: Neville's Algorithm is used to approximate $f(0)$ using the values $f(-2)$, $f(-1)$, $f(1)$, and $f(2)$. If $f(-1)$ is understated by 2 and $f(1)$ is overstated by 3, determine the error in the original calculation of the interpolating polynomial used to approximate $f(0)$. What is the error in the approximation?

Answer: \dfrac{2}{3}


Question: Let $A \in M(n,\mathbb{R})$ be the permutation matrix defined as $A = (e_n \ e_1 \ \ldots \ e_{n-1})$, where $e_i$ is the $i$-th standard basis vector of $\mathbb{R}^n$. Given that $A^n = I$ and $n$ is odd, determine if $A$ can have a repeated eigenvalue in $\mathbb{C}$. Provide a yes or no answer.

Answer: No


Question: Determine the center of the algebra $PB(H)P$, where $P$ is a projection on a Hilbert space $H$. Express your answer in terms of $P$.

Answer: \mathbb{C}P


Question:
Let $0<\alpha<1$ and $1\leq p<\infty$. Suppose $f\in L^p(\mathbb{R}^n)$ satisfies
\[ \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{|f(x)-f(y)|^p}{|x-y|^{n+\alpha p}}dxdy<\infty \]
Is it true that
\[ \int_{B(0,r)}\frac{|f(x+h)-f(x)|^p}{|h|^{n+\alpha p}}dh\to 0\quad\text{as}\quad r\to 0 \]
for almost every $x\in\mathbb{R}^n$?

Answer: Yes


Question: Evaluate the integral \( \int_0^1 \frac{x}{\sqrt{x(1-x)}} \, dx \).

Answer: \dfrac{\pi}{2}


Question:
Consider the linear transformation defined by
\[u(x) = \begin{bmatrix} x_1 - \bar{x} \\ x_2 - \bar{x} \\ \vdots \\ x_m - \bar{x} \end{bmatrix},\]
for all \(x \in \mathbb{R}^m\), where \(\bar{x} = (1/m) \sum x_i\). Find the matrix \(A\) such that \(u(x) = Ax\), and determine the dimension of the range of \(A\).

Answer: m - 1


Question:
In a category \( \mathfrak{A} \), consider the following commutative diagram of morphisms:
\[
\begin{CD}
A @>{f}>> B\\
@V{h}VV @V{k}VV\\
C @>{g}>> D
\end{CD}
\]
Assume that \( k \circ f = g \circ h \) and there exists a unique morphism \( w: B \to C \) such that \( w \circ f = h \). Can it be deduced that \( g \circ w = k \)?

Answer: No


Question: Given the sequence defined by \( z_{n+1} = \frac{1}{2}\left(z_n + \frac{1}{z_n}\right) \) for \( n \in \mathbb{N} \cup \{0\} \) and \( -\frac{\pi}{2} < \arg(z_0) < \frac{\pi}{2} \), determine \( \lim_{n \rightarrow \infty} z_n \).

Answer: 1


Question:
Given the vector \( v = \left( \frac{1}{2}, \frac{1}{2}, \frac{1}{2} \right) \) and the sequence of vectors defined by:
\[
\begin{aligned}
w_1 &= (e, e+2, e-2), \\
w_n &= v \times w_{n-1} + (2, -4, 2) \quad (n \geq 2),
\end{aligned}
\]
where \( e \) is Euler's number, find the value of \( \lim_{n \to \infty} (2, -3, 1) \cdot w_{2n-1} \).

Answer: 12


Question: Given a differentiable function \( f(x,y) = z \) representing a three-dimensional frictionless surface, with gravity acting in the \(-z\) direction, consider a sphere of arbitrary mass and small radius tangent to a point \( P_0(x_0,y_0,z_0) \) on the surface. The sphere has an initial velocity vector \( \langle \frac{dx_{sphere}}{dt}, \frac{dy_{sphere}}{dt} \rangle \). For any point \( P_1(x_1,y_1,z_1) \) on the surface, is it possible to choose \( \frac{dx_{sphere}}{dt}|_{t=0} \) and \( \frac{dy_{sphere}}{dt}|_{t=0} \) such that the sphere becomes tangent to \( P_1 \) after some time? Ignore air resistance.

Answer: Yes


Question: Consider a series \( \sum\limits_{k=0}^\infty a_k \) with positive terms \( a_k > 0 \) for all \( k \). Let \( s_n \) and \( s_m \) be the partial sums of this series with \( m < n \). If the sequence \( \left\{ \frac{s_m}{s_n} \right\} \) converges to 1, does it imply that the series \( \sum\limits_{k=0}^\infty a_k \) converges?

Answer: No


Question: Determine whether the product \( \prod_{i=1}^n \left(1+\frac{|x_i|^{\frac{1}{n}}}{n}\right) \) converges to \( e \) as \( n \to \infty \), given that \((x_i)\) is a sequence of non-zero real numbers and \((|x_i|)\) is bounded with \(\delta \leq |x_i| \leq M\) for all \(i \in \mathbb{N}\), where \(\delta > 0\).

Answer: e


Question:
Calculate the covariance between the random variables X and Y given their joint distribution:
\[ f(x, y) = 3 \min\{x, y\} \text{ if } 0 < x, y < 1 \]
and 0 otherwise.

Answer: \dfrac{3}{320}


Question: Suppose $B$ and $F$ are closed manifolds with $\pi_{1}(B) = \{1\}$. Additionally, there is a surjective homomorphism $H: \pi_{2}(B) \rightarrow \pi_{1}(F)$. Determine whether there exists an orientable $F$-bundle over $B$, denoted by $E$, such that $\pi_{1}(E) = \{1\}$. (Note: Orientable means that the transition functions are in the orientation-preserving homeomorphisms of $F$. A closed manifold is connected, compact, and oriented.)

Answer: Yes


Question: Let $(f_n)_{n\ge 1}$ be a sequence of differentiable functions such that $f_n \xrightarrow{L^p} f$. Does it follow that $f_n' \xrightarrow{L^p} f'$?

Answer: No


Question:
Given the random variables $X, X_1, X_2, I_1,$ and $I_2$ which are independent, where $X, X_1,$ and $X_2$ have the same distribution and $I_1, I_2 \sim \text{Bernoulli}\left(\frac{1}{2}\right)$, and the relationship $X = 2 + I_1X_1 + I_2X_2$, determine the correct sign in the quadratic solution for the probability-generating function $M_X(s)$ of $X$ given by:

$$M_X(s) = \frac{2-s^2 \pm \sqrt{1-s^2}}{s^2}.$$


Question: Given a Brownian motion \(B_t\) on a probability space \((\Omega, \mathcal{F}, \mathcal{Q})\), consider the stochastic process \(X_t\) defined by the stochastic differential equation \(dX_t = \mu \, dt + \sigma \, dB_t\), where \(\mu \in \mathbb{R}\) and \(\sigma > 0\). Show that the quadratic variation of \(X\), denoted \(\langle X, X \rangle_t\), is \(\sigma^2 t\). Additionally, demonstrate that this result holds for every probability measure \(\mathcal{P}\) that is absolutely continuous with respect to \(\mathcal{Q}\).

Answer: \sigma^2 t


Question: Determine how many zeros of the polynomial \( z^{5}+2z^{4}+z^{3}+20z^{2}+3z-1=0 \) lie inside the circle \(|z|=2\) using Rouché's Theorem.

Answer: 2


Question: What is the expected percentage of composite numbers of the form \( S_p = 12 \cdot p \cdot (2p+1) + 1 \) such that \( 2^{S_p-1} \equiv 1 \pmod{S_p} \) up to some arbitrary upper bound \( n \)?

Answer: 0\%


Question:
Evaluate the limit:

\[ \lim_{x \to \infty}\left(\frac {\displaystyle\int_{x^{2}}^{2x}t^{4}e^{t^{2}}dt}{e^{x}-1-x - \frac{x^2}{2}- \frac{x^3}{6}-\frac{x^4}{24}}\right) \]

Answer: -\infty


Question: Consider the sequence of polynomials \( p_n(z) = z^2 + \frac{1}{n^2(n^2-1)}z - \frac{1}{n^2(n^2-1)} \). Let \( B = \{ b_n \mid p_n(b_n) = 0, n \in \mathbb{N} \} \) be the set of roots of these polynomials, regarded as a sequence in the cofinite topology on \( \mathbb{R}^2 \). Determine the limit points of the sequence \( B \).

Answer: \mathbb{R}^2


Question:
Consider the probability space \((\Omega,\mathcal{F},P)\) and the filtration \(\mathcal{F}_t\). The Ito integral \(I(f)(\omega)=\int_S^T f(t,\omega)dB_t(\omega)\) is defined on a space \(\mathcal{V}(S,T)\) of functions satisfying the following conditions:

1. \((t,\omega)\to f(t,\omega)\) is \(\mathcal{B}\times\mathcal{F}\) measurable, where \(\mathcal{B}\) is the Borel sigma algebra on \([0,\infty)\).
2. \(f(t,\cdot)\) is \(\mathcal{F}_t\) adapted.
3. \(E[\int_S^Tf^2(t,\omega)dt]<\infty\).

The Ito integral of \(f\) is defined by \(\int_S^T f(t,\omega)dB_t(\omega):=\lim_{n\to \infty}\int_S^T \phi_n (t,\omega) dB_t(\omega)\) in \(L^2(P)\), where \(\{\phi_n\}\) is a sequence of elementary functions such that \(E\Big[\int_S^T (f(t,\omega)-\phi_n(t,\omega))^2 dt\Big]\to 0\) as \(n\to\infty\).

Now, consider the space \(\mathcal{W}\) of adapted elementary functions such that \(E[\int_S^T\phi^2(t,\omega)dt]<\infty\) and define the inner product \(\langle \phi,\psi\rangle_{\mathcal{W}}=E[\int_S^T\phi \psi dt]\). Is \(\mathcal{V}\) the closure of \(\mathcal{W}\) with respect to the norm defined by this inner product?

Answer: Yes


Question: Determine the number of continuous involutions on the set of real numbers, where an involution is a function \( f \) such that \( f = f^{-1} \).

Answer: \mathfrak{c}


Question: Let \( f(z) = z^{1/3} \) be the branch of the cube root with the domain defined by \( 0 < \theta < 2\pi \), \( z \neq 0 \) (i.e., the branch cut is along the ray \( \theta = 0 \)). Find \( f(-i) \).

Answer: i


Question: Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed random variables with $X_i > 0$, $E[X_i] = \mu$, and $\text{Var}(X_i) = \sigma^2$. Find the expected value $E\left(\frac{X_1 + X_2 + \cdots + X_m}{X_1 + X_2 + \cdots + X_n}\right)$, where $m < n$. Provide your answer in terms of $\mu$, $\sigma^2$, $m$, and $n$. \( \text{Assume } m < n. \)

Answer: \dfrac{m}{n}


Question: Determine the integer \( D \) such that the quadratic subfield of \( \mathbb{Q}(\zeta_7) \), where \( \zeta_7 = e^{i2\pi/7} \) is a 7th root of unity, can be expressed as \( \mathbb{Q}(\sqrt{D}) \).

Answer: -7


Question: Is the forgetful functor from groups to monoids right adjoint?

Answer: Yes


Question: Evaluate the infinite series \( \sum_{k=1}^{\infty}\frac{18[(k-1)!]^2}{(2k)!} \).

Answer: \pi^2


Question: Find the length of the curve \( r(t) = e^{-t} (\cos(t) \mathbf{i} + \sin(t) \mathbf{j}) \) for \( t \in [0, \infty) \).

Answer: \sqrt{2}


Question: Calculate the work done by the vector field \( \vec{F}(x,y) = xy^2 \hat{i} + yx^2 \hat{j} \) along the path from \((0,0)\) to \((1,0)\) and then to \((1,1)\).

Answer: \dfrac{1}{2}


Question: Let $P$ and $Q$ be linear operators on an $n$-dimensional vector space $V$ such that $P^2 = P$, $Q^2 = Q$, and $(P+Q)^2 = P+Q$. Prove that $PQ = QP = 0$.

Answer: PQ = QP = 0