Question: Is it possible to divide the segments formed by extending the edges of a cube inside a sphere into two groups with equal sum of lengths?

Answer: Yes


Question: Determine whether there exists a faithful linear representation of the additive group of integers. Specifically, does there exist a homomorphism \( \phi : \mathbb{Z} \rightarrow GL_{n}(\mathbb{C}) \) that is injective?

Answer: Yes


Question: Find the value(s) of \(n\) such that \(2^{200} - 2^{192}\cdot31 + 2^n\) is a perfect square.

Answer: 198


Question: Let $T$ and $S$ be diagonalizable linear operators on $\mathbb{R}^n$ such that $TS = ST$. Let $E$ be an eigenspace of $T$. Is the restriction of $S$ to $E$ diagonalizable?

Answer: Yes


Question: In a one-dimensional random walk starting at the origin, the probability of moving right is \( \frac{1}{3} \) and the probability of moving left is \( \frac{2}{3} \). What is the probability that the walk returns to the origin?

Answer: \dfrac{2}{3}


Question: Let $AB$ be an arc of a circle. Tangents are drawn at $A$ and $B$ to meet at $C$. Let $M$ be the midpoint of arc $AB$. A tangent drawn at $M$ meets $AC$ and $BC$ at $D$ and $E$, respectively. Evaluate the limit: $$\lim_{AB \to 0}\frac{\Delta ABC}{\Delta DEC}.$$

Answer: 4


Question: Let $R$ be a commutative Noetherian ring and $M$ be a finitely generated reflexive $R$-module. If $N \cong R$, is the canonical map $d_{M,N}: M \to \operatorname{Hom}_R(\operatorname{Hom}_R(M,N),N)$ an isomorphism?

Answer: Yes


Question: A Jeep needs to cross a 1000-mile wide desert. The Jeep consumes one gallon of fuel per mile and can carry up to 1000 gallons at a time. There are 3000 gallons of fuel available at the starting base camp. Fuel can be dropped off at any point in the desert and picked up later. What is the maximum amount of fuel that can be transported to the camp on the other side of the desert?

Answer: \dfrac{1600}{3}


Question: Evaluate the line integral \( \int_{C} y^2dx + z^2dy + x^2dz \) using Stokes' Theorem, where \( C \) is the curve of intersection of the sphere \( x^2 + y^2 + z^2 = a^{2} \) and the cylinder \( x^2 + y^2 = az \) with \( a > 0 \) and \( z \ge 0 \). The curve \( C \) is oriented anticlockwise when viewed from the origin.

Answer: 0


Question: Let $X$ be a random variable following a Poisson distribution with parameter $\lambda > 0$. If $\delta(X)$ is an unbiased estimator of $g(\lambda) = e^{-\lambda}(3\lambda^2 + 2\lambda + 1)$, find the value of $\sum_{k=0}^\infty \delta(k)$. \( \text{(Hint: Use the property of unbiased estimators)} \)

Answer: 9


Question: Consider an irreducible element \( f \in K[[X_1,\ldots]] \). Let \( f_m \) be the image of \( f \) under the natural quotient ring homomorphism \( K[[X_1,\ldots]] \twoheadrightarrow K[[X_1,\ldots,X_m]] \). Is \( f_m \) also irreducible for sufficiently large \( m \)?

Answer: Yes


Question: Evaluate the integral \( \int_{|z|=2}\frac{e^{3z}}{(z-1)^3}dz \) using the residue theorem.

Answer: 9\pi i e^3


Question: Find the exponential generating function for the number of different ways to seat \( n \) people around a round table, denoted as \( r(n) \).

Answer: -\ln(1 - x)


Question:
Calculate the value of the following limit:
\[ \lim_{n\to\infty} \left(1+\frac{1}{n^2}\right)\left(1+\frac{2}{n^2}\right)\dotsm \left(1+\frac{n}{n^2}\right) \]
Use the hint that the inequality \(x - \frac{x^2}{2} < \log(1+x) < x\) holds for any positive number \(x\).

Answer: \sqrt{e}


Question: Let $R$ be a commutative ring with unity and of odd prime characteristic $p$. Suppose $n > 1$ is an integer such that $(a+b)^n = a^n + b^n$ for all $a, b \in R$, meaning the map $f : R \to R$ given by $f(a) = a^n$ for all $a \in R$ is a ring homomorphism. Is it true that $n$ is a power of $p$?

Answer: Yes


Question: Evaluate the limit: \[ A = \lim_{x \rightarrow 0} \left(\frac{(1+x)^{1/x} - e + \frac{ex}{2}}{x^2}\right) \]

Answer: \dfrac{11e}{24}


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

Answer: e


Question: Evaluate the limit: \[ \lim_{n\to \infty}\frac{1}{n}\sum_{i=1}^n\left\lfloor\sqrt{\frac{4i}{n}}\right\rfloor \] where \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \).

Answer: \dfrac{3}{4}


Question: Find the minimum value of the function \( H(x) = \frac{(x+\frac{1}{x})^6-(x^6+\frac{1}{x^6})-2}{(x+\frac{1}{x})^3+(x^3+\frac{1}{x^3})} \) for \( x > 0 \) without computing the derivative.

Answer: 6


Question: Evaluate the line integral \( \oint_C \mathbf{F} \cdot d\mathbf{r} \) using Stokes' Theorem, where \( S \) is the surface defined by the graph of \( f(x,y) = 4x - 8y + 30 \) over the rectangle \( R = \{ (x,y) \mid -2 < x < 3, 0 < y < 2 \} \). The vector field is given by \( \mathbf{F}(x,y,z) = -x^2 \mathbf{i} + xz \mathbf{j} + yx \mathbf{k} \). The curve \( C \) is the boundary of \( S \), oriented counterclockwise when viewed from above.

Answer: 160


Question: The expressions $3x+3y-1$, $4x^2+y-5$, and $4x+2y$ represent the sides of an equilateral triangle. Determine the integer closest to the area of this triangle.

Answer: 85


Question: Determine the cardinality of the set \([0, \Omega) \setminus (\bigcup \{ U_\lambda : \lambda < \Omega \text{ is limit} \})\), where \(\Omega\) is the first uncountable ordinal with the order topology, and for each limit ordinal \(\lambda < \Omega\), \(U_\lambda\) is an open neighborhood of \(\lambda\).

Answer: \aleph_0


Question: Evaluate the integral \( \int_{\mathbb{R}} (F(x+a) - F(x)) \, dx \) for a general distribution function \( F \) and any \( a \in \mathbb{R} \).

Answer: a


Question: Determine the number of ways to roll a sum of 8 using two dice: one 4-sided and one 6-sided.

Answer: 3


Question: Evaluate the integral: \[ \int_{0}^{\pi} \frac{2\cos(x)-\cos(2021x)-2\cos(2022x)-\cos(2023x)+2}{1-\cos(2x)}\,\textrm{d}x \]

Answer: 2022\pi


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

Answer: 1


Question: Find the convolution of two Heaviside step functions, one centered at \( \frac{1}{2} \) and the other at \( -\frac{1}{2} \).

Answer: t \, H(t)


Question: Two numbers $a$ and $b$ are chosen independently and uniformly from the set \( \{1, 2, \ldots, N\} \). Let \( p_N \) be the probability that \( a^2 + b^2 \leq N^2 \). Determine the value of \( \lim_{N \to \infty} p_N \).

Answer: \dfrac{\pi}{4}


Question: Calculate the residue of \( \frac{\tan(z)}{z^3} \) at \( z = 0 \).

Answer: 0


Question: Does there exist a continuous function from some open subset of $\mathbb{R}$ to $\mathbb{R}$ that is nowhere monotonic?

Answer: Yes


Question: Suppose $R$ is an integral domain and $M$ is a finitely generated module over $R$. Does $M$ necessarily have finite rank? Justify your answer.

Answer: Yes


Question: Find the indicial equation in terms of \( r \) for the differential equation: \[ xy'' - 4y' + 5xy = 0 \] given that \( x = 0 \) is a regular singular point.

Answer: r(r - 5) = 0


Question: Given a set of numbers \( A = \{10, 15, 20, 13, 2, 1, 44\} \), and two integers \( K = 4 \) and \( X = 14 \), select exactly \( K \) different numbers from the set. For each selected number, replace it with the result of the XOR operation between the number and \( X \). You can perform this operation as many times as needed. What is the maximum possible sum of the numbers in the set after performing these operations?

Answer: 129


Question: Could the complex zeros of the Riemann zeta function be of the form \( s=0.5+ik \) with \( k \) a positive integer?

Answer: No


Question:
Evaluate the limit of the integral of the function defined as follows:

\[ f(x) = \frac{1}{\left(\sum_{i=0}^{n} x^i\right)^2 + 1} \]

Find:

\[ \lim_{n \to \infty} \int_{-\infty}^{\infty} f(x) \, \text{dx} \]

Answer: 2 - \arctan 2


Question: Does there exist a countable separating family of bounded continuous functions for the space of measures \( \mathcal{M}_1 \) on a Polish space \((E, \mathcal{A})\)?

Answer: Yes


Question: Given the joint density function \( f_{X,Y}(x,y)=\frac{1}{x}\cdot\exp\left(-\frac{4x^2+y}{2x}\right)\cdot I_{x>0, y>0} \), calculate the expected value \( E[XY] \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{n\to\infty} \sin^2{\left(\frac{1}{n^{48}}\right)}\left((n+\frac{1}{n^3})^{100}-(n^{32}+n^{10}+1)^3-n^{100}\right). \]

Answer: 99


Question: Consider the sequence \( \{a_n\} \) defined by \( a_{n+1} = \arctan(\sec(a_n)) \) with \( a_1 = \frac{\pi}{6} \). Find the positive integer \( m \) such that \( \sin(a_1) \sin(a_2) \cdots \sin(a_m) = \frac{1}{100} \).

Answer: 3333


Question: Find the value of \(x\) that satisfies the equation: \[ \sec^{-1}\sqrt{5} + \csc^{-1}\frac{\sqrt{10}}{3} + \cot^{-1}\frac{1}{x} = \pi. \]

Answer: 1


Question: Solve the differential equation using the power series method: \[ tx''(t) - tx'(t) - x(t) = 0, \ x(0)=0, \ x'(0)=1 \] Assume \( x(t) = \sum_{n=0}^{\infty}a_nt^n \) and find the function \( x(t) \).

Answer: t e^t


Question: Evaluate the limit \( \lim_{n \to \infty}(2@3@\ldots@n) \), where the operation \( a@b = \frac{a+b}{ab+1} \).

Answer: 1


Question: Let $a,b \in \mathbb{R}$ with $a < b$. Consider a sequence $(f_n)_{n \in \mathbb{N}}$ of continuous functions on $[a,b]$ that converges weak-* to $f \in \mathrm{L}^\infty([a,b])$. Is it true that $f$ is continuous?

Answer: No


Question: Let $x_1, x_2, x_3$ be the roots of the polynomial $x^3 - x^2 - 1 = 0$, where $x_1$ is the real root. Compute the limit: \[ \lim_{n \to \infty} (x_2^n + x_3^n). \]

Answer: 0


Question: In a locally compact Hausdorff measure space \((X, \mu)\), if a set \(E\) satisfies \(\mu(X \setminus E) = 0\), must \(E\) be dense in \(X\)? Provide a justification for your answer.

Answer: No


Question: Determine the relationship between the constants \(a\) and \(b\) such that the center of curvature of the helix \(\alpha(t) = (a\sin t, a\cos t, bt)\) is contained within the same cylinder as the helix itself.

Answer: a = \pm b


Question: How many ways can 4 managers and 3 employees be arranged around a round table such that no 3 managers sit together?

Answer: 288


Question: Given the function \( f(x) = \int_{a(x)}^{b(x)} \sin{\sqrt{t}}\,\mathrm{d}t \), where \( a(x) = \frac{\pi^2}{4} + \cos(3x+\frac{\pi}{2}) \) and \( b(x) = \frac{25 \pi^2}{4} + 2x^2 \), compute \( f^{\prime}(0) \).

Answer: 3


Question:
Determine the characteristic curve of the partial differential equation \((2x+u)u_x + (2y+u)u_y = u\) that passes through the point \((1,1)\). Choose the correct option from the following:

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

Answer: 1


Question: Let $A, B \in M_n(\mathbb{R})$ be matrices in the space of all square matrices, and let $\lVert \cdot \rVert$ be a norm over this space. Prove that there exists a constant $C > 0$ such that $\lVert AB \rVert \le C \lVert A \rVert \lVert B \rVert$ for all such norms.

Answer: C


Question:
Find the natural number \( n \) that satisfies the equation:
\[ \dfrac{1}{2}C_{2n}^1 - \dfrac{2}{3} C_{2n}^2 + \dfrac{3}{4} C_{2n}^3 - \dfrac{4}{5} C_{2n}^4 + \cdots - \dfrac{2n}{2n+1} C_{2n}^{2n} =\dfrac{1}{2013}. \]

Answer: 1006


Question: Evaluate the triple integral \( \iiint_E yz \, dV \) where \( E \) is the region above the plane \( z = 0 \), below the plane \( z = y \), and inside the cylinder \( x^2 + y^2 = 9 \).

Answer: \dfrac{162}{5}


Question: Evaluate the limit \( \lim_{x\to 0} x^{-3}\int_0^{x^2}\sin{(\sqrt t)}dt \).

Answer: \dfrac{2}{3}


Question: Let $V$ and $W$ be two $k$-vector spaces of the same dimension, and let $K/k$ be any field extension. If $V\otimes_k K \cong W\otimes_k K$ as $K$-vector spaces, does it follow that $V \cong W$ as $k$-vector spaces?

Answer: Yes


Question: Let $X$ be a complex smooth projective variety with trivial topological Euler characteristic $\chi_{\text{top}}(X)=0$. Assume that $D$ is a smooth irreducible divisor in the linear system $|K_X|$ of the canonical divisor $K_X$ of $X$. Determine if $\chi_{\text{top}}(D)=0$.

Answer: 0


Question: For sufficiently large $n$, does there exist a tournament of size $n$ such that any subtournament of size at least $\sqrt{n}$ contains no nontrivial automorphism, and there exists no node with outdegree $n-1$?

Answer: Yes


Question:
Consider a metric space $(X, d)$ and a metric $p$ on $X$ defined by $p(x, y) = \min \left\{ \frac{1}{2}, d(x, y) \right\}$ for $x, y \in X$. Let $\tau_1$ and $\tau_2$ be the topologies on $X$ induced by the metrics $d$ and $p$, respectively. Determine which of the following statements is true:

(a) $\tau_1$ is a proper subset of $\tau_2$.
(b) $\tau_2$ is a proper subset of $\tau_1$.
(c) Neither $\tau_1 \subseteq \tau_2$ nor $\tau_2 \subseteq \tau_1$.
(d) $\tau_1 = \tau_2$.

Select the correct option.

Answer: D


Question: Evaluate the limit: $$\lim_{x \rightarrow 4} \left( \frac{1}{13 - 3x} \right) ^ {\tan \frac{\pi x}{8}}$$

Answer: e^{-\frac{24}{\pi}}


Question: Calculate $31^6 \mod 189$ using modular arithmetic techniques without the aid of a calculator.

Answer: 127


Question:
Given the function $f(x)$ satisfying the properties:
\[f(x+y)+f(x-y)=2\bigl(f(x)+f(y)\bigr)\]
and \[f(1)=1\], find the value of $f\left(\frac{22}{7}\right)$. \(\text{Express your answer as a single number.}\)

Answer: \dfrac{484}{49}


Question: Determine the number of distinct terms in the expansion of \((1 + a^3 + a^{-3})^{100}\).

Answer: 201


Question:
Suppose $X$ and $Y$ have a jointly continuous distribution with joint density:
$$f(x,y)= \frac{1}{2\pi \sqrt{1-p^2}} e^{ \frac{-(x^2 - 2pxy + y^2)}{2(1-p^2)}}$$
where $p$ is a constant for which $|p| < 1$. Find $E(X^2Y^2)$. Express your answer in terms of $p$.

Answer: 1 + 2p^2


Question: Evaluate the contour integral \( \int_C \frac{8-z}{z(4-z)} \, dz \), where \( C \) is the circle of radius 7 centered at the origin, oriented negatively (clockwise).

Answer: -2\pi i


Question: A subset of \( \{1, 2, 3, \dots, 20\} \) containing 10 elements is chosen at random. Determine the expected value of the largest element in this subset.

Answer: \dfrac{210}{11}


Question: If $A \oplus B \simeq A' \oplus B$ for abelian groups $A$, $A'$, and $B$, does it necessarily follow that $A \simeq A'$?

Answer: No


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

Answer: \dfrac{4}{e}


Question:
Given the function \( f(x) = \begin{cases} x-1 & x \in \mathbb{Q} \\ 0 & x \in \mathbb{Q}^{c} \end{cases} \), determine if the following statements are sufficient to conclude that \( \forall x_n \to 1 \implies f(x_n) \to f(1) \) for any sequence \( x_n \) of real numbers:

1. \( \forall x_r \to 1 \implies f(x_r) \to f(1) \) for sequences \( x_r \) of rational numbers.
2. \( \forall x_i \to 1 \implies f(x_i) \to f(1) \) for sequences \( x_i \) of irrational numbers.

Provide a justification for your conclusion.

Answer: Yes


Question: Let $G$ be a non-Abelian group with order 6. Prove that the center of $G$, denoted $Z(G)$, is trivial, i.e., $Z(G) = \{1\}$.

Answer: \{1\}


Question: Compute the flux of the vector field \( F(x,y,z) = (y, x, x^3) \) across the upper hemisphere of the unit sphere \( x^2 + y^2 + z^2 = 1 \) with the outward normal. Verify if the flux is zero.

Answer: 0


Question: Given the functions \( f(n) = n^3(5+2\cos(2n)) \) and \( g(n) = 3n^2+4n^3+5n \), determine the asymptotic relationship between \( f(n) \) and \( g(n) \) using the appropriate notation: little-o, little-omega, or big-theta. Specifically, find the correct symbol to complete the expression \( f(n) \in \_(g(n)) \).

Answer: \Theta


Question: Determine whether the expression \((ab+1)^{a^n} - 1\) is always divisible by \(a^{n+1}b\) for all positive integers \(a, b, n\).

Answer: Yes


Question:
Solve the following system of simultaneous congruences:
\[ 2x \equiv 1 \pmod{5} \]
\[ 3x \equiv 9 \pmod{6} \]
\[ 4x \equiv 1 \pmod{7} \]
\[ 5x \equiv 9 \pmod{11} \]

Answer: 653


Question: Is the assignment from schemes to the category of strictly locally ringed topoi fully faithful?

Answer: Yes


Question: Consider a scenario where \( n \) people each write their name on a card and place it in a box. Each person, one by one, draws a card from the box. If a person draws their own card, they put it back and draw again until they get a different card. After drawing a different card, they return their own card to the box. What is the probability that the last person to draw a card ends up with their own card? Assume \( n = 31 \).

Answer: \dfrac{1}{31}


Question: Evaluate the integral: $$\int_0^1 \frac{1}{(1-x^6)^{1/6}} \, dx$$

Answer: \dfrac{\pi}{3}


Question: Is an epimorphic endomorphism of a Noetherian commutative ring necessarily an isomorphism? Justify your answer.

Answer: Yes


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

Answer: e^{-\frac{1}{3}}


Question: Let $X$ be a topological space and $V \subseteq U \subseteq X$. If $V$ is connected in the subspace topology inherited from $X$, is $V$ also connected in the subspace topology inherited from $U$?

Answer: Yes


Question: Find the greatest value of $x^3y^4$ given that $2x + 3y = 7$ and $x \geq 0$, $y \geq 0$. Use the AM-GM inequality to solve the problem.

Answer: \dfrac{32}{3}


Question: Given that $a$, $b$, and $c$ are positive numbers satisfying $a + b + c = abc$, find the maximum value of the expression \[ \frac{1}{\sqrt{1+a^2}} + \frac{1}{\sqrt{1+b^2}} + \frac{1}{\sqrt{1+c^2}}. \]

Answer: \dfrac{3}{2}


Question: Does $SO(7)$ have a universal covering by $SU(6)$?

Answer: No


Question: Find the moment-generating function (mgf) of a random variable $X$ given that for all positive integers $r$, the even moments are \( E(X^{2r}) = \frac{(2r)!}{2^r r!} \) and the odd moments are \( E(X^{2r-1}) = 0 \).

Answer: e^{t^2/2}


Question: Given the matrix \( M = \begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{bmatrix} \), find the product of its positive eigenvalues.

Answer: 6


Question: Let \( \{X_n\} \) be a sequence of real random variables such that \( \liminf_{n\to \infty}E({X_n}^2)<\infty \). Prove that there exists an integrable random variable \( X \) and a subsequence \( \{n_k\} \) such that \( X_{n_k}\implies X \) as \( k\to \infty \) and \( \lim_{k\to\infty}E(X_{n_k})=E(X) \). Here, \( \implies \) denotes weak convergence, or convergence in distribution.

Answer: X


Question: Verify that the Fourier transform of the constant function \( f \equiv 1 \) is the Dirac delta function \( \delta(x) \) by showing that the integral \( \int_\mathbb{R} \int_\mathbb{R} \exp(i\lambda x)\phi(x)\, dx \, d\lambda \) equals \( \phi(0) \) for any Schwartz function \( \phi \).

Answer: \phi(0)


Question: Let $X$ be an open, simply connected subset of $\mathbb{R}^2$ containing the point $0$. For every pair of points $p, q \in X \setminus \{0\}$, is there an open, simply connected subset $A_{pq} \subseteq X \setminus \{0\}$ such that $p, q \in A_{pq}$?

Answer: Yes


Question: Evaluate \( \lim_{n \rightarrow \infty} \int_0^n \frac{(1 -\frac{x}{n})^n}{ne^{-x}}\,dx \).

Answer: 0


Question: Evaluate the surface integral \( \iint_S x \, \mathrm{d}S \), where \( S \) is the part of the parabolic cylinder \( z = x^2/2 \) that lies inside the cylinder \( x^2 + y^2 = 1 \) and in the first octant.

Answer: \dfrac{\pi}{8}


Question:
Consider a principal $SO(9)$ bundle over $S^8$. These bundles correspond to elements in $[S^8, BSO(9)] \cong \pi_7(SO(9)) \cong \mathbb{Z}$. For a chosen bundle $\xi$, consider the long exact sequence of homotopy groups:
\[ \cdots \to \pi_8(S^8) \xrightarrow{\partial} \pi_7(SO(9)) \to \cdots \]
Both $\pi_8(S^8)$ and $\pi_7(SO(9))$ are isomorphic to $\mathbb{Z}$. Determine the integer $n$ such that the boundary homomorphism $\partial$ is a multiplication by $n$ map. This integer $n$ depends on the classifying map of the principal bundle $\xi$. What is the value of $n$?

Answer: 1


Question:
Consider the sequence of functions defined by \( f_n(x) = 1 \) if \( x \in [-n-2, -n) \) and \( f_n(x) = 0 \) otherwise. Let \( \alpha = \int_{-\infty}^\infty \lim_{n\rightarrow \infty} f_n(x) \, dx \) and \( \beta = \lim_{n\rightarrow \infty} \int_{-\infty}^\infty f_n(x) \, dx \). Determine the correct values of \( \alpha \) and \( \beta \) from the following options:

(A) \( 0 < \alpha < 1, \beta = 1 \)

(B) \( \alpha = 0, \beta = \infty \)

(C) \( \alpha = \beta = 0 \)

(D) \( \alpha = 0, \beta = 2 \)

Answer: D


Question: Does there exist a decreasing sequence of sets \( \{A_n\} \) of real numbers, where \( A_{n+1} \subseteq A_n \) and \( m^*(A_n) < \infty \) for all \( n \in \mathbb{N} \), such that \( m^*(\cap_{n \in \mathbb{N}} A_n) < \lim_{n \to \infty} m^*(A_n) \)? Here, \( m^* \) denotes the Lebesgue outer measure on the real line.

Answer: Yes


Question: Given the composite number \( N = 1270000^{16384} + 1 \), which has 100,005 digits and no prime factor below \( 2 \times 10^{13} \), determine the expected number of digits of the smallest prime factor of \( N \).

Answer: 14


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

Answer: 1


Question: Compute \( f^{(n)}(0) \) for the function \( f(x) = x^n e^x \), where \( n \in \mathbb{N} \) is fixed.

Answer: n!


Question:
Given the joint distribution of random variables $X$ and $Y$:
\[
f(x, y) = \begin{cases}
2, & 0 < x \leq y < 1, \\
0, & \text{otherwise}
\end{cases}
\]
Determine the correlation coefficient between $X$ and $Y$. \( \rho(X, Y) \)

Answer: \dfrac{1}{2}


Question: Evaluate the definite integral: \[ \int_0^{4\pi} \frac{dx}{5+4\cos x} \]

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


Question: Determine the probability, under the Bernoulli(\(\frac{1}{2},\frac{1}{2}\)) product measure, of the set of infinite binary sequences \(x_1,x_2,x_3,\ldots\) that satisfy \(x_{2n} = x_{2n+1}\) for all \(n \ge 1\). Does this set have positive probability?

Answer: 0


Question: Let X and Y be two independent standard Gaussian variables. Calculate the probability that X is greater than 100 times Y, i.e., find \( P(X > 100Y) \).

Answer: \dfrac{1}{2}


Question: Evaluate the expression $I_{10} + \frac{8}{9} I_8$, where $I_n = \int_0^\pi \sec^n x \, dx$. Use the reduction formula for $\sec^n x$ to find the result.

Answer: 0


Question: Calculate the variance of the hitting time \( \tau = \inf \{t>0: |B_t|=1\} \) for a reflected Brownian motion \( Z_t = |B_t| \), where \( B_t \) is a standard Brownian motion.

Answer: \dfrac{2}{3}


Question: Evaluate the integral: \[ \int_{-\infty}^\infty\int_1^2\frac{y}{\sqrt{2\pi}}e^{-x^2y^2/2}\,\mathrm dy\,\mathrm dx. \]

Answer: 1


Question: Let $G$ be a non-solvable group of order 120 with a center $Z(G)$ such that $|Z(G)| = 2$. If $G/Z(G) \cong A_5$, prove that $G \cong SL(2,5)$. Provide a proof for this conclusion.

Answer: SL(2,5)


Question: Let $K \subset \mathbb{R}^{n}$ be a closed cone (not necessarily convex) and $y \in K^{*}$. Prove that the orthogonal projection of $y$ onto $K$ is unique and equal to zero, where $K^{*} = \{d \in \mathbb{R}^{n} \mid \langle d,y\rangle \leq 0 \text{ for each } y \in K\}$.

Answer: 0


Question:
Given two positive continuous random variables X and Y that are approximately normally distributed with E(X) = 50, sd(X) = 6 and E(Y) = 30, sd(Y) = 4, calculate Pr(X/Y > 2). Choose the correct answer from the following options:

0.023
0.159
0.238
0.460
0.841

Answer: 0.159


Question: Given that the positive divisors of a number $n$ are $d_1, d_2, \ldots, d_r$ such that $1 = d_1 < d_2 < \ldots < d_r = n$, and that $(d_7)^2 + (d_{15})^2 = (d_{16})^2$, find all possible values of $d_{17}$.

Answer: 28


Question: Solve the exponential equation: \(5^x - 4^x = 1\)

Answer: 1


Question: Evaluate the limit: $$\lim_{x \to 0^+} \left [ \ln({x^3+2x^2+x})+ \frac{2}{x} \right ]$$

Answer: \infty


Question: Let X be a two-digit positive integer and Y be a three-digit positive integer. If X is increased by Y% and Y is decreased by X%, they become equal. How many pairs (X, Y) satisfy this condition?

Answer: 3


Question: Evaluate the limit \( \lim_{n \rightarrow \infty} \frac{\ln(n!)}{n^n} \) using the squeeze theorem.

Answer: 0


Question: If $f(x) = 5x^2 - 2kx + 1 < 0$ has exactly one integral solution, find the sum of all positive integral values of $k$.

Answer: 9


Question: If $A$ is a matrix and $v$ is a vector such that $Av=0$, does this imply that there exists a non-zero projection matrix $P$ such that $AP=0$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{1-\cos{(1-\cos{(1-\cos x)})}}{x^8} \] without using Maclaurin series.

Answer: \dfrac{1}{128}


Question: Find all pairs of positive integers \((a, b)\) such that \(a^b - b^a = 3\).

Answer: (4, 1)


Question:
Let $S_{10}$ denote the symmetric group of permutations of 10 symbols. Determine the number of elements in $S_{10}$ that commute with the permutation $(1\ 3\ 5\ 7\ 9)$. Choose the correct answer from the following options:

(a) $5!$
(b) $5 \cdot 5!$
(c) $5! \cdot 5!$
(d) $\frac{10!}{5!}$

Answer: B


Question: Find the value of \( n \) such that the limit \( \lim_{x\to 0} \frac{(\cos x-1)(\cos x-e^x)}{x^n} \) is a finite non-zero number.

Answer: 3


Question: How many sequencing runs are required to analyze 15 million unique DNA molecules from a pool of 20 million, given that each run can analyze 10 million DNA molecules and the selection probabilities follow a negative binomial distribution with parameters \(p=0.5\) and \(r=0.5\)?

Answer: 4


Question: Find the continuous function \( f: \mathbb{R} \to \mathbb{R} \) that satisfies the equation \( f(x+y) + f(x-y) = 2f(x) + 2f(y) \) for all real numbers \( x \) and \( y \).

Answer: f(x) = ax^2


Question: Let $(X, \mathcal{A}, \mu)$ be a measure space with finite measure. Suppose $f_n : (X, \mathcal{A}) \to (\mathbb{R}, \mathcal{B}(\mathbb{R}))$ are measurable functions for every $n \in \mathbb{Z}_+$ such that $f_n \to 0$ pointwise and $f_n \geq f_{n+1}$ for every $n$. Does this imply that $f_n \to 0$ in measure? Justify your answer.

Answer: Yes


Question: Determine the number of sequences of 6 positive integers \(a_1, a_2, a_3, a_4, a_5, a_6\) such that \(a_1 = 1\), \(a_6 = 2000\), and for each \(2 \leq n \leq 6\), \(a_n\) is divisible by \(a_{n-1}\).

Answer: 2450


Question: Determine the radius of convergence for the power series solution of the differential equation \( u''+(x+1)u'-u=0 \) with initial conditions \( u(-1)=2 \) and \( u'(-1)=0 \).

Answer: \infty


Question:
Let $T=\{z\in \mathbb{C}:|z|=1\}$ and $D=\{z\in \mathbb{C}:|z|<1\}$. Define $R=\{z_0\in T: \text{there exists a holomorphic function } f \text{ in an open neighborhood } U_{z_0} \text{ of } z_0 \text{ such that } f(z)=\sum_{n=0}^{\infty}z^{4n}\}$. Determine which of the following statements is true:

1. $T\subset R$
2. $R$ contains infinitely many points of $T$.

Choose the correct statement.

Answer: B


Question: Given that a function $f(x)$ is defined on the interval $[0,1]$ and satisfies the following conditions: \( \int_{0}^{1} (f(x))^2 \, dx = 4 \), \( \int_{0}^{1} f(x) \, dx = 1 \), and \( \int_{0}^{1} x \cdot f(x) \, dx = 1 \), find the value of \( \int_{0}^{1} (f(x))^3 \, dx \).

Answer: 10


Question: Given that 71 is a prime number and \(a\) is an integer such that \(71 \nmid a\), determine the number of positive integer solutions to the congruence \(x^7 \equiv a \pmod{71}\) that are less than 71.

Answer: 7


Question: Find the magnitude \(|z|\) of the complex number \(z\) that satisfies the equation \(z^2(1-|z^2|)=-20\).

Answer: \sqrt{5}


Question: Evaluate the integral \( \int_M \sqrt{u + v} \, d\lambda^2 \) where \( M = \{ (u,v) \in \mathbb{R}^2 \mid 0 < u + v < 1, 0 < 2u - 3v < 4 \} \). Use the change of variables given by the linear transformation \( \begin{pmatrix} u \\ v \end{pmatrix} \mapsto \begin{pmatrix} 1 & 1 \\ 2 & -3 \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix} \). Determine the appropriate region \( M' \) for the new variables and compute the integral over this region.

Answer: \dfrac{8}{15}


Question: Evaluate the integral \( \iiint_{x^2+y^2+z^2 \le z}\sqrt{x^2+y^2+z^2}\,dx\,dy\,dz \) using spherical coordinates. Provide your answer to two decimal places.

Answer: 0.31


Question: Let $S$ be the square $\{x + iy: |x| < 1, |y| < 1\}$ and $f:S \rightarrow S$ a holomorphic function such that $f(0)= 0$ and $f'(0) = 1$. Find $f''(0)$. Use Cauchy's integral formula if necessary.

Answer: 0


Question: For a nonempty set $A$ and a positive integer $k$ with $k \leq |A|$, the symmetric group $S_A$ acts on the set of all subsets of $A$ of cardinality $k$. Determine for which values of $k$ the action of $S_A$ on $k$-element subsets is faithful.

Answer: k \neq |A|


Question: Use differentials to approximate the value of \( h(1.02, 1.99) \), where \( h = f \circ \vec{g} \), \( f(u,v) = 3u + v^2 \), \( \vec{g}(1,2) = (3,6) \), and the Jacobian of \( \vec{g} \) at \( (1,2) \) is \( \begin{pmatrix} 2 & 1 \\ 3 & 5 \end{pmatrix} \).

Answer: 45.21


Question: Let $f: \mathbb{R}^n \to \mathbb{R}$ be a continuous function that vanishes at infinity. Is it true that $f$ is integrable over $\mathbb{R}^n$, i.e., \( \int_{\mathbb{R}^n} |f| \, dm_n < +\infty \)?

Answer: No


Question: Determine the number of complex numbers $z \in \mathbb{C}$ such that $(2z + i\overline{z})^3 = 27i$ and $\text{Re}(z) \geq \text{Im}(z)$. Provide your answer as a single integer.

Answer: 2


Question: Evaluate the integral: $$\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)} + \sqrt{\ln(x+3)}}.$$

Answer: 1


Question: Evaluate the limit: $$\lim_{n\to\infty}\sum_{i=1}^n \frac{1}{\sqrt i}$$

Answer: \infty


Question: Find the supremum of the function \( f(x) = \frac{x^2 e^{-n/x}}{n^2+x^2} \) for \( x \in (0, +\infty) \).

Answer: 1


Question: In \(\Delta ABC\), if \(\frac{b+c}{a} \leq \csc \frac{A}{2}\), and \(\csc^n \frac{A}{2} \csc^n \frac{B}{2} \csc^n \frac{C}{2} = 2^{kn}\) for all integers \(n \geq 1\), find the value of \(k\).

Answer: 3


Question: Determine the exact value of the Hausdorff measure \( \mathcal{H}_{2\alpha}(\mathcal{C} \times \mathcal{C}) \) for the two-dimensional Cantor set, where \( \mathcal{C} \) is the classical Cantor set with Hausdorff dimension \( \alpha = \ln 2 / \ln 3 \).

Answer: 2^{\ln 2 / \ln 3}


Question: Evaluate the expression \( \epsilon^{\mu \nu \alpha \beta}\epsilon_{\mu \nu \alpha \beta} \) in four dimensions.

Answer: -24


Question:
Given the following transition matrix for a Markov chain with state space $S=\{1,2,3,4,5\}$:
\[
\begin{pmatrix}
0 & 0 & 1 & 0 & 0 \\
0 & 0 & \frac{4}{5} & \frac{1}{5} & 0 \\
0 & \frac{1}{6} & \frac{2}{3} & 0 & \frac{1}{6} \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1
\end{pmatrix}
\]
Calculate the probability that the Markov chain is absorbed at state 4.

Answer: \dfrac{1}{6}


Question: Find the smallest prime number that does not divide the number \( x = 9 + 9^2 + 9^3 + \, \ldots \, + 9^{2010} \).

Answer: 17


Question: Find the limit: \[ \lim_{x\to\infty} \left((x^3-x^2+\frac{x}{2})e^{\frac{1}{x}} -\sqrt{x^6+1}\right) \]

Answer: \dfrac{1}{6}


Question: Calculate the volume of the region defined by the inequalities: \( x^2 + y^2 + z^2 \le 2x \) and \( z \ge \sqrt{x^2 + y^2} \).

Answer: \dfrac{3\pi - 8}{9}


Question: Integrate the function \( f(x,y) = x \) over the region inside the disc \( x^2 + y^2 \leq 4 \) and outside the disc \( (x-1)^2 + y^2 = 1 \).

Answer: -\pi


Question: A player plays 9 games each day with his computer. In each game, he will win, draw, or lose with probabilities \(\frac{1}{2}\), \(\frac{1}{3}\), and \(\frac{1}{6}\) respectively. A day is considered “great” if he wins all 9 games. Approximate the probability that within 128 days, he experiences exactly three “great days.”

Answer: 0.002


Question: Evaluate the integral \( \int_{[0,1]^n} \max{(x_1,x_2,x_3, \cdots,x_n)} \,dx_1dx_2\cdots dx_n \).

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


Question: Suppose $C$ is a simple closed contour and for each $z_0$ inside $C$, the following holds: \[ f(z_0) = \frac{1}{2\pi i} \int_C \frac{f(z)}{z-z_0} \, dz. \] Does it follow that $f$ is analytic inside $C$? Justify your answer.

Answer: Yes


Question: Given a nonsingular matrix $A \in M_n(\mathbb{F})$, is there a polynomial $f(x) \in \mathbb{F}[x]$ such that $f(A) = I$, where $I$ is the identity matrix?

Answer: Yes


Question: Given the line \( L = 3x - 2y - 5 = 0 \) and the matrix \( A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \), find the equation of the line \( P \) such that the image of \( P \) under the transformation represented by \( A \) is \( L \).

Answer: 4x + y - 5 = 0


Question: Evaluate the integral \( \int_{S} (x + 2y - z) \, dV \) over the tetrahedron \( S \) with vertices \((0,0,0), (1,2,3), (0,1,2), (-1,1,1)\). You may use a suitable linear transformation as a coordinate change.

Answer: \dfrac{1}{6}


Question: Evaluate the limit: \[ \lim_{a \to 0} \dfrac{ \int^a_0 \ln(1+\tan a\tan x)\ \mathrm{d}x}{a^3} \]

Answer: \dfrac{1}{2}


Question: Determine whether there exists an analytic function \( f = u + iv \) defined on the unit disk such that \( u_x = 0 \) but \( f \) is not constant. Justify your answer.

Answer: Yes


Question: Determine whether the following statement is true: \( \sum_{t \in \mathbb{F}_{p^n}} \zeta^{Tr(t)} = 0 \), where \( \zeta = e^{2\pi i/p} \) and \( Tr: \mathbb{F}_{p^n} \rightarrow \mathbb{F}_{p} \) is the trace map defined by \( Tr(a) = a + a^p + a^{p^2} + \ldots + a^{p^{n-1}} \). Justify your answer.

Answer: True


Question: Let $R$ be a commutative ring with no zero divisors. Determine if the function $f: R \to S$ defined by $f(a) = \frac{a}{1}$ can serve as a monomorphism embedding $R$ into an integral domain $S$.

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{\infty} \frac{x \sin x}{1 + x^2} \, dx \) using the method of residues.

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


Question: Determine the values of \(a > 0\) for which the series \(\sum_{n=1}^{\infty} \left(a^n \cdot \sum_{k=1}^{n} \frac{1}{k}\right)\) converges.

Answer: (0, 1)


Question:
Calculate the area defined by the region where \( x+y \geq 1 \) and \( x^2+y^2 \leq 1 \) using the double integral:
\[
 \iint \limits_{D}xy^2 \,dx\,dy
\]
Find the value of the integral over the domain \( D \).

Answer: \dfrac{1}{20}


Question: Compute the limit \(\lim_{x \to 0} \left(a^x + b^x - c^x\right)^{\frac{1}{x}}\) where \(0 < a, b, c \in \mathbb{R}\).

Answer: \dfrac{ab}{c}


Question: Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists a constant $C > 0$ such that for every rotation $R \in SO(3)$, the area of every connected component of the set $\{x \in S^2 : u(x) > u(Rx)\}$ is at least $C$. Is $u$ necessarily a constant function?

Answer: Yes


Question:
Evaluate the expression:
\[ \frac{16\displaystyle\int_0^\pi\int_0^1x^2\cdot\operatorname{sin}(y)\,dxdy\,\left(\frac{\displaystyle\sum_{n=-\infty}^{\infty}\frac{1}{1+n^2}}{\operatorname{coth}(\pi)}\right)^2}{\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2}}+5 \]

Answer: 69


Question: Find a pair of positive integers \((m, n)\) such that \(m\), \(n\), and \(m+n\) are all perfect squares.

Answer: (9, 16)


Question: Evaluate whether the limit \( \lim\limits_{x \rightarrow \infty}{F(x)} \) is always zero, where \( F(x) = \int_{x}^{x^2} f(t) \, dt \) and \( f(t) \) is a Riemann-integrable function with \( \int_{-\infty}^{\infty} f(t) \, dt = 1 \).

Answer: 0


Question: Evaluate the second derivative of the integral \( \int_0^1 e^{tf(x)} \, dx \) with respect to \( t \), evaluated at \( t = 0 \), where \( f(x) \) is defined as follows: Fix \( n \). If \( x \in (0,1] \) can be written in binary in non-terminating fashion as \( 0.a_1a_2a_3\ldots a_n\ldots \), then \( f(x) \) is the number of \( 1 \)'s among \( \{a_1,\ldots,a_n\} \) minus the number of \( 0 \)'s among the same set.

Answer: n


Question: Determine whether the eigenvalues of the Householder reflector \( F = I - 2qq^* \) in \( \mathbb{C}^{m \times n} \) are always \( \pm 1 \). Justify your answer.

Answer: \pm 1


Question:
Let $H$ be a Hilbert space and $E = (e_j)_{j \in J} \subset H$ a subset of elements in $H$ with $J$ countable. The associated frame operator $S$ is defined by
\[ S:H \to H, \quad Sf=\sum_{j \in J} \langle f,e_j\rangle e_j. \]
Assume that $S$ is a bounded and invertible operator. Can we conclude that there exist constants $0 < A \leq B$ such that the inequality
\[ A\|f\|^2 \le \langle Sf,f \rangle \leq B \|f\|^2 \]
holds for all $f \in H$?

Answer: Yes


Question: Find a measure \( \mu \) such that \( \int_\mathbb{R} \phi \, d\mu_n \to \int_\mathbb{R} \phi \, d\mu \) as \( n \to \infty \) for every continuous function \( \phi \) on \( \mathbb{R} \), where \( f_n(x) = n \cdot 1_{(0,1/n)} \) and \( \mu_n(E) = \int_E f_n(x) \, dx \).

Answer: \delta_0


Question: Evaluate the double integral \(\iint\limits_{x^2+y^2\ge k}\frac{\exp(-(x^2+y^2)/2)}{2\pi}dxdy\).

Answer: e^{-k/2}


Question: Find the maximum value of the expression \( \sqrt{x+27} + \sqrt{13-x} + \sqrt{x} \) for \( 0 \le x \le 13 \).

Answer: 11


Question:
Let $S$ be the set of all tuples $(x, y)$ with $x, y$ as non-negative real numbers satisfying $x+y=2n$, for a fixed $n \in \mathbb{N}$. Determine the supremum value of $x^2y^2(x^2+y^2)$ on the set $S$. Choose the correct answer from the following options:

- $3n^2$
- $2n^6$
- $4n^6$
- $n^6$

Answer: B


Question: Given a sum of a sequence and the number of iterations \( T \), determine the base number \( x \) such that the sequence \( x^2 \) repeated \( T \) times and summed results in the given sum. For example, if the sum is 126 and \( T = 6 \), find the value of \( x \).

Answer: \sqrt{21}


Question:
Find the value of \( n \) given that \( x = \frac{\log n}{\log(577+408\sqrt{2})} \), where \( x \) is the greatest root of the equation:
\[(577+408\sqrt{2})^x + (577-408\sqrt{2})^x = \frac{226}{15}.\]

Answer: 15


Question: Find the order of the conjugacy class of the matrix \( A = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} \) in the group \( GL_2(\mathbb{F}_5) \).

Answer: 30


Question: Evaluate whether the integral \( \int_{\tau} f\,dz \) is zero, where \( f \) is analytic on \( \Omega \) and \( \tau \) is the curve defined by \( \tau(t) = \cos(t) + i\sin(t) \) for \( 0 \leq t \leq \pi \) and \( \tau(t) = \cos(t) - i\sin(t) \) for \( \pi \leq t \leq 2\pi \).

Answer: 0


Question: Given that \(a + b + c = 1\), \(a^2 + b^2 + c^2 = 2\), and \(a^3 + b^3 + c^3 = 3\), find the value of \(a^5 + b^5 + c^5\).

Answer: 6


Question: Given the vectors \(\mathbf{v}_1 = (1, 2, 0, 2)\), \(\mathbf{v}_2 = (1, 1, 1, 0)\), and \(\mathbf{v}_3 = (2, 0, 1, 3)\), find a fourth vector \(\mathbf{v}_4\) such that the set \(\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4 \}\) forms a basis for \(\mathbb{R}^4\).

Answer: (0, 0, 0, 1)


Question: Simplify the expression \( \sqrt[3]{2+\sqrt{-121}} + \sqrt[3]{2-\sqrt{-121}} \) to find its value, assuming only basic formal rules of manipulation.

Answer: 4


Question: Evaluate the double integral \( \int \int_R (x-y) \, dA \) over the region \( R = \{ x \leq y \leq x + 2; -x \leq y \leq 2-x \} \). Determine the correct limits of integration and compute the integral.

Answer: -2


Question: If a focal chord with positive slope \(m\) of the parabola \(y^2 = 16x\) touches the circle \(x^2 + y^2 - 12x + 34 = 0\), find the value of \(m\).

Answer: 1


Question: Consider a ring $R$ and the prime radical topology defined on it. Is the ring $(R, +, \cdot)$ a topological ring with respect to this topology?

Answer: Yes


Question: Let $X_n$ be a sequence of random variables such that $X_n \stackrel{d}{\to} N(0,1)$, where $N(0,1)$ is a standard Gaussian random variable. Determine whether it is true that $\frac{X_n}{n^\alpha} \stackrel{P}{\to} 0$ for every $\alpha > 0$. Provide a justification for your answer.

Answer: Yes


Question: Find the derivative with respect to $x$ of the function $f(x) = \arctan \left( \frac{\cos x}{1+\sin x} \right)$. Express your answer in terms of $x$.

Answer: -\dfrac{1}{2}


Question: Compute the limit without using L'Hospital's Rule: \[ \lim_{x\to1}\frac{\sqrt[3]{x+7}+2\sqrt{3x+1}-6}{\sqrt[4]{x}-1} \]

Answer: \dfrac{19}{3}


Question: Let \( f: C[-1,1] \to \mathbb{R} \) be a functional defined by \( f(x) = \int_{-1}^0 x(t) \, dt - \int_0^1 x(t) \, dt \). Find \( \Vert f \Vert \) with respect to the supremum norm defined on \( C[-1,1] \).

Answer: 2


Question: Compute \( \lim_{n\to \infty }\frac{1}{n}\sum_{k=1}^n\left(1+\frac{k}{n^2}\right)^n \).

Answer: e - 1


Question: Given a list of N numbers, if we perform random permutations each time and check whether the list is sorted, what is the expected number of permutations required to sort the list?

Answer: N!


Question: Let $X$ be a surface that cannot be embedded into $\mathbb{R}^n$. Is it true that the connected sum $X \# X$ is also not embeddable into $\mathbb{R}^n$? Provide a justification for your answer.

Answer: Yes


Question: Verify that the values \(x = -1\), \(y = -3\), and \(z = 5\) satisfy the following system of equations: \[ x+y+z=1 \] \[ x^2+y^2+z^2=35 \] \[ x^3+y^3+z^3=97 \]

Answer: 97


Question: Determine the value of $t \in \mathbb{C}$ for which the projective curve $\mathcal{C}_{t}$ given by $X^{2}+Y^{2}+Z^{2}+t(X+Y+Z)^{2}=0$ has a singular point.

Answer: -\dfrac{1}{3}


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

Answer: \dfrac{4}{e}


Question: Find the value of $b > 0$ such that the inequality $b^x \geq x + 1$ holds for all real numbers $x$.

Answer: e


Question: Let $n \gg 1$ be a fixed number (e.g., $n=10^{10}$), and $k \gg 1$ a natural number. Let $a$ and $b$ be two permutations from the symmetric group $S_k$. Suppose that for every word $w(x,y)$ of length $\le n$, the permutation $w(a,b)$ has a fixed point. Is it true that every permutation in the group generated by $a$ and $b$, denoted $\langle a, b \rangle$, has a fixed point?

Answer: No


Question: In triangle ABC, vertex A is located at (1, 1) and the orthocenter is at (2, 4). The sides AB and BC belong to the family of straight lines given by the equation \( ax + by + c = 0 \), where \( a, b, c \) are in arithmetic progression. If the coordinates of vertex C are \( (h, k) \), find the value of \( 2h + 12k \).

Answer: 14


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

Answer: n < 2


Question: For which integer value of \( a \) does the ratio \( \frac{I_1}{I_0} = \frac{3\sqrt{2}}{4} \), where \( I_0 = \iiint_G \,dx\,dy\,dz \) and \( I_1 = \iiint_G z\,dx\,dy\,dz \), and the region \( G \) is defined as \( G = \{ (x,y,z) : x^2 + y^2 + z^2 \leq a^2, 0 \leq z \leq \sqrt{x^2 + y^2} \} \)?

Answer: 4


Question: Let \( S \) be a subgroup of the group of positive rational numbers under multiplication, which is dense in the positive reals. Define the S-restricted irrationality measure by confining \( p/q \) to \( S \). If \( S \) is finitely generated, particularly if \( S \) is generated by the first \( n \) primes, determine the S-restricted irrationality measure \( \xi \) such that almost all \( x \) have measure \( \xi \).

Answer: 1


Question: Consider the linear embedding of $\mathbb{C}\mathbb{P}^1$ into $\mathbb{C}\mathbb{P}^n$ for $n > 1$, given by the map $[x:y] \mapsto [x:y:0:\dots: 0]$. Is it true that for any open neighborhood $U$ of $\mathbb{C}\mathbb{P}^1$ in the analytic topology, there exists a smaller neighborhood $V$ such that $H^i(V, \mathcal{O}) = 0$ for all $i > 0$, where $\mathcal{O}$ is the structure sheaf?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \rightarrow 0} \frac{(1-\cos x)^2}{\log (1 + \sin^4x)} \]

Answer: \dfrac{1}{4}


Question: Let $A$ be a $117 \times 117$ matrix such that for any vector $(x_1, x_2, \\ldots, x_{117})$, the product $(x_1, x_2, \\ldots, x_{117})A = (x_{117}, x_1, x_2, \\ldots, x_{116})$. Find $\det(A)$.

Answer: 1


Question: Given the differential equation \( ty'' + 3y' = -te^t \), where \( y_1 \) and \( y_2 \) are linearly independent solutions, and the Wronskian \( W(y_1, y_2)(1) = 2 \), find the value of \( W(y_1, y_2)(2) \).

Answer: \dfrac{1}{4}


Question: Find all positive integer solutions \((a, b, c)\) to the equation \(5^a + 4^b = 3^c\).

Answer: (1, 1, 2)


Question: Can the friendship graph $F_n$ be determined by its adjacency spectrum? The friendship graph $F_n$ is constructed by coalescing $n$ copies of the cycle graph $C_3$ with a common vertex. A graph $G$ is said to be determined by its adjacency spectrum if, whenever $Spec(G) = Spec(H)$ for some graph $H$, then $G \cong H$. Determine if $F_n$ is determined by its adjacency spectrum.

Answer: Yes


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

Answer: \dfrac{2}{9}


Question: Compute the limit: $$\lim_{x\to0^+}\frac{2\arctan(x)\sqrt{1-x^2}+\frac{x^3-2x^4}{\ln(2x+2x^2)}-\ln(1+2x+2x^2)}{e^{x-\frac{x^2}2}-\cos(x^2)+\tan(2x^4-x)+x^2e^{-\frac1{x}}}$$

Answer: \dfrac{1}{2}


Question: Let \( f: \mathbb{C} \to \mathbb{C} \) be a meromorphic function analytic at \( 0 \) satisfying \( f\left(\frac{1}{n}\right) = \frac{n}{2n+1} \) for all \( n \ge 1 \). Determine if \( f(2) = \frac{1}{4} \).

Answer: \dfrac{1}{4}


Question: Suppose $E$ and $E'$ are two degree-$n$ extensions of $\mathbb{F}_p$. They are both splitting fields of $x^{p^n}-x$ and are isomorphic. Is it possible to obtain an isomorphism $E \to E'$ that fixes the base field $\mathbb{F}_p$?

Answer: Yes


Question: Find all functions \( f: \mathbb{R} \to \mathbb{Z}^+ \) such that for all real numbers \( x \) and \( y \), the equation \( f^{f(x)}(y) = f(x)f(y) \) holds, where \( f^n(x) \) denotes the \( n \)-th iteration of \( f \) applied to \( x \).

Answer: 1


Question:
Given a normed linear space $X$ over a field $K$, an arbitrary indexing set $I$, a family of elements $\{f_\alpha: \alpha \in I\} \subset X$, and a family of scalars $\{c_\alpha: \alpha \in I\} \subset K$, determine if there exists exactly one bounded linear functional $f' \in X'$ such that:

1. $f'(f_\alpha) = c_\alpha$ for all $\alpha \in I$.
2. $\|f'\| \le M$ for some $M \geq 0$.

This existence and uniqueness hold if for every finite subfamily $J \subset I$ and every choice of scalars $\{\beta_\alpha : \alpha \in J\} \subset K$, the following inequality is satisfied:
\[\left|\sum_{\alpha \in J}\beta_\alpha c_\alpha\right|\le M\left\|\sum_{\alpha\in J} \beta_\alpha f_\alpha\right\|_X.\]

Answer: Yes


Question: Let $X$ be a complete Hausdorff locally convex topological vector space, and let $X''=(X',\beta(X',X))'$ be its standard bidual. Consider the weak topology $\sigma(X,X')$ on $X$ and the weak* topology $\sigma(X'',X')$ on $X''$. Suppose $A$ is an absolutely convex closed subset of $X$ such that its $\sigma(X'',X')$-closure in $X''$, denoted $\overline{A}^{\sigma(X'',X')}$, is compact in $(X'',\sigma(X'',X'))$. Is the weak closure $\overline{A}^{\sigma(X,X')}=A$ compact in $(X,\sigma(X,X'))$?

Answer: No


Question:
Determine the largest lower bound, independent of \(\lambda\), for the expression:
\[
\frac{1+|\lambda|^2}{2} + \text{Re}\lambda - \frac{2\pi}{n+2}\text{Im}\lambda
\]
where \(|\lambda| < 1\), \(\text{Re}\lambda < 0\), \(\text{Im}\lambda > 0\), and \(n\) is a positive integer. You may choose \(n\) to be as large as necessary for the problem to make sense, but aim for the smallest possible lower bound on \(n\).

Answer: 0


Question: Is there a lattice homomorphism from the complete lattice of ideals of a complete lattice $L$, denoted by ${\cal Id}(L)$, onto $L$?

Answer: Yes


Question:
Compute the expression \(L - R_1 - R_2\), where
\begin{align}
L&=2^{3 / 2} \pi^{4} \frac{1}{4} \sum_{n, m, k=1}^{\infty} n^{2} m a_{n} a_{m} b_{k}\left(\delta_{m, n+k}+\delta_{n, m+k}-\delta_{k, n+m}\right), \\
R_{1}&=2^{3 / 2} \pi^{4} \frac{1}{4} \sum_{n, m, k=1}^{\infty} n^{3 / 2} m^{3 / 2} a_{n} a_{m} b_{k}\left(\delta_{m, n+k}+\delta_{n, m+k}+\delta_{k, n+m}\right), \\
R_{2}&=\frac{1}{2} 2^{3 / 2} \pi^{4} \frac{1}{4} \sum_{n, m, k=1}^{\infty} n^{1 / 2} m^{1 / 2} k^{2} a_{n} a_{m} b_{k}\left(\delta_{m, n+k}+\delta_{n, m+k}-\delta_{k, n+m}\right),
\end{align}
and the \(\delta\) denote the Kronecker delta.

Answer: 0


Question: A number of women proposed once each, of whom one-eighth were widows. Consequently, a number of men were to be married, of whom one-eleventh were widowers. Of the proposals made to widowers, one-fifth were declined. All the widows were accepted. Thirty-five forty-fourths of the widows married bachelors. One thousand two hundred and twenty-one spinsters were declined by bachelors. The number of spinsters accepted by bachelors was seven times the number of widows accepted by bachelors. How many women proposed?

Answer: 11616


Question:
Determine the number of integer solutions \((i, j)\) to the equation:
\[3i^2 + 2j^2 = 77 \cdot 6^{2012}.\]

Answer: 8


Question: Calculate the limit \( \lim_{n\to\infty}\sum_{0\leqslant k\leqslant 2n}\frac{k}{k+n^2} \) using Riemann sums.

Answer: 2


Question: In a topological space $X$, can the boundary of a subset $A \subset X$ contain an open set?

Answer: Yes


Question: Evaluate the contour integral \( \oint_C \frac{dz}{z-2} \), where \( C \) is the square with vertices at \( 3 \pm 3i \) and \( -3 \pm 3i \).

Answer: 2\pi i


Question: Let $K$ be a number field, $\mathcal{O}_K$ the ring of integers, and $\mathcal{O}_K^\times$ the unit group of $\mathcal{O}_K$. Using Dirichlet's theorem, we know that $\mathcal{O}_K^\times$ is a finitely generated group. Is the quotient group $\mathcal{O}_K^\times / (\mathcal{O}_K^\times)^m$ finite?

Answer: Yes


Question: Find the maximum value of the function \( f(z) = \frac{1}{z - \frac{i}{2}} \) on the set \( \{ z : 1 \leq |z| \leq 2 \} \).

Answer: 2


Question: Find \( \lim_{n\rightarrow \infty}\sum_{k=1}^n\frac{(k+1)^l}{k^{l+1}} \) for \( l \geq 0 \).

Answer: \infty


Question: Let \( w \) and \( z \) be complex numbers such that \( w = \frac{1}{1-z} \) and \( |z|^2 = 1 \). Find the real part of \( w \).

Answer: \dfrac{1}{2}


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

Answer: \dfrac{1}{e}


Question: A man can row at 3 miles per hour and run at 5 miles per hour. He is currently 5 miles out to sea and wants to reach a point on the coast that is 13 miles along the shore from his current position. Determine the point on the coast where he should land to minimize his total travel time.

Answer: \dfrac{15}{4}


Question: Evaluate the surface integral \( \iint_S F \cdot n \, dS \) over the unit sphere \( S \) in \( \mathbb{R}^3 \), where \( F(x) = \nabla V(x) \) and \( V(x) = \frac{1}{|x|} \).

Answer: -4\pi


Question:
Determine the nature of the sequence \(a_n = \frac{e^{n!\tan (n\pi)}-1+n^2\sin n}{\log (1+3^{n\log n})\log (n+5^n)}\) for \(n \in \mathbb{N}\). Does the sequence have:

- A minimum but no maximum
- Neither maximum nor minimum
- Both maximum and minimum
- A maximum but no minimum

Select the correct option.

Answer: C


Question: Determine the radius of convergence of the power series \( p(x) = \sum b_n x^n \), given that the series \( \sum b_n \) is conditionally convergent but not absolutely convergent.

Answer: 1


Question: Evaluate the integral \( \int_{-100}^{100} \lfloor x^3 \rfloor \, dx \).

Answer: -100


Question:
Let \( \{a_n:n\ge 1\} \) be a sequence of real numbers such that the series \( \sum a_n \) converges and the series \( \sum |a_n| \) diverges. Determine the radius of convergence \( R \) of the power series \( \sum a_n x^n \). Choose the correct option:

- \(0 < R < 1\)
- \(1 < R < \infty\)
- \(R = 1\)
- \(R = \infty\)

Answer: C


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

Answer: \dfrac{4}{\pi}


Question: What is the topological dimension of the Cantor set?

Answer: 0


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

Answer: I_0(2) - 1


Question: Given the cubic equation \(x^3 + x + m = 0\) with roots \(x_1, x_2, x_3\), find the value of \(m\) such that \(x_1^5 + x_2^5 + x_3^5 = 10\).

Answer: 2


Question: Let $S$ be a ring with identity (not necessarily commutative) and $f: M_{2}(\mathbb{R}) \to S$ be a non-zero ring homomorphism, where $M_{2}(\mathbb{R})$ is the ring of all $2 \times 2$ matrices. Does $S$ have infinitely many nilpotent elements?

Answer: Yes


Question: Evaluate \( \frac{1}{\pi}\int_{-\pi}^{\pi}\left[D_{100}(t)\right]^{2}dt \), where \( D_{m}(t)=\frac{1}{2}+\sum_{n=1}^{m}\cos{nt} \) is Dirichlet's kernel for \( m = 100 \).

Answer: \dfrac{201}{2}


Question: Determine whether the following statement is true or false: If \( \{s_n\}_{n=1}^\infty \) is a sequence in \( \mathbb{R} \) that satisfies \( |s_{n+1} - s_n| \to 0 \), then \( \{s_n\} \) is a Cauchy sequence.

Answer: B


Question: Let $X \subset k^n$ and $Y \subset k^m$ be two connected algebraic varieties in the Zariski topology. Is the Cartesian product $X \times Y \subset k^{n+m}$ connected in the Zariski topology of $k^{n+m}$?

Answer: Yes


Question:
Evaluate the integral:
\[ I = \int_{eV/2}^{\infty} d\epsilon_1 \int_{-\infty}^{-eV/2} d\epsilon_2 \int_{0}^{\infty} d\epsilon' \int_{-\infty}^{0} d\epsilon \, \delta(\epsilon_2 - \epsilon_1 + \epsilon - \epsilon') \]
where \( \delta(x) \) is the Dirac delta function.

Answer: 0


Question: Compute the value of \(i^i\).

Answer: e^{-\pi/2}


Question: Given the probability measure \( \mathbf{P}_X [A] := \frac{1}{2} \delta_0 (A) + \frac{1}{2} \int_{A \cap (0, \infty)} e^{-t} dt \) for \( A \in \mathcal{B} (\mathbb{R}) \), find the expectation \( \mathbf{E} [X] \).

Answer: \dfrac{1}{2}


Question: Let $A$ be a $10 \times 10$ matrix such that $A^{10,000} = 0$. Prove that $A^{10} = 0$.

Answer: A^{10} = 0


Question: Evaluate the contour integral \( \oint _{C} \frac{e^{3z}}{(z-\ln{2})^{4}}dz \), where \( C \) is the square with vertices \( \pm1 \pm i \).

Answer: 72\pi i


Question: Find the real solution to the equation: \[ \sin\left(\frac{x}{x^2+1}\right) + \sin\left(\frac{1}{x^2+x+2}\right) = 0. \]

Answer: -1


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

Answer: \pi


Question:
Let $V$ be a closed subspace of $L^2[0,1]$ and $f,g \in L^2[0,1]$ be given by $f(x)=x$ and $g(x)=x^2$. If $V^{\perp}=\text{span}\{f\}$ and $Pg$ is the orthogonal projection of $g$ onto $V$, find $(g-Pg)(x)$ for $x \in [0,1]$. Choose the correct answer:

- $\frac{3}{4}x$
- $\frac{1}{4}x^2$
- $\frac{3}{4}x^2$
- $\frac{1}{4}x$

Answer: A


Question: Find all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(x) + x = 2f(2x) \) for all real numbers \( x \).

Answer: \dfrac{x}{3}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{(\sqrt[n]{(7^n+n)}-\frac{1}{7})^n}{7^n-n^7} \]

Answer: 0


Question: Evaluate the contour integral \( \int_\beta \frac{e^z}{e^z-\pi} \mathrm{d}z \), where \( \beta \) is a loop in the annulus \( \{z:10<|z|<12\} \) that winds \( N \) times about the origin in the counterclockwise direction, with \( N \) being an integer.

Answer: 0


Question: Is it possible to find a symmetric matrix $S$ that is not positive semidefinite, such that there exist two different indices $i_1 \neq i_2$ for which the submatrices $A_{-i_1}$ and $A_{-i_2}$ are positive semidefinite?

Answer: Yes


Question: Given the square matrix \(A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -2 & 4 \end{pmatrix}\), find the values of \(c\) and \(d\) such that the equation \(6A^{-1} = A^2 + cA + dI\) holds. Determine \((c, d)\).

Answer: (-6, 11)


Question: Let \( X_1, X_2, \ldots \) be independent random variables such that \( \mathbb{P}(X_i = 2) = \mathbb{P}(X_i = 0) = \frac{1}{2} \). Define \( Y_n = \prod_{i=1}^{n} X_i \) and let \( \tau = \min \{ n \geq 1 : Y_n = 0 \} \). Calculate \( \mathbb{E}(Y_{\tau - 1}) \).

Answer: \infty


Question: Solve the congruence equation $x^{113} \equiv 2 \pmod{143}$. Given that $143 = 13 \times 11$, use the Chinese Remainder Theorem and Fermat's Little Theorem to find the solution.

Answer: 84


Question: Find the value of $p$ such that the area $A$ of the triangle formed by the x-axis, y-axis, and the tangent line to the curve $y = x^p$ (where $p < 0$) at $x = c$ is independent of $c$. What does it mean for $A$ to be independent of $c$?

Answer: -1


Question: Consider a sequence \( \{ \varphi_n \} \) that is bounded in \( L^\infty(0,\infty; H_0^1(0,1)) \). Does there exist a function \( \varphi \in L^\infty(0,\infty; H_0^1(0,1)) \) and a subsequence \( \{ \varphi_{n_k} \} \) such that \( \varphi_{n_k} \to \varphi \) in the weak* topology of \( L^\infty(0,\infty; H_0^1(0,1)) \) and for almost every \( t \in (0,\infty) \), \( \varphi_{n_k}(t) \to \varphi(t) \) weakly in \( H_0^1(0,1) \)?

Answer: Yes


Question: Given the joint density function \( f(x,y,z,u) = \frac{24}{(1+x+y+z+u)^5} \) for \( x > 0, y > 0, z > 0, u > 0 \), find the probability \( P(X+Y+Z+U \ge 1) \).

Answer: \dfrac{15}{16}


Question: Simplify the expression \( \sqrt[4]{161-72 \sqrt{5}} \) and express it in the form \( a + b\sqrt{5} \), where \( a \) and \( b \) are integers.

Answer: -2 + \sqrt{5}


Question:
Let $f_n(x)$, for $n \geq 1$, be a sequence of continuous non-negative functions on $[0,1]$ such that \( \lim_{n \to \infty} \int_0^1 f_n(x)\,dx = 0 \). Determine which of the following statements is true:

A. $f_n \rightarrow 0$ uniformly on $[0,1]$.
B. $f_n$ may not converge uniformly but converges to 0 pointwise.
C. $f_n$ will converge pointwise and the limit may be non-zero.
D. $f_n$ is not guaranteed to have a pointwise limit.

Select the correct statement.

Answer: D


Question: Evaluate the sum \(\sum\limits_{(m,n) \in D,m < n} \frac{1}{n^2 m^2}\), where \(D = \{ (m,n) \in (\mathbb{N}^*)^2 \mid \gcd(m,n) = 1\}\).

Answer: \dfrac{3}{4}


Question: Let $a_i, b_i, r_i, s_i$ be positive integers for $i \in \{1, 2\}$, with $r_i$ and $s_i$ being non-zero. Define $a = \left(\frac{1}{a_1}, \frac{1}{a_2}\right)$, $b = \left(\frac{1}{b_1}, \frac{1}{b_2}\right)$, $r = (r_1, r_2)$, $s = (s_1, s_2)$, and $c = \left(\frac{1}{a_1 + b_1}, \frac{1}{a_2 + b_2}\right)$. Let $\langle -, - \rangle$ denote the standard inner product on $\mathbb{R}^2$. If $\langle a, r \rangle < 1$ and $\langle b, s \rangle < 1$, does it imply that $\langle c, r + s \rangle \leq 1$?

Answer: Yes


Question: Let \(X\sim\mathrm{Exp}(1)\) and \(Y\sim\mathrm{Exp}(2)\) be independent random variables. Let \(Z = \max(X, Y)\). Calculate \(E(Z)\).

Answer: \dfrac{7}{6}


Question: If \((A\otimes B)^{-1}=(A\otimes B)^{\dagger}\), does this imply that both matrices \(A\) and \(B\) are unitary?

Answer: No


Question: Find the total derivative at 0 of a function \( f: \mathbb{R}^p \to \mathbb{R}^q \) that satisfies \( \|f(x)\| \leq \|x\|^2 \).

Answer: 0


Question: Compute the Lebesgue measure of the set \( S = \{(x, y, z) \in \mathbb{R}^3 : x \geq 1, \ y^2 + z^2 \leq \frac{1}{x^2} \} \).

Answer: \pi


Question: Calculate the limit: $$\lim_{n\to\infty}\sum_{k = 0}^{n}\left(\frac{k\binom{n}{k} }{n2^n+k}\right) $$

Answer: \dfrac{1}{2}


Question: Determine the units in the ring \( R = \mathbb{Z}[\frac{1 + \sqrt{-15}}{2}] \).

Answer: \pm 1


Question: For two ordinals $A$ and $B$, if $|A| \nleq |B|$, can one conclude $|A| > |B|$ in terms of cardinality?

Answer: Yes


Question: Determine whether the following statement is true or false: A non-commutative ring (not necessarily with a multiplicative identity) of order $n$ exists if and only if there exists a prime $p$ such that $p^2$ divides $n$. Provide a justification for your answer.

Answer: True


Question: Determine whether the set of numbers whose sum of digits is prime has zero natural density. Specifically, evaluate if \( \lim_{n \to +\infty} \frac{A(n)}{n} = 0 \), where \( A(n) \) is the number of elements in the set that are less than or equal to \( n \).

Answer: 0


Question: Given a set $S$ of size $n=6$ and a $k$-multicombination of size $k=9$, where each element can appear no more than $n$ times, determine the position of the combination $\{1,1,1,1,1,1,2,3,3\}$ in the ordered list of all such combinations. The total number of combinations is 1876. What is the position of the combination $\{1,1,1,1,1,1,2,3,3\}$?

Answer: 1847


Question: Determine the number of distinct ways to color a 3x4 rectangle using two colors, such that each color appears exactly 6 times. Two colorings are considered equivalent if one can be obtained from the other by permuting the rows or by cyclically permuting the columns.

Answer: 48


Question: Let \(A\) and \(B\) be square matrices of the same order such that \(A^2 = I\) and \(B^2 = I\), where \(I\) is the identity matrix. If \(\det(A) + \det(B) = 0\), find the value of \(\det(A + B)\).

Answer: 0


Question: Evaluate the contour integral \( \oint_{C} \frac{z}{(z+1)(z-1)^2} \, dz \), where the contour \( C \) is given by \( C = \{ 3\cos(t) + 2i\sin(t) : 0 \leq t \leq 2\pi \} \).

Answer: 0


Question: Calculate the area of the largest rectangle that can be inscribed in an ellipse using integrals. Express the maximum area in terms of the ellipse's semi-major axis \(a\) and semi-minor axis \(b\).

Answer: 2ab


Question: Calculate \((1567)^5 \mod 2881\).

Answer: 2008


Question: Determine the number of distinct values of \(z^{\frac{m}{n}}\) when \(m\) and \(n\) are integers such that \(n \neq 0\) and \(m\) and \(n\) are relatively prime. Express your answer as a single integer.

Answer: n


Question: Find an explicit isomorphism between the rings \( \Bbb F_5[x]/(x^2+x+2) \) and \( \Bbb F_5[x]/(x^2+4x+2) \).

Answer: x \mapsto x + 4


Question: Evaluate the limit: $$\lim_{n\to \infty}\frac{(1)(2)(3)\cdots (n)}{(n+1)(n+2)(n+3)\cdots(2n)}.$$

Answer: 0


Question: Suppose that two measurable functions, $g: \mathbb{R} \to \mathbb{R}$ and $h: \mathbb{R} \to \mathbb{R}$, have Lebesgue points at every $x \in \mathbb{R}$. Can the product $gh$ have points that are not Lebesgue points?

Answer: No


Question:
Find the positive integer \( n \) such that the following equation holds:
\[ \lfloor \sqrt{9n^2+1}\rfloor + \lfloor\sqrt{9n^2+2}\rfloor + \cdots + \lfloor\sqrt{9n^2+24n}\rfloor = 349. \]

Answer: 2


Question: For which values of $p \in [1, \infty]$ does the sequence $f_n(x) = \chi(x/n)f(x)$ converge to $f$ in the $L^p(\mathbb{R}^n)$ norm, where $f \in L^p(\mathbb{R}^n)$ and $\chi \in C^0(\mathbb{R}^n)$ is a compactly supported function such that $\chi \geq 0$ and $\chi(x) = 1$ for $|x| \leq 1$?

Answer: [1, \infty)


Question: Find the coordinates of the point on the line segment joining the points \((-3, 7, -13)\) and \((-6, 1, -10)\) that is nearest to the intersection of the planes \(3x-y-3z+32=0\) and \(3x+2y-15z=8\).

Answer: (-6, 1, -10)


Question: Determine the positive integers \(k\) for which the series \(\sum_{n=1}^{\infty}\frac{(n!)^2}{(kn)!}\) is convergent.

Answer: k \geq 2


Question: A cylinder has a height of 6 and a radius of 3, with the centers of its bases at (0,0,0) and (0,0,6). Determine the volume of the smallest convex polyhedron that encloses all integer lattice points inside the cylinder.

Answer: 144


Question:
Let $K(x)$ be a bounded and continuous function, and let $\phi_n \to \delta$ weakly in the space of tempered distributions, where $\phi_n$ is a Schwartz function. Determine whether the following limit holds:
\[
\lim_{n \to \infty} \int_{\mathbb{R}} \phi_n(x) (K(x) - K(0)) \, dx = 0.
\]

Answer: 0


Question: Given a list of numbers \([a_1, a_2, a_3, \ldots, a_n]\) and a target value \(t\), determine the number of non-negative integer solutions to the equation \(x_1a_1 + x_2a_2 + \ldots + x_na_n = t\), where each \(x_i\) is a non-negative integer.

Answer: [z^t] \prod_{i=1}^{n} \frac{1}{1 - z^{a_i}}


Question: Let $V$ be a finite-dimensional inner product vector space over $\mathbb{R}$, and let $T : V \to V$ be a normal operator (i.e., $T T^\ast = T^\ast T$) whose characteristic polynomial splits over $\mathbb{R}$. Determine whether $T$ admits an orthonormal basis of eigenvectors.

Answer: Yes


Question: Calculate the angle of rotation \(\theta\) between two points \(A\) and \(B\) on a helix that wraps around a cylinder with a diameter of \(2r\) and a length \(L\). The helix makes an acute angle \(\alpha\) with the vertical axis of the cylinder. Given \(\alpha = 0.3^\circ\), \(L = 1.5\), and \(r = 0.2\), find the angle \(\theta\) in degrees, noting that \(\theta\) can exceed \(360^\circ\) if the helix completes more than one full revolution around the cylinder.

Answer: 2.25


Question:
Determine the number of possible values of \(\theta\), where \(0 < \theta < \pi\), for which the system of equations
\[ (y + z)\cos 3\theta = xyz\sin 3\theta \]
\[ x\sin 3\theta = \frac{2\cos 3\theta}{y} + \frac{2\sin 3\theta}{z} \]
\[ xyz\sin 3\theta = (y + 2z)\cos 3\theta + y\sin 3\theta \]
has a solution \((x_0, y_0, z_0)\) with \(y_0z_0 \neq 0\).

Answer: 3


Question: Calculate the expectation \( E\left(\int^1_0 W(s)\,ds \int^1_0 tW(t)\,dt\right) \), where \( W(t) \) is a Wiener process. The integrals are evaluated from 0 to 1.

Answer: \dfrac{5}{24}


Question: Let $\mathcal{X}$ be a finite set, and consider the function class $\mathcal{F}_k=\{f: \vert \text{supp}(f)\vert \leq k\}$, where the support of a binary function $f: \mathcal{X} \rightarrow \{0,1\}$ is defined as $\text{supp}(f)=\{x\in\mathcal{X}: f(x)=1\}$. For any $k\leq \vert \mathcal{X}\vert$, find the VC dimension of $\mathcal{F}_k$. Express your answer as $VC(\mathcal{F}_k)$. \(\mathcal{X}\) is a finite set with $n$ elements.

Answer: k


Question: Evaluate the limit \( \lim_{n\to \infty} nI_n \), where \( I_n=\int_0^1 \frac{x^n}{x^2+2019}\,\mathrm{d}x \).

Answer: \dfrac{1}{2020}


Question: Determine the natural numbers \( n \) for which \( 3^n + 5^n \) is divisible by \( 3^{n-1} + 5^{n-1} \).

Answer: 1


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

Answer: \dfrac{1}{2}


Question: Determine whether the following statement is true or false: If \( \sum C_n 6^n \) converges, then \( \sum C_n (-6)^n \) also converges. Consider the case where \( C_n \) are not necessarily positive.

Answer: B


Question: Let $\mathbb{D}$ denote the open unit disk of the complex plane and let $f : \mathbb{D} \rightarrow \mathbb{C}$ be an analytic function. Assume that $f$ has non-tangential limits on a dense set on $\partial\mathbb{D}$ such that the values can be extended to a continuous function on $\partial\mathbb{D}$. Is it sufficient to conclude that $f$ can be extended to a continuous function on $\overline{\mathbb{D}}$ that is analytic on its interior?

Answer: Yes


Question: Let $X \in M_n(\mathbb{C})$ be a matrix and $m > n$ a positive integer. If $X^m = O_n$, does it necessarily follow that $X^n = O_n$? Justify your answer.

Answer: Yes


Question: Does the 14-dimensional compact Lie group $G_2$ contain a nontrivial normal subgroup? Specifically, does $G_2$ contain a normal subgroup isomorphic to $\mathbb{Z}_2$?

Answer: No


Question: Let $X$ be a topological space. Consider the projection function $f: X^2 \to X$ defined by $f(x, y) = x$. Is the function $f$ continuous?

Answer: Yes


Question: Compute the limit: \[ \lim_{x\to\infty} \left( \frac{x^2+x+1}{3x^2+2x+1} \right) ^{\left( \frac{3x^2+x}{x-2} \right)} \]

Answer: 0


Question: Does there always exist an isometric embedding from a finite metric space $X$ to $\mathbb{R}^4$?

Answer: No


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

Answer: 0


Question: Let \( \{ a_{n} \} \) be a sequence of real numbers. Define the set \( A \) as the set of all real numbers \( x \) that belong to infinitely many open intervals \( (a_{n}, a_{n} + 2^{-n}) \). Determine if the Lebesgue measure \( \mu(A) = 0 \). Justify your conclusion.

Answer: 0


Question:
Given a prime number $p$ and an integer $n > 1$, define the expression:
\[ F(n) = \frac{n^p - 1}{n - 1} = n^{p-1} + n^{p-2} + \dots + 1 \]
Is every prime factor of $F(n)$ either equal to $p$ or congruent to $1 \pmod{p}$?

Answer: Yes


Question: Assume we have a sequence of non-negative values \(x_n\) in \(\ell_1\) that converges to \(c\) in \(\ell_1\). Is it true that this sequence also converges in \(\ell_p\) for any \(p > 1\)? Justify your answer.

Answer: Yes


Question: Let $X$ be a smooth projective variety with a polyhedral finitely generated effective cone $Eff(X)$. Consider a birational automorphism $f: X \dashrightarrow X$ that is an isomorphism in codimension one, meaning neither $f$ nor $f^{-1}$ contracts any divisor. Does $f$ necessarily map a divisor generating an extremal ray of $Eff(X)$ to a divisor with the same property? In other words, does the automorphism induced by $f$ on $Pic(X)$ preserve the set of extremal rays of $Eff(X)$?

Answer: Yes


Question: Find the hexadecimal value of A that satisfies the equation: \((7EFEFEFF + A) \oplus (A \oplus FFFFFFFF) = 81010100\), where '+' denotes bitwise addition and '\oplus' denotes bitwise XOR.

Answer: 81010100


Question: Let $G$ be a finite non-abelian group such that all proper subgroups of $G$ are cyclic. Does this imply that $G$ has exactly one prime divisor? Answer yes or no.

Answer: No


Question: Given the function \( f(x) = 3x \int_{4}^{x^2} e^{-\sqrt{t}} \, dt \), find \( f'(2) \).

Answer: \dfrac{24}{e^2}


Question: Calculate the limit: $$\lim_{n \to \infty} \left(\frac{n}{n+3}\right)^\sqrt{n(1+n)}$$

Answer: e^{-3}


Question: Find the remainder when \(7^{7^{7}}\) is divided by 32.

Answer: 23


Question:
Consider a set $G$ with the following properties:

1. $G$ is closed under an operation $\circ$.
2. $G$ is associative under $\circ$.
3. There exists an element $e \in G$ such that for every $g \in G$, $e \circ g = g$.
4. For every $g \in G$, there exists $g^{-1} \in G$ such that $g \circ g^{-1} = e$.

Does $G$ form a group under these conditions?

Answer: Yes


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

Answer: \infty


Question: Evaluate the definite integral \( \int_{0}^{2\pi}\frac{\cos(t)}{\cos(t) + i \sin(t)}\;dt \).

Answer: \pi


Question: Calculate the flux of the vector field \( F(x,y,z) = \langle x,y,z \rangle \) across the surface of the cone defined by \( z^2 = x^2 + y^2 \) for \( 0 \leq z \leq 1 \), with normal vectors pointing upward.

Answer: 0


Question: How many balls of radius \(\frac{1}{2}\) are needed to completely cover a ball of radius 1 in \(\mathbb{R}^3\)?

Answer: 13


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

Answer: e^{-3}


Question: Given the polynomial equation \( f(x) = x^4 - Lx^3 - 2x^2 + 6x - 4 = 0 \), find the value of the parameter \( L \) such that the sum of two of its roots is \(-1\).

Answer: 1


Question: Let $A$ and $B$ be independent random variables uniformly distributed over the interval $(0, 1)$. Divide the interval $(0, 1)$ into three segments using $A$ and $B$ as dividing points. Do the lengths of these three segments necessarily have the same distribution?

Answer: Yes


Question: Determine the value of \(\alpha\) for which the series \(\sum_{n=1}^{\infty} \left(n^{1/n} - 1\right)^{\alpha}\) converges.

Answer: \alpha > 1


Question: Determine the number of group homomorphisms from \( S_3 \) to \( S_3 \).

Answer: 10


Question: What is the probability that for two randomly chosen natural numbers \(a\) and \(b\), the fraction \(\frac{a}{b}\) is a natural number?

Answer: 0


Question: Does every commutative monoid admit a translation-invariant measure when equipped with the Borel $\sigma$-algebra $\mathcal B(T)$ derived from its canonical pre-order topology?

Answer: No


Question: Given a polynomial $f(x)$ of degree 9 such that $f(k) = k$ for all $k \in \{1, 2, \ldots, 10\}$, find the value of $f(100)$. \( \boxed{} \)

Answer: 100


Question: An urn contains 18 differently-colored balls. You draw one ball at a time, with replacement. How many draws are required to have a 90% probability of obtaining at least one ball of each color?

Answer: 91


Question:
Let \(T: \mathbb{R}^n \to \mathbb{R}^m\) be a linear map such that \(\langle Tx, Ty \rangle_m = \langle x, y \rangle_n\) for all vectors \(x, y \in \mathbb{R}^n\), where \(\langle , \rangle_n\) and \(\langle , \rangle_m\) denote the standard inner products on \(\mathbb{R}^n\) and \(\mathbb{R}^m\), respectively. Which of the following is necessarily true?\
(a) \(n \geq m\) \
(b) \(n \leq m\) \
(c) \(n = m\) \
(d) The map \(T\) is onto.

Answer: B


Question: Evaluate the line integral of the function \( f(x, y, z) = \sqrt{1 + 4x^2z^2} \) over the curve of intersection of the surfaces \( x^2 + z^2 = 1 \) and \( y = x^2 \).

Answer: 3\pi


Question: Find the minimum value of \( q \) such that the distance between the point \( P(u,v) \) and the curve \( x^2 + 4x + y^2 = 0 \) is equal to the distance between \( P(u,v) \) and the point \( M(2,0) \), given that \( u^2 - \frac{v^2}{q} = 1 \).

Answer: 3


Question:
Determine which of the following subsets of sequences of roots of unity is countable:

1. The set $A$ of all sequences $(x_n) \in S(X)$ such that $(x_n)$ is eventually constant.
2. The set $B$ of all sequences $(x_n) \in S(X)$ such that $x_n = 1$ whenever $n$ is a prime number.
3. The set $C$ of all sequences $(x_n) \in S(X)$ such that each $x_n$ is a 26th root of unity.
4. The set $D$ of all sequences $(x_n) \in S(X)$ such that $x_{2n} = 1$ for all $n \geq 1$.

Which subset is countable?

Answer: A


Question: Evaluate the surface integral \( I = \iint_S \vec{A} \cdot d\vec{s} \) over the entire surface of the region above the \(xy\)-plane bounded by the cone \(x^2 + y^2 = z^2\) and the plane \(z = 4\), where \(\vec{A} = 4xz \hat{i} + xyz^2 \hat{j} + 3z \hat{k}\).

Answer: 320\pi


Question:
Assume $X$ is a vector space contained in $H^{1}(\mathbb{R}^d)$, and consider the orthogonal complements $X^{\perp_{L^2}}$ and $X^{\perp_{H^{-1}}}$ defined as follows:

- $X^{\perp_{L^2}} := \{ \xi \in L^2; \langle \xi, x \rangle_{L^2} = 0 \ \forall x \in X \}$
- $X^{\perp_{H^{-1}}} := \{ \xi \in H^{-1}; \langle \widehat{\xi}, \widehat{x} \rangle_{H^{-1},H^1} = 0 \ \forall x \in X \}$

If $X^{\perp_{L^2}}$ is a finite-dimensional space, does it imply that $X^{\perp_{H^{-1}}} = X^{\perp_{L^2}}$?

Answer: Yes


Question:
Prove that the following series converges to 1:
\[ \sum_{j=1}^{\infty} \frac{e^{-j}j^{j-1}}{j!} = 1. \]

Answer: 1


Question: Find the constant \(a\) such that the Laplace equation \(u_{xx} + u_{yy} = 0\) for \(1 < x < 2, 1 < y < 2\) with boundary conditions \(u_x(1,y) = y\), \(u_x(2,y) = 5\), \(u_y(x,1) = a\frac{x^2}{7}\), and \(u_y(x,2) = x\) has a solution.

Answer: 15


Question: Solve for \(x\) in the congruence equation \(11^x \equiv 21 \pmod{71}\).

Answer: 37


Question: Evaluate the limit: \[ \lim_{n\to\infty} \int_{0}^{\infty} \frac{x^n}{1+x^n} \, dx \]

Answer: \infty


Question:
Consider i.i.d. random variables $X_{1}, X_{2}, \ldots$, such that $X_{i}$ has a uniform distribution over the interval $(0, 1)$. Find the limit:
\[\lim_{n \to \infty} P\left(\sum_{i=1}^{n} X_i \leq 2 \sum_{i=1}^n{X_{i}^2}\right).\]

Answer: 1


Question: Determine the number of real solutions to the equation \( \sin 14x - \sin 12x + 8\sin x - \cos 13x = 4 \) within the interval \([0, 2\pi]\).

Answer: 2


Question: Let $T = A + iB$, where $A$ and $B$ are self-adjoint operators on a Hilbert space $H$. Compute the adjoint $T^*$ in terms of $A$ and $B$.

Answer: A - iB


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function and \( g: \mathbb{R} \to [0, \infty) \) be a function such that \( f(x) + f''(x) = -xg(x)f'(x) \) for all \( x \in \mathbb{R} \). Is it true that \( f \) is bounded, i.e., there exists \( M \ge 0 \) such that \( |f(x)| \le M \) for all \( x \in \mathbb{R} \)?

Answer: Yes


Question:
Given the system of equations:
\[\begin{cases}
2^x+3^y=5^z \\
2^y+3^z=5^x \\
2^z+3^x=5^y
\end{cases}\]
where \(x, y, z \in \mathbb{R}\), determine if it implies that \(x = y = z = 1\).

Answer: x = y = z = 1


Question: Find the 2001st derivative of the function \( f(x) = \frac{x^{5}}{x^{6}+1} \) at \( x = 0 \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{x - \sin x - \cos^{-1}(e^{-\frac{x^4}{2}})}{x^2} \]

Answer: -1


Question: Find the positive integer \( n \) such that \( n < 6(1-1.001^{-1000}) < n+1 \).

Answer: 3


Question: Find the volume of the solid enclosed by the parabolic cylinder \( y = 16x^2 \) and the planes \( z = 3y \) and \( z = 2 + y \) by subtracting two volumes.

Answer: \dfrac{4}{15}


Question: Determine the dimension of \( \mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q} \) as a vector space over \( \mathbb{Q} \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \to \infty}\lim_{n \to \infty} \frac{e^x}{x^n} \]

Answer: 0


Question: Find the minimum value of \( P = ab + bc + 3ca + \frac{3}{a+b+c} \) given that \( a, b, c \) are non-negative real numbers such that \( a^2 + b^2 + c^2 = 3 \).

Answer: \sqrt{3}


Question: Determine if there exists a real number greater than 16 that can be expressed as \(A^B = B^A\) for two distinct real numbers \(A\) and \(B\).

Answer: Yes


Question: Find the exact curvature of the curve given by the vector function \( \mathbf{r}(t) = (\sinh(t), \cosh(t), t) \) at the point \((0, 1, 0)\).

Answer: \dfrac{1}{2}


Question: Let \( \mathcal{U} \) be a selective ultrafilter and \( \omega = \coprod_{i<\omega}S_i \) be a partition of \( \omega \) with small sets \( S_i \notin \mathcal{U} \). All \( S_i \) are infinite. Does there exist a system of bijections \( \varphi_i: \omega \to S_i \) such that for any big set \( B \in \mathcal{U} \) and any system of big sets \( \{B_i\}_{i\in B} \), the set \( \cup_{i\in B}\varphi_i(B_i) \) is big?

Answer: No


Question: Find the area of the surface S, which is the portion of the cylinder defined by \(x^2+y^2=a^2\) that is bounded below by the \(xy\)-plane and above by the cone \(z=\sqrt{(x-a)^2+y^2}\).

Answer: 8a^2


Question:
Given a semicircle with diameter \( AB = 2 \), let \( D \) and \( C \) be points on the tangents at \( A \) and \( B \) respectively, such that \( \overline{AD} = \overline{CB} = 1 \). Let \( P \) be a point on the semicircle such that \( P\hat{A}B = x \). Solve the equation:
\[ \overline{PC}^2 + 2 \overline{PD}^2 + 2\overline{AP}^2 = 7. \]

Answer: \dfrac{\pi}{4}


Question:
Calculate the expected value of the random variable $Xe^X$, where $X$ follows a standard normal distribution $N(0,1)$. Use integration by parts to evaluate the integral:
\[ E(Xe^X) = \int_{-\infty}^{\infty} xe^x \frac{1}{\sqrt{2\pi}} e^{-0.5x^2} \, dx \]

Answer: \sqrt{e}


Question:
Solve the system of non-linear differential equations given by:
\[
\begin{array}{rcl}
\displaystyle \frac{dx}{dt}&=&y^{2} \\
\displaystyle \frac{dy}{dt}&=&x^{2}
\end{array}
\]

Answer: y^3 - x^3 = C


Question: Let $f$ and $g$ be two homomorphisms from $\mathrm{SO}(3)$ to $PGL(2,\mathbb{C})$. Determine whether there exists an element $S \in \mathrm{PGL}(2,\mathbb{C})$ such that for all $X \in \mathrm{SO}(3)$, the equation $g(X) = Sf(X)S^{-1}$ holds.

Answer: Yes


Question: Find all integer solutions $(x, y)$ to the equation $x^5 - 1 = y^2$. Provide your answer as a set of ordered pairs.

Answer: (1, 0)


Question:
Evaluate the following limit as \( x \to \infty \):
\[
\lim_{x \to \infty} \frac{\left(1 + x\right)\left(1 + 1/x\right)^{2x} - x\,\mathrm{e}^{2}}{\mathrm{e}^{2} - \left(1 + 1/x\right)^{2x}}
\]

Answer: \dfrac{1}{6}


Question: Find a non-zero solution to the equation \(3\sin(\sqrt{x})=x\).

Answer: 2.966


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty}\int_0^{2 \pi} \frac{\sin(nx)}{x^2 +n^2}dx \]

Answer: 0


Question: Evaluate the limit \( \lim_{x\to 0} \int _0 ^ {x} (1- \tan (2t) ) ^ {\frac{1}{t}}\ dt \).

Answer: 0


Question: Determine the Picard group of the ring $R = \bar{K} \otimes_K \bar{K}$, where $K$ is a field.

Answer: 0


Question: Find the value of \( \lim_{x \to \pi/2} \left\lfloor \frac{x - \pi/2}{\cos x} \right\rfloor \).

Answer: -2


Question:
Determine the correct formula for the projection of a matrix $X$ into the row space of a matrix $Y$. Is it:

1. \( \hat{X} = XY^T(YY^T)^{-1}Y \)

or

2. \( \hat{X} = Y^T(YY^T)^{-1}YX \)?

Answer: 1


Question: Calculate the limit: \[ \lim_{x \to 0} \frac{\frac{\mathrm{d}}{\mathrm{d}x} (e^{\sec x})}{\frac{\mathrm{d}}{\mathrm{d}x} (e^{\sec 2x})} \]

Answer: \dfrac{1}{4}


Question: Using Taylor's theorem, determine the largest positive real value \( r \) for which the Maclaurin polynomial \( P_6(x) \) of \( f(x) = (\sin(x))^2 \) has an error of less than 0.2 on the interval \( 0 < x < r \).

Answer: \sqrt[8]{63}


Question:
Suppose \( f \) is a differentiable function for which \( \lim_{x \to \infty} f(x) \) and \( \lim_{x \to \infty} \frac{df(x)}{dx} \) both exist and are finite. Which of the following must be true?

(A) \( \lim_{x \to \infty} \frac{df(x)}{dx} = 0 \)
(B) \( \lim_{x \to \infty} \frac{d^2f(x)}{dx^2} = 0 \)
(C) \( \lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{df(x)}{dx} \)
(D) \( f \) is a constant function
(E) \( \frac{df(x)}{dx} \) is a constant function

Answer: A


Question: Let $O_1$, $O_2$, and $O_3$ be circles of radius 1, each intersecting the others. Consider a point $P$ on circle $O_2$ and a point $Q$ on circle $O_3$. Determine the maximum value of the dot product $\vec{O_1P} \cdot \vec{O_1Q}$. Provide your answer with justification.

Answer: 9


Question: Evaluate \( g\left(\frac{1}{3}\right) \) for the function \( g(x) = \sum_{n=0}^{\infty} \lceil \frac{n+1}{2} \rceil x^n \). Verify if \( g(x) = \frac{1}{x^3-x^2-x+1} \).

Answer: \dfrac{27}{16}


Question: Given the function \( f(z) = \sqrt[3]{z^2 - 1} \), where \( f(3) = 2 \), find the value of \( f(-3) \) by considering the analytic continuation of \( f \) along the semicircle \( |z| = 3 \) in the lower half-plane.

Answer: -1 - i\sqrt{3}


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

Answer: e - 1


Question: Find all real numbers $a > 0$ such that for all $x > 0$, the inequality $a^x \geq x^a$ holds.

Answer: e


Question: Find the minimum value of the expression \( \frac{x^4 + y^4 + z^2}{xyz} \) for real numbers \(x\), \(y\), and \(z\).

Answer: 2\sqrt{2}


Question: Evaluate the integral: \[ \int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}} \, dx \]

Answer: 1005


Question: Suppose $f : (a,b) \to \mathbb{R}$ is a function such that $f'$ exists and is bounded on $(a,b)$. Is $f$ uniformly continuous on $(a,b)$? Justify your answer.

Answer: Yes


Question: Solve for \(x\) in the equation: \[ \frac{\sin (10^\circ) \sin (30^\circ)}{\sin 40^\circ \sin (80^\circ-x^\circ)} = \frac{\sin 20^\circ}{\sin x} \]

Answer: 60


Question: Determine the cardinality of the set of all limited sets of lattice points, where a limited set is defined as a set of points that can be contained within a circle.

Answer: \aleph_0


Question: Given a frequency response $H(f)$, find the inverse Fourier transform of $|H(f)|^2$ in terms of a convolution.

Answer: h(t) * h^*(-t)


Question: Let $f_n:E \to \mathbb{R}$ be a sequence of functions converging in measure to $f:E \to \mathbb{R}$, where $m(E)< \infty$. Let $\eta >0$. Prove or disprove that $\lim_{n \to \infty} m\{x:|f_n(x)-f(x)||f_n(x)|> \eta\}=0$.

Answer: 0


Question: Determine the center of the dihedral group $D_6$ and show that it is isomorphic to $\mathbb{Z}_2$. The group $D_6$ is defined by the presentation $\langle a, b \mid a^6 = b^2 = e,\, ba = a^{-1}b \rangle$. Provide a proof without appealing to the geometric interpretation of $D_6$ as symmetries of a hexagon.

Answer: \mathbb{Z}_2


Question: Given a discrete random variable \( \xi \) that takes values greater than 1, with probability mass function \( P(\xi = k) = \frac{C}{k(k+1)(k+2)} \) for \( k = 2, 3, 4, \ldots \), find the expected value \( E(\xi) \).

Answer: 4


Question: Find the limit: \[ \lim_{x \to 0} \left( \int_{0}^{x} e^{t^2} \, dt \right)^{1/x} \]

Answer: 0


Question: Determine the topological space that the quotient space $D^2/S^1$ is homeomorphic to, where $D^2$ is the unit disk in $\mathbb{R}^2$ and $S^1$ is its boundary. Is it $S^2$? Justify your answer.

Answer: S^2


Question: A stock's price follows a model where, if the current price is \( s \), after one period, it will be either \( us \) with probability \( p \) or \( ds \) with probability \( 1 - p \). Assuming successive movements are independent, approximate the probability that the stock's price will be up at least 30% after the next 1000 periods, given \( u = 1.012 \), \( d = 0.990 \), and \( p = 0.52 \).

Answer: 0.999


Question: Let $k$ be an algebraically closed field, and $A$ a finitely generated $k$-algebra. Consider two distinct closed points $x$ and $y$ of $\mathrm{Spec}(A)$ with corresponding maximal ideals $\mathfrak{m}_x$ and $\mathfrak{m}_y$. Define $I = \mathfrak{m}_x \cap \mathfrak{m}_y$ and let $B$ be the $k$-subalgebra of $A$ generated by $k$ and the elements of $I$. Is $B$ a finitely generated algebra over $k$?

Answer: Yes


Question: Compute the expected value \( \mathrm{E}\left[\sum _{i=1}^X I_{\{Y_i\leq Y_{n+1}\}}\right] \), where \( Y_1,\dots,Y_{n+1} \) is a random sample from a continuous distribution function \( F \), and \( X \sim \mathrm{Uniform}\{1,\dots,n\} \) is independent of the \( Y_i \)'s.

Answer: \dfrac{n+1}{4}


Question: Find the zeros of the function \( h(z) = \frac{1}{3}e^z - z \) inside the unit disk \( |z| \leq 1 \) using Rouche's Theorem.

Answer: 1


Question: Find the area of the image of the region \( \{0 < x, y < 1\} \) under the transformation \( f(x,y) = (e^{x+y}, e^{x-y}) \).

Answer: e^2 - 1


Question: Let $f$ be a meromorphic function and $a$ a pole of $f$. Let $U$ be a neighborhood of $a$. Is it true that $f(U)$ contains the set $\{z \mid |z| > r\}$ for some $r$?

Answer: Yes


Question: Find the difference between the greatest and smallest values of $S(n,k,5)$ for $k=0,1,2,3,4$, where $$S(n,k,m) = \sum_{r=0}^{\left\lfloor\frac {n-k}{m}\right\rfloor}\binom{n}{k+rm}.$$

Answer: F_{n+1}


Question: Find the volume of the solid with height given by the function \( h(x,y) = xy \) and base \( D \), where \( D \) is the region bounded by the curves \( y = x + 2 \) and \( y = x^2 \).

Answer: \dfrac{45}{8}


Question: Consider a matrix $A \in \mathbb{F}^{m \times n}$ where $m < n$. Determine if the matrix $A^tA$ has a zero eigenvalue with multiplicity $n-m$. Provide a justification for your answer.

Answer: No


Question:
Given sequences \( a_n \) and \( b_n \) of positive numbers such that \( \lim_{n\to\infty}a_n^n = a \) and \( \lim_{n\to\infty}b_n^n = b \), where \( a, b \in (0, \infty) \), and constants \( p \) and \( q \) such that \( p + q = 1 \), find the limit:
\[ \lim_{n\to\infty}(pa_n + qb_n)^n. \]

Answer: a^p b^q


Question: Consider a hypercube of dimension $2n$ with $2^{2n}$ vertices, where each vertex is represented by a binary string $(a_1,a_2,\ldots,a_{2n})$. Two vertices are connected by an edge if they differ in exactly one coordinate. Initially, there is one ant at each vertex. Each ant aims to move to the vertex $(a_{n+1},a_{n+2},\ldots,a_{2n},a_1,a_2,\ldots,a_n)$ by fixing the foremost differing bit at each step. Each edge can be traversed by only one ant per second, and each traversal takes one second. Determine the minimum time required for all ants to reach their destinations.

Answer: 2n


Question: Evaluate the integral \( \int_{\gamma} \frac{|dz|}{z} \) where \( \gamma \) is the unit circle.

Answer: 0


Question: Let $R$ be a Noetherian domain of dimension at least 1, and let $T = R[[X]]$, where $X$ is an indeterminate. For each prime ideal $\mathfrak{p}_i$ of height one in $R$, define $\mathfrak{q}_i = \mathfrak{p}_i[[X]]$. Consider the set $S = T \setminus \bigcup_{i \ge 1} \mathfrak{q}_i$. Determine if the dimension of the localization $T_S$ is 1.

Answer: 1


Question: Let $C$ be a projective nonsingular irreducible curve, and let $D$ be a divisor on $C$ with $l(D) = n > 0$ and $L(D) = \langle f_1, \ldots, f_n \rangle$. Consider the map $\varphi_D : C \to \mathbb{P}^{n - 1}$ defined by $\varphi_D(P) = (f_1(P) : \cdots : f_n(P))$. If $D$ and $D'$ are linearly equivalent divisors, does there exist a projective linear transformation $\psi : \mathbb{P}^{n - 1} \to \mathbb{P}^{n - 1}$ such that $\varphi_{D'}(C) = \psi(\varphi_D(C))$? Answer yes or no.

Answer: Yes


Question: Calculate the limit: \[ \lim_{n \to \infty} \int_0^1 \frac{n \cos(x)}{1+x^2n^{1.5}}\,dx \]

Answer: \infty


Question: Given the function \( f(x) = x^3 - 12x^2 + 69x + 6 \), compute the value of \( \int_{64}^{104} f^{-1}(x) \, dx \).

Answer: \dfrac{235}{4}


Question: Suppose $H$ is an indefinite quaternion algebra over $\mathbb{Q}$. Determine whether there are infinitely many quadratic fields that can be embedded into $H$. Answer 'finite' or 'infinite'.

Answer: infinite


Question: Evaluate the double integral \( \int\int_{D}\frac{1}{x^2+y^2}dx \, dy \) over the domain \( D=\{(x,y)\in \mathbb{R}^2 :1\leq x^2+y^2\leq4,0\leq y \} \).

Answer: \pi \ln 2


Question: Assume the series \( \sum |f_n| \) is uniformly Cauchy. Does this imply that the series \( \sum f_n \) is also uniformly Cauchy? Justify your answer.

Answer: Yes


Question: Evaluate the line integral \( \int_C (3xy^2 \, dx + 8x^3 \, dy) \) where \( C \) is the boundary of the region between the circles \( x^2 + y^2 = 1 \) and \( x^2 + y^2 = 64 \), oriented positively.

Answer: 24570\pi


Question: Approximate the probability that a simple symmetric random walk \( \{X_n\} \) with \( p = 1/2 \) satisfies \( X_n \in [-3\sqrt{n}, 2\sqrt{n}] \) for large \( n \). Use the central limit theorem to express this probability in terms of a standard normal distribution.

Answer: \Phi(2) - \Phi(-3)


Question: Evaluate the limit: \[ \lim_{\substack{x \rightarrow 2 \\ y \rightarrow 1}} \frac{\arcsin(xy-2)}{\arctan(3xy-6)} \]

Answer: \dfrac{1}{3}


Question: Evaluate the integral \( \int_C \frac{1}{2z+1} \, dz \), where \( C \) is the square with corners at \( 1+i \) and \( -1-i \).

Answer: \pi i


Question: Solve the nonlinear partial differential equation \(xu_x+yu_y+uu_z = 0\) subject to the condition \(u(x,y,0) = xy\) for \(x > 0\) and \(y > 0\), using the method of characteristics.

Answer: u = xy e^{-2z/u}


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

Answer: 0


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

Answer: e^2


Question: Given that \( \int_{-x}^{x} f(t) \, dt = x^3 - x^2 + x + 1 \), find the value of \( f(-2) + f(2) \).

Answer: 9


Question: Determine whether the following statement is true or false: Let \( f(x, y) \) be a function of two variables defined on a neighborhood \( (0,0) \subseteq \mathbb{R}^2 \). If there exists a partial derivative of the function \( f \) at the point \( (0,0) \), then the function is continuous at the point \( (0,0) \).

Answer: B


Question: Find the multiplicative inverse of \( x \) in the quotient ring \( \mathbb{Z}_2[x]/(f(x)) \), where \( f(x) = x^3 + x^2 + 1 \).

Answer: x^2 + x


Question: Evaluate the integral \( \int_{\Gamma_r} \frac{1}{z} \, dz \) where \( \Gamma_r \) is a closed curve around the origin. Use the parameterization \( \Gamma_r(t) = Re^{it} \) for \( t \in [0, 2\pi] \).

Answer: 2\pi i


Question: Given a set \( S \) of size 100, find the maximum size of a collection \( G \) of subsets of \( S \), each of size 5, such that any two different subsets in \( G \) have different intersections of size 4. What is the maximum size of \( G \)?

Answer: 2800


Question: Let $f$ be a real trigonometric polynomial of degree $N$ with a constant term of $0$. Determine a lower bound on the measure $\mu$ of the set $\{ t \in S^1 : f(t) \geq 0 \}$, independent of the coefficients of the polynomial.

Answer: \pi


Question: Evaluate the integral \( \int_{-a\pi}^{a\pi} \frac{\cos^5(x)+1}{e^x+1}dx \), where \( a \in \mathbb{N} \).

Answer: a\pi


Question: Determine whether the following statement is true or false: A function \( f: X \times X \to A \) is continuous if and only if the functions \( f_{1,x}(y) = f(x,y) \) and \( f_{2,x}(y) = f(y,x) \) are continuous for all \( x \in X \).

Answer: B


Question: Calculate the limit of the function \( f(x) = \frac{x^{2}i+e^{x}}{2^{x}+e^{x}i} \) as \( x \to \infty \).

Answer: -i


Question: Determine whether the following statement is true or false: There exists a function \( f: \mathbb{R} \to \mathbb{R} \) such that for every \( a \in \mathbb{R} \) and \( \epsilon > 0 \), there is a \( \delta > 0 \) such that if \( |f(x) - f(a)| < \epsilon \), then \( |x - a| < \delta \).

Answer: True


Question: Evaluate the limit \( \lim_{n\rightarrow\infty}(a_n)^{-1/n^2} \), where \( a_n = \sum_{k=1}^n \left(1 + \frac{k}{n^2} \right)^k \).

Answer: 1


Question: Find the value of \( x \) such that \( \sum\limits_{n=0}^{57} \frac{x^n}{n!} = 1.586 \times 10^{15} \).

Answer: 35


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty}(\sin(\sqrt[3]{n^3-9})-\sin(n)) \]

Answer: 0


Question: Find the value of \(x\) to three decimal places for the equation \(4^x = \frac{5}{x}\).

Answer: 1.095


Question: Let \( \{ X_\alpha \}_{\alpha \in \Lambda} \) be a collection of simply connected spaces. Prove or disprove that the fundamental group of the product space \( \prod_{\alpha \in \Lambda} X_\alpha \) is trivial, i.e., \( \pi_1(\prod_{\alpha \in \Lambda} X_\alpha) = 0 \).

Answer: \pi_1\left( \prod_{\alpha \in \Lambda} X_\alpha \right) = 0


Question:
Given three lights in a line, each can be in one of three states: OFF, LIGHT RED, and DARK RED, cycling in that order. There are three switches:
- Switch A advances the cycle for the first two lights.
- Switch B advances the cycle for all three lights.
- Switch C advances the cycle for the last two lights.

Starting with all lights OFF, determine if it is possible to use the switches in some order to achieve the configuration: OFF-LIGHT RED-DARK RED.

Answer: Yes


Question: Find the kernel of the ring homomorphism \( f: R \rightarrow R/I \times R/J \) defined by \( f(a) = (a+I, a+J) \), where \( I \) and \( J \) are ideals of a ring \( R \).

Answer: I \cap J


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

Answer: e - 2


Question: Consider two bit strings of even length L: the string consisting entirely of 1s, denoted as 11...1, and the alternating string 1010...10. Determine whether the Kolmogorov Complexity of the string 11...1 is less than that of the string 1010...10. Assume the Kolmogorov Complexity is defined as the length of the shortest string <M,w> such that the Turing machine M on input w halts with the bit string on its tape, independent of any particular programming language.

Answer: Yes


Question: Given the recursive sequence defined by \( a_{n+1} = \frac{a_n}{1 + \frac{1}{n} + a_n + a_n^3} \) with \( a_1 > 0 \), find \( \lim_{n \to \infty} a_n \).

Answer: 0


Question:
Given the sequence \( x_n \) where each term comes from the set \( \{2, 3, 6\} \), the following conditions hold:

1. \( x_1 + x_2 + x_3 + \cdots + x_n = 633 \)
2. \( \frac{1}{{x_1}^2} + \frac{1}{{x_2}^2} + \frac{1}{{x_3}^2} + \cdots + \frac{1}{{x_n}^2} = \frac{2017}{36} \)

Determine the value of \( n \).

Answer: 262


Question: Calculate the expected value \( E[X(u)X(u+v)X(u+v+w)] \) for a Brownian motion \( X(t) \), where \( 0 < u < u+v < u+v+w \).

Answer: 0


Question: Find all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \( xf(x) + f(1-x) = x^3 - x \) for all real numbers \( x \).

Answer: x^2 - x


Question: Evaluate the expression: \[ \tan\left(\sum_{r=1}^{\infty} \arctan\left(\frac{4}{4r^2 +3}\right)\right) \]

Answer: 2


Question:
Given the equations:
\[ x^{2} - 2px -5q = (x-m)(x-n) \]
and
\[ x^{2} - 2mx -5n = (x-p)(x-q) \]
where \( p \neq q \neq m \neq n \), find the value of \( p+q+m+n \).

Answer: 30


Question: Find the Laurent series expansion of \( e^{\sin z} \) at \( z = 0 \) and determine the radius of convergence.

Answer: \infty


Question: Let $G$ be an abelian and periodic group. Define $O_{\pi}(G) = \langle N : N \trianglelefteq G \text{ and } N \text{ is a } \pi \text{-subgroup} \rangle$. If $G/O_{2'}(G)$ is a $2$-group, is $G$ necessarily a $2$-group?

Answer: No


Question: Let $V$ be a smooth algebraic variety defined over the complex numbers, and suppose that $G$ admits a free discrete action on $V$ such that $V/G$ is compact and Kähler (or algebraic). Is it true that $G$ is virtually abelian, meaning it contains an abelian subgroup of finite index?

Answer: Yes


Question:
Determine whether the following equality holds for the Dirac delta function:
\[ f(x) = \int_{-\infty}^{\infty} f(x') \delta(x-x') \, dx' = \int_{-\infty}^{\infty} f(x') \delta(x'-x) \, dx' \]

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to\infty} n \left( \frac{1+2^k+3^k+\ldots+n^k}{n^{k+1}} - \frac{1}{k+1} \right) \]

Answer: \dfrac{1}{2}


Question: Find the orthogonal trajectories of the family of curves given by the equation \(3x^2 - y^2 = c\).

Answer: x y^3 = C


Question: Is a Lebesgue measurable subgroup of \(\mathbb{R}\) necessarily a Borel subset of \(\mathbb{R}\)?

Answer: Yes


Question: Let $[a,b]$ be a closed bounded interval on the real line, and let $S$ be a set containing all real-valued monotone functions on $[a,b]$. Suppose $S$ satisfies the properties that if $f, g \in S$, then $f+g \in S$, and if $f \in S$ and $c \in \mathbb{R}$, then $cf \in S$. Is it true that $S$ contains all functions of bounded variation on $[a,b]$?

Answer: Yes


Question: Determine the number of ideals in the quotient ring \( \mathbb{R}[x]/\langle x^2-3x+2 \rangle \).

Answer: 4


Question: Given that $f$ is an analytic function from $|z|<3$ to $\mathbb{C}$ and $f\left(1+\frac{\sqrt{2}i}{n}\right)=-\frac{2}{n^2}$ for all positive integers $n$, find the value of $f(\sqrt{2})$. Assume $f(z)=(z-1)^2$ based on the identity theorem.

Answer: 3 - 2\sqrt{2}


Question: Find the minimum value of \( M \) such that there exist real numbers \( a, b, c \) satisfying the inequality \( \left|4x^3 + ax^2 + bx + c\right| \leq M \) for all \( x \) in the interval \([-1, 1]\).

Answer: 1


Question: Does a probability distribution exist that has infinite variance but a finite mean?

Answer: Yes


Question:
A function \( f: \mathbb{N^+} \rightarrow \mathbb{N^+} \), defined on the set of positive integers \( \mathbb{N^+} \), satisfies the following properties:
\[f(n)=\begin{cases} f(n/2) & \text{if } n \text{ is even}\\ f(n+5) & \text{if } n \text{ is odd} \end{cases}\]
Let \( R=\{ i \mid \exists{j} : f(j)=i \} \) be the set of distinct values that \( f \) takes. What is the maximum possible size of \( R \)?

Answer: 2


Question: Determine the splitting field of the polynomial \((x^3+x^2+1)(x^3+x+1)(x^2+x+1)\) over the finite field \(\mathbb{F}_2\).

Answer: \mathbb{F}_{2^6}