Question: Given two prime numbers \( p \) and \( q \) such that \( 3 \) does not divide \( p-1 \) nor \( q-1 \), and let \( n = pq \). Determine the number of elements in the multiplicative group \( \mathbb{Z}_n^{*} \) that can be expressed as \( b^3 \) for some integer \( b \).

Answer: (p-1)(q-1)


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

Answer: -3\pi i


Question: Calculate the residue at $z = 0$ for the function $g(z) = \frac{1 - \cos z}{z^4}$. Provide your answer as $\text{Res}(g, 0)$.

Answer: 0


Question: Compute \( \lim_{n\to\infty}\int_0^1\frac{x\sin{nx}}{1+x^2n^6}dx \).

Answer: 0


Question: Calculate the flux of the vector field \( \mathbf{F} = (-2x, -2y, xy) \) through the region \( D = \{(x,y,z) \mid y^2+z^2 \le 3|x|, (x-2)^2+y^2+z^2 \le 4\} \) using the divergence theorem. The divergence of \( \mathbf{F} \) is \( \mathrm{div}(\mathbf{F}) = -4 \). Set up and evaluate the integral to find the flux.

Answer: -42\pi


Question: Evaluate the limit: \[ \lim_{x\to0} \frac{\sin(\pi\sqrt{\cos x})}{x} \]

Answer: 0


Question:
Given the recursive sequences defined by:
\[ B_n = \sum_{i=0}^{n-1} (B_i B_{n-i-1}) \quad \text{with initial conditions} \quad B_1 = 1, \; B_0 = 1 \]
and
\[ M_n = \sum_{i=1}^{n-1} (M_i M_{n-i}) \quad \text{with initial conditions} \quad M_2 = 1, \; M_1 = 1 \]
prove that \( B_{n-1} = M_n \) for all \( n \geq 2 \).

Answer: B_{n-1} = M_n


Question: Find the imaginary part of the holomorphic function \( f(z) = z\overline{z} + iv(x,y) = x^2 + y^2 + iv(x,y) \).

Answer: -2xy


Question: A gambler plays a game where they win $1 with probability \( \frac{2}{3} \) and lose $1 with probability \( \frac{1}{3} \). The gambler's goal is to win $2 over their initial fortune. What is the minimum initial fortune required for the probability of achieving this goal to be at least 0.99?

Answer: 7


Question:
Three individuals are running a one-kilometer race. The completion time for each individual is a random variable. Let $X_1$, $X_2$, and $X_3$ be the completion times, in minutes, for persons 1, 2, and 3, respectively. The distributions are as follows:

- $X_1$: uniform distribution on the interval [2.9, 3.1]
- $X_2$: uniform distribution on the interval [2.7, 3.1]
- $X_3$: uniform distribution on the interval [2.9, 3.3]

The three completion times are independent of one another. Find the expected value of the latest completion time, $Y = \max\{X_1, X_2, X_3\}$, rounded to the nearest tenth.

Answer: 3.1


Question: Find the derivative with respect to the parameter $A$ of the solution to the differential equation \( \ddot{x} = \dot{x}^2 + x^3 \) with initial conditions \( x(0) = 0 \) and \( \dot{x}(0) = A \), evaluated at \( A = 0 \).

Answer: t


Question: Given that $\det(A) \neq 0$ for a $6 \times 6$ matrix $A$, and $2A + 7B = 0$, calculate the value of $6\det(2(A^t)^2B^{-1}A^{-1})$. Assume $B$ is invertible.

Answer: 705894


Question: Find the value of \( A \) such that the volume of the solid generated by rotating the region bounded by \( y = 1 + \sqrt{x}e^{x^2} \), \( y = 1 \), and \( x = a \) (where \( x = a \) is to the right of the y-axis) around the line \( y = 1 \) is equal to \( 2\pi \).

Answer: \sqrt{\ln 3}


Question: Determine the number of pairs of vectors \( \vec{V}_1 = a_1 \mathbf{i} + b_1 \mathbf{j} + c_1 \mathbf{k} \) and \( \vec{V}_2 = a_2 \mathbf{i} + b_2 \mathbf{j} + c_2 \mathbf{k} \) such that \( \vec{V}_1 \) and \( \vec{V}_2 \) are perpendicular, where all components \( a, b, c \) are elements of \( \{-2, -1, 1, 2\} \).

Answer: 288


Question: Evaluate the integral \( \int_0^{\infty} \frac{\sin^2 x}{x^2} \mathrm{d}x \) using the residue theorem. Consider the function \( \frac{e^{2iz}-1-2iz}{z^2} \) and integrate it within a semicircle in the complex plane.

Answer: \dfrac{\pi}{2}


Question: Determine the values of $a \in \mathbb{R}$ for which the linear operator $\delta_a: \mathbb{R}[X] \to \mathbb{R}$, defined by $\delta_a(P) = P(a)$ for all polynomials $P \in \mathbb{R}[X]$, is continuous when $\mathbb{R}[X]$ is equipped with the norm $||P||_\infty = \sup_{x \in [0, 1]} |P(x)|$. For such $a$, compute the norm of the operator $\delta_a$. Express the norm as $|||\delta_a||| = \sup_{P \neq 0} \frac{|P(a)|}{||P||_\infty}$.

Answer: 1


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{\lceil 4^{n+\frac{\log \frac{8}{3}}{\log 4}} \rceil}{\lceil 4^{n+0.707519} \rceil} \] Determine if this limit equals 1.

Answer: 1


Question: Let \(U\) be a finite set, \(D\) a set, and \(F = \{(X_\lambda, Y_\lambda)\}_{\lambda \in \Lambda}\) a finite family of pairs of subsets of \(U\). Suppose \(a \in U \setminus \cup_{\lambda \in \Lambda} (X_\lambda \cup Y_\lambda)\) and \(K \subseteq U\) satisfies that for any finite subset \(I\) of \(D^U\), \(I \vDash F\) implies \(P(I, K, U)\). Determine if \(a \in K\).

Answer: a \in K


Question: Determine the least number of imaginary roots for the equation \(x^{10} - 4x^6 + x^4 - 2x - 3 = 0\).

Answer: 4


Question:
Find the asymptotic equivalent of the numbers \( b_{n,k} \) as \( n \to +\infty \) with \( k \) fixed, where \( b_{n,k} \) are defined by the equation:
\[ X^n = \sum_{k=0}^n b_{n,k}\binom{X}{k} \]
and \( \binom{X}{n} = \frac{1}{n!}\prod_{k=0}^{n-1}(X-k) \).

Answer: k^n


Question: Solve the equation \(\sqrt{\sqrt{x + 5} + 5} = x\) and express the solution in the form \(\frac{a + \sqrt{b}}{c}\), where \(a\), \(b\), and \(c\) are natural numbers. Then, find the value of \(a + b + c\).

Answer: 24


Question: Evaluate the surface integral \( \iint_S \bar{F} \cdot \bar{N} \, dS \) for the vector field \( \bar{F} = (4x^{3}+y^{2}-z , x^{2}+y^{3}+z , x^{3}+z+1) \) over the surface \( S \) defined by \( z = 4 - 4x^2 - y^2 \) with \( z \geq 0 \), where the normal vector \( \bar{N} \) is pointing upward. Consider using the divergence theorem to simplify the calculation.

Answer: 22\pi


Question: In the alphanumeric equation \((T+H+I+N+K)^3 = THINK\), determine the value of \(K\). The answer will be one of 1, 2, 3, 4, or 5.

Answer: 3


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function such that \( f(0) = 0 \), \( f(1) = 1 \), and \( |f'(x)| < 2 \) for all \( x \in \mathbb{R} \). If \( a \) and \( b \) are real numbers such that the set of possible values of \( \int_{0}^{1} f(x) \, dx \) is the open interval \((a, b)\), find the value of \( b-a \).

Answer: \dfrac{3}{4}


Question: Let \( f \in L^1(\mathbb{R}^n) \) be a non-negative function such that \( \|f\|_{L^1} = 1 \). Prove that \( \sup_{\xi \in \mathbb{R}^n} |\mathcal{F}(f)(\xi)| = 1 \) and that this supremum is attained exactly at \( \xi = 0 \).

Answer: 1


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

Answer: \dfrac{\pi}{4}


Question: Find the coefficient of \(x^{33}\) in the expansion of \((x+x^3)(1+5x^6)^{-13}(1-8x^9)^{-37}\).

Answer: 83019300


Question: An urn contains balls of $n = 10,000,000$ different colors, with $1,000$ balls of each color, making a total of $10^{10}$ balls. If you draw $10^8$ balls, what is the probability that you have drawn at least 90% of the different colors?

Answer: 1


Question: Consider the following constructions: Let $A=\bigcup_{n=0}^\infty A_n$ where $A_0=\emptyset$ and $A_{n+1}=P(A_n)$. Let $B=\bigcup_{n=0}^\infty B_n$ where $B_0=\{\emptyset\}$ and $B_{n+1}=\{P(X):X\in B_n\}\cup\{X\setminus Y:X,Y\in B_n\}$. Determine if $A=B$. Note: $P(X)$ denotes the power set of $X$.

Answer: A = B


Question: Find an integer \( x \) such that \( 1 < x < 2^{32767} - 1 \) and \( 2^{32767} - 1 \) is divisible by \( x \).

Answer: 127


Question: Determine if the function \( \sum_{n=1}^{\infty} \cos \frac{n \pi x}{l} \) is periodic. If it is periodic, find its period.

Answer: 2l


Question: Let $z$ be on the curve $\arg\left(\dfrac{z-z_1}{z+z_1}\right)=\dfrac{\pi}{2}$. If $\min\left(\arg\left(\dfrac{z}{z_1}\right),~\pi-\arg\left(\dfrac{z}{z_1}\right)\right)$ lies in $\left[\dfrac{\pi}{6},~\dfrac{\pi}{3}\right]$, then find $\left(\max\left(\lvert z-z_1\rvert\right)-\min\left(\lvert z+z_1\rvert\right)\right)$. Choose the correct option: (A) $2\lvert z_1 \rvert$, (B) $\left(\dfrac{\sqrt{2}-\sqrt{3}+1}{\sqrt{2}}\right)\lvert z_1 \rvert$, (C) $\lvert z_1\rvert\sqrt{2}$, (D) $\dfrac{\lvert z_1\rvert}{\sqrt{2}}$.

Answer: C


Question: Calculate \( 50! \pmod{53} \) using Wilson's Theorem.

Answer: 26


Question:
Let \( f : \mathbb{S}^2 \to \mathbb{R} \) be the smooth real-valued function that selects the \( z \)-coordinate of a point \( p \in \mathbb{S}^2 \). Consider the coordinate chart \( \{\mathbb{S}^2 - N, \sigma_N\} \) on \( \mathbb{S}^2 \), where \( \sigma_N \) is the stereographic projection from the north pole. The differential \( df_p \) in these coordinates is given by:
\[
df_p = \left( \frac{4x}{(x^2 + y^2 + 1)^2} dx + \frac{4y}{(x^2 + y^2 + 1)^2} dy \right)\Bigg|_{\sigma_N(p)}.\]
Evaluate \( df_{(0,0,-1)}(v) \) for the vector \( v = (1,1,0) \) based at the point \( p = (0,0,-1) \).

Answer: 0


Question: Solve the differential equation $x^4y^{\prime\prime} = (y-xy^\prime)^3$ with the initial conditions $y(1) = 1$ and $y^\prime(1) = 1$. Determine the solution using these initial conditions.

Answer: y = x


Question: Find the coefficient of \(\frac{1}{n^4}\) in the series \(S = \sum_{r=1}^\infty \frac{2^r}{r(3n+2)^r}\).

Answer: -\dfrac{4}{81}


Question: Evaluate the limit: \[ \lim_{x\rightarrow -1^+}\sqrt[3]{x+1}\ln(x+1) \]

Answer: 0


Question: Determine the number of ways to distribute 25 donuts to 4 police officers such that each officer receives at least 3 but no more than 7 donuts. Use generating functions to find the solution.

Answer: 20


Question: Determine the degree of the field extension \(\mathbb{Q}(\sqrt{1 + \sqrt{3}}):\mathbb{Q}\).

Answer: 4


Question: If a function is Lebesgue integrable, can it have a set of discontinuity points with positive measure?

Answer: Yes


Question: Show that the infinite product \( \prod_{n=1}^{\infty} \frac{R^{2^n}+z^{2^n}}{R^{2^n}} \) converges for \(|z|<R\), and prove that \( \prod_{n=1}^{\infty} \frac{R^{2^n}+z^{2^n}}{R^{2^n}} = \frac{R}{R-z} \).

Answer: \dfrac{R}{R - z}


Question: Calculate the limit: \[ \lim_{n\to\infty}\frac{n!\cdot e^n}{n^n} \]

Answer: \infty


Question: Determine whether the sequence \(a_n = \frac{1}{n}[n\beta] + n^2\beta^n\), where \(0 < \beta < 1\), converges. If it converges, find the limit.

Answer: \beta


Question: Evaluate the limit: \[ \lim_{x\rightarrow\infty} \frac {(3x^2-4)  \left( \sqrt [3]{2x^2+1}+1 \right)^2}{ (2x-1) \left( 4-\sqrt {8x^3-2} \right)x^{3/2}} \] without using L'Hôpital's rule.

Answer: 0


Question: Find the limit of \( \frac{\log(n^n)}{\log((2n)!) } \) as \( n \to \infty \).

Answer: \dfrac{1}{2}


Question: Find the maximum value of \( \left|e^{i\theta}-2\right|+\left|e^{i\theta}+2\right| \) for \( 0 \le \theta \le 2\pi \).

Answer: 2\sqrt{5}


Question: Determine the orientation of a unit cube in \(\mathbb{R}^3\) such that its projection onto the \(Oxy\)-plane has the maximal possible area. Provide the orientation that achieves this maximal projection area.

Answer: \sqrt{3}


Question: Let $X$ and $Y$ be topological spaces, and let $f: X \rightarrow Y$ be a function. Suppose $V \subset f(X)$ and $f^{-1}(V) = U$. Is it true that $V = f(U)$? Justify your answer.

Answer: Yes


Question: Let \((X_n)\) be a sequence of random variables on a shared probability space. Prove that if \(\mathbb{P}(\lim_{n \to \infty} X_n = \beta) = 1\), then \(\lim_{n \to \infty} \mathbb{P}(X_n > \alpha) = 1\) for all \(\alpha < \beta\).

Answer: 1


Question: Find the surface area of the part of the plane \(3x + 2y + z = 5\) that lies inside the cylinder \(x^2 + y^2 = 4\).

Answer: 4\sqrt{14}\pi


Question: If $L_1L_2$ is a regular language, is $L_2L_1$ necessarily regular as well? Provide a justification for your answer.

Answer: No


Question: Determine the radius of convergence for the series \( \sum_{n=0}^{\infty} a_n (x-3)^n \), where the series is given by \( 1 - x^2 + x^4 - x^6 + \cdots \).

Answer: \sqrt{10}


Question: Let $f_k$ be a sequence of non-negative functions from $L_2(\Omega)$, where $\Omega$ is a bounded open set. Assume that $f_k\to f$ weakly in $L_2$ and strongly in $L_p$, for all $p<2$. Assume also that $f_k^2\to F$ weakly in $L_1$. Does it imply that $F=f^2$? Provide a justification for your answer.

Answer: F = f^2


Question:
Determine the radius of convergence for the series representation of the function \( \arcsin(x) \):
\[ \arcsin(x) = \sum_{k=0}^\infty \binom{-1/2}{k}(-1)^k\frac{x^{2k+1}}{2k+1} \]

Answer: 1


Question: Find \( \lim_{n\to\infty} 2^n a_n \) where \( a_1 = 1 \) and \( a_{n+1} = \frac{\sqrt{a_{n}^2 + 1} - 1}{a_n} \).

Answer: \dfrac{\pi}{2}


Question:
Find the smallest positive integer \( x \) that satisfies the following system of congruences:
\[
x \equiv 32 \pmod{81} \\
x \equiv 59 \pmod{64}
\]

Answer: 3515


Question: Determine the condition on the rational numbers \( \frac{a}{b} \) and \( \frac{c}{d} \) such that the functions \( \sin\left(2\pi\frac{a}{b} x\right) \) and \( \sin\left(2\pi\frac{c}{d}x\right) \) are equal for all \( x \).

Answer: \frac{a}{b} = \frac{c}{d}


Question:
For what natural number \( n \) does the equation \([\log_2 1] + [\log_2 2] + [\log_2 3] + \dotsb + [\log_2 n] = 1538\) hold true? Choose the correct answer from the following options:

A. 255
B. 256
C. 254
D. 313

Answer: A


Question: Are there infinitely many pairs of rational numbers $(a, b)$ such that $a^3+1$ is not a square in $\mathbf{Q}$, $b^3+2$ is not a square in $\mathbf{Q}$, and $b^3+2 = x^2(a^3+1)$ for some $x$ in $\mathbf{Q}$?

Answer: Yes


Question: Determine the value of $c$ such that $f(x,y) = ce^{-2x^2-8y^2}$ is a probability density function.

Answer: \dfrac{4}{\pi}


Question:
Find the limit as \( n \to \infty \) of the following expression:
\[
\frac{e^{1/n}}{n^2} + 2\frac{e^{2/n}}{n^2} + 3\frac{e^{3/n}}{n^2} + \ldots + n \frac{e}{n^2}
\]

Answer: 1


Question: Consider an ellipse given by \(\frac{x^2}{4} + y^2 = 1\). A point \(P\) on the ellipse has an eccentric angle of \(45^\circ\). Through this point, variable pairs of chords are drawn such that they are perpendicular to each other and meet the ellipse at points \(A\) and \(B\). If the line joining \(A\) and \(B\) always passes through a fixed point \(Q(a, b)\), find the value of \(a^2 + b^2\).

Answer: \dfrac{9}{10}


Question: Consider the region defined by \((x,y,z)\) such that \(x^2+y^2<1, x>0, 0 \le z \le 5\). This describes one half of a cylinder. A wedge is cut from this half-cylinder along the plane \(z=2x\). Find the volume of the wedge.

Answer: \dfrac{4}{3}


Question: Find the smallest natural number \( n \) such that \( n^{25} \equiv 37 \pmod{83} \).

Answer: 69


Question: Determine the Galois group of the polynomial $4x^4 + 5x^3 - 9$ over the field of rational numbers $\mathbb{Q}$.

Answer: S_3


Question: Evaluate the triple integral \( \iiint \sqrt{x^2+y^2} \, dx \, dy \, dz \) using cylindrical coordinates, where the region of integration is bounded by the surfaces \( x^2+y^2=z^2 \), \( z=1 \), and \( z=2 \). Determine the correct limits of integration for \( r \), \( \phi \), and \( z \).

Answer: \dfrac{5\pi}{2}


Question: Determine if there exists a natural number \( n \) other than 2 such that the homotopy group \( \pi_n(G) \) is trivial for every Lie group \( G \).

Answer: No


Question: Evaluate the limit: \[ \lim _{x\to \infty }\frac{\left(\int _{10}^x\frac{e^{\sqrt{t}}}{t}dt\right)^2}{e^x} \]

Answer: 0


Question: Calculate the volume of the convex hull of the points \([0,0,0], [1,0,0], [0,2,0], [0,0,3]\) using Fubini's theorem.

Answer: 1


Question:
Determine the dimension of the image of the map:
\[ H^{1}(G_{S}, Ad(\bar{\rho})) = H^{1}(G_{S}, \mathbb{F}_{p}) \oplus H^{1}(G_{S}, Ad^{0}(\bar{\rho})) \rightarrow H^{1}(I_{p}, Ad^{0}(\bar{\rho})) \rightarrow H^{1}(I_{p}, \mathbb{F}_{p}(\omega^{k-1})) \]
where \( Ad^{0}(\bar{\rho}) \cong \mathbb{F}_{p} \oplus \mathbb{F}_{p}(\omega^{k-1}) \oplus \mathbb{F}_{p}(\omega^{1-k}) \) as an \( I_p \) module.

Answer: 1


Question:
Given the following integrals for a continuous function \( f(x) \):
\[ I = \int_0^1 x f(x) \, dx = \frac{1}{6} \]
\[ J = \int_0^1 (f(x))^2 \, dx = \frac{1}{12} \]
Determine the value of \( f\left(\frac{1}{2}\right) \).

Answer: \dfrac{1}{4}


Question: Find the radius of the inscribed sphere of a pyramid in \(\mathbb{R}^5\) with its vertex at \((1,0,0,0,0)\). The base of the pyramid is a regular 4-dimensional simplex lying in the hyperplane \(x_1=0\), which has an inscribed sphere of radius 1 centered at \((0,0,0,0,0)\).

Answer: \sqrt{2} - 1


Question: Determine whether the following statement is true or false: Assume that $f:[0,1]\to [0,1]$ is a continuous function such that $$\sup_t\lim\sup_{s\to t}\frac{|f(s)-f(t)|}{|t-s|}<\infty,$$ then $f$ is Lipschitz continuous.

Answer: True


Question: Find the limit: \[ \lim_{n \to \infty} \left(\frac{n^2+2}{n^2+1}\right)^{3n^2+1/n} \] Use the fact that \( \lim_{n \to \infty} (1+1/n)^n = e \).

Answer: e^3


Question: Given that the graph of the continuous function \( y = f(x) \) is symmetric with respect to the origin for all real numbers \( x \), and \( f(x) = \frac{\pi}{2} \int_1^{x+1} f(t) \, dt \) with \( f(1) = 1 \), find the value of \( \pi^2 \int_0^1 x f(x+1) \, dx \).

Answer: 2\pi - 4


Question: Find the minimal extension field of \(\mathbb{F}_2\) that contains an element of order 21.

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


Question:
Consider the system of differential equations given by \( y'_1 = y_2 \) and \( y'_2 = ay_1 + by_2 \), where \( a, b \in \mathbb{R} \). Determine which of the following conditions ensures that every solution \( y(x) \to 0 \) as \( x \to \infty \):

(a) \( a < 0, b < 0 \)
(b) \( a < 0, b > 0 \)
(c) \( a > 0, b > 0 \)
(d) \( a > 0, b < 0 \)

Answer: a


Question: Determine the degree of the field extension \( \mathbb{Q} ( \zeta_{14} + \zeta_{14}^9 + \zeta_{14}^{11}) \) over \( \mathbb{Q} \).

Answer: 2


Question: Let $F$ be a non-archimedean local field with finite field $\mathbb{F}_q$ of prime characteristic $p$, and let $L$ be the completion of the maximal unramified extension of $F$. Denote by $\mathcal{O}$ the valuation ring of $L$ and by $\varpi$ a uniformizer of $L$. Set $G = \mathrm{GL}_n$. Let $I$ be the inverse image of the subgroup of lower triangular matrices under the map $G(\mathcal{O}) \rightarrow G(\overline{\mathbb{F}}_q)$, $\varpi \mapsto 0$. Consider the Iwahori decomposition $G(L) = \bigcup_{w \in \tilde{W}} I\tilde{w}I$. If $w = w_1w_2$ with $\text{length}(w_1) + \text{length}(w_2) = \text{length}(w)$, does the equality $Iw_1Iw_2I = IwI$ hold?

Answer: Yes


Question:
Evaluate the line integral \( \int_C y \, dx + x^2 \, dy \) where the path \( C = C_1 + C_2 \) consists of:

- \( C_1 \): the straight line segment from the origin \((0,0)\) to the point \((2,18)\).
- \( C_2 \): the path of the parabola \( y = -x^2 + 8x + 6 \) from the point \((2,18)\) to the point \((5,21)\).

Answer: \dfrac{225}{2}


Question: Find the least possible value of \( N \) such that the error in approximating the integral \( \int_0^1 3e^{x^2} \, dx \) using Simpson's Rule is less than or equal to \( 1 \times 10^{-9} \). Use the error bound formula for Simpson's Rule: \( \text{Error}(S_N) \le \frac{K(b-a)^5}{180N^4} \), where \( K \) is the maximum value of the fourth derivative of the integrand on the interval \([a, b]\).

Answer: 244


Question: Is there a finite dimensional local ring with infinitely many minimal prime ideals? Alternatively, is there a ring with a prime ideal \(\mathfrak{p}\) of finite height such that the set of minimal prime sub-ideals of \(\mathfrak{p}\) is infinite? Assume the ring is commutative with unity, and the dimension refers to Krull dimension. A local ring is defined as a ring with exactly one maximal ideal.

Answer: Yes


Question: If a polynomial $g(x)$ satisfies the equation $x \cdot g(x+1) = (x-3) \cdot g(x)$ for all $x$, and it is given that $g(3) = 6$, find the value of $g(25)$. \( \boxed{?} \)

Answer: 0


Question: Given the equations \(a^4+8b=4(a^3-1)-16\sqrt{3}\) and \(b^4+8a=4(b^3-1)+16\sqrt{3}\), find the value of \(a^4+b^4\).

Answer: 56


Question: Given the temperature on a hot surface is described by the function \( T = 100e^{-(x^2+y^2)} \), and a bug follows the trajectory \( r(t) = \langle t\cos(2t), t\sin(2t) \rangle \), determine the rate at which the temperature is changing as the bug moves. Use the chain rule to find \( \frac{dT}{dt} \) and simplify to show that the rate of change is \( -200te^{-t^2} \).

Answer: -200te^{-t^2}


Question: Let $X = C([0,1])$ be the space of continuous functions on $[0,1]$ with the uniform norm $\|f\| = \sup_{x \in [0,1]} |f(x)|$. Define the operator $T: X \to X$ by \( T(f)(x) = f(x) - \int_0^1 f(s) \, ds \). Find the operator norm $\|T\|$. \( \|T\| = \sup \{ \|Tf\| : f \in X, \|f\| \le 1 \} \).

Answer: 2


Question: Determine the minimum sample size \( n \) required such that for a random sample \( X_1, X_2, \ldots, X_n \) drawn from a uniform distribution on the interval \([0, \theta]\), the probability \( \Pr(|\max\{X_1, X_2, \ldots, X_n\} - \theta| \leq 0.1\theta) \geq 0.95 \) holds for all possible values of \( \theta \).

Answer: 29


Question: Calculate the greatest common divisor (GCD) of the numbers $5^{2^{303} - 1} - 1$ and $5^{2^{309} - 1} - 1$. Provide your answer as a single number.

Answer: 78124


Question: Verify whether the solution \( T(n) = \Theta(n) \) is correct for the recurrence relation \( T(n) = T(n/2) + n(2 - \cos n) \) using the Master Theorem.

Answer: \Theta(n)


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

Answer: \dfrac{1}{e^3}


Question: Calculate the value of the following limit: $$\lim_{x \rightarrow 0}\frac{1}{x}\int_{x}^{2x}e^{-t^{2}}dt$$

Answer: 1


Question: If a group $G$ has only the trivial group and $G$ itself as its subgroups, can we be sure that $G$ has prime order?

Answer: Yes


Question: Find the sum of the orders modulo 83 for all elements in the set \( \{1, 2, 3, \ldots, 82\} \). Include each order as many times as it appears in the set.

Answer: 4923


Question: Determine the values of $z$ for which the sequence $z_n = nz^n$ converges.

Answer: |z| < 1


Question: Evaluate the limit: $$\lim_{x \to +0}\frac{1}{x}{\int^{2022x}_{0}{t\,\sqrt{|\cos(\frac{1}{t})|}\,dt}}$$

Answer: 0


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

Answer: 0


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

Answer: 4\pi i


Question: In a shuffled deck of 52 cards, what is the probability that the thirteenth spade appears before the thirteenth diamond when the cards are dealt one by one?

Answer: \dfrac{1}{2}


Question: Determine whether the sequence \( \{a_n\} \), which is bounded above and satisfies the condition \( a_{n+1} - a_n > -\frac{1}{n^2} \) for all natural numbers \( n \), is convergent. Answer true or false.

Answer: True


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

Answer: \dfrac{1}{3}


Question: Determine the number of terms needed in the series expansion of \( \int_{0}^{1} \frac{e^x-1}{x} \, dx \) to approximate the integral with an error less than or equal to \( 10^{-6} \).

Answer: 8


Question: Let $A$ be an $n \times n$ matrix with complex entries, and let $A^*$ denote its conjugate transpose. Given that $A^* = A^7$, prove that $A^8 = I$, where $I$ is the identity matrix.

Answer: A^8 = I


Question:
Let \(0 = t_0 < t_1 < \cdots < t_n = T\) be a partition of the interval \([0, T]\). Denote \(\Delta t_k \equiv t_{k+1} - t_k\) for every \(0 \leq k < n\). Assume \(\lim_{n \to \infty} \Delta t_k = 0\). Find the limit of \(S_n\) as \(n \to \infty\), where:
\[S_n \equiv \sum_{k=0}^{n-1} (\Delta t_k)^2\]

Answer: 0


Question: Prove that if an infinite series \( \sum a_n = S > 0 \) converges and \( a_n > 0 \), then \( \lim_{n \to \infty} n(\prod_{i=1}^n a_i)^{1/n} = 0 \).

Answer: 0


Question: Compute the divergence of the cross product of the gradients of two smooth scalar functions \( f \) and \( g \). That is, find \( \text{div}(\nabla f \times \nabla g) \).

Answer: 0


Question: Given that \( \sum\limits_{k=1}^{\infty}a_k = S \), determine the sum of the series \( a_4 + a_3 + a_2 + a_1 + a_8 + a_7 + a_6 + a_5 + \dots \). Assume the series converges. What is the sum in terms of \( S \)?

Answer: S


Question: Is the norm \( \langle \dot{\gamma}, \dot{\gamma} \rangle \) of the velocity vector along a geodesic constant?

Answer: Yes


Question: Determine the Euler characteristic of the edge word $abc^{-1}b^{-1}da^{-1}d^{-1}c$. Recall that the Euler characteristic is given by $X = V - E + F$, where $V$, $E$, and $F$ represent the number of vertices, edges, and faces, respectively.

Answer: -2


Question: Determine whether the inequality \( \Vert f\Vert_{L^\infty}^2 \leq \Vert f\Vert_{L^2}\Vert f'\Vert_{L^2} \) holds for all functions \( f \in H^1(\mathbb{R}) \).

Answer: Yes


Question: Determine whether there exists a constant \( M > 0 \) such that the inequality \[ \left| \sum_{n=0}^{\infty} \frac{(-1)^n x_{2n}}{\sqrt{n!}} \right| \le M \sqrt{\sum_{n=0}^{\infty}|x_n|^2} \] holds for all sequences \( \{x_n\} \in \mathcal{l}^2 \), where \( x_n \in \mathbb{K} \).

Answer: \sqrt{e}


Question: Let \( X_1, \dotsc, X_n \) be independent, zero mean random variables. Define \( Y_k = \alpha^{n-k}X_k \). Is the sequence \( \{Z_k\} \) with \( Z_k = \sum_{i=1}^k Y_i = \sum_{i=1}^k \alpha^{n-i}X_i \) a martingale?

Answer: Yes


Question: There are 7 dwarfs, D1 through D7, and 7 tasks, T1 through T7, in a mine. Each dwarf can perform exactly one task. However, D1 cannot perform T1, D4 cannot perform T2, D5 cannot perform T6, and D6 cannot perform T3 or T7. Determine the number of ways to assign the tasks to the dwarfs under these constraints.

Answer: 2364


Question: Evaluate the polynomial \(2x^3 + 2x^2 - 7x + 72\) at \(x = \frac{3-5i}{2}\), where \(i = \sqrt{-1}\).

Answer: 4


Question: Given the sequence \(a_n\) defined by the initial conditions \(a_1 = 1\), \(a_2 = 4\), \(a_3 = 5\), and the recurrence relation \(a_n + a_{n-1} + a_{n-2} + a_{n-3} = n^2\) for all \(n \geq 4\), find the sum of the digits of \(a_{2021}\).

Answer: 19


Question: Evaluate the infinite series: \[ \sum\limits_{n=1}^{\infty} \frac{6^n}{2^{1 + 2n} + 3^{1 + 2n} - 5 \cdot 6^n} \]

Answer: 2


Question: Find the function $f(x)$ such that $f(0) = 1$ and \[ f(x) = f'(x) + f''(x) + f'''(x) + ''''(x) + 	ext{...} \] where the series continues indefinitely.

Answer: e^{x/2}


Question: Determine the number of zeros of the function \( f(z) = 2z^5 + 8z - 1 \) within the annular region \( \{ 1 < |z| < 2 \} \).

Answer: 4


Question: Find the point $P(x, y, z)$ on the curve $z^2 = x^2 + y^2$ that is nearest to the point $A(2, 4, 0)$. Then, determine the distance from this point $P$ to the origin.

Answer: \sqrt{10}


Question: Determine whether there exists a prime number $p$ such that the $p$-adic numbers $\mathbb{Q}_p$ contain the third root of unity $\zeta$, where $\zeta^3 = 1$. Provide a justification for your answer.

Answer: Yes


Question: Consider the function \( f(x) \), which is \( 2\pi \)-periodic and defined as follows: \( f(0) = 1 \) and for every \( 0 \neq x \in [-\pi, \pi] \), \( f(x) = 1 + \sin \frac{\pi^2}{x} \). Determine whether the Fourier series of \( f(x) \) converges at \( x=0 \). If it does converge, find the value of the Fourier series at that point.

Answer: 1


Question: Find the kernel of the differential operator \( T \colon \mathcal{P}_2(\mathbb{C}) \longrightarrow \mathcal{P}_3(\mathbb{C}) \) defined by \( T(p(t)) = p(t) + t^2 p'(t) \), where \( p(t) \) is a quadratic polynomial with complex coefficients.

Answer: \{0\}


Question: Given the polynomial \(P(x) = ax^3 + bx^2 + cx + d\) with real coefficients \(a, b, c,\) and \(d\) each less than or equal to 12, and the conditions \(P(2) = 2\), \(P(4) = 4\), and \(P(6) = 6\), find the maximum possible value of \(P(10)\).

Answer: 58


Question: Evaluate the expression $A_{lm}x^lx^m$ given that the tensor $A_{lm}$ is skew-symmetric.

Answer: 0


Question: Evaluate the residue of the function \( f(z) = \frac{e^{z^2}}{z^5(1-z^2)} \) at \( z=1 \).

Answer: -\dfrac{e}{2}


Question: Find the limit \( \lim_{x \to \infty} \left( e^{1/x}(x^3-x^2+x/2)-(x^3+x^6)^{0.5} \right) \).

Answer: -\dfrac{1}{3}


Question: Let $q$ be a power of a prime. Consider a subset $S \subseteq \mathrm{P}^2 \mathbf{F}_q$ such that for every homography $g \in \operatorname{PGL}(3,q)$, the intersection $gS \cap S$ is non-empty. Is it possible for $|S|$ to be less than $1+q$?

Answer: No


Question: Consider a quadratic polynomial \( f(x) = x^2 + \cdots \) with \( f(1) = 0 \). Divide \( \{f(x)\}^3 \) by \( x^2 + 1 \), and let the quotient \( Q(x) \) be \( x^4 + ax^3 + bx^2 + cx + d \). Find \( |13a + 7b + 3c + d| \).

Answer: 21


Question: Given the implicit function defined by the equation \( y - \sin y = x \), find the value of the third derivative \( y'''(\pi) \).

Answer: \dfrac{1}{16}


Question: Evaluate the second mixed partial derivative \( \frac{\partial^2 f}{\partial x \partial y}(0,0) \) for the function \( f(x,y) = \begin{cases} \frac{xy(x^2-y^2)}{x^2+y^2} & \text{if }(x,y) \neq (0,0) \\ 0 & \text{otherwise} \end{cases} \).

Answer: 1


Question: Determine the smallest integer \( n \) such that for any two sets \( \{ x_1, x_2, \ldots, x_n \} \) and \( \{ y_1, y_2, \ldots, y_n \} \) of \( n \) distinct points on the unit circle \( \mathbb{S}^1 \), there exists a unique automorphism \( f: \mathbb{D} \rightarrow \mathbb{D} \) satisfying \( f(x_i) = y_i \) for all \( i \).

Answer: 3


Question:
Use the predictor-corrector method with the following second-order multi-step methods to approximate \( y(0.3) \) for the initial value problem \( y'(t) = 3ty, y(0) = -1 \), using a step size of \( h = 0.1 \):

1. Predictor: \( \omega_{i+1} = \omega_i + \frac{h}{2}(3f_i - f_{i-1}) \)
2. Corrector: \( \omega_{i+1} = \omega_i + \frac{h}{2}(f_{i+1} + f_i) \)

Use Euler’s method to start the process. Compute the approximation to \( y(0.3) \).

Answer: -1.1284


Question: Find the $n$-th derivative of the function $f(x) = x^{n-1} \log x$. Choose the correct option: \(\text{A) } \frac{(n-1)!}{x}\), \(\text{B) } \frac{n}{x}\), \(\text{C) } (-1)^{n-1}\frac{(n-1)!}{x}\), \(\text{D) } \frac{1}{x}\).

Answer: A


Question: Evaluate the integral: \[ \lim_{\epsilon\to 0^+} \int_{\Lambda/s}^\Lambda \int_{-\infty}^\infty \frac{i}{y^2-x^2+i\epsilon} \, dy \, dx = \pi\log s, \] where \( s \) and \( \Lambda \) are positive constants.

Answer: \pi \log s


Question: Let $I$ be a bounded connected subset in $\mathbb{R}$, and let $f: I \rightarrow \mathbb{R}^k$ be a differentiable function. Does the boundedness of $f'$ imply that $f$ is also bounded? Provide a justification for your answer.

Answer: Yes


Question: Determine all symmetric real \( n \times n \) matrices \( A \) such that the zero solution of the differential equation \( \dot{x} = A^2x \) is stable.

Answer: 0


Question: Is the following inequality true for independent zero mean random elements in a separable Hilbert space $\mathbb{H}$? Given $X_1, \ldots, X_n$ are independent $\mathbb{H}$-valued random elements with $\|X_i\| \le s$ for each $1 \le i \le n$, and $S_n = X_1 + \ldots + X_n$, does the inequality \[ P(\|S_n\| \ge t) \le Ce^{-c\frac{t^2}{ns^2}} \] hold for $t > 0$, where $C$ and $c$ are positive constants?

Answer: Yes


Question: Determine the number of odd permutations of order 4 in the symmetric group $S_6$. Provide your answer as a single integer.

Answer: 90


Question:
Consider the parabola given by \( y = x^2 \) and the circle given by \( x^2 + y^2 = 2 \) in the standard embedding plane \( \{[x:y:1]\mid x,y\in\mathbf{R} \}\subset \mathbf{RP}^2 \). Determine the number of projective transformations \( t_A \) that satisfy the following conditions:

1. Maps the parabola onto the circle.
2. Leaves the \( Y \)-axis invariant.
3. Leaves the intersection points of the parabola and circle invariant.

How many such transformations exist?

Answer: 2


Question:
Let $\mu(n)$ be the Möbius function, $\phi(n)$ Euler's totient function, $\sigma(n)$ the sum of divisors function, and $\tau(n)$ the number of divisors function. Define the set $S_N$ for a natural number $N$ by:
\[S_N = \{ (m,n) \in \mathbb{N} \times \mathbb{N} \mid m \ne n, \, \mu(m)=\mu(n), \, \phi(m)=\phi(n), \, \sigma(m)=\sigma(n), \, \tau(m)=\tau(n) \text{ and } \max \{ m,n \} \le N \} .\]
Determine the size of the set $S_N$. \( \boxed{?} \)

Answer: 0


Question: How many $4 \times 4$ matrices with entries from $\{0, 1\}$ have an odd determinant?

Answer: 20160


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

Answer: 0


Question: Let \( f : \mathbb{R} \to \mathbb{R} \) be a smooth (infinitely differentiable everywhere) function such that \( f'(0) = 1 \) and \( |f^{(n)}(x)| \le 1 \) for all \( x \in \mathbb{R} \) and for all \( n \ge 0 \). Is it true that \( f(x) = \sin x \) for all \( x \in \mathbb{R} \)?

Answer: \sin x


Question: Does there exist a bijection \( f \) between an open set \( V \) in \( \mathbb{R}^{r+1} \) and another open set \( W \) in \( \mathbb{R}^{r+1} \) such that for some \( (y_1, y_2, \ldots, y_{r+1}) \in V \), \( f(y_1, y_2, \ldots, y_{r+1}) = (0, 0, \ldots, \alpha) \) with \( \alpha \neq 0 \)?

Answer: Yes


Question: In a model \( \mathbf{M} = (\mathbf{W}, \mathbf{R}, \mathbf{V}) \) for a set of atomic formulae \( \Omega \), consider possible worlds \( \mathbf{w}, \mathbf{w'} \in \mathbf{W} \) with an access relation satisfying \( (\mathbf{w}, \mathbf{w'}), (\mathbf{w'}, \mathbf{w}) \in \mathbf{R} \), and \( p \in \Omega \). The necessity operator \( \Box \) and possibility operator \( \Diamond \) are defined such that if a proposition \( \phi \) is true at \( \mathbf{w_2} \), then \( \Diamond{\phi} \) is true at \( \mathbf{w_1} \), and if \( \Box \phi \) is true at \( \mathbf{w_1} \), then \( \phi \) is true at \( \mathbf{w_2} \). Can it be the case that \( p \) is true at \( \mathbf{w} \) and \( \lnot p \) is true at \( \mathbf{w'} \) with \( \mathbf{w} \) and \( \mathbf{w'} \) having access to each other?

Answer: Yes


Question: Evaluate the Lebesgue integral \( \int f \, d\mu \) where \( f(x) = 0 \) if \( x \in \mathbb{Q} \) and \( f(x) = 1 \) otherwise, and \( \mu \) is a measure such that \( \mu(\{y\}) = \frac{1}{2} \) if \( y = 1 \) or 2, and \( \mu(\mathbb{R}) = 1 \).

Answer: 0


Question: Evaluate the line integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \) where \( \mathbf{F} = x^2y^3\mathbf{i} + yz^2\mathbf{j} + zx^2\mathbf{k} \) and the curve \( C \) is given by \( \mathbf{r}(u) = 3u^2\mathbf{i} + u\mathbf{j} + 2u^3\mathbf{k} \). The curve \( C \) is parameterized from point \( A(3,-1,-2) \) to point \( B(3,1,2) \).

Answer: 12


Question: Find the function \( f(x) \) that satisfies the equation \( f(x) + f(y) = f \left( \frac{x+y}{1-xy} \right) \) and the condition \( f'(0) = 5 \).

Answer: 5 \arctan x


Question: Calculate the sum of the infinite series: \[ \sum_{n=1}^{\infty} \frac{n^4}{4^n} \]

Answer: \dfrac{380}{81}


Question: Let \( f: [0,1] \rightarrow [0,1] \) be a function with a continuous derivative. Is the second derivative \( f'' \), if it exists, necessarily Lebesgue measurable?

Answer: Yes


Question: Find the coefficient of \(x^{17}\) in the expansion of \((x^2 + x^3 + x^4 + x^5 + x^6 + x^7)^3\).

Answer: 15


Question: A monkey is typing randomly on a typewriter with three keys: A, B, and C. Each key is pressed with equal probability, independently of previous presses. Let \( T \) be the random time at which the sequence "ABC" first appears in the typed sequence. What is the expected value of \( T \)?

Answer: 27


Question: Calculate the flux of the vector field \( F(x,y,z) = (x,y,z) \) across the surface defined by \( x^2 - y^2 + z^2 = 0 \) with \( y \in [0, 1] \).

Answer: 0


Question: Determine the function space on a Riemannian manifold \(M\) for which the Dirichlet energy \(\varepsilon(f,g) = \int_M \langle \nabla f, \nabla g \rangle \, dvol\) is a strongly local, regular, and tight Dirichlet form. Is it \(W^{1,2}(M)\), \(W^{1,2}_{loc}(M)\), or \(W^{1,2}_0(M)\)?

Answer: W^{1,2}_0(M)


Question: Given the force of interest \( F(t) = \frac{0.02 + 0.01t}{1 + 0.02t + 0.005t^2} \), where \( t \) is the number of years starting from March 31, 2001, calculate the accumulated value of two investments: $100 invested on March 31, 2002, and $200 invested on March 31, 2006. Find the total accumulated value of these investments on March 31, 2010.

Answer: 413.41


Question: Given the differential equation \( y/t + te^{-t} = y' \), find \( \displaystyle \lim_{t\rightarrow \infty} \frac{y}{t} \).

Answer: 0


Question: Find the function $f(x) \in \operatorname{span}(1, \sin(x), \cos(x))$ that minimizes the integral \( \int_{-\pi}^{\pi} |f(x) - \sin(2x)|^2 \, dx \).

Answer: 0


Question: Determine the condition on the positive integers $k$ and $n$ such that there exist positive integer solutions $x$ and $y$ to the equation $y(n-x) = (k+nx)$.

Answer: n \geq 2


Question: Let $A$ and $B$ be two metric spaces, and let $f: A \to B$ be an isometry. If $C \subset A$ is dense in $A$, is $f(C)$ dense in $B$? Provide a justification for your answer.

Answer: No


Question: Find the value of \( a \) that minimizes the integral \( \int_{-1}^{1} \left(x^3-p_2(x) \right)^2\ dx \), where \( p_2(x) \) is a polynomial of degree 2 that interpolates the function \( f(x)=x^3 \) at the points \( x=-1, x=0, x=a \) with \( 0<a<1 \).

Answer: \dfrac{3}{4}


Question: What is the minimum number of elementary row or column transformations needed to find the inverse of a given matrix using Gauss-type methods?

Answer: n^2


Question: Calculate the difference $A - B$, where $A$ is the number of 6-digit tickets (from 000000 to 999999) that are considered lucky, and $B$ is the number of tickets that are considered medium. A ticket is lucky if the sum of its first three digits equals the sum of its last three digits. A ticket is medium if the sum of all its digits equals 27.

Answer: 0


Question: Let $f$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}^+$ such that $f(x+y) = f(x)f(y)$ for all $x, y \in \mathbb{R}$. Find the analytic expression for $f(x)$. \( \text{(Hint: Consider the properties of exponential functions.)} \)

Answer: e^{kx}


Question: Let $A$ be a random variable uniformly distributed over the interval $(0, 2)$, and $B$ be a random variable uniformly distributed over the interval $(1, 3)$. Define $W = \min(A, B)$ and $Z = \max(A, B)$. Find the covariance $\text{Cov}(W, Z)$. Express your answer as a single number.

Answer: \dfrac{25}{576}


Question: A company has a manager and $n \geq 3$ employees. The manager flips a coin $2n$ times: each time he flips heads, he gives one task to one of his employees, chosen randomly and uniformly. Let $X_1$ and $X_2$ be the number of tasks given to employee 1 and employee 2, respectively. Find the covariance $\text{Cov}(X_1, X_2)$. Provide your answer as a single numerical value.

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


Question: In a maximal planar graph, are two consecutive neighbors of a vertex necessarily adjacent? Provide a justification for your answer.

Answer: Yes


Question: Find \( \lim_{\alpha \to 0} \int_0^{\overline{w}(\alpha)} F^\alpha(w)(1-F^\alpha(w)) \, dw \), where \( \overline{w}(\alpha)=\frac{2\alpha\pi}{1+\alpha} \) and \( F^\alpha(w) = \frac{1-\alpha}{2\alpha} \frac{w}{\pi-w} \).

Answer: 0


Question: Given a matrix $A \in \Bbb R^{n \times n}$ such that $A^2 = -I$, find $\det(A)$. Assume $n$ is even.

Answer: 1


Question: Evaluate the integral \( \int_{\mathbb{R}^3} e^{-|x|} \, dx \), where \(|x| = \sqrt{x_1^2 + x_2^2 + x_3^2}\).

Answer: 8\pi


Question: Given two continuous functions \( f, g: \mathbb{R} \to \mathbb{R} \), for each \( t \in \mathbb{R} \), there exists a number \( h(t) \) such that \( e^{f(t)} - e^{g(t)} = e^{h(t)}(f(t) - g(t)) \) by the mean value theorem. Can the numbers \( h(t) \) be chosen such that \( h: \mathbb{R} \to \mathbb{R} \) is a continuous function?

Answer: Yes


Question: Let \((X_i)_{i\in \mathbb{N}}\) be iid random variables such that there exists \(\alpha>0\) where \(\mathbb{P}\left(X_1\in [x,x+1]\right)\leq \alpha\) for all \(x\in \mathbb{R}\). Assume \(\alpha\) is small enough. Does there exist a universal constant \(C>0\) such that \[\mathbb{P}\left(\sum_{i=1}^N X_i\in [x,x+1]\right)\leq \frac{C\alpha}{\sqrt{N}}\] for all \(x\in\mathbb{R}\) and \(N\in \mathbb{N}^*\)?

Answer: Yes


Question: Find the coefficient of \(x^{12}\) in the expansion of \((1+x^{3} +x^{6} +x^{9} +\ldots)^{7}\).

Answer: 210


Question: Find the function \( f(x) \) that satisfies the equation \( f(2x) - f(x) = x^3 \), given that \( f(x) \) is continuous.

Answer: \dfrac{x^3}{7}


Question: Calculate the limit: \[ \lim\limits_{x\to \infty} \left( \mathrm{e}^{\sqrt{x+1}} - \mathrm{e}^{\sqrt{x}} \right) \]

Answer: \infty


Question: Let G be the Galois group of a field with nine elements over its subfield with three elements. Determine the number of orbits for the action of G on the field with 9 elements.

Answer: 6


Question:
Find a closed form for the limit:
\[ f(m) = \lim_{n\to\infty} \left(\prod_{i=1}^n \frac{m+i}{n}\right)^{1/n} \]
Assume \(m\) is a positive real number.

Answer: \dfrac{1}{e}


Question: A physical system consists of four identical particles. The total energy of the system is \(4E_0 > 0\). Each particle can have an energy level equal to \(kE_0\) for \(k \in \{0, 1, 2, 3, 4\}\). A particle with energy \(kE_0\) can occupy any one of the \(k^2 + 1\) distinct energy states at that energy level. How many distinct energy configurations can the system have, considering the particles obey Bose-Einstein statistics?

Answer: 72


Question: Find the limit: \( \lim_{n \to \infty} \left( \sqrt{n^3 + 6n - 1} - \sqrt{n^3 - 10n - 11} \right) \).

Answer: 0


Question: Evaluate the definite integral: $$\int _{-\pi }^{\pi }\left(\frac{x^3+\cos \left(x\right)+1}{1+\cos \left(x\right)}\right)\,\mathrm{d}x$$

Answer: 2\pi


Question: For two numbers that are independently exponentially distributed with rate 3, calculate the expected value of the smaller number given that both numbers are greater than \( \frac{1}{3} \).

Answer: \dfrac{1}{2}


Question: Let $P_{n}$ be the vector space of all polynomials with real coefficients and degree at most $n$. Define the linear transformation $T: P_n \rightarrow P_{n+1}$ by $T(p)(x) = p'(x) - \int_0^x p(t) \, dt$. Determine the dimension of the null space of $T$.

Answer: 0


Question: Evaluate the integral: \[ \int_0^{2\pi} x \frac{\cos x}{2-\cos^2 x} \, dx \]

Answer: 0


Question: Show that \( \lim_{n \to \infty}\frac{c_n}{4^n} = 0 \), where \( c_n \) represents the Catalan numbers.

Answer: 0


Question: Determine the largest natural number $n$ such that for any subset of $A = \{1, 2, 3, \ldots, 2003\}$ with $n$ elements, the complementary set contains at least one pair of consecutive numbers.

Answer: 1000


Question: Given that $1 = \lim_{k\rightarrow \infty} (a_k + a_{k-1})$ and for all $k$, $a_k \geq a_{k-1} \geq 0$, determine if $\lim_{k\rightarrow \infty} a_k = \frac{1}{2}$.

Answer: \dfrac{1}{2}


Question: Let \(R\) be an \(\mathbb{R}\)-algebra. If \(R \otimes_\mathbb{R} \mathbb{C}\) is a finitely generated \(\mathbb{C}\)-algebra, is \(R\) necessarily a finitely generated \(\mathbb{R}\)-algebra? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the line integral \( \oint_C (e^{x^3} \vec{i} + y^3 \vec{j} + e^{z^2} \vec{k})\cdot d\vec{r} \), where \( C \) is the intersection of the surfaces \( z = x^2 + y^2 \) and \( z = x - 3y \).

Answer: 0


Question: Find the remainder when $a$ is divided by $18$, given that $\gcd(a^{226} + 4a + 1, 54) = 3$. Assume $b = a^{226} + 4a + 1$. Since $b$ is odd, $a$ must be even. Also, $3 \nmid a$. Using Fermat's theorem, show that $a \equiv 1 \pmod{3}$. Determine the possible values of $a \pmod{9}$, given that $a \equiv 0 \pmod{2}$ and $a \equiv 1 \pmod{3}$. What is the remainder of $a$ when divided by $18$?

Answer: 4


Question: In a convex quadrilateral ABCD, diagonals AC and BD intersect at point P. Given that PA = 1, PB = 2, PC = 6, and PD = 3, let O be the circumcenter of triangle PBC. If OA is perpendicular to AD, determine the circumradius of triangle PBC.

Answer: 3


Question: Compute the treewidth of the complete bipartite graph $K_{m,n}$ using the chordal graph characterization. Recall that the treewidth $tw(G)$ is given by $\min \{\omega(H)-1 : G\subseteq H \ \wedge \ H \ \text{is chordal}\}$. Determine the treewidth of $K_{m,n}$.

Answer: \min(m, n)


Question: Consider the set \( A = \{0, 1, \ldots, 100\} \) with the partial order relation defined by divisibility: for all \( a, b \in A \), \( a \mid b \) if there exists \( k \in \mathbb{N} \) such that \( b = ka \). Determine the maximal element of \( A \) with respect to this partial order relation.

Answer: 100


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{dt}{t^2} \delta(\cos t) \).

Answer: 1


Question: Given a matrix \( A \) such that \( A^4 = I_n \), \( A \neq I_n \), \( A^2 \neq I_n \), and \( A^3 \neq I_n \), determine which power of \( A \) is equal to \( A^{-1} \). Prove your answer.

Answer: A^3


Question: Consider a shuffled deck of $n$ cards numbered $1, 2, \ldots, n$. A match occurs when card number $i$ is in position $i$. For example, in the sequence $1, 4, 3, 2, 5$, there are 3 matches. Let $X$ be a random variable representing the number of matches in the deck. Calculate $\text{Var}(X)$, the variance of $X$.

Answer: 1


Question: Let $X$ and $Y$ be Hilbert spaces, and let $U \in \mathcal{B}(X,Y)$ be a surjective bounded linear operator. If the adjoint $U^{*} \in \mathcal{B}(Y,X)$ satisfies $U^{*} U = I_X$, does it follow that $U U^{*} = I_Y$? Provide a proof or counterexample.

Answer: Yes


Question: Suppose a non-zero ring homomorphism $k \to k'$ factors through a direct sum of finitely many artinian local rings $A_i$, i.e., $k \to \oplus A_i \to k'$. Is it true that the homomorphism $k \to k'$ factors through one of the $A_i$? Here, $k$ and $k'$ are fields.

Answer: Yes


Question: Determine whether every topological space can be structured as a right topological semigroup. A triple \((X, \tau, \ast)\) is a right topological semigroup if for every \(x, y, z \in X\), it holds that \((x \ast y) \ast z = x \ast (y \ast z)\) and for every \(y \in X\), the map \(\ast_y: X \to X\) given by \(\ast_y(x) = x \ast y\) is \(\tau\)-continuous. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the line integral \( \oint_C (3y \, dx + x^2 \, dy) \) where \( C \) is the path consisting of the line segment from \((0, 0)\) to \((1, 0)\), the line segment from \((1, 0)\) to \((1, 1)\), and the curve \( y = x^2 \) from \((1, 1)\) to \((0, 0)\). Verify the result using Green's theorem.

Answer: -\dfrac{1}{2}


Question:
Let $X$ and $Y$ be normed linear spaces, and let $T : X \to Y$ be a bijective linear map with a closed graph. Which of the following statements is true?

1. The graph of $T$ is equal to $X \times Y$.
2. $T^{-1}$ is continuous.
3. The graph of $T^{-1}$ is closed.
4. $T$ is continuous.

Answer: 3


Question:
Verify that the following 12 elements generated by $a = (12)(34)$ and $b = (123)$ in the alternating group $A_4$ are distinct:

1. $a^0 b^0 = 1$
2. $a^1 b^0 = (12)(34)$
3. $a^0 b^1 = (123)$
4. $b^2 = (132)$
5. $ab = (243)$
6. $ab^2 = (143)$
7. $ba = (134)$
8. $b^2a = (234)$
9. $bab = (124)$
10. $bab^2 = (14)(32)$
11. $b^2 a b = (13)(24)$
12. $b^2 a b^2 = (142)$

Show that these elements are distinct and confirm that they form the group $A_4$. What is the order of $A_4$?

Answer: 12


Question: Let \( A \) be a ring and \( M \) be an \( A \)-module. Consider an \( A \)-linear map \( \phi: M \to M \). Is the representation matrix \( N \) of this \( A \)-linear map composed of entries from the ring \( A \)? In other words, is the representation matrix of an \( A \)-linear map in \( M(A) \)?

Answer: Yes


Question:
Evaluate the limit: $$\lim_{n\to\infty} \left[ \frac{1}{2}\tan\frac{x}{2} + \frac{1}{2^2}\tan\frac{x}{2^2} + \frac{1}{2^3}\tan\frac{x}{2^3} + \cdots + \frac{1}{2^n}\tan\frac{x}{2^n} \right]$$

Choose the correct answer from the following options:

A: $-\cot x$

B: $-\sin x$

C: $\cot x+\frac{1}{x}$

D: $-\cot x+\frac{1}{x}$

Answer: D


Question: What is the minimal order of a homogeneous linear ordinary differential equation with constant coefficients that admits the solution \( y(t) = t^2 e^{-t/3} \cos(2t) \)?

Answer: 6


Question: Let $f: I \rightarrow \mathbb{R}$ be a function defined on an interval $I$. Suppose for all $t_1, t_2, t_3 > 0$ with $t_1 + t_2 + t_3 = 1$ and all $x_1, x_2, x_3 \in I$, the inequality \[ f(t_1x_1 + t_2x_2 + t_3x_3) \leq t_1 f(x_1) + t_2 f(x_2) + t_3 f(x_3) \] holds. Does this condition imply that $f$ is convex, meaning that for all $t_1, t_2 > 0$ with $t_1 + t_2 = 1$ and all $x_1, x_2 \in I$, the inequality \[ f(t_1x_1 + t_2x_2) \leq t_1 f(x_1) + t_2 f(x_2) \] holds?

Answer: Yes


Question: Let $S=\{a_1,...,a_p\}$ be a finite subset of the compact interval $[a,b]$. Consider a bounded function $f:[a,b]\rightarrow \mathbb{R}$ such that $f(x)=0$ for all $x\in[a,b]\setminus S$. Prove that $f$ is Riemann integrable on $[a,b]$ and that \( \int_a^b f(x)\,dx=0 \).

Answer: 0


Question: Determine whether there are any local extrema of the function \( f(x,y) = \sin(xy) \) within the domain \( x \in (0, \pi) \) and \( y \in (0, 1) \).

Answer: Yes


Question:
Determine for which of the following statements the equality holds:

1. \( \{f(f^{-1}(t)) \mid t \in N\} = N \)
2. \( M = \{f^{-1}(f(t)) \mid t \in M\} \)

where \( M \subset \text{domain } f \) and \( N \subset \text{Im } f \).

Answer: 1


Question: Determine all real values of \( \alpha \) such that the integral \( \int_{0}^{\infty} x(x^2/2 + 1)^{\alpha} \, dx \) converges.

Answer: (-\infty, -1)


Question: Evaluate the contour integral \( \int_{|z|=4} \frac{\sin z}{z(z-2i)}\, dz \).

Answer: \pi i \sinh 2


Question: Determine the value of $k \in [0, 1]$ that minimizes the probability of winning the following game: You are given an infinite sequence of independent $U(0, 1)$ random variables $X_1, X_2, \dotsc$. At any point, you can choose to stop or reveal the next number in the sequence. The objective is to stop exactly before the sum of the revealed numbers exceeds $k$. What value of $k$ minimizes the probability of winning?

Answer: 2 - \sqrt{2}


Question: Given the joint density function \( f(y_1, y_2) = \frac{1}{8} y_1 e^{-(y_1 + y_2)/2} \) for \( y_1 > 0 \) and \( y_2 > 0 \), and \( f(y_1, y_2) = 0 \) elsewhere, find the expected value of \( D = 50 + 2Y_1 + 4Y_2 \).

Answer: 66


Question: Given that $T_0$ is the minimum variance unbiased estimator (MVUE) for $\gamma(\theta)$ and $T_1$ is another unbiased estimator for $\gamma(\theta)$ with an efficiency of 0.0169, find the correlation coefficient between $T_0$ and $T_1$. \( \text{corr}(T_0, T_1) = ? \)

Answer: 0.13


Question: Evaluate the limit: \[ \lim_{n\to\infty} \frac{1^{99} + 2^{99} + \cdots + n^{99}}{n^{100}} \]

Answer: \dfrac{1}{100}


Question: A 1000 by 1000 grid is to be tiled using s-tiles, where each s-tile is a path of squares that makes two turns in opposite directions and has a length of at least 4. What is the maximum area that can be covered by these s-tiles on the grid?

Answer: 1000000


Question: Given the joint probability density function \( f(x,y) = 9x^2y^2 \) for \( 0 < y < 1 \) and \( -y < x < y \), find the probability \( P(X > Y^2) \).

Answer: \dfrac{1}{6}


Question: Minimize the function \( c = 5x^2 - 80x + y^2 - 32y \) subject to the constraints \( x, y \geq 0 \) and \( x + y \geq 20 \) using the Karush-Kuhn-Tucker (KKT) conditions.

Answer: -576


Question: Let $V$ be a 4-dimensional vector space over the complex numbers, and let $S$ be a subspace of the endomorphisms of $V$ such that the elements of $S$ commute. If there exists an element in $S$ that has at least two distinct eigenvalues, is the dimension of $S$ at most 4? Justify your answer.

Answer: 4


Question:
Given a dictionary \( D^{m \times n} \) of \( n \) basis vectors of size \( m \) and an input vector \( x^{m \times 1} \), find a vector \( \alpha^{n \times 1} \) that minimizes the following constrained optimization problem for some constant \( L \):
\[
\min_{\alpha} ||x-D\alpha||_2^2
\]
subject to:
\[\sum_{i=1}^m\alpha_i=1\ \text{and}\ \forall{i}:\alpha_i\ge0\]
\[||\alpha||_0 \le L\]

Answer: \alpha


Question: Determine whether the following statement is true or false: If $Y$ is an infinite set and $|X| < |Y|$, then $|Y^X| \leq |Y|$. Justify your answer.

Answer: False


Question: Prove that \( \lim_{x \to 1} (x^3 - 2x^2 + 4x + 2) = 5 \) using an \( \epsilon-\delta \) argument.

Answer: 5


Question: Evaluate the integral \( \int_0^{2 \pi } \frac{e^{e^{-i \theta }}}{e^{4 i \theta }} \, d\theta \).

Answer: 0


Question: Let \( I \subset \mathbb{R} \) be an open interval and let \( u': \overline{I} \to \mathbb{R} \) be a function such that \( u' \in L^1(I) \). Can we conclude that the function \( u \), defined by \( u(x) = \int_{c}^{x} u'(z) \, dz \) for some constant \( c \in I \), is continuous on \( I \)?

Answer: Yes


Question: Find the value of \( n \) such that \((m-1)(m+3)(m-4)(m-8) + n\) is a perfect square for all integer values of \( m \).

Answer: 196


Question: Let \( H \) be the space of all absolutely continuous functions \( f:[0,1] \rightarrow \mathbb{C} \) such that \( f(0)=0 \) and \( f' \in L^2(0,1) \), with the inner product defined by \( \langle f, g \rangle = \int_0^1 f'(t) \overline{g'(t)} \, dt \) for \( f, g \in H \). For a fixed \( 0 < t \le 1 \), define the linear functional \( L: H \rightarrow \mathbb{C} \) by \( L(h) = h(t) \). Determine the norm \( \|L\| \) of this functional and find the vector \( h_0 \in H \) such that \( L(h) = \langle h, h_0 \rangle \) for all \( h \in H \).

Answer: \sqrt{t}


Question: Consider a symmetric random walk on the integers \( \mathbb{Z} \) where \( X_i \) are independent random variables such that \( P(X_i = 1) = P(X_i = -1) = 0.5 \) for all integers \( i \) with \( 1 \leq i \leq n \). Define \( S_n = X_1 + X_2 + \cdots + X_n \). Now consider \( 2^n \) independent copies of \( S_n \) and let \( Z_n \) be their maximum. Determine \( \lim_{n \to \infty} \frac{Z_n}{n} \).

Answer: 1


Question: Find the sum of the infinite series \( \sum_{n = 1}^{+\infty}\frac{n(n+1)}{2^n} \).

Answer: 8


Question: Determine the number of solutions to the equation $cz^n = e^z$ in the region $G = \{z \in \mathbb{C} : \Re(z) < 1\}$, where $c \in \mathbb{C}$ and $|c| > e$. Use Rouché's theorem to justify your answer.

Answer: n


Question: If 10 values are sampled from a uniform distribution on $[0,1]$, and the average $\hat{u}$ of the 9 smallest values is computed, what is the expectation of $\hat{u}$?

Answer: \dfrac{5}{11}


Question: Calculate the limit \( \lim _{n\to \infty }n\sin\left(\frac{1}{\lceil{\frac{n}{2}}\rceil}\right) \) using Taylor expansion.

Answer: 2


Question: Determine the number of real roots of the equation \(2x^5 - 8x + 7 = 0\).

Answer: 1


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\left(\frac{\sin ^3n^2-5\cdot \sin ^2n^2+3}{\sqrt{\ln \left( \left| \dfrac{1}{\tan ^{19}e^ {- n }}+3\right| \right)+\sin \left( \dfrac{1}{\tan ^{24}e^ {- n }} \right)}}\right) \]

Answer: 0


Question: Use Parseval's identity to evaluate the integral \( \int_{-\pi}^{\pi} \sin^4 x \, dx \).

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


Question: Find the greatest common divisor of the following expressions: \( x+y+z, x^2+xy+z^2, y^2+yz+z^2, z^2+zx+x^2 \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{(x,y)\rightarrow(0,0)} \frac{x^3}{y^4+2\sin^2{x}} \] Determine whether the limit exists, and if so, find its value.

Answer: 0


Question: Let \( A \) be a symmetric positive semidefinite matrix. Determine if the inequality \( A^2 \succeq A \) holds when \( A \succeq I \).

Answer: A^2 \succeq A


Question: Let $U$ be a bounded open set in $\mathbb{R}^N$, and let $f_n: U \rightarrow \mathbb{R}$ be a sequence of positive Lebesgue-measurable functions such that $\int_U f_n \to 1$. Let $b: U \rightarrow \mathbb{R}$ be a continuous function such that $0 < b < 1$ in $U$. Determine whether $\int_U b f_n \to 1$ is possible.

Answer: Yes


Question: Does there exist a space $X$ and two loops $f, g$ based at $x_0 \in X$ such that $f$ and $g$ are freely homotopic but not based point homotopic?

Answer: Yes


Question: Determine the smallest integer \( k \) such that if the Fourier coefficients \( a_n \) and \( b_n \) of a function \( f \) satisfy \( |a_n| \leq Cn^{-k} \) and \( |b_n| \leq Cn^{-k} \) for some constant \( C \), then \( f \) is in \( C^1(T) \).

Answer: 3


Question: Determine whether the series \( \sum_{n=1}^\infty \sin z^n \) is uniformly absolutely convergent on compact subsets of the unit disk.

Answer: Yes


Question: Suppose \(\sum_{n=0}^{\infty} a_n\) converges. Prove that the power series \(\sum_{n=0}^{\infty} a_n z^n\) has a radius of convergence \(R \geq 1\).

Answer: R \geq 1


Question: If $f: \mathbb{R} \to \mathbb{R}$ is a uniformly continuous function, does there exist a function $a: \mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, the inequality $|f(x) - f(y)| \leq a(|x-y|)$ holds, where $a$ is independent of $x$ and $y$ and $\lim_{h \to 0} a(h) = 0$?

Answer: Yes


Question: Find the volume of the solid obtained by revolving the region bounded by the circle \(x^2 + y^2 \leq 1\) around the line \(x + y = 2\).

Answer: 2\sqrt{2}\pi^2


Question: Determine the values of \( \alpha > 0 \) for which the integral \( \int_{0}^{\infty}\exp\left ( \frac{1}{2}x-x^\alpha \right )\,\mathrm{d}x \) converges.

Answer: \alpha \geq 1


Question: Evaluate the integral \( I(z) = \int_0^{2\pi} \frac{e^{-it} \, dt}{e^{it} - z} \) for \( |z| < 1 \). Determine if \( I(z) = 0 \) for these values of \( z \).

Answer: 0


Question: There are n balls in an urn, each labeled with a unique number from 1 to n. You randomly pick k balls one by one without replacement. What is the probability that the label on the kth ball is larger than the labels on all previously picked balls?

Answer: \dfrac{1}{k}


Question: Determine the norm of the continuous linear operator defined by \((T u)(t) := u(\sqrt{t})\) for \(0 \le t \le 1\), where \(T : L^1[0, 1] \to L^1[0, 1]\).

Answer: 2


Question: Evaluate the infinite series given by the summation \( \sum_{k=0}^{\infty} \frac{(2k+1)!}{(k!)^2} \left(\frac{1}{8}\right)^k \).

Answer: 2\sqrt{2}


Question: Find the least value of the expression \(x^{2017} + y^{2017} + z^{2017} - 2017xyz\), where \(x, y, z \geq 0\).

Answer: -2014


Question: Consider a cube with an inscribed octahedron formed by connecting the centers of the cube's faces. Inside this octahedron, inscribe another cube by connecting the centers of the octahedron's faces. Determine the ratio of the side lengths of the inner cube to the outer cube.

Answer: \dfrac{1}{3}


Question: Let $f$ be a real-valued continuous function on the interval $[0,1]$ satisfying the estimate \( \left|\int_0^1 f(t) e^{st}dt\right|\le Cs^{\frac{1}{2}},\quad s>1, \) where the constant $C$ is independent of $s$. Can we conclude that $f$ is identically zero on $[0,1]$?

Answer: Yes


Question: Given a compact line segment $[a,b] \subset \mathbb{R}^n$ and an open cover of the form $[a,b] \subset \bigcup_{x \in [a,b]} B(x, \rho_x)$, where $B(x, \rho_x)$ is a nonempty open ball centered at $x$, determine if it is always possible to construct a finite subcover $[a,b] \subset \bigcup_{k=1}^{n} B(x_k, \rho_{x_k})$ such that $x_{k+1} \in B(x_k, \rho_{x_k})$ for all $k \in \{1, \ldots, n-1\}$. Provide a justification for your answer.

Answer: Yes


Question: Let \( f: \mathbb{D} \rightarrow \mathbb{C} \) be an univalent function such that \( f(0) = 0 \) and \( f'(0) = 1 \). Find the minimum area of the image of \( \mathbb{D} \) under \( f \).

Answer: \pi


Question:
Let \( f(x+iy) = u(x,y) + iv(x,y) \) be a holomorphic function. Given that:
1. \( \text{Im}(f'(x+iy)) = 6x(2y-1) \)
2. \( f(0) = 3-2i \)

Find \( f(1+i) \).

Answer: 5 + 2i


Question: Express the series \( \sum_{n\geq1} \left(\sum_{k=1}^{n-1} t_{k-1}t_{n-k-1}\right)x^n \) in terms of the generating function \( T(x) = \sum_{n \geq 0} t_nx^n \).

Answer: x^2 T(x)^2


Question: Let $f$ be a real-valued differentiable function on $(0,1)$ such that \( \sup_{0<r<1} |f(r)| < \infty \) and \( \liminf_{r \to 0+} f'(r) \geq 0 \). Does this imply that $f(0+)$ exists?

Answer: Yes


Question: Let \( f : \mathbb{C} \to \mathbb{C} \) be an analytic function such that for an infinite subset \( A \) of \([0,1]\), we have \( f(A) \subseteq \mathbb{R} \). Does this imply that for any \( x \in \mathbb{R} \), \( f(x) \) is real?

Answer: Yes


Question: Consider the following procedure: Start with a prime number \( p \). At each step \( n \), calculate \( p^n \) and add a number from the set \( \{1, 2, \ldots, p^{n-1}\} \) to obtain a new number. Check if this new number is prime. Continue this process for each \( n \). Does there exist a prime \( p \) such that this procedure fails to produce a prime number after a finite number of steps?

Answer: Yes


Question: Let \( \{X_n\} \) be a sequence of i.i.d. exponentially distributed random variables with mean 1. Define \( M_n = \max(X_1, X_2, \ldots, X_n) \). Find a sequence \( \{a_n\} \) such that \( P[M_n - a_n \leq x] \) converges to a non-degenerate distribution function \( G(x) \) as \( n \to \infty \), for every \( x \in \mathbb{R} \) where \( G(x) \) is continuous.

Answer: \ln n


Question: Let $p$ be a prime. If $G$ is a normal subgroup of $SL(2, \mathbb{Z}/p\mathbb{Z})$, is $G$ also normal in $GL(2, \mathbb{Z}/p\mathbb{Z})$? Provide a justification for your answer.

Answer: Yes


Question: Let $G$ be a group and $f$ be a class function on $G$. If $f(g) = f(h)$ for elements $g, h \in G$, are $g$ and $h$ necessarily in the same conjugacy class?

Answer: No


Question: Determine the number of zeros of the function \( J(x) = 3^x - 1 - 2x^2 \) given that \( 1.09 < \ln 3 < 1.1 \).

Answer: 3


Question: Find the derivative of the function \( y = \int_{0}^x t f(x^2-t^2) \, dt \) with respect to \( x \), where \( f \) is a differentiable function.

Answer: x f(x^2)


Question: Find the asymptotic equivalent of the function \( f(t) = -t \sqrt{n} + n\log\left(1+ \frac{t}{\sqrt{n}}\right) \) as \( n \to +\infty \).

Answer: -\dfrac{t^2}{2}


Question: Determine whether the quotient space $G(\mathscr A)/G_{0}(\mathscr A)$, where $T=\mathbb{R}/\mathbb{Z}$ is the circle group, $\mathscr{A}=C(T)$ is the set of continuous functions on $T$, $G(\mathscr{A})$ is the set of invertible elements in $\mathscr{A}$, and $G_{0}(\mathscr{A})$ is the identity component of $G(\mathscr{A})$, is equal to the integer group.

Answer: \mathbb{Z}


Question: Determine whether the following statement is true or false: For every natural number $n$, there exists a natural number $N$ such that for any subset $A$ of $\{1, 2, \ldots, N\}$, either $A$ or its complement contains an arithmetic progression of length $n$. Provide a justification for your answer.

Answer: True


Question: Calculate the value of the wedge product \( 3e_1 \wedge 4e_3((1, \alpha, 0), (0, \beta, 1)) \), where \( e_1 = (1,0,0) \), \( e_2 = (0,1,0) \), \( e_3 = (0,0,1) \), and \( \alpha, \beta \) are irrational numbers.

Answer: 12


Question: Consider the two-dimensional lattice $G=(\mathbb{Z}^2,\mathbb{E}^2)$, where the edge set $\mathbb{E}^2$ consists of pairs of nearest neighbors in the $\ell^1$ norm in $\mathbb{Z}^2$. Let $V\subset\mathbb{Z}^2$ be an infinite subset such that the induced subgraph $G[V]$ is connected. Define $\Lambda_n=([-n,n]\times[-n,n])\cap \mathbb{Z}^2$ as a sequence of squares on the lattice. Suppose that $$ \limsup_{n\to\infty} \frac{|V\cap \Lambda_n|}{|\Lambda_n|}=0. $$ Is it true that for independent bond percolation with parameter $p$ on $G[V]$, there is almost surely no infinite cluster for any $p\in [0,1)$? Provide a justification for your answer.

Answer: Yes


Question: Let $\Omega = \{(x, y)\in [0, 1]^2 \mid x\geq y\}$ and $P$ be the uniform probability measure on $\Omega$. Define the $\mathcal{B}(\Omega)$-measurable random variables $X_1, X_2:\Omega\to\mathbb{R}$ by $X_1(x, y)=x$ and $X_2(x, y)=y$ for every $(x, y)\in \Omega$. Find the explicit formula for the conditional expectation $E[X_2\mid \sigma(X_1)]$ in terms of $X_1$ and $X_2$. Here, $\sigma(X_1)=\{X_1^{-1}(A)\mid A \in \mathcal{B}(\mathbb{R})\}$. Provide your answer as a function of $X_1$. \( \boxed{\text{Your answer here}} \)

Answer: \dfrac{X_1}{2}


Question: Given an acute angle in the plane and a point $A$ located inside that angle, construct an equilateral triangle $ABC$ such that vertex $B$ is on one side of the angle and vertex $C$ is on the other side. Provide a step-by-step construction method.

Answer: ABC


Question: Let \( H = \begin{pmatrix} A_{i \times i} & B_{i \times n-i} \\ C_{n-i \times i} & D_{n-i \times n-i} \end{pmatrix} \in M_n(\mathbb{F}) \) where \( \mathbb{F} \) is a field. Suppose that \( \text{rank}(H) \geq n-i \) and there exist matrices \( X \in M_{n-i \times i}(\mathbb{F}) \) and \( Y \in M_{n-i \times n-i}(\mathbb{F}) \) such that \( XB + YD = I_{n-i \times n-i} \). Is it true that \( \text{rank}\left(\begin{pmatrix} B \\ D \end{pmatrix}\right) = n-i \)?

Answer: Yes


Question: Evaluate the infinite series \( \sum_{n=1}^{\infty} a_n \) given that the sequence \( a_n \) satisfies the recurrence relation \( \frac{a_n}{a_{n-1}} = \frac{(n-1)(2n-1)}{(n+2)(2n-3)} \) with \( a_1 = \frac{2}{9} \).

Answer: 1


Question: Evaluate the limit \( \lim_{n\to \infty}\int_{0}^{1}x^nf(x)\,dx \) where \( f:[0,1]\to \mathbb{R} \) is a continuous function.

Answer: 0


Question: Calculate the area enclosed by the polar curve \( r = \sqrt{|\sin(n\theta)|} \) over the interval \( 0 \leq \theta \leq 2\pi \). Verify that the area is constant for any integer \( n \).

Answer: 2


Question: Consider the set $A \subset 2^{\omega}$ of all non-eventual-constant sequences. Define the left translation operator $\operatorname{S}$ such that for a sequence $x = \{b_i\}_{i \in \omega}$, $\operatorname{S}{x} = \{c_i\}_{i \in \omega}$ where $c_i = b_{i+1}$. Determine if there exists a non-empty subset $B$ of $A$ such that for every $y \in B$, $\operatorname{S}{y} \in A \setminus B$. Provide a justification for your answer.

Answer: Yes


Question:
Find all positive integers \( n \) such that there exist primes \( p \) and \( q \) satisfying the equations:
\[ n = p(p^2 - p - 1) = q(2q + 3) \]

Answer: 2015


Question:
The function \( f:\mathbb{R}\rightarrow \mathbb{R} \) is defined as follows:
\[
f(x)=\begin{cases} 3x^2 & \text{if } x \in \mathbb{Q} \\
-5x^2 & \text{if } x \notin \mathbb{Q} \end{cases}
\]
Which of the following is true?
(A) \( f \) is discontinuous at all \( x \in \mathbb{R} \)
(B) \( f \) is continuous only at \( x=0 \) and differentiable only at \( x=0 \).
(C) \( f \) is continuous only at \( x=0 \) and nondifferentiable at all \( x \in \mathbb{R} \).
(D) \( f \) is continuous at all \( x \in \mathbb{Q} \) and nondifferentiable at all \( x \in \mathbb{R} \).
(E) \( f \) is continuous at all \( x \notin \mathbb{Q} \) and nondifferentiable at all \( x \in \mathbb{R} \).

Answer: B


Question: Let $f(x)$ be a polynomial on $\mathbb{R}$ such that $f(2x) = f'(x)f''(x)$. Find the value of $f(3)$. \(\text{Note: } f'(x) \text{ and } f''(x) \text{ are the first and second derivatives of } f(x).\)

Answer: 12


Question: Prove analytically that 0 is the only zero of the function \( \sin(2x) + 2x \).

Answer: 0


Question: Determine the number of prime numbers $p$ such that $p-1$ is squarefree.

Answer: \infty


Question: Find the coefficient of \(x^{11}\) in the expansion of \((x+x^2+x^3+x^4+x^5)^7(1+x+x^2+x^3+\dots)^4\).

Answer: 1001


Question: Evaluate the integral of the function \( f(x, y, z) = \sqrt{3x^2 + 3y^2 + z + 1} \) over the surface defined by \( z = x^2 + y^2 \) for the region where \( 1 \leq x^2 + y^2 \leq 4 \).

Answer: 33\pi


Question: Find the value of the limit: \[ \lim_{n\to \infty}\frac{(2n)!}{4^n(n!)^2} \]

Answer: 0


Question: Find the coordinates of point $P$ on the graph $y = ae^{-x}$ such that the perimeter of the rectangle $OAPB$, where $A$ and $B$ lie on the axes, is minimized. Express your answer in terms of $a$.

Answer: (\ln a, 1)


Question: Evaluate the integral \( \iint_{x^2+y^2<\sqrt{x}}\sqrt{x^2+y^2}\,dx\,dy \).

Answer: \dfrac{2}{3}


Question: Evaluate the integral \( \int_{0}^{\pi}\int_{y}^{\pi}\frac{\sin(x)}{x}\,\mathrm{d}x\,\mathrm{d}y \).

Answer: 2


Question: Find the minimum value of the function \( y = \frac{1}{2}(e^x - e^{-x}) + \frac{1}{2}n(e^x + e^{-x}) \), where \( n \) is a constant.

Answer: \sqrt{n^2 - 1}


Question: Given the joint probability density function \(f_{X,Y}(x,y)=\begin{cases}\tfrac{8}{3}(xy) & \text{for } 0<x<1 , x<y<2x \\ 0 & \text{elsewhere}\end{cases}\), calculate the covariance of \(X\) and \(Y\).

Answer: \dfrac{28}{675}


Question: Is it possible to calculate \( \mathbb{E}[X^2 Y] \), where \( X \) and \( Y \) are standard normal random variables with a correlation \( \rho \)?

Answer: 0


Question: Let \(A\) be a compact nonempty subset of \(\mathbb{R}^k\). Define \(\delta = \sup \{d(x, y): x, y \in A\}\), where \(d(x, y)\) is the standard metric. Prove that there exist points \(x_0, y_0 \in A\) such that \(d(x_0, y_0) = \delta\).

Answer: d(x_0, y_0) = \delta


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

Answer: -\dfrac{1}{2}


Question: Find the number of elements of order 3 in the symmetric group \( S_6 \).

Answer: 80


Question: How many vertex-labeled graphs with 50 vertices are there such that every vertex has an odd degree? Provide your answer as a power of 2.

Answer: 2^{1176}


Question: Determine whether the ring \( \frac{\mathbb{Z}_p[[X]] \otimes_\mathbb{Z} \mathbb{Q}_p}{(X-p)^r} \) is principal for all integers \( r \geq 1 \).

Answer: Yes


Question: Use Green's theorem to calculate the area enclosed by the circle \(x^2 + y^2 = 16\).

Answer: 16\pi


Question: Calculate the volume between the surfaces defined by the equations \(x^2 + y^2 = z\) and \(2x + 2y + z = 2\). Determine the correct order of subtraction for the integration to find the volume.

Answer: 8\pi


Question: Compute the flux of the vector field \( F = \langle 4x, 4y, 0 \rangle \) through the surface \( S \), which is the part of the surface \( z = 25 - (x^2 + y^2) \) above the disk of radius 5 centered at the origin, oriented upward.

Answer: 2500\pi


Question: Determine the smallest cubic graph that contains a square-free 2-factor, where a square-free 2-factor is a 2-factor with no cycles of length 4.

Answer: 6


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

Answer: \dfrac{\pi}{2}


Question: Evaluate the limit \( \lim_{n\to \infty} \sum_{k=2}^n \log_{\frac{1}{3}} \left(1-\frac{2}{k(k+1)}\right) \).

Answer: 1


Question: Given the recurrence relation \( a_{n+2} = \sqrt{a_{n+1} \cdot a_{n}} \) with initial conditions \( a_0 = 2 \) and \( a_1 = 8 \), determine the value of \( a_2 \).

Answer: 4


Question: Evaluate the expression: \[ \frac{\sum\limits_{n=1}^{99} \sqrt{10 + \sqrt{n}}}{\sum\limits_{n=1}^{99} \sqrt{10 - \sqrt{n}}} \]

Answer: 1 + \sqrt{2}


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

Answer: \infty


Question: Find the residue of the function \( \frac{1}{z^{100} - 1} \) at \( z = 1 \).

Answer: \dfrac{1}{100}


Question: Find the smallest integer $n$ such that the alternating group $A_n$ contains an element of order 4.

Answer: 6


Question: Given the implicit curve \( y^2 x + a = x^2 + y^2 \) and the tangent line \( y = \frac{3}{2} x - 2 \), find the value of the parameter \( a \).

Answer: 3


Question: Determine the radius of convergence of the power series \( \sum_{n=1}^\infty a_n z^n \) where \( a_n = (\log n)^2 \).

Answer: 1


Question: Is it true that for any smooth complex projective Fano threefold $X$, the class $c_2(X)$ can be realized as an effective curve?

Answer: Yes


Question: Given the quadratic function \( f(x) = ax^2 + bx + c \) with \( a < b \) and \( f(x) \ge 0 \) for all \( x \in \mathbb{R} \), find the minimum value of \( \frac{a+b+c}{b-a} \).

Answer: 3


Question: Consider a normed space $X$ and a Banach space $Y$ over $\mathbb{R}$. Let $D \subseteq X$ be dense, and let $T_n \in \mathcal{L}(X,Y)$ be a bounded sequence such that $T_n z$ converges for all $z \in D$. There exists $T \in \mathcal{L}(X,Y)$ such that $T_n x \rightarrow Tx$ for all $x \in X$. Is it possible that $\|T_n\|_{\mathcal{L}} \not\rightarrow \|T\|_{\mathcal{L}}$?

Answer: Yes


Question: Determine the invariant factors of the finitely generated Abelian group $M$ given by the generators $x, y, z$ and the relations $4x + y + 2z = 0$, $5x + 2y + z = 0$, and $6y - 6z = 0$. Provide your answer as a list of invariant factors.

Answer: [3, 6]


Question: Determine the number of elements of order 6 in the group \(D_{12} \times Z_2\).

Answer: 6


Question: How many arrangements of the letters in "DISAPPEARANCES" satisfy the following conditions: (i) there are at least two letters between each 'A', (ii) the arrangement ends with an 'A', and (iii) the consonants are not in alphabetical order?

Answer: 179607780


Question:
Consider the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space $E$, denoted by $\mathcal{L}(E)$. On $\mathcal{L}(E)^2$, we define two norms:
\[ N_1(A,B) = \sup\{\|Ax\|^2 + \|Bx\|^2 : x \in E, \|x\| = 1\} \]
and
\[ N_2(A,B) = \|A\|^2 + \|B\|^2. \]
It is known that $N_1(A,B) \leq N_2(A,B)$. Determine if the equality $N_1(A,B) = N_2(A,B)$ holds when $A$ and $B$ are normal operators on $E$ and they commute, i.e., $AB = BA$. Is this claim correct?

Answer: No


Question: Find all real values of \( a \) such that the function \( f(z) = |i + z|^2 + az + 3 \), where \( z \) is a complex number, satisfies the condition: if \( f(u) = 0 \) for some complex number \( u \), then \( f(\overline{u}) = 0 \) as well, where \( \overline{u} \) is the complex conjugate of \( u \).

Answer: \mathbb{R}


Question: Evaluate the limit: \( \lim_{x\to\infty} (x!*e^{-x^2}) \).

Answer: 0


Question: Let $F$ be a finite field of order 32. Determine the number of solutions $(x, y) \in F \times F$ for the equation $x^2 + y^2 + xy = 0$. Provide your answer as a single integer.

Answer: 1


Question: A father has a pie made for his two sons. Eating more than half of the pie will give indigestion to anyone. While he is away, the older son helps himself to a piece of the pie. The younger son then comes and has a piece of what is left by the brother. Assume that the size of each of the two pieces eaten by the sons is random and uniformly distributed over what is currently available. What is the expected size of the remaining piece given that no son has indigestion?

Answer: \dfrac{2 - \ln 2}{4 \ln 2}


Question: Find \( \lim_{n \rightarrow \infty} \left( \frac{1}{\sqrt{n}} \sum_{k=1}^n \left| \frac{a_k}{\sqrt{k}} \right| \right) \), where \(a_n\) converges to \(a \in \mathbb{R}\).

Answer: 2|a|


Question: Determine whether every unitary matrix can be expressed as the exponential of a skew-adjoint matrix.

Answer: Yes


Question: Find the greatest natural number that divides \(n^2(n^2 - 1)(n^2 - n - 2)\) for all natural numbers \(n\).

Answer: 48


Question: Evaluate the limit: \[ \lim_{x \to 0^+} (\ln\cot(x))^{\tan(x)} \]

Answer: 1


Question: In triangle ABC, which is right-angled at B and inscribed in the parabola $y^2 = 4x$, find the minimum length of the hypotenuse AC. The points are given as $A((t_1)^2, 2t_1)$, $B((t_2)^2, 2t_2)$, and $C((t_3)^2, 2t_3)$. Given that $\overrightarrow{AB}$ is perpendicular to $\overrightarrow{BC}$, the product of their slopes is $-1$. Thus, $(t_1 + t_2)(t_2 + t_3) = -4$. Determine the minimum length of AC.

Answer: 8


Question: Is the composition $R_{\alpha} \circ T$ ergodic for (Lebesgue) almost every $\alpha \in \mathbb{R}$, where $T$ is a Lebesgue measure-preserving automorphism of the unit interval $I$ and $R_{\alpha}$ denotes rotation by $\alpha$?

Answer: Yes


Question: Determine the radius of convergence of the series \( \sum_{n=0}^\infty 4^{(-1)^n n} z^{2n} \).

Answer: \dfrac{1}{2}


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

Answer: \dfrac{2}{e}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\sum_{i=1}^{n}{\frac{2n}{(n+2i)^2}}. \]

Answer: \dfrac{2}{3}


Question: Let \( f: X \rightarrow X \) be a continuous map on the compact metric space \( X \). Prove that there exists a subset \( A \subset X \) such that \( f(A) = A \).

Answer: A


Question: In an isosceles right triangle $ABC$ with $AB = AC = 3$, let $M$ be the midpoint of the hypotenuse $\overline{BC}$. Points $I$ and $E$ are on sides $\overline{AC}$ and $\overline{AB}$, respectively, such that $AI > AE$ and quadrilateral $AIME$ is cyclic. Given that the area of triangle $EMI$ is 2, express the length $CI$ as $\frac{a-\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers with $b$ not divisible by the square of any prime. Find the value of $a + b + c$. \(\text{Note: Use properties of cyclic quadrilaterals and triangles to solve.}\)

Answer: 12


Question: There are 100 countries participating in an olympiad. Each country is willing to communicate in exactly \( n \) languages. If each set of 20 countries can communicate in at least one common language, and no language is common to all 100 countries, what is the minimum possible value of \( n \)?

Answer: 20


Question: Determine whether the sequence of functions \( \{f_n(x) = x^n(1-x) \} \) converges uniformly to 0 on the interval \([0,1]\).

Answer: Yes


Question: Let $X_i$ be iid random variables with a uniform distribution on $(0,100)$. Find the expected number of random variables $N$ such that $\sum_{i=1}^{N} X_i \geq 1000$, where $N = \min \{ n : \sum_{i=1}^{n} X_i \geq 1000 \}$.

Answer: \dfrac{62}{3}


Question: Find the residue at $0$ of the function \( f(z)=\frac{1+iz-e^{iz}}{z^3} \) on \( \mathbb{C} - \{0\} \).

Answer: \dfrac{1}{2}


Question: Suppose X and Y are independent Poisson random variables with respective parameters \(\lambda\) and \(2\lambda\). Find \(E[Y - 2X \mid X + Y = 10]\).

Answer: 0


Question: Find the value of the following limit: \[ \lim\limits_{n \rightarrow \infty}  \sum\limits_{k=1}^n \arctan\left(\frac{1}{\sqrt{k^2+1}}\right) \]

Answer: \infty


Question: In how many ways can 10 different people choose to visit up to 3 destinations each from a set of 20 different destinations, given that some people may choose not to travel at all and the order of visits is not important?

Answer: 1351^{10}


Question: Prove that the infinite product \( \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2+\sqrt{2}}}{2} \cdot \frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2} \cdots \) equals \( \frac{2}{\pi} \).

Answer: \dfrac{2}{\pi}


Question: Consider a series of independent, identically distributed random variables $X_1, X_2, \ldots$, each following an exponential distribution with parameter $\lambda$. Determine the expected value of the index $n$ such that $X_n$ is the first variable with a value greater than or equal to a constant $Y$.

Answer: e^{\lambda Y}


Question:
Determine the point at which the function \( f(z) = |z|^2 + i\bar{z} + 1 \) is differentiable. Choose from the following options:

- \( i \)
- \( 1 \)
- \( -i \)
- No point in \( \mathbb{C} \)

Answer: C


Question: Let \( A = k[x,y] \) where \( k \) is an algebraically closed field, and let \( M = A/(xy) \) be an \( A \)-module. Calculate the support of \( M \), denoted as \( \text{Supp}(M) = \{ P \in \text{Spec}(A) : M_P \neq 0 \} \), where \( M_P = S^{-1}M \) and \( S = A \setminus P \).

Answer: V(x) \cup V(y)


Question: Determine whether the Riemann integral of the function \( g: [0,1] \to (0,1) \) defined by \( g(x) = 1 \) for \( 0 < x \le 1 \) and \( g(x) = 0 \) for \( x = 0 \) exists. If it exists, calculate its value.

Answer: 1


Question:
Given the group $G$ generated by the permutations $\sigma$ and $\tau$, where:
\[\sigma = ( 1, 2, 3, 4, 5 ) ( 6, 10 ) ( 7, 11 ) ( 8, 12 ) ( 9, 13 )\]
\[\tau = ( 2, 5 ) ( 3, 4 ) ( 6, 7, 8, 9, 10, 11, 12, 13 )\]
The orders of these permutations are 10 and 8, respectively. The subgroup $H = \langle \sigma \rangle$ and $K = \langle \tau \rangle$ are subgroups of $G$. Given that $\langle H \cap K \rangle$ has order 2, determine the order of the group $G$.

Answer: 40


Question: Let $f(x)$ be a uniformly continuous function such that $f(x) > 0$ for all $x \geq 0$ and $\int_0^{\infty} f(x) \, dx \leq M$ for some constant $M$. Does it follow that $\lim_{x \to \infty} f(x) = 0$? Justify your answer.

Answer: Yes


Question: Is it always true that for every blocking flow $f$ in an acyclic directed graph $G=(V,E)$ with source $s \in V$ and sink $t \in V$, the value of $f$ satisfies $\text{value}(f) \geq \frac{c}{n}$, where $c$ is the value of a maximum $s$-$t$-flow and $n=|V|$?

Answer: Yes


Question: What is the probability that the pattern "101" will appear infinitely many times in an endless series of independent Bernoulli trials, where the probability of getting a "1" is \( p \) and the probability of getting a "0" is \( q = 1 - p \)?

Answer: 1


Question: Let \(X_1, X_2, \ldots, X_n\) be a random sample from a Bernoulli distribution with parameter \(\theta\). Find the UMVUE of \(\theta(1-\theta)\). Given a sample size of \(n=10\) with observations \(1, 1, 1, 1, 1, 0, 0, 0, 0, 0\), compute the value of this estimator.

Answer: \dfrac{5}{18}


Question: Determine whether there exists a smooth function \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that \( f(x) = 0 \) for all \( x \leq 0 \), \( f(x) = 1 \) for all \( x \geq 1 \), and \( f \) is monotonically increasing.

Answer: Yes


Question: Solve the equation \(-2^x + 3^{x/2} + 1 = 0\) for integer values of \(x\) without using numerical methods.

Answer: 2


Question: Find the 2002th term of a binary sequence defined as follows: Start with $S_1 = 0$. For $k \ge 2$, obtain $S_k$ from $S_{k-1}$ by replacing every 0 with 01 and every 1 with 001. The sequence satisfies $S_k = S_{k-1}S_{k-2}S_{k-1}$ for all $k \ge 3$. What is the 2002th term of this sequence?

Answer: 0


Question: Let \( \mathbb{R} \) be the set of all real numbers, and let \( \tau \) be a topology on \( \mathbb{R} \) consisting of \( \mathbb{R}, \emptyset \), and all open intervals \((-n,n)\) for any positive integer \( n \). Given the set \( A = (-\infty, 4) \), determine the derived set (limit points) of \( A \) in this topology.

Answer: \mathbb{R}


Question: Consider the vector \((1\ 2\ 2\ 3\ 3\ 3)^T\) in the vector space \(\mathbb{R}^6\). The symmetric group \(S_6\) acts on \(\mathbb{R}^6\) by permuting the coordinates. Determine the order of the stabilizer of this vector under the given action of \(S_6\).

Answer: 12


Question: Suppose that $C$ is a convex subset of a normed vector space $X$. Let $f:C\to \mathbb{R}$ be a continuous function such that there exists a $\beta\in \left(0,1\right)$ for which \( f(\beta x+(1-\beta)y)\leq \beta f(x)+(1-\beta)f(y) \) for all $x,y\in C$. Is $f$ necessarily convex?

Answer: Yes


Question: Find the value of the expression \( \arctan \frac{p-q}{1+pq} + \arctan\frac{q-r}{1+qr} + \arctan\frac{r-p}{1+rp} \) given that \( p > q > 0 \) and \( pr < -1 < qr \).

Answer: \pi


Question: Evaluate the limit: \[ \lim_{x \rightarrow \infty}\left(\frac{2}{\pi}\arctan(x)\right)^{\frac{x^2}{1+2x}} \] without using L'Hôpital's rule.

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


Question: Evaluate the limit: \[ L = \lim_{n \to \infty} \int_0^1 x^{10} \sin(nx) \, dx \]

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to 0} \int_0^x \frac{\sin^2(u)}{\sin(x^3)} \, du \]

Answer: \dfrac{1}{3}


Question: Evaluate the limit: \[ \lim _{x\to -\infty }\frac{3x^7-25x^5+e^{-\frac{x^3}{(1-x)^3}}}{-5x^7+111! \cdot x} \]

Answer: -\dfrac{3}{5}


Question: Determine the maximum value of the ratio \( \frac{r}{R} \) for an arbitrary triangle, where \( r \) is the incircle radius and \( R \) is the circumcircle radius.

Answer: \dfrac{1}{2}


Question: Determine whether the following statement is true or false: Let $A$ be an Artinian ring and $M$ an $A$-module. The module $M$ is faithful if and only if there exists an exact sequence of the form $0\rightarrow A \rightarrow M^r$ for some natural number $r$. Provide a justification for your answer.

Answer: True


Question: Consider a sequence of complex Hilbert spaces \( H_n \) such that \( H_{n+1} \subsetneq H_n \) and \( \bigcap_{n=1}^\infty H_n = \{v_0\} \). Let \( T_1: H_1 \to \mathbb{C} \) be a continuous linear functional with \( T_1(v_0) = 0 \). For each \( n \in \mathbb{N} \), define \( T_n: H_n \to \mathbb{C} \) as the restriction of \( T_1 \) to \( H_n \). Determine whether \( \lim_{n \to \infty} \lVert T_n \rVert = 0 \).

Answer: 0


Question: Evaluate the limit of the integral: \[ \lim_{q \to 0}\int_0^1 \frac{1}{qx^3+1} \, \operatorname{d}\!x \]

Answer: 1


Question: Given that \( Q(x) = f(x^3) + xg(x^3) \) is divisible by \( x^2 + x + 1 \), find the value of \( Q(1) \).

Answer: 0


Question: Evaluate the double integral \( \iint (x^2 \cdot y + y \cdot \sin(x^8)) \, dx \, dy \) over the region \( D = \{(x,y) : x^2 + y^2 \leq 2, y > 0\} \).

Answer: \dfrac{8\sqrt{2}}{15}


Question: Evaluate the limit: \[ \lim_{(x,y) \rightarrow (0,0)} \frac{x^3y+xy^2}{x^2+y^2} \]

Answer: 0


Question: Solve the equation \( t \log t = 10 \) for \( t \).

Answer: 10


Question: Given the function \( f(x) = \int_0^{\ln x} \frac{1}{\sqrt{4+e^t}} \, dt \) for \( x > 0 \), find the value of \((f^{-1})'(0)\).

Answer: \sqrt{5}


Question: Find a function $f \in L^2(0,1)$ such that $\int_0^1 x f(x) \, dx = 1$ and $f$ has the minimal norm in $L^2(0,1)$. Determine the function $f$.

Answer: 3x


Question: Let $V$ be a 10-dimensional real vector space and $U_{1}, U_{2}$ be two linear subspaces such that $U_{1} \subseteq U_{2}$, $\dim U_{1} = 3$ and $\dim U_{2} = 6$. Let $E$ be the set of all linear maps $T: V \rightarrow V$ which have $U_{1}$ and $U_{2}$ as invariant subspaces (i.e., $T(U_{1}) \subseteq U_{1}$ and $T(U_{2}) \subseteq U_{2}$). Calculate the dimension of $E$ as a real vector space.

Answer: 67


Question:
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for every \( x, y \in \mathbb{R} \), the following equation holds:
\[
f\left(x^2+f(y)\right)=y+f(x)^2
\]

Answer: f(x) = x


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

Answer: 3


Question: Let \( f(x) = \left(\int_{0}^{x} e^{-t^2} dt\right)^2 \) and \( g(x) = \int_{0}^{1} \frac{e^{-x^2(1+t^2)}}{1+t^2} dt \). Find the value of \( f'(\sqrt{\pi}) + g'(\sqrt{\pi}) \).

Answer: 0


Question: Let $S$ be the set of all values of $a$ for which the function \( f(x) = \frac{4a-7}{3}x^3 + (a-3)x^2 + x + 5 \) is monotonic for all \( x \in \mathbb{R} \). Determine the set $S$. \( \boxed{[2,8]} \)

Answer: [2, 8]


Question: Given a random variable $X$ with an unconditional mean of 0, and the conditions $E[XY]=E[X|Y]=0$ and $E[XZ] \neq 0$, determine whether it follows that $E[XYZ]=0$. Justify your answer.

Answer: No


Question: Maximize the quotient \( \frac{||f||}{||f'||} \) of Euclidean norms for a function \( f \) defined on \([0,1]\), where \( f \) vanishes on the boundary. Assume \( f \) is continuously differentiable. Here, the norm is defined as \( ||g||^2 = \int_0^1 g(x)^2 \, \mathrm{d}x \).

Answer: \dfrac{1}{\pi}


Question: In a Poisson process with a rate of 0.1 observations per minute, calculate the expected time of the 2nd arrival, given that the 2nd arrival occurs within the first 2.5 minutes.

Answer: 1.63


Question: Determine whether the functional \( f: l^1 \to \mathbb{R} \) defined by \( f(a_1, a_2, a_3, \ldots) = \frac{2a_1}{3} + \frac{4a_2}{9} + \frac{8a_3}{27} + \ldots + \frac{2^n a_n}{3^n} + \ldots \) is a bounded linear functional. If it is, calculate \( ||f|| \).

Answer: \dfrac{2}{3}


Question:
Determine the number of real roots of the equation \(x^{13} - e^{-x} + x - \sin{x} = 0\). Choose one of the following options:

- No real root
- More than two real roots
- Exactly two real roots
- Exactly one real root

Answer: D


Question: Given the set $X=\{a,b,c,d\}$ and the topology $\tau=\{\emptyset, \{a,b\}, \{a,b,c\}, \{c,d\}, X\}$, find the connected component $C(a)=\bigcup \{A \mid a \in A , A \subset X, \text{$A$ connected}\}$.

Answer: \{a, b\}


Question: In a tutorial session, the time a student waits to ask a question is exponentially distributed with an expectation of 10 minutes. Assuming that waiting times are independent and only one student can ask a question at a time, calculate the expected number of students who will have their questions answered before a student appears who has to wait more than 20 minutes (including this student).

Answer: e^2


Question: What is the probability that four randomly dropped disks within a finite area are collinear? Assume each drop is independent and identically distributed, and the area is large enough to prevent overlapping disks.

Answer: 0


Question: Determine whether the probability mass function \( f(y)=\frac{\mu^y}{(y!)^\theta z(\mu,\theta)} \), where \( z(\mu,\theta) = \sum_{i=0}^{\infty}\frac{\mu^i}{(i!)^\theta} \), belongs to the exponential family.

Answer: Yes


Question: Evaluate the surface integral \( \iint_S (x^2z+y^2z)\,dS \) where \( S \) is the part of the plane \( z=4+x+y \) that lies inside the cylinder \( x^2+y^2=4 \).

Answer: 32\sqrt{3}\pi


Question: In a triangle with integer side lengths X, Y, and 20, where X > Y > 20, let x, y, and h be the altitudes to sides X, Y, and 20, respectively. If x + y = h, determine the perimeter of the triangle.

Answer: 101


Question: Calculate the surface integral \( \iint_{S} \text{curl}(\vec{F}) \cdot \vec{dS} \) for the vector field \( \vec{F} = -y\vec{i} + x\vec{j} + xyz\vec{k} \), where \( S \) is the part of the sphere \( x^2 + y^2 + z^2 = 25 \) below the plane \( z = 4 \) and is oriented positively. The correct answer is \(-18\pi\).

Answer: -18\pi


Question: Consider the metrics on the set of real numbers defined by \(d(x,y)=|x-y|\) and \(\tilde{d}(x,y)=\frac{|x-y|}{\sqrt{(x^2+1)(y^2+1)}}\). It is known that \((\mathbb{R},d)\) is complete, but \((\mathbb{R}, \tilde{d})\) is not. Additionally, \(\tilde{d}(x,y) \leq d(x,y)\) for all \(x,y \in \mathbb{R}\), implying that if a sequence \(x_n\) converges to \(x\) with respect to \(d\), it also converges to \(x\) with respect to \(\tilde{d}\). Is there a sequence that converges with respect to \(\tilde{d}\) but does not converge with respect to \(d\)?

Answer: No


Question: Can you find an example of an unbounded linear operator on a Banach space whose rank is finite? Specifically, consider two Banach spaces $X$ and $Y$, and a linear map $T: X \to Y$ such that $T(X) \subseteq Y$ is of finite dimension. Is it possible for $T$ to be unbounded?

Answer: Yes


Question: Using Green's Theorem, determine the simple closed curve \( C \) for which the line integral \( \oint_C F \cdot dr \) (with positive orientation) attains the largest positive value, where \( F(x,y) = (y^3-6y)i + (6x-x^3)j \). Compute this largest possible value.

Answer: 24\pi


Question: Find the minimal integer \( n \) such that there exists a \([n, n-5]\) cyclic binary code with the generator polynomial \( g(x) = 1 + x^4 + x^5 \).

Answer: 21


Question: Solve the equation \( \arccos \frac{1-x}{1+x} - \arcsin \frac{2 \sqrt{x}}{1+x} = 0 \) for \( x \in \mathbb{R} \).

Answer: [0, 1]


Question: Calculate the residues at all singularities of the function \( f(z) = e^{z^2 + \frac{1}{z^2}} \).

Answer: 0


Question: Evaluate \( \int_0^\infty \frac{x-\sin x}{x^3} \, dx \) in terms of \( A \), where \( A = \int_0^\infty \frac{\sin^3 x}{x} \, dx \).

Answer: A


Question: Find the operator norm \( \|T\| \) for the operator \( T: C[-1,1] \rightarrow \mathbb{R} \), defined by \( T_g(f) = \int_{-1}^1 g(t)f(t) \, dt \), where \( g(t) = t \) for all \( t \in [-1,1] \) and \( f(t) = t^{\frac{1}{n}} \) with \( n \) being a large odd integer. The norm of \( f \) is given by \( \|f\| = \max_{t \in [-1,1]} |f(t)| \). Use the definition of the operator norm: \( \|T\| = \sup_{\|f\|=1} \|T_g(f)\| \).

Answer: 1


Question: Evaluate the infinite series: \[ \sum_{r=1}^{\infty} \frac{1 \cdot 3 \cdot \ldots (2r-1)}{r!}\left(\frac{2}{5} \right)^{r} \]

Answer: \sqrt{5} - 1


Question: Determine whether the expression \(78n^2 + 72n + 2018\) can be expressed as the sum of four perfect cubes for any integer \(n\).

Answer: Yes


Question: Given matrices $A$ and $B$ such that $A^2 = I$, $B^2 = I$, and $(AB)^2 = I$, prove or disprove that $AB = BA$.

Answer: AB = BA


Question: Compute the surface integral \( \iint_{S}{x^2dS} \), where \( S \) is the portion of the cylinder \( x^2+y^2=4 \) between the planes \( z=0 \) and \( z=x+3 \).

Answer: 24\pi


Question:
Evaluate the limit of the sequence \((I_n)_{n \geq 1}\) defined by:
\[ I_n = \int_0^1 \frac{x^n}{4x + 5} \, dx \]
Find:
\[ \lim_{n \to \infty} nI_n \]

Answer: \dfrac{1}{9}


Question: Determine the value of the real parameter \( t \) for which the function \( (x_1, x_2),(y_1, y_2) \to (x_1, x_2) \begin{pmatrix} 6 & e^{2t} \\ 2e^t-1 & 2 \end{pmatrix} \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} \) is a scalar product in \( \mathbb{R}^2 \).

Answer: 0


Question: Evaluate the line integral \( \int_{\gamma} \frac{-y\,dx + x\,dy}{x^2 + y^2} \), where \( \gamma \) is the ellipse given by \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \) with counter-clockwise orientation.

Answer: 2\pi


Question: Is there an infinite non-abelian group in which all irreducible representations are one-dimensional?

Answer: No


Question: Given a bivariate random variable \( A = (A_1, A_2)^T \) with a Gaussian distribution on \( \mathbb{R}^2 \), zero mean, and covariance matrix \( \begin{pmatrix} 1 & -0.4 \\ -0.4 & 1 \end{pmatrix} \), let \( B = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \) and \( C = \begin{pmatrix} 2 \\ 1 \end{pmatrix} \). Define \( X = B^T A \) and \( Y = C^T A \). Find the covariance of \( X \) and \( Y \).

Answer: 2


Question: Given the parametric equations \( x = 2t + t^2 \sin\left(\frac{1}{t}\right) \) for \( t \neq 0 \) and \( x = 0 \) for \( t = 0 \), and \( y = \frac{\sin(t^2)}{t} \) for \( t \neq 0 \) and \( y = 0 \) for \( t = 0 \), find \( \frac{dy}{dx} \) at \( t = 0 \).

Answer: \dfrac{1}{2}


Question: Find a basis of solutions for the system of differential equations given by \( y' = \begin{pmatrix} 3x-1 & x-1 \\ -x-2 & x-2 \end{pmatrix} y \), where one solution is \( y = \begin{pmatrix} y_1 \\ -y_1 \end{pmatrix} \). Determine the function \( y_1 \).

Answer: e^{x^2}


Question: Determine whether the sequence of functions \(f_n(x) = nx^n(1-x)\) converges pointwise on the interval \([0,1]\). If it converges, find the limit function.

Answer: 0


Question: Find the minimal polynomial of the linear transformation \( T: \mathbb{C}_n[x] \to \mathbb{C}_n[x] \) defined by \( T(p) = p' + p \).

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


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

Answer: -\dfrac{2}{3}


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

Answer: 0


Question: Find all positive integers \( n \) such that \( 2n^3 + 5 \mid n^4 + n + 1 \).

Answer: \emptyset


Question: Evaluate the limit \( \lim_{n\to\infty} \left(\frac{1}{n}\cdot\frac{2}{n}\cdot\frac{3}{n}\cdots\cdots\frac{n}{n}\right)^\frac{1}{n} \) by expressing it as a Riemann integral.

Answer: \dfrac{1}{e}


Question: Let $n$ be a positive integer. Suppose that $2^n$ and $5^n$ begin with the same digit. What is this digit?

Answer: 3


Question: Let $(X,d)$ be a compact metric space. Let $f:X \to X$ be a function such that $d(f(x),f(y)) < d(x,y)$ for all $x, y \in X$ with $x \neq y$. Can $f$ be surjective? Justify your answer.

Answer: No


Question: Evaluate the limit: \[ \lim _{x \to 0} \left(\frac{ \sin x}{x}\right)^{1/x} \]

Answer: 1


Question: Find the set of real values of \(x\) for which \[ x^{(\log_{10}x)^2-3\log_{10}x+1} > 1000. \]

Answer: (1000, \infty)


Question: Evaluate the limit: \[ \lim_{x \to y} \frac{x^y - y^x}{x^x - y^y} \] Choose the correct answer from the following options: \[(a) \ \frac{1 - \log x}{1+\log x} \quad (b) \ \frac{1-\log y}{1+\log y} \quad (c) \ \frac{\log x - \log y}{\log x + \log y} \quad (d) \ \text{None of these}\]

Answer: B


Question: Evaluate the integral \( \int_0^1 A(x) \, dx \), where the function \( A: [0,1] \rightarrow \mathbb{R} \) is defined by \( A(x) = 1 \) if \( x \) is an algebraic number and \( A(x) = 0 \) otherwise.

Answer: 0


Question: Evaluate the contour integral \( \int_{\gamma}\frac{\sin(z)}{z}dz \) where \( \gamma:[0,2\pi]\rightarrow\mathbb{C} \) is given by \( \gamma(t)=e^{it} \).

Answer: 0


Question: Let $G$ be a group with subgroups $H$ and $K$ such that $HK = G$. Given that $|G| = n$ and $\gcd([G:H], [G:K]) = 1$, prove that $G = HK$.

Answer: G = HK


Question: Determine whether there exists an isometric embedding of the space $L^1(a,b;H^*)$ into the dual space of $L^{\infty}(a,b;H)$, where $H$ is a separable Hilbert space and $H^*$ denotes its dual.

Answer: Yes


Question:
Suppose $S$ is a collection of subsets of the natural numbers $\{1,2,3,\ldots\}$ such that:

1. If $A \in S$ and $B \subset A$, then $B \in S$ (i.e., $S$ is closed under taking subsets).
2. If $A \in S$ and $B \in S$, then $A \cup B \in S$ (i.e., $S$ is closed under taking finite unions).
3. If $F \subset \{1,2,3,\ldots\}$ is finite, then $F \in S$ (i.e., $S$ contains all finite sets of numbers).

Consider a set $A \subset \{1,2,3,\ldots\}$ such that for each infinite subset $B \subset A$, there exists an infinite subset $C \subset B$ such that $C \in S$. Must $A \in S$?

Answer: No


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

Answer: 1


Question: On a surface where the metric is given by \( ds^2 = du^2 + dv^2 \), find the angle between the lines \( v = u \) and \( v = -u \). Use the parametrizations \( u(t) = t, v(t) = t \) and \( u(s) = s, v(s) = -s \) to determine the tangent vectors and calculate the angle between these curves.

Answer: \dfrac{\pi}{2}


Question: Given that $W(t)$ is a standard Brownian motion, calculate the probability that $W(2) < 0$ given that $W(1) > 0$. Express your answer as a numerical value for $\mathbb{P}[W(2)<0|W(1) > 0]$. Use the joint distribution of $(W(1), W(2)) \sim N((0,0), \begin{bmatrix} 1 & 1 \\ 1 & 2 \end{bmatrix})$.

Answer: 0.25