Question: Does hypergraph isomorphism preserve the size of edges? Specifically, if an edge in a hypergraph contains a certain number of vertices, must its corresponding edge in an isomorphic hypergraph contain the same number of vertices?

Answer: Yes


Question: Given the pair of straight lines represented by the equations \(x^2 - 2pxy - y^2 = 0\) and \(x^2 - 2qxy - y^2 = 0\), where each pair bisects the angle between the other pair, find the value of \(pq\).

Answer: -1


Question: Let $f: [0, 1] \rightarrow \mathbb{R}^+$ be a continuous density function with cumulative distribution function $F$. Assume $f > 0$ over $(0, 1)$ and that the function \( \frac{1 - F(x)}{f(x)} \) is decreasing. Determine if \( \lim_{x \rightarrow 1} \frac{1 - F(x)}{f(x)} = 0 \) when $f(1) = 0$. Justify your answer.

Answer: 0


Question: Let $X$ be a Polish space and $f: X \rightarrow \mathbb{R}$ be a function such that for any real number $d$, the pre-image $f^{-1}(-\infty, d)$ is open in $X$. For a real number $c$, consider a sequence $(c_n)$ converging to $c$ with $c < c_n$ for all $n$. Define $A_n = (-\infty, c_n)$ for each $n \in \mathbb{N}$. Is the set $f^{-1}(-\infty, c] = \cap_{n \in \mathbb{N}} f^{-1}(-\infty, c_n)$ closed in $X$?

Answer: Yes


Question: Let $G$ be a group of order 12. Does the group of automorphisms of $G$, denoted as $\mathrm{Aut}(G)$, contain an element of order 5?

Answer: No


Question:
Determine the minimal number of time steps required to reach the state $x_f = \begin{bmatrix} 1 & 1 & 0 \end{bmatrix}^T$ from the initial state $x_0 = \begin{bmatrix} 0 & 0 & 0 \end{bmatrix}^T$ in the discrete-time state-space system given by:

$$x(t+1) = Ax(t) + Bu(t), \quad y(t) = Cx(t)$$

where

$$A = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 1 & 1 \\ 2 & 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}.$$

Answer: 2


Question: Find the radius of the circle formed by the intersection of the plane $x + 2y - z = 4$ and the sphere $x^2 + y^2 + z^2 - x + z - 2 = 0$.

Answer: 1


Question: In an acute-angled triangle \( \triangle ABC \) with \( \angle A = 20^\circ \), let \( D, E, F \) be the feet of the altitudes from vertices \( A, B, C \) respectively, and let \( H \) be the orthocenter of the triangle. Calculate the value of the expression \( \frac{AH}{AD} + \frac{BH}{BE} + \frac{CH}{CF} \).

Answer: 2


Question:
Compute the radius of convergence of the power series \( \sum_{n \ge 0} f_n(z) \), where each function \( f_n: \mathbb{C} \rightarrow \mathbb{C} \) is defined by:
\[ f_n(z) = \frac{1}{p_n}[z(1-z)]^{4^n} \]
and \( p_n \) is the largest absolute value of a coefficient in the expansion of \([z(1-z)]^{4^n}\).

Answer: 1


Question: In a complete, separable, compact geodesic metric space $(M,d)$, is it true that for any point $x \in M$, there exists an $\varepsilon' > 0$ such that all $\varepsilon$-balls around $x$ for $\varepsilon < \varepsilon'$ are convex?

Answer: Yes


Question: Let \( f(x) \) be a function that is defined and has a continuous derivative on \((2, \infty)\). Assume that \( f(a) = b \), \( \vert f(x) \vert < x^9 + 9 \), and \( \int_{4}^{\infty} f(x) e^{-x/5} \, dx = -2 \). Determine the value of \( \int_{4}^{\infty} f'(x) e^{-x/5} \, dx \).

Answer: -\dfrac{2}{5}


Question:
Consider the continuous convex functions \(g_i(x):\mathbb{R}^{n} \to \mathbb{R}\) for \(i=1,\ldots,n\). Define \(g_{\rm max}(x) = \max_{i=1,\ldots,n}\{g_i(x)\}\). Let the feasible sets be:
\[\mathcal{F}_1 = \{x \in \mathbb{R}^n: g_i(x)\leq0 \ \forall i \}\]
and
\[\mathcal{F}_2 = \{x \in \mathbb{R}^n: g_{\rm max}(x)\leq0 \}\]
Determine which of the following statements is true:
1) \(\mathcal{F}_1 \subseteq \mathcal{F}_2\)
2) \(\mathcal{F}_1 \supseteq \mathcal{F}_2\)
3) \(\mathcal{F}_1 = \mathcal{F}_2\)

Answer: 3


Question: Let $\alpha,\beta$ be two complex numbers with $\beta\ne 0$ and $f(z)$ a polynomial function on $\mathbb{C}$ such that $f(z)=\alpha$ whenever $z^5=\beta$. Determine the degree of the polynomial $f(z)$. Justify your answer.

Answer: 5


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

Answer: 1


Question: Is there a group $G$ with elements $a$ and $b$ such that the orders $o(a)$ and $o(b)$ are finite, but the order $o(ab)$ is infinite?

Answer: Yes


Question: Determine whether the symmetric group \( S_6 \) has an abelian Sylow 2-subgroup.

Answer: No


Question: Find all natural numbers \( n \) such that \( 3^n + 5^n \) is divisible by \( 3^{n-1} + 5^{n-1} \).

Answer: 1


Question: Under the mapping \( w = \frac{1}{z + 2 + 3i} \), where \( z = x + iy \), the image of the line \( x + y = 1 \) is a circle minus a point \( w_0 \). Given that the circle is \( (x - \frac{1}{12})^2 + (y + \frac{1}{12})^2 = \frac{1}{72} \), find the point \( w_0 \).

Answer: 0


Question:
Calculate the phase margin \( \phi \) of the second-order system given by the transfer function:
\[ F(j\omega) = \frac{1}{(1-\frac{\omega^2}{\omega_f^2}) + j\omega\frac{2\xi}{\omega_f}} \]
where \( \xi < 1 \). Find \( \phi = \text{Arg}(F(j\omega)) \) for \( \omega = 2\omega_f \) and \( \xi = 0.25 \).

Answer: -161.57^\circ


Question: Let $X$ be a random variable uniformly distributed in the unit ball. Consider the random variable $Y = X + (0, \rho, 0)$, where $\rho \neq 0$. Is the probability density function (pdf) of $Y$ still constant inside the translated region?

Answer: Yes


Question: Calculate the Ramsey number \( R(T, K_{1,n}) \) where \( T \) is a tree with \( m \) vertices and \( m-1 \) is a divisor of \( n-1 \), with \( m, n \ge 2 \).

Answer: m + n - 1


Question: Given a real analytic function $f$ that is bounded on the real line $\mathbb{R}$, determine whether there always exists a holomorphic function $h$ that is bounded on a given finite open strip containing $\mathbb{R}$, such that $h = f$ on the real line.

Answer: No


Question: Determine the radius of convergence of the power series \( \sum_{n=0}^{\infty} n!x^{n^2} \).

Answer: 1


Question: For a given integer \(k\), is there a known upper bound on the number of vertices for the forbidden minors of graphs with treewidth at most \(k\)?

Answer: 2^{O(k^2)}


Question:
A system consists of 7 components with independent, identically distributed lifetimes. The system will operate until any one of the components fails. If the lifetime \( X \) of each component has the probability density function:
\[
f(x) =
\begin{cases}
\frac{3}{x^4},  & \text{for } x > 1 \\
0, & \text{otherwise}
\end{cases}
\]
what is the expected lifetime until failure of the system?

Answer: \dfrac{21}{20}


Question: Consider the Fibonacci sequence $F_n = 1, 1, 2, 3, 5, \ldots$ (starting with $n=1$) and the Catalan sequence $C_n = \frac{1}{n+1}\binom{2n}{n}$. Define $B_z$ as the number of integers $n \leq z$ for which $\gcd(F_n, C_n) = 1$. Does the ratio $\frac{B_z}{z}$ converge as $z$ approaches infinity? If so, to what value does it converge?

Answer: 1


Question:
Determine whether the following limit exists:
\[\lim_{x\to +\infty}\dfrac {\cos^5x\sin^5x} {x^8\sin^2x-2x^7\sin x\cos^2x+x^6\cos^4x+x^2\cos^8x}\]

Answer: 0


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

Let $H$ be a monoid such that $H = H^\times A H^\times$ for some finite $A \subseteq H$, where $H^\times$ is the group of units of $H$. If $H$ is duo (i.e., $aH = Ha$ for every $a \in H$), then $H$ is also unit-duo (i.e., $aH^\times = H^\times a$ for every $a \in H$).

Answer: True


Question: Calculate the limit \[ L = \lim_{x \to 0^+} \left(2\sin\left(\sqrt{x}\right) + \sqrt{x}\sin\left(\frac{1}{x}\right)\right)^x. \]

Answer: 1


Question: Construct an open set $E \subset [0,1]$ that is dense in $[0,1]$ and has Lebesgue measure $m(E) = \alpha$, where $0 < \alpha < 1$. Provide a detailed construction of such a set.

Answer: E


Question:
Let \(A=\{a_{ij}\}\) be a \(3\times 3\) matrix, where
\[
a_{ij}=\begin{cases}
(-1)^{j-i}&\text{if } i<j,\\
2&\text{if } i=j,\\
(-1)^{i-j}&\text{if } i>j,
\end{cases}
\]
then find the value of \(\det(3\,\text{adj}(2A^{-1}))\).

Answer: 108


Question: Are epimorphisms in the category of magmas always surjective? Provide a justification for your answer.

Answer: Yes


Question: Is it true that for any probability measure \( \mu \) on \((\mathbb{R},\mathcal{B}(\mathbb{R})))\), there exists a probability space \((\Omega,\Sigma,\mathbb{P}))\) and a random variable \(X\) defined on this space such that \( P(X \in B) = \mu(B)\) for every Borel set \(B\)?

Answer: Yes


Question: Determine the number of homomorphisms from the symmetric group $S_4$ to the cyclic group $\mathbb{Z}_6$. Provide your answer as a single integer.

Answer: 2


Question: Determine the quotient space of \(\mathbb{C}^n\) under the antipodal action given by the \(\mathbb{Z}_2\) action \([z_0, z_1, \ldots, z_n] \mapsto [-z_0, -z_1, \ldots, -z_n]\). Is this quotient space homeomorphic to \(\mathbb{C}^n\)?

Answer: \mathbb{C}^n


Question: Find all prime numbers \( p \) such that \( p^2 \mid 11^{p^2} + 1 \).

Answer: 3


Question: Evaluate the limit \( \lim_{s \to \infty} s^n P(|Y| > s) \), where \( Y \) is a random variable such that \( E[|Y|^n] < \infty \) for \( n > 0 \).

Answer: 0


Question: Compute the limit \( \lim_{n \to \infty} \int_{0}^{1} \frac{e^{-(n + x^2)}}{n^2 + x} \, dx \).

Answer: 0


Question: Determine all real numbers \(x\) for which the series \(\sum_{n=1}^\infty \left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)\frac{\sin nx}{n}\) converges.

Answer: \mathbb{R}


Question: Calculate the derivative of the expression \( x^T A x \) with respect to the vector \( x \), where \( x \) is a vector and \( A \) is a matrix. Verify that the derivative is \( 2Ax \).

Answer: 2Ax


Question: Determine the degree of the field extension \([\mathbb{Q}(2^{1/5}, e^{2\pi i/5}) : \mathbb{Q}]\).

Answer: 20


Question: Determine whether the following statement is true or false: "If $A \subset B \subset X$ with $B$ equipped with the subspace topology, then $A$ is connected in $B$ if and only if $A$ is connected in $X$."

Answer: A


Question: If the gradient of a function $f(x,y,z)$, denoted as $\nabla f(x,y,z)$, is always parallel to the vector $xi + yj + zk$, prove that $f$ has equal values at the points $(0,0,a)$ and $(0,0,-a)$. Determine if this statement is true or false.

Answer: True


Question: Verify Stokes' theorem for the vector field \( \mathbf{v} = y\unit{i} + 2x\unit{j} + z\unit{k} \) where the path \( C \) is a circle \( x^2 + y^2 = 1 \) in the \( xy \) plane, and the surface is the plane area bounded by \( C \).

Answer: \pi


Question: Prove that for \( x > 0 \), the equation \( \sin{\pi x} = \frac{\text{imag}((-x)^x)}{\text{real}(x^x)} \) holds true.

Answer: \sin{\pi x}


Question: Evaluate the limit \( \lim_{n\rightarrow \infty }\int \frac{n\sin(x/n)}{x(1+x^2)}\,dx \) using the Lebesgue Dominated Convergence Theorem. Use the hint: \( \int \frac{1}{1+x^2}\,dx=\pi \).

Answer: \pi


Question: Calculate the expected present value of a continuous payment stream of £190 per annum for 20 years, beginning at time 0. For the first 12 years, the force of interest is 5%. After 12 years, the interest rate changes to 3% with probability 0.25, remains at 5% with probability 0.5, and increases to 7% with probability 0.25. What is the expected present value of this payment stream?

Answer: 2403.38


Question:
Consider a 2D version of the Cantor set on the plane $[0,1] \times [0,1]$. Define $C_n$ as the set obtained by removing the middle third strip of height 1 and width $1/3$ from each interval of $C_{n-1}$. Specifically, let:

$$ C_0 = [0,1] \times [0,1],$$
$$ C_1 = \left(\left[0,\frac{1}{3}\right] \times \left[0,1\right]\right) \cup \left(\left[\frac{2}{3},1\right] \times \left[0,1\right]\right)$$

and continue this process iteratively. Calculate the Lebesgue measure of the set $C_n$ as $n$ approaches infinity. Is the measure of this 2D Cantor set zero?

Answer: 0


Question: Find the length of the curve defined by \( y(x) = \int_1^x \sqrt{t^3 - 1} \, dt \) for \( 1 \leq x \leq 4 \).

Answer: \dfrac{62}{5}


Question: Consider a vector space $X$ over the reals with a separating family of seminorms $P$. For each $p \in P$ and $\epsilon > 0$, define the set \( V(p, \epsilon) = \{ x \in X : p(x) < \epsilon \} \). If $P$ induces a locally convex topology $\tau$ on $X$, is it true that the sets $V(p, \epsilon)$ are open in the topology $\tau$?

Answer: Yes


Question: Let $R$ be a commutative ring with unity, and let $I$ and $J$ be ideals in $R$. For a prime ideal $P$, define $I_P = (R - P)^{-1}I$ as the localization of $I$ at $P$. If $I_P = J_P$ for all prime ideals $P$, does it follow that $I = J$? Provide a proof or counterexample.

Answer: I = J


Question: For the curve defined by \(x^3 + y^3 = c^3\), the tangent at the point \((a, b)\) passes through the point \((a_1, b_1)\). Given that \(\frac{a_1}{a} + \frac{b_1}{b} = -k\), find the value of \(k\). Assume that \((a_1, b_1)\) also lies on the curve.

Answer: 1


Question:
Find the limit of the series as \( n \to \infty \):
\[ u_n(x) = \frac{1}{x+1} + \frac{2}{(x+1)(x+2)} + \frac{3}{(x+1)(x+2)(x+3)} + \cdots + \frac{n x^{n-1}}{(x+1)(x+2)\cdots(x+n)} \]
Evaluate \( \lim_{n \to \infty} u_n(x) \).

Answer: 1


Question:
Derive \(\neg R\) from the following premises using logical rules:

1. \(R \leftrightarrow (R \lor (P \land \neg P))\)
2. \(R \leftrightarrow \neg P\)
3. \(\neg P \to (P \leftrightarrow (Q \to Q))\)
4. \(P \to Q\)

Answer: \neg R


Question: Evaluate the limit \( \lim_{x\to\infty}\cos\left(\frac{2}{x}\right)^{x^2} \) using L'Hospital's Rule, without variable substitution.

Answer: e^{-2}


Question: Given that $X$ and $Y$ are independent and identically distributed random variables with $P(X+Y=0)=1$, determine the characteristic function of $X$.

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \to 0}\left(x-\ln(x+1)+\cos(xe^{-x})\right)^{\frac{9\cot^{3}(x)}{2}} \]

Answer: e^3


Question: Show that \( \lim_{n \to \infty} \int_0^1 f_1(x)f_2(x)\cos(2\pi nx) \, dx = 0 \) for functions \( f_1 \in L^{3/2}([0,1]) \) and \( f_2 \in L^3([0,1]) \).

Answer: 0


Question: Let $\omega$ be a complex fifth root of unity. Define $b_k = \sum_{j=0}^4 j\omega^{-kj}$ for $0 \le k \le 4$. Find the value of $\sum_{k=0}^4 b_k \omega^{k}$.

Answer: 5


Question: Evaluate the line integral \( \int_l \sqrt{x^2+y^2} \, dl \) where the curve \( l \) is defined by the equation \( x^2+y^2=ax \).

Answer: 2a^2


Question: Let $V$ be a compactly-supported smooth vector field on $\mathbb{R}^2$, whose zeros inside some open neighbourhood of the closed unit disk $\mathbb{D}^2$ are isolated. Does there exist a sequence of vector fields $V_n \in C^\infty \cap L^{2}$ on $\mathbb{R}^2$, such that $V_n \to V$ in $L^2$ and $V_n$ do not vanish on $\mathbb{D}^2$?

Answer: Yes


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous odd function that vanishes at exactly one point and satisfies $f(1) = \frac{1}{2}$. Define $F(x) = \int_{-1}^{x} f(t) \, dt$ and $G(x) = \int_{-1}^{x} t |f(f(t))| \, dt$ for all $x \in [-1, 2]$. Given that $\lim_{x \to 1} \frac{F(x)}{G(x)} = \frac{1}{14}$, find the value of $f\left(\frac{1}{2}\right)$. \(\text{Note: } f(x) \text{ is odd and continuous.}\)

Answer: 7


Question: Determine the values of $a$ for which both roots of the quadratic equation \( z^2 + 2a(1+i)z + (4 + 2a^2i) = 0 \) lie in the first quadrant of the complex plane.

Answer: (-\infty, -2)


Question: A point P moves inside a triangle formed by vertices A(0,0), B(1,√3), and C(2,0), satisfying the conditions PA ≤ 1, PB ≤ 1, or PC ≤ 1. If the area bounded by the curve traced by P is equal to \(\frac{a\pi}{b}\), find the minimum value of \(a+b\).

Answer: 3


Question: Find the arc length of the curve $C$ parametrized by \( \alpha(t) = (5\cos t - \cos 5t, 5\sin t - \sin 5t) \) for \( t \in [0, 2\pi] \). Use the identities \( \cos 4t = \cos t \cos 5t + \sin t \sin 5t \) and \( \sin^2 t = \frac{1 - \cos 2t}{2} \) to assist in your calculation.

Answer: 40


Question: Let $X_{\bullet}$ be a simplicial set and let $\Sigma X_{\bullet}$ denote its simplicial suspension. If $X_{\bullet} \to Y_{\bullet}$ is a fibration, is $\Sigma X_{\bullet} \to \Sigma Y_{\bullet}$ also a fibration? Answer yes or no.

Answer: No


Question: Evaluate the expression: \[ \sum_{cyc} \tan^{-1}\left(\sqrt{\frac{x(x+y+z)}{yz}}\right) \] and determine its value.

Answer: \pi


Question: Find the inverse of the polynomial \(2x^2 + x + 1\) in the quotient ring \(\frac{\mathbb{Z}_3[x]}{\langle x^3 + 2x + 2 \rangle}\).

Answer: x^2


Question: Calculate the number of open subsets of the real line, denoted as \( o(\mathbb{R}) \).

Answer: 2^{\aleph_0}


Question: Find the smallest prime number that divides the expression \(n^2 + 5n + 23\) for some integer \(n\).

Answer: 17


Question: Determine whether an ambiguous context-free language can also be a linear context-free language in the Chomsky classification of formal languages.

Answer: Yes


Question: Let $X$ and $Y$ be independent random variables, each uniformly distributed on the interval $(0,1)$. Define the random variable $Z = \frac{\min\{X,Y\}}{\max\{X,Y\}}$. Determine the distribution of $Z$ by calculating $\mathbb{P}(Z \le z)$ for $0 \le z \le 1$. Provide the cumulative distribution function of $Z$.

Answer: z


Question: What is the expected number of tosses needed to get 3 consecutive heads when tossing a fair coin?

Answer: 14


Question: If $A$ is a diagonalizable matrix of rank 1 in $\mathbb{C}_{n \times n}$, must $A$ be normal? Justify your answer.

Answer: No


Question: Determine whether the growth in the number of decimal digits of the factorial function is exponential or sub-exponential. For example, 7! = 5040, which has 4 digits.

Answer: sub-exponential


Question: Find the value of $a$ for which the integral \[ \int \limits^{\infty }_{0}\frac{dx}{a^{2}+(x-\frac{1}{x})^{2}} =\frac{\pi}{5050} \] holds, given that $a^{2}\geq0$. \(a\) is a real number.

Answer: 2525


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{n! F_n(1)}{\ln n} \] where $F_0(x) = \ln x$ and for $n \geq 0$ and $x > 0$, $F_{n+1}(x) = \int_0^x F_n(t) \, dt$.

Answer: -1


Question: Evaluate the integral \( \int_{-1}^1 \frac{\cos x}{a^x+1} \mathrm{d}x \) where \( a \) is a real parameter with \( a \geq 1 \).

Answer: \sin 1


Question: Find the degree of the field extension $[\mathbb{Q}(\alpha):\mathbb{Q}(\alpha^2 + 3)]$, where $\alpha$ is a root of the polynomial $P(x) = x^4 + 9x^2 + 15$.

Answer: 2


Question: Find the variance of the cumulative distribution function of a normal distribution, where $X \sim N(\mu,\sigma^2)$ and $\Phi(x)$ is the cumulative distribution function of this normal distribution.

Answer: \dfrac{1}{12}


Question:
Consider the heat diffusion problem described by the function:
\[ u(x,t)= \frac {a_0}2 + \sum_{n=1}^{\infty} a_n \cos\left(\frac {n\pi  x}{L}\right) e^{-\frac {4\pi^2n^2t}{L^2}} \]
Find \( \lim\limits_{t\to\infty} u (x, t) \).

Answer: \dfrac{a_0}{2}


Question: Evaluate the limit \( \lim_{n\to\infty} \int_{\mathbb{R}} e^{-nx^2} f(x) \, dx \) using the dominated convergence theorem, given that \( f \in L^1(\mathbb{R}) \).

Answer: 0


Question: Let \( f \) and \( g \) be continuous functions over the reals such that \( g(x) = \int_0^x f(y)(y-x) \, dy \) for all \( x \), and \( g \) is three times continuously differentiable. What is the greatest integer \( n \) such that \( f \) is \( n \) times continuously differentiable?

Answer: 1


Question: Let $X$ be a locally compact Hausdorff space, and let $K \subset U$ where $K$ is compact and $U$ is open. Find a function $f: X \to \mathbb{R}$ with compact support such that $1_K \leq f \leq 1_U$ on $X$. Given an open set $V$ with $K \subset V \subset \overline{V} \subset U$ where $\overline{V}$ is compact, construct a function $f: \overline{V} \to [0,1]$ that is $1$ on $K$ and $0$ on $\overline{V} \setminus V$. Extend $f$ to $X$ by setting $f = 0$ on $X \setminus \overline{V}$. Is this extended function $f$ continuous on $X$?

Answer: Yes


Question:
Let $f(z)$ and $g(z)$ be entire functions such that $g(z)$ does not vanish anywhere in the complex plane and $|f(z)| \leq |g(z)|$ for all $z$. Which of the following statements is definitely true?

1) $f$ does not vanish anywhere.
2) $f(0) = 0$.
3) $f$ is a constant function.
4) $f(z) = C \cdot g(z)$ for some constant $C$.

Select the correct option.

Answer: D


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{x \cos x - \sin x}{x^3} \] without using l'Hôpital's rule or Taylor series.

Answer: -\dfrac{1}{3}


Question:
Consider the limit \( \lim_{n \to \infty} \int_0^n e^{-px} \cos(ax)\, dx.\) Determine the correct statement about the values of \(p\) and \(a\) for which the limit exists:

(a) Exists for infinitely many values of \(p\).
(b) Exists for finitely many values of \(a\).
(c) Exists for every real \(p\) but not for every value of \(a\).
(d) Exists for every real \(p\) and real \(a\).
(e) None of the above.

Answer: A


Question: Find the smallest positive integer \( m \) such that \((49^{13})^m \equiv 49 \pmod{155}\).

Answer: 7


Question: Evaluate the limit \( \lim_{n\to \infty} \sum_{r=1}^{n}\frac{1}{n+r} \) by expressing it as a Riemann sum and then as a definite integral. Find the value of the limit.

Answer: \ln 2


Question: Evaluate the limit: $$\lim_{x\to\infty}\frac{(x+1)^{2/3}-(x-1)^{2/3}}{(x+2)^{2/3}-(x-2)^{2/3}}$$

Answer: \dfrac{1}{2}


Question: Let $A = (a_{ij})$ be an $n \times n$ invertible matrix over $\mathbb{C}$, and let $D = \text{diag}(a_{11}, a_{22}, \dots, a_{nn})$ be the diagonal matrix with the same diagonal entries as $A$. Suppose $A - D$ is nilpotent and $A$ has the property that $a_{ij} \neq 0 \Rightarrow a_{ji} = 0$. Is it true that $D$ is invertible?

Answer: Yes


Question: If \( v_1, v_2, \ldots, v_n \) are orthonormal vectors in \( \mathbb{R}^n \), what is the matrix \( v_1v_1^T + v_2v_2^T + \cdots + v_nv_n^T \)?

Answer: I


Question: Determine whether the degree of the field extension \([\mathbb{Z}_2[x]/f\mathbb{Z}_2[x]: \mathbb{Z}_2]\) is equal to the degree of the irreducible polynomial \(f\) in \(\mathbb{Z}_2[x]\).

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty} \sum_{r=0}^n \frac{1}{{n \choose r}}. \]

Answer: 2


Question: Let $X$, $Y$, and $Z$ be independent random variables, each distributed uniformly on the interval (0,1). Calculate the probability $\textrm{P}(YZ < X)$. Provide the setup for the integration needed to find this probability.

Answer: \dfrac{3}{4}


Question: Let $\mathbb{T}^n$ be an $n$-dimensional torus and $M \subseteq \mathbb{T}^n$ be a submanifold homeomorphic to $\mathbb{T}^n$. Is it necessarily true that $M = \mathbb{T}^n$?

Answer: Yes


Question: Consider a compact metric space $X$ and a closed relation $\sim$ on $X$. The quotient space $X/\sim$ is compact Hausdorff and metrizable. Is the Hausdorff metric on the equivalence classes $[x] \in X/\sim$, which are compact subsets of $X$, a metric compatible with the topology of $X/\sim$?

Answer: No


Question: Given an i.i.d. sequence $X_1, X_2, \ldots$ uniformly distributed on $[0,1]$, determine $\lim_{n \to \infty} \sqrt[n]{X_1 \cdots X_n}$ with probability 1.

Answer: \dfrac{1}{e}


Question: Let $u \in L^{p}(\Omega)$, where $\Omega$ is a bounded subset of $\mathbb{R}^n$. Given that $q+1<p$ and $p \geq 2$, determine if the functional $v \mapsto \int_{\Omega} u^q v$ is continuous over $W_0^{1,p}(\Omega)$. Provide a justification for your answer.

Answer: Yes


Question: Let $a, b, c$ be non-negative real numbers such that $a + b + c = 4$. Find the minimum value of the expression $\sqrt{2a+1} + \sqrt{2b+1} + \sqrt{2c+1}$. \( \text{Provide your answer in the form } \boxed{\text{answer}}. \)

Answer: 5


Question: Evaluate the expression \( \sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}} \). Determine the value of this infinite nested radical.

Answer: 7


Question: Calculate the limit: \[ \lim_{x\to 0} \frac{\arctan(x)-\arcsin(x)}{x^3} \] without using L'Hôpital's rule.

Answer: -\dfrac{1}{2}


Question: Evaluate the integral \( \int_{S^2} \frac{1}{\sqrt{(x-a)^2+(y-b)^2+(z-c)^2}} \, dS \) over the unit sphere \( S^2 \) in \( \mathbb{R}^3 \), given that \( a^2 + b^2 + c^2 < 1 \).

Answer: 4\pi


Question: Find the average value of the function \( f(x,y,z) = xyz \) over the tetrahedron with vertices \((0,0,0), (1,0,0), (1,1,0), \text{and } (1,1,1)\).

Answer: \dfrac{1}{8}


Question: Find the sum of the infinite series \( \sum_{k=1}^{\infty} \frac{k(k+1)3^k}{k!} \).

Answer: 15e^3


Question: In how many ways can 20 candy canes be distributed to 7 children if each child must receive at least one candy cane, and two of the children are twins who must receive the same amount?

Answer: 4956


Question: Determine whether there exists a function \( f: X \to X \) such that for all \( x \in X \), \( x = f(f(x)) \) and \( x \neq f(x) \), where \( X \) is the set of all triplets \((x_1, x_2, x_3)\) with \( x_i \in \{0, 1, \ldots, 255\} \).

Answer: Yes


Question: Given that \( \int_{0}^{e^2} \frac{f(\ln(\sqrt{x}))}{x}dx = 2e \), find the value of \( \int_{0}^{1}f(x)\ dx \).

Answer: e


Question: Determine whether the following statement is true or false: For every linear transformation \( \phi : V^* \to W^* \), there exists a linear transformation \( \psi: W \to V \) such that \( \psi^* = \phi \), where \( V \) and \( W \) are finitely dimensional vector spaces over the field \( K \).

Answer: True


Question: Evaluate the infinite product \( \prod_{p \text{ prime}} \frac{p}{p-1} \).

Answer: \infty


Question: Find the value of \(a > 0\) such that the family of ellipses \(x^2 + ay^2 = r^2\) is orthogonal to the parabola \(y = 5x^2\), where \(r > 0\).

Answer: 2


Question: Let \(x_n\) be the unique solution of the equation \(\left( 1+\frac{1}{n}\right)^{n+x_n}=1+\frac{1}{1!}+\frac{1}{2!}+\dots+\frac{1}{n!}\). Find \(\lim_{n \to \infty} x_n\).

Answer: \dfrac{1}{2}


Question: Find the radius of convergence of the series \( \sum_{k=1}^{\infty}\frac{(-5)^k}{(k+1)!}\left(\frac x2\right)^{2k-1} \).

Answer: \infty


Question: Let $g:B(x, 2r) \to \mathbb R$ be a real analytic function, and let $u$ be the harmonic function in $B(x, r)$ such that $u=g$ on $\partial B(x, r)$. Is it possible to extend $u$ to a harmonic function $\tilde u$ in $B(x, (1+\delta)r)$ for some $\delta >0$, such that $\tilde u = u$ in $\overline{B(x,r)}$ and $\tilde u$ is harmonic in $B(x, (1+\delta)r)$?

Answer: Yes


Question:
Find a closed form for the infinite series:
\[ \sum_{i=0}^{\infty}\left[\frac{1}{i!}\left(\frac{e^2 -1}{2} \right)^i \prod_{j=0}^{i}(x-2j) \right]\]

Answer: x e^{x - 2}


Question: Solve the differential equation \( yy'' = (y')^2 - y' \) with the initial conditions \( y(0) = 1 \) and \( y'(0) = 2 \).

Answer: 2e^x - 1


Question:
Let $X$ be a topological space and $A, C \subseteq X$. Determine whether the following identity is true:
\[(C \cap \mathrm{Cl}(A)) \cup (C \cap \mathrm{Cl}(X \setminus A)) = C \cup (C \cap \partial A)\]

Answer: True


Question: Find the Lebesgue measure of the set \( A \subset [0,1] \), where \( A \) consists of numbers in the interval \([0,1]\) whose decimal representation has the digit 2 occurring before the digit 3. For example, \(0.1213 \in A\), \(0.1231 \in A\), \(0.1211 \in A\), but \(0.1111 \notin A\), \(0.1321 \notin A\), and \(0.1312 \notin A\). Determine the measure of this set.

Answer: \dfrac{1}{2}


Question:
Let \( y(x) \) be a function satisfying the integral equation:
\[
y(x) = 1 + 2 \int_{0}^{x} u^3 y(u) \, du
\]
Determine if it is true that \( y(\sqrt{2}) < 4 \).

Answer: False


Question: Calculate the following limit: $$\lim_{n \rightarrow \infty} \int_{0}^{\infty} \exp(-nx(\sin(x))^2)\,dx$$

Answer: 0


Question: Find the real value of \(x\) that satisfies the equation: \[2^{144} - \frac{1}{2^{144}} = 8^x - 8^{-x}\]

Answer: 48


Question: Determine whether the statement is true: If $q \equiv 3 \pmod{4}$, then ${q+1 \choose 2}$ is semiprime if and only if $q = 3$. Provide a proof or counterexample.

Answer: True


Question: Suppose a rational constant \( k = \frac{p}{q} \) can be represented by a non-alternating convergent infinite series whose partial sums are rational. If a partial sum \( s_k = \frac{a}{b} \), does the inequality \( b < q \) always hold? Answer yes or no.

Answer: No


Question: Is it true that the free group $F_k$ on $k$ generators, where $k$ is a finite number, requires at least $k$ elements to generate it? In other words, does every generating set of $F_k$ have a cardinality of at least $k$?

Answer: Yes


Question: Calculate the sum: \[ \cos\left(\frac{2\pi}{2n+1}\right) + \cos\left(\frac{4\pi}{2n+1}\right) + \cdots + \cos\left(\frac{2n\pi}{2n+1}\right) \] for any natural number \( n \).

Answer: -\dfrac{1}{2}


Question: Evaluate the limit: $$\lim_{x\to0}\frac{\big(\frac{1}{x}\big)^{\sin x}-\big(\frac{1}{\sin x}\big)^x}{x}$$

Answer: 0


Question: Evaluate the limit as $n$ approaches infinity of the integral: $$\int_0^\infty \frac{\cos(nx)}{1+x^n}\,dx$$

Answer: 0


Question: Find the number of idempotent elements (i.e., elements $a$ such that $a^2 = a$) in the quotient ring $R/I$, where $R = \{ a + b\sqrt{2} \mid a, b \in \mathbb{Z} \}$ and $I = \langle 4 + \sqrt{2} \rangle$. Assume $R/I \simeq \mathbb{Z}_{14}$. Determine the number of idempotent elements in $R/I$. \( \text{Note: The characteristic of } R/I \text{ is 14.} \)

Answer: 4


Question:
The lifetime (in hours) $X$ of a certain electronic component is a random variable with the density function:
\[ f(x) = \begin{cases}
 \frac{ e^{-\frac{x}{100}} }{100} & \text{ , } x > 0 \\
 0 & \text{elsewhere}
\end{cases} \]
Three of these components operate independently in a piece of equipment. The equipment fails if at least two of the components fail. Find the probability that the equipment operates for at least 200 hours without any failure.

Answer: 3e^{-4} - 2e^{-6}


Question: Determine whether the constant $c$ in the inequality \( \prod_{i=1}^{n}\frac{p_i}{p_i-1} < e^c \cdot (\log(p_n)) \cdot (1+\frac{1}{\log_2(p_n)}) \) is equal to the Euler-Mascheroni constant \( \gamma \).

Answer: \gamma


Question: Evaluate the integral \( \int_{C} \frac{z}{z^2+1} \, dz \), where \( C \) is the path \( |z-1| = 3 \), using the Residue Theorem.

Answer: 2\pi i


Question: Find the smallest positive integer $k$ such that $(3^3 + 4^3 + 5^3) \cdot k = a^n$ for some positive integers $a$ and $n > 1$.

Answer: 1


Question: Let $G$ be a finite group of order $n$, and $R$ a ring on which $n$ is invertible. Is taking $G$-invariants an exact functor on the category of $R[G]$-modules?

Answer: Yes


Question: Consider a subscheme $X$ of a smooth variety $Y$. Suppose that for every point $x \in X$, the fiber $(C_{X/Y})_x$ of the canonical morphism $C_{X/Y} \to X$ over $x$ is isomorphic to the affine space $T_xY/T_xX$. Is it true that $X$ must be smooth?

Answer: Yes


Question: Let $F$ be a finitely generated free group, and let $H_1, \dots, H_n$ be finitely generated subgroups of infinite index in $F$. Given $\epsilon > 0$, determine if there must exist an epimorphism $\phi \colon F \to G$ to a finite group such that \[ \frac{|\phi(H_1 \cdots H_n)|}{|G|} < \epsilon. \] Here, $H_1 \cdots H_n = \{h_1 \cdots h_n : h_1 \in H_1, \dots, h_n \in H_n\}$. Equivalently, is the Haar measure of the closure of $H_1 \cdots H_n$ in the profinite completion of $F$ equal to $0$?

Answer: Yes


Question: Calculate the sum: \[ \sum_{k=1}^n (-1)^k {n\choose k}\frac{1}{k+1} \]

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


Question: Calculate the flux of the vector field \( \vec{F} = 2y\vec{j} + z\vec{k} \) over the surface \( S \), which is the portion of the surface defined by \( y = \ln(x) \) in the first octant. The projection of \( S \) onto the \( xz \)-plane is the rectangle defined by \( 1 \le x \le e \) and \( 0 \le z \le 1 \). The unit normal vector \( \vec{n} \) to \( S \) points outwards from the \( xz \)-plane. Compute the flux of \( \vec{F} \) over \( S \) in the direction of \( \vec{n} \).

Answer: 2


Question: Let \(X\) and \(Y\) be two independent random variables, each uniformly distributed on \((-1, 1)\). Calculate the expected value \(E(|X+Y|)\).

Answer: \dfrac{2}{3}


Question: Determine whether the following statement is true: For a symmetric, square matrix $A$, the $L^2$ operator norm $||A||_2$ is less than or equal to the $L^\infty$ operator norm $||A||_\infty$. The $L^2$ operator norm is defined as $||A||_2 := \sup_{||x|| =1}||Ax||$, and the $L^\infty$ operator norm is defined as $||A||_\infty := \max_i \sum_{j}|A_{i,j}|$. Provide a justification for your answer.

Answer: True


Question: Determine whether the sequence \( \left\{ \left(\frac{\ln(n+1)}{\ln n}\right)^n \right\} \) is convergent. If it is convergent, find its limit.

Answer: 1


Question: In the category of quasi-projective varieties, consider a surjective regular map \( \pi: X \to Y \). Is it true that for any regular map \( \rho: X \to Z \), a map of sets \( f: Y \to Z \) such that \( \rho = f \circ \pi \) is necessarily a regular map?

Answer: No


Question:
Determine the Neumann Stability condition for the numerical scheme given by:
\[
u_{j+1,m} = \frac{1}{2}\sigma(\sigma-1)\nu_{j,m+1}-(\sigma^1-1)\nu_{j,m}+\frac{1}{2}\sigma(\sigma+1)\nu_{j,m-1}\]
where \(\sigma = \frac{c\Delta t}{\Delta x}\) and the differential equation is \(u_t+cu_x = 0\). Find the condition on \(\sigma\) that ensures stability, which should match the CFL condition.

Answer: 0 \leq \sigma \leq 1


Question:
Given \( f'(x) = \frac{192x^3}{2+\sin^4(\pi x)} \) for all \( x \in \mathbb{R} \) with \( f(0.5) = 0 \), determine the possible values of \( m \) and \( M \) such that \( m \leq \int_{0.5}^{1} f(x) \, dx \leq M \). Choose the correct pair from the following options:

(a) \(13, 24\)

(b) \(0.25, 0.5\)

(c) \(-11, 0\)

(d) \(1, 12\)

Answer: D


Question: Determine the signature of the bilinear form represented by the matrix \( M = \begin{pmatrix} 1 & -1 & 2 \\ -1 & 3 & 1 \\ 2 & 1 & 1 \end{pmatrix} \).

Answer: (2, 1)


Question: Consider a measure space $(\Omega, \mathcal{F}, \mu)$ and a sub-$\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$. Let $\mu|_\mathcal{G}$ be the restricted measure on $\mathcal{G}$, so that $(\Omega, \mathcal{G}, \mu|_\mathcal{G})$ is also a measure space. Suppose $f: \Omega \rightarrow [0,+\infty[$ is a $\mathcal{G}$-measurable function. For any $B \in \mathcal{G}$, consider the integrals $\int_B f \, d\mu$ in the measure space $(\Omega, \mathcal{F}, \mu)$ and $\int_B f \, d\mu|_\mathcal{G}$ in the measure space $(\Omega, \mathcal{G}, \mu|_\mathcal{G})$. Are these integrals equal for all $B \in \mathcal{G}$?

Answer: Yes


Question: Determine whether the following proposition about cyclic groups is true or false: A group $G$ is a cyclic group if and only if for every subgroup $H$ of $G$, there exists an integer $k$ such that $H = \{a^k : a \in G\}$.

Answer: True


Question: Compute \(5^{10,000} \equiv \mod 52\).

Answer: 1


Question: Find the limit, as \( n \to \infty \), of the expected distance between two points chosen uniformly at random within a unit edge-length hypercube in \( \mathbb{R}^n \).

Answer: \infty


Question: Evaluate the limit: \[ \lim_{x \to \infty} \left( \int_0^{\pi /6} (\sin t)^x \, dt \right)^{1/x} \]

Answer: \dfrac{1}{2}


Question: Find the maximum value of $4x - 3y - 2z$ subject to the constraint $2x^2 + 3y^2 + 4z^2 = 1.$

Answer: 2\sqrt{3}


Question: Determine the points of continuity for the function \( f(x) = \sum_{n=1}^{\infty} \frac{\sin nx}{n^{1/2}} \).

Answer: \mathbb{R} \setminus \{2\pi k \mid k \in \mathbb{Z}\}


Question: Find a function \( f(x) \) that satisfies the integral equation: \[ f(x) = \int_0^x f(s) \, \mathrm{d}s \]

Answer: 0


Question: Evaluate the limit: $$\lim_{n\to\infty} \int_0^{\infty} \left(1+\frac{x}{n}\right)^{-n}\sin \frac{x}{n}\,dx.$$

Answer: 0


Question: Evaluate the expression: $$\frac{1}{2^{101}}\sum_{k=1}^{51} \sum_{r=0}^{k-1}\binom{51}{k}\binom{50}{r}$$

Answer: \dfrac{1}{2}


Question: Calculate the limit: $$ \lim_{x\to \pi}\frac{\cos2x+2(\pi-x)\sin x-1}{(\cos2x+\cos x)^2} $$ as \( x \to \pi \).

Answer: \dfrac{4}{27}


Question: Find the derivative with respect to $x$ of the integral $\int_0^x f(x-s)\,ds$. Use appropriate calculus techniques to determine the expression for the derivative.

Answer: f(x)


Question: Evaluate the sum \( \displaystyle\sum_{r=1}^{4} \frac{1}{2-\alpha^r} \), where \( \alpha^k \) for \( k=0,1,2,3,4 \) are the fifth roots of unity.

Answer: \dfrac{49}{31}


Question: Given a full rank matrix $X$ of size $n \times p$ with $n > p$, where $X = UDV^T$ is the skinny SVD and $X = QR$ is the skinny QRD, show that $U^TQQ^TU = Q^TUU^TQ = I$. Use the relationship between these decompositions to prove the required identities.

Answer: I


Question: Evaluate the limit \( \lim_{x\to 0^+}\sum_{n=2020}^{\infty}\frac{(-1)^n}{\log^x(n)} \).

Answer: \dfrac{1}{2}


Question: Let $f(x) = o(\sqrt{x})$ as $x\rightarrow\infty$. Define $x^*(a)$ as the minimizer of the expression $f(x) + a^{3/2}/x$. As $a\rightarrow\infty$, determine if $x^*(a) = \omega(a)$, meaning that the minimizer grows super-linearly in $a$. Provide a justification for your answer.

Answer: Yes


Question: Prove that \( \operatorname{exp}\left(\sum_{n=1}^\infty (-1)^{n-1}\frac{z^n}{n}\right)=z+1 \) for all \( z\in\mathbb{C} \) where \( |z|<1 \).

Answer: z + 1


Question:
Solve the matrix equation for $B$ given the following:

$$M^TB^T \Sigma B - 2 \Sigma BMB = 0$$

where $M$ is a $1 \times k$ matrix with feature means, $B$ is a $k \times 1$ matrix with feature coefficients, and $\Sigma$ is a $k \times k$ symmetric covariance matrix.

Answer: B = 0


Question: Let $p$ be a prime number and $u$ an integer such that $u^2 \equiv -1 \pmod{p}$. Define the additive subgroup $S$ of $\mathbb{Z} \times \mathbb{Z}$ as follows: $$S = \{ (a,b) \in \mathbb{Z} \times \mathbb{Z} : b \equiv ua \pmod{p} \}.$$ Determine the index of $S$ in $\mathbb{Z} \times \mathbb{Z}$. \( \boxed{\text{Index}} \)

Answer: p


Question: Find the first digit of the number \(2^{4242}\) without using a calculator.

Answer: 9


Question: Calculate the approximate probability that the sum of 100 independent and identically distributed random variables, each with expected value 0 and variance 1, falls within the interval \((-10, 10)\). Use the hint that the cumulative distribution function of a standard normal variable at 1 is 0.8413.

Answer: 0.6826


Question: Determine the maximum value of \( a \) such that the function \( a \log(x+1) \) does not exceed \( \sqrt{x} \) for any \( x \geq 0 \).

Answer: 1.242


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a function such that \( f'' \) is continuous on \( \mathbb{R} \) and \( f(0) = 1, f'(0) = 0, f''(0) = -1 \). Find the value of \( \lim_{x \to \infty} \left(f\left(\frac{\sqrt{2}}{x}\right)\right)^x \).

Answer: 1


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

Answer: 2


Question: Find the particular integral of the differential equation \( y''' + 3y'' + 3y' + y = 30e^{-x} \) using the method of undetermined coefficients.

Answer: 5x^3 e^{-x}


Question: Evaluate the expression \( y(x) = x^{x^{x^{x^{\cdots}}}} \) for \( x = \sqrt{2} \).

Answer: 2


Question: Let $X$ be a path-connected topological space and $Y$ be an open connected subset of $X$. Is $Y$ necessarily path-connected?

Answer: No


Question: Evaluate the integral \( \int_0^\infty \frac{\sin^2x}{x^2}dx \) using the Residue Theorem.

Answer: \dfrac{\pi}{2}


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

Answer: 2\pi


Question: Determine whether it is possible for an even semi-perfect number \( n \) to have an aliquot sum of its divisors that does not include \( \frac{n}{2} \).

Answer: Yes


Question: Calculate the value of the following limit as \( n \to \infty \): \[ \lim _{n\to\infty} \left(1-\frac{1}{\sqrt{2}}\right) \cdots \left(1-\frac{1}{\sqrt{n+1}}\right) \]

Answer: 0


Question: In a normed linear space $F$ with a norm that is not strictly convex, can we always find distinct elements $f_1$ and $f_2$ such that $\lVert f_1 \rVert = \lVert f_2 \rVert = 1$ and $\lVert f_1 + f_2 \rVert = 2$? Assume $F$ is not the trivial space containing only the zero element.

Answer: Yes


Question: Evaluate the surface integral \( \iint_{S} \vec{F} \cdot \vec{n} \, ds \) for the vector field \( \vec{F} = x \vec{i} - y \vec{j} + (z^{2} - 1) \vec{k} \), where \( S \) is the surface bounded by the cylinder \( x^{2} + y^{2} = 4 \) and the planes \( z = 0 \) and \( z = 1 \).

Answer: 4\pi


Question: Is it possible to construct a group epimorphism \( \varphi : \mathsf{B}_2(C_2) \to G \) such that the order of \( s = \varphi(\sigma) \) is at least 3? If yes, what is the minimum order of \( G \) for which this occurs? If not, provide a justification.

Answer: 3


Question: Is the ideal \((1+x^2, x^3)\) in \(\mathbb{Q}[x]\) a principal ideal? Justify your answer.

Answer: Yes


Question: Calculate the expected value \( \mathbb{E}(\exp(1/X)) \) where \( X \sim N(0, \sigma_x) \).

Answer: \infty


Question: Consider a function $f(x)$ that takes a finite real value for all $x$ on the closed interval $[a,b]$. Is there always a sub-interval $[a',b']$ with $a < a' < b' < b$ such that there is an upper bound for $f(x)$ on this sub-interval?

Answer: No


Question: Solve the equation for \( x \): \[ \frac{\pi}{4} = \frac{e^x - e^{-x}}{2} + \arctan(x+1) \]

Answer: 0


Question: Let $(M, g)$ be a smooth compact Riemannian manifold, and consider the function $p(x, \xi) = |\xi|_g = \sqrt{g^{ij}(x) \xi^i \xi^j}$ on $T^*M \setminus 0$. Let $\omega = d\xi \wedge dx$ be the canonical symplectic form. Suppose $b \in C^\infty (T^*M \setminus 0)$ and fix $\lambda > 0$. Determine if the following integral necessarily equals zero: $$\int_{p < \lambda}\{p, b\} \, dx \, d\xi = 0$$ where $\{p, b\}$ denotes the Poisson bracket and $dx \, d\xi = \omega^n$. Provide a justification for your answer.

Answer: 0


Question:
Identify which of the following definite integrals is equivalent to \( \int_{\pi/4}^{\pi/2}(2\csc(x))^{17}dx \):

(A) \( \int_{0}^{\ln(1+\sqrt2)}2(e^u+e^{-u})^{16}du \)
(B) \( \int_{0}^{\ln(1+\sqrt2)}2(e^u+e^{-u})^{17}du \)
(C) \( \int_{0}^{\ln(1+\sqrt2)}2(e^u-e^{-u})^{17}du \)
(D) \( \int_{0}^{\ln(1+\sqrt2)}2(e^u-e^{-u})^{16}du \)

Choose the correct option.

Answer: A


Question: Let $f(x)$ be a polynomial of degree $6$ that is divisible by $x^3$ and has a point of extremum at $x = 2$. If $f'(x)$ is divisible by $1+x^2$, find the value of $\frac{3f(2)}{f(1)}$. \( \text{Express your answer as a simplified fraction.} \)

Answer: 16


Question:
Find the integer value of $x$ that satisfies the following system of congruences:

1. $x \equiv 7 \pmod{11}$
2. $x \equiv 1 \pmod{5}$
3. $x \equiv 18 \pmod{21}$

Answer: 711


Question: If \( \lim_{h \to 0}[f(x+h)+f(x-h)-2f(x)] = 0 \) for every \( x \in \mathbb{R} \), does it follow that \( f \) is continuous? Justify your answer.

Answer: No


Question: Consider the submersion theorem: Let $U$ be an open subset of $\Bbb R^n$ and $F: U\to \Bbb R^m$ be a smooth function where $m < n$. If the set of all $p$ in $U$ where $F(p)=0$ has no critical points, then this set is an $(n-m)$-dimensional manifold. Would this theorem still hold if we replace $0$ with any other real number?

Answer: Yes


Question: Determine the image of the line $y = mx$ under the exponential transformation.

Answer: r = e^{\theta/m}


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

Answer: \dfrac{1}{3}


Question: Let $V$ be a real $n$-dimensional vector space, and let $g$ and $h$ be two inner products on $V$. For a fixed $1 \le k \le n-1$, denote by $\text{Gr}_k(V)$ the Grassmannian of $k$-dimensional subspaces of $V$. Define the orthogonal complement map $\perp_g: \text{Gr}_k(V) \to \text{Gr}_{n-k}(V)$ with respect to $g$ by $\perp_g(W) = W^\perp$. Suppose $\perp_g = \perp_h$. Is it true that $g = \lambda h$ for some $\lambda \in \mathbb{R}$?

Answer: Yes


Question: Find the remainder when \(5,000,000^{500,000,000,000}\) is divided by the prime number \(10^6 + 3\).

Answer: 225


Question: Is the Schwartz space $S(\mathbb{R}^n)$ dense in the Sobolev space $H^1(\mathbb{R}^n)$ with respect to the $\lVert \cdot \rVert_{H^1}$-norm?

Answer: Yes


Question: Find the coefficient $a_5$ in the power series expansion of the integral \( \int_0^x\left(e^{-t^2}+\cos t\right)dt = \sum_{n=1}^\infty a_n x^n \).

Answer: \dfrac{13}{120}


Question: Find the absolute value of the complex exponential expression \( e^{iaRe^{ix}} \), where \( a \) is a constant. Verify that the absolute value is \( e^{-aR\sin x} \).

Answer: e^{-aR\sin x}


Question: Find the value of $x$ in the interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ that satisfies the equation: $$2^{\sin 3x} - 8^{\sin x} = \sin^3{x}.$$

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\rightarrow 1^+} \frac{\sin(x^3-1)\cos\left(\frac{1}{1-x}\right)}{\sqrt{x-1}} \]

Answer: 0


Question: A fair six-sided die is rolled repeatedly. Let \( X \) be the number of the roll where the outcome changes for the first time, and \( Y \) be the number of the roll where the outcome changes for the second time. Calculate the covariance \( \text{cov}(X, Y) \).

Answer: \dfrac{6}{25}


Question: Compute \( \frac{d}{dx}\int_a^x g(x,t)\,dt \) where \( g(x,t) = f(x-t) \).

Answer: f(x - a)


Question: Let \( n \) be a positive integer. Determine the largest integer \( f(n) \) such that there exists a polynomial \( p \in \mathbb{Z}[x, y] \) satisfying \((x+2y)^{2n} - x^{2n} = 2^{f(n)} p\).

Answer: 2 + \nu_2(n)


Question: Find the greatest value of \( a \) such that the limit \( \lim_{h \to 0} \left(\frac{\sinh(ah)}{h+\sinh(h)}\right)^{1+\sqrt{1+h}} = \frac{1}{4} \).

Answer: 1


Question: Let \((X, \mathcal{A}, \mu)\) be a measure space, and let \((u_j)_{j \in \mathbb{N}}\) be a sequence of integrable functions in \(\mathcal{L}^p(\mu)\). Suppose that \(\lim_{j \rightarrow \infty} \int |u_j|^p \, d\mu = \int |u|^p \, d\mu\) for some measurable function \(u\). Is it possible that \(\lim_{j \rightarrow \infty} \int |u_j|^p \, d\mu = \int |u|^p \, d\mu = \infty\)?

Answer: Yes


Question: Find the residue of the function \( f(z) = \frac{1}{z^3 \sin z} \) at \( z = 0 \).

Answer: 0


Question: Which of the following is the correct definition of a torsion subgroup? (A) A subgroup which is torsion. (B) A collection of all elements in a group G whose order is finite. Choose the correct option.

Answer: B


Question: Calculate the value of \(\sum_{X,Y\subseteq\Omega}P(X\cup Y)-P(X\cap Y)\) where \(P\) is the uniform probability distribution on \(\Omega=\{1,2,3,4,5,6\}\).

Answer: 2048


Question: Calculate the volume of the solid obtained by rotating the region bounded by the curves \(y = x - x^2\) and \(y = 0\) about the line \(x = 2\).

Answer: \dfrac{\pi}{2}


Question: Calculate \( \lim_{n\to\infty}n\left((1+\frac{1}{n})^n-e\right) \).

Answer: -\dfrac{e}{2}


Question: The number 2001 can be expressed as the difference of squares $x^2 - y^2$ in four different ways, where $x$ and $y$ are positive integers. Find the sum of all possible values of $x$. \( \boxed{} \)

Answer: 1440


Question: Consider the set \( A = \left\{ \frac{(-1)^n+1}{n+1} + (-1)^n \mid n \in \mathbb{N} \right\} \). Find an element \( x_{\varepsilon} \in A \) such that for any \( \varepsilon > 0 \), \( 3 - \varepsilon < x_{\varepsilon} \leq 3 \).

Answer: 3


Question: Find the coefficient of \(x^{24}\) in the expansion of \(\left(1 - x\right)^{-1} \left(1 - x^2\right)^{-1} \left(1 - x^3\right)^{-1}\).

Answer: 61


Question: Evaluate the integral \( I(x) = \int_0^x \frac{\sin(t)}{t} \, dt \) and find \( I(1) \) correct to three decimal places.

Answer: 0.946


Question: Determine which is larger: $e^{\pi}$ or $\pi^e$. You may use calculus to justify your answer.

Answer: e^{\pi}


Question: Let \(a, b, c \in \mathbb{R}\) such that \(a + b + c = 3\). Find the minimum value of \(\frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}} + \frac{1}{\sqrt{c}}\).

Answer: 3


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

Answer: \dfrac{1}{2}


Question:
Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that $f(i)=0$ for all integers $i$. Determine which of the following statements is always true:

A) The image of $f$ is closed in $\mathbb{R}$.
B) The image of $f$ is open in $\mathbb{R}$.
C) $f$ is uniformly continuous.
D) None of the above.

Answer: D


Question: Determine the range of integer values of $a$ for which the inequality \( \frac{x^2+a^2}{a(4+x)} \ge 1 \) holds for all \( x \in (-1, 1) \).

Answer: [5, \infty)


Question: Find the volume of the region in the first octant enclosed by the planes $x=0$, $z=0$, $y=0$, $y=2$, and the parabolic cylinder $z=3-x^2$. Determine the volume of this region.

Answer: 4\sqrt{3}


Question: Suppose there are 100 people on an airplane, each with an assigned seat. If they all sit randomly, what is the probability that at least one person sits in their assigned seat?

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


Question: In a group \(G\), let \(a\) and \(b\) be elements such that \(|a| = 2\), \(b \neq e\), and \(aba = b^2\). Determine the order of \(b\).

Answer: 3


Question: Calculate \( \iint z \, dS \) where \( S \) is the upper hemisphere of radius \( a \).

Answer: \pi a^3


Question:
Given a random variable \(X\) with range \(\{1,2,3,4,\dots\}\) and probabilities defined by \(P(X=k) = \frac{3^{ck}}{k!}\) for \(k \in \{1,2,3,4,\dots\}\), determine the value of the constant \(c\). Choose the correct option:

(i) \(\frac{1}{2} \log(\log 2)\)
(ii) \(\log_3(\log 2)\)
(iii) \(\frac{\log_e (\log 2)}{\log_3 e}\)
(iv) \(\log_2(\log 3)\)

Answer: ii


Question: Determine the number of unitary homomorphisms \( \phi : \mathbb{Z}[X]/(X^3+3X+5) \longrightarrow \mathbb{R} \).

Answer: 1


Question: In a $5 \times 5$ grid, there is one correct square in each row. You start by picking any of the five squares in the first row. From each square, you can move forward, forward and up, or forward and down to the next row. The correct path is guaranteed to be contiguous. What is the probability that you can traverse the grid from the first row to the last row in one try without selecting any wrong squares?

Answer: \dfrac{1}{259}


Question: Let $f(x)$ be an irreducible quartic polynomial over $\mathbb{Q}$. Let $L$ be the splitting field of $f(x)$. Suppose $\mathbb{Q}(\alpha) \cap \mathbb{Q}(\beta)=\mathbb{Q}$ for any two distinct roots $\alpha$ and $\beta$ of $f(x)$. Determine the Galois group of $f(x)$ over $\mathbb{Q}$.

Answer: S_4


Question:
Given the equations with positive integer variables $a$, $b$, and $c$:
\[ 5a + 5b + 2ab = 92 \]
\[ 5b + 5c + 2bc = 136 \]
\[ 5c + 5a + 2ac = 244 \]
find the value of $7a + 8b + 9c$. Provide your answer as a single integer.

Answer: 172


Question: Let $C$ be the field of complex numbers, and let $f \in C[x,y]$ be a polynomial such that there exists $g \in C[x,y]$ with $\text{Jac}(f,g) \in C^{*}$, meaning the determinant of the Jacobian matrix of the polynomials $f$ and $g$ is a nonzero constant. Is it true that $f$ is irreducible?

Answer: Yes


Question:
Consider a continuous $2\pi$-periodic function $f : \mathbb{R} \to \mathbb{R}$ of bounded variation, with its Fourier series given by:
\[ f(\theta) = \frac{a_0}{2} + \sum_{n = 1}^{\infty} (a_n\cos{n\theta} + b_n\sin{n\theta}) \]
The real part of the series can be expressed as:
\[ F(z) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n-jb_n)z^n \]
Is the function $F(z)$ holomorphic on the unit disc for any such function $f(\theta)$?

Answer: Yes


Question: Consider the coupon collector's problem where there are N different coupons, each drawn independently with equal probability and with replacement. After collecting at least one of each coupon, let $X_i$ be the random variable representing the number of times coupon $i$ has been drawn. Determine the expected value $\mathbb{E}[X_i]$ for each coupon $i$.

Answer: H_N


Question: Determine the Lebesgue measure of the set \( \left\{ \frac{x}{y} : x, y \in I_\alpha^* \right\} \), where \( I_\alpha \subset [0,1] \) is an \( \alpha \)-Cantor set of Lebesgue measure \( \alpha \) and \( I_\alpha^* = I_\alpha \setminus \{0\} \).

Answer: \infty


Question: Calculate \( \lim_{\lambda \rightarrow \infty} T_{\frac{\sin(\lambda x)}{x}} \) in the sense of distributions \( \mathcal{D}'(\mathbb{R}) \).

Answer: \pi \delta_0


Question:
Consider a square in chain complexes over a field $k$:
\[
\begin{array}{ccc}
A & \xrightarrow{f} & B \\
\downarrow{g} & & \downarrow{h} \\
C & \xrightarrow{k} & D
\end{array}
\]
If $A$ is chain equivalent to a direct sum of chain complexes $\bigoplus_i A_i$, where each summand $A_i$ is concentrated in exactly one degree and any two $A_i$'s are in different degrees, is the square equivalent (in the diagram category) to a direct sum of squares with the top left corner being $A_i$? Provide a justification for your answer.

Answer: Yes


Question: Compute the integral \( \int_{|z|=1} \operatorname{Re}(z)e^{z^2} \, dz \). Hint: Find a holomorphic function \( f \) such that \( f(z) = \operatorname{Re}(z) \) when \( |z| = 1 \).

Answer: \pi i


Question:
Given the equations:
\[ a \sec(200°) - c \tan(200°) = d \]
and
\[ b \sec(200°) + d \tan(200°) = c \]
where \(a, b, c, d \subseteq \mathbb{R}\), find the value of:
\[ \frac{(a^2 + b^2 + c^2 + d^2) \sin(20°)}{bd - ac}. \]

Answer: 2


Question: Find all positive integer solutions \((a, n)\) with \(n > 1\) to the equation \((a+1)^n = a^{n+2} + (2a+1)^{n-1}\).

Answer: (1, 2)


Question: Consider a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Let $\{A_n\}_{n=1}^{\infty}$ be a sequence of independent events defined on $(\Omega, \mathcal{F}, \mathbb{P})$ such that $\mathbb{P}(A_n)<1$ for all $n$ and $\mathbb{P}(\bigcup_{n=1}^{\infty}A_n)=1.$ Find the value of $\mathbb{P}(\bigcap_{i=1}^{\infty}\bigcup_{n=i}^{\infty}A_n).$

Answer: 1


Question:
Determine which of the following mappings the function \( f(z) = \frac{2z+1}{5z+3} \) performs:

1. \( H^+ \to H^+ \) and \( H^- \to H^- \)
2. \( H^+ \to H^- \) and \( H^- \to H^+ \)
3. \( H^+ \to L^- \) and \( H^- \to L^+ \)
4. \( H^+ \to L^+ \) and \( H^- \to L^- \)

Here, \( H^+ = \{z : y > 0\} \), \( H^- = \{z : y < 0\} \), \( L^+ = \{z : x > 0\} \), and \( L^- = \{z : x < 0\} \). The symbol \( \to \) denotes "onto". Additionally, the \( X \) axis is mapped to the curve \( \frac{2x+1}{5x+3} \). Choose the correct mapping.

Answer: 1


Question: Evaluate the double integral \( \iint_D (x+y) \, dx \, dy \), where \( D \) is the region bounded by the circle \( (x-1)^2 + (y-1)^2 = 2 \).

Answer: 4\pi


Question: Find the sixth derivative of \( y = e^x \left( 3\sin\left(\frac{x}{\sqrt{3}}\right) + 2\cos(\sqrt{3}x) \right) \) with respect to \( x \) at \( x = 0 \).

Answer: 128


Question: Let \(F \subseteq E\) be a field extension and let \(\alpha \in E\) be transcendental over \(F\). Prove that the degree of the field extension \([F(\alpha) : F(\alpha^3)]\) is 3.

Answer: 3


Question: Find the limit, if it exists, of the function \( \frac{x^3 + y^3}{x^2 + y^2} \) as \((x,y)\) approaches \((0,0)\).

Answer: 0


Question: Find the polynomial \( Q(x) \) such that \( Q(x)^3 = 27 + 108x + 90x^2 - 80x^3 - 60x^4 + 48x^5 - 8x^6 \).

Answer: -2x^2 + 4x + 3


Question: Evaluate the limit: \[ \lim_{N\to\infty} \frac{\sum_{n=0}^N \sin^4 (5n)}{N} \]

Answer: \dfrac{3}{8}


Question: Find the minimum value of \( \frac{7x^2 - 2xy + 3y^2}{x^2 - y^2} \) for two positive numbers \( x \) and \( y \) such that \( x > y \).

Answer: 2 + 2\sqrt{6}


Question: Given that both $n^3 + 3$ and $n^5 + 5$ are divisible by a prime $p$, determine how many distinct prime values $p$ can take, where $n$ is an integer.

Answer: 2


Question: Evaluate the limit: \[ \lim_{n\to\infty} \frac {e^{1/n}  + e^{2/n} + \ldots + e^{n/n}}{n}. \]

Answer: e - 1


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

Answer: e^{-a^2}


Question: Evaluate the sum \( \sum_{n=1}^{\infty} \left(\frac{2}{n}\right)^n \) up to 5 decimal places.

Answer: 3.37058


Question: Evaluate the integral \( \int_{0}^1 x \bigg\lceil \frac{1}{x} \bigg\rceil \left\{ \frac{1}{x} \right\}\, dx \), where \( \lceil x \rceil \) is the ceiling function, and \( \left\{x\right\} \) is the fractional part function.

Answer: \dfrac{1}{2}


Question: Find the value of $m-b$ where $y=mx+b$ is the line tangent to the parabola $y=x^2$ at the point $(1,1)$. Express your answer as a single number.

Answer: 3


Question: Let $(X, d)$ be a compact metric space, and let $f, g: X \to X$ be maps on this space. The Hausdorff distance between the images $f(X)$ and $g(X)$ is denoted by $d_H(f(X), g(X))$, and the $C^0$-distance between the maps $f$ and $g$ is denoted by $d_{C^0}(f, g)$. If $d_{C^0}(f, g) < \delta$, is it true that $d_H(f(X), g(X)) < \delta$? Provide a justification for your answer.

Answer: Yes


Question: Is a field with a cyclic multiplicative group necessarily finite? Provide a justification for your answer.

Answer: Yes


Question: In a finite field of order $2^n$ where $n$ is odd, suppose $a$ and $b$ are elements such that $a^2 + ab + b^2 = 0$. Is it necessary that both $a$ and $b$ must be zero?

Answer: Yes


Question: Let $A, B, C$ be subgroups of a group $D$ such that $A \le B$, $A \cap C = B \cap C$, and $AC = BC$. Prove that $A = B$.

Answer: A = B


Question: Consider an experiment where 1000 green points are sampled from a $N(-1, 1)$ distribution and 1000 blue points are sampled from a $N(1, 1)$ distribution. If a point is selected at random from these 2000 points, what is the probability that the point is green?

Answer: \dfrac{1}{2}


Question: Evaluate the expression \( y = \sin^{-1}(\sin 8) - \tan^{-1}(\tan 10) + \cos^{-1}(\cos 12) - \sec^{-1}(\sec 9) + \cot^{-1}(\cot 6) - \csc^{-1}(\csc 7) \). If \( y \) simplifies to \( y = a\pi + b \), find the value of \( a - b \).

Answer: 53


Question: Consider a projection valued measure $E$ from the complex plane into the projections on a Hilbert space $\mathcal{H}$, with compact support $K \subseteq \mathbb{C}$. Determine the spectrum of the normal operator $\int_K z \, dE(z)$ in $\mathcal{B}(\mathcal{H})$. \( \mathcal{B}(\mathcal{H}) \) denotes the bounded linear operators on $\mathcal{H}$. \( K \) is the compact support of the measure.

Answer: K


Question:
Consider the equation with multiple radicals:
\[ \frac{f(x) + \sqrt{ g(x)}}{h(x)} = \frac{k(x) + \sqrt{ l(x)}}{m(x)} \]
where \(f(x)\), \(g(x)\), \(h(x)\), \(k(x)\), \(l(x)\), and \(m(x)\) are quadratic polynomials, and the discriminants of \(l(x)\) and \(g(x)\) are non-zero. Determine if it is possible to simplify this equation to a quartic equation in \(x\) (i.e., of order 4 or lower).

Answer: Yes


Question: Evaluate the integral \( \iiint 6z\,dx\,dy\,dz \) over the region \( \Omega = \{ (x,y,z) \in \mathbb{R}^3 : |x+y| \le z \le |x|+|y| \le 1 \} \). Determine the limits of integration for \( x, y, \) and \( z \) and compute the integral.

Answer: 1


Question: If \(x = \sqrt{2} + \sqrt{3} + \sqrt{6}\) is a root of the polynomial \(x^4 + ax^3 + bx^2 + cx + d\), where \(a, b, c,\) and \(d\) are integers, find \(|a + b + c + d|\).

Answer: 93


Question: Find the maximum value of the expression \( y = 2(a-x)(x+\sqrt{x^2+b^2}) \).

Answer: a^2 + b^2


Question: Let $K$ be a finite $d$-dimensional simplicial complex that is homotopy equivalent to a wedge of $k$ $d$-spheres. Does $K$ necessarily contain a $d$-sphere as a subcomplex?

Answer: Yes


Question:
Given a real polynomial $P(x)$ of degree 4, with the following conditions:

1. $P(1) = 0$.
2. $P(2) = 3$ and $P'(2) = 0$.
3. $P(3) = 3$ and $P'(3) = 0$.

Compute the value of $P(4)$.

Note: You do not need to determine the entire polynomial $P(x)$, only the value at $x = 4$.

Answer: 0


Question: Classify the singularities and determine the radius of convergence of the function \( f(z) = \frac{e^{-z^2}}{(z-2)} \) at \( a=0 \).

Answer: 2


Question: Determine the number of elements in the set \( A = \{ x \in \mathbb{R} \mid f(x) = 1 \} \), where the function \( f: \mathbb{R} \to \mathbb{R} \) is defined by \( f(x) = 3^{x^3-3x} - 3^{x+1} + x^3 - 4x \). Find \( |A| \).

Answer: 3


Question: Determine the order of the pole of the function \( \frac{1}{(2\cos z - 2 + z^2)^2} \) at \( z=0 \).

Answer: 8


Question: Find the minimum value of the function \( f(x,y,z) = x^2 + y^2 + z^2 \) subject to the constraints \((x-2)^2 + y^2 + z^2 = 1\) and \(x + z = 3\) using Lagrange multipliers.

Answer: 5


Question:
Using the divergence theorem, find the flux of the vector field \( \mathbf{F} = \left(x(z - 1), y(z - 1), xy\right) \) over the region defined by the inequalities:
\[
(z-1)^2 \le 4-x^2-y^2, \quad (z-1)^2 \ge x^2+y^2, \quad z \ge 1
\]

Answer: 4\pi


Question: Determine whether the subset \( H = \{ f \in E \mid \exists n_f \in \mathbb{N}_* : f = 0 \text{ on } [-n_f, +n_f] \} \) is a vector subspace of the vector space \( E \) of all functions from \( \mathbb{R} \) to \( \mathbb{R} \). Justify your answer rigorously.

Answer: H


Question:
Calculate \( I'(0) \) for the function \( I(x) = \int_{0}^{2x} f(x-t) \, dt \), where
\[
f(x) = \begin{cases} \frac{\ln(1+\sqrt{2}x+x^2)}{x} & x \neq 0 \\ 0 & x = 0 \end{cases}
\]

Answer: 2\sqrt{2}


Question: Find the greatest common divisor (gcd) of the elements 4 and 6 in the ring $\mathbb{Z}[\sqrt{2}]$. Provide your answer as an element of $\mathbb{Z}[\sqrt{2}]$.

Answer: 2


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

Answer: e^6


Question: What is the remainder when \(6^{273} + 8^{273}\) is divided by 49?

Answer: 0


Question: Find the coefficient of $x^6$ in the expansion of the expression \( \left[(1+x)(1+x^2)^2(1+x^3)^3 \cdots (1+x^n)^n\right] \).

Answer: 28


Question: Find the remainder when \(15^{40}\) is divided by 1309.

Answer: 1


Question: Let $f: X \to Y$ be a finite, surjective morphism of normal algebraic varieties, and let $D$ be a Weil divisor on $Y$. If the pullback divisor $f^*(D)$ is Cartier, is the divisor $D$ also Cartier?

Answer: No


Question: Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be continuous on $[0,1]$. Consider $z\in(0,1)$ and suppose that $f$ is differentiable and convex on $[0,z]$ and $[z,1]$. If $f'$ (i.e., $\frac{df}{dx}$) is continuous at $z$, determine whether $f$ is convex on $[0,1]$. Is this proposition true?

Answer: True


Question:
Consider the unit circle \(\mathbb{S}^1 \subset \mathbb{R}^2\) and let \(\mu\) be an arbitrary Borel probability measure on \(\mathbb{S}^1\) whose support is not a finite set. Is it possible to find an orthonormal basis \(\{f_n\}_{n\geq 1}\) for \(L^2(\mathbb{S}^1,\mu)\) such that for every \(g\in L^2(\mathbb{S}^1,\mu)\), the following holds?
\[
\lim_{n \rightarrow \infty} \int_{\mathbb{S}^1} f_n(z)~ g(z)~ d\mu(z)=0.
\]

Answer: Yes


Question: Find the minimum value of the expression \(|z+1|+|z-1|+|z-i|\), where \(z\) is a complex number.

Answer: 1 + \sqrt{3}


Question: Let \(a, b, c, d\) be positive real numbers such that \(a + b + c + d = 1\). Find the minimum value of the expression \(\frac{1}{a} + \frac{4}{b} + \frac{9}{c} + \frac{16}{d}\).

Answer: 100


Question: Let \(\alpha\) and \(\beta\) be two isomorphic ordinals. Prove that \(\alpha = \beta\).

Answer: \alpha = \beta


Question: Given two independent normally distributed random variables, X and Y, with means \( E[X] = 1 \) and \( E[Y] = 10^{-3} \), and variances \( \text{Var}(X) = (0.12)^2 \) and \( \text{Var}(Y) = (0.05 \times 10^{-3})^2 \), find the variance of the random variable \( Z = \frac{X}{Y} \). Assume that X and Y are independent. What is \( \text{Var}(Z) \)?

Answer: 16900


Question: Does there exist an increasing function \( g \) such that \( g' = 0 \) almost everywhere, but \( g \) is not constant on any open interval?

Answer: Yes


Question: Given two non-negative lists $x$ and $y$ of the same length, if the mean of $x$ equals the mean of $y$, the standard deviation of $x$ equals the standard deviation of $y$, and the product of their standard deviations equals their covariance, does it follow that $x = y$? Justify your answer.

Answer: Yes


Question: A point \((X, Y)\) is uniformly distributed on the unit square \([0,1]^2\). Let \(\theta\) be the angle between the x-axis and the line segment connecting \((0,0)\) to the point \((X,Y)\). Find the expected value \(E[\theta]\).

Answer: \dfrac{\pi}{4}


Question: Evaluate the integral \( \int_C \tan z \, dz \), where \( C \) is the positively oriented circle \( |z|=2 \).

Answer: -4\pi i


Question: Compute the limit: $$\lim_{n \to \infty} \left(\dfrac{n}{(n+1)^2}+\dfrac{n}{(n+2)^2}+\cdots+\dfrac{n}{(n+n)^2}\right).$$

Answer: \dfrac{1}{2}


Question: A continuous random variable $X$ takes positive values only and is such that $P(X > x) = k(2-x)$ for $0 \leq x \leq 2$. Find the expected value of $X$.

Answer: 1


Question: Determine the interval for \( m \in \mathbb{R} \) such that the polynomial \( 2x^4 + x^3 + mx^2 + x + 2 = 0 \) has only real roots.

Answer: (-\infty, -6]


Question: Calculate the value of the Riemann-zeta function at \( s = 0 \), i.e., find \( \zeta(0) \).

Answer: -\dfrac{1}{2}


Question: Evaluate the integral \( \int \int_B \frac{\sqrt[3]{y-x}}{1+y+x} \, dx \, dy \) over the triangular region \( B \) with vertices \((0,0), (1,0), (0,1)\).

Answer: 0


Question:
Consider a connected, compact 2-dimensional submanifold $S \subset \mathbb{R}^3$ with a one-to-one Gauss map $N$. Given that the Gauss curvature $K$ satisfies $K(p) \neq 0$ for all $p \in S$, we have:
\[ \int_S K(p) \, dp = 4 \pi \]
Does this conclusion still hold if we do not assume $K(p) \neq 0$ for all $p \in S$?

Answer: 4\pi


Question:
Determine whether there exists a finite minimal nonsolvable group $G$ with the following properties:
1. $G$ has only one proper normal subgroup $N$.
2. $N$ is an elementary abelian 2-group.
3. $C_{G}(N) = N$.
4. $\frac{G}{N} \cong A_{5}$.

Answer: Yes


Question: For \(\alpha \in (0,\frac{3}{2})\), define \(x_n=(n+1)^{\alpha}-n^{\alpha}\). Determine \(\lim_{n \to \infty}x_n\). Choose the correct option: \(\text{A) 1 for all } \alpha\), \(\text{B) 1 or 0 depending on the value of } \alpha\), \(\text{C) 1 or } \infty \text{ depending on the value of } \alpha\), \(\text{D) 1, 0 or } \infty \text{ depending on the value of } \alpha\).

Answer: D


Question: Consider a connected complex manifold $X$ with a complex submanifold $Y$ such that a function $f:Y \to \mathbb{C}$ is holomorphic if and only if $f$ is constant. If $Y$ is not a closed submanifold, can $X$ have non-constant holomorphic functions?

Answer: Yes


Question: Find the least positive integer \( x \) that satisfies both congruences: \( 3x \equiv 11 \pmod{17} \) and \( 5x \equiv 9 \pmod{23} \).

Answer: 287


Question: Consider independent random variables $X_i$ for $i = 1, 2, \ldots$, each uniformly distributed on the interval $[0, 1]$. Determine the probability that the sum \( \sum_{n=1}^\infty \prod_{i=1}^n X_i \) is finite.

Answer: 1


Question: Determine the values of \( \lambda > 0 \) for which the set \( I_{\lambda} \), consisting of all irrational numbers \( \rho = [a_{1}, a_{2}, \ldots, a_{n}, \ldots] \in (0,1) \) with continued fraction partial quotients satisfying \( a_{n} \leq \text{const} \cdot n^{\lambda} \), has a positive Lebesgue measure.

Answer: \lambda > 1


Question: Let $I=[t_{0},t_{1}]\subset \mathbb{R}$ be fixed and $\Omega \subset \mathbb{R}^{n}$. Suppose $f$ and $\psi$ are smooth functions, and $\psi(t_{0},x)= \psi(t_{1},x) = 0$ for every $x \in \mathbb{R}^{n}$ and $\psi(t,x)|_{\partial \Omega} = 0$ for every $t \in I$. If \(\int_{I}\int_{\Omega}f(t,x)\psi(t,x)\,dx\,dt = 0\) for every $\psi$ satisfying these conditions, prove or disprove that $f(t,x) \equiv 0$.

Answer: f(t,x) \equiv 0


Question:
Given a distributive and complemented lattice $(L, \leq_L, 0, 1)$ that is dense as an order, and subsets $U, V \subset L \setminus \{0\}$ such that:

1. $(U, \leq_L)$ and $(V, \leq_L)$ are totally ordered sets.
2. $\inf (U \cup V) = 0$.
3. For all $a, b \in U \cup V$, $\inf (a, b) \neq 0$.

Assuming $|V|$ and $|U|$ are regular cardinals, does there exist a subset $T \subset L \setminus \{0\}$ that is totally ordered by $\leq_L$, such that $\inf T = 0$ and $|T| \leq |U \cup V|$?

Answer: Yes


Question:
Consider the sequence of polynomials \( \{f_n(x)\}_{n \in \mathbb{N}} \) defined inductively by:
\[
f_1(x) = (x-2)^2, \\
f_{n+1}(x) = (f_n(x) - 2)^2 \text{ for all } n \geq 1.
\]
Let \( a_n \) and \( b_n \) denote the constant term and the coefficient of \( x \) in \( f_n(x) \), respectively. Determine the correct pair \( (a_n, b_n) \) from the following options:

\(\text{(A)}\ a_n = 4, b_n = -4^n\)

\(\text{(B)}\ a_n = 4, b_n = -4n^2\)

\(\text{(C)}\ a_n = 4^{(n-1)!}, b_n = -4^n\)

\(\text{(D)}\ a_n = 4^{(n-1)!}, b_n = -4n^2\)

Answer: A


Question: Compute the value of the line integral \( \oint_{\partial\Sigma} \frac{xdy-ydx}{x^{2}+y^{2}} \) where \( \partial\Sigma \) is a positively oriented, piecewise smooth, simple closed curve in the plane that encloses the origin \((0,0)\).

Answer: 2\pi


Question: Let $V$ be the vector space of real $n \times n$ symmetric matrices. Determine the dimension of the orthogonal complement of the subspace $W$ of $V$, where $W$ consists of matrices with trace zero, relative to a positive definite scalar product.

Answer: 1


Question: Solve the equation $e = xe^x$ for $x$.

Answer: 1


Question: Evaluate the limit without using L'Hôpital's Rule or Taylor Series: \[ \lim\limits_{x\rightarrow 0^+}{\dfrac{e^x-\sin x-1}{x^2}} \]

Answer: \dfrac{1}{2}


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

Answer: \dfrac{3}{10}


Question: Given the function \( f(x) = x + 2x^3 + x^5 \), evaluate \( (f^{-1})'(4) \).

Answer: \dfrac{1}{12}


Question: Prove that \( \sum_{k=1}^{n} k \binom{n}{k} = n \cdot 2^{n-1} \) for \( n \in \mathbb{N} \).

Answer: n \cdot 2^{n-1}


Question:
Given the sequence \( \{x_n\} \) defined by \( x_1 = 5 \) and \( x_{n+1} = x_n^2 - 2 \) for all \( n \geq 1 \), find the limit:
\[ \lim_{n \to \infty} \frac{x_{n+1}}{x_1 x_2 \cdots x_n}. \]

Answer: \sqrt{21}


Question: Determine the smallest integer \( n \) such that the Maclaurin series approximation for \( \arctan(1) \) satisfies \(|4P_n(1) - \pi| < 10^{-3}\), where \( P_n(x) = \sum_{i=1}^{n} (-1)^{i+1} \frac{x^{2i-1}}{2i-1} \).

Answer: 2000


Question: Given a subset $S \subset B(H)$ that does not contain the identity operator $Id_H$, does there exist a projection $p$ such that the dimension of $pSp$ is finite but not zero?

Answer: No


Question: Determine the number of prime numbers of the form \(a^{k/2} + b^{k/2}\) for positive integers \(a\), \(b\), and \(k > 1\).

Answer: \infty


Question: Find the values of $x$ that satisfy the equation $\frac{11}{2} x - \cos x = 0$. Provide all solutions.

Answer: 0.1789


Question: Prove that if for all primes $p$ which divide $n$, $p^6$ also divides $n$, then there exist positive integers $x$ and $y$ such that $n = x^3 y^4$.

Answer: n = x^3 y^4


Question: If $f: R \to S$ is an isomorphism between rings $R$ and $S$, both with identity, does it follow that $f(1_R) = 1_S$? Provide a proof or counterexample.

Answer: f(1_R) = 1_S


Question: Let \( f(z) \) be a function that is analytic in the unit disk \( U = \{z:|z|<1\} \) and continuous on the closed unit disk. Suppose that \( \frac{f(z)}{z^2} \) can be extended to be analytic in the open unit disk \( U \), including at the origin. Given that \( |f(z)| \leq 6 \) in the closed unit disk, determine the maximal possible value of \( f(0.4 + 0.5i) \).

Answer: \dfrac{123}{50}


Question: Do there exist two integer polynomials $P(x, y)$ and $Q(x, y)$ that touch at an irrational point $(a, b)$, where at least one of $a$ or $b$ is irrational?

Answer: Yes


Question: Calculate the variance of the Ito integral \( \int_{0}^{t} W^{2}_{s} \, dW_{s} \).

Answer: t^3


Question: Given a positive sequence \(a_k\) that converges to zero, i.e., \(a_k \geq 0\) for all \(k \in \mathbb{N}\) and \(\lim_{k\rightarrow \infty} a_k = 0\), does there exist another sequence \(b_k\) such that for every \(k > 0\), \(a_{k+p} < b_k\) for all \(p > 0\), and \(\lim_{k\rightarrow \infty} b_k = 0\)?

Answer: Yes


Question: Let \( f:[2,4]\to[3,5] \) be a bijective decreasing function. Find the value of \( \int_{2}^{4}f(t) \, dt - \int_{3}^{5}f^{-1}(t) \, dt. \)

Answer: 2


Question:
Apply the Leibniz rule for differentiation under the integral sign twice to evaluate the following expression:
\[ \frac{\textrm{d}^2}{\textrm{d}x^2}\int_a^x g(s)(x-s)\,ds \]
Show that the result is \( g(x) \).

Answer: g(x)


Question: Consider a sequence of i.i.d. random variables $X_k$, $k = 1, 2, \dots$, each with finite second moments. Let $N_k \geq 1$, $k = 1, 2, \dots$, be a sequence of random variables taking integral values, such that $\lim_k N_k = \infty$ almost surely, and assume each $N_k$ is independent of the $X_k$'s. Define $S_k := \sum_1^{N_k} X_k$. Does $(S_k - \mu N_k)/\sigma\sqrt{N_k}$ converge in distribution to the standard normal variable as $k \to \infty$, where $\mu = \mathbb{E}[X_k]$ and $\sigma^2 = \mathbb{V}[X_k]$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} n \left((n^5 +5n^4)^{1/5} - (n^2 +2n)^{1/2}\right) \]

Answer: -\dfrac{3}{2}


Question: Determine whether every subgroup of order $p$ in a $p$-group is normal, where $p$ is a prime number. Provide a justification for your answer.

Answer: No


Question:
Consider three independent random variables uniformly distributed between 0 and 1. What is the probability that the middle value of the three lies between \(a\) and \(b\), where \(0 \leq a < b \leq 1\)? Choose the correct answer from the following options:

A) \(3 (1 - b) a (b - a)\)

B) \(3 ((b - a) - (b^{2}- a^{2})/2)\)

C) \(6 (1 - b) a (b - a)\)

D) \((1 - b) a (b - a)\)

E) \(6 ((b^{2}- a^{2})/ 2 - (b^{3} - a^{3})/3)\)

Answer: E


Question: Consider a hyperelliptic Riemann surface \( \mathcal{R} \) of genus \( g \geq 1 \). Determine whether the only rational functions on \( \mathcal{R} \) with \( \leq g \) poles are the liftings of rational functions on \( \mathbb{C} \) to \( \mathcal{R} \). Equivalently, verify if all principal divisors on \( \mathcal{R} \) of the form \( \sum_{j=1}^m P_j - \sum_{j=1}^m Q_j \) with \( m \leq g \) are involution-symmetric, meaning each point is included along with its image by the hyperelliptic involution.

Answer: Yes


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

Answer: e^2


Question: Compute the volume of the region \( E = \{(x,y,z) \in \mathbb{R}^3 \mid (x,y) \in D \text{ and } 0 \leq z \leq \sqrt{1-x^2-y^2}\} \), where the region \( D = \{(x,y) \in \mathbb{R}^2 \mid x^2+y^2 \leq 1\} \).

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


Question: Determine the cardinality of the set \( A = \{ t \in [-1,1] : f(t) = t \} \), where \( f : [-1,1] \to [-1,1] \) is a function such that \( f \in C^1[-1,1] \), \( |f(t)| \leq 1 \), and \( |f'(t)| \leq \frac{1}{2} \) for all \( t \in [-1,1] \).

Answer: 1


Question: Calculate the mass of a solid hemisphere with a radius of 2 units, where the base of the hemisphere lies in the xy-plane and the center of the base is at the origin. The density of the solid is given by the function \( \rho(x,y,z) = xyz \). Use spherical coordinates to evaluate the triple integral and determine the mass of the hemisphere.

Answer: 0


Question: Let $f$ be a continuous function on the interval $[0,1]$. Given that $\int_{0}^{1} f(x) x^n \, dx = 0$ for all non-negative integers $n$, prove that $\int_{0}^{1} f^2(x) \, dx = 0$. Provide a detailed justification for your answer.

Answer: 0


Question: Find the sum of all possible values of the constant \( k \) such that the graph of the parametric equations \( x = 2 + 4\cos s \) and \( y = k - 4\sin s \) intersects the graph of the parametric equations \( x = 1 + \cos t \) and \( y = -3 + \sin t \) at only one point.

Answer: -12


Question: Find the number of surjective functions \( f: A \to B \), where \( A = \{1,2,3,4,5,6\} \), \( B = \{1,2,3,4\} \), and \( f(1) > f(2) > f(3) \).

Answer: 148


Question: In a Banach space, it is known that for a convex subset, the weak-closure and strong-closure coincide. Does this result still hold for a subset that is convex only for combinations with rational coefficients?

Answer: Yes


Question:
Estimate the upper bound of the number of solutions \(N_{\sqrt{2}}(1)\) for the inequality:
\[ \left \lvert \sqrt{2} - \frac{p}{q} \right \rvert < \frac{1}{q^{3}} \]

Answer: 3


Question: Determine whether the cyclotomic polynomial \( \Phi_r(X) \) can always be represented with coefficients in the finite field \( F_p \), where \( F_p \) is a field with prime characteristic \( p \).

Answer: Yes


Question: Find the limit: \( \lim_{n \to \infty} \sup \cos \frac{\pi \cdot n}{4} \)

Answer: 1


Question: Evaluate the limit: \[ \lim _{n\to \infty }\left(n\left(\frac{\left(\frac{(\ln n)^{10}}{n^{1.1}}\right)}{\frac{(\ln(n+1))^{10}}{(n+1)^{1.1}}}-1\right)\right) \]

Answer: \dfrac{11}{10}


Question: Evaluate the limit: $$ \lim_{n\to\infty} \frac{1}{(\ln n)^2} \sum_{i=1}^n \ln (i^{1/i}) .$$

Answer: \dfrac{1}{2}


Question: Calculate the limit: \[ \lim_{x \to \infty} \left((x-1)e^{1/x} - x\right) \]

Answer: 0


Question: Evaluate the double integral \( \iint_D 2xy\exp(y^2)\,dxdy \) over the region \( D \) using polar coordinates.

Answer: 0


Question: Express \( \left(\frac{i-\sqrt{3}}{i-1}\right)^{24} \) in algebraic form.

Answer: 4096


Question:
A stock trader has 100 shares of a company and uses a random strategy for buying or selling shares based on the outcome of a die roll at each trading session. The strategy is as follows:

- Outcome = 1: Do nothing
- Outcome = 2 or 4: Buy one share
- Outcome = 3 or 5: Sell one share
- Outcome = 6: Buy two shares

Calculate the approximate probability that after 18 independent trading sessions, the trader will have more than 110 shares of the company.

Answer: 0.169


Question: Does every continuous function $f: I \subset \mathbb{R} \to \mathbb{C}$ on a closed interval $I$ admit a modulus of continuity?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to0}\frac{\sin(3x^2)}{\log\cos(2x^2-x)} \]

Answer: -6


Question: Given the differential equation \( x' + x^3 = 2x \), find the differential equation for \( v = \frac{1}{x^2} \).

Answer: v' + 4v = 2


Question: Let \( f : [-1,1] \rightarrow \mathbb{R} \) be a continuous function satisfying the functional equation \( f(2x^2-1) = (x^3+x)f(x) \). Determine the value of \( \lim\limits_{x \to 0}\dfrac{f(\cos x)}{\sin x} \).

Answer: 0


Question: Find the greatest common divisor \( \gcd(m, n) \) given that \( (m^2 - mn - n^2)^2 = 1 \).

Answer: 1


Question: Find the derivative of the inverse function of \( f(x) = x - \tan x \) at the point \( (\pi/4) - 1 \). Assume the domain of \( f(x) \) is \((-\pi/2, \pi/2)\) to ensure it is one-to-one.

Answer: -1


Question: Solve the equation: \(2^x + x = 4\).

Answer: 1.386


Question: Compute the limit: \[ \lim_{z \to 0} \left(\dfrac{1}{z^2} - \dfrac{1}{\sin^2 z}\right) \]

Answer: -\dfrac{1}{3}


Question: Calculate the residue of the function \( \frac{f'(z)}{f(z)} \) at \( z = w \), where \( f(z) = \sum_{n=m}^\infty a_n (z-w)^n \) with \( 0 < | z-w | < r \) and \( a_m \neq 0 \).

Answer: m


Question: Let $X$ be a random variable with a uniform distribution on the interval $(0,1)$, denoted as $X \sim U(0,1)$. The conditional distribution of $Y$ given $X=x$ is uniform on the interval $(0,x)$, denoted as $Y|X=x \sim U(0,x)$. Calculate the probability $P(X+Y \geq 1)$. Provide your answer in terms of known constants or functions.

Answer: 1 - \ln 2


Question: In a tournament with 13 players, matches are played in groups of 4 players, and there are 13 rounds in total. Each player skips one round, and in the other 12 rounds, they participate in matches with 3 other players. Is it possible to arrange the matches so that each player competes against every other player exactly three times throughout the tournament?

Answer: Yes


Question: Determine whether a number of the form $10^x + 1$ can ever be a perfect square. Justify your answer.

Answer: No


Question: Evaluate the limit \( \lim_{n \to \infty} \int_0^\infty n e^{-\frac{2n^2x^2}{x + 1}} \, dx \) and show that it equals \( \infty \).

Answer: \infty


Question: In the tetrahedron $OABC$, where $|OA|=|OB|=|OC|$, let $D$ and $E$ be the midpoints of segments $AB$ and $AC$, respectively. If $\alpha=\angle(DOE)$ and $\beta=\angle(BOC)$, find the ratio $\beta/\alpha$. Assume $|BC|=2|DE|$ due to the similarity of triangles $\Delta ABC$ and $\Delta ADE$. Determine the value of $\beta/\alpha$. \(\text{Express your answer as a simplified fraction.}\)

Answer: \dfrac{3}{2}


Question: Determine whether the sequence of functions \( f_n(x) = \sum_{k=0}^n x^k \) converges uniformly on the interval \([0, 3/4]\).

Answer: Yes


Question: Find the supremum of the set \( S = \{ x \ge 0 \mid \sum_{n=1}^{\infty} x^{\sqrt{n}} < \infty \} \).

Answer: 1


Question: Find the minimal polynomial over \( \mathbb{Q} \) for the number \( \sqrt{9-4\sqrt{2}} \), given that the minimal polynomial of \( 9-4\sqrt{2} \) is \( x^2-18x+49 \).

Answer: x^2 + 2x - 7


Question: Calculate the volume of the region that lies above the $x-y$ plane, under the surface defined by $r^2 = 2z$, and inside the cylinder $r = 2$. Use cylindrical coordinates to set up the integral.

Answer: 4\pi


Question: Compute the cardinality of the set \(\{(\aleph_{\omega_1})^\kappa : 1 \leq \kappa < \aleph_{\omega_1}\}\), given that for every \( \alpha, \beta < \omega_1 \), there exists a \( \gamma < \omega_1 \) such that \( \aleph_{\alpha}^{\aleph_\beta} = \aleph_\gamma \).

Answer: 1


Question:
Let $x_1 > 0$ and consider the recurrence relation:
\[ x_{n+1} = x_n + \frac{n^2}{x_1 + x_2 + \ldots + x_n} \]
Determine the nature of convergence for the sequence \( \frac{x_n}{n} \).

Answer: \sqrt{2}


Question: Determine whether the statement $((\omega+1) \cdot \omega) \in (\omega \cdot (\omega +1))$ is true, given that $(\omega+1) \cdot \omega = \omega^2$ and $\omega \cdot (\omega +1) = \omega^2 + \omega$. Justify your answer.

Answer: True


Question: Find a second degree polynomial $t(x)$ such that for every second degree polynomial $r(x)$, there exist constants $a$, $b$, and $c$ satisfying the equation $r(x) = a \cdot p(x) + b \cdot q(x) + c \cdot t(x)$, where $p(x) = x^2 + 5x - 3$ and $q(x) = 4x^2 + 18x + 4$. Determine $t(x)$. \(t(x)\) should be a second degree polynomial.

Answer: x^2


Question: Determine the real numbers \(a\) for which the series \(\sum \frac{\sin(ka)}{\log(k)}\) converges.

Answer: \mathbb{R}


Question:
Let \( u \) be a solution of the differential equation \( y' + xy = 0 \) and \( \phi = u\psi \) be a solution of the differential equation \( y'' + 2xy' + (x^2 + 2)y = 0 \) such that \( \phi(0) = 1 \) and \( \phi'(0) = 0 \). Determine \( \phi(x) \) from the following options:

(A) \( (\cos^2(x))e^{-\frac{x^2}{2}} \)
(B) \( (\cos(x))e^{-\frac{x^2}{2}} \)
(C) \( (1+x^2)e^{-\frac{x^2}{2}} \)
(D) \( (\cos(x))e^{-x^2} \)

Answer: B


Question:
Let \( P(x) = 5x^{2015} + x^2 + x + 1 \) be a polynomial with roots \( x_1, x_2, x_3, \ldots, x_{2015} \). Find the value of the expression:
\[ \left(\prod_{i=1}^{2015} (1+x_i)\right) \left( \sum_{i=1}^{2015} \frac{1}{1-x_i} \right). \]

Answer: \dfrac{5039}{5}


Question: Solve the system of equations for $z \in \mathbb{C}$ given by: \[ \begin{cases} mz^{m+1} - (m+1)z^m + 1 = 0 \\ nz^{n+1} - (n+1)z^n + 1 = 0 \end{cases} \] where $m \neq n$ and $m, n \in \mathbb{N}$. Determine if $z=1$ is the only solution.

Answer: 1


Question: Let $a$ and $b$ be positive integers such that $a \mid b^2$, $b^2 \mid a^3$, $a^3 \mid b^4$, and so on, such that $a^{2n-1} \mid b^{2n}$ and $b^{2n} \mid a^{2n+1}$ for all positive integers $n$. Is it true that $a = b$?

Answer: a = b


Question: Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $p \in M$. Let $0 \le k \le d$ be fixed. Does there exist an open neighborhood $U$ of $p$ that admits a non-zero harmonic $k$-form, i.e., $\omega \in \Omega^k(U)$ satisfying $d\omega = \delta \omega = 0$? Consider the cases where $2 \le k \le d-2$. Provide a justification for your answer.

Answer: Yes


Question: Let $f$ and $g$ be functions analytic at $a \in \mathbb{C}$, with the power series expansions $$f(z) = \sum_{n=0}^{\infty} a_n (z-a)^n$$ and $$g(z) = \sum_{n=0}^{\infty} b_n (z-a)^n.$$ Is it true that the power series expansion of the product $f(z)g(z)$ at $a$ is given by $$f(z)g(z) = \sum_{n=0}^{\infty} \left( \sum_{i=0}^{n} a_i b_{n-i} \right) (z-a)^n?$$

Answer: Yes


Question:
Consider the sequence of real numbers \((a_{n})\) defined by the recurrence relation:
\[ \frac{n(n+2)}{(n+1)}a_{n}-\frac{n^2-1}{n}a_{n+1}=n(n+1)a_{n}a_{n+1} \]
with \(a_{1} > 0\). Let \((b_{n})\) be a convergent sequence of real numbers. Calculate:
\[ \lim_{n\rightarrow\infty}\frac{\sum_{i=1}^{n} a_{i}b_{i}}{n} \]

Answer: 0


Question: Assume $A$ is a symmetric positive semidefinite matrix with diagonal zero and all other entries less than one. Also, assume $D$ is a diagonal matrix with all diagonal entries positive and less than $1$. Are the diagonal entries of $[D+A]^{-1}$ all positive?

Answer: Yes


Question: Let \( f:[0,\infty)\to\mathbb{R} \) be a continuously differentiable function such that \( \lim _{x\to \infty} f'(x)=\ell \) for a real number \( \ell \). Find \( \lim _{x \to \infty} \frac{f(6x)}{x} \).

Answer: 6\ell


Question: Determine whether there exists a function \( f \in C^{\infty}(\mathbb{R}^2) \) that has an infinite number of strong local maxima and no local minima.

Answer: Yes


Question: Prove that the sequence \( a_n = \frac{2^n}{n^4} \) diverges to infinity as \( n \to \infty \).

Answer: \infty


Question:
Let $N$ be a homogeneous Poisson process with intensity $\lambda$. Compute the probability:
$$P[N(5)=2 \, | \, N(2)=1,N(10)=3].$$

Answer: \dfrac{15}{32}


Question: If $Q^TQ = I$, does it necessarily follow that $QQ^T = I$? Here, $I$ is the identity matrix and $Q$ is an $m \times m$ matrix.

Answer: Yes


Question: In a finite-dimensional vector space $V$, suppose $\lambda_j, j=1,\cdots,p$ are the eigenvalues of a linear operator $A$ with multiplicities $m_1,\cdots,m_p$. Let $k_j$ be the smallest index such that $\text{Ker}(A-\lambda_j)^{k_j} = \text{Ker}(A-\lambda_j)^{k_j+1}$. Is it true that the subspaces $\text{Ker}(A-\lambda_j)^{k_j}$ for $j=1,\cdots,p$ span the entire space $V$?

Answer: Yes


Question: Evaluate the sum \( \sum_{n\ge 0}\frac{1}{4^n(n+1)}\binom{2n}{n} \).

Answer: 2


Question: Find the inverse of the polynomial $x^2 + 1$ in the field $\frac{\mathbb{Z}_3[x]}{m(x)}$, where $m(x) = x^3 + 2x + 1$. Use the Euclidean Algorithm and Bezout's Identity to determine the inverse.

Answer: 2x^2 + x + 2


Question: Determine if there is a known bound for the VC-dimension of the set of binary classifiers \(\mathcal{H} = \{\boldsymbol{x}\mapsto\boldsymbol{1}\hspace{-0.9ex}\mathrm{1}(\boldsymbol{h}^\mathsf{T}\boldsymbol{x}\ge 0)\mid  \boldsymbol{h} \in \mathcal{C} \}\), where \(\mathcal{C}\) is a pointed, salient, full-dimensional cone in \(\mathbb{R}^d\), that is significantly better than the VC-dimension of the set of all linear classifiers in \(\mathbb{R}^d\) (i.e., \(d\)).

Answer: d


Question: In a symmetric random walk on an integer number line, starting from the origin, what is the probability that the particle hits -9 before +6? Assume the probability of moving left or right at each step is \( \frac{1}{2} \).

Answer: \dfrac{2}{5}


Question: In an election, candidate A receives 60 votes and candidate B receives 40 votes. What is the probability that candidate A always leads candidate B during the counting of the votes?

Answer: \dfrac{1}{5}


Question: Given that for each \(a \in (0,1)\), the limit \(\lim_{h \to 0^+} \int_{h}^{1-h} t^{-a} (1-t)^{a-1} dt\) exists and is denoted by \(g(a)\), and that \(g(a)\) is differentiable on \((0,1)\), find the value of \(g'\left(\frac{1}{2}\right)\).

Answer: 0


Question: Determine the values of $d$ for which the field $\mathbb{Q}(\sqrt{d})$ has an integral basis of the form $\alpha, \alpha'$ where $\alpha'$ is the conjugate of $\alpha$. Provide the condition on $d$.

Answer: d \equiv 1 \pmod{4}


Question: Let $G$ be a torsion-free, finitely generated abelian group. Suppose that $f : G \to G$ is a surjective homomorphism. Does it follow that $f$ is an isomorphism?

Answer: Yes


Question: Consider a map $\Phi$ from $\mathbb{R}^d$ onto itself, where $d \geq 1$. The map $\Phi$ is bi-Lipschitz continuous, has an almost everywhere constant Jacobian, and maps the unit ball onto the unit cube. Is the Jacobian of $\Phi^{-1}$ bounded almost everywhere?

Answer: Yes


Question: Suppose $R$ is a commutative ring such that for every ascending chain of ideals $I_1 \subseteq I_2 \subseteq I_3 \subseteq \cdots$, there exist infinitely many $n \in \mathbb{N}$ such that $I_n = I_{n+1}$. Does it follow that $R$ is Noetherian?

Answer: Yes


Question: Given the joint probability density function \( f_{X,Y}(x,y) = x + y \) for \( 0 < x < 1 \) and \( 0 < y < 1 \), find the conditional probability \( P(Y > 1/3 \mid X = 1/2) \).

Answer: \dfrac{7}{9}


Question: Given four random variables W, X, Y, and Z, where W and X have the same distribution, and Y and Z have the same distribution, and knowing that E(W+Z) = 0, can we conclude that E(X+Y) = 0, even if EX and EY may not exist?

Answer: No


Question: Determine the number of roots of the equation \(x^3 + x^2 + x + 1 = 0\) modulo 340.

Answer: 18


Question: In the icosahedron graph $G$, are there two non-adjacent vertices such that contracting them does not change the Hadwiger number of the graph?

Answer: Yes


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

Answer: 1


Question:
Determine the smallest degree of the polynomial \( q(x) \) such that the function \( f(x) \) defined by
\[
f(x) = \begin{cases}
0,  & \text{if } x \le 0, \\
q(x),  & \text{if } 0 < x < 1, \\
x, & \text{if } x \ge 1
\end{cases}
\]
is differentiable for all \( x \in \mathbb{R} \).

Answer: 3


Question: Find the smallest positive integer \( x \) such that \( 15625 \mid (1024x - 8404) \).

Answer: 3121


Question: Does the maximum modulus principle apply to antiholomorphic functions?

Answer: Yes


Question: Given that \( \frac{z_1 - 2z_2}{2 - z_1\bar{z_2}} \) is unimodulus and \(z_2\) is not unimodulus, find \(|z_1|\).

Answer: 2


Question:
Let $X$ be a topological space and $U$ be a proper dense open subset of $X$. Determine which of the following statements is correct:

(A) If $X$ is connected, then $U$ is connected.
(B) If $X$ is compact, then $U$ is compact.
(C) If $X \setminus U$ is compact, then $X$ is compact.
(D) If $X$ is compact, then $X \setminus U$ is compact.

Choose the correct statement.

Answer: D


Question: If $D$ is a division ring and $D^*$ is a finitely generated group, is $D^*$ necessarily an abelian group?

Answer: Yes


Question: Let \( f(x) = \sqrt{-x^2 + 20x + 400} + \sqrt{x^2 - 20x} \). How many elements in the range of \( f \) are integers?

Answer: 9


Question:
Let $Z_1, Z_2, \ldots, Z_7$ be independent standard normal random variables, each distributed as $N(0,1)$. Calculate the probability that the following inequality holds:
\[ P(Z_1^2 + Z_2^2 + Z_3^2 \leq 4.943(Z_4^2 + Z_5^2 + Z_6^2 + Z_7^2)) \]

Answer: 0.95


Question: Find the stationary point of the functional \( J[y]=\int \left( x^2y'^2+2y^2 \right) dx \) subject to the boundary conditions \( y(0)=0 \) and \( y(1)=2 \).

Answer: 2x


Question: Find the extreme values of the function \( f(x,y,z) = x^2 + y^2 + z^2 \) subject to the constraint \( xy + yz + zx = 3a^2 \).

Answer: 3a^2


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

Answer: \alpha > 0


Question: Find the value of \( k \) such that the function \( f(x) = \frac{x}{x^2 + k} \) satisfies \( f(x) \neq f(x+k) \) for any value of \( x \).

Answer: (-4, 0)


Question: Evaluate the integral \( \int \frac{\csc^2 x - 2020}{\cos^{2020} x} \, dx \) and express it in the form \( \frac{A f(x)^B}{g(x)^{2020}} + C \), where \( f(\frac{\pi}{6}) = \sqrt{3} \). Then, find the value of \( A^2 + B^2 + f(-\frac{\pi}{4}) \).

Answer: 1


Question: Consider a $2^n$ by $2^n$ bipartite graph with at least $(1-\epsilon)2^{2n}$ edges. For any $\epsilon > 0$ and sufficiently large $n$, determine if it is always possible to find a $2^{(1-f(\epsilon))n}$ by $2^{(1-f(\epsilon))n}$ complete bipartite subgraph, where $\lim_{\epsilon\rightarrow0^+}f(\epsilon)=0$. Provide a justification for your answer.

Answer: Yes


Question: Find the limit \( \lim_{n \to \infty}\frac{x_n}{n^{1/2}} \) where the sequence \( x_n \) is defined as follows: \( x_1 = 1, x_2 = 2, x_3 = 2, x_4 = 3, x_5 = 3, x_6 = 3, x_7 = 4, x_8 = 4, x_9 = 4, x_{10} = 4, \ldots \).

Answer: \sqrt{2}


Question: Consider a square matrix of size n, where the entries are filled with consecutive integers from 1 to n^2 in row-major order. Does the antidiagonal of this matrix always contain at least one prime number for n > 1?

Answer: Yes


Question: Let $(X, \tau)$ and $(Y, \sigma)$ be $T_2$-spaces such that there exist injective lattice homomorphisms $f: \tau \to \sigma$ and $g: \sigma \to \tau$. Does this imply that $(X, \tau)$ is homeomorphic to $(Y, \sigma)$?

Answer: No


Question: At an airport, the service time for checking a passenger's passport and ticket at the gate follows an exponential distribution with a mean of 10 seconds. Assuming the service times for different passengers are independent, approximate the probability that the total service time for 100 passengers is less than 15 minutes.

Answer: 0.1587


Question: Solve for $x$ in terms of $k$ in the equation $x + e^x = k$.

Answer: x = k - W(e^k)


Question: Determine whether the sequence \(x_n = (n+i\pi)^n n^{-n + 1/n}\) converges, and if it does, find its limit.

Answer: -1


Question: Solve the differential equation: \[ f'''(x) = f(x)f'(x)f''(x) \]

Answer: f(x) = Ax + B


Question: Evaluate the integral \( \int_C \bar{z} \, dz \), where \( C \) is the unit circle traversed in a counterclockwise direction.

Answer: 2\pi i


Question: Let $f$ be a uniformly continuous function defined on the open ball $B(0,1)$ in $\mathbb{R}^n$. Does there exist a function $g$ that is continuous on the closed ball $\bar{B}(0,1)$ such that $f = g$ for every $x \in B(0,1)$?

Answer: Yes


Question: Evaluate \( \cos(z) \) given that \( z = i \log(2+\sqrt{3}) \).

Answer: 2


Question: Consider a Markov process on the state space $S = \{1,2\}$ with a positive transition matrix, meaning all entries are strictly positive. Let $S_\infty = \times_{t=1}^\infty S$ be the underlying probability space, and let $\mathbb{P}$ be the probability measure on $S_\infty$ induced by the Markov transition matrix. For each $n$, define $F_n = \{ s \in S_\infty : \text{there are at most $n$ consecutive occurrences of $1$ or $2$} \}$. Determine if $\lim_{n \to \infty} \mathbb{P}(F_n) = 1$.

Answer: 0


Question: Find the maximum value of the function \( F(x,y,z) = \frac{x(1-x)y(1-y)z(1-z)}{1-(1-xy)z} \) on the interval \([0,1]\times[0,1]\times[0,1]\).

Answer: 17 - 12\sqrt{2}


Question: Evaluate the integral \( \int \int_E f(x,y) \, dx \, dy \) where \( f(x,y) = \begin{cases} (x-1/2)^{-3} & \text{if } 0 < y < |x-1/2| \\ 0 & \text{else} \end{cases} \) and \( E = [0,1] \times [0,1] \).

Answer: 0


Question: Given that \( z \) is a complex number satisfying \(|z - 3i| = 3\) and \( \theta = \arg{z} \) with \( 0 < \theta < \frac{\pi}{2} \), compute all possible values of \( \cot{\theta} - \frac{6}{z} \).

Answer: i


Question: Find the maximum value of the function \( g(x,y) = e^{-3(x-1)^2} + e^{-3y^2} \) subject to the constraint \( \frac{(x-1)^2}{4} + \frac{y^2}{4} \le 1 \).

Answer: 2


Question: Let $A$ be a positive definite matrix and $B$ be a positive semi-definite matrix. Under what conditions does the equality $\det(A+B) = \det(A) + \det(B)$ hold?

Answer: B = 0


Question: Determine the radius of convergence of the power series \( \sum_{n=0}^{\infty} a_n 2^n z^n \), given that \( \sum |a_n| < \infty \) and \( \sum n|a_n| = \infty \).

Answer: \dfrac{1}{2}


Question: Evaluate the triple integral \( \iiint \frac{1}{\sqrt{x^2+y^2+(z-2)^2}} \, dV \) over the region defined by \( x^2+y^2+z^2 = 1 \) using spherical coordinates.

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


Question: Evaluate the limit: $$\lim_{u,v\to 0,0}\frac{|u^2+u^2v+v^2u+2vu|}{\sqrt{u^2+v^2}}.$$ Prove that this limit equals zero.

Answer: 0


Question: Find the range of values of $k$ such that the cubic equation $x^3 - hx^2 + kx - 9 = 0$ has only one real root, which is $x = 1$. Express your answer as an interval or union of intervals.

Answer: (3, 15)


Question: Calculate the volume of the region bounded by the surfaces and curves given by \( z = 2x^2 + 2y^2 \), \( x^2 + y^2 = 2x \), and \( z = 0 \).

Answer: 3\pi


Question: Evaluate the integral \( \int_{-1}^{1} \left[ x^{2012} \sin(x) + \frac{1}{x^{2} + 1} \right] \, \mathrm{d}x \).

Answer: \dfrac{\pi}{2}


Question: Two boys are sent from a baker's shop and a butcher's shop, respectively, to deliver messages to each other. They pass each other at a point 720 yards from the baker's shop. After stopping for 10 minutes at their destinations, they return and pass each other again at a point 400 yards from the butcher's shop. Assuming each boy travels at a constant speed, calculate the distance between the baker's shop and the butcher's shop.

Answer: 1760


Question: Consider an unending sequence of independent trials, where each trial is equally likely to result in any of the outcomes 1, 2, or 3. Given that the outcome of the first trial is 1, and outcome 3 is the last of the three outcomes to occur, find the conditional probability that the outcome of the second trial is 1.

Answer: \dfrac{1}{3}


Question: Let \( F \in \mathbb{C}[X,Y] \) be an irreducible polynomial and \( n \in \mathbb{N} \), \( n \ge 1 \), with \( p_i \in \mathbb{C}[X] \) for \( 0 \le i \le n \), such that \[ F(X,Y) = \sum\limits_{i=0}^{n} p_i(X) Y^{n-i}. \] Suppose \( x \in \mathbb{C} \) is such that \( p_0(x) \neq 0 \), so \( F(x,Y) \) has \( n \) zeros, and one of these \( n \) zeros is a single one (i.e., there exists \( y \in \mathbb{C} \) such that \( F(x,y) = 0 \) and \( \frac{\partial F}{\partial Y}(x,y) \neq 0 \)). Is it possible for the polynomial \( F(x,Y) \) to have a multiple zero?

Answer: Yes


Question: Determine whether every open subset of \(\mathbb{C}\) can be expressed as a countable union of disjoint connected open sets. That is, if \(A \subset \mathbb{C}\), can \(A = \bigcup_{i=1}^{\infty} U_i\) where \(U_i\) are pairwise disjoint, connected, open sets?

Answer: Yes


Question: Is the commutator subgroup $[S_n, S_n]$ of the symmetric group $S_n$ generated by elements of the form $[\tau_1, \tau_2]$, where $\tau_1$ and $\tau_2$ are transpositions?

Answer: Yes


Question: Evaluate the contour integral \( \oint_{|z|=1}\frac{\sin(z)}{z^2}\,dz \).

Answer: 2\pi i