Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{n^n}{e^n n!} \] Choose the correct answer from the following options: \( A. 0 \), \( B. 1 \), \( C. e \), \( D. \sqrt{e} \), \( E. \infty \).

Answer: A


Question: In \( \triangle PQR \), if \( 3\sin P + 4\cos Q = 6 \) and \( 4\sin Q + 3\cos P = 1 \), find the measure of angle \( R \).

Answer: 30^\circ


Question:
Let $p \ge 3$ be a prime, and let $V$ be a crystalline 2-dimensional representation of $G_{\mathbb{Q}_p}$ with a lattice $T$ in $V$. Assume the following conditions on $V$:

- $V$ is irreducible.
- $\operatorname{Fil}^0 \mathbb{D}_{\mathrm{cris}}(V)$ is 1-dimensional.
- None of the eigenvalues of Frobenius on $\mathbb{D}_\mathrm{cris}(V)$ are integral powers of $p$.
- The Hodge filtration of $V$ has length $< (p-1)$.

Let $\omega$ be a $\mathbb{Z}_p$-basis of the tangent space $t_T = \mathbb{D}(T) / \operatorname{Fil}^0 \mathbb{D}(T)$. The Tamagawa number of $T$ over $K_n = \mathbb{Q}_p(\mu_{p^n})$ is given by
\[ \operatorname{Tam}^0_{K_n, \omega}(T) = \frac{[H^1_f(K_n, T) : \exp(\mathcal{O}_{K_n} \omega)]}{[\mathbb{D}(T) : (1- \varphi) \mathbb{D}(T)]} \]
where $[ A : B ]$ is a generalized index.

Is it true that under these hypotheses, this Tamagawa number is always 1?

Answer: 1


Question: Given the joint density function \( f(x,y) = xy \) for \( 0 \leq x \leq 2 \) and \( 0 \leq y \leq 1 \), find the probability \( P\left(\frac{X}{2} \leq Y \leq X\right) \).

Answer: \dfrac{3}{8}


Question: For which values of \( n \) does the cyclotomic extension \( \mathbb{Q}(\zeta_n) \) over \( \mathbb{Q} \) contain the imaginary unit \( i \)? Determine the condition on \( n \) such that \( i \in \mathbb{Q}(\zeta_n) \).

Answer: 4 \mid n


Question: Solve for \(x\) in the equation involving the Gaussian error function: \[ \frac{2}{\sqrt{\pi}}\int\limits_{0}^{x}e^{-t^2}dt = 0.7 \]

Answer: 0.733


Question: Consider the matrix product \(A^iB^{ij}C^{jk}D^k\), where repeated indices are summed. Given that \(B^{ij}\) is symmetric and \(C^{jk}\) is skew-symmetric, determine whether this product vanishes.

Answer: 0


Question: Is the support of a smooth function always open? Let \((X,\tau)\) be a topological manifold and \(f:X\to\Bbb R\) a smooth function. Determine if the support of \(f\), defined as \(\text{supp}(f)=\{x\in X\mid f(x)\neq 0\}\), is open in \(\tau\).

Answer: Yes


Question: A company has two electric generators. The time until failure for each generator follows an exponential distribution with a mean of 10 hours. The company will begin using the second generator immediately after the first one fails. What is the variance of the total time that the generators produce electricity?

Answer: 200


Question: Suppose \((a+i)^{a+i} \equiv a^a \pmod{p}\) for all integers \(a = 1, 2, \ldots\). Prove that \(i\) is divisible by \(p(p-1)\).

Answer: p(p-1)


Question: Let $V$ and $W$ be finite-dimensional inner product spaces. Suppose $T:V \to W$ is an injective linear transformation and $T^*:W \to V$ is its adjoint, satisfying \( \langle Tv,w \rangle = \langle v,T^*w \rangle \) for all $v \in V$ and $w \in W$. Is the composition $T^*T:V \to V$ necessarily an isomorphism?

Answer: Yes


Question: Determine whether the sequence of functions \( f_n(x) = \left(x - \frac{1}{n}\right)^2 \) converges pointwise on the interval \([0,1]\). If it does, find the limit function.

Answer: x^2


Question: Suppose that a sequence of sets $\{A_n : n \in \mathbb{N}\}$ is increasing, and let $A = \bigcup_{n=1}^\infty A_n$. If $A$ is measurable, $\mu(A) > 0$, and $\mu$ is an atomless measure, does there exist an $n \in \mathbb{N}$ and a measurable set $B$ such that $B \subset A_n$ and $\mu(B) > 0$?

Answer: Yes


Question: Calculate the sum \( \sum^{20}_{k=1}\frac{1}{x_k-x_k^2} \) where \( x_k \) are the roots of the polynomial \( P(x)=x^{20}+x^{10}+x^5+2 \).

Answer: 7


Question: Evaluate the integral \( \int_{\frac{1}{10}}^{\frac{3}{2}} \frac{1}{k} \, da \).

Answer: \dfrac{7}{5k}


Question: Let $V_0$ be any arbitrary finite-dimensional $sl_{\ell +1}(\mathbb{C})$ module, and let $V$ be the standard $(\ell+1)$-dimensional representation of $sl_{\ell+1}(\mathbb{C})$. Determine if there exists an integer $m \in \mathbb{N}$ such that $V_0$ occurs as an irreducible summand of $V^{\otimes m}$, up to isomorphism of $sl_{\ell+1}(\mathbb{C})$ modules.

Answer: Yes


Question: Let $H$ be a separable Hilbert space. Let $A$ be a maximal abelian von Neumann subalgebra of $B(H)$, and $B$ an abelian von Neumann algebra with $A \cap B = \mathbb{C}I$, where $I$ is the identity element of $B(H)$. Does there exist another maximal abelian von Neumann subalgebra of $B(H)$, say $C$, such that $C \supseteq B$ and $A \cap C = \mathbb{C}I$?

Answer: Yes


Question: Let $X$ be a real finite-dimensional linear space, and let $B: X \times X \rightarrow \mathbf{R}$ be a bilinear symmetric non-degenerate form. Suppose $M$ and $N$ are totally isotropic subspaces of the same dimension $r$ such that $M \cap N = \{0\}$. Given linearly independent vectors $e_1, \ldots, e_s$ in $M$ and $f_1, \ldots, f_s$ in $N$ (where $s < r$) with $B(e_i, f_j) = \delta_{i,j}$ for $i, j = 1, \ldots, s$, is it possible to extend $e_1, \ldots, e_s$ to a basis $e_1, \ldots, e_r$ of $M$ and $f_1, \ldots, f_s$ to a basis $f_1, \ldots, f_r$ of $N$ such that $B(e_i, f_j) = \delta_{i,j}$ for $i, j = 1, \ldots, r$?

Answer: Yes


Question: Evaluate the limit \( \lim_{x \to 0}{\left(\frac{\tan x}{x}\right)}^{1/{x^2}} \) using l'Hôpital's Rule.

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


Question:
Consider the set \( S = \{ z \in \mathbb{C} \mid (z+i)^n = (z-i)^n \} \) where \( n \in \mathbb{N}, n \geq 2 \). Determine which of the following statements about \( S \) is true:

A) \( S \) has \( n \) elements for all \( n \in \mathbb{N}, n \geq 2 \).

B) \( S = \{ (-1)^{n-1} + \cot \frac{k \pi}{n} \mid 1 \leq k \leq n-1; k \in \mathbb{N} \} \).

C) \( S = \{ \cot \frac{k \pi}{n} \mid 1 \leq k \leq n-1; k \in \mathbb{N} \} \).

D) \( S \cap \mathbb{R} \) has no more than 2 elements for all \( n \in \mathbb{N} \).

Answer: C


Question: Determine the cardinality of the set \(E = \{(x, y) : x, y > 0 \text{ and } x+y, xy \in \mathbb{Q}\} \subseteq \mathbb{R}^2\).

Answer: \aleph_0


Question:
Find an explicit form or a recurrence relation for the coefficients \(C_n(k)\) in the polynomial \(P_n(x) = \left(x\frac{d}{dx}\right)^n f(x) = xP_{n-1}'(x)\), where \(P_0 = f_0 = f\) and \(f_n(x) = \frac{d}{dx}f_{n-1}(x)\). The coefficients are defined by \(P_n(x) = \sum_{k=1}^{n}C_n(k)x^kf_k(x)\). Given the following values for \(C_n(k)\):

- \(n=1\): \(C_1(1)=1\)
- \(n=2\): \(C_2(1)=1,\ C_2(2)=1\)
- \(n=3\): \(C_3(1)=1,\ C_3(2)=3,\ C_3(3)=1\)
- \(n=4\): \(C_4(1)=1,\ C_4(2)=7,\ C_4(3)=6,\ C_4(4)=1\)
- \(n=5\): \(C_5(1)=1,\ C_5(2)=15,\ C_5(3)=25,\ C_5(4)=10,\ C_5(5)=1\)
- \(n=6\): \(C_6(1)=1,\ C_6(2)=31,\ C_6(3)=90,\ C_6(4)=65,\ C_6(5)=15,\ C_6(6)=1\)

Determine a pattern or formula for \(C_n(k)\).

Answer: C_n(k) = S(n, k)


Question: Evaluate the sum of the infinite series: \[ \frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{4}{32} + \frac{5}{64} + \cdots \]

Answer: 1


Question: Let $a, b, c$ be positive real numbers such that $a + 2b + 3c = 26$ and $a^2 + b^2 + c^2 = 52$. Find the largest possible value of $a$. Use the Cauchy-Schwarz inequality to verify your result.

Answer: \dfrac{26}{7}


Question: Let $f :\mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function and define $u: \mathbb{R}^n \rightarrow \mathbb{R}$ by $u(x) = f(r)$, where $r = \lVert x\rVert_2$, for all $x \in \mathbb{R}^n$. Calculate \( \sum _{j=1}^{n} \left(\frac{\partial u}{\partial x_j}\right)^2 \) in terms of $f$ and $r$ only.

Answer: (f'(r))^2


Question: Determine whether a smooth surjective map $f: X \to Y$ between smooth manifolds, with connected and compact fibers, is necessarily a closed map.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to \infty} \left(\frac{\ln(x^2+3x+4)}{\ln(x^2+2x+3)}\right)^{x\ln(x)} \]

Answer: \sqrt{e}


Question: Let $\beta \in \mathbb{F}_{16}$ be an element whose minimal polynomial over $\mathbb{F}_{2}$ is $x^4 + x + 1$. Determine the minimal polynomial of $\beta^7$ over $\mathbb{F}_{2}$.

Answer: x^4 + x^3 + 1


Question: Let $R$ be a principal ideal domain (PID) and $M$ a free $R$-module of finite rank $a$. If $N$ is a free $R$-submodule of $M$ with $\operatorname{rank}(N) = a$, is it true that $N = M$? Justify your answer.

Answer: No


Question: Determine whether the following statement is true or false: An integrable sub-martingale $S_t$ with $\mathbb{E}(S_t)$ being a constant is a martingale.

Answer: True


Question: Evaluate the integral using the Cauchy Principal Value: \[ \int_{-\infty}^\infty \frac{\cos(ax) - \cos(bx)}{x^2} \, dx \]

Answer: \pi(|b| - |a|)


Question: Determine whether there exists a continuous surjection from the open unit square \((0,1) \times (0,1)\) to the closed unit square \([0,1] \times [0,1]\).

Answer: Yes


Question: Compute the residue of the function \( \frac{(\sin(z))^2}{(\sinh(z))^5} \) at \( z=0 \).

Answer: -\dfrac{7}{6}


Question: Determine the smallest integer $n$ such that the alternating group $A_n$ contains an element of order 15.

Answer: 8


Question: Evaluate the integral \( \int_c \left( e^{z^2} \left( \frac{1}{z^2} - \frac{1}{z^3} \right) \right) \, dz \) where \( c \) is the unit circle.

Answer: -2\pi i


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

Answer: -1


Question: Let \( \omega \) be a complex number which satisfies the equation \( i \omega = \frac{u^2 -1}{u^2 +1} \) for some suitable complex number \( u \). Find the value of \( \tan^{-1}{\omega} \).

Answer: -i \ln u


Question: At CH Company, Joan has a secretary, Teresa, and three other administrative assistants. Seven accounts must be processed. In how many ways can Joan assign the accounts so that each assistant works on at least one account, and Teresa's work includes the most expensive account?

Answer: 2100


Question: In a Frechet space with a countable family of semi-norms \(\{P_i\}\), consider a series \(\sum_k x_k\). If \(\sum_k P_i(x_k) < \infty\) for every semi-norm \(P_i\), does the series \(\sum_k x_k\) converge in the Frechet topology?

Answer: Yes


Question: Compute the limit: $$\lim_{x\to 0} \frac{\sqrt{\cos x} - \sqrt[3]{\cos x}}{\sin^2 x}$$ without using l'Hôpital's rule.

Answer: -\dfrac{1}{12}


Question: Let $K$ be an algebraically closed field. Consider a simple module $M$ over the ring $K \times K \times \cdots \times K$ with $n$ copies of $K$. If $n \geq 2$, is the dimension of $M$ equal to 1?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\int_0^x \frac{t\,dt}{\cos t}}{\sin^2(x)} \]

Answer: \dfrac{1}{2}


Question: Determine the largest interval where the initial value problem \( t\dfrac{dy}{dt} = t + \left|y\right| \) with the initial condition \( y(-1) = 1 \) has a unique solution.

Answer: (-e, 0)


Question: Let $X$ be a Hilbert space and $T: X \to X$ be a bounded linear operator. Suppose $\lambda$ is a scalar such that $\|T\| = |\lambda|$. Prove that if there exists $x \in X$ such that $T^*x = \bar{\lambda} x$, then $Tx = \lambda x$. Use the Cauchy-Schwarz inequality and consider $\langle Tx, x \rangle$ in your proof.

Answer: Tx = \lambda x


Question:
Calculate the time derivative of the total kinetic energy, \( \frac{dT}{dt} \), for a fluid with velocity \( \underline{u} \), pressure \( p \), and constant density \( \rho \) inside a closed region with a fixed solid boundary. The velocity \( \underline{u} \) satisfies the Euler Equations:
\[ u_t + (u \cdot \nabla )u = \frac{-1}{\rho} \nabla p \]
\[\nabla \cdot u = 0 \]
Given that the total kinetic energy \( T \) is defined as:
\[ T = \frac{1}{2}\rho\iiint_Vu^2dV \]
Use vector identities and the divergence theorem to find \( \frac{dT}{dt} \).

Answer: 0


Question: Is the set of all finite subsets of the positive integers, \( \mathbb{Z}_+ \), countable?

Answer: Yes


Question: Determine the range of the function \( f(x) = |x-1| + |x-a| + |x| + |x+1| + |x+2a-21| \), where \( a \in \mathbb{R} \) and \( x \in \mathbb{R} \).

Answer: [12, \infty)


Question: In triangle \( \Delta ABC \), the altitudes \( AD \), \( BE \), and \( CF \) are drawn. Given that \( \angle FDE = 60^{\circ} \) and \( FE = 8 \text{ cm} \), find the length of side \( BC \) in centimeters.

Answer: 16


Question: Compute the remainder of the power tower \(2^{3^{4^{5^{6^{7}}}}} \mod 9\).

Answer: 8


Question: Calculate the flux of the vector field \( A = (yz, xz, xy) \) through the surface \( x+y+z=1 \) where \( x, y, z \ge 0 \), with the normal vector chosen such that \( \hat{n} \cdot e_z > 0 \).

Answer: \dfrac{1}{8}


Question: Find \( \lim_{n\to\infty}\int_{0}^{\pi}x\left|\sin 2nx\right|\,dx \).

Answer: \pi


Question: Find the polynomial that generates the kernel of the evaluation map \( \phi_a : \mathbb{R}[x] \to \mathbb{R} \), where \( \phi_a(f) = f(a) \) for a fixed \( a \in \mathbb{R} \).

Answer: x - a


Question: Consider the sequence defined by Graham's number, where $G_1 = 3 \uparrow \uparrow \uparrow \uparrow 3$, $G_2 = \underbrace{3 \uparrow \ldots \uparrow 3}_{G_1 \ \text{times}}$, and so on, up to $G_{64}$. Now, define a similar sequence starting with $H_1 = 4 \uparrow \uparrow \uparrow \uparrow 4$. At which index $n$ does $H_n$ first become greater than $G_n$?

Answer: 1


Question: Find all pairs of natural numbers \((x, y)\) that satisfy the equation \[ x \cdot 2^{x-1} + 1 = y^2. \]

Answer: (5, 9)


Question: Find all real numbers \( x \) such that the sequence \( (x_n)_{n=1}^\infty \), defined by \( x_1 = e^x \) and \( x_{n+1} = e^{x^{x_n}} \), converges to 2.

Answer: \sqrt{\ln 2}


Question: Let \( A \) be a \( 6 \times 6 \) diagonal matrix with characteristic polynomial \( x(x+1)^2(x-1)^3 \). Find the dimension of the set \( S = \{ B_{6\times 6} : AB = BA \} \).

Answer: 14


Question: Let $f$ be a continuous function on $[0,1]$. Evaluate the limit: \[ \lim_{n\to \infty}\int_{0}^{1}nx^nf(x)\,dx. \]

Answer: f(1)


Question: Given the parametric equations \( x = \cos^{-1} (8t^4 - 8t^2 + 1) \) and \( y = \sin^{-1} (3t - 4t^3) \), find \( \frac{dy}{dx} \) for \( 0 < t < \frac{1}{2} \).

Answer: \dfrac{3}{4}


Question: Does there exist a function $f: \mathbb{R}^n \to \mathbb{R}^m$ such that there is a point $x_0 \in \mathbb{R}^n$ where $\frac{\partial f_i}{\partial x_j}(x_0)$ exists for all $i,j$, but $f$ is discontinuous at $x_0$?

Answer: Yes


Question: In the acute triangle $ABC$, the altitudes $AH$ and $BN$ are drawn. The line through $H$ is extended to intersect the circumcircle of triangle $ABC$ at point $P$. Given that $	riangle AON = 30^	ext{°}$, where $O$ is the orthocenter of triangle $ABC$, find the measure of $	riangle PBN$.

Answer: 120^\circ


Question: Evaluate the double integral \( \iint_{x+y\leq1, \,x\geq0, \, y\geq0} \frac{1}{\sqrt{1-x-y}}\, dxdy \).

Answer: \dfrac{4}{3}


Question: Is it possible to generate a smooth function $f: \Bbb R \to \Bbb R$ from a given histogram $(x_1,...,x_N), x_i \in \Bbb Z$, such that $f(i)=x_i$, $f \in C^\infty$, and $f(\pm\infty)= 0$? Note: Interpolating polynomials are not suitable as they diverge at infinity.

Answer: Yes


Question: Consider the Boolean algebra of regular open sets in \( \mathbb{R}^n \), denoted as \( R(\mathbb{R}^n) \), where the operations are defined as \( U \vee V = (\overline{U \cup V})^\circ \) and \( U \wedge V = U \cap V \). Is there a countably additive measure on this algebra, specifically if we define the measure \( m(U) = \frac{1}{2}\mu(U^\circ) + \frac{1}{2}\mu(\overline{U}) \) for \( U \in R(\mathbb{R}^n) \)?

Answer: No


Question:
Given an arithmetic sequence \(\{a_n\}\) with first term \(a_1 = 9\) and common difference \(d\), where \(d \in \mathbb{N}\), determine the sum of the first 10 values of \(d\) that satisfy the following conditions:

1. For each term \(a_n\), the maximum integer \(p \ge 0\) such that \(\frac{a_n}{3^p} \in \mathbb{N}\) is defined as \(b_n\).
2. It is given that \(b_3 = 1\) and \(b_{13} = 2\).

Find \(\sum_{k=1}^{10} d_k\), where \(\{d_k\}\) is the sequence of \(d\) values in ascending order that satisfy the above conditions.

Answer: 435


Question: Let $f$ be an entire function with order of growth $\rho_0$. If $\rho < \rho_0$, does there exist a sequence $z_n$ such that for all $A, B > 0$, $f$ does not satisfy the inequality $|f(z_n)| \leq Ae^{B|z_n|^\rho}$?

Answer: Yes


Question: Evaluate the limit: \( \lim_{x \to 0^-} \frac{\mathrm{e}^{\frac{1}{x}}}{x^2} \)

Answer: 0


Question:
Given the conditional probability density function (pdf) of \( X \) given \( Y = y \):
\[
  f_{X|Y=y}(x) =
\begin{cases}
1/y^2,  & \text{for } 0 \leq x \leq y^2 \\
0, & \text{otherwise,}
\end{cases}
\]
and the marginal density of \( Y \):
\[
  f_Y(y) =
\begin{cases}
4y^3,  & \text{for } 0 \leq y \leq 1 \\
0, & \text{otherwise.}
\]
Calculate the expectation \( E(XY) \) by determining the appropriate limits and evaluating the double integral of the joint pdf.

Answer: \dfrac{2}{7}


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\ln(\tan 2x)}{\ln(\tan 3x)} \]

Answer: 1


Question: Construct a \( \varepsilon - \delta \) proof to show that \( \lim_{x \to 27} 2x^{2/3} = 18 \).

Answer: 18


Question: Consider \(\tau\) and \(\tau'\) in the upper half plane such that \(j(g \tau) = j(g \tau')\) for all \(g \in GL_2^{+}(\mathbb{Q})\), where \(j\) is the modular \(j\)-function and \(GL_2^{+}(\mathbb{Q})\) acts as Mobius transformations. Determine if \(\tau = \tau'\).

Answer: \tau = \tau'


Question: Consider a random walk where $X_1, X_2, \ldots$ are independent and identically distributed random variables with a Bernoulli$(\frac{1}{2})$ distribution, meaning $P(X_i = 0) = P(X_i = 1) = \frac{1}{2}$. Define $S_0 = 0$ and $S_n = \sum_{i=1}^n X_i$. Let $\tau_{1000} = \inf\{ n \ge 1 : S_n = 1000 \}$. Find the expected value $\mathbb{E}[\tau_{1000}]$. Provide your answer as a single numerical value or state if it is infinite.

Answer: 2000


Question:
Calculate the total variation of the function \( f(x) \) on the interval \([0, 2\pi]\), where
\[
 f(x) =
 \begin{cases}
 \sin(x), & \text{if } x \in [0, \pi) \\
 2, & \text{if } x = \pi \\
 \cos(x), & \text{if } x \in (\pi, 2\pi]
 \end{cases}
\]

Answer: 9


Question: Determine whether the following statement is equivalent to saying that $S$ is finitely generated as an $R$-algebra: "For some $n \in \mathbb{N}$, there exists a surjective ring homomorphism from $R[t_1,\dots,t_n]$ to $S$."

Answer: Yes


Question: Determine the least ordinal \( \alpha \) such that there is no order-embedding of \( \alpha \) into any bounded interval of an ordered field \( F \).

Answer: \omega_1


Question: Find the limit \( \lim_{|n|\rightarrow\infty}\cos(in) \), where \( i=\sqrt{-1} \).

Answer: \infty


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty}\sqrt{\left(1+\frac{1}{n}+\frac{1}{n^3}\right)^{5n+1}} \]

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


Question: Evaluate the integral \( \int_{-1}^{1} x^2 P_{2n-1}(x) \, dx \), where \( P_{2n-1}(x) \) is the Legendre polynomial of degree \( 2n-1 \).

Answer: 0


Question: Find the sum of the infinite series: $$\sum_{n=1}^{\infty}\frac{(7n+32)3^n}{n(n+2)4^n}$$

Answer: \dfrac{33}{2}


Question: Let $\hat{X}\rightarrow X$ be a degree 10 connected covering space where $X$ is a bouquet of 3 circles. Determine the rank of the free group $\pi_{1}(\hat{X})$. What is the rank of this free group?

Answer: 21


Question: How many 10-digit numbers (allowing the initial digit to be zero) use exactly 5 distinct digits?

Answer: 1285956000


Question: Determine whether the infinite product \( \prod_{n=1}^\infty \frac{2}{1+2^{1/2^n}} \) equals \( \ln 2 \).

Answer: \ln 2


Question: Given the conditions $x + y + z = 0$, $x + 1 > 0$, $y + 1 > 0$, and $z + 4 > 0$, find the maximum value of the expression \[ Q = \frac{x}{x+1} + \frac{y}{y+1} + \frac{z}{z+4}. \]

Answer: \dfrac{1}{3}


Question: Evaluate the integral: \[ I = \int_{-\infty}^{\infty} \frac{x^2 \, dx}{1 + x^6} \] Simplify your answer to a real number.

Answer: \dfrac{\pi}{3}


Question: How many parabolas with integer coefficients of the form \( y = ax^2 + bx + c \) can have an equilateral triangle with vertices on the intersection of the parabola and the x and y axes?

Answer: 4


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\sqrt[n]{\sum^{n}_{k=1}\left(k^{999}+\frac{1}{\sqrt{k}}\right)} \]

Answer: 1


Question: Evaluate the expression \(2016^{2016} + 2018^{2016} \pmod{2017^{2016}}\).

Answer: 2


Question: Evaluate the limit: \[ \lim_{x \rightarrow \frac{1}{4} \pi } \frac{\tan x - \cot x}{x - \frac{1}{4} \pi } \] without using L'Hôpital's rule.

Answer: 4


Question: Does there exist a function $f(x)$ such that $\int_{-\infty}^{\infty} |f(x)| \, dx < \infty$ and $f(x)$ is unbounded?

Answer: Yes


Question: Does there exist a finite, simple, undirected, connected graph $G=(V,E)$ such that there is a pair of non-adjacent vertices $\{v, w\}$ where collapsing $v$ and $w$ increases the chromatic number, but for all pairs of non-adjacent vertices $\{a, b\}$, adding an edge between $a$ and $b$ does not increase the chromatic number?

Answer: Yes


Question: Let \( f \) be a differentiable function such that \( \nabla f(1,1)=(1,-3) \) and \( \nabla f(1,0)=(-3,3) \). Define the function \( h(x,y)=f(x^2,x^2-y^2) \). Find \( ||\nabla h(1,1)|| \).

Answer: 6


Question: Let $X$ be an affine variety and $Y$ be a non-empty open subset of $X$ such that $Y \subsetneq X$. Is it true that the closure of $Y$, denoted $\overline{Y}$, is equal to $X$?

Answer: Yes


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

Answer: 0


Question:
Calculate the variance of the function \( f(X) = 100 \times \exp(x) + 100 \times \exp(2x) \) using the formula for variance:
\[
\operatorname{Var}(f(X))=[f'(EX)]^2\operatorname{Var}(X)+\frac{[f''(EX)]^2}{4}\operatorname{Var}^2(X)+\tilde{T}_3
\]
where the expected value of \( x \) is \( 0.05 \) and the standard deviation is \( 0.1 \). Assume \( x \) follows a normal distribution.

Answer: 1071.30


Question: Compute the flux of the vector field \( F = (x, y, z^4) \) across the cone defined by \( z = \sqrt{x^2 + y^2} \) for \( z \in [0, 1] \) in the downward direction.

Answer: \dfrac{\pi}{3}


Question: Is every ideal of \(\mathbb{Z}[x]\) finitely generated?

Answer: Yes


Question: Determine a tight asymptotic upper bound for the expected maximum load \( \mathbb{E}(X_n) \) when \( n \) balls are thrown into \( n \) bins. Each ball is placed in the least full of two bins chosen uniformly and independently at random, with the bins selected with pairwise independence.

Answer: O(\log \log n)


Question: Evaluate the integral \( \int_C e^{\pi(x+iy)} \, dz \), where \( C \) is the square with vertices \( 0, 1, i, 1+i \) traversed once in the counterclockwise direction.

Answer: 0


Question:
Determine the interval in which the series \(\sum_{n=0}^{\infty}(1-x)^n\) is uniformly convergent. Choose from the following options:

1. \(\frac{1}{4} \leq x \leq \frac{3}{4}\)
2. \(0 < x < 1\)

Answer: 1


Question: Determine whether \(x^3 - x + 30\) belongs to the radical of the ideal \(I = \langle 240, x^3 - x^2 \rangle\) in \(\mathbb{Z}[x]\). Specifically, does there exist \(f, g \in \mathbb{Z}[x]\) and \(n \in \mathbb{N}\) such that \(240f(x) + (x^3 - x^2)g(x) = (x^3 - x + 30)^n\)?

Answer: Yes


Question: Evaluate the integral \( \int_{S_1^+(0)}\left(1 - \frac{1}{z^2}\right)e^{z + 1/z}dz \) over the unit circle centered at the origin. Determine if the integral is zero by considering the presence of an antiderivative and the singularity at zero.

Answer: 0


Question: Let $X$ and $Y$ be two independent random variables, each uniformly distributed on the interval $[0,1]$. Define $U = \min(X,Y)$ and $V = \max(X,Y)$. Determine the joint density function of $U$ and $V$ for $0 < u < v < 1$. Provide your answer in terms of $u$ and $v$.

Answer: 2


Question: Let \( f \) be a continuous function on \( \mathbb{R} \) such that \( \int_0^{\infty} |f(x)| \, dx < +\infty \). Evaluate the limit \[ \lim_{n \to +\infty} \int_0^{\infty} \left| f\left(x + \frac{1}{n}\right) - f(x) \right| \, dx. \]

Answer: 0


Question: Simplify the expression \( \arctan 8 + \arctan 2 + \arctan \frac{2}{3} \) and express your answer in terms of \( \pi \).

Answer: \pi


Question:
Let $f$ be a continuously differentiable real-valued function on $[a,b]$ such that $|f'(x)| \leq K$ for all $x \in [a,b]$. For a partition $P=\{a=a_{0} <a_{1}<...<a_{n}=b \}$, let $U(f,P)$ and $L(f,P)$ denote the Upper and Lower Riemann Sums of $f$ with respect to $P$. Determine which one of the following statements is correct:

A. $|L(f,P)| \leq K(b-a) \leq |U(f,P)|$.
B. $U(f,P)-L(f,P) \leq K(b-a)$.
C. $U(f, P)- L(f,P) \leq K\|P\|$, where $\|P\| =\max_{0\le i\le n-1} (a_{i+1} - a_{i} )$.
D. $U(f,P)-L(f,P) \leq K\|P\|(b-a)$.

Choose the correct option.

Answer: D


Question: Evaluate the integral \( I = \int^{\infty}_0 y^{\frac{1}{2}} e^{-\frac{1}{2}y} \, dy \).

Answer: \sqrt{2\pi}


Question: Consider the set of $r \times n$ matrices with real coefficients, identified with $\mathbb{R}^{rn}$. Let $P_n$ be the space of symmetric positive semi-definite matrices, with topology induced by restriction from $\mathbb{R}^{d(d+1)/2}$. Is the map $A \mapsto A^TA$ proper?

Answer: Yes


Question: Find the last three digits of \(3^{3^{3^3}}\).

Answer: 387


Question: Given a map $A: \mathbb{R}^2 \rightarrow \mathbb{R}$ with the properties $A(-x,x) = -A(x,x)$ and $A(x+y,x) = A(x,x)+ A(y,x)$, determine if these conditions are sufficient to conclude that $A( \lambda x+y ,x ) = \lambda A(x,x)+ A(y,x)$ for any scalar $\lambda$.

Answer: No


Question: Calculate the flux of the vector field \( \bar{V} = z\bar{i} + x\bar{j} + y\bar{k} \) through the surface defined by \( x^2 + y^2 + z^2 = 1 \), \( x = 0 \), \( y = 0 \), \( z = 0 \) with \( x > 0 \), \( y > 0 \), \( z > 0 \) using Gauss's theorem.

Answer: 1


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

Answer: 3


Question: Find the area of the region in the first quadrant bounded by the parabolas $y=x^2$, $y=2x^2$, $x=y^2$, and $x=2y^2$ using a change of variables.

Answer: \dfrac{1}{12}


Question: Find the maximum value of the function \( f(x) = \int_0^1 e^{|t-x|} \, dt \) for \( 0 \leq x \leq 1 \).

Answer: e - 1


Question:
Determine the correct expression for the Fourier transform of the Fourier transform of \( f(t) \):

\[ \hat{\hat{f(t)}} = f(-t) \] or \[ \hat{\hat{f(t)}} = 2\pi f(-t) \]?

Justify your answer using the definition of the Fourier transform.

Answer: 2\pi f(-t)


Question: Find a continuous bounded function \( f: \mathbb{R} \to \mathbb{R} \) such that the expression \[ \int_{-\infty}^{+\infty}\frac{1}{\sqrt{2\pi t}}e^{-(y-x)^2/2t}f(y)dy -f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} e^{-y^2/2}(f(x+\sqrt{t}y)-f(x))dy \] does not uniformly converge to 0 as \( t \to 0^+ \).

Answer: \sin(x^2)


Question: Determine the projective dimension of the ring $A = \mathbb{Z}_l[T,T^{-1}]$, where $l$ is a prime number and $\mathbb{Z}_l$ is the ring of $l$-adic integers.

Answer: 2


Question: If a quotient group $G/H$ has a section that is a subgroup of $G$, must $G$ be a semidirect (or direct) product of $H$ and this section?

Answer: Yes


Question: Let $M$ be a connected orientable smooth manifold. Determine whether $M$ must have exactly two orientations. Justify your answer.

Answer: Yes


Question: Determine the values of \( \alpha \ge 0 \) for which the series \( \sum_{n=2}^{\infty} \frac{1}{n^\alpha \log_2(n)} \) converges.

Answer: \alpha > 1


Question: Given the constraints \(x + y \geq a\) and \(x - y \leq -1\), find the value of \(a\) such that the minimum value of \(z = x + ay\) is 7.

Answer: 3


Question: Let $f$ be an analytic function on the closed unit disk $\overline{B(0,1)}$ such that $|f(z)| < 1$ for $|z|=1$. Determine the number of solutions (counting multiplicities) to the equation $f(z) = z^n$ within the unit disk.

Answer: n


Question: Calculate the limit: \[ \lim_{x \to \infty} \frac{f^{-1} (8x) - f^{-1} (x)}{x^{\frac{1}{3}}} \] where \( f(x) = 8x^3 + 3x \).

Answer: \dfrac{1}{2}


Question: Evaluate the limit \( \lim_{x \to \infty} \frac{a^x - b^x}{a^{x+1} - b^{x+1}} \), where \( a \) and \( b \) are real constants such that \( a > b > 0 \).

Answer: \dfrac{1}{a}


Question: Evaluate the limit: \[ \lim_{n\to \infty} \sum_{k=1}^n k \sin \frac{k}{n^3} \]

Answer: \dfrac{1}{3}


Question: Calculate the line integral \( \oint_C \frac{1}{x^2 + y^2} \, dx + 2y \, dy \) where \( C \) is the circle defined by \( x^2 + y^2 = 1 \). Use parameterization to solve the integral without applying Green's Theorem.

Answer: 0


Question: Compute the path integral \( \int_{\gamma} z^2 \, dz \) where \( \gamma : [0, 1] \to \mathbb{C} \) is given by \( \gamma(t) = 2 + e^{2\pi i t} \).

Answer: 0


Question: Can all the roots of the quintic equation \( ax^5 + bx^2 + c = 0 \), where \( a, b, c \) are real numbers and \( a, c \neq 0 \), be real numbers? Provide a justification for your answer.

Answer: No


Question:
Given the center of projection at (0, 0, -5) and the view plane as the UV plane containing the center, find the perspective projection of the world coordinate point (10, -20, -10) onto the view plane using the transformation matrix:

\[
\begin{pmatrix}
1 & 0 & 0 &  0 \\
0 & 1 & 0 &  0 \\
0 & 0 & 0 &  0 \\
0 & 0 & 0.2 & 1
\end{pmatrix}
\]

Calculate the projected coordinates \((x', y', z')\) on the view plane.

Answer: (-10, 20, 0)


Question: Determine the number of positive integers \( n \leq 2020 \) that satisfy the congruence \( n^2 \equiv 444 \pmod{1000} \).

Answer: 8


Question: Let $f$ be a continuous function on $\mathbb{R}$ such that $f(0) = f(2)$. Determine whether the following statement is true or false: There exists $\alpha \in [0, 1]$ such that $f(\alpha) = f(\alpha + 1)$. Provide your answer as either "True" or "False".

Answer: True


Question: Evaluate the integral \( \int_0^{\pi/2} \sin^7(\theta) \cos^5(\theta) \, d\theta \).

Answer: \dfrac{1}{120}


Question: Do the divided powers of elements of the lattice $V$, generated by $x_1, \ldots, x_n$ in the divided-power ring $A = \mathbb{Z} \langle x_1, \ldots, x_n \rangle$, generate $A$ as an abelian group?

Answer: No


Question: Given the joint probability mass function $p(x,y)$ for $x \in \{0, 1\}$ and $y \in \{0,1,2\}$, where $3p(1,1) = p(1,2)$ and $p(1,1)$ maximizes the variance of $XY$, calculate the probability that $X$ or $Y$ is zero.

Answer: \dfrac{23}{49}


Question: Find the value of \( x \) such that \( 547^{384^{273}} \equiv x \pmod{11} \).

Answer: 4


Question: Consider the expression \((1 + x + x^2)^n = C_0 + C_1x + C_2x^2 + \cdots + C_{2n-1}x^{2n-1} + C_{2n}x^{2n}\), where \(n\) is a positive integer. Find the value of \(C_0 + C_3 + C_6 + C_9 + C_{12} + \cdots\).

Answer: 3^{n-1}


Question: Given a triangle \(\triangle PQR\) with sides 16, 30, and 34, let the incircle touch the sides of \(\triangle PQR\) at points \(X, Y,\) and \(Z\). The ratio of the area of triangle \(\triangle XYZ\) to the area of triangle \(\triangle PQR\) can be expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\).

Answer: 20


Question: Consider a connected reductive group $G$ over a field $k$, with a maximal split torus $S$ and a relative root system $\Phi = \Phi_k(S,G)$. Given a choice of positive roots $\Phi^+$, determine whether there exists a one-parameter subgroup $G_m \rightarrow S$ that acts on each positive root space as multiplication by $t^k$, where $k$ is a positive integer for each positive root space.

Answer: Yes


Question: Let $f(x) = \sin(\pi x/2)$ for $x \in \mathbb{R}$. For $y \in \mathbb{R}$, consider the sequence $\{x_{n}(y)\}_{n \geq 0}$ defined by $x_{0}(y) = y$ and $x_{n+1}(y) = f(x_{n}(y))$ for all $n \geq 0$. Define $g(y) = \lim_{n \to \infty} x_{n}(y)$. Evaluate the integral $\int_{0}^{3} g(y) \, dy$. \[2ex]

Answer: 1


Question: For any subset $S \subseteq \{1, 2, \ldots, 15\}$, a number $n$ is called an anchor for $S$ if both $n$ and $n + \#(S)$ are elements of $S$. For example, in the set $S = \{4, 7, 14\}$, the number 4 is an anchor because $4 \in S$ and $4 + \#(S) = 7 \in S$. If $S$ is randomly chosen from all $2^{15}$ subsets of $\{1, 2, \ldots, 15\}$, what is the expected value of the number of anchors of $S$?

Answer: \dfrac{13}{8}


Question:
In the boolean lattice $B_n$ of rank $n$, each edge (interval of cardinality 2) is labeled with either $\alpha$ or $\beta$. The labeling satisfies the following conditions:

1. For every maximal chain, exactly one edge is labeled by $\alpha$.
2. For every element $a \in B_n$, $a \neq \hat{1}$, at most one edge $[a,x]$ is labeled by $\alpha$, where $x$ is an atom of the interval $[a, \hat{1}]$.

Determine if there exists a coatom $y$ of $B_n$ such that the edge $[y,\hat{1}]$ is labeled by $\alpha$. Provide a justification for your answer.

Answer: Yes


Question: Find the length of the curve given by the equation \(6xy = 3 + x^4\) between \(x = 1\) and \(x = 2\).

Answer: \dfrac{17}{12}


Question: Compute the integral of the curl of the vector field \( F = (x^2+y-4, 3xy, 2xz+z^2) \) over the surface defined by \( x^2+y^2+z^2=16, z\geq 0 \).

Answer: -16\pi


Question: Determine whether the set \( \mathbb{R}^n \cap \{(x_1,...,x_n) : x_1^2 + x_2^3 + \ldots + x_n^{n+1}=1 \} \) is a differentiable submanifold of \( \mathbb{R}^n \).

Answer: Yes


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be defined as \( f(x) = \begin{cases} x^2 - 2x, & x \leq 0 \\ -x^2 - 2x, & x > 0 \end{cases} \). Find \((f^{-1})'(3)\).

Answer: -\dfrac{1}{4}


Question: Let $a < b$ and $f : (a,b) \rightarrow \mathbb{R}$ be a differentiable function with a continuous derivative $f' : (a,b) \rightarrow \mathbb{R}$. Let $x_0 \in (a,b)$, and let $(u_n)$ and $(v_n)$ be two sequences converging to zero such that $u_n > v_n$ and $u_n + x_0, v_n + x_0 \in (a,b)$. Show that \[ \lim_{n \rightarrow \infty} \frac{f(x_0 + u_n) - f(x_0 + v_n)}{u_n - v_n} = f'(x_0). \]

Answer: f'(x_0)


Question:
If $\alpha_i, i=0,1,2,...,n-1$ are the nth roots of unity, evaluate the sum $\sum_{i=0}^{n-1} \frac{\alpha_i}{3-\alpha_i}$. Choose the correct answer from the following options:

A) $\frac{n}{3^n-1}$

B) $\frac{n-1}{3^n-1}$

C) $\frac{n+1}{3^n-1}$

D) $\frac{n+2}{3^n-1}$

Answer: A


Question:
Let \( f: X \to \mathbb{C} \) be an integrable function in a measure space \((X, \mathfrak{M}, \mu)\) such that \( \int |f| \, d\mu < \infty \) and \( |f(x)| \leq 1 \) for all \( x \in X \). Compute the limit:
\[ \lim_{n\to \infty} \int \left( \frac{f^n}{1 + n |f|} \right) \, d\mu. \]

Answer: 0


Question: Let \( X \) be a random variable with the probability density function \( f(x) = 2e^{-2x+2} \) for \( x \geq 1 \) and \( f(x) = 0 \) otherwise. Determine the moment generating function \( M_X(\theta) \) for \( X \), and specify the values of \( \theta \) for which \( M_X(\theta) \) is defined. Use \( M_X(\theta) \) to calculate \( E(X^2) \).

Answer: \dfrac{5}{2}


Question: Consider a matrix with dimensions 8x10, containing numbers from 1 to 80. You randomly pick two distinct numbers to form an ordered pair. What is the expected number of picks required such that every number from 1 to 80 is used at least once in the pairs?

Answer: 199


Question: Determine whether any square matrix $A$ can be expressed as a product of triangular matrices. Specifically, can $A$ be written as $\prod_{i=1}^k B_i$, where each $B_i$ is either a lower or an upper triangular matrix for some natural number $k$?

Answer: Yes


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

Answer: 0


Question: Determine the norm of the linear functional \( L(f) = f(1) - f(0) \) for \( f \in C^0([0,1],\mathbb{R}) \) with the supremum norm \( \|\cdot\|_{\infty} \).

Answer: 2


Question: Determine the number of integers $X$ such that $X^2 \equiv X \pmod{10^n}$.

Answer: 4


Question: Given two dependent standard normal variables $X_1$ and $X_2$ with $Cov(X_1, X_2) = a$, find the covariance $Cov(X_1^2, X_2^2)$. Express your answer in terms of $a$.

Answer: 2a^2


Question: Given that $b_n$ is a decreasing and positive sequence with $\sum b_n$ converging, and $1/a_n$ is a decreasing and positive sequence with $\sum 1/a_n$ diverging, determine whether $\lim \limits_{n \to \infty} a_n b_n = 0$ must hold.

Answer: 0


Question: Find the probability density function (PDF) of the random variable \( Z = \frac{X+Y}{2} \), where \( X \) and \( Y \) are independent exponential random variables with mean 1. Use the Jacobian method to determine the PDF of \( Z \).

Answer: 4 z e^{-2 z}


Question: Given a prime number \( p = 4k + 1 \), determine whether there exists an integer \( x \) such that \( x^2 + 1 \) is a quadratic nonresidue modulo \( p \).

Answer: Yes


Question: Consider the cubic polynomial \(P(x) = x^3 - ax^2 + bx + c\). If the equation \(P(x) = 0\) has integral roots and \(P(6) = 3\), find the sum of all possible values of \(a\).

Answer: 51


Question: Determine the radius of convergence of the power series \( \sum_{k=1}^{\infty} \frac{(2k-1)^{2k-1}}{2^{2k}(2k)!}x^k \).

Answer: \dfrac{4}{e^2}


Question: Evaluate the limit: $$\lim_{(x,y) \to (0,0)}(1+yx^2)^{(1/(x^2+y^2))}.$$

Answer: 1


Question: Let $X$ and $Y$ be random variables representing the incomes of people in two cities, with probability density functions $f(x) = \frac{2}{x^3}$ for $1 < x < \infty$ and $g(y) = \frac{3}{y^4}$ for $1 < y < \infty$, respectively. Compute the probability $P(X < Y)$.

Answer: \dfrac{2}{5}


Question: Consider two functions $f$ and $g$ mapping $\mathbf{R}^n$ to $\mathbf{R}$. Suppose $f$ is differentiable at a point $c$ and $f(c) = 0$, while $g$ is continuous at $c$ but not necessarily differentiable. Can the derivative of the product $fg$ at $c$ be expressed as $f'(c)g(c)$? Justify your answer.

Answer: f'(c)g(c)


Question: Let \((f_n)\) and \((g_n)\) be sequences of nonnegative functions in \(L^1(\mathbb{R})\) such that \(f_n \to 0\) and \(g_n \to 0\) almost everywhere. Show that \(\int_A \frac{2f_n g_n}{1+f_n^2+g_n^2} \to 0\) as \(n \to \infty\), where \(A \subset \mathbb{R}\) is a set of finite measure.

Answer: 0


Question: Evaluate the line integral \( \int_C (z^2, xz, 2xy) \cdot dr \), where \( C \) is the curve formed by the intersection of the surface \( z = 1 - y^2, z \ge 0 \) and the plane \( 2x + 3z = 6 \), oriented anti-clockwise.

Answer: 4


Question: Given three independent exponential random variables with means 1, 2, and 3, calculate the expected value of the minimum of these variables, \( E(\min(X_1, X_2, X_3)) \).

Answer: \dfrac{6}{11}


Question: Find the limit \( t \) of the sequence \( \{a_n\} \) given that \( \lim_{n \to \infty}\left|a_n+3\left(1-\frac{2}{n}\right)^n\right|^{\frac{1}{n}}=\frac{3}{5} \).

Answer: -\dfrac{3}{e^2}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\int_0^1 n\ln\left(1+\frac{1}{n\sqrt{x}}\right)\, dx \]

Answer: 2


Question: Is it possible to have an open set $G$ containing all the rationals in the interval $[0,1]$ such that the Lebesgue outer measure $m^*(G) < \frac{1}{2}$?

Answer: Yes


Question: Calculate the integral of the function \( \frac{x^2}{\sin(x)} \) over the unit circle in the positive direction using the residue theorem.

Answer: 0


Question: Find the solution of the first-order partial differential equation \( u_x u_y = u \) with the initial condition \( u(x,0) = 0 \) in the domain \( x \geq 0 \) and \( y \geq 0 \).

Answer: 0


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

Answer: \dfrac{1}{e}


Question: Evaluate the limit \( \lim_{t \to 0^{+}} \sup_{x \in [0, \infty)} |e^{-t^2-2tx}f(x+t)-f(x)| \), where \( f \in C_0([0, \infty)) = \{f \in C([0, \infty)): \lim_{x \to \infty} f(x)=0\} \). Show that this limit is equal to zero.

Answer: 0


Question: Is the group of $n$-th roots of unity $G = \{x \in \mathbb{F} : x^n = 1\}$ cyclic when $\mathbb{F}$ is an infinite field?

Answer: Yes


Question: Find the kernel of the homomorphism \(\phi: \mathbb{Z} \times \mathbb{Z} \rightarrow (\mathbb{Z}/2\mathbb{Z}) \times \mathbb{Z}\) defined by \(\phi(x,y) = ([x+y]_2, 2x-y)\). Show that the kernel is isomorphic to \(\mathbb{Z}\).

Answer: \mathbb{Z}


Question: Find the function \( f(x) \) that satisfies the following conditions: \( f(x+h) = e^{x} f(h) + e^{h} f(x) \) and \( \lim_{x \rightarrow 0} \frac{f(x)}{x} = 0 \).

Answer: 0


Question: Consider the polynomial \( f(x) = x^7 - 4x^3 + x + 1 \). Determine how many of its roots lie within a unit distance from the origin in the complex plane. Note that repeated roots are counted with multiplicity.

Answer: 3


Question:
Suppose that for constant coefficients $A_n$ (where $n=1,2,3,...$), the following Fourier series holds:
\[
A_1\sin\left(\frac{\pi x}{L}\right) + A_2\sin\left(\frac{2\pi x}{L}\right) + A_3\sin\left(\frac{3\pi x}{L}\right) + \cdots = 0
\]
for $x \in (0, L)$. Can we conclude that $A_n = 0$ for all $n = 1, 2, 3, ...$?

Answer: Yes


Question: Determine the values of \( k \) for which the inequality \( (\cos A - \frac{1}{5})(\cos B - \frac{1}{5})(\cos C - \frac{1}{5}) \leq k \) holds for all triangles \( ABC \).

Answer: \dfrac{4}{125}


Question: Calculate the expected value of the determinant of the matrix \( \begin{bmatrix} x_1 & x_2 & x_3 \\ x_2 & x_4 & x_5 \\ x_3 & x_5 & x_6 \end{bmatrix} \), where \(x_1, x_2, x_3, x_4, x_5, x_6\) are independently drawn from a distribution with expected value \(m\) and variance \(\sigma^2\). Express your answer as a function of \(m\) and \(\sigma^2\).

Answer: -3 m \sigma^2


Question:
Calculate the product of all primitive n-th roots of unity for each integer \( n \geq 3 \). Specifically, evaluate the expression:
\[ \prod_{w \in G_n^*} w \]
where \( G_n^* \) denotes the set of primitive n-th roots of unity.

Answer: 1


Question: Given the basis \(B = \{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3, \mathbf{e}_4\}\) for \(\mathbb{R}^4\), where \(\mathbf{e}_1 = (1, 0, 0, 0)\), \(\mathbf{e}_2 = (1, 1, 0, 0)\), \(\mathbf{e}_3 = (1, 1, 1, 0)\), and \(\mathbf{e}_4 = (1, 1, 1, 1)\), and the dual basis \(B^\perp = \{\mathbf{e}^1, \mathbf{e}^2, \mathbf{e}^3, \mathbf{e}^4\}\), where \(\mathbf{e}^1 = (1, -1, 0, 0)\), \(\mathbf{e}^2 = (0, 1, -1, 0)\), \(\mathbf{e}^3 = (0, 0, 1, -1)\), and \(\mathbf{e}^4 = (0, 0, 0, 1)\), find the dot product \(\mathbf{u} \cdot \mathbf{v}\) where \(\mathbf{u} = \mathbf{e}^1 - \mathbf{e}^2 + \mathbf{e}^3 - \mathbf{e}^4\) and \(\mathbf{v} = (1, 2, 3, 4)\). Use the covariant components of \(\mathbf{v}\) and the contravariant components of \(\mathbf{u}\) to compute the dot product.

Answer: -5


Question: Let $H_1$ and $H_2$ be two distinct subgroups of a finite group $G$, each of order 2. Let $H$ be the smallest subgroup containing both $H_1$ and $H_2$. Is it necessary that the order of $H$ is among 2, 4, or 8? Justify your answer.

Answer: No


Question: Find the singular part of the pole at $z=0$ for the function $f(z)= \frac{\pi^2}{\sin^2(\pi z)}$. Express your answer as a function of $z$.

Answer: \dfrac{1}{z^2}


Question: Determine the limit of the sequence \( \frac{1}{e^n/n^n - 2(n!)/n^n} \) as \( n \to \infty \).

Answer: -\infty


Question: Determine the order of the pole of the function \( \frac{\sin(x^2)}{(x-7)^9(x^5+x+1)^3} \) at \( x = 7 \).

Answer: 9


Question: Given the parabola with a focus at \((-1, -1)\) and a tangent line at the point \((7, 13)\) described by the equation \(y = 3x - 8\), find the length of the latus rectum of the parabola.

Answer: \dfrac{4\sqrt{65}}{13}


Question: Let V be the vector space of all real polynomials p(x) of degree ≤ n. Define the linear transformation T: V → V by T(p) = q, where q(t) = p(t+1). Determine the eigenvalues and the eigenvectors of the transformation T.

Answer: 1


Question: Let $m$ be the inverse function of $h(x) = 3x + \cos(2x)$. Find $m'\left(\frac{3\pi}{4}\right)$. Use the formula for the derivative of an inverse function, $m'(y) = \frac{1}{h'(m(y))}$, to solve the problem.

Answer: 1


Question: Consider Riemannian manifolds $(M,g)$ and $(N,g')$ with an isometry $f: M \to N$. Let $X, Y, Z$ be smooth vector fields on $M$. Determine if the following statement is true: \[ X\langle Y, Z\rangle_p = df_p(X)\langle df_p(Y), df_p(Z)\rangle_{f(p)} \] for any point $p \in M$. Provide a justification for your answer.

Answer: True


Question: Solve the polar Laplace equation with the initial condition \( u(a,\theta) = 0 \).

Answer: 0


Question: Calculate the rate of heat flow inward across the cylindrical surface defined by \( y^2 + z^2 = 324 \) and \( -10 \le x \le 3 \), given the temperature function \( u(x,y,z) = 4y^2 + 4z^2 \) and the conductivity constant \( K = 5 \). Use the formula \( \iint_S F \cdot dS = -K \iint_S \nabla u \cdot dS \) to find the solution.

Answer: 336960\pi


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

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


Question: Find the limit of the sequence \( \lim_{n\rightarrow \infty}{\sqrt[n]{1^n+2^n+\ldots+n^n}} \) using the sandwich rule.

Answer: \infty


Question: Find the number of pairs of natural numbers \((m, n)\) such that \(m + n = \text{gcd}(m, n) + \text{lcm}(m, n) = 9!\).

Answer: 317


Question: Calculate \( \int_C (1-4x) \, dx + 2x \, dy + (1-5y) \, dz \) using Stokes' Theorem, where \( C \) is the intersection curve between the surfaces \( z = 3x \) and \( x^2+y^2 = 1 \), oriented counterclockwise as seen from above.

Answer: 17\pi


Question: Find the sum of the digits of the number \( a \) such that \( 101a = 6539 \int_{-1}^1 \frac{x^{12} + 31}{1 + 2012^x} \, dx \).

Answer: 5


Question: Given $n$ independent and identically distributed random variables $X_1, X_2, \dots, X_n$, what is the probability that $X_1 < X_2 < \cdots < X_n$?

Answer: \dfrac{1}{n!}


Question: Let \( R \) be a Noetherian domain and \( t \in R \) be a non-zero, non-unit element. Determine whether the following statement is true: \( \bigcap_{n \ge 1} t^nR = \{0\} \).

Answer: True


Question: Evaluate the definite integral: $$\int_a^b \frac{dx}{\sqrt{(x-a)(b-x)}}$$

Answer: \pi


Question: Evaluate the generalized hypergeometric function $_1F_2(a_1;b_1,b_2;z)$ when $a_1 = 0$. What is the result?

Answer: 1


Question: Determine the radius of convergence of the series \( \sum_{n=1}^\infty \frac{2^n x^{n!}}{n} \).

Answer: 1


Question: Given the equations \(a+b+c = 0\), \(a^3 + b^3 + c^3 = 12\), and \(a^5 + b^5 + c^5 = 40\), find the value of \(a^4 + b^4 + c^4\).

Answer: 8


Question: Compute the limit using L'Hospital's rule: \[ \lim_{x\rightarrow 0^+} \left(x \int ^1_x \frac{\cos t }{t^2} \, dt\right) \]

Answer: 1


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

Answer: 1


Question: Given that the function \(f(x) = e^x + x\) is one-to-one, find \((f^{-1})(e+1)\).

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


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

Answer: \sin 1


Question: Let $(R, \mathfrak{m})$ be a Noetherian local ring of dimension $d > 0$. The Hilbert-Samuel multiplicity of $R$ is given by \( e(R) = (d-1)! \lim_{n \to \infty} \frac{\mu(\mathfrak{m}^n)}{n^{d-1}} \). Is it true that $e(R) > 0$ for any such ring?

Answer: Yes


Question:
Calculate the determinant of the matrix \( \begin{vmatrix} n & 0 & 0 & \ldots & 0 & 0 \\
0 & 0 & 2 & \ldots& 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & \ldots & 0  &n-1 \\
0 &1 & 0 &\ldots &  0 & 0  \end{vmatrix} \).

Answer: (-1)^n n!


Question: If $H_1$ and $H_2$ are two semi-positive definite Hermitian matrices, $U_1$ and $U_2$ are two unitary matrices, and $A$ is a square matrix, determine if the equation $H_1AH_2=U_1AU_2$ implies that $H_1AH_2=A$. Is this statement true?

Answer: True


Question: Evaluate the integral: \[ \int_{-2\pi}^{2\pi}(1−u_0(x))\sin(x/2)(\delta(x + \pi) + \delta(x−\pi))\mathrm{d}x \]

Answer: -1


Question: Identify the group given by the presentation \( \langle a, b, c, d, e \mid cde = 1 \rangle \).

Answer: F_4


Question: In a linear regression model given by \( Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i \) for \( i = 1, \dots, n \), where \( \hat \beta_0 \) and \( \hat \beta_1 \) are the least squares estimates of \( \beta_0 \) and \( \beta_1 \), respectively, calculate the expected value \( E[(\hat \beta_1 - \beta_1)X_1] \).

Answer: 0


Question: Evaluate the integral \( f(n) = \int_0^\infty t^{n-1} e^{-t} \, dt \) in terms of \( n \).

Answer: \Gamma(n)


Question:
Determine the value of $k$ such that the highest temperature $T(x,y,z) = kxyz^2$ on the surface of the sphere $x^2 + y^2 + z^2 = a^2$ is $2a^4$. Choose the correct answer from the following options:
(a) 8
(b) 12
(c) 16
(d) 32

Answer: C


Question: Let $p, q, r \in [1, \infty)$ with $p < q$. Suppose $L^p(\mathbb{R}) \cap L^q(\mathbb{R}) \subseteq L^r(\mathbb{R})$. Is it true that $r \in [p, q]$?

Answer: Yes


Question: Compute the volume of the solid \( E \) where the points \((x,y,z)\) of this solid projected on the plane \( z=0 \) are in the parallelogram with vertices \((2,1),(6,-1),(7,0),(3,2)\) and \( 0 \le z \le e^{x+2y} \). Verify if the integral \( \int_{2}^{7} \int_{-0.5x+2}^{-0.5x+3.5} \int_{0}^{e^{x+2y}} dz \, dy \, dx \) correctly computes this volume. The result given in the book is \( 2(e^7-e^4) \).

Answer: 2(e^7 - e^4)


Question: Provide an example where \(E\left[\lim_{n \to \infty}X_n\right] \neq \lim_{n \to \infty}E\left[X_n\right]\), with \(E\) representing expectation and \(X_n\) being a random variable. Explain why this discrepancy occurs.

Answer: E\left[\lim_{n \to \infty}X_n\right] \neq \lim_{n \to \infty}E\left[X_n\right]


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

Answer: -\pi


Question: Consider the linear transformation \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^2 \) defined by \( T\left(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\right) = \begin{bmatrix} xy \\ xz \end{bmatrix} \). Determine the dimension of the kernel of \( T \).

Answer: 2


Question: Given a countable collection \( \{ A_i \} \) of subsets of \([0,1]\), can the collection \( \{ \cap_{i \in J} A_i \}_{J \subset \mathbb{N}, |J| = \infty} \) be uncountably infinite?

Answer: Yes


Question: Assume that a function $f:[0,1]\to [0,1]$ is Hölder continuous with exponent $1/3$. For every $s\in [0,1]$, it holds that \(\limsup_{x\to s}\frac{\lvert f(x)-f(s)\rvert}{\lvert x-s\rvert^{1/2}}<\infty.\) Does this imply that \(\sup_{x\neq s}\frac{\lvert f(x)-f(s)\rvert}{\lvert x-s\rvert^{1/2}}<\infty?\)

Answer: Yes


Question: Determine the number of field extensions between \( \mathbb{Q}(\zeta^3) \) and \( \mathbb{Q}(\zeta) \), where \( \zeta \) is a primitive 12th root of unity.

Answer: 2


Question: Let \( G \) be the Galois group of a field with 9 elements over its subfield with 3 elements. Determine the number of orbits for the action of \( G \) on the field with 9 elements.

Answer: 6


Question: Given two random variables $X$ and $Y$ with a correlation of $\rho = 1$, both having a mean of $100$ and a standard deviation of $40$, and two other random variables $U$ and $V$ with a correlation of $\rho = 0.8$, both having a mean of $0$ and a standard deviation of $20$, compute the correlation of $A = X + U$ and $B = Y + V$. Assume $\text{cor}(X,U) = \text{cor}(X,V) = \text{cor}(Y,U) = \text{cor}(Y,V) = 0$. Find $\text{cor}(A,B)$. \( \text{cor}(A,B) = ? \)

Answer: 0.96


Question: Find the value of \( p \) that satisfies the equation: \[ \frac{d}{dp}\left[2\int_0^1(p-y)^2{y^{x-1}(1-y)^{-x}}dy\right]=0 \]

Answer: x


Question: Compute the integral \( \int_{-\infty}^\infty e^{-x^2/2}\,dx \) using the Gamma function.

Answer: \sqrt{2\pi}


Question: Let $T_1$ and $T_2$ be two unitary operators. Determine whether the spectrum of $T_1 + T_2$ is contained within the closed disc of radius 2.

Answer: Yes


Question: Let \( A \) be a diagonal matrix with the characteristic polynomial \( P(x) = (x-15)(x-14)^2(x-13)^3\ldots(x-2)^{14}(x-1)^{15} \). Determine the dimension of the set \( V \) of all \( 120 \times 120 \) matrices that commute with \( A \).

Answer: 1240


Question: Find the minimum value of the function \( \frac{\sin 5x}{\sin^5 x} \).

Answer: -4


Question: Find the Laplace transform of the derivative of the Dirac delta function, \( \mathscr{L}\{\delta'(t-1)\} \).

Answer: s e^{-s}


Question: Evaluate the limit \( \lim_{t \rightarrow 0} \phi(t) \) for the matrix function \( \phi(t) = \frac{1}{t}(e^{tA} - I) \), where \( A \in \mathbb{R}^{n \times n} \).

Answer: A


Question: Evaluate the limit: \[ \lim_{x\to \infty} \left(\sqrt[3]{x^{3}+3x^{2}}-\sqrt{x^{2}-2x}\right) \] without using L'Hopital's rule or Taylor series.

Answer: 2


Question: Find the coordinates of the closest point on the cylinder defined by the equation \(y^2 + z^2 = 25\) to the point \((2,9,12)\) using Lagrange multipliers.

Answer: (2, 3, 4)


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

Answer: e^2


Question: Let $f:X\rightarrow Y$ be a morphism of normal projective varieties. Let $S\subseteq X$ be a surface admitting a morphism $g:S\rightarrow C$ to a curve $C$ such that any fiber of $g$ is a curve. Assume that the general fiber of $g$ is contracted to a point by $f$. Is it true that any fiber of $g$ is contracted by $f$?

Answer: Yes


Question: Evaluate the double sum: \[ \sum_{n = 2}^{\infty} \sum_{m = 2}^{\infty} m^{-n} \]

Answer: 1


Question: Let $U$ be an orthogonal matrix with real spectra. Given that the spectrum of $U$ is $\{-1,1\}$, determine if $U^{2}$ is similar to the identity matrix.

Answer: Yes


Question: Given that \(p+q+r=1\), \(p^2+q^2+r^2=2\), and \(p^3+q^3+r^3=3\), find the value of \(p^4+q^4+r^4\).

Answer: \dfrac{25}{6}


Question: Determine the value of \( \lim_{n\to\infty} \frac{f(n)}{10^n} \), where \( f(n) \) denotes the first \( n \) digits of the decimal expansion of \( (10^n-1)^{10^n-1} \).

Answer: \dfrac{1}{e}


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

Answer: e^{-8}


Question: Compute \((101^{24} \times 97^{25}) \mod 493\).

Answer: 345


Question: Evaluate the second-order partial derivative \( \frac{\partial^2}{\partial \bar{z}\,\partial z}\log|z|^2 \) for the function \( z \mapsto \log|z|^2 \) defined on \( \mathbb{C} - \{0\} \).

Answer: 0


Question: Given constants \(a\) and \(b\) such that \(a \ge b > 0\), define the sequence \(\{a_n\}\) by \(a_1 = a+b\) and \(a_n = a_1 - \frac{ab}{a_{n-1}}\) for \(n \ge 2\). Determine \(\lim_{n\to\infty}{a_n}\).

Answer: a


Question: Can a simple abelian variety have a Picard number of \(n^2-1\)?

Answer: Yes


Question: Consider an open complete manifold $M$ with Ricci curvature $\ge 0$. Let $p$ be a fixed point in $M$ and $B(p, r)$ be the distance ball of radius $r$ in $M$. For any given $R>0$, determine whether there exists a constant $c=c(p,R)>0$ such that $\text{Area}(\partial B(p, r)) \ge c(p,R)$ for any $r>R$. Is this statement true?

Answer: Yes


Question: Let $X$ be a topological space such that $X \times \mathbb{R}$ is diffeomorphic to $S^n \times \mathbb{R}$, where $S^n$ is the $n$-sphere. Determine if $X$ is homeomorphic to $S^n$. Provide a justification for your answer.

Answer: Yes


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

Answer: 1


Question: Determine the number of $\sigma$-algebras that can be formed on a nonempty set $X$ with $m$ members.

Answer: B_m


Question: Let $A$ be a local ring with maximal ideal $\mathcal{m}$. Is every $A$-module homomorphism from $\mathcal{m}$ to $A/\mathcal{m}$ equal to zero?

Answer: No


Question: Find \( \cos(z) \) given that \( z = i\ln(2-\sqrt{3}) \).

Answer: 2


Question: In triangle \(ABC\), angle \(C\) is \(30^\circ\). The triangle is inscribed in a circle with center \(O\) and radius 9 cm. Determine the radius \(R\) of the circle that is tangent to segments \(AO\) and \(BO\), as well as the arc \(AB\).

Answer: 3


Question: Given that \( a + b = k\pi \), determine the value of \( k \) if the parameter \( t \) is in the interval \( (0, 2\pi) \) such that the inequality \( \frac{-2x^2 + 5x - 10}{(\sin t) x^2 + 2(1+ \sin t )x + 9\sin t + 4} > 0 \) holds for all real values of \( x \). The range of \( t \) is \((a, b)\) and \( a + b = k\pi \). Find \( k \).

Answer: 3


Question: Determine the degree of the field extension \( [ \mathbb{Q} (\sqrt{2}, e^{2\pi i/3}) : \mathbb{Q}] \).

Answer: 4


Question: Evaluate the series \( \sum_{n=1}^{\infty} \frac{1}{n^5} \) up to the second decimal place.

Answer: 1.04


Question: Find a generator for the ideal $I = \{a + bi \in \mathbb{Z}[i] \mid a \equiv b \pmod{2}\}$ in the ring $R = \mathbb{Z}[i]$. Provide your answer as a single complex number.

Answer: 1 + i


Question: Given a differentiable function \( f \) on the reals with \( f(1) = 1 \), \( f(2) = 3 \), and \( f(3) = 3 \), determine if there necessarily exists a point \( c \) in the interval \((1,3)\) such that \( f'(c) = \frac{1}{2} \).

Answer: Yes


Question: Evaluate the line integral \( \oint_{C} -y^3dx+x^3dy+z^3dz \) using Stokes' Theorem, where \( C \) is the intersection of the cylinder \( x^2 + y^2 = 1 \) and the plane \( x+y+z=1 \). The orientation of \( C \) is counter-clockwise in the \( xy \)-plane.

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


Question: Find the cardinality of the conjugacy class of the permutation \(\sigma = (1,3,5)(2,9)(4,12)(6,11)(7,10,8)\) in the symmetric group \(S_{12}\).

Answer: 554400


Question: Consider the partial differential equation \( u_t = i u_x \) where \( u(x,t):\mathbb{R}^2\to\mathbb{C} \) is a \( C^1 \) function. Are there any non-trivial solutions for which \( \int_\mathbb{R} |u(x,t)|^2 \, dx < \infty \) and \( \frac{d}{dt} \int_\mathbb{R} |u(x,t)|^2 \, dx = 0 \)?

Answer: No


Question: Find the volume of the region bounded between the surfaces \( z = (x^2 + y^2)^2 \) and \( z = x \). Use cylindrical coordinates to set up the integral.

Answer: \dfrac{\pi}{12}


Question: Given the polynomials \(f(x) = x^5 + x^4 + 4x^3 + 3x^2 + 3x\), \(g(x) = x^5 + 4x^3 + 2x^2 + 3x + 6\), and \(p(x) = x^2 + 3\), determine the valuation \(v_p(fg)\) of the product \(fg\) with respect to \(p\).

Answer: 2


Question: Determine the value of \(s\) in the series solution \(y = \sum_{n=0}^{\infty}a_nx^{n+s}\) for the differential equation \(y''+\frac{y'}{x^2}-\frac{a^2}{x^2}y = 0\) using the Frobenius method, where \(a \ge 0\). Verify if the relation \(x^{n+s-3}(a_{n-1}(n-1+s)(n+s-2) + a_n(n+s) + a_{n-1}) = 0\) is correct.

Answer: 0


Question: Calculate the work done by the force field \( F(x, y, z) = \langle 3x^2+yz+y, xz+x+z^2, xy+2yz+2z \rangle \) on a particle moving along the path \( C \), which consists of the line segment from \((2, 3, 0)\) to \((5, 7, 9)\) followed by the line segment from \((5, 7, 9)\) to \((1, 1, 2)\).

Answer: -2


Question: Let $X$ and $Y$ be independent and identically distributed random variables, each following a uniform distribution on the interval $(0,1)$. Define $Z = \frac{X}{Y}$. Determine the joint probability density function $f_{XZ}(x,z)$.

Answer: \dfrac{x}{z^2}


Question: Given that the differential equation \( t^2 y'' - 2y' + (3 + t)y = 0 \) has \( y_1 \) and \( y_2 \) as a fundamental set of solutions, and \( W(y_1, y_2)(2) = 3 \), find \( W(y_1, y_2)(4) \).

Answer: 3\sqrt{e}


Question: Consider the differential equation \( m \ddot{x} + c \dot{x} = - \nabla f(x) \), where \( m > 0 \) and \( c \geq 0 \). Given that \( f \) is a convex function with compact level sets, determine whether the solution \( x(t) \) remains bounded.

Answer: Yes


Question:
Evaluate the limit:
\[ \lim_{x\to 1}\left(\frac{x^{x^{x{^{.^{.^{.}}}}}}-1}{x-1}\right)^{\displaystyle\frac{1}{\sqrt{x-1}}} \] where the exponentiation is iterated n times.

Answer: 1


Question: Determine whether the stochastic process \(X_t=\int_0^t \left(a+b\frac{u}{t}\right)\,dW_u\), which has variance \(t\) and is normally distributed for \(a^2+\frac{b^2}{3}+ab=1\), is continuous at \(t=0\). Consider the case where \(b=0\).

Answer: Yes


Question:
Let $E$ be an infinite-dimensional complex Hilbert space with inner product $\langle\cdot\;| \;\cdot\rangle$ and norm $\|\cdot\|$. Let $\mathcal{L}(E)$ be the algebra of all bounded linear operators on $E$. Consider $M \in \mathcal{L}(E)^+$, meaning $M^* = M$ and $\langle Mx\;, \;x\rangle \geq 0$ for all $x \in E$. Assume $M$ is an injective operator on $E$, so $x \longmapsto \langle Mx\;, \;x\rangle$ is a norm on $E$.

Define the sets:
$$S(0,1) = \{x \in E;\;\|x\| = 1\}$$
and
$$S_M(0,1) = \{x \in E;\;\langle Mx\;, \;x\rangle = 1\}$$

Is $S_M(0,1)$ always homeomorphic to $S(0,1)$ even if $E$ is an infinite-dimensional complex Hilbert space?

Answer: Yes


Question: Determine the highest order of derivative of the function \( f(x) = x^5\sin\left(\frac{1}{x}\right) \) at \( x = 0 \). The function is defined as \( f(x) = x^5\sin\left(\frac{1}{x}\right) \) for \( x \neq 0 \) and \( f(x) = 0 \) for \( x = 0 \).

Answer: 2


Question: Consider the system of differential equations given by \( \dot x = -y \) and \( \dot y = x \). Using Euler's method with a time-step \( h = 1/n \), starting from the initial point \( (x_0, y_0) = (1, 0) \), find the limit \( \lim\limits_{n \to \infty} (x_n^n, y_n^n) \).

Answer: (\cos 1, \sin 1)


Question: Determine the minimum number of throws required to ensure that no boxes are empty when distributing balls into boxes.

Answer: n


Question: Let $f: \mathbb{R} \to [0,1]$ be a Lipschitz continuous function with $f(0) = 0$. Assume there does not exist $\delta > 0$ such that $f(x) = 0$ for all $x \in [0, \delta]$. Determine whether there must exist a strictly decreasing sequence $x_n \to 0$ such that $f(x_n) > 0$ and $f'(x_n) > 0$. Provide a justification for your conclusion.

Answer: Yes


Question: Compute the triple integral \( \iiint_V (z+2)\, dx\, dy\, dz \) where \( V \) is the region bounded by the planes \( z=1 \), \( z=3 \), and the paraboloid \( \frac{x^2}{25}+\frac{y^2}{9}=z \).

Answer: 250\pi


Question:
Given the conditions \( x + y + z = 1 \), \( x^2 + y^2 + z^2 = 9 \), and \( x^3 + y^3 + z^3 = 1 \), calculate the value of \( \frac{4}{{x^4} + {y^4} + {z^4}} \). Choose the correct answer from the following options:

A) \( \frac{1}{33} \)
B) \( \frac{2}{33} \)
C) \( \frac{4}{33} \)
D) \( \frac{16}{33} \)
E) \( \frac{64}{33} \)

Answer: C


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

Answer: Yes


Question: Let $X = \text{Spec}(A)$ be an affine scheme and $U = \text{Spec}(R)$ be an affine open subset of $X$. Is it true that $R$ is a localization of $A$, i.e., $R = S^{-1}A$ for some multiplicative subset $S \subset A$?

Answer: Yes


Question: Given $n \geq 1$ permutations of the string $abcd$, construct bipartite graphs $G_{a,b}$, $G_{b,c}$, $G_{c,d}$, and $G_{d,a}$ as follows: For $G_{a,b}$, the $n$ vertices on one side are labeled with the sets containing $a$ and the letters after it in each permutation, and the $n$ vertices on the other side are labeled with $b$ and the letters before it in each permutation. There is an edge between two vertices if their sets overlap. Construct $G_{b,c}$, $G_{c,d}$, and $G_{d,a}$ similarly. Is it true that among $G_{a,b}$, $G_{b,c}$, $G_{c,d}$, and $G_{d,a}$, at least one must have a perfect matching?

Answer: Yes


Question: How many natural numbers $n$ not greater than $10000$ are there such that $2^n - n^2$ is divisible by $7$?

Answer: 2858


Question: In triangle ABC, AB = 10 cm and AC = 12 cm. The incenter (I) and the centroid (B) lie on the same line parallel to side BC. Determine the length of side BC.

Answer: 11


Question: Find the maximal value of the sum \( \sum_{n=0}^{\infty} a_n \), where \( a_n \) is the distance from a real number \( a \) to the nearest number of the form \( \frac{m}{2^n} \) with \( m \in \mathbb{Z} \).

Answer: \dfrac{2}{3}


Question: Let $k$ be a field and $R$ be the exterior ring over $k^d$, which is the $k$-algebra generated by elements $x_1, \ldots, x_d$ satisfying $x_ix_j = -x_jx_i$. Is the ring $R$ Artinian?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{(x,y)\rightarrow (0,0)} \frac{(xy)^3}{x^2+y^6} \]

Answer: 0


Question: Let \( f \) be a continuous function satisfying \( f(x+y) = f(x) + f(y) + f(x) \cdot f(y) \) for all real numbers \( x \) and \( y \), and \( f'(0) = -1 \). Find the value of \( \int_0^1 f(x) \, dx \).

Answer: -\dfrac{1}{e}


Question: Determine the radius of convergence of the power series \( f(x) = \sum_{k=1}^{\infty} \frac{x^{2k}}{(4+(-1)^k)^{3k}} \).

Answer: 3\sqrt{3}


Question: Let $B$ be a base that is greater than $50^{50}$, and let $N = (11_B)^{50}$. Determine the third digit (from the right) of $N$ when expressed in base $B$. Express this digit as a base-10 number.

Answer: 1225


Question:
Let \( f: [0,1] \to [0,\infty) \) be a continuous function such that \((f(t))^2 < 1 + 2\int_{0}^{t} f(s) \, ds\) for all \( t \in [0,1] \). Determine which of the following statements is true:

(a) \( f(t) = 1 + t \) for all \( t \in [0,1] \)
(b) \( f(t) < 1 + t/2 \) for all \( t \in [0,1] \)
(c) \( f(t) > 1 + t \) for all \( t \in [0,1] \)
(d) \( f(t) < 1 + t \) for all \( t \in [0,1] \)

Answer: D


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{(2n+1)!!}{2^nn!(2n-1)} = 0. \]

Answer: 0


Question: Determine the number of real-valued solutions for the equation \(2^x + 2^{-x} = 2 - (x - 2)^2\).

Answer: 0


Question: Determine whether it is possible to partition the set of real numbers \( \mathbb{R} \) into uncountably many subsets \( \{S_i: i \in I\} \) such that each subset \( S_i \) is countably infinite and if \( w \) is a limit point of \( S_i \), then \( w \notin S_i \).

Answer: Yes


Question: Is the additive group of $p$-adic numbers, $Q_p$, residually finite? A group $G$ is said to be residually finite if the intersection of all subgroups of finite index is the trivial subgroup.

Answer: No


Question: Using Chebyshev's Inequality, show that for a fair die thrown $n$ times, the probability that the number of sixes lies between $\frac{1}{6}n - \sqrt{n}$ and $\frac{1}{6}n + \sqrt{n}$ is at least $\frac{31}{36}$. Assume $X$ is the random variable representing the number of sixes thrown.

Answer: \dfrac{31}{36}


Question: Is it possible for two topological spaces on the real numbers, \((\mathbb{R}, \mathcal{T})\) and \((\mathbb{R}, \mathcal{T}')\), to be homeomorphic if their topologies \(\mathcal{T}\) and \(\mathcal{T}'\) are different?

Answer: Yes


Question: Suppose $V$ is a Hilbert space and $T: V \rightarrow V$ is a linear map. Given that there exists a constant $C$ such that \(|\langle T(v), w \rangle| \leq C\|v\| \|w\|\) for all $v, w \in V$, is $T$ necessarily a bounded operator?

Answer: Yes


Question: Let $k$ be an algebraically closed field, $O = k[[\pi]]$, and $F = k((\pi))$. Consider a faithful representation $G \rightarrow GL_n$ of a semisimple group. Suppose $A, B \in G(O) \cap G(F)^{rs}$, where $rs$ denotes regular semisimple elements, and there exists $g \in GL_n(O)$ such that $gAg^{-1} = B$. Does there exist an element $g_1 \in G(O)$ such that $g_1Ag_1^{-1} = B$?

Answer: Yes


Question: Calculate the surface area of the cone defined by \( z^2 = x^2 + y^2 \) that is bounded above by the disc \( x^2 + y^2 \leq 4 \).

Answer: 4\sqrt{2}\pi


Question: Calculate the limit: \[ \lim_{z \to 0} \left( \frac{\Re(z^3)}{|z|^2} \right) \] where \( z \in \mathbb{C} \).

Answer: 0


Question: Find the number $n$ that satisfies the following conditions: $n$ is close to $10000$, ends in $101$ in binary, ends in $2$ in both base $7$ and base $11$, and ends in $21$ in base $3$. What is this number?

Answer: 9781


Question: Given a graph where each node has at most $e$ edges and the shortest distance between any two nodes is at most $d$, determine the maximum number of nodes $M$ the graph can have. For instance, if $(e,d) = (2,6)$, then $M = 13$. Find $M$ for the case where $(e,d) = (3,2)$. Provide your answer as a single number.

Answer: 10


Question: Let \( f \in L^p(\mathbb{R}) \) and \( E_n \) be measurable sets of finite measure such that \( \lim_{n \to \infty} m(E_n) = 0 \). Define \( g_n = m(E_n)^{-1/q} \chi_{E_n} \), where \( 1 < p, q < \infty \) and \( \frac{1}{p} + \frac{1}{q} = 1 \). Show that \( \lim_{n \to \infty} \int f g_n = 0 \).

Answer: 0


Question: Evaluate the limit: \[ \lim \limits_{\theta \to \frac{\pi}{2}} \tan^2(\theta)[1-\sin(\theta)] \]

Answer: \dfrac{1}{2}


Question:
Let $n \geq 3$ be an integer. For a permutation $P = (a_1, a_2, \ldots, a_n)$ of $(1, 2, \ldots, n)$, define $f_P(x) = a_n x^{n-1} + a_{n-1} x^{n-2} + \cdots + a_2 x + a_1$. Let $S_P$ be the sum of the roots of $f_P(x) = 0$, and let $S$ denote the sum over all permutations $P$ of $(1, 2, \ldots, n)$ of the number $S_P$. Determine which of the following is true:

A) $S < -n!$
B) $-n! < S < 0$
C) $0 < S < n!$
D) $n! < S$

Answer: A


Question: Find the maximum value of the function \( f(x) = \tan^{-1}\left(\dfrac{(\sqrt{12}-2)x^2}{x^4+2x^2+3}\right) \).

Answer: \dfrac{\pi}{12}


Question: What is the smallest dimension $d$ such that a simple random walk on $\mathbb{Z}^d$ is transient?

Answer: 3


Question: Determine the set of integers \( x \) such that \( \frac{7x+2}{x+5} \) is a perfect cube.

Answer: -38


Question:
For which value of \( a \) does the following linear programming problem have an optimal solution?

\[\begin{align}
\max \quad & 2x_{1}+6x_{2}+3x_{3}, & \\
\text{s.t.} \quad & -3x_{2}+a x_{3} \geq2, \\
& x_{1}+5x_{2}+2x_{3} =2, \\
& x_{1},x_{2},x_{3} \geq 0
\end{align}\]

Answer: a \geq 2


Question: Evaluate the integral \( \int_{-1}^{1} e^{-5x} \delta'(x) \, dx \).

Answer: 5


Question: Evaluate the complex line integral \( \int_{C} \frac{1}{2z-1}dz \) where \( C \) is the unit circle oriented counter-clockwise. Verify if the method of using the parameterization \( z(t) = e^{it} \) for \( 0 \leq t \leq 2\pi \) and the principal branch of the logarithm is correct, leading to the result \( i\pi \).

Answer: i\pi


Question: Consider a triangle $ABC$ where $M$ is the midpoint of side $BC$. Given that $\angle MAC = \angle ABC$ and $\angle BAM = 105^{\circ}$, find the measure of $\angle ABC$. Solve the equation $$(4+\sqrt{3})\cos2x = 2 + \sin2x$$ to find the value of $x = \angle ABC$. Provide your answer in degrees.

Answer: 30


Question: A mouse enters a series of rooms, each time having a probability of \( \frac{1}{2} \) to go right or left. The mouse forgets its previous path each time it enters a new room, allowing it to potentially walk indefinitely. However, if the mouse enters a room with a cat, it will be eaten and stop moving. Calculate the expected number of rooms the mouse enters before being eaten by the cat.

Answer: 2


Question: Find the closest point on the parabola given by the equation \(2x^2 - 4xy + 2y^2 - x - y = 0\) to the line \(9x - 7y + 16 = 0\). Use the distance formula \(d = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}\) to determine the distance from a point \((x_0, y_0)\) to the line \(ax + by + c = 0\).

Answer: (3, 5)


Question: Is $577^3$ the smallest cube that can be expressed as the sum of three positive cubes in six different ways? Verify if $577^3$ satisfies this condition.

Answer: 577^3


Question: Given a Noetherian domain $A$ of dimension 1 and a non-zero prime ideal $p$, determine the maximal ideal of the Noetherian local domain $A/p$.

Answer: (0)


Question: Consider a polynomial \( P_n(x) \) of degree \( n > 0 \). Determine if the equation \( P_n(x) = e^{ax} \), where \( a \neq 0 \), can have at most \( n+1 \) real solutions. Justify your answer.

Answer: n+1


Question: Find the locus of the foot of the perpendicular drawn from the focus to a variable tangent of the parabola \((2x-y+1)^2=\frac{8}{\sqrt{5}}(x+2y+3)\).

Answer: x + 2y + 3 = 0


Question:
Given a matrix \( \mathbf{X} \in \mathbb{C}^{m \times 2} \) with full column rank, determine the matrix that completes the equality:
\[ \mathbf{I}_{m} - \mathbf{X} \left( \mathbf{X}^{*} \mathbf{X} \right)^{-1} \mathbf{X}^{*} = \mathbf{P} \]
where \( \mathbf{P} \) is the orthogonal projection matrix onto the orthogonal complement of the column space of \( \mathbf{X} \). Choose the correct option:

(a) \( \mathbf{I}_{m} \)
(b) \( \mathbf{0} \)
(c) \( \mathbf{M} \)
(d) None of the options will work.

Answer: D


Question: For a Brownian bridge defined by \( X_t = B_t - tB_1 \), where \( B_t \) is standard Brownian motion for \( 0 \leq t \leq 1 \), find the conditional expectation \( \mathbb{E}(X_s | X_t = 1) \) for \( 0 \leq s < t \leq 1 \).

Answer: \dfrac{s}{t}


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

Answer: 1


Question: If the dual space $V^*$ of a vector space $V$ is finite dimensional, is the vector space $V$ also finite dimensional?

Answer: Yes


Question: Find the most negative value of the function \( w = \cos{x} \cos{y} \cos(x+y) \) for \( 0 \leq x, y < \frac{\pi}{2} \).

Answer: -\dfrac{1}{8}


Question: A board has six coins arranged such that every alternate coin shows heads and tails initially. A die is thrown, and the coin corresponding to the number rolled is flipped. Compute the expected number of die throws required for all coins to show the same side.

Answer: 37


Question:
Let $D$ be the open unit disc in the complex plane and $U = D \setminus \{\frac{-1}{2}, \frac{1}{2}\}$. Define $H_{1} = \{f: D \rightarrow \mathbb{C} \mid f \text{ is holomorphic and bounded}\}$ and $H_{2} = \{f: U \rightarrow \mathbb{C} \mid f \text{ is holomorphic and bounded}\}$. Consider the map $r: H_1 \rightarrow H_2$ given by $r(f) = f|_U$, the restriction of $f$ to $U$. Determine the nature of the map $r$:

A. Injective but not surjective.
B. Surjective but not injective.
C. Both injective and surjective.
D. Neither injective nor surjective.

Select the correct option.

Answer: C


Question: Let $F:(0,\infty) \to [0,\infty)$ be a continuous function satisfying $F(1)=0$, which is strictly increasing on $[1,\infty)$, and strictly decreasing on $(0,1]$. Suppose also that $F|_{(1-\epsilon,1+\epsilon)}$ is convex and smooth for some $\epsilon>0$. Is it true that $F$ is convex at every point $y \in (1-\epsilon,1)$?

Answer: Yes


Question: Determine whether $S = \operatorname{SL}(n, \mathbb{R}) = \{ A \in M_{n\times n}(\mathbb{R}) \mid \det(A) = 1 \}$ is a regular submanifold of the $n^2$-dimensional $C^\infty$ manifold $M = M_{n\times n}(\mathbb{R})$. Is $S$ a regular submanifold?

Answer: Yes


Question: Determine whether the following statement is true or false: For any increasing function \( f : [a,b] \rightarrow \mathbb{R} \), the set of discontinuities has measure zero. Provide a justification for your answer.

Answer: A


Question: Given that \( f'(k/x)f(x) = x \), determine the expression for \( f(k/x)f'(x) \).

Answer: \dfrac{k}{x}


Question:
Determine if the following expression is a valid discrete representation of the delta function:
\[ \delta(n) = \lim_{\epsilon\rightarrow 0} \left [ \sum_{n=-\infty}^\infty  \exp \left (- \frac{n^2}{2 \epsilon} \right ) \right ]^{-1} \exp \left (- \frac{n^2}{2 \epsilon} \right ) \]

Answer: Yes


Question: Determine the number of functions from the set of integers \( \mathbb{Z} \) to itself.

Answer: 2^{\aleph_0}


Question: Find the value that the solution to the logistic equation \( \frac{dP}{dt} = 3P(4 - P) \) approaches as \( t \to \infty \), given the initial condition \( P(0) = 2 \).

Answer: 4


Question: How many sequences of positive integers contain all positive integers?

Answer: 2^{\aleph_0}


Question:
Which of the following statements does not ensure the convergence of a real sequence \(\{a_n\}\)?

A. \(|a_n - a_{n+1}|\) goes to zero as \(n\) goes to infinity.

B. \(\sum\limits_{n=1}^\infty |a_n - a_{n+1}|\) is convergent.

C. \(\sum\limits_{n=1}^\infty na_n\) is convergent.

Answer: A


Question: Evaluate the limit: $$\lim_{x \rightarrow 0_+}\frac{\sin \sqrt{x}}{x^2}\left(\sqrt{x+2x^2}-\sqrt{2\sqrt{1+x}-2}\right)$$

Answer: \dfrac{9}{8}


Question:
If \(a, b, c\) are distinct numbers such that \(a^2 - bc = 2014\), \(b^2 + ac = 2014\), and \(c^2 + ab = 2014\), compute \(a^2 + b^2 + c^2\). Choose the correct answer from the following options:

(A) 4030
(B) 4028
(C) 4026
(D) 4000
(E) 2014

Answer: B


Question: Given the joint probability density function $f(x,y) = cxy^{-2}$ for $0 < x < 1$ and $y > 1$, where $c$ is a constant, compute the expected value $E[Y]$ of the random variable $Y$. Note that $f(x,y) = 0$ otherwise.

Answer: \infty


Question: Evaluate the line integral \( \int_{c} (3a(x^2+y^2)-y^3)dx+3xy(2a-y)dy \), where \( c \) is the upper half of the circle defined by \( y = a + \sqrt{a^2-x^2} \).

Answer: 6a^4


Question: Evaluate the limit of the definite integral: \[ \lim_{n\to\infty}\int_{-\sqrt n}^{\sqrt n}\left(1-\frac{x^2}{2n}\right)^n \,\mathrm{d}x. \]

Answer: \sqrt{2\pi}


Question: Evaluate the limit: \( \lim_{x \to \infty}(x-\ln^3(x)) \)

Answer: \infty


Question: Evaluate the limit: \( \lim_{x\to1} x^{\cot{\pi x}} \) without using L'Hospital's rule or derivatives.

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


Question: Evaluate the limit: $$\lim_{x\to0} {\frac{\frac{1}{1+x^3} + \frac{1}{3}\log{\left(1+3x^3\right)}-1}{2\sin{\left(3x^2\right)}-3\arctan{\left(2x^2\right)}}}$$ without using L'Hôpital's rule.

Answer: \dfrac{1}{2}


Question: Find the mean number of records set by the first $n$ jumpers, where each jumper's height is an independent and identically distributed continuous random variable. What is the behavior of this mean as $n$ approaches infinity?

Answer: \ln n


Question: Find all prime numbers \( p \) such that there exist positive integers \( x \) and \( y \) satisfying the equations \( p+1=2x^2 \) and \( p^2+1=2y^2 \).

Answer: 7


Question: Determine the degree of the field extension \([\mathbb{Q}(\sqrt[4]{2},i\sqrt[4]{2}):\mathbb{Q}]\).

Answer: 8


Question: Determine the dimension of the 2 by 2 orthogonal group, \( O(2) = \{ A \in M(2,2,\mathbb{R}): AA^T = I \} \), where \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \). Given the equations \( a^2 + b^2 = 1 \), \( c^2 + d^2 = 1 \), and \( ac + bd = 0 \), find the dimension of this group.

Answer: 1


Question: Let \( f \) be a function in \( L^1(\mathbb{R}) \). Is it true that there exists a sequence \( \{a_n\} \) such that \( a_n \to \infty \) and \( \lim_{n \to \infty} f(a_n) = 0 \)?

Answer: Yes


Question: Find the function $f(x)$ such that, for $x \ge 0$, the arc length of $f(x)$ over the interval $[0, x]$ is given by $2x + f(x)$. Determine $f(x)$.

Answer: -\dfrac{3}{4}x


Question: Determine the exact value of \( a \) for which the solutions of the equation \( y^2 - (1+x)x^2 = a \) in the interval \(-1 \leq a \leq 1\) undergo a transformation such that the loop on the left-hand side of the \( y \)-axis degenerates to a point and vanishes. Provide your answer as a precise value of \( a \).

Answer: -\dfrac{4}{27}


Question: Determine the degree of the field extension \(\mathbb{Q}(\sqrt{1 + \sqrt{2}})\) over \(\mathbb{Q}\).

Answer: 4


Question: Find all integer solutions to the equation: \[ \biggl\lfloor{\frac{x}{1!}\biggr\rfloor} + \biggl\lfloor{\frac{x}{2!}\biggr\rfloor} + \cdots + \biggl\lfloor{\frac{x}{10!}\biggr\rfloor} = 1001 \]

Answer: 584


Question: Determine the number of $10 \times 10$ matrices $A$ with entries from the set \( \{0, 1, 2, 3\} \) such that the matrix $AA^T$ has zeros on its diagonal. Choose the correct answer from the following options: \( (4^3)^{10} \), \( (4^2)^{10} \), \( 4^{10} \), or \( 1 \).

Answer: 1


Question:
Consider the function \( f: \mathbb{R}^{2} \to \mathbb{R}^{2} \) given by \( f(x,y) = \left(e^{x+y}, e^{x-y}\right) \). Find the area of the image of the region \( \{(x,y) \in \mathbb{R}^{2} \mid 0 < x, y < 1\} \) under the mapping \( f \). Choose the correct answer from the following options:

(1) 1
(2) \( e-1 \)
(3) \( e^{2} \)
(4) \( e^{2}-1 \)

Answer: D


Question: Find the sum of the infinite series: \[ 2^2 + \frac{3^2}{2!} + \frac{4^2}{3!} + \cdots \]

Answer: 5e - 1


Question: What is the cardinality of the class of all rational Cauchy sequences that converge to $0$ in $\mathbb{R}$?

Answer: 2^{\aleph_0}


Question: Determine the asymptotic relationship between the functions \( f(n) = 2^n n \) and \( g(n) = \frac{3^n}{n^2} \) by evaluating the limit \( \lim_{n \to \infty} \frac{f(n)}{g(n)} \).

Answer: 0


Question:
Determine whether there exists a continuous function, other than $f(x)=1$, that satisfies the following conditions for all $x \in [0,1)$:
\[\begin{align*}
&f(x) \geq 0, \\
&f(0) = 1, \\
&f(x) \geq f(2x), \\
&f(x) = f\left(x + \frac{1}{2}\right).
\end{align*}\]

Answer: f(x) = 1


Question: Determine which is greater: $600!$ or $300^{600}$.

Answer: 300^{600}


Question:
Determine whether the following inequality is true for a sequence of random variables $X_k$:
\[ E\left[\min_{1\le k\le n}X_k\right] \ge \min_{1\le k\le n} E[X_k] \]
provided that the left-hand side exists.

Answer: False


Question: Evaluate the limit: $$ \lim_{(x,y) \to (0,0)}\frac{\sin(x^2+y^2)}{\ln(3x^2y^2+1)} $$

Answer: \infty


Question: Let \( g(x) = C \) be a constant function. Determine whether the Riemann-Stieltjes integral \( \int_a^b f \, dg \) exists for any function \( f \).

Answer: 0


Question: For $a > 0$ and $0 < x_0 < \frac{2}{a}$, let the sequence $(x_n)_{n \in \mathbb{N}_0}$ be defined by the recurrence relation $x_{n+1} = x_n(2 - ax_n)$. Show that the sequence $(x_n)$ converges and determine its limit.

Answer: \dfrac{1}{a}


Question: Given a matrix $A \in M_3(\mathbb{Z}_{11})$ with $\det A = 4$, calculate $\det((3A)^{-1})$. Note: $(3A)^{-1} = \frac{1}{3}A^{-1}$.

Answer: 5


Question: Solve for $x$ if $x \in \mathbb{R}^{+}$ in the equation: $$\tan^{-1}(2x+1) + \tan^{-1}(2x-1) = \tan^{-1}2$$

Answer: \dfrac{1}{2}


Question: Determine the number of solutions to the equation \( \frac{1}{(x-3)^3} - 2x - \cos x = 0 \).

Answer: 2


Question: Use polar coordinates to show that the fixed point of the dynamical system \(x' = x - y - x^3\) and \(y' = x + y - y^3\) is less than 2 units away from the origin \((0,0)\).

Answer: 2


Question: Determine the cardinality of the subset of the interval [0,1] consisting of infinite decimal expansions that use only the digits 2 and 5.

Answer: \mathfrak{c}


Question: Let \(k\) be an algebraically closed field, and let \(q(x) = \frac{r(x)}{s(x)}\) be a nonzero rational function in \(k(x)\), where \(r\) and \(s\) are coprime polynomials and \(d = \deg r \ge \deg s\). Determine whether the degree of the field extension \([k(x):k(q(x))]\) is equal to \(d\).

Answer: d


Question: Compute the limit: \[ \lim_{n\to\infty}\int_{0}^{\infty}\frac{\sin\left(\frac{x}{n}\right)}{\left(1+\frac{x}{n}\right)^{n}}\,dx. \]

Answer: 0


Question: Express $abab^3$ in standard form for the group $Dic_6 = \langle a, b \mid a^4 = e, a^2 = b^6, aba^3 b = e \rangle$. Use the group relations to simplify the expression.

Answer: a^2 b^2


Question: Evaluate the limit: $$\lim_{(x,y)\to(0,0)}\frac{(2x+y)\sin(x+y)}{(x+y)\sin(2x+y)}$$

Answer: 1


Question: Determine whether it is true that you win a contest with probability at least \( \frac{1}{2} \), regardless of the number of participants \( n \geq 1 \). The contest has two rounds: you are automatically in the second round, while \( n \) other people are in the first round. Each of the \( n \) other people proceeds to the second round with probability \( \frac{1}{n} \), independently. If there are \( k \) people in the second round, you win the contest with probability \( \frac{1}{k} \).

Answer: True


Question: Let \( x \in \mathbb{R}^n \) be a vector with entries \( x_1 \geq x_2 \geq \cdots \geq x_n \). Decompose this vector into vectors \( x^1 = (x_1, \ldots, x_m, 0, \ldots, 0) \), \( x^2 = (0, \ldots, 0, x_{m+1}, \ldots, x_l, 0, \ldots, 0) \), and so on, such that the following conditions hold: \( 2x_m > x_1 \), \( 2x_{m+1} < x_1 \), \( x_{m+1} < 2x_l \), and \( x_{l+1} > 2x_{l+1} \). Determine the number of such vectors \( x^i \) in the decomposition.

Answer: 2


Question: Evaluate the limit: $$\lim_{x \to 0} \frac{x^{6000} - (\sin x)^{6000}}{x^2(\sin x)^{6000}}$$

Answer: 1000


Question: Determine the order of growth of the complex polynomial \( p(z) = a_nz^n + \ldots + a_1z + a_0 \).

Answer: 0


Question: Calculate the surface integral \( \iint_S \mathbf{F} \cdot \mathbf{n}\, d \sigma \) where \( \mathbf{F} = xy\mathbf{i} + z^2\mathbf{j} + y\mathbf{k} \) and \( S \) is the exterior side of the cylinder \( y^2 + z^2 = 9 \), bounded by \( x=0 \), \( x=1 \), \( y=0 \), and \( z=0 \) in the first octant.

Answer: \dfrac{27}{2}


Question:
Determine whether the Poincaré inequality holds for the space \( H = W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \), where \( \Omega \) is an open and bounded subset in \( \mathbb{R}^2 \). The inequality is given by:
\[ \|u\|_{0,\Omega} \leq C_p\|\Delta u\|_{0,\Omega}, \quad \forall u \in H, \]
where \( C_p > 0 \) is the constant in the Poincaré inequality on \( \Omega \).

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\rightarrow\infty}\left(\Gamma\left(\frac{1}{x}\right)\right)^{-1}\int_{0}^{x}\frac{|\sin(t)|}{t}\,dt \]

Answer: 0


Question: Let $E$ and $F$ be fields such that $E$ is an extension field of $F$ with $[E:F] = n$. Let $M_k(F)$ denote the ring of $k \times k$ matrices over $F$. Does there exist a homomorphism from $M_k(E)$ to $M_{nk}(F)$?

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to0}\left(\frac{1+\ln(1-x^2)}{2x+1-\sin(x)}\right)^{\frac{1}{x^2}}$$

Answer: 0


Question: Evaluate the line integral \( \int_\gamma e^{x^2-y^2}(\cos(2xy) \, dx + \sin(2xy) \, dy) \), where \( \gamma \) is the unit circle oriented in the positive direction.

Answer: 0


Question:
Suppose $X$ is a topological space of infinite cardinality which is homeomorphic to $X \times X$. Determine which of the following statements is true:

1. $X$ is not connected.
2. $X$ is not compact.
3. $X$ is not homeomorphic to a subset of $\mathbb{R}$.
4. None of the above.

Answer: D


Question: Given that the $p$-th, $q$-th, and $r$-th terms of an arithmetic progression (AP) are $a$, $b$, and $c$ respectively, and the $p$-th, $q$-th, and $r$-th terms of a geometric progression (GP) are also $a$, $b$, and $c$ respectively, find the value of the expression $a^b b^c c^a - a^c b^a c^b$. Assume the first term and common difference of the AP are $m$ and $d$, and the first term and common ratio of the GP are $k$ and $t$. Evaluate the expression based on these conditions.

Answer: 0


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

Answer: 1


Question: Is the tensor product $R[x] \otimes_k R[x]$ a free module over $R \otimes_k R$, where $R$ is a $k$-algebra?

Answer: Yes


Question:
Determine whether there exist bases for \(\mathbb{R}^5\) and \(\mathbb{R}^4\) such that the matrix
\[
A = \begin{bmatrix}
1 & 3 & 0 & 0 & 1 \\
1 & 1 & 1 & 1 & -1 \\
-1 & -3 & -1 & -1 & -1 \\
1 & 5 & -1 & -1 & 5
\end{bmatrix}
\]
represents the same homomorphism \(f : \mathbb{R}^5 \to \mathbb{R}^4\) as the matrix
\[
\begin{bmatrix}
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0
\end{bmatrix}.
\]

Answer: Yes


Question: Calculate the expectation \( \mathbf{E}\left[ W_{(W_t)^2} \right] \) for a one-dimensional standard Brownian motion \( W \) defined under a stochastic basis with probability \( \mathbf{Q} \) and filtration \( (\mathscr{F})_{t \in \mathbf{R}_+} \).

Answer: 0


Question: Find the constant \(c\) such that \(\max_{a\le n/2}\frac{C_n^a}{\sum_{k=0}^{\lfloor{a/3}\rfloor}C_n^k}=(c+o(1))^n.\)

Answer: \dfrac{4}{3}


Question: Find the directional derivative of the function \( f(x,y,z) = \sqrt{x^2 + y^2 + z^2} \) at the point \((0,0,0)\) in the direction of the vector \( v = \left(\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right) \).

Answer: 1


Question:
Determine the radius of convergence for the Maclaurin series:
\[
(-1)^{n+1} \frac{\left(\left|\frac{1-x}{1+x}\right|-1\right)^n}{n}
\]

Answer: \dfrac{1}{3}


Question: Find the value of \( \lim_{n\to\infty} \frac{1}{n+1}\sum_{k=1}^n \frac{k}{1 + \sqrt{2} +\sqrt{3} +\cdots+\sqrt{k+1}} \).

Answer: 0


Question: Find the remainder when \(666^{666}\) is divided by 1000.

Answer: 16


Question: Given that \( f(x)=\frac{e^{\tan x} -e^x+\ln (\sec x +\tan x)-x}{\tan x -x} \) is a continuous function at \( x=0 \), find \( f(0) \).

Answer: \dfrac{3}{2}


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

Answer: 0


Question: Suppose $M$ is a Riemannian manifold such that each point has a coordinate chart into $\mathbb{R}^n$ that is an isometry, preserving inner products. Does this imply that $M$ is locally flat, meaning it has vanishing Gaussian curvature?

Answer: Yes


Question: Find all natural numbers \(n\) such that \(\sum_{d|n}{d^4} = n^4 + n^3 + n^2 + n + 1\).

Answer: p^4


Question: Find the maximum likelihood estimator (MLE) of \(\theta\) for the probability distribution given by \(f_\theta(x) = \frac{2x}{\theta^2}\) for \(0 < x < \theta\), and \(f_\theta(x) = 0\) otherwise.

Answer: \max \{ X_1, X_2, \ldots, X_n \}


Question: Consider the function \( f(x,y) = \begin{cases} x^2+2x+2y & \text{ for } (x,y)\neq (0,0) \\ y^2 & \text{ for } (x,y)=(0,0) \end{cases} \). Determine if this function is derivable at the point \((0,0)\).

Answer: Yes


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

Answer: -\dfrac{1}{48}


Question: Find all functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that for all natural numbers \( m \) and \( n \), the following equation holds: \[ \gcd(f(m),n) + \text{lcm}(m,f(n)) = \text{lcm}(f(m),n) + \gcd(m,f(n)) \]

Answer: f(n) = n


Question: Determine whether the number of prime ideals in the ring \( \mathbb{Z}_3[x] \oplus \mathbb{Z}_5 \) is the same as the number of prime ideals in the ring \( \mathbb{Z}_3 \oplus \mathbb{Z}_5 \). Answer with 'Yes' or 'No'.

Answer: No


Question: Let \(f\) be a non-decreasing and continuous function on \([0,1]\) such that \(\int_0^1 f(x) \, dx = 2 \int_0^1 x f(x) \, dx\). Given that \(f(1) = 10.5\), find the value of \(f(0) + f(0.5)\).

Answer: 21


Question: Can a group $G$ have a maximal subgroup and simultaneously have a subgroup $H$ that is not contained in any maximal subgroup?

Answer: Yes


Question: Find the largest integer \( m \) such that \( a^{12} \equiv 1 \pmod{m} \) for all integers \( a \) that are relatively prime to \( m \).

Answer: 65520


Question: Decompose the symmetric polynomial \( \Sigma{x_1^2x_2^2x_3^2} \) into elementary symmetric polynomials.

Answer: \sigma_3^2


Question: Let \( f \) be a differentiable function satisfying the functional equation \( f(xy) = f(x) + f(y) + \frac{x+y-1}{xy} \) for \( x, y > 0 \), and \( f'(1) = 2 \). Find the value of \( \lfloor f(e^{100}) \rfloor \), where \( \lfloor k \rfloor \) denotes the greatest integer less than or equal to \( k \).

Answer: 99


Question: Determine the number of solutions of the equation \( a - z - e^{-z} = 0 \) in the half-plane \( \{ z \in \mathbb{C} : \text{Re}(z) > 0 \} \), where \( a \in \mathbb{R} \) and \( a > 1 \).

Answer: 1


Question: Let \(a, b \in \mathbb{N}\), with \(a \neq b\). The quadratic equations \((a-1)x^2 -(a^2 + 2)x +a^2 +2a=0\) and \((b-1)x^2 -(b^2 + 2)x +b^2 +2b=0\) have a common root. Find the value of \(\frac{ab}{5}\).

Answer: \dfrac{8}{5}


Question: Let $f$ be a function in $C^{\infty}$ with compact support, and let $A$ be a self-adjoint operator. If $\mathrm{supp}(f) \cap \sigma(A) = \varnothing$, prove that $f(A) = 0$.

Answer: f(A) = 0


Question: Are projections of Lebesgue-measurable sets also Lebesgue-measurable?

Answer: No


Question: Find the mass of a wire shaped as a semicircle described by the parametric equations \(x = 1 + \cos(t), y = \sin(t)\), where \(t\) ranges from \(0\) to \(\pi\). The density at any point \((x, y)\) on the wire is directly proportional to the distance from the \(y\)-axis, with a constant of proportionality of 3. Round your answer to the nearest tenth.

Answer: 9.4


Question: Given the recurrence relation \(a_n = 2a_{n-1} - a_{n-2} - 2\) for \(1 < n < 50\), with initial conditions \(a_0 = 0\) and \(a_{50} = a_{49} + 1\), find the value of \(a_{17}\).

Answer: 1411


Question: Let \( A \in M_n \) be a normal matrix with a polar decomposition \( A = PU \), where \( U \) is a unitary matrix and \( P \) is a positive semidefinite matrix. Is it true that \( PU = UP \)?

Answer: Yes


Question:
Evaluate the integral \( \frac{1}{a+b} \int_{a}^{b} x \left[ f(x) + f(x+1) \right] dx \) given that \( f(a+b+1 -x) = f(x) \) where \( a \) and \( b \) are positive real numbers. Determine which of the following expressions it is equal to:

1. \( \int_{a-1}^{b-1} f(x) \, dx \)
2. \( \int_{a+1}^{b+1} f(x+1) \, dx \)
3. \( \int_{a-1}^{b-1} f(x+1) \, dx \)
4. \( \int_{a+1}^{b+1} f(x) \, dx \)

Answer: 4


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

Answer: \infty


Question: Given \( f(x) = \int x(x^2-a)^2 \, dx \) and \( f(a) = 7 \), find the value of \( f(-a) \).

Answer: 7


Question: Evaluate the integral \( \int_0^{\pi} \frac{x}{1+\sin x} \, dx \).

Answer: \pi


Question: Sandy purchases a perpetuity-immediate with annual payments starting at 100, increasing by 10 each year. Danny purchases a perpetuity-due with annual payments of 180. If the present values of both perpetuities are equal at an effective interest rate i > 0, calculate i.

Answer: \dfrac{\sqrt{34} - 4}{18}


Question: Determine the Euler characteristic of the surface \( M=\left\{(x,y,z); \sqrt{x^2+y^2}=1+z^{2n}, 0< z< 1\right\} \).

Answer: 0


Question: Let \( h_n \) be the number of ways to pack a bag with \( n \) fruits, where the fruits can be apples, oranges, bananas, or pears. The conditions are: an even number of apples, at most two oranges, a multiple of three bananas, and at most one pear. Find an explicit formula for \( h_n \).

Answer: n + 1


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

Answer: 9\pi i


Question: Is there a nonempty perfect set in \( \mathbb{R}^1 \) that contains no rational numbers?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to \infty} \frac{x^2}{2^{\sqrt{\log x}}} \]

Answer: \infty


Question: Find the degree of the splitting field of the polynomial \(x^{616} - 1\) over the field \(\mathbb{F}_{2}\).

Answer: 30


Question: Calculate the degree of the field extension \(\mathbb{Q}(\zeta_{10})\) over \(\mathbb{Q}\).

Answer: 4


Question: Given that $n(\hat{f}(x)-f(x)) \sim N(\mu,\sigma)$ and $\mu$ cannot be estimated, determine if the bias of the confidence interval for $\hat{f}(x)$ is $\mu n^{-1}$.

Answer: \mu n^{-1}


Question: Does there exist a unique cyclic subgroup of order \( n \) in the quotient group \( G = (\mathbb{Q}/\mathbb{Z}, +) \)? Justify your answer.

Answer: Yes


Question: Evaluate the limit or state that it does not exist: \[ \lim_{x\rightarrow 0} x^{\frac{1}{3}} (-1)^{\big[\big[\frac{1}{x}\big]\big]} \]

Answer: 0


Question: Determine the points at which the function \( f(x,y) = \begin{cases} xy\sin\left(\frac{1}{x^2+y^2}\right), & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases} \) is differentiable.

Answer: \mathbb{R}^2


Question: Evaluate the integral \( \int_{0}^1 \int_{0}^{1} \int_{x^2}^{1}  12xze^{zy^2} \, dy \, dx \, dz \) using a change of variables.

Answer: 3(e - 2)


Question: Compute the limit: \[ \lim_{x\to 0}\left(\frac{\cot(x)}{x^3}-\frac{1}{x^4}+\frac{1}{3x^2}\right) \]

Answer: -\dfrac{1}{45}


Question: Determine whether the limit \( \lim_{n\to\infty}\frac{n!}{\prod_{i=1}^n(2i-1)} \) equals zero.

Answer: 0


Question: Determine whether the limit \( \lim_{n\rightarrow\infty}\sin\left(\pi\sqrt[3]{n^{3}+1}\right) \) exists. If it does, find its value.

Answer: 0


Question: Evaluate the integral \( \int_{B(0,r)} \frac{1}{|x|^{n-1}} \, dx \) and determine if it can be expressed as a constant times the radius \( r \).

Answer: \frac{2\pi^{n/2}}{\Gamma(n/2)} r


Question: Given that \( \text{det}\left[\begin{matrix}a & 1 & c\\ b & 1 & d\\ e & 1 & f \end{matrix}\right]= -3 \) and \( \text{det}\left[\begin{matrix}a & 1 & c\\ b & 2 & d\\ e & 3 & f \end{matrix}\right]= 5 \), find the determinant of the matrix \( \text{det}\left[\begin{matrix}a & -4 & c\\ b & -7 & d\\ e & -10 & f \end{matrix}\right] \).

Answer: -12


Question: Consider a function $f$ such that \( f(x)f(y) = f(xy) + f\left(\frac{x}{y}\right) \). Find the value of \( f\left(\frac{3-2\sqrt{2}}{\sqrt{2} + 1}\right) - f\left(\frac{3+2\sqrt{2}}{\sqrt{2} - 1}\right) \).

Answer: 0


Question: Calculate the average squared distance between two points chosen randomly on the disk \( D = \{(x,y) \in \mathbb{R}^2 \mid x^2 + y^2 \leq 1\} \). Use the integral formula: \[ \frac{1}{m(D)^2} \int \int_D \int \int_D ((x_1 - x_2)^2 + (y_1 - y_2)^2) \, dx_1 \, dy_1 \, dx_2 \, dy_2 \] where \( m(D) \) is the Lebesgue measure of \( D \).

Answer: 1


Question: Given the equation \(x^3 = (3+4i)\overline{x}\), where \(x\) is a complex number, find the sum of the moduli of its roots.

Answer: 4\sqrt{5}


Question: Jo practices her tennis serve by serving until she achieves 50 successful serves. Each serve is successful with a probability of 0.4, independently of previous serves. What is the approximate probability that she will need more than 100 serves to accomplish her goal?

Answer: 0.974


Question: Evaluate the limit: \[ \lim_{x\rightarrow0}\frac{1-\frac{1}{2}x^2-\cos\left(\frac{x}{1-x^2}\right)}{x^4} \]

Answer: \dfrac{23}{24}


Question: Suppose $X$ is a Banach space, and $(T_{n})_{n\geq1}$ is a sequence of linear maps such that $T_{n}(x) \to T(x)$ for all $x \in K$, where $K \subset X$. Does it follow that $T_{n}(x) \to T(x)$ for all $x \in \text{span}(K)$? Justify your answer.

Answer: Yes


Question: Consider the curve on the $xy$-plane given by \( \gamma(t) = \left(\log\left(\cot\left(\frac{t}{2}\right) \right) - \cos(t), \sin(t),0\right) \), where \( t \in (0, \frac{\pi}{2}) \). Let \( S \) be the surface of revolution obtained by rotating this curve around the \( x \)-axis. Compute the volume of the region bounded by \( S \) and the \( yz \)-plane.

Answer: \dfrac{\pi}{3}


Question: Determine the order of the point \( M(0,1) \) on the curve defined by the polynomial \( f(x,y) = (x^2-1)^2 - y^2(3-2y) \).

Answer: 2


Question: Given that the service time follows an exponential distribution with a rate of 12 customers per hour, calculate the probability that a customer waits for less than 3 minutes.

Answer: 0.4512