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

Answer: e


Question: Evaluate the limit: \[ \lim\limits_{x\rightarrow0} \frac{e^{-\frac{x^2}{2}}-\cos x}{x^3\sin x} \]

Answer: \dfrac{1}{12}


Question: Find the value of \(a\) such that \(\lim_{x\to\infty}\left(\frac{x+a}{x-a}\right)^x = e\).

Answer: \dfrac{1}{2}


Question:
Find the limit of the sequence as \( m \to \infty \):
\[ \lim_{m \to \infty} \left(\frac{2m^2 + m + 4}{2m^2 + 3m + 5}\right)^{3m-3} \]

Answer: e^{-3}


Question: Does there exist a commutative ring $R$ with an idempotent endomorphism $f : R \rightarrow R$ that cannot be expressed as $f(x) = sx$ for some idempotent $s \in R$?

Answer: Yes


Question: A fair coin is tossed 10 times. Let $X$ denote the number of heads observed, and let $Y = X^2$. Calculate the covariance $\text{Cov}(X, Y)$. Express your answer as a single numerical value.

Answer: 25


Question: Let $A$ be a linear operator on a finite-dimensional complex vector space with minimal polynomial $m_A(\lambda) = (\lambda - 2)^2$. Determine the minimal polynomial of the operator $A^2$. \( \text{Provide your answer in the form of a polynomial.} \)

Answer: (\lambda - 4)^2


Question: Consider 4 different letters: a, b, c, and d. You have between 4 and 12 letters, with at least 1 of each letter. In how many ways can you choose the letters, considering that the repetition of specific letters is irrelevant? Use generating functions to solve this problem.

Answer: 495


Question: Evaluate the integral \( \int_{C} \frac{\cos(z)}{z^{2}+1} \, dz \), where the contour \( C \) is the circle of radius 5 centered at the origin.

Answer: 0


Question: Determine the values of $x$ for which the series \( \sum_{n=1}^{\infty} \frac{1}{n^x}\arctan\left(\left(\frac{x-4}{x-1}\right)^n\right) \) is convergent.

Answer: (1, \infty)


Question: Find the smallest integer \( n \) such that \( n! \) is divisible by \( 13^{28} \).

Answer: 338


Question: Find the value of the ratio \( \frac{\int_{0}^{\pi}x^3\log(\sin x) \, dx}{\int_{0}^{\pi}x^2\log(\sqrt{2}\sin x) \, dx} \).

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


Question: How many divisors of \(9!\) can be expressed in the form \(x^2 y^3\), where \(x\) and \(y\) are natural numbers?

Answer: 28


Question: Given that \( \frac{\cos x+\cos y+\cos z}{\cos(x+y+z)}=\frac{\sin x+\sin y+\sin z}{\sin(x+y+z)}=k \), find the value of \( \cos(x+y)+\cos(y+z)+\cos(z+x) \).

Answer: k


Question: Find the arithmetic average of all solutions \(x \in \mathbb{R}\) to the equation \([x] + [2x] + [3x] = 4x\), where \([x]\) denotes the integer part of \(x\).

Answer: \dfrac{5}{12}


Question: Given an unbiased estimator $\hat{k}$ and a likelihood function $L$, with $\ell = \ln L$, find the covariance $\operatorname{Cov}\left( \frac{d \ell}{d k}, \hat{k} \right)$. Assume the estimation method is maximum likelihood or another method, and that $E(\hat{k}) = k$ and $E\left( \frac{d \ell}{d k} \right) = 0$. Provide a detailed solution.

Answer: 1


Question: Consider an analytic function $f(z)$ defined outside a closed circular region $|z|=R$, with its expression at $z=\infty$ given by $f(z)=a_0+a_1/z+a_2/z^2+\cdots$ where $a_0\neq 0$. If $f$ is continuous on $|z|=R$, can the maximum modulus of $f$ be attained on $|z|=R$?

Answer: Yes


Question: Compute the value of $528^{843^{418}} \pmod{25}$. Provide a general method for finding the modulo of numbers in the form $a^{b^c} \pmod{m}$.

Answer: 8


Question: Given that the bisectors of two lines \(L_1\) and \(L_2\) are represented by the equation \(3x^2 - 8xy - 3y^2 + 10x + 20y - 25 = 0\), and that the line \(L_1\) passes through the origin, determine the equation of the line \(L_2\).

Answer: x + 2y - 5 = 0


Question:
Suppose $o_1, o_2, \dots, o_n$ are $n$ objects, each colored with a single color. For each object $o_i$, let $c_i$ be the number of objects having the same color as $o_i$. Consider the monic polynomial with roots $-c_1, -c_2, \dots, -c_n$:
\[ x^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \]
If exactly $k$ colors are used, determine which of the following statements is always true:

A. $k > a_0$
B. $ka_0 = a_1$
C. $k = a_{n-1}$
D. $ka_1 = a_0$

Answer: B


Question: Determine the number of zeros of the function \( \sin(z) - 100z^9 \) within the unit disk \( |z| < 1 \).

Answer: 9


Question: Let $k$ be a field, and let $X/k$ be a stable curve. Determine whether it is always possible to find a deformation $\mathcal{X}/k[[t]]$ such that $\mathcal{X}$ is regular.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to0^+}\frac{1}{\sin^2x}\int_{\frac{x}{2}}^x\sin^{-1}t\,dt$$

Answer: \dfrac{3}{8}


Question: Compute the Cheeger constant of the unit disc in \(\mathbb{R}^2\), defined as \(\inf_E\frac{S(E)}{\min(V(A),V(B))}\), where \(E\) is a 1-dimensional submanifold dividing the disc into disjoint subsets \(A\) and \(B\), \(S(E)\) denotes the 1-dimensional volume (length) of \(E\), and \(V\) denotes 2-dimensional volume (area).

Answer: \dfrac{4}{\pi}


Question: Calculate the limit \( \lim_{n \to \infty }{\frac{{[n(n+1)(n+2)...(2n-1)]}^\frac{1}{n}}{n}} \) using the Riemann sum approach. Use the integral representation \( \int_0^1 f(x)\,dx=\lim_{n\to\infty}\frac{1}{n} \sum_{k=1}^n f\left(\frac{k}{n}\right) \) to assist in your calculation.

Answer: \dfrac{4}{e}


Question: Find the limit of the function $f(x) = x^x$ as $x$ approaches 0.

Answer: 1


Question: Show that \( \int_{0}^{\epsilon} \frac{1}{x} \, dx = +\infty \) for all \( \epsilon > 0 \).

Answer: +\infty


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

Answer: 1


Question: Let \( f:[0,1] \to \mathbb{R} \) be a differentiable function such that \( \sup_{x \in [0,1]} |f'(x)| \) is finite. Is it true that \( \int_0^1 f'(x) \, dx = f(1) - f(0) \)?

Answer: Yes


Question: Calculate the value of \(14^{(2017^{2017})} \mod 60\).

Answer: 44


Question: Determine the minimum expected number of bits required to encode sequences of length 200 from the typical set $S$, where $S$ consists of all sequences $x$ with at most three 1s. Assume $x \in \{0,1\}^{200}$ and $P(0) = 0.99$. Use $\log_{2}|S|$ to encode such sequences.

Answer: 20.35


Question: Compute the integral \( \int_{0}^{+\infty}\frac{x^2\mathrm{d}x}{e^{x}-1} \) analytically and verify that it equals \( 2\zeta(3) \).

Answer: 2\zeta(3)


Question:
Evaluate the limit of the following triple integral as \( m \to \infty \):
\[
\lim_{m\to\infty} \int_{\frac{1}{m}}^1 \int_0^\pi \int_{-\pi}^\pi \frac{r^6 \sin^4(a) \cos^2(a) \cos^2(b) \sin^2(b)}{r^{17/2}} \, db \, da \, dr
\]

Answer: \infty


Question: The monthly demand for the Instant Pie Maker is given by \(D(x,y)= \frac{1}{125}xe^{xy/1000}\) units, where \(x\) dollars are spent on infomercials and \(y\) dollars are spent on in-person demonstrations. If \(t\) months from now \(x=20+t^{2/3}\) dollars are spent on infomercials and \(y=t\ln(1+t)\) dollars are spent on demonstrations, at approximately what rate will the demand be changing with respect to time 8 months from now? Round your answer to 3 decimal places.

Answer: 0.027


Question: Let $S$ be a countable well-ordered set which is unbounded (i.e., it has no maximum). Does there exist an unbounded increasing sequence in $S$? Answer yes or no.

Answer: Yes


Question: Let $p \in (1, \infty)$. Prove that if $f \in L^p(\mathbb{R})$, then \( \lim_{x \rightarrow \infty} \int_{x}^{x+1} f(t) \, dt = 0 \).

Answer: 0


Question:
Determine the number of real roots of the polynomial equation \(1 + a_1x + a_2x^2 + \ldots + a_nx^n = 0\) given that \(|x| < \frac{1}{3}\) and \(|a_i| < 2\) for all \(i = 1, 2, 3, \ldots, n\). Choose the correct option:

(a) \(n\) if \(n\) is even
(b) 1 if \(n\) is odd
(c) 0 for any natural number \(n\)
(d) none of these

Answer: C


Question: Evaluate the integral \( \int_{C}\frac{\tan z}{z^2}dz \), where \( C \) is the unit circle.

Answer: 2\pi i


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

Answer: 1


Question: Find the value of \(x\) such that \(2^{35} \equiv x \pmod{561}\).

Answer: 263


Question: Evaluate the expression \( \frac{\sum_{k=0}^{6}\csc^2\left(a+\frac{k\pi}{7}\right)}{7\csc^2(7a)} \) where \( a = \frac{\pi}{8} \).

Answer: 7


Question: Evaluate the sum \( \sum_{k=1}^{101} \frac{k(k-1)}{2}\cos\left[\frac{k(k-1)\pi}{2}\right] \).

Answer: 5100


Question:
Verify the correctness of the following solution for the expectation $E[e^{2 \sqrt{2}(W_T-W_t)}]$, where $W_s$ is a Wiener process, $t$ is fixed, and $T > t$. Using Ito's lemma, the differential equation for $m(T) = E(Z_T)$ is derived as:

$$m'(T) = 2\sqrt{2}m(T)$$
$$m(t) = 1$$

The proposed solution is $E(Z_T) = e^{2\sqrt{2}(T-t)}$. Is this solution correct?

Answer: e^{4(T - t)}


Question: Evaluate the limit \( \lim_{n\to \infty}\frac{\ln(a^n+b^n+c^n)}{\sqrt{n^2+n+1}} \) where \( 0<a<b<c \).

Answer: \ln c


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

Answer: \dfrac{1}{36}


Question: Let $W(t)$ be a Brownian motion with $W(0) = 0$. Determine the expected value $E[\cos{W(t)} + \sin{W(t)}]$. Provide your answer in terms of $t$.

Answer: e^{-t/2}


Question: Suppose $C$ is a simple closed curve in the first quadrant. Given the vector field $F = \left(\frac{y^2}{2} + x^2y, -x^2 + 8x\right)$, determine the curve $C$ that produces the maximum amount of work done by $F$. What is the maximum value of this work?

Answer: \dfrac{136}{5}


Question: Determine the possible values of $k \in \mathbb{N}$ such that the identity \(1^k + 2^k + \cdots + n^k = (1 + 2 + \cdots + n)^{k-1}\) holds for all $n \in \mathbb{N}$. Provide your answer as a set of values for $k$.

Answer: 3


Question: Given that \( \frac{3 \pi}{4}<\theta<\pi \) and \( \sqrt{2 \cot \theta+\frac{1}{\sin ^{2} \theta}}=k-\cot \theta \), find the value of \( k \).

Answer: -1


Question: In Mr. Fat's combinatorics class, there are ten girls and four boys. In how many ways can these students sit around a circular table such that no boys are next to each other?

Answer: 1828915200


Question: Evaluate the line integral \( \int_C (xy^2+1)\,dx + x^2y\,dy \) where \( C \) is any curve from \((1,-2)\) to \((3,3)\).

Answer: \dfrac{81}{2}


Question: Let \( n \in \mathbb{N} \) and let \( A, B \in M_n \) be Hermitian matrices. If all eigenvalues of \( A \) and \( B \) are positive and \( A^k = B^k \) for some \( k \in \mathbb{N} \), prove that \( A = B \).

Answer: A = B


Question: Is there only one counterexample in the complete graph $K_5$ for the Ramsey number $R(3,3)$?

Answer: Yes


Question: Calculate the probability that when 36 people are randomly split into 12 groups of 3, and then split again into 12 groups of 3, at least one group will have the same members as in the first grouping.

Answer: \dfrac{12}{595}


Question: Compute the fourth moment of a standard Wiener process, \( W_t \), i.e., find \( E[W_t^4] \).

Answer: 3t^2


Question: Let \(w_1, \ldots, w_n\) be a basis of \(\mathbb{R}^n\). For each subset \(S_j\) of \(\{1, 2, \ldots, n\}\), define \(H_j = \text{span}\{w_i : i \in S_j\}\). Does there exist a basis \(v_1, \ldots, v_n\) of \(\mathbb{R}^n\) such that \(H_j^{\perp} = \text{span}\{v_i : i \in S_j^c\}\) for all \(j\)?

Answer: Yes


Question: If \( R_1 \subset R_2 \) are two real closed fields, where \( R_2 \) is an extension of \( R_1 \), does \( R_1 \) always contain the algebraic elements of \( R_2 \)?

Answer: Yes


Question: Find the non-zero function \( q(x) \) that is orthogonal to all polynomials of degree less than or equal to \( n \) with respect to the weight function \( w(x) = \sqrt{1-x^2} \) over the interval \([-1, 1]\).

Answer: U_{n+1}(x)


Question: Let $f$ be a continuous real function on $\mathbb{R}^1$, such that $f'(x)$ exists for all $x \neq 0$ and $f'(x) \to 3$ as $x \to 0$. Does it follow that $f'(0)$ exists? Justify your answer.

Answer: Yes


Question: Determine the radius of convergence of the power series \( P(x) = x - \frac{1}{3}x^3 + \frac{1}{5}x^5 - \frac{1}{7}x^7 + \cdots \).

Answer: 1


Question: Find the maximum possible constant \( c \) such that the inequality \( \int_0^{2\pi} (u'(x))^2 \, dx \geq c\int_0^{2\pi} (u(x))^2 \, dx \) holds for all functions \( u \in C^1[0,2\pi] \) with \( u(0) = u(2\pi) = 0 \).

Answer: \dfrac{1}{4}


Question: Calculate the integral \( \int_{c(0,1)} \sin\left(\frac{1}{z}\right) \, dz \) using the Laurent series expansion. Express your answer in terms of \( \sin(1) \).

Answer: 2\pi i


Question: Determine if the limit \( \lim_{n\rightarrow \infty }\int_{-\infty}^{\infty}\frac{(\sin(x))^{n}}{x^2+1}d\lambda(x) \) exists, where \( \lambda \) is the Lebesgue measure on \( \mathbb{R} \). If it exists, calculate its value.

Answer: 0


Question: Find the smallest contraction constant \( \alpha \in (0, 1) \) such that the function \( T : X \to X \) defined by \( T(x) = \frac{x}{2} + \frac{1}{x} \) is a contraction map on the set \( X = \{ x \in \mathbb{R} : x \ge 1 \} \). Show that \( |T(x) - T(y)| \le \alpha |x - y| \) for all \( x, y \in X \).

Answer: \dfrac{1}{2}


Question: Consider the space of all Lipschitz continuous functions on $[0,1]$ equipped with the infinity norm $\|\cdot\|_\infty$. Let $f:[0,1] \to \mathbb{C}$ be Lipschitz. Is the operator $L(f)=\int_{0}^{1}f \, dx$ continuous on this space?

Answer: Yes


Question: Prove that \(\lim_{n \to \infty} \frac{n^3-3n^2+4n-7}{2n^3-10n^2+1} = \frac{1}{2}\) using the \((\epsilon, N)\) definition of a limit. Specifically, find an \(N \in \mathbb{N}\) such that for all \(\epsilon > 0\), there exists \(N > 0\) such that for all \(n \geq N\), \[\left|\frac{n^3-3n^2+4n-7}{2n^3-10n^2+1} - \frac{1}{2} \right| < \epsilon.\]

Answer: \frac{1}{2}


Question: Find the third term in the Taylor series expansion of \( f(x) = \frac{(1+x)^4}{(1+2x)^3(1-2x)^2} \) around \( x = 0 \).

Answer: 10


Question: Find all prime numbers $p$ such that there exists a finite field $GF(p^n)$ containing a primitive 9th root of unity. Recall that for a primitive 9th root of unity to exist, the order of the multiplicative group of the field, $p^n - 1$, must be divisible by 9. Determine the primes $p$ for which this condition holds.

Answer: p \neq 3


Question: Let \( A \) be a subset of \( \mathbb{R} \) such that every point in \( A \) is isolated from the right, meaning that for every \( a \in A \), there exists \( \epsilon > 0 \) such that \( A \cap [a, a+\epsilon) = \{ a \} \). If \( \bar{A} \) denotes the closure of \( A \) in \( \mathbb{R} \), is it true that the Lebesgue measure \( |\bar{A}| \) of \( \bar{A} \) is always zero?

Answer: Yes


Question: Let \( A \) be an \( n \times n \) diagonal matrix with real entries and characteristic polynomial \((x-a)^p(x-b)^q\), where \( a \) and \( b \) are distinct real numbers. Define \( V = \{ B \in M(n, \mathbb{R}) : AB = BA \} \). What is the dimension of the subspace \( V \) of matrices that commute with \( A \)?

Answer: p^2 + q^2


Question: Find the minimal polynomial of the linear transformation \( T: M_{4\times 4}^{\mathbb{R}} \to M_{4\times 4}^{\mathbb{R}} \) defined by \( T(M) = -2M^t + M \).

Answer: x^2 - 2x - 3


Question: In triangle \(\triangle ABC\), vertices \(A(0,a)\) and \(B(0,b)\) are fixed, where \(a, b > 0\), and vertex \(C(x,0)\) is variable. Determine the value of \(x\) that maximizes the angle \(ACB\).

Answer: \pm \sqrt{ab}


Question: Compute the directional derivative of the function \( f(x,y) = \begin{cases} \frac{x^3}{x^2+y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} \) at the point \((0,0)\) in the direction of the unit vector \( u = (u_1, u_2)^T \).

Answer: u_1^3


Question: Let $X$ and $Y$ be independent random variables, each uniformly distributed on $[0,3]$. Calculate the conditional expectation $E(Y| X<1 \cup Y<1 )$. Provide your answer in the form $\boxed{\text{answer}}$.

Answer: \dfrac{11}{10}


Question: Determine whether the function \( f(z) = \frac{e^z-1}{\sin z} \) has a holomorphic antiderivative on the unit circle.

Answer: Yes


Question: Does there exist an unbounded operator $P$ on some Banach space $X$ such that $\text{Dom}(P) = X$ and $P^2 = P$?

Answer: Yes


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

Answer: 9 - 6\gamma


Question: Let \( f: \mathbb{D} \setminus \{0, 1/2, 1/3, \dots \} \) be a bounded analytic function. Can \( f \) necessarily be extended to an analytic function on \( \mathbb{D} \)?

Answer: Yes


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

Answer: 0


Question: Determine whether the following claim is true or false: If a symmetric matrix $A$ satisfies $A^k = I$ for some integer $k > 0$, then $A^2 = I$. Provide a proof if the claim is true, or a counterexample if it is false.

Answer: True


Question: Given the recurrence relation \(x_{n+1} = 4x_n - x_{n-1}\) for \(n \geq 1\), with initial conditions \(x_0 = 1\) and \(x_1 = 2\), find an odd prime factor of \(x_{2015}\).

Answer: 181


Question: Compute the limit: \[ \lim_{n \to \infty} \frac{5 \cdot 9 \cdot 13 \cdot \dots \cdot (4n+1)}{7 \cdot 11 \cdot 15 \cdot \dots \cdot (4n+3)}. \]

Answer: 0


Question: Evaluate the integral \( \int_C \text{Re }z\,dz \) where \( C \) is the lower half of the circle of radius 4 centered at the origin, from \(-4\) to \(4\).

Answer: 8\pi i


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty} n \sum ^{2n}_{k=n}\frac{1}{k^2} \]

Answer: \dfrac{1}{2}


Question:
Consider a sequence of polynomials with real coefficients defined by:
\[ p_0(x) = (x^2 + 1)(x^2 + 2) \cdots (x^2 + 1009) \]
with subsequent polynomials defined by \( p_{k+1}(x) = p_k(x+1) - p_k(x) \) for \( x > 0 \).

Find the least integer \( n \) such that:
\[ p_n(1) = p_n(2) = \cdots = p_n(5000). \]

Answer: 2018


Question: Evaluate the definite integral \( \int_{-10}^{10} \sin(x^3) \, dx \).

Answer: 0


Question: Let $T$ be a nonzero linear functional on a normed space $X$. Is it true that $T$ is not constant on each ball $B(x,r)$ in $X$?

Answer: Yes


Question: Determine the value of \( \alpha \) such that the integral \( \int_{0}^{\infty} x^{\alpha}\sin\left(\frac{1}{x}\right)dx \) converges.

Answer: (-2, 0)


Question: Does there exist a non-linear polynomial \( P(x) \in \mathbb{Q}[X] \) such that for every rational number \( y \), there exists a rational number \( x \) satisfying \( y = P(x) \)?

Answer: No


Question: Find the point $x^*$ such that $\lambda \sum_{\text{left}} m_i |x_i-x^*| = \mu \sum_{\text{right}} m_i |x_i-x^*|$, where $\lambda + \mu = 1$, for a given set of points with coordinates and masses \{($x_i$, $m_i$)\}. Provide an efficient algorithm or method to calculate this shifted center of mass.

Answer: x^*


Question: Given that for real numbers \(a\), \(b\), and \(c\), the sequence \(S_k\) is defined as \(S_k = \frac{a^k + b^k + c^k}{k}\) for non-negative integers \(k\), and \(S_1 = 0\), find the value of \(\frac{S_5 S_2}{S_7}\).

Answer: 1


Question: Evaluate the double integral \( \int\int_{R} xy \, dx \, dy \) over the region \( R \) bounded by the lines \( y = x \), \( y = 2x \), and \( x + y = 2 \) using the change of variables \( u = \frac{y}{x} \) and \( v = \frac{1}{x} \).

Answer: \dfrac{13}{81}


Question: Consider a 0-dimensional smooth manifold $M$ and a smooth manifold $N$ (which may have a boundary). Is any map $f: M \to N$ smooth?

Answer: Yes


Question: Let $h \in \text{Aut}(G)$ be an automorphism that preserves all subgroups of $G$, meaning $h(U) = U$ for every subgroup $U$ of $G$. If $\alpha$ is any automorphism of $G$, is it true that $\alpha h \alpha^{-1}$ also preserves all subgroups of $G$?

Answer: Yes


Question: Let $V$ be a vector space of dimension at least 3 over an infinite field $k$. Given any 1-dimensional subspaces $W_1, W_2, \ldots, W_n$ of $V$, does there exist a 2-dimensional subspace $W$ of $V$ such that $W \cap (\cup_{i=1}^n W_i) = \{0\}$?

Answer: Yes


Question: Assume we have a centrally symmetric convex set $K \subset \mathbb{R}^n$ with $\text{Vol}(K)=1$. For every direction $u$, it is given that $\text{Vol}(K \Delta R_u(K)) < \epsilon$, where $A \Delta B$ is the symmetric difference and $R_u(K)$ is the reflection of $K$ with respect to $u^\perp$. Does this imply that $K$ is close (in terms of $\epsilon$) to a Euclidean ball in the same metric?

Answer: Yes


Question: Find the Laurent series and the residue around $z_0 = 0$ for the function $f(z) = \frac{1}{1 - \cos z}$. Provide the first few terms of the series.

Answer: 0


Question: Let $f(x) = ax^2 + bx + c$ be a quadratic function such that $a + b + c = 3$ and $f(x+y) = f(x) + f(y) + xy$ for all real numbers $x$ and $y$. Calculate the value of $\sum_{n=1}^{10} f(n)$. \( \text{Express your answer as a single number.} \)

Answer: 330


Question: Determine whether the set \( F = \{ f \in C([a,b]) : f(t) > 0 \text{ for all } t \in [a,b] \} \) is open in the topology \( \mathcal{O}_{d_{\infty}} \), where the metric \( d_{\infty}(f,g) = \sup\{|f(t)-g(t)| : t \in [a,b]\} \).

Answer: Yes


Question: Evaluate the contour integral \( \int_{C} \frac{z^3}{2z-i} \, dz \), where \( C \) is the unit circle.

Answer: \dfrac{\pi}{8}


Question: Find the angle between the vector \((2,2,1,1)\) and the linear subspace spanned by the vectors \((3,4,-4,-1)\) and \((0,1,-1,2)\).

Answer: \dfrac{\pi}{3}


Question:
Determine the values of $x$ for which the Maclaurin series expansion of $4\cos(x)\ln(1+x)$ converges. The series up to the 8th order is given by:
\[ T_p = 4x - 2x^2 - \frac{2}{3}x^3 + \frac{3}{10}x^5 - \frac{1}{4}x^6 + \frac{31}{140}x^7 - \frac{37}{180}x^8 + O(x^9) \]

Answer: (-1, 1]


Question: Determine whether there exist integers \(a, b, c\) such that \(a^2 + b^2 = c^2 + 1\) with \(a \neq 0\) and \(b \neq 0\).

Answer: Yes


Question: Evaluate the area of the surface of the cylinder given by the equation \(x^2 + y^2 = 2x\), bounded by \(z = 0\) and \(z = \sqrt{x^2 + y^2}\), using a double integral. Express your answer in terms of polar coordinates if preferred.

Answer: 8


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

Answer: 1


Question: Apply the ratio test to the sequence \( \frac{\sqrt{n^n}}{2^n} \) and determine the limit as \( n \to \infty \).

Answer: \infty


Question: Suppose $X$ is a compact, connected $n$-dimensional orientable homology manifold. Does $X$ have a fundamental class?

Answer: Yes


Question: Determine the conditional expectation \( \mathbb{E}(X|\mathcal{G}) \) where \( X(\omega) = \cos(\pi \omega) \) and \( \mathcal{G} \) is the \( \sigma \)-algebra of countable and co-countable subsets of \((0,1)\). Assume \( \lambda \) is the Lebesgue measure restricted to \((0,1)\).

Answer: 0


Question:
Given the differential equation \( xy' = y - xe^{\frac{y}{x}} \) with the initial condition \( y(e) = 0 \), determine the value of \( u(e^e) \). Choose the correct answer from the following options:

(a) \( u(e^e) = e^e \)

(b) \( u(e^e) = 2^e \)

(c) \( u(e^e) = -e^e \)

(d) \( u(e^e) = e^2 \)

(e) \( u(e^e) = e^{-e} \)

Answer: C


Question: Find the volume of the region lying below the plane \( z = 3 - 2y \) and above the paraboloid \( z = x^2 + y^2 \).

Answer: 8\pi


Question: Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space and $(\mathcal F_k)_{k\in\mathbb N}$ a filtration of $\mathcal F$ such that $\mathcal F=\sigma(\mathcal F_k\mid k\in\mathbb N).$ Let $X:\Omega\rightarrow\mathbb R^d$ be $\mathcal F$-measurable. If for some $k_0\in\mathbb N$ we have $E[X\mid\mathcal F_{k_0}]=0$ $\mathbb P$-a.s., does it follow that $E[X\mid\mathcal F]=0$ $\mathbb P$-a.s.?

Answer: No


Question: Apply Stokes' theorem to evaluate the line integral of the vector field \( \vec{F}(x,y,z) = 3z\vec{i} + 5x\vec{j} + 2y\vec{k} \) over the positively oriented circle \( C \) defined by \( x^2 + y^2 = 9 \) in the \( xy \)-plane. The surface \( \sigma \) is the part of the paraboloid \( z = 9 - x^2 - y^2 \) with \( z \ge 0 \), oriented upwards.

Answer: 45\pi


Question: Consider a positive odd integer \(N > 1\) of the form \(N = T^2 + 27U^2\), where \(T, U \in \mathbb{Z}\) and \(T, U \neq 0\), which cannot be divided by 3. Suppose \(N\) can be divided by a prime number \(p\) such that \(p \equiv 1 \pmod{3}\). Is it possible that \(p = T_0^2 + 27U_0^2\) for some integers \(T_0, U_0\)?

Answer: Yes


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

Answer: -\dfrac{2}{3}


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

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


Question: Evaluate the limit: \[ \lim_{x \to 0} x^2 \left(1 + 2 + 3 + \cdots + \left\lfloor \frac{1}{|x|} \right\rfloor \right). \]

Answer: \dfrac{1}{2}


Question: Find the closest point on the line \(y=\frac{1}{e}x+e^2+1\) to the curve \(y=\ln(x)\).

Answer: (0, e^2 + 1)


Question: Let $G$ be an uncountable group, $H$ a countable subgroup of $G$, and $g \in G \setminus H$. Is there necessarily an element $x \in G \setminus H$ such that $g \notin \langle H \cup \{x\} \rangle$?

Answer: Yes


Question: Evaluate the limit: $$\lim_{n \rightarrow \infty} \int_{2n\pi}^{2(n+1)\pi} x\ln(x)\cos x\,\mathrm{d}x.$$

Answer: 0


Question: For a given prime number \( p \), determine if there exists a positive integer \( n \) such that \( p \mid b_n \) in the expression \((1+\sqrt{2})^n = a_n + b_n\sqrt{2}\), where \( a_n \) and \( b_n \) are positive integers.

Answer: Yes


Question: Consider a sample $X_1, X_2, \ldots, X_{100}$ from a Poisson distribution with parameter $\lambda = 3.5$. Calculate the approximate probability that the total sum $T = \sum_{i=1}^{100} X_i$ exceeds 360.

Answer: 0.287


Question: Find the minimum value of \( n \) such that \( \binom{2n}{n} \) is divisible by 200.

Answer: 13


Question: Given that $a_2 = 2$ and $a_{mn} = a_m a_n$ for coprime integers $m$ and $n$, prove that $a_n = n$ for all natural numbers $n$.

Answer: a_n = n


Question: Determine the radius of convergence of the power series \(\sum_{n=1}^{\infty}\frac{W_n}{n^{\alpha}}x^n\), where \(W_n = \int_{0}^{\frac{\pi}{4}}\tan^{n}t\,dt\). Express your answer in terms of \(\alpha\).

Answer: 1


Question: Determine whether the following statement is true or false: "Every invertible matrix $A \in \mathbb{Q}^{n \times n}$ is similar to a diagonal matrix over $\mathbb{C}$."

Answer: B


Question: Find an unbiased estimator for the expression \(3\lambda + \lambda^2\) given that \(Y\) follows a Poisson distribution with parameter \(\lambda\) and \(C = 2Y + Y^2\).

Answer: 2Y + Y^2


Question: Evaluate the contour integral \( \int_{\partial B_1(2i)} \frac{e^{z^2}}{2i-z} \, dz \) using Cauchy's integral formula, where \( B_1(2i) \) is the open ball of radius 1 centered at \( 2i \).

Answer: -2\pi i e^{-4}


Question:
Given that the variables $x$ and $y$ satisfy the equation $f(x,y) = x^5 + xy^4 + y + x^2 = 0$, approximate $y$ in a neighborhood of $(0,0)$ using the implicit function theorem. Choose the correct approximation from the following options:

(i) $1+x+x^2$
(ii) $x+2x^2$
(iii) $-2x^2$
(iv) $-x^2$

Answer: iv


Question: Consider the equation \(z^4 - 6z + 3 = 0\). Determine how many of its complex solutions satisfy the condition \(1 < |z| < 2\).

Answer: 3


Question: Compute \( \lim\limits_{n\to\infty}\int\limits_0^{\infty} \frac{1}{x^{3/2}}\sin\left(\frac{x}{n}\right) dx \).

Answer: 0


Question:
Let $A \subseteq B \subseteq B(H)$ be an inclusion of $C^*$-algebras where $H$ is a Hilbert space. Assume the following conditions:

- $B$ is a von Neumann algebra with $A'' = B$.
- The inclusion $A \subseteq B(H)$ is non-degenerate.
- $B$ contains $\operatorname{id}_H$, but $A$ does not.

Let $\omega \in B^*$ be a normal state and $(H_\omega, \pi_\omega, \xi_\omega)$ be the associated GNS-triplet. If $\{u_\lambda\}_{\lambda\in \Lambda}$ is an approximate unit for $A$, determine whether
\[\lim_{\lambda \in \Lambda}\|\pi_\omega(u_\lambda)\xi_\omega-\xi_\omega\| = 0.\]

Answer: 0


Question: Evaluate the limit: $$ \lim_{(x,y)\to(1,1)} \frac{(x-1)^{4/3}-(y-1)^{4/3}}{(x-1)^{2/3}+(y-1)^{2/3}} $$

Answer: 0


Question: Determine the measure of the subset of the interval \([0,1]\) consisting of all numbers that do not have the digit 4 in their decimal expansion.

Answer: 0


Question: Determine the number of zeros of the polynomial \( f(z) = z^6 - 5z^4 + 3z^2 - 1 \) within the closed disk \( |z| \leq 1 \).

Answer: 4


Question: How many trees can be formed on the vertex set $V=\{1,2,3,4,5,6,7,8,9\}$ such that the degree of vertex 4 is exactly 5?

Answer: 17920


Question: Does there exist a noncommutative ring $R$ without an identity, but with an element $e \in R$ such that $ex = x$ for all $x \in R$, while $xe \neq x$ for some $x \in R$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x^2+y^2 \to +\infty}\frac{\log{\left(x^2+e^{|x|+|y|}\right)}}{x^2y^2+|x|+|y|} \] in the domain \(\Bbb D=[1, +\infty[ \times [1, +\infty[\).

Answer: 0


Question: Given that there are $k$ equally frequent colors, and we do not know $k$, we examine four smarties and observe that they are red, green, red, and orange. Find the maximum likelihood estimate (MLE) for the number of colors, $k$. Assume that the probability of observing any specific sequence of colors is equally likely.

Answer: 3


Question: Let \( F = \langle a, b \rangle \) be a free group. If \( N \) is the normal subgroup of \( F \) generated by \( a^2 \) and \( b^2 \), find the index of \( N \) in \( F \).

Answer: \infty


Question: Determine the degree of the field extension \(\mathbb{Q}(\sqrt{3}, \sqrt{5})\) over \(\mathbb{Q}\).

Answer: 4


Question: Consider the fixed point iteration defined by \( x_{n+1} = 2 \sin x_n + \tan x_n - 3x_n \), where \( g(x) = 2 \sin x + \tan x - 3x \). Determine the order of convergence at the fixed point \( \alpha = 0 \). (Hint: Using Taylor's series might be easier.)

Answer: 5


Question: Evaluate the line integral \( \int_c \frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy \) along the curve \( c(t) = (1+2\cos t, \sin t) \) for \( 0 \leq t \leq 2\pi \).

Answer: 2\pi


Question: Determine whether the integral \( \int_0^\infty x^2 e^{-ax} \, dx \) converges, and if so, find its value.

Answer: \dfrac{2}{a^3}


Question: Find the codimension of the subspace $c_0$ within the space $c$, where $c$ is the set of all convergent sequences and $c_0$ is the set of sequences tending to zero. Specifically, determine $\text{dim} \ c/c_0$. Provide a justification for your answer.

Answer: 1


Question: Let \(A\) be a 2x2 orthogonal matrix such that \(AA^T = I\) and \(\det(A) < 0\). Find \(\det(I + A)\).

Answer: 0


Question:
Let $f$ be an analytic function such that the real part of $f$, denoted as $\Re(f)$, is non-negative. Determine which of the following statements is true:

1. $\Im(f)$ is constant.
2. $\Im(f) \ge 0$.
3. $f$ is constant.
4. $\Re(f) = |z|$.

Select the correct statement.

Answer: 3


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

Answer: 2


Question: Compute the limit \( \lim_{z \to 0} \frac{z^2\sin(z)-z\sin(z^2)}{z^5} \) using the Taylor series expansions of the functions \( f(z) = z^2\sin(z) \) and \( g(z) = z\sin(z^2) \) centered at \( z_0 = 0 \).

Answer: -\dfrac{1}{6}


Question: Evaluate the integral \( \int_{-1}^1 g(x)\,dx \), where \( g(x) = \frac{d}{dx} \left( \frac{1}{1+2^{1/x}} \right) \). Assume that the function is not defined at \( x = 0 \). Can the integral still be evaluated to \( \frac{2}{3} \) under Riemann's approach?

Answer: \dfrac{2}{3}


Question:
For positive real numbers $p$ and $q$, determine the condition under which the series \[ \frac{2^p}{1^q} + \frac{3^p}{2^q} + \frac{4^p}{3^q} + \ldots \] is convergent. Choose the correct option:

(a) $p < q - 1$
(b) $p < q + 1$
(c) $p \ge q - 1$
(d) $p \ge q + 1$

Answer: A


Question:
Consider a random walk on the integer lattice \(\mathbb{Z}^2\) starting at the origin \((0,0)\). At each step, the walk moves with equal probability to one of the four adjacent lattice points (up, down, left, or right). Let \(r\in\mathbb{N}\) be a fixed integer, and define the square \(S = \{(i,j) \in \mathbb{Z}^2 : ||(i,j)||_1 \leq r\}\) as the region "close to the origin". Define the event \(A_n\) as the random walk being within the square \(S\) for more than half of its \(n\) steps. Determine the value of the limit:
\[
\lim_{n\to\infty} \mathbb{P}(A_n)
\]

Answer: 0


Question: Consider a random sample $X_1, X_2, \ldots, X_n$ from a uniform distribution $U(\theta, \theta + 1)$. We want to test the hypotheses $H_0: \theta = 0$ versus $H_1: \theta = \theta_1$ for some $0 < \theta_1 < 1$. Using Neyman-Pearson's lemma, find the critical value $k$ for the likelihood ratio test $\Lambda \leq k$ such that the test has size $\alpha$, i.e., $P_{\theta = 0}(\Lambda \leq k) = \alpha$. Assume $\Lambda = \frac{1}{I_{(\theta_1, \infty)}(y_1)}$, where $y_1 = \min(x_1, \ldots, x_n)$. Determine the value of $k$.

Answer: 1


Question: Evaluate the limit: $$\lim \limits_{x\rightarrow 0} \left(\frac{1}{x^2}-\frac{1}{\sin^2x}\right)$$ without using L'Hôpital's rule or Taylor series.

Answer: -\dfrac{1}{3}


Question: Let $H \leq G$. Define a map $\psi: N(H) \rightarrow \text{Aut}(H)$ given by $\psi(g) = \phi_g$, where $\phi_g$ is the inner automorphism of $H$ induced by $g$: $\phi_g(h) = ghg^{-1}$ for all $h \in H$. Find the kernel of the homomorphism $\psi$. \( N(H) = \{g \in G \mid gH = Hg\} \) and \( \text{Aut}(H) = \{gHg^{-1} \mid g \in G\} \).

Answer: C_G(H)


Question: Is there a countable family of open subsets of \( \mathbb{R} \) or \( [0,1] \) such that each rational number belongs to only finitely many of the open sets, while each irrational number belongs to infinitely many of the sets?

Answer: Yes


Question: Assume we have two sequences of random variables \(\{X_{n}\}\) and \(\{Y_{n}\}\) defined on the same probability space such that \(X_{n} \stackrel{d}{\to} X\) and \(Y_{n} \stackrel{d}{\to} Y\), where \(X\geq_{s}Y\) (stochastically greater). Is it true that \(P[X_{n} \geq Y_{n}] \to 1\) as \(n\to\infty\)?

Answer: No


Question: Let $A$ be a nondegenerate skew-symmetric matrix over the complex field $\mathbb{C}$. Is there an invertible matrix $P$ such that \( P^{T}AP=\begin{bmatrix} 0 & I_{\ell} \\ -I_{\ell} & 0 \end{bmatrix} \), where $P^{T}$ is the transpose of $P$ and $I_{\ell}$ is the $\ell$ by $\ell$ identity matrix?

Answer: Yes


Question: Consider a real-analytic function $f(x,y,t):[-1,1]^3 \to \mathbb{R}$. For any fixed $x,y$, the function $f(x,y;t)$ is not constant on $[-1,1]$. The zeros of a non-constant real-analytic function of one variable are isolated, so we define $N(x,y)$ as the number of zeros of $f(x,y;t)$ on $[-1,1]$. Is there a constant $C$ such that $$\sup_{(x,y)\in[-1,1]^2}N(x,y)\le C$$ holds? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_C \frac{x}{x^2+y^2}dx + \frac{-y}{x^2+y^2}dy \), where \( C \) is the path determined by the square with vertices \((1,1),(-1,1), (-1,-1), (1,-1)\) in the counterclockwise direction.

Answer: 0


Question: Let $A$ be a ring and $M$ an $A$-module. Suppose the map \( M \otimes M^\vee \rightarrow \operatorname{End}(M) \) is an isomorphism, where \( M^\vee = \operatorname{Hom}(M, A) \). Is the canonical map \( M \rightarrow (M^\vee)^\vee \) necessarily an isomorphism?

Answer: Yes


Question: Calculate the mass of the region \( R \) enclosed by the hyperbolas \( x^2 - y^2 = 1 \), \( x^2 - y^2 = 6 \), \( xy = 3 \), and \( xy = 8 \) in the first quadrant. The density \( \delta(x,y) \) is proportional to the square of the distance from the origin, with a proportionality constant of 2. Use the change of variables \( u = x^2 - y^2 \) and \( v = xy \) to evaluate the integral \( m = \int \int_R \delta(x,y) \, dA \).

Answer: 25


Question: Solve the congruence equation: \( x^{17} \equiv 243 \pmod{257} \).

Answer: 28


Question: Determine the value of \( r \) in the logistic map bifurcation diagram that corresponds to \( r = 1 + \sqrt{6} \).

Answer: 1 + \sqrt{6}


Question:
Determine all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy Jensen's functional equation:
\[ f\left(\frac{x+y}{2}\right) = \frac{f(x)+f(y)}{2} \]
for all \( x, y \in \mathbb{R} \).

Answer: f(x) = ax + b


Question: Let \( f :[0,1] \rightarrow \mathbb{R} \) be a continuous function that is differentiable on \((0,1)\) and satisfies \( f(0)=f(1)=0 \). Determine whether there exists a point \( x_0 \in (0,1) \) such that \( f(x_0)=f'(x_0) \).

Answer: Yes


Question: Determine the values of \(\alpha > 0\) for which the series \(\sum_{n=1}^\infty \frac{1}{n^\alpha \ln(1+n^\beta)}\) converges, given that \(\beta > 1\).

Answer: \alpha > 1


Question: Suppose $\mathfrak{A}$ is a Boolean algebra and $\mathfrak{J}$ is a chain of ideals in $\mathfrak{A}$ ordered by inclusion, such that none of its elements is countably generated. Is the union $\bigcup \mathfrak{J}$ countably generated?

Answer: No


Question: Evaluate \( \lim_{r \to \infty} \int_{0}^{\pi} e^{-r\cos^2(\theta)} \, d\theta \).

Answer: 0


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{\sin (5x)}{1+(x-\frac{\pi}{2})^2} \, dx \) using a complex integral theorem.

Answer: \pi e^{-5}


Question: Evaluate the integral \( I(x) = \int_{0}^{\frac{\pi}{2}}{\frac{du}{x^2\cos^2u+\sin^2u}} \).

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


Question: Let $A$ be a $3 \times 3$ matrix with eigenvalues $1$, $2$, and $3$. Find $\det(B)$ where $B = A^2 + A^T$. \(A^T\) denotes the transpose of matrix \(A\).

Answer: 144


Question: Is $V_k$ a transitive model of ZFC when $k$ is inaccessible? Justify your answer.

Answer: Yes


Question: Solve the congruence \(2^x \equiv 5 \pmod{13}\) for \(x\).

Answer: 9


Question: Find all positive integers \(k\) such that \(7 \times 2^k + 1\) is a perfect square.

Answer: 5


Question: Does the group $\text{Sym}(\omega)$, consisting of all bijections $f: \omega \to \omega$ with composition as the group operation, have $2^{\aleph_0}$ pairwise non-isomorphic subgroups?

Answer: Yes


Question: Find the quotient group $N(D_4)/D_4$, where $N(D_4)$ is the normalizer of the subgroup $D_4$ in the dihedral group $D_{16}$. Provide the structure of the quotient group.

Answer: \mathbb{Z}_2


Question: Calculate \( F(3500) \) using the formula \( F(n) = \sum_{d\mid n} \mu(d)d \), where \( \mu \) is the Möbius function.

Answer: -24


Question: How many ways are there to place nine different rings on four fingers, assuming the order of rings on each finger does not matter?

Answer: 262144


Question: Determine whether the following statement is true: \( \lim_{z \rightarrow 0} f(z) = 100i \quad \Leftrightarrow \quad \lim_{z\rightarrow \infty} f(1/z) = 100i. \)

Answer: True


Question: Let $f \in C^{\infty}(\mathbb{R};\mathbb{R})$ be a bounded and strictly monotonically increasing function, whose $n^{th}$-derivative does not vanish except possibly at a finite number of points. For every $n \in \mathbb{N}$, is the set of derivatives $\{f^n\}_{n \in \mathbb{N}}$ linearly independent?

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{\infty} e^{-2ix} \, dx \).

Answer: -\dfrac{i}{2}


Question:
Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a continuous map such that $f(x) = 0$ only for finitely many values of $x$. Which of the following statements is true?

1. Either $f(x) \leq 0$ for all $x$ or $f(x) \geq 0$ for all $x$.
2. The map $f$ is onto.
3. The map $f$ is one-to-one.
4. None of the above.

Answer: 1


Question: Let $G$ be a group and $H$ a normal subgroup of $G$ with index $p$, where $p$ is a prime number. Suppose $K$ is a subgroup of $G$ that is not contained in $H$. Is it true that $G = HK$?

Answer: Yes


Question: Evaluate the sum of the series: \[ \sum_{n=0}^{\infty} (n\bmod 3)\cdot  2^{-n} \]

Answer: \dfrac{8}{7}


Question: Suppose $f(z)$ is a twice differentiable function on the interval $[0,1]$ with $f(0)<0$, $f(1)>0$, and $f''(z)<0$ for all $z$ in $(0,1)$. Prove that there is a unique $x \in (0,1)$ such that $f(x)=0$.

Answer: x


Question: Given a financial market with a bond $A$, a stock $S$, and call options $f$ on the stock, consider two times $0$ and $T$ with two possible outcomes for $S$ at time $T$. The bond and stock values are $A(0)=100$, $A(T)=120$, $S(0)=50$, $S^u(T)=70$, $S^d(T)=30$. Construct a portfolio consisting of shares of $S$ and call options $f$ with strike $K=50$ and expiration $T$ such that the portfolio value at time $T$ is $120$ regardless of the stock's movement. What is the price of the call option?

Answer: 12.5


Question: Determine whether the series \( a_n = \frac{1}{\sqrt{n}} \left( 1 + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}} \right) \) converges or diverges as \( n \to \infty \). If it converges, find its sum.

Answer: 2


Question:
Find the value of \( \theta \) that maximizes the area given by the formula:
\[ A = r^2(\sin\theta\cos\theta-\sqrt{3}\sin(\theta)^2) \]

Answer: \dfrac{\pi}{12}


Question: Derive \((\forall x) Lx\) from the premise \(- (\exists x) -Lx\) using primitive rules of logic.

Answer: (\forall x) Lx


Question: In an acute-angled triangle $ABC$, points $D$, $E$, and $F$ are located on sides $BC$, $CA$, and $AB$, respectively. The line $AD$ is perpendicular to $BC$, $AE = EC$, and $CF$ bisects angle $C$ internally. The line $CF$ intersects $AD$ and $DE$ at points $M$ and $N$, respectively. Given that $FM = 2$, $MN = 1$, and $NC = 3$, determine the perimeter of triangle $ABC$. Provide your answer as a single numerical value.

Answer: 12\sqrt{3}


Question: Solve the equation: \[ \log_3\left(\frac{1}{\sqrt{\log_3(x)}}\right) = \log_9\left(\log_9\left(\frac{x}{3}\right)\right) \]

Answer: 9


Question: Let $b$ be an integer greater than 2, and let $N_b = 1_b + 2_b + \cdots + 100_b$, where the sum contains all valid base $b$ numbers up to $100_b$. Compute the number of values of $b$ for which the sum of the squares of the base $b$ digits of $N_b$ is at most 512.

Answer: 30


Question: Determine for which values of \( \alpha > 1 \) the series \( \sum_{n=1}^{\infty}\left[1-\left(1-\frac{1}{n^\alpha}\right)^{2n}\right] \) converges.

Answer: \alpha > 2


Question: Determine the fundamental group of a Klein bottle with two points removed.

Answer: \mathbb{Z} * \mathbb{Z} * \mathbb{Z}


Question: Determine whether the function \( f(y) = \sum_{k=1}^{\infty}\frac{1}{k^2}\frac{|y- q_k|}{\sqrt{1 + (y-q_k)^2}} \) is Lipschitz with respect to \( y \), where \( q_k \) is a sequence of rational numbers.

Answer: Yes


Question: Let $p(x)$ and $q(x)$ be two polynomials such that $p(2)=5$, $p(3)=12$, and $p(q(x))=p(x)q(x)-p(x)$. Find the value of $q(10)$.

Answer: 114


Question: Determine whether the only integer solution to the equation \(77x^{12}-49y^{12}+2z^{12}=63xyz^{10}\) is the trivial solution \((0,0,0)\).

Answer: (0, 0, 0)


Question: Find the value of \( k \) such that \( \lim_{p\rightarrow 1^{+}} \sum_{n=1}^{\infty} \frac{1}{n^p} = k \).

Answer: \infty


Question: What is the limit of $\Im{(z_n)}$ as $n$ goes to infinity, where $z_n$ are the non-trivial zeros of the Riemann zeta function with positive imaginary part, sorted by ascending imaginary part?

Answer: \infty


Question: Identify the space for which the set \( \{1, \sin(x), \cos(x), \sin(2x), \cos(2x), \ldots\} \) forms a basis.

Answer: L^2([-\pi, \pi])


Question: Let $X$ and $Y$ be independent and identically distributed random variables with support $[0, \infty)$. Is the function $f(x) = \Pr(X+Y \le x | X \le x)$ increasing for $x > 0$?

Answer: Yes


Question: Calculate the expectation of $X$, the number of failures until the $r$-th success, in an infinite series of Bernoulli experiments with success probability $p$ and failure probability $q = 1 - p$. Express your answer in terms of $r$, $p$, and $q$.

Answer: \dfrac{rq}{p}


Question: Find the number of ordered triplets \((x, y, z)\) such that \(x, y, z > 0\) and the following inequalities are satisfied: \(x(1-y) > \frac{1}{4}\), \(y(1-z) > \frac{1}{4}\), and \(z(1-x) > \frac{1}{4}\).

Answer: 0


Question: Determine the cardinality of the set of equivalence classes of well-orders on \( \mathbb{N} \) under the equivalence relation \( R \equiv R' \leftrightarrow (\mathbb{N},R) \cong (\mathbb{N},R') \), where \( R \) and \( R' \) are well-orders on \( \mathbb{N} \).

Answer: \aleph_1


Question: Given that $f(x)$ is a quadratic function with $f(0) = 1$, and the integral \( \int \frac{f(x)}{x^2(x+1)^3} \, dx \) is a rational function, find the value of $f'(0)$. \( \boxed{} \)

Answer: 3


Question: Does a function $f \in L^1(\mathbb{R}, m)$, where $m$ is the Lebesgue measure, necessarily have compact support? Justify your answer.

Answer: No


Question: Evaluate the limit: $$\lim_{x \to 0^{+}} (\sin x)^{\cos x}\left(\frac{\cos^{2}x}{\sin x} - \sin x\log(\sin x)\right)$$

Answer: 1


Question: If $\hat{\theta}$ is a consistent estimator based on an i.i.d. random sample $X_1, \ldots, X_n$ with $\text{var}(\hat{\theta}) < \infty$ for all $n$, does it follow that $\lim_{n\to\infty} \text{var}(\hat{\theta}) = 0$?

Answer: No


Question: Calculate the intersection point of the tangent planes to the surface $S$ in $\mathbb{R}^3$ given by the graph of $f(x, y) = xy^2 − 2y^2 + e^x$ at the points $(x, y) = (0, 0)$, $(0, 1)$, and $(0, 2)$. Provide your answer as a point in 3D space.

Answer: (-2, 0, -1)


Question: Evaluate the limit \( \lim_{n\rightarrow\infty}\frac{1}{n}(|\sin 1|+|\sin 2|+\ldots +|\sin n|) \).

Answer: \dfrac{2}{\pi}


Question: Find the closest distance from the origin to the surface defined by the equation \(x^2 - z^2 - 1 = 0\) using the method of Lagrange multipliers.

Answer: 1


Question: Find all pairs \((n, p)\) of natural numbers \(n\) and prime numbers \(p\) that satisfy the equation \(p(p-1) = 2(n^3 + 1)\). Calculate the sum \(\sum (n_i + p_i)\) for all such pairs \((n_i, p_i)\).

Answer: 147


Question: Find the integral solution \((x, y)\) of the equation \(x^2 - y^3 = 1\) where \(x > 1\) and \(y > 1\).

Answer: (3, 2)


Question: Determine the minimum number of black boxes required to pinpoint the location of the origin of a signal that caused spaceships to explode. Assume the signal spreads through space at a constant velocity uniformly in every direction, and each black box contains the spatial coordinates of the ship and the time it received the signal.

Answer: 4


Question:
Let \( f \) be a differentiable function satisfying the equation:
\[ \int_0^{f(x)} f^{-1}(t) \, dt - \int_0^x (\cos t - f(t)) \, dt = 0 \]
and \( f(\pi) = 0 \). Define \( g(x) = f(x) \) for all \( x \in \mathbb{R}_0 = \mathbb{R} \cup \{0\} \).

If \( \int_0^{\infty} (g(x))^3 \, dx = A \) and \( \int_0^{\infty} \frac{1-g(x)}{x^2} \, dx = \frac{kA}{k+1} \), find the value of \( k \).

Answer: 2


Question: Evaluate the line integral \( \oint_C (x-z) \, dx + (x + y) \, dy + (y+z) \, dz \) where \( C \) is the ellipse formed by the intersection of the plane \( z=y \) with the cylinder \( x^2 + y^2 = 1 \), oriented counterclockwise as viewed from above, using Stokes' Theorem.

Answer: 2\pi


Question:
Find the partial derivative \( \frac{\partial X}{\partial x_0}(t,0) \) for the function \( X: \mathcal{R} \times \mathcal{R} \rightarrow \mathcal{R} \) defined by the differential equation:
\[ \frac{dX}{dt}(t,x_0)=X(t,x_0)+\sin(X^2(t,x_0)), \]
with the initial condition \( X(0,x_0)=x_0. \)

Answer: e^t


Question: Determine whether there exists a Noetherian local domain $A$ with a prime ideal $P$ such that $\operatorname{ht}P + \dim A/P < \dim A$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_\gamma \frac{dz}{(z-\frac{1}{2}-i)(z-1-\frac{3i}{2})(z-1-\frac{i}{2})(z-\frac{3}{2}-i)} \), where \( \gamma \) is the polygon with vertices \( 0, 2, 2+2i, 2i, 0 \).

Answer: 0


Question: Determine the maximum length needed for a random character password, using any of the 95 ASCII printable characters, to be more secure against a guessing program than a pass-phrase consisting of three to five random words chosen from a pool of one million words. Assume the guessing program cycles between guessing random words and random characters, and the pass-phrase has a total of $10^{6\cdot3} + 10^{6\cdot4} + 10^{6\cdot5}$ combinations.

Answer: 16


Question:
Determine the correct well-formed formula (WFF) for the following statement: "A person may steal something if the person is a thief and the person likes the thing and the thing is valuable." Choose between the two options:
1. ∀x ∃y (person(x) ∧ steal(x,y) → thief(x) ∧ likes(x,y) ∧ valuable(y))
2. ∀x ∃y (person(x) ∧ thief(x) ∧ likes(x,y) ∧ valuable(y) → steal(x,y))

Which formula correctly represents the given statement?

Answer: 2


Question: The random variables Y and X have a bivariate normal distribution with means 1 and 2, variances 2 and 5, and covariance 3. Compute the squared correlation between Y and E[Y|X].

Answer: \dfrac{9}{10}


Question: Compare the values of $P(28, 19, 1, 2)$ and $P(30, 10, 2, 7)$, where $P(n, k, a, b)$ denotes the number of arrangements of $n$ indistinguishable balls into $k$ distinct buckets such that there are no less than $a$ and no more than $b$ balls in each bucket. Which value is higher?

Answer: P(28, 19, 1, 2)


Question: Let $A$ be a $6 \times 6$ matrix with characteristic polynomial $x(x+1)^2(x-1)^3$. Find the dimension of the centralizer $\gamma = \{B \in M_6(\mathbb{R}) : AB = BA\}$.

Answer: 14


Question: Evaluate the integral by changing the order of integration: \[ \int^{\infty}_{0} \int^{x}_{0} x e^{-x^2/y} \, dy \, dx \]

Answer: \dfrac{1}{2}


Question: Use Stokes' Theorem to find the line integral of the vector field \( \langle xz, 7x+2yz, 3x^2 \rangle \) around the path \( C \) determined by the circle \( x^2 + y^2 = 9, z=3 \).

Answer: 63\pi


Question:
Find the minimum value of \(x_1 + x_2 + x_3 + \dots + x_{10}\) given that the real numbers \(x_1, x_2, \dots, x_{10}\) satisfy the equation:
\[ \sqrt{x_1 - 1^2} + 2 \sqrt{x_2 - 2^2} + 3 \sqrt{x_3 - 3^2} + \dots + 10 \sqrt{x_{10} - 10^2} = \frac{x_1 + x_2 + \dots + x_{10}}{2}. \]

Answer: 770


Question: Find the number of integer solutions to the equation \((x_1)(x_2)(x_3)(x_4) = 2016\), where each \(x_i\) is an integer.

Answer: 17920


Question: Let \( P(x) = 1 + x + x^2 + x^3 + x^4 + x^5 \). Determine the remainder when \( P(x^{12}) \) is divided by \( P(x) \).

Answer: 6


Question: Evaluate the limit of the series: \[ \sum_{n=1}^{\infty}\frac{n^2}{n!} \]

Answer: 2e


Question: Evaluate the contour integral \( \int_C f(z) \, dz \) where \( f(z) = 1 \) for \( y < 0 \) and \( f(z) = 4y \) for \( y > 0 \). The contour \( C \) is the arc from \( z = -1-i \) to \( z = 1+i \) along the curve \( y = x^3 \).

Answer: 2 + 3i


Question: An explorer is trapped in a cave with three tunnels. The first tunnel leads to the surface after 1 day, the second leads back to the cave in 4 days, and the third leads back to the cave in 2 days. The explorer chooses the first, second, and third tunnels with probabilities 0.2, 0.3, and 0.5, respectively. Given that the expectation of the number of days until the explorer is free has been calculated, compute the variance of the number of days until the explorer is free.

Answer: 155


Question: Find the integer value of $m$ that satisfies the congruence $108 \equiv m^{37} \pmod{143}$. Provide a step-by-step solution.

Answer: 69


Question: Find the residue of the function \( f(z) = \frac{e^{4z} - 1}{\sin^2(z)} \) at \( z = 0 \).

Answer: 4


Question: A convex quadrilateral has sides of lengths 10, 6, 8, and 2, in that order. If the angle between the diagonals is \( \frac{\pi}{4} \), find the area of the quadrilateral.

Answer: 31


Question: Compute the dimension of the $k$-vector space $k[x,z]/(x^{2}+1,z^{2})$, where $\mathrm{char}(k) \neq 2$. Provide your answer as a single integer.

Answer: 4


Question: Determine whether 3 is ever a seventh power modulo a prime $p$ if $p \equiv 1 \pmod{7}$. Provide a justification for your answer.

Answer: Yes


Question: Given a non-zero vector \( v \) and a positive definite matrix \( M \), consider the vector \( M^k v \) obtained by multiplying \( v \) by \( M \) \( k \) times. Determine if the angle \( \theta \) between \( v \) and \( M^k v \) is always less than 90 degrees, i.e., if \( v^T M^k v > 0 \) for all positive integers \( k \).

Answer: Yes


Question: Determine the limit of the sequence \( x_n = n\log n\left(\frac{\log(n+2)}{\log (n+1)}-1\right) \).

Answer: 1


Question: A line through the origin intersects the lines \( \frac{x-2}{1} = \frac{y-1}{-2} = \frac{z+1}{1} \) and \( \frac{x-\frac{8}{3}}{2} = \frac{y+3}{-1} = \frac{z-1}{1} \) at points \(P\) and \(Q\) respectively. Calculate \((PQ)^2\).

Answer: 6


Question: A manufacturer of car radios ships them to retailers in cartons of $n$ radios. The profit per radio is $\$59.50$, minus a shipping cost of $\$25$ per carton, so the profit is $59.5n-25$ dollars per carton. To promote sales by assuring high quality, the manufacturer promises to pay the retailer $\$200X^2$ if $X$ radios in the carton are defective. Suppose radios are produced independently and that $5\%$ of radios are defective. How many radios should be packed per carton to maximize the expected net profit per carton?

Answer: 50


Question: Given the joint density function \( f(x,y) = 6x^2y \) for \( 0 \leq x \leq 1 \) and \( 0 \leq y \leq 1 \), compute the probability \( P(X+Y>1) \). Verify if the integral setup \( \int_{0}^{1} \int_{y}^{1-y} 6x^2y \, dx \, dy \) correctly represents this probability.

Answer: \dfrac{9}{10}


Question: Consider a non-homogeneous differential equation $Ly = f$ in the ring of exponential sums $\mathbb{C}[e^{\lambda x} \mid \lambda \in \mathbb{C}]$, where $f = e^{b_i x}$. If $b_i$ is a root of the characteristic equation of the associated homogeneous equation with multiplicity $M$, what is the form of the particular solution? Is it $y(x) = Cx^M e^{b_i x}$ or $y(x) = e^{b_i x} (A_0 + A_1 x + \dots + A_M x^M)$?

Answer: y(x) = Cx^M e^{b_i x}


Question: Is there a number $K$ such that for any natural number $n$, at least one of the numbers $n, n+1, \ldots, n+K-1$ is a product of two distinct primes?

Answer: No


Question: Evaluate the limit: \[ \lim_{L \rightarrow \infty} (\ln{L})^2\left[1-\left(1-e^{-\frac{1}{2}(\sqrt{L}-2)}\right)^L\right] \] and show that it approaches zero.

Answer: 0


Question: Evaluate the line integral \( \int_\gamma \omega \) where \( \omega(x, y) = x^3 \, dx + \frac{x}{x^2+y^2} \, dy \) and \( \gamma: [0, 2\pi] \to \mathbb{R}^2 \setminus \{(0, 0)\} \) is given by \( \gamma(t) = (\cos t, \sin t) \).

Answer: \pi


Question: Is the canonical representation of the group $SO(3,1)$ on a 4-dimensional complex vector space irreducible?

Answer: Yes


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

Answer: \dfrac{3}{2}


Question: Reduce the lambda calculus expression \((\lambda x. (\lambda y. x \, y) \, x) \, (\lambda z.p)\) to its simplest form.

Answer: p


Question: Let \( A \in M_n(\mathbb{C}) \) be a matrix such that \( A = A^* \) and \( x \neq 0 \implies \langle Ax, x \rangle > 0 \). For such a matrix, \( A^z \) can be defined for \( z \in \mathbb{C} \). Determine whether the map \( \mathbb{C} \to \mathbb{C} : z \mapsto (A^z)_{i,j} \) is analytic.

Answer: Yes


Question: Find the volume of the solid obtained by revolving the area enclosed by the curve \(27ay^{2} = 4(x-2a)^{3}\), the \(x\)-axis, and the parabola \(y^{2} = 4ax\) about the \(x\)-axis.

Answer: 80\pi a^3


Question: For a zebra group of order $2^8p^b$, where $p$ is a prime, determine the maximum number of abelian 2-group stripes that can appear in its chief series.

Answer: 8


Question: Evaluate the integral: \[ \int_0^{2\pi} \frac{1}{2} \sin(t) (1- \cos(t)) \sqrt{\frac{1}{2} - \frac{1}{2} \cos(t)}\,dt \]

Answer: 0


Question: Given that the Fourier transform of a function $f(x)$, defined over $(-\infty, \infty)$, is approximately $g(w) \approx 1-w^2$ for small $w$ ($w \ll 1$), but $g(w)$ over the full range of $w$ is not provided, can we calculate the value of \(\int_{-\infty}^\infty x^2f(x) \,dx\)?

Answer: 2


Question: Evaluate the integral: \[ \int_{-\infty}^{\infty} \frac{\sin^2x}{x^2} e^{ix} \, dx \]

Answer: \dfrac{\pi}{2}


Question: Evaluate the integral \( \int_{\gamma} \omega \), where \( \gamma \) is the circumference with center \( C(2,0) \) and radius \( 1 \), and \( \omega = \frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy \). Use the parametrization \( \gamma \colon t \mapsto (2+\cos t,\sin t) \) to compute the integral.

Answer: 0


Question: What is the minimal number of vertices required for a triangulation of the Klein bottle?

Answer: 6


Question: Compute the limit \(\lim_{x\to 0^{+}}(\cos x-1)e^{\cot x}\).

Answer: -\infty


Question: Does there exist a positive continuous function $f(x)$ on $\mathbb{R}$ such that $f(x) \to 0$ as $x \to \pm \infty$, but $f^p$ is not integrable for any $p > 0$?

Answer: Yes


Question: Let $\kappa \geq \aleph_0$ be an infinite cardinal, and let $\mathcal{A}$ be a collection of subsets of $\kappa$ such that for all $A \in \mathcal{A}$, $|A| = \kappa$, and for $A, B \in \mathcal{A}$ with $A \neq B$, $|A \cap B| < \kappa$. Does there exist a set $D \subseteq \kappa$ such that for all $A \in \mathcal{A}$, $A \cap D \neq \emptyset$ and $A \not\subseteq D$?

Answer: Yes


Question: Consider two unit vectors $u$ and $v$, with the angle between them denoted as $\theta$. Prove or disprove the claim that \[ \lim_{\theta \to 0} \frac{\theta}{|u - v|} = 1. \]

Answer: 1


Question: Given the function \( f(x) = \frac{e^{(x-1)^2}-1}{(x-1)^{2}} \), find the value of \( f''(1) \).

Answer: 1


Question: Find the minimum value of \((1+a^2)(1+b^2)(1+c^2)\) given that \(a, b, c \geq 0\) and \(ab + bc + ac = 27\).

Answer: 676


Question: For every positive integer \( n \in \mathbb{N} \), does there exist a finite, simple, undirected graph \( G = (V, E) \) such that \( G \) does not have a complete minor with more than \( \frac{\delta(G)}{n} \) vertices, where \( \delta(G) \) denotes the minimum degree of \( G \)?

Answer: Yes


Question: Compute the limit \( \lim_{n \rightarrow \infty} \binom{n}{n-k} p^n (1-p)^n \) where \( 0 \leq p \leq 1 \) and \( 0 \leq n-k \leq n \).

Answer: 0


Question: Let \( p \) be a prime number. Determine the largest integer \( m \) such that the set \( \{1, \zeta_p, \zeta_p^2, \ldots, \zeta_p^m\} \) is linearly independent over \( \mathbb{Q} \), where \( \zeta_p \) is a primitive \( p \)th root of unity.

Answer: p-2


Question: Evaluate the integral \( \int_0^{2\pi} e^{-a\cos (\theta-b)} \, d\theta \), where \( a \) and \( b \) are real constants.

Answer: 2\pi I_0(a)


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

Answer: 39


Question: Find the power series expansion of the function \( f(x) = \frac{1}{1+x+x^2+x^3} \) around \( x_0 = 0 \) and determine its radius of convergence.

Answer: 1


Question: Find the potential function for the conservative vector field \( F(x,y,z) = (2xyz^2, x^2z^2 + z\cos(yz), 2x^2yz + y\cos(yz)) \). Use this potential function to compute the line integral of \( F \) along the line segment from \((0,0,0)\) to \((5,0,5)\).

Answer: 0


Question:
Consider two real polynomials $f(x)$ and $g(x)$ of degrees $m > 1$ and $n > 0$, respectively, that satisfy the equation $f(x^2 + 1) = f(x)g(x)$ for every $x \in \mathbb{R}$. Determine which of the following statements is correct:

1. $f$ has exactly one real root $x_0$ such that $f'(x_0) \neq 0$.
2. $f$ has exactly one real root $x_0$ such that $f'(x_0) = 0$.
3. $f$ has $m$ distinct real roots.
4. $f$ has no real root.

Select the correct option.

Answer: D


Question: Let $R$ be a commutative ring with unity having at most 5 distinct ideals (including $\{0\}$ and $R$ itself). Is it true that $R$ is a principal ideal ring, meaning every ideal of $R$ is principal?

Answer: Yes


Question: Given that $Z = X + Y$ with $X \sim N(0,9)$ and $Y \sim N(0,4)$, and you have observed $Z = 3$, find the most probable value of $X$ that would have resulted in this observation.

Answer: \dfrac{27}{13}


Question: Given an affine real algebraic variety $X$ defined as the zero locus of $k$ degree-two polynomials in $\mathbb{R}^N$, and a map $F : X \to \mathbb{R}$ that projects onto one of the coordinates, determine an asymptotic upper bound on the number of critical values of $F$ in terms of $k$ and $N$. Assume $X$ is compact in the subspace topology. Is it true that if $k \approx N$, the number of critical values of $F$ is bounded by $e^{cN}$ for some constant $c$?

Answer: e^{O(N)}


Question: Determine whether there exist subsets \(A_1, A_2, \ldots\) of \(\mathbb{N}\) such that for any two natural numbers \(i\) and \(j\), the size of the intersection \(|A_i \cap A_j|\) is equal to the greatest common divisor \((i,j)\).

Answer: Yes


Question: Let $E \subset [0,1]$ be a set with Lebesgue measure $|E| = 0$. If $f(x) = x^3$, prove that the Lebesgue measure of the image $|f(E)| = 0$.

Answer: 0


Question: Find the coefficient of \(x^{20}\) in the expansion of the generating function \(g(x) = \frac{5(1-x^5)^7}{(1-x)^{2}}\).

Answer: -175


Question: Evaluate the limit: \[ \lim_{x,y\to\infty}\dfrac{(xy)^{x^2}}{(x^2+y^2)^{x^2}} \]

Answer: 0


Question: Given matrices $A$ and $B$, compute the gradient $\nabla_X g$ of the scalar field $g(X) = \operatorname{tr}(AXB)$. Express your answer in terms of $A$ and $B$.

Answer: A^\top B^\top


Question: Determine the dimension of the vector space $H_n$ of all $n \times n$ matrices $A = (a_{ij})$ with entries in $\Bbb{R}$ such that $a_{ij} = a_{rs}$ whenever $i+j = r+s$ for $i,j,r,s = 1, 2, \ldots, n$. Express your answer as a function of $n$.

Answer: 2n - 1


Question: Using Green's Theorem, find the area of the ellipse defined by \( \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 \), where \( a, b > 0 \).

Answer: \pi ab


Question: Is there a nontrivial group $G$ such that $[G,G]=G$ and $g^3=1$ for every $g \in G$?

Answer: No


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

Answer: 8


Question: Let $K$ be a semi-simple quadratic extension over $\mathbb{Q}$ and consider the regular representation $\rho: K \to M_2(\mathbb{Q})$. Compute the index of $\rho(K^\times)$ in the normalizer of $\rho(K^\times)$ in $GL_2(\mathbb{Q})$. Justify your answer.

Answer: 2


Question: Evaluate the integral \( \int_0^y w^{a-1}(1-w)^{b-1} \, dw \).

Answer: B(y; a, b)


Question: Let $E$ be an open set in $[0,1]^n$. Is it possible for the Lebesgue measure of $E$ to differ from the Lebesgue measure of its closure $\bar{E}$?

Answer: Yes


Question: Given two primes \( p \) and \( q \) such that \( p = 2q + 1 \) and \( q \equiv 3 \pmod{5} \), determine if 5 is a primitive root modulo \( p \).

Answer: Yes


Question: Determine the superior limit (lim sup) of the sequence \(a_n = \cos\left(\frac{2n^2+1}{3n}\pi\right)\).

Answer: 1


Question: A partially silvered mirror covers a square area with vertices at $(\pm 1, \pm 1)$. The fraction of incident light it reflects at a point $(x, y)$ is given by $\frac{(x-y)^2}{4}$. Assuming a uniform intensity of incident light, calculate the total fraction of light reflected by the mirror.

Answer: \dfrac{1}{6}


Question: Determine the number of ways to distribute 12 indistinguishable balls into 11 distinguishable bins such that no bin contains more than 2 balls.

Answer: 24068


Question: Compute the dimension of the Zariski tangent space at the origin of $\operatorname{Spec} A$, where $A = k[w, x, y, z] /(wz - xy)$. Determine the dimension of the cotangent space $\mathfrak m / \mathfrak m^2$, where $\mathfrak m$ is the unique maximal ideal of $A_{(w, x, y, z)}$.

Answer: 4


Question: Are there infinitely many primes $p \equiv 1 \pmod{8}$ such that $p$ can be represented as $x^2 + 64y^2$?

Answer: Yes


Question: Solve for \( r \) given that \( \frac{\log(1+r)}{r} = 0.4181 \).

Answer: 3.703


Question:
Determine whether there exists a unique value \(x_m > 0\) such that the matrix \(C(x) = A - S(x)\) is positive semi-definite and singular, where \(A\) is a real symmetric positive-definite matrix and \(S(x)\) is given by:
\[
S(x)=\begin{bmatrix}
2x & -2x \\
-2 x & 4 x-x^2
\end{bmatrix}
\]
for \(0 < x\). Specifically, verify if for any \(0 < x < x_m\), \(C(x)\) is positive-definite, and for any \(x > x_m\), \(C(x)\) has at least one negative eigenvalue.

Answer: Yes


Question: Determine whether for any sequence of real numbers \( \{a_k\}_{k\geq 1} \), there exists a smooth function \( f: (-\epsilon, \epsilon) \to \mathbb{R} \) (for some positive \( \epsilon \)) such that for every positive integer \( k \geq 1 \), \( f^{(k)}(0) = a_k \).

Answer: Yes


Question: Let $G$ be a non-abelian group. Define $C_G(x)$ as a maximal centralizer if $x \in G - Z(G)$ and there is no $g \in G - Z(G)$ such that $C_G(x) < C_G(g)$. Let $C$ be the intersection of all maximal centralizers of $G$. Is $C$ necessarily abelian?

Answer: Yes


Question: Show that the sequence \( b_n = \frac{\int_{0}^\epsilon \cos^n(x) \, dx}{\int_{\epsilon}^{1/2} \cos^n(x) \, dx} \) tends to infinity as \( n \to \infty \) for every \( \epsilon \) such that \( \frac{1}{2} > \epsilon > 0 \).

Answer: \infty


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sum_{k=1}^{2n} \binom{2n}{k} \frac{1}{4^n} \]

Answer: 1


Question: Alice and Bob are playing a coin flipping game where they flip coins until one of them wins. Alice wins when the sequence "h, h, h" appears, and Bob wins if the sequence "t, h, t" appears. Find the expected number of coin flips, $E[X]$, until one of them wins.

Answer: \dfrac{35}{6}


Question: Determine the value of \( n \) such that the limit \( \lim\limits_{x \to \infty}\frac {x^x}{(x!)^n} \) changes from diverging to converging. Here, \( x! \) refers to \( \Gamma(x+1) \).

Answer: 1


Question: Given the joint probability density function \(f(x,y) = 8xy\) for \(0 < x < y < 1\) and \(0\) otherwise, calculate the expected value \(E(XY^3)\).

Answer: \dfrac{1}{3}


Question: Find the residue of the function \( f(z) = \frac{1}{\sin(z) - 1} \) at \( z = \frac{\pi}{2} \).

Answer: 0


Question:
Find the smallest disk in the complex plane of the form \(|z - 1| < r\) that contains all the eigenvalues of the matrix:
\[
\begin{bmatrix}
1 & -2 & 3 & -2 \\
1 & 1 & 0 & 3 \\
-1 & 1 & 1 & -1 \\
0 & -3 & 1 & 1
\end{bmatrix}
\]

Answer: 4


Question: Does there exist a function \( f: \mathbb{R} \to \mathbb{R} \) such that \( f \) is continuous on a dense \( G_{\delta} \) set and discontinuous almost everywhere on \( \mathbb{R} \)? Provide a justification for your answer.

Answer: Yes


Question: A structure with a design life of 50 years is planned for a site where high intensity earthquakes may occur with a return period of 100 years. The structure is designed to have a 0.98 probability of not suffering damage within its design life. Assuming that the occurrence of high intensity earthquakes at the site follows a Bernoulli sequence and that damage effects between earthquakes are statistically independent, what is the probability of damage to the structure under a single earthquake?

Answer: 0.0404


Question:
Evaluate \( \lim_{x \to 0^-} f([x - \tan x]) \) for the function \( f(x) \) defined as:
\[
f(x) = \begin{cases}
x + 3 & \text{if } x \in (-2,0) \\
4 & \text{if } x = 0 \\
2x + 5 & \text{if } 0 < x < 1
\end{cases}
\] where \([\cdot]\) denotes the greatest integer function.

Answer: 4


Question: Prove that \( \sum\limits_{k=0}^n (2k+1) \binom{n}{k} = 2^n(n+1) \).

Answer: 2^n(n+1)


Question:
Let \( \mu \) be a finite measure on \((\mathbb{R}, \mathcal{B}(\mathbb{R}))\) and define \( F_\mu : \mathbb{R} \rightarrow [0,\infty) \) by \( F_\mu(x) = \mu((-
fty, x]) \) for \( x \in \mathbb{R} \). Find the limit \( \lim_{x\rightarrow -\infty}F_\mu(x) \).

Answer: 0


Question: Find all harmonic functions \( \phi \) in the unit disk \( D = \{ z \in \mathbb{C} : |z| < 1 \} \) that satisfy \( \phi(\frac{1}{2}) = 4 \) and \( \phi(z) \ge 4 \) for all \( z \in D \).

Answer: 4


Question: Find the quadratic Taylor approximation of the function \( f(x,y) = \int_0^{x+y^2} e^{-t^2} \, dt \) about the point \((0,0)\).

Answer: x + y^2


Question: Given the characteristic polynomial of a linear operator \( S \) as \( p(z) = (z-1)^2(z-2)^2 \), determine the minimal polynomial of \( S \).

Answer: (z - 1)^2(z - 2)^2


Question:
Given that $a+b+c+d=3$ and $a^2+b^2+c^2+d^2=45$, find the value of the expression:
\[ \frac{a^5}{(a-b)(a-c)(a-d)} + \frac{b^5}{(b-a)(b-c)(b-d)} + \frac{c^5}{(c-a)(c-b)(c-d)} + \frac{d^5}{(d-a)(d-b)(d-c)} \]

Answer: 27


Question: Given the differential equations \( f'(x) = -\frac{f(x)}{\sqrt{(f(x))^2+(g(x))^2}} \) and \( g'(x) = 1 - \frac{g(x)}{\sqrt{(f(x))^2+(g(x))^2}} \) for all \( x \in \mathbb{R} \), with initial conditions \( g(0) = 0 \) and \( f(0) = 10 \), and knowing that \( \lim_{x\to\infty}f(x) = 0 \), find the value of \( \lim_{x\to\infty}g(x) \).

Answer: 5


Question: Find the maximum value of \( M = |z^3 - z + 2| \) where \( z \) is a complex number such that \( |z| = 1 \).

Answer: \sqrt{13}


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

Answer: -\dfrac{4}{3}


Question: Evaluate the infinite series: \( \sum_{k = 1}^{\infty} \frac{k^2}{k!} \). Show that it equals \( 2e \).

Answer: 2e


Question: Find all natural numbers such that when the first digit is moved to the end, the resulting number is 3.5 times the original number.

Answer: 153846


Question: Given elements $a, b, c$ in a unital non-commutative ring, where $c$ is an inverse of $1-ab$ such that $c(1-ab) = 1$, find an inverse for $1-ba$.

Answer: 1 + bca


Question: Given that \( \mathbb{E}[X] = 10 \), find a lower bound for the expression \( 0.8\mathbb{E}[X^2] + 0.2\mathbb{E}[X^4] \) in terms of \( \mathbb{E}[X] \).

Answer: 2080


Question: Compute the limit of the integral: \[ \lim_{n\rightarrow\infty}\int_0^{\infty}\left(1+\frac{x}{n}\right)^{-n}\sin\left(\frac{x}{n}\right)dx. \]

Answer: 0


Question: Evaluate \( \lim_{n \to \infty} \frac{s_6(n) s_4(n)}{n^2} \), where \( s_k(n) \) denotes the number of digits in \((k+2)^n\) in base \(k\).

Answer: \dfrac{3}{2}


Question: Evaluate the limit as \( x \to 0 \) of the function \( \frac{\ln(1+x) - \sin(x) - \cos(x) + 1}{\arctan(x) - x} \) using Maclaurin series expansions. Provide your answer as a simplified fraction.

Answer: -\dfrac{3}{2}


Question: Evaluate the integral \( \int_C \nabla \varphi \cdot \tau \, ds \), where \( C = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1\} \) is the unit circle in the \( xy \)-plane, and \( \varphi(x, y, z) = x^2 + y^3 + z^4 \). The unit tangent vector \( \tau \) is tangent to \( C \) in the \( xy \)-plane and points left as we move clockwise along \( C \).

Answer: 0


Question: Determine the values of \(a \in \mathbb{R} \setminus \{0\}\) for which the series \(\sum_{n \geq 2} \left(1 - \frac{1}{\log(n^a)}\right)^n\) converges.

Answer: a > 0


Question: In how many ways can 6 people buy ice cream bowls from a store that sells 5 different flavors, such that each flavor is chosen at least once? Each bowl contains three scoops of ice cream, and the order of scoops does not matter. Use the inclusion-exclusion principle to find the solution.

Answer: 1528224670


Question: Consider the symmetrizing map \( S: \mathbb{Q}[x_1,...,x_n] \rightarrow \mathbb{Q}[f_1,...,f_n] \) defined by \( f \mapsto \frac{1}{n!}\sum_{\sigma\in S_n}\sigma\cdot f \), where \( f_i \) are the elementary symmetric polynomials and the action \( \cdot \) is the permutation of variables. If there exists a \( \sigma\in S_n \) such that \( \sigma\cdot f = -f \), then \( S(f)=0 \). Determine whether the converse is true: if \( S(f)=0 \), does there exist a \( \sigma\in S_n \) such that \( \sigma\cdot f = -f \)?

Answer: No


Question:
Is the mean value theorem for integration applicable in two dimensions? Specifically, for a continuous function \( G: B^2 \to \mathbb{R} \) defined over a connected, convex, closed subset of the plane such as the unit disc, does there exist a point \((x_0, y_0) \in B^2\) such that:
\[ G(x_0, y_0) = \frac{1}{\text{Area}(B^2)} \int_{B^2} G(x, y) \, dxdy \]?

Answer: Yes


Question: Let \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) be a \( 2 \times 2 \) matrix over a field \( k \) such that \( A^2 = 0 \). Consider the polynomials \( a^2 + bc, \ c(a+d), \ b(a+d), \ d^2 + bc \) in \( a, b, c, d \). Let \( I \) be the ideal generated by these polynomials. Determine the smallest integer \( r \) such that \( (a+d)^r \in I \).

Answer: 3


Question: Let \( p: (\tilde{X}, \tilde{x}_0) \rightarrow (X, x_0) \) be a covering map where the fundamental group of \( X \) at \( x_0 \) is \( \pi_1(X, x_0) = \mathbb{Z} \) and \( p^{-1}(x_0) \) is finite. Determine the fundamental group of \( \tilde{X} \) at \( \tilde{x}_0 \).

Answer: \mathbb{Z}


Question: Three points are placed independently and at random in a unit square. What is the expected value of the area of the triangle formed by these three points?

Answer: \dfrac{11}{144}


Question: Find the sum \( S = \sum_{i,j,k \ge 0, i+j+k=17} ijk \). Use a bijection to solve the problem.

Answer: 11628


Question: Determine the number of non-isomorphic three-dimensional simple Lie algebras over the rational numbers.

Answer: \infty


Question: Given a 3x3 real symmetric matrix M with distinct eigenvalues \(\alpha_1 < \alpha_2 < \alpha_3\), consider the matrix \(S = (M-\alpha_{2}I)(M-\alpha_{3}I)\). How many columns of S will be non-zero vectors? Provide a justification for your answer.

Answer: 1


Question: Given a plane \( P \) in \( \mathbb{R}^4 \) that includes the points \((0,0,0,0)\), \((1,0,-1,1)\), and \((0,1,1,2)\), find the point on the plane that is closest to the point \((-5,4,0,0)\).

Answer: (-2, 1, 3, 0)


Question: Determine the number among $0, 1, 2, \ldots, 60$ to which $10^{60 \cdot 2^{100} + 2}$ is congruent modulo $61$. Provide your answer as a single number.

Answer: 39


Question: Does every non-null two-sided ideal of the universal enveloping algebra $U(\mathrm{gl}(n,\mathbb{C}))$ have a nonzero intersection with its center?

Answer: Yes


Question: Let $X$ and $Y$ be independent random variables, each following an exponential distribution with parameter $\alpha > 0$. Determine the probability density function (PDF) of the random variable $Z = \frac{X}{X+Y}$. Provide your answer in terms of $\alpha$.

Answer: 1


Question: Evaluate the limit \( \lim_{n \to \infty}\frac{x_n}{\prod_{i=1}^{n-1} x_i} \) where \( x_n = x_{n-1}^2 - 2 \) and \( x_1 = 5 \).

Answer: \sqrt{21}


Question: Let \((A,+,\cdot)\) be a unital ring with \(|A| \ge 4\). If for all distinct elements \(x, y \in A \setminus \{0,1\}\), either \(x^2 = y\) or \(y^2 = x\) holds, determine the cardinality \(|A|\) of the ring.

Answer: 4


Question: Do there exist CW-complexes $X$ and $Y$ that are not homeomorphic, but such that $X \times I$ and $Y \times I$ are homeomorphic, where $I$ is the unit interval $[0, 1]$?

Answer: Yes


Question: Let $K$ be a field and suppose that $F$ is Galois over $K$ with $[F:K]=8$. Also suppose that $E$ is an intermediate field which is not Galois over $K$ and $[E:K]=4$. Determine the Galois group of $F$ over $K$.

Answer: D_4


Question: Calculate the mean time to hit state 2 in a two-state Markov chain with the transition matrix \( \begin{pmatrix} 1-p & p \\ 0 & 1 \end{pmatrix} \), starting from state 1. The mean time is given by the series \( \sum_{n=1}^\infty n(1-p)^{n-1}p \). Find the value of this series.

Answer: \dfrac{1}{p}


Question: Determine the number of roots of the equation \(z^4 + z^3 + 1 = 0\) that lie in the first quadrant of the complex plane using Rouche's Theorem.

Answer: 1


Question: Solve the recurrence relation: \( T_n = \sqrt{n} T_{\sqrt{n}} + 1 \).

Answer: \Theta(n)


Question: Determine the dimension of the space of Jacobi fields $J$ along a geodesic $\gamma : [0,a] \to M$ such that $J(0)=0$ and $\langle J,\gamma' \rangle(t) \equiv 0$. Assume $M$ is an $n$-dimensional manifold.

Answer: n - 1


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

Answer: -2\pi i


Question: If the order of a group $G$, $|G|$, and the order of its automorphism group, $|{\rm Aut}(G)|$, are relatively prime, must $G$ be cyclic? Justify your answer.

Answer: Yes


Question: Determine the length of the longest word that can be accepted by a deterministic finite automaton (DFA) with the following properties: it has 4 states and does not accept any word of length 8, 9, 10, 11, or 12.

Answer: 7


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

Answer: 0


Question: Provide an example of a function \( f: \mathbb{R} \to \mathbb{R} \) that is injective and bounded, but whose inverse \( f^{-1} \) is not injective or bounded.

Answer: f(x) = \arctan(x)


Question: Find the residue of the function \( f(z) = \frac{e^{\pi z}}{(z-i)^2} \) at the point \( z_0 = i \).

Answer: -\pi


Question:
Given two dependent random variables $X$ and $Y$, with the conditions:
- If $Y > a$, then $f = P(X > a)$
- If $Y \le a$, then $f = P(X > b)$

Determine which of the following formulas correctly represents $f$:
1. $f = P(X > a)P(Y > a) + P(X > b)P(Y \le a)$
2. $f = P(X > a, Y > a) + P(X > b, Y \le a)$

Here, $a, b > 0$. Choose the correct formula.

Answer: 2


Question: Let \( X \sim \exp(1) \). Calculate \( E[X^n] \) for every \( n \in \mathbb{N} \).

Answer: n!


Question:
Given a measure \( \mu \) that satisfies \( \mu((a,b]) = F(b) - F(a) \) for a specified increasing, right-continuous function \( F \), find \( \mu(\mathbb{R}) \). Define \( F : \mathbb{R} \to \mathbb{R} \) by
\[F(x) := \begin{cases} 0 & \quad x < 1 \\ \frac{2x}{1+x} & \quad x \ge 1.\end{cases}\]
Then \( F \) is increasing and right continuous. Suppose that \( \mu \) is a measure that satisfies \( \mu((a,b]) = F(b) - F(a) \), where \( a<b \). Find \( \mu(\mathbb{R}) \).

Answer: 2


Question:
Find the coefficient of $x^{11}$ in the expansion of $(x+x^2+x^3+x^4+x^5)^4(1+x+x^2+
\ldots)^4$. Express your answer as a single integer.

Answer: 3288


Question: Determine whether there exists a simple graph with \( n \) vertices, where \( n \geq 3 \), such that the degree of every vertex is at least \((n-1)/2\) and the graph does not have a Hamiltonian circuit.

Answer: Yes


Question: Determine whether the following proposition is true: If $u \in L^2(\Omega)$ satisfies $\int_\Omega uv = 0$ for all $v \in H^1_0(\Omega)$, then must $u = 0$? Here, $H^1_0(\Omega)$ denotes the elements of the Hilbert space $H^1(\Omega)$ with null trace on $\partial\Omega$. Provide a justification for your answer.

Answer: True


Question: Calculate the limit \( \lim_{t\to 0^+} \frac{e^{-x^2/4t}}{t^{3/2}}. \)

Answer: 0


Question: Let $G$ be a group of order $p^n$, where $p$ is a prime number, and let $H$ be a subgroup of $G$ with order $p^{n-2}$ that is not normal in $G$. Determine whether the index $[G:N_G(H)]$ is equal to $p$.

Answer: p


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

Answer: 1


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

Answer: e


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{1}{2n}\log\binom{2n}{n} \]

Answer: \log 2


Question: Determine the maximum radius of the disk centered at 0 in which the function \( f(z) = \cos\left(\frac{1}{z-1}\right) \) can be expressed as a power series.

Answer: 1


Question: Evaluate the limit: $$L=\lim_{x\to 0}\frac{\sin(\ln(x+1))-\ln(\sin(x)+1)}{\sin^4\left(\frac{x}{2}\right)}$$

Answer: \dfrac{4}{3}


Question: Evaluate the integral \( \int_0^{2\pi} \frac{1}{\sin^4 x + \cos^4 x} \, dx \).

Answer: 2\sqrt{2}\pi


Question: Let $P_1, \dots, P_k$ be polynomials over $\mathbf{C}$, with no two of them being proportional. Determine whether there exists an integer $N$ such that $P_1^N, \dots, P_k^N$ are linearly independent.

Answer: Yes


Question: Evaluate the line integral \(\int_{l:|x|+|y|=a} x y \, ds\), where \(l\) is the path defined by \(|x| + |y| = a\).

Answer: 0


Question: Determine whether the statement \((\log_2 n)^{\sqrt{ \log_2 n}} = \mathcal{O}(n^{50})\) is true or false.

Answer: True


Question: Suppose $\mathfrak m$ and $\mathfrak n$ are infinite cardinals. Does $2^{\mathfrak m} = 2^{\mathfrak n}$ imply $\mathfrak m = \mathfrak n$?

Answer: No


Question: A line $l$ makes an angle of $60^\circ$ with a line $m$ and $45^\circ$ with a line $n$, where $m$ and $n$ are perpendicular lines on a plane $\alpha$. Determine the angle between the line $l$ and the plane $\alpha$ without using vectors.

Answer: 30^\circ


Question: Evaluate the double integral \(\displaystyle \iint_R (x^2+y^2) \, dx \, dy\), where \(R\) is the region in the first quadrant bounded by the curves \(xy=2\), \(xy=4\), \(x^2-y^2=3\), and \(x^2-y^2=5\). Use the transformation \(u=xy\) and \(v=x^2-y^2\) to express \(x^2+y^2\) in terms of \(u\) and \(v\), and evaluate the integral over the region \(1 \le u \le 3\), \(1 \le v \le 5\).

Answer: 4


Question:
Find the value of the infinite product:
\[ P = \sqrt{\frac{1}{2}} \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}} \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}}} \ldots \]

Answer: \dfrac{2}{\pi}


Question: Determine which function grows faster as $n$ approaches infinity: $f(n) = n!$ or $g(n) = n^{\log_2 n}$.

Answer: n!


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

Answer: f(x) = c


Question: Let $f: X \to Y$ be a proper, flat morphism of projective schemes where $Y$ is an irreducible, non-singular surface. Assume there exists a Zariski open subset $U$ of $Y$ whose complement is of codimension 2 such that the total space $f^{-1}(U)$ is non-singular and $f: f^{-1}(U) \to U$ has geometrically irreducible fibers. Denote by $\eta$ the generic point of $Y$. Suppose there exists a morphism from $\eta$ to the generic fiber $X_{\eta}$. Does there exist a rational section of $f$?

Answer: Yes


Question: Can a group of order 42 be a subgroup of SL(4,C)?

Answer: Yes


Question: Determine the number of solutions of the equation \( z^7 - 5z^4 + iz^2 - 2 = 0 \) within the region \(|z| < 1\) using Rouche's theorem.

Answer: 4


Question:
Identify the incorrect step in the following process of finding the quotient of the free abelian group $\langle a,b,c\rangle/\langle -b+c-a,b+c-a\rangle$. The expected result is $\mathbb{Z}\oplus\mathbb{Z_2}$, but the process below leads to $\mathbb{Z}\oplus\mathbb{Z_2}\oplus\mathbb{Z_2}$. Determine which step is incorrect:

1. Start with the relations $-b+c-a=0$ and $b+c-a=0$.
2. Derive $2c=2a$ and $2b=0$.
3. Conclude the group is $\langle a-c,b,c\rangle/\langle 2(a-c),2b\rangle$.
4. Simplify to $\langle d,b,c\rangle/\langle2d,2b\rangle\simeq\mathbb{Z}\oplus\mathbb{Z_2}\oplus\mathbb{Z_2}$.

Answer: 3


Question: Calculate the line integral of the function \( w = x + y^2 \) along the path from point \( A(0,0) \) to point \( B(1,1) \).

Answer: \dfrac{5\sqrt{2}}{6}


Question: Consider the function \( f : \mathbb{N} \to \mathcal{P}(\mathbb{N}) \) defined as \( f(n) = \emptyset \) for all \( n \in \mathbb{N} \). In the context of Cantor's theorem, the set \( D \) is defined as \( \{ x \in \mathbb{N} \mid x \notin f(x) \} \). Determine the set \( D \) without using set-builder notation. Is it clear why \( f(n) \neq D \) for any \( n \in \mathbb{N} \)?

Answer: \mathbb{N}


Question: Evaluate the limit \( \lim_{n \to \infty} A_n^{\frac{1}{n}} \), where \( A_n = \int_{0}^{1}\frac{dt}{(1+t)^n} \) and for \( n \neq 1 \), \( A_n = \frac{1}{(1-n)2^{n-1}} + \frac{1}{n-1} \).

Answer: 1


Question: Given a real-valued function $f$ that satisfies the relation \( f(x)f(y) = f(2xy+3) + 3f(x+y) - 3f(y) + 6y \) for all real numbers $x$ and $y$, find the value of $f(8)$. \( \)

Answer: 19


Question: Does there exist a Lebesgue nonmeasurable set $A$ such that all its measurable subsets are null sets? In other words, is there a nonmeasurable set $A$ for which every Lebesgue measurable subset $E \subset A$ satisfies $\mu(E) = 0$, where $\mu$ is the Lebesgue measure?

Answer: Yes


Question: Determine the norm of the linear functional \( \ell^{2} \ni \{x_n\}^{\infty}_{n=0} \rightarrow \sum^\infty_{n=0}\frac{x_n}{\sqrt{2^n}}\in \mathbb{K} \).

Answer: \sqrt{2}


Question: Let $S_4$ be the group of permutations on the set $\{1,2,3,4\}$, and let $G = S_4 \oplus \mathbb{Z}_4$. Determine the order of the largest cyclic subgroup of $G$. \( \text{Provide your answer as a single integer.} \)

Answer: 12


Question: Calculate \( \lim_{x \rightarrow \infty} \int_{x}^{+\infty} \frac{e^{-t}}{t} \, dt \).

Answer: 0


Question: Given that $x, y, z$ are positive real numbers satisfying $\frac{1}{3} \le xy + yz + zx \le 3$, determine the range of possible values for the product $xyz$. Express your answer as an interval.

Answer: (0, 1]


Question: Evaluate the integral \( \iint_D 4(x^2 + y^2) \, dx \, dy \) over the region \( D = \{ |z-1| \leq 1 \} \).

Answer: 6\pi


Question: Let F be a countable field and B an infinite set. Let \((F^B)_0\) be the set of all functions with finite support from F to B. Determine whether \(|(F^B)_0| = |B|\).

Answer: |(F^B)_0| = |B|


Question: Evaluate the limit: \[ \lim_{N \rightarrow \infty} \sum_{n=1}^{N}\frac{a_n}{n} \] where \( a_n = \int_{0}^{\pi/2}(1-\sin(t))^{n}\sin(2t)\, dt \).

Answer: \dfrac{1}{2}


Question: Determine whether the following statement is true or false: Any uniformly continuous function \( f : (X,d) \rightarrow (Y,d) \) maps bounded sets to bounded sets.

Answer: False


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: In a game involving two players, a fair coin is tossed repeatedly until one player wins. Player A wins if the sequence (Z, K, Z) appears, while player B wins if the sequence (K, K, K) appears. Calculate the probability that player A wins the game.

Answer: \dfrac{7}{12}


Question: Evaluate the limit: \[ L = \lim_{n \to \infty} \sqrt[n^2]{\sqrt{3!!}\cdot \sqrt[3]{5!!} \ldots \sqrt[n]{(2n-1)!!}} \]

Answer: 1


Question: Find the coefficient of \(x^{200}\) in the generating function \((1 + x + x^2 + \cdots + x^{200})(1 + x^2 + x^4 + \cdots + x^{200})\).

Answer: 101


Question: Find the curvature of the vector function \( r(t) = \langle 3t, 2\sin t, 2\cos t \rangle \) at the point \( \left( \frac{5\pi}{2}, 1, -\sqrt{3} \right) \).

Answer: \dfrac{2}{13}


Question: Three 6-sided fair dice are rolled in 10 independent throws. Let A be the number of times all the sides are the same, and let B be the number of times only two sides are the same. Calculate the expected value of 6AB.

Answer: \dfrac{25}{4}


Question: Let $k$ be a field, and $A$ and $B$ be $k$-algebras such that $B$ is integral over $A$, and both $B$ and $A$ are integral domains. Denote $F(B)$ and $F(A)$ as the fields of fractions of $B$ and $A$, respectively. Is $F(B)$ an algebraic extension of $F(A)$?

Answer: Yes


Question: Let A be a symmetric matrix such that \( A^2 = A \). Prove that A is a projection matrix.

Answer: A


Question:
If $x, y, z$ are positive real numbers satisfying $xyz = 32$, find the minimum value of the expression:
\[ x^2 + 4xy + 4y^2 + 2z^2 \]

Answer: 96


Question: Let $f: \Omega \to \mathbb{C}$ be a continuous function. Define \( g(z) = \int_\rho^z f(\zeta)\,d\zeta \) for \( z \in \Omega \). Show that for all \( z \in \Omega \), \( \frac{g(z)-g(z_0)}{z-z_0} - f(z_0) \to 0 \) as \( z \to z_0 \).

Answer: 0


Question: Determine the number of integers \( m \leq 1000 \) for which the sum \( \sum\limits_{k=1}^m \left\lfloor \frac{m}{k} \right\rfloor \) is even. Here, \( \lfloor x \rfloor \) denotes the floor function, which gives the largest integer less than or equal to \( x \).

Answer: 495


Question: Show that the total variation of the function \( f(t) = \frac{\sin(\pi t)}{\pi t} \) is infinite. Specifically, evaluate the integral \( \int_{-\infty}^{\infty} \left| \frac{\pi t \cos(\pi t) - \sin(\pi t)}{\pi t^2} \right| \, dt \) and demonstrate that it diverges.

Answer: \infty


Question: Suppose that $P(x)$ and $Q(x)$ are polynomials in $\mathbb{R}[x]$ such that $P(P(x)) = Q(x)^2$. Does it necessarily follow that there exists a polynomial $R(x) \in \mathbb{R}[x]$ such that $P(x) = R(x)^2$? Justify your answer.

Answer: Yes


Question: Let $D \subset \mathbb{R}^{2}$ be a bounded open set with a $C^{1}$ boundary and finite perimeter $P$. Let $f : \mathbb{R}^{2} \to \mathbb{R}$ be a $C^{1}$ function satisfying $|f(x,y)| \leq 1$ for all $(x,y) \in D$. Show that \[ \left| \int\int_{D} \frac{\partial f}{\partial y}(x,y) \, dx \, dy \right| \leq P. \]

Answer: P


Question: Evaluate the limit: $$\lim_{x\rightarrow\infty} x^{b-a} \frac{\Gamma(x+a+1)}{\Gamma(x+b+1)}$$

Answer: 1


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

Answer: \dfrac{1}{6}


Question: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function, and define the set $A$ by \[ A = \{ y \in \mathbb{R} : y = \lim_{n \rightarrow \infty} f(x_n) \text{ where } x_n \text{ diverges to } +\infty \}. \] Determine which of the following properties the set $A$ satisfies: \(\text{A. Connected, B. Compact, C. Singleton, D. None of the above}\).

Answer: A


Question: Calculate the following limit: \[ \lim_{n\to\infty}n^{2020}\int^{\pi/2}_0 \left[(\sin x)^n \cdot (\pi/2-x)^n\right] \,dx \]

Answer: 0


Question: Determine the number of distinct colorings of the vertices of a cycle graph $C_{8}$ using 10 colors, where each color can be used multiple times. Two colorings are considered the same if one can be transformed into the other by an isometry of the octagon. Use Burnside's Lemma to find the number of distinct colorings.

Answer: 6278140


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

Answer: e - 1


Question: Determine the minimum number of pairs \((x, R(x))\) required to fully characterize a rotation \(R\) in \(\mathbb{R}^n\).

Answer: n-1


Question: Let \( y \) be a solution to the differential equation \((1-x^2)y''-2xy'+6y=0\). Given that \( y(1)=2 \), find the value of the integral \( \int_{-1}^1 y^2 \, dx \).

Answer: \dfrac{8}{5}


Question: Find the remainder when $2^{1990}$ is divided by 1990.

Answer: 1024


Question:
Consider the function \( f(x,y) = \frac{xy^2}{x^2+y^4} \) for \( x \neq 0 \) and \( f(x,y) = 0 \) for \( x = 0 \). Determine which of the following statements is true about the function at the origin:

1. It is continuous but not differentiable at the origin.
2. It is differentiable at the origin.
3. It has all first order partial derivatives at the origin.
4. It does not have all first order derivatives at the origin.

Evaluate the limit \( (x,y) \rightarrow (0,0) \) along the curve \( y = m\sqrt{x} \) and use it to determine the correct statement.

Answer: 3


Question: Evaluate the limit: \[ \lim_{n \to \infty} n^{1/n} \] using the fact that \( \log(n) << n \).

Answer: 1


Question: Compute the limit: \[ \lim_{n\rightarrow\infty} \left(\sqrt[n]{1}+\sqrt[n]{2}+\cdots+\sqrt[n]{2007}-2006\right)^n \]

Answer: 2007!


Question: Determine the number of non-negative integer solutions \((N, D, Q)\) to the equation \(0.05N + 0.1D + 0.25Q = 3\).

Answer: 205


Question: Evaluate \( \lim_{n \to \infty} U_n \) where \( U_n = \int_{n^2+n+1}^{n^2+1} \frac{\tan^{-1}x}{x^{0.5}} \, dx \) without performing the integration.

Answer: \dfrac{\pi}{2}


Question: Solve the integral equation \( F(t) = a\sin{t} - 2 \int^t_0 F(u) \cos(t - u) \, du \).

Answer: a t e^{-t}


Question: Let $f:(0,1) \to \mathbb{R}$ be a $C^1$ function such that $f|_{(0,1/2]}=0$. Assume there exists a strictly decreasing sequence $t_n \to 1/2$ such that for each $n$, $f$ is affine on $(t_n-\epsilon_n,t_n+\epsilon_n)$ for some $\epsilon_n>0$. Determine if $f|_{(0,1/2+\epsilon]}=0$ for some $\epsilon>0$.

Answer: Yes


Question: Find the vector \( \mathbf{C} \) that bisects the angle between the vectors \( \mathbf{A} = (2, -3, 6) \) and \( \mathbf{B} = (-1, 2, -2) \), given that the norm of \( \mathbf{C} \) is \( 3\sqrt{42} \).

Answer: (-3, 15, 12)


Question: Given the joint density function \( p(x,y) = 120xy(1-x-y)I \{x \geq 0, y \geq 0, x+y \leq 1 \} \), find the correlation coefficient of \( Z = Y - E(Y|X) \) and \( X \).

Answer: 0


Question: Determine whether the following statement is true or false: Let $(X,d)$ be a locally compact separable metric space and $\mu$ be a positive Radon measure on $(X,d)$. Then there exists a sequence of open subsets $(U_n)$ of $X$ such that $U_n\subseteq U_{n+1}$, $0<\mu(U_n)<\infty$ for all $n\ge 1$, and $ \bigcup_{n=1}^\infty U_n=X$. Provide a justification for your answer.

Answer: True


Question: Determine the smallest positive integer \( m \) such that the symmetric group \( S_n \) can be embedded into the general linear group \( GL_m(F) \), where \( F \) is a field.

Answer: n - 1


Question: Let $f(x)$ be a continuous real-valued function on $\mathbb{R}$. If $f(x)$ is differentiable at every $x \neq 0$ and $\lim_{x\rightarrow0}f'(x)$ exists, does it imply that $f(x)$ is differentiable at $x=0$? Provide a justification for your answer.

Answer: Yes


Question: Let \( f \in \mathbb{Z}[x] \) be an irreducible polynomial over \( \mathbb{Q}[x] \) with a leading coefficient of 1. If \( \omega \in \mathbb{C} \) is such that \( f(\omega) = 0 \), is the minimal polynomial of \( \omega \) over \( \mathbb{Q} \) equal to \( f \)? Justify your answer.

Answer: Yes


Question:
Let $A$ be a commutative ring, $M$ a module over it, and $k_1, k_2$ fields such that we have the following maps $A \to k_1 \to k_2$. Construct the natural map:
\[ f: M\otimes_A k_1 \to M\otimes_A k_2. \]
If $m\in M$ is such that $m\otimes 1 = 0 \in M\otimes_A k_2$, does it follow that $m\otimes 1 = 0\in M\otimes_A k_1$? Answer yes or no.

Answer: Yes


Question: What is the smallest dimension $n$ such that the special orthogonal group $SO(3)$ can be topologically embedded into $\\mathbb{R}^n$?

Answer: 4


Question: Given a finite field \( K \) with \( |K| = q \) and an irreducible polynomial \( f \) with a root \( \alpha \) and degree \( n \), determine if \( |K(\alpha)| = q^n \). Justify your answer.

Answer: q^n


Question: Four tigers in a reserve forest are monitored using geo tags. The waiting times for responses from the 4 tigers follow an independent and identically distributed exponential distribution. If the system must locate all 4 tigers within 5 minutes, what is the maximum expected response time for each geo tag that ensures a location for all four tigers within 5 minutes or less with at least 90% probability?

Answer: 1.37


Question: Prove that for a prime number \( p \) such that \( p \equiv \pm 2 \pmod{5} \) and \( p \equiv 3 \pmod{4} \), the following congruence holds: \[ F_{\frac{p^2+1}{2}} \equiv p-1 \pmod{p} \] where \( F_n \) denotes the \( n \)-th Fibonacci number.

Answer: F_{\frac{p^2+1}{2}} \equiv p-1 \pmod{p}


Question: Determine the number of positive integers \( x \) such that \( x \leq 9,999,999 \) and the sum of the digits of \( x \) is 31.

Answer: 512365


Question: Evaluate the limit: \[ \lim\limits_{x \to 0} \frac{x \cos(x) - \sin(x)}{x \sin^2(x)} \]

Answer: -\dfrac{1}{3}


Question: Evaluate the limit: \[ \lim_{n \to \infty} 2^{n + 2} \times (x^{2^n} - x^{2^{n+1}}) = 0 \] for \( x \in (0,1) \).

Answer: 0


Question: Evaluate the contour integral \( \oint_{|z-1|=2} \frac{e^{-z}}{z^2} \, dz \) using Cauchy's integral formula.

Answer: -2\pi i


Question: A needle 2.5 cm long is dropped onto a piece of paper with parallel lines drawn 2.25 cm apart. What is the probability that the needle will lie entirely between two lines?

Answer: 0.314


Question: Find the sum of the digits in the number 100!.

Answer: 648


Question: Calculate the limit: \[ \lim_{n\to\infty}\left(\left(\frac{9}{4}\right)^n+\left(1+\frac{1}{n}\right)^{n^2}\right)^{\frac{1}{n}} \]

Answer: e


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

Answer: (-1, \infty)


Question: Determine the existence and compute the limit \( \lim_{n \rightarrow \infty} \int_{(0,\infty)}\left(\frac{ne^x+1}{ne^{2x}+4x^2}\right)d\lambda(x) \).

Answer: 1


Question: Calculate the integral \( I = \int_{-\infty}^\infty f(x) \partial_x^2 f(x) \, dx \) where \( f(x) = e^{-\frac{|x|}{a}} \) and \( \partial_x = \frac{\partial}{\partial x} \). Verify the result using both direct calculation and integration by parts.

Answer: -\dfrac{1}{a}


Question:
Find the coefficient of $x^{10}$ in the expansion of the expression:
\[
(1-x^7)(1-x^8)(1-x^9)(1-x)^{-3}.
\]

Answer: 47


Question: Let $T$ be a positive operator on a Hilbert space $H$, and let $P$ be a projection on $H$. Determine whether the inequality $T \ge PTP$ holds.

Answer: No


Question: Consider the set $E_r = \{P \in \mathbb{R}[x_1, x_2, \dots, x_d] \mid \deg(P) \le r\}$. If $V \in E_r$ and $e^{-V(x)} \in L^2(\mathbb{R}^d)$, prove that $\lim_{|x| \to \infty} V(x) = +\infty$.

Answer: \lim_{|x| \to \infty} V(x) = +\infty


Question: Let $X$ be a random variable such that $X \sim \text{Poisson}(\lambda)$ and $Y|X \sim \text{Binomial}(x+1,p)$. Find $\text{Cov}(X,Y)$. Provide your answer as a single numerical value.

Answer: p\lambda


Question: Is it true that if \( A \in GL(n,\mathbb{Z}_p) \), then the order of \( A \) cannot exceed \( p^n - 1 \)?

Answer: True


Question: Let $X \sim \mathrm{Exp}(1)$ and $Y \sim \mathrm{Exp}(2)$ be independent random variables. Let $Z = \max(X, Y)$. Calculate the expected value of $Z$. \( \boxed{\frac{7}{6}} \)

Answer: \dfrac{7}{6}


Question: Convert the 8-bit floating-point number represented by the bit pattern \( 0101 1110_2 \) from Format A to Format B. Format A has 3 exponent bits with a bias of 3 and 4 fraction bits. Format B has 4 exponent bits with a bias of 7 and 3 fraction bits. Show your steps and verify the conversion.

Answer: 01001111


Question: Evaluate the sum of the infinite series: \[ \sum_{n=1}^{\infty} \frac{1}{n} \sin\left(\frac{n\pi}{3}\right) \]

Answer: \dfrac{\pi}{3}


Question: Evaluate the infinite series \( \sum_{n=0}^{\infty} \frac{(-1)^n F_{2n}}{9^n} \), where \( F_{n} \) is the \( n^{th} \) Fibonacci number.

Answer: -\dfrac{9}{109}


Question: Determine the number of zeros of the function \( f(z) = z^{10} + 10ze^{z+1} - 9 \) within the unit disc \( |z| < 1 \).

Answer: 1


Question: Prove or disprove the statement: If $A$ is a Hermitian matrix (i.e., $A^* = A$) and $A^m = 0$ for some positive integer $m$, then $A = 0$.

Answer: A = 0