Question: Determine the value of the term: $$\inf \bigg\{\sup \bigg\{ \frac{2n}{2+n(-1)^n}:n>k\bigg\}:k\in\mathbb N\bigg\}$$

Answer: 2


Question: Given a closed set $C$ in $\mathbb{R}^m \times \mathbb{R}^n$, determine if the set $\{y \in \mathbb{R}^n: (x_0, y) \in C\}$ is closed for any $x_0 \in \mathbb{R}^m$. Justify your answer.

Answer: Yes


Question: Given the curve \( z = x^3 \), rotate this curve towards the y-axis about the z-axis to express it as \( z = f(x, y) \). Write \( z = f(x, y) = f(x^3, \theta) \). Determine the expression for \( z \) when \( \theta = 90^\circ \).

Answer: z = y^3


Question:
Evaluate the limit \( \lim_{n \to \infty} \int_0^1 x^n e^x \, dx \), given the expression for the integral:
\[ \int_0^1 x^n e^x \, dx = (-1)^{n+1} n! \left( 1 - e \sum_{k=0}^n \frac{(-1)^k}{k!} \right) \]

Answer: 0


Question: Evaluate the triple integral \( \iiint_D (x^2+y^2) \, dxdydz \) over the region \( D=\{(x,y,z); 0 \leq x^2 +y^2 \leq z^2, \, 0 \leq z\leq 1 \} \). Use the transformation \( x = r \cos \theta, y = r \sin \theta \) with the functional determinant \( r \). The integral simplifies to \( \iint_E r(r^2 \cos^2 \theta + r^2 \sin^2 \theta) dr d \theta \) over the region \( E=\{(r, \theta); \, 0 \leq r \leq z, \, 0 \leq \theta \leq 2 \pi \} \). Compute the value of this integral.

Answer: \dfrac{\pi}{10}


Question: Determine the degree of the differential equation \( \left(\frac{d^2y}{dx^2}\right)^{\frac{3}{2}} + \left(\frac{dy}{dx}\right)^{\frac{2}{3}} + y^2 = 0 \) after eliminating the radicals.

Answer: 9


Question: Calculate the limit: $$\lim_{x\to 0}\frac{1-\cos x\times (\cos4x)^{\frac{1}{4}}\times (\cos5x)^{\frac{1}{5}}}{x^2}$$

Answer: 5


Question: If the continuous function \( f: \mathbb{R} \to \mathbb{R} \) satisfies \( f \circ f \circ f(x) = x \), does it necessarily follow that \( f(x) = x \) for all \( x \in \mathbb{R} \)? Answer "yes" or "no" and justify your answer.

Answer: yes


Question: Given a set of $2^n$ numbers from $1$ to $2^n$, where $n$ is an integer, a process is repeated $n$ times: divide the numbers into two halves and flip a coin with probability $p$ of heads. If the coin lands tails, discard that half; otherwise, proceed with the half that landed heads. What is the average number of numbers left after all steps? Verify if the average is $2^n p^n$.

Answer: 2^n p^n


Question: Let $A$ be a real, square, and nonsingular matrix such that $\det (A^3 - 3A + 2I) = 0$. Determine an eigenvalue of the matrix $A^2 + A^{-2}$.

Answer: 2


Question: Assume $R$ is a commutative Noetherian ring with unity, and consider a family of ideals $\{I_k\}_{k\in\mathbb{N}}$ in $R$ such that $I_k I_j \subset I_{k+j}$. Define the ring $T = R + I_1 X + I_2 X^2 + \cdots$ (a subring of $R[X]$). Does $T$ being Noetherian imply the existence of an ideal $J \subset R$ satisfying $I_k = J^k$ for sufficiently large $k$?

Answer: No


Question: In the ring of continuous functions $C[0,1]$, determine if a function $f \in C[0,1]$ with an uncountable number of zeros can be a zero divisor.

Answer: Yes


Question: Determine the values of $a \in \mathbb{R}$ for which the function \[ f(x,y) = \begin{cases} \dfrac{xy^a}{x^2+y^2}&(x,y) \neq (0,0)\\ \ 0&(x,y) = (0,0) \end{cases} \] is continuous.

Answer: a > 1


Question: Evaluate the integral: $$\int_{-\infty}^{+\infty}e^{-\frac{1}{x^{2}}}\, dx$$

Answer: \infty


Question: Given that $r$ and $s$ are roots of the cubic equation $x^3 - cx + d = 0$, and $r \neq s$, find an algebraic expression for the third root $t$ such that $t = r + s$.

Answer: 0


Question: Find the sum of the series \( \sum_{k=1}^\infty a_k \) where \( a_k = \int_k^{k+1} xe^{-x} \, dx \).

Answer: \dfrac{2}{e}


Question: Determine the number of equivalence classes of propositional terms, using a set of 5 propositional variables, that evaluate to TRUE under exactly 13 distinct valuations.

Answer: \dbinom{32}{13}


Question: Evaluate the limit \( \lim_{x\rightarrow 0}\frac{(e^{x^2}-1)(\cos(x^4)-1)}{\sin ^6(x)-\sin ^6(x^3)} \).

Answer: 0


Question: Is every finite subgroup of order $n$ of the multiplicative group of non-zero complex numbers, $\mathbb{C}^*$, of the form $A_n=\{z \in \mathbb{C}^* : z^n=1\}$?

Answer: Yes


Question: Determine whether the function \( g : (X, \tau) \to (\mathbb{R}, \tau_u) \) defined by \( (t,k) \mapsto t+k+1 \) is continuous, given that \( (X, \tau) \) is equipped with the product topology.

Answer: Yes


Question: Determine whether the following statement is true or false: For functions $f(x) > 0$ and $F(x) = \int_0^x f(t) dt$, if $\int_0^\infty f(x)^p e^{-x}dx < \infty$ for $p>1$, then $\int_0^\infty F(x)^p e^{-x}dx < \infty$.

Answer: True


Question: Evaluate the Beta function \( B\left(\frac{1}{2},\frac{1}{2}\right) \) using the substitution \( t = \cos^2\theta \).

Answer: \pi


Question: Determine the number of terms required from the series \( \sum_{n=0}^\infty \frac{(-1)^{n}}{(2n+1)n!} \) to approximate \( \int_{0}^{1}e^{-x^2} \, dx \) with an error less than \( 10^{-6} \).

Answer: 9


Question: In the group $\mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$, where $m$ and $n$ are coprime, is the element $(\bar{1}, \bar{1})$ a generator of the group?

Answer: Yes


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

Answer: \infty


Question: Determine the number of 4-digit numbers such that if any single digit is removed, the resulting 3-digit number divides the original 4-digit number.

Answer: 14


Question: Evaluate the limit: \[ \lim_{n\to \infty} \frac{(3^3)^{\sqrt n}}{(1+(15)^{-3^3})^n} \]

Answer: 0


Question: Given a real function $f$ such that its second derivative $f''$ is greater than or equal to a constant $C > 0$, determine whether it follows that $\lim_{x\to +\infty}f'(x) e^{-f(x)}=0$.

Answer: 0


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

Answer: 1


Question: Calculate the expected value of the length of the longest increasing subsequence of a random permutation of \(n\) elements, divided by \(\sqrt{n}\).

Answer: 2


Question: Determine the number of integer-sided right triangles whose side lengths are given by the binomial coefficients \( \binom{x}{2}, \binom{y}{2}, \binom{z}{2} \).

Answer: 0


Question: In a group of 50 people, what is the minimum number of rotations required to seat them at tables of 5 so that each person meets every other person at least once?

Answer: 13


Question: Evaluate the limit: $$\lim _{x\to 0}\left(\frac{1-\cos \left(x\right)\sqrt{\cos \left(2x\right)}\cdot \sqrt[3]{\cos \left(3x\right)}}{x^2}\right)$$

Answer: 3


Question: Determine whether the spectral norm is preserved when two matrices are similar. Specifically, if \( A = PBP^{-1} \) where \( P \) is not necessarily a unitary matrix, does \( A \) have the same spectral norm as \( B \)? Provide a justification for your answer.

Answer: No


Question: Determine the minimum value of \( k \) such that the inequality \( \sin{a} \leq \frac{a}{\sqrt{1+\frac{a^2}{k}}} \) is satisfied for all \( a > 0 \).

Answer: 3


Question: Determine whether all continuous homomorphisms from the unit circle \(\mathbb{S}^{1}\) to itself are of the form \(z \mapsto z^{n}\) for some \(n \in \mathbb{Z}\).

Answer: Yes


Question: If X and Y are independent binomial random variables with parameters n and p, find the conditional expected value of X given that X + Y = m.

Answer: \dfrac{m}{2}


Question: A differentiable function \( f(x) \) satisfies the inequality \(|f(x)-f(y)| \le (x-y)^2\) for all real numbers \( x \) and \( y \). Given that \( f(0) = 0 \), determine the range of possible values of \( f(1) \).

Answer: 0


Question:
Determine the fixed points of the group action of $H$ on $G/H$, where $H$ is a subgroup of $G$, $|G| = p^2$, $|H| = p$, and $p$ is a prime, given the action:
$$ H \times G/H \rightarrow G/H, \quad (h, gH) \rightarrow h * (gH) = (hg)H $$

Answer: G/H


Question: Compute the quadratic variation of the Ornstein-Uhlenbeck process.

Answer: \sigma^2 t


Question: Maximize the expression \(x_2 - x_1 + y_1 - y_2\) subject to the constraints \(x_1^2 + y_1^2 = 1\) and \(x_2^2 + y_2^2 = 1\).

Answer: 2\sqrt{2}


Question: Evaluate the definite integral: $$\int_{0}^{\frac{\pi}{2}} \cot\left(\frac{x}{2}\right) (1-\cos^4x)\,dx.$$

Answer: \dfrac{25}{12}


Question: Given a measurable set $A$ with measure $\mu(A)=0$ in a measurable space $X$, determine if its characteristic function is Lebesgue integrable.

Answer: Yes


Question: Given distinct real numbers $a, b, c$ such that $a^3 = 3(b^2 + c^2) - 25$, $b^3 = 3(c^2 + a^2) - 25$, and $c^3 = 3(a^2 + b^2) - 25$, find the value of $abc$. \( \boxed{} \)

Answer: 2


Question: Compute the Lebesgue outer measure $\mu^*$ of the set $A = \left\{\left(1 + \frac{1}{n}\right)^n \mid n \in \mathbb{N}\right\}$ on the interval $I_0 = [a, b]$. Is the Lebesgue outer measure of $A$ equal to $0$?

Answer: 0


Question: Calculate the 25th derivative of the function \( f(x) = \cos(x^3) \) at \( x = 0 \), denoted as \( f^{(25)}(0) \).

Answer: 0


Question: Determine the signature of the symmetric bilinear form on $\mathbb{R}^n$ defined by $e_i \cdot e_j = 0$ when $i = j$, and $e_i \cdot e_j = 1$ when $i \neq j$. Provide your answer as a pair of integers representing the number of positive and negative eigenvalues.

Answer: (1, n-1)


Question: Determine whether any square complex matrix can be decomposed into the form $A=U\Lambda U^{-1}$, where $\Lambda$ is triangular, even if $A$ is not diagonalizable.

Answer: Yes


Question: Evaluate the integral \( \int_{-1}^{1} x P_{n+1}(x)P_n'(x) \, dx \) without using the orthogonality property of Legendre polynomials.

Answer: 0


Question: Determine whether the statement "${n \choose j}$ is a multiple of $j$ if and only if $n$ and $j$ are coprime" is true or false.

Answer: False


Question: Calculate \(\lim_{r\rightarrow 0^+}\int_{D_r} \frac{x+1}{y} \ dx \ dy\) where \(D_r=\{(x,y)\in \mathbb{R}^2:x^2+y^2\ge r^2,0\le x\le y^2,r\le y \le 1 \}\). Determine if this is an improper integral.

Answer: \dfrac{5}{8}


Question:
Let $n = P_1^{r_1} \cdot P_2^{r_2} \cdot \dots \cdot P_k^{r_k}$, where $P_1, P_2, \dots, P_k$ are distinct primes. Define the function $f(n) = r_1^{P_1} \cdot r_2^{P_2} \cdot \dots \cdot r_k^{P_k}$. Determine which of the following statements is true:

1. The function $f$ is one-to-one.
2. The function $f$ is surjective.
3. If there exist $a, b$ such that $f(a) = m$ and $f(b) = n$, then there exists a $c$ with $f(c) = mn$.
4. $f(m) \cdot f(n) \ge f(mn)$.
5. $f(m) \cdot f(n) \le f(mn)$.

Choose the correct statement.

Answer: 3


Question: Given two complex sequences $(a_n)$ and $(b_n)$, if $|a_n - b_n| \to 0$ as $n \to \infty$, does it imply that $| a_n^n - b_n^n | \to 0$ as $n \to \infty$? Answer 'yes' or 'no' and justify your answer.

Answer: No


Question: Find the function $f(x)$ that satisfies the integro-differential equation: $$\int_0^x f(u)du - f'(x) = x.$$ Note that the initial condition $f(0)$ is not provided.

Answer: C(e^x + e^{-x}) + 1


Question: Find the value of $x$ in the range $0 \leq x \leq 143$ that satisfies the congruence $x^{103} \equiv 2 \pmod{143}$. Provide your answer as a single integer.

Answer: 128


Question: Determine the number of connected components of the set \( L \cap \Delta \), where \( L = \alpha b \) is a line in \( \mathbb{R}^n \) with \( \alpha \in \mathbb{R} \) and \( b \in \mathbb{R}^n \) is fixed, and \( \Delta = \{ x \in \mathbb{R}^n : \pi(x) \text{ has roots in the open unit disk of } \mathbb{C} \} \). Assume \( n > 2 \).

Answer: 1


Question: On a $98 \times 98$ chessboard, the squares are colored alternately black and white. A move consists of selecting a rectangular subset of the squares (with boundaries parallel to the sides of the board) and changing their color. Determine the smallest number of moves required to make all the squares black.

Answer: 98


Question: Find a function $f$ with norm 1 in $L^1_{[0,1]}$ such that the norm of its antiderivative $Tf$ is also 1, where $T$ is the integration operator.

Answer: 1


Question: Let $\mathcal{H}^{\oplus n}$ denote the n-dimensional Hilbert space. We define a linear operator $A=[a_{i,j}]: \mathcal{H}^{\oplus n} \to \mathcal{H}^{\oplus n}$ by \[ A\begin{bmatrix} h_{1}\\ \vdots\\ h_{n} \end{bmatrix}= \begin{bmatrix} \sum_{k=1}^{n}a_{1k}h_{k}\\ \vdots\\ \sum_{k=1}^{n}a_{nk}h_{k}. \end{bmatrix}. \] Define $A^t := [a_{j,i}].$ Are the operator norms of $A$ and $A^t$ the same?

Answer: Yes


Question:
Determine which of the following statements about the partial differential equation \(xU_x + yU_y = 0\) with the boundary condition \(U(x, y) = x\) on the unit circle \(x^2 + y^2 = 1\) is true:

1. There exists a unique solution for all \(x, y \in \mathbb{R}\).
2. There exists a bounded solution in \(\{ (x, y) \in \mathbb{R}^2 : (x, y) \neq (0, 0) \}\).
3. There exists an unbounded solution in \(\{ (x, y) \in \mathbb{R}^2 : (x, y) \neq (0, 0) \}\).
4. There exists a unique solution in \(\{ (x, y) \in \mathbb{R}^2 : (x, y) \neq (0, 0) \}\), but the solution is unbounded.

Given \(U(\cos(\theta), \sin(\theta)) = \cos(\theta)\) for all \(\theta\), verify the correct statement.

Answer: 2


Question: Given a holomorphic function $g: \mathbb{C} \to \mathbb{C}$, find the derivative of $g \bar{g}$ with respect to $z$, where $\bar{g}$ denotes the complex conjugate of $g$. Use the derivative definition: $\frac{\partial}{\partial z} = \frac{1}{2}(\frac{\partial}{\partial x} - i \frac{\partial}{\partial y})$. Provide your answer as an expression.

Answer: g' \overline{g}


Question: Determine the order of the special linear group $SL_2(\mathbb{F}_3)$. Provide your answer as a single integer.

Answer: 24


Question: Determine whether the set defined by the constraints $p_0(a) = 1$ and $|p_t(a)| \le 1$ for all $t \in [0,1]$, where $p_t(a) = \sum_k a_k t^{k-1}$, is a convex set.

Answer: Yes


Question:
Let $f: [0,1] \times [0,1] \to \mathbf{R}$ be an infinitely differentiable function that is non-negative and has finitely many zeros. Consider an arbitrary zero $(a, b) \in [0,1] \times [0,1]$ of $f$. The Taylor series of $f$ centered at $(a, b)$, denoted by $\widetilde{f}(x, \xi)$, is given by:

$$ \widetilde{f}(x, \xi) = \sum_{0 \leq i+j< \infty} \frac{\partial^{i+j}f(a,b)}{\partial x^i \partial \xi^j}\frac{(x-a)^i(\xi-b)^j}{i!j!} $$

Assume that this series converges for points within a ball $B_{\delta}$ with radius $\delta > 0$ and $f(x, \xi) = \widetilde{f}(x, \xi)$ for those points. Moreover, the first non-zero derivative of $f$ is the second partial derivative with respect to $\xi$, i.e., $\frac{\partial^2}{\partial \xi^2}f(a,b) \neq 0$, and there exists an $n \geq 1$ such that $\frac{\partial^n}{\partial x^n}f(a,b) \neq 0$. Determine if there exist positive constants $c$, $k$, and $\widetilde{\delta}$ such that for $(x, \xi) \in B_{\widetilde{\delta}}$, the following inequality holds:

$$ |f(x,y)| \geq c \left((x-a)^2 + (\xi-b)^2\right)^{\frac{k}{2}} $$

Answer: Yes


Question:
Let $\vec{i},\vec{j},\vec{k}$ be the orthogonal unit vectors along the axes of $\mathbb{R}^3$. Define $\vec{v} = -\vec{j} + \vec{k}$. Consider the linear transformation $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ given by $f(U) = \vec{v} \times U + 2U$ for each $U \in \mathbb{R}^3$. Determine which of the following statements is true:

(a) $f$ is surjective but not injective
(b) $f$ is injective but not surjective
(c) $t(t^2 + t - 3)$ is the characteristic polynomial of $f$
(d) $f$ is injective

Select the correct option.

Answer: D


Question: Determine the values of $q$ for which the series \( \sum_{n=2}^\infty \frac{(-1)^n}{n^2 \log^q n \log \log n} \) converges.

Answer: (-\infty, \infty)


Question: Find a number such that when its digits are reversed and multiplied by the original number, the product consists only of the digit 1. If no such number exists, prove it.

Answer: 1


Question: Determine the dimension of \(\mathcal{L}_F(K)\) as a vector space over \(F\), given that \(K/F\) is a finite-dimensional field extension with \([K:F]=n\).

Answer: n^2


Question: In the group ring \( \mathbb{Z}[S_3] \), find the coefficient of the identity element \( e \) in the expansion of \( (e + (12))^2 \).

Answer: 2


Question: Evaluate the limit: $$\lim\limits_{h\to0} \frac{\tan(a+2h)-2\tan(a+h)+\tan a}{h}$$ using L'Hopital's rule or other appropriate techniques.

Answer: 0


Question: Find the degree of the field extension \( \mathbb{Q}(i,\sqrt{2},\sqrt{3}) \) over \( \mathbb{Q}(i,\sqrt{2}) \).

Answer: 2


Question: Evaluate the sum \( \tan^2(1^\circ) + \tan^2(3^\circ) + \tan^2(5^\circ) + \tan^2(7^\circ) + \ldots + \tan^2(89^\circ) \).

Answer: 4005


Question: Evaluate the triple integral \( \iiint_V y^2\exp(2xy)\, dx\,dy\,dz \) where the volume \( V \) is bounded by the surfaces \( x=0 \), \( y=1 \), \( z=0 \), \( y=x \), and \( z=8 \).

Answer: e^2 - 3


Question: Given an invertible matrix $A$, if there exists a row $a_i$ and a column $a^j$ such that deleting both the row and column from $A$ results in an invertible matrix, can the inverse of the reduced matrix be derived from the original inverse of $A$ using a specific operation similar to Theorem 3.1? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit \( \lim\limits_{x\to 0} \frac{\sin{(x^2)}-\sin^2{(x)}}{\sin{(x^2)}\sin^2{(x)}} \) using Taylor's Theorem and little-o notation.

Answer: \dfrac{1}{3}


Question: Determine the values of $p$ for which the triangle inequality fails for the function $\lVert (x_1,x_2) \rVert_p = (|x_1|^p+|x_2|^p)^{1/p}$ on $\mathbb{R}^2$. Provide your answer as a range or set of values for $p$.

Answer: (0, 1)


Question: Let \( P(z) \) be a polynomial with complex coefficients. Consider an arc large enough to contain all the simple poles of \( \frac{1}{P(z)} \). Is the integral of \( \frac{1}{P(z)} \) around this arc always zero?

Answer: No


Question: Suppose $f: \mathbb{R}^2 \to \mathbb{R}$ is a continuous function. Let $Z$ be the set of $z \in \mathbb{R}$ such that there exists a sequence $(x_i, y_i) \in \mathbb{Z}^2$ with $f(x_i, y_i) \to z$. Determine whether $Z$ is a closed subset of $\mathbb{R}$.

Answer: Yes


Question: Calculate the dual norm \(\|T\|_{E^*}\) of the linear functional \(T: V \rightarrow \mathbb{R}\), where \(V\) is the vector space of continuous functions \(v:[0,1] \rightarrow \mathbb{R}\) with \(v(0) = 0\), equipped with the supremum norm \(\|v\| = \sup_{x \in [0,1]} |v(x)|\). The functional \(T\) is defined by \(T(v) = \int_{0}^{1} v(t) dt\). It is given that \(\|T\|_{E^*} = 1\).

Answer: 1


Question: Calculate the volume under the surface defined by the function \( f(x,y) = |x-y| \) over the triangular region with vertices at \((0,0)\), \((l,0)\), and \((0,l)\).

Answer: \dfrac{l^3}{6}


Question: Given the function \( f(x) = \sqrt{(x^2 - 2x + 2)(-x^2 + 3x + 10)} - \sqrt{(-x^2 + x - 1)(x^2 + x - 6)} \) with its domain \( D = [a, b + c] \), find the value of \( \frac{a^3 + b^3 + c^3}{abc} \).

Answer: 3


Question: Suppose $T: V \rightarrow V$ is a diagonalizable linear transformation with distinct eigenvalues, and let $\{e_i : i \in I\}$ denote an eigenbasis for $V$. If $W$ is an invariant subspace of $T$, does there exist a subset $J \subset I$ such that $W = \text{span}\{e_i : i \in J\}$? Justify your response.

Answer: Yes


Question: Evaluate the limit \( \lim_{x \to 0} \frac{\ln(1+x) - \sin(x) - \cos(x) + 1}{\arctan(x) - x} \) using Maclaurin series expansions.

Answer: -\dfrac{3}{2}


Question: Determine whether the expression \( \frac{1^\alpha + 2^\alpha + \cdots + n^\alpha}{n^{\alpha+1}} \) diverges to infinity as \( n \to \infty \) for \( \alpha \in (-\infty, -1) \).

Answer: \infty


Question: Find the vector \( w \in W \) that is closest to \( (0,2,1,0) \), where \( W = \text{span}\left(\{(1,-1,0,0),(1,2,0,-1),(1,0,0,1)\}\right) \subseteq \mathbb{R}^4 \).

Answer: (0, 2, 0, 0)


Question: Determine the rate of divergence of the series \( \sum_{j=0}^{\infty} \frac{1}{\sqrt{j}} \) as \( n \to \infty \). Express the rate of divergence of the partial sums \( S_n \) in the form \( O(n^{\alpha}) \) for some \( \alpha > 0 \).

Answer: O(n^{1/2})


Question: Evaluate the integral \( \int_D \frac{e^{1/w^2}}{w-2}dw \), where \( D \) is the annulus centered at the origin with inner radius 1 and outer radius 3.

Answer: 2\pi i e^{1/4}


Question:
Find the limit of the following integral as \( n \to \infty \) using the Dominated Convergence Theorem:
\[ \lim_{n \to \infty} \int_1^2 \left(1 - \frac{1}{n}\right)x^{-2} \, dx \]

Answer: \dfrac{1}{2}


Question: Show that the function \( f(x,y) = \frac{xy}{\sqrt{x^2+y^2}} \) is continuous at the point \((0,0)\), given that \( f(0,0) = 0 \).

Answer: 0


Question: Evaluate the sum of the shifted sinc function: $$g(y) = \sum_{n=-\infty}^\infty \frac{\sin(\pi(n - y))}{\pi(n-y)}.$$

Answer: 1


Question: In triangle $XYZ$, angle $XZY$ is $45$ degrees, and angle $ZYX$ is $80$ degrees. Let $P$ be the foot of the perpendicular from $X$ to $YZ$, $Q$ the center of the circle circumscribed around triangle $XYZ$, and $R$ the other end of the diameter which goes through $X$. Determine the measure of angle $XRP$ in degrees.

Answer: 45


Question: Solve for $x$ in the equation $x^2e^{4x}=4$ using the Lambert W function.

Answer: \dfrac{W(4)}{2}


Question: Determine the radius of convergence of the power series \( \sum_{n=0}^{\infty} \frac{n}{5^{n-1}} z^{\frac{n(n+1)}{2}} \).

Answer: 1


Question: For a connected matrix Lie group $G$, determine if there exists a neighborhood $U$ of the identity element $I \in G$ such that the map $\exp_I(.)$ restricted to this neighborhood is a bijection.

Answer: Yes


Question: Find the dimension of the null space of a linear transformation $T \in \wedge^{n-2}V^*$, where $V$ is a vector space with dimension $n$ and $T \neq 0$. Provide your answer as a single integer.

Answer: 2


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

Answer: \dfrac{\pi}{4}


Question:
Calculate the error bound for the Hermite interpolation of the function \( f(x) = 3xe^x - e^{2x} \) using the interpolation points \( x_0 = 1 \) and \( x_1 = 1.05 \). The Hermite interpolating polynomial is given by:

\[ H(x) = 0.7657893864 + 1.5313578773(x-1) - 2.770468386(x-1)^2 - 4.83859508(x-1)^2(x-1.05) \]

Use the error bound formula:

\[ E = \frac{f^3(\xi)}{3!} \cdot (x-1)(x-1.05) \]

Determine the maximum of the expression \((x-1)(x-1.05)\) and calculate the error bound.

Answer: 0.00319


Question: Evaluate the surface integral of the vector field \( \mathbf{F}(x,y,z) = xy\hat{i} + x^2\hat{j} + 2\hat{k} \) over the closed surface enclosed by the cone \( z=\sqrt{x^2+y^2} \) and the plane \( z=4 \), using the Divergence Theorem. The divergence of \( \mathbf{F} \) is \( \nabla \cdot \mathbf{F} = y \).

Answer: 0


Question: Given the functions \( x(s,t) = st \), \( y = y(s,t) \) with \( y(1,1) = 1 \) and \( \frac{\partial y}{\partial t}(1,1) = 2 \), and \( u = x^2 - y^2 \), \( v = v(x,y) \) with \( \frac{\partial v}{\partial y}(1,1) = -3 \). Define \( f(s,t) = u(x(s,t),y(s,t)) \) and \( g(s,t) = v(x(s,t),y(s,t)) \). Given that \( \frac{\partial f}{\partial s}(1,1) = -4 \), \( \frac{\partial f}{\partial t}(1,1) = -2 \), and \( \frac{\partial g}{\partial s}(1,1) = -1 \), find the value of \( \frac{\partial g}{\partial t}(1,1) \).

Answer: 2


Question: Let \(X = C^1([0,1], \mathbb{R})\) be the space of real-valued continuous functions defined on \([0,1]\), differentiable and having a continuous derivative on \([0,1]\). Define a function from \(X \times X\) into \(\mathbb{R}^{+}\) by \(d(f,g) = \sup_{x \in [0,1]} | f'(x) - g'(x)|\) where \(f '\) stands for the derivative of \(f\). Is \(d\) a metric on \(X\)?

Answer: No


Question:
Confirm the correctness of the following estimate for a function \( g(t,x) \leq 0 \) for all \( x \in \mathbb{R} \) and \( t \geq 0 \), where \( m(t) = \inf_{x\in\mathbb{R}}g(t,x) \) and \( h(y) \) is a positive function such that \( \int_0^{\infty}{h(x)dx} = 1 \):

$$\int_0^{\infty}{h(y)g(t,x)^2dx} \leq \sup_{x\in\mathbb{R}}|g(t,x)|^2\int_0^{\infty}{h(y)dx} = (-m(t))^2 = m(t)^2.$$

Answer: m(t)^2


Question: Given $x_1, x_2, \dots, x_n, y_1, y_2, \dots, y_n$ such that $x_i^2 + y_i^2 = 1$ for all $i$, find the maximum value of the sum \( \sum_{1 \leq i < j \leq n} a_i a_j (x_i y_j - x_j y_i)^2 \) over all $a_1, a_2, \dots, a_n \geq 0$ such that \( \sum_{i=1}^n a_i = 1 \).

Answer: \dfrac{1}{4}


Question: Determine whether a symmetric matrix can have a largest eigenvalue that is smaller than one of its diagonal entries. Provide a justification for your answer.

Answer: No


Question: Find the remainder when $314^{420}$ is divided by 1001.

Answer: 1


Question: Define the sequence $\{a_i\}_{1}^{\infty}$ by the formula: $$a_i = \left\lfloor \dfrac{10^{i+1}}{7}\right\rfloor-100\left\lfloor\dfrac{10^{i-1}}{7}\right\rfloor.$$ Determine the maximum value of $a_i$ for $i \in \mathbb{N}$. \(\boxed{\text{Your answer here}}\)

Answer: 85


Question: Evaluate the limit of the expression \(\frac{x^x}{x!}\) as \(x\) approaches infinity.

Answer: \infty


Question: Find the remainder when $7^{23456789}$ is divided by 100.

Answer: 7


Question: Determine whether the set of continuous functions on the interval $[a,b]$ that satisfy the inequality $|f(x)| \leq 2+f(x)^2$ is a complete metric space with respect to the sup norm metric $\rho_{\infty}$.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to\pi} \frac{1}{x-\pi} \left(\sqrt{\frac{4\cos^2x}{2+\cos x}} - 2\right)$$

Answer: 0


Question: Determine whether the zero matrix is the only real-valued matrix that is both symmetric and nilpotent. Provide a justification for your answer.

Answer: Yes


Question: Let $g:\mathbb{R}\to\mathbb{R}$ be an injective function such that $$(g\circ g)(x)=16^x-1$$ for all $x\in\mathbb{R}$. If $g(0) < g(1)$, determine the possible values of $g(0)$. \(\boxed{?\)}

Answer: 0


Question:
Suppose \(\sum_{j=1}^{\infty}a_j\) is a convergent series with non-negative terms. Determine whether the following limit is true:
\[\lim_{n\to \infty}S_n=0,\qquad S_n:=\sum_{j=1}^{n-1} \frac{a_j}{n-j}.\]

Answer: 0


Question:
Compute \( \lim_{x \to -1} f(x) \) for a function \( f: \mathbb{R} \to \mathbb{R} \) given by the following limit equation:

\[ 4 = \lim_{x \to -1} \left( \frac{f(x)+2}{x+1} - \frac{x}{x^2-1} \right) \]

Answer: -\dfrac{3}{2}


Question: Determine the total number of homomorphisms from a free group of rank 2 to the group of integers modulo 4.

Answer: 16


Question: Evaluate the limit: $$\lim\limits_{x \to +\infty} \left(\frac{(\ln(x+2))^x}{(3x)^{x/2}}\right)$$ without using Taylor expansions.

Answer: 0


Question: Let $L = \mathbb{F}_3[X]/(X^3 + X^2 + 2)$ be a field with $27$ elements. Determine whether there exists a finite field extension $M \supseteq L$ with degree $n \geq 2$. Provide a justification for your answer.

Answer: Yes


Question: Determine the field of fractions of the quotient ring \( \mathbb{Z}[X]/(X^2+1) \).

Answer: \mathbb{Q}(i)


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

Answer: \dfrac{2}{5}


Question: Determine whether the set of $n \times n$ invertible matrices with entries from a commutative ring $R$ with identity forms a group under matrix multiplication.

Answer: Yes


Question: Find the coefficient of \(x^{25}\) in the expansion of \((x^2+x^4)(1+3x^6)^{-11}(1-5x^8)^{-25}\).

Answer: 0


Question: Do matrices in $GL_{2n+1}(\mathbb{R})$ always have at least one eigenvector?

Answer: Yes


Question: Determine the number of subsets of the set \( A = \{1, 2, 3, \ldots, 35\} \) that contain exactly 26 elements and have a sum of elements divisible by 5.

Answer: 14121492


Question: Verify the trigonometric identity \(1+z+z^2+z^3+z^4=0\) given \(z=e^{2\pi i/5}\) without using a calculator.

Answer: 0


Question: Determine whether there exists a sequence of integers that contains a convergent subsequence but does not converge itself. Provide a justification for your answer.

Answer: Yes


Question: What is the group of symmetries of the Euclidean n-cube generated by reversions and transpositions?

Answer: B_n


Question: Evaluate the integral \( \int_0^\infty x^2 e^{-2x} \, dx \) using the Gamma function.

Answer: \dfrac{1}{4}


Question: Evaluate the double integral \(\int^{4}_{0}\int_{0}^{\sqrt{4x-x^2}}\sqrt{x^2+y^2}\, dy\, dx\) by converting it into polar coordinates. The region is defined by the equation \((x-2)^2 + y^2 = 4\). Determine the limits for \(r\) and \(\theta\) and express the integral in polar form.

Answer: \dfrac{128}{9}


Question: Calculate the value of $i^i$, where $i$ is the imaginary unit.

Answer: e^{-\pi/2}


Question: Evaluate the limit \( \lim_{x\to 1^-} \int_0^1 \left( \frac{2}{\sqrt{(1-t^2)(1-xt^2)}} -\frac{x}{1-xt} \right)\, dt \).

Answer: 4 \ln 2


Question: Determine whether the set \( \left\{ \frac{m^3}{n^3} : m,n \in \mathbb{N} \right\} \) is everywhere dense in the set of positive rational numbers.

Answer: Yes


Question: Determine whether the following statement is true or false: If a sequence $\{g_n\}$ in $L^2(\Omega,\mathcal{A},\mu)$ converges to $g$ in the $L^2$-norm and $\int g_n\,d\mu=C$ for some constant $C$ and all $n$, then $\int g \, d\mu=C$. Provide a justification for your answer.

Answer: False


Question: Determine the fundamental group of the space of matrices of the form \( \begin{pmatrix} a & b & c \\ -b & a & d \\ 0 & 0 & 1 \end{pmatrix} \), where \( a^2 + b^2 = 1 \).

Answer: \mathbb{Z}


Question: Consider a norm \( \| \cdot \| \) on \( \mathbb{R}^n \) that is smooth, meaning its restriction \( \| \cdot \|: \mathbb{R}^n \setminus \{ 0 \} \to \mathbb{R} \) is a smooth map. Determine whether the differential of this norm at every non-zero point is nonzero.

Answer: Yes


Question: Let \( f:\mathbb{R}\rightarrow \mathbb{R} \) be a continuous function satisfying \( x+\int_{0}^{x}f(t)dt=e^x-1 \) for all \( x\in \mathbb{R} \). Determine the interval representing the set \( \{x \in \mathbb{R}: 1\le f(x)\le 2\} \).

Answer: [\ln 2, \ln 3]


Question: Determine the subset $A \subset [0,8]^2$ with the largest Lebesgue measure such that $A$ and its translation by the vector $(3,4)$ do not intersect, i.e., $(A+(3,4)) \cap A = \emptyset$. What is the measure of this subset?

Answer: 44


Question:
Find the function $y(x)$ that is differentiable for all $x \in \mathbb{R} - \{0\}$ and satisfies the integral equation:
\[ x\int_{1}^{x} y(t) \,dt = (x+1) \int_{1}^{x} t \,y(t) \,dt \]

Answer: 0


Question: Let $F$ be a field of characteristic $0$ and let $D$ be the formal derivative map, defined as \( D(a_0 + a_1x + a_2x^2 + \dots + a_nx^n) = a_1 + 2a_2x + \dots + na_nx^{n-1} \). Find the image of $F[X]$ under $D$.

Answer: F[X]


Question: Let $\phi: A \to B$ be a ring homomorphism where $B$ is finitely generated as an $A$-module. Consider a prime ideal $\mathfrak{p} \subseteq A$ and let $K$ be the field of fractions of the quotient ring $A/\mathfrak{p}$. Determine whether the ring $B \otimes_A K$ has only finitely many prime ideals.

Answer: Yes


Question: Determine whether a non-semisimple Lie algebra can have an invariant inner product. Provide a yes or no answer.

Answer: Yes


Question: Solve for \(x\) in the equation: \[ \sqrt{x + \sqrt{4x + \sqrt{16x + \sqrt{64x + 5}}}} - \sqrt{x} = 1 \]

Answer: \dfrac{1}{16}


Question: Let $f : [a,b] \to \mathbb{R}$ be a continuous function. Define the function $g : (c, \infty) \to \mathbb{R}$ by \[ g(x) = \int_a^b f(t) \, t^{x-1} \, dt. \] Is $g$ necessarily a differentiable function on $(c, \infty)$?

Answer: Yes


Question: Calculate the determinant $D_n$ of an $n \times n$ matrix with the following structure: the first and last rows have 1 in the first position, 2 in the second, and 0s in the remaining entries; the second and penultimate rows have 5s in the second position, 8s in the third, and 3s in the fourth; and all other rows follow a pattern similar to the central 3 diagonals (5s, 8s, and 3s).

Answer: 0


Question: Determine the minimum value of \( a > 0 \) such that \( x^a \ge \log_e(x) \) holds for all \( x > 0 \).

Answer: \dfrac{1}{e}


Question: Let $f(x,y)$ be a function satisfying the condition \( f(x,y) = f(2x + 2y, 2y - 2x) \) for all \(x, y \in \mathbb{R}\). Define a function \(g(x) := f(2^x, 0)\). Determine if \(g(x)\) is periodic. If it is periodic, find its period.

Answer: 12


Question: Let $T : X \rightarrow \mathbb{R}$ be a non-bounded linear operator. Determine whether there exists a sequence $\{x_n\} \subseteq X$ such that $\|x_n\| \to 0$ and $Tx_n \to \alpha$ for some $\alpha \in \mathbb{R}$.

Answer: Yes


Question: Find a differentiable function that serves as a lower bound for \( \min(a,b) \) given \( a, b \geq 0 \) and \( a + b = c \), where \( c \) is a constant.

Answer: \dfrac{ab}{c}


Question: Let $X$ be a $p \times q$ matrix with characteristic polynomial $(x-c)^r(x-d)^s$, where $c$ and $d$ are distinct complex numbers. Determine the dimension of the complex vector space $W$ of all $q \times p$ matrices $Y$ such that $XY=YX$. Provide your answer as a single integer.

Answer: r^2 + s^2


Question: Determine the lower bound for the index of the commutator subgroup of a non-abelian p-group using the conjugacy class equation.

Answer: p^2


Question: Determine the Krull dimension of the ring \( \frac{\mathbb{K}[X_1,X_2]}{(X_1X_2)} \), where \( \mathbb{K} \) is a field.

Answer: 1


Question: Simplify the expression \( \sqrt[3]{19\sqrt{5} + 56} + \frac{11}{\sqrt[3]{19\sqrt{5} + 56}} \) without using the cubic equation \( z^3 - 33 z - 112 = 0 \) or any other prior knowledge about the value of the expression.

Answer: 7


Question: Evaluate the limit: \( \lim_{x\rightarrow\infty}2x(a+x(e^{-a/x}-1)) \).

Answer: a^2


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

Answer: \sqrt{e}


Question: Find the value of $f(3)$ for a function $f : \mathbb{R} \to \mathbb{R}$ with a continuous derivative such that $f(\sqrt{2}) = 2$ and satisfying the property $$f(x) = \lim_{t \to 0}\dfrac{1}{2t}\int_{x-t}^{x+t}sf'(s)\,ds \ \text{for all} \ x \in \mathbb{R}.$$ The given options are: (a) $\sqrt{3}$, (b) $3\sqrt{2}$, (c) $3\sqrt{3}$, (d) $9$. Choose the correct option.

Answer: B


Question: Evaluate the limit: $$ \lim_{x \to\infty} \frac{\log\log x }{ \sqrt{\log x}} $$

Answer: 0


Question: If two solutions $u(t)$ and $v(t)$ of the differential equation $x' = f(t, x)$ intersect at a point $(t_0, x_0)$, are their tangent lines at that point identical?

Answer: Yes


Question: Consider a noncompact symmetric space $X = G/K$, where $G$ is a semisimple, noncompact, connected Lie group with finite center, and $K$ is the stabilizer of a fixed point in $X$. Given that the dimension of $X$ is $n$, the volume of a geodesic ball $B_X(x, r)$ in $X$ grows as $|B_X(x, r)| \sim r^n$ for small radii $r > 0$. Let $\Gamma$ be a discrete, torsion-free subgroup of $G$, and consider the locally symmetric space $M = \Gamma \backslash G/K$. Does the volume of small geodesic balls $B_M(\tilde{x}, r)$ in $M$ grow as $|B_M(\tilde{x}, r)| \sim r^n$ for every $\tilde{x} \in M$ and sufficiently small $r > 0$?

Answer: Yes


Question: Find the last eight digits of the binary representation of \(27^{1986}\).

Answer: 11011001


Question: Find the Taylor series expansion of the operator \( D = e^{bx \frac{d}{dy}} \) when it operates on an infinitely differentiable function \( g(y) \).

Answer: \sum_{n=0}^{\infty} \frac{(bx)^n}{n!} g^{(n)}(y)


Question: Calculate the asymptotic relative efficiency of a Poisson process for $t=1$, using the information provided for $t=0$. Provide your answer as a numerical value or expression.

Answer: \infty


Question: Suppose $g$ is a holomorphic function on $\mathbb{C} \backslash L$, where $L = \{1, 2, \dots\}$, and is bounded. If $g(0) = 2$, find all possible functions $g$. Provide your answer in the form of a function or expression.

Answer: 2


Question: In a casino game involving fair coin flips, the game ends when the sequence $HTH$ appears. Using the concept of martingales and the optional stopping theorem, calculate the expected number of flips required for the sequence $HTH$ to occur.

Answer: 10


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:
Determine the number of distinct parse trees for the expression $a+b*c-d$ using the following grammar:

$$\begin{align}
E &\to E + T \mid E - T \mid T \\
T &\to T * F \mid T / F \mid F \\
F &\to (E) \mid id
\end{align}$$

Answer: 1


Question: In a commutative integral domain with unity, consider an irreducible element $a$ such that $a = bc$, where $b$ is an associate of $a$. Does it logically follow that $c$ must be a unit? Provide a justification for your answer.

Answer: Yes


Question: Determine whether there exists a function \( f: \mathbb{C} \to \mathbb{C} \), expressed as \( f = u + iv \), such that the partial derivatives \( u_x, u_y, v_x, \) and \( v_y \) exist and satisfy the Cauchy-Riemann equations at \((0,0)\), but \( f \) is not differentiable at \((0,0)\).

Answer: Yes


Question: Evaluate the limit: $$\lim_{h\rightarrow 0} \frac{\ln(1 + hx) - hx}{h^{2}}$$

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


Question: Evaluate the limit \( \lim_{n\to\infty} n\int_{1}^{2} \frac{1}{x^2(1+x^n)}dx \).

Answer: \ln 2


Question: Determine whether the following statement is true or false: For a finite group $G$ and elements $g_1, g_2 \in G$, if every representation $\rho: G \to GL(V)$ with $V$ a finite-dimensional complex vector space satisfies $\chi_\rho(g_1) = \chi_\rho(g_2)$, where $\chi(g) := \text{trace}(\rho(g))$, then $g_1$ and $g_2$ belong to the same conjugacy class.

Answer: True


Question: Determine if the sequence of random variables $\{Y_n\}$, where $Y_n = X_n/\sqrt{1+1/n}$, satisfies the Central Limit Theorem.

Answer: Yes


Question: Find a generator for the cyclic group of units in \(\mathbb{Z}/(2017\mathbb{Z})\).

Answer: 5


Question: Evaluate the limit: \[ \lim_{x\to 0}\dfrac{e^{-x^2/2}-\cos(x)}{x^3\sin(x)} \]

Answer: \dfrac{1}{12}


Question: Determine whether the limit \( \lim_{n \to \infty} \sum_{k=1}^n\ln\left(1-\frac{x^2\sin^2k}{2n}\right) \) exists for \( x \in \mathbb{R} \). If it exists, find its value.

Answer: -\dfrac{x^2}{4}


Question: Evaluate \( \lim_{n \to \infty} \int_0^1 \frac{nx^{n-1}}{1+x^2} \, dx \).

Answer: \dfrac{1}{2}


Question: Determine whether the polar decomposition of a bounded operator is unique if and only if the operator is invertible. Answer true or false.

Answer: False


Question: Determine the saddle point of the function \( f(x,y) = h(x) + yx \), where \( h(x) \) is a nonlinear function. The saddle point is the point where \( f \) is a maximum with respect to \( x \) and a minimum with respect to \( y \).

Answer: (0, -h'(0))


Question: Find all possible values of $x$ such that \( \tan^{-1}(x+1) + \tan^{-1}(x) + \tan^{-1}(x-1) = \tan^{-1}(3) \). Consider the domain restrictions of the inverse tangent function.

Answer: 5 - \sqrt{19}


Question:
Determine if the function \( g(x) \) is Riemann integrable on the interval \([0,1]\), where \( g(x) \) is defined as:

\[ g(x) = \begin{cases} 1/q & \text{ if } x=p/q \text{ is rational and in lowest terms}; \\
0 & \text{ if }x\text{ is irrational}. \end{cases} \]

Answer: Yes


Question: Given a field extension $K/F$ with $[K:F] = p$, where $p$ is a prime number, and an element $a \in K \setminus F$, determine the possible value of $[F(a):F]$. \( \text{Provide your answer as a single number.} \)

Answer: p


Question:
Evaluate the integral \( \iiint yz \,dV \) over the tetrahedron with vertices at \( A(0,0,0), B(1,1,0), C(1,0,0), D(1,0,1) \). The limits of integration are given by:
\[
\int_0^1\int_0^x\int_0^{x-y} yz\;dzdydx
\]

Answer: \dfrac{1}{120}


Question: Determine whether the following statement is true or false: If a function $f$ is continuous on $[0,1]$, differentiable on $(0,1)$, and its derivative $f'$ is continuous on $(0,1)$, then the total variation of $f$ on $[0,1]$ is equal to the integral of the absolute value of $f'$ on $[0,1]$. Provide a proof or counterexample.

Answer: True


Question: Find the last two digits of the infinite tetration \(256^{256^{256^{\ldots}}}\), where the number of 256's in the sequence is odd.

Answer: 56


Question: Determine the conditions on the constant $c$ such that the process $X_t = Y\cos(ct)$, where $Y$ is a random variable with mean zero and variance $\sigma^2$, is stationary.

Answer: c = 0


Question: Evaluate the integral \( \int_{0}^{2\pi} \frac{1}{5-3\sin(\theta)} \, d\theta \) using complex analysis techniques.

Answer: \dfrac{\pi}{2}


Question: Can the total sum of geometric multiplicities for an $n \times n$ matrix be less than $n$?

Answer: Yes


Question: Determine the values of $n$ for which the alternating group $A_n$ has a subgroup of index 4.

Answer: 4


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

Answer: \infty


Question: Determine the number of field homomorphisms from a finite field with nine elements to a finite field with four elements.

Answer: 0


Question: Let $x_n$ be a real-valued sequence such that $x_n \to 0 \pmod{a}$ for every real $a > 0$. Determine whether it follows that $x_n \to 0$.

Answer: Yes


Question: Determine the limit of the expression \( \frac{x^{3x}}{(2x+1)^{2x+1}} \) as \( x \to \infty \).

Answer: \infty


Question: Evaluate the limit: $$\lim_{x\to 1^+}\frac{\sqrt{x+1}+\sqrt{x^2 -1}-\sqrt{x^3+1}}{\sqrt{x-1}+\sqrt{x^2 +1}-\sqrt{x^4+1}}$$

Answer: \sqrt{2}


Question: Rewrite the expression \((A^{-1}B^{-1})^T(X+C^T)(B^TA^T)^{-1}\) without using inverses, given that \(A^TB=I\).

Answer: A B (X + C^\top) B A


Question: Calculate the limit: \( \lim_{n \to \infty} \left( 1 + \frac{2}{n} \right)^{n^2} e^{-2n} \).

Answer: e^{-2}


Question: Find the dimension of the vector space \( \mathbb{R}(x) \) over the field \( \mathbb{R}(x^3) \).

Answer: 3


Question: Evaluate the line integral \( \int_C y^2 \,dx + x\,dy \) using Green's theorem, where curve \( C \) is a square with vertices at \((\pm2, 0)\) and \((0, \pm2)\), traced counterclockwise. Verify if the result is 8 as suggested by the textbook.

Answer: 8


Question: Given a projective curve \( C \) in \( \mathbb{P}_\mathbb{C}^2 \), if a point \( p \in C \) has multiplicity \( m \), and \( \mathcal{O}_p(C) \) is the local ring of \( C \) at \( p \) with its maximal ideal \( \mathfrak{m} \), find the dimension of \( \mathfrak{m}^k/\mathfrak{m}^{k+1} \) over \( \mathbb{C} \) for \( k < m \).

Answer: k + 1


Question: Does there exist a probability distribution defined on $x \in (0, \infty)$ such that $p(x) = p(1/x)$ for all $x$, and satisfying the normalization condition $\int_0^\infty p(x) \, dx = 1$?

Answer: Yes


Question: Determine the number of ways to distribute 15 distinct balls into 5 indistinguishable boxes.

Answer: 255514355


Question: Compute the double integral \(\int_{0}^{\infty} \int_{0}^{x} \frac{h(x)}{x} \, dy \, dx\), where \(h(x)\) is a non-negative function with \(\int_{0}^{\infty} h(x) \, dx = 1\). Explain the choice of bounds for the integrals.

Answer: 1


Question: Evaluate the integral \( I = \int_{0}^{\pi/2} \lfloor \cot(x) \rfloor \, dx \).

Answer: \infty


Question: Evaluate the integral \( I = \int_0^{\infty} \frac{\log\left(x + \frac{1}{x}\right)}{1+x^2}\,dx \) given the integral identity \( \int_0^{\frac{\pi}{2}} \log(\sin x)\,dx = -\frac{\pi}{2} \log 2 \).

Answer: \pi \log 2


Question: Given that $u \in L^\infty(\mathbb{R}, \mathbb{R}^n)$, does the set of limit points of $u(x)$ as $x \to \infty$ necessarily contain at least one point?

Answer: Yes


Question: Prove that if $a$ and $b$ are positive integers such that $ab \mid a^2 + b^2$, then $a = b$.

Answer: a = b


Question: In a triangle with integer sides \(a > b > c\), the remainders when \(3^a, 3^b, 3^c\) are divided by 10000 are identical. Find the smallest possible perimeter of such a triangle, given that \(3^a \equiv 3^b \equiv 3^c \pmod{10000}\).

Answer: 3003


Question: Determine whether the following statement is always true: Let $A \in GL_n( \mathbb{Z})$. If $\det(A^2)= \det(A)$, then $A^2 \in \{A, A^{-1}, -A, -A^{-1}\}$. Note: $GL_n(\mathbb{Z})$ is the set of all invertible $n \times n$ matrices over $\mathbb{Z}$. Answer 'yes' if the statement is always true, otherwise answer 'no'.

Answer: no


Question: Can the complement of the zero set of a non-trivial holomorphic function in a connected open subset of \(\mathbb{C}^n\) be disconnected if \(n > 1\)?

Answer: No


Question: Consider a measurable space $(\Omega, \mathscr{F})$ with two probability measures $P$ and $Q$, where $Q$ is absolutely continuous with respect to $P$ (denoted $Q \ll P$) and has the Radon-Nikodym derivative $\frac{\text{d}Q}{\text{d}P} = f$. Let $E^P$ and $E^Q$ denote expectations with respect to $P$ and $Q$, respectively. Given an integrable random variable $X$ and a sub-$\sigma$-algebra $\mathscr{G} \subset \mathscr{F}$, the equation \[ E^Q(X|\mathscr{G})E^P(f|\mathscr{G}) = E^P(Xf|\mathscr{G}) \] holds $Q$-almost surely. Determine if this equation also holds $P$-almost surely.

Answer: Yes


Question: Compute the divergence of the force field \(\vec{F}(x,y,z) = (\sin(y+z),\cos(x+z),\sin(x+y))\). Then, evaluate the surface integral \(\int\int_S \vec{F} \cdot \hat{n} \, dA\), where \(S\) is the sphere with center at the origin and radius 1, and \(\hat{n}\) is the outward unit normal vector to the sphere.

Answer: 0


Question: Use the Seifert-van Kampen theorem to determine the fundamental group of the unit disk with the boundary identification where each point is identified with the point that is \(\pi/n\) radians away.

Answer: \mathbb{Z}_{2n}


Question: Does there exist an operator $A$ in a Hilbert space such that the composition $PAP$ is invertible, where $P$ is a non-trivial projection operator ($P \neq I$ and $P^2 = P$)?

Answer: No


Question: Evaluate the expression \(16 \cos \frac{2 \pi}{17} \cos\frac{4 \pi}{17} \cos\frac{8 \pi}{17} \cos\frac{16 \pi}{17}.\)

Answer: -1


Question: Determine whether the following statement is true or false: If $X$ is a topological space with $X = A \cup B$, where $A$ and $B$ are closed subsets, and $f : X \rightarrow Y$ is an injective mapping such that the restrictions $f_{|A} : A \to f(A)$ and $f_{|B} : B \to f(B)$ are homeomorphisms, then $f : X \rightarrow Y$ is an embedding, meaning that $f : X \to f(X)$ is a homeomorphism.

Answer: B


Question: Are two finite fields, each with $p^n$ elements, isomorphic if they are constructed using different irreducible polynomials of degrees $n$ and $m$ where $n = mt$ and $m, t > 1$? Provide a justification for your answer.

Answer: Yes


Question:
Find the determinant of the infinite matrix \(M\) with the following repeating pattern:

$$M=\begin{bmatrix}1&0&1&0&\cdots\\ 0&1&0&1&\cdots\\ 1&0&1&0&\cdots\\ 0&1&0&1&\cdots\\ \vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix}$$

Answer: 0


Question: Suppose $X$ and $Y$ follow a bivariate normal distribution with parameters $\mu_X = \mu_Y = 0$, $\sigma_X^2 = \sigma_Y^2 = 1$, and correlation coefficient $\rho_{X,Y} \neq 0$. Determine the correlation coefficient between $XY$ and $X$, denoted as $\rho(XY, X)$. Use the hint that a linear transformation can make $X$ and a new variable $Z$ independent.

Answer: 0


Question: Evaluate the sum \( C_n = \sum_{k=0}^{n} \cos\left(\dfrac{2k\pi}{2n+1}\right) \).

Answer: \dfrac{1}{2}


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

Answer: \dfrac{\pi}{2}


Question: Find the area of the largest rectangle that can be inscribed in the ellipse given by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

Answer: 2ab


Question: Evaluate the expression $23^{54^{26}} \bmod 45$.

Answer: 1


Question:
Evaluate the limit:

$$ \lim_{x \to \infty} {\displaystyle{\large\left(1 + x\right)\left(1 + \frac{1}{x}\right)^{2x} - x\,{\rm e}^{2}} \over \displaystyle{\large{\rm e}^{2} - \left(1 + \frac{1}{x}\right)^{2x}}} $$

Answer: \dfrac{1}{6}


Question: In a Hilbert space \( \mathcal{H} \), does every element \( x \) have a corresponding element \( y \) in a closed subspace \( M \) such that the difference \( x - y \) lies in the orthogonal complement \( M^{\perp} \)?

Answer: Yes


Question: Evaluate the limit: $$\lim _{n\to \infty }\left(\frac{\left(4n-4-2a_n\right)}{\pi }\right)^n$$ where $$a_n=\int _1^n\:\frac{2x^2}{x^2+1}dx$$.

Answer: e^{-\frac{4}{\pi}}


Question: Let $H$ be a Hilbert space. Consider a sequence $\{a_n\} \subseteq H$. If for every $x \in H$, the inner product $\langle x, a_n \rangle \to 0$ as $n \to \infty$, does it follow that $a_n \to 0$ in $H$ as $n \to \infty$?

Answer: No


Question: Calculate the value of the following expression: $$ \log_{10}(\cot(1^\circ)) + \log_{10}(\cot(2^\circ)) + \cdots + \log_{10}(\cot(89^\circ)) $$ where all logarithms are base 10.

Answer: 0


Question:
Determine the correct option for the minimal polynomial of a 3x3 real matrix $A$ that satisfies $A^8 = I$ from the following choices:
1. The minimal polynomial has degree 2.
2. The minimal polynomial has degree 3.
3. $A = I$ or $A = -I$.
4. There are uncountably many such $A$.

Consider the implications of $A = I$ and $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$ to eliminate incorrect options.

Answer: 4


Question: Given the function \( f = 1 + \sum_{k=0}^{100}\frac{(-1)^{k+1}}{(k+1)!}x(x-1)\cdots (x-k) \), let \( S \) be the sum of all real roots of \( f \) and \( T \) be the sum of all real roots of \( f' \). Find the value of \( S-T \).

Answer: 51


Question:
Find the coefficient of $x^{n-1}$ in the Chebyshev polynomial of the first kind $T_n(x)$, where the polynomials are defined by the recurrence relations:

$$ T_0(x)=1, \quad T_1(x)=x, \quad T_n(x)=2x\cdot T_{n-1}(x)-T_{n-2}(x) $$

Answer: 0


Question: Find the minimum value of \( \frac{a}{b + 1} + \frac{b}{a + 1} + \frac{1}{a + b} \) given that \( a \) and \( b \) are positive numbers such that \( a + b \le 1 \).

Answer: \dfrac{5}{3}


Question: Find the infimum of the function \( g(y) = \frac{1}{2}\left(\sum_{i=1}^{m}\frac{x_i^2}{y_i} - 1^Ty\right) \) over the set of positive vectors \( y \).

Answer: -\infty


Question: Determine a formula to calculate the number of distinct accepted $2 \times n$ matrices, where the first row is a permutation of the numbers $1$ to $n$, and the second row is also a permutation such that no numbers are the same within rows or columns, and no column has identical numbers.

Answer: n! \cdot !n


Question: Evaluate the limit of the sequence \( y_n = \sqrt[n]{x_n} \), where \( x_n = \frac{(3n)!}{(n!)^3} \).

Answer: 27


Question: Evaluate the complex line integral \( \int_{C}\mbox{Im}(z^2)dz \) around the circle of radius 1 centered at the origin in the complex plane, oriented counterclockwise.

Answer: 0


Question: Calculate the conditional expectation $E(g(x,y)|H)$, where $f(x,y)=x^2+y^2$, $g(x,y)=x-y$, and $H=\sigma(X-Y)$. Provide your answer as an expression.

Answer: X - Y


Question: Consider a measure space \((X\times Y, \mathcal{M}\times\mathcal{N}, \mu\times\nu)\). Let \(E=\{(x,y):x+y=1\}\) be a line in \(X\times Y\). Determine \((\mu\times\nu)(E)\).

Answer: 0


Question: Determine the limits of $c$ as $\frac{A}{B}$ approaches infinity, given the equation $\frac{C}{4} + 2Ac^3 - c^2C - 2Ac^2C = 0$ and the constraints $A > 0$, $-\frac{1}{2} < A < 0$, or $A \leq -\frac{1}{2}$.

Answer: 0


Question: Given a finite group $G$, a subgroup $H$ of $G$, and a normal subgroup $N$ of $G$, where $|H| = n$ and $|G| = m|N|$, with $m$ and $n$ coprime, determine the order of the element $hN$ in the quotient group $G/N$ for an element $h \in H$.

Answer: 1


Question: Determine which function grows faster: \((n!)^{(n-1)!}\) or \(((n-1)!)^{(n!)}\). Use logarithms to justify your answer.

Answer: ((n-1)!)^{n!}


Question: Determine if the following calculation is valid: \[\int\limits_{ - \infty }^\infty x^2 e^{-x^2/2} \, dx = -xe^{-x^2/2} \Bigg|_{-\infty}^{\infty} + \int\limits_{-\infty}^{\infty} e^{-x^2/2} \, dx,\] which simplifies to \(0 + \sqrt{2\pi}\).

Answer: \sqrt{2\pi}


Question:
A sequence \( \{b_n\} \) is defined by the recurrence relation \( b_n = 2b_{n-1} - b_{n-2} + b_{n-3} \) with initial conditions \( b_1 = 1 \), \( b_2 = 1 \), and \( b_3 = 2 \). Calculate the value of the infinite series:
\[
b_1 + \frac{b_2}{3} + \frac{b_3}{3^2} + \cdots
\]

Answer: \dfrac{21}{11}


Question: Find the solution to the equation \(x^4 + x + 3 = 0\) modulo \(27\).

Answer: 24


Question: Suppose that $f(x)$ is a periodic function satisfying the equation: $$\forall x :f(x+346)=\frac{1+f(x)}{1-f(x)}.$$ Determine the minimum period of $f(x)$. \(\boxed{\text{Answer}}\)

Answer: 1384


Question: Given a sequence $(a_n)$ such that for every positive integer function $f(n)$, the difference $a_{n+f(n)} - a_n$ converges to $0$ as $n$ approaches infinity, determine whether this implies that the sequence $(a_n)$ is convergent.

Answer: Yes


Question: Determine whether a compact Lie group can have a non-compact Lie subgroup. Answer with 'Yes' or 'No'.

Answer: Yes


Question: In the floor topology on \( \mathbb{R} \), is every point of \( \mathbb{R} \) a limit point of \( \mathbb{R} \)?

Answer: Yes


Question: Determine all values of $c$ for which the series \( \sum\limits_{n=1}^{\infty} \left(\frac{c}{n}-\frac{1}{n+1}\right) \) converges.

Answer: 1


Question: Find the 39th derivative of the function \( f(x) = e^{-x^2} \) at \( x = 0 \) using the Taylor series centered at \( x = 0 \).

Answer: 0


Question: Calculate the total surface area of the two cylinders given by the equations \(y^2 + z^2 = 1\) and \(x^2 + y^2 = 1\).

Answer: 16


Question: An ellipse has a circle inscribed within it, touching the ellipse at its vertices and center. The circle's diameter is 7. The foci of the ellipse are also the vertices of an equilateral triangle, whose third vertex lies on the minor axis of the ellipse. If the height of the triangle is equal to the length of the semi-minor axis, determine the eccentricity of the ellipse.

Answer: \dfrac{1}{2}


Question: Determine the maximum number of elements in the set \( \mathcal{S} \) of all linear maps \( \mathbf{L} (\mathbb{C}^3) \) where no two maps are similar and satisfy the equation \( \mathbf{A^{10} - 3A^9 + 2A^8 = 0} \).

Answer: 14


Question: If $(x'_n)$ is a Cauchy sequence in the quotient space $X/Y$, is the corresponding sequence $(x_n)$ necessarily a Cauchy sequence in the normed space $X$? Answer with 'yes' or 'no' and justify your answer.

Answer: No


Question: Determine the number of elements in the ring \( \mathbb{Z}_2[X]/(x^5+x+1) \), given that \( x^5 + x + 1 = (x^2 + x + 1)(x^3 + x^2 + 1) \).

Answer: 32


Question: Calculate the area that lies inside the cardioid \( r = 1 + \sin \theta \) and outside the circle \( r = 7 \), where \( \theta \) is measured from the positive x-axis in polar coordinates.

Answer: 0


Question: Suppose $X$, $Y$, and $Z$ are independently and identically distributed (iid) random variables following an exponential distribution with parameter $a > 0$ ($Exp(a)$). Calculate the probability $P(X < Y + Z)$. Express your answer in terms of $a$. \( \boxed{?} \)

Answer: \dfrac{3}{4}


Question: Given a simple closed curve with winding number zero around each of several red points, is it possible to homotope the curve to a loop that does not intersect any of the red points?

Answer: Yes


Question: Evaluate the limit: $$\lim_{n \to \infty}\sum_{k=1}^n \frac{(-1)^{k}}{2k-1}$$

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


Question: Determine the minimum value of $a$ that results in a stable positive population of horseshoe crabs in the model $y' = -\frac{1}{4} + y - \frac{y^2}{a}$, where $y$ represents the number of kilocrabs.

Answer: 1


Question:
Determine the set of points of discontinuity, \( D_f \), for the function \( f: \mathbb{R} \to \mathbb{R} \) defined by:
\[ f(x) = \begin{cases}
0 & x \notin \mathbb{Q}\\
\frac{1}{n} & x = r_n \in \mathbb{Q}
\end{cases} \]

Answer: \mathbb{Q}


Question: How many invertible elements are there in the matrix ring \( M = \left\{ \left( \begin{matrix} a & b \\ c & d \end{matrix} \right)  \;\middle\vert\;\; a,b,c,d \in \mathbb{Z}_3 \right\} \)?

Answer: 48


Question: Suppose $X$ and $Y$ are topological spaces and $f: X \to Y$ is a continuous map. Let $\mathbb{k}$ be a field, and $i \ge 1$ an integer. If the induced linear map on homology $f_* : H_i ( X, \mathbb{k}) \to H_i ( Y, \mathbb{k})$ is surjective, does it necessarily follow that the induced linear map on cohomology $f^* : H^i ( Y, \mathbb{k}) \to H^i ( X, \mathbb{k})$ is injective? Provide a justification for your answer.

Answer: Yes


Question: Let \( p(x) \) be a polynomial of degree strictly less than 100. If \( \frac{d^{100}}{d x^{100}} \frac{p(x)}{x^2+1} =\frac{f(x)}{g(x)} \) for some polynomials \( f(x) \) and \( g(x) \), what is the smallest possible degree of \( f(x) \) when \( p(x) \) does not have \( x^2+1 \) as a factor?

Answer: 100


Question: Determine the order of the quaternion group $Q_8$ given by the presentation $$ Q_8 = \langle x, y, z : xz = zx, yz = zy, xy = yxz, x^4 = y^4 = z^2 = 1 \rangle. $$ Also, identify and eliminate any redundant elements in its representation.

Answer: 8


Question: Evaluate the integral: $$\int_0^\pi \frac{dx}{1+3\sin^2x}$$

Answer: \dfrac{\pi}{2}


Question: In the symmetric group $S_{10}$, consider the permutation $\sigma=(1\ 3\ 5\ 7\ 9)$. Determine the number of elements in $S_{10}$ that commute with $\sigma$. Provide your answer as a single integer.

Answer: 600


Question: Find the radius of convergence for the power series \( \sum_{k=1}^{\infty} \sqrt[k!]{k!} x^k \).

Answer: 1


Question: Calculate the remainder when \(256^{478}\) is divided by 1027.

Answer: 399


Question: Does the condition $\lim_{n\to\infty}S_n \leq S$ imply that $S_n < S$ for all $n$?

Answer: No


Question: Calculate the area between the curves $\ln x$, $|\ln x|$, $\ln |x|$, and $|\ln |x||$ over the interval $[0, 1]$. Provide your answer as a single numerical value.

Answer: 2


Question: Let $X_t$ be a Brownian motion. Calculate the probability that $X_2 > 1$ given that $X_1 = 0$. Express your answer as a single probability value.

Answer: 0.1587


Question: Determine whether the following statement is true or false: If $f$ is continuous on an open interval $(a, b)$, then there exists a point $y \in (a, b)$ such that $f(y) = \sup \{f(x) : x \in (a, b)\}$.

Answer: B


Question:
Determine all positive integers \(n\) such that:

\[ \left\lfloor\frac{n}{2}\right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor = n^2 \]

Answer: 24


Question: If $F$ is an isomorphic function, is its inverse $F^{-1}$ also isomorphic? Provide a justification for your answer.

Answer: Yes


Question: Determine whether the convergence of the series \(\sum_{i=1}^\infty \frac{a_i}{i}\), where \((a_i)_{i=1}^\infty\) is a sequence of positive real numbers, implies that the sequence has Cesaro mean zero, i.e., \( \lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n a_i = 0 \).

Answer: Yes


Question: Find the first homology group of a 3-dimensional sphere with $m$ cylindrical holes drilled through it.

Answer: \mathbb{Z}^m


Question: Determine whether the sequence \(X_n = \frac{n+1}{n^2+1} + \frac{n+2}{n^2+2} + \cdots + \frac{n+n}{n^2+n}\) is bounded above. Provide a justification for your answer.

Answer: Yes


Question: Determine the increasing function $f: \mathbb{R} \to \mathbb{R}$ that satisfies $f(f(x)) = f(x)$ for all $x \in \mathbb{R}$. What is the function $f$?

Answer: f(x) = x


Question: Find the derivative of the inverse function of \( g(x) = 2x^5 + 5x^3 + 2 \) at the point where \( x = 2 = g(1) \).

Answer: \dfrac{1}{25}


Question: In a Hausdorff space $X$, given a point $x \in X$ and an open neighborhood $U$ containing $x$, determine if there exists another open set $V$ containing $x$ such that the closure of $V$, denoted as $\overline{V}$, is a subset of $U$. Answer with 'Yes' or 'No'.

Answer: No


Question: Evaluate the limit \( \lim \limits_{n \to \infty} \int_3^4 (-x^2 + 6x - 8)^{\frac{n}{2}} \, dx \).

Answer: 0


Question: Solve the equation \((x+1)\cdot \sqrt{x+2} + (x+6)\cdot \sqrt{x+7} = (x+3)\cdot (x+4)\) in the set of real numbers.

Answer: 2


Question: Evaluate the limit \( \lim_{x\to 0} \frac{1}{x^3}\int_{0}^{x} \sin^{2}(3t)dt \) using a method other than L'Hôpital's rule.

Answer: 3


Question: Let two points \((x_1, y_1)\) and \((x_2, y_2)\) on the curve \(y = x^4 - 2x^2 - x\) have a common tangent line. Determine the value of \(|x_1| + |x_2| + |y_1| + |y_2|\).

Answer: 4


Question: Determine whether there exists a normal subgroup of order 4 in the symmetric group \(\mathfrak{S}_4\).

Answer: Yes


Question: Determine if the following limit holds true when $x$ is replaced by its integer part $[x]$: \[ \lim_{x\rightarrow -\infty} \left(1+3 \sin \frac{1}{x}\right)^{[x]} = e^{3}. \]

Answer: e^3


Question: Let $(X,d)$ be a metric space. Let $B_\epsilon(x)$ be the open ball of radius $\epsilon$ centered at $x$. For $x\in X$ and $\epsilon>0$, suppose that $V$ is an open set in $X$ with $V\subseteq B_\epsilon(x)$. If $V$ and $B_\epsilon(x)$ share the same nonempty boundary, are they necessarily the same set?

Answer: Yes


Question: Evaluate the surface integral \(\int_{S}\frac{dxdy}{z}+\frac{dydz}{x}+\frac{dxdz}{y}\) over the surface of the sphere \(S\) with radius \(R\) and outward orientation, without using the Gauss-Ostrogradsky formula.

Answer: 4\pi R


Question: Suppose $X_i$ are independent and identically distributed random variables. If $S_n/n \rightarrow c$ almost surely for some constant $c$, does it imply that $E|X_1|<\infty$? Answer yes or no.

Answer: Yes


Question: Given the recurrence relation \(a_{n+1}=2a_n - n^2 + n\) with \(a_1 = 3\), determine the value of \(\left|\dfrac{a_{20} - a_{15}}{18133}\right|\) where the sequence \(a_n\) is defined by the relation.

Answer: 28


Question: For a continuous function $f$, if the quadratic variation $\mathcal{Q}_f$ is absolutely continuous with density $q_f$, and $S$ is the set of points where $f$ is differentiable, determine whether $q_f(x) = 0$ for all $x \in S$. Provide a justification for your answer.

Answer: Yes


Question:
Consider the function \( f: \mathbb{R^2} \to \mathbb{R^2} \) defined as \( f(x, y) = (x^2 - y^2, 2xy) \). Determine the nature of the image of the open ball \( B(P, \frac{1}{2}) \), where the center \( P = (0, 1) \) and the radius is \( \frac{1}{2} \). Is the image:

A. a closed set
B. an open set
C. neither open nor closed set
D. an unbounded set

Find the correct option.

Answer: B


Question: Let \( A \) be a \( 3 \times 3 \) matrix with real entries that commutes with all \( 3 \times 3 \) matrices with real entries. What is the maximum number of distinct eigenvalues that the characteristic polynomial of \( A \) can have? Describe such matrices.

Answer: 1


Question: Can two distinct piecewise differentiable simple closed curves in \(\mathbb{R}^2\) intersect in an infinite set of points if they are not identical over any range of the parameter \(t\)? Answer yes or no.

Answer: No


Question: Assume that $f$ is a continuous function on $[a,+\infty)$ and differentiable on $(a,+\infty)$. If $\lim_{x\to+\infty} f(x)=f(a)$, prove that there exists $\xi>a$ such that $f'(\xi)=0$.

Answer: 0


Question: Find the values of $n$ for which the differential of the square map $P:SO(n) \to SO(n)$ is not an isomorphism.

Answer: n \geq 3


Question: Determine the cumulants of odd order for a compound Poisson distribution where the sum is given by \( Z = \sum_{i=1}^{K} r_i \), and \( r_i \) are independent and identically distributed random variables following a normal distribution \( N(0,1) \).

Answer: 0


Question: Let $Y$ be a random variable following a standard normal distribution, $Y \sim N(0,1)$. Define $X$ as the absolute value of $Y$, such that $X = |Y|$. Calculate the covariance between $X$ and $Y$, denoted as $\text{Cov}(X,Y)$. \( \boxed{} \)

Answer: 0


Question: Determine the radius of convergence of the series \( \sum_{n=0}^{\infty} a_n x^n \), where \( a_n = 1 \) if \( n \) is a factorial (i.e., \( n = k! \) for some integer \( k \)), and \( a_n = 0 \) otherwise.

Answer: 1


Question: Let $x_1, x_2, x_3$ be the roots of the equation $x^3 - x^2 + ax - a = 0$. Calculate $x_1^{2003} + x_2^{2003} + x_3^{2003}$ in terms of $a$.

Answer: 1


Question: Find the value of \( b \) such that \( 0 \leq b < 29 \) and \( 3^{(3^{505})} \equiv b \pmod{29} \).

Answer: 27


Question: Evaluate the double integral \( \iint_A x^2+4y^2\ dx \,dy \) where \( A \) is the region bounded by the parabolas \( y=x^2 \) and \( x=y^2 \).

Answer: \dfrac{3}{7}


Question: Let $F \subseteq E \subseteq K$ be a chain of fields, where the characteristic of $F$ is neither $2$ nor $3$. Assume that $z \in K$ exists with $K = E(z)$ and $a \in E$ such that the minimal polynomial of $z$ over $E$ is $X^3 - a$. Moreover, let $E = F(x)$, where $[E : F] = 2$ and $x$ is a root of $X^2 + X + 1$ (implying that $x$ is a primitive cube root of unity). If both $K/E$ and $E/F$ are Galois extensions, determine if we can choose $a$ such that $a \in F$. Provide an example if possible.

Answer: Yes


Question: Determine the set of all real values of $a$ such that the union of intervals \( \bigcup_{n \in \mathbb{N}} \left[1 - \frac{1+a}{n}, 2 + \frac{a}{n}\right] \) is equal to the half-open interval \([1, 2)\).

Answer: -1


Question: Is it possible for a non-open set in a topological space \( \mathbf{Y} \) to have a preimage that is an open set in another topological space \( \mathbf{X} \) under a continuous function \( \mathbf{F}: \mathbf{X} \to \mathbf{Y} \)?

Answer: Yes


Question: Evaluate the integral \( \int_{0.1}^{1} |\pi x^{-2} \sin(\pi \cdot x^{-1})| \, dx \) and find the result.

Answer: 18


Question: Determine whether the following statement is true or false: For a continuous random variable $X$ with density $f_X$, if $X$ has the same distribution as $-X$, then the density of $|X|$ is given by $f_{|X|}(x) = 2f_X(x)$ for all $x \in \mathbb{R}$. Provide your answer as either "True" or "False."

Answer: False


Question:
For non-negative integers $n$, define a function $h_n(x)$ as follows:

$$h_n\colon [0, 1]\to\mathbb{R}, \quad h_n(x)=\int_0^x h_{n-1}(t)dt\quad (n>0),\quad h_0(x) = x^2$$

Compute the sum $\displaystyle\sum_{n=0}^\infty h_n(x)$. Express your answer in terms of $x$. \(\boxed{\text{Your answer here}}\)

Answer: 2(e^x - x - 1)


Question: Suppose \( \{a_n\} \) is a monotonically decreasing sequence such that \( a_n > 0 \) for all \( n \in \mathbb{N}^+ \) and \( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = 0 \). Determine whether \( \lim_{n \to \infty} a_n = 0 \).

Answer: 0


Question: Evaluate the integral: \[ I = \int_{0}^{\infty} \frac{1-e^{-t}}{t} \sin{t} \, dt \]

Answer: \dfrac{\pi}{4}


Question: If a sequence of functions \((f_n)_{n = 1}^\infty\) converges uniformly on a domain \(D\) to a function \(f: D \to \mathbb{C}\), does the sequence \((f_n)\) also converge uniformly to \(f\) on any subset \(S \subseteq D\)?

Answer: Yes


Question: Evaluate the contour integral \( \oint_\gamma \frac{z}{z^2+1} \, dz \) using the residue theorem, where \( \gamma \) is the contour that bounds the rectangle with vertices at \((-1, -2)\), \((1, -2)\), \((1, 2)\), and \((-1, 2)\), traced counterclockwise.

Answer: 2\pi i


Question: Can the sequence \((1, 2^3, 3^8, 4^{27}, \ldots, n^{n^3}, \ldots)\) be expressed as a linear combination of the sequences \((1, 2^n, 3^n, \ldots)\)?

Answer: No


Question: Find the remainder when \(2^{1990}\) is divided by \(1990\).

Answer: 1024


Question: Determine whether a nonempty set can have a metric that generates the indiscrete topology, which consists only of the empty set and the set itself.

Answer: Yes


Question: Given that $X_1, X_2, X_3, X_4$ are independent standard normal random variables, each $X_i \sim N(0, 1)$, consider the random variable $Y = X_1^2 + X_2^2 + X_3^2 + X_4^2$. Determine the probability $P(Y \leq 3)$. Express your answer as a decimal rounded to four decimal places.

Answer: 0.4422


Question: Given a nonnegative matrix \( A \) and its largest eigenvalue \( h \), determine if the inverse of \( (A - g I) \) is a nonpositive matrix when \( h < g \), with \( g \) being a scalar. Answer with 'Yes' or 'No'.

Answer: Yes


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

Answer: 0


Question:
Find an approximate value of \( N \) that satisfies the inequality:

\[ \left(\dfrac{1}{2}\right)^{1000} \sum_{x=N+1}^{1000} {1000 \choose x} < 0.01 \]

using the normal approximation to the binomial distribution.

Answer: 537


Question: Find the gradient of the function \( f(x) = \frac{1}{2}\|G(x)\|_2^2 \) in terms of \( G(x) \) and \( G'(x) \), where \( G(x): \mathbb{R}^n \rightarrow \mathbb{R}^n \) is a twice continuously differentiable function.

Answer: G'(x)^\top G(x)


Question: Determine whether the statement is true: If $n \ge 2c\ln(c)$ for any constant $c>0$, then $n\ge c\ln(n)$, where $n$ is a positive integer.

Answer: True


Question: Evaluate the sum \(\sum^{\infty}_{p\in \mathbb P} \frac{\ln(p)}{p^{s+1}}\) where \(s > 1\) and \(\mathbb P\) is the set of prime numbers.

Answer: -\frac{\zeta'(s+1)}{\zeta(s+1)}


Question: Evaluate the integral \( \int_{-\infty}^{\infty}\delta(y-x)\delta(y-z)\,dy \) in a single dimension.

Answer: \delta(x - z)


Question: Given a non-bipartite, undirected, connected graph $G(V,E)$, a new graph $G'(V', E')$ is constructed as follows: $V' = V \times V = \{(v_i,v_j) \mid v_i, v_j \in V\}$ and $E' = \{((v_i,v_j),(v_k,v_l)) \mid (v_i,v_k),(v_j,v_l) \in E\}$. Is $G'$ guaranteed to be connected?

Answer: Yes


Question: Find the upper bound for the integral \( \left\lvert\int_\gamma \frac{\sin(z)}{z} \,dz\right\rvert \) where \(\gamma\) is a path that traces the circle of radius 2 centered at the origin once.

Answer: 0


Question:
Determine whether the following equation is true for all positive integers \(n\):

\[E[\:\{\:(X+Y) - E(X+Y)\:\}^n\:] = \sum_{k=0}^n \binom{n}{k} E[\:\{X - E(X)\:\}^k\:] E[\:\{\:Y - E(Y)\:\}^{n-k}\:]\]

where \(X\) and \(Y\) are independent random variables.

Answer: True


Question: Let $L_2 = \{x(t) \in C[0,1] : x(t) = at^2, a \in \mathbb{R}\}$ be a linear subspace of $C[0,1]$. Given $g_0 \in L_2^*$ defined by $g_0(x) = x'(0)$, find a functional $g \in (L_2 + \text{span}(t))^*$ such that $g|_{L_2} = g_0$ and $||g|| = ||g_0||$. Provide your answer as the explicit form of the functional $g$. \( \text{Note: } ||g|| \text{ denotes the norm of the functional.} \)

Answer: g(x) = 0


Question: Evaluate the integral \( \int_0^{2\pi} e^{e^{ix}} \, dx \).

Answer: 2\pi


Question: In triangle $ABC$, given that $\sin B = -\cos C$, determine the minimum value of $\frac{a^2 + b^2}{c^2}$, where $a$, $b$, and $c$ are the side lengths opposite angles $A$, $B$, and $C$, respectively.

Answer: 4\sqrt{2} - 5


Question: In trapezoid $ABCD$, the angle bisectors of the exterior angles at vertices $A$ and $D$ intersect at point $M$, and the angle bisectors of the exterior angles at vertices $B$ and $C$ intersect at point $K$. Given that the distance $MK$ is 15 cm, determine the perimeter of trapezoid $ABCD$. Express your answer in centimeters.

Answer: 30


Question: Suppose that \(\int_{0}^{\infty}f(x)dx<\infty\) and \(f(x)\geq0\) for all \(x\geq0\). Prove that \(\lim_{t \to \infty}\int_{\frac{t}{2}}^{t} f(x)\,dx=0\).

Answer: 0


Question: Find the value of $x$ such that $2^{33} \equiv x \pmod{4725}$ using the Chinese Remainder Theorem and the prime factorization $4725 = 3^3 \times 5^2 \times 7$. Calculate $2^{33} \pmod{3^3}$, $2^{33} \pmod{5^2}$, and $2^{33} \pmod{7}$ to determine $x$. Provide a simplified approach for these calculations.

Answer: 2717


Question: Find the maximum value of the expression \(3x^2 + 2\sqrt{2}xy\) given the constraint \(x^4 + y^4 = 1\), where both \(x\) and \(y\) are positive.

Answer: 2\sqrt{5}


Question: Rewrite the Jacobian determinant $u_{x} v_{y}-u_{y} v_{x}$ in terms of the partial derivatives of the function $f(z)$ using the Cauchy-Riemann equations and the definition of the complex derivative.

Answer: |f'(z)|^2


Question: Find the maximum value of the function \( f(x) = \frac{\sin(nx)}{\sin(x)} \) in terms of \( x \), where \( n \) is a positive integer greater than 0.

Answer: n


Question: Solve for \( x \) in the equation: \[ 3^{\frac{x-1}{2}} - 2^{\frac{x+1}{3}} = 2^{\frac{x-2}{3}} + 3^{\frac{x-3}{2}} \]

Answer: 5


Question: Is it correct to claim that the limit of the form $0^0$ always lies between $[0,1]$ without considering the specific functions involved? Answer "yes" or "no" and justify your answer.

Answer: no


Question: Consider a set $A$ and the collection $P(A)$ of partitions of $A$. The refinement order on $P(A)$, denoted by $\lesssim$, is defined such that for any partitions $\pi$ and $\sigma$ in $P(A)$, $\pi \lesssim \sigma$ if for each block $x$ in $\pi$, there exists a block $y$ in $\sigma$ with $x \subseteq y$. The join $\pi \vee \sigma$ of two partitions $\pi$ and $\sigma$ is constructed by defining an equivalence relation on the blocks of $\pi \cup \sigma$, where blocks $x$ and $y$ are equivalent if there exists a sequence of blocks connecting them with non-empty intersections. This construction extends to arbitrary nonempty subsets $S$ of $P(A)$. Determine if this construction extends to the case where $S$ is empty, and if so, describe the join in this scenario.

Answer: \{\{a\} \mid a \in A\}


Question: Evaluate the integral: $$\frac{3}{4\pi}\int\int\int\limits_{\mathbb{R^3}}e^{-\left(\sqrt{x^2+y^2+z^2}\right)^3}dV$$

Answer: 1


Question: Evaluate the triple integral \( \iiint_E y\,dV \) where \( E \) is the region bounded by the paraboloid \( x=4y^2+4z^2 \) and the plane \( x=4 \).

Answer: 0


Question: Verify that the function \( f(x) = \frac{x}{1+|x|} \) satisfies the Lipschitz condition globally.

Answer: 1


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f'(x) > x^2$ for all $x < 0$. Determine the limit $\lim_{x \to -\infty} f(x)$. \( \text{Provide your answer in the form of a single value or expression.} \)

Answer: -\infty


Question: For a complete metric space $(A_1, d_1)$, determine whether the completion of the set $A_1$ with respect to a different metric $d_2$ is always complete. Answer true or false.

Answer: A


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left(\frac{a_n}{1} + \frac{a_{n-1}}{2} + \frac{a_{n-2}}{2^2} + \ldots + \frac{a_1}{2^{n-1}}\right) \] given that \( \lim_{n \to \infty} a_n = a \).

Answer: 2a


Question: Evaluate the limit: $$\lim _{N\rightarrow \infty } \left( \,{\frac {{N}^{3}}{8\sum _{k=0}^ {N-1}({k}^{2}+\left\lfloor k/2\right\rfloor)}} \right) $$

Answer: \dfrac{3}{8}


Question: Evaluate the integral: $$ \int_0^{\frac{\pi}{2}}{\sqrt{\sin(2\theta)} \cdot \sin(\theta)\,d\theta} $$

Answer: \dfrac{\pi}{4}


Question: Evaluate the limit \( \lim_{n \to \infty} \int_0^1 \left( \frac{n}{1+n^2x^2} - \frac{ne^x\cos x}{1 + n^2x^2} \right) \, dx \).

Answer: 0


Question: Determine the number of distinct subgroups of the group \( \mathbb{Z}_{13} \times \mathbb{Z}_{13} \). Consider both cyclic and non-cyclic subgroups.

Answer: 16


Question: Let $X$ be a matrix defined as $X = [x_{ij}]$, where $x_{ij}=c_id_j$ for $1 \leq i\leq n$ and $1 \leq j\leq n$, and $c_i, d_j \in \mathbb{R}$. If $X^4=81X$, find the trace of $X$, denoted as $\operatorname{Tr}(X)$. \( \operatorname{Tr}(X) = \boxed{?} \)

Answer: 3\sqrt[3]{3}


Question: Determine the order of the subgroup $H$ generated by $\{i, e^{\frac{2i \pi}{5}}, -1\}$ in the multiplicative group $\mathbb{C}^\times$. Express your answer as $|H| = \boxed{\text{your answer}}$. Assume the neutral element is $1$, and the inverses are $i^{-1} = -i$, $(e^{\frac{2i \pi}{5}})^{-1} = e^{\frac{-2i \pi}{5}}$, and $(-1)^{-1} = -1$. Verify if $|H| = 6$.

Answer: 20


Question: Does the existence of the prime pair 89 and 97, which differ by 8, refute the proof that there are infinitely many pairs of consecutive primes with a difference of no more than 6?

Answer: No


Question: Find the value of \(k\) such that the equation \((x^2 - 1)(x^2 - 4) = k\) has four distinct equidistant real roots, where \(k\) is a real, non-zero number.

Answer: \dfrac{7}{4}


Question: Determine the group that is isomorphic to the group $G$ using the concept of semidirect products.

Answer: S_3


Question: Given two random variables $X$ and $Y$ with joint probability density function (PDF) \( f_{x,y}(x,y) = 2 \) for \( 0 < x < 1 \) and \( 0 < y < x \), and 0 otherwise, calculate the probability density function (PDF) of the random variable \( Z = X - Y \).

Answer: 2(1 - z)


Question: Given a polynomial \( f(x) \) with integer coefficients such that the equation \( f(x) = 37 \) has 5 distinct integer roots, determine the number of integer roots of the equation \( f(x) = 41 \).

Answer: 0


Question: Find the value of $x$ such that $2^{33} \equiv x \pmod{4725}$. Use the Chinese Remainder Theorem to solve the problem.

Answer: 2717


Question: Compute \( \int_{0}^{\pi} f(x) \, dx \) where \( f(x) = \begin{cases} \sin x & \text{if } x \text{ is irrational} \\ \cos x & \text{if } x \text{ is rational} \end{cases} \).

Answer: 2


Question: Determine the number of terms required in the Taylor series expansion of $\arctan x$ around $0$ to approximate $\pi$ to 10 decimal places when $x = 1$. Provide your answer as an integer.

Answer: 40000000000


Question: Find the fixed points of the action of the group \(G = \{(t, \frac{1}{t}) \in (\mathbb{C}^*)^2 : t^d = 1\}\) on \(\mathbb{C}^2\) given by \((t, \frac{1}{t}) \cdot (x, y) = (tx, \frac{1}{t}y)\).

Answer: (0, 0)


Question: Are two matrices with the same determinant always row-equivalent? Answer yes or no.

Answer: No


Question: For which values of $c$ is the matrix \( A = \begin{bmatrix} i & c & 1 \\ 0 & i & 2i \\ 0 & 0 & 1 \end{bmatrix} \) complex diagonalizable?

Answer: 0


Question: Determine whether the operator \(S:C[0,1]\rightarrow C[0,1]\) defined by \((Sx)(t)=\int^1_t x(s)ds, \ \ \forall t\in[0,1]\) is compact.

Answer: Yes


Question: If a set \( A \) is infinite, does there exist a subset of \( A \) that is equivalent to \( A \)?

Answer: Yes


Question: Determine whether an isomorphism between fundamental groups preserves conjugacy classes, even when the base points of the groups are different.

Answer: Yes


Question: Find the fundamental group of the open disk $D^2$.

Answer: 0


Question: Find the coefficient of $z^5$ in the Taylor series expansion of $f(z) = (\\sin(z))^3$. \(\)

Answer: -\dfrac{1}{2}


Question: Determine the maximum number of elements $k$ in a subset $A = \{a_{1}, ..., a_{k}\} \subset S$, where $S = \{1,2,...,2005\}$, such that the sum of any two distinct elements $a_{i} + a_{j}$ is not a multiple of 125.

Answer: 998


Question: Given the joint probability density function (PDF) of the random variables $X$ and $Y$ as $$f_{X,Y}(x,y) = e^{-y}\mathcal{I}_{(0<x<y<\infty)},$$ find the numerical value of the probability $P(X+Y \le 1)$. Express your answer as a single number.

Answer: 0.1548


Question: Determine the range of a uniform distribution from which N random numbers are drawn such that, on average, half of the unique numbers drawn are smaller than M/2. Assume that only unique numbers smaller than M/2 are counted.

Answer: [0, M]


Question: Let $f: \Omega \rightarrow \mathbb{R}^n$ be a linear operator, where $\Omega$ is a bounded open subset of $\mathbb{R}^n$. Let $A = \{a_{ij}\}$ be the matrix of the linear operator. Is it true that $||f|| = |\det(A)|$?

Answer: No


Question: Evaluate the expression \( \frac{\partial f'}{\partial f} \) given that \( f: \mathbb{R} \to \mathbb{R} \) and \( f' = \frac{df}{dt} \).

Answer: 0


Question: Given positive integers $n$ and $k$, consider the sequence $(n+2)^{n+2}, (n+4)^{n+4}, (n+6)^{n+6}, \ldots, (n+2k)^{n+2k}$. Determine the maximum possible value of $k$ such that all terms in this sequence end with the same digit in their decimal representation.

Answer: 3


Question: In a directed graph, vertex \( u \) has an out-degree of \( k \) and an in-degree of 0, vertex \( v \) has an in-degree of \( k \) and an out-degree of 0, and all other vertices have equal in-degree and out-degree. What is the minimum number of arc-disjoint directed paths from \( u \) to \( v \) in this graph?

Answer: k


Question: In a class, every group of three students contains a pair of friends and a pair of enemies. Determine the maximum number of students that can be in the class.

Answer: 5


Question: Given the function $f(x+y) = f(x)f(y)$ for all $x$ and $y$, with $f(5) = 2$ and $f'(0) = 3$, find the value of $f'(5)$.

Answer: 6


Question:
Determine the value of the parameter \( h \) such that Matrix A is similar to Matrix B, where:

Matrix A:
\[
\begin{bmatrix}
3 & 2 & 0 \\
h & 1 & 5 \\
0 & 0 & 1
\end{bmatrix}
\]

Matrix B:
\[
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 3
\end{bmatrix}
\]

Answer: 0


Question: Determine the order of the element $32$ in the multiplicative group $(\mathbb{Z}/2015\mathbb{Z})^\times$. Find the smallest positive integer $x$ such that $32^x \equiv 1 \pmod{2015}$, given that $2015 = 5 \cdot 13 \cdot 31$. Express your answer as a single integer.

Answer: 12


Question: Given a matrix $A = I + G$, where $G_{ii} = 0$ for all $i$, $G_{ij} \in \{0, a\}$ with $a \in [0, 1[$, $G$ is symmetric, and $A$ is positive definite. The inverse of $A$, denoted by $B$, is also symmetric positive definite with $B_{ii} > 1$ for all $i$. Prove or disprove that $B_{ii} > B_{ij}$ for all $i \neq j$. \(\text{True or False?}\)

Answer: True


Question: Calculate the surface integral of the function \( f(x,y,z) = x^2 + y^2 + z^2 \) over the portion of the sphere \( x^2 + y^2 + (z + 1)^2 = 1 \) that lies below the plane \( z = 0 \).

Answer: 8\pi


Question: Calculate the expected number of steps in the Cantor set construction before a uniformly random number in $[0,1]$ is removed, given the integral of the function $f$ related to this process.

Answer: 3


Question: In a regular 11-gon $A_1 A_2 \dotsb A_{11}$ inscribed in a circle of radius 2, a point $P$ lies at a distance of 3 from the circle's center. Calculate the sum of the squares of the distances from $P$ to the vertices, i.e., $PA_1^2 + PA_2^2 + \dots + PA_{11}^2$. Provide your answer as a single number.

Answer: 143


Question:
Evaluate the limit using the Lebesgue dominated convergence theorem:
\[\lim_{n\to\infty} \int_0^{\infty} \left(1+\frac{x}{n}\right)^{-n}\sin \frac{x}{n}\,dx\]
Is the evaluation correct that the limit is \(0\)?

Answer: 0


Question: Find the splitting field of the polynomial \( p(x) = x^4 + x + 1 \) over the Galois Field GF(2). Determine the degree of this splitting field.

Answer: 4


Question:
Determine the correct solution to the differential equation \(\frac{\partial^2u}{\partial x\partial t} = e^{-t}\cos x\) given the conditions \(\frac{\partial u}{\partial t} = 0\) at \(x = 0\) and \(u = 0\) at \(t = 0\). Choose the correct option from the following:

(a) \((1 - e^{-t})\sin x\)
(b) \((-e^{-t})\sin x + g(x)\)
(c) \((1 + e^{-t})\cos x\)
(d) \((-e^{-t})\sin x\)

Answer: a


Question: Find the minimum value of the expression \( \frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_n}{a_1} \) for a sequence of positive real numbers \( a_1, a_2, \ldots, a_n \).

Answer: n


Question: Let $a$ and $b$ be positive integers such that the range of the function \( y = \frac{x^2 + ax + b}{x^2 + 2x + 3} \) as $x$ varies over all real numbers is the interval $-5 \leq y \leq 4$. Find $a + b$. \( \boxed{} \)

Answer: 23


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

Answer: \dfrac{1}{2}


Question: A cylindrical tank with hemispherical ends is to be constructed to hold a given volume of liquid. If the cost of making the hemispherical ends is twice as much as the cost of making the cylindrical part, determine the ratio of the height to the radius of the cylinder that minimizes the total cost of the tank.

Answer: 4


Question: Let \( \mu \) and \( \nu \) be measures with \( \nu(\Omega) = 3 \). Suppose \( g \) is the Radon-Nikodym derivative of \( \nu \) with respect to \( \mu + \nu \). Calculate \( \mu(\{x: g(x) > 2\}) \).

Answer: 0


Question: Let $n \in \mathbb{N}$, $1 \le p < n$, and $c > 0$. Determine if the inequality \( \lVert f \rVert_{L^q(\mathbb{R}^n)} \le c \lVert \nabla f \rVert_{L^p(\mathbb{R}^n)} \) for all \( f \in C_0^\infty(\mathbb{R}^n) \) implies that \( q = \frac{np}{n-p} \).

Answer: q = \dfrac{np}{n - p}


Question: Determine whether there exists a non-linear matrix group that has a normal subgroup which is linear.

Answer: Yes


Question: Calculate the integral \( \int_{-2}^{2}(g^{'}(x))^{2} \, dx \), where \( g(x) = |x+1| \) and \( g^{'} \) is the weak derivative of \( g \).

Answer: 4


Question: Given two permutations \(\sigma = ( 1, 2, 3, 4, 5 ) ( 6, 10 ) ( 7, 11 ) ( 8, 12 ) ( 9, 13 )\) and \(\tau = ( 2, 5 ) ( 3, 4 ) ( 6, 7, 8, 9, 10, 11, 12, 13 )\) with orders 10 and 8 respectively, let \(H = \langle \sigma \rangle\) and \(K = \langle \tau \rangle\). Given that \(G\) is generated by \(\sigma\) and \(\tau\), and that \(\langle H\cap K \rangle\) has order 2 (contains elements \(()\) and \(( 6, 10 )( 7, 11 )(8, 12)(9, 13)\)), determine the size of \(G\).

Answer: 40


Question: Let $X$ be a compact subset of the complex plane and let $D^1(X)$ be the set of functions with continuous derivatives on $X$. Suppose $(f_n)$ is a sequence in $D^1(X)$ that converges uniformly to $f \in D^1(X)$. If the derivatives $f_n'$ converge pointwise to a function $g$ on $X$, does it follow that $g = f'$? Provide a justification for your answer.

Answer: No


Question: Let $T = \left\{(\lambda_n)_{n} \in l^2 : \sum_{n=1}^{\infty} \frac{\lambda_n}{\sqrt{n}} = 0\right\}$. Determine $T^{\perp}$, the orthogonal complement of $T$ in $l^2$. Assume $\mathbb{K} = \mathbb{R}$. Provide your answer in terms of a description of the subspace $T^{\perp}$.

Answer: \{0\}


Question:
Determine if the substitution \( W_t = \tilde{W_t} - \theta t \) is valid for the integral transformation:

\[ \int_{t=a}^{t=b}{\theta}dW_t=\int_{t=a}^{t=b}{\theta}d\tilde{W_t}-\int_{t=a}^{t=b}{\theta}^2dt \]

Answer: Yes


Question:
Let $f:[a,b] \to \mathbb{R}$ be a function that is continuous at $x_0 \in [a,b]$. Determine whether the following limit holds:
\[ \lim_{h\to0} \frac{\int_{x_0}^{x_0+h}f(x)\,dx}{h} = f(x_0). \]

Answer: f(x_0)


Question: Let $X = \{(x,y) : x^2 + y^2 < 9\}$ and $K = \{(x,y) : 4 \leq x^2 + y^2 \leq 5 \text{ or } 7 \leq x^2 + y^2 \leq 8\}$. Determine the number of relatively compact connected components of $X \setminus K$ within $X$. Provide your answer as a single integer.

Answer: 2


Question: Does the construction of the fundamental group of a connected space require the axiom of choice? Answer yes or no.

Answer: No


Question: Determine whether the limit \( \lim _{x\to 0} \frac{1}{x} \int_0^x \left|\cos \frac{1}{t} \right| dt \) exists. If it exists, find its value.

Answer: \dfrac{2}{\pi}


Question: Determine all natural numbers \( n \) for which \( n! + n^2 + 2 \) is a prime number.

Answer: 3


Question: Find the maximum vertical distance between two points on an ellipse with major axis 10 units and minor axis 6 units, after it has been rotated by 30 degrees.

Answer: 2\sqrt{13}


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

Answer: 1


Question: Given the sequence \( \{a_n\} \) defined by \( a_1 = 1 \) and \( a_{n+1} = (1 + a_1)(1 + a_2)\cdots(1 + a_n) \), find the value of \( \sum_{n=1}^\infty \frac{1}{1 + a_n} \).

Answer: 1


Question: Evaluate the limit of the series: \[ \lim_{n\to\infty}\frac{1^p + 2^p + \ldots + n^p}{n^{p+1}} \]

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


Question: If $A_1 \supseteq A_2 \supseteq A_3 \supseteq...\supseteq A_n$ are all sets containing an infinite number of elements, determine whether $\bigcap_{n=1}^{\infty} A_n$ is infinite as well. Answer with 'True' or 'False'.

Answer: False


Question:
Evaluate the double limit:
\[ \lim_{n\to \infty}\lim_{m\to \infty} \frac{1+\sqrt[n]{1^n+2^n}+\sqrt[n]{2^n+3^n}+\cdots+\sqrt[n]{(m-1)^n+m^n}}{m^2} \]

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_{0}^{\pi/2}\frac{x\cdot \sin(2x)}{(1+\cos(4x^2))}\,dx \). Choose the correct answer from the following options: A) \( \pi/2 \), B) \( \pi^2/4 \), C) \( \pi/8 \), D) \( \pi^2/16 \).

Answer: D


Question: Evaluate the integral \( \int _{0}^{\pi/2} \sqrt{\sin {2x}} \cdot \sin{x} \, dx \) using appropriate trigonometric manipulations and special functions.

Answer: \dfrac{\pi}{4}


Question: Find the last three digits of the product \( n = 3 \times 7 \times 11 \times 15 \times \cdots \times 2003 \).

Answer: 875


Question: Find the value of $\beta^*$ that minimizes the Kullback-Leibler divergence when the observed random variables $X_1$ and $X_2$ are independent and normally distributed with means $\mu_1$ and $\mu_2$, and variances $\sigma_1^2$ and $\theta\sigma_1^2$, respectively.

Answer: \theta


Question: Evaluate the improper integral \( \int_{0}^{\infty} \frac{\ln(x)}{1+x^2}dx \). Transform the limits of integration to facilitate the evaluation.

Answer: 0


Question: Let \( M \) be a metric space that is connected, locally connected, and contains more than one point. Do these conditions on \( M \) imply that \( M \) is always arc-wise connected? Answer with 'Yes' or 'No' and justify your answer.

Answer: Yes


Question: Find an integer x such that \( x^3 \equiv 2 \pmod{59} \).

Answer: 38


Question: Find the largest integer $a$ such that the inequality \(x^2 - ax - \ln(x) + e^{\sin(x)} - 1 > 0\) holds true for all \(x > 0\). Given that \(\sin(1) \approx 0.84\) and \(\ln(2) \approx 0.693\), determine the equivalent form of the inequality and solve for $a$.

Answer: 2


Question: Determine the values of \( n \) for which an \( n \times n \) board can be completely covered by T-shaped tetrominoes without overlapping or extending off the board.

Answer: n \equiv 0 \pmod{4}


Question: Given a measure space \((X,\mu)\) and a function \(f\in L^1(X)\), determine whether it is true that for every \(\varepsilon>0\), there exists a set \(F\subset X\) with \(\mu(F)<\varepsilon\) such that \(f\) is essentially bounded on \(X\setminus F\).

Answer: True


Question: Find an expression for the function \( h(z) = \sum_{n=1}^{\infty}\frac{(z-2)^n}{n} \) as an elementary function. Consider using techniques similar to those used for the related function \( f(z) = \sum_{n=1}^{\infty} n(z-2)^n = \frac{z+3}{(z+2)^2} \). Knowledge of logarithms in the complex plane may be required.

Answer: -\ln(3 - z)


Question: Evaluate the limit: $$\lim_{x \to 0^+} ((1+x)^x - 1)^x$$

Answer: 1


Question: Using the random walk approach, calculate the expected time for a mouse to reach one of the three exits in a maze, where each exit is an absorbing state.

Answer: 4


Question: Evaluate the integral \( \int\limits_{0}^{\pi} \cos(x) F(\sin(x)) \, dx \), where \( f(t) = e^{\sin(t)} \) and \( F(x) = \int\limits_{0}^{x} f(t) \, dt \).

Answer: 0


Question: Let U denote the set of all nxn matrices A with complex entries such that A is unitary. Determine whether U, as a topological subspace of $\mathbb{C}^{n^2}$, is: (a) compact but not connected, (b) connected but not compact, (c) connected and compact, or (d) neither connected nor compact.

Answer: C


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

Answer: 160


Question: Let $V$ be a finite dimensional vector space. Determine whether the statement $End(V \otimes V) = End(V) \otimes End(V)$ is true or false.

Answer: True


Question: Evaluate the integral that corresponds to the following Riemann sum: $$\lim\limits_{m\to\infty}\frac{1}{m}\sum_{n=1}^m\sin\left({\frac{3\pi n}{m}}\right)$$

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


Question: Given the joint probability density function \( f(x,y,z) = kxyz^2 \) for \( 0<x<1 \), \( 0<y<1 \), and \( 0<z<2 \), find the probability that the sum of two independent random variables \( X \) and \( Y \) is less than a third independent random variable \( Z \).

Answer: \dfrac{13}{20}


Question: Evaluate the limit: $$\lim_{x\rightarrow\infty} \frac{\int_0^x \cos(t^2) dt - \int_0^\infty \cos(t^2) dt}{x^3}$$

Answer: 0


Question: Determine whether the equation $b^2 c - a^2 = d^3$ can be solved given the conditions $b^2c > a^2$, $b > 0$, $c > 0$, and $a$, $b$, $c$, $d$ are rational numbers.

Answer: Yes


Question: Find the set of natural numbers \( r \) such that the set \( R = \{ m + nr\sqrt{3} \mid m, n \in \Bbb Z \} \) forms a subring of \( \Bbb Q[\sqrt{3}] \).

Answer: \mathbb{N}


Question: Compute the double integral \(\iint_{\Omega} (x+y^2) \, dx \, dy\) where \(\Omega\) is the parallelogram defined by the boundaries \(x+y=0\), \(x+y=1\), \(2x-y=0\), and \(2x-y=3\). Use a change of variables to simplify the computation of this integral.

Answer: \dfrac{22}{27}


Question: Determine whether the following statement is true or false: A symmetric operator $T$ with a dense domain is self-adjoint if and only if it is bounded.

Answer: False


Question: Determine whether a function with an isolated singularity can have a Laurent series expansion about that singularity.

Answer: Yes


Question: Determine the area of the largest rectangle that can be inscribed in the ellipse \( \frac{x^2}{2} + \frac{y^2}{6} = 1 \).

Answer: 4\sqrt{3}


Question: Determine the number of distinct ring homomorphisms from the ring of real numbers \((\mathbb{R}, +, \times)\) to itself.

Answer: 1


Question: Find the minimum value of the expression \(x+y+z+\frac{1}{x} + \frac{1}{y} +\frac{1}{z}+1\), where \(x, y, z\) are positive real numbers.

Answer: 7


Question: Find the harmonic conjugate of the harmonic function \( u(x,y) = \sinh(x)\cos(y) \).

Answer: \cosh(x)\sin(y)


Question: Does the convergence of the series \(\sum_{n=1}^{\infty}(a_{n+1}-a_n)\) imply the convergence of the sequence \(a_{n}\)? Provide a justification for your answer.

Answer: Yes


Question: In the ring of integers modulo 23, \( M = (\mathbb{Z}_{23}, +, \cdot) \), determine whether there exists a subset \( T \subseteq M \) with \(|T| = 22\) such that \((T, \cdot)\) forms a group.

Answer: Yes


Question: Consider a permutation \( \sigma \in S_n \) and an integer \( k \) such that \( 1 \leq k \leq n \). Calculate the probability that the first cycle of \( \sigma \) has length \( k \), assuming a uniform distribution over all permutations.

Answer: \dfrac{1}{n}


Question: Is the sequence of halved unit vectors in $\ell^1$ totally bounded?

Answer: No


Question: In triangle $\Delta ABC$, with sides $a = 3$, $b = 4$, and the circumradius $R$ being minimum, find the value of $\left[2rR\right]$, where $\left[.\right]$ denotes the greatest integer function, and $r$ is the inradius.

Answer: 3


Question: Evaluate the integral \( \int_{-\infty}^{\infty} dk \left[ \Im\left( \frac{1}{(\omega+i\eta+k)(\omega+i\eta+k+q)} \right) \right] \), where \( \eta \) is an infinitesimal parameter.

Answer: 0


Question: Given a matrix \( A \in M_3(\mathbb{Q}) \) such that \( \det(A^2 - 2023I_3) = 0 \), find \( \operatorname{Tr}(A^{2023}) \).

Answer: 0


Question: Find all real numbers $x$ such that $2^x + 3^x - 4^x + 6^x - 9^x = 1$. Let $a = 2^x$ and $b = 3^x$. Rewrite the equation as $a + b - a^2 + ab - b^2 = 0$. Further simplify to $(a - b)^2 + (a - 1)^2 + (b - 1)^2 = 0$. Solve for $x$.

Answer: 0


Question: Find all extreme values of the function $f(x, y, z) = x^2 + y^2 + z^2$ subject to the constraint $x^2 + 2y^2 - z^2 - 1 = 0$ using Lagrange multipliers.

Answer: \dfrac{1}{2}


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

Answer: 0


Question: Let \( f(x) = 17x^{12} - 124x^9 + 16x^3 - 129x^2 + x - 1 \). Determine \( \frac{d}{dx}(f^{-1}(x)) \) at \( x = -1 \), if it exists.

Answer: 1


Question: Evaluate the integral \( \int_C \frac{\tan z}{z} \, dz \) using the residue theorem, where \( C \) is the circle centered at the origin with radius 2.

Answer: 0


Question: Find the residue of the function \( \frac{z+1}{(z^2+4)^2} \) at the pole \( z_0 = 2i \).

Answer: -\dfrac{i}{32}


Question: Evaluate the integral of the complex function: \( \int \cos(z)\,dz \) where \( z \) is a complex number.

Answer: \sin(z) + C


Question: Let $m$ be an odd prime. Consider the set $A = \{0, 1, 2, \ldots, 2m-1\}$, which consists of all remainders modulo $2m$. Determine the number of elements $x$ in $A$ such that $x^2 \equiv 1 \pmod{2m}$.

Answer: 2


Question: Given that \(a < b\) and \(\frac{f(b)/b}{f(a)/a}<2\), find a tighter upper bound for the expression \(\frac{a}{b}+\frac{b}{a}\).

Answer: \dfrac{5}{2}


Question: Find the limit of the expression as \( n \) approaches infinity: \[ \lim_{n \to \infty} \frac{(2n-1)!}{(2n)^n} \]

Answer: \infty


Question: Determine the period of the cycle of numbers that are coprime to 9 when you repeatedly take the digit sum of their powers.

Answer: 6


Question: Suppose $g$ is a function with a power series representation at $0$ that converges on $(-2, 2)$. If there is a sequence of distinct elements $(y_n)$ such that $y_n \to 0$ and $g(y_n) = 1$ for all $y_n$, can the function $g$ be uniquely determined?

Answer: 1


Question: Compute the residue of the function \( f(z) = \frac{1+z}{1-\sin z} \) at the pole \( z = \frac{\pi}{2} \), which is of order 2, without using the residue calculation formula or L'Hospital's rule.

Answer: 2


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

Answer: 6e^2


Question: Find the inverse Laplace transform of \( F(s) = \frac{1}{s(\cosh{\sqrt{2s}}-1)} \). If an analytic expression is not possible, provide an asymptotic expansion for \( f(t) = \mathcal{L}^{-1}[F](t) \) as \( t \to \infty \).

Answer: t - \dfrac{1}{6}


Question: Determine whether there exists a function \( g : [0,1) \rightarrow \Re^+ \) such that \( \|g\|_2 = 1 \) and \( \|g\|_a \geq 1 \) for some \( a \in (2,3] \).

Answer: Yes


Question: Evaluate the limit \(\lim_{n\to \infty} \left(\binom{2n}{n}\right)^{\frac{1}{n}}\) without using the Limit as a Sum or the Stirling Approximation.

Answer: 4


Question: Evaluate the integral \(\int_0^{\pi/2} (\cos x)^{2011} \sin (2013 x) \, dx \) using the recursive relation \(I(m,n)= \frac {1}{m+n} + \frac{m}{m+n}I(m-1,n-1)\), where \(I(m,n) =\int_0^{\pi/2} (\cos x)^{m} \sin (n x) \, dx \). Verify that the value of the integral is \(\frac{1}{2012}\).

Answer: \dfrac{1}{2012}