Question: Find the remainder when the sum of all possible remainders of numbers of the form $3^n$ (where $n$ is a positive integer) divided by $500$ is itself divided by $500$.

Answer: 0


Question: Evaluate the contour integral \( \int_\gamma \frac{\mathrm{Re} (z)}{z - \beta} dz \), where \( \beta \) is a complex number such that \( 0 < |\beta| < 4 \) and \( \gamma \) is the circle oriented in the negative sense, centered at the origin with radius 5.

Answer: - \pi i \beta


Question: Find the minimal polynomial of the matrix \(\begin{bmatrix}5&0&0&0\\ 0&5&0&0\\ 0&0&5&1\\ 0&0&0&5\end{bmatrix}\) by determining its Jordan canonical form.

Answer: (x - 5)^2


Question: Evaluate the sum \( \sum_{\textbf{t} \in \mathbb{Z}_{2}^{d}}(-1)^{\textbf{t}\cdot(\textbf{s} + \textbf{r})} \) when \( \textbf{s} \neq \textbf{r} \in \mathbb{Z}_{2}^{d} \).

Answer: 0


Question: Is it possible to approximate any bounded operator on a separable Hilbert space uniformly with operators of finite rank?

Answer: No


Question: Compute the limit \( \lim_{z\to 0}z^3\Gamma(z)\Gamma(z-1)\Gamma(z-2) \) using the functional relation of the Gamma function.

Answer: -\dfrac{1}{2}


Question: If $f(x)$ and $g(x)$ are continuous and differentiable functions such that $f(x) > g(x) > 0$ for all $x$, does the convergence of $\int_{-\infty}^{+\infty} f(x) \, dx$ imply the convergence of $\int_{-\infty}^{+\infty} g(x) \, dx$? Provide a justification for your answer.

Answer: Yes


Question:
Compute the limit of the hypergeometric function as follows:

$$\lim_{{x\rightarrow+\infty}}\left({}_{2}F_{1}\left(1,m-\frac{1}{\xi}+1;m+2;\frac{x\xi}{\mu\xi-\sigma}\right)\right)$$

Find the result that converges when $\xi<\frac{1}{m}$. Begin by considering the case $m=1$ and then generalize your solution.

Answer: 0


Question: Evaluate the limit: \[ \lim_{\substack{x\to 2 \\ y\to 1}} \frac{\tan(y-1)\sin^2(2 y - x)}{(x-2)^2 +(y-1)^2} \]

Answer: 0


Question: Let $x_n \to x$ with $x < 0$. Determine whether it is true that $nx_n \to -\infty$. Justify your answer.

Answer: -\infty


Question: Find the remainder when the binary message 010100011010000 is divided by the polynomial divisor $x^{14} + x^{12} + x^8 + x^7 + x^5$ using the generator polynomial $x^5 + x^4 + x^2 + 1$ in the field $GF(2)$. What is the correct remainder?

Answer: 00001


Question: Does a closed set in a metric space necessarily contain all its limit points? Provide a justification for your answer.

Answer: Yes


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

Answer: \infty


Question: Compute \((3^{999^{100}} + 7^{960^{961}}) \bmod 225.\)

Answer: 154


Question: Evaluate the infinite series: $$\sum_{n=1}^{\infty}\frac{5}{n(n+2)}.$$

Answer: \dfrac{15}{4}


Question: Evaluate the limit: $$\lim_{x\to\pi}\frac{\log_{\cos^2x}\sin x}{\log_{\cos^2(x/4)}\sin(x/4)}$$ using appropriate logarithmic manipulations and L'Hôpital's Rule.

Answer: \infty


Question: Determine whether the expression $10^n + 4$ can be a perfect square for any positive integer $n$.

Answer: No


Question:
Let $G$ be a group of order 30. Determine which of the following statements could be false:

1. $G$ has a subgroup of order 10.
2. $G$ has a normal subgroup of order 3.
3. $G$ has an element of order 5.
4. $G$ has a normal subgroup of order 6.

Answer: 4


Question: Determine the number of ten-digit positive integers such that each digit from 0 to 9 appears exactly once, the first digit is odd, and five even digits appear in five consecutive positions.

Answer: 72000


Question: For a random variable X that follows a Uniform distribution on the interval (1, 9.9) and a conditional random variable Y given X = x that follows a Normal distribution with mean 1.4 and variance x^2, calculate the covariance Cov(X, Y).

Answer: 0


Question: Simplify the expression \( F(n) = \sum_{d\mid n} \mu(d)d \) using the Mobius Inversion Formula or other results, and compute \( F(3500) \).

Answer: -24


Question: Determine the number of distinct circular bracelets that can be formed using $6n$ blue beads and $3$ red beads, where $n$ is a positive integer. Note that the beads are indistinguishable when flipped.

Answer: 3n^2 + 3n + 1


Question: Find the maximum value of the function \( f(x,y) = x+y \) subject to the constraints \( x^2+y^2 \leq 4, x \geq 0, y \geq 0 \) using the Kuhn-Tucker conditions.

Answer: 2\sqrt{2}


Question: Consider the sequence \((a_n)\) defined by \(a_1 = 3\) and \(a_{n+1} = 3^{a_n}\) for all \(n\). Determine the last two digits of \(a_n\).

Answer: 87


Question: Evaluate the limit: $$\lim_{x\rightarrow\infty} \frac {(3x^2-4) \left( \sqrt [3]{2x^2+1}+1 \right)^2}{ (2x-1) \left( 4-\sqrt {8x^3-2} \right)x^{3/2}}$$

Answer: 0


Question: Find the ring to which the quotient ring $\mathbb{Z}[x,y,z]/(x-y, x^3-z)$ is isomorphic.

Answer: \mathbb{Z}[x]


Question: Evaluate the limit: $$\lim_{(x,y)\to (0,0)} \frac{x^2y^2 + 1}{x^4 + y^4}$$

Answer: \infty


Question: Is it possible to extend a smooth function defined on a closed subset of Euclidean space to a smooth function on the entire space? Provide a yes or no answer.

Answer: Yes


Question: Consider a sequence of random variables $V_n = V(W_1,...,W_n)$, where $W_i$ are iid random variables. Let $W$ be another random variable such that $W \sim W_i$. Suppose we have a non-negative function $f(V_n,W)$. If $P_W\{f(V_n,W) > \epsilon\} \rightarrow 0$ in probability as $n\rightarrow \infty$ for any fixed $\epsilon >0$, does this imply that $|f(V_n,W)| \rightarrow 0$ in probability?

Answer: Yes


Question: Given $\theta \in V \subseteq \mathbb{R}^k - \{0\}$ such that $H\theta = 0$ for some full rank $d \times k$ matrix $H$, with $d \leq k$. Determine if there exists a full rank $d \times k$ matrix $A$ such that $A\theta = r$ for a given $r \in \mathbb{R}^d$. Answer "yes" or "no."

Answer: yes


Question: Find all positive integers \( n \geq 2 \) such that the inequality \((x_1 + x_2 + \cdots + x_n)^2 \geq c_n(x_1^2 + x_2^2 + \cdots + x_n^2)\) holds for all non-negative real numbers \(x_1, x_2, \ldots, x_n\), where \(c_n\) is a constant depending on \(n\).

Answer: n \geq 2


Question: Find the Fourier series of the function \( f(t) = \begin{cases}5\sin t & 0\leq t\leq\pi\\0 & \pi \leq t \leq 2\pi \end{cases} \) with period \( T=2\pi \) and angular frequency \( \omega=1 \). Calculate the Fourier coefficients and determine the correct value for \( a_1 \).

Answer: 0


Question: Determine the radius of convergence for the power series \(\sum_{n=1}^\infty \frac{(\sqrt{n^2+n} - \sqrt{n^2+1})^n}{n^2} x^n\) and analyze the convergence at the endpoints of the interval of convergence.

Answer: 2


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

Answer: 1


Question: Determine whether the limit \( \lim_{(x,y)\to(0,0)}\dfrac{x^3 y^2}{x^4+y^4} \) exists.

Answer: 0


Question: Let $V$ be a vector space over a field $F$ with dimension $n$, and let $W$ be a subspace of $V$ with dimension $m < n$. Define $E = \{T: V \to V \mid T \text{ restricted to } W \text{ is zero}\}$. Determine the dimension of $E$ as a subspace of $L(V,V)$. \( \text{Express your answer in terms of } n \text{ and } m. \)

Answer: n(n - m)


Question: Determine whether the following statement is true or false: Let $M$ be a smooth manifold and $p \in M$. Then $M - \{p\}$ is orientable if and only if $M$ is orientable.

Answer: True


Question: In a vector space $V$ with dimension 5 over a field $F$, consider a subspace $W$ with dimension 2. Let $S$ be the set of all linear transformations $T:V\to V$ such that $T$ restricted to $W$ is the zero transformation. Determine the dimension of $S$ as a subspace of the space $L(V,V)$ of all linear transformations from $V$ to $V$. \( \text{Provide your answer as a single integer.} \)

Answer: 15


Question: Evaluate the integral \( \oint_{|z|=2} \frac{f^{\prime}(z)}{f(z)} \, dz \) for the function \( f(z)=\frac{2 z^{4}-7 z^{3}-2 z-i}{(z-i)^{2}\left(3 z^{2}-z+9\right)^{2}} \).

Answer: -6\pi i


Question: How many distinct outcomes are possible when rolling 50 identical six-sided dice such that each of the numbers 1, 2, 3, 4, 5, and 6 appears an even number of times?

Answer: 142506


Question: What is the probability that a randomly chosen number from the interval $[0, 1]$ is rational?

Answer: 0


Question: Determine the number of automorphisms of the cyclic group $\mathbb{Z}_{30}$ that satisfy $\phi(15) = 15$. Provide your answer as a single integer.

Answer: 8


Question: Determine whether a polyhedron and its mirror image can have the same net. Provide a yes or no answer.

Answer: Yes


Question: Evaluate the double integral \( \iint_A y \, dA \) where \( A \) is the region bounded by the curves \( y=0 \), \( y^2=4x \), and \( y^2=5-x \).

Answer: 5


Question: Is a topological space with a finite topology separable if it contains an uncountable set that is not a subset of any closed set?

Answer: Yes


Question: Evaluate the limit: $$\lim_{ x \to 1}\frac{\sqrt[n]{x^n-1}}{\sqrt[n]{nx}-\sqrt[n]{n}-\sqrt[n]{nx-n}}$$

Answer: -1


Question: Let $V$ be a Banach space and $W$ be a subspace of $V$. If $V/W$ is a Banach space, is $W$ necessarily a closed subspace of $V$? Provide a justification for your answer.

Answer: Yes


Question: Let $\left({z_n}\right)$ be a sequence of distinct points in $D\left(0,1\right)=\{{z \in \mathbb{C} :\vert{z}\vert \lt{1}} \}$ with $\lim_{n\to\mathbb{\infty}} {z_n}=0$. Consider the following statements P and Q.\nP: There exists a unique analytic function $f$ on $D\left(0,1\right)$ such that $f\left({z_n}\right)=\sin\left({z_n}\right)$ for all $n $.\nQ: There exists an analytic function $f$ on $D\left(0,1\right)$ such that $f\left({z_n}\right)=0$, if $n$ is even and $f\left({z_n}\right)=1$, if $n$ is odd.\nDetermine which of the statements P and Q are true.

Answer: P


Question: Evaluate the expression \(\sqrt{2+\sqrt{5}-\sqrt{6-3\sqrt{5}+\sqrt{14-6\sqrt{5}}}}\). Simplify the expression step by step to find its value.

Answer: 2


Question:
Given real numbers x, y, z, and w that satisfy the following equations:

1. \(x^2 + 5z^2 = 10\)
2. \(yz - xw = 5\)
3. \(xy + 5zw = \sqrt{105}\)

Find the value of \(y^2 + 5w^2\).

Answer: 23


Question: Evaluate the integral \( \int _{0}^{1} \frac{7x^4+5x^2+4x}{\sqrt {x^3+x+1}} \, dx \).

Answer: 2\sqrt{3}


Question:
Compute the limit of the following product as \( n \) approaches infinity:

\[ \lim_{n \to \infty} \prod_{k=3}^{n+2} \left( 1 - \frac{2}{k} \right)^{\frac{k}{n}} \]

Answer: e^{-2}


Question: Find the highest power of 3 that divides the sum \( S = \sum_{k=1}^{2048} \binom{2048}{k} 3^k \).

Answer: 1


Question: Let \( a \) be a positive real number and define the set \( M_a = \{ z \in \mathbb{C}^* : |z + \frac{1}{z}| = a \} \). Determine the product of the minimum and maximum values of \( |z| \) for \( z \in M_a \).

Answer: 1


Question: Evaluate the limit: $$\lim_{x\to \pi}\left(\frac{1}{x-\pi}\right)\int_{\pi}^{x}\frac{\cos t}{1-\cos t}\,dt.$$

Answer: -\dfrac{1}{2}


Question: Find a possible lower bound for the mutual information $I(X;Y)$ given that $I(X;Z) = 0$.

Answer: 0


Question: Evaluate the integral \( \int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ \mathrm{d}x \) for \( a > 0 \).

Answer: \pi a


Question: Determine whether a finite dimensional $k$-algebra $A$ (where $k$ is an algebraically closed field of characteristic $p$) can have an infinite socle series.

Answer: No


Question: Find the sixth derivative of the function \( f(x) = x^4e^{x^2} \) using the Maclaurin series.

Answer: 720


Question: Differentiate the expression \( \frac{\partial(x^T Bx)}{\partial x} \) where \( x \) is a column vector and \( B \) is a square matrix.

Answer: (B + B^T)x


Question: Find the expression for \(\langle [W(s) + W(t) - 2W(0)]^2 \rangle\), where \(W(t)\) is a Wiener process and \(s \leq t\).

Answer: 3s + t


Question: Determine whether the following claim is true or false: If matrices $AB$ and $A+B$ are nilpotent and upper triangular, then matrices $A$ and $B$ are also nilpotent.

Answer: True


Question: Evaluate the integral \( \int_{0}^{\infty} \frac{x^8 - 1}{x^{10} + 1} \, dx. \)

Answer: 0


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

Answer: 1


Question: Evaluate the integral: $$ \int_{0}^{\infty} \frac{\sin^4x}{x^4}dx. $$

Answer: \dfrac{\pi}{3}


Question:
Calculate the arc length of the curve formed by the intersection of the sphere \(x^2+y^2+z^2 = \frac{1}{4}\) and the cylinder \(4x^2+5z^2=1\), using the parametrization:
\[x = \frac{1}{2} \sin(t)\]
\[z = \frac{1}{\sqrt{5}} \cos(t)\]

Answer: \pi


Question:
Determine the value of \( k \) for the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfies the equation:

\[ f(x+y) = e^{y}f(x) + e^{x}f(y) + k, \quad \forall x,y \in \mathbb{R}. \]

Answer: 0


Question: Evaluate the triple integral \( \iiint_{D} z(x^{2}+y^{2}+z^{2}) \, \mathrm{d}x \, \mathrm{d}y \, \mathrm{d}z \), where the region \(D\) is defined by \(x^{2}+y^{2}+z^{2} \leq 2z\).

Answer: \dfrac{8\pi}{3}


Question: Suppose that $f$ is a continuous function on the interval $[a,b]$ and that $f(x)$ is always a rational number. Can $f(x)$ be non-constant?

Answer: No


Question: Given $N \geq 9$ distinct positive real numbers in the range $[0,1)$, such that the sum of any 8 of them plus one of the remaining numbers is an integer, determine the maximum possible value of $N$. \( \boxed{} \)

Answer: 9


Question: Find the inverse function of the transformation $T:(0,\infty)^2 \rightarrow T((0, \infty)^2)$ defined by $T(x,y)=\left( \frac{y^2}{x},\frac{x^2}{y} \right)$, and compute the Jacobian determinant of the inverse function.

Answer: -\dfrac{1}{3}


Question:
Determine whether there exists a function \( f: \mathbb{N} \rightarrow \mathbb{N} \) that satisfies the following conditions:
1. \( \sqrt{f(x)} \geq \frac{f(x) + f(1)}{2} \) for some \( x \in \mathbb{N} \).
2. \( \frac{f(n)}{f(1)} = 2n - f(1)^2 \) for all \( n \geq 2 \).

Answer: Yes


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

Answer: \sqrt{10}


Question: Determine the values of $p$ for which the inequality \[ \frac{3(pq^2+p^2q+3q^2+3pq)}{p+q} > 2p^2q \] holds for all $q > 0$. Express your answer in interval notation in decimal form.

Answer: [0, 3)


Question: Determine whether every point in a perfect set in \( \mathbb{R} \) is a condensation point. Answer with 'True' or 'False'.

Answer: True


Question: Evaluate the surface integral \( \iint_S H(x,y,z) \, dS \) where \( H(x,y,z) = yz \) and \( S \) is the part of the sphere \( x^2 + y^2 + z^2 = 4 \) that lies above the cone \( z = \sqrt{x^2 + y^2} \).

Answer: 0


Question: Determine whether the following statement is true or false: If a function $f$ is Riemann-integrable on the interval $[a, b]$, then the set of points where $f$ is continuous is dense in $[a, b]$. Provide a justification for your answer.

Answer: True


Question: Determine the optimal reserve price \( r \) that maximizes the seller's expected revenue in a second-price sealed-bid auction with two buyers, where each buyer's valuation is independently drawn from \( \{0, 1, 2\} \) with equal probability.

Answer: 2


Question: Let $X_1, X_2, \ldots, X_n$ be $n$ independent and identically distributed random variables. Calculate the probability $P(X_n > X_1 \mid X_2 = \max(X_2, X_3, \ldots, X_{n-1}))$.

Answer: \dfrac{1}{2}


Question: Find the value of $(g^{-1})'(2)$ where $g(t) = t^5 + t^3$. Express your answer as a single number.

Answer: \dfrac{1}{8}


Question: Determine whether the process \( \int_0^t (t-s) \, dB_s + \int_s^t (t-u) \, dB_u \) is Gaussian.

Answer: Yes


Question: Let $X$ and $Y$ be locally compact Hausdorff open subsets of the real line. If there is a homeomorphism $\phi: X \to Y$, is the number of connected components of $X$ equal to the number of connected components of $Y$?

Answer: Yes


Question: Consider a function $f:\mathbb{R} \to \mathbb{R}$. If the limit $\lim_{n\to\infty} n(f(c+1/n)-f(c))$ exists for some $c\in \mathbb{R}$, does it imply that $f$ is continuous at $c$? Provide a justification for your answer.

Answer: No


Question: Let $\{C_\alpha\}_{\alpha \in A}$ be objects in a concrete category, and let the product object $C = \Pi C_{\alpha \in A}$ exist with corresponding projections $\{\pi_\alpha\}$. If $x, y \in C$ and $\pi_\alpha (x) = \pi_\alpha (y)$ for all $\alpha$, does it follow that $x = y$? Answer yes or no.

Answer: Yes


Question: Consider the initial value problem: \(\left\{ \begin{array}{@{}l} x'(t) = \sin (x(t)),\ t\in\mathbb{R}\\ x(0)=x_0 \in \left]0,\pi\right[ \end{array} \right.\). Determine \(\lim\limits_{t \to +\infty} x(t)\).

Answer: \pi


Question: Determine if there can be more than three consecutive prime numbers in the sequence defined by $p_1$, $2p_1+1$, $2(2p_1+1)+1$, ..., where each term is prime and $p_1 \equiv 1 \pmod{3}$. Provide a justification for your answer.

Answer: No


Question: Are two polynomials $f$ and $g$ in $\mathbb{Q}[x]$ with a common real root $\alpha \in \mathbb{R}$ guaranteed to have a common divisor $h \in \mathbb{Q}[x]$ with $\deg(h) \geq 1$?

Answer: Yes


Question: Calculate the Lebesgue integral \( \int_{\mathbb{R}} f(x) \, \mu(dx) \), where \( \mu \) is a Borel measure on \( \mathbb{R} \) defined by \( \mu(x)=\begin{cases} 2x+1 & x \leq 2 \\ 2x^2 & x > 2\ \end{cases} \) and \( f(x)=\begin{cases} 1 & x\in \mathbb{Z}\cap [-1,3] \\ 2 & x\in [-1,3] \setminus\mathbb{Z} \\ 0 & x\notin [-1,3] \ \end{cases} \).

Answer: \dfrac{112}{3}


Question: Find a polynomial $g \in \mathbb{Q}[X]$ such that $\mathbb{Q}(\sqrt{2} + i \sqrt{5})$ is the splitting field of $g$ over $\mathbb{Q}$. The minimal polynomial of $\sqrt{2} + i \sqrt{5}$ over $\mathbb{Q}$ is $x^4+6x^2+49$. What is $g$?

Answer: x^4 + 6x^2 + 49


Question: Determine whether the following statement is true or false: In any metric space, a non-empty open set without any isolated points is also closed.

Answer: B


Question: What is the smallest possible order of a finite group $G$ that contains elements of all orders from 1 to 12?

Answer: 27720


Question: Compute the limit \( \lim \limits_{n \to \infty} (n!)^{1/(n!)} \) analytically.

Answer: 1


Question: Determine whether it is always true that for a non-constant entire function, there exist curves along which the function is unbounded, even if infinity is not an asymptotic value.

Answer: Yes


Question: Calculate the fundamental group \(\pi_1(Y)\) of the space \(Y\), where \(Y\) is the quotient space obtained from \(X = S^2 \times [0,1]\) by identifying each point \(x \in S^2 \times \{1\}\) with its antipodal in \(S^2 \times \{0\}\).

Answer: \mathbb{Z}


Question: Determine the maximum percentage increase in the number of basic solutions for a bounded linear programming problem when an additional constraint is added.

Answer: 100\%


Question: Find the maximum absolute value of the elements of the matrix \( A = (a_{ij})_{i,j \in [1,M]} \), where \( a_{ij} = \cos\left(\frac{2\pi ij}{M}\right) \).

Answer: 1


Question: Evaluate the contour integral \(\oint_{C} \frac{e^z}{z-\pi i}\ \mathrm dz\) where \(C\) is the ellipse \(|z - 2| + |z+2| = 6\).

Answer: 0


Question: Evaluate the limit \( \lim_{x \to 0}\frac{\sqrt[3]{\cos x}-\sqrt{\cos x}}{x^2} \) without using L'Hôpital's rule or the Taylor expansion.

Answer: \dfrac{1}{12}


Question: Evaluate the limit of the function \( f(x,y) = (x+y^2)\sin\left(\frac{1}{xy}\right) \) as \((x,y)\) approaches \((0,0)\).

Answer: 0


Question: Find a tight bound for the recurrence relation: \[ T(n) = T\left(\left\lfloor \frac{n}{2} \right\rfloor \right) + T\left(\left\lfloor \frac{n}{4} \right\rfloor \right) + T\left(\left\lfloor \frac{n}{8} \right\rfloor \right) + n \]

Answer: \Theta(n)


Question: Suppose \(A\) is an \(n \times n\) matrix with complex entries, and its characteristic polynomial is identical to its minimal polynomial, given by \(p_{A}(t) = m_{A}(t) = (t - \lambda)^{n}\). Determine the dimension of the eigenspace corresponding to the eigenvalue \(\lambda\).

Answer: 1


Question: What is the cardinality of the set of all sequences of real numbers that have a finite number of terms?

Answer: \mathfrak{c}


Question: Is the topology induced by the Riemannian metric $d_M$ on a Riemannian manifold $M$ the same as the manifold's inherent topology as a topological space?

Answer: Yes


Question: Consider an $m \times n$ ($m,n \ge 3$) square grid placed on a Klein bottle. Each square contains a non-mirror symmetric marker. Define a position as an arrangement of these markers on the Klein bottle. An operation is defined as follows: for each square, if the symbol on the square matches the orientation of 3, 6, 7, or 8 of its Moore neighbors, do nothing; otherwise, replace it with its mirror reflection. Determine if there exists a still-life position, where the operation leaves the position unchanged.

Answer: Yes


Question: Determine the limit of the imaginary part of $(-1)^x \Gamma(x)$ as $x$ approaches $0$. \( \lim_{x \to 0} \Im((-1)^x \Gamma(x)) \)

Answer: \pi


Question: Determine whether it is possible to find rational numbers $a_i$, $b_i$, $p_i$, and $q_i$ such that $f(\pi) = \sum_{i=1}^M a_i \pi^{p_i}$ and $g(\mathrm{e}) = \sum_{i=1}^N b_i \mathrm{e}^{q_i}$ are both nonzero and $\dfrac{f(\pi)}{g(\mathrm{e})}$ is rational.

Answer: Yes


Question: If a function \( f: S \times S \rightarrow S \) is continuous, does it imply that for each \( a \in S \), the map \( f_a(b) = f(a, b) \) is also continuous, where \( S \) is equipped with the product topology?

Answer: Yes


Question: Evaluate the integral \( \int_{-\frac{2\pi}{3}}^{\frac{2\pi}{3}} -\frac{4\cos(x)}{1 + e^{-4x}} \, dx \) using the properties of even and odd functions.

Answer: -2\sqrt{3}


Question: Let $T: H \rightarrow H$ be a self-adjoint operator with a pure point spectrum, represented as $T(x) = \sum_{n=0}^{\infty} \lambda_n \langle x , x_n \rangle x_n$. If $V \subset H$ is a finite-dimensional subspace such that $T(V) \subset V$, is $T|_V$ a diagonalizable matrix when viewed as an operator on $V$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to -\infty} \frac{\sqrt[n]{x+1} + \sqrt[n]{x+2} - 2\sqrt[n]{x-3}}{\sqrt[n]{x-1} + \sqrt[n]{x-2} - 2\sqrt[n]{x+3}} \] for \( n \in \mathbb{N} \), \( n > 2 \), \( n \) odd.

Answer: -1


Question: Is the hypothesis that a group $G$ is abelian necessary for a group with a totally ordered subgroup lattice to be isomorphic to $\mathbb{Z}(p^k)$ for some prime $p$ and $k \in \{1,2,...\}\cup\{\infty\}$?

Answer: No


Question: Find the natural number $n$ for which $3^n + 5^n$ is divisible by $3^{n-1} + 5^{n-1}$.

Answer: 1


Question: Given the function \( g = \sqrt{1+\dot{x}^2} \), where \( \dot{x} \) is a function of time, verify that the equation \( \frac{\partial g}{\partial x} - \frac{d}{dt} \left[ \frac{\partial g}{\partial \dot{x}} \right] = 0 \) simplifies to \( \ddot{x} = 0 \).

Answer: \ddot{x} = 0


Question: Evaluate the complex line integral \( \int_\gamma \frac{e^{z}}{\sin z} dz \), where \(\gamma\) is the circle centered at the origin with radius 1, traversed once in the counterclockwise direction.

Answer: 2\pi i


Question: Calculate the second mixed partial derivative \( \frac{\partial^2 g}{\partial x \partial y} \) of the function \( g(x,y)= \begin{cases} \frac{x^2y(x^2-y^2)}{x^2+y^2} &\text {if }(x,y) \neq (0,0) \\ 0 & \text{otherwise}\end{cases} \) at the point (0,0).

Answer: 0


Question: If $A$ is a Lebesgue-measurable subset of $\mathbb{R}$ and $B$ is a Lebesgue non-measurable subset of $\mathbb{R}$ such that $B \subset A$, determine if it is possible that $m^*(A \cup B) = m^*(A)$. Provide a justification for your answer.

Answer: Yes


Question: Find the rate of change of temperature with respect to time as a bug moves along the path given by the parametric equations $r(t) = \langle t\cos(2t), t\sin(2t) \rangle$ on a surface where the temperature is modeled by $T(x, y) = 100e^{-(x^2 + y^2)}$. Express your answer in terms of $t$. \( \frac{dT}{dt} \)

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


Question: Determine the approximate proportion of line segments with irrational lengths in an n by n lattice as n approaches infinity.

Answer: 1


Question: Given that when the polynomial $P(x+4)$ is divided by $P(x)$, the remainder is $3x+m$, and when $P(x)$ is divided by $P(x+4)$, the remainder is $nx-6$, compute the value of $m-n$ assuming that $P(x)$ and $P(x+4)$ have the same degree.

Answer: 9


Question: Determine the value of $t \in \mathbb{R}$ such that the set $A = \{ (x_1, x_2, x_3, x_4) \mid 3x_1 - 2x_2 + x_3 + x_4 - t(x_1^2 + x_4^2) = 0 \}$ forms a subspace of $\mathbb{R}^4$. \( \boxed{t} \)

Answer: 0


Question: In Rudin's proof that a subalgebra $\mathscr{A}$ of $C(X,\mathbb{R})$ on a compact space $X$ is dense in $C(X,\mathbb{R})$ under the uniform topology if it separates points and vanishes nowhere, identify the step that relies on the Hausdorffness condition of $X$. The proof consists of the following steps: (1) $|f|\in \overline{\mathscr{A}}$ for all $f\in \overline{\mathscr{A}}$, (2) $\max(f,g),\min(f,g)\in \overline{\mathscr{A}}$ for all $f,g\in \overline{\mathscr{A}}$, (3) existence of $g\in \overline{\mathscr{A}}$ such that $g(x)=f(x)$ and $g(x)>f(x)-\epsilon$ for any $f\in C(X,\mathbb{R}), x\in X, \epsilon >0$, and (4) existence of $h\in \overline{\mathscr{A}}$ with $||h-f||_\infty <\epsilon$ for any $f\in C(X,\mathbb{R}), \epsilon>0$. Which step relies on the Hausdorffness of $X$?

Answer: 3


Question: Determine whether the following statement is true or false: If $F$ is a finite field and $K$ is a finite field extension of $F$ of degree 2, then the minimal polynomial of any element $a \in K \setminus F$ over $F$ splits completely over $K$. Provide a justification for your answer.

Answer: True


Question: Find the eigenvalues of the operator \( Q \) on \( L^2[-1,1] \), given by \( Q(f(x)) = \int_{x+1}^1 f(y)dy \).

Answer: 0


Question: Evaluate the limit: $$\lim_{x\rightarrow 0} \frac{(a+x)^x-a^x}{x^2}$$ where \(a\) is a positive constant.

Answer: \dfrac{1}{a}


Question: Evaluate the integral \( \int_{-\infty}^{\infty}\frac{x^2}{1+x^6} \, dx \) using a suitable substitution.

Answer: \dfrac{\pi}{3}


Question: Find the limit of \( y(x) \) as \( x \to \infty \), where \( y(x) = 10 + \int_0^x \frac{22(y(t))^2}{1 + t^2}\,dt \).

Answer: \infty


Question: Find the radius of convergence of the power series \( \sum_{n=1}^{\infty} \frac{(n!)^2}{(2n)!} x^{2n} \) using the Cauchy-Hadamard Theorem.

Answer: 2


Question: Find the Taylor series of \( f(z) = \frac{z(z-3)}{(z-1)} \) centered at \( z=3 \) and determine its radius of convergence.

Answer: 2


Question: Determine an upper bound for the sequence \( \frac{F_{n+1} + F_n}{F_{n+1} - F_n} \), where \( F_n \) represents the Fibonacci sequence.

Answer: 5


Question: Evaluate the integral \( \int_{C} f(z) \, dz \) over the upper half-circle \( C \) with \( 0 \leq \arg z \leq \pi \), starting from \( z=1 \), using the parametrization \( z = e^{ix} \).

Answer: i\pi


Question: Three fair twenty-sided dice are rolled and arranged in decreasing order. The expected value of the largest die can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are relatively prime positive integers. Calculate \( p+q \).

Answer: 1319


Question: Given the equation \( z^c = \exp(w) \), where \( z = 1+i \) and \( c = 1-i \), find the value of \( \text{Re}(w) + \text{Im}(w) \).

Answer: \dfrac{\pi}{2}


Question: Calculate the expected value of the minimum distance between a randomly chosen point inside a circle of radius 1 and either the center or the circle boundary. The distance to the center is denoted as $X$, and the distance to the circle boundary is $1 - X$. Provide your answer as a single numerical value.

Answer: \dfrac{1}{4}


Question: Let $f(x)$ be a real-valued strictly convex function on $[0, 1]$. For any integer $k$ between $0$ and some positive integer $n$, let $x_k = k/n$. Consider the average value function \( g(n) = \frac{1}{n+1} \sum_{k=0}^{n} f(x_k) \). Determine whether \( g(n) \) is a decreasing function as \( n \) increases.

Answer: Yes


Question: Find all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \( f(x + y) = x + f(y) \) for all \( x, y \in \mathbb{R} \). What is the general form of such functions?

Answer: f(x) = x + c


Question: Consider a measure space $(X, M, \mu)$ and a function $f: X \rightarrow \mathbb{C}$ in $L^{\infty}(\mu)$. Define $A_f$ as the set of all averages \( \frac{1}{\mu(E)} \int_{E} f \, d\mu \) where $E \in M$ with $0 < \mu(E) < \infty$. Determine if there exists a non-trivial measure space $X$ such that $A_f$ contains more than one point for some function $f$. Provide a justification for your answer.

Answer: Yes


Question: Consider the alternating group $A_5$. What is the probability that the subgroup generated by two randomly chosen elements $a$ and $b$ in $A_5$, denoted by $\langle a, b \rangle$, is a cyclic group?

Answer: \dfrac{3}{40}


Question: Let $f: G \to G$ be a bijective function on a finite simple graph that preserves adjacency. Is $f$ automatically a graph automorphism?

Answer: Yes


Question: Is a bounded function $f: S \to \mathbb{R}$, defined on a Jordan measurable set $S \subset \mathbb{R}^n$, Riemann integrable over $S$ if and only if the set $D$ of its discontinuities has Lebesgue measure zero in $\mathbb{R}^n$?

Answer: Yes


Question: Determine the structure of the group \( G = \mathbb{Z} \times \mathbb{Z} / H \) where \( H = \langle (4, -3) \rangle \), and decide if it is torsion-free.

Answer: \mathbb{Z}


Question: Determine whether there exists a positive integer $n$ such that $n^n + (n + 1)^n$ is divisible by 1987.

Answer: Yes


Question: Given two positive real numbers $A$ and $l$, such that $4\pi A \leq l^2$, determine whether it is always possible to find a plane, simple, closed curve with length $l$ that encloses a region with area $A$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the flux of the vector field \( F(x,y)=\langle 2x-y, x+3y\rangle \) out of the ellipse centered at the origin with semi-major axis 2 and semi-minor axis 1.

Answer: 10\pi


Question: What property of the function \(\arctan(x)\) restricts the convergence of its power series to the interval \(-1 \leq x \leq 1\), unlike the power series of functions like \(e^x\), \(\sin(x)\), and \(\cos(x)\), which converge for all real numbers?

Answer: [-1, 1]


Question:
Given the equations:
\[ a + b = 6 \]
\[ ax + by = 10 \]
\[ ax^2 + by^2 = 24 \]
\[ ax^3 + by^3 = 62 \]
Find the value of \( ax^4 + by^4 \).

Answer: 162


Question: Does the ratio \( \frac{f(n+1)}{f(n)} \) converge as \( n \to \infty \) for all linear recursions \( f(n) \)?

Answer: No


Question: Determine whether $x^p$ is algebraic over the field $(K(x))^p$ when $K$ is a field with characteristic $p > 0$.

Answer: Yes


Question: Simplify the expression: \( \sqrt[3]{18+5\sqrt{13}}+\sqrt[3]{18-5\sqrt{13}} \).

Answer: 3


Question: Evaluate the surface integral \(\int_S z \, dS\) where \(S = \{ (x, y, z) : x^2 + y^2 + z^2 = a^2, z \ge 0, a > 0 \}\) using Gauss's Divergence Theorem.

Answer: \pi a^3


Question: Find the minimum value of \(|z+2i|+|z-2i|+|z+8i|+|z-4i|\) for a complex number \(z\).

Answer: 16


Question: Find the sum of the series \( \sum_{n=1}^{\infty}(-1)^{(n+1)}\frac{2n+1}{3^n} \).

Answer: \dfrac{5}{8}


Question: Evaluate the limit \( \lim_{n\rightarrow\infty}(1+2^n+3^n+\ldots+n^n)^{1/n}\sin(1/n) \).

Answer: 1


Question: Determine the possible values of $n$ for which an automorphism $\sigma$ of $\mathbb{Q}(\omega)$ can map $\omega$ to $\omega^n$, where $\omega = e^{\frac{2\pi i}{10}}$. Provide your answer as a set of integers.

Answer: \{1, 3, 7, 9\}


Question: Determine whether there exist functions \( f, g : \mathbb{N} \to \mathbb{R}^+ \) and a constant \( c \in \mathbb{R} \setminus \{0\} \) such that the following series all equal zero simultaneously: \[ S_0 = \sum_{n=1}^{\infty} \frac{(-1)^n}{f(n)} = 0, \] \[ S_1 = \sum_{n=1}^{\infty} \frac{(-1)^n}{f(n)} \cdot g(n)^c = 0, \] \[ S_2 = \sum_{n=1}^{\infty} \frac{(-1)^n}{f(n)} \cdot \frac{1}{g(n)^c} = 0. \]

Answer: Yes


Question: When three fair six-sided dice are rolled to obtain values $a$, $b$, and $c$, what is the probability that the inequality \( \frac{a}{b} + \frac{b}{c} + \frac{c}{a} \leq \frac{a+b}{b+c} + \frac{b+c}{c+a} + \frac{c+a}{a+b} \) holds true?

Answer: \dfrac{1}{36}


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

Answer: 3 + \sqrt{29}


Question: Evaluate the surface integral \(\int_S (x^2 + y^2) \, d\sigma\), where \(S\) is the sphere of radius 1 centered at the origin \((0,0,0)\).

Answer: \dfrac{8\pi}{3}


Question: Determine whether a continuous function $g:[c,d]\to\mathbb{R}$ that maps every point in $[c,d]$ to a rational number must be constant. Answer with 'true' or 'false'.

Answer: true


Question: Suppose $f: \mathbb{B}^n \to \mathbb{R}^n$ is a continuous function, where $\mathbb{B}^n$ is the $n$-dimensional unit ball. The restriction of $f$ to the boundary, $g := f|_{\mathbb{S}^{n-1}}$, does not include $0$ in its image, and $g: \mathbb{S}^{n-1} \to \mathbb{R}^n \setminus\{0\}$ is homotopic to the identity map. Determine whether it is necessarily true that $f(x) = 0$ for some $x \in \mathbb{B}^n$. Provide a justification for your answer.

Answer: Yes


Question: Determine the number of ways to choose distinct subsets \(A_0, A_1, \ldots, A_n\) of \(\{1, 2, \ldots, n\}\) such that \(A_0 \subset A_1 \subset \ldots \subset A_n\).

Answer: n!


Question: Evaluate the definite integral \( \int_{-\pi/2}^{\pi/2} \sqrt{\sin^3x - \sin^5x} \, dx \).

Answer: \dfrac{4}{5}


Question: Let $N$ be a subspace of a topological space $M$. If for every subset $S \subset N$, the openness of $S$ in $N$ is equivalent to the openness of $S$ in $M$, does it follow that $N$ is an open subset of $M$? Answer yes or no and justify your answer.

Answer: Yes


Question: Estimate the value of \( \int^{1}_{-1} \left(\frac{\sin{x}}{x}\right)^{300} dx \) to one significant figure.

Answer: 0.3


Question: Find the value to which the sum \( \frac{1}{n^3}\sum_{i=1}^n (2i^2 - i) \) converges as \( n \to \infty \).

Answer: \dfrac{2}{3}


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

Answer: 1


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

Answer: \dfrac{4}{e}


Question: Find the number of pairs of positive integers \((x, y)\) that satisfy the equation \((x+2)! + (y+2)! = x^3 y^3\).

Answer: 0


Question: Given an infinite stack of squares, find the ratio of the sum of the areas of all the squares to the area of a circle that circumscribes the largest square.

Answer: \dfrac{4}{\pi}


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty}\frac{(3n)!}{(n!)^3} \] without using Stirling's approximation.

Answer: \infty


Question:
Two subgroups H and K of a group G have orders 12 and 30, respectively. Determine which of the following could NOT be the order of the subgroup of G generated by H and K:

A. 30
B. 60
C. 120
D. 360
E. Countable infinity

Answer: A


Question: Calculate the remainder when \( 37^{877} \) is divided by 323.

Answer: 284


Question: Determine the equation of a curve such that its tangent lines form a triangle with the x-axis and y-axis, and the area of this triangle is $2a^2$. What is the equation of this curve?

Answer: xy = a^2


Question: Determine the complexity of an algorithm that tests whether a given polynomial is a square in a finite field with small characteristic.

Answer: O(n^2)


Question: Calculate the surface integral \( \iint_S 2x \,dS \) over the surface \( S \) defined by the portion of the paraboloid \( z = 9 - x^2 - y^2 \) that lies above the \( xy \)-plane, oriented upwards.

Answer: 0


Question: Determine whether there exists a polynomial with integer coefficients such that the ratio of its values at consecutive positive integers is always a perfect square.

Answer: Yes


Question: Determine the number of distinct spanning trees in the complete graph $K_9$ such that vertex $1$ has a degree of $4$. Use Prufer codes to find the solution.

Answer: 143360


Question: Determine whether the following statement is true or false: If $f_n \to f$ almost everywhere, then $f_n \to f$ in measure.

Answer: False


Question: Consider the function \( g(x) = \frac{1}{\sqrt{1-x^2}} \) defined on the interval \([0,1]\) with the Lebesgue measure. Find \( \lim_{p \to \infty} \|g\|_p \).

Answer: \infty


Question: Evaluate the limit: $$\lim_{x\to 2}\left(\frac{x^2-4}{x-2}\right)^{\large\left(\frac{x^3-6x^2+11x-6}{x^2-3x+2}\right)}$$

Answer: \dfrac{1}{4}


Question: Find the set $T$ of complex numbers defined by \[ T = \left\{ z \in \mathbb{C} \mid \left|\frac{z - i}{z + i}\right| = 1 \right\}. \]

Answer: \mathbb{R}


Question:
Consider the initial value problem (IVP) given by the heat equation $u_t = u_{xx}$ for $x \in \mathbb{R}$ and $t > 0$, with the initial condition:
\[ u(x,0) = \begin{cases} 1, & x \in [0,1] \\ 0, & \text{otherwise} \end{cases} \]
Determine the value of \( \lim_{t \to 0^+} u(1,t) \).

Answer: \dfrac{1}{2}


Question:
If $A$ and $B$ are real matrices of order $n$ such that $\det A>0$ and $\det B<0$, and $C(t)=tA+(1-t)B$ for $t\in[0,1]$, determine which of the following statements is true:

1) $C(t)$ is non-singular for all $t$ in $(0,1)$.
2) $C(t)$ is singular for a finite number of $t$ in $(0,1)$.
3) $C(t)$ is singular for an infinite number of $t$ in $(0,1)$.

Select the correct statement.

Answer: 2


Question: In a finite affine plane over the finite field $\mathbb{F}_q$ where $q = 2^n$, consider the case where $q = 4$. Is it possible to find a set of 6 points $\{P_1, P_2, \ldots, P_6\}$ such that no three points lie on the same line, and for any odd $i$, the vectors $\overrightarrow{P_iP_{i+1}}$ and $\overrightarrow{P_1P_2}$ are equal? It is known that this is not possible for $q = 4$. What is the maximum number of points with these properties, and how does this relate to hyperovals in the projective plane $PG(2,q)$?

Answer: 4


Question: Evaluate the double integral \( \iint_{D} xy \, dx \, dy \) over the region \( D \), where \( D \) is the unit circle defined by \( x^2 + y^2 = 1 \).

Answer: 0


Question:
Determine whether the solutions of the following system of equations can be expressed solely in terms of $a$:
\[ x^2 + y^2 + xy = a, \]
\[ x^2 + z^2 + zx = a, \]
\[ y^2 + z^2 - yz = a. \]

Answer: Yes


Question: Is it possible to construct three non-isomorphic graphs such that one graph is planar, another is 2-colorable, and the third contains a subgraph isomorphic to a subdivision of $K_{3,3}$?

Answer: Yes


Question: Let \(f(x)=ax^3+bx^2+cx+d\) and \(g(x)=x^2+x-2\). Given that \(\lim_{x \to 1}\frac{f(x)}{g(x)}=1\) and \(\lim_{x \to -2}\frac{f(x)}{g(x)}=4\), find the value of \(\frac{c^2+d^2}{a^2+b^2}\).

Answer: 16


Question: Determine whether the sequence of functions \( f_n(x) = \frac{x^n}{\sqrt[4]{n}} \) converges uniformly for \( x \in [0,1] \). If it converges, find the function \( f(x) \) to which the sequence converges.

Answer: 0


Question: Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of independent random variables, each uniformly distributed over the interval [0, 1]. Define $Y_n = (\prod^n_{i=1} X_i)^{\frac{1}{n}}$. Use the Central Limit Theorem to approximate $\mathbb{P}(Y_{50} \geq \mathbb{E}(Y_{50}))$. Compute this probability.

Answer: 0.472


Question: What is the name of the smallest ordinal number that is not ordinally similar to any subset of any chain of an infinite partially ordered set $S$?

Answer: \omega_1


Question: Find the volume of the region $K$ in $\\mathbb{R}^3$ defined by the inequalities: $$K = \{(x, y, z) \mid x \ge y^2, x - y \le 2, 0 \le z \le x\}.$$

Answer: \dfrac{36}{5}


Question: Calculate the limit: $$\lim_{x \to 0} \frac{\sqrt{1+ \tan x} - \sqrt{1+ \sin x}}{x^3}$$ without using L'Hospital's rule.

Answer: \dfrac{1}{4}


Question: Calculate the sum of the infinite series: \[ \sum_{n=0}^\infty \frac{5^{2n-5^n}}{1+5^{-5^n}} \]

Answer: \dfrac{91}{521}


Question: Find the derivative of the function \( g(x) = x^T B x \), where \( B \) is a symmetric matrix and \( x \) is a vector.

Answer: 2Bx


Question: Find the almost sure limit of the sequence \( \frac{\eta_n}{n} \), where \( \eta_n \) is a Poisson random variable with parameter \( \lambda = n \).

Answer: 1


Question: Evaluate the integral: \[ \int_{0}^{\infty} x^{3}e^{-x^2}dx \]

Answer: \dfrac{1}{2}


Question: Find the remainder when \(3^{3^{3^{\:\!\phantom{}^{.^{.^.}}}}}\) is divided by 46.

Answer: 41


Question: Find an example of a closed connected smooth manifold $M$ such that $\pi_1(M) = \pi_2(M) = \pi_4(M) = 0$ and $\pi_3(M) = \mathbb{Z}$. If such a manifold exists, determine its minimum possible dimension.

Answer: 6


Question: Evaluate the integral \( \int\limits_{L} z \cos(2\pi z \bar{z}) \, dz \), where \( L = \{|z| = 1; \quad 0 \le \arg z \le \pi \} \).

Answer: 0


Question: Given the polynomial \( p(x) = x^4 + ax^3 + bx^2 + cx + d \) with conditions \( p(1) = 10 \), \( p(2) = 20 \), and \( p(3) = 30 \), find the value of \( \frac{p(12) + p(-8)}{10} \).

Answer: 1984


Question: Find the maximum value of the function \( f(x) = \frac{\sin x}{x} \) for \( x \neq 0 \) without using L'Hopital's rule.

Answer: 1


Question: Find the derivative of the function \(f(x)=\begin{cases} x^2\sin(\frac{1}{x}), &x \neq 0 \\ 0, & x=0 \end{cases}\) at \(x=0\).

Answer: 0


Question: Evaluate the limit: \[\lim _{(x,y) \to (1,0)} (1+xy)^{f(x,y)},\] where \[f(x,y)=\frac{1}{x^2+y^2}.\]

Answer: 1


Question:
Find the coefficient of $x$ in the determinant of the matrix:
\[ \begin{pmatrix} x & (1+\sin x)^3 & \cos x \\ 1 & \ln(1+x) & 2 \\ x^2 & 1+x^2 & 0 \end{pmatrix} \]

Answer: -2


Question: Find the function $f(x)$ that satisfies the equation $f(x+y) + f(x-y) = 2f(x) + 2f(y)$ for all real numbers $x$ and $y$, given that $f: \Bbb{R} \to \Bbb{R}$ is continuous.

Answer: f(x) = ax^2


Question: Evaluate the limit: $$\lim_{x\to 0} \frac{\sin(x) - x + \frac{x^3}{6}}{x^5}$$ without using L'Hopital's Rule.

Answer: \dfrac{1}{120}


Question: Determine whether the series \(1 + \frac{1}{2} -\frac{1}{3} + \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \dots\), where the generic term is given by \(a_n = \left\{\begin{array}{ll} -\frac{1}{n}, & \text{if } 3 \mid n \\ \frac{1}{n}, & \text{otherwise}\end{array}\right.\), converges. If it diverges, specify whether it diverges to \(\infty\) or exhibits no limit.

Answer: \infty


Question: Find a formula to calculate the sum of quadratic residues modulo a prime $q$ when $q \equiv 1 \pmod{3}$. Provide the formula.

Answer: 0


Question: What fraction of the natural numbers are powers of 2?

Answer: 0


Question: Determine the limit of the sequence \(a_n = n^2 \log\left(\cos \frac{1}{n}\right)\) using only techniques related to sequences, without employing l'Hospital's rule, Taylor series, or derivatives.

Answer: -\dfrac{1}{2}


Question: Evaluate the expression: $$T = \sin^{-1}\cot\left(\cos^{-1}\left(\sqrt{\frac{2 + \sqrt{3}}{4}}\right) + \cos^{-1}\left(\frac{\sqrt{12}}{4}\right) + \csc^{-1}\left(\sqrt{2}\right)\right) $$

Answer: 0


Question: Determine whether the following statement is true or false: Given a decreasing positive sequence $\{a_n\}$ such that $\sum_{n=1}^{\infty} a_n$ diverges, and a function $f: \mathbb{N} \rightarrow \mathbb{N}$ with $\lim\limits_{n\to\infty} f(n) = \infty$, if $\sum_{n=1}^{\infty} a_{[f(n)]}$ converges, then $\sum_{n=1}^{\infty} \frac{a_n}{f(n)}$ also converges.

Answer: True


Question: Determine the radius of convergence of the Taylor series for the function \( g(z) = \frac{1}{1+e^{-z}} \) centered at \( z=0 \) without evaluating the Taylor series.

Answer: \pi


Question: If a surface is oriented by an outward-pointing normal vector in the sense of Folland's Advanced Calculus, does this correspond to an A) orientation-preserving, B) orientation-reversing, or C) unrelated orientation in the sense of Spivak's Calculus on Manifolds? Justify your answer.

Answer: A


Question: Evaluate the integral \(\int \mathrm{d}x \, f(x) \, \delta(\mathrm{e}^x)\) for a suitable function \(f(x)\).

Answer: 0


Question: Determine the radius \( R \) of a sphere such that the center of mass of the solid region is located at \( z = 2 \).

Answer: 2


Question: Let $a: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$ be a bilinear operator. Determine if there exists a square matrix $A \in \mathbb{R}^{n \times n}$ such that for all $u, v \in \mathbb{R}^n$, $a(u, v) = v^t A u$, where $v^t$ is the transpose of the vector $v$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_0^{\infty} \frac{\sin^3{x}}{x} \, dx \) given that the Dirichlet integral \( \int_0^{\infty} \frac{\sin{x}}{x} \, dx=\frac{\pi}{2} \).

Answer: \dfrac{\pi}{4}


Question: Can a Riemann integrable function on a bounded interval have uncountably many points of discontinuity? Provide a justification for your answer.

Answer: Yes


Question: Use Stokes' Theorem to transform the surface integral \( \int_S \text{rot}\vec{F}\cdot \vec{dS} \) into a line integral, and then calculate the line integral for the vector field \( \vec{F}(x,y,z)=(y,z,x) \). The surface \( S \) is the part of the paraboloid \( z=1-x^2-y^2 \) with \( z\ge 0 \), and the normal vector has a non-negative \( z \) component.

Answer: -\pi


Question: Given a prime number $p$ such that $p-1$ is divisible by $4$, let $K=\mathbb{Q}_p$ and consider the unramified extension $F/K$ of degree $4$. Determine whether $F/K$ always admits a generator of the form $\sqrt[4]{a}$, where $a \in \mathbb{Z}$. In other words, can we express $F$ as $K(\sqrt[4]{a})$ for some integer $a$?

Answer: Yes


Question: Find the order of the normalizer of the subgroup of elementary abelian 2-groups in the affine general linear group $AGL(3,2)$ within the symmetric group $S_8$. Provide your answer as a single integer.

Answer: 1344


Question: Evaluate the integral \( \int_0^\infty \frac{x^3}{(x^8+1)^2} \, dx \) using a suitable substitution.

Answer: \dfrac{\pi}{16}


Question: Evaluate the limit: $$\lim_{x\to \infty} \int_{x-k}^{x} g(t) \, dt$$ where $g: [0, \infty) \rightarrow \mathbb{R}$ is a function such that $\lim_{x\to \infty} g(x) = n$ for some $n \in \mathbb{R}$ and $k > 0$ is fixed.

Answer: nk


Question:
Solve the system of equations:

$$\large \left\{ \begin{align} \sqrt{3(x - y)^2 - 4x + 8y + 5} - \sqrt x = \sqrt{y + 1}\\ x^2y + y^2 - 3xy - 3x + 7y + 8 = 2x\sqrt{y + 3} \end{align} \right.$$

Answer: (2, 1)


Question:
Given the system of equations:

$$x + y\sqrt{x} = \frac{95}{8}$$
$$y + x\sqrt{y} = \frac{93}{8}$$

where $x, y \in \mathbb{R}$, find the value of $\sqrt{xy}$.

Answer: \dfrac{15}{4}


Question: Find a tight upper bound for the matrix norm \( \| I - \frac{x x^T}{\|x\|_2^2} \| \), where \( x \) is a column vector.

Answer: 1


Question: Given that \(\lim _{n \rightarrow \infty} \int_{0}^{a_{n}} x^{n} \, dx=2\) with \(a_n>0\), find \(\lim _{n \rightarrow \infty} a_n\).

Answer: 1


Question: Solve for integer $x$ in the equation: $$3^{3x} - 3^x = (3x)!$$

Answer: 2


Question: Given linearly independent vectors $a_1, \ldots, a_m \in \mathbb{R}^n$, let $S$ be the vector space defined as $S = \{x \in \mathbb{R}^n : a_i \cdot x = 0 \text{ for all } 1 \leq i \leq m\}$. Determine if $n - m$ is always the largest possible number of linearly independent solutions to $a_i \cdot x = 0$, or provide a counterexample.

Answer: n - m


Question: Let $f(x)=x^{3}+x$ and $g(x)=x^{3}-x$ for all $x \in \mathbb{R}$. If $f^{-1}$ denotes the inverse function of $f$, find the derivative of the composite function $g \circ f^{-1}$ at the point $2$. \( \text{Calculate } \left( g \circ f^{-1} \right)'(2). \)

Answer: \dfrac{1}{2}


Question: Evaluate the limit: $$\lim_{\theta\to 0} \dfrac {\sin\left ( \sin a \theta \right ) }{\theta}$$ and determine if $$\lim_{\theta\to 0} \dfrac {\sin\left ( \sin \left ( \sin \left ( \sin \cdots a \theta \right ) \right )\cdots \right ) }{\theta} =a.$$

Answer: a


Question:
Solve the boundary value problem for the differential equation:

$$y'' + (y')^2 + y = \ln(x)$$

with the boundary conditions $y(1) = 0$ and $y(2) = \ln(2)$. Find the function $y(x)$ that satisfies these conditions.

Answer: \ln(x)


Question: Find the value of the infinite product \( \prod_{i=1}^\infty\frac{2i-1}{2i} \).

Answer: 0


Question: In a tetrahedron $ABCD$, the six edges have lengths $7, 13, 18, 27, 36$, and $41$ units. Given that edge $AB$ is $41$ units long, find the length of edge $CD$. Provide your answer as a single number.

Answer: 13


Question: Find the smallest positive solution for the intersection of the polar curves given by \( R_1 = \cos(2\theta) \) and \( R_2 = \sin(3\theta) \).

Answer: \dfrac{\pi}{10}


Question: What is the term for a structure similar to a semiring, but where associativity is not required for multiplication?

Answer: non-associative semiring


Question: Evaluate the limit: $$\lim_{x\to 0}\left(\frac{1}{\log_{10}{5} + \log_{10}{x}} - \frac{1}{\log_{x}{5} + \log_{x}{x}}\right)$$

Answer: -1


Question: Find a function of three variables \( G(x, y, z) \) such that the hyperboloid \( x^2 - y^2 - z^2 = 1 \) is a level surface of \( G(x, y, z) \).

Answer: G(x, y, z) = x^2 - y^2 - z^2


Question: In complex analysis, can a multivalued function associate an infinite number of values with a single argument?

Answer: Yes


Question: Determine whether there exists a non-continuous idempotent linear operator $P: X \to X$ on an infinite-dimensional Banach space $X$ such that $P^2 = P$, and there is a sequence $\{x_n\}$ in $X$ with $\lim\limits_{n \to \infty} \|x_n\| = 0$ and $\lim\limits_{n \to \infty} \|P x_n\| = \infty$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit \( \displaystyle\lim_{n \to \infty}\, \int_{0}^{1}{\frac{x^{n + 1}}{1 + x}}\,\mathrm{d}x \).

Answer: 0


Question: Determine the total number of spanning trees in the undirected labelled graph $Q_3$, which is a 3-dimensional hypercube.

Answer: 384


Question: Determine whether the converse of the statement "If a connected Lie group or algebraic group G is perfect, then it is semisimple" is true or false.

Answer: True


Question: Determine the number of real solutions to the equation \( \sin(\sin(\sin(\sin(\sin(x))))) = \frac{x}{3} \).

Answer: 3


Question: For a seminorm $p$ on a vector space $V$, determine whether it is always true that there exist distinct vectors $x_1$ and $x_2$ in $V$ such that $p(x_1) = p(x_2) = 0$, given that $p$ is not a norm.

Answer: Yes


Question: Evaluate the limit \( \lim_{n \to \infty} T_n \), where \( T_n = \int_0^1 \frac{x^n}{(1+x)^2} \, dx \).

Answer: 0


Question: Identify a function for which the Taylor series expansion does not represent the function within an interval containing the expansion point.

Answer: e^{-1/x^2}


Question: Find the maximum area of a hexagon that can be inscribed in the ellipse given by the equation \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \).

Answer: 18\sqrt{3}


Question: Let $G$ be a finitely generated abelian group, $H \leq G$, and $S = \{g_1, \dots, g_n\}$ be a finite generating set for $G$. Define $G’$ as the set of nonnegative linear combinations of elements of $S$, i.e., $G’ = \{a_1g_1 + \dots + a_ng_n \mid a_1,\dots, a_n \in \mathbb{N}\}$. Let $H’= G’ \cap H$. Is $H’$ finitely generated as a monoid? In other words, is there a finite subset $S’$ of $H’$ such that every element of $H’$ is a nonnegative linear combination of elements of $S’$?

Answer: Yes


Question: Find the minimal polynomial of \( \frac{\omega+\omega^{-1}}{2} \), where \( \omega \) is the primitive 5th root of unity, without using Chebyshev polynomials.

Answer: 4x^2 + 2x - 1


Question: Determine whether the order of the inner automorphism corresponding to an element $g$ in a group always divides the order of $g$. Provide a proof or counterexample.

Answer: Yes


Question: Determine the cardinality of the union of disjoint sets, each having the cardinality of the real numbers. Provide a proof without using the Schröder-Bernstein theorem.

Answer: \mathfrak{c}


Question: Evaluate the triple integral \( \iiint x\ \text{d}x \) over the region bounded by the surfaces \( z = x^2 + y^2 \) and \( 4x + z = -2 \).

Answer: -4\pi


Question: In classical propositional logic, consider three conjunctive forms: \(\alpha\), \(\beta\), and \(\varphi\). If \(\alpha\) and \(\beta\) have two or more disagreements over the same propositional variables, can their disjunction \((\alpha \lor \beta)\) be equivalent to \(\varphi\), which is also in conjunctive form?

Answer: Yes


Question: If $f(x)g(x) \to \infty$ and $f(x) \to 0$, determine whether $|g(x)| \to \infty$.

Answer: Yes


Question: Let $f$ be a measurable function on a measure space $(X, \mathcal{F}, \mu)$ and $p \in [1, \infty]$ such that $||f||_p := \left(\int_X |f|^p \, d\mu\right)^{1/p} < \infty$. Is it true that there exists a sequence of simple functions $\{g_n\}$ such that $||g_n - f||_p \to 0$ as $n \to \infty$?

Answer: Yes


Question: Evaluate the sum: $$ \frac{1}{\log_{2}50!} + \frac{1}{\log_{3}50!} + \frac{1}{\log_{4}50!} + \dots + \frac{1}{\log_{50}50!} $$ and verify that it equals 1.

Answer: 1


Question: Let $2 \leq n \leq \omega$ and $F$ be a set of size $\leq n$. Let $\mathbb{P}$ be a poset and $Q \subseteq \mathbb{P}$ an $n$-linked subset. If $\dot{a}$ is a name for a member of $\mathbb{P}$ such that $\Vdash \dot{a} \in F$, prove that there exists a $c \in F$ such that $p \nVdash \dot{a} \neq c$ for all $p \in Q$. Provide the element $c$.

Answer: c


Question: Evaluate the double integral of the function \( f(x,y) = (x+y)^9(x-y)^9 \) over the region defined by \( 0 \leq x \leq \frac{1}{2} \) and \( x \leq y \leq 1-x \).

Answer: -\dfrac{1}{400}


Question: Calculate the flow of the vector field \(\mathbf{F}(x, y, z) = \frac{1}{(x^2 + y^2 + z^2)^{\frac{3}{2}}} (x, y, z)\) out of a sphere with radius 10 and center at the origin.

Answer: 4\pi


Question: Determine the range of the expression \( \sin \alpha \cdot \sin \beta + \sin \beta \cdot \sin \gamma + \sin \gamma \cdot \sin \alpha \), where a line makes angles \( \alpha, \beta, \gamma \) with the positive x, y, and z axes, respectively.

Answer: [1, 2]


Question: Determine the values of $n$ for which the Galois field with $2^n$ elements has a non-cyclic additive group.

Answer: n \geq 2


Question: Determine whether the function \( g:[0,1]\to \mathbb{R} \) defined by \[ g(x)=\begin{cases} \frac{1}{m} & \text{for } x = \frac{1}{m} \text{ where } m \in \mathbb{N} \\ x & \text{otherwise} \end{cases} \] is Riemann integrable. If it is, calculate the value of \( \int_{0}^{1}g(x)\,dx \).

Answer: \dfrac{1}{2}


Question: In the context of category theory, consider objects $X$ and $Y$ in a category $D$, and let $X \times Y$ be their product with projections $\pi_1$ and $\pi_2$. If there exists a map $h$ from an object $Z$ to $X \times Y$ such that the diagram commutes, is $h$ necessarily a morphism in the category $D$? Provide a justification for your answer.

Answer: Yes


Question: Find a formula for \( S(n) \), which represents the number of ways to express the natural number \( n \) as a sum of natural numbers.

Answer: 2^{n-1}


Question: Given a deck of $n$ cards numbered $1$ to $n$, calculate the probability of forming the maximum possible number of piles when drawing cards according to a specified rule.

Answer: \dfrac{1}{n!}


Question: Determine whether the following statement is true or false: If a theory is inconsistent, then all statements in the language can be derived from that theory, making the theory necessarily complete.

Answer: A


Question: Find the maximum value of the function \( f(x,y,z) = (x-1)^2 + (y-1)^2 + (z-1)^2 \) subject to the constraint \( D(f) = \{(x,y,z): x^2+y^2+z^2 \leq 2, z \leq 1\} \).

Answer: 5 + 2\sqrt{6}


Question: Determine the number of equivalence relations on a set with 7 elements such that the equivalence relation contains exactly 29 ordered pairs.

Answer: 21


Question: Given a scalar random variable $y = z + v$, where the probability density function (pdf) of $v$ is $p_v(t) = \frac{t}{2}$ for $t \in [0, 2]$ and the pdf of $z$ is $p_z(t) = 2t$ for $t \in [0, 1]$, both zero elsewhere, compute the maximum a posteriori (MAP) estimate of $y$ given a measurement of $y = 2.5$. The pdf of $y$ is given as $\frac{y^3}{6}$ on the interval $[0, 2]$ and zero elsewhere.

Answer: 2


Question: Determine whether there exist three natural numbers $a$, $b$, and $z$ such that $a^2 + b^2 = 2z^2$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether an irreducible polynomial $f \in \mathbb{Q}[x]$ can have an irrational multiple root in $\mathbb{C}$. Provide a justification for your answer.

Answer: No


Question: Given a set of points \(X\) in \(\mathrm{R}^2\), let \(f(X)\) denote the geometric median of \(X\). If \(X\) and \(X'\) differ by exactly one point, determine the maximum possible value of \(|f(X) - f(X')|\).

Answer: \infty


Question: Find the maximum value of the function \( f(x, y, z) = 4x^{1/4}y^{1/4}z^{1/4} \) subject to the constraints \( x + y = 3 \) and \( y + z = 3 \) using the Lagrange multiplier method.

Answer: 4\sqrt{2}


Question: Find the last two digits of the number $7^{7^{2023}}$.

Answer: 43


Question: Evaluate the integral \[ I = \int_{0}^{2\pi} \frac{\cos^2(\theta) - \sin^2(\theta)}{\sin^4(\theta) + \cos^4(\theta)} \, d\theta. \]

Answer: 0


Question: Given a point $(X, Y)$ randomly selected from the unit disk with a uniform distribution, find the probability distribution of $R^2$, where $R$ denotes the distance of the point from the origin. The joint distribution of $X$ and $Y$ is $f_{X,Y}(x,y) = \frac{1}{\pi}$ for all $(x,y)$ in the disk $C = \{(x,y) \in \mathbb{R}: x^2 + y^2 \leq 1\}$. Determine the distribution of $R^2$.

Answer: 1


Question: Consider the continuous function $f:\Bbb{R}\to\Bbb{R}$ defined by $xf(x) = e^x-1$. Determine the limit \[\lim _{n\to \infty }nf^{\left(n\right)}\left(x\right)\] where $f^{\left(n\right)}\left(x\right)$ denotes the $n$th derivative of $f$. Provide your answer as a single value or expression.

Answer: e^x


Question: Determine whether there exist infinitely many positive integers $k$ such that $1093$ divides $3^k - 1$. If so, specify the conditions that $k$ must satisfy for $3^k - 1$ to be divisible by $1093$.

Answer: k \equiv 0 \pmod{7}


Question: Determine whether a countable collection of open intervals in \(\mathbb{R}\) can be rearranged to satisfy the following conditions: (i) The intervals are disjoint. (ii) There exists a positive minimum distance between all intervals. (iii) The total measure of the intervals remains finite and unchanged.

Answer: Yes


Question: Find a real, continuous function \( f \) that satisfies the functional equation \( f(ab+cd) = f(a) + f(b) + f(c) + f(d) \) for all real numbers \( a, b, c, \) and \( d \).

Answer: 0


Question:
Given the integral equation:

$$ y(x)= f(x) + \int_{0}^x \sin(x-t)y(t) dt $$

Which option correctly represents the solution for $y(x)$?

(a) $y(x) = f(x) + \int_{0}^x (x-t)f(t) dt$

(b) $y(x) = f(x) - \int_{0}^x (x-t)f(t) dt$

(c) $y(x) = f(x) - \int_{0}^x \cos(x-t)f(t) dt$

(d) $y(x) = f(x) - \int_{0}^x \sin(x-t)f(t) dt$

Answer: a


Question: Given the equation $x^2 - 3y^2 = 2023$ with integer solutions for $x$ and $y$, find the remainder when $x^2 + y^2$ is divided by 4.

Answer: 3


Question: Consider a Hilbert space $H$ with an orthonormal basis $\{b_k\}_{k \in \mathbb{N}_0}$. Let $T: H \to H$ be a bounded linear operator such that $T(b_k) = c_k b_k$ for some $c_k \in \mathbb{R}$. If $f: \mathbb{R} \to \mathbb{C}$ is a bounded continuous function, determine whether $f(T)$ is uniquely determined by its action on the basis vectors, i.e., $f(T)(b_k) = f(c_k)b_k$. Provide a justification for your answer.

Answer: Yes


Question:
Given the conditions:

(a) \( \lim\limits_{h\to 0}|f(x+h)-f(x-h)|=0 \)

(b) \( \lim\limits_{h\to 0}|f(x+h)+f(x-h)-2f(x)|=0 \)

for every \( x \in \mathbb{R} \), does it imply that the function \( f \) is differentiable?

Answer: No


Question: Let $g(x)$ be a function such that $g(x) = g\left(\frac{25}{x}\right)$ for all positive $x$. Given that \(\int_{1}^{5} \frac{g(x)}{x} \, dx = 3\), find the value of \(\int_{1}^{25} \frac{g(x)}{x} \, dx\).

Answer: 6


Question:
Evaluate the sum of the absolute squares of the coefficients \(c_n\) given by the integral
\[ c_n = \frac{1}{\sqrt{2\pi}} \int_{-\pi}^{\pi} e^{i(n-i)x} \, dx \]
for \(n \in \mathbb{Z}\). Which of the following is the correct result?

(a) \(\cosh(\pi)\)
(b) \(\sinh(\pi)\)
(c) \(\cosh(2\pi)\)
(d) \(\sinh(2\pi)\)

Answer: D


Question: Evaluate the limit: \[ \lim_{n \to \infty} n^2 \int_{n}^{5n}\frac{x^3}{1+x^6}dx. \]

Answer: \dfrac{12}{25}


Question: Given two random variables $X$ and $Y$ that are normally distributed with mean $0$ and variance $1$, and they are correlated with a correlation coefficient $\rho$ where $-1 \leq \rho \leq 1$, find the expected value of $X + 2Y$ conditioned on $X - Y = a$. Provide your answer in terms of $a$ and $\rho$. \( \mathbb{E}[X + 2Y | X - Y = a] = \boxed{?} \)

Answer: -\dfrac{a}{2}


Question:
Calculate the determinant of the matrix $3\,\text{adj}(2A^{-1})$ for the given $3\times 3$ matrix $A$:
$$A=\left[\begin{array}{ccc} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{array}\right].$$

Answer: 108


Question: Evaluate the sum \( \sum_{i=1}^{2021} \frac{z_i^3}{z_i+1} \), where \( z_1, z_2, \ldots, z_{2021} \) are the roots of the polynomial \( z^{2021} + z - 1 = 0 \).

Answer: 1347


Question: Calculate the sum of the determinants of all possible 9x9 matrices that can be formed using each of the numbers from 1 to 9 exactly once in each matrix.

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to 0}\frac{\sin^2(3x)-3x\tan(3x)}{(3x)^4}$$

Answer: -\dfrac{2}{3}


Question: A uniform container with a height of 6 meters is completely filled with water. It has three identical drainage pipes: pipe A at the base, pipe B at 5 meters above the base, and pipe C at an unknown height between A and B. Each pipe can individually empty the water above it in T minutes. When all three pipes work together, they can empty the entire container in \( \frac{2}{3}T \) minutes. Assuming no pressure variations, find the height of pipe C above the base.

Answer: 4


Question:
Find the minimum positive integer \( n \) such that there exist real numbers \( x_1, x_2, \ldots, x_n \) satisfying the following equations:
\[
\sum_{i=1}^{n}\sin x_i=0,
\]
\[
\sum_{i=1}^{n}i\cdot\sin x_i=2019.
\]

Answer: 90


Question: Determine the number of critical points where the function \( f(x) = \begin{cases} x^2 e^{-x^2-x^{-2}} & \text{if } x \neq 0, \\ 0 & \text{if } x = 0 \end{cases} \) has a local extremum.

Answer: 3


Question: Let $f(z)=\sum_{n=0}^{\infty}a_nz^n$ be a formal power series with radius of convergence $R(f)=1$. Define $g(z)=\sum_{n=0}^{\infty}s_nz^n$, where $s_n=a_0+a_1+\cdots+a_n$. If $h(z)=\sum_{n=0}^{\infty}\frac{s_n}{n+1}z^n$, determine the radius of convergence of $h(z)$.

Answer: 1


Question:
Calculate the determinant of the matrix:
\[ M = \begin{bmatrix} \sin^2 A & \cot A & 1 \\
\sin^2 B & \cot B & 1 \\
\sin^2 C & \cot C & 1 \end{bmatrix}, \]
where $A$, $B$, and $C$ are the angles of a triangle.

Answer: 0


Question: In triangle $ABC$ with sides $AB = \sqrt{17}$, $BC = 5$, and $CA = 4\sqrt{2}$, points $D, E, F$ lie on lines $AB, BC$, and $CA$, respectively, forming triangle $DEF$ inside $ABC$. If the minimum perimeter of triangle $DEF$ can be expressed as $\left(\frac{q}{p}\right)\sqrt{34}$, where $p$ and $q$ are coprime positive integers, find the value of $p + q$. \(\boxed{}\)

Answer: 37


Question: Consider linear transformations $A$ and $B$ on a finite-dimensional vector space $V$, where $A \neq 0$, $B \neq 0$, $A^2 = B^2 = 0$, and the dimensions of their ranges are equal. Are $A$ and $B$ necessarily similar?

Answer: Yes


Question: Given a function $f$ defined for natural numbers $x, y \geq 4$ such that $f(x+y) = f(xy)$ for all $x, y$, and $f(8) = 9$, determine the value of $f(n)$ for all $n \geq 8$. Provide a rigorous proof for your conclusion.

Answer: 9


Question: Determine whether every infinite planar zigzag path, starting at \((0,0)\) and alternating between moving 1 unit up and 1 unit right, contains at least \(n\) collinear lattice points for any given positive integer \(n\).

Answer: Yes


Question: Given a process with output $XY^3$, where $X$ and $Y$ are mutually dependent, $Y$ follows a uniform distribution $U(0,2)$, and conditional on $Y=y$, $X$ has an exponential distribution with mean $y$. Calculate the expected output of the process using the law of total expectation.

Answer: \dfrac{16}{5}


Question: Consider a disconnected topological space with a finite number of connected components. Can this space be partitioned into a finite number of proper open subsets?

Answer: Yes


Question: Determine the order of the largest cyclic subgroup in the automorphism group of $\mathbb{Z}_{720}$, denoted as $\text{Aut}(\mathbb{Z}_{720})$. Provide your answer as a single integer.

Answer: 12


Question: Find the value of $t$ for which the tangent line to the curve $\vec{r}(t) = \langle t^3 + 3t, t^2 + 1, \ln(1 + 2t) \rangle$, for $0 \leq t \leq \pi$, is orthogonal to the plane $15x + 4y + 0.4z = 10$. The derivative of $\vec{r}(t)$ is $\vec{r}'(t) = (3t^2 + 3)\vec{i} + 2t\vec{j} + \left(\frac{2}{1 + 2t}\right)\vec{k}$ and the normal vector to the plane is $\vec{n} = \langle 15, 4, 0.4 \rangle$. Determine the value of $t$.

Answer: 2


Question: Calculate the variational derivative of the expression $-f\nabla^2f$ with respect to $f$. Express your answer as $\frac{\delta(-f\nabla^2f)}{\delta f}$. What is the result?

Answer: -2\nabla^2 f


Question: What is the probability of selecting the same real number more than once when an infinite number of random real numbers are chosen from a continuous probability distribution?

Answer: 0


Question: Suppose $Y$ and $Z$ are $n \times n$ matrices, and there exists a unitary matrix $X$ such that $X^* Y X$ and $X^* Z X$ are upper triangular. Determine the eigenvalues of the matrix $YZ - ZY$. \( \text{Provide your answer in the form of a set of eigenvalues.} \)

Answer: \{0\}


Question:
Calculate the limit:
\[\lim_{x\to 0} \frac{f(x^2) - f(0)}{\sin^2(x)}\]
where \(f(x)\) is a differentiable function with \(f'(0) = 1\).

Answer: 1


Question: Evaluate the integral \( \int_{C}\cos\left(\frac{z}{2}\right)\,dz \) where the curve \( C \) is parameterized by \( \gamma(t) = t + i\sqrt{\pi^2 - t^2} \) for \( -\pi \leq t \leq \pi \).

Answer: 4


Question: A pitcher contains $X + 1$ blue balls and $Y + 1$ red balls. $X$ and $Y$ are independent random variables with $X \sim \mathrm{Poisson}(n)$ and $Y \sim \mathrm{Poisson}(3n)$. Determine the limit: \[ \lim_{n\to \infty}\mathbb{P} (X = 2Y). \]

Answer: 0


Question: Evaluate the expression: \( \sin\dfrac{2\pi}{11}-\sin\dfrac{\pi}{11}-\sin\dfrac{3\pi}{11} \)

Answer: -\dfrac{1}{2}


Question: Given that $y_1 = \sin(t^2)$ is a solution to the differential equation $t y'' - y' + 4t^3 y = 0$, use the method of reduction of order to find a second linearly independent solution.

Answer: \cos(t^2)


Question: Evaluate the limit \( \lim_{x\to 0} (\sec x)^x \) without using L'Hôpital's rule or the continuity of \( e^x \).

Answer: 1


Question: Let $X$ be a positive random variable on $[0, \infty)$ with finite expected value, $\mathbb{E}(X) < \infty$. Prove or disprove that $\lim_{N\to\infty}\mathbb{E}(X\,\mathbb{I}_{\{X>N\}}) = 0$. \(\mathbb{I}_{\{X>N\}}\) is the indicator function that is 1 if \(X > N\) and 0 otherwise.

Answer: 0


Question: Find the value of $x$ in terms of $a$, $b$, and $c$ if the following equation holds: $$\frac{x-a^2}{b+c} + \frac{x-b^2}{c+a} + \frac{x-c^2}{a+b} = 4(a+b+c).$$

Answer: (a + b + c)^2


Question: Determine whether the following statement is true for symmetric positive definite matrices of the same size $A_1, A_2, \ldots, A_m$: $$ Tr(\prod_{i=1}^m A_i) \leq \prod_{i=1}^m Tr(A_i)$$

Answer: True


Question: Given a parallelepiped with volume \( V_1 \) determined by vectors \( \mathbf{u}, \mathbf{v}, \mathbf{w} \), find the volume of a second parallelepiped determined by vectors \( \mathbf{u} + a\mathbf{v}, \mathbf{v} + b\mathbf{w}, \mathbf{w} + c\mathbf{u} \) in terms of \( V_1, a, b, \) and \( c \).

Answer: |1 + abc| V_1


Question: Find the 100th derivative of the function \( g(x) = (x^3 -4x + 7)^{30} \) without computing all the intermediate derivatives.

Answer: 0


Question: Determine whether the following statement is true or false: If \( \sum_{i=0}^\infty a_{i_n} y^i \rightarrow \sum_{i=0}^\infty a_{i} y^i \) for all \( y \in [0,1] \), does it imply that \( a_{i_n} \to_{n \to \infty} a_{i} \) for all \( i \), given that \( 0 \leq a_i,a_{i_n} \leq 1 \) for all \( i,n \in \mathbb{N} \) and \( \sum_{i=0}^\infty a_i = \sum_{i=0}^\infty a_{i_n} = 1 \) for all \( n \in \mathbb{N} \)?

Answer: True


Question:
Find the value of $f(0)$ for the polynomial $f(x)$ of least degree with a leading coefficient of 1, given that $f(1)=5$, $f(2)=4$, $f(3)=3$, $f(4)=2$, and $f(5)=1$.

Express your answer as a number in a box: \( \boxed{\text{your answer}} \)

Answer: -114


Question: Find the equation of the locus of the foot of the perpendicular drawn from the origin to a line that makes intercepts on the coordinate axes, given that the square of the length of the perpendicular is equal to the product of the lengths of the intercepts.

Answer: x^2 + y^2 - xy = 0


Question: Determine whether it is possible to construct a sequence of simple measurable functions $s_n(x)$ such that $s_n(x) \geq 0$ and $s_n(x) \nearrow |f(x)|$ for an $L^1$-integrable function $f$, even if $|f(x)|$ is unbounded.

Answer: Yes


Question: Determine whether the following statement is true or false: For any morphism of schemes $f: X \to Y$ and any open subset $U$ of $X$, if $f$ is quasi-compact and $U$ contains the pre-image of all generizations of some $y \in Y$, then there exists an open subset $V$ of $Y$ containing $y$ such that $f^{-1}(V) \subset U$. Provide a justification for your answer.

Answer: True


Question: Determine whether the following statement is true or false: If $X_i$ are independent and identically distributed random variables with $X_i \sim U(0,1)$, then $P(\sum_{i=1}^\infty X_i < \infty) = 0$. Provide a justification for your answer.

Answer: True


Question: Find the values of $q$ for which the curve $y = x^p \sin \frac{\pi}{x}$, where $0 < x \leq 1$, has finite length.

Answer: q > 1


Question: Let $M$ be an oriented, compact, connected $n$-dimensional smooth manifold with boundary. Determine whether the $n$-th singular homology of $M$, denoted by $H_n(M)$, is always zero. Provide a justification for your answer.

Answer: 0


Question: Given two lists, $L_1$ and $L_2$, where $L_1$ contains the dot products of 100 vectors $x_1, x_2, ..., x_{100}$ without self-pairs, and $L_2$ is $L_1$ with all signs flipped, determine which list, $L_1$ or $L_2$, consists of the correct dot product pairs.

Answer: L_1


Question:
Let $M$ be the number of $2023$-digit numbers such that the leading digit is $3$ and there are an even number of $7$'s. Find the remainder when $M$ is divided by $1000$.

Calculate the remainder of $M$ modulo 1000.

Answer: 232


Question: In triangle \(\triangle ABC\), point \(I\) is the incenter. The areas of \(\triangle IBC\), \(\triangle ICA\), and \(\triangle IAB\) are 28, 30, and 26 square units, respectively. Determine the value of \(AC^2 - AB^2\).

Answer: 56


Question: Express the matrix \( \begin{pmatrix}1 & 1 \\ 0 & 1\end{pmatrix} \) as a product of the generators \( a, b, a^{-1}, b^{-1} \) of the group \( SL(2,\mathbb{Z}) \), where \( a = \begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix} \) and \( b = \begin{pmatrix}1 & 1 \\ -1 & 0\end{pmatrix} \).

Answer: a b^{-1}


Question: Determine whether it is possible to find a $G_\delta$ dense subset of an open interval $I$ in $\mathbb{R}$. Provide a justification for your answer.

Answer: Yes


Question: Given that \( x^{x^{x^{16}}} = 16 \), find the value of \( x^{x^{x^{12}}} \).

Answer: 2


Question: Compute the integral \( \int_{|z|=2}\frac{f'(z)}{f(z)}dz \), where \( f(z) = 1 + \frac{1}{z}+\frac{1}{2!z^2}+\dots+\frac{1}{n!z^n} \). Given that \( f \) has no zeros with \( |z|\geq 2 \), apply the Argument Principle to determine the value of the integral.

Answer: 0


Question: Calculate the volume of the region enclosed by the planes $z=0$, $y=-1$, $y=1$, and the parabolic cylinder $z=1-x^2$. Provide your answer as a single numerical value.

Answer: \dfrac{8}{3}


Question: Evaluate the integral \( \int_{0}^{2\pi} e^{\sin\theta} \sin(\cos\theta) \, d\theta \).

Answer: 0


Question: Let $f$ be a nonnegative continuous decreasing function on $[0,\infty)$. Suppose $\frac{f(x)}{\sqrt{x}}$ is integrable. Prove that $\sqrt{x}f(x) \to 0$ as $x\to \infty$.

Answer: 0


Question: Determine whether the following statement is true or false: If the derivative of a function approaches zero as \( x \) approaches infinity, then the ratio of the function to \( x \) also approaches zero as \( x \) approaches infinity.

Answer: True


Question: Evaluate the limit: $$\lim_{n\to\infty}\sqrt[n]{1^{\pi}+2^{\pi}+\cdots+n^{\pi}}$$

Answer: 1


Question: Determine if the function defined by the series \( \sum_{n=1}^{\infty}\frac{e^{-\sqrt{n}x^{2}}}{n^{2}} \) is differentiable at \( x = 0 \).

Answer: Yes


Question: Calculate the Euler characteristic of a surface formed by joining two projective planes, each with two punctures, along the punctures.

Answer: -2


Question: Determine the number of group homomorphisms from $\mathbb{Z}_6$ to $S_3$.

Answer: 6


Question: Determine whether the condition \([1-x_n]^{n}\to 1\) implies that \(n x_n\to 0\) as \(n\to\infty\).

Answer: Yes


Question: Determine the limit of the expression \( \frac{f(x)-T_{n, x_0}f(x)}{(x-x_0)^n} \) as \( x \) approaches \( x_0 \).

Answer: 0


Question: Find the limit of the sequence \( x_n = \frac{a^n}{(1+a)(1+a^2)\cdots(1+a^n)} \) as \( n \to \infty \), where \( a > 0 \).

Answer: 0


Question: Find the limit of the expression \( \frac{\sin(2\pi x) - 2\pi x}{2\sin^2(\pi x)} \) as \( x \to 0 \) using Taylor series expansion.

Answer: 0


Question: Suppose $X$ is a compact topological space and $A \subset C(X)$ is a linear subspace of the space of continuous functions on $X$. Assume that for any $x, y \in X$ with $x \neq y$ and any $a, b \in \mathbb{R}$, there exists a function $f \in A$ such that $f(x) = a$ and $f(y) = b$. Additionally, assume that if $g \in A$, then $g^3, g^4, g^5, \ldots \in A$. Determine whether it is true that $g^2 \in A$ for all $g \in A$. Provide a justification for your answer.

Answer: Yes


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

Answer: 0


Question: Find the area enclosed between the graphs of the inverse functions of \( f(x) = \frac{1}{2} x(x+1) \) and \( g(x) = \frac{1}{2} x^2(x+1) \) for \( x \in [0,1] \).

Answer: \dfrac{1}{8}


Question: Determine the number of solutions to the equation \( x^{10} = \begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix} \) in the set of 2x2 real matrices \( M_{2}(\mathbb{R}) \). Provide a brief explanation for your answer.

Answer: \infty


Question: Determine the number of homomorphisms from the dihedral group $D_5$ to the symmetric group $S_3$. Provide your answer as a single number.

Answer: 4


Question: Let $f_n$ be a sequence of bounded measurable functions on a probability space. Suppose that for any $\epsilon > 0$, there exists no set $E$ of positive measure such that $\lim\sup\limits_{n\to\infty}|f_n(x)|>\epsilon$ for all $x\in E$. Does this imply that $f_n$ converges to $0$ almost everywhere?

Answer: Yes


Question: An insurance policy covers two employees of a company, reimbursing no more than one loss per employee per year. The policy reimburses the full amount of the loss up to a company-wide maximum of $8000. The probability of an employee incurring a loss is 40%, independent of the other employee. Each loss amount is uniformly distributed between $1000 and $5000. Given that one employee has incurred a loss exceeding $2000, calculate the probability that the total losses will exceed the reimbursements.

Answer: \dfrac{2}{51}


Question: Consider a set of vectors $\{ x_i \}_{i=1}^{m}$ in $\mathbb{R}^d$ that span a subspace $S$ of dimension $n$. Let $\mathcal{S}$ be the set of matrices $A \in \mathbb{R}^{d\times d}$ that map each vector $x_i$ to itself. Determine the cardinality of $\mathcal{S}$ in terms of $n$ and $d$. Express your answer as a function of $n$ and $d$.

Answer: d(d - n)


Question: Calculate the sum of the infinite series: \[ \frac{3}{0!}+ \frac{7}{1!}+ \frac{13}{2!} + \frac{21}{3!}+\cdots \]

Answer: 8e


Question: Evaluate the limit: $$\lim_{x\to 0} \frac{a^x\sin(bx)-b^x\sin(ax)}{\tan(bx)-\tan(ax)}$$ where $a \neq b$.

Answer: 1


Question: Evaluate the limit \( \lim_{R \to \infty} \int_{-R}^{R} \int_{-R}^{R} e^{-(x^2+y^2)} \, dx \, dy \) using an approximation for the error function.

Answer: \pi


Question: Given the functional equation \( f(f(x)) = x^2 - 5x + 9 \), determine the value of \( f(3) \).

Answer: 3


Question: Determine the values of \(\alpha\) for which the improper integral \(\iint_{\Bbb R^2}\frac{dxdy}{(1+x^2+xy+y^2)^\alpha}\) converges.

Answer: \alpha > 1


Question: Determine the order of the poles for the function \( f(z) = \frac{3z - 1}{z^2 + 2z + 5} \) at the singularities \( z = -1 + 2i \) and \( z = -1 - 2i \).

Answer: 1


Question: Determine whether the following statement is true or false: "If a function $f$ is continuously differentiable on $\mathbb{R}$ and $f(x) \rightarrow 1$ and $f'(x) \rightarrow b$ as $x \rightarrow \infty$, then the value of $b$ must be $1$." Provide a proof or a counterexample.

Answer: B


Question:
Determine if the dimension of the vector space $W = \operatorname{span}(w_1,w_2,w_3,w_4)$ can be uniquely determined, given that any three of the vectors $v_1,v_2,v_3,v_4$ are linearly independent over $\mathbb{C}$, and the vectors $w_1, w_2, w_3, w_4$ are defined as follows:

\begin{align*}
w_1 &= iv_1 - v_2,\\
w_2 &= v_1 - iv_4,\\
w_3 &= iv_2 - v_3,\\
w_4 &= v_2 + iv_3 - 2v_4.
\end{align*}

Answer: 4


Question: Given a sequence of real numbers \(\{a_n\}\), where \(A_N = \sum_{n=1}^N a_n\) represents the \(N\)th partial sum, and the sequence \(\{A_N\}\) of partial sums is positive but oscillates and does not converge. Moreover, the limit \(\lim_{K \to \infty} \frac{1}{K+1} \sum_{k=1}^K A_k = L\) exists with \(L < \infty\). Is the sequence of partial sums \(\{A_N\}\) necessarily bounded? Answer yes or no.

Answer: No


Question: Determine whether the following mathematical identity is true for all positive integers $a, b, c,$ and $n$, where $n$ is a positive integer divisor of both $a^b$ and $a^c$: \[((a^b \mod n) \cdot (a^c \mod n)) \mod n = a^{b+c} \mod n.\]

Answer: True


Question: Let $X$ be an integral scheme with generic point $\eta$ and $U \subset X$ an open subscheme. If the restriction of structure sheaves $\mathcal{O}_X(X) \xrightarrow{\sim} \mathcal{O}_X(U)$ is an isomorphism, does it imply that $U = X$? Answer yes or no and justify your answer.

Answer: No


Question: Let $Z$ be the subset of the binary space $B$ consisting of all points that are eventually $0$. Describe the set of all limit points of $Z$ in the metric space $(B,d)$, where $d$ is defined as follows: for $x = (x_1, x_2, x_3, \ldots)$ and $y = (y_1, y_2, y_3, \ldots)$ in $B$, $d(x, y) = 0$ if $x = y$, and $d(x, y) = 1/n$ if $x \neq y$ and $n = \min\{i \mid x_i \neq y_i\}$. Provide a detailed description of the limit points.

Answer: B


Question: Calculate \( \lim_{n \rightarrow \infty} \int_0^\infty \frac{x^n}{ x^{(n+3)}+1} \, dx \).

Answer: \dfrac{1}{2}


Question: Given complex numbers $z_1 = a_1$, $z_2 = a_2 e^{2\pi i/3}$, and $z_3 = a_3 e^{-2\pi i/3}$ with $|a_1| \leq A$, $|a_2| \leq A$, and $|a_3| \leq A$, determine the maximum value of $|z_1 + z_2 + z_3|$, where $A$ is a positive real number.

Answer: 3A


Question: Evaluate the limit: $$\lim \limits_{x \to 0^+} \big(1+x^3 \ln x\big)^{\frac{1}{\sin x}}$$

Answer: 1


Question: Determine whether the following statement is true or false: In an additive category, an object is a zero object if and only if the diagonal morphism is an isomorphism.

Answer: True


Question: Let $f$ be a continuous function on $[0,1]$ with $f(0)=1$. Define $H(a)=\frac{1}{a} \int_{0}^{a}xf(x)dx$. Find $\lim_{a\to 0} H(a)$. \(\boxed{\text{Answer}}\)

Answer: 0


Question: Calculate the volume of the solid generated by revolving the region under the first arc of the cycloid, given by the parametric equations \(x(t) = a(t - \sin t)\) and \(y(t) = a(1 - \cos t)\), about the \(x\)-axis. Use the formula \(V = \pi \int [f(y)]^2 \, dy\) to find the volume.

Answer: 5\pi^2 a^3


Question: Find all integer solutions \((a, b, c)\) such that \(3^a + 5^b = 7^c\).

Answer: \emptyset


Question: Determine the real values of $a$ such that the equation $x^3 - 3x + a = 0$ has three distinct real roots.

Answer: (-2, 2)


Question: Determine whether an algebra with finitary operations can be infinite if it is generated by a finite set of generators. Provide a justification for your answer.

Answer: Yes


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

Answer: 1


Question: Determine whether the series \( \sum_{n=1}^\infty \frac{2^n+n^2+n}{2^{n+1}n(n+1)} \) converges. If it converges, find its limit as \( n \to \infty \).

Answer: 1


Question: Given that \( A(\text{Adj}A) = I \det A \), calculate the determinant of matrix \( A \) if the determinant of its adjugate is 4. Why is the answer \(-2\)?

Answer: -2


Question: Calculate the expected number of arrivals within an interval of size $u$ for a point process where the time between successive events, $T$, has a first moment $E(T) = t$. Assume the start of the observation window, $J$, is chosen uniformly at random from $(0, m)$, with $m \to \infty$. Provide your answer in terms of $u$ and $t$.

Answer: \dfrac{u}{t}


Question: How many distinct arrangements of the letters in the word "MATHEMATICS" exist if both 'T's appear before both 'A's, both 'A's appear before both 'M's, or both 'M's appear before the 'E'?

Answer: 2835360


Question:
Calculate the probability that, out of 10 points uniformly distributed within the interval $(0, T)$, exactly $m$ points lie within another interval $(0, X)$, where $X$ is a random variable uniformly distributed over $(0, T)$. Express your answer in terms of $m$ and $T$.

\( \text{Probability} = \boxed{\text{[Your Answer]}} \)

Answer: \dfrac{1}{11}


Question: Determine if there exists a positive integer $n$ such that no number in the set \( \{n, n+1, n+2, \ldots, n+2017\} \) can be expressed as the sum of two squares of integers.

Answer: Yes


Question: Is the product space \( \prod\limits_{n \in \mathbb{N}} A_n \) compact when each \( A_n = \{0, 1\} \) is given the discrete topology? Justify your answer using Tychonoff's theorem.

Answer: Yes


Question:
Calculate the limit \( \lim_{R\rightarrow \infty}I_R \), where \( I_R \) is defined as the double integral
\[ I_R=\int\int_{D_R}\frac{x^2\cos y}{x^2+y^2+1}dxdy \]
with \( D_R \) given by
\[ D_R=\{(r\cos \theta, r\sin \theta):0\leq r\leq R,0\leq \theta\leq \frac{\pi}{R^2} \} \]
Determine if the limit is 0 or, if not, find the correct limit and its relation to \( \pi \).

Answer: \dfrac{\pi}{2}


Question: Given the equation \(x^4 - x^3 - 1 = 0\) with roots \(\alpha, \beta, \gamma, \delta\), use the substitution \(y = x^3\) to find the exact value of \(\alpha^6 + \beta^6 + \gamma^6 + \delta^6\).

Answer: 7


Question: Find the residue of the function \( f(z) = \frac{z}{1-\cos(z)} \) at \( z=2\pi k \), where \( k \in \mathbb{Z} \).

Answer: 2


Question: Let \( f \) be a continuous function on the interval \([0,1]\). Evaluate the limit: \[ \lim_{n\to \infty} (n+1) \int_0^1 x^n f(x) \, \mathrm{d}x \]

Answer: f(1)


Question: Evaluate whether \( \lim\limits_{n\rightarrow \infty} ((1-p)^{n^{4/3}}) = 0 \), given that \( p \in (0,1) \) is a function of \( n \) such that \( p = f(n) \) and \( pn \geq m \) for all \( n \in \mathbb{N} \) with \( m > 0 \).

Answer: 0


Question:
Evaluate the value of the following binomial sum:

$$\sum_{r=0}^{k}(-1)^r2^{k-r}\binom{20}{r}\binom{20-r}{20-k}$$

Choose the correct answer from the options below:
(a) 7
(b) 8
(c) 10
(d) 20

Answer: D


Question:
Consider a sequence $c_n$ defined as follows: $c_1 = c_2 = 1$, and $c_{n+1}$ is randomly chosen from the interval $[c_n, c_{n-1}]$ with a uniform distribution. Determine the value of the limit:

$$\lim_{n\to\infty}\sqrt[n]{\frac{c_{n+1}}{c_1}}$$ if it exists.

Answer: 1


Question: Determine the conditions under which the field extension \( \mathbb{C}(X)/ \mathbb{C}(Y) \) is a Galois extension, where \( Y = X^3 + 3X^2 + aX \) and \( a \in \mathbb{C} \).

Answer: a = 3


Question: A fair six-sided die is tossed continuously, with two faces each bearing brown, black, and white colors. Let the random variable $Y$ denote the number of tosses required to observe all three colors. Specifically, $Y$ takes on the value $k$ if exactly two colors appear in the first $k-1$ tosses and all three colors appear in the first $k$ tosses. What is the expected value of $Y$, denoted as $E[Y]$? Use the formula $E[Y]=\sum_{k=0}^{\infty} P(Y>k)$ to compute this value, considering the probability of not observing all colors in the first $k$ tosses.

Answer: \dfrac{11}{2}


Question: Evaluate the limit \[ L = \lim_{n\rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} (-1)^k f\left(\frac{k}{n}\right) \] where $f$ is a continuous function on the closed interval $[0,1]$. Consider how the alternating term $(-1)^k$ affects the Riemann sum.

Answer: 0


Question: Eliminate \(\alpha\) from the following equations: \(\cos(\theta-\alpha)=\lambda\sin2\theta\) and \(\sin(\theta-\alpha)=\lambda\cos2\theta\).

Answer: \lambda^2 = 1


Question: Determine whether there exists a subset of the rational numbers that is both dense in the real numbers and has an infinite complement within the rational numbers.

Answer: Yes


Question:
Calculate the determinant of the matrix:
\[\begin{vmatrix} 1^k & 2^k & 3^k & \cdots & n^k \\
2^k & 3^k & 4^k & \cdots & (n+1)^k\\
3^k & 4^k & 5^k & \cdots & (n+2)^k\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
n^k & (n+1)^k & (n+2)^k & \cdots & (2n-1)^k \end{vmatrix}\]
where \(2 \leq n\) and \(0 \leq k \leq n-2\). Determine the value of the determinant.

Answer: 0


Question: Determine the number of endomorphisms on the direct sum of the groups \( \mathbb{Z}_2 \) and \( \mathbb{Z}_4 \).

Answer: 32


Question: Find the angle between the tangent lines to the curves \(y = x^2 + 1\) and \(x^2 + y^2 = 1\) at their intersection point.

Answer: 0


Question: Provide an example of a Young function \( \Phi: \mathbb{R} \to \mathbb{R}^+ \) that is convex, satisfies \( \Phi(0) = 0 \), \( \Phi(-x) = \Phi(x) \), and \( \lim_{x \to \infty} \Phi(x) = +\infty \), but does not satisfy the \( \Delta_2 \) condition, which is \( \Phi(2x) \leq K\Phi(x) \) for some \( x \geq x_0 \geq 0 \) and a constant \( K > 0 \).

Answer: e^{|x|} - 1


Question: Given a $4 \times 6$ matrix $A$, consider the set $\mathcal{B} = \left\{\begin{pmatrix} 1 \\ -1 \\ 2 \\ 3 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ -1 \\ 1 \\ 1 \\ -2 \end{pmatrix}\right\}$, which forms a basis for the null space of $A$. Let $\vec b \in \operatorname{Col}A$ and $\vec x = \begin{pmatrix} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 2 \end{pmatrix}$ be a particular solution to $\operatorname{proj}_{\operatorname{Col}A}\vec b = \begin{pmatrix} 1 \\ 2 \\ -1 \\ 1 \end{pmatrix}$. Determine the sum of the components of $\vec b$. \( \boxed{?} \)

Answer: 3


Question: Find the number of elements in the automorphism group of \( \mathbb{Z}_{p^{n}} \) for an odd prime \( p \) and \( n \geq 2 \).

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


Question: Let $f$ be an entire function such that $f(z^2) = f(z)^2$ for all $z \in \mathbb{C}$ and $f(0) = 1$. Prove that $f$ is a constant function.

Answer: 1


Question: Evaluate the improper integral \( \int_{0}^{\infty} \frac{\sin{x}}{x}\,dx \).

Answer: \dfrac{\pi}{2}


Question: Evaluate the limit \( \lim_{n \to \infty} \int_{0}^{\pi} \frac{e^x \cos(nx)}{1+e^{2x}} \, dx \) using a geometric series and Euler's formula.

Answer: 0


Question: Evaluate the limit of the sum \( \sum_{n=1}^\infty \frac{\cos(n)}{n^k} \) as \( k \to 0^+ \) using the polylogarithm function.

Answer: -\dfrac{1}{2}


Question: Determine whether the following property of the cosine integral function \(\text{Ci}(z)\) is valid for all \(z\) with positive real part: \(\pi - i\text{Ci}(2\pi-iz) + i\text{Ci}(-2\pi+iz) = 0\), where \(\text{Ci}(z) = -\int_z^\infty \frac{\cos(t)}{t} dt\).

Answer: 0


Question:
Determine whether the following statement is true:

$$\frac{1}{(2\pi)^n} \lim_{\epsilon \rightarrow 0} \sum_{z\in \mathbb{Z}^n} \int_{\mathbb{R}^n} f\left( \frac{x}{\epsilon} \right) e^{iz (x-a\epsilon)} dx = f(a)$$

where $f \in S(\mathbb{R}^n)$. Justify your answer.

Answer: True


Question: Evaluate the infinite series \( \sum_{n=1}^\infty \frac{(2n+3)^2}{4^n} \).

Answer: \dfrac{305}{27}


Question: Given two $n \times n$ matrices $A$ and $B$ with $\operatorname{rank}(A) = n$ and $\operatorname{rank}(B) = n-1$, determine if $\operatorname{rank}(AB) = \operatorname{rank}(BA) = n-1$.

Answer: Yes


Question: Let $X$ be a Polish space and $(\mu_n)_{n \geq 1}$ be a sequence of Borel probability measures on $X$. Assume that for every continuous bounded function $f : X \to \mathbb{R}$, the limit $L(f) := \lim_{n \to \infty} \int_X f \, d \mu_n$ exists. Does there exist a Borel probability measure $\mu$ such that $L(f) = \int_X f \, d \mu$ for all such functions $f$?

Answer: Yes


Question: Calculate the line integral \( \oint_L p \, dx + q \, dy \) for the vector field \( F = [p(x, y), q(x, y)] = [y^3 + e^{x^2}, x^3 + \tan^2 y] \), where \( L \) is the positively oriented circle defined by \( x^2 + y^2 + 4y = 0 \). Use Green's theorem to perform the calculation.

Answer: -48\pi


Question: Given a differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $x^2f'(x) \to 0$ as $x \to \infty$, determine whether it implies that $f(x) \to c \in \mathbb{R}$ as $x \to \infty$. Provide a justification for your answer.

Answer: Yes


Question: In an equilateral triangle $ABC$, three lines are drawn perpendicular to each side, intersecting within the triangle. Blue points are placed such that any line connecting two blue points is parallel to one of the sides of the triangle. Each perpendicular line divides the parallel lines it intersects into three segments, with each segment containing at most one blue point. What is the maximum number of blue points that can be placed within the triangle?

Answer: 9


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

Answer: -\dfrac{7}{45}


Question: Determine the cardinality of the set \( S = \{ (x, y, z) \mid x^2 + y^2 + z^2 = 2^{2018}, xyz \in \mathbb{Z} \} \).

Answer: 6


Question: Evaluate the limit: $$\lim _{x\to \infty \,}\left(\frac{\left(\cos ^2\left(x^3\right)+x\right)}{x!}\right)\,$$ as \(x\) approaches infinity.

Answer: 0


Question: Evaluate the integral: \[ \int_{\mathbb{R}^2} e^{-(4x^2 + 4xy + 5y^2)} \, d(x,y) \]

Answer: \dfrac{\pi}{4}


Question: Let $G$ be a finite abelian group with elements $a_1, a_2, \dots, a_n$. Prove that if $G$ has more than one element of order 2, then $a_1 a_2 \dots a_n = 1$. \(\text{(where 1 is the identity element of the group)}\)

Answer: 1


Question: Calculate the work done by the vector field \( \vec{G}=(x^2y+z^2)\vec{i}+(xy^2-z)\vec{j}+(yz+x)\vec{k} \) in moving a particle along the curve formed by the intersection of the plane \( x+y+z=1 \) and the cylinder \( x^2+y^2=4 \).

Answer: 12\pi


Question: Simplify the expression \(x = (10 + \sqrt{108})^{1/3} - (-10 + \sqrt{108})^{1/3}\) to show that \(x = 2\).

Answer: 2


Question: Determine whether the following statement is true or false: If a function $f: [a,b] \to \mathbb{R}$ is such that for a partition $a = c_0 < c_1 < \cdots < c_m = b$, the restrictions of $f$ to each subinterval $[c_{i-1},c_i]$ are Riemann integrable, then $f$ is Riemann integrable on $[a,b]$. Provide a justification for your answer.

Answer: A


Question: Consider a rectangle tiled with dominos. Is it always possible to transform this tiling into any other domino tiling of the same rectangle by repeatedly performing the following move: pick two dominos that share a long edge and rotate them in place by 90 degrees?

Answer: Yes


Question: Suppose $f:E\rightarrow E'$ is a linear embedding between two locally convex topological vector spaces. Given an open, convex subset $A\subseteq E$ with $0\in A$, determine if there exists an open, convex set $A'\subseteq E'$ such that $f(A)=f(E)\cap A'$.

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{2\pi} e^{-\sin\theta} \cos(\theta-\cos\theta) \, d\theta \).

Answer: 0


Question: Find the function $f(x)$ that satisfies the following conditions: \( \int_0^1 f(x) \, dx = 3 \) and \( \int_0^1 x f(x) \, dx + 1 = 3 \), and minimizes the value of \( \int_0^1 f^2(x) \, dx \).

Answer: 6x


Question: Find the number of different sets of 3 regular polygons that can be placed around a point such that their interior angles add up to 360 degrees. This is equivalent to finding the number of integer solutions for (a, b, c) that satisfy the equation \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{2} \).

Answer: 10


Question: Given a linear transformation \( T: V \to V \) that is a projection (i.e., \( T^2 = T \)), find the expression for \( e^T \).

Answer: I + (e - 1)T


Question: Evaluate the limit: \[ \lim_{x \to 0} \left( 1 + \tan 3x \right) ^{\dfrac{1}{\sin x}} \]

Answer: e^3


Question: Determine whether the limit \( \lim_{(x,y)\to(0,0)} \frac{3x^2y}{x^2+y^2} \) exists.

Answer: 0


Question: Evaluate the limit: $$ \lim_{n \to \infty} \frac{1}{n}\sum_{k=1}^{n} \cos\left(\sin\left(\frac{1}{k}\right)\right) $$

Answer: 1


Question: Determine the values of \( n \) for which the last digit of \( 2^{2^{n-1}(2^n+1)} - 1 \) is 5.

Answer: n \geq 3


Question: Evaluate the limit: $$\lim_{x \to \frac{\pi}{2}} \left| \frac{1}{(1+\sin x)\ln(\sin x)} + \frac{1}{\cos^2x} \right| $$

Answer: \dfrac{1}{4}


Question: Find the number of solutions to the equation $\sin(x) = \frac{x}{2}$ using Rolle's theorem.

Answer: 3


Question: Identify the only multiplicative functional on the algebra $M_n(\mathbb{C})$ for $n > 1$.

Answer: 0


Question: Let $f$ be a continuous function on $[0,1]$ such that \( \int_a^b f(x)\,dx =0 \) for all $a$, $b$ belonging to $[0,1]$. Determine the nature of $f$ on the interval $[0,1]$.

Answer: 0


Question: In a complete graph $G$ with 25 vertices, each edge is colored either red or blue. Given that there are no red triangles in $G$, determine the largest integer $m$ such that a blue complete subgraph $K_m$ is guaranteed to exist. Use Ramsey theory to find your answer.

Answer: 7


Question: Determine the maximum number of triangles that can be formed using up to 25 nodes and up to 40 edges, allowing for overlapping triangles.

Answer: 90


Question: Let $X, Y, Z$ be metric spaces with $Y$ compact. Suppose $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are functions such that $g$ is a continuous bijection. If the composition $g \circ f$ is uniformly continuous, does it imply that $f$ is also uniformly continuous? Provide a justification for your answer.

Answer: Yes


Question:
Suppose $A$ and $B$ are two real matrices of size $p \times q$ satisfying the following conditions:
\[ \sum_{i=1}^p \sum_{j=1}^q A_{i,j}^2 = 1, \quad \sum_{i=1}^p \sum_{j=1}^q A_{i,j} = 0, \quad \sum_{i=1}^p \sum_{j=1}^q B_{i,j}^2 = 1, \quad \sum_{i=1}^p \sum_{j=1}^q B_{i,j} = 0 \]
Additionally, for the $i$-th column of $A$ denoted as $a_i$, it holds that:
\[ (a_i - a_j) = c_{ij} (b_i - b_j), \quad c_{ij} > 0, \quad \forall i, j \]
where $a_i - a_j$ and $b_i - b_j$ are parallel and have the same direction for all $i$ and $j$. Determine if $A = B$. Provide a justification for your answer.

Answer: A = B


Question: Evaluate the integral \( \int_{0}^{\infty} \frac{\ln x}{1+x^2} \, \mathrm{d}x \).

Answer: 0


Question: Determine if the integral \( \int_{-R}^{R}\int_{-\sqrt{R^2-x^2}}^{\sqrt{R^2-x^2}}\frac{R}{\sqrt{R^2 - x^2-y^2}}dydx \) can be expressed in terms of the natural logarithm without involving the imaginary unit \(i\).

Answer: 2\pi R^2


Question: Evaluate the integral \(\int_{|z|=1}\frac{1}{z^{50}+1}dz\) using the residue at infinity theorem.

Answer: 0


Question: Find the residue of \( \frac{z^2}{(z-1)(z-2)(z-3)} \) at \( \infty \).

Answer: -1


Question: Find the probability distribution of the random variable $Y$, where $Y$ is the minimum distance from the origin to any of 10 points uniformly distributed within a circle of radius 1.

Answer: 20y(1 - y^2)^9


Question: Given the function $f(x,y,z)=\exp(xyz)$ and the condition $f(x,y,g(x,y))=2$ with $g(1,1)=\ln2$, derive an equation to calculate the partial derivative $\frac{\partial f}{\partial z}$ at the point $(1,1,\ln 2)$. What is the value of this partial derivative?

Answer: 2


Question: Find the quadratic approximation of the function \( g(x,y) = e^x \ln(2x+y+1) \) at the origin, and use it to approximate \( g\left(\frac{1}{10},\frac{1}{5}\right) \).

Answer: \dfrac{9}{25}


Question: Find a lower bound for the norm \( \|g\| \) where \( g \) is defined on \( C[-2, 2] \) by the expression: \[ g(x) = \int_{-2}^1 x(t) \, dt - \int_1^2 x(t) \, dt. \]

Answer: 4


Question: Apply the operator \( \hat{u} \) to the derivative \( \dfrac{\partial f(x)}{\partial x} \), given that \( \hat{u}(f(x)) = f(u(x)) \) for any function \( f(x) \) over the real numbers \( \mathbb{R} \).

Answer: \dfrac{\partial f}{\partial x}(u(x))


Question: Determine whether the $n$-sphere given by the equation $x_1^2 + \cdots + x_n^2 - 1 = 0$ is a rational variety in $\mathbb{A}^n$ for all $n \geq 2$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the value of \(i^i\).

Answer: e^{-\pi/2}


Question: Fifteen balls, consisting of 3 balls each of 5 different colors, are to be arranged in a triangular formation. Determine the number of distinct arrangements possible if rotations of the triangle are considered equivalent.

Answer: 56056080


Question: Show that the derivative of the function \( f(M) = \text{Tr}(M \log M - M) \) with respect to the positive definite matrix \( M \) is \( \log M \).

Answer: \log M


Question: Determine whether the order of a matrix $A \in GL(n,\mathbb{Z}_p)$ can exceed $p^n - 1$ if the matrix is not diagonalizable.

Answer: No