Question: If the orbit of every nonzero vector in a vector space \(X\) is unbounded under the operator \(\mathcal{T}\), does it necessarily follow that the orbit of every nonzero vector in \(X\) is also unbounded under the adjoint operator \(\mathcal{T^*}\), even if the kernel of \(\mathcal{T^*}\) is infinite dimensional?

Answer: No


Question: Determine if the function \( f(x,y) = \dfrac{\sin(xy)}{\sqrt{x^2 + y^2}} \) can be extended to be continuous on the entire plane \( \mathbb{R}^2 \). If it can, specify the condition required for continuity at the origin.

Answer: 0


Question: Given the function $f(x)$ with positive values such that $f(1)+f(2)=10$ and $f(a+b)=f(a)+f(b) + 2\sqrt{f(a)\cdot f(b)}$, determine the value of $f(2^{2017})$. \(\boxed{\text{Your answer here}}\)

Answer: 2^{4035}


Question: Let $n_{1}, n_{2}, \ldots, n_{k}$ be a sequence of $k$ consecutive odd integers. If $n_{1} + n_{2} + n_{3} = p^3$ and $n_{k} + n_{k-1} + n_{k-2} + n_{k-3} + n_{k-4} = q^4$, where $p$ and $q$ are primes, find the value of $k$. Given that $3n_{1} + 6 = p^3$ and $5n_{k} - 20 = q^4$, and the second sum can also be written as $5n_{1} + 10k - 30 = q^4$. What is $k$?

Answer: 62


Question: Evaluate the integral \( \int_{-\infty}^\infty \frac{\sin x}{x-i} \, dx \) and verify that it equals \( \frac{\pi}{e} \).

Answer: \dfrac{\pi}{e}


Question: A company employs 3 workers, each of whom stays on the job for an exponentially distributed time with a mean of one year before quitting. When a worker quits, it takes an exponentially distributed time with a mean of 1/10 of a year to hire a replacement. The company earns $1000 per day with 3 workers, $800 per day with 2 workers, and $500 per day with 0 or 1 worker. Calculate the company's long-run average daily revenue.

Answer: 943.28


Question:
Given $A = 17$, $B = 34$, and $C = 87$, solve the system of equations:
$$ Ax + sy + tz = 0 $$
$$ rx + By + tz = 0 $$
$$ rx + sy + Cz = 0 $$
Assuming $r \neq A$, $x \neq 0$, and the denominators in the expression $\frac{A}{A-r} + \frac{B}{B-s} + \frac{C}{C-t}$ are valid, determine the value of the expression.

Answer: 2


Question:
Let $D = \{z \in \mathbb{C} : |z-1| < 1 \}$. Consider the analytic function $f(z)$ on $D$ such that $f(1) = 1$ and $f(z) = f(z^2)$ for all $z \in D$. Determine which of the following statements is not true:

1) $f(z) = [f(z)]^2$ for all $z \in D$
2) $f(\frac{z}{2}) = \frac{1}{2}f(z)$ for all $z \in D$
3) $f(z^3) = [f(z)]^3$ for all $z \in D$

Answer: B


Question: Let \( S_n = \left\{ \frac{2^n+1}{2^n}, \frac{2^n+3}{2^n}, \frac{2^n+5}{2^n}, \ldots, \frac{2^{n+1}-1}{2^n} \right\} \), which contains fractions with denominator \( 2^n \) and numerators as all odd numbers between \( 2^n \) and \( 2^{n+1} \). If \( G(n) \) is the geometric mean of these numbers, find \( \lim_{n\to\infty}G(n) \).

Answer: \dfrac{4}{e}


Question: Determine which function is asymptotically greater as \( n \to \infty \): \( f(n) = 3^{2^n} \) or \( g(n) = n! \times n^3 \).

Answer: f(n)


Question: Evaluate the limit: $$\lim_{x \rightarrow 0} \frac{\int_{0}^{2 \sin x} \cos(t^2) \, dt}{2x}.$$

Answer: 1


Question:
Evaluate the limit of the expression as \( n \to +\infty \):
\[ \lim_{n\to +\infty}\left[\frac{\frac{a_n}{s_n}}{-\ln\left(1-\frac{a_n}{s_n}\right)}\right] \]
where \( a_n = \sqrt{n} \) and \( s_n = a_1 + a_2 + \ldots + a_n \).

Answer: 1


Question: If the determinant of a matrix $A$ is given by $e^{\text{tr}(B)}$, is it necessarily true that $A = e^{B}$? Provide a justification for your answer.

Answer: No


Question: Determine the set of real values of $x$ that satisfy the inequality: $$x^{(\log_{10}x)^2-3\log_{10}x+1} > 1000.$$

Answer: (1000, \infty)


Question: Calculate the area of the region enclosed by the curves $xy=4$, $xy=8$, $xy^3=5$, and $xy^3=15$. Provide your answer as a single numerical value.

Answer: 2 \ln 3


Question: Find the area of the region in the plane defined by the inequalities \( 0 < x < 1 \) and \( 0 < y < x^2 \).

Answer: \dfrac{1}{3}


Question: Is every fully characteristic subgroup of a group normal?

Answer: Yes


Question: Evaluate the limit: $$\lim \limits_{n \to \infty}\left(1-e^{\left(\frac{t+ \log^2(n)}{t-\log(n)}\right)}\right)^n$$

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


Question: Find the greatest common divisor of all numbers of the form $p^{12} - 1$, where $p$ is a prime number with a units digit of one.

Answer: 65520


Question: Evaluate the limit: $$\lim_{n\to\infty}\sqrt{n}(\sqrt[n]{3}-\sqrt[n]{2})$$ without using L'Hopital's rule.

Answer: 0


Question: Determine the time at which the solution of the differential equation \( u_t + uu_x = 0 \) with the initial condition \( u(x,0) = \cos{(\pi x)} \) becomes multi-valued.

Answer: \dfrac{1}{\pi}


Question: Provide an example of a non-commutative algebra that has a maximal ideal $M$ such that $A/M$ is not a division algebra, and $M$ is not the unique nontrivial two-sided ideal.

Answer: M_2(F) \times F


Question: Consider a sequence of independent and identically distributed (i.i.d.) random variables $U_1, U_2, \ldots, U_n$, each following a uniform distribution on the interval $[0, 1]$. Determine the limit of the probability that the sum of these random variables is less than or equal to $\frac{3n}{4}$ as $n$ approaches infinity.

Answer: 1


Question: Compute \( \lim\limits_{(x,y)\to (0,0)}{\sin(xy)-xy\over x^2y} \) using Taylor's theorem.

Answer: 0


Question: Find the splitting field of the polynomial \( f(x) = x^3 - x + 1 \) over the field \( \mathbb{Z}_5 \).

Answer: \mathbb{F}_{25}


Question: Is it possible to construct a function $f$ that is continuous on $[0,\pi]$ and positive on $[\epsilon,\pi-\epsilon]$ for some $\epsilon>0$, represented as a Fourier series of the form $f=\sum_{k=3}^\infty{a_k\cos(kx)+b_k\sin(kx)}$?

Answer: Yes


Question: Consider algebraically independent irreducible homogeneous polynomials $P_1, P_2, \ldots, P_k$ in $n$ variables over the complex numbers. If the variety defined by the ideal $I=(P_1, \ldots, P_k)$ is reducible, determine whether there exists a point $b = (b_1, b_2, \ldots, b_n)$ such that $P_i(b) = 0$ for all $i \in [k]$ and the rank of the Jacobian matrix of $P_1, P_2, \ldots, P_k$ at $b$ is $k$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether there exists a topological manifold that is not a smooth manifold.

Answer: Yes


Question: Find the curl of the cross product of two vectors: \( \nabla \times (\mathbf{k} \times \mathbf{r}) \), where \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \) is a position vector and \( \mathbf{k} \) is the unit vector in the z-direction.

Answer: 2\mathbf{k}


Question: Determine the number of elements in the subgroup of $H = \{a, b, b^2, b^3, c, bc, b^2c, b^3c\}$ generated by $b$ and $c$, given that $O(b)=4$, $O(c)=2$, and $bc=cb^3$. \(H\) is a group with these elements and relations.

Answer: 8


Question: Determine the domain of convergence for the series \( \sum_{n = 1}^{\infty} \frac{\cos(nz)}{n^2} \).

Answer: \mathbb{R}


Question: Let $X$, $Y,$ and $Z$ be random variables. Suppose that $X$ and $Z$ are independent, and also suppose that $Y$ and $Z$ are independent. Determine whether it follows that $\mathrm{cov}(XY,Z)=0$. Provide a justification for your answer.

Answer: No


Question: Simplify \( \frac{1}{x^2 + x} \) in \( \mathbb{Z}_2[x] \) over \( \mathbb{Z}_2[x]/(x^3 + x + 1) \).

Answer: x + 1


Question: Let $R$ be a commutative ring and $M, N$ be finitely generated $R$-modules that are isomorphic. If the projective dimension of $M$ is finite, is the projective dimension of $N$ also finite?

Answer: Yes


Question: Given that $f(x)$ is a continuous and differentiable function satisfying $f(x+y)=f(x)f(y)$ for all real numbers $x$ and $y$, and $f'(0)=3$, calculate $f^{(5)}(0)$. Provide your answer as a single number.

Answer: 243


Question: Evaluate the integral \( \int_{-\pi}^{\pi} \frac{\sin\left(e^{ix}\right)}{e^{ix}} \, dx \).

Answer: 2\pi


Question: Find the value of \(\omega_0\) in the set of complex numbers such that the substitution \(z = \omega - \omega_0\) transforms the cubic equation \(z^3 + Az^2 + Bz + C = 0\) into \(\omega^3 - m\omega - n = 0\), where \(A\), \(B\), \(C\), \(m\), and \(n\) are real numbers.

Answer: \dfrac{A}{3}


Question: Given a compact metric space $X$, an open set $O \subsetneq X$, and a compact set $C \subset O$, determine whether it is possible to construct a sequence of compact sets $C_i$ such that $C \subsetneq C_i \subset O$ and $\bigcup_{i\in\mathbb{N}} C_i = O$. Answer 'yes' or 'no' and justify your answer.

Answer: yes


Question: Find a closed-form expression for the series \( F(s) = \sum_{n>0} \frac{s^n}{n} \) for \( |s| < 1 \). Use the harmonic number \( h_n = \sum_{i=0}^n \frac{1}{i} \), where \( f_0 = 0 \) and \( f_n = \frac{1}{n} \), and the recurrence relation \( h_n = h_{n-1} + f_n \) to manipulate the series and find a solution.

Answer: -\ln(1 - s)


Question:
Calculate the limit using Taylor's series:

$$\lim_{x \to \infty}x-x^2\ln\left(1+\frac{1}{x}\right)$$

Do not use L'Hospital's rule.

Answer: \dfrac{1}{2}


Question: Evaluate the limit: $$\lim_{n\to\infty}\int_{n}^{n+1} e^{-x^2}dx $$.

Answer: 0


Question: In an equilateral triangle with an area of 1, color parts of the triangle red, blue, and green such that each color forms exactly one connected region strictly inside the triangle, and no line parallel to any side contains points of multiple colors. Determine the maximum possible value of the minimum area among the red, blue, and green regions.

Answer: \dfrac{1}{4}


Question: Compute the dimension of the space of global sections of the tangent bundle of a genus $g$ smooth projective curve $C$, denoted as $h^0(T_C)$. Provide your answer as a single integer.

Answer: 0


Question: Determine the values of $p$ for which the $p$-norm of a matrix is invariant under orthogonal transformations. Specifically, find the values of $p$ such that for any orthogonal matrix $A$ and any matrix $X$, the equality $\|AX\|_p = \|X\|_p$ holds.

Answer: 2


Question: Two teams, Team A and Team B, each consist of 10 people. Team A selects and removes n people from its team, where 1 ≤ n ≤ 9. Team B then selects m people from its team to replace with m of the n people removed from Team A, where 0 ≤ m ≤ n. Determine the number of distinct possible combinations of teams after these changes, and find the remainder when this number is divided by 1000.

Answer: 696


Question: Let $X$ be an infinite set with the co-finite topology, and let $\varphi: G \times X \to X$ be a continuous group action. Consider a finite open cover $\mathcal{U} = \{U_1, U_2, \ldots, U_n\}$ of $X$. Define $A = \cup_{i=1}^n (X - U_i)$, which is a finite set. Determine whether there exist distinct elements $x, y \in U_i$ for some $i \in \{1, 2, \ldots, n\}$ such that $(Gx \cup Gy) \cap A = \emptyset$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the magnitude of the tangential component of the vector field \(\vec{A} = (y^3+z^3) \hat{i} + (x^3+z^3) \hat{j} + (y^3+x^3) \hat{k}\) at the point \(\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right)\) on the surface of the unit sphere \(x^2+y^2+z^2=1\).

Answer: 0


Question: Evaluate the double integral \( \int \int_R (2x-y) \, dA \) over the region \( R = \{ 2x \leq y \leq 2x +4; -2x \leq y \leq 4-2x \} \) without using a substitution.

Answer: -8


Question: Let \(u(x,y)=x^3+ax^2y+bxy^2+2y^3\) be a harmonic function and \(v(x,y)\) be its harmonic conjugate. If \(v(0,0)=1\), find \(|a+b+v(1,1)|\).

Answer: 10


Question: Consider the sequence $a_n$ defined by $a_{n+1} = \frac{1}{a_n} + \frac{1}{n + k}$. Determine if the subsequence of $a_n$ consisting of terms where $a_n \geq x_n$ converges, where $x_n$ is the positive root of $f_{n}(x) = (n+k)x^2 - x - (n+k)$. Provide a justification for your answer.

Answer: 1


Question:
Determine the dimension of the null space of the following $n \times n$ matrix:
$$\left( \begin{array}{cccccc}
1-n & 1 & \dots & \dots & \dots & 1 \\
1 & 1-n & 1 & \dots & \dots & 1 \\
\vdots & 1 & \ddots & \ddots & & \vdots \\
\vdots & \vdots & \ddots & \ddots & \ddots & \vdots \\
\vdots & \vdots && 1 & 1-n & 1 \\
1 & 1 & \dots & \dots & 1 & 1-n
\end{array} \right)$$

Answer: 1


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

Answer: 1 - e


Question: Let $G$ be a subgroup of $GL_n(\mathbb{F}_p)$, and let $P$ be a Sylow $p$-subgroup of $G$. Determine whether all non-trivial elements of $P$ have order $p$.

Answer: No


Question: Solve the equation: \(10^{x^2+x} + \log(x) = 10^{x+1}\)

Answer: 1


Question:
Evaluate the limit of the expression as \( n \) approaches infinity:
\[ \lim_{n\to\infty}\left(1+\frac{1}{3}\right)\left(1+\frac{1}{3^2}\right)\left(1+\frac{1}{3^4}\right)\cdots\left(1+\frac{1}{3^{2^n}}\right) \]

Answer: \dfrac{3}{2}


Question: Consider a symmetric random walk $S_n$ on $\mathbb{Z}$. Let $a$ and $b$ be positive integers, and let $\tau$ be the first time the walk reaches $b$ or $-a$. Compute the expected value of $S_\tau^3$ given that the walk reaches $b$ before $-a$. Provide your answer as a single number.

Answer: b^3


Question: A group G has order 30, and its center Z(G) has order 5. Determine the commonly known group that G/Z(G) is isomorphic to.

Answer: S_3


Question: Find the sum of all values of $n$ such that the function \( g(x) = 100|f(x)| - \sum_{k=1}^n |f(x^k)| \) is differentiable over \( \mathbb{R} \), where \( f(x) = e^{x+1} - 1 \) and \( n \in \mathbb{N} \).

Answer: 39


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

Answer: 0


Question: Determine the number of zeros of the function \( F(x) = \int_0^x e^t(t^2 - 3t - 5)dt \) within the interval \((0, 4)\).

Answer: 0


Question: In triangle $ABC$, all sides have integral lengths. Vertex $A$ is at the origin $(0,0)$, vertex $B$ is on the line through $(0,0)$ and $(3,6)$, and vertex $C$ is on the line through $(0,0)$ and $(2,-1)$. Two sides of the triangle are prime numbers with a difference of 50. What is the minimum possible length of the third side?

Answer: 60


Question: Given a matrix $A$ such that $A = A^{-1}$, find the inverse of the matrix $(A + A^{-1})$. Provide your answer in terms of $A$.

Answer: \dfrac{1}{2}A


Question: Solve the equation $i \cos(z) + z^2 + 3z + i\sin(z) = i$ over the complex numbers.

Answer: 0


Question: Calculate the sum \( \sum_{n=3, n \text{ odd}} (\zeta(n) - 1) \), where \( \zeta(n) \) is the Riemann zeta function.

Answer: \dfrac{1}{4}


Question:
Prove that for a positive integer $n$, the following identity holds for the Gamma function:

$$\int_{0}^{\infty} e^{-t^{2}} t^{2n+1} \, dt = \frac{n!}{2}$$

Answer: \dfrac{n!}{2}


Question: Let $n$ be an integer such that $n \geq 2$. Find the greatest possible value of the expression \[ E = \frac{a_1}{1+a_1^2} + \frac{a_2}{1+a_2^2} + \ldots + \frac{a_n}{1+a_n^2} \] given that the positive real numbers $a_1, a_2, \ldots, a_n$ satisfy $a_1 + a_2 + \ldots + a_n = \frac{n}{2}$. Provide your answer as a single value.

Answer: \dfrac{2n}{5}


Question: Determine whether the following statement is true: "For all positive integers \(n > 1\), \(2^n \not \equiv 1 \pmod{n}\)."

Answer: True


Question: Evaluate the limit of the series: $$\lim_{n\to\infty}\sum_{j=1}^{n^2}\frac{n}{n^2+j^2}$$

Answer: \dfrac{\pi}{2}


Question: Calculate the limit: \[ \lim_{{x \to 0}} \frac{\cos(\sin x) - \cos x}{x^4} \]

Answer: \dfrac{1}{6}


Question: Find the maximum value of the function \( f(x,y,z) = x^4 - 6x^2y^2 + y^4 \) subject to the constraint \( x^2 + y^2 + z^2 \leq 1 \).

Answer: 1


Question: Determine the limit of the sequence using L'Hopital's rule and the Euler-Mascheroni constant, given that the sequence does not necessarily converge.

Answer: \gamma


Question: Calculate the greatest common divisor (GCD) of the expressions \(5^{100} - 1\) and \(5^{120} - 1\).

Answer: 5^{20} - 1


Question: In a real vector space $V$ of dimension $2m$ with a non-singular quadratic form $Q$, if $Q$ is hyperbolic, determine the dimension of the subspace on which $Q$ vanishes.

Answer: m


Question: Let \( h(z)=\frac{f(z)}{f'(z)} \) where \( f \) is holomorphic and \( 0 \) is a zero of \( f \) of order \( n \geq 1 \). Find the value of \( h'(0) \) in terms of \( n \).

Answer: \dfrac{1}{n}


Question: Determine the total number of automorphisms of the symmetric group $S_6$, considering both inner and outer automorphisms.

Answer: 1440


Question: Evaluate the integral \( \int_{|z-i|=3} \frac {dz}{(z^2+2)(z+1)} \) using the residue theorem.

Answer: 0


Question: Given a function \(f: \mathbb{R} \to \mathbb{R}\) that satisfies the equation \[f(2xy) + f(f(x + y)) + f(x + y) = xf(y) + yf(x)\] for all real numbers \(x\) and \(y\), determine the sum of all possible absolute values \(|f(48)|\).

Answer: 186


Question: Given $n \times n$ matrices $L$ and $M$ defined as follows: $L_{ij} = 1$ if $i + j \geq n + 1$ and is zero otherwise; $M_{ij} = \min(i,j)$ for $1 \leq i, j \leq n$. It is known that $M = L^2$. Determine whether it is always true, for arbitrary $n$, that $L$ can be expressed as a polynomial in $M$ with integral coefficients.

Answer: Yes


Question: Let \(f\) be a real function and let \(A\) be a countable dense subset of \(\mathbb{R}\). Assume that the function \(f\) is continuous on \(A\). For any \(x \in A\) and \(\delta > 0\), does there exist \(y \in \mathbb{R} \setminus A\) such that \(y\) is in the \(\delta\)-neighborhood of \(x\)?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to\infty} (x!*e^{-x^3}) \]

Answer: 0


Question: Calculate the flux of the vector field \( \vec{F} = x^2\widehat{i} + y^2\widehat{j} + z^2\widehat{k} \) through the lateral surface of the cylinder defined by \( x^2 + y^2 = 4 \) with boundaries \( z = 0 \) and \( z = 8 \). Exclude the flux through the top and bottom surfaces of the cylinder.

Answer: 0


Question: Calculate the expected number of spins required to obtain three consecutive 'e's on an unbiased wheel labeled with the vowels (a, e, i, o, u).

Answer: 155


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

Answer: \sqrt{2}


Question: Determine the number of conjugacy classes in a group of order $625$ if its center has order $25$. Provide your answer as a single integer.

Answer: 145


Question: Determine whether there exists an idempotent semiring where the natural order relation is not a total order, but the underlying set is totally ordered.

Answer: Yes


Question: Given the recurrence relation \( f(x) = x\sqrt{-2x-1+f(x+1)} \), find the value of \( f(1) \).

Answer: 1


Question: Given that $\sup_n E|X_n|^{2+\sigma} < \infty$ for $\sigma > 0$, determine whether the sequence $\{|X_n|\}$ is uniformly integrable.

Answer: Yes


Question: Are the roots of all polynomials with rational coefficients dense in the real numbers?

Answer: Yes


Question: Find the limit: \( \lim_{{n\rightarrow \infty}} \frac{(n!)^2}{(2n)!} \).

Answer: 0


Question: Find the determinant of the matrix \((m+1)I_m - \begin{bmatrix}1\\ 1\\ \vdots\\ 1\end{bmatrix}\begin{bmatrix}1& 1& \cdots& 1\end{bmatrix}\) using the Sylvester determinant theorem.

Answer: (m+1)^{m-1}


Question: Suppose $X$ is an infinite countable bounded subset of $\mathbb{R}$. Is the statement "The closure of $X$ must be uncountable" true or false?

Answer: B


Question: Determine whether there exists a non-transitive embedding of the symmetric group $S_4$ into the symmetric group $S_8$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the sequence of functions \(f_n(x) = \frac{x}{n}e^{-\frac{x}{n}}\) converges uniformly on every compact interval \([a, b] \subset \mathbb{R}\).

Answer: Yes


Question: Evaluate the limit: $$\lim_{x \to 1}\frac{\sqrt{x^4 + 1} - \sqrt{2}}{\sqrt[3]{x} - 1}$$ without using derivatives.

Answer: 3\sqrt{2}


Question: Let $L(X)$ denote the Banach algebra of all bounded linear operators on a Banach space $X$. If $T \in L(X)$ is invertible, is it true that $||T^{-1}|| = ||T||^{-1}$? Provide a justification for your answer.

Answer: No


Question: Determine the interval of convergence for the series \( \sum_{n=0}^{\infty} \frac{(n!)^3}{(3n)!}\left(\frac{z}{3}\right)^n \).

Answer: (-81, 81)


Question: Calculate the arc length of the curve given by the equation \(x^{2/3} + y^{2/3} = 1\) using integration, considering both the positive and negative values of \(x\).

Answer: 6


Question: Compute the expectation \( M_T = \mathbb{E}\left[W_T\int_0^T t \, dW_t \right] \) where \( 0 < t < T \) and \( W = (W_t)_{t \geq 0} \) is a standard Brownian motion starting at 0.

Answer: \dfrac{T^2}{2}


Question: In triangles XYZ and XWP, point X is common to both. A circle passes through points Y, Z, W, and P. Given that angle YXZ is 30 degrees, determine the measure of angle XWP.

Answer: 30


Question: Determine whether the inequality \(|x + y|^\alpha \leq |x|^\alpha + |y|^\alpha\) holds for all real numbers \(x\) and \(y\) when \(\alpha \in [0, 1)\).

Answer: Yes


Question: Is the following statement a correct analogous result for monotonically decreasing functions? Let $f$ be monotonically decreasing on $(a, b)$. Then $f(x+)$ and $f(x-)$ exist at every point $x$ of $(a,b)$. More precisely, $$ \inf_{a<t<x} f(t) = f(x-) \geq f(x) \geq f(x+) = \sup_{x<t<b} f(t).$$ Furthermore, if $a<x<y<b$, then $$f(x+) \geq f(y-).$$

Answer: Yes


Question: Find the number of integers $n$ such that $2000 \leq n \leq 2010$ and $7$ divides $\left\lfloor \frac{\sqrt{2}}{4}(1+\sqrt{2})^n \right\rfloor$. Provide your answer as a single integer.

Answer: 5


Question:
Find the limit of the expression as \( m \) approaches infinity:

\[ \lim_{m\rightarrow\infty}\left(\frac{1}{m}\sum_{j=1}^{m}\sqrt[j]{j+1}\right) \]

Answer: 1


Question: Let $S$ be a countably infinite set with a strict order relation such that there exists an element between any two distinct elements. Must $S$ be order-isomorphic to a subset of the rational numbers under their natural order?

Answer: Yes


Question: Does there exist a sequence of functions $g_{n}:[0,1] \to \mathbb{R}$, $n \in \mathbb{N}$, that are not continuously differentiable but converge uniformly to a function that is continuously differentiable?

Answer: Yes


Question: Evaluate the limit \( \lim_{n\to \infty} n \sin \left (2\pi \sqrt{1+n^2}\right ) \), where \( n \in \mathbb{N} \).

Answer: \pi


Question:
Calculate the expected value of the random variable $X$ given the joint probability density function:
\[f(x,y) = \begin{cases} \frac{2y}{x^2}e^{-x} & ,0 < y < x<+ \infty \\ 0 & , \text{otherwise}\end{cases}\]
using a single integral.

Answer: 1


Question: Let A be a set with 12 elements, and let f be a bijection from A to A such that for each a in A, f(f(f(a))) = a and f(f(a)) ≠ a. How many such functions exist?

Answer: 246400


Question: Find the number of integer solutions to the equation \(5^x - 2^y - 1 = 0\), where \(x\) and \(y\) are real numbers.

Answer: 1


Question: Let $L: \mathbb{C}^3 \rightarrow \mathbb{C}^3$ be a linear map defined as $Lu = \langle v, u \rangle v$, where $u \in \mathbb{C}^3$ and $v \in \mathbb{C}^3$ is a non-zero vector with norm $||v|| = 1$. Does there exist a basis of $\mathbb{C}^3$ for which the matrix of $L$ is diagonal?

Answer: Yes


Question: Calculate the area of the region defined by the inequality \( \sin^2(\pi x) + \sin^2(\pi y) \le 1 \) for \( x, y \in [-1, 1] \).

Answer: 2


Question: Let \(x = [1/n, 1/n, \dots, 1/n]\) and \(y = [y_1, \dots, y_n]\) be probability vectors on \(n\) elements, where \(\sum x_i = \sum y_i = 1\) and \(x_i, y_i \geq 0\). Determine if it is always true that \(x\) is majorized by \(y\).

Answer: Yes


Question: Determine the Krull dimension of the injective hull $E_R(R/\mathfrak{m})$ of the residue field $R/\mathfrak{m}$ in a Noetherian local ring $(R,\mathfrak{m})$. Provide your answer as a single number.

Answer: 0


Question:
Consider the ring of symmetric polynomials $\Lambda_N=\mathbb{C}[x_1, \ldots, x_N]^{S_N}$ in $N$ variables and its subspace of homogeneous symmetric polynomials of degree $M$, denoted by $\Lambda_N^M$. A polynomial $f\in \Lambda_N$ is said to have a $(k,r)$-clustering property if it satisfies the following condition for some $g\in \mathbb{C}[Z, x_{k+1}, \ldots, x_N]$:

$$ f(\underbrace{Z,Z,\cdots, Z}_{k\text{ times}}, x_{k+1},\ldots, x_N)=\prod_{i=k+1}^N(Z-x_i)^r g(Z,x_{k+1}, \ldots, x_N) $$

Let $V_{N,M}^{(k,r)}$ be the $\mathbb{C}$-vector space spanned by homogeneous symmetric polynomials of degree $M$ with the $(k,r)$-clustering property. For $N=nk$ and $M=n(n-1)nk/2$, where $k+1$ and $r-1$ are coprime, determine if the following statement is true:

$$ \dim V^{(k,r)}_{N,M}=1 $$

It is known that $V_{N,M}^{(k,r)}\neq \{0\}$ because the Jack polynomial $P^{\alpha}_{\Lambda}(x_1, \ldots, x_N)$, with $\Lambda$ being a specific partition and $\alpha=-\frac{r-1}{k+1}$, is well-defined and belongs to $V_{N,M}^{(k,r)}$. Is this polynomial the only one with the given properties?

Answer: 1


Question: Let $V$ and $W$ be vector spaces over a field $\mathbb{F}$, and let $T: V \to W$ be a linear map. Let $T^{\vee}: W^{\vee} \to V^{\vee}$ denote the dual map of $T$, where $V^{\vee}$ and $W^{\vee}$ are the dual spaces of $V$ and $W$, respectively. Determine whether the statement $\dim \ker T = \dim \ker T^{\vee}$ is true or false.

Answer: False


Question: Determine whether the following statement is true or false: Integrability of a function on a sub-$\sigma$-algebra of a $\sigma$-finite measure space implies integrability on the original $\sigma$-algebra.

Answer: True


Question: Determine if there exists a polynomial $g(x) \in \mathbb{C}[x]$ such that $g(x)^2 = x^4 + 2x^2 + 1$. Provide a justification for your answer.

Answer: Yes


Question:
Determine whether the following two infinite series are equal:

1. \( \sum_{k=1}^\infty \frac{2}{k} \)

2. \( \sum_{k=1}^\infty \frac{k+1}{\frac{k^2 + k}{2}} \)

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to 0}\dfrac{e^{5x^2}-e^{-5x^2}}{e^{2x^2}-e^{-2x^2}}$$ using the factorization method.

Answer: \dfrac{5}{2}


Question:
Determine whether the following inequality holds for continuous functions $f(x)$ and $g(x)$, given that $\int_0^1 f(x) \, dx = 0$:

$$\left( \int_0^1 f(x)g(x) \, dx \right)^2 \leq \left( \int_0^1 g(x)^2 \, dx - \left( \int_0^1 g(x) \, dx \right)^2 \right) \left( \int_0^1 f(x)^2 \, dx \right)$$

Is this statement true or false?

Answer: True


Question: Let $\psi: V \rightarrow V$ be a linear transformation and let $\psi^n$ be the n-th power of $\psi$. Consider a polynomial $P(x) = a_nx^n + \ldots + a_0$. Define $P(\psi) = a_n\psi^n + \ldots + a_1\psi + a_0$. Is $P(\psi)$ a linear transformation?

Answer: Yes


Question: An analog clock with a broken 12-hour mechanism advances the hour hand forward by one hour with a probability of 0.99 every sixty minutes, while it remains stationary with a probability of 0.01. These movements are independent across hours. At noon, what is the probability that the hour hand will point at 3 as N hours elapse, and how does this probability converge as N approaches infinity?

Answer: \dfrac{1}{12}


Question: Let \(X\) be a topological space, and consider \(V \subseteq U \subseteq X\). If \((V, \tau_V)\) is connected, where \(\tau_V\) is the subspace topology inherited from \(X\), is \((V, \tau')\) also connected, where \(\tau'\) is the subspace topology inherited from \(U\)? Provide a justification for your answer.

Answer: Yes


Question: Find the Fourier transform of the Laplacian operator in polar coordinates, given by \( \nabla^2 = \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} \). Assume no angular dependence.

Answer: -k^2


Question: Determine a generator of the ideal generated by $3$ and $2-2\sqrt{-2}$ in ${\mathbb{Z}[\sqrt{-2}]}$. \( \boxed{\text{Your answer here}} \)

Answer: 1 - \sqrt{-2}


Question: Evaluate the integral: $$2\int_0^{2\pi}\sqrt{a^2-\sin^2t}\cdot(-\sin t)\, dt$$

Answer: 0


Question: Compute the line integral \( \int_{C} -y^3 \, dx + x^3 \, dy - z^3 \, dz \) directly, where \( C \) is the intersection of the cylinder \( x^2 + y^2 = 1 \) and the plane \( x+y+z = 1 \). Verify that the result is \( \frac{3\pi}{2} \).

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


Question: Find the residue of \(\frac{\cosh z}{z^4}\) at \(z=0\).

Answer: 0


Question: Evaluate the integral: $$\int_{-1}^1 \frac{x^2-1}{x^4+2x^2+1}~ dx$$

Answer: -1


Question: Find the general solution to the partial differential equation \( u_x + u_y = 2 \) with the initial condition \( u(y, \frac{y}{3}) = 2y \) using the method of characteristics.

Answer: 2y


Question: Determine whether the following statement is true or false: If the derivative of a function \(f_1\) is smaller than that of another function \(f_2\) for \(x > 0\), then \(f_1\) is always less than \(f_2\) for \(x > 0\), given that both \(f_1(x)\) and \(f_2(x)\) are strictly positive.

Answer: B


Question: Determine if the function \( y(x) = \ln\left(\frac{1+\sqrt{1-x^2}}{x}\right) - \sqrt{1-x^2} \), defined on the interval \((0,1]\), has a tangent line at the endpoint \(x=1\). If a tangent line exists, find its equation.

Answer: y = 0


Question: Calculate the volume in the first octant enclosed by the cylinder \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) and below the plane \( z = 3x \) using a double integral.

Answer: a^2 b


Question: Let $X=(X_1,X_2,X_3)$ be a continuous random vector with the joint probability density function given by \( f(x_1,x_2,x_3)= 12x_2 \) for \( 0<x_3<x_2<x_1<1 \) and \( 0 \) elsewhere. Calculate the probability of the event \( B = \{ x_3 \leq 1/3 \} \).

Answer: \dfrac{49}{81}


Question: Determine the order of the center \( Z(G) \) of a group \( G \) of order 8, which is generated by elements \( a \) and \( b \), satisfying the relations \( a^4 = b^2 = 1 \) and \( ba = a^3b \).

Answer: 2


Question: Determine the number of 9-letter palindromic words that can be formed using the letters A, B, C, D, and E, given the following conditions: the letter A appears exactly three times, no more than two different letters are used, no letter appears next to itself, and each letter appears no more than twice in the word.

Answer: 0


Question: Let $f(k)$ be a non-negative integer-valued function such that $f(k) \geq p$ for at least $t$ values of $k$. Define $h(x)$ as the number of distinct values of $s$ for which $x-s$ is in the set where $f(k) \geq p$. Determine the lower bound on the sum $\sum_{x} h(x)$. Provide your answer as a single expression or value.

Answer: t^2


Question: Determine whether there exists a non-constant continuous function \( f : \mathbb{R} \to \mathbb{R} \) such that \( f(\mathbb{R} \setminus K) \) is not open for every compact set \( K \) in \( \mathbb{R} \).

Answer: Yes


Question: Determine the number of non-similar matrices with the characteristic polynomial \((\lambda-1)^4(\lambda-2)^3\).

Answer: 15


Question: Given the complex number $z$ such that $|z - \frac{1}{z}| = 2$, determine the maximum value of $|z|$.

Answer: 1 + \sqrt{2}


Question: Determine the number of Hamilton cycles in the complete bipartite graph $K_{n,n}$. Express your answer in terms of $n$. \( \boxed{n!(n-1)!} \)

Answer: n!(n-1)!


Question: In a topological space, let $A \subset B \subset C$. If $A$ is compact as a subspace of $C$, is $A$ also compact as a subspace of $B$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the expression \(2^n \prod_{k=0}^{n-1}\cos(2^k\theta)\) given that \((2^n + 1)\theta = \pi\).

Answer: 1


Question: Let $G$ be the group of $2 \times 2$ non-singular matrices over the field $\mathbb{Z}_3$ under multiplication. Define a map $f: G \to \mathbb{R}^*$ by $f(A) = |A|$, where $|A|$ denotes the determinant of matrix $A$. Determine the order of the kernel of $f$. \(\mathbb{R}^*\) is the group of non-zero real numbers under multiplication.

Answer: 24


Question: Evaluate the definite integral \( \int_0^1 f(x) \, dx \), where \( f(x) = \begin{cases} 2x\sin\left(\frac{1}{x}\right) - \cos\left(\frac{1}{x}\right) & : x \in (0,1] \\ 0 & : x = 0 \end{cases} \). Determine if the integral is defined, and if so, find its value.

Answer: \sin 1


Question: Find the limiting value of the ratio \( \lim_{r\to\infty}\frac{f(r)}{r^2} \), where \( f(r) \) is the number of integer lattice points contained within a circle of radius \( r \) centered at the origin.

Answer: \pi


Question: Calculate the expected number of rolls of a fair six-sided die until the sum of the outcomes is a multiple of 3.

Answer: 3


Question: Find the limit of the sequence $x_n$ defined as follows: $$x_n = \left[ \left(\frac{2}{1}\right) \left(\frac{3}{2}\right)^2 \left(\frac{4}{3}\right)^3 \ldots \left(\frac{n+1}{n}\right)^n \right]^{\frac{1}{n}}$$

Answer: e


Question: Evaluate the double integral \( \mathop{\int\!\!\! \int}_{R_{1}}f(x,y)\,dA \) for the rectangle \( R_{1}=[0, 5] \times [-4, 4] \), given that \( \mathop{\int\!\!\! \int}_{R_{2}}f(x,y)\,dA=10 \) for the rectangle \( R_{2}=[0, 5] \times [0, 4] \), \( \mathop{\int\!\!\! \int}_{R_{3}}f(x,y)\,dA=24 \) for the rectangle \( R_{3}=[-5, 0] \times [-4, 0] \), and \( f(-x,y)=-f(x,y) \) for all \( (x,y) \).

Answer: -14


Question: Evaluate the improper integral: $$ \int_{-\infty}^{\infty} \left(\frac{\arctan(x)}{x}\right)^2 \, dx $$.

Answer: 2\pi \ln 2


Question: Evaluate the line integral \( \int_c xdx + ydy + zdz \) where \( c \) is the intersection of the cylinder \( x^2 + y^2 = 1 \) and the sphere \( x^2 + y^2 + z^2 = 9 \), oriented clockwise when viewed from the positive z-axis.

Answer: 0


Question:
Determine which of the following sets are homeomorphic to a line:

A. $\{(x,y)\in \mathbb R^2:xy=0,x+y\geq0\}$

B. $\{(x,y)\in \mathbb R^2:xy=1,x+y\geq 0\}$

Answer: B


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: Describe the spectrum of the $C^*$-algebra $C(K) \oplus C(K')$, where $K$ and $K'$ are compact Hausdorff spaces.

Answer: K \sqcup K'


Question: Determine the number of positive integers \( N \) between 3 and 1000 (inclusive) for which the following condition holds: For any set \( \{a_i\}_{i=1}^N \) of \( N \) real numbers such that \( a_1 + a_2 + \cdots + a_N = 0 \), it must always be true that \( a_1a_2 + a_2a_3 + \cdots + a_{N-1}a_N + a_Na_1 \leq 0 \).

Answer: 2


Question: Consider a real vector space $V$ with a scalar product $g$, a symmetric bilinear function such that the mapping $h: V \to V^*$, defined by $h(v) = g(v, \cdot)$, is an isomorphism. An orthonormal basis in this context consists of vectors $u_i$ ($i=1,\dots,n$) satisfying $g(u_i, u_j) = 0$ for all $i \neq j$ and $g(u_i, u_i) = \pm 1$. For a linear transformation between two orthonormal bases, does the matrix representation have unit length orthogonal columns in the general scalar product case? If not, can we conclude that the determinant of such a transformation is $\pm 1$?

Answer: \pm 1


Question: Let $D$ be a simply connected domain in the complex plane $\mathbb{C}$ with an analytic boundary. Suppose there is a simple curve $\Gamma$ connecting two distinct boundary points $a$ and $b$, dividing $D$ into two disjoint regions $D_1$ and $D_2$. Given a function $f: \overline{D} \to \overline{\mathbb{C}}$ that is holomorphic on $\overline{D} \setminus \{a\}$ with an essential singularity at $z=a$, and $|f| < \tau$ in $D_1$ with $\tau > 0$, and $|f| = \tau$ on $\Gamma$, determine if there exists a subdomain $D_3 \subset D_2$, with $\Gamma$ as part of its boundary, such that $|f| > \tau$ in $D_3$. Provide a justification for your answer.

Answer: Yes


Question: Let $f, g, h: \mathbb{R} \to \mathbb{R}$. Determine whether the following statement is true: If $fh = gh$, then $f|h| = g|h|$, where $|z| = z^+ + z^-$ and $z^+, z^-$ are the positive and negative parts of a function $z$, respectively.

Answer: True


Question: Evaluate the limit: $$\lim_{(x,y)\to(1,2)} [\ln(5-(x^2+y^2))]\sqrt{y^2-4}.$$

Answer: 0


Question: Find the maximum value of \(a^2 + b^2\) given that the polynomial \(x^4 + ax^3 + 3x^2 + bx + 1\) is non-negative for all real numbers \(x\).

Answer: 40


Question: In the context of topological groups, determine whether the statement is true or false: A subgroup $H$ is open if and only if $H$ has finite index in the parent group $G$. Provide a justification for your answer.

Answer: False


Question: Find the least value of \( K > 0 \) such that the inequality \( |\sin^2x - \sin^2y| \leq K|x - y| \) holds for all \( x, y \in \mathbb{R} \).

Answer: 1


Question: Determine the number of real solutions of the equation \( xe^{\sin x} = \cos x \) in the interval \( (0, \frac{\pi}{2}) \).

Answer: 1


Question: Determine whether there exists an arc $A \subseteq [0,1]^2$ that connects the intervals $I_0 = \{0\} \times [0,1]$ and $I_1 = \{1\} \times [0,1]$, while avoiding the union of the arcs $\bigcup_{n \in \omega} A_n$, where each $A_n$ is an arc with endpoints in $I_0$ and $I_1$ and $A_i \cap A_j = \varnothing$ for $i \neq j$. Assume $A_0 = [0,1] \times \{0\}$ and $A_1 = [0,1] \times \{1\}$. Provide a justification for your answer.

Answer: Yes


Question: Find the limit of the sequence \(a_n = \frac{\sqrt[n]{\binom{2n+3}{n+2}}}{\sqrt[n]{(n+2)!}}.\)

Answer: 0


Question: Evaluate the integral \[ I(a) = \int_{-\frac{1}{a}}^{\frac{1}{a}} (2x^6 - 2x^4 + 3) \sin^{-1}(ax) \, dx \] for \( a > 0 \).

Answer: 0


Question: If $\{f_n(x)\}$ is a sequence of real-valued Riemann integrable functions that converges to a function $f(x)$, is $f(x)$ necessarily a Riemann integrable function? Provide a justification for your answer.

Answer: No


Question:
Evaluate the limit of the following function as \( x \) approaches infinity, assuming \( x \) is even (\( x = 2n \), where \( n \) is a positive integer):
\[
\lim_{x \to +\infty} \frac{\Gamma \left( \frac{1}{2} - \frac{x}{2} \right)\Gamma \left( 1 + \frac{x}{2} \right)}{\Gamma \left( \frac{1}{2} + \frac{x}{2} \right)\Gamma \left( \frac{x}{2} \right)}
\]

Answer: 0


Question: Consider a number field $F$ and its adele ring $\mathbb{A}$. Let $\mu_2$ denote the group scheme of 2nd roots of unity, defined over $F$. Is the set difference $\mu_2(F) \backslash \mu_2(\mathbb{A})$ compact? Justify your answer, considering that for all finite places $v$ of $F$, the special maximal compact subgroup of $\mu_2(F_v)$ is $\mu_2(F_v)$ itself.

Answer: Yes


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

Answer: 0


Question: Determine if the process \(W(t) = t^2 Z(t) - 2\int_0^t sZ(s) \, ds\), where \(Z(t)\) is a standard Brownian motion, is a martingale. Verify if \(E[W(t+1)|F(t)] = W(t)\), given that \(F(t)\) is a filtration process.

Answer: E[W(t+1)|F(t)] = W(t)


Question: Evaluate the integral \( \int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt \).

Answer: \dfrac{\pi}{2}


Question: Evaluate the improper integral \( \int_{1}^{\infty}\frac{x^3}{e^x}dx \).

Answer: \dfrac{16}{e}


Question: Determine the equation of a plane that contains the line of intersection of the planes $x + y - z = 1$ and $2x - y + 3z = 2$, and is perpendicular to the plane $3x - 2y + z = 4$. Provide your answer in the form $ax + by + cz = d$.

Answer: x + y - z = 1


Question: Determine whether the statement is true or false: If $H$ is a subgroup of $G$ with index $2$, then $\{a^2 \mid a \in G\} = H$. Provide a justification for your answer.

Answer: False


Question: Evaluate the double integral \( \iint_D \frac{1}{\sqrt{x^2+y^2}}\,dx\,dy \), where \( D \) is the region defined by \( 1 \le x^2 + y^2 \le 4 \). Transform the integral using polar coordinates and find the resulting integral.

Answer: 2\pi


Question: Let $n > 1$ be a non-square odd integer. Determine whether there necessarily exists a prime $p$ such that $p + n$ is a perfect square.

Answer: Yes


Question: Find a point C on a circle in 3D space, centered at O (0, 0, 0), such that the distance from C to the center is equal to the distance from C to both points A (1, 5, 2) and B (5, 2, 1).

Answer: (2, 3, -1)


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

Answer: \dfrac{1}{2}


Question: Determine the largest possible value of the greatest common divisor (GCD) of 49 natural numbers whose sum is 540.

Answer: 10


Question: Evaluate the limit \( L = \lim_{n \to \infty} \int_{0}^{\infty} \frac{1}{1+x^n} \, dx \) and identify the type of integral problem this represents.

Answer: 1


Question: Determine the values of $a$ for which the integral \( \int_{a-2}^{a+2} \frac{1}{2+x^4} \, dx \) reaches its maximum value.

Answer: 0


Question: Determine whether the following statement is true or false: If \( \int_1^\infty f(x)\,dx \) converges, then \( \int_1^\infty f(x)g(x)\,dx \) converges for any function \( g(x) \) such that \( |g(x)| \leq 1 \).

Answer: B


Question: Given that \( \int_{0}^{\infty} \left(\frac{\sin x}{x}\right)^3 \, dx = A \), find the value of \( k \) such that \( \int_{0}^{\infty} \frac{x - \sin x}{x^3} \, dx = kA \).

Answer: \dfrac{2}{3}


Question: In triangle $ABC$, $AD$ is the angle bisector. The perpendicular bisector of $AD$ intersects the extended side $BC$ at point $F$, and intersects $AB$ and $AC$ at points $E$ and $G$ respectively. Given that $\frac{AC}{AB}=\frac{3}{4}$, determine the ratio $\frac{CF}{BF}$.

Answer: \dfrac{9}{16}


Question: Assuming the axiom of choice, determine if the collection of all subsets of an infinite set $A$ that have the same cardinality as $A$ has the same cardinality as the power set of $A$. Provide your answer as either 'yes' or 'no'.

Answer: yes


Question:
In a Hilbert space, let $A_1, B_1, A_2, B_2$ be open sets satisfying the following conditions:

1. $A_1 \cap B_1 = A_2 \cap B_2 = \emptyset$.
2. $A_1 \cup B_1 = A_2 \cup B_2$.
3. $A_1$ and $A_2$ have the point $0$ in common.
4. $A_1$ and $A_2$ are both connected sets.

Determine if $A_1 = A_2$.

Answer: A_1 = A_2


Question: What is the expected number of coin tosses needed to achieve a run of "k" successive heads, assuming each coin toss is independent?

Answer: 2^{k+1} - 2


Question: Find the distribution of the ratio $Y/X$ where $X$ and $Y$ are random variables uniformly distributed over the region $A = \{(x,y) \in \mathbb{R^2}: 0 \leq y \leq x \leq 1\}$ with joint density $d\mu_{(X,Y)}(x,y) = 2\mathbf{1}_A(x,y)dxdy$. Provide the distribution of $Y/X$.

Answer: U(0,1)


Question: Evaluate the limit: $$\lim_{x\to 0}\frac {1}{x}\int_x^{2x}e^{-t^2}dt$$

Answer: 1


Question: Calculate the sum of the infinite series: \[ \sum_{n=1}^{\infty} \frac{n^2+2n+3}{2^n} \]

Answer: 13


Question: Find the general solution of the differential equation \((2y+e^{2y})dx+(4xy+2xe^{2y})dy=0\) using an integrating factor.

Answer: x e^{2y} = C


Question: Consider an even-dimensional hypercube. Is it possible to orient the edges of this hypercube such that the resulting directed graph is vertex transitive?

Answer: Yes


Question:
Evaluate the following alternating series:
\[ \sum_{r=0}^{n} (-1)^r \binom{n}{r}\frac{1+r\ln 10}{(1+\ln 10^n)^r} \]

Answer: 0


Question:
Evaluate the limit using L'Hopital's Rule:

$$\lim_{x\to 0}\frac{\sinh x-x\cosh x+\frac{x^3}{3}}{x^2\tan^3x}$$

Answer: -\dfrac{1}{30}


Question: Determine the radius of convergence of the power series \( \sum\limits_{n=1}^{\infty} \tau(n^{25})Z^n \), where \( \tau(n) \) represents the number of divisors of \( n \).

Answer: 1


Question:
Given continuous random variables $X$ and $Y$ with the following probability density functions (pdfs) and cumulative distribution functions (cdfs):

$$f_X(x) = \begin{cases}e^{-x} & x>0 \\ 0 & x \leq 0\end{cases}\quad\quad F_X(x) = \begin{cases}1-e^{-x} & x > 0 \\ 0 & x \leq 0 \end{cases}$$

$$f_Y(y) = \begin{cases}\frac{1}{2\sqrt{y}} & 0<y<1 \\ 0& \text{otherwise}\end{cases}\quad\quad F_Y(y) = \begin{cases}\sqrt{y}&0<y<1\\ 0 & \text{otherwise} \end{cases}$$

Determine the function $Y = g(X)$, if it exists, solely based on the distributions of $X$ and $Y$. It is suggested that the answer is:

$$Y = (1 - e^{-X})^2$$

Answer: Y = (1 - e^{-X})^2


Question: Find the number of non-abelian groups of order $p^n$ (where $n > 2$) that cannot be expressed as a direct product of any of their two subgroups.

Answer: 2


Question: Determine the number of solutions for the equation \( \cot^{-1}{\sqrt{4-x^2}} + \cos^{-1}{(x^2-5)} = \frac{3\pi}{2} \), considering the principal values of the inverse trigonometric functions.

Answer: 2


Question: In a commutative ring $R$ with unity, if a subring $S$ of $R$ also contains the unity, is it necessarily true that $R = S$? Provide a justification for your answer.

Answer: No


Question: Find the abscissa of convergence for the Dirichlet series: $$\sum_{m,n\in\mathbb{Z}}'\frac{m^6+n^6}{(m^2+n^2)^s}$$ where the prime on the summation symbol indicates that the term $(0,0)$ is omitted.

Answer: 4


Question: Resolve the expression: $$\sum_{d|n}\frac{1}{d}\sum_{d'|d}d'\mu(d')$$

Answer: \dfrac{1}{n}


Question: Determine whether the following statement is true or false: The intersection of an open subset of a metric space with any subset of that space is always open in the subspace topology.

Answer: A


Question: Consider a sequence of one-to-one conformal mappings $\{f_n\}$, each mapping the open unit disk $\mathbb{D}$ onto the right half-plane $\mathbb{A} = \{z \in \mathbb{C} : \mathfrak{R} z > 0\}$. Assume that $f_n$ converges uniformly to a function $f$ on compact subsets of $\mathbb{D}$, but $f$ is not one-to-one. Determine the real part of $f(0)$. \( \mathfrak{R} f(0) = \boxed{?} \)

Answer: 0


Question: Determine whether the sum \( \sum_{k=1}^n k^{2m} \) is divisible by \( 2n+1 \) for all natural numbers \( m \).

Answer: No


Question: Let \(A\) be an uncountable Borel subset of \(\mathbb{R}^n\). Does there exist a Lebesgue measurable set \(B \subset A\) that has zero Lebesgue measure and is not a Borel set?

Answer: Yes


Question: In the symmetric group $S_{10}$, determine the number of permutations $\sigma$ such that $\sigma\beta\sigma^{-1} = \delta$, where $\beta = (1,3,5,7)(2,4,6,8,10)(9)$ and $\delta = (1,2,3,4)(5,6,7,8,9)(10)$. Provide your answer as a single integer.

Answer: 20


Question: Are hyperbolic triangle groups hyperbolic according to Gromov's definition?

Answer: Yes


Question: Let $A$ and $B$ be $n$-by-$n$ positive semidefinite matrices such that $A^2 = B^2$. Prove that $A = B$.

Answer: A = B


Question: Is there a countably infinite subset of the set of irrational numbers?

Answer: Yes


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

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

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

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

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

Answer: A


Question: In an acute triangle $\triangle ABC$ with side lengths $a$, $b$, and $c$, it is given that $\frac{b}{a} + \frac{a}{b} = 6\cos C$. Find the value of the expression $\frac{\tan C}{\tan A} + \frac{\tan C}{\tan B}$.

Answer: 4


Question: Find the conditional expectation $E[X|XY]$ where $X$ and $Y$ are independent standard normal random variables.

Answer: 0


Question: Find the sum of all possible integer values of $a$ that satisfy the equation \((a-1)^{p+1}+a^p=(a+1)^{p-1}\), where $p$ is an odd prime and $p\nmid (a-1)$.

Answer: 2


Question: Compute the limit: $$\lim_{n \to \infty}\prod_{k=1}^n \left(\frac{k}{n}\right)^{1/n}$$

Answer: \dfrac{1}{e}


Question: In a topological space $(X, \tau)$ where for every $x \in X$, the intersection of all open sets containing $x$ is $\{x\}$, does every convergent sequence in $X$ have a unique limit? Answer yes or no and justify your answer.

Answer: No


Question: Evaluate the line integral \(\int_C \cos(x)\,dy - \sin(y)\,dx\) where \(C\) is the triangle with vertices \((0,0)\), \((1,0)\), and \((1,1)\), traversed counterclockwise.

Answer: 1 - \sin(1)


Question:
Given a sequence of positive continuous functions $\{ g_n \}$ on the interval $[0,1]$, a positive Borel measure $\mu$ on $[0,1]$, and the following conditions:

1. $\lim_{n\to \infty} g_n (x) = 0$ almost everywhere with respect to a measure $m$.
2. $\int_I g_n dm = 1$ for all $n$.
3. $\lim_{n\to \infty} \int_I fg_n dm = \int_I f d\mu$ for every $f\in C(I)$.

Determine if it follows that $\mu \perp m$. Provide a justification for your answer.

Answer: \mu \perp m


Question: Consider a graph $G$ with vertices labeled by the integers $1, 2, \dots, 99$. Two distinct vertices $i$ and $j$ are connected by an edge if and only if their greatest common divisor is at least $3$. Determine the chromatic number of the graph $G$. \( \boxed{} \)

Answer: 33


Question: Evaluate the triple integral \( \iiint_D y \, dV \), where \( D \) is the region bounded below by the plane \( z = x + 1 \), above the \( xy \)-plane, and between the cylinders \( x^2 + y^2 = 1 \) and \( x^2 + y^2 = 9 \). Determine the bounds for this integral in cylindrical coordinates and compute the integral.

Answer: 0


Question: Determine the norm of the linear operator $T: L_p[0,1] \to L_p[0,1]$ defined by $Tf(x) = xf(x)$ for $p \in [1, \infty)$. It is given that $\|T\| \leq 1$. What is the exact value of $\|T\|$?

Answer: 1


Question: Calculate the line integral \( \int_N h(x,y) \, dS(x) \) where \( h(x,y) = x^2y \) and \( N \) is the boundary of the square with vertices \((0,0), (2,0), (2,2), (0,2)\).

Answer: \dfrac{40}{3}


Question: Calculate the line integral of the vector field \( G(x,y,z) = (x-y)i + (y-z)j + (z-x)k \) along the circular arc in \( \mathbb{R}^3 \) on a plane parallel to the xy-plane, centered at the origin with radius \( b \). The arc follows the path from \((b,0,c)\) to \((0,b,c)\) to \((-b,0,c)\) and back to \((b,0,c)\).

Answer: \pi b^2


Question: Is it possible for $\sup \{\langle p,x \rangle, x \in S_+\} = 1$ under the same conditions as in the original question?

Answer: Yes


Question: In any metric space, does the diameter of a set remain the same for both open and closed balls?

Answer: No


Question: Find the remainder when \(13^{2023}\) is divided by \(2000\).

Answer: 1797


Question: Find the distance from the point \((1, 3, 2)\) to the line of intersection of the planes \(x + y + 2z = 1\) and \(x - z = 2\).

Answer: \sqrt{10}


Question: Find the form of the polynomial $Q(x)$ that satisfies the equation $Q(x+y) + Q(x-y) = 2Q(x) + 2Q(y)$. Provide your answer in terms of the degree and coefficients of $Q(x)$.

Answer: Q(x) = ax^2


Question: Consider a sequence of uniformly continuous functions on a compact set that converges uniformly. Determine whether the sequence is bounded.

Answer: Yes


Question: Evaluate the integral \( \int _{-\pi }^{\pi }\left(\frac{x^3+\cos \left(x\right)+1}{1+\cos \left(x\right)}\right)\,\mathrm{d}x \).

Answer: 2\pi


Question: Given a sequence of non-negative terms \((a_n)\), if the arithmetic mean \(\frac{1}{N}\sum_{n=1}^{N}a_n\) converges, does it imply that \(\lim_{n \rightarrow \infty} \frac{a_n}{n} = 0\)? Provide a justification for your answer.

Answer: Yes


Question: Find the greatest integer that divides $p^8 - 1$ for every prime number $p > 7$. Express your answer as an integer.

Answer: 480


Question: Let $A$, $B$, and $C$ be subspaces of a vector space $V$. Determine whether the equality $V = (A\cap(B+C)) \cap (C \cap (B + A)) = A \cap C$ holds. Justify your answer.

Answer: A \cap C


Question: Does there exist a closed, smooth planar curve and a specific direction such that, when a finite opening is cut in the curve and a photon is launched through this opening into the interior along the specified direction, the photon will perpetually bounce off the curve's boundary, remaining confined within the interior?

Answer: Yes


Question: Find the sum of the ninth powers of the roots of the polynomial equation \(x^9+7x-2=0\).

Answer: 18


Question: How many non-isomorphic abelian groups of order 2700 are there, ensuring that all factors have prime power order?

Answer: 12


Question: Find the minimum value of $xy+yz+xz$ given that $x, y, z$ are real numbers satisfying $x^2+y^2+z^2=1$.

Answer: -\dfrac{1}{2}


Question: Compute the valuation \( v_{\pi_L}(1-\zeta^{a-1}) \) in the cyclotomic field \( L=\mathbb{Q}[\zeta] \), where \( \pi_L=1-\zeta \) is the ideal above \( p \) and \( (a,p)=1 \).

Answer: 1


Question: Evaluate the limit \( \lim_{x \to 0} \frac{(\sin(x-x^2))^2 - x^2 + 2x^3}{\sin^{4}(2x)} \) using Taylor series.

Answer: \dfrac{1}{24}


Question: Is every finite-dimensional normed space complete?

Answer: Yes


Question: In any category, determine if the coproduct of an initial object with itself is always isomorphic to the initial object.

Answer: Yes


Question: Determine the angle between the integral curves of the differential equations \( x' = x - t^2 \) and \( x' = 2x + t^2 \) at the point \( M = (1, -2) \).

Answer: 0


Question: Determine the number of values of 'a' such that the equations $x^3 + ax + 1 = 0$ and $x^4 + ax^2 + 1 = 0$ have a common root.

Answer: 1


Question: Evaluate the surface integral \(\int_S \mathbf{F}\cdot\mathbf{dS}\) where \(\mathbf{F}=\langle 7x,2y,5z\rangle\) and \(S\) is the boundary surface of a region \(V\) with unit volume.

Answer: 14


Question: Determine if it is true that the binomial coefficient \( {q+1 \choose 2} \) is semiprime if and only if \( q = 3 \), given that \( q \equiv 3 \pmod{4} \).

Answer: True


Question: Evaluate the limit: $$\lim_{x\to \infty} \frac{4x^3-3\sin{2\pi x}}{\sqrt{x^6+\cos{3\pi x}}}$$

Answer: 4


Question: Evaluate the sum \( \binom{p+1}{2} - \binom{p+1}{3} + \cdots - \binom{p+1}{p} + \binom{p+1}{p+1} \) for an odd prime \( p \).

Answer: p


Question: Evaluate the limit: $$\ell =\lim_{x\to 0}\frac{\sin^{200}(x)}{x^{199}\sin(4x)}.$$ Determine if this limit exists, and if so, find its value.

Answer: \dfrac{1}{4}


Question: Find the derivative of the function defined by the infinite continued fraction \( f(x) = x + \frac{1}{x + \frac{1}{x + \frac{1}{x + \ldots}}} \) at \( x = \frac{3}{2} \).

Answer: \dfrac{4}{5}


Question: Determine the number of equivalence relations on a set of size 7 such that the total number of equivalence classes is 29.

Answer: 21


Question: In the context of group theory, consider a subgroup $H$ of a group $G$ with order $p^k$, where $p$ is a prime number and $k \geq 1$. Does every element of $H$ necessarily have an order that is a power of $p$? Provide a justification for your answer.

Answer: Yes


Question: Determine whether there exist distinct lambda terms $M$ and $B$ such that $M \neq B$ and the normal form of $MB$ is the same as the normal form of $(MB)(MB)$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit of the sequence \(I_n\) defined by \[I_n=\int_0^1 \frac{x^n}{\sqrt{1+x^2}} \, dx\] as \(n\) approaches infinity.

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\left(\frac{1}{(2n+1)^2}+\frac{1}{(2n+3)^2}+ \ldots +\frac{1}{(4n-1)^2}\right). \]

Answer: 0


Question: Find the value of \( a \) such that \( a > 0 \) and \( \lim_{x \to a}\frac{x^x-a^x}{x-a} = \lim_{x \to a}\frac{a^x-x^a}{x-a} \).

Answer: e^2


Question: Determine whether the statement "For all positive integers b, there exists a positive integer c such that for all positive integers d, there exists a positive integer a satisfying \( \frac{a}{b} = \frac{c}{d} \)" is true or false. Justify your answer.

Answer: False


Question: Evaluate the integral: $$\int_{0}^{\pi} \frac{2\cos(x)-\cos(2021x)-2\cos(2022x)-\cos(2023x)+2}{1-\cos(2x)}\,\textrm{d}x$$ and verify that it equals $2022\pi$. Provide a brief explanation of your approach.

Answer: 2022\pi


Question: Can a unital abelian Banach algebra be embedded unitally into an abelian semi-simple Banach algebra?

Answer: No


Question: Can all elements of an $n \times n$ matrix $A$ be permuted using only pre- and post-multiplication by permutation matrices?

Answer: No


Question: Determine whether the power series \( \sum_{n=1}^{\infty}{n^n x^n} \) converges only at \( x=0 \).

Answer: Yes


Question: Evaluate the limit of the improper integral: \[ \lim_{n \to \infty} \int_0^{\infty} \frac{1}{1+x^n} \, dx. \]

Answer: 1


Question: Determine the number of distinct ways to form a committee of 17 politicians, where politicians are indistinguishable within their parties (Democrats, Republicans, and Independents). Each party must have at least two representatives, and no single party should hold a majority of the committee seats. Provide your answer as a single number.

Answer: 33


Question: Find the supremum of the expression \(|\cos(px) - \cos(qx)|\) for \(x \in [-1, 1]\), where \(p\) and \(q\) are natural numbers.

Answer: 2


Question: Evaluate the limit: $$ \lim_{n \rightarrow \infty} \left( \frac{n}{n-1} \right)^2 \left( \frac{1}{2} \right)^n \sum_{i=1}^n \binom{n}{i} \cdot \frac{i-1}{i} $$

Answer: 1


Question: Evaluate the integral \( \int_2^{10} g(y) \, dy \), where \( f(x) = x + x^3 \), \( g(y) = f^{-1}(x) \), and \( dy = 1 + 3x^2 \).

Answer: \dfrac{51}{4}


Question: Let $G$ be a group with subgroups $H$ and $K$. Suppose an automorphism $\phi$ of $G$ satisfies $\phi(H) \subseteq H$. Does it necessarily follow that $\phi(H) = H$? Provide a justification for your answer.

Answer: No


Question: Determine whether the integral \( \int_\pi^{+\infty}\frac{\sin t}{t}\mathrm dt \) is positive, negative, or zero.

Answer: Negative


Question: Determine all natural numbers \(n\) for which there exist \(n\) distinct natural numbers \(a_1, a_2, \ldots, a_n\) such that \(\sum_{i=1}^n \frac{1}{a_i^3} = 1\).

Answer: 1


Question: Determine whether the space of all polynomials of degree at most $n$ with real or complex coefficients, defined on a closed interval $[a,b]$, is complete with respect to the norm induced by the inner product \( \langle x, y \rangle = \int_a^b x(t) \overline{y(t)} \, \mathrm{d}t \).

Answer: Yes


Question:
Consider the quadratic equation $p(x) = ax^2 + bx + c = 0$ with real coefficients. If this equation has purely imaginary roots, what can be said about the roots of the equation $p(p(x)) = 0$? Choose the correct option:

a) only purely imaginary roots
b) all real roots
c) two real and two imaginary roots
d) neither real nor purely imaginary roots

Given that the roots of $p(x)$ are conjugate pairs, let them be $ki$ and $-ki$ where $k$ is a real number.

Answer: D


Question:
Let \( f:\mathbb{R}\rightarrow\mathbb{R} \) be a continuous function satisfying \[ f(x)=e^{\frac{x^2}{2}}+\int_0^x tf(t)\mathrm{d}t \] Determine which of the following is correct for \( f(\sqrt{2}) \):

(A) \( 5<f(\sqrt{2})<6 \)
(B) \( 2<f(\sqrt{2})<3 \)
(C) \( 3<f(\sqrt{2})<4 \)
(D) \( 4<f(\sqrt{2})<5 \)

Answer: A


Question: Determine whether there exist two real-valued functions $f_1$ and $f_2$ defined on a closed interval $[a, b]$ such that they agree at infinitely many points and disagree at infinitely many points within the interval.

Answer: Yes


Question: Find the upper bound for \( y \in \mathbb{R}^+ \) such that for all positive real numbers \( a, b, c, \) and \( d \) where \( b = n \cdot a \) and \( d = n \cdot c \), the inequality \( \frac{a+c}{b+d} \geq y \cdot \left( \frac{a}{b} + \frac{c}{d} \right) \) holds.

Answer: \dfrac{1}{2}


Question: Let $\pi_1, \pi_2, \ldots, \pi_n$ be a random permutation of the numbers $1, 2, \ldots, n$. Given that $\pi_k > \pi_1, \pi_k > \pi_2, \ldots, \pi_k > \pi_{k-1}$, find the probability that $\pi_k = n$.

Answer: \dfrac{k}{n}


Question: Let $(X, \mu)$ be a probability measure space. Suppose $f_n \to f$ pointwise and $f_n$ is dominated by some $\mu$-integrable function. Let $(A_n)_n$ be a sequence of measurable sets such that $\mu(A_n) \to 0$. Determine whether it is true that $\int_X f_n 1_{A_n} \, d\mu \to 0$. Provide a justification for your answer.

Answer: 0


Question: Given two numbers $x \in (0,1)$ and $y \in (0,1)$, determine whether the supremum of the expression \( \sup_{x,y\in (0,1)}\min_{n\ge 1} \frac{1-x}{(1-xy)x^{n-1}(1-y^n)} \) is finite or infinite.

Answer: \infty


Question: Provide an example of a sequence of numbers \( v_n \) such that its mean tends to a limit, but \( v_n \) itself does not converge to a limit.

Answer: (-1)^n


Question: Find the simplest possible relation between the holomorphic functions $f$ and $g$ in a certain region of the complex plane, given that $$|f(z)|^2 + |f(z)| = |g(z)|^2 + |g(z)|.$$

Answer: f(z) = e^{i\theta} g(z)


Question: Calculate the volume of the region in \( \mathbb{R}^n \) bounded by the hyperplanes \( x_i = 0 \) for each \( i \) and \( x_1 + x_2 + \cdots + x_n = 1 \).

Answer: \dfrac{1}{n!}


Question: Use the divergence theorem to evaluate the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{s} \), where \( \mathbf{F}(x,y,z) = x^3 \hat{\imath} + y^3 \hat{\jmath} + z^3 \hat{k} \) and \( S \) is the surface of the solid bounded by the cylinder \( x^2 + y^2 = 1 \) and the planes \( z = 0 \) and \( z = 2 \).

Answer: 11\pi


Question: Let \( \{U_\alpha\} \) be an open cover of the set \( \{x \in \mathbb{R}^n : \|x\| = 1 \} \), where \( n \ge 2 \). Determine whether there exists a \( \delta > 0 \) such that \( \{U_\alpha\} \) is also an open cover of the set \( \{ x \in \mathbb{R}^n : 1-\delta < \|x\| < 1 + \delta \} \).

Answer: Yes


Question: Determine whether there exists a natural number $N$, depending on $n \in \mathbb{N}$, such that every subset $A$ of $\{1, 2, \ldots, N\}$ ensures that either $A$ or its complement contains an arithmetic progression of length $n$.

Answer: Yes


Question: Given a Markov chain with state space $E = \{1, 2, 3, 4\}$ and transition matrix $$P = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0.4 & 0.6 & 0\\ 0.8 & 0 & 0.2 & 0\\ 0.2 & 0.3 & 0 & 0.5\\ \end{bmatrix},$$ compute the expected value of the product $E[f(X_5)f(X_6)|X_4 = 4]$, where $f$ assigns values $2, 4, 7,$ and $3$ to states $1, 2, 3,$ and $4$, respectively.

Answer: 14.41


Question: Evaluate the line integral \( \int_{\gamma} \alpha \) where \( \alpha = dx_1 - \frac{1}{x_1} dx_2 \) and \( \gamma(t) = (t^2, t^3) \) for \( t \in [0, 1] \). Determine if this integral can be evaluated solely based on the endpoint coordinates of \( \gamma \).

Answer: -2


Question: Suppose that \( T_1 \) is a \( \alpha \times 100\% \) lower confidence limit for \( \theta \) and \( T_2 \) is a \( \alpha \times 100\% \) upper confidence limit for \( \theta \). Further assume that \( P(T_1<T_2)=1 \). Determine a \( (2\alpha-1) \times 100\% \) confidence interval for \( \theta \).

Answer: [T_1, T_2]


Question: Evaluate the integral of the function \( f(x) \) over the interval \([0,1]\), where \( f(x) = 0 \) on the Cantor set \( C \) and \( f(x) = x \) on \([0,1] \setminus C\). Is the conclusion that the integral of \( f(x) \) is \( \frac{1}{2} \) valid, given that \( f(x) = x \) almost everywhere on \([0,1]\)?

Answer: \dfrac{1}{2}


Question: Given Stirling's approximation for $n!$ as $n! \sim c\sqrt{n}\left(\frac{n}{e}\right)^n$, find the constant $c$ such that $c = \sqrt{2\pi}$.

Answer: \sqrt{2\pi}


Question:
Given the equations:
1) \(x^{2}+y^{2}+z^{2}=6\)
2) \(w^{3}+z^{3}=5xy+12\)

Find \(\frac{\partial x}{\partial w}\) at a constant \(z\) at the point \((x,y,z,w) = (1,-2,1,1)\).

Answer: -\dfrac{2}{5}


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

Answer: -4\pi i


Question: Let $V$ be the region defined by $\frac{1}{4} \leq x^2 + y^2 + z^2 \leq 1$ in $\mathbb{R}^3$. Consider the vector field $\vec{F} = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{(x^2 + y^2 + z^2)^2}$ for $(x, y, z) \in V$. The surface $S$ is the part of the boundary of $V$ given by $x^2 + y^2 + z^2 = \frac{1}{4}$, with $\hat{n}$ denoting the outward unit normal vector. Compute the surface integral $\int\int_S \vec{F} \cdot \hat{n} \, dS$. \(\text{Express your answer as a single number.}\)

Answer: -8\pi


Question: Determine whether the sequence \( y_n = \frac{n^n}{n!} \) converges or diverges. If it converges, find the limit.

Answer: \infty


Question: Determine whether the sequence \( \left(\sum_{i=0}^{j}\frac{(-1)^{i}}{(2i+1)!}r^{2i+1}\right)_{j\in\mathbb{N}} \) is a Cauchy sequence.

Answer: Yes


Question: Find the minimum value of $x^4 + y^4 + z^4$ given that $yz + zx + xy = 3a^4$.

Answer: 3a^8


Question: Find the number of group homomorphisms from the multiplicative group of units modulo 12, denoted as $\mathbb{Z}_{12}^{\times}$, into the multiplicative group of units modulo 8, denoted as $\mathbb{Z}_8^{\times}$. Provide your answer as a single integer.

Answer: 16


Question:
Determine the number of integer solutions for the equation \(x_1 + x_2 - x_3 + x_4 - x_5 = 3\) subject to the following constraints:

\[ \begin{aligned}
&0 < x_1 \le 6, \\
&-8 \le x_2 < -2, \\
&x_3 \le 1, \\
&3 < x_4, \\
&2 \le x_5 \le 8
\end{aligned} \]

Answer: 2016


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

Answer: 1


Question: Evaluate the limit of \( f(x,y) = \frac{x^5 - y^7}{x^4 + y^4} \) as \((x,y)\) approaches \((0,0)\).

Answer: 0


Question: Find the greatest common divisor of \(3^{2023}+1\) and \(3^{2022}+1\) using modular arithmetic.

Answer: 2


Question: Evaluate the integral \( \int_{-1}^{1} \log(1-u^2)\,\mathrm{d}u \) using integration by parts.

Answer: 4 \ln 2 - 4


Question: Determine whether the following statement is true or false: If a sequence of continuous functions $f_n$ converges to $f$ in $L^1$, then $f_n$ converges to $f$ in $L^2$.

Answer: B


Question: Let $f$ be an entire function such that $f(0)=1$ and $|f(z)-e^z\cdot\sin 2z|<4$ for all $z$ in $\mathbb{C}$. Determine the value of $f(1)$. \( \boxed{} \)

Answer: 1 + e \sin 2


Question: Determine the relationship between the length of a space diagonal in an icosahedron and the radius of the sphere in which it is inscribed.

Answer: d = 2R


Question: Find the image of the set $A = \mathbb{R}^+ \times \mathbb{R}^+$ under the function $f(x, y) = (x - y^2)(y - x^2)$.

Answer: \mathbb{R}


Question: Determine the distribution of $X_2 - X_1$ as $n$ approaches infinity, where $X_1 \sim \text{Beta}(1, n)$ and $X_2 \sim \text{Beta}(n, 1)$ are independent random variables.

Answer: 1


Question: Determine whether the following statement is true or false: If for all closed intervals $[a, b] \subset \mathbb{R}$, a function $f$ is uniformly continuous on $[a, b]$, then $f$ is uniformly continuous on $\mathbb{R}$. Provide a justification for your answer.

Answer: False


Question: If a sequence of random variables \(\{X_n\}\) converges almost surely to a constant \(c\), does the sequence \(\{X_n^3\}\) converge almost surely to \(c^3\)? Answer "Yes" or "No" and justify your answer.

Answer: Yes


Question: Calculate the conditional expectation \( E(Y|X) \) given the joint probability density function \( f_{XY}(x,y) = 2e^{-(x+y)} \) for \( 0 < x < y \). The conditional density function is \( f_{Y|X}(y|x) = e^{x-y} \) for \( 0 < x < y \).

Answer: X + 1


Question: Given the sequence $\{a_n\}$ defined by $a_1 = 10$ and the recursive relation $a_n = \sqrt{\frac{a_{n-1}^2 + a_{n+1}^2}{2}}$ for $n = 2, 3, 4, \dots$, where all terms $a_n$ are positive integers, determine the general term $a_n$.

Answer: 10


Question: Provide an example of an entire function $g$ such that $g(n) = 2n$ for all natural numbers $n$. An entire function is a function that is holomorphic over the entire complex plane.

Answer: 2z


Question: Derive the gradient of \( \beta = x^TA^Tx \) with respect to \( x \), where \( x \) is an \( n \times 1 \) vector and \( A \) is an \( n \times n \) matrix that does not depend on \( x \).

Answer: (A + A^\top)x


Question: Evaluate the limit \( \lim_{{n \to \infty}} n\int_1^2 \frac{dx}{x^2(1+x^n)} \) without using Taylor expansion.

Answer: \ln 2


Question: Count the number of different sequences of match results that could lead to team A winning the NBA Finals, where the series ends as soon as team A wins 4 matches and team B has won at most 3 matches. Assume each match results in a victory for one of the two teams, with no ties.

Answer: 35


Question: Given a matrix $A$ and a vector $x \in \mathbb{R}^n$, if the norm of the time-varying vector $x(t) = e^{At}x$ decreases with time ($\frac{d}{dt}||x(t)|| < 0$) for all non-zero vectors $x$, does $||x(t)||$ converge to zero exponentially fast? Specifically, do there exist constants $\beta > 0$ and $\alpha > 0$ such that, for $||x|| \leq 1$, the inequality $$||x(t)|| \leq \beta e^{-\alpha t}$$ holds?

Answer: Yes


Question: Determine whether the following statement is true or false: A function with an integrable derivative is absolutely continuous.

Answer: B


Question: Determine the number of integer solutions \((x, n)\) to the equation \(x^2 + 217 = 3^n\), where \(n\) is a positive integer.

Answer: 0


Question: Find the remainder when \(20!\) is divided by 487.

Answer: 406


Question: Given non-negative real numbers $p$, $q$, and $r$ such that $p + q + r = 3$ and each number is less than or equal to $1.5$, find the maximum value of the expression $p^3 + q^3 + r^3 + 4pqr$. \( \text{Express your answer as a single number.} \)

Answer: \dfrac{243}{32}


Question: Is a well-ordered set of real numbers with the usual order always countable?

Answer: Yes


Question: Determine whether there exists a group with elements $a$ and $b$ such that the cyclic subgroups generated by $ab$ and $ba$ are equal ($\langle ab \rangle = \langle ba \rangle$), but the cyclic subgroups generated by $a$, $b$, and $ab$ are distinct.

Answer: Yes


Question: Determine the spectrum of a self-adjoint idempotent element in a unital $C^*$-algebra.

Answer: \{0, 1\}


Question: Calculate the sum \( \sum _{k=1}^n\:\begin{pmatrix}4n\\ 4k-1\end{pmatrix} \).

Answer: 4^{2n -1}


Question: Calculate the limit: \[ \lim_{n\to\infty}\left(\frac{1}{2}+\frac{3}{2^2}+\cdots+\frac{2n-1}{2^n}\right). \]

Answer: 3


Question: Let $(v_n)$ be a real sequence such that $v_n - v_{n+1} \sim \dfrac{1}{n^2}$. If $(v_n)$ converges to $0$, find a simple sequence equivalent to $(v_n)$. Provide your answer as a function of $n$.

Answer: \dfrac{1}{n}


Question: Determine whether the following statement is true or false: If \( \lim\limits_{x\to\infty}\int_a^x f(x,t)\,dt=0 \) for all \( a \), then \( \lim\limits_{x\to\infty}f(x,t)=0 \) for all \( t \), assuming \( f(x,t) \) is continuous in \( x \) and \( t \).

Answer: B


Question: Determine the number of matrices \( \mathbf{A} \in \mathbb{Z}^{m \times n}_q \) such that the nullspace of \( \mathbf{A} \) contains both vectors \((1,0,\ldots,0)\) and \((0,1,\ldots,0)\).

Answer: q^{m(n - 2)}


Question: Evaluate the limit: $$\lim _{t\to 0}\frac{\sqrt{1+t^2}-\sin{t}}{t^4}$$

Answer: \infty


Question: Given a positive function \( f \in \mathcal{S}(\mathbb{R}) \), find the value of \( \xi \) at which the function \( g(\xi) = |\hat{f}(\xi)|^2 \) attains its maximum, where \( \hat{f} \) denotes the Fourier transform of \( f \).

Answer: 0


Question: Find the coefficient of $x^3$ in the Maclaurin series expansion of $\frac{1}{1+\ln(1-x)}$. \( \)

Answer: \dfrac{7}{3}


Question: Determine whether an infinite set can be partitioned into an infinite number of infinite sets.

Answer: Yes


Question: A financial contract has a payoff of either 100 or 0 units at time N. Given that the price of this contract, S(n), is a martingale for any time 0 < n < N, and S(0) is initially valued at 47 units, determine the probability that the contract will be worth 100 units at maturity, time N.

Answer: 0.47


Question: Suppose $f(x)$ is a polynomial function that is continuous for all real numbers. Given that $f(2x) = f'(x) f''(x)$, determine the value of $f(3)$. \( \boxed{} \)

Answer: 12


Question: Let $g : \mathbb S^1 \to \mathbb R$ be a smooth function with no critical points, where $\mathbb S^1$ is the unit circle in $\mathbb R^2$. Let $\psi:\mathbb D^2 \to \mathbb R$ be the harmonic function corresponding to the Dirichlet problem imposed by $g$. Determine whether $d\psi$ is nonzero almost everywhere in $\mathbb D^2$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit of the function \( f(x,y) = \frac{xy}{|x|+|y|} \) as \( (x,y) \) approaches \( (0,0) \). Determine whether the limit exists and, if so, find its value.

Answer: 0


Question: Determine the total number of maximal ideals in the ring \( \mathbb{Q}[x]/\langle x^4-1\rangle \).

Answer: 3


Question: Solve the following system of equations for \(x\) and \(y\): \[\begin{cases}\sqrt{x^2+(y-2)(x-y)}\pm\sqrt{xy}=2y\\\sqrt{xy+x+5}-\dfrac{6x-5}{4}=\dfrac{1}{4}\left(\sqrt{2y+1}-2\right)^2\end{cases}\]

Answer: (4, 4)


Question: Let G be a group and K a normal subgroup in G of finite index. Determine whether the set X of all automorphisms \( \phi \) of G that fix K (meaning \( \phi(K)=K \)) has finite index in \( \text{Aut}(G) \).

Answer: No


Question: Given a circle \( O \) and two points \( A \) and \( B \) outside of circle \( O \), such that the line segment \( AB \) does not intersect circle \( O \), determine the number of circles that can be drawn tangent to circle \( O \) and passing through points \( A \) and \( B \).

Answer: 2


Question:
Evaluate the following limit as $x$ approaches $0$:
\[ \lim_{x \to 0} \left[ \ln \left( 1-(\cos x)^2 \right) - \ln x^2 \right] \]
Is the approach of moving the limit inside the logarithm valid, considering the continuity of the logarithm function for $x > 0$?

Answer: 0


Question: Determine whether multiple directed graphs can exist with the same in-degree and out-degree totals and the same number of edges.

Answer: Yes


Question: Find the area of the smallest trapezoid that can have an inscribed circle with a radius of 8 units.

Answer: 256


Question: Let $f:\Bbb{R}\to\Bbb{R}$ be a positive, increasing function such that $$\lim_{x\to\infty} \frac{f(3x)}{f(x)}=1.$$ Find the value of the limit $$\lim_{x\to\infty} \frac{f(2x)}{f(x)}.$$

Answer: 1


Question: Find the minimum period of the periodic function $g(x)$ given that for all $x$, \( g(x+2023) = \frac{1-g(x)}{1+g(x)} \).

Answer: 4046


Question: Determine the number of zeros of the function \( g(z) = z^4 - 2z - 2 \) within the annulus \( \frac{1}{2} < |z| < \frac{3}{2} \) using Rouche's Theorem.

Answer: 4


Question:
Find a closed-form analytic expression for the function \( f(x) \) given by:
\[
f(x) = \int_{0}^{2\pi } \left[ H\left(x + \left| \cos\left( \phi + \beta\right) \right| \right) - H\left(x - \left| \cos\left(\phi + \beta\right) \right| \right) \right] d\phi,
\]
where \(-1 \leq x \leq 1\), \(0 \leq \beta \leq \pi\), and \(H\) is the Heaviside step function.

Answer: 4 \arccos(|x|)


Question: Determine whether the function \( f(x) = \frac{1}{2\pi} \int_{S_1(0)} \Vert x - y \Vert \, d \lambda^1(y) \), where \( x \in B_1(0) \) and \( S_1(0) \) is the unit sphere in \( \mathbb{R}^2 \), is convex.

Answer: Yes


Question: Calculate the limit: \[ \lim_{x \rightarrow +\infty} \frac{\sqrt x + x^3 -\sqrt{x^6-2x^4} -x}{\log(1+e^{3\sqrt x})} \]

Answer: \dfrac{1}{3}


Question: Calculate the surface area of the portion of the plane $z = x + 3$ that is directly above the region in the $xy$-plane bounded by the circle $x^2 + y^2 = 1$. Provide your answer as a single numerical value.

Answer: \pi \sqrt{2}


Question: What is the degree of a hypersurface defined by a given homogeneous polynomial?

Answer: d


Question: Determine whether a hyperkähler structure can exist on a manifold of two real dimensions.

Answer: No


Question: Determine whether every integer $n$ can be expressed as the difference of two sums of cubes, $(a^3+b^2+c^2)-(d^3+e^2+f^2)$, where $a$, $b$, $c$, $d$, $e$, and $f$ are integers.

Answer: Yes


Question: In triangle ABC, where \(\angle A = 90^{\circ}\), M is the midpoint of BC. A point D is chosen on AC such that AD = AM. The circumcircles of triangles AMC and BDC intersect at C and P (other than C). Determine the ratio \(\frac{\angle ACB}{\angle PCB}\).

Answer: 2


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

Answer: 0


Question: Let $x > 0$. Solve the inequality: $$ \Gamma(x) \le \frac{1}{x} $$

Answer: (0, 1]


Question: Find the 1-dimensional Lebesgue measure of the set \( \Gamma^1 := \{ \{at\} \; : \; t \in \mathbb{R} \} \subset \mathbb{R} \), where \( a \) is an irrational number.

Answer: 1


Question: Evaluate the limit: $$\lim_{x\to0}\frac{x^2}{\ln(\int_{0}^{x^\frac{2}{3}}e^{\frac{1}{2}x^2}\,dx+1-x^\frac{2}{3})}$$

Answer: 6


Question: Determine the minimum order of a subgroup of the symmetric group $S_n$ that acts transitively on a set of $n$ points.

Answer: n


Question: Determine the number of roots of the function \( h(x) = x - e^{-Bx^2}\sin(x) \) over the real numbers, where \( B > 0 \).

Answer: 1


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

Answer: 6


Question: Evaluate the limit \(\lim_{n\to \infty}\left(\dfrac{f\left(a-\frac{2}{n}\right)}{f(a)}\right)^{1/n}\) where \(f:\mathbb{R}\to \mathbb{R}\), \(a\in \mathbb{R}\), \(f(a)>0\), and \(f\) is differentiable at \(a\).

Answer: 1


Question: Find the largest integer that is guaranteed to divide the product of $n$ consecutive integers starting from any positive integer $k$.

Answer: n!


Question: Consider the ring \( R = \left\{ \begin{pmatrix} a & a & a \\ a & a & a \\ a & a & a \end{pmatrix} \mid a \in \mathbb{R} \right\} \). Determine the number of ideals in this ring.

Answer: 2


Question: Does the Uniform Cauchy Criterion guarantee that a sequence of functions $f_n: E \to \Bbb{R}$ converges uniformly to a function $f: E \to \Bbb{R}$?

Answer: Yes


Question: Evaluate the integral of the function \( g(x,y)= \frac{\beta^2}{y}x^{-\beta-1} \mathbb{1}_{1\leq y\leq x} \) over its support to determine if it is a density function, where \( \beta >0 \).

Answer: 1


Question: Let \( f(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \frac{x^5}{5} \) and let \( g(x) = f^{-1}(x) \). Find \( g'''(0) \).

Answer: 1


Question: Consider the group $X=\langle h_1, h_2\rangle$ acting on $\mathbb{R}$, where $h_1(x)=-x$ and $h_2(x)=-x+1$. Find the order of the element $h_1 \cdot h_2$ in the group $X$. \(h_1 \cdot x = h_1(x)\).

Answer: \infty


Question: Provide an example of a non-commutative ring $A$ over a field $k$ that has no two-sided ideals and has infinite dimension over $k$. If such an example does not exist, explain why.

Answer: A_1(k)


Question: Determine whether the following statement is true or false: If $(f^2)'$ and $f'$ are in $L^1(0,\infty)$, then the limit of $f^2(x)$ as $x$ approaches infinity exists, and if $f$ is in $L^2$, then the limit of $f(x)$ as $x$ approaches infinity is 0.

Answer: True


Question: Determine the rank of the $n \times n$ matrix $A$ where the entry $a_{i,j} = \frac{1}{i+j}$ for all $1 \leq i, j \leq n$. Provide your answer as a single integer.

Answer: n


Question: If \( S \) is a regular and compact surface, does \( S \) have a finite number of connected components? Provide a justification for your answer.

Answer: Yes


Question: Given a function $f(x)$ such that $|f(x)| \le 1$ for all $x \in \mathbb{R}$, and the function $$g(x) = \frac{e^{f(x)} - e^{|f(x)|}}{e^{f(x)} + e^{|f(x)|}},$$ determine the maximum value of $g(x)$ if $f(x)$ covers the entire range $[-1, 1]$. \( \text{Provide your answer as a single number.} \)

Answer: 0


Question: Given the equation \(\sum_{r=0}^{50}z^r=0\), express the sum \(\sum_{k=1}^{50}\frac{1}{1-z_k}\) in terms of \(z\).

Answer: 25


Question: Suppose a solution of the differential equation \((xy^3 + x^2y^7)\frac{dy}{dx} = 1\) satisfies the initial condition \(y \left(\frac{1}{4}\right)=1\). Find the value of \(\dfrac{dy}{dx}\) when \(y = -1\). Choose from the following options: (A) \(\frac{4}{3}\) (B) \(-\frac{4}{3}\) (C) \(\frac{16}{5}\) (D) \(-\frac{16}{5}\).

Answer: D


Question: Evaluate the limit: $$\lim_{x \to \frac{\pi}{2}} \left(\tan \left(\frac{\pi}{4}\sin x\right)\right)^{\left(\frac{1}{\tan(\pi \sin x)}\right)}.$$

Answer: \sqrt{e}


Question: Determine whether the expression \( \frac{\binom{n}{1} a_1 + \binom{n}{2} a_2 + \cdots + \binom{n}{n} a_n}{2^n} \) converges to \( z \) as \( n \to \infty \), given that \( z \) is the limit of the sequence \( \{a_n\} \).

Answer: z


Question: A quadrilateral is inscribed in an ellipse given by the equation \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\). One of the diagonals of the quadrilateral is fixed along the line \(y=mx\). Find the maximum possible area of the quadrilateral if its other diagonal is a conjugate diameter of the ellipse.

Answer: 2ab


Question: Evaluate the integral \( \int_{-\pi}^{\pi}\ln(4e^2+e^2+4e^2\cos(\theta))d\theta \) when \( a = 2e \).

Answer: 4\pi(1 + \ln 2)


Question: Determine the number of points in the interval $(-\infty, \infty)$ where the graphs of $y = x^3$ and $y = x \cos x + \sin x$ intersect.

Answer: 3


Question: Determine the infimum of the sequence \( \frac{l_n}{n} \), where \( l_n = \sum_{k=1}^{n}\frac{1}{k} \) and \( l_n \sim \log n \).

Answer: 0


Question: Evaluate the function \( f(a,b) = \min\left(\max\left(a^2+b, b^2+a\right)\right) \) for all real numbers \( a \) and \( b \).

Answer: -\dfrac{1}{4}


Question: Determine the limit \( \lim_{n \rightarrow \infty} G_n \), where \( G_n \) is the geometric mean of the set \( \{\sin (\frac{\pi}{2} \frac{k}{n}) : 1 \leq k \leq n\} \) for \( n \geq 1 \).

Answer: \dfrac{1}{2}


Question: Evaluate the expression \( z = \sqrt{i} - \sqrt{-i} \) and find its value.

Answer: i\sqrt{2}


Question: A triangle is formed by selecting three lattice points (points with integer coordinates) at random with replacement within the square \([-99, 100] \times [-99, 100]\). Determine the probability that the area of this triangle (which may be degenerate) is an integer. Express this probability as a fraction \(\frac{m}{n}\), where \(m\) and \(n\) are coprime positive integers, and calculate \(m + n\).

Answer: 13


Question: Compute the operator norm of the integral operator \( T \) defined by \( Tf(x) = \int_0^1 \frac{f(y)}{1+x^2+y^2} \, dy \), where the norm is given by \( \|f\|_\infty = \sup_{x \in [0,1]} |f(x)| \).

Answer: \dfrac{\pi}{4}


Question: Determine the multiplicity of the point \( Q = (0:1:0) \) for the curve given by the equation \( G = X^4Z + X^4Y + Y^4Z \) using the method of partial derivatives.

Answer: 1


Question: Given the function $f(z) = z^{2} + z + 1$, find the radius of convergence of its reciprocal $g(z) = \frac{1}{f(z)} = \sum_{n=0}^{\infty} a_n z^{n}$. Compute the coefficients $a_0$, $a_1$, and $a_2$ in the Taylor expansion of $g(z)$ around $z=0$. The expansion is given by $g(z) = \frac{1}{ z^{2} + z +1} \approx 1 - z + \mathcal{O}(z^{3})$, implying $a_0 = 1$, $a_1 = -1$, and $a_2= 0$. Determine the radius of convergence.

Answer: 1


Question: Find the norm of the linear operator \( A: l_2 \rightarrow l_2 \) defined by \( A(x_1,x_2,...) = (x_1, \frac{1}{2}x_2,...,\frac{1}{2^{n-1}}x_n,...) \).

Answer: 1


Question:
Find the product of \(\tan x\) and \(\cot x\) using the series definitions:
\[\tan(x) = \sum_{n=1}^{\infty} \frac{B_{2n}(-4)^n(1-4^n)}{(2n)!} x^{2n-1}\]
\[\cot(x) = \sum_{n=0}^{\infty} \frac{(-1)^n 2^{2n} B_{2n}}{(2n)!} x^{2n}\]

Answer: 1


Question: Given n balls of different colors arranged in a row, determine the expected number of swaps needed to rearrange them such that they satisfy a condition defined by a function f. Assume that at least one arrangement exists that satisfies the condition, and a function g provides all possible arrangements. What is the expected number of swaps required?

Answer: n - H_n


Question: Find the sum of the series \( \sum\limits_{n=1}^{\infty} \frac{(n+1)^3}{2^n} \) using a recursive method.

Answer: 51


Question: Find the area of a right triangle if the ratio of its legs is 1.05 and the difference between the radii of its inscribed and circumscribed circles is 17 units.

Answer: 840


Question: Determine whether for any two points \(x\) and \(y\) in the Cantor set, there exists a Möbius transformation \(h: \mathbb{C} \to \mathbb{C}\) such that \(h(x) = y\).

Answer: Yes


Question:
Calculate the limit as \(x\) approaches 0:

\[\lim_{x \to 0} \frac{x - \sin^{(150)}(x)}{x^3}\]

where \(\sin^{(150)}(x)\) represents the sine function applied 150 times recursively to \(x\).

Answer: 25


Question: Find the last two digits of the number $9^{9^9}$. Provide your answer as a two-digit number.

Answer: 89


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

Answer: x^2


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

Answer: \dfrac{4}{e}


Question: Determine whether a function $f(x_{1},x_{2},\cdots,x_{n})$ with partial derivatives of bounded variation must belong to the Sobolev space $W^{1,\infty}$. Provide a justification for your answer.

Answer: No


Question: Evaluate the limit: $$\lim_{x \to 1} \frac{\int_1^x \frac{t^{p-1}}{t^2 \ln t} \, dt}{\frac{1}{x-1}}$$ where $p$ is a positive integer.

Answer: 0


Question: Find the residue of the function \( \frac{\sin(z)}{4(z-\frac{\pi}{6})^{3}} \) at the pole \( z=\frac{\pi}{6} \).

Answer: -\dfrac{1}{16}


Question: Given non-negative numbers $x, y, z$ such that $x + y + z \leq \frac{1}{2}$, find the maximum value of the expression \[ S = 2(x + y + z) - 3(xy + yz + zx) + 4xyz. \]

Answer: 1


Question: Determine whether a non-principal ideal can be contained within a principal ideal such that the minimum size of a generating set for the non-principal ideal is larger than that of the principal ideal.

Answer: Yes


Question: Let $(X,d)$ and $(Y,D)$ be two metric spaces, and let $f: X \to Y$ be a homeomorphism. Define a new metric $p$ on $X$ by $p(x,y) = D(f(x),f(y))$ for all $x, y \in X$. Are the metrics $d$ and $p$ equivalent on $X$?

Answer: Yes


Question: Determine whether the function \( f(v) = v^T Q v \) is strongly convex with respect to the norm \( ||v|| = \sqrt{v^T Q v} \), given that \( Q \) is a positive definite matrix.

Answer: Yes


Question: Determine the natural number values of $k$ for which the identity \(1^k + 2^k + \cdots + n^k = (1+2+\cdots+n)^{k-1}\) holds for all natural numbers $n$.

Answer: 3


Question: Calculate the determinant of the matrix \(C=\begin{pmatrix}1 &0&c&0\\0&1&0&c\\c&0&1&0\\0&c&0&1\end{pmatrix}\) by computing its eigenvalues.

Answer: (1 - c^2)^2


Question:
Find all functions \( g: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the functional equation:
\[
g(g(x)+y^2+1)+2y=x+(g(y+1))^2
\]
for all \( x, y \in \mathbb{R} \).

Answer: g(x) = x


Question: Determine the value of \( a \) for which the set of vectors \( \{ u_1+4u_2, u_2+3u_3, u_3+3u_4, u_4+au_1 \} \) is linearly dependent, given that \( u_1, u_2, u_3, u_4 \) are linearly independent vectors in \( \mathbb{R}^n \).

Answer: \dfrac{1}{36}


Question: Determine whether the sum of an uncountable number of zeros is always zero.

Answer: 0


Question: Suppose $X_1, X_2, \ldots, X_n$ are independent random variables uniformly distributed on the interval $[0, 1]$. Find the expected value of the ratio of the second largest to the largest of these random variables.

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


Question: For a given \(\epsilon > 0\), determine if all zeros of the function \(g_{n}(z) = \sum_{k=0}^{n} \frac{1}{k!z^{k}}\) are contained within the disk \(D(0; \epsilon)\) for sufficiently large \(n\).

Answer: Yes


Question: Compute the integral \( \int_{|z|=1} \text{Re}(z)e^{z^2} \, dz \) using Cauchy's theorem by finding a holomorphic function \( f \) such that \( f(z) = \text{Re}(z) \) on the unit circle \( |z|=1 \).

Answer: \pi i


Question: Compute the integral \( \int_0^1 \frac{\log(x)\log(1-x)}{x} \, dx \) and determine if it equals Apéry's constant, \( \zeta(3) \).

Answer: \zeta(3)


Question: Is it always possible to select 11 numbers from any permutation of the sequence $1, 2, 3, \ldots, 101$ such that the selected numbers form either an increasing or decreasing sequence?

Answer: 11


Question: Determine whether the following statement is true or false: An abelian group is uniquely divisible by a prime number $p$ if and only if it can be represented as a module over the ring $\mathbb{Z}[\frac{1}{p}]$. Provide a justification for your answer.

Answer: True


Question: Compute the integral \( \int_0^\infty \frac{\sin(x)}{x}dx \) using complex integration with the function \( f(z) = \frac{e^{iz}}{z} \).

Answer: \dfrac{\pi}{2}


Question: Determine whether the following statement is true or false: A $\mathcal C^1$ function $f:\mathbb R^d\rightarrow \mathbb R$ that is convex is strictly convex if and only if $\nabla f$ is injective.

Answer: True


Question: Determine whether polynomials of the form \( \sum_{n=-N}^n a_n z^n \) can approximate any function in \( C(S^1,\mathbb{C}) \).

Answer: Yes


Question: Can the Petersen graph be used as a counterexample to refute the assertion that "Every graph $G$ without an induced $K_{3,3}$ or $K_5$ subgraph is planar"?

Answer: Yes


Question: For any integer $n \ge 1$, determine whether at least one of the orthogonal groups $O^+(2n,2)$ or $O^-(2n,2)$, consisting of $2n \times 2n$ matrices over the field of order 2, contains an element of order $2n+1$. Provide a justification for your answer.

Answer: Yes


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

Answer: [-1, 0]


Question: Given that $x^2 \equiv y^2 \pmod {p^r}$ and $p$ is a prime not dividing $x$ or $y$, does it follow that $p^r$ does not divide any linear combination of $x$ and $y$?

Answer: No


Question:
Let $A$ and $B$ be linear operators on a finite-dimensional vector space $V$ over $\mathbb{R}$ such that $AB=(AB)^2$. If $AB$ is invertible, determine which of the following statements is false:
(a) $AB = BA$
(b) $\operatorname{tr}(B)$ is non-zero
(c) $0$ is an eigenvalue of $A$
(d) $1$ is an eigenvalue of $B$

Answer: C


Question: Find all positive integer solutions \((a, b, c)\) such that for the numbers \(x = 60a + 13b\) and \(y = 60a + 11c\), the equation \(4x^2 - y^2 = 2020\) holds. Prove that no other solutions exist.

Answer: (4, 1, 24)


Question: Find the maximum value of \(|z^2 - 2iz + 1|\) given that \(|z| = 3\).

Answer: 14


Question: Given the determinant \( A = \begin{vmatrix} f(x) & g(x) & h(x) \\ i(x) & j(x) & k(x) \\ l(x) & m(x) & n(x) \end{vmatrix} \), where \( f(x), g(x), h(x), i(x), j(x), k(x), l(x), m(x), n(x) \) are functions of \( x \), determine the power of \( x-a \) in the factorization of \( A \) if substituting \( x = a \) makes exactly three rows identical.

Answer: 2


Question: Let \( E \) and \( F \) be Banach spaces. Suppose \( f: [0,1] \times E \to F \) is continuously differentiable. Determine if the function \( g: E \to C^1([0,1],F) \), defined by \( y \mapsto f(\cdot,y) \), is differentiable. Specifically, verify if the derivative \( \partial g(y)h \) at a point \( y \) in \( E \) with respect to the direction \( h \) in \( E \) is given by \( (\partial g(y)h)(t) = \partial_2f(t,y)h \) for all \( t \) in \( [0,1] \).

Answer: (\partial g(y)h)(t) = \partial_2f(t,y)h


Question: Determine the cardinality of the set of all geometric surfaces, assuming the general continuum hypothesis (GCH). If \(\aleph_1\) is the cardinality of the set of all points and \(\aleph_2\) is the cardinality of the set of all curves, is the cardinality of all surfaces \(\aleph_3\)?

Answer: \aleph_3


Question:
Determine whether there exists a set of numbers $S = \{s_1, s_2, \ldots, s_n\}$ such that the average of the numbers in $S$ equals the sum of the proper divisors common to all elements of $S$. In other words, can a "perfect set" exist where:

$$\frac{s_1+s_2+\cdots+s_n}{n} = \sum_{d\mid s_1\wedge d\mid s_2\wedge d\mid s_3\wedge \cdots \wedge d\mid s_n,\, d\ne s_1,\, d\ne s_2,\, \cdots,\, d\ne s_n}d$$

Answer: Yes


Question: Let \( m \) be the number of positive integers \( n \), where \( 1 \le n \le 2005 \), such that the polynomial \( x^{2n} + 1 + (x + 1)^{2n} \) is divisible by \( x^2 + x + 1 \). Find the remainder when \( m \) is divided by 1000.

Answer: 337


Question:
Let $V$ be the space of real polynomials in one variable $t$ of degree less than or equal to three. We define an inner product on $V$ as follows:

$$ \langle p, q \rangle = p(1)q(1) + p'(1)q'(1) + p''(1)q''(1) + p'''(1)q'''(1). $$

Given the polynomials $p(t) = -4 + 9t - 6t^2 + t^3$ and $q(t) = 3 - 6t + 3t^2$, determine the angle between them using this inner product.

Answer: \dfrac{3\pi}{4}


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

Answer: 0


Question: Determine the number of non-singular matrices of the form \(A=\begin{bmatrix} 1 & a & b\\ w&1 &c \\ w^2 &w & 1 \end{bmatrix}\) where \(a, b, c\) are elements of the set \(S=\{w, w^2\}\), and \(w\) is a complex cube root of unity with \(w \neq 1\).

Answer: 2


Question: Evaluate the limit: $$ \lim_{x\rightarrow \infty}\frac{\ln(1+\sin(2x))}{x} $$

Answer: 0


Question: Find the limit of the sequence \( \lim_{n \to \infty} \sqrt[n]{\frac{n^n}{n!}} \).

Answer: e


Question:
Find the determinant of an \((n-1)\times (n-1)\) matrix where the diagonal entries are \(n-1\) and all off-diagonal entries are \(-1\).

\[
\begin{pmatrix}
n-1 & -1 & \cdots & -1 \\
-1 & n-1 & \cdots & \vdots \\
\vdots & \vdots & \ddots & \vdots \\
-1 & \cdots & \cdots & n-1
\end{pmatrix}
\]

Answer: n^{n-2}


Question: Let $R$ be a commutative ring with unity. Determine if a submodule of a finitely generated $R$-module can be isomorphic to a quotient of $R$. Provide a justification for your answer.

Answer: Yes


Question: Determine the distance from the center of a homogeneous spherical planet at which the gravitational force is at a maximum.

Answer: R


Question: In measure theory, determine whether the following statement is true or false: If the preimage of every measurable set is measurable for a function $f$, then $f$ is measurable.

Answer: A


Question: Calculate the determinant of the matrix expression \( \det \left((A^2 B^{-1})^{-1}+BC \right) \) given that \( \det(A) = 2 \), \( \det(B) = -8 \), and \( \det(A^2 C+I)=1 \), where \( A, B, C \in \mathbb{R}^{n \times n} \).

Answer: -2


Question: If a topological space is homeomorphic to a metric space, and another topological space is homeomorphic to the first space, is the second space also metrizable?

Answer: Yes


Question: Consider a sequence in $L^2(\mathbb{R}^n)\cap L^4(\mathbb{R}^n)$. If the sequence is bounded in $L^4$ and converges in $L^2$, does the limit belong to $L^4(\mathbb{R}^n)$? Answer yes or no.

Answer: Yes


Question: Calculate the expected squared distance of the endpoint from the origin for a random walk consisting of n line segments, each of unit length.

Answer: n


Question: Determine the positive value of $c$ such that the equation $\log x = cx^4$ has exactly one real solution for $x$. Provide your answer as a single number.

Answer: \dfrac{1}{4e}


Question: Determine the value of \( C \) such that the equation \( \log x = Cx^4 \) has exactly one root.

Answer: \dfrac{1}{4e}


Question: Determine the coefficient of $z$ in the Laurent series expansion of $\frac{e^z}{z-1}$.

Answer: -2


Question: Evaluate the limit: $$\lim_{(x,y)\to(1,1)} \dfrac{(x-y)(x+y-2)}{\sqrt{(x-1)^2+(y-1)^2}}$$

Answer: 0


Question: Evaluate the expression \( \sqrt{1+ 2 \sqrt{1+3 \sqrt{1+\dots\infty}}} \).

Answer: 3


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

Answer: e


Question: Evaluate the sum of the series: \[ \sum_{n=0}^{\infty} \arctan\left(\frac{1}{n^2+n+1}\right) \]

Answer: \dfrac{\pi}{2}


Question: Let $G$ be a group of order 30 with a center $Z(G)$ of order 5. Determine the isomorphism class of the quotient group $G/Z(G)$. To which commonly known group is $G/Z(G)$ isomorphic?

Answer: S_3


Question: Evaluate the limit \( \lim_{x \to \infty}\int_0^x{e^{-x^2+t^2}}\,dt \).

Answer: 0


Question: Evaluate the limit \(\lim_{n\to\infty}\left(\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{2n}\right)\) using sequences, without employing integration techniques.

Answer: \ln 2


Question: Find the limit superior (limsup) of the expression \( \sqrt[n]{(n+1)^{n+1}e^{-(n+1)^{1.002}}} \) as \( n \to \infty \).

Answer: 0


Question: Determine the remainder when \( \prod_{k=0}^{n-1}\left(p^n-p^k\right) \) is divided by \( n! \), where \( p \) is a prime number and \( n \) is an integer greater than 1, using only arithmetical arguments.

Answer: 0


Question: Does there exist a model of ZF¬C in which there is a function \( f: \mathbb{R} \to \mathbb{R} \) such that \( f \) is sequentially continuous at some \( a \in \mathbb{R} \) but not \( \epsilon-\delta \) continuous?

Answer: Yes


Question: Evaluate the limit \( \lim_{x \to 0}{\frac{\sin{2x}-2x}{x^3}} \) without using L'Hopital's rule or Taylor series.

Answer: -\dfrac{4}{3}


Question: Determine whether the matrix representation of the operator $T-S$ with respect to the basis $B$ is diagonal, given that $B = (v_1, v_2, ..., v_n)$ is a set of linearly independent eigenvectors of $T$, and $S(v) = \alpha v$ for $\alpha \neq 0$. Assume $T$ is diagonalizable with eigenvalues $\lambda_1, \lambda_2, ..., \lambda_n$ and $S$ has $\alpha$ as its diagonal entries in the basis $B$. Answer "Yes" if it is diagonal, otherwise "No".

Answer: Yes


Question:
Evaluate the limit of the complex sequence \( W_n \) defined by:

\[ W_n = \left(1 - \frac{2i}{n} \right)^n \]

Answer: e^{-2i}


Question: Determine the possible values of \( n \) for which the set \( S_n \), the collection of all functions that are continuously differentiable up to order \( n \) from \([0,1]\) to \( \mathbb{R} \), is not an integral domain. Note that \( S_1 \) and \( S_2 \) are not the answer.

Answer: n \geq 3


Question: Find the values of $c$ for which the series \( \sum_{t=1}^{\infty} t^2 c^{t-1} \) converges.

Answer: (-1, 1)


Question: Find the value of $x+y$ given that \((x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3.\)

Answer: 0


Question:
Determine whether the following limit in probability holds for a sequence of independent and identically distributed positive random variables $X_1, \dots, X_n$, each following a $\chi^2(n)$ distribution:

$$\lim_{n \to \infty} E\left(\frac{n^2X_iX_j}{\left(\sum_{k=1}^n X_k\right)^2}\right) = 1 \quad \forall i, j$$

Note that the distribution of the random variables depends on $n$. Provide a justification for your answer.

Answer: 1


Question: Determine the value of $t > 0$ that minimizes the length of the arc of the curve \( y = x^3 + x^2 - \frac{29x}{2} + 1 \) over the interval \( x \in [t, t+1] \).

Answer: \dfrac{4}{3}