Question: Let $M \subset \mathbb{R}^3$ be a regular surface and $A \in M$ a point where the Gaussian curvature $K$ and the mean curvature $H$ satisfy $K \neq 0$ and $H = 0$. Let $\alpha, \beta: (-1,1) \to M$ be regular curves such that $\alpha(0) = \beta(0) = A$ and the angle between their tangents at this point is not zero. Also, $\alpha''(0), \beta''(0) \in T_A M$. Find the angle between the curves at point $A$.

Answer: \dfrac{\pi}{2}


Question: Verify the property of the inverse Fourier Transform: Show that applying the inverse Fourier Transform to the Fourier Transform of a function \( f(t) \) returns the original function \( f(t) \).

Answer: f(t)


Question: Let \( f_n: \mathbb{R} \to \mathbb{R} \) be a sequence of continuous functions converging uniformly on compact sets to \( f: \mathbb{R} \to \mathbb{R} \). Let \((X_n)_{n\in\mathbb{N}}\) be a collection of independent and identically distributed random variables that are integrable. Determine whether \( \lim_{n\to\infty} (f_n(X_n) - f(X_n)) = 0 \) with probability one.

Answer: 0


Question: Compute the integral \( \int_{\gamma(1;1/2)} \frac{\cos(\pi z)}{z^{2}-3z+2} \, dz \) using Cauchy's integral formula, where \( \gamma(1;1/2) \) is the circle centered at 1 with radius 1/2.

Answer: 2\pi i


Question: Determine the codimension of the variety defined by the zero set of the determinants of all $m \times m$ submatrices of a symbolic $m \times n$ matrix, where $m \leq n$. Express your answer in terms of $m$ and $n$.

Answer: n - m + 1


Question: Let $H$ be a subgroup of a group $G$. Construct an injective function from $G/H$ (the set of left cosets of $H$) to $H\setminus G$ (the set of right cosets of $H$). Note that $|G/H|=|H\setminus G|$. Provide the function and justify its injectivity.

Answer: gH \mapsto Hg^{-1}


Question: Given the function \( f(n) \) defined as follows: \( f(n) = n - 2 \) for \( n > 3000 \) and \( f(n) = f(f(n+5)) \) for \( n \leq 3000 \), find the value of \( f(2022) \).

Answer: 3001


Question: Find the coefficient of $z^4$ in the power series expansion of $\frac{z}{\sin z}$.

Answer: \dfrac{7}{360}


Question: Provide an example of a primary ideal in a commutative ring with identity that is not a prime ideal.

Answer: (4)


Question: Compute the following limit analytically: \[ \lim_{n \to \infty} \sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}{\binom{n-k}{k}\frac{1}{2^{n-k}}}. \]

Answer: \dfrac{2}{3}


Question: Determine whether the following statement is true or false: Every subset of a measurable set is measurable with respect to the same measure.

Answer: B


Question: Do the functions of the form $\psi(x)\phi(y)$ span $L^2(\mathbf{R}^6)$?

Answer: Yes


Question: Determine whether every even perfect number satisfies one of the following congruences: \(n \equiv 1 \ (\text{mod 5})\) or \(n \equiv 3 \ (\text{mod 5})\).

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to0} |x|^{\frac{3}{2}}\ln|x| \]

Answer: 0


Question: Solve the differential equation \( y + xy' = a(1 + xy) \) with the initial condition \( y\left(\frac{1}{a}\right) = -a \).

Answer: -\dfrac{1}{x}


Question: Let \( f:[a, b] \rightarrow \mathbb{R} \) be a Riemann integrable function. If the set \( X = \{ x \in [a, b] ; f(x) \neq 0 \} \) has an empty interior, prove that \( \int_{a}^{b} |f(x)| \, dx = 0 \).

Answer: 0


Question:
Given a sequence \( \{a_n\} \) defined by the recurrence relation \( a_n = a_{n-1} + 2a_{n-2} - a_{n-3} \) with initial conditions \( a_1 = a_2 = \frac{a_3}{3} = 1 \), find the value of the infinite series:
\[
a_1 + \frac{a_2}{2} + \frac{a_3}{2^2} + \cdots
\]

Answer: 8


Question: Find the surface area of the cylinder defined by the equation \(x^2 + y^2 = 2x\), which is bounded by the cone \(z = \sqrt{x^2 + y^2}\) and the plane \(z = 0\).

Answer: 8


Question: Let $X$ be a standard normal random variable with mean $0$ and variance $1$. Let $\Phi$ be the cumulative distribution function of $X$. Find the expectation of $\Phi(X)$. Express your answer as a single value.

Answer: \dfrac{1}{2}


Question: Consider the sequence defined by the following rule: A natural number $n$ is a member of the sequence if, when taking all divisor pairs of $n$ (excluding $1$ and $n$ itself), dividing the larger of each pair by the smaller, and multiplying the results, the product equals $n$. For example, for $n=24$, the divisor pairs are $(2,12)$, $(3,8)$, and $(4,6)$. Dividing the larger by the smaller gives $6$, $8/3$, and $3/2$, and their product is $24$. The sequence starts with $1, 24, 30, 40, 56, 64, 70, 105, 135, 154, \\ldots$. With the exception of $1$ and $p^6$ (where $p$ is prime), all members of this sequence appear to have exactly 8 divisors. Is it true that for all $n$ in this sequence, the number of divisors $\sigma_0(n) = 8$?

Answer: Yes


Question: Find the relative maximum and minimum values of the function \( f(x, y) = x^4 + y^4 - 2x^2 + 4xy - 2y^2 \).

Answer: -8


Question: Is a commutative ring $R$ with $1$ indecomposable as an $R$-module over itself? Answer yes or no.

Answer: No


Question: Evaluate the limit: \[ \lim_{x\to 2} \left(\frac{e^{x+1}-e^3}{\ln(x)-\ln (2)}\right) \] without using L'Hôpital's Rule.

Answer: 2e^3


Question: Show that the linear operator $T: C(\mathbb{R}) \to C(\mathbb{R})$ defined by $T(u(x)) = \int_0^x u(s) \, ds$ does not admit eigenvalues.

Answer: T


Question: Determine the interval of convergence for the power series: $$\sum_{n=1}^\infty \left(\sum_{k=1}^{n}\frac{1}{k}\right)x^n$$

Answer: (-1, 1)


Question: Let \( f: C \to G \) be a covering map of graphs. If \( T \) is a tree in \( C \), is \( f(T) \) a tree in \( G \)? Provide a justification for your answer.

Answer: No


Question: Find the limit of \( n \sin (2 \pi \cdot e \cdot n!) \) as \( n \) approaches infinity.

Answer: 2\pi


Question: In an obtuse triangle $ABC$ with obtuse angle at $B$, the incenter $I$ and the centroid $G$ are such that $IG \parallel BC$. The perimeter of the triangle is $144$. Determine the maximum integer value of the segment $IG$.

Answer: 7


Question: Suppose \( f \) is a continuous function on \( \mathbb{R} \) such that \( f(x+1) = f(x) \) for all \( x \). Does this imply that \( f \) is uniformly continuous on \( \mathbb{R} \)? Justify your answer.

Answer: Yes


Question:
Given the Legendre differential equation:
\[ (1-x^2)y'' - 2xy' + n(n+1)y = 0 \]
find the value of \( n \) such that the \( n^{th} \) degree polynomial solution \( y_n(x) \) satisfies \( y_n(1) = 3 \) and the integral
\[ \int_{-1}^{1} [y_n^2(x) + y_{n-1}^2(x)] \, dx = \frac{144}{15}. \]

Answer: 2


Question: Determine whether the Stieltjes integral \( \int_{0} ^{4} x^{2} d([x^{2}]) \) exists. If it exists, calculate its value.

Answer: 136


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

Answer: 2


Question: Find the value of \( f^{-1}(3) \) for the function \( f(x) = x^5 + x^3 + x \).

Answer: 1


Question: Evaluate the integral \( \int_0^\pi \int_0^\pi |\cos(x+y)| \, dx \, dy \).

Answer: 2\pi


Question:
Find the value of the limit:
\[ \lim_{n\rightarrow\infty}\left[\frac{f(x)-n}{(f(x))^2-\frac{n^3(n+2)}{4}}\right] \]
where \( f:(0,\infty)\rightarrow\mathbb{N} \) is defined as
\[ f(x)=\left[\frac{x^2+x+1}{x^2+1}\right]+\left[\frac{4x^2+x+2}{2x^2+1}\right]+\left[\frac{9x^2+x+3}{3x^2+1}\right]+\cdots+\left[\frac{(nx)^2+x+n}{nx^2+1}\right] \]
and \([.]\) denotes the greatest integer function.

Answer: 2


Question: Person A chooses 60 numbers randomly from the set \( \{1, 2, \ldots, 100\} \). Person B chooses 30 numbers randomly from the set \( \{1, 2, \ldots, 50\} \). What is the expected value of the number of numbers chosen by person A that are not chosen by person B?

Answer: 42


Question: Simplify the following fraction: \[ \frac{-a^3 (c^2 - b^2) + b^3 (c^2 - a^2) - c^3 (b^2 - a^2)}{(c-b)(c-a)(b-a)} \]

Answer: -ab - bc - ca


Question: Let \( f(x) = \frac{1}{2x+1} \) for \( x \in [0,1] \). Apply the Mean Value Theorem on the interval \([0,x]\) where \( x \in (0,1) \), to find a point \( c(x) \in (0,x) \). If \( l = \lim_{x\to 0} \frac{c(x)}{x} \), determine the value of \( l \).

Answer: \dfrac{1}{2}


Question: Jimmy Butler, a shooting guard, makes 30% of his three-point shot attempts. In any game, the number of three-point shots he attempts follows a Poisson distribution with a mean of 5. What is the probability that he makes exactly 2 three-point shots in a random game?

Answer: 0.251


Question: Let \( p(x) \) be a polynomial with integer coefficients. Let \( a, b, c \) be distinct integers. Is it possible that \( p(a) = b, p(b) = c, p(c) = a \)? Answer yes or no.

Answer: No


Question: Calculate the flux of the vector field \( F = (x^3, 4y^3, z^3) \) through the surface defined by \( x^2 + 4y^2 + z^2 = 1 \).

Answer: \dfrac{6\pi}{5}


Question: Consider an i.i.d. discrete random sequence \(\{X_i\}\) with \(EX_1 \neq 0\). Define \(S_n := \sum\limits_{i=1}^n X_i\) and let \(R\) be the set of recurrent values for \(S_n\). A value \(x\) is recurrent for \(S_n\) if for every \(\varepsilon > 0\), \(P( |S_n - x| < \varepsilon\; \text{infinitely often})=1\). Prove that the set \(R\) is empty.

Answer: R = \emptyset


Question: Evaluate the integral \( \int_C \cot z\ \mathrm{d}z \), where \( C(t) = 2e^{it} \) and \( t \in [0, 2\pi] \), using the residue theorem.

Answer: 2\pi i


Question: Find the greatest common divisor (GCD) of the expressions \( \frac{a^{2m+1}+1}{a+1} \) and \( a+1 \).

Answer: 1


Question: Determine the number of distinct values of the expression \( \frac{k}{3} + \frac{m}{8} \) that are less than 6, where \( k \) and \( m \) are positive integers.

Answer: 126


Question: In Terence Tao's Analysis I, a number \(x\) is defined as eventually \(\epsilon\)-close to a sequence \((a_n)\) if there exists an \(N\) such that for all \(n \geq N\), \(d(x, a_n) < \epsilon\). A number \(x\) is continually \(\epsilon\)-adherent to \((a_n)\) if it is \(\epsilon\)-adherent to \((a_n)_{n=N}^{\infty}\) for every \(N\). Is it true that if \(x\) is eventually \(\epsilon\)-close to \((a_n)\), then \(x\) is continually \(\epsilon\)-adherent to \((a_n)\)? Justify your answer.

Answer: Yes


Question: Let $I$ be the identity operator in $l^2(\mathbb{Z})$ and $B$ be a compact operator in $l^2(\mathbb{Z})$. Suppose $I + B$ is invertible. Define $B^N$ as a truncated version of $B$ such that its elements are the same as $B$ when both $i$ and $j$ are less than $N$, and equal to zero when at least one of $i$ or $j$ are greater than or equal to $N$. Is $I + B^N$ invertible for sufficiently large $N$?

Answer: Yes


Question:
Let $\mathcal{H}$ be a Hilbert space with an orthonormal basis $(e_k)_{k \in \mathbb{N}}$. For vectors $x, y \in \mathcal{H}$, suppose:
\[ \langle x, e_k \rangle = \frac{1}{2^k}, \quad \langle y, e_k \rangle = (-1)^k \frac{1}{2^k} \quad \text{for all } k \in \mathbb{N}. \]
Calculate the inner product $\langle x, y \rangle$. Given that $\|x\| = \|y\| = \sqrt{\frac{1}{3}}$, find $\langle x, y \rangle$. \( \boxed{} \)

Answer: -\dfrac{1}{5}


Question: Calculate the limit: $$\lim _{x\to 0}\left(\frac{xF\left(x\right)}{e^{x^2}}\right)$$ where $F$ is a primitive function of $e^{x^2}$.

Answer: 0


Question:
Given a Gaussian distribution \(\mathcal{N}(\mu,\sigma^2)\) with unknown \(\mu\) and known \(\sigma^2 = 0.1\), and a single data point \(x = 2\), determine the posterior mean \(\mu_1\) using the formula:
\[\mu_1 = x - (x - \mu_0)\frac{\sigma^2}{\sigma^2 + \sigma_0^2}\]
where \(\mu_0\) and \(\sigma_0^2\) are the mean and variance of the prior distribution. Assume \(\mu_0 = 0\) and \(\sigma_0 = \infty\). What is the value of \(\mu_1\)?

Answer: 2


Question: Find the gradient \( \frac{df}{dX} \) of the function \( f(X) = \text{trace} \left( X^T A X \right) \).

Answer: (A + A^\top)X


Question: Evaluate the integral \( \int_C \frac{2z^2-5z+1}{z-1} \, dz \) where \( C \) is the path defined by \(|z-1|=1\).

Answer: -4\pi i


Question: Evaluate the limit \( \lim_{x \to \pi/2} (\tan x)^{\tan(2x)} \) without using l'Hôpital's rule.

Answer: 1


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

Answer: e


Question: Determine the number of group homomorphisms from \((\mathbb{Z}/3, +)\) to the symmetric group \(S_4\).

Answer: 9


Question: Calculate the sum of the infinite series: \[ \sum_{k = 0}^{\infty} \frac{(-1)^k}{2k + 1}. \]

Answer: \dfrac{\pi}{4}


Question: Determine the values of $a$ for which the series \( \sum\limits_{n=1}^\infty \frac{\log(1 + 1/n)}{n^a} \) converges.

Answer: a > 0


Question: Find the probability density function of the random variable $X$ if the expected value of $X^n$ is $n!$ for all positive integers $n$.

Answer: e^{-x}


Question: Determine the number of real solutions to the equation \( \log_3\frac {2x^2+3x+3}{5} = \frac {1}{\log_{2x^2+3x+9}9} \).

Answer: 2


Question: Find the condition on \(x\) such that \(\lim_{n\to \infty }\left(f(x) n^x\right) =1\) holds, where \(x\in \mathbb{R}\).

Answer: 0


Question: Find the last three digits of \(9^{9^{9^9}}\).

Answer: 289


Question: Let $V$ be a vector space over a field $F$ with dimension $n$, and let $W$ be a subspace of $V$ with dimension $m < n$. Define $E = \{T: V \to V \mid T \text{ restricts to } W \text{ as zero} \}$. Determine the dimension of $E$ as a subspace of $L(V,V)$, the set of all linear transformations from $V$ to $V$. What is $\dim E$?

Answer: n(n - m)


Question: Is it necessary for the limit \( \lim_{x \rightarrow \infty} f(x) = 0 \) for the integral \( \int_0^{\infty} |f(x)| \, dx \) to converge?

Answer: No


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

Answer: 2


Question: Determine the number of values of the parameter \( m \) such that the system of equations \( \begin{cases} 2(x+y) + mxy = 5 \\ (m-1)(x+y) + xy = 1 \\ 3(x+y) - xy = m+1 \end{cases} \) has real solutions \((x, y) \in \mathbb{R} \times \mathbb{R}\).

Answer: 1


Question: Determine the maximum number of 1's that can be present in an invertible matrix \( A \in M_n(F) \) where all entries are either 0 or 1.

Answer: n^2 - n + 1


Question: Determine the degree of the splitting field of the polynomial $x^{p^2} - 2$ over $\mathbb{Q}$, where $p$ is a prime number.

Answer: p^3(p - 1)


Question: Determine the value of \(a+b\) such that all roots of the polynomial \(f(x) = x^4 - 8x^3 + ax^2 + bx + 16\) are positive.

Answer: -8


Question: Evaluate the surface integral \( \iint_S F \cdot n \, dS \) using the Divergence Theorem, where \( S \) is the surface of the cylinder defined by \( x^2 + y^2 \leq 1, 0 \leq z \leq 1 \) and \( F(x,y,z) = (x^2y, z-xy^2, z^2) \).

Answer: \pi


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

Answer: \infty


Question: Calculate the expected value of the absolute sum of two independent random variables $x$ and $y$, each uniformly distributed between -1 and 1. Specifically, find $E(|x + y|)$.

Answer: \dfrac{2}{3}


Question: Find the remainder when the number of 2017-digit numbers with a leading digit of 2 and an odd number of 9's is divided by 1000.

Answer: 672


Question: Evaluate \( \lim_{x \to 0^+} \sum_{n=1}^\infty \sin(\sqrt{n}) e^{-nx} \).

Answer: 0


Question: Find the limit of the sequence \(a_n\) defined by the recurrence relation \(a_{n+1} = a_n + \frac{a_n - 1}{n^2 - 1}\) for \(n \geq 2\), with initial conditions \(a_1 = 0\) and \(a_2 = 100\).

Answer: 199


Question: Suppose $f$ is a polynomial of degree 7 which satisfies $f(1) = 2$, $f(2) = 5$, $f(3) = 10$, $f(4) = 17$, $f(5) = 26$, $f(6) = 37$, and $f(7) = 50$. Find the value of $f(0) + f(8)$. \( \boxed{} \)

Answer: 66


Question: Determine whether the square of every positive integer can appear as one of the squares in the representation of some prime number \(p\) of the form \(4k+1\) as a sum of two squares.

Answer: Yes


Question: Given the probability function \( P(x=n) = \frac{4}{(n^2 -1)n} \) for \( n = 2, 3, 4, 5, \ldots \), find the limit \( \lim\limits_{n\to \infty} P(x \leq 2n \mid n<x) \).

Answer: \dfrac{3}{4}


Question: Determine all values of the parameter $a$ such that the equation $x^3 - ax^2 + 3x - 2 = 0$ has no solutions for $0 < x < 2$. Provide your answer as a set of values for $a$.

Answer: [3, \infty)


Question: Determine the values of \( y \) for which the polynomial \( f(x) = -x^3 + 3x - y \) has three distinct real roots.

Answer: (-2, 2)


Question: Evaluate the integral \( I = \int_{0}^{\pi} \frac{\sin(x)\ln(x)}{\ln(\pi x - x^2)} \, dx \).

Answer: 1


Question: Determine the number of real solutions to the equation \( f(f(f(f(x)))) = 0 \), where \( f(x) = x^2 + 10x + 20 \).

Answer: 2


Question: Determine the number of automorphisms of order 12 in the group \( \mathbb{Z}_{105} \).

Answer: 16


Question:
Let \( A \in \mathbb{R}^{n \times n} \) be a positive semidefinite matrix. Determine whether the following limit holds:
\[ \lim_{\lambda \to \infty} (A + \lambda I)^{-1} = \mathbf{0}. \]

Answer: 0


Question: An urn contains 12 white balls and 8 black balls. Balls are drawn one by one without replacement. What is the expected number of draws required to obtain the first black ball?

Answer: \dfrac{7}{3}


Question: Evaluate the limit: \[ \lim_{R\rightarrow \infty} \int_{C_R} \frac{6z^6 + 5z^5}{z^6 + z^5 + 10}dz, \] where \( C_R=\{z \in \mathbb{C} : |z|=R  \} \) for \( R>0 \).

Answer: -2\pi i


Question:
Given that \(x, y, z > 0\), solve the following system of equations to find the value of \(x+y+z\):
\[\begin{cases}
\dfrac{x}{xy-z^2}=-\dfrac{1}{7}\\
\dfrac{y}{yz-x^2}=\dfrac{2}{5}\\
\dfrac{z}{zx-y^2}=-3
\end{cases}\]
Show that \(x+y+z=6\).

Answer: 6


Question: A stick of unit length is cut into three pieces of lengths $X$, $Y$, and $Z$, with $X \le Y \le Z$. First, the stick is cut into two pieces at a randomly chosen point uniformly distributed along its length. Then, the smaller of these two pieces is cut again at a randomly chosen point uniformly distributed along its length. These two cuts are conducted independently. Find the covariance $\text{Cov}(X,Y)$.

Answer: \dfrac{5}{2304}


Question: Let $f:\mathbb{R}^2\to\mathbb{R}$ be a continuous function such that \(\lim _{x\to +\infty}f(x,y)=+\infty\) for each fixed \(y\in\mathbb{R}\). Let \(\mathcal{I}\subset\mathbb{R}\) be a compact set and define \(g:\mathbb{R}\to\mathbb{R}\) by \(g(x)=\inf_{y\in\mathcal{I}}f(x,y)\). Determine whether \(\lim _{x\to +\infty}g(x)=+\infty\).

Answer: +\infty


Question: Calculate the length of the arc traveled by a particle moving with speed \(s'(t)=|t\sin t|\) in the direction \((\sin t \tanh t, \cos t \tanh t, \operatorname{sech} t)\) from \(t=0\) to \(t=\pi\).

Answer: \pi


Question:
Consider a sequence \( V_N \) of subspaces of \( \ell^N_1 \) such that \( \dim V_N = N - n \) and \( n \) is \( \mathsf{o}(N) \). Determine if there exist constants \( K_1, K_2 > 0 \) such that for sufficiently large \( N \), the following holds:
\[
\sup_{x \in V_N, \|x\|_1 \leq K_1} \|x\|_\infty \geq K_2.
\]

Answer: Yes


Question: Find the radius of convergence of the series \(\sum_{n=1}^\infty{\frac{(n!)^3 \cdot x^n}{(3n)!}}\). Use the ratio test to determine the correct inequality for \(|x|\).

Answer: 27


Question: Consider a geodesically convex Riemannian manifold $M$, where for any two points $p, q \in M$, there is a unique minimizing geodesic connecting them. For a point $p \in M$, is the function $f(x) = (d(x, p))^2$ smooth, where $d$ is the distance function on $M$?

Answer: Yes


Question: Prove that 19 cannot be expressed as the sum of the squares of two rational numbers.

Answer: 19


Question: Is the space of simple predictable processes, topologized by uniform convergence, metrizable?

Answer: Yes


Question: Evaluate the sum \(\sum_{j=1}^\infty a_j\) or determine that it diverges, given the formula for its \(n\)-th partial sum: \[ S_n = \frac{n^2-4n+5}{n^3+7n-9}. \]

Answer: 0


Question: In how many ways can 5 married couples be seated around a circular table such that man 1 is seated next to his wife, and man 3 is not seated next to his wife?

Answer: 60480


Question: Determine the third homotopy group of the space $S^2 \times S^1$, where $S^n$ denotes the $n$-dimensional sphere.

Answer: \mathbb{Z}


Question: Given the equation for experience required for a level in an online game: \[ \frac{50}{3} (x^3-6x^2+17x-12) = 37800 \] solve for the level \( x \).

Answer: 15


Question: Evaluate the factorial of \(-\frac{1}{2}\) using the Gamma function.

Answer: \sqrt{\pi}


Question: Find the volume of the wedge intercepted between the cylinder \(x^2+y^2=2ax\) and the planes \(z=x\) and \(z=2x\) using cylindrical coordinates.

Answer: \pi a^3


Question: Find the splitting field of the polynomial \( f(x) = x^{p^8} - 1 \) over the field \( \mathbb{Z}_p \).

Answer: \mathbb{Z}_p


Question: Show that the function \( f(z) = \frac{\pi^2}{\sin^2(\pi z)} \) has a pole of order 2 at \( z = n \).

Answer: 2


Question:
Compute the winding number of the curve \( \alpha(t) = ((2 \cos t - 1) \cos t, (2 \cos t - 1) \sin t) \) with respect to the point \( p = \left(\frac{1}{2}, 0\right) \). The winding number is given by:
\[
i_p(\alpha) = \frac{1}{2\pi i} \int_{0}^{2\pi} \frac{\alpha'(t)}{\alpha(t) - p} \, dt\]
Use complex analysis techniques to determine the winding number without performing explicit calculations. The known answer is 2.

Answer: 2


Question: What is the image of \(\zeta_3\) under the non-identity embedding of \(\mathbb{Q}(\zeta_3)\) in \(\mathbb{C}\)?

Answer: \zeta_3^2


Question: Evaluate the expression \( \frac{f(3)f(5)f(7)f(9)f(11)f(13)}{f(2)f(4)f(6)f(8)f(10)f(12)} \) where \( f(n) = n^4 + n^2 + 1 \).

Answer: 61


Question: A die is rolled 100 times, and the sum of the numbers rolled is recorded as X. A coin is tossed 600 times, and the number of heads is recorded as Y. Calculate the probability that X is greater than Y, denoted as P(X > Y).

Answer: 0.991


Question:
Calculate the variance of the probability density function given by:
\[ f(x) = 0.8 \left( \frac{1}{5\sqrt{2\pi}} e^{-\frac{1}{2} \left( \frac{x-50}{5} \right)^2} \right) + 0.2 \left( \frac{1}{8\sqrt{2\pi}} e^{-\frac{1}{2} \left( \frac{x-60}{8} \right)^2} \right) \]
This function is a mixture of two normal distributions: \(0.8N(\mu=50, \sigma^2=25)\) and \(0.2N(\mu=60, \sigma^2=64)\). The expectation is calculated as 52. Determine the correct variance of this distribution.

Answer: 48.8


Question: Find the shortest distance from the point \((1, -1, 1)\) to the set of points defined by the equation \(z = xy\).

Answer: \sqrt{2}


Question: Determine whether the Kullback-Leibler divergence between the standard Gaussian distribution \( f(x) = \frac{1}{\sqrt{2\pi}}\exp{(-\frac{x^2}{2})} \) and the standard Cauchy distribution \( g(x) = \frac{1}{\pi(1+x^2)} \) is finite.

Answer: Finite


Question: A curve is defined by the parametric equation \((x, y, z) = (-2 + 3t, 1 + 3t^2, 2t-3t^3)\). Find the coordinates of the unique point \(P\) on the curve such that the tangent line at \(P\) passes through the point \((-8, 10, 8)\).

Answer: (-5, 4, 1)


Question: Determine whether the limit \( \lim_{x \to y^-} \sum_{\{n : r_n \in (x, y)\}} \frac{1}{2^n} \) is equal to zero, where \( \{r_n\}_{n=1}^\infty \) is an enumeration of the rational numbers.

Answer: 0


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

Answer: a > 1


Question: Find the smallest positive value of \( x \) such that \( \cos(2 \pi x) \), \( \cos(4 \pi x) \), \( \cos(8 \pi x) \), \( \cos(16 \pi x) \), and \( \cos(32 \pi x) \) are all nonpositive.

Answer: \dfrac{21}{64}


Question: Transform the curve given by the equation \(x^3 + y^3 - 3xy = 0\) into the form \(r(t)\), where \(r = \sqrt{x^2 + y^2}\) and \(t = \frac{y}{x}\). Then, calculate the area \(B\) described by the curve for \(x, y > 0\).

Answer: \dfrac{3}{2}


Question: Determine the Taylor series for \( f(x) = x^3 \cdot \ln{\sqrt{x}} \) around the point \( a = 1 \) and find its radius of convergence.

Answer: 1


Question: Evaluate the limit \( \lim_{\rho \to 0}\frac{1}{\rho^2}\left|u(0,0) -\frac{1}{\pi\rho^2}\int\int_{B_{\rho}}u(x,y)\,dx\,dy\right| \), where \( \rho>0 \), \( B_\rho=\{(x,y)\in\mathbb{R^2}\mid x^2+y^2\leq \rho^2 \} \), and \( u(x,y) \) is a \( C^3 \) function satisfying \( u_{xx}(0,0)+u_{yy}(0,0)=0 \).

Answer: 0


Question: Consider a continuous surjection \( g: (0,1) \to [0,1] \). Determine whether the preimage of \([0,1]\) under \(g\) is open in \((0,1)\).

Answer: Yes


Question:
Determine which of the following partitions of the set $A = \{P, Q, U, S, T\}$ is consistent with the equivalence relation of homeomorphism, where $P = (0,1)$, $Q = [0,1)$, $U = (0,1]$, $S = [0,1]$, and $T = \mathbb{R}$. Choose the correct partition:

1. $\{ P, Q, U, S\}, \{T\}$
2. $\{P, T\}, \{Q, U\}, \{S\}$
3. $\{P, T\}, \{Q\}, \{U\}, \{S\}$

Answer: 2


Question: Find the maximum value of \( \left| \frac{1-(-x^2)^{n+1}}{1+x^2} \right| \) for \( -1 \le x \le 1 \).

Answer: 1


Question: Find all pairs of positive integers \((x, y)\) such that \(2^x + 3^y\) is a perfect square.

Answer: (4, 2)


Question: Customers arrive at a bank according to a Poisson process with rate \( \lambda \) per minute. Given that two customers arrived in the first 4 minutes, what is the probability that one arrived in the first minute and the other arrived in the last minute?

Answer: \dfrac{1}{8}


Question: Is it possible to embed a surface with fundamental group \(\mathbb{Z}_2\) in \(\mathbb{R}^3\) without self-intersection?

Answer: No


Question: Given a function $f(x)$ with $f'(0) = 2$ and $f(0) = 1$, use the Maclaurin expansion of $f(x)$ to calculate $\lim_{x \to 0} (f(x))^{1/x}$. Assume $f(x)$ is differentiable at $x=0$.

Answer: e^2


Question: Find all prime numbers \( p \) such that \( p^2 \mid 5^{p^2} + 1 \).

Answer: 3


Question:
Consider a field $K$ with characteristic 2 and let $V = K^n$. The determinant function $D: V^n \rightarrow K$ is characterized by the following properties:

1. $D$ is linear in each row.
2. Switching two rows changes the sign of $D$.
3. $D(e_1, \dots, e_n) = 1$.

Given these properties, determine the value of $D(a_1, \dots, b, \dots, b, \dots, a_n)$ when the characteristic of $K$ is 2.

Answer: 0


Question: Determine the dimension of the quotient space of the vector space of all finite closed connected intervals of \( \mathbb{R} \) under the equivalence relation defined by the evaluation map \( E([b,c]) = \int_b^c 1 \, dx \).

Answer: 1


Question: Evaluate the integral: \[ \int_{-\infty}^{\infty} \exp\left(-\frac{1}{2} y^2\right) \, dy \]

Answer: \sqrt{2\pi}


Question: Determine whether an unbounded non-orthogonal projection exists in a Hilbert space. A non-orthogonal projection is defined as an idempotent operator, meaning $T^2 = T$. Can such an operator be unbounded?

Answer: Yes


Question: Let $G$ be a finite group and $g \in G$ an element with order $o(g)$. If $r$ is a divisor of $o(g)$, what is the order of $g^r$?

Answer: \dfrac{o(g)}{r}


Question:
Explain why the following calculation of the limit is incorrect:
\[
\lim_{n\to \infty}\sqrt[n]{n!} =
\lim_{n\to \infty}\sqrt[n]{n(n-1)(n-2)\cdots(1)} =
\lim_{n\to \infty}\sqrt[n]{n} \cdot \lim_{n\to \infty}\sqrt[n]{n-1} \cdot \lim_{n\to \infty}\sqrt[n]{n-2}\cdots \lim_{n\to \infty}\sqrt[n]{1} = 1 \cdot 1 \cdot 1 \cdot 1 \cdots 1 = 1
\]
Therefore, \(\lim_{n\to \infty} \sqrt[n]{n!}=1\). However, WolframAlpha gives infinity as the limit. Why is the calculation incorrect?

Answer: \infty


Question: Let $R$ be a commutative Noetherian ring and $G_0(R)$ the Grothendieck group of finitely generated $R$-modules. If $M$ and $N$ are finitely generated $R$-modules such that $[M] = [N]$ in $G_0(R)$, is it true that there exist short exact sequences of finitely generated $R$-modules $0 \to A \to B \to C \to 0$ and $0 \to A \to B' \to C \to 0$ such that $M \oplus B \cong N \oplus B'$?

Answer: Yes


Question: Let $(R, \mathfrak{m}) \xrightarrow{\phi} (S,\mathfrak{n})$ be a flat homomorphism of local rings such that $\mathfrak{n} = \mathfrak{m}S + xS$ for some $x \in \mathfrak{n} \setminus \mathfrak{n}^2$. Suppose $J$ is an ideal of $R$ with $\text{depth}_R(R/J) = 0$. Determine if $\text{depth}_S\left(\frac{S}{JS+xS}\right) = 0$. Assume $\phi$ is injective if necessary.

Answer: 0


Question: Prove that \( S_n = \sum_{k=1}^{n} H_k = (n+1)H_n - n \), where \( H_k \) are the harmonic numbers.

Answer: (n+1)H_n - n


Question: Given the joint probability density function for continuous random variables $X$ and $Y$ as $f(x) = \frac{3}{2}x$ for $0 < x < 2$, $0 < y < 1$, and $x < 2y$, calculate the probability $\Pr(x < 1.5 \mid y > 0.5)$. Provide your answer as a single probability value.

Answer: \dfrac{23}{28}


Question: Evaluate the line integral \( \int_{C_1} \frac{y\vec{e_1}-x\vec{e_2}}{x^2+y^2} d\vec{r} \), where \( C_1 \) is a circle of radius 3 centered at the origin in \( \mathbb{R}^2 \).

Answer: -2\pi


Question: Evaluate the limit: \[ \lim_{x \to +\infty} \frac{\left(x+\sqrt{x^2-1}\right)^n+\left(x-\sqrt{x^2-1}\right)^n}{x^n}, \quad n \in \mathbb{N}. \]

Answer: 2^n


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

Answer: \dfrac{1}{e}


Question: Determine whether the following statement is true or false: There exists a sequence of Lebesgue integrable functions \( f_n: \mathbb{R} \rightarrow \mathbb{R}^+ \) such that \( f_n \rightarrow 0 \) uniformly on every compact subset \( K \subseteq \mathbb{R} \), but \( \int_\mathbb{R} f_n \, d\lambda \geq 1 \) for every \( n \in \mathbb{N} \).

Answer: True


Question: Let $H$ be an infinite-dimensional Hilbert space over $\mathbb{C}$. Consider a sequence $\{v_n\}_{n \in \mathbb{N}} \subset H$ of linearly independent vectors such that $v_n \to u$. For each $m \in \mathbb{N}$, let $V_m = \operatorname{span} \{v_n\}_{n \geq m}$ and $P_m$ be the orthogonal projection onto $V_m$. Is it true that for all $v \in V_1$, \( \lim_{m \to \infty} P_m(v) = a \cdot u \) in the $H$-norm, where $a \in \mathbb{C}$?

Answer: Yes


Question: Let $X$ be a random variable with probability density function \( f(x)=\frac{1}{2} e^{-|x|},\; -\infty<x<\infty. \) Calculate the expected value of \( \lfloor X \rfloor \), where \( \lfloor X \rfloor \) denotes the greatest integer less than or equal to \( X \).

Answer: -\dfrac{1}{2}


Question: Consider a differentiable function \(f(x)\) satisfying the integral equation \(\int_{0}^{1} f(x)(x-f(x))\,dx=\frac{1}{12}\). Find the nearest integer less than or equal to \(\frac{1}{f'(1)}\).

Answer: 2


Question: Let $H$ be an arbitrary Hilbert space. Suppose that for some sequence $\{a_n\} \subseteq H$ we have \(\forall x\in H,\quad x\cdot a_n \to 0\in\mathbb{K}.\) Does it follow that $a_n\to 0$ in $H$?

Answer: No


Question: Approximate \( \sqrt{8} \) using the Taylor series expansion of \( \sqrt{1+x} \) around zero up to the fourth term. Use the Lagrange Remainder to ensure the approximation is accurate to within \( 10^{-4} \).

Answer: 2.8284


Question: Evaluate the integral \( \int_{0}^{1}\frac{x^{2015}-1}{\ln x}dx \).

Answer: \ln 2016


Question: Determine whether there exists a natural number \( n_0 \) greater than 2 and a concave polygon \( p \) such that \( p \) can be tiled with \( n_0 \) congruent connected parts that are not polygons.

Answer: Yes


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

Answer: \dfrac{\pi}{2}


Question: Let \(a_n\) be the sequence defined by \(a_1 = 3, \ a_{n+1} = 3^{a_n}\). Let \(b_n\) be the remainder when \(a_n\) is divided by 100. Find \(b_{2004}\).

Answer: 87


Question: Given the equation \((x+1)^4 + (x+3)^4 = 4\), find the sum of all non-real solutions.

Answer: -4


Question: State whether the following statement is true or false: An entire function $f: \mathbb{C} \to \mathbb{C}$ satisfying $f(z) \to 0$ as $z \to \infty$ is a constant.

Answer: True


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

Answer: \dfrac{\pi}{4}


Question: Let $A \in M_{n}(\mathbb{C})$. Prove or disprove: If $-2 \in \sigma(A^2 + 2A)$, then $-4 \in \sigma(A^4)$. Here, $\sigma$ denotes the spectrum of the matrix.

Answer: -4 \in \sigma(A^4)


Question: Evaluate the integral \( \int_{|z|=1} \overline{z}^2 \sin z \, \mathrm{d}z \).

Answer: 2\pi i


Question: Evaluate the integral \( \int_{-\pi}^\pi \sin(2\cos\theta)\cos((2m+1)\theta)\,d\theta \) using complex analysis methods, where \( m \in \mathbb{N} \).

Answer: 2\pi (-1)^m J_{2m+1}(2)


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

Answer: \dfrac{1}{2}


Question: Determine the dimension of the subspace $W$ of the vector space $V$ of all infinitely differentiable functions on $\mathbb{R}$, where $W$ consists of functions that are solutions to the initial value problem (IVP): \[ f'' + 4f = 0, \quad f(\pi) = 0. \]

Answer: 1


Question: Find the value of $x+y$ if $x, y \in \mathbb{N}$ and $x^2 + y^2 = 2019^2$. Provide your answer as a single integer.

Answer: 2811


Question: Determine whether the following statement is true or false: If $v_1, v_2, \dots, v_k$ are linearly independent vectors in $\mathbb{R}^n$ and $u$ is any nonzero vector in $\mathbb{R}^n$, then the matrices $uv_1^T, uv_2^T, \dots, uv_k^T$ are linearly independent.

Answer: True


Question: Determine the radius of convergence of the power series \( \frac{\sin^2(z)}{z} \).

Answer: \infty


Question: Let $A$ and $B$ be two positive-definite self-adjoint operators on a Hilbert space $H$. If both $A$ and $B$ have discrete spectra, does the operator $AB$ also have a discrete spectrum?

Answer: Yes


Question: For any integer \( n \ge 2 \), consider the group \( \mathbb{Z}_{p^n} \oplus \mathbb{Z}_p \). Determine the number of cyclic subgroups of order \( p \). Here, \( p \) is a prime number.

Answer: p + 1


Question: Find the derivative of the expression \( \text{trace} (\Sigma^{\frac{1}{2}} S^{-1}\Sigma^{\frac{1}{2}}) \) with respect to the matrix \( \Sigma \), where both \( \Sigma \) and \( S \) are positive definite matrices.

Answer: S^{-1}


Question:
Consider the ordinary differential equation $$(3y^2-x)f(x+y^2)+2y(y^2-3x)f(x+y^2)y'=0$$ where $f$ is an arbitrary continuously differentiable function. If this differential equation is exact, determine which of the following statements is true:

1) $xf'(x)-3f(x)=0$
2) $xf'(x)+3f(x)=0$
3) $f'(x)+3f(x)=0$
4) $f'(x)-3f(x)=0$

Answer: 2


Question: Let $A$ be the point with coordinates $(1, 0)$ in $\mathbb{R}^2$. Another point $B$ is chosen randomly over the unit circle. What is the expected value of the area of the triangle $OAB$?

Answer: \dfrac{1}{\pi}


Question: Let $p$ be a prime number and $n$ a natural number. Determine the value of $\sum_{i=1}^{p-1} i^n \pmod{p}$. Consider the cases where $p > 2$ and $n$ is odd.

Answer: 0


Question: Solve the equation over the real numbers: \((x^2+x+1)^{1/3}+(2x+1)^{1/2}=2\).

Answer: 0


Question: Evaluate the outward flux \( \iint_S \mathbf{F} \cdot d\mathbf{S} \) where \( \mathbf{F}(x,y,z) = \frac{1}{(x^2+y^2+z^2)^{3/2}}\langle x,y,z\rangle \) and \( S \) is the outward oriented surface given by \( 9x^2+4y^2+16z^2=144 \).

Answer: 4\pi


Question: Let \( A \) be a square matrix of odd size with real entries, satisfying the equation \( A^3 + 2A^2 + 2A + I_n = 0_n \), where \( I_n \) is the identity matrix and \( 0_n \) is the zero matrix. Prove that \( \det(A) = -1 \).

Answer: -1


Question: Given the functional equation \(2f(x+2) + 3f(x-2) = x + 5\), find the value of \(f(1)\).

Answer: \dfrac{32}{25}


Question: Let $f \in \mathbb{C}(X)^\times$. Determine whether the following statement is true: $$\sum_{v \in \mathbb{CP}^1} \text{ord}_v(f) = 0,$$ where $\text{ord}_v$ denotes the order of zero of $f$ at $v$. Justify your answer.

Answer: True


Question: Determine whether the following statement is correct: \( \sum\limits_{n = 1}^{p-1}{(n^2+n)} \equiv \frac{p(p-1)}{2} \pmod{p} \) for a prime number \( p \equiv 7 \pmod{8} \).

Answer: Correct


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0}(-1+\cos x)^{\tan x} \]

Answer: 1


Question: A tree T has four vertices of degree 3, six vertices of degree 4, one vertex of degree 5, and eight vertices of degree 6. No other vertices of T have a degree of 3 or more. How many leaf vertices does T have?

Answer: 53


Question: Determine whether the following statement is true: If a functor $F$ from the category of groups $\mathcal{G}$ to the category of groups with a distinguished element $\mathcal{G}_o$ satisfies $F(G)=(G,x)$ for all $G \in \mathcal{G}$, then $x$ must be the identity element of $G$. Provide a justification for your answer.

Answer: True


Question: Let $X$ be a completely regular $T_1$ space, and let $A$ and $B$ be disjoint closed subsets of $X$, where $A$ is compact. Determine whether there exists a continuous function $f: X \to [0,1]$ such that $f(A) = \{0\}$ and $f(B) = \{1\}$. Provide a proof or counterexample.

Answer: Yes


Question: Calculate the Lebesgue measure (volume) of the 3-simplex defined by the set \( S_3 = \{(x,y,z) \in \mathbb{R}^3 : x+y+z \leq 1, x \geq 0, y \geq 0, z \geq 0 \} \).

Answer: \dfrac{1}{6}


Question: Is an anti-unitary operator $J$ on a complex Hilbert space $\mathscr{H}$, with $J^2 = I$, always diagonalizable by an orthonormal basis?

Answer: Yes


Question: Is there a closed form expression for the sum \(\sum_{i=0}^n\binom{n}{i}i^2\)? Answer with 'yes' or 'no'.

Answer: yes


Question: What is the largest possible number of inversions in a sequence of positive integers whose sum is 2014?

Answer: 507024


Question: An infinite increasing list of numbers has the property that the median of the first $n$ terms equals the $n$th odd positive integer. Determine how many numbers in the list are less than 2018.

Answer: 505


Question: Determine the range of values for \(a\) such that the function \(f(x) \propto x^{-a}\) is a valid probability density function for a continuous random variable \(X\) with range \([x_l, \infty)\), where \(x_l > 0\).

Answer: a > 1


Question: Evaluate the integral \( \int_\gamma \frac{dz}{z-2} \) where \( \gamma \) is the circle \( |z|=3 \).

Answer: 2\pi i


Question: Determine the total number of 3x3 matrices with real entries such that the characteristic polynomial of their adjoint matrix is \( \lambda^3 - \lambda^2 + \lambda + 1 = 0 \).

Answer: 0


Question: Given a sequence \(a_n\) that converges to \(l\), prove that the sequence \(b_n = \frac{1}{n} \sum_{k=1}^{n} a_k\) also converges to \(l\).

Answer: l


Question: Evaluate the limit \( \lim_{n \rightarrow \infty} a_n \) where \( a_n = \left(\frac{3n^2}{3n^2-1}\right)^{2n^2+3} \).

Answer: e^{\frac{2}{3}}


Question: Consider the symmetric group $S_3$ on the objects $x_1, x_2, x_3$. Given an infinite sequence of permutations $\sigma_1, \sigma_2, \sigma_3, \ldots$ in $S_3$, determine whether there exists a positive integer $N$ such that the sequence of products $\sigma_{N}, \sigma_{N+1}\sigma_{N}, \sigma_{N+2}\sigma_{N+1}\sigma_{N}, \ldots$ has an infinite subsequence whose terms fix $x_1$. Is this statement true?

Answer: True


Question: Determine the number of solutions to the equation \(e^{\tan x} = \sin x + \cos x\) in the interval \([-\pi, \pi]\).

Answer: 2


Question: The numbers \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{2018}\) are written on a blackboard. John repeatedly chooses any two numbers, say \(x\) and \(y\), erases them, and writes the number \(x + y + xy\) on the board. He continues this process until only one number remains. What is the possible value of the final number left on the board?

Answer: 2018


Question: Determine the number of distinct real roots of the equation \(13x^{13} - e^{-x} - 1 = 0\).

Answer: 1


Question: Determine the values of \( \alpha \in \mathbb{R} \) for which the function \( \frac{1}{1+\lvert\lvert x\rvert\rvert^{\alpha}} \) is integrable on \( \mathbb{R}^n \setminus \{0\} \), where \( \lvert\lvert \, .\rvert\rvert \) is a norm on \( \mathbb{R}^n \).

Answer: \alpha > n


Question: Find the exact value of the limit: \[ \lim_{x \to \infty} \left(1 + \frac{1}{2x} - \frac{15}{16x^2}\right)^{6x}. \]

Answer: e^3


Question: Determine the number of real solutions of the equation \(3x^5 + 2x^4 + x^3 + 2x^2 - x - 2 = 0\).

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \to 0^{+}} \frac{\ln(x+\arccos^{3}x)-\ln x}{x^{2}}. \]

Answer: +\infty


Question: Calculate \( \lim \limits_{x \to 0} \ x^2\Big(1+2+3+\ldots+\Big[\frac{1}{\vert x \vert}\Big]\Big) \), where \([.]\) represents the greatest integer function.

Answer: \dfrac{1}{2}


Question: Is there a formal proof that for a Galton-Watson tree with an arbitrary offspring distribution having an expected value of 1 and finite nonzero variance, the ratio of expected diameter to expected height converges to \( \frac{4}{3} \)?

Answer: \dfrac{4}{3}


Question: Find the sum of the series \( \sum_{k=1}^n 2^k \binom{n}{k} \).

Answer: 3^n - 1


Question: Find the value of \(a\) for which the integral \(\int_{0}^{\infty}\frac{dx}{a^2+\left(x-\frac{1}{x}\right)^2}\) equals \(\frac{\pi}{5050}\) for \(a \geq 2\).

Answer: 2525


Question: Let $G$ be a finite non-abelian group. Determine whether it is true that if $32$ does not divide the order of $G$, then $G$ has at least one abelian centralizer.

Answer: True


Question: Calculate the limit: \[ \lim_{n\to\infty}\int_{0}^{1}\frac{\sqrt{n}}{1+n\ln(1+x^2)}\,dx \]

Answer: \dfrac{\pi}{2}


Question: Evaluate the complex integral \( \frac{1-|a|^{2}}{\pi}\int_{|z|=1}\frac{|dz|}{{|z+a|}^{2}} \), where \( a \in \mathbb{C} \) with \( |a|<1 \).

Answer: 2


Question: Determine the number of pairwise non-isomorphic abelian groups of order 67500, given that the prime factorization is \(67500 = 2^2 \times 3^3 \times 5^4\).

Answer: 30


Question:
Given a random variable X with the density function:
\[ F_X(x) = \begin{cases} 2x^{-2}, & x \in (1,2) \\
0, & \text{otherwise} \end{cases} \]
Calculate the variance of the expression \(3X^2 - 5\).

Answer: 6


Question: Calculate the volume of the finite body $K$ that is bounded by the surfaces $z = 2 - x^2 - y^2$ and $z = y^2$. Express your answer in terms of $\\pi$. \[ \text{Volume} = \boxed{\pi\sqrt{2}} \]

Answer: \pi\sqrt{2}


Question: Find the smallest positive integer \( n \) such that the symmetric group \( S_n \) contains an element of order greater than \( 2n \).

Answer: 9


Question: Consider a square matrix \(A\) of dimension \(n \times n\) where \(n = 2, 3, 4, \ldots\). The matrix is hollow (all elements on the principal diagonal are zero), symmetric, and all off-diagonal elements are equal to \(a\). Determine if the spectral radius (maximum eigenvalue) of this matrix is equal to \((n-1) \times a\).

Answer: (n-1)a


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

Answer: \pi


Question: Evaluate the definite integral: \[ \int_{e}^{4+e} (3x - \lfloor 3x \rfloor) \, dx \]

Answer: 2


Question: Determine the value of \( N \) such that when \((a+b+c+d+1)^N\) is expanded and like terms are combined, the resulting expression contains exactly 1001 terms, each involving all four variables \(a, b, c,\) and \(d\) to some positive power.

Answer: 14


Question:
Find the integer value of $r$ that maximizes the expression:
\[ \binom{20}{r}\binom{20}{0} + \binom{20}{r-1}\binom{20}{1} + \cdots + \binom{20}{0}\binom{20}{r}. \]

Answer: 20


Question: Determine the number of positive integer solutions to the equation \( x^2 + y^2 + z^2 - xz - yz = 1 \).

Answer: 1


Question: Two numbers $m$ and $n$ are chosen independently from a normal distribution with probability density function $f(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2}$. What is the probability that $n \ge 2m$?

Answer: \dfrac{1}{2}


Question: Find the minimum value of \( y = \frac{1}{2}(e^x - e^{-x}) + \frac{n}{2}(e^x + e^{-x }) \) for \( n \geq 2 \) using algebra.

Answer: \sqrt{n^2 - 1}


Question: Given that $X$ and $Y$ are independent and identically distributed standard normal random variables, i.e., $X, Y \sim \mathcal{N}(0,1)$, determine the probability $Pr(X > 0, X + Y < 0)$. Provide your answer as a single probability value.

Answer: \dfrac{1}{8}


Question: Let $f \in B[a,b]$ be a function with only countably many points of discontinuity. Does it follow that $f \in \overline{S[a,b]}$, where $S[a,b]$ denotes the collection of all step functions on $[a,b]$? Assume the supreme norm is used.

Answer: No


Question: Let $G_1$ and $G_2$ be finitely presentable groups, and let $f : G_1 \rightarrow G_2$ be a surjective homomorphism. Denote the $k$th term of the lower central series of $G_i$ by $\gamma_k(G_i)$. Assume that $f$ induces an isomorphism $G_1 / \gamma_k(G_1) \rightarrow G_2 / \gamma_k(G_2)$ for all $k \geq 1$, and that the intersection of the lower central series of $G_i$ is trivial for $i=1,2$. Is $f$ necessarily an isomorphism?

Answer: Yes


Question: Given \( a_n = \frac{1}{n} \sqrt[3]{n^{3}+n^{2}-n-1} \), find the smallest positive integer \( k \geq 2 \) such that \( a_2 a_3 \cdots a_k > 4 \).

Answer: 254


Question: Compute the curve integral \( \int_{|z|=3}\dfrac{dz}{\sin^2 z} \) using the residue theorem.

Answer: 0


Question: Let $f: Y \rightarrow X$ be a continuous function, with $\mathcal O_Y$ a sheaf of rings on $Y$. Suppose for some $p \in X$, there exists an open neighborhood $U_0$ of $p$ such that $f^{-1}U_0$ is empty. Prove that the stalk $(f_{\ast} \mathcal O_Y)_p$ is the zero ring.

Answer: 0


Question: Determine the operator norm \( \|T\| \) for the operator \( T: \ell^p \to \ell^p \) defined by \( Tx = (c_1 x_1, c_2 x_2, \ldots) \), where \( c_k \in \mathbb{C} \) and \( \lim_{k \to \infty} c_k = 0 \), for \( 1 \le p \le \infty \).

Answer: \sup_{k} |c_k|


Question: In how many ways can 4 people be selected from 100 people sitting at a circular table such that no two selected people are adjacent?

Answer: 3460375


Question: Evaluate the integral \( \int_{\pi/2}^{3\pi/2}(4\sin 3x - 3 \cos 4x)\,dx \) using Riemann sums.

Answer: 0


Question:
If \( a, b, c \) are roots of the equation \( x^3 - 3x^2 + 2x + 4 = 0 \), find the value of \( y \) at \( x = 2 \) for the expression:
\[
y = 1 + \frac{a}{x-a} + \frac{bx}{(x-a)(x-b)} + \frac{cx^2}{(x-a)(x-b)(x-c)}
\]

Answer: 2


Question: In \( \triangle{ABC} \), which is isosceles with \( AB = AC \), point \( P \) is located inside the triangle such that \( \angle{BCP} = 30^{\circ} \), \( \angle{APB} = 150^{\circ} \), and \( \angle{CAP} = 39^{\circ} \). Determine the measure of \( \angle{BAP} \).

Answer: 13


Question: Determine the Lebesgue measure of Luzin's non-Borel set of reals, defined as follows: For $x \in [0,1]$, let \( x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\,}}}} \) be the continued fraction expansion of $x$. Let $A$ be the set of numbers $x \in [0,1]$ whose corresponding sequence $a_0, a_1, a_2, \cdots$ admits an infinite subsequence $a_{k_1}, a_{k_2}, a_{k_3}, \cdots$ such that for each $i$, $a_{k_{i+1}}$ is divisible by $a_{k_i}$. What is the Lebesgue measure of the set $A$?

Answer: 1


Question: Let $\Omega$ be a smooth plane domain of unit area. Given $u(x,y) = 3x^2 + y^2$, compute the integral \( \int_{\partial \Omega} \frac{\partial u}{\partial n} \, ds \), where \( \frac{\partial u}{\partial n} \) denotes the outer normal derivative on the boundary \( \partial \Omega \).

Answer: 8


Question:
Given that $E[X]=\theta(n^{1.5})$ and $Var[X]=\theta(n^{2.5})$ for a non-negative random variable $X$, use Chebyshev's inequality to determine the order of $P(X=0)$. Choose the correct option:

A: $P(X=0)=O(1/n^2)$
B: $P(X=0)=O(1/n^{1.5})$
C: $P(X=0)=O(1/n)$
D: $P(X=0)=O(1/n^{0.5})$

Answer: D


Question: Let $n$ be an odd natural number, and let $1 \leq t, k < n$. Consider the sets $A = \{ a_1, a_2, \ldots, a_t \}$, $B = \{ b_1, b_2, \ldots, b_{n-t} \}$, $C = \{ c_1, c_2, \ldots, c_k \}$, and $D = \{ d_1, d_2, \ldots, d_{n-k} \}$, which are subsets of $\{ 1, 2, \ldots, n \}$ such that $A \cup B = C \cup D = \{ 1, 2, \ldots, n \}$ and $A \cap B = C \cap D = \emptyset$. Additionally, $A \neq C$ and $A \neq D$. Determine whether the subgroup generated by the permutations $(a_1, a_2, \ldots, a_t)(b_1, b_2, \ldots, b_{n-t})$ and $(c_1, c_2, \ldots, c_k)(d_1, d_2, \ldots, d_{n-k})$ is the symmetric group $S_n$. Assume $n$ is odd.

Answer: S_n


Question: Evaluate the surface integral \( \int_S (\text{Curl } \vec{F}) \cdot d\vec{S} \), where \( \vec{F}(x, y, z) = (z, x, y) \) and \( S \) is the portion of the surface \( x^2 + y^2 + z = 1 \) lying above the xy-plane, oriented upward.

Answer: \pi


Question:
Evaluate the line integral of the vector field \( \vec{F}(x, y, z) = 2y \vec{i} + x \vec{j} + z \vec{k} \) over the path \( \Gamma_1 \cup \Gamma_2 \), where \( \Gamma_1 \) is the path \( x = z = y^2 \) from \((0, 0, 0)\) to \((1, 1, 1)\), and \( \Gamma_2 \) is the line segment from \((1, 1, 1)\) to \((2, 3, 4)\). Compute:
\[ \int_{\Gamma_1 \cup \Gamma_2} ( \vec{F} \cdot \vec{dr} ) \]

Answer: \dfrac{50}{3}


Question: Let \( A = \{1, 2, 3, \ldots, 9\} \) and let \( f: A \rightarrow A \) be a bijection such that \( f^n = f \) for the smallest positive integer \( n \), where \( f^n \) denotes the \( n \)-fold composition of \( f \) with itself. Determine the largest possible value of \( n \).

Answer: 21


Question: Let \( H = \left \{ x \in \mathbb{R}^{4} \mid x_2 - x_3 + 2x_4 = 0 \right \} \). Find a real number \( a \) and vector subspaces \( S \) and \( T \) such that \( \dim(S) = \dim(T) \), \( S + T^\perp = H \), and \( S \cap T = \langle (1, a, 0, -1) \rangle \).

Answer: 2


Question: Determine whether there exists an Artinian ring with exactly two prime ideals such that their product is non-zero and their sum equals the ring itself. Provide a justification for your answer.

Answer: Yes


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

Answer: 8


Question: Determine the modulus of continuity of the function \( f(x) = \frac{1}{x} \) on the interval \( I = (0, 1) \).

Answer: \infty


Question: In triangle $ABC$, point $A$ is located at $(3,4)$, and the equation of the angle bisector of angle $B$ is $y = x$. The orthocenter of the triangle is at $(2,2)$. Determine the coordinates of point $B$, denoted as $(h,k)$. Provide your answer as an ordered pair $(h,k)$.

Answer: \left( \dfrac{10}{3}, \dfrac{10}{3} \right)


Question: Consider the recursive function defined by \( C_n = C_{n-1} + iC_{n-2} \), with initial conditions \( C_1 = 1 \) and \( C_2 = 1 \). If \( C_{10} \) is expressed in the form \( a + bi \), find the value of \( b \).

Answer: -12


Question: Evaluate the integral in a closed form: \[ \int_{25\pi/4}^{53\pi/4} \frac{1}{(1+2^{\sin x})(1+2^{\cos x})}\,dx \]

Answer: \dfrac{7\pi}{4}


Question: Given the XOR result X = 0000 0011, determine if it is possible to uniquely identify the original values A and B, where A and B are 7-bit ASCII characters ranging from 32 to 126. Calculate the number of combinations a computer would need to try to find the correct values of A and B.

Answer: 94


Question: Given the permutation \( a = (1\ 6)(2\ 5\ 7)(3\ 8\ 4\ 9) \) in the symmetric group \( S_9 \), determine the number of permutations \( b \in S_9 \) that commute with \( a \), i.e., for which \( a \circ b = b \circ a \).

Answer: 24


Question: Evaluate the product: \( \prod_{n=1}^{9} \sin(20n - 10^\circ) \).

Answer: \dfrac{1}{256}


Question: Find the remainder when the polynomial \(x^{2015} + x^{2016}\) is divided by \(x^2 + x + 1\).

Answer: -x


Question: Determine the number of roots of the polynomial \( z^6 - 5z^2 + 8z + 2 \) that lie within the closed unit disk \( |z| \leq 1 \).

Answer: 1


Question: Determine the number of inequivalent cusps of the modular group \(\Gamma_0(4)\).

Answer: 3


Question: Find the analytic function \( f(z) = u(x,y) + iv(x,y) \) such that \( u(x,y) = \sin(x^2-y^2)\cosh(2xy) \) and \( f(z) \) is analytic over all \( \mathbb{C} \).

Answer: \sin(z^2)


Question: Find the maximum value of the expression \( \sum_{1 \le i < j \le n} x_i x_j \), subject to the constraint \( x_1 + x_2 + \cdots + x_n = 1 \).

Answer: \dfrac{n-1}{2n}


Question: Find the integer solutions of the equation \(3\sqrt{x + y} + 2\sqrt{8 - x} + \sqrt{6 - y} = 14\).

Answer: (4, 5)


Question: Find the remainder when \(1111^{2019}\) is divided by \(11111\).

Answer: 11101


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

Answer: \lambda^2


Question: Find the value of \( f^{(11)}(2) \) given the Taylor series expansion \( \sum (-\frac{2^n}{3^{n+1}})(x-2)^{2n+2} \).

Answer: 0


Question: Consider a sequence \(a_n\) such that \(a_n \rightarrow \infty\) and a sequence \(x_n\) such that \(x_{a_n} \rightarrow 0\). If the sequence \(b_n\) is of the same order of magnitude as \(a_n\) (that is, \(b_n = O(a_n)\) and \(a_n = O(b_n)\)), does \(x_{b_n} \rightarrow 0\)? Answer yes or no and justify your answer.

Answer: No


Question:
Let \( f \) be a non-negative function defined on the interval \([0,1]\). If \( \int_0^x \sqrt{1-(f'(t))^2} \, dt = \int_0^x f(t) \, dt \) for \( 0 \leq x \leq 1 \) and \( f(0) = 0 \), determine which of the following is true:

(A) \( f(\frac{1}{2}) < \frac{1}{2} \) and \( f(\frac{1}{3}) > \frac{1}{3} \)
(B) \( f(\frac{1}{2}) > \frac{1}{2} \) and \( f(\frac{1}{3}) > \frac{1}{3} \)
(C) \( f(\frac{1}{2}) < \frac{1}{2} \) and \( f(\frac{1}{3}) < \frac{1}{3} \)
(D) \( f(\frac{1}{2}) > \frac{1}{2} \) and \( f(\frac{1}{3}) < \frac{1}{3} \)

Answer: C


Question: Evaluate the integral \( \int_T (x^2 + y^2) \, dx \, dy \), where \( T = \{(x, y) : x^2 + y^2 < 1\} \).

Answer: \dfrac{\pi}{2}


Question: Determine the number of points on the elliptic curve $E$ given by $y^2 = x^3 + bx + c$ over the integers modulo $221 = 13 \cdot 17$. There exist points $P$ and $Q$ on $E$ such that $11P = \mathcal{O}$ and $7Q = \mathcal{O}$. What is the number of points on $E$?

Answer: 154


Question: Let $a_1 = 1$ and $a_n = n(a_{n-1} + 1)$ for $n = 2, 3, \ldots$. Define \( P_n = \left(1 + \frac{1}{a_1}\right)\left(1 + \frac{1}{a_2}\right) \cdots \left(1 + \frac{1}{a_n}\right) \) for $n = 1, 2, \ldots$. Find \( \lim_{n \to \infty} P_n \).

Answer: e


Question: Find the value of the determinant of the Hessian matrix, D(0,3), using the second partial derivative test for the function \( g(x,y) = e^{3x^2 - 6y^2 + 36y} \). The critical point is at \((0,3)\).

Answer: -72 e^{108}


Question: If \( \sqrt{9-8\cos40} = a + b\sec40 \), where \( a \) and \( b \) are rational numbers, find \( \lvert a+b \rvert \).

Answer: 2


Question: Determine the minimum number of equispaced points needed in the interval \([0, \frac{\pi}{2}]\) to perform piecewise linear interpolation of the sine function such that the maximum error does not exceed \(10^{-4}\).

Answer: 57


Question: Given the equations \(\alpha^3 - 6\alpha^2 + 13\alpha = 1\) and \(\beta^3 - 6\beta^2 + 13\beta = 19\), where \(\alpha\) and \(\beta\) are real numbers, find the value of \(\alpha + \beta\).

Answer: 4


Question:
Consider the function \(f:\mathbb{R}^2\to\{1,2,...,2012\}\) that satisfies the following rule: If \(a<b<c\), then \(f(a,c)=f(b,c)=f(a,b)\) and \(f(b,a)=f(c,a)=f(c,b)\). Let \(x_1,x_2,...,x_{2010}\) be a sequence of real numbers, which are all different. Find the number of possible ordered \(2010^2\)-tuples
\[\Bigl(f(x_1,x_1),f(x_1,x_2),\ldots,f(x_1,x_{2010}),f(x_2,x_1),f(x_2,x_2),\ldots,f(x_2,x_{2010}),\ldots,f(x_{2010},x_{2010})\Bigr)\]

Answer: 2012^{2012}


Question: Let \( \mathcal{F} \) be a family of subsets of a finite set \( X \), where \( |X| \geq 5 \). The family \( \mathcal{F} \) has the property that for every three elements \( a, b, c \in X \), there exists a set \( F \in \mathcal{F} \) such that \( F \subseteq X \setminus \{a, b, c\} \). Must there always exist three sets \( F, G, H \in \mathcal{F} \), at least two of which are inclusion-maximal in \( \mathcal{F} \), and three points \( f, g, h \in X \) such that \( f \in F \setminus (G \cup H) \), \( g \in G \setminus (F \cup H) \), and \( h \in H \setminus (F \cup G) \)?

Answer: Yes


Question: Does the property that "closed bounded subsets of \(\Bbb{R}\) are compact" imply the least upper bound property of \(\Bbb{R}\)? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the sum: \[ \cos\left(\frac{2\pi}{2021}\right) + \cos\left(\frac{4\pi}{2021}\right) + \cdots + \cos\left(\frac{2020\pi}{2021}\right) \]

Answer: -\dfrac{1}{2}


Question: Determine the number of maximal ideals in the ring \( \mathbb{Z}_5[x]/\langle (x+1)^2(x+2)^3 \rangle \).

Answer: 2


Question:
Given the following cubic spline that interpolates the function \(f\):
\[
\begin{cases}
(x+3)^3-9(x+3)^2+22(x+3)-10, & -3 \leq x < -1, \\
(x+1)^3-3(x+1)^2-2(x+1)+6, & -1 \leq x < 0, \\
ax^3+bx^2+cx+d, & 0 \leq x < 2, \\
(x-2)^3+6(x-2)^2+7(x-2), & 2 \leq x \leq 3
\end{cases}
\]
It is given that \(f(-3) = -10\), \(f(-1) = ?\), \(f(0) = ?\), \(f(2) = 0\), and \(f(3) = 14\).

Determine the value of \(c\) in the cubic spline for the interval \(0 \leq x < 2\).

Answer: -5


Question: If a function $f: \mathbb{R} \to \mathbb{R}$ has an extremum at the point $x$ and is differentiable in some neighborhood of $x$, does the derivative $f'(x)$ change sign at $x$?

Answer: No


Question:
Determine which of the following expressions is valid for the Fourier transform of the product of two shifted functions, where \( \mathcal{F} \) denotes the Fourier transform and \( \circ \) denotes convolution:

1. \[ \mathcal{F}\{c(x-x_0)b(x-x_0)\} = \mathcal{F}\{c(x-x_0)\} \circ \mathcal{F}\{b(x-x_0)\} = \left[e^{-2ix_0y}C(y) \right] \circ \left[e^{-2ix_0y}B(y)\right] = \left(e^{-2ix_0y}\right)^2\left[C(y) \circ B(y)\right] \]

2. \[ \mathcal{F}\{c(x-x_0)b(x-x_0)\} = \mathcal{F}\{(cb)(x-x_0)\} = e^{-2ix_0y}\left(\mathcal{F}\{(cb)(x)\}\right) = e^{-2ix_0y}\left[C(y) \circ B(y)\right] \]

Explain why the chosen expression is valid and identify any errors in the other expression.

Answer: 2


Question: Consider the ring of formal power series $R[[x]]$. Define a metric on $R[[x]]$ for two series $\sum a_{n}x^{n}$ and $\sum b_{n}x^{n}$ as $d((a_n),(b_n))=2^{-k}$, where $k$ is the smallest natural number such that $a_{k} \neq b_{k}$, or zero if no such $k$ exists. Let $\{x_{i}: i \in \mathbb{N}\}$ be a sequence of elements of $R[[x]]$. For each natural number $j$, define $G(x_{i},j)$ as the set of all natural numbers $i$ such that $x_{i}(j) \neq 0$. Assume $G(x_{i},j)$ is finite. Define the formal series $\sum_{i=0}^{\infty} x_{i}$ by $(\sum_{i=0}^{\infty} x_{i})(j) = \sum_{i \in G(x,j)} x_{i}(j)$. Let $S_{n}=\sum_{i=0}^{n} x_{i}$. Does the sequence $\{S_{n}: n \in \mathbb{N}\}$ converge to $\sum_{i=0}^{\infty} x_{i}$ as $n \rightarrow \infty$ with respect to the defined metric?

Answer: Yes


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

Answer: 1


Question: Let $f: [0,1] \to \mathbb{R}$ be a continuous function, and let $G \subset \mathbb{R}^2$ be the graph of $f$. Consider the set $G + G$, which is the algebraic sum of the graph with itself. If $G + G$ is measurable and has positive measure, does it necessarily have a nonempty interior?

Answer: Yes


Question: Evaluate the multivariable limit: \[ \lim_{(x,y)\to(0,0)}\frac{\sqrt{4-x+y}-2}{7x-7y} \]

Answer: -\dfrac{1}{28}


Question: Evaluate the limit: $$\lim_{n\rightarrow\infty}\left(\frac{\pi}{2^n}\left(\sum_{j=1}^{2^n}\sin\left(\frac{j\pi}{2^n}\right)\right)\right)$$

Answer: 2


Question: Calculate the length of the module $A = \mathbb{R}[X] / (X^3 - X^2 + X -1)$ as an $\mathbb{R}[X]$-module.

Answer: 2


Question: Find the number of integer solutions to the equation \(q_1 + q_2 + q_3 + q_4 + q_5 + q_6 = 26\) where each variable satisfies \(1 \leq q_i \leq 6\).

Answer: 2247


Question: Evaluate the double integral \( \int_0^3 \! \int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}} \sqrt{9-x^2-y^2}\,dy\,dx \) using a geometric formula.

Answer: 9\pi


Question: Compute the limit: \[ \lim_{n \to \infty} \int_{0}^{\infty} \frac{n \sin(x/n)}{x(1+x^2)} \, dx \]

Answer: \dfrac{\pi}{2}


Question:
Find the smallest integer \( x \) that satisfies the following system of congruences:
\[ 77x \equiv 1 \pmod{144} \]
\[ 67x \equiv 3 \pmod{77} \]

Answer: 4997


Question: Given that \(7x^2 - 40xy + 7y^2 = (|(x - y)| + 2)^3\) and \(x-y\equiv a\pmod{13}\), find the value of \(a\).

Answer: 4


Question: Evaluate the limit: \[ \lim_{n\to\infty}\sum_{k=0}^n\frac{^nC_k}{n^k(k+3)} \]

Answer: e - 2


Question: Determine whether the operation \( f \odot g = f\left(\int_0^x g(y)dy\right) g(x) \) defines a group law on the set \( G = \left\{ f : [0,1] \to (0,\infty) \text{ Lebesgue measurable and } \int_{[0,1]} f d\mu = 1 \right\} \) with identity \( 1 \) and inverse \( \frac{1}{f(F^{-1}(x))} \), where \( F(x) = \int_0^x f(y)dy \).

Answer: Yes


Question: Find the number of solutions to the equation \( f(z) = z^n \) where \( f: \overline{\mathbb{D}} \to \mathbb{C} \) is continuous, analytic in \( \mathbb{D} \), and satisfies \(|f(z)|<1\) for \(|z|=1\). Here, \( n \) is a positive integer and \( \mathbb{D} \) is the unit disk.

Answer: n


Question: Evaluate the limit \( \lim_{\textbf{X} \rightarrow \infty} \left(\sum_i (\ln(X_i))^a\right)^{1/a} \) where \( a \) is a negative constant. Determine whether the limit is 0 or infinity.

Answer: \infty


Question: Find the value of the infinite series: \[ \frac{1}{2\cdot4} + \frac{1\cdot3}{2\cdot4\cdot6} + \frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot8} + \cdots \]

Answer: \dfrac{1}{2}


Question: Find the value of \( a \geq 0 \) such that \( X + aY \) is independent of \( X - aY \), given that \( X \) and \( Y \) are bivariate normal with equal variance and covariance matrix \( \Sigma = \begin{bmatrix}1 & \rho \\ \rho & 1\end{bmatrix} \).

Answer: 1


Question: Compute the Riemann-Stieltjes integral \( \int_{-1}^{1} \cos x \ \mathsf{d}g(x) \), where \( g(x) = -\mathsf{1}_{[-1,0]}(x) + 3 \cdot \mathsf{1}_{(0,1]}(x) \).

Answer: 4


Question: Determine the number of elements in the quotient ring \( \Bbb Z_5[i]/\langle 1+i\rangle \).

Answer: 1


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

Answer: \dfrac{1}{6}


Question: Find a harmonic conjugate of the harmonic function \( u(x,y) = x^3 - 3xy^2 \). Then, express the resulting analytic function in terms of the complex variable \( z \).

Answer: z^3


Question: Evaluate the limit: \[ \lim_{x\to 0}\left(\frac{1}{1+ \tan x}\right)^{1/x^2}e^{1/x} \]

Answer: \sqrt{e}


Question: Evaluate the surface integral \( \iint_S \sqrt{1+4x^2} \, dS \) where the surface \( S \) is defined by the boundaries \( x=0, x=2, y=0, y=3 \) and \( z=x^2 \).

Answer: 38


Question: Given the parametric equations \( x(t) = 2t^2 + t - 1 \) and \( y(t) = t^2 - 3t + 1 \), find the value of \( t \) where the slope of the tangent line to the curve is 4.

Answer: -\dfrac{1}{2}


Question: Find the ordered triple of numbers $(a, b, c)$ with $0 < a < b < c < 10$ that satisfies the equation $$\arctan(a) + \arctan(b) + \arctan(c) = \pi.$$

Answer: (1, 2, 3)


Question: Find the minimum value of the function \( f(x, y) = \frac{xy}{2} + \frac{500}{x} + \frac{500}{y} \) over the domain \( D = \{(x, y) \in \mathbb{R}^2 : 1 \leq x \leq 1000, 1 \leq y \leq 1000\} \).

Answer: 150


Question: Evaluate the limit: $$\lim\limits_{x \to 0} \frac{x^4\cos(\frac{1}{x})} {\sin^3x}.$$

Answer: 0


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

Answer: 14121492


Question:
Let $X=l^2(\mathbb{Z})$ be the Hilbert space of all real sequences $x= (x_{k})_{k}$ $(k\in \mathbb{Z})$ for which the norm $\|x\|= \left( \sum_{k=-\infty}^{\infty} |x_{k}|^{2}\right)^{\frac{1}{2}}$ is finite. Denote by ${e_{k}:k\in\mathbb{Z}}$ the canonical basis in $X$, and define $L: X\longrightarrow X$ by
$$L
\left(\sum_{k=-\infty}^{\infty}x_{k}e_{k}
\right)=\sum_{k=-\infty}^{\infty}(x_{k}-x_{k-1})e_{k}$$
Determine the norm $\|L\|$.

Answer: 2


Question: Find all rational numbers $q \in (0,1)$ such that $\sin\left(\frac{\pi}{2}q\right)$ is a rational number.

Answer: \dfrac{1}{3}


Question: Given the functional equation \( f(x)f(1/x) = f(x) + f(1/x) \) and the condition \( f(3) = 28 \), find the value of \( f(4) \).

Answer: 65


Question: Determine the set of real numbers $x \in \mathbb{R}$ for which the power series \[ x + \sum_{k \geq 2} (-1)^{k-1} \frac{2k x^{2k-1}}{(2k-1)!} \] converges.

Answer: \mathbb{R}


Question:
Given the recursive sequences defined by $A_1=1$, $B_1=2$ and
\[ \begin{align*} A_{k+1} &= A_k\cdot (p_{k+1}-1)+B_k, \\ B_{k+1} &= B_k\cdot p_{k+1}, \end{align*} \]
where $p_k$ is the $k$-th prime number, prove that
\[ \lim_{n\to\infty}\frac{A_n}{B_n}=1. \]

Answer: 1


Question: Determine the radius of convergence of the function \( f(z) = \frac{\sin(z)}{z} \) for any point \( z_0 \in \mathbb{C} \setminus \{0\} \).

Answer: \infty


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{\int_0^{x^2} (e^{t^2}-1)~dt}{\log(1+2x^6)} \]

Answer: \dfrac{1}{6}


Question: In a family of 4, each member exercises at least one of the 7 deadly sins, with no sin being shared between two distinct members. Determine the number of ways to assign the sins to the family members such that no single member exercises both greed and gluttony.

Answer: 6840


Question: In a branching process where the mean of the offspring distribution is exactly 1, is the expected extinction time always infinite?

Answer: Yes


Question: Seven thieves steal diamonds and hide in the jungle. During the night, two thieves try to divide the diamonds between themselves but find one extra diamond. They wake a third thief, but still have one extra diamond. This pattern continues until they wake the seventh thief, at which point the diamonds are evenly distributed. How many diamonds did they steal in total?

Answer: 301


Question: Find all positive integers $n$ that are almost perfect and have the property that there exists an integer $k > 1$ such that $n^k$ is also almost perfect.

Answer: 2^k


Question: How many ways are there to arrange the letters in the word 'INTELLIGENT' such that there are at least two consecutive pairs of identical letters?

Answer: 286920


Question: Find \( \lim_{n\to\infty}\left(\frac{\binom{3n}{n}}{\binom{2n}{n}}\right)^{\frac{1}{n}} \).

Answer: \dfrac{27}{16}


Question: Evaluate the line integral \( \int_\gamma \left(\frac{1}{x+1}-\frac{y}{(x+y)^2}\right)dx+\left(\frac{1}{y+1}-\frac{x}{(x+y)^2}\right)dy \) where \( \gamma \) is the path along the circle from \((1,0)\) to \((0,1)\).

Answer: 0


Question: Determine whether the limit \( \lim_{m,n\to\infty} a_{mn} \) exists for the double sequence given by \[ a_{mn} = \frac{m^2n^2}{m^2+n^2}\left(1-\cos\left(\frac{1}{m}\right)\cos\left(\frac{1}{n}\right)\right). \]

Answer: \dfrac{1}{2}


Question: Given that $a + b + c + d = 4$, find the minimum value of the expression $\Sigma_{cyc}\frac{a}{b^3+4}$.

Answer: \dfrac{2}{3}


Question: Find all entire functions $f(z)$ such that $f(x) = e^x$ for all real numbers $x$. Is $f(z) = e^z$ the only solution?

Answer: e^z


Question: Compute \( \lim_{n\to\infty}\sqrt[3]{8n^3+4n^2+n+11}-\sqrt{4n^2+n+9} \).

Answer: \dfrac{1}{12}


Question: Calculate the limit: \[ \lim_{x\to\infty}\frac{\gamma(x,-c\sqrt{x}+x)}{\Gamma(x,c\sqrt{x}+x)} \] where \( \gamma(a,b) \) and \( \Gamma(a,b) \) are the lower and upper incomplete gamma functions, respectively.

Answer: 1


Question: Is there a rational polynomial \( f(x) \) of degree 100 such that its remainder after division by \( g(x) = x^{60} - 2 \) and \( h(x) = x^{49} + 3 \) are the same?

Answer: No


Question: Prove that \( \pi^e < e^\pi \) by considering the decreasing nature of the function \( x^{1/x} \) for \( x > 0 \).

Answer: \pi^e < e^\pi


Question: Determine the value of $b$ for which the series \( \sum_{r=2}^{\infty}\left(\frac{8}{3(br+1)}-\frac{3}{2(r+1)}+\frac{1}{6(r-1)}\right) \) is convergent using the integral test.

Answer: 2


Question: Given the polynomial equation \(x^4-(a+b+c+d)x^3-(a+b+c)x^2-(a+b)x-a=0\), where \(a, b, c,\) and \(d\) are natural numbers, determine the number of integer roots of the equation.

Answer: 0


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

Answer: 2\pi


Question: Determine the number of integral points on the hyperbola \(x^2 - y^2 = (2000)^2\). An integral point is a point where both coordinates are integers.

Answer: 98


Question: Let $f: D^2 \to S^2$ be a local diffeomorphism between the closed unit disk and the unit sphere. Is $f$ necessarily injective?

Answer: Yes


Question: The random variables $X$ and $Y$ are both $Be(1/2) = Bin(1,1/2)$-distributed and have a correlation coefficient of $1/2$. Determine the probability $P(X=Y)$. Provide your answer as a single probability value.

Answer: \dfrac{3}{4}


Question:
Find all integer solutions \((x, y)\) to the equation:
\[ 5^{2x} - 3 \cdot 2^{2y} + 5^x \cdot 2^{y-1} - 2^{y-1} - 2 \cdot 5^x + 1 = 0. \]

Answer: (2, 4)


Question: Determine the value of \( \lambda \in \mathbb{R} \) for which the improper integral \( I \) exists, where \[ I = \int_0^\infty \left( \frac{x+1}{3x^2+\lambda} - \frac{\lambda}{2x+1} \right) dx. \]

Answer: \dfrac{2}{3}


Question: Find all polynomials \( P(x) \in \mathbb{R}[X] \) such that \( P(x) \in \mathbb{Q} \) if and only if \( x \in \mathbb{Q} \).

Answer: P(x) = ax + b


Question: Given that $X_1$ and $X_2$ follow a normal distribution $N(\mu, \Sigma)$ with mean $\mu = \begin{bmatrix} 2 \\ 1 \end{bmatrix}$ and covariance $\Sigma = \begin{bmatrix} 4 & -4 \\ -4 & 8 \end{bmatrix}$, find $\mathbb{E}[X_1 \mid (X_1 + X_2)^2]$. Assume $X_1$ and $X_2$ are jointly normal.

Answer: 2


Question: Consider the rectangular hyperbola \(xy = c^2\). Normals at points \(P, Q, R,\) and \(S\) on the curve are concurrent and meet at point \(O(h, k)\). Find the value of \(OP^2 + OQ^2 + OR^2 + OS^2\).

Answer: 3(h^2 + k^2)


Question: Is the closure of every bounded convex set with non-empty interior in \(\mathbb{R}^n\) (where \(n > 1\)) homeomorphic to a closed ball \(B[a,r] = \{x \in \mathbb{R}^n : d(x,a) \le r\}\), where \(a \in \mathbb{R}^n\) and \(r > 0\)?

Answer: Yes


Question: Calculate the volume of the solid bounded by the cylinder $x^2 + y^2 = 4$ and the planes $y + z = 4$ and $z = 0$. Use a volume integral to find the solution.

Answer: 16\pi


Question: Let $L$ be a finite-dimensional restricted Lie algebra over a field of characteristic $p > 0$. An element $x$ of $L$ is called $p$-nilpotent if $x^{[p]^k} = 0$ for some positive integer $k$. If $L$ is nilpotent as an ordinary Lie algebra and has a basis consisting of $p$-nilpotent elements, is every element of $L$ $p$-nilpotent?

Answer: Yes


Question: The reflection of the hyperbola \(xy = 4\) in the line \(x - y + 1 = 0\) is given by the equation \(xy = mx + ny + l\). Find the value of \(m + n + l\).

Answer: 5


Question: Suppose set \( A \) is contained within the unit square in \( \mathbb{R}^2 \), and the projection of \( A \) onto any line outside the unit square is not Lebesgue measurable in \( \mathbb{R} \). Does this imply that \( A \) is not Lebesgue measurable in the plane?

Answer: Yes


Question: An inspector selects every \( n \)-th item in a production line for a detailed inspection. The time between item arrivals follows an exponential distribution with a mean of 1 minute. It takes 2 minutes to inspect an item. Determine the smallest value of \( n \) such that the probability of completing the inspection before the arrival of the next item that requires inspection is at least 90%.

Answer: 5


Question: Let \( \Phi: \mathbb{R}^{n-1} \to \mathbb{R} \) be a vector function. Suppose there exists a set \( S_1 \subset \mathbb{R}^{n-1} \) on which the gradient \( G = \nabla\Phi \) is unbounded. Does this necessarily imply that there exists a set \( S_2 \) on which the Hessian determinant \( H = \det \left( \frac{\partial^2 \Phi}{\partial x_i \partial x_j} \right) \) or at least one of its eigenvalues is non-vanishing?

Answer: Yes


Question: Given the differential equation \( \frac{dx}{dy} \cdot \frac{d^2y}{dx^2} = k \frac{dy}{dx} \), where \( k \) is a constant, find the value of the gradient \( \frac{dy}{dx} \) at the point where \( y = 2 \). It is known that the gradient \( \frac{dy}{dx} = 1 \) at the point \( (0,1) \).

Answer: e^k


Question: Find the probability density function (p.d.f) of the random variable $Z = XY$, where $X$ and $Y$ are independent and uniformly distributed over the interval (0,1).

Answer: -\ln z


Question:
Consider two estimators \(\hat{\theta}_{1}\) and \(\hat{\theta}_{2}\) for the parameter \(\theta\) of a distribution \(P_{\theta}\), where \(\theta \in \mathbb{R}\). These estimators satisfy the following conditions:

1. \(E_{X|\theta}(\hat{\theta}_{1}-\theta)^{2} = E_{X|\theta}(\hat{\theta}_{2}-\theta)^{2}\)
2. \(P_{\theta|X}(\hat{\theta}_{1} = \hat{\theta}_{2}) = 0\) for all \(\theta \in \mathbb{R}\).

Is there an estimator \(\hat{\theta}\) such that \(E_{X|\theta}(\hat{\theta}-\theta)^{2} < E_{X|\theta}(\hat{\theta}_{1}-\theta)^{2}\) for all \(\theta \in \mathbb{R}\)?

Answer: Yes


Question: Given a $3 \times 3$ real matrix $A$ such that the rank of $A^3$ is 2, determine the rank of $A^6$. Justify your answer.

Answer: 2


Question: Sam has $255$ cakes, each labeled with a unique non-empty subset of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Each day, he chooses one cake uniformly at random from the cakes not yet eaten. He eats that cake and all remaining cakes labeled with a subset of that cake. The expected number of days that Sam eats a cake before all cakes are gone can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime natural numbers. Find $p+q$. \( \boxed{} \)

Answer: 213


Question: Evaluate \( \lim_{n \to \infty} \int_{0}^{1} f(x) \sin(nx) \, dx \) where \( f \) is continuously differentiable over \([0,1]\). Use integration by parts and the fact that there exists a constant \( M \) such that \( M \geq |f'(x)| \) for all \( x \in [0,1] \).

Answer: 0


Question: Determine whether the ring $R$ of all functions $f : \Bbb{R} \to \Bbb{R}$, which are continuous outside $(-1,1)$, is isomorphic to the ring $S$ of all functions $f : \Bbb{R} \to \Bbb{R}$, which are continuous outside a bounded open interval containing zero. Is $R \cong S$?

Answer: R \not\cong S


Question: Evaluate the limit: \[ \lim_{x\to -\infty} \frac{\ln (1+e^x)}{x} \] without using L'Hospital's Rule.

Answer: 0


Question: Determine the necessary and sufficient condition for the matrix \( A = I - 2xx^T \) to be orthogonal, where \( x \) is a non-zero column vector in \( \mathbb{R}^n \).

Answer: x^T x = 1


Question: Calculate the flux of the vector field \( F = (x^2 y, y^2 z, z^2 x) \) through the hemisphere \( M = \{(x, y, z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 = 1, z \geq 0\} \). Use the parameterization in polar coordinates: \( \psi(\varphi, \theta) = (\cos(\varphi)\sin(\theta), \sin(\varphi)\sin(\theta), \cos(\theta)) \) with \( \varphi \in [0, 2\pi] \) and \( \theta \in [0, \frac{\pi}{2}] \).

Answer: 0


Question: Find the limit \( \lim_{n\to \infty}\frac{1}{n+1} \sum\limits_{j=0}^{n}x(j) \), where \( x(j) = 1 \) with probability \( p \) and \( x(j) = 0 \) with probability \( q \), and \( p + q = 1 \).

Answer: p


Question: Consider the map \( \Phi: \mathcal{P}(C([0,T],\mathbb{R}^d)) \to \mathcal{P}(\mathbb{R}^d) \) defined by \( \mu \mapsto \mu_t \), where \( \mu_t \) is the marginal of \( \mu \) at time \( t \). Is the map \( \Phi \) continuous?

Answer: Yes


Question: Is it true that every finite rank Euclidean building can be biLipschitz embedded into a finite product of metric \(\mathbb{R}\)-trees with the condition that \(p=1\) in the biLipschitz inequality?

Answer: Yes


Question:
Calculate the value of the infinite summation \( \sum\limits_{j=0}^\infty S_j \), where \( S_j \) is the area of the region bounded by the \( x \)-axis and the curve \( ye^x = \sin x \) for \( j\pi \leq x \leq (j+1)\pi \). Choose the correct answer from the options below:

(A) \( \dfrac{e^\pi(1+e^\pi)}{2(e^\pi-1)} \)

(B) \( \dfrac{1+e^\pi}{2(e^\pi-1)} \)

(C) \( \dfrac{1+e^\pi}{(e^\pi-1)} \)

(D) \( \dfrac{e^\pi(1+e^\pi)}{(e^\pi-1)} \)

Answer: B


Question: Find the integer part of the sum \( \sum_{k=2}^{9999} \frac{1}{\sqrt{k}} \).

Answer: 197


Question: Find all compact self-adjoint operators $T: H \to H$ such that $T^{2017} = 0$. Provide a description or characterization of these operators.

Answer: 0


Question: Solve for \(x\) in the equation \((1+2i)^{(2x+6)}=(-11-2i)^{(x+1)}\).

Answer: 3


Question: Compute the limit \( \lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^{n^2} \).

Answer: \infty


Question: Does there exist an additive function \( \phi: \mathbf{R} \to \mathbf{R} \) that is not of the form \( cx \) for some real number \( c \)?

Answer: Yes


Question: A coin is flipped 1000 times, and it lands on heads every time. Assuming the coin has a bias $p$ for heads, where $p \in [0, 1]$, determine the most likely value of $p$.

Answer: 1


Question: Let \((B_t)_{t \geq 0}\) be a standard Brownian Motion. Define the stopping time \(T = \inf\{t \geq 0 : B_t = at - b\}\) for some positive constants \(a, b > 0\). Calculate \(E[T]\).

Answer: \dfrac{b}{a}


Question: Determine the values of \( n \) for which the equation \( x^4 + x^3 + x^2 + x + 1 = 0 \) has roots in the finite field with \( 3^n \) elements.

Answer: n \equiv 0 \pmod{4}


Question: Find the derivative of the complex conjugate function with respect to the complex variable: \( \frac{\mathrm{d} z^*}{\mathrm{d} z} \), where \( z^* = x - iy \) and \( z = x + iy \).

Answer: 0


Question: Suppose \( f_n \) is a sequence of functions defined on a set \( S \) that converges uniformly to a function \( f \). If \( x_n \) is any sequence in \( S \), does \( f_n(x_n) - f(x_n) \to 0 \) as \( n \to \infty \)? Justify your answer.

Answer: Yes


Question: In an equilateral triangle $XYZ$, find the ratio $\frac{a}{b}$, where $b = \frac{YZ}{2}$. Express $b$ in terms of $a$ and determine if this ratio has any connection to the Golden ratio.

Answer: \sqrt{3}


Question: Determine the region of convergence for the series \( \sum_{n=1}^{\infty} \frac{(-1)^n n (x-1)^n}{(n^3+1)(3^n+1)} \).

Answer: [-2, 4]


Question: Given two independent and identically distributed random variables $X_1$ and $X_2$ from a distribution with cumulative distribution function $F$ and support on $[0,1]$, compute the probability $P(X_{(1)} < x < X_{(2)})$, where $X_{(1)}$ and $X_{(2)}$ are the minimum and maximum of $X_1$ and $X_2$, respectively, in terms of $F$. Express your answer in terms of $F(x)$.

Answer: 2F(x)(1 - F(x))


Question: Given that \( \sum_{k=1}^n (2k-1)a_k = n(n+1)(4n-1) \), find the value of \( a_n \).

Answer: 6n


Question: Evaluate the limit \( \lim_{n\to\infty} \int_{-\infty}^{\infty} x^{2n}e^{(-1/2)ax^2}dx \).

Answer: \infty


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

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \to \infty} x^2 \left( 2017^{\frac{1}{x}} - 2017^{\frac{1}{x+1}} \right) \]

Answer: \ln 2017


Question:
Let $a \neq b \in \mathbb{C}$ and define $U = \mathbb{C} \setminus [a,b]$. Consider a cycle $\Gamma$ in $U$. Determine whether the following equality holds:
\[ \int_{\Gamma} \frac{1}{(z-a)(z-b)} \, dz = 0 \]

Answer: 0


Question: Given the function \( f(x) = \frac{2x^3 + 7x}{x^2 + 4} \), which is invertible, calculate the derivative \((f^{-1})'\left(\frac{-9}{5}\right)\).

Answer: \dfrac{25}{47}


Question: Calculate the volume of the solid $D$ defined by the inequalities: \(x^2+y^2+(z-1)^2 \le 1\) and \(z^2 \le x^2+y^2\).

Answer: \dfrac{\pi}{3}


Question: Evaluate the limit \( \lim\limits_{x\to 0}\frac {2\sin(x)-\arctan(x)-x\cos(x^2)}{x^5} \) using power series.

Answer: \dfrac{19}{60}


Question: Compute the integral \( \int_{-\infty}^{\infty} [F(x+10) - F(x)] \, dx \), where \( F \) is a distribution function.

Answer: 10


Question: Determine the number of pairs of positive constants \(a\) and \(b\) such that \(P(a < X < b) = 0.95\), where \(X\) follows a chi-squared distribution with \(n\) degrees of freedom.

Answer: \infty


Question: Given a normal element $x$ in a $C^*$ algebra with spectrum $\{-1, 1\}$, determine if there exist two non-trivial projections $p$ and $q$ such that $pq = 0$. Provide a justification for your answer.

Answer: Yes


Question: Determine which ordinal is larger: \(\tau_1\), the supremum of ordinals writable by iterated Infinite Time Turing Machines, or \(\tau_2\), the smallest \(\Sigma_2^1\)-reflecting ordinal.

Answer: \tau_2


Question: Given that the quadratic equation \( az^2 + bz + 1 = 0 \) has complex coefficients \( a, b, z \in \mathbb{C} \) with \( |a| = \frac{1}{2} \), and one of its roots \( \alpha \) satisfies \( |\alpha| = 1 \), find the value of \(|a\bar{b} - b|\).

Answer: \dfrac{3}{4}


Question: Let $P$ and $Q$ be distinct Sylow $p$-subgroups of a group $G$ such that $|P \cap Q|$ is maximal. Prove that the normalizer $N_G(P \cap Q)$ contains more than one Sylow $p$-subgroup, and that each pair of these Sylow $p$-subgroups intersects in $P \cap Q$. Additionally, show that $N_G(P \cap Q) \neq P \cap Q$.

Answer: N_G(P \cap Q) \neq P \cap Q


Question: Minimize the function \( f(x, y, z) = x^2 + y^2 + z^2 \) subject to the constraint \( 4x^2 + 2y^2 + z^2 = 4 \). Find the minimum value of the function.

Answer: 1


Question:
Calculate the limit:
\[ \lim_{(x,y)\to(0,0)}\frac{1-\sqrt{1-x^2-y^2}}{2x^2+y^2} \]
subject to the restriction \(|y|<x^2\).

Answer: \dfrac{1}{4}


Question: Given that $a + \frac{1}{b} = b + \frac{1}{c} = c + \frac{1}{a}$ for distinct real numbers $a$, $b$, and $c$, find the value of $abc$. \( \text{Express your answer as a single number.} \)

Answer: -1


Question: Let $M$ and $N$ be two closed subspaces of a Hilbert space $H$ such that $N \subset M$. If $P_M$ and $P_N$ are the orthogonal projections onto $M$ and $N$, respectively, determine the subspace of $H$ on which $P_M - P_N$ is an orthogonal projection.

Answer: M \ominus N


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

Answer: (-1, 1)


Question: Determine the number of distinct real roots of the equation \(f(f(x)) = 0\), where \(f(x) = x^3 - 3x + 1\).

Answer: 7


Question:
Let \( f:(-1,1) \to \mathbb{R} \) be a function such that \( f^{(n)}(x) \) exists and \( |f^{(n)}(x)| \leq 1 \) for every \( n \geq 1 \) and for every \( x \in (-1,1) \). Determine where \( f \) has a convergent power series expansion:

(a) every \( x \in (-1,1) \)
(b) every \( x \in (-1/2,0) \) only
(c) no \( x \in (-1,1) \)
(d) every \( x \in (0,1/2) \) only

Choose the correct option.

Answer: a


Question: Given the initial list of numbers: $48, 99, 120, 165, 273$, determine the maximum possible size of the list after playing the game optimally. In the game, a player can choose any two numbers $a$ and $b$ from the list, compute $c = |a - b|$, and add $c$ to the list if $c$ is non-zero and not already in the list. What is the maximum score a player can achieve?

Answer: 91


Question: Find the limit of the sequence defined by the recurrence relation \(a_{n+1}=\frac{1+\sqrt{4a_{n}+9}}{2}\) with the initial condition \(a_0=3\).

Answer: 1 + \sqrt{3}


Question: Find all functions \( f : \mathbb{Q} \to \mathbb{Q} \) such that \( f(1) = 2 \) and \( f(xy) = f(x)f(y) - f(x+y) + 1 \) for all \( x, y \in \mathbb{Q} \).

Answer: f(x) = x + 1


Question:
Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f'(x) > f(x) \) for all \( x \in \mathbb{R} \) and \( f(0) = 1 \). Determine the interval in which \( f(1) \) lies:

(a) \((0, e^{-1})\)

(b) \((e^{-1}, \sqrt{e})\)

(c) \((\sqrt{e}, e)\)

(d) \((e, \infty)\)

Answer: D


Question: Determine the value of $f(d)$, the smallest possible size for an efficient substitute for $\Omega = \{0,1\}^d$, where $d$ is an integer $\geq 2$. An efficient substitute $\Omega'$ satisfies $p(\Omega',i,j,A)=p(\Omega,i,j,A)$ for all $i,j$ and $A$, with $p(\Omega,i,j,A)$ being the probability that $(x_i,x_j) \in A$ for a randomly selected $(x_1,x_2, \ldots, x_d) \in \Omega$. The trivial bound is $f(d) \leq 2^d$, and it is known that $f(3)=4$. What is $f(d)$ in general?

Answer: 2^{\lceil \log_2 (d+1) \rceil}


Question: Let \( A = (a_{ij})_{i,j=1,\dots,n} \) be a matrix such that \( a_{ij} \ge 0 \) for all \( i,j = 1, \dots, n \), and \( A \) is positive definite. Let \( I \) be the identity matrix, and \( \pmb{1} \) the vector containing only ones. Suppose that the solution \( x = (x_i)_{i=1,\dots,n} \) to the system of linear equations \( (A+I)x=\pmb{1} \) is non-negative, i.e., \( x_i \ge 0 \) for \( i=1,\dots,n \). Define \( f(t) = \frac{e^t + 1}{e^t - 1} \). Is the solution \( x \) to the system of linear equations \( (A+f(t)I)x=\pmb{1} \) non-negative for every \( t>0 \)?

Answer: Yes


Question: How many possible outcomes are there when rolling 10 dice, each showing a number from 1 to 6?

Answer: 60466176


Question: Find the values of \(\lambda\) for which the operations \(a \oplus b = a + b - 6\) and \(a \otimes b = ab + \lambda a + \lambda b + 42\) induce a ring structure \((\mathbb{Z}, \oplus, \otimes)\).

Answer: -6


Question: Given that \(\arg(z^{1/3}) = \frac{1}{2} \arg(z^2 + \bar{z}z^{1/3})\), find the value of \(|z|\), where \(z\) is a complex number.

Answer: 1


Question:
Determine the values of \(a_1, a_2, \ldots, a_n\) for which the following \(n \times n\) matrix is nonsingular:
\[
A=\begin{bmatrix}
a_1 & 1 & 0 & \cdots & 0 & 0 \\
a_2 & 0 & 1 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
a_{n-2} & 0 & 0 & \cdots & 1 & 0 \\
a_{n-1} & 0 & 0 & \cdots & 0 & 1 \\
a_{n} & 0 & 0 & \cdots & 0 & 0 \\
\end{bmatrix}
\]

Answer: a_n \neq 0


Question: Evaluate the limit: \[ \lim_{x \to \infty} x^a a^x \] where \( 0 < a < 1 \).

Answer: 0


Question: Is it true that for a von Neumann algebra $A$, the second dual space $A^{**}$ is equal to $A$?

Answer: No


Question: Let \( M \) be a \( 2 \times 2 \) matrix with entries in the ring \( \mathbb{C}[x][[y]] \) and a constant determinant. Given \( k > 0 \), does there exist a matrix \( N \) with entries in \( \mathbb{C}[x, y] \) such that \( M \mod (y^{k+1}) = N \mod (y^{k+1}) \) and \( \det M = \det N \)?

Answer: Yes


Question: Determine if the solution to the differential equation \( x''+tx'+\frac{1}{1+t+t^2}x=0 \) with initial conditions \( x(1)=0 \) and \( x'(1)=1 \) is analytic at \( t_0=1 \) and if its radius of convergence \( R > 1 \).

Answer: Yes


Question: Determine whether it is possible for two different sets, each containing $n$ unique positive integers, to have the same sum when the squares of their elements are added. Assume $n > 1$. Provide a justification for your answer.

Answer: Yes


Question: Use the Argument Principle or Rouche's Theorem to determine the number of zeros of the polynomial \( z^2 - z + 2 \) inside the unit disk \( |z| < 1 \).

Answer: 0


Question: The plane is partitioned into parabolas such that each point belongs to exactly one parabola. Determine if it necessarily follows that all the parabolas have their axes in the same direction.

Answer: Yes


Question:
Determine whether the following binomial congruence is true or false:

$$\binom{k+sp}{j}\equiv\binom{k}{j}\pmod{p}$$

where $p \ge 1$, $s \ge 1$, $k \ge 1$, and $p \not\mid j$. Consider the case where $k=5$, $j=4$, $s=1$, and $p=3$. Verify if the congruence holds for these values.

Answer: False


Question: Find a constant \( c > 0 \) such that the series \( \sum_{n=1}^{\infty} b_n = c \sum_{n=1}^{\infty} \frac{1}{n^2} \) satisfies \( b_n \ge \frac{n+17}{6n^3+4n^2+5} \) for all \( n \in \mathbb{N} \).

Answer: \dfrac{6}{5}


Question: Find the probability density function (pdf) of the random variable \( Z = \frac{9}{(X+1)^2} \) given that the pdf of \( X \) is \( f_X(x) = \frac{x + 1}{4} \) for \( 0 < x < 2 \) and \( 0 \) otherwise.

Answer: \dfrac{9}{8 z^2}


Question: Compute the sum \( \sum^n_{k=0}(-1)^k\binom{n}{k}k^n \). Verify if it equals \((-1)^n n!\).

Answer: (-1)^n n!


Question: Use Green's Theorem to calculate the work done by the force field \( \mathbf{F}(x,y)=x(x+y)\mathbf{i}+xy^2\mathbf{j} \) as a particle moves along the path from the origin along the \( x \)-axis to \((1,0)\), then along the line segment to \((0,1)\), and back to the origin along the \( y \)-axis.

Answer: -\dfrac{1}{12}


Question: Let \(f\) be an analytic function in the upper half-plane such that \(|f(z)| < 1\) for all \(z\) in the upper half-plane, and \(f(i) = 0\). Determine the maximum possible value of \(|f(2i)|\).

Answer: \dfrac{1}{3}


Question: Determine whether the graph of the function \( f:(a, b) \to \mathbb{R} \), where \( f \) is continuous and \( \lim\limits_{x\to a}f(x)=\infty \) and \( \lim\limits_{x\to b}f(x)=\infty \), is closed in \( \mathbb{R}^2 \).

Answer: Yes


Question: Compute the value of the infinite product: \[ \prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1} \]

Answer: \dfrac{2}{3}


Question: Given the differential equation \( \frac{dy}{dx} = \frac{x^2 - y^2}{x^2 + y^2} \), let \( c_1: y = f(x) \) and \( c_2: y = g(x) \) be two integral curves of this equation. A line passing through the origin meets \( c_1 \) at \( P(x_1, y_1) \) and \( c_2 \) at \( Q(x_2, y_2) \). Find the value of \( \frac{f'(x_1)}{g'(x_2)} \).

Answer: 1


Question: Evaluate the integral \( \int_{-\infty}^{\infty}\frac{x^2 \cos(x)}{(x^2+1)^2}dx \) using the method of residues.

Answer: 0


Question: Compute the contour integral \( A = \int_{\partial \mathcal{D}(0,1)} \frac{8z^2-8z+1}{4z^3-8z^2+z-2} \, dz \).

Answer: 2\pi i


Question: Let $G$ be a finite group with normal subgroups $M$ and $N$ such that $M \cap N = \{e\}$, where $e$ is the identity element. Suppose $G$ has a subgroup of index 2. Must there exist a subgroup of index 2 in $G$ that contains either $M$ or $N$?

Answer: Yes


Question: In a random triangle where all pairs of angles have the same joint distribution, determine the correlation between two of the angles, assuming the variance of the angles is nonzero.

Answer: -\dfrac{1}{2}


Question: Calculate the integral \( \int_S f(x,y,z) \, dV \) over the tetrahedron \( S \) with vertices \((0,0,0), (0,1,2), (1,2,3), (-1,1,1)\), where \( f(x,y,z) = x + 2y - z \).

Answer: \dfrac{1}{6}


Question: Consider a branching process $X=\{X_n, n=0,1,\dotsc\}$ where $X_n=\sum\nolimits_{i = 1}^{{X}_{n-1}}{Z_i }$, $X_0=1$, and let $Z_i$ be such that $P[{Z_i=0]}=1/2$, $P[Z_i=1]=1/4$, $P[Z_i=2]=1/4$. Find the probability of extinction $\pi_0=P[\bigcup_n(X_n=0)\mid X_0=1]$. \(\pi_0\) is the probability that the process eventually reaches zero individuals.

Answer: 1


Question: Find all functions \( f : \mathbb{N}_0 \to \mathbb{N}_0 \) such that for all \( x, y \in \mathbb{N}_0 \), the equation \( x f(y) + y f(x) = (x + y) f(x^2 + y^2) \) holds.

Answer: f(x) = c


Question: Determine the values of \(a\) for which the matrix \(A = \begin{pmatrix} 2 & 0 & 0 \\ a & 2 & 0 \\ a+3 & a & -1 \end{pmatrix}\) can be diagonalized.

Answer: 0


Question: Consider an infinite set $X$ with the co-finite topology and a continuous group action $\varphi:G\times X\to X$. Let $\mathcal{U}=\{U_1, U_2, \ldots, U_n\}$ be a finite open cover of $X$. Define $A=\cup_{i=1}^n X-U_i$, which is a finite set. Determine if there exist distinct elements $x$ and $y$ in some $U_i$ for $i\in \{1, 2, \ldots, n\}$ such that $(Gx\cup Gy)\cap A=\emptyset$. Answer with 'Yes' or 'No'.

Answer: No


Question: Determine whether the following statement is true or false: If $E(Z\mid X)=0$, then $E(Z\mid X,Y)=0$. Provide a justification for your answer.

Answer: False


Question:
Let \( f, g: [0,1] \to \mathbb{R} \) be continuous functions. Prove that there exists \( \alpha \in (0,1) \) such that
\[
\int_0^1 f(x) \, dx \int_0^1 g(x) \, dx = g(\alpha) \int_0^\alpha f(x) \, dx + f(\alpha) \int_0^\alpha g(x) \, dx.
\]

Answer: \alpha


Question: Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function such that $f(x)f(f(x)) = 1$ for all $x \in \mathbb{R}$. Given that $f(1000) = 999$, find $f(500)$. \( \text{Let } f : \mathbb{R} \to \mathbb{R} \text{ be continuous, with } f(x)f(f(x)) = 1 \text{ for all } x \in \mathbb{R}. \text{ If } f(1000) = 999, \text{ find } f(500). \)

Answer: \dfrac{1}{500}


Question: Given an analytic function \( F(s) \) on the right half-plane, which is zero everywhere except at \( s=0 \), determine the value of \( F(0) \).

Answer: 0


Question: Is it possible to select a finite number of terms from the sequence \( \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \frac{1}{9}, \frac{1}{11}, \ldots \) such that their sum equals one?

Answer: Yes


Question: Is the sequence $\sqrt{p}-\lfloor\sqrt{p}\rfloor$, where $p$ runs over the primes, dense in the interval $[0,1]$?

Answer: Yes


Question: Given an antichain \(\{v^i\}_{i \in[d]} \subseteq \mathbb{Z}^n_+\) with \(d < n\), where an antichain is defined with respect to the component-wise order such that for any \(i \neq j \in [d],\) there exist indices \(t_1, t_2 \in [n],\) such that \( v^i_{t_1} > v^j_{t_1}\) and \(v^i_{t_2} < v^j_{t_2}\). Does there always exist a hyperplane \(\alpha^T x = \beta\) containing all these integer points, with \(\alpha_i > 0\) for any \(i \in [n]\)?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to\infty} \left(\frac{1}{x}\frac{a^x - 1}{a - 1} \right)^\frac{1}{x}, \] where \(0 \leq a\) and \(a \neq 1\).

Answer: \max\{a, 1\}


Question: Given the equation \( \int_{\pi/2}^{x} \sqrt{3-2\sin^2 t} \, dt + \int_{0}^{y}\cos t \, dt = 0 \), find \( \frac{dy}{dx} \) at \( x=\pi \) and \( y = \pi \).

Answer: \sqrt{3}


Question: Let $M$ be a connected, compact, oriented manifold of dimension $n<7$. If any two maps $M \to M$ having equal degrees are homotopic, must $M$ be diffeomorphic to the $n$-sphere? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim _{x\to 0^+}\left(\left(\left(1+x\right)^x-1\right)^x\right) \] as \( x \) approaches 0 from the positive numbers.

Answer: 1


Question:
Solve the system of differential equations \( \frac{dy}{dx} = 2y+z \) and \( \frac{dz}{dx}=3y \). Determine which of the following relations is satisfied by the solution:

(a) \((y + z)e^{3x} = k\)

(b) \((3y + z)e^{-3x} = k\)

(c) \((3y + z)e^{3x} = k\)

(d) \((y - z)e^{3x} = k\)

Answer: B


Question: If a function \( f: [a, b] \to \mathbb{R} \) is Lipschitz continuous and its derivative \( f' \) exists and is uniformly continuous on \([a, b]\), does the second derivative \( f'' \) exist for all \( x \in [a, b] \)?

Answer: No


Question: In a laboratory, a chemist adds sulfuric acid to flasks held by four groups, each consisting of 3 people. For the first group, he adds $a_1$, $a_2$, and $a_3$ drops to the flasks such that $a_1 + a_2 + a_3 = 4$. For the second group, he adds $b_1$, $b_2$, and $b_3$ drops such that $b_1 + b_2 + b_3 = 3$. For the third group, he adds $c_1$, $c_2$, and $c_3$ drops such that $c_1 + c_2 + c_3 = 2$. For the fourth group, he adds $d_1$, $d_2$, and $d_3$ drops such that $d_1 + d_2 + d_3 = 1$. Assuming the drops are of equal size and randomly distributed, what is the probability that the flask with the most acid belongs to the first group? Express your answer as a percentage.

Answer: 50\%


Question: Determine the Galois group of the polynomial $x^5 - 5$ over the field of rational numbers $\mathbb{Q}$.

Answer: F_{20}


Question: Let $(X, \mathcal{U})$ be a compact and Hausdorff uniform space, and let $X$ be totally disconnected. For $D \in \mathcal{U}$, a sequence $\{x_i\}_{i \in \mathbb{Z}}$ is called a $D$-chain if $(x_i, x_{i+1}) \in D$ for all $i \in \mathbb{Z}$. Given $E \in \mathcal{U}$, is there a $D \in \mathcal{U}$ such that for every $D$-chain $\{x_i\}_{i \in \mathbb{Z}}$, we have $(x_i, x_j) \in E$ for all $i, j \in \mathbb{Z}$? Answer true or false.

Answer: True


Question: Show that there exists a linear operator \( L: \mathbb{R}^m \rightarrow \mathbb{R}^m \) such that \( L(v) = -v \) and \( L(v_i) = v_i \) for \( i = 1, \ldots, m-1 \). Additionally, demonstrate that \( L \) corresponds geometrically to a reflection in the subspace spanned by \( \{v_1, \ldots, v_{m-1}\} \).

Answer: L


Question: Compute the group of automorphisms $\text{Aut}(S)$ where $S = \mathbb{Q}[x]/(x^2)$. Express your answer in terms of a known group.

Answer: \mathbb{Q}^\times


Question: Determine the maximum number of vertices that can exist in the intersection of a triangle with a box in 3D space.

Answer: 6


Question: Compute the integral \( \int_0^1 g(z) \, dz \), where \( g(z) = z \int_1^N [t] e^{2 \pi i t z} \, dt \) and \( N \) is a fixed positive integer.

Answer: 0


Question: Does there exist a finite topological space with a fundamental group isomorphic to \(\mathbb{Z}_2\)?

Answer: Yes


Question: Given a unit vector $u \in \mathbb{R}^n$, let $v$ be sampled uniformly at random from the unit sphere centered at $0$ in $\mathbb{R}^n$. Determine if the probability that $u$ and $v$ are orthogonal is zero.

Answer: 0


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

Answer: 4\pi i


Question: Determine whether the following statement is true or false: For a sequence of normal random variables with mean 0 and variance $n$, for any $x \in \mathbb{R}$, \[ \lim_{n\to \infty} \int_{-\infty}^{x}\frac{1}{\sqrt{2\pi n}}e^{-\frac{y^2}{2n}}dy=\frac{1}{2}. \]

Answer: True


Question: Find the sum of all positive integers \(k\) for which the inequality \(5x^2-2kx+1<0\) has exactly one integral solution.

Answer: 9


Question: In triangle \(ABC\), where \(AB = 8\), \(AC = 7\), and \(BC = 10\), let \(I\) be the incenter. If \(AI\) intersects \(BC\) at \(K\) and the circumcircle of \(\triangle ABC\) at \(D\), find \(\frac{DK}{DI}\).

Answer: \dfrac{2}{3}


Question: Find the critical points of the function \( f(x, y) = xy \) in the interior of the set \( S = \{(x,y) \in \mathbb{R}^2 \mid x^2 + 4y^2 \leq 1\} \).

Answer: (0, 0)


Question: Determine the set of all values of \( p \) such that for every \( q > 0 \), the inequality \( \frac{3(pq^2+p^2q+3q^2+3pq)}{p+q} > 2p^2q \) holds. Express your answer in interval notation in decimal form.

Answer: [0, 3)


Question: Determine the values of \(x \in \mathbb{R}\) for which the series \(\sum_{k=1}^\infty \frac{(1+(-1)^k)e^{ikx}}{k^2}\) converges.

Answer: \mathbb{R}


Question: Let \( \mathbf{G}: \mathbb{R}^2 \to \mathbb{R}^2 \) be defined by \( G(x,y) = (\cos(x-y), \sin(x+y)) \). Assume \( f: \mathbb{R}^2 \to \mathbb{R} \) is differentiable with gradient \( \nabla f(0,1) = (2,3) \). Define \( h: \mathbb{R}^2 \to \mathbb{R} \) by \( h(x) = f(\mathbf{G}(\mathbf{x})) \). Find the gradient \( \nabla h\left( \frac{\pi}{2}, 0 \right) \).

Answer: (-2, 2)