Question: Evaluate the limit: \[ \lim_{x \to \infty} \sqrt{x} \left( \sqrt[3]{x+1} - \sqrt[3]{x-1} \right) \]

Answer: 0


Question: Find the auxiliary equation for the ordinary differential equation with constant coefficients: \((x^2D^2 + xD + 1)y = \sin(2\log x)\sin(\log x)\).

Answer: m^2 + 1 = 0


Question: Evaluate the limit: \[ \lim_{x \to 0} \left(\dfrac{1}{\tan^2 x}-\dfrac{1}{x^2} \right) \]

Answer: -\dfrac{2}{3}


Question: Determine the minimum sample size required such that the probability of at least two people being a match at all six genetic markers exceeds 0.001, given that the probability of a match at each individual marker is 1/9.

Answer: 34


Question: Find the limit: \[ \lim_{x \to \infty} (x!)^{1/x} \]

Answer: \infty


Question: Find the length of the polar curve given by \( r = \sqrt{1 + \cos(2\theta)} \) for \( 0 \leq \theta \leq \frac{\pi\sqrt{2}}{4} \).

Answer: \dfrac{\pi}{2}


Question: Let $A$ be a proper infinite subset of a set $X$. If $x$ and $y$ are two distinct elements of $X$ that are not in $A$, and we define $B = \{x, y\} \cup A$, what is the cardinality of $B$ in terms of the cardinality of $A$?

Answer: |A|


Question: Consider a regular pentagon circumscribed in a circle. Connect each vertex of this pentagon to every other non-adjacent vertex with a straight line segment to form a pentagram, which contains a smaller pentagon. What is the ratio of the area of the original (large) pentagon to the smaller one in terms of the golden ratio?

Answer: \phi^4


Question: Find the longest geometric progression with a common ratio greater than 1 that can be formed from the set \( \{100, 101, 102, \ldots, 1000\} \). What is the length of this progression?

Answer: 6


Question: Find the expectation \( \mathbb{E}[X_t] \) where \( X_t = \sin(B_t) \) and \( B_t \) is a standard Brownian motion for \( t \geq 0 \).

Answer: 0


Question: Determine the values of \( k \) for which the equation \( x^3 - 3x^2 + 6x + k = 0 \) has three real roots.

Answer: \emptyset


Question: Verify that the distance from the function \( f = x \) to the set \( Y = \{ f \in C[0,1] : \int_0^1 f = 0 \} \) is 0.5, given that \( Y \) is a closed subset of \( X = \{ f \in C[0,1] : f(0) = 0 \} \).

Answer: 0.5


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

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


Question: Provide an example of a field that properly contains the field of complex numbers \( \mathbb{C} \).

Answer: \mathbb{C}(t)


Question: Let $T$ be a continuous linear operator. Suppose $(u_n)$ is a sequence that converges weakly to $u$, denoted as $(u_n) \rightharpoonup u$. Additionally, assume $T(u_n) \rightharpoonup T(u)$ and there exists a subsequence $(u_{n_k})$ such that $T(u_{n_k}) \rightarrow T(u)$. Does it follow that $T(u_n) \rightarrow T(u)$?

Answer: No


Question: Let $a, b, c, d$ be a permutation of the numbers $1, 9, 8, 4$. Define $n = (10a + b)^{10c + d}$. Calculate the probability that $1984!$ is divisible by $n$. Use Fermat's Little Theorem to assist in your calculations.

Answer: \dfrac{5}{6}


Question: Let \( A \) be a C* algebra of operators on a Hilbert space \( H \). Determine if there exists an \( x \) in \( H \) such that the set \( Ax \) is dense in \( H \) but not equal to the whole \( H \).

Answer: Yes


Question: Evaluate the limit: \[ \lim_{t\to 0}\left(\frac{1}{\ln(1 + t)}+\frac{1}{\ln(1-t)}\right). \]

Answer: 1


Question: Determine the limit of \( \frac{\pi(x)}{x} \) as \( x \to \infty \), where \( \pi(x) \) is the prime counting function.

Answer: 0


Question: Consider a sequence $(r_n)$ resulting from infinite coin flips, where $R_n=1$ if $r_n$ is a head and $R_n=-1$ if $r_n$ is a tail. Determine if $P\left(\sum \frac{R_n}{n} < \infty\right) = 1$. Provide a justification for your answer.

Answer: 1


Question: Consider a locally path connected space $X$. If every open subset $U \subseteq X$ is semi-locally simply connected, does it follow that $X$ is locally simply connected?

Answer: Yes


Question: Find parametric equations for a unit circle with a speed of \( e^t \), starting from \( x=1 \), \( y=0 \). Determine when the circle is completed.

Answer: \ln(2\pi + 1)


Question: Using the generating function \( \sum_{n=0}^{\infty}P_n(x)r^n=(1-2rx+r^2)^{-\frac{1}{2}} \), find the value of \( P_n(1) \).

Answer: 1


Question: Evaluate the expression \( \log \left| 1 + \alpha + \alpha^2 + \alpha^3 - \frac{1}{\alpha} \right| \), where \( \alpha \) is a fifth root of unity.

Answer: \log 2


Question:
Verify that for \(n = 2^kN\), where \(N\) is odd, the following identity holds:
\[ \sum_{d\mid n}(-1)^{n/d}\phi(d) = \sum_{d\mid 2^{k-1}N}\phi(d) - \sum_{d\mid N}\phi(2^kd) = 0. \]

Answer: 0


Question:
Find the kernel of the Vandermonde matrix \( A \) given by
\[
A = \begin{pmatrix} x_1^0 & x_1^1 & \ldots & x_1^n \\ x_2^0 & x_2^1 & \ldots & x_2^n \\ \vdots & \vdots & \ldots & \vdots \\ x_m^0 & x_m^1 & \ldots & x_m^n \end{pmatrix}
\]
where \( n < m - 1 \) and the \( x_i \) are pairwise distinct.

Answer: \{\mathbf{0}\}


Question: Find the minimum value of the product \(abc\) if the quadratic equation \(ax^2 - bx + c = 0\) has two distinct roots in the interval \((0, 1)\), where \(a\), \(b\), and \(c\) are natural numbers.

Answer: 25


Question:
Find the maximum value of $x^2 + y^2$ given that $(x, y)$ satisfy the following equations:
\[ 2x^2 + 5xy + 3y^2 = 2 \]
\[ 6x^2 + 8xy + 4y^2 = 3 \]
Note: Calculus is not allowed.

Answer: \dfrac{5}{2}


Question: Determine whether the following statement is true or false: "Any bounded sequence in $L^4[0,1]$ has a convergent subsequence in $L^2[0,1]$."

Answer: B


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

Answer: 1


Question:
Determine whether the following inequality is true for all real numbers $0 < r < 1$ and $t \geq 0$:
\[ \int_t^{t+r} \sin(x)\, dx \leq \int_{\frac{\pi}{2}-\frac{r}{2}}^{\frac{\pi}{2}+\frac{r}{2}}\sin(x)\, dx. \]

Answer: True


Question: Let $E \subset \mathbb{C}$ be a set such that for any sequence of distinct elements $(e_n)_{n \in \mathbb{N}}$ from $E$, $e_n \to 0$ in norm. Is $E$ necessarily countable? Provide a justification for your answer.

Answer: Yes


Question: Let $S$ be an algebraic smooth surface over $\\mathbb{C}\\). Suppose there is a fibration $p: S \rightarrow C$ onto a smooth curve, and let $f$ be a fiber of this fibration. Let $K$ be a canonical divisor on $S$ such that $K \cdot f = 0$. Determine whether each $m$-canonical map $\varphi_{mK}: S \rightarrow \mathbb{P}^N$, when defined, will contract $f$ to a point.

Answer: Yes


Question: Find the number of ways to make change for a dollar using generating functions. Specifically, determine the coefficient of the \(x^{100}\) term in the expansion of the generating function \(\frac{1}{(x-1)(x^5-1)\cdots(x^{50}-1)}\).

Answer: 292


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

Answer: 23


Question: Evaluate the limit: \[ \lim _{n\to \infty }n \int_{-1}^0(x + e^x)^{n}dx. \]

Answer: \dfrac{1}{2}


Question:
Given the joint probability distribution of $X$ and $Y$:
\[ f(x,y) = \begin{cases}
  e^{-(x+y)}, & x > 0, y > 0 \\
  0, & \text{otherwise}
\end{cases} \]
compute the probability $P(Y > X+1)$.

Answer: \dfrac{1}{2e}


Question: Evaluate the limit: \[ \lim_{x\to 0}\left(\log\frac{1}{x}\right)^x \]

Answer: 1


Question: Evaluate \(1234^{1234} \pmod{5379}\). Note that \(5379 = 3 \times 11 \times 163\).

Answer: 4603


Question: Find the derivative of the matrix product $\mathbf{A}\mathbf{B}\mathbf{c}$ with respect to the matrix $\mathbf{B}$, where $\mathbf{A}$ is an $n\times m$ matrix, $\mathbf{B}$ is an $m\times k$ matrix, and $\mathbf{c}$ is a $k\times 1$ vector.

Answer: \mathbf{c}^\top \otimes \mathbf{A}


Question: Find the maximum value of the function \( f(x) = \int^{x}_{0} \sqrt{(x^2-x)^2+y^4}~dy \) for \( 0 \leq x \leq 1 \).

Answer: \dfrac{1}{3}


Question: Is the space \(X = \prod_{t \in \mathbb{Z}} \mathbb{R}\), with the product topology \(T\), completely metrizable by a metric \(d\) such that \(d(\tau x, \tau y) = d(x, y)\) for the shift map \(\tau\) defined by \(\tau x = (\ldots, x_0, x_1, x_2, \ldots)\)?

Answer: No


Question: A triangle is inscribed in a circle with radius 1. What is the maximum value of the sum of the squares of the sides of the triangle?

Answer: 9


Question: Find a matrix $X \in M_n(\mathbb{C})$ such that the linear functional $f(A) = \text{tr}(XA)$ on $M_n(\mathbb{C})$ preserves matrix multiplication.

Answer: 0


Question: Find the minimum natural number \( n \) such that the expression \( f(n) = \sqrt{100+\sqrt{n}} + \sqrt{100-\sqrt{n}} \) is an integer.

Answer: 6156


Question: Compute the limit \( \lim_{n\to\infty}{(\sqrt[n]{e}-\frac{2}{n})^n} \).

Answer: \dfrac{1}{e}


Question: Calculate the integral: \[ \int_{0}^{2} \sqrt{1+x^3} \, dx \]

Answer: 3.241


Question: For the function \( u(x, y) = \sinh x \cos y \), find the conjugate harmonic function \( v(x, y) \).

Answer: \cosh x \sin y


Question: In a compact metric space $X$, consider a finite subset $F \subset X$ that is $\epsilon$-equidistant, meaning that the distance $d(x,y) = \epsilon$ for all distinct $x,y \in F$. For a given $\epsilon > 0$, does there exist a maximum number $N$, depending on both $\epsilon$ and $X$, such that the cardinality of any $\epsilon$-equidistant set $F$ is at most $N$? Provide a proof or counterexample.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to\infty}\left(\frac1{x^2\sin^2\frac 1x}\right)^\frac 1{x\sin\frac 1x-1}$$ without using L'Hospital's Rule or Series expansion.

Answer: e^{-2}


Question: Determine the number of orientations of a smooth manifold with \(n\) maximal connected components.

Answer: 2^n


Question: Let $X$ and $Y$ be orthogonal skew symmetric matrices. Determine if $X$ and $Y$ are orthogonally conjugate, i.e., if there exists an orthogonal matrix $U \in O(n)$ such that $UX = YU$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the double integral \( \iint_\Omega \sqrt{x^2+y^2} \,\mathrm{d}x\mathrm{d}y \), where \( \Omega \) is the region defined by \((x-1)^2+y^2 \leq 1\) and \(0 \leq y\).

Answer: \dfrac{16}{9}


Question:
Let \( A \) and \( B \) be \( n \times n \) matrices. Which of the following is equal to \( \text{trace}(A^2 B^2) \)?

(i) \( (\text{trace}(AB))^2 \)
(ii) \( \text{trace}(AB^2 A) \)
(iii) \( \text{trace}((AB)^2) \)
(iv) \( \text{trace}(BABA) \)

Answer: ii


Question: Find all real harmonic functions $u$ on the unit disk $D$ centered at $0$ in the complex plane such that $u(0) = 0$ and $u^2$ is also harmonic on $D$.

Answer: 0


Question: Let \( A \) be an integral domain. For every maximal ideal \( m \) in \( A \), consider \( A_m \) as a subring of the quotient field \( K \) of \( A \). Prove that \( \bigcap_{m \text{: maximal ideal}} A_m = A \), where the intersection is taken over all maximal ideals \( m \) of \( A \).

Answer: A


Question: Find the value of \( \lim_{x \to 0} \frac{4\cos^2(f(x)) - 1}{1 - x^2} \), given that the function \( f(x) \) satisfies \( f(x) = f(2x) \) for all \( x \in \mathbb{R} \) and \( f(2017) = \frac{\pi}{4} \).

Answer: 1


Question: Evaluate the integral \( \int_{\gamma} \frac{e^{2 \pi z}}{(z+i)^3}dz \) using the Cauchy Integration Formula, where \( \gamma(t)=2e^{it}, t \in [0,2 \pi] \). Determine if the calculation \( \int_{\gamma} \frac{f(z)}{z-0}dz = 2 \pi i f(0) = 0 \) is correct.

Answer: 4\pi^3 i


Question: A stock market trader buys 100 shares of stock A and 200 shares of stock B. Let X and Y be the price changes of stock A and B, respectively, over a certain time period. Assume that the joint probability density function (PDF) of X and Y is uniform over the set of integers (x, y) satisfying −2 ≤ x ≤ 4 and −1 ≤ y − x ≤ 1. Find the expected value of the trader's profit.

Answer: 300


Question: Using the Fundamental Theorem of Line Integrals, evaluate \( \int_{C} e^x \, dy + e^{x}y \, dx, \) where \( C \) is the parabola parameterized by \( r(t)=\langle t+1,t^2 \rangle \) for \( t\in[-1,3]. \)

Answer: 9e^4 - 1


Question: Determine the fundamental group of the space \((S^1 \times S^1)/(S^1 \times \{x\})\), where \(x\) is a point in \(S^1\).

Answer: \mathbb{Z}


Question: Determine the radius of convergence for the function \( \frac{1}{z^2 + 2z + 2} \).

Answer: \sqrt{2}


Question: Find the surface area of the solid of revolution obtained by rotating the function \( x=\frac{1}{15}(y^2+10)^{3/2} \) from \( y=2 \) to \( y=4 \) about the \( x \)-axis.

Answer: 36\pi


Question: Find the minimum value of the function \( f(x) = \max_{t \in [0,1]} |t^2 - tx| \) for \( x \in (0,1) \).

Answer: 3 - 2\sqrt{2}


Question:
Assume that \( f \) is a non-negative real function, and let \( a > 0 \) be a real number. Define \( I_a(f) \) as follows:
\[ I_a(f) = \frac{1}{a}\int_{0}^{a} f(x) \, dx \]
Suppose \( \lim_{x \rightarrow \infty} f(x) = A \) exists. Determine whether \( \lim_{a \rightarrow \infty} I_a(f) = A \) is always true. Provide a proof or a counterexample to support your conclusion.

Answer: A


Question: Given a linear transformation \( T: M^R_{2x2} \rightarrow R_4[x] \) defined by the following mappings: \( T\begin{pmatrix} 2 & 3 \\ 1 & 0 \end{pmatrix} = x^2 \), \( T\begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} = 3x - 4 \), and \( T\begin{pmatrix} 0 & 2 \\ 4 & 5 \end{pmatrix} = 2x^2 - 7 \), find \( T\begin{pmatrix} 5 & 0 \\ -10 & -13 \end{pmatrix} \). Assume the given matrices are linearly independent.

Answer: -4x^2 + 3x + 17


Question: A cube is to be colored using 6 distinct colors such that no two adjacent faces share the same color. How many distinct ways can this be done?

Answer: 30


Question: Let $k$ be a number field, and $F/k$ a finite extension. Is it possible to find a countable family of extensions $k_i/k$ of degree 2 and a place $v_i$ of $k_i$ such that if $v$ is the place of $k$ lying below $v_i$, then $[k_{v_i}:k_v] = 2$, where $k_{v_i}$ and $k_v$ are the completions of $k_i$ and $k$ at $v_i$ and $v$, respectively? Furthermore, for some place $w$ of $F$ lying above $v$, is it possible that $F_w = k_v$?

Answer: Yes


Question: Evaluate the limit \( \lim_{n\to \infty} \int_{1}^{\pi}\frac{\cos(\frac{x}{n})}{1-e^{-xn}}dx \).

Answer: \pi - 1


Question: Find the sum of all integer values of \( a \) such that \( a(x^2+x-1) \leq (x^2+x+1)^2 \) for all real numbers \( x \).

Answer: 36


Question: If a vector space over the complex numbers has dimension $n$, is it possible to redefine the operations of addition and scalar multiplication such that the dimension of the vector space changes?

Answer: Yes


Question: Consider a connected undirected simple non-planar graph $G$ with 15 vertices. If removing any edge from $G$ results in a planar graph, how many edges does $G$ have?

Answer: 40


Question: Evaluate the limit \( \lim_{n\to\infty} \sum_{k=1}^{n} \sin\left(\frac{(2k-1)a}{n^2}\right) \) by expressing it as a Riemann sum and finding the corresponding integral, where \( a \in \mathbb{R} \).

Answer: a


Question:
Calculate the first partial derivative with respect to $x$ of the function \( f(x,y) \) at the point \((0,0)\), where
\[
f(x,y) =
\begin{cases}
0 & (x,y)=(0,0)\\
\frac{xy}{|x|+|y|} & (x,y) \neq (0,0)
\end{cases}
\]

Answer: 0


Question: Determine the minimal possible order of a group $G$ that contains a subset $A \subset G$ with $|A| = 2n$, such that for every $a \in A$, there exists a unique $b \in A$ with $[a, b] \neq e$. Provide your answer as a single integer.

Answer: 6^n


Question: Find all strictly increasing functions \( f : \Bbb{Z} \rightarrow \Bbb{Z} \) such that \( f(f(x)) = x + 2 \) for all integers \( x \).

Answer: f(x) = x + 1


Question: Is the statement "If $\gcd(y,z)=1$, then $yz$ is perfect if and only if $D(y)D(z)=2s(y)s(z)$" always true for odd perfect numbers $q^k n^2$? Here, $D(x) = 2x - \sigma(x)$ is the deficiency of $x$, $s(x) = \sigma(x) - x$ is the sum of the aliquot divisors of $x$, and $\sigma(x)$ is the sum of divisors of $x \in \mathbb{N}$, the set of positive integers.

Answer: Yes


Question: Given that \( f(\phi) = \Delta \phi \), where \( \Delta \phi = \nabla \cdot \nabla \phi \), find the expression for \( \frac{df(\phi)}{d\phi} \).

Answer: \Delta


Question: Calculate the minimum distance from the curve \( f(x) = \begin{pmatrix} \cos(\pi x) \\ \sin(\pi x) \\ 1-x^2 \end{pmatrix} \) to the origin in \( \mathbb{R}^3 \).

Answer: 1


Question: Determine whether the sequence \( \frac{2^n - 3n^3}{1-4^n} \) converges or diverges using the ratio test.

Answer: 0


Question: Compute \( \lim\limits_{n\to \infty} \int\limits_0^1 x^{2019} \{nx\} \, dx \), where \( \{a\} \) denotes the fractional part of the real number \( a \).

Answer: \dfrac{1}{4040}


Question: Use Logarithmic Differentiation to find \(\frac{d}{dx} (x^{{x}^{x}})\) at \(x=1\).

Answer: 1


Question: Determine the number of integers in the range from 1 to \(10^9\) that are not perfect squares, perfect cubes, or perfect fifth powers.

Answer: 999967355


Question: Determine whether the hyperoctahedral group, defined as the wreath product of $S_2$ and $S_n$ or equivalently $G = S_2^n \rtimes S_n$, has only 3 maximal normal subgroups. Consider $G$ as a subgroup of $S_{2n}$ with its natural action on $[2] \times [n]$. The group $G$ has two known normal subgroups of index two: $N_1$, the preimage of $A_n$ under the quotient map $G \to S_n$, and $N_2$, the intersection in $S_{2n}$ of $G$ and $A_{2n}$. A third normal subgroup of index two arises from the diagonal in the Klein group $G/(N_1 \cap N_2)$. Are there any additional maximal normal subgroups in $G$?

Answer: 3


Question: Let $F$ be a finite Galois extension of the rational function field $\mathbb{Q}(x)$. Let $k$ be the field of constants of $F$, defined as the algebraic closure of $\mathbb{Q}$ in $F$. Is $k$ necessarily a Galois extension of $\mathbb{Q}$? Provide a justification for your answer.

Answer: Yes


Question: Find the 9th derivative of \( \frac{\cos(5 x^2)-1}{x^3} \) and evaluate it at \( x=0 \) using the Maclaurin Series.

Answer: -7875000


Question: Consider the upper half-plane \( \mathbb{H} = \{ x \in \mathbb{R}^n : x_n > 0 \} \). Let \( u \in C^2(\mathbb{H}) \cap C(\bar{\mathbb{H}}) \) be a bounded harmonic function such that \( u \leq 0 \) on \( \partial\mathbb{H} = \{ x_n = 0 \} \). Determine if it is possible to conclude that \( u \leq 0 \) in all of \( \mathbb{H} \) for \( n \geq 3 \).

Answer: Yes


Question: A ladder is being moved through a corridor in the first quadrant of the $x$-$y$ plane. The ladder is represented by the line $y = mx + c$, where $m < 0$ and $c > 0$. The length of the ladder is $L$. The goal is to find the maximum length $L$ such that the ladder just touches the corner of the corridor as it clears it. Given the coordinates of the corner are $(a, b)$, find the maximum length of the ladder when $a = 8$ and $b = 6$. Express your answer to the nearest whole number.

Answer: 20


Question: Determine the sign of the map induced by the covering map \( p: S^n \to \mathbb{R}P^n \) on homology, specifically \( p_{*}: H_n(S^n, R) \to H_n(\mathbb{R}P^n, R) \), where \( R \) is an arbitrary ring. Given that \( H_n(S^n, R) \cong R \) is generated by the class \([\sigma_n^{(1)} + \sigma_n^{(2)}]\) and \( H_n(\mathbb{R}P^n, R) \cong R \) is generated by the class \([\tau_n]\), find the sign in the expression \( p_{*} [\sigma_n^{(1)} + \sigma_n^{(2)}] = \tau_n \pm \tau_n \).


Question: Given the fixed point iteration \( p_n = \frac{p_{n - 1}^2 + 3}{5} \), which converges for any initial \( p_0 \in [0, 1] \), estimate an expression for the number of iterations \( n \) required to achieve an absolute error \( \left| p_n - p \right| < 10^{-4} \) when \( p_0 = 1 \). Use the error bound \( \left| p_n - p \right| \leq k^n\max\{ p_0 - a, b - p_0 \} \), where \([a, b]\) is the interval of convergence and \( k \) is the bound on the derivative of the function within this interval.

Answer: 11


Question: Find the unique polynomial \( r(x) \) such that for all \( p(x) \in \mathbb{P}^2 \), the functional \( f(p(x)) = p'(-15) + 8p(-1) \) can be expressed as \( \langle p, r \rangle = \int_0^1 p(x)r(x)dx \). Here, \( \mathbb{P}^2 \) is the space of polynomials of degree less than two.

Answer: -132x + 74


Question: Evaluate the limit: \[ \lim_{x \to \infty} \frac{\left(\int_0^x e^{t^2} \, dt\right)^2}{\int_0^x e^{2t^2} \, dt} \]

Answer: 0


Question: Let \( A \in M_{5\times 6}(\mathbb{R}) \) be a matrix with rank 4. Consider the block matrix \( D = \begin{pmatrix} I_5 & A \\ A^T & 0 \end{pmatrix} \). Determine the rank of \( D \).

Answer: 9


Question: A consumer is searching for 1 unit of a product across stores with prices 1, 2, 3, ..., each occurring with probabilities 1/2, 1/4, 1/8, ..., 1/2^n respectively. The consumer incurs a search cost of 1 for each store visited. Determine the price at which the consumer should stop searching.

Answer: 2


Question: Determine the intervals where the function \(f(x) = (x+\frac{1}{x})^{x}\) is increasing and where it is decreasing.

Answer: (0, \infty)


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

Answer: \dfrac{1}{3}


Question: Calculate the unconditional variance of the random variable $N$, where $N|\Lambda$ follows a binomial distribution with parameters $\Lambda$ and $q = 0.4$, and $\Lambda$ has a probability function defined by $p(1) = p(2) = p(3) = p(4) = 0.25$. Provide your answer as a numerical value.

Answer: 0.8


Question: Is it true that for all Sylow subgroups $P$ of a nonabelian simple group $G$, the inequality $|P|^2 < |G|$ holds?

Answer: True


Question: Calculate the limit: $$ \lim_{n\rightarrow\infty}\frac{2^n + n\sin{n}}{\log_2{n} + e^n} $$

Answer: 0


Question: Find the minimum value of the perimeter of a triangle whose area is 3 cm².

Answer: 6\sqrt[4]{3}


Question: Evaluate the surface integral \( \int \int_S z \, dS \) where \( S \) is the surface of the hemisphere defined by \( x^2 + y^2 + z^2 = a^2 \) with \( z \geq 0 \).

Answer: \pi a^3


Question: Evaluate the limit for any natural number \( p \): \[ \lim_{n\to\infty} n^{p+1} \int_{0}^{1} e^{-nx} \ln (1+x^p) \space dx. \]

Answer: p!


Question: Determine whether the sequence of functions \( f_n(x) = x^n - x^{2n} \) converges for \( x \in (0,1) \).

Answer: 0


Question: Determine the order of the quotient ring $\mathbb{Z}[i]/(1+i)$ and prove that it is isomorphic to a field of that order.

Answer: 2


Question: Determine whether the following statement is true or false: For positive integers $m$ and $n$, if \(\phi(mn)=\phi(m)\) and \(n>1\), then \(n=2\) and \(m\) is odd, where \(\phi\) denotes the Euler totient function. Provide a justification for your answer.

Answer: True


Question: Compute the definite integral \(\int_0^1(-1)^{\left\lfloor\frac{1}{x}\right\rfloor}\,\mathrm{d}x\) and verify that the solution is \(\boxed{1-2\ln2}\).

Answer: 1 - 2 \ln 2


Question: Let \((x_0, y_0)\) be the solution of the following equations: \((2x)^{\ln 2} = (3y)^{\ln 3}\) and \(3^{\ln x} = 2^{\ln y}\). Find the value of \(x_0\).

Answer: \dfrac{1}{2}


Question: Find all integer solutions \((x, y, z)\) to the Diophantine equation: \[ x^2 + y^2 = 3z^2 \]

Answer: (0, 0, 0)


Question: Find the value of \( \beta \in \mathbb{R} \) for which the process \( 2W_t^3 + \beta t W_t \) is a martingale, where \( W_t \) is a standard Wiener process.

Answer: -6


Question: Consider the function \( f(x) = x^3 + ax^2 + bx + c \), where the coefficients \( a, b, \) and \( c \) are determined by rolling a six-sided die three times. What is the probability that \( f(x) \) is an increasing function?

Answer: \dfrac{4}{9}


Question: In a group of 10 chickens, 9 are real and each weighs 2kg, while 1 is a false chicken that weighs either 1kg or 3kg, each with a probability of \(\frac{1}{2}\). A machine randomly selects each chicken with a probability of \(\frac{1}{2}\) for weighing, and after weighing, all chickens are returned. Three weightings are performed, resulting in sets weighing 11kg, 12kg, and 13kg, respectively. What is the probability that the false chicken weighs 3kg?

Answer: \dfrac{3}{5}


Question: Let $K$ be an abelian number field and $H(K)$ be the Hilbert class field of $K$. Assuming that $H(K)$ is abelian over $\mathbb{Q}$, do $K$ and $H(K)$ have the same conductor?

Answer: Yes


Question: Find the largest constant \( k \) such that \[ \frac{kabc}{a+b+c} \leq (a+b)^2 + (a+b+4c)^2 \] for all positive \( a, b, c \).

Answer: 100


Question: Find all finite groups that have exactly two conjugacy classes.

Answer: \mathbb{Z}_2


Question: Let $p \ge 3$ be a prime. Consider the set $U(p^n)$, which consists of all invertible elements of $\mathbb{Z}/p^n\mathbb{Z}$. Determine whether $1+p$ is an invertible element in $U(p^n)$. If it is invertible, find its order in $U(p^n)$. Provide your answer with justification.

Answer: p^{n-1}


Question: Let $D$ be a unique factorization domain (UFD) with infinitely many maximal ideals. Determine whether $D$ has infinitely many irreducible elements that are pairwise non-associate. Justify your answer.

Answer: Yes


Question: Evaluate the volume of the region between the cylinders defined by \(x^2 + y^2 = 1\) and \(x^2 + y^2 = 4\), and between the plane \(z = x + 2\) and the \(xy\)-plane. Determine why the volume is equal to \(0\).

Answer: 0


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

Answer: \dfrac{1}{e}


Question: Let $M$ be a smooth manifold and $N$ a submanifold of $M$. Consider vector fields $X_1, \\ldots, X_k \in \Gamma(TM)$ on $M$ that restrict to vector fields on $N$. For a smooth function $f \in C^\infty(M)$, we obtain two smooth functions on $N$: one by restricting the derivative $X_1 \ldots X_k(f)$ on $M$ to $N$, and the other by taking the derivative of $f|_N$ in $N$. Determine whether these two functions coincide, i.e., does $$(X_1 \ldots X_k(f))|_N = X_1|_N \ldots X_k|_N(f|_N)$$ always hold?

Answer: Yes


Question: Determine whether the congruence $x^2 \equiv 3 \pmod{10007}$ has a solution using quadratic reciprocity.

Answer: Yes


Question: Determine the number of different 6-digit codes that can be formed using the digits 4, 6, and 9, with the following conditions: the code must use all the digits, it ends in 4, and 4 and 9 are never consecutive.

Answer: 38


Question: Consider the cubic equation \(2^{k + 1} x^3 + 3x^2 - d = 0\), where \(d, k \in \mathbb{Z}\) and \(d \gg 2^{k + 1}\). Given that the discriminant \(\Delta < 0\), there is one real root and two imaginary roots. Determine if the real root can be a positive integer.

Answer: Yes


Question: Find the value of \( f'(2) \) where \( f(x) = \lim_{N \to \infty} \sum_{n=1}^{N} \arctan\left(\frac{x}{n(n+1)+x^2}\right) \).

Answer: \dfrac{1}{5}


Question: Let $p$ be an odd prime, and consider two matrices $A, B \in GL_n(\mathbb{Z}_p)$, each of finite order $m$. If the reductions of $A$ and $B$ modulo $p$ are conjugate in $GL_n(\mathbb{F}_p)$, are $A$ and $B$ conjugate in $GL_n(\mathbb{Q}_p)$?

Answer: Yes


Question: Prove that the automorphism group of the cyclic group \(\mathbb{Z}_{49}\) is isomorphic to \(\mathbb{Z}_{42}\).

Answer: \mathbb{Z}_{42}


Question: Find the minimum integer value of \( k \) such that the equation \( e^x = kx^2 \) has exactly three real solutions.

Answer: 2


Question: Evaluate the integral: \[ \int^{\infty}_0 \frac{1}{\sqrt{2\pi}} x^2 \cdot \exp\left(-\frac{x^2}{2}\right) \, dx \]

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_{-\infty}^0 xe^{-4x} \, dx \).

Answer: -\infty


Question: Is it true that for a non-amenable discrete group \(\Gamma\), there exists a generating set \(S\) such that the critical probability for percolation \(p_c(\Gamma,S)<\frac{1}{2}\)?

Answer: Yes


Question: Determine the interval of convergence for the power series \( \sum_{n=1}^\infty \left(\frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{n}\right)x^n \).

Answer: (-1, 1)


Question:
Consider the set \(\mathcal{S}_X\) of trigonometric polynomials \(f(t) = \sum_{|k| \leq X} c_k e^{2\pi i kt}\) on the circle \(\mathbb{T} = \mathbb{R}/\mathbb{Z}\) with degree \(\leq X\), such that \(f(0) = 1\) and \(c_0 = 0\). Define
\[ M_X(f) = \sup_{\mathbb{T} \setminus [-\frac{1}{X},\frac{1}{X}]} |f|. \]
Let \(B_X = \inf_{f \in \mathcal{S}_X} M_X(f)\). Determine whether the limit \(\lim_{X \to \infty} B_X\) is strictly positive or zero.

Answer: 0


Question: Consider the series \(\sum_{k = 1}^{\infty}\frac{a_k}{k}\), where each term satisfies \(0 < a_k < B\) for a strictly positive number \(B\). If this series diverges, does it imply that the sequence \(a_k\) becomes constant, i.e., \(a_k = a\) for all \(k \geq K\), where \(K\) is a finite natural number?

Answer: No


Question: Evaluate the line integral \( \int_{\gamma} \frac{1}{z-a} \, dz \), where \( \gamma = a + Re^{it} \) for \( 0 \leq t \leq 2\pi \), and \( a \) is a complex number.

Answer: 2\pi i


Question:
Find the area of the surface defined by the mapping \( \sigma(u,v) = (u,v,uv) \) over the domain \( A = \{(u, v) \in \mathbb{R}^2 : u^2 + v^2 < 3\} \). Choose the correct answer from the following options:

(A) \( \frac{14}{3} \)
(B) \( \frac{14\pi}{3} \)
(C) \( 9\pi \)
(D) \( 9 \)

Answer: B


Question: Let $A$ be a $C^*$-algebra and $	au: A \to \mathbb{C}$ a bounded functional. Let $u = [u_{i,j}] \in M_n(A)$ be a unitary matrix and consider the matrix $m = [\tau(u_{i,j})] \in M_n(\mathbb{C})$. Find an estimate for $\|m\|$ in terms of $\|\tau\|$. For instance, is it true that $\|m\| \le \|\tau\|$?

Answer: \|m\| \le \|\tau\|


Question: Calculate the value of \( 6239^5 \mod 15367 \).

Answer: 8700


Question: Consider five numbers $a_1, a_2, a_3, a_4, a_5$ such that $a_1, a_2, a_3$ are in arithmetic progression (AP), $a_2, a_3, a_4$ are in geometric progression (GP), and $a_3, a_4, a_5$ are in harmonic progression (HP). Determine whether $\ln a_1, \ln a_3, \ln a_5$ form an arithmetic progression (AP), geometric progression (GP), or harmonic progression (HP).

Answer: AP


Question: Find all integer solutions to the equation \( y^3 - 1 = x^4 + x^2 \).

Answer: (0, 1)


Question: Does the Witten-Reshetikhin-Turaev invariant detect the hyperelliptic involution on a genus 2 surface? Specifically, if $-I \in \mathrm{Mod}(\Sigma_2)$ is the hyperelliptic involution on the genus 2 surface, is there any $U \in \mathrm{Mod}(\Sigma_2)$ for which $Z(M_U) \neq Z(M_{-U})$?

Answer: Yes


Question: Evaluate the integral: \[ \int_{-\pi}^{\pi} \frac{\cos^2(x)}{1+a^x} \, dx \] given that \[ \int_{-\pi}^{\pi} \frac{\cos^2(x)}{1+a^x} \, dx = \int_{-\pi}^{\pi} \frac{a^x\cos^2(x)}{1+a^x} \, dx. \]

Answer: \dfrac{\pi}{2}


Question: Let $A$ be an abelian group. Does $A \otimes \Bbb Q = 0$ imply that $A$ is finite?

Answer: No


Question: Evaluate the improper integral \( \int_{0}^{\infty} \frac{1}{(x^2+1)^2} \, dx \) using the method of residues.

Answer: \dfrac{\pi}{4}


Question: Given that \( \int_0^{\pi}(f(x) + f''(x)) \sin x \, dx = 2 \) and \( f(\pi) = 1 \), find \( f(0) \).

Answer: 1


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

Answer: f(x) = ax^2


Question: Determine whether the function \( \sigma : \mathbb{R}^{m \times n} \rightarrow \mathbb{R}^m \), which maps a matrix \( X \in \mathbb{R}^{m \times n} \) to its ordered singular values \( \sigma_1, \cdots, \sigma_m \), is continuous.

Answer: Yes


Question: Given a Poisson point process $X$ with rate $\lambda = 1$, let $X_1$ be the number of points in the interval $[0,3]$ and $X_2$ be the number of points in the interval $[2,4]$. Calculate the covariance $\operatorname{Cov}(X_1, X_2)$. Provide your answer as a single numerical value.

Answer: 1


Question: Given the function \( f(x) = \int_0^{g(x)}(1+t^3)^{-\frac{1}{2}} \, \mathrm{d}t \) where \( g(x) = \int_0^{\cos x}(1+\sin (t^2))\,\mathrm{d}t \), find \( f'\left(\frac{\pi}{2}\right) \).

Answer: -1


Question: Evaluate the double integral \( \iint \delta (ax^2+by-c) \, dx \, dy \).

Answer: \infty


Question: Find a power series expression \( \sum_{n=0}^\infty A_n z^n \) for \( \frac{1}{z^2-\sqrt2 z +2} \) and determine its radius of convergence.

Answer: \sqrt{2}


Question: Evaluate the limit: \[ \lim_{\varepsilon\to 0}\int_{-1}^1 \frac{1}{\sqrt{2\pi \varepsilon}} e^{-\frac{x^2}{2\varepsilon}} \ell(x) \,dx, \] where \( \ell(x) \) is a smooth and bounded function (\( \ell \in C^\infty \)). Consider the behavior in the neighborhood of 0.

Answer: \ell(0)


Question: If a function $f(x)$ is differentiable at $x = a$, is it necessarily differentiable in an open interval including $x = a$?

Answer: No


Question: If $f$ is a differentiable function and $\lim_{x \to \infty} f(x) = M$, does this imply that $\lim_{x \to \infty} f'(x) = 0$? Justify your answer.

Answer: No


Question: Determine whether every unique factorization domain (UFD) is noetherian on principal ideals.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to 0} \left(1+\frac{1}{x^\frac{1-a}{a}}\right)^{\frac{a}{1-a}} \left(x^{\frac{1}{a}}+x \right) \] where \(0 < a < 1\).

Answer: 1


Question: Find the function \( f(x) \) that satisfies the functional equation \( f\left(\frac{x}{y}\right) = \frac{f(x)}{f(y)} \) for all \( x, y \) such that \( f(y) \neq 0 \), and the derivative condition \( f'(1) = 2 \).

Answer: x^2


Question: Evaluate the sum \(\sum_{t=0}^{1000} (-1)^t \binom{2000}{2t}\).

Answer: 2^{1000}


Question: Let $a \in \Bbb Z$ such that $\gcd(9a^{25} + 10, 280) = 35$. Find the remainder of $a$ when divided by 70.

Answer: 65


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

Answer: 1


Question: Given N trucks, each assigned a number from 1 to N, assume the prior distribution of N is proportional to \( \frac{1}{x} \) for \( x = 1, \ldots, 500 \). Find the posterior mean of N when observing a truck numbered 50, assuming the likelihood is \( \frac{1}{N} \) for \( N \geq 50 \) and 0 otherwise.

Answer: 127.1


Question: Suppose $Y_k = \sum_{i=1}^k X_i$ for $k = 1, \ldots, n$ are jointly Gaussian random variables. Are the random variables $X_1, \ldots, X_n$ jointly Gaussian?

Answer: Yes


Question:
Given a random variable \(X\) which is non-negative and integer-valued, with the probability generating function:
\[G_X(s) = e^{s-1}\]
Find the probability \(P(X < 2)\).

Answer: \dfrac{2}{e}


Question: In the symmetric group S2020, a permutation \( \sigma \) has an order of 2019. What is the maximum number of fixed points that \( \sigma \) can have?

Answer: 1344


Question: Determine the number of distinct terms in the expansion of \((x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8)^4\).

Answer: 25


Question: Given a segment AB, construct a point C on the segment AB using only a straightedge and compass such that \( \frac{AC}{CB} = \frac{\phi}{2} \), where \( \phi \) is the golden ratio (\( \phi = 1.61803\ldots \)).

Answer: C


Question: Let $R$ be a finite-dimensional semisimple $k$-algebra, where $R$ is not necessarily commutative, and let $M$ be an $R$-bimodule with finite dimension over $k$. Define $M^{\ast} = \text{Hom}_{R}(M_{R}, R_{R})$, the dual right module of $M$. Determine $\dim_{k} M^{\ast}$. Is $\dim_{k} M^{\ast}$ always equal to $\dim_{k} M$?

Answer: \dim_{k} M^{\ast} = \dim_{k} M


Question: Evaluate the integral of the function \( f(z) = \frac{z^2}{z^2+2z+2} \) around the contour \( C \), where \( C \) is the circle centered at the origin with radius 2.

Answer: -4\pi i


Question:
Find a general formula for \( E(X^t) \) when \( X \) has a log-normal distribution. Start with the integral:
\[ E(X^t) = \int_0^\infty x^t \frac{1}{\sqrt{2\pi}x} e^{-\ln(x)^2/2} \, dx \]
Show that the solution to this integral is \( e^{t^2/2} \).

Answer: e^{t^2/2}


Question: Compute \( \lim_{n \to \infty} \mathbb{P}\{S_n \leq n\} \) where \( S_n = X_1 + \ldots + X_n \) and each \( X_i \sim \text{Poisson}(1) \) are independent and identically distributed random variables.

Answer: \dfrac{1}{2}


Question: Let $f : \mathbb{R}_+ \rightarrow \mathbb{R}_+$ be an increasing continuous function such that $f(0) = 0$ and $\beta \geq 0$. Given that \( \lim_{x \to +\infty}{\frac{\int_{0}^{x}{f(t)\mathrm{d}t}}{x f(x)}}=\frac{1}{1 + \beta} \), does this imply that there exists some \( \lambda \geq 0 \) such that \( \lim_{x \rightarrow +\infty}{\frac{f(x)}{x^{\beta}}} = \lambda \)?

Answer: No


Question: Let $X$ be a finite set and $X^*$ be the set of all non-empty proper subsets of $X$. Consider an increasing function $f: X^* \to X^*$ such that there exists some $A \in X^*$ for which $|f(A)| \neq |A|$. Is it true that $f$ must have a fixed point? (An increasing function means that if $A \subseteq B$, then $f(A) \subseteq f(B)$.)

Answer: Yes


Question: Find the permutation \( \beta \) in the symmetric group \( S_7 \) such that \( \beta^8 = (1\ 5\ 4\ 3\ 6) \).

Answer: (1\ 4\ 6\ 5\ 3)


Question: Solve the equation \( \frac{7x^2 - x + 4}{\sqrt{3x^2 - 1} + \sqrt{x^2 - x} - x\sqrt{x^2 + 1}} = 2\sqrt{2} \) over the real numbers.

Answer: -1


Question:
Let $A$ be a $3 \times 3$ matrix and $\vec{x}, \vec{y}, \vec{z}$ be vectors in $\mathbb{R}^3$. Given the equations:
\[ A\vec{x} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \quad A\vec{y} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad A\vec{z} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \]
find the determinant of the matrix $A$. \( \boxed{} \)

Answer: 0


Question: Maximize \( P = a^2 + b^2 + c^2 + ab + ac + bc \) for real numbers \( a, b, c \) that satisfy \( a + b + c = 6 \) and \( 0 \leq a, b, c \leq 4 \).

Answer: 28


Question: Find the radius of convergence for the series \( \sum_{n\geq 1}\left(\frac{x}{\sin n}\right)^{n} \).

Answer: 0


Question: If a matrix $A \in \mathbb{R}^{m \times n}$ is real, does there exist a singular value decomposition (SVD) $A = U\Sigma V^T$ where both $U$ and $V$ are real matrices?

Answer: Yes


Question: Calculate the covariance \( \text{Cov}(X_i, X_j) \) for \( i \neq j \) when sampling 5 numbers without replacement from the set \( \{1, 2, \ldots, 10\} \).

Answer: -\dfrac{11}{12}


Question: Find a formula for the number of elements of order 2 in the group $D_m \times D_n$, where $m$ is an even integer greater than 2 and $n$ is an odd integer greater than 2. Here, $D_r$ denotes the dihedral group of order $2r$, which is the symmetry group of a regular $r$-gon.

Answer: mn + m + 2n + 1


Question:
Determine the norm of the continuous linear operator \( S \) defined by:
\[ S{u} = \sum_{n=1}^{\infty} \frac{(-1)^{n} U_{n}}{n} \]
where \( U \in \ell^{1} \), and \( \ell^{1}=\{ U=(U_{n})_{n \in \mathbb{N}} \subset \mathbb{R}~ \text{such that} ~ \sum_{n=1}^{\infty} | U_{n} | < \infty \} \).

Answer: 1


Question: Evaluate the limit: $$\lim_{x\to0}\left|\frac{\Gamma(x)}{\Gamma(-x)}\right|$$

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \rightarrow 0}\frac{x^{2}\ln\left(1+2x\right)}{2\sin\left(x\right)\left[ \cos\left(3x\right)-1\right]} \] without using L'Hôpital's rule.

Answer: -\dfrac{2}{9}


Question:
Given that the function \( f(x) \) and the product \( f(x)g(x) \) belong to the Sobolev space \( \mathcal{W}^{s+1} \) with \( s \ge 1 \), and \( g \in \mathbb{L}^{\infty}(\mathbb{R}_+) \) where \( \mathbb{R}_+ = [0,\infty) \), and for all \( k \ge 0 \), \( \int x^k f(x) \, dx < \infty \) and \( 0 < \int f(x)g(x) \, dx < \infty \), determine whether the following limit holds:
\[ \lim_{x\rightarrow \infty} x f(x) g(x) = 0 \]

Answer: 0


Question: Determine the values of \( b \) for which the equation \(-x^3 + 3x^2 + 9x - 11 = 9x + b\) has three distinct solutions.

Answer: (-11, -7)


Question:
Let $a, b, c$ be positive integers such that $0 < a, b, c < 11$. If $a, b, $ and $c$ satisfy the following system of congruences:
\[
\begin{align*}
3a+b+c&\equiv abc\pmod{11} \\
a+3b+c&\equiv 2abc\pmod{11} \\
a+b+3c&\equiv 4abc\pmod{11} \\
\end{align*}
\]
find the sum of all possible values of $abc$. \(\boxed{\text{Answer}}\)

Answer: 198


Question: Evaluate the limit: \[ \lim_{n\to\infty} \int_{-\pi}^{\pi} |\cos(nx)| \, \mathrm{d}x. \]

Answer: 4


Question: Calculate the work done by the vector field \( f(x,y,z) = (x,z,2y) \) along the curve formed by the intersection of the surfaces \( x^2 + y^2 = 1 \) and \( z = x^2 - y^2 \), traversed in the anti-clockwise direction as viewed from the point (0,0,100).

Answer: 0


Question: Find the last four digits of the number $2^{3^{4^5}}$. Provide your answer as a four-digit number.

Answer: 0352


Question: Consider a surface $S$ from which two discs have been removed. If the boundary circles of these two discs are glued together, is the resulting object a surface?

Answer: Yes


Question: Calculate \(2^{731} \mod 645\).

Answer: 8


Question: Let \( \Omega \subseteq \mathbb{R}^{n} \) be an open set and let \( f, g: \Omega \to \mathbb{R} \) be \( C^{k} \) functions, where \( k \ge 0 \). Suppose \( \int_{\Omega} f(x)g(x) \, dx = 0 \) for every \( g \) with compact support. Does it follow that \( f \equiv 0 \) on \( \Omega \)?

Answer: Yes


Question: Let $S = \{ x \in \mathbb{R}^d : ||x||_2 = 1 \}$ be the $d$-dimensional unit sphere, where $||x||_2$ is the Euclidean norm. Given $\epsilon > 0$ and an arbitrary point $s \in S$, determine if there exists an $\alpha > 0$ and a $k \in \mathbb{Z}^d \setminus \{0\}$ such that the distance between $\alpha s$ and $k$ is less than $\epsilon$. In other words, can every point on the unit sphere be scaled such that its distance to a non-zero integer vector is arbitrarily small?

Answer: Yes


Question: Find \( \epsilon > 0 \) such that for all partitions \( P \) of \([1,2]\) with mesh \( \lambda(P) < \epsilon \), the inequality \(|U_{f,P} - L_{f,P}| < 0.01\) holds, where \( U_{f,P} \) and \( L_{f,P} \) are the upper and lower Darboux sums of the function \( f(x) = \frac{1}{x} \) on \([1,2]\).

Answer: 0.01


Question: Evaluate the limit \( \lim_{n \to \infty} \int_{0}^{1} x^2 \left(1+\frac{x}{n}\right)^n dx \).

Answer: e - 2


Question: Determine the cardinality of $\aleph_0^{\aleph_0}$.

Answer: 2^{\aleph_0}


Question: Evaluate the limit: \[ \lim_{n \to \infty}\sum_{k=1}^{n}\frac{(k-1)^7}{n^8} \]

Answer: \dfrac{1}{8}


Question:
Compute the flux of the vector field \( \vec{F} \) through the surface \( S \), where \( \vec{F} : U \subseteq \mathbb{R}^3 \to \mathbb{R}^3 \) is given by:
\[
\vec{F} = \left( \frac{x}{(x^2 + y^2 + z^2)^{3/2}}, \frac{y}{(x^2 + y^2 + z^2)^{3/2}}, \frac{z}{(x^2 + y^2 + z^2)^{3/2}} \right)
\]
where \( U = \mathbb{R}^3 \setminus \{(0, 0, 0)\} \). The surface \( S \) is a sphere of radius 12345 centered at the origin, with a small section chopped off from the top and replaced by a flat disk, ensuring \( S \) remains closed. Assume \( S \) is oriented with an outward-pointing normal. Calculate the flux of \( \vec{F} \) through \( S \).

Answer: 4\pi


Question: Consider a vector $v$ obtained by summing $k$ vectors of the form $(0,0,\ldots,0, -n, *,*,\ldots,*)$, where "*" represents either $0$ or $1$, and the position of the $-n$ entry can vary for each vector. The sum of all entries of $v$ is required to be zero. Determine if it is possible for the ratio $$\frac{\|v\|_1}{k\cdot n}$$ to be arbitrarily small.

Answer: Yes


Question: Find all functions \( f: \mathbb{N}^+ \to \mathbb{R} \) such that for a given positive integer \( n \), the equation \( f(m+k) = f(mk-n) \) holds for all positive integers \( m \) and \( k \) with \( mk > n \).

Answer: f(x) = c


Question: Find the minimum value of the function \( f: [0,\frac{\pi}{2}]^3 \to \mathbb{R}^+ \) defined by \( f(\theta_1,\theta_2,\theta_3) = |2+e^{i\theta_1}+e^{i\theta_2}+e^{i\theta_3}| \).

Answer: \sqrt{13}


Question: Suppose that $A$ is an $n \times n$ matrix containing $0$ in its numerical range. Is it true that $0$ is also contained in the numerical range of $UAU^*$ for some unitary matrix $U$? Provide a justification for your answer.

Answer: Yes


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

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{+\infty} \frac{M}{a}x \big(1-\exp(-\frac{x}{a})\big)^{M-1}\exp(-\frac{x}{a})dx \).

Answer: a \sum_{k=1}^{M} \frac{1}{k}


Question: Evaluate the contour integral \( \int_{\gamma}{\frac{e^{z^{2}}}{z-1}dz} \), where \( \gamma \) is the rectangle with vertices at \( (0, -1), (3, -1), (3, 1), (0, 1) \).

Answer: 2\pi i e


Question: Let $y \in \mathbb{R}^n \setminus \{0_n\}$ and let $X \subset \mathbb{R}^n$ be a compact polytope. Determine whether there exists $(x_0, \lambda_0) \in X \times \mathbb{R}_+$ such that the projection of $\lambda y$ onto $X$ is $x_0$ for all $\lambda \geq \lambda_0$. Justify your answer.

Answer: Yes


Question:
Compute the expected value of the piecewise function \( u(x,y) \) given two independent continuous random variables \( x \) and \( y \) with uniform distribution in the range \([0,b]\). The joint PDF is \( f(x,y)=\frac{1}{b^2} \). The piecewise function is defined as:

\[
u(x,y) = \begin{cases}
0 & \text{if } x,y < b/2 \\
b/2 & \text{if } (y<b/2 \text{ and } x > b/2) \text{ or } (x < b/2 \text{ and } y > b/2) \\
x & \text{if } y,x>b/2 \text{ and } y>x \\
y & \text{if } y,x>b/2 \text{ and } x>y
\end{cases}\]

Calculate \( E(u(x,y)) = \int_0^b\int_0^b \frac{u(x,y)}{b^2} \, dx \, dy \).

Answer: \dfrac{5b}{12}


Question: Determine the values of $p$ for which the series \( \sum_{n=2}^{\infty} \frac{\sin(\frac{\pi}{n})}{n^p} \) converges.

Answer: p > 0


Question: Calculate the symmetric-decreasing rearrangement of the function \( f(x) = x \) on the interval \([0, 10]\), with \( f(x) = 0 \) elsewhere.

Answer: 10 - 2|x - 5|


Question: Consider a square matrix with orthogonal, non-zero rows that are not orthonormal. Must each row have exactly one non-zero element for the columns of the matrix to be orthogonal as well? Answer yes or no.

Answer: No


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

Answer: -\dfrac{1}{4}


Question: Calculate the value of \(13^{498} \mod 997\).

Answer: 1


Question: Determine the values of \( q \ge 1 \) for which the function \( f(x) = |x-a|^q \) is strictly convex, where \( a \in \mathbb{R} \) is fixed.

Answer: q > 1


Question: Find the value of \( n \) such that \( \sum_{r=0}^{n}(2r+1)\binom{n}{r}=2^{n+4} \).

Answer: 15


Question: Evaluate the limit \( \lim_{h \to 0} \frac{f(a-h^2)-f(a)}{h} \) where \( f \) is a function differentiable at \( a \).

Answer: 0


Question: Find the value of the convergent series: \[ \sum_{n=0}^\infty 3^{n-1}\sin^3\left(\frac{\pi}{3^{n+1}}\right) \]

Answer: \dfrac{\pi}{12}


Question: Find the value of the fractional part of \( \frac{3^{1001}}{82} \).

Answer: \dfrac{3}{82}


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

Answer: \infty


Question: Determine whether there exists a continuous onto function from the interval $[0,1)$ to the interval $(0,1)$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the sequence defined by \(a_{n+2} = \sqrt{a_n} + \sqrt{a_{n+1}}\) with initial conditions \(a_1 > 0\) and \(a_2 > 0\) converges or diverges. If it converges, find the limit.

Answer: 4


Question: Determine the type of conic section formed by the intersection of the plane \(2x + y + z - 2 = 0\) and the cone \(x^2 + y^2 = z^2\).

Answer: hyperbola


Question: Evaluate the double integral \( \int_{B} \int \sin(y + x) \, dB \) over the triangular region \( B \) with vertices \((0, 0)\), \((\pi, 0)\), and \((\pi/2, \pi/2)\).

Answer: \dfrac{\pi}{2}


Question: Find the residue of the function \( e^{\left(\frac{1}{z^2}\right)} \) at the point \( z = 0 \).

Answer: 0


Question: Evaluate the ordinal expression $1^\omega$ using the definition of ordinal exponentiation for limit ordinals.

Answer: 1


Question: Determine the limiting distribution of the sequence \( U_n = \sqrt{\frac{1}{n}\sum\limits_{i=1}^n Y_i^2} \), where \( Y_1, Y_2, \ldots \) are independent and identically distributed Poisson random variables with mean 1.

Answer: \sqrt{2}


Question:
Evaluate the limit of the sequence \( S_n = \int_{0}^{1} \frac{nx^{n-1}}{1+x}dx \) as \( n \to \infty \). Choose the correct answer from the following options:

- \(0\)
- \(\frac{1}{2}\)
- \(1\)
- \(+\infty\)

Answer: B


Question: Determine the Krull dimension of the ring \( A = \mathbb{Q}[\pi,\sqrt{11},x,y]/I \), where \( I = \langle x^2-y^3+xy+3 \rangle \).

Answer: 2


Question: Evaluate the integral \( \int\limits_{|x|=2}\frac{x}{\cos (x)}\mathrm{dx} \) using complex analysis techniques.

Answer: -2\pi^2 i


Question: Evaluate the complex integral \( \int_{|z|=2} \frac{z^3}{z^2-2z+1} \, dz \).

Answer: 6\pi i


Question: Given that \( X_1, X_2, \ldots, X_n \sim N(\mu, 1) \), consider the estimator \( \hat{g} = \overline{X}^2 \) for the mean squared \( g(\mu) = \mu^2 \). Calculate \( E_{\mu}(\overline{X}^2) - \mu^2 \).

Answer: \dfrac{1}{n}


Question: Compute the integral of \( z^2 + z \overline{z} \) over the contour \( C = \{ z \mid |z| = 1 \} \), where \( \overline{z} \) is the complex conjugate of \( z \).

Answer: 0


Question: Solve the equation \(2^{-3x^3+5x^2-x}=\frac{x^2+1}{x}\) for all possible values of \(x\).

Answer: 1


Question: Determine whether the following statement is true or false: For a polynomial \( f \in \mathbb{Z}[x] \), is it true that \( \deg(\gcd_{\mathbb{Z}_q}(f, x^p - 1)) \geq \deg(\gcd_{\mathbb{Q}}(f, x^p - 1)) \)? Here, \( \mathbb{Z}_q = \mathbb{Z}/q\mathbb{Z} \) for some prime \( q \).

Answer: True


Question: Find the exponential generating function for the sequence \( t_n \), where \( t_n \) represents the number of ways a teacher can divide \( n \) students into groups, assigning one student as president and another as vice president within each group.

Answer: e^{x^2 e^x}


Question: Let $R$ be an integral domain with a total ordering. Determine if the field of fractions $F$ of $R$ can be given a total ordering that is naturally induced by the ordering of $R$.

Answer: Yes


Question: Let \( \alpha \) be a root of the polynomial \( f(x) = x^2 - x + 2 \) over the field \( \mathbb{F}_5 \). Consider the field extension \( \mathbb{F} = \mathbb{F}_5(\alpha) \). Determine the order of \( \alpha \) in the multiplicative group \( \mathbb{F}^* \).

Answer: 24


Question: Let $f$ be a continuous function on the interval $[0,1]$. Define the sequence $S(n) = \int_0^1 x^n n f(x) \, dx$. Determine the limit of $S(n)$ as $n$ approaches infinity.

Answer: f(1)


Question: Compute the distance between the unilateral shift operator $S$ and the set of compact operators in a Hilbert space, denoted as $d(S, \mathbb{K}(\mathcal{H})) = \inf\{\|S-K\| : K \in \mathbb{K}(\mathcal{H})\}$, where $S \in \mathbb{B}(\mathcal{H})$ is a unilateral shift and $\mathbb{K}(\mathcal{H})$ is the set of compact operators.

Answer: 1


Question: Find the quotient group \( \frac{\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}}{\langle(1,1,1),(1,3,2)\rangle} \).

Answer: \mathbb{Z}


Question:
Compute the Fourier transform of the function \( f(t) = 1 - t^2 \) for \(|t| < 1\) and \(f(t) = 0\) elsewhere. Use the result to evaluate the integral:
\[
\int_{-\infty}^{\infty} \frac{\sin t - t \cos t}{t^3} \, dt.
\]

Answer: \dfrac{\pi}{2}


Question: Let $X$ be a smooth projective curve of genus $g > 2$ over an algebraically closed field $k$. Determine whether there exists a line bundle $L$ on $X$ of degree $(g-1)$ such that $H^0(X, L) = 0$. Provide a justification for your answer.

Answer: Yes


Question: What is the value of $1^i$?

Answer: 1


Question: Evaluate the double integral \( \int\int_R \frac{1}{x} \, dx \, dy \) over the region \( R \), which is a circular disc in \( \mathbb{R}^2 \) with radius \( a \) and center \( (a,0) \). Use polar coordinates to express the integral and determine the appropriate limits for \( \theta \).

Answer: 2\pi a


Question:
An array $\mathbb{T}$ has elements $T_{ijkl}$ where $i,j,k,l=1,2,3,4$. It is given that
$$T_{ijkl}=T_{jikl}=T_{ijlk}=-T_{klij}$$
for all values of $i,j,k,l$. Determine the number of independent components in this array.

Answer: 45


Question: Evaluate the limit: $$\lim_{n\to\infty}\left(\dfrac{(2n)!}{n^n\cdot n!}\right)^{1/n}$$

Answer: \dfrac{4}{e}


Question: Let $S$ be the $n$-simplex defined as \( S = \{ x \in \mathbb{R}_+^n \mid \sum_{i=1}^n x_i = 1 \} \), and let $E$ be a linear subspace of codimension 2 in $\mathbb{R}^n$ such that $E \cap \mathring{S} \neq \varnothing$. For a face $F$ of $E \cap S$, there exists a unique face $G$ of $S$ such that $\mathring{F} \subset \mathring{G}$ and $E \cap G = F$. Assume that in $\text{Aff}(G)$, $\text{Aff}(F)$ has codimension 2. If a face $G'$ of $S$ satisfies $G' \cap E = F$, does it follow that $G' = G$?

Answer: Yes


Question: Find the best approximation of the polynomial \( t^3 + t^2 + t + 1 \) using polynomials from the subspace \( M \) of \( P_4 \), where \( M = \{ p \in P_4 : \deg{p} \leq 2, p(t) = p(-t) \} \). The scalar product is defined as: \( \langle q, p \rangle = \int_{-1}^{1} p(t) \cdot q(t) \, dt \).

Answer: t^2 + 1


Question:
Consider the functions \( f_1(x) = \psi(x)\cos(\phi(x)) \) and \( f_2(x) = \psi(x)\sin(\phi(x)) \) defined on \((0, \infty)\), where \( \psi: [0, \infty) \to [0, \infty) \) is a smooth, strictly increasing function with \( \psi(0) = 0 \) and \( \psi'(0) > 0 \), and \( \phi: (0, \infty) \to \mathbb{R} \) is smooth. Assume \( \lim_{x \to 0^+} \phi'(x)\psi(x) = 0 \). Extend \( f_1 \) and \( f_2 \) continuously to zero by setting \( f_i(0) = 0 \). Can the following properties hold simultaneously?

1. \( f_i \) are infinitely (right) differentiable at \( x = 0 \).
2. All the (right) derivatives of \( f_i \) of even order vanish at zero.
3. At least one of the \( f_i'(0) \) is non-zero.

Answer: Yes


Question: A huge pie is divided among $N$ guests. The first guest receives $\frac{1}{N}$ of the pie. Guest number $k$ receives $\frac{k}{N}$ of what remains, for all $1 \leq k \leq N$. A guest is considered fortunate if their share of the pie is strictly greater than the average share, which is $\frac{1}{N}$ of the original pie. Let $f(N)$ represent the number of fortunate guests out of the total $N$ guests. Determine the value of \( \lim\sup_{N\to\infty}\frac{f(N)}{N} \).

Answer: 0


Question: Consider a random walk on a finite state space $0, 1, \\ldots, N$ with absorbing barriers at states $0$ and $N$. At each state $1, \\ldots, N-1$, the probability of moving to the adjacent states is $\\frac{1}{2}$ each. At states $0$ and $N$, the process is absorbed, meaning it stays in the same state with probability $1$. How many stationary measures does this Markov chain have?

Answer: 2


Question: Find the maximum value of the product $xyz$ given that $x, y, z$ are integers and $x + y + z = 3n$, where $n$ is a constant integer.

Answer: n^3


Question:
Find all polynomials \(P(x)\) such that for all real numbers \(x, y, z\) satisfying \(xy + yz + zx = 0\), the following equation holds:
\[ P\left((x - y)^2\right) + P\left((y - z)^2\right) + P\left((z - x)^2\right) = 18P\left(\left(\frac{(x + y + z)^2}{3}\right)^2\right) \]

Answer: 0


Question: Evaluate the double integral \( \int_{0}^1\int_0^{\sqrt{2y-y^2}}\ dxdy \) using polar coordinates and find its value.

Answer: \dfrac{\pi}{4}


Question: Compute \( \lim_{n \to \infty} \int_{0}^{\pi/3} \frac{1}{1+\tan^n(x)}\,dx \).

Answer: \dfrac{\pi}{4}


Question: Find the minimal polynomial of $\zeta_9 + \zeta_9^{-1}$ over $\mathbb{Q}$, given that the degree of $\mathbb{Q}(\zeta_9 + \zeta_9^{-1})$ over $\mathbb{Q}$ is 3.

Answer: x^3 - 3x + 1


Question: Suppose we have a unit square ABCD. Arbitrarily pick a point E within the interior of the unit square. Let the line through E parallel to AB intersect AD at F, and let the line through E parallel to BC intersect DC at G. What is the expected value of the area of the rectangle EFDG?

Answer: \dfrac{1}{4}


Question: Evaluate the integral \( I = \int_{-r}^r \int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}} \sqrt{1 - \frac{x^2 + y^2}{x^2 + y^2 - r^2}} \, dy \, dx \) using an appropriate substitution and the identity \( \frac{1}{\sqrt{1 - x^2}} = \frac{d}{dx} \arcsin(x) \).

Answer: 2\pi r^2


Question: Let $u: M \rightarrow N$ be an $A$-module homomorphism, and let $N'$ be the image of $u$. Suppose $a$ is an ideal of $A$, and the induced map $\bar{u}: M/aM \rightarrow N/aN$ is surjective. Prove that the module $N/N'/a(N/N')$ is the zero module.

Answer: 0


Question: Determine the upper bound of the magnitude of \(|e^{\sin(z)}|\) over the line segment from \(z = 0\) to \(z = i\).

Answer: 1


Question: Given $n$ independent Gaussian random variables $x_i \sim N(0,1)$ and constants $c_i \geq 0$, let the set $K$ contain $k$ indices corresponding to the smallest $c_i$. What is the probability that $\sum_{i \in K} c_ix_i \leq 0$?

Answer: \dfrac{1}{2}


Question: Determine the degree of the map \( \bar{r} : \mathbb{CP}^n \rightarrow \mathbb{CP}^n \) induced by \( r(z_0, z_1, \ldots, z_n) = (-z_0, z_1, \ldots, z_n) \) on \( \mathbb{C}^{n+1} \). Consider the cases for even and odd \( n \).

Answer: 1


Question: Calculate \( \lim_{n\to\infty}\left( \frac{4n^2+5n-6}{4n^2+3n-10}\right)^{3-4n} \).

Answer: e^{-2}


Question: Let $(M, d)$ be a metric space, $K$ a compact set, and $G$ an open set such that $K \subset G$. Is it true that there exists an $\epsilon > 0$ such that $K \subset K_\epsilon \subset G$? Here, $K_\epsilon = \{x \in M : d(x, K) \leq \epsilon\}$ and $d(x, K) = \inf \{d(x, k) : k \in K\}$.

Answer: Yes


Question: Find the value of \(i^i\), where \(i\) is the imaginary unit.

Answer: e^{-\pi/2}


Question: Find the limit of the sequence: \( \lim_{n \to \infty} (\sqrt[3]{1-n^3} + n) \)

Answer: 0


Question: Determine whether the following statement is true or false: In any compact metric space \( X \), every sequence \( \{x_n\} \subset X \) has an accumulation point in \( X \), but not necessarily a limit in \( X \).

Answer: A


Question: Compute the integral \( \int_{-\infty}^{+\infty} 2^{-4^t}(1-2^{-4^t})\,dt \).

Answer: \dfrac{1}{2}


Question: Evaluate the limit \( \lim_{n\to\infty} \sum\limits_{i=1}^{n^2} \frac{e^{i/n}}{ne^n} \) by recognizing it as a Riemann sum of an integral.

Answer: 1


Question: Let $D \in \mathbb{R}^{n,n}$ be a diagonal matrix with non-zero diagonal entries $d_{ii}$. Let $\tilde{D} \in \mathbb{R}^{n,n}$ be another diagonal matrix with the same diagonal entries as $D$, but possibly permuted. Does there exist a unitary matrix $U$ such that $UDU^H = \tilde{D}$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0^+} \frac{\ln(x)}{\ln(\sin x)} \] without using l'Hôpital's rule.

Answer: 1


Question: Express the conjugate \( \bar{z} \) in terms of \( z \) for the equation \( z^2 = \bar{z} \).

Answer: \bar{z} = z^2


Question: Consider a hash function \( H(x) \rightarrow \{ 0,1 \}^{160} \) that maps inputs to 160-bit strings. What is the minimum number of attempts required to find a collision, i.e., two different inputs \( x_1 \) and \( x_2 \) such that \( H(x_1) = H(x_2) \)?

Answer: 2^{80}


Question: Calculate the expected stopping time $E[S]$ for the first occurrence of the sequence HTH in a series of fair coin tosses using the optional stopping theorem.

Answer: 10


Question: Is every countably compact subset of a Hausdorff space closed?

Answer: No


Question: Maximize the function \( f(x,y,z) = xy + z^2 \) subject to the constraints \( 2x - y = 0 \) and \( x + z = 0 \).

Answer: \infty


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

Answer: \infty


Question:
Find the norm of the bounded compact operator \( A \) defined on the space \( C[0,1] \) by:
\[ Ax(t) = 2x(0) - tx(1), \quad t \in [0,1] \]
What is \( \|A\| \)?

Answer: 3


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

Answer: e^2


Question: Determine whether there exists a star-Lindelöf space that is not DCCC.

Answer: Yes


Question: A gun is located at the origin of an infinite number line and starts shooting bullets along the positive x-axis at a rate of one bullet per second. Each bullet's velocity is randomly chosen from a uniform distribution between 0 and 1 m/s. If two bullets collide, they explode and disappear. What is the probability that at least one bullet will travel infinitely without colliding with another bullet?

Answer: 0


Question: Let \( X \sim \mathcal{N}(0,1) \) be a standard normal random variable. Compute \( \mathbb{E}[\cos(X)] \).

Answer: e^{-1/2}


Question: Determine whether the following implication is true: If \( u(x,t) \in W^{1,1}([0,T],L^2(\Omega^d)) \), then \( u \in L^{\infty}([0,T],L^2(\Omega^d)) \). Here, \( \Omega \subset \mathbf{R}^d \) (\(d=2,3\)) is a domain with a smooth boundary \( \Gamma \).

Answer: True


Question: Use Green's Theorem to compute the value of the line integral \( \int_{\gamma} y\,dx + x^2\,dy \), where \( \gamma \) is the circle given by \( g(t) = (\cos t, \sin t), 0 \leq t \leq 2\pi \).

Answer: -\pi


Question: Calculate the limit: \[ \lim_{n\to \infty}\frac{(-1)^n\cdot 6^n-5^{1+n}}{5^n-(-1)^{n+1}\cdot 6^{n+1}} \]

Answer: \dfrac{1}{6}


Question: Find the natural number \( n \) such that the volume of the solid formed by revolving the function \( f(x) = \cos(n \arccos(x)) \) around the x-axis over the interval \([-1, 1]\) is \( \frac{14\pi}{15} \).

Answer: 2


Question: Find the sum of all natural numbers $n$ such that the function $g(x) = 100|e^{x+1} - 1| - \sum_{k=1}^n |e^{x^k + 1} - 1|$ is differentiable over the entire real line $\mathbb{R}$. Here, $n \in \mathbb{N}$. Provide your answer as a single number.

Answer: 39


Question: Find the value of $a$ such that \( \lim_{x \to 0^+} \ln(x\ln a)\ln\bigg(\frac{\ln(ax)}{\ln(\frac{x}{a})}\bigg)=6 \).

Answer: e^3


Question: Given a distribution with probability density function \( f(x) = \frac{2x}{49} \), find the probability that the 5th smallest observation from a sample exceeds 0.01.

Answer: 1


Question: Calculate the conditional expectation \( \mathbb{E}(X \mid XY) \) where \( X \) and \( Y \) are independent standard normal random variables.

Answer: 0


Question: What is the smallest possible value of the correlation \( \rho \) in an \( n \times n \) correlation matrix where the correlation between any pair of two random variables is \( \rho \)?

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


Question: Given the probability density function \( f(x;\lambda) = 1 - \frac{2}{3}\lambda + \lambda\sqrt{x} \) for \( 0 \le x \le 1 \) and 0 otherwise, find the maximum likelihood estimate of the parameter \( \lambda \) based on two independent observations \( x_1 = \frac{1}{4} \) and \( x_2 = \frac{9}{16} \).

Answer: -3


Question: Let \( f(x,y): \mathbb{R}^2 \rightarrow \mathbb{R} \) be a smooth (\( {\cal C}^\infty \)) function such that \( f(0,y)=0 \) for all \( y \in \mathbb{R} \) and \( f(x,y)>0 \) for \( x \neq 0 \). Determine whether there exists \( y_0 \in \mathbb{R} \), \( \varepsilon >0 \) such that \( f(x,y_0)+\int_0^x \frac{\partial f}{\partial y}(s,y_0)ds >0 \) for \( |x|<\varepsilon, x \neq 0 \).

Answer: Yes


Question: There are 100 people in a queue waiting to enter a hall with exactly 100 seats numbered from 1 to 100. The first person in the queue enters the hall and chooses any seat to sit in. Each subsequent person, from the 2nd to the 100th, will sit in their corresponding seat number if it is vacant; otherwise, they will choose any unoccupied seat. Determine the total number of ways the 100 seats can be filled such that the 100th person occupies seat number 100.

Answer: 2^{98}


Question: Evaluate the line integral \( \int_C \vec{F} \cdot d\vec{r} \) using Stokes' theorem, where \( \vec{F} = (xz, xy, y^2) \). The curve \( C \) is the boundary of the surface of the cylinder \( z = 4-x^2 \), bounded by the planes \( x=2 \) and \( y=3 \) in the first octant.

Answer: 45


Question:
Let $A$ and $B$ be $
C$-algebras, which are also integral domains, and suppose there is an injective ring homomorphism $f: A \to B$. Assume that $f$ is a finite morphism, meaning it induces a finite $A$-module structure on $B$. Let $M$ be a finitely generated $A$-module, and consider $m \in M$ such that there exists $m' \in M$ and $b \in B$ for which $m \otimes 1 = m' \otimes b$ in $M \otimes_A B$. Does this imply that $b$ belongs to the image of $A$ under $f$?

Answer: No


Question: Given a Fourier sine series $A\sin x + B\sin 2x + C\sin 3x + \cdots$ that represents the function $x$ on the interval $[0, \pi]$ and $[-\pi, 0]$, determine the sum of the series at $x = \pi$. Assume the series represents a periodic "sawtooth function" with period $2\pi$. What is the value of the series at $x = \pi$?

Answer: 0


Question: Find the probability density function of the random variable $Z = XY$, where $X$ and $Y$ are independent and identically distributed random variables with density $f(x) = 3x^2$ for $x \in (0, 1)$. Use the product distribution formula to determine $f_Z(z)$. Note that $X, Y > 0$. Verify the convergence of the integral used in the calculation.

Answer: -9 z^2 \ln(z)


Question: Evaluate the limit: $$\lim_{x\to0} \frac{(1+x)^{1/x}-e}{x}$$

Answer: -\dfrac{e}{2}


Question: Let \( f(x) = x^3 + x \) and \( g(x) = x^3 - x \) for all \( x \in \mathbb{R} \). Find the derivative of the composition function \( g \circ f^{-1} \) at the point \( x = 2 \).

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int^a_0{\cfrac{dx}{x + \sqrt{a^2 - x^2}}} \).

Answer: \dfrac{\pi}{4}


Question: In triangle $ABC$ inscribed in a circle, the perpendicular bisector of $AB$ intersects $BC$ at $M$ and the extension of $AC$ at $N$. Given that $O$ is the center of the circle and $OH = OM = MN$ where $H$ is the midpoint of $AB$, calculate the measure of angle $OCB$.

Answer: 18.5^\circ


Question:
Let \( Q(b) \) denote the right-tail probability of a standard Gaussian random variable. Suppose that \( X \) and \( Y \) are jointly-distributed Gaussian random variables with \( X \sim N(3, 8) \) and \( Y \sim N(10, 5) \). If the correlation coefficient between \( X \) and \( Y \) is 0.8, find the value of \( b \) such that \( P[Y > 0 | X = 3] = Q(b) \). Choose from the following options:

- \(-7.453\)
- \(-4.431\)
- \(2.639\)
- \(-4.671\)

Answer: -7.453


Question: Let $R$ be a commutative ring with unity, and let $A, B \in \operatorname{Mat}(n, R)$ be invertible matrices. Suppose $A^kB^l = B^lA^k$ for all natural numbers $k, l > 1$. Does it follow that $AB = BA$?

Answer: Yes


Question: Determine the number of linear conditions that must be added to the set \( \mathcal{T} \) to ensure that \(|M_{\gamma_{1}}(Q, \mathcal{T'})| < |M_{\gamma_{2}}(Q, \mathcal{T'})|\) holds for any choice of linear conditions, where \( \mathcal{T'} \) is a subset of \( \mathcal{T} \) defined by these linear conditions. Assume \( 0 < \gamma_1 < \gamma_1 + 1 \leq \gamma_2 \).

Answer: \gamma_2 - \gamma_1


Question: Company A runs buses between New York and Newark, with a bus leaving New York every half hour starting from 0:00, 24 hours a day. Company B also runs buses on the same route, with their buses leaving New York on average twice per hour, but not necessarily on a fixed schedule. Assuming passengers arrive at the bus stop uniformly at random and do not plan their journeys, determine if passengers using buses from Company A wait shorter on average than those using buses from Company B.

Answer: Yes


Question:
Evaluate the integral \( \int_{0}^1 \left( \int_0^{1} f(x,y) \, dx \right) dy \), where the function \( f(x,y) \) is defined as follows:

\[ f(x,y) = \begin{cases} \frac{1}{2}, & \text{if } x \text{ is rational} \\ y, & \text{if } x \text{ is irrational} \end{cases} \]

Find the value of the integral.

Answer: \dfrac{1}{2}


Question: Determine whether the hyperbolic dodecahedral space admits a Heegaard splitting of genus 3.

Answer: No


Question:
Let \(g(x) = \int_x^{x^2} \frac{1}{\ln t} \, dt\) for \(x > 0\) and \(x \neq 0\). Given the inequalities:

1. \(t - \frac{t^2}{2} \leq \ln(1+t) \leq t\)
2. \(\frac{1}{t - \frac{t^2}{2}} = \frac{1}{t} + \frac{\frac{1}{2}}{1 - \frac{t}{2}}\)

Prove that \(g(x)\) has a finite limit as \(x \to 1\).

Answer: \ln 2


Question: Given the matrix \(A=\begin{bmatrix} 4 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 2 \end{bmatrix}\), find the maximum value of \(\frac{|(Ax,x)|}{(x,x)}\), where \((.,.)\) denotes the dot product of vectors. The maximization is performed over all vectors \(x=\begin{bmatrix}x_1 & x_2 & x_3\end{bmatrix}^T \in \mathbb{R}^3\) such that \(\sum_{i=1}^{3}x_i=0\).

Answer: 3


Question: Evaluate the integral \( \int_{|z|=2} \frac{1}{z^2-1} \, dz \) where the circle \(|z|=2\) is oriented counterclockwise.

Answer: 0


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

Answer: \sqrt{2}


Question:
Find all stationary points of the system of nonlinear ordinary differential equations given by:
\[ \frac{dx}{dt} = -y + ax^3 \]
\[ \frac{dy}{dt} = x + ay^3 \]

Answer: (0, 0)


Question: Given a sequence of non-negative, identically distributed random variables $X_i$ with expectation $\mu = \mathbb{E}[X_i]$, determine the expected stopping time $\tau$ for the sum $\sum_{i=1}^k X_i$ to first reach or exceed a threshold $t$. Express $\mathbb{E}[\tau]$ in terms of $t$ and $\mu$.

Answer: \dfrac{t}{\mu}


Question: Let \( X \) be a Poisson-distributed random variable with parameter \( \lambda \). Calculate the expected value \( E[2^{-X}] \).

Answer: e^{-\lambda/2}


Question: Find the number of non-negative integral solutions for the equation $x + 2y + 3z = 33$.

Answer: 108


Question: Find the minimum value of the function \( f = \sin\theta_1 + \sin\theta_2 \), given that \( \theta_1 + \theta_2 + \phi = \pi \) and \( 0 < \phi < \pi \).

Answer: \sin \phi


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

Answer: f(x) = -x


Question: Find the length $L$ of the catenary $C$ given by the equation $y = \cosh(x)$ over the interval $\log(2) \leq x \leq \log(3)$. Express your answer in terms of known functions or constants.

Answer: \dfrac{7}{12}


Question: Find the coefficient of \(x^{21}\) in the expansion of \((x^3 + x^4 + x^5 + \ldots + x^{10})^4\).

Answer: 204


Question: Does a countable product of topological spaces, each having a countable basis, have a countable basis?

Answer: Yes


Question: Given a triangle ABC with points D, E, and F on sides BC, AC, and AB respectively, such that BD : DC = 1 : 1, CE : EA = 1 : 3, and AF : FB = 1 : 4. A line parallel to AB is drawn from D to G on side AC. Lines DG and EF intersect at X. If the area of triangle ABC is 120, find the area of triangle DEX.

Answer: 13


Question: Determine the integer values of \( n \) for which the inequality \( n^{\sqrt{n+2}} > (n+1)^{\sqrt{n+1}} \) holds.

Answer: n \geq 9


Question: Does there exist a holomorphic function that is not identically zero and has an uncountable number of zeroes?

Answer: No


Question: Determine the number of distinct cyclic subgroups of order 10 in the group \( \mathbb{Z}_{30} \oplus \mathbb{Z}_{120} \).

Answer: 18


Question: If matrix $A$ is $5 \times 4$ and matrix $B$ is $4 \times 5$, and matrix $B$ has at least one linearly dependent row, does the product matrix $AB$ have at least one linearly dependent row?

Answer: Yes


Question: Consider a finite-dimensional Lie group $G$ and two conjugacy classes $H$ and $I$ of isomorphic subgroups of $G$. Is there a finite-dimensional Lie overgroup of $G$ that fuses $H$ and $I$ into a single conjugacy class?

Answer: No


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

Answer: 1


Question: Let \(\alpha\) be a root of the polynomial \(\alpha^3 + \alpha^2 + 1 = 0\) over \(\mathbb{F}_5\). Express \(2\alpha(\alpha + 1)^{-1}\) as a polynomial of \(\alpha\) with degree at most 2 and coefficients in \(\mathbb{F}_5\).

Answer: 2\alpha^2 + 2


Question: Calculate the integral \( \int_{-\infty}^{\infty} \frac{x \sin(\pi x)}{(x-3)(x-2)} \, dx \) using the Residue Theorem.

Answer: -5\pi


Question: Find the smallest positive integer \( n \) such that \( \left(\frac{1-i}{\sqrt{2}}\right)^n = 1 \).

Answer: 8


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

Answer: 0


Question: Determine whether there exist integers \( n \neq 0,1 \) such that \( \pi_n(S^2) = 0 \).

Answer: No


Question: Solve the recursive relation: \( T(n) = 2nT(\sqrt{n}) + 1 \).

Answer: \Theta(n^2 \log n)


Question: The duration $Y$ of long-distance telephone calls (in minutes) is a random variable with the following properties: $P(Y=3)=0.2$ and $P(Y=6)=0.1$. Otherwise, $Y$ has a continuous density function given by \( f(y)= \begin{cases} (1/4)ye^{-y/2}, & y>0 \\ 0, & \text{elsewhere.} \end{cases} \) Find the expected duration of a randomly selected long-distance call.

Answer: 4


Question: Is the number of conjugacy classes of a group $G$ equal to the number of elements in its abelianization $G/N$, where $N$ is the normal subgroup generated by all commutators $g^{-1}h^{-1}gh$?

Answer: No


Question: Let $x_1, x_2, x_3$ be the roots of the equation $x^3 - x^2 - 1 = 0$, where $x_1$ is the real root. Compute the limit: $$\lim_{n\to\infty} (x_2^n + x_3^n).$$

Answer: 0


Question: Given a $3\times3$ matrix $A$ such that \[\mathrm{adj}(A) = \begin{pmatrix}3 & -12 & -1 \\ 0 & 3 & 0 \\ -3 & -12 & 2\end{pmatrix},\] find the value of $\det(A)$.

Answer: 3


Question: Let $f$ and $g$ be independent and identically distributed (iid) random variables, and let $h$ be linearly dependent on $f$ and $g$. If $f$, $g$, and $h$ all have the same distribution, can we conclude that they are all normally distributed?

Answer: No


Question: Find the minimal positive integer \( n \) such that the polynomial \( g(x) = x^{15} + x^{14} + 1 \) divides \( x^n - 1 \) over the field \( \mathbb{Z}_2 \).

Answer: 32767


Question: Let \(\{z_1,z_2,\ldots,z_9\}\) be the set of the tenth roots of unity, excluding 1. Calculate the value of the following double sum: \[ \sum_{r=0}^{37}\sum_{i=1}^{9}z_i ^r \]

Answer: 2


Question: Given that \(a + b + c = 1\), \(a^2 + b^2 + c^2 = 2\), and \(a^3 + b^3 + c^3 = 3\), find the value of \(a^4 + b^4 + c^4\).

Answer: \dfrac{25}{6}


Question: Determine the value of \( n > 1000 \) that maximizes the probability \( P(x=10) = \frac{\binom{1000}{10}\binom{n-1000}{140}}{\binom{n}{150}} \), where \( n \) is the total number of identical balls in a box. Initially, 1000 balls are marked and returned to the box, and then 150 balls are randomly selected. Find the value of \( n \) that makes this probability largest.

Answer: 15000


Question: Determine whether there exists a non-zero integer \( n \) such that \( A + nB \) is invertible, given that \( A \) is an invertible \( 3 \times 3 \) matrix and \( B \) is any \( 3 \times 3 \) matrix.

Answer: Yes


Question: Determine the asymptotic bound for the expression \( \sum_{i=0}^{\log(n)} 2^{i} \sum_{k=0}^{\frac{n}{2^{i}}} (k+2)^{2} + \theta(n) \).

Answer: \Theta(n^3)


Question: Evaluate the limit: \[ \lim_{x \to 0^{+}} \frac{\sqrt{x}-\sqrt[3]{x}}{\sqrt[5]{x}-\sqrt[7]{x}} \]

Answer: 0


Question: Determine the integrating factor \(\mu(x,y) = x^m y^n\) for the non-exact differential equation \( ydx + (2x - ye^y)dy = 0 \). The integrating factor should make the equation exact.

Answer: y


Question: Given the matrix \( A = \begin{pmatrix} 3 & 0 & 1 \\ -1 & 2 & -1 \\ -2 & -2 & 1 \end{pmatrix} \) and a diagonalizable matrix \( X \) of order 3 such that \( AX = XA \), find the maximum value of \( \det(AX) \) as a function of \( \text{Tr}(AX) = d \). Assume all eigenvalues of \( X \) are positive.

Answer: \dfrac{d^3}{27}


Question: Compute the expected value \( \mathbb{E}[X 1_{X \geq \omega}] \) for a random variable \( X \) with cumulative distribution function \( F(x) \) defined as follows: \( F(x) = 0 \) if \( x < 1 \), \( F(x) = 1 - 1/x^p \) if \( 1 \leq x < \omega \), and \( F(x) = 1 \) if \( x \geq \omega \), where \( 0 < p < 1 \) and \( \omega > 1 \).

Answer: \omega^{1 - p}


Question: Determine whether the metric space \(X = \prod_{i=1}^{\infty} \mathbb{R}\), with the metric defined by \(d(f,g) = e^{-k}\) where \(k = \min \{ i : f(i) \neq g(i) \}\), is complete.

Answer: Yes


Question:
Find all real number pairs \((x, y)\) that satisfy the following system of equations:
\[
\begin{align*}
\log_3{x} + \log_2{y} &= 2, \\
3^{x} - 2^{y} &= 23.
\end{align*}
\]

Answer: (3, 2)


Question: Evaluate the integral \( \int_C \frac{z + i}{z} \, dz \) where \( C \) is the positively oriented unit circle \( |z|=1 \) in the complex plane.

Answer: -2\pi


Question: Find an analytical expression for the integral \( \int_{0}^{2\pi} e^{x\cos \theta + y\cos \theta} d\theta. \) You may use the fact that \( \int_{0}^{2\pi} e^{x\cos \theta } d\theta = 2\pi I_0(x) \), where \( I_0 \) is the modified Bessel function.

Answer: 2\pi I_0(x + y)


Question: Find the minimum value of \( \int_0^1 (f''(x))^2 \, dx \) for functions \( f \) that are twice continuously differentiable on the interval \([0, 1]\), satisfying the conditions \( f(0) = f(1) = 0 \) and \( f'(0) = 2 \).

Answer: 12


Question: Given a bounded convex open set $\Omega$ in $\mathbf{R}^n$ and a sequence of convex functions $P_n$ such that $||P_n||_{L^2(\Omega)} \leq C$ for all $n$, determine if there exists a subsequence that is uniformly convergent on each compact subset of $\Omega$. Justify your answer.

Answer: Yes


Question: Compute the Lie bracket of the vector fields \(\xi = \left(\frac{x}{\sqrt{x^2 + y^2}}, \frac{y}{\sqrt{x^2+y^2}} \right)\) and \(\eta = (-y, x)\) in \(\mathbb{R}^2\).

Answer: (0, 0)


Question:
Consider the system of partial differential equations in cylindrical coordinates:
\[ \partial_r b_2 - \partial_{\theta}b_1 = 0, \\
\partial_{\theta} b_3 - \partial_{z}b_2 = 0, \\
\partial_rb_3 - \partial_{z}b_1 = \xi(r, z), \]
where $\xi(r, z)$ is a given function of $r$ and $z$, and $b_1, b_2, b_3$ are unknown functions of $(r, \theta, z)$. Assume that this system has a unique solution. Determine whether $b_2 = 0$ for this solution.

Answer: b_2 = 0


Question: Determine the value of $f(0)$ such that the function $f(x) = \left[\frac{3 \sin x}{x}\right]$ is continuous at $x=0$, where $[.]$ denotes the greatest integer function and $x \neq 0$. Provide your answer as a single integer.

Answer: 2


Question: Consider a separable infinite-dimensional Banach space $B$ and a continuous linear injective map $f: E \rightarrow F$, where $E$ is a separable nuclear space and $F$ is a separable Banach space, both infinite-dimensional. Let $\otimes_{\epsilon}$ denote the injective tensor product of locally convex spaces (LCS) and $\hat{\otimes}_{\epsilon}$ its completion. Is the map $1_{B}\hat{\otimes}_{\epsilon} f: B\hat{\otimes}_{\epsilon} E \rightarrow B\hat{\otimes}_{\epsilon} F$ a continuous linear injective map?

Answer: Yes


Question: Determine whether there exists a function \( f(x) \) on \( \mathbb{R} \) that is discontinuous at every point, but \( |f(x)| \) is continuous on \( \mathbb{R} \).

Answer: Yes


Question: Let $X \subseteq \mathbb{P}^{n-1}_{\mathbb{C}}$ be a projective variety. Determine the codimension of $X$ in terms of its dimension. Is it $(n - \text{dim } X)$ or $(n-1 - \text{dim } X)$?

Answer: n - 1 - \dim X


Question: Evaluate the principal value of the integral \( PV \int_{-\infty}^{\infty} \frac{1}{(x^2+1)(x^2+2x+2)} \, dx \) using the Cauchy Principal Value method.

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


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

Answer: p > 2


Question: Consider the measurable spaces $(A, \mathcal{E}_A)$, $(B, \mathcal{E}_B)$, and $(C, \mathcal{E}_C)$. Let $f$ be a Markov kernel from $(A, \mathcal{E}_A)$ to $(B, \mathcal{E}_B)$, and let $g$ be a Markov kernel from $(A, \mathcal{E}_A)$ to $(C, \mathcal{E}_C)$. Define $h_a$ as the unique measure on $(B \times C, \mathcal{E}_B \otimes \mathcal{E}_C)$ such that \( h_a(E_B \times E_C) = f(a, E_B) \cdot g(a, E_C) \) for all $E_B \in \mathcal{E}_B$ and $E_C \in \mathcal{E}_C$. If we write $h(a, E) = h_a(E)$, determine whether $h$ is a Markov kernel from $(A, \mathcal{E}_A)$ to $(B \times C, \mathcal{E}_B \otimes \mathcal{E}_C)$. Specifically, is the map $a \mapsto h_a(E)$ a real-valued measurable function for all $E \in \mathcal{E}_B \otimes \mathcal{E}_C$?

Answer: Yes


Question: Let \(\Omega\) be a \(C^2\) domain with \(0 \in \partial \Omega\) and \(e_n\) orthogonal to the boundary of \(\Omega\) at \(0\). In a neighborhood of \(0\), express \(x_n\) as \(x_n = \psi(x_1, x_2, \ldots, x_{n-1})\) for a \(C^2\) function \(\psi\). Is it sufficient to conclude that \(\psi_i(0) = 0\) for \(i = 1, 2, \ldots, n-1\)?

Answer: Yes


Question: For $p \in (0,1)$, if $f, g \in L^p$, is it true that $f+g \in L^p$?

Answer: Yes


Question: True or false: If $f: \mathbb{R}^n \to \mathbb{R}^m$ is a continuous function and $A$ is a convex set in $\mathbb{R}^n$, then $f(A)$ is convex in $\mathbb{R}^m$.

Answer: B


Question: Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a function such that $|f|$ is measurable. Is $f$ necessarily measurable? Answer "True" or "False".

Answer: B


Question: Determine whether the following statement is true or false: Assume $A$ is an $m \times n$ matrix and $B$ is an $m \times p$ matrix. If $X$ is an $n \times p$ unknown matrix, then the system $A^TAX = A^TB$ always has a solution.

Answer: True


Question: Find the remainder when \( \displaystyle \sum^{2014}_{r=0}\sum^{r}_{k=0}(-1)^k(k+1)(k+2)\binom{2019}{r-k} \) is divided by 64.

Answer: 62


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

Answer: \dfrac{1}{e}


Question: Evaluate the limit: \[ \lim_{x\to0}\ln(e + 2x)^\frac{1}{\sin x} \]

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


Question: If the distinct nonzero numbers $x(y - z)$, $y(z - x)$, and $z(x - y)$ form a geometric progression with common ratio $r$, find the equation that $r$ satisfies.

Answer: r^2 + r + 1 = 0


Question: Evaluate the integral \( \int_{C} \frac{e^{iz}}{z^3} \, dz \) where \( C \) is the circle \( |z| = 2 \) traversed once in the positive direction, using the Cauchy Integral Formula for derivatives.

Answer: -\pi i


Question: Find all integer solutions to the equation \(a^2 + b^2 + c^2 = a^2 b^2\).

Answer: (0, 0, 0)


Question: Calculate the surface integral \( \oint_S \ \vec{F} \cdot \vec{dS} \) for the vector field \( \vec{F} = x^2\hat{a}_x + y^2\hat{a}_y + (z^2-1)\hat{a}_z \). The surface \( S \) is defined by the cylindrical coordinates \( r = 2; 0 < z < 2; 0 \leq \Phi \leq 2\pi \).

Answer: 0


Question: In the space $X = \{0,1\}^{\mathbf{N}}$ with the product topology, does there exist a second category set $S \subseteq X$ with empty interior?

Answer: Yes


Question: Let $p_1, \ldots, p_n$ be a finite sequence of nonconstant polynomials with integer coefficients. Determine whether there exists a finite sequence of integers $x_1, \ldots, x_n$ such that the integers $p_1(x_1), \ldots, p_n(x_n)$ have a nontrivial common divisor.

Answer: Yes


Question: Given algebraic numbers $a$ and $b$, determine if all algebraic conjugates of $a+b$ can be expressed as $a'+b'$, where $a'$ and $b'$ are algebraic conjugates of $a$ and $b$, respectively.

Answer: Yes


Question: Let $X$ be an inner product space. Prove that if $\|\lambda x + (1-\lambda)y\| = \|x\|$ for every $\lambda \in [0,1]$, then $x = y$.

Answer: x = y


Question: Compute the residue of the differential form \( \frac{dz}{w^3} \) at the point \( p \) on the Riemann surface defined by \( w^3 = z(z-1)(z-2) \), where \( w = 0 \) and \( z = 1 \).

Answer: -3


Question: Compute the limit: \[ \lim_{n \to \infty}\left(\frac {\sqrt[n]{a} + \sqrt[n]{b}}{2}\right)^{n} \] where \(a, b > 0\).

Answer: \sqrt{ab}


Question: Consider a continuous map $f : X \rightarrow Y$ between topological spaces. The function $\mathcal{O}(f) : \mathcal{O}(Y) \rightarrow \mathcal{O}(X)$ is defined by taking preimages and is known to be a frame homomorphism, preserving finite meets and arbitrary joins. Determine whether $\mathcal{O}(f)$ can fail to preserve pseudocomplements, where the pseudocomplement of an open set $U$ is defined as the interior of its complement.

Answer: Yes


Question:
A function \( f \) is even. It is known that there are four values \( x_1, x_2, x_3, x_4 \) satisfying \( f\left(\frac{x+1}{3-x}\right) = f(x) \). What is the value of \( x_1 + x_2 + x_3 + x_4 \)?

A) 0
B) 2
C) 4
D) 6

Answer: D


Question:
A straight line L intersects perpendicularly both the lines:
\[\frac{(x+2)}{2} = \frac{(y+6)}{3}=\frac{(z-34)}{-10} \]
and
\[\frac{(x+6)}{4}=\frac{(y-7)}{-3}=\frac{(z-7)}{-2}\]
Find the square of the perpendicular distance from the origin to the line L.

Answer: 5


Question: Determine the values of $p > 0$ for which the integral \( \int_{\mathbb{R}} \frac{1}{(1+|x|)^p} \, d\mathcal{L}(x) \) is finite.

Answer: p > 1


Question: Let $T\colon X\to Y$ be a bounded linear operator between Banach spaces. If $T$ is an isomorphism onto its range, must the bidual operator $T^{**}\colon X^{**}\to Y^{**}$ also be an isomorphism onto its range?

Answer: Yes


Question: Determine the conditions on \(m\) and \(n\) such that the integral \(\int_{0}^{\infty} \frac{x^{n}}{(1+x)^{m}}dx\) converges, where \(m,n \geq 0\) and \(m,n \in \mathbb{R}\).

Answer: m > n + 1


Question: Let $f$ be an entire function such that $g(x, y) = |f(x+iy)|$ is integrable over $\mathbb{R}^2$. Prove that $f \equiv 0$.

Answer: 0


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

Answer: 0


Question: Evaluate the limit: \[ \lim_{N\rightarrow\infty}N\sum^{N}_{k=2}\left(\frac{k-1}{N}\right)^{N^2} \]

Answer: 0


Question:
Evaluate the limit using the Central Limit Theorem:
\[ \lim_{n\to\infty}p^{n}\sum_{k \geqslant{n(p^{-1}-1)}}^{\infty}\binom{n+k-1}{n-1}(1-p)^{k}. \]

Answer: \dfrac{1}{2}


Question: Find the range of the function \( f(x) = x\sqrt{x} + \frac{1}{x\sqrt{x}} - 4\left(x + \frac{1}{x}\right) \), where \( x > 0 \).

Answer: [-10, \infty)


Question: Let \( X \) be an \( n \times (k+1) \) matrix such that \( X^TX \) is invertible. Determine the rank of the matrix \( I - X(X^TX)^{-1}X^T \), where \( I \) is the \( n \times n \) identity matrix.

Answer: n - k - 1


Question:
In the cube $ABCD.EFGH$, the point $M$ is on the edge $AD$ such that $|AM|=2|MD|$. Calculate the tangent of the angle between the planes $BCF$ and $BGM$. Choose the correct answer from the options below:

(A) $3 \sqrt{2}$
(B) $2 \sqrt{2}$
(C) $\frac{3}{2} \sqrt{2}$
(D) $\frac{1}{2} \sqrt{2}$
(E) $\sqrt{2}$

Answer: C


Question: Let $M=\{(x,y,z) \in \mathbb{R}^3 : x+y=5, x+z=\cos^2y\}$ be a submanifold of $\mathbb{R}^3$. Consider the point $p=(5,0,-4)$ and the tangent vector $v=(-C,C,C) \in T_{(5,0,-4)}M$. Define the smooth map $F:M \rightarrow S^1$ by $F(x,y,z) = \left(\frac{x}{\sqrt{x^2+y^2}},\frac{y}{\sqrt{x^2+y^2}}\right)$. Let $\omega = -\frac{1}{x}dy$ be a 1-form on $S^1$ in a neighborhood of $(1,0)$. Compute $(F^{*}\omega)_p(v)$. Express your answer in terms of $C$.

Answer: -\dfrac{C}{5}


Question: Consider the sequence \( \{u_n\} \) defined by the initial condition \( u_0 \in [-1, 1] \) with \( u_0 \neq 0 \) and the recursive relation \( u_{n+1} = 2^{u_n} - u_n - 1 \). Determine the limit of the sequence \( \{u_n\} \) as \( n \to \infty \).

Answer: 0


Question: Does there exist a domain $R$ with fraction field $K$, and an element $x \in K \setminus R$, such that for any maximal ideal $\mathfrak{m}$ in $R[x]$, there exists an element $a \in R$ with $x-a \in \mathfrak{m}$?

Answer: Yes


Question: Suppose $A = \{a_n \mid n \in \Bbb N \}$ is a subset of $\Bbb R$ such that $A$ has an infinite linearly independent subset over $\Bbb Q$. Is $A$ dense in $\Bbb R$?

Answer: No


Question: Evaluate the limit: \[ \lim_{n\to\infty} \exp(-n^2) \sum_{j=n}^{4n} \frac{n^j}{j!} \]

Answer: 0


Question: Determine the number of ways to distribute 15 fruits to 6 people such that each person receives at least 1 fruit and no more than 3 fruits.

Answer: 50


Question: In the game of coins, there are two bowls. The first player places 100 coins in the first bowl. The second player must decide how many coins to place in the second bowl. Each player, starting with the first, can make one of the following moves: take any number of coins from the first bowl, take any number of coins from the second bowl, or take the same number of coins from both bowls. The winner is the player who leaves no coins in either bowl after their move. How many coins should the second player place in the second bowl to guarantee a win with optimal play?

Answer: 162


Question: Determine whether a closed form formula exists for the series \( \sum_{k=0}^{n-1} \binom{n+k}{n-k-1} \).

Answer: F_{2n}


Question: Let $E = \{E_1, \ldots, E_k \}$ be a set of events in a probability space, and let $p = \sum \Pr(E_i)$. For a fixed $n \leq k$, define $P_n$ as the event that some $n$ independent events from $E$ occur. Show that the probability of $P_n$ is bounded by $\Pr(P_n) \leq \frac{p^n}{n!}$.

Answer: \Pr(P_n) \leq \dfrac{p^n}{n!}


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\left( x^2 \cos^2 x - \sin^2 x \right)\left( x^3 - \sin^3 x \right)}{\left( e^x + e^{-x} - x^2 - 2 \right)^2 \sin x} \]

Answer: -48


Question:
Let \( X_1, X_2, \ldots, X_n \) be a random sample from a \( N(2\theta, \theta^2) \) population, where \( \theta > 0 \). Which of the following is a consistent estimator for \( \theta \)?

(A) \( \dfrac{1}{n}\sum_{i=1}^{n}X_i \)

(B) \( \left(\dfrac{5}{n}\sum_{i=1}^{n}X_i^{2}\right)^{\frac{1}{2}} \)

(C) \( \dfrac{1}{5n}\sum_{i=1}^{n}X_i^{2} \)

(D) \( \left(\dfrac{1}{5n}\sum_{i=1}^{n}X_i^{2}\right)^{\frac{1}{2}} \)

Answer: D


Question: Determine the values of \(\alpha\) for which the series \(\sum_{k=1}^\infty \frac{1}{(k+1)[\ln(k+1)]^\alpha}\) converges.

Answer: \alpha > 1


Question: Evaluate the integral \( \int_{\gamma}\frac{z^2-1}{z^2+1}dz \) where \( \gamma \) is a circle of radius 2 centered at 0.

Answer: 0


Question: If $X$ is an infinite set, is it always possible to define a group structure on $X$ such that every element has order at most 2?

Answer: Yes


Question: Simplify the expression \(-\sqrt{\lambda}t - \lambda(1-e^{\frac{t}{\sqrt{\lambda}}})\) and show that it approaches \(\frac{t^2}{2}\) as \(\lambda\) approaches infinity.

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


Question: Let $H_{1}$ and $H_{2}$ be finite-dimensional Hilbert spaces with dimensions $n_{1}$ and $n_{2}$, respectively. Consider the linear operator $T: H_{1} \to H_{2}$. Suppose for any orthonormal bases $\{ e_{i} \}$ for $H_{1}$ and $\{ f_{j} \}$ for $H_{2}$, the inequality \(|\langle f_{j}, Te_{i} \rangle| \leq (1-\varepsilon) \cdot ||e_{i}|| \cdot ||f_{j}||\) holds for some $\varepsilon > 0$. Is $T$ a contraction, i.e., does $||T|| \leq 1$?

Answer: Yes


Question: Find the Laplace transform of the derivative of the Dirac delta function, \( \delta'(t) \).

Answer: s


Question: Evaluate the limit: $$\lim_{x\to 0}\frac{x^2}{\sqrt[5]{1+5x}-1-x}$$ without using L'Hopital's rule or Taylor series.

Answer: -\dfrac{1}{2}


Question: Consider a sequence of real numbers \((a_n)\). Suppose that for every function \(f\) from the positive integers to the positive integers, the difference \(a_{n+f(n)} - a_n\) tends to 0 as \(n\) tends to infinity. Does this imply that the sequence \((a_n)\) is convergent?

Answer: Yes


Question: Find the order of the subgroup of \( \mathbb{C}^\times \) generated by \( \{i, e^{\frac{2i \pi}{5}}, -1\} \).

Answer: 20


Question: Let $\{a_n:n \geq 1\}$ be a sequence of real numbers such that $\sum_{n=1}^{\infty} a_n$ is convergent and $\sum_{n=1}^{\infty} |a_n|$ is divergent. Determine the radius of convergence $R$ of the power series $\sum_{n=1}^{\infty} a_n x^n$. What is $R$?

Answer: 1


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

Answer: \dfrac{36}{5}


Question: Evaluate the limit: \[ \lim_{x \to a} \frac{\tan x - \tan a}{\ln x - \ln a} \] where \( a \) is an unknown constant.

Answer: a \sec^2 a


Question: For the function \( k(x) = 4\sqrt{x} + \frac{2}{\sqrt{x}} \) on the interval \([\frac{1}{4}, 1]\), find the value \( c \) that satisfies the Mean Value Theorem.

Answer: \dfrac{1}{2}


Question: Compute the surface integral of the vector field \( F(x,y,z) = (xy,-x^2,x+z) \) over the surface \( S \), which is the portion of the plane \( 2x+2y+z=6 \) in the first octant (where \( x, y, z \geq 0 \)).

Answer: \dfrac{27}{4}


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

Answer: 0


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

Answer: \dfrac{7}{24}


Question: Given the polynomial function \( y = x^5 + x^3 + x + 1 \), find \( f^{-1}(-41) \) assuming the function is one-to-one.

Answer: -2


Question: Assume $f(x) \in L^1([0,1])$ and let the Fourier coefficients be $\{a_n\}_{n=-\infty}^{\infty}$. If the partial sum $S_n(x) = \sum_{k=-n}^{n} a_k e^{ikx}$ converges pointwise almost everywhere on $[0,1]$, does it necessarily converge to the original function $f(x)$?

Answer: Yes


Question: Find the value of \( \alpha \in \mathbb{R} \) such that the transformation \( v = e^{\alpha x} u \) eliminates the first derivative term \( v_x \) in the equation \( u_t = u_{xx} + cu_x + au \), where \( a, c \in \mathbb{R} \), on the interval \((-L, L)\) with homogeneous Dirichlet boundary conditions. Assume \( u \in L^2([-L, L]) \) and \( c \neq 0 \).

Answer: \dfrac{c}{2}


Question: Let \( X \) be a random variable with the characteristic function \( \varphi_{X}(t) = \frac{1}{7}\left(2+e^{-it}+e^{it}+3e^{2it}\right) \). Determine the probability \( \mathbb{P}(-1\leqslant X\leqslant\frac{1}{2}) \).

Answer: \dfrac{3}{7}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{\sqrt{n+1}+\sqrt{n+2}+\dots+\sqrt{2n}}{\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}} \]

Answer: 2\sqrt{2} - 1


Question: Find the real scalar value of $k$ such that the complex number \( z = \frac{2}{1+ki} - \frac{i}{k-i} \) lies on the line \( y = 2x \).

Answer: -2


Question: Determine whether the following statement is true or false: Let $f(x)$ be an integrable function on $[a,b]$. Suppose $|f(x)| \geq 1$ on $[a,b]$. Then, $\frac{1}{f(x)}$ is also integrable on $[a,b]$. Provide a justification for your answer.

Answer: A


Question: Let $X$ be an affine variety over an algebraically closed field $k$. Does the ring of regular functions $k[X]$ always have a countable basis?

Answer: Yes


Question: Given the double summation \( \sum_{i=1}^{n}\sum_{j=i}^{i+1}(3i+j) \), change it to the form \( \sum_{j}^{}\sum_{i}^{}(3i+j) \) and calculate the result.

Answer: 4n^2 + 5n


Question: Find all integer pairs \((x, y)\) such that \(x^3 = y^3 + 2y + 1\).

Answer: (1, 0)


Question: Let $S$ be a closed orientable surface of genus 3, and let $R \rightarrow S$ be a degree 2 covering map. Determine the genus of the surface $R$. \( \text{(Hint: Use the Euler characteristic as a topological invariant.)} \)

Answer: 5


Question: Let $X, Y, Z$ be independent and identically distributed standard normal random variables. Calculate the probability $P(X > YZ)$. Express your answer as a single probability value.

Answer: \dfrac{1}{2}


Question: Calculate the limit: \[ \lim_{k \to \infty}(1+2^{k+1})^{(2^{k-2})-2}\cdot\frac{(2^k-1)^2}{(2^k-1)!!} \]

Answer: 0


Question: Let $a_j, b_j, c_j, d_j > 0$ with $d_j \le c_j$ for $j = 1, \ldots, n$. Suppose \( \prod_{j=1}^k \frac{a_j}{b_j} \le \prod_{j=1}^k \frac{c_j^2}{d_j^2} \) for each \( k = 1, \ldots, n \). Does this imply that \( \frac{\sum_{j=1}^n a_j}{\sum_{j=1}^n b_j} \le \left(\frac{\sum_{j=1}^n c_j}{\sum_{j=1}^n d_j}\right)^2 \)?

Answer: No


Question: Verify the trigonometric identity: \( \sum_{i=1}^{n-1} \cos \frac{2ik\pi}{n}\sin \frac{2il\pi}{n}=0 \) for \( 1\leq k,l \leq n-1 \) and \( k,l \in \mathbb{N} \).

Answer: 0


Question: Determine the number of elements in the set \( \{z \in \mathbb{C} : z^{60} = -1 , z^k \neq -1, 0<k<60 \} \).

Answer: 32


Question: If \( \int_a^b f(x) \, dx \le \int_a^b g(x) \, dx \), is it true that there exists some \( c \in [a, b] \) such that \( f(c) \le g(c) \)? Justify your answer.

Answer: Yes


Question: Determine whether there exists a matrix $A$ of infinite order in the group $\{ A \in GL_2(\mathbb{R}) : A^T = A^{-1} \}$. Provide a justification for your answer.

Answer: Yes


Question: Determine if the zero function is the only solution to the Euler-Lagrange equation derived from the functional \( I[y]=\int_{a}^{b}(y'^2+y^4)dx \) with boundary conditions \( y(a)=0 \) and \( y(b)=0 \).

Answer: Yes


Question: Calculate the limit: \[ \lim_{x \to 0}((-2^x+1)^{\sin(x)}) \]

Answer: 1


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

Answer: 3e


Question: What is the maximum number of perfect squares that can be in an arithmetic progression of positive integer terms of length 10?

Answer: 5


Question: Evaluate the integral \( \int_{0}^{\infty} \left( \int_{0}^{\infty} x e^{-x^2(1+y^2)} \, d\lambda(y) \right) \, d\lambda(x) \) using Tonelli's theorem.

Answer: \dfrac{\pi}{4}


Question: Find the last three digits of the sum \( S = \sum_{i=2}^{10^7} (i^7 + i^5) \).

Answer: 998


Question: Evaluate the limit: \( \lim_{x \rightarrow 0} \frac{2\exp\left(-x^{-2}\right)}{x^{3}} \)

Answer: 0


Question: Consider a finite set $X$ and a topology $T$ on $X$. Given that $T$ is both union-closed and intersection-closed, determine whether $T$ satisfies Frankl's union-closed set conjecture.

Answer: Yes


Question: A man weighs 90 kg. If he takes no calories for 14 days, his mass reduces by 20%. How long will it take him to reduce his mass by the same amount if he consumes 3/4 of the calories required to maintain his mass at 90 kg? Assume the daily calorie intake is a fixed quantity \( C \) and is modeled by the differential equation \( \frac{dt}{dm} = \frac{1}{a(C-bm)} \), where \( a \) and \( b \) are constants. Calculate the time in days.

Answer: 101


Question: Evaluate the expression $E[E[X|Y]|X]$. What is the result?

Answer: X


Question: Consider the function \( f(x) \) defined on the interval \([0, 1]\) as follows: \( f(x) = x \) if \( x \) is rational, and \( f(x) = 1-x \) if \( x \) is irrational. Determine the point within the interval \([0, 1]\) at which the function \( f \) is continuous.

Answer: \dfrac{1}{2}