Question: Let \( f \) be a differentiable function such that both \( f \) and \( f' \) are in \( L^1(\mathbb{R}) \). Prove that \( \int_{\mathbb{R}} f'(x) \, dx = 0 \).

Answer: 0


Question:
Given a $10 \times 10$ matrix $A = [a_{ij}]$ with the following properties:

1. The set of elements $\{a_{ij} \mid 1 \le i, j \le 10\} = \{1, 2, \ldots, 100\}$.
2. For any $p, q \in \mathbb{N}$ where $1 \le p, q \le 10$, the sum of the elements in each column $\sum_{i=1}^{10} a_{ip}$ is equal to the sum of the elements in each row $\sum_{j=1}^{10} a_{qj}$.
3. The determinant of $A$ is non-zero, $\det(A) \neq 0$.

Find the sum of the elements of the inverse matrix $A^{-1}$. \( \boxed{} \)

Answer: \dfrac{2}{101}


Question: Determine whether there are infinitely many positive integers $n$ that are not $p$-leading for any odd prime $p$. Specifically, is the set \( S = \mathbb{N}\setminus \bigcup_{p\geq 3} \bigcup_{d\geq 1} [p^d - p^{d-1},p^d) \) infinite?

Answer: Yes


Question: Let \( f: [0,1] \to [0,1] \) be a continuous bijection, and let \( g: [0,1] \to \mathbb{R} \) be a continuous function such that \( \int_0^1 g(x)(f(x))^{6n} \, dx = 0 \) for all integers \( n \ge 0 \). Is it true that \( g(x) = 0 \) for all \( x \in [0,1] \)?

Answer: Yes


Question: Evaluate the surface integral \( \int_{S}\nabla\times\vec{F}\cdot{\rm d}\vec{S} \) using Stokes' Theorem, where \( \vec{F}(x,y,z) = y^2\vec{i} + 2xy\vec{j} + xz^2\vec{k} \) and \( S \) is the surface of the paraboloid \( z = x^2 + y^2 \) bounded by the planes \( x=0, y=0 \), and \( z=1 \) in the first quadrant, with the surface pointing upward.

Answer: \dfrac{2}{7}


Question: Evaluate the integral: \[ \int_0^{2\pi} \log|e^{i\theta} - 1| \, d\theta. \]

Answer: 0


Question:
If you sum a certain number of rank-1 matrices:
\[X = u_1 u_1^T + u_2 u_2^T + \cdots + u_N u_N^T\]
where the vectors \(u_1, u_2, \ldots, u_N\) are linearly independent, is the resulting matrix \(X\) guaranteed to have rank \(N\)?

Answer: Yes


Question: Find a linear polynomial $A(x) \in (\Bbb Z/5\Bbb Z)[x]$ such that $\bar{A}(x) = 2x^4 + x^3 + 2x + 1$ in the quotient ring $(\Bbb Z/5\Bbb Z)[x]/(x^2 + x + 2)$. Provide your answer as a polynomial of degree 1.

Answer: 2x + 2


Question: Let \( f(x, y) \) be a homogeneous function of degree 3. Given that \( f'_x(2, 1) = 1 \), \( f(1, 0.5) = 2 \), \( f(4, 2) = 128 \), and \( f(2, 1) = 16 \), compute \( f'_y(4, 2) \).

Answer: 184


Question: Evaluate the integral \( \int_0^{\pi/2} \frac{1}{1+(\tan x)^e} \, dx \).

Answer: \dfrac{\pi}{4}


Question:
Determine the norm of the linear functional \( l(f) = \int_{0}^{\pi} f(x) \sin^{2}(x) \, dx \) in the Banach space \( C[0,\pi] \) with the supremum norm. Choose the correct answer from the following options:

A. 1
B. \( \frac{\pi}{2} \)
C. \( \pi \)
D. \( 2\pi \)

Answer: B


Question: Evaluate the integral \( \iint \operatorname{curl}(F) \, dS \) using Stoke's Theorem, where \( F(x,y,z) = \langle x^2z^2,y^2z^2,xyz\rangle \) and \( S \) is the part of the paraboloid \( z=x^2+y^2 \) that lies inside the cylinder \( x^2+y^2=4 \), oriented upward.

Answer: 0


Question: Find the norm of the linear functional \( l: C[0,\pi] \to \mathbb{R} \) defined by \( l(f) = \int_{0}^{\pi} f(x)\sin^2(x) \, dx \).

Answer: \dfrac{\pi}{2}


Question: Determine whether the following statement about field extensions is true or false: Let $F$ be a field. If $[F(a) : F] = 4$ and $[F(b) : F] = 6$, then $[F(a,b) : F] = 24$. Provide a justification for your answer.

Answer: False


Question: Is it true that for the Lie algebra \(\mathfrak{gl}(n)\), the trace of the commutator of any two elements is zero, i.e., \(\operatorname{Tr}([x, y]) = 0\) for all \(x, y \in \mathfrak{gl}(n)\)?

Answer: True


Question:
Find the total number of solutions to the following system of modular equations:
\[ a^2+bc \equiv a \pmod{37} \]
\[ ba+bd \equiv b \pmod{37} \]
\[ ca+cd \equiv c \pmod{37} \]
\[ bc+d^2 \equiv d \pmod{37} \]
\[ ad-bc \equiv 1 \pmod{37} \]

Answer: 1


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

Answer: 0


Question: Find the radius of convergence of the power series \( \sum\limits_{n=1}^{\infty} a_n Z^n \), where \( a_n \) is the number of divisors of \( n^{50} \).

Answer: 1


Question: Evaluate the integral \( \int_0^{2\pi} \frac{d \theta}{5-3 \cos \theta} \).

Answer: \dfrac{\pi}{2}


Question: Evaluate the integral: \[ \int_0^1 \left(\sqrt[3]{(1-x^7)} - \sqrt[7]{(1-x^3)} \right) \, dx \]

Answer: 0


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

Answer: 1


Question: Evaluate the sum \( S = \sum_{x=0}^n\sum_{y=0}^{n-x}\sum_{z=0}^{n-x-y} \binom{n}{x} \binom{n-x}{y} \binom{n-x-y}{z} \) and express it in terms of \( n \).

Answer: 4^n


Question: In a hospital, the number of new Covid-42 patients arriving follows a Poisson process with an average rate of 2 new cases per day. Given that exactly 2 new patients arrived within a 12-hour period, what is the expected interval of time between the two arrivals?

Answer: 4


Question: Given the derivative of a function $f(x)$, where \( f'(x) = 2x^2 + 5ax + 3a^2 + 2 \), find all possible values of the parameter \( a \) such that the function \( f \) is increasing for all \( x \in \mathbb{R} \).

Answer: [-4, 4]


Question: Find the smallest positive integer \( n \) such that \( S(n^2) = S(n) - 7 \), where \( S(N) \) denotes the sum of the digits of \( N \).

Answer: 149


Question: Evaluate the integral \( \int_{-3}^{+1} (x^3 - 3x^2 + 2x - 1) \delta(x+2) \, dx \).

Answer: -25


Question: Determine the minimum number of times you need to spin a bottle to have at least a 50% chance of pointing to each of the 10 people in a circle at least once.

Answer: 27


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

Answer: -I_0(2\alpha)


Question: Let $A \subset B$ be a finite extension of graded rings, meaning $B$ is finite as a graded $A$-module. Consider the induced finite morphism $\operatorname{Proj} B \to \operatorname{Proj} A$. Is this morphism necessarily surjective?

Answer: Yes


Question: Find the limit of the sequence \( \sqrt[n]{\left| \frac{1}{n^{10}3^{n}} - \frac{n^{10}}{e^n} \right|} \) as \( n \to \infty \).

Answer: \dfrac{1}{e}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{{n^2 \choose n}}{{n^2+n-1 \choose n}}. \]

Answer: \dfrac{1}{e}


Question: Player A has £1 and Player B has £99. They flip a coin, and the loser pays the other £1. What is the expected number of games they will play before one player is bankrupt?

Answer: 99


Question: Calculate the arc length of the curve defined by the equation \(6xy - y^4 = 3\) from the point \(\left(\frac{19}{12}, 2\right)\) to the point \(\left(\frac{14}{3}, 3\right)\).

Answer: \dfrac{13}{4}


Question: Compute the correlation coefficient \( r(X_{(1)},X_{(3)}) \) for the order statistics of three independent random variables \( X_1, X_2, \) and \( X_3 \), each following an \( \text{Exp}(1) \) distribution.

Answer: \dfrac{2}{7}


Question: Evaluate the integral \( K = \iiint_{P}(x^2+\sin(xy^2)+ y^2) \,dx\,dy\,dz \) where \( P = \{(x,y,z) : 0 \leq z \leq 2 -|x| -|y|\} \).

Answer: \dfrac{64}{15}


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\frac{1}{n}\int^{\pi}_{0}\lfloor n \sin x \rfloor \, dx \]

Answer: 2


Question:
Suppose \( f \) and \( g \) are entire functions and \( g(z) \neq 0 \) for all \( z \in \mathbb{C} \). If \( |f(z)| \leq |g(z)| \) for all \( z \in \mathbb{C} \), which of the following statements is true?

1. \( f \) is a constant function.
2. \( f(0) = 0 \).
3. For some \( C \in \mathbb{C} \), \( f(z) = C g(z) \).
4. \( f(z) \neq 0 \) for all \( z \in \mathbb{C} \).

Answer: 3


Question: Solve for \( n \) in the equation: \[ \frac{365!}{365^{n}(365-n)!} = 0.5 \]

Answer: 23


Question: Determine the number of equations required to enforce that a matrix $M$ of size $n \times n$ has a rank of at most $n-2$. Specifically, identify the necessary conditions on the submatrices of $M$ to achieve this rank condition.

Answer: n^2


Question: Evaluate the limit: \[ \lim_{t \to 0} \frac{1}{t} \int_t^{2t} \frac{\ln(1+x)}{\sin x} \, dx. \]

Answer: 1


Question: Find the greatest common divisor of the Gaussian integers $16+7i$ and $10-5i$ using the Euclidean algorithm in the ring of Gaussian integers $\mathbb{Z}[i]$. Use the Euclidean norm $N(a+bi) = a^2 + b^2$ for $a+bi \in \mathbb{Z}[i]$.

Answer: 1 + 2i


Question: Evaluate the line integral \( I = \oint_{\gamma} \frac{dz}{(z^2+4)^2} \), where \( \gamma \) is the circle defined by \( |z-i|=2 \).

Answer: \dfrac{\pi}{16}


Question: Evaluate the integral \( \int_{-2}^{2} \frac{\sin^2 x}{\left[\frac{x}{\pi}\right] + \frac{1}{2}} \, dx \), where \([\cdot]\) denotes the greatest integer function.

Answer: 0


Question:
Determine whether there exists a natural number \( n > 1 \) that satisfies the following conditions:

- \( n \) is not squarefree.
- \( n \) is a strong Fermat pseudoprime to bases 2 and 3.

Assume that the only Wieferich primes are 1093 and 3511. Can it be shown that such a number \( n \) cannot exist?

Answer: No


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

Answer: -\dfrac{1}{2}


Question: Let \( f \in L^1(\mathbb{R}) \) and \( a > 0 \). Prove that \( \lim_{a \to 0} a \cdot \mu (\{ x \in \mathbb{R} : |f(x)| > a \}) = 0 \).

Answer: 0


Question: Consider a series of random variables defined as follows: let $a_1 = 1$, and for each $n \geq 2$, $a_n$ is either $a_{n-1}$ or $\frac{1}{2} a_{n-1}$, each with equal probability. What is the probability that the series $\sum_{i=1}^{\infty} a_i$ converges?

Answer: 1


Question: Given a random variable \(X\) such that \(-1 \leq X \leq \frac{1}{2}\) and \(\mathbb{E}[X] = 0\), find the maximal value of \(\mathbb{E}[X^2]\).

Answer: \dfrac{1}{2}


Question: Let $R: \mathbb{C} \to \mathbb{C}$ be a rational function, and for some $a \in \mathbb{C}$, define the function \( S(z) = \frac{zR(z) - aR(a)}{z-a} \). If $R(z)$ does not have any roots inside the unit circle, can $S(z)$ have a root inside the unit circle?

Answer: Yes


Question: Determine the radius of convergence of the power series obtained by the Taylor expansion of \( f(z) = \frac{(z+20)(z+21)}{(z-20i)^{21} (z^2 +z+1)} \) about \( z = 0 \).

Answer: 1


Question: Given that \(X\sim\mathrm{Uni}[0,3]\) and \(Y\sim\mathrm{Uni}[0,5]\), let \(Z=\ln\left(\frac{X^2}{Y}\right)\). Find \(\mathrm{Var}(Z)\).

Answer: 5


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

Answer: \ln 3


Question:
Given a continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ with a continuous derivative, it is known that:
\[ (f(1))^2 - (f(0))^2 = 0 \]
\[ (f(1))^3 - (f(0))^3 = 54 \]
Find the value of:
\[ \int_0^1 (f(x))^4 \cdot f'(x)\, dx. \]

Answer: \dfrac{486}{5}


Question: Prove using the epsilon-delta definition that \( \lim_{z \to \infty} \frac{4z^2}{(z-1)^2} = 4 \).

Answer: 4


Question: Find all real numbers \(x\) that satisfy the equation \(|x^2 - \lfloor x \rfloor| = 1\), where \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \(x\).

Answer: \sqrt{2}


Question: Consider a connected Riemannian manifold $M$ with boundary. Define the "distance to boundary" function $d(p) = d(p, \partial M)$, which is the infimum of lengths of (piecewise) smooth paths connecting $p$ with some point in $\partial M$. Determine whether there always exists a neighborhood $U$ of $\partial M$ such that $d$ is smooth in $U$. Justify your answer.

Answer: Yes


Question: Evaluate the integral \( \int_{C} f(z) \, dz \) where \( f(z) = x^2 + iy^2 \) and \( z = x + iy \). The path \( C \) is given by \( z(t) = t + it \) for \( 0 \leq t \leq 1 \).

Answer: \dfrac{2}{3}i


Question: Determine the asymptotic complexity of the recurrence relation \( T(n) = T\left(\frac{3}{5} n + 1\right) + n^2 \) using Big O notation.

Answer: O(n^2)


Question: Calculate the work done by the vector field \( \mathbf{F}(x,y,z)=(2x\sin(\pi y)-e^z)\mathbf{i}+(\pi x^2 \cos (\pi y)-3e^z)\mathbf{j}-xe^z \mathbf{k} \) along the curve \( C \) from \((0,0,0)\) to \((1,1,\ln(2))\), where \( C \) is defined by the surfaces \( z=\ln(1+x) \) and \( y=x \).

Answer: -\dfrac{13}{2}


Question: Determine the degree of the expression \(1+\sqrt[3]{2}+\sqrt[3]{4}\) over \(\mathbb{Q}\).

Answer: 3


Question: Given that \( \text{cov}(a, b) = 0 \), where \( a \in \mathbb{R}^p \) has mean 0 and covariance \( \Sigma_a \), and \( b \in \mathbb{R}^n \) has mean 0 and covariance \( \Sigma_b \), and \( C \in \mathbb{R}^{n \times p} \) is a matrix of constants, compute \( \text{E}[b^TCa] \).

Answer: 0


Question: Evaluate the integral \( \int_0^{2\pi}\frac{1}{3-2\cos \theta +\sin\theta}\,\mathrm d\theta \) by converting the trigonometric functions to their complex form and using the residue theorem.

Answer: \pi


Question: Evaluate the integral \( \iint_{\Sigma} \langle x, y^3, -z\rangle \cdot d\vec{S} \) using the Divergence Theorem, where \( \Sigma \) is the sphere \( x^2 + y^2 + z^2 = 1 \) oriented outward.

Answer: \dfrac{4\pi}{5}


Question: Determine the number of elements in the quotient ring \( \mathbb{Z}[\sqrt{-19}]/I \), where \( I=(18+\sqrt{-19}, 7) \).

Answer: 7


Question: Find the least integral value of \(a\) for which all the roots of the equation \(x^4 - 4x^3 - 8x^2 + a = 0\) are real.

Answer: 0


Question: Let $F:=\{U_j\subseteq \mathbb{R}^d\}_j$ be a countable disjoint collection of bounded open connected sets such that the complement $U_j^c$ is connected for each $j$. Assume that $X:=\overline{\bigcup_{j}U_j}$ is connected. Is the set $X\setminus \left(\bigcup_{j}U_j\right)$ connected?

Answer: Yes


Question: Determine the values of \( \alpha \in \mathbb{R} \) for which the matrix \( B = \begin{pmatrix} A & -A \\ -A & \alpha A \end{pmatrix} \) is positive definite, given that \( A \in \mathbb{R}^{n \times n} \) is symmetric positive definite.

Answer: \alpha > 1


Question: Find a number \( x \) such that for all integers \( a \) in the set \( \{0, 1, 2, \ldots, 1926\} \), the congruence \( a^x \equiv a \pmod{1926} \) holds. Exclude \( x = 1 \) from consideration.

Answer: 319


Question: Find the sum of the series \( \sum_{k=0}^{n/2} C(n-k, k) \), where \( C(n, k) \) denotes the number of combinations of \( n \) items taken \( k \) at a time.

Answer: F_{n+1}


Question: Given a \(1 \times n\) rectangle divided into \(n\) unit squares, each square is colored either Red, Green, or Blue. Let \(f(n)\) denote the number of colorings where the Red color appears an even number of times. Calculate the value of \(\frac{f(9)}{f(3)}\).

Answer: 703


Question: Consider a directed graph G with n vertices labeled 0, 1, 2, ..., n-1. Each vertex is connected to the next vertex, and vertex 0 has a directed edge to itself and to vertex 1, while vertex n-1 is only connected to vertex 0. Starting at vertex 0, you take random steps with uniform probability until you cover all vertices, stopping when you reach vertex n-1. What is the big-O notation for the average length of a path that starts and ends at vertex 0?

Answer: O(n)


Question: A tower is constructed with a square base and square horizontal cross-sections. Viewed from any direction perpendicular to a side, the tower has base $y = 0$ and profile lines $y = (x - 1)^2$ and $y = (x+1)^2$. Find the volume of the solid.

Answer: \dfrac{2}{3}


Question:
Let \( a \in \mathbb{C} \) be a complex number such that \( a^n = 1 \) and \( a^m \neq 1 \) for \( m = 1, 2, \ldots, n-1 \). Find the value of the following sum:
\[ \sum_{k=1}^{n-1} \ln|1-a^k| \]

Answer: \ln n


Question: Let $A$ be an $n \times n$ matrix of rank $r$. Determine the dimension of the set $U = \{X \in M_{nn}: AX=0\}$. Express your answer in terms of $n$ and $r$.

Answer: n(n - r)


Question: Calculate the limit of the sequence \( \frac{3+(-1)^n+9\sqrt{n}-7n^5-2[\sqrt[3]{n}]n}{(3n-1)(n-2)(2n-3)(n-4)(4n-5)+2^{-n}} \) as \( n \to \infty \).

Answer: -\dfrac{7}{24}


Question: Evaluate the limit of the sequence: $$ \lim_{n\to\infty}\frac{1}{n}\left[\log\left(\frac{n+1}{n}\right)+\log\left(\frac{n+2}{n}\right)+\dots+\log\left(\frac{n+n}{n}\right)\right] $$

Answer: 2 \ln 2 - 1


Question: Evaluate the contour integral \( \int_{|z|=4} \tan z \, dz \).

Answer: -4\pi i


Question: Find the constant \( c \) that minimizes the integral \( \int_0^1 |e^x - c| \, dx \).

Answer: \sqrt{e}


Question:
Determine whether the following optimization problem has a solution for some fixed integer \( n \ge 1 \) and \( a > 0 \):
\[
\min_{(x_1, x_2, \ldots, x_n)} \sum_{i=1}^n \sum_{k=1}^n e^{-\frac{(x_i-x_k)^2}{2}}
\]
subject to the constraint \(|x_i| \le a\) for all \(i \in \{1, 2, \ldots, n\}\).

Answer: Yes


Question: Given a function \( f: \mathbb{R} \to [-1,1] \) that is differentiable up to the second order and satisfies the condition \((f(0))^2 + (f'(0))^2 = 4\), determine if there must exist a point \( c \in \mathbb{R} \) such that \( f(c) + f''(c) = 0 \). Is this statement true or false?

Answer: True


Question:
Given that the series \( \sum_{n=1}^\infty c_n \) converges, where each \( c_n \) may be positive or negative, determine which of the following statements must always be true:

I) \( \sum_{n=1}^\infty (-1)^n c_n \) converges.

II) \( \sum_{n=1}^\infty \frac{c_n}{2^n} \) converges.

III) \( \sum_{n=1}^\infty 2^n c_n \) converges.

Select the correct statement(s).

Answer: II


Question: Consider a sequence of positive increasing harmonic functions $v_{R}$ defined on $B_{R}\setminus \overline{B}_{1}$, with the condition $v_{R}=0$ on $\partial B_{1}$. Given that for $R>r>1$, the inequality $$\min_{\partial B_{r}}v_{R}\leq C(r)$$ holds, where $C(r)$ is a function depending only on $r$, determine if there exists a subsequence $R_{k}$ such that $v_{R_{k}}$ converges to a positive harmonic function $v$ in $\mathbb{R}^{n}\setminus \overline{B}_{1}$. Provide a justification for your answer.

Answer: Yes


Question: Given the functional equation \(2f(x) + f(-x) = \frac{1}{x}\sin\left(x-\frac{1}{x}\right)\), find the value of the integral \(\int_{\frac{1}{e}}^e f(x)\,dx\).

Answer: 0


Question: Let \( f(z) = \bar{z} \). Using the definition \( D_{\alpha}f(z) = \lim_{r\to 0}\frac{f(z+re^{i\alpha})-f(z)}{re^{i\alpha}} \), compute \( D(z) = \frac{\max_{\alpha}|D_{\alpha}f(z)|}{\min_{\alpha}|D_{\alpha}f(z)|} \). Find the value of \( D(z) \).

Answer: 1


Question: Given vectors \(a\) and \(b\) such that \(a \times b = 2i + j - k\) and \(a + b = i - j + k\), find the least value of the magnitude of vector \(a\). Here, \(i\), \(j\), and \(k\) are unit vectors in the directions of the \(x\), \(y\), and \(z\) axes, respectively, and \(\times\) denotes the cross product.

Answer: \sqrt{2}


Question: Let \( \varepsilon > 0 \) and define the set \( A = \bigcup_{j=1}^\infty (x_j - \varepsilon, x_j + \varepsilon) \) where \( x_j \in \mathbb{R} \). Suppose that \( A \cap [0,1] \) is dense in \([0,1]\). Prove that the Lebesgue measure of \( A \cap [0,1] \) is 1.

Answer: 1


Question:
Let $A$ be a ring with $1$, possibly commutative. Consider the commutative monoid $\mathrm{Proj}(A)$ of isomorphism classes $\langle M \rangle$ of finitely generated projective modules, with the operation $\langle M \rangle + \langle N \rangle = \langle M \times N \rangle$. Define $K_0(A)$ in two ways:

1. As the quotient of the free abelian group on $\mathrm{Proj}(A)$ by the relations $\langle M_2 \rangle = \langle M_1 \rangle + \langle M_3 \rangle$ for short exact sequences $0 \to M_1 \to M_2 \to M_3 \to 0$.

2. As the Grothendieck group of the monoid $\mathrm{Proj}(A)$, the quotient of $\mathrm{Proj}(A) \times \mathrm{Proj}(A)$ by the equivalence relation $$(\langle M_1 \rangle, \langle N_1 \rangle) \sim (\langle M_2 \rangle, \langle N_2 \rangle) \iff \exists \ \langle P \rangle \in \mathrm{Proj}(A), \langle M_1 \rangle + \langle N_2 \rangle + \langle P \rangle = \langle M_2 \rangle + \langle N_1 \rangle + \langle P \rangle.$$

Are these two constructions of $K_0(A)$ isomorphic?

Answer: Yes


Question: Let \(p\) be a Pythagorean prime. Does there necessarily exist an integer \(n\) such that \(n^2 \equiv (p-1) \mod p\)? Provide a justification for your answer.

Answer: Yes


Question: Provide an example of an $F$-algebra $A$ with an antisymmetric product $\cdot$ such that for all $v,w \in A$, $v \cdot w = -w \cdot v$, but there exist $u,v,w \in A$ for which the Jacobi identity $u \cdot (v \cdot w) + v \cdot (w \cdot u) + w \cdot (u \cdot v) \neq 0$. This algebra should not be a Lie algebra.

Answer: A


Question:
Let $X$ be a compact metric space, and $T$ a uniquely ergodic measure preserving transformation on $X$, with associated invariant ergodic probability measure $\mu$. Assume $\mu$ is non-atomic and supp $\mu = X$. Given a positive real-valued continuous function $f$ on $X$, define the error function $E_n: X \times \mathbb{R}^+ \to \mathbb{R}$ by
\[ E_n(x, r) := \frac{1}{\mu(B_r (x))} \int_{B_r (x)} |A_n f - Cf| \, d\mu, \]
where $A_n f := \frac{1}{n}\sum_{k=0}^{n-1} T^k f$ and $Cf := \int_X f \, d\mu$. Define also for each $\delta > 0$, the set $S_\delta := \{ (x, r) \mid \ (x, r) \in X \times \mathbb{R}^+, \ \mu (B_r (x)) \geq \delta \}$. Prove or disprove that for all $\delta > 0$, we have
\[ \limsup_{n \to \infty} \sup_{(x_1, r_1), (x_2, r_2) \in S_\delta} \frac{E_n (x_1, r_1) - E_n (x_2, r_2) }{E_n (x_1, r_1) + E_n (x_2, r_2)}  = 0. \]
Note: By convention, set $\frac{0}{0} = 0$.

Answer: 0


Question: Find the number of ordered triples \((a, b, c)\) such that the least common multiple (LCM) of \((a, b)\) is 432, the LCM of \((b, c)\) is 72, and the LCM of \((c, a)\) is 324.

Answer: 0


Question: If $V \neq L$, does it necessarily follow that the set of natural numbers $\omega = \{0, 1, 2, 3, \ldots\}$ has nonconstructible subsets?

Answer: No


Question: Does there exist a real-valued function that is continuous everywhere on the real line and differentiable at exactly one point?

Answer: Yes


Question: Given the equation \(x^4 + 3\cos(ax^2 + bx +c) = 2(x^2-2)\) with parameters \(a, b, c \in (2,5)\), it has two solutions. Determine the maximum value of \(\frac{ac}{b^2}\).

Answer: 1


Question: Find the maximum value of \(x^{\ln y}\) given that \(x > 1\), \(y > 1\), and \((\ln x)^2 + (\ln y)^2 = \ln x^2 + \ln y^2\).

Answer: e^4


Question: Determine if the expression \( \sum_{k=1}^n{{n}\choose{k}}\left (1+\frac{c}{2n}\right)^{n-k}o\left(\frac{1}{n}\right)^k \) is equal to \( o(1) \), where \( c \) and \( n \) are finite, positive real numbers, and \( o \) denotes the little-oh notation.

Answer: o(1)


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

Answer: e


Question: Given the cubic equation $x^3 + px - q = 0$, where $p, q \in \mathbb{R}$, find the value of the expression $$(\alpha + \beta)(\beta + \gamma)(\alpha + \gamma)\left(\frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2}\right),$$ where $\alpha, \beta, \gamma$ are the roots of the equation.

Answer: -\dfrac{p^2}{q}


Question: In 3-dimensional projective space, a degenerate conic $C$ is defined as $C = lm^T + ml^T$, where $l$ and $m$ are different lines. Determine the rank of the conic $C$.

Answer: 2


Question: Let $A$ and $B$ be $C^*$-algebras that admit faithful representations on separable Hilbert spaces. Does the $C^*$ tensor product $A \otimes_\beta B$ also admit a faithful separable representation?

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{1}\int_{0}^{\sqrt{1-x^2}} \sqrt{1-x^2-y^2} \, dy \, dx \) by changing to polar coordinates.

Answer: \dfrac{\pi}{6}


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

Answer: [-1, 1)


Question: Evaluate the limit: \[ \lim _{x\to 0}\left(\frac{xe^x-2+2\cos x-x}{\left(\sin x\right)^2 \tan\left(2x\right)+xe^{-\frac{1}{\left|x\right|}}}\right) \] without using l'Hospital's rule.

Answer: \dfrac{1}{4}


Question: Calculate the triple integral \( I = \int_{R} (x-y) \,dV \) over the region \( R = \{(x,y,z) \in \mathbb{R^3} \mid 1 \leq x^2 + y^2 \leq 16, 0 \leq z \leq y + 4\} \). Express your answer to three decimal places.

Answer: -200.277


Question: Determine the number of solutions for the equation \(x^2 = a + ib\), where \(a\) and \(b\) are real numbers.

Answer: 2


Question: Evaluate the double integral \( \int\int_D f(x, y) \, dA \) where \( f(x, y) = \begin{cases} 5e^{x^2} & \text{if } y \leq x \\ 5e^{y^2} & \text{if } y > x \end{cases} \) and \( D = [0, 9] \times [0, 9] \).

Answer: 5(e^{81} - 1)


Question: Determine the number of 3-Sylow subgroups in a group $G$ of order 126. Let $|G| = 126 = 2 \cdot 3^2 \cdot 7$. Use the Sylow theorems to find the possible values for the number of 3-Sylow subgroups, $s_3$. Additionally, consider a surjective group homomorphism $\varphi: G \rightarrow H$ where $H$ is a subgroup of order 14. What is the value of $s_3$?

Answer: 1


Question: Given a quadratic polynomial \(ax^2 + bx + c\) with positive coefficients \(a, b, c\) in arithmetic progression, and integer roots \(\alpha\) and \(\beta\), find the value of \(\alpha + \beta + \alpha \beta\).

Answer: 7


Question: Evaluate the integral \( I = \int_{0}^{\pi} \sin(x)\cos(x)\cos(2x)\cos(4x)\cdots \cos(2^{n-1}x) \, \mathrm{d}x \).

Answer: 0


Question: Let \( f \) be a continuous integrable function on \( \mathbb{R} \) such that either \( f(x) > 0 \) or \( f(x) + f(x + 1) > 0 \) for all \( x \in \mathbb{R} \). Determine whether \( \int_{-\infty}^\infty f(x)\,dx > 0 \) is true or false.

Answer: True


Question:
Find the sum of the series \( \sum_{n=1}^{\infty} \frac{2^n n^2}{n!} \). Given that \( x\cdot e^x=\sum_{n\geq 1} \frac{x^n}{(n-1)!} \), determine the value to which the series converges. Choose from the following options:

a. \( e^2 \)
b. \( 2e^2 \)
c. \( 4e^2 \)
d. \( 6e^2 \)

Answer: D


Question:
Determine the power of 2 that divides \(\lceil(3+\sqrt{5})^{2n}\rceil\). Choose the correct option:

A. \(2^{n+1}\)
B. \(2^n\)
C. \(2^{n-1}\)
D. Not divisible by 2

Answer: A


Question: Calculate the following limit: \[ \lim \limits_{x \to 0^+} \frac{\int_{0}^{x} (e^{t^2}-1)dt}{{\int_{0}^{x^2} \sin(t)dt}} \]

Answer: \infty


Question: Determine the value of \( \ell(n) \), the minimal number of edges in a complete linear hypergraph on \( n \) vertices, where \( n > 2 \). A complete linear hypergraph \( H = (V, E) \) satisfies the following conditions: \(|e| \geq 2\) for all \( e \in E \), \(|e_1 \cap e_2| = 1\) for all distinct \( e_1, e_2 \in E \), and for all \( v \in V \), \(|\{e \in E : v \in e\}| \geq 2\).

Answer: n


Question: Find the values of $a$ and $b$ such that the set $G = \{ z \in \mathbb{C} \ | \ az^{n}=b \}$, where $a \in \mathbb{C}^{*}$ and $b \in \mathbb{C}$, forms a subgroup of $(\mathbb{C}^{*}, \cdot)$. The answer should be $a = b$.

Answer: a = b


Question: Determine the best known estimation for \( \limsup \left| \frac{p_{n+1}}{n+1} - \frac{p_n}{n} \right| \), where \( p_k \) is the \( k \)-th prime number.

Answer: 0


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

Answer: 0


Question: If a sequence of measurable functions \(f_n\) converges to \(f\) almost everywhere on a set \(E\), does it imply that \(|f_n|\) converges to \(|f|\) almost everywhere on \(E\)?

Answer: Yes


Question: Define a sequence of sets \( \{S_i\}_{i=0}^{\infty} \) such that \( S_0 = \{0\} \) and \( S_i = \bigcup\limits_{k=0}^{i-1} \{S_k\} \). Find the number of brackets in the power set of \( S_4 \) if the empty set is written as \( \{\} \).

Answer: 272


Question: Let $k$ be a field of characteristic 0, and let $\bar{k}$ be an algebraic closure of $k$. Consider a $k$-algebraic group $G$. A twist of $G$ is defined as a $k$-algebraic group $H$ such that $G \times_k \bar{k} \cong H \times_k \bar{k}$. Determine whether there exists a finite field extension $K/k$ such that $G \times_k K \cong H \times_k K$. Provide a justification for your answer.

Answer: Yes


Question: Determine the set of all possible values of the series \( \sum_{n\geq 1}\frac{1}{x_n} \) where \( (x_n)_{n\geq 1} \) is an increasing sequence of positive natural numbers and the series converges.

Answer: (0, \infty)


Question: Find the slope of the curve at the point where the surface given by \( z = x^3 + xy^2 \) intersects the plane \( 2x - 2y = 1 \) at \( x = 1 \) and \( y = \frac{1}{2} \).

Answer: \dfrac{17}{4}


Question:
Given the following logical statements:

- If Mr. M is guilty, then no witness is lying unless he is afraid.
- There is a witness who is afraid.

Which of the following statements is true?

1. Mr. M is guilty.
2. Mr. M is not guilty.
3. From these facts, one cannot conclude that Mr. M is guilty.
4. There is a witness who is lying.
5. No witness is lying.

Use the predicates:
- $G$: Mr. M is guilty
- $W(x)$: $x$ is a witness
- $L(x)$: $x$ is lying
- $A(x)$: $x$ is afraid

Choose the correct statement based on the given information.

Answer: 3


Question: If $f:\mathbb{R}^n\to\mathbb{R}$ is a continuous function, does Lebesgue's differentiation theorem hold at all points? Specifically, does the following limit hold everywhere: $$\lim_{r\to0}\frac{1}{|B(x,r)|}\int_{B(x,r)}f(y) \, dy=f(x)$$

Answer: Yes


Question:
Does a mean value formula exist for separately harmonic functions over the product of spheres in \( \mathbb{R}^4 \)? Specifically, consider the Laplacian \( \Delta = \frac{\partial^2}{\partial\alpha^2} + \frac{\partial^2}{\partial\beta^2} + \frac{\partial^2}{\partial\gamma^2} + \frac{\partial^2}{\partial\delta^2} \) and a function \( f: \mathbb{R}^4 \times \mathbb{R}^4 \rightarrow \mathbb{R} \) that is separately harmonic, meaning \( \Delta_1 f = \Delta_2 f = 0 \). Consider two 3-dimensional spheres \( \partial B_{r_1}(x_1) \) and \( \partial B_{r_2}(x_2) \), where \( x_i = (\alpha_i, \beta_i, \gamma_i, \delta_i) \). Is there a mean value formula of the form:
\[
f(x_1,x_2) = \frac{1}{\text{Area}(\partial B_{r_1}(x_1) \times \partial B_{r_2}(x_2))} \int_{\partial B_{r_1}(x_1) \times \partial B_{r_2}(x_2)} f(y_1, y_2) \, d\sigma(y_1) \times d\sigma(y_2)?
\]

Answer: Yes


Question: Find the limit: \[ \lim_{ x \to a }\left(\frac{f(x)}{g(x)}-\frac{h(x)}{k(x)}\right) \] where \( f(x), g(x), h(x), k(x) \) are polynomials, and it is given that \( \lim_{ x \to 1 }\left(\frac{1}{x-1}-\frac{3x-1}{x^2-1}\right) = -1 \).

Answer: -1


Question: Evaluate the integral \( \int_{|z| = \pi} \tan(nz) \, dz \) using the residue theorem, where \( n \in \mathbb{N} \).

Answer: -4\pi i


Question: A point moves on a circle with a radius of 2 meters at an angular velocity of \(\frac{1}{2} \text{ rad}\cdot s^{-1}\), starting at an angle of \(\frac{\pi}{4}\). Determine the time \(t\) within the interval \([0, \frac{\pi}{2}]\) when the cosine of the angle between the tangent at \((x(t), y(t))\) and the line \(y = \left( \frac{-1}{\sqrt{3}} \right) x + 7\) is maximized. The coordinates of the point are given by \(x(t) = 2\cos\left(\frac{1}{2}t + \frac{\pi}{4}\right)\) and \(y(t) = 2\sin\left(\frac{1}{2}t + \frac{\pi}{4}\right)\).

Answer: \dfrac{\pi}{6}


Question: Given a random variable $X$ with probability density function (pdf) \( f_X(x) = \begin{cases} \frac{3}{x^4} & \text{if } x > 1 \\ 0 & \text{if } x \leq 1 \end{cases} \), find a function of a uniform random variable \( U \sim \text{Unif}(0,1) \) that has the same distribution as \( X \).

Answer: (1 - U)^{-1/3}


Question: Given the change of variables \(x = u + v\) and \(y = uv^2\), which transforms \(f(x,y)\) into \(g(u, v)\), compute the value of \(\frac{\partial^2 g}{\partial v \partial u}\) at the point \((u, v) = (1, 1)\). Assume that at this point, \(\frac{\partial f}{\partial y} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x} = 1\).

Answer: 8


Question: Evaluate the limit \( \lim_{n \to \infty} n a_n \), where \( a_n = L(L(\cdots L(\frac{17}{n})\cdots)) \) with \( n \) applications of the function \( L(x) = x - \frac{x^2}{2} \).

Answer: \dfrac{34}{19}


Question: Find all elements of order 7 in the semi-direct product group $H \rtimes_\alpha G$, where $H = \langle h \rangle$ with $o(h) = 7$, $G = \langle g \rangle$ with $o(g) = 3$, and the homomorphism $\alpha: G \rightarrow \text{Aut}(H)$ is defined by $\alpha(g)(h) = h^2$. Determine the elements $(h^i, g^j)$ such that their order is 7.

Answer: (h^i, e)


Question: Consider a well-ordered set $A$ with a strict well-order $<_A$. Let $B$ be the set of all bijections from $A$ to $A$. Define a relation $<_B$ on $B$ such that $b <_B a$ if and only if there exists $c \in A$ such that $b(c) <_A a(c)$ and for all $d <_A c$, $b(d) = a(d)$. Is the relation $<_B$ a strict well-order on $B$?

Answer: Yes


Question: Determine the order of the quotient group \( \frac{\mathbb{R}^*}{G^*} \), where \( \mathbb{R}^* \) is the group of all non-zero real numbers under multiplication, and \( G^* \) is the subgroup of \( \mathbb{R}^* \) consisting of all squares of real numbers.

Answer: 2


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

Answer: \dfrac{1}{e}


Question: Given that $x^{x^4} = 4$, find the value of $x^{x^2} + x^{x^8}$.

Answer: 258


Question: What is the value of \( \ln \left(e^{2 \pi i}\right) \)?

Answer: 0


Question: Determine the infimum of the set \( \left\{ \frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} : a,b,c \in \mathbb{R}^+ \right\} \).

Answer: \dfrac{3}{2}


Question: Determine whether the function \( f : \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R} \times \mathbb{R} \), defined by \( f((x_1,x_2),(y_1,y_2)) = (x_1,y_1) \), is continuous. Consider \( \mathbb{R} \) with the standard topology of open intervals, and equip \( \mathbb{R}^2 = \mathbb{R} \times \mathbb{R} \) and \( \mathbb{R}^2 \times \mathbb{R}^2 \) with the product topologies.

Answer: Yes


Question: Consider the linear equation \(Ax = b\), where \(A\) is a matrix and \(b\) and \(x\) are vectors. Suppose there exists a vector \(x_S\) that solves this equation (i.e., \(Ax_S = b\)) and \(x_S\) has \(k\) entries of value 0, making it a sparse solution. Let \(x_m\) be the solution with the smallest \(\ell_1\) norm among all solutions. Does \(x_m\) necessarily have at least \(k\) entries of value 0?

Answer: No


Question: Determine the number of points where the graph of the function \( y = f(x) \), defined by the condition \(|f(x)-f(y)| \leq \frac{1}{2}|x-y|\) for all real numbers \(x\) and \(y\), intersects the line \( y = x \).

Answer: 1


Question: Prove that \( \int_{a}^{b} \left\lfloor x \right\rfloor \, dx + \int_{a}^{b} \left\lfloor -x \right\rfloor \, dx = a-b \).

Answer: a - b


Question: Calculate the flux of the vector field \( \mathbf{F}(x,y,z) = \mathbf{i} + \mathbf{j} + \mathbf{k} \) across the surface \( \sigma \), which is the portion of the cone \( z = \sqrt{x^2 + y^2} \) between the planes \( z = 3 \) and \( z = 6 \), oriented by downward unit normals.

Answer: -27\pi


Question: Evaluate the line integral \( \int_C \frac{1}{z-a} \, dz \), where \( C \) is the circle defined by \( |z-a| = r \), without using parametrization.

Answer: 2\pi i


Question: Evaluate the integral \( \int_{0}^{\sqrt{3\pi}}\int_{y}^{\sqrt{3\pi}}\cos 7x^2\;dxdy \) by reversing the order of integration.

Answer: 0


Question: Classify the singularity of the function \( f(z) = \frac{\sin z}{\cos(z^3) - 1} \) at \( z = 0 \) and calculate the residue at this point.

Answer: -\dfrac{1}{60}


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\sum^{n}_{r=1}\frac{1}{f(r)g(r)} \] where the functions \( f \) and \( g \) are defined as follows: \( f(n) \) is the smallest prime greater than \( n+1 \), and \( g(n) \) is the greatest prime less than or equal to \( n+1 \).

Answer: \dfrac{1}{2}


Question: Evaluate the contour integral \( \int_{|z-2| = 3} e^{1/z} \, dz \). Can Cauchy's theorem be applied to simplify this integral?

Answer: 2\pi i


Question: Determine the value of \(a \in \mathbb{R}\) such that the sequence \(a_n = 1^9 + 2^9 + \cdots + n^9 - an^{10}\) is convergent.

Answer: \dfrac{1}{10}


Question:
Calculate the Lévy distance between the distribution functions of two Bernoulli random variables $X$ and $Y$ with parameters $p$ and $q$, respectively. The Lévy distance is defined as:
\[ d_L(F_X, F_Y) := \inf \{ \varepsilon > 0 \mid F_Y(x - \varepsilon) - \varepsilon < F_X(x) < F_Y(x+  \varepsilon) +  \varepsilon \} \]
Determine $d_L(F_X, F_Y)$ when $p \neq q$. Assume $d_L(F_X, F_Y) = 0$ when $p = q$. Provide your answer as a single numerical value.

Answer: |p - q|


Question: Determine whether the linear functional \( \ell^2 \ni (x_n) \rightarrow 2x_{1} + 28x_2 + 35x_{3} \) is bounded, and find its norm.

Answer: \sqrt{2013}


Question: Given a sequence $(a_k)$ such that $0<\liminf |a_k|\le\limsup |a_k|<\infty$, determine the radius of convergence of the power series $\sum a_k X^k$. Provide your answer as a single number.

Answer: 1


Question: Evaluate the triple integral \( \iiint_D |z| \, dx \, dy \, dz \) over the domain \( D = \{(x,y,z) \in \mathbb{R}^3 : x^2+y^2-16 \leq z \leq 4 - \sqrt{x^2+y^2} \} \).

Answer: 704\pi


Question: Find the polynomial \( p(y) \) such that \( \int_{-1}^{1} p(y) \, dy = 0 \) and \( \int_{-1}^{1} y p(y) \, dy = 1 \).

Answer: \dfrac{3}{2}y


Question: Find \( \limsup\left(\frac{\sin n}{n}+\cos\frac{\pi n}{3}\right) \).

Answer: 1


Question: Find the operator norm \( \|A\|_o \) of the matrix \( A = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix} \). The operator norm is defined as \( \|A\|_o = \sup \|Ax\|_2 \), where the supremum is taken over all \( x \in \mathbb{R}^3 \) with \( \|x\|_2 = 1 \). Here, \( ||.||_2 \) denotes the Euclidean norm in \( \mathbb{R}^3 \).

Answer: 2


Question: Suppose that the number $x$ satisfies the equation $x + x^{-1} = 3$. Compute the value of $x^7 + x^{-7}$. \( \boxed{} \)

Answer: 843


Question: Given the matrix \( A = \begin{bmatrix}0&0&1&0\\0&0&0&1\\-37&3&-4&0\\3&-37&0&-4\end{bmatrix} \), find the eigenvalue associated with the eigenvector \( \begin{bmatrix}-4i\\4i\\24+8i\\-24-8i\end{bmatrix} \).

Answer: -2 + 6i


Question:
Prove the identity for the sum of powers:
\[ S_n^p := 1^p + 2^p + \ldots + n^p \]
Show that:
\[ (p+1)S_n^p + \binom{p+1}{2}S_n^{p-1} + \binom{p+1}{3}S_n^{p-2} + \ldots + S_n^0 = (n+1)^{p+1} - 1 \]

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


Question: Find the smallest integer $k$ such that $2^{k!} \geq 2^{n-k}$ for a given fixed integer $n$.

Answer: k


Question: Find the sum of the series \( \sum_{n=2}^{\infty} \frac{\binom{n}{2}}{4^n} \) using generating functions.

Answer: \dfrac{4}{27}


Question: What is the maximum distance between any two positions of a Rubik's cube, denoted as \( \max_{A,B \in P} d(A,B) \), where \( d(A,B) \) is the minimal number of moves required to reach position \( B \) from position \( A \)?

Answer: 20


Question: Let $A=\mathbb{C}$ and $E=F=C[0,1]$. Are $E=F$ Hilbert $\mathbb{C}^*$-modules?

Answer: No


Question: Does a continuous function \( f: \mathbb{R} \to \mathbb{R} \) satisfying \( f(f(x)) = x \) for all \( x \in \mathbb{R} \) necessarily have a fixed point?

Answer: Yes


Question: Evaluate the limit: \[ \lim _{x\to 0+}\left(\frac{\left(e^{-\frac{1}{x^2}}\cos \left(\log _e\left(x\right)\right)+\cos \left(\arctan \left(x\right)\right)-e^{-\frac{x^2}{2}}\right)}{\log _e\left(1+x^2\right)-\sin \left(x^2\right)}\right) \]

Answer: -\dfrac{1}{2}


Question: Calculate the area enclosed by the polar curve \( r = 2 + 2\sin(\theta) \) but outside the curve \( r = 4\sin(\theta) \).

Answer: 2\pi


Question: Let \( f_n \in C(a,b) \) and suppose \( f_n \rightrightarrows f \) on \([a+\delta,b-\delta]\) for all \( \delta > 0 \). Assume the improper integrals \( \int_a^b f_n \) and \( \int_a^b f \) exist, and \( \lim_{n\to\infty} \int_a^b f_n = \int_a^b f \). Prove that \( \lim_{n\to\infty} \int_a^b |f_n-f| = 0 \).

Answer: 0


Question: Let \(f\) be a function defined on the interval \((a,b)\) such that \(f' + f^2 + 1 \ge 0\). Given that \(\lim_{x\to a}f(x)=\infty\) and \(\lim_{x\to b}f(x)=-\infty\), determine the possible interval for \(b-a\).

Answer: [\pi, \infty)


Question: Evaluate the sum \( \sum_{k=2}^{n}{(k-1)(n-k)!{n \choose k}} \) using properties of binomial coefficients.

Answer: n! - 1


Question: Let $T : V \to V$ be a linear map and $H \le V$ a subspace such that $T_H = \operatorname{id}_H$, meaning $T(v) = v$ for all $v \in H$. Additionally, $T$ induces the identity transformation on $V / H$, i.e., $\overline{T}(v + H) = v + H$ or equivalently $Tv - v \in H$ for all $v \in V$. Determine if $\det T = 1$.

Answer: 1


Question: Is the function \( f(x) = x \) always continuous when \( f \) maps from \( \mathbb{R} \) with an arbitrary metric \( \rho \) to \( \mathbb{R} \) with the Euclidean metric?

Answer: No


Question: Determine the number of natural number solutions to the equation \(x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 24\) subject to the condition \(x_1 + x_2 + x_3 > x_4 + x_5 + x_6\).

Answer: 15312


Question: Determine the dimension of the set of all solutions to the differential equation \( y'' + ay' + by = 0 \) with the boundary conditions \( y(0) = y(1) \), where \( a \) and \( b \) are positive real numbers.

Answer: 1


Question: Given that the area under the curve of the function \( f(x) \) from \( x = 1 \) to \( x = 5 \) is 8 square units, find the area under the curve of the inverse function \( f^{-1}(x) \) from \( x = 1 \) to \( x = 5 \). Assume \( f(x): \mathbb{R}^+ \to \mathbb{R}^+ \) is an invertible function with \( f'(x) > 0 \) and \( f''(x) > 0 \) for all \( x \in [1,5] \), and that \( f(1) = 1 \) and \( f(5) = 5 \).

Answer: 16


Question: The series \(\sum a_n x^n\) and \(\sum b_n x^n\) have radii of convergence of 1 and 2, respectively. Determine the radius of convergence \(R\) of the series \(\sum a_n b_n x^n\).

Answer: 2


Question: Calculate the limit: \[ \lim_{x \to 0} \frac{e^{\arctan(x)} - e^{\arcsin(x)}}{1 - \cos^3(x)} \] without using L'Hôpital's rule.

Answer: 0


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty}\; \frac{1}{n}\;\sum_{k=1}^{\Big\lfloor\frac{n}{2}\Big\rfloor} \cos\Big(\frac{k\pi}{n}\Big) \]

Answer: \dfrac{1}{\pi}


Question: Let \((a_{n})\) and \((b_{n})\) be sequences of real numbers such that \((2+i)^n = a_{n} + b_{n}i\) for all integers \(n \geq 0\), where \(i = \sqrt{-1}\). Calculate the value of \(\sum_{n=0}^{\infty} \frac{a_{n}b_{n}}{7^n}\).

Answer: \dfrac{7}{16}


Question: Identify the geometric object obtained by gluing together all pairs of antipodal points of an n-sphere.

Answer: \mathbb{RP}^n


Question:
Determine whether the following statement is true or false:

Let \( f, g: [a, b] \to \mathbb{R} \) be continuous and either both monotone increasing or both monotone decreasing. If Chebyshev's integral inequality holds as equality, i.e.,
\[
\left(\int_a^b f(x) \, \mathrm{d}x\right)\left(\int_a^b g(x) \, \mathrm{d}x\right) = (b-a)\int_a^b f(x) g(x) \, \mathrm{d}x,
\]
then either \( f \) is constant or \( g \) is constant.

Answer: True


Question: Let $X$ and $Y$ be random variables uniformly distributed over the triangle $A = \{(x,y) \in \mathbb{R}^2 : 0 \leq y \leq x \leq 1\}$. The joint density is given by $d\mu_{(X,Y)}(x,y) = 2\mathbf{1}_A(x,y)dxdy$. Determine the probability distribution of the random variable $Z = Y/X$. Provide your answer in terms of a probability density function or cumulative distribution function.

Answer: 1


Question: Find the greatest common divisor (GCD) of the complex numbers \(4 + 22i\) and \(17 + i\) in the Gaussian integers \(\mathbb{Z}[i]\).

Answer: 1 + 3i


Question:
Determine whether the following statement is true for any languages \( L_1 \) and \( L_2 \):
\[ L_{1}^{*} \cap L_{2}^{*} = (L_{1}^{*} \cap L_{2}^{*})^{*} \]

Answer: True


Question: Let $A$ be a positive-definite real matrix such that $x^T A x > 0$ for every nonzero real vector $x$. Does it follow that $\mathrm{det}(A) > 0$?

Answer: Yes


Question: Simplify the expression: \( \sqrt{14} - \sqrt{16 - 4 \sqrt{7}} \)

Answer: \sqrt{2}


Question: Determine whether the interval $[0,1)$ with the Sorgenfrey topology is Lindelöf.

Answer: Yes


Question: Let $G$ be a nilpotent group and $H$ a subgroup of $G$ such that $H[G,G] = G$. Prove that $H = G$.

Answer: H = G


Question: Compute the limit: \( \lim_{n\to\infty} \{ (\sqrt{2}+1)^{2n} \} \), where \( \{x\} \) denotes the fractional part of \( x \).

Answer: 1


Question: Calculate the residue of the function \( \frac{z\ln(z)}{(z^2 +1)^3} \) at \( z=0 \).

Answer: 0


Question: Find the area of the region bounded by the polar curve \( r^2 = a^2 \cos(2\theta) \).

Answer: a^2


Question: Given that $x, y, z$ are positive real numbers satisfying the equations \( \frac{x}{y} + \frac{y}{x} + \frac{x}{z} + \frac{z}{x} + \frac{y}{z} + \frac{z}{y} = 2018 \) and \( \sqrt{\frac{x}{y}} + \sqrt{\frac{y}{z}} + \sqrt{\frac{z}{x}} = 17 \), find the value of \( \sqrt{\frac{y}{x}} + \sqrt{\frac{z}{y}} + \sqrt{\frac{x}{z}}. \)

Answer: 43


Question: Suppose $T: L^2(\mathbb{R}) \to L^2(\mathbb{R})$ is a continuous, injective linear operator such that $T(\lambda f + g) = \lambda T(f) + T(g)$ for $f,g \in L^2(\mathbb{R})$ and $\lambda \in \mathbb{R}$, and $T(f) = T(g)$ implies $f = g$ almost everywhere. Given a sequence $h_n$ and a function $h$ such that $\|T(h_n) - T(h)\|_2 \to 0$, it is known that there exists a subsequence $T(h_{n_k})$ such that $T(h_{n_k}) \to T(h)$ almost everywhere. Does this imply that $h_{n_k} \to h$ almost everywhere?

Answer: No


Question: Evaluate the limit: \[ \lim_{j\to0}\lim_{k\to\infty}\frac{k^j}{j!\,e^k} \]

Answer: 0


Question: Find the coefficient of $x^{20}$ in the expansion of $(x^{1} + x^{2} + x^{3} + x^{4} + x^{5} + x^{6})^{10}$. Provide your answer as a single integer.

Answer: 85228


Question: Find a $3 \times 3$ matrix $A$ that is invariant under any rotation, i.e., satisfies $AR = RA$ for any rotation matrix $R$. What is the form of such a matrix?

Answer: \lambda I


Question: Consider the set of numbers formed by cyclic permutations of the digits \(a, b, c, d, e\), where each digit is a different non-zero digit. Define \(N(a,b,c,d,e)\) as the number of these permutations that are divisible by 41. Determine how many distinct values \(N(a,b,c,d,e)\) can take.

Answer: 2


Question: Calculate $\text{Hom}(\mathbb{Z}^n, \mathbb{Z})$, where $\mathbb{Z}^n = \bigoplus_{i=1}^n \mathbb{Z}_i$. Express your answer in terms of a known group or module.

Answer: \mathbb{Z}^n


Question: Consider a measurable space $(\Omega, \mathcal{F})$ with a filtration $\{\mathcal{F}_n\}_{n \in \mathbb{N}}$ such that $\mathcal{F}_n \uparrow \mathcal{F}$. Let $\mathcal{C}$ be a convex set of mutually absolutely continuous probabilities on $(\Omega, \mathcal{F})$ generated by finitely many extreme points $P_1,...,P_n$. Suppose $\{R_n \}_{n \in \mathbb{N}}$ is a sequence of probability measures on $(\Omega, \mathcal{F}_n)$, and for all $Q \in \mathcal{C}$, $R_n \ll Q|_{\mathcal{F}_n}$ for all $n$. Let $Y^Q_n = dR_n/dQ|_{\mathcal{F}_n}$ be the Radon-Nikodym derivative. Assume that for all $Q \in \mathcal{C}$, $\{Y_n^Q \}_{n \in \mathbb{N}}$ is a martingale with respect to $Q$. Given that $Y_n^Q \to Y^Q_\infty$ almost surely for any $Q \in \mathcal{C}$, does it follow that $\sup_Q |Y^Q_n - Y^Q_\infty| \to 0$ almost surely as $n \to \infty$? Assume the filtration is simple, such as each $\mathcal{F}_n$ being generated by a finite measurable partition, and that $\{ Y_n^Q\}$ is uniformly integrable for all $Q \in \mathcal{C}$. Determine if the uniform convergence holds.

Answer: Yes


Question: Determine the positive integer $n$ for which any continuous map from $S^n$ to $S^1 \times S^1$ is nulhomotopic.

Answer: 2


Question: Determine if there exists a universal constant that can upper bound the series \( \sum_{i=1}^\infty \frac{i^2(\ln K)^2}{K^{10i}} \) for sufficiently large values of \( K > 0 \).

Answer: Yes


Question: Evaluate the closed contour integral \( \int_{|z|=1} \frac{z}{2z^2+1} \, dz \) where the contour is the unit circle.

Answer: \pi i


Question: Determine the flow rate of water across a net submerged in a river. The velocity vector field of the river is given by \( \vec{v}=(x-y)\vec{i}+(z+y)\vec{j}+z^2\vec{k} \). The net is described by the surface \( y=\sqrt{1-x^2-z^2}, y\geq 0 \), and is oriented in the positive \( y \)-direction. Calculate the flow rate by evaluating the surface integral \( \iint_{S}\vec{v}\,d\vec{S} \).

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


Question: Evaluate the integral \( \int_{-\infty}^\infty \frac{1}{(x^2+1)^3} \, dx \).

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


Question: Determine the value of \(\alpha > 0\) for which the function \(f(z) = |z|^\alpha z\) belongs to the class \(C^m(\mathbb{C}, \mathbb{C})\), where \(m\) is a positive integer.

Answer: \alpha > m - 1


Question: Given a cubic Bezier curve, determine the parameter value \( t \) at a specific point \( B \) on the curve.

Answer: t


Question: Given an open set \( \mathcal{U} \) in \( \mathbb{R}^n \) and a point \( x \in \mathcal{U} \), suppose there exists \( r > 0 \) such that the closed ball \( \bar{B}(x,r) \) is contained in \( \mathcal{U} \). Is it true that there exists \( R > r \) such that the open ball \( B(x,R) \) is also contained in \( \mathcal{U} \)?

Answer: Yes


Question: Let \( \psi: V \times \cdots \times V \to \mathbb{R} \) be an antisymmetric \( k \)-tensor on \( V \), which is \( n \)-dimensional. Prove that if \( k > n \), then \( \psi \) is the trivial tensor that sends every element to 0.

Answer: 0


Question: Find the number of integer solutions to the equation \(x_1 + x_2 + x_3 + x_4 = 30\) subject to the constraints: \(1 \leq x_1 \leq 5\), \(4 \leq x_2 \leq 9\), \(6 \leq x_3 \leq 11\), and \(10 \leq x_4 \leq 15\).

Answer: 125


Question: If $(M,\cdot)^{\times} = \{ 1 \}$, can the monoid ring $k[M]$ contain an invertible element $z \in GL(k[M])$ such that the coefficient of $z$ at $1$ is zero?

Answer: No


Question: Find the largest possible area of a rectangle that can be inscribed in the ellipse given by \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \).

Answer: 12


Question: Determine whether the following statement is true or false: Let \( f \) be a non-negative continuous function on \( \mathbb{R} \) such that \( \int_{0}^{\infty} f(x)\,dx \) is finite. Then \( \lim_{x\to \infty} f(x) = 0 \).

Answer: B


Question: Is the weak $\sigma$-$(V,V')$ topology on a normed space $V$ Hausdorff? Justify your answer.

Answer: Yes


Question:
Let $R$ be an integral domain such that every non-zero prime ideal is maximal in $R[x]$. Choose the correct statement:

(a) $R$ is a field.
(b) $R$ contains $\mathbb{Z}$ as a subring.
(c) $R$ contains $\mathbb{F}_p$ as a subring for some prime $p$.

Select the correct option.

Answer: a


Question: Evaluate the limit \( \lim_{ x \rightarrow a}\;\frac{a^x-x^a}{x-a} \).

Answer: a^a (\ln a - 1)


Question: Determine the smallest positive value of \(k\) for which the equation \(e^x = kx^2\) has exactly three solutions.

Answer: \dfrac{e^2}{4}


Question: Evaluate the integral \( \int_{-\pi}^{\pi} \frac{\cos(x)}{\sqrt{4+3\sin(x)}} \, dx \). Determine if the integral equals zero due to symmetry.

Answer: 0


Question: A company manufactures light bulbs with lifetimes that are normally distributed with a mean of 3 months and a variance of 1 month. A consumer wants to ensure that a succession of these bulbs lasts at least 40 months with a probability of at least 0.9772. What is the smallest number of bulbs the consumer should purchase to achieve this?

Answer: 16


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\sqrt{1 + x\sin x} - \sqrt{\cos x}}{x\tan x} \]

Answer: \dfrac{3}{4}


Question: Determine the number of distinct prime divisors of the integer \(5685858885855807765856785858569666876865656567858576786786785^{22}\) given that it has 6,436,343 divisors.

Answer: 5


Question: Determine whether the following statement is true or false: If $A$ and $B$ are $3 \times 3$ matrices such that they have the same determinant, characteristic polynomial, trace, eigenvalues, rank, and minimal polynomial, then $A$ and $B$ are similar.

Answer: True


Question: Joe has 1729 unit cubes, which are initially unpainted. He randomly forms two cubes of side lengths 9 and 10 or 1 and 12, and dips them into white paint. This process is repeated, with the side lengths being chosen randomly each time, until every side of every unit cube is painted. After how many repetitions is the expected number of painted faces closest to half of the total possible painted faces?

Answer: 7


Question: Let $(M,d)$ be a metric space. Define $d_1(x,y) = \sqrt{d(x,y)}$. Is the metric $d_1$ topologically equivalent to the metric $d$ on the space $M$? Justify your answer.

Answer: Yes


Question: In triangle \(ABC\), the equation of side \(BC\) is \(2x - y = 3\). The circumcenter and orthocenter of the triangle are \((2, 4)\) and \((1, 2)\) respectively. Find the value of \(\tan B \tan C\).

Answer: 3


Question: Find the maximum value of the function \( f(x) \), where \( f(x) \) is a twice differentiable function satisfying the differential equation \( f(x) + f''(x) = -x|\sin x|f'(x) \) for \( x \geq 0 \). Given that \( f(0) = -3 \) and \( f'(0) = 4 \), determine the maximum value of \( f(x) \).

Answer: 5


Question: Calculate the volume of the solid that lies below the surface \( z = 3x^2 + y^2 \) and above the region in the xy-plane bounded by the curves \( y = x \) and \( y^2 - y = x \).

Answer: \dfrac{144}{35}


Question: Calculate \( \int_0^\infty \frac{\sin x}{x} \, dx \) by integrating a suitable function along the specified paths, where \( R \) and \( \varepsilon \) are the radii of the outer and inner semicircles, respectively.

Answer: \dfrac{\pi}{2}


Question: Determine the cardinality of the set represented by \( \omega \uparrow^\omega \omega \), where \( \uparrow \) denotes Knuth's up-arrow notation.

Answer: \aleph_0


Question: Derive the characteristic function for a normal distribution with mean 0 and variance 2, denoted as $N(0,2)$. Provide the resulting expression.

Answer: e^{-t^2}


Question: Find the infimum of the set \( \{ \lambda \in \mathbb{R} : x^2y'' - 3xy' - \lambda y = 0, \lim_{x \to \infty} y(x) = 0 \} \). Assume \( x > 0 \) and use the substitution \( x = e^t \) to solve the differential equation.

Answer: 0


Question: Evaluate the integral \( \int_{|z|=1} \frac{\sin(z^2)}{(\sin(z))^2} \, dz \) where \( z \in \mathbb{C} \).

Answer: 0


Question: Let F be a cyclic extension of K with degree \( p^n \), where \( p \) is a prime number. Suppose L is an intermediate field such that \( F = L(u) \) and L is cyclic over K with degree \( p^{n-1} \). Prove that \( F = K(u) \).

Answer: F = K(u)


Question:
Circles are drawn passing through the origin O to intersect the coordinate axes at points P and Q such that \((m)(OP)+(n)(OQ)=k\). Determine the fixed point that satisfies all such circles.

(A) \((m,n)\)
(B) \(\left(\dfrac{m^2}{k},~\dfrac{n^2}{k}\right)\)
(C) \(\left(\dfrac{mk}{m^2+n^2},~\dfrac{nk}{m^2+n^2}\right)\)
(D) \((k,k)\)

Answer: C


Question: Calculate \(\lim_{x \to 0}{\frac{1-\left(\frac{\log(1+x)}{x}\right)^n}{1-\frac{\log(1+x)}{x}}}.\)

Answer: n


Question: Prove that the $\mathbb{K}$-algebra $\mathbb{K}[s, t]/\langle s-t \rangle$ is isomorphic to $\mathbb{K}[t]$, where $\mathbb{K}$ is a field.

Answer: \mathbb{K}[t]


Question: If \( \alpha, \beta, \gamma \) are the roots of the polynomial \( x^3 + x^2 - x + 1 = 0 \), find the value of \( \prod \left( \frac{1}{\alpha^3} + \frac{1}{\beta^3} - \frac{1}{\gamma^3} \right) \).

Answer: -7


Question: Evaluate the contour integral \( \int_C \bar{z}^3 \,\mathrm{d}z \) where the curve \( C: [0,\frac{\pi}{2}] \to \mathbb{C} \) is given by \( C(t) = 2\exp(-it) \). Verify if the result is 16.

Answer: 16


Question: Evaluate the contour integral \( \int_\gamma \frac{\cos^2 z}{z^2} \, dz \), where \( \gamma \) is the unit circle in the complex plane, traversed in the anti-clockwise direction.

Answer: 0


Question: Given the numbers $1, 2, 3, \ldots, 101$ arranged in any order, is it always possible to remove $90$ numbers such that the remaining $11$ numbers are either in increasing or decreasing order?

Answer: Yes


Question: Calculate the norm \( \lVert \left[ \sin\frac{x}{2} \right] \rVert_Q \) in the quotient space \( Q = L^\infty(\mathbb{R})/S \), where \( S \) is the subspace of \( 2\pi \) periodic functions in \( L^\infty(\mathbb{R}) \). The norm is defined as \( \lVert [f] \rVert_{Q} = \inf \left\{ \lVert f + g \rVert_\infty: g \in S \right\} = \text{dist}\left(f, S \right) \).

Answer: 1


Question: Given that \((x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3\), find the value of \(x+y\).

Answer: 0


Question: Calculate the integral \( \int_\mathcal{C} j^*\alpha \), where \( M = \mathbb{R}^2\backslash\{0\} \), \( \alpha = \frac{x\cdot dy-y\cdot dx}{x^2+y^2} \), and \( \mathcal{C} = \{(f(e^{i\theta})\cos\theta,f(e^{i\theta})\sin\theta);\theta\in\mathbb{S}^1\} \), with \( f:\mathbb{S}^1\to (0,1) \) being a smooth function. Here, \( j:\mathcal {C}\to  M \) is the inclusion map.

Answer: 2\pi


Question:
Let \( X \) be the space \( \mathbb{R}^3 \) with the \( l^1 \) norm and let \( Y \) be \( C[0,2] \) with the usual supremum norm. Define \( T:X\to Y \) by:
\[ Tx(t)=T(x_1,x_2,x_3)(t)=x_1+x_2t+x_3t^2 \]
Find the operator norm of \( T \).

Answer: 4


Question: Solve the differential equation: \[ y' = \frac{2xy}{x^2-y^2} \]

Answer: x^2 + y^2 = C y


Question: Let $G=(V_1 \cup V_2,E)$ be a finite bipartite graph where every vertex in $V_1$ has degree at least $r \le |V_1|$. Given that $G$ has a perfect matching, prove that there are at least $r!$ perfect matchings in $G$.

Answer: r!


Question:
A spaceship travels randomly between three planets: A, B, and C. The journey starts at planet A and ends when the spaceship reaches planet C. The transition probabilities and travel times (in months) between the planets are given in the matrices below. Calculate the expected time (in months) for the spaceship to reach planet C starting from planet A.

**Probability Matrix**

| Origin | Destination A | Destination B | Destination C |
|--------|---------------|---------------|---------------|
| A      | 0.05          | 0.90          | 0.05          |
| B      | 0.50          | 0.50          | 0.00          |
| C      | 0.00          | 0.00          | 1.00          |

**Travel Time Matrix (in months)**

| Origin | Destination A | Destination B | Destination C |
|--------|---------------|---------------|---------------|
| A      | 5             | 2             | 10            |
| B      | 3             | 3             | 0             |
| C      | 0             | 0             | 0             |

Answer: 159


Question: Let \( T : \mathbb{R}^n \to \mathbb{R}^n \) be an orthogonal transformation. Suppose \( v_1 \) and \( v_2 \) are eigenvectors of \( T \) with distinct eigenvalues \( \lambda_1 \) and \( \lambda_2 \), respectively. Prove that \( v_1 \cdot v_2 = 0 \).

Answer: v_1 \cdot v_2 = 0


Question: Given that 3 is a generator of \( U(14) \) and \( \phi \) is an automorphism of \( U(14) \) with \( \phi(3) = 5 \), determine \( \phi(11) \).

Answer: 9


Question: Evaluate the limit \( \lim_{n\to \infty}\left(\dfrac{1}{\sqrt{n^2}}+\dfrac{1}{\sqrt{n^2+1}}+\cdots+\dfrac{1}{\sqrt{n^2+2n}}\right) \) using the Squeeze Theorem.

Answer: 2


Question: Determine the value of \( k \) such that the function \( f(x) = x \left( \left\lfloor \frac{1}{x} \right\rfloor + \left\lfloor \frac{2}{x} \right\rfloor + \cdots + \left\lfloor \frac{8}{x} \right\rfloor \right) \) for \( x \neq 0 \) and \( f(x) = 9k \) for \( x = 0 \) is continuous at \( x = 0 \). Here, \( \lfloor . \rfloor \) denotes the greatest integer function.

Answer: 4


Question: Calculate the limit: $$ \lim_{x\rightarrow +\infty}{(x-\sqrt{x-1}\ln(x^2+e^{\sqrt{x}}))} $$

Answer: \dfrac{1}{2}


Question: Let $X$ be a rigid Calabi-Yau threefold. Does $X$ have only finitely many automorphisms?

Answer: Yes


Question: Given a positive semi-definite matrix \( A \), determine if it can be decomposed as \( A = U^T U \).

Answer: Yes


Question: Suppose $x_1, \dots, x_n$ and $y_1, \dots, y_n$ are independent random variables, each distributed as $N(0,1)$. Define $z_i = x_i y_i$ for each $i$. Are the random variables $z_1, \dots, z_n$ independent from one another?

Answer: Yes


Question: Determine the minimum number of real variables required to describe all 300 by 300 matrices that are their own inverse.

Answer: 45000


Question: A vector in 110-dimensional Euclidean space $\mathbf{R}^{110}$ makes equal acute angles with the positive directions of the 110 coordinate axes. What is the approximate measure of that angle in degrees?

Answer: 84.5


Question: Determine the number of pairs \((p, q)\) in \(\mathbb{R}^2\) such that the polynomial \(x^4 + px^2 + q\) is divisible by \(x^2 + px + q\).

Answer: 5


Question: Suppose $M$ is a finitely generated module over a commutative ring $R$ such that $M \oplus R \cong R$. Is it true that $M = 0$? Justify your answer.

Answer: 0


Question: Calculate the integral \( \int_{[0,\frac{1}{4}]\times[0,2\pi]} x\cos(xy)\cos(2\pi x)\ \text{d}x\wedge\text{d}y \).

Answer: \dfrac{1}{4\pi}


Question: Let $X_1, X_2, \ldots, X_{100}$ be independent random variables each having the standard normal distribution. Calculate the approximate probability $P(80 < \sum_{j=1}^{100} X_j^2 < 120)$. Express your answer as a single probability value.

Answer: 0.843


Question: Provide an example of a $T_1$ crowded topological space $X$ that is not a countable union of co-dense sets.

Answer: \mathbb{R}


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

Answer: 1


Question: Use Stokes' Theorem to evaluate \( \iint (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, ds \), where \( \mathbf{F} = \langle z, -2x, xy \rangle \) and \( S \) is the surface defined by \( z = 4 - x^2 - y^2 \) above the xy-plane.

Answer: -8\pi


Question: Evaluate the limit: \[ \lim_{n\to \infty} n^2 \int_0^1 \left(\frac{1}{1+x^2}\right)^n dx \]

Answer: \infty


Question: Evaluate the flux of the vector field \( F(x,y,z)=(z^2-x,-xy,3z) \) across the surface \( S \) of the solid bounded by the equations \( z=4-y^2, x=0, x=3 \), and \( z=0 \), with the normal vector pointing outward. Compute \( \iint_S F \cdot n \,ds \) without using the Divergence Theorem.

Answer: 16


Question: Let $H$ be a real Hilbert space, and let $B : H \times H \to \mathbb{R}$ be a bilinear and symmetric form. Suppose there exists a constant $C$ such that for all $x \in H$, $|B(x,x)| \le C \|x\|^2$. Must $B$ be continuous? Provide a justification for your answer.

Answer: Yes


Question: In triangle $\Delta ABC$, let $O$ be a point inside the triangle. Extend $AO, BO, CO$ to meet $BC, CA, AB$ at $D, E, F$ respectively. Given that $AO = 30$, $FO = 20$, $BO = 60$, $DO = 10$, and $CO = 20$, find the length of $EO$. \(\boxed{\text{EO}}\)

Answer: 20


Question: Evaluate the limit: \[ \lim_{n \to \infty} n((n+1)^{\frac{1}{100}}-n^{\frac{1}{100}}) \]

Answer: \infty


Question: Find the number of solutions in non-negative integers to the equation \(x_1 + x_2 + x_3 + x_4 + x_5 = 9\) where \(x_1, x_2, x_3, x_4, x_5 \neq 1\). Use generating functions to determine the coefficient of \(x^9\) in the expression \((1 + x^2 + x^3 + x^4 + \dots)^5\).

Answer: 185


Question: Evaluate the integral \( \int_C \left(1+ \frac{2}{z}\right) dz \) along the contour \( C(\theta)=e^{i\theta}, 0 \le \theta \le \pi \), where \( C \) is the unit half-circle in the upper plane centered at 0. Use an appropriate branch of the logarithmic function to find the value of the integral.

Answer: -2 + 2\pi i


Question: Let $V$ be the vector space of all $2\times 2$ matrices over the field of real numbers and let \(B=\begin{pmatrix}2&-2\\-1&1\end{pmatrix}.\) Let $W$ be the subspace of $V$ consisting of all $A$ such that $AB=0.$ Let $f$ be a linear functional on $V$ which is in the annihilator of $W.$ Suppose that $f(I)=0$ and $f(C)=3,$ where $I$ is the $2\times 2$ identity matrix and \(C=\begin{pmatrix}0&0\\0&1\end{pmatrix}.\) Find $f(B).$

Answer: 0


Question:
Consider the linear system of first-order differential equations given by \( y' = Ay + h \), where
\[
A = \begin{bmatrix} 1 & 1 \\ 4 & -2 \end{bmatrix}
\] and \[
h = \begin{bmatrix} 3t+1 \\ 2t+5 \end{bmatrix}.
\]
Suppose \( y(t) \) is a solution such that \( \lim_{t\to\infty}\frac{y(t)}{t}=k \in \mathbb{R}^2 \). Determine the value of \( k \) from the following options:

1. \( \begin{bmatrix} \frac{-4}{3} \\ \frac{-5}{3} \end{bmatrix} \)
2. \( \begin{bmatrix} \frac{4}{3} \\ \frac{-5}{3} \end{bmatrix} \)
3. \( \begin{bmatrix} \frac{2}{3} \\ \frac{-5}{3} \end{bmatrix} \)
4. \( \begin{bmatrix} \frac{-2}{3} \\ \frac{-5}{3} \end{bmatrix} \)

Answer: 1


Question: Evaluate the limit: \[ \lim _{x\to \infty }\left(\frac{\left(\int _0^x e^{t^2} \, dt\right)^2}{\int _0^x e^{2t^2} \, dt}\right) \]

Answer: 0


Question: Determine whether the function \( f(z) = \frac{\sin(z)}{z} \) for \( z \neq 0 \) and \( f(0) = 1 \) is entire by analyzing its Maclaurin series representation.

Answer: Yes


Question: Determine whether there exists a field extension $E$ containing $S$ such that $E$ has finite dimension when considered as a vector space over $S$. Here, $S$ is a subset of the positive real numbers $\mathbb{R}^+$, and $(S, \times)$ is a group. Additionally, $(S, \times, *)$ is a field with the operation $m: S \times S \to S$ defined by $(e^a, e^b) \mapsto e^{ab}$ for $a, b \in \mathbb{Q}$. Provide a justification for your answer.

Answer: Yes


Question: For which values of \(a\) does the linear transformation \(L: \mathbb{R}^3 \rightarrow \mathbb{R}^3\) satisfy \(v_1, v_2 \in \ker(L)\) and \(L(v_3) = (1, 1, 1)\), where \(v_1 = (1, a, a^2)\), \(v_2 = (a^2, 1, a)\), and \(v_3 = (a, a^2, 1)\)?

Answer: a \neq 1


Question: Evaluate the integral \( \int_{[0,1]} f \, \, d\lambda \), where \( f(x) = e_2(x) \) and \( e_k(x) \) is the \( k \)-th digit after the decimal point of \( x \).

Answer: \dfrac{9}{2}


Question: Determine the number of idempotents in the ring \( \mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{Z}[i] \), where \( \mathbb{Z}[i] = \{a+bi : a,b \in \mathbb{Z} \} \) is the ring of Gaussian integers.

Answer: 2


Question: Calculate the flux of the vector field \( \vec{F} = xy\hat{i} + y^2\hat{j} + zy\hat{k} \) through the pyramid with vertices \((0,0,0)\), \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\). Use both direct calculation and the divergence theorem to verify your result.

Answer: \dfrac{1}{6}


Question:
Let $Y$ and $X$ be independent centered normal random variables with variances $\frac{\sigma^2}{1-\theta^2}$ and 1, respectively. Compute the expectation:
\[E\left[\frac{YX}{Y^2 + (\theta Y+X)^2}\right]\]
where $\theta$ is a constant.

Answer: 0


Question: Solve the equation: \( \left(x-\dfrac{1}{6} \right)\sqrt{4x+1}+\left(x+\dfrac{1}{6} \right)\sqrt{1-2x}+\dfrac{7x+2}{x+2}=\left(x+1 \right)^{2} \).

Answer: 0


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

Answer: \dfrac{1}{e}


Question:
Determine the type of singularity at \( z=0 \) for the function \( \exp\left(\frac{\cos(z)-1}{z^2}\right)^2 \). Choose one of the following options:
1. Removable singularity
2. Pole of order 2
3. Pole of order 4
4. Essential singularity

Answer: 1


Question: Show that \( R(x) = o(x^3) \) given that \[ R(x) = 4! \, x^4 \int _0^{\infty} \frac{1}{(1+xt)^{5}}e^{-t} \, \, dt. \]

Answer: R(x) = o(x^3)


Question: Let $M$ be a connected $(d \geq 2)$-dimensional smooth closed manifold. Does there exist a class $C^1$-diffeomorphism $\phi$ from $M$ onto itself such that $(M,\phi)$ is a topologically mixing discrete dynamical system?

Answer: Yes


Question: Find the codimension of the subspace \( A = \overline{\operatorname{span}}\{S^n(1,4,-1,0,0,\ldots):n=0,1,2,\ldots\} \) in \( \ell_2 \), where \( S \) is the right shift operator defined by \( Se_i = e_{i+1} \).

Answer: 1


Question: Let \( f: (0,\infty )\rightarrow \mathbb{R} \) and \( F(x) = \int_{0}^{x}tf(t)dt \). If \( F(x^2)= x^{4} + x^{5} \), find the value of \( \sum_{r=1}^{12}f(r^{2}) \).

Answer: 219


Question: Let $f:[0,1]\to\mathbb{R}$ be a bounded function. For a given $\epsilon > 0$, determine whether the set $S:=\{x\in[0,1]:f(x)>\sup f-\epsilon\}$ can have a Lebesgue measure of $0$. Provide a justification for your answer.

Answer: Yes


Question: Let $K$ be a field of characteristic zero and $X$ be a projective variety over $\text{Spec} K$. Denote by $\bar{K}$ the algebraic closure of $K$. Let $\bar{X}$ be the fiber product $X \times_K \bar{K}$. Suppose that $\bar{X}$ is a smooth Fano variety. Is $X$ also a Fano variety?

Answer: Yes


Question: Let \( f: \mathbb{R}^2 \rightarrow \mathbb{R} \) be a differentiable function such that \( \nabla f(2,3) = (3,4) \). Define \( g: \mathbb{R}^2 \rightarrow \mathbb{R} \) by \( g(x,y) = f(x^2 - y + 2, y^3 - x + 3) \). Find the partial derivative \( g_x(1,1) \).

Answer: 2


Question: Let $X$ and $Y$ be independent random variables, each with an exponential distribution with rate parameter 1. Define $U = e^X$ and $V = X + Y$. Calculate the conditional expectation $E(U \mid V = 1)$. Provide your answer in terms of $e$. \( \text{(The answer is } e-1 \text{)} \)

Answer: e - 1


Question: Given an arithmetic sequence \(\{a_n\}\), where the sum of the first \(m\) terms is \(S_m = \frac{m}{n}\) and the sum of the first \(n\) terms is \(S_n = \frac{n}{m}\) with \(n \neq m\), determine whether \(S_{m+n}\) is greater than 4.

Answer: S_{m+n} > 4


Question: Evaluate the limit \( \lim_{x \to 0^+} \left [ \frac{\pi}{2} + \tan x - \arctan \frac{1}{x} \right]^{\frac{1}{\ln x}} \) without using de L'Hopital's rule.

Answer: e


Question: Does the function \( f(z) = \sin\left(\frac{1}{z}\right) \) violate Riemann's theorem for isolated singularities at \( z = 0 \)? Answer Yes or No.

Answer: No


Question: Determine the number of solutions \((x, y)\) to the equation \(x^2 - y^2 = 1\) where \(x, y \in \mathbb{F}_{p^n}\), given that \(p > 2\) is an odd prime and \(\mathbb{F}_{p^n}\) is the field with \(p^n\) elements.

Answer: p^n - 1


Question: Suppose $P_1$ and $P_2$ are two probability measures on $(\Omega = \{0,1\}^{\infty}, \mathcal{F})$. If the induced probability measures on $(\{0,1\}^k, \mathcal{F}_k)$ by $P_1$ and $P_2$ are the same for each $k$, are $P_1$ and $P_2$ the same measures on $(\Omega, \mathcal{F})$? Justify your answer.

Answer: Yes


Question: Assuming a Poisson process with a rate of 2.5 mistakes per year, calculate the probability that an individual dies before reaching the age of 67.2 years, given that death occurs after 196 mistakes. Use the gamma distribution with parameters 196 and 2.5 to find this probability.

Answer: 0.0228


Question:
Find the intersection of the sets \(A\) and \(B\), where:
\[A = \{ \lfloor n\sqrt{2} \rfloor \mid n \in \mathbb{N}^* \}\]
\[B = \{ \lfloor n(2+\sqrt{2}) \rfloor \mid n \in \mathbb{N}^* \}\]
Determine the common elements in these sets.

Answer: \emptyset


Question: Does there exist a sequence \(a_n\) consisting of every natural number such that for each positive integer \(N\), the sum \(\sum\limits_{k=1}^N a_k\) is divisible by \(N\)?

Answer: Yes


Question: Let $A$ be a positive definite bilinear form defined on a real vector space. Determine whether $A$ satisfies the inequality $2A(u,u)^{\frac{1}{2}} A(v,v)^{\frac{1}{2}} \ge A(u,v) + A(v,u)$ for all vectors $u$ and $v$ in the space.

Answer: Yes


Question: Maximize the function \( c = \frac{(2-a_0) + \sqrt{-a_0(3a_0-4)}}{2} \) and find the corresponding value of \( c \) without using calculus.

Answer: \dfrac{4}{3}


Question: Solve the equation: \[ x\sqrt{x^2+5} + (2x+1)\sqrt{4x^2+4x+6} = 0. \]

Answer: -\dfrac{1}{3}


Question: In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $(a^2 + b^2) : c^2 = 2019 : 1$, find the ratio $(\cot A + \cot B + \cot C) : (\cot A + \cot B)$. Provide your answer as a simplified ratio.

Answer: 1010:1


Question: Calculate the sum of the series \( S = \sum_{n=1}^{243} \frac{1}{n^{4/5}} \) and find the value of \( \lfloor S \rfloor \), where \( \lfloor \cdot \rfloor \) denotes the greatest integer function.

Answer: 10


Question: Let $R$ be a ring (unital, not necessarily commutative), and let $M$ be a finitely presented left $R$-module. Suppose $m_1, \ldots, m_n \in M$ generate $M$, determining a surjection $f: R^n \to M$. Is the kernel of $f$ finitely generated?

Answer: Yes


Question: Let \(X_1, X_2, \ldots, X_n\) be independent and identically distributed observations from a normal distribution with mean \(\mu\) and variance \(\sigma^2\), where \(\sigma^2 > 0\) is known and \(\mu\) is an unknown real number. Consider the parameter of interest \(g(\mu) = 2\mu\) and the statistic \(T(X_1, X_2, \ldots, X_n) = X_1^2 + 2X_3 - X_4^2\). Compute the conditional expectation \(E(T | \bar{X})\), where \(\bar{X}\) is the sample mean of \(X_1, X_2, \ldots, X_n\).

Answer: 2\bar{X}


Question: Let $A$ and $B$ be two positive semi-definite $n \times n$ matrices such that \( \mathrm{Range}(B^{1/2}AB^{1/2}) = \mathrm{Range}(B) \) and \( \mathrm{Rank}(A) = \mathrm{Rank}(B) = n-1 \). Determine whether there exists a real symmetric matrix $X$ such that $A = XBX$. Provide a justification for your answer.

Answer: Yes


Question: Find a nonzero solution \((x, y, z) \in \mathbb{F}_7^3\) to the quadratic equation \(2x^2 + y^2 + 2yz + 2z^2 = 0\) over the finite field \(\mathbb{F}_7\).

Answer: (1, 1, 1)


Question:
Let $a > 1$ be a fixed positive integer and $f(x)$ be a polynomial with integer coefficients and a positive leading coefficient. Define the set $S$ as follows:
\[
S = \{ n \in \mathbb{Z}^+ : n \mid a^{f(n)} - 1 \}.
\]
Determine the density of the set $S$, i.e., compute
\[
\lim_{n \to \infty} \frac{|S \cap \{1, 2, \dots, n\}|}{n}.
\]

Answer: 0


Question: Choose any 9 distinct integers. These 9 integers can be arranged to form 9! determinants, each of order 3. What is the sum of these 9! determinants?

Answer: 0


Question:
Let $f:\mathbb{R}\to\mathbb{R}$ be a function satisfying the following conditions:
\[\begin{gather}
f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2} \\
f(0)=1 \\
f'(0)=-1 \\
\end{gather}\]
Find the value of $f(2)$. \(\boxed{}\)

Answer: -1


Question: Determine whether there exists a constant \( C > 0 \) such that for all complex numbers \( z_1 \) and \( z_2 \), the inequality \[ 4|z_1|^2 |z_2|^2 + |z_1^2 - z_2^2|^2 \ge C (|z_1|^2 + |z_2|^2)^2 \] holds.

Answer: 1


Question: Find \( f(5) \) given that the function \( f \) satisfies the equation \[ f(x) + f\left(\frac{1}{1-x}\right) = x \] for all \( x \in \mathbb{R} \) where \( x \neq 0, 1 \).

Answer: \dfrac{121}{40}


Question: Calculate the following limit: $$\lim_{x \to +\infty}{x^{3/2}(\sqrt{x+1}+\sqrt{x-1}-2\sqrt{x})}$$

Answer: -\dfrac{1}{4}


Question: Verify whether the toric variety corresponding to an n-simplex Delzant polytope is the projective space \( \mathbb{P}^n \).

Answer: \mathbb{P}^n


Question: Find the probability that $A > B + C$ if $A$, $B$, and $C$ are independent and uniformly distributed over the interval $[0, 1]$. Express your answer as a fraction.

Answer: \dfrac{1}{6}


Question: Evaluate the limit: \[ \lim_{n \to \infty}\;\frac{ 1+ \frac{1}{2}+\cdots+\frac{1}{n}}{ (\pi^n + e^n)^{1/n} \ln{n} } \]

Answer: \dfrac{1}{\pi}


Question: Evaluate the integral: $$\int_0^{\infty}\dfrac{\ln(x+\frac{1}{x})}{1+x^2}\cdot dx$$

Answer: \pi \ln 2


Question: Evaluate the definite double integral: $$\int_0^1\int_0^1\ln(|x-y|)\,\mathrm{d}x\,\mathrm{d}y.$$

Answer: -\dfrac{3}{2}


Question: Find the derivative of the function \( f(x) = x^TAx \), where \( A \) is an \( n \times n \) symmetric matrix and \( x \in \mathbb{R}^n \).

Answer: 2Ax


Question: Suppose $Y$ is uniformly distributed on $(0,1)$ and the conditional distribution of $X$ given $Y=y$ is uniform on $(0,y)$. Find the expected value $E[X]$.

Answer: \dfrac{1}{4}


Question: Does there exist a poset whose quasiwidth strictly exceeds its width?

Answer: Yes


Question: Determine if there exists a largest nested prime number, and if so, find it.

Answer: 739397


Question: Find the smallest positive integer \( n \) such that \( 2^{10n} \) does not have 1 as its first digit.

Answer: 30


Question: Consider a cube with corners labeled as coordinates in a 3D space. Starting from the corner at (0,0,0), each second you move to a random neighboring corner with equal probability. What is the expected number of seconds required to reach the opposite corner at (1,1,1)?

Answer: 10


Question: Let \( f : [0,1] \rightarrow [-1,1] \) be a non-zero function such that \( f(2x) = 3f(x) \) for \( x \in \left[0, \frac{1}{2}\right] \). Evaluate \( \lim_{x \to 0^+} f(x) \).

Answer: 0


Question: Find the coefficient of \(X^8\) in the power series expansion of \((1 - 3X^4)^{-6}\).

Answer: 189


Question: Evaluate the integral: $$ \frac{\exp\left(-\frac{N}{2}\right)}{2\pi N} \iint \exp\left(-\left( \frac{x^2-2xy-2Ny+2y^2}{2N}\right)\right) \,dx\,dy $$ where $N > 0$. Provide your answer in terms of $N$.

Answer: 1


Question:
Find the sequence $a_n$ of natural numbers such that $a_1 = 10$ and for $n = 2, 3, \ldots$, the following holds:
\[ a_n^2 = \frac{a_{n-1}^2 + a_{n+1}^2}{2}. \]

Answer: 10


Question: Find the value of the expression \( \sqrt{1+\sqrt{\frac{1}{2}+\sqrt{\frac{1}{8}+\sqrt{\frac{1}{128}+\cdots}}}} \). Each term is of the form: \( \frac{2^{2^r}}{2^{2^{r}+2^{r-1}+\cdots+2+1}} = \frac{1}{2^{2^{r-1}+\cdots+2+1}} \).

Answer: \sqrt{2}


Question: On a bounded domain $\Omega$ in $\mathbb{R}^n$ with a $C^1$ boundary, does every function $f \in C_c^{\infty}(\Omega)$ belong to a Sobolev space $W^{k,p}(\Omega)$ for $1 \le p < \infty$? Answer yes or no.

Answer: Yes


Question: Determine the values of \( \alpha \) for which the series \( \sum_{n=1}^{\infty} \frac{n^2+n-1}{\sqrt{n^\alpha+n+3}} \) converges.

Answer: \alpha > 6


Question: Suppose that $X_1, X_2, \ldots, X_n$ are independent and identically distributed random variables with characteristic function $\chi(t) = e^{-|t|^{1.9}}$. Determine the weak limit of $n^{-5/9} S_n$ as $n$ approaches infinity, where $S_n = \sum_{k=1}^n X_k$. What can be said if the exponents 1.9 and 5/9 are replaced by arbitrary values?

Answer: 0


Question: In triangle ABC, point H is the orthocenter, and point C is the centroid of triangle ABH. Determine the smallest angle in triangle ABC. Choose from the following options: 60°, 30°, 45°, or ∠ACB.

Answer: 30^\circ


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{\sinh(x)}{\cos(x)-1} \int_{0}^{x} \sqrt{e^t-t^4} \, dt \]

Answer: -2


Question: Let $\phi: A \to \mathbb{C}$ be a state on a unital $C^*$-algebra $A$, and let $a \in A$ be a positive element with $\phi(a) = 0$. Is it true that $\phi(b^*ab) = 0$ for every $b \in A$?

Answer: No


Question:
Determine which set A forms a free submonoid of \( \{a,b,c,d,e,f,g\}^* \):

A) \( A = \{ ae, b,c,de\} \)

B) \( A = \{ ade, ddbee,dfc,dgd\} \)

C) \( A = \{ a, ab,bc,c\} \)

D) \( A = \{ ab, ba ,ca\} \)

E) \( A = \{ ab, abc,cde ,de\} \)

Select the correct option.

Answer: A


Question: Determine the largest integer $n$ such that $99^{(n+1)} > 100^n$ without using logarithms.

Answer: 457


Question: Is the Lebesgue measure of the boundary of a bounded Lipschitz domain in \(\mathbb{R}^n\) zero?

Answer: Yes


Question: Calculate the value of \( \int_{-\infty}^{\infty} y(t) \, dt \), where \( y(t) = \int_{-\infty}^{t} g(\tau) \, d\tau \) and for a function \( g(t) \), it is given that \( \int_{-\infty}^{\infty} g(t)e^{-j\omega t} \, dt = \omega e^{-2\omega^2} \) for any real value \( \omega \).

Answer: -j


Question: Given the differential equation \( y'' + py = 0 \) with boundary conditions \( y(0) = 2 \) and \( y(\pi) = 3 \), where \( p \in (1, a) \), determine the maximum value of \( a \).

Answer: 4


Question: Evaluate the limit: \[ \lim_{x \to 0} \left( \frac{\cos x}{\cos 2x} \right)^{\frac{1}{x^2}}. \]

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


Question: Find the limit: \[ \lim_{n\to\infty}\sqrt{n^3}(\sqrt{n+1}+\sqrt{n-1}-2\sqrt{n}) \] without using L'Hopital's rule, derivatives, or integrals.

Answer: -\dfrac{1}{4}


Question: Evaluate the limit \( \lim_{n\to\infty} \frac{n^n}{3^n} a_n \) for the power series \( f(z)= \sum_{n=0}^{\infty } a_n z^n \) given that \( |f(z)| \leq M e^{|z|} \).

Answer: 0


Question: What is the minimum number of interactions required for 16 people, each with a distinct piece of information, to share all their information such that everyone ends up with all 16 pieces?

Answer: 28


Question: Evaluate the integral \( \int_{-L}^L e^{i\frac{(m-n)\pi}{L}x} \, dx \), where \( 0 < L \in \mathbb{R} \) and \( m, n \in \mathbb{N} \).

Answer: 2L \delta_{mn}


Question: Find the intersection point of the projective lines corresponding to \(y = mx\) and \(y = mx + c\) (where \(c \neq 0\)) in the projective plane, given the embedding \((x,y) \to [1,x,y]\).

Answer: [0, 1, m]


Question: Given a strictly increasing sequence of natural numbers $(p_n)$ and a positive sequence $(\epsilon_n)$ decreasing to $0$, let $(x_n)$ be a sequence in the unit sphere of $l_2$ with the property that for any $j \in \mathbb{N}$ and for any $1 \leq k \leq j$, \( \sum_{i>p_k}|(x_j,e_i)|^2<\epsilon_k \). Determine whether there exists a subsequence of $(x_n)$ that converges strongly to some $x \in S(l_2)$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the equation \( \frac{\log(a)}{\log(b)} = \log(a^{\frac{1}{\log(b)}}) \) holds true. Justify your answer.

Answer: True


Question: Find the value of $b$ such that \( \lim_{x \to 0} \left(\frac{ae^x-b\cos x+ce^{-x}}{x\sin x}\right) = 2 \).

Answer: 2


Question: A mouse is placed in a maze consisting of 9 rooms, with connections between the rooms indicated by gaps in the walls. The mouse moves randomly between rooms, including the possibility of staying in the same room. If there are $k$ possible moves from a room, each move is equally likely. Determine the long-term expected fraction of time the mouse spends in each room.

Answer: \dfrac{k_i}{\sum_{j=1}^{9} k_j}


Question: Evaluate the limit \( \lim_{x\rightarrow \frac{\pi}{2}}-(x-\frac{\pi}{2})\cdot\frac{\ln(\cos(x))}{x^2} \).

Answer: 0


Question: Compute the limit: \[ \lim_{x \to 0^+} \frac{\arctan x - x}{x^2} \] without using Taylor's formula or L'Hôpital's rule.

Answer: 0


Question: Find all real-valued bounded measurable functions \( f(x) \) such that \( \int_1^2 x^n f(x) \, dx = 0 \) for all non-negative integers \( n = 0, 1, 2, \ldots \).

Answer: 0


Question: Find the natural number $k < 117$ such that $2^{117} \equiv k \pmod{117}$. Determine the value of $k$. \(k = 44\)

Answer: 44


Question: Express the sum of the series \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} \) in terms of \( s \), where \( s = \sum_{n=1}^{\infty} \frac{1}{n^2} \).

Answer: -\dfrac{s}{2}


Question: Determine whether the series \( \sum_{n=0}^{\infty} \frac{(-1)^n (2n)!x^n}{(n!)^2 4^n} \) converges or diverges.

Answer: (-1, 1]


Question: What is the unit object in the monoidal category \( \operatorname{Vect(k)} \), where \( \operatorname{Vect(k)} \) is the category of vector spaces over a field \( k \)?

Answer: k


Question: Let \( L \) be the field generated by \( \omega \) and \( 2^{1/3} \) over \( \mathbb{Q} \), where \( \omega \neq 1 \) is a complex number such that \( \omega^3 = 1 \). Determine the number of subfields \( K \) of \( L \) such that \( \mathbb{Q} \subsetneq K \subsetneq L \).

Answer: 4


Question: Consider a graph \( G \) where the set of vertices is the power set of \( \{0,1,2,3,4,5,6,7,8,9\} \). Two vertices \( A \) and \( B \) are connected by an edge if and only if the symmetric difference \( |A \Delta B| \le 2 \). Determine the total number of edges in the graph \( G \).

Answer: 28160


Question: Calculate the line integral \( Z = \int_L (x-y) \, dx + x \, dy \) along the parabola \( y = 4 - x^2 \) from point \( A(1, 3) \) to point \( B(2, 0) \). Then, compute the value of \( H = \sqrt[5]{90876845839099} \cdot (Z + 0.5) \cdot 3 + 1 \).

Answer: -8046


Question: Determine whether there exists an integer \( n \) such that the number \( 2^n \) begins with the digits 9786543120.

Answer: Yes


Question: Evaluate the double integral \( \int\limits_{-1}^1\int\limits_{-\sqrt{1-y^2}}^0\frac{1}{(1+x^2+y^2)^2}dxdy \) by converting it to polar coordinates and then calculating the result.

Answer: \dfrac{\pi}{4}


Question: Evaluate the limit: \[ \lim_{x \to 1^{-}} (1 - x^2)^{\frac{1}{\log(1 - x)}} \]

Answer: e


Question: Evaluate the double integral \( \iint_R 8(x^2+y^2)(x^2-y^2) \, dA \), where \( R \) is the region in the first quadrant of the \( xy \)-plane bounded by the circles \( x^2+y^2=1 \), \( x^2+y^2=4 \), and the lines \( y=x+1 \) and \( y=x-1 \).

Answer: 0


Question:
Let $E$ be a measure zero set of $\mathbb{R}$. Is there a finite positive measure set $F$ such that for every $x \in E$, the following conditions hold?
\[ \liminf_{r\to 0} \frac{m(F\cap B_{r}(x))}{m(B_{r}(x))}=0 \quad \text{and} \quad \limsup_{r\to 0} \frac{m(F\cap B_{r}(x))}{m(B_{r}(x))}=1 \]

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\rightarrow\infty}\frac{\ln(1+x+x^\frac{1}{3})}{\ln(1+x^\frac{1}{4}+x^\frac{1}{3})} \]

Answer: 3


Question: A stained glass window consists of nine squares of glass arranged in a 3x3 grid. Of these nine squares, \( k \) are red, and the rest are blue. A set of windows is created such that any possible window can be formed in exactly one way by rotating and/or flipping one of the windows in the set. If the total number of red squares in the set exceeds 100, find the value of \( k \).

Answer: 5


Question:
For any nonempty sets $A,B \subseteq \mathbb{R}$, let $A \cdot B$ be the set defined by \( A \cdot B = \{xy : x \in A \wedge y \in B\} \). If $A$ and $B$ are nonempty bounded sets and if $\sup A > \sup B$, what is $\sup(A \cdot B)$?

(A) $\quad\sup(A)\sup(B)$
(B) $\quad\sup(A)\inf(B)$
(C) $\quad\max\{\sup(A)\sup(B),\ \inf(A)\inf(B)\}$
(D) $\quad\max\{\sup(A)\sup(B),\ \sup(A)\inf(B)\}$
(E) $\quad\max\{\sup(A)\sup(B),\ \inf(A)\sup(B),\ \inf(A)\inf(B)\}$

Answer: E


Question: Prove that \( \lim_{x \to \infty} \frac{e^x}{x^3} = +\infty \) without using derivatives, Taylor series, or l'Hopital's rule.

Answer: +\infty


Question: Let $X$ be an infinite dimensional Banach space, and let $X^{\mathrm{*}}$ be the space of linear continuous functionals on $X$. Assume $X^{\mathrm{*}}$ is separable, and let $(x^{\mathrm{*}}_n)_{n \in \mathbb{N}}$ be a dense subset of $X^{\mathrm{*}}$. Determine whether the intersection $\bigcap_{n=0}^{\infty} \mathrm{ker}(x^{\mathrm{*}}_n)$ is trivial.

Answer: \{0\}


Question: Let $G$ be a transitive subgroup of the symmetric group $S_n$. Calculate the value of \( \frac{1}{|G|} \sum_{g \in G} f(g) \), where \( f(g) = |\{ 1 \le i \le n \mid g(i) = i \}| \).

Answer: 1


Question:
Find the value of \(a\) such that the following limit exists and is different from zero:
\[ \lim_{x\to0}\frac{\int_0^{x^2}\frac{\log(1+t)-at}{t}dt}{(1-\cos(x/2))^2} \]

Answer: 1


Question:
Determine whether there exist natural numbers $a$ and $b$ such that $b > a$ and the following condition is violated:
\[
0 < \left| \sum_{k=n+1}^{\infty} \frac{a^k}{b^k} \right| > \frac{a^n}{b^n}
\]

Answer: Yes


Question: Find the sum of the infinite series whose general term is given by \( \frac{n^2+2}{5^n} \).

Answer: \dfrac{31}{32}


Question:
Can the n-th derivative of a Schwartz function go to infinity as n approaches infinity? Consider the semi-norm defined for a function \( \varphi: \mathbb{R}^M \to \mathbb{R}^N \) as:
\[ \|\varphi(x)\|_{\alpha,\beta} = \sup_{x \in \mathbb{R}^M} |x^{\alpha} D^{\beta} \varphi(x)| \,, \]
where \( \alpha \) and \( \beta \) are multi-indices on \( \mathbb{R}^N \) and \( D^{\beta} \) is the derivative with respect to the multi-index \( \beta \). A function \( \varphi(x) \) belongs to the Schwartz space if this norm is finite. Determine if there exists a Schwartz function such that \( \|\varphi(x)\|_{\alpha,\beta} \to \infty \) as \( \alpha \to \infty \) or \( \beta \to \infty \).

Answer: Yes


Question: Find the area of the figure bounded by the curve \((x^2+y^2)^3=x^4+y^4\) using polar coordinates.

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


Question: Let $A(t)$ be an $n \times n$ continuously differentiable matrix such that $\det A(t) = 1$ for all $t$ and $A(0) = E$, where $E$ is the identity matrix. Show that $\operatorname{tr} A'(0) = 0$.

Answer: 0


Question: Let \( f(z) \) be a function such that \( |f(z)| \leq \frac{1}{\sqrt{1-|z|^2}} \) for all \( z \) in the open unit disk \( \mathbb{D} \). Prove that \( |f'(0)| \leq 2 \).

Answer: 2


Question: Let $\{ Y_n \}_{n \in \mathbb{N}_0}$ be a family of random variables such that $\mathbb{P}(Y_0=0)=1$. For each $n \in \mathbb{N}$, $Y_n$ is conditionally uniformly distributed on the interval $(y_{n-1}, y_{n-1}+1)$ given $\{ Y_1=y_1, \ldots, Y_{n-1}=y_{n-1} \}$. Determine the expected value $\mathbb{E}(Y_n)$. \( \text{Assume } Y_0 = 0. \)

Answer: \dfrac{n}{2}


Question: Find the smallest constant \( k > 0 \) such that \[ \frac{ab}{a+b+2c} + \frac{bc}{b+c+2a} + \frac{ca}{c+a+2b} \leq k(a+b+c) \] for every \( a, b, c > 0 \).

Answer: \dfrac{1}{4}


Question: Given that $X$ and $Y$ are independent and identically distributed standard normal random variables, find the conditional covariance $Cov(XY|X+Y>0)$. Express your answer in terms of known constants or functions.

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


Question: Does every invertible complex matrix have a non-zero eigenvalue? Justify your answer.

Answer: Yes


Question: Given a smooth function \( f: \mathbb{R}^n \rightarrow \mathbb{R} \) and a linear transformation \( \phi \in GL(n) \), find the Hessian matrix \( H_{f \circ \phi} = \left(\frac{\partial^2 (f \circ \phi)}{\partial x_i \partial x_j}\right)_{ij} \).

Answer: \phi^\top H_f(\phi(x)) \phi


Question: Find the coefficient of \(x^k\) in the expansion of \((x^2 - \frac{1}{x})^{100}\).

Answer: (-1)^{\frac{200 - k}{3}} \dbinom{100}{\frac{200 - k}{3}}


Question: Determine the maximal number of real intersections of the unbounded components of two plane cubic curves, $C_1$ and $D_1$, in the projective plane.

Answer: 9


Question: Determine whether there exists a model of set theory that includes the Axiom of Choice but does not have a universal well-order, where a universal well-order is defined as a class-function that bijects the universe $V$ with the class of ordinals $ORD^V$. Provide a justification for your answer.

Answer: Yes


Question: Is the set of purely real square matrices that are complex-diagonalisable dense in the set of real matrices?

Answer: Yes


Question: Compute the contour integral \( \oint_{|z|=4} \frac{e^{1 /(z-1)}}{z-2} \, dz \) using the residue theorem.

Answer: 2\pi i


Question: Provide an example of an ideal $I$ in the polynomial ring $R[x]$ such that the set $X = \{ x \in R : f(x) = 0, \forall f \in I \}$ is empty.

Answer: (x^2 + 1)


Question: Let \((x_n)_{n=1}^{\infty}\) be a periodic sequence of real numbers, and let \(f: \mathbb{N} \to \mathbb{N}\) be a polynomial function. Is the sequence \((x_{f(n)})_{n=1}^{\infty}\) also periodic? Justify your answer.

Answer: Yes


Question: Determine whether every symmetric matrix with a zero diagonal can be expressed in the form $X^t \Omega X$, where $X$ is a $2n \times 2n$ matrix and $\Omega = \begin{bmatrix} 0 & 1_n \\ 1_n & 0 \end{bmatrix}$, with entries in a field $F$ of characteristic $2$. Provide a proof or counterexample.

Answer: Yes


Question: Calculate the limit: \[ \lim_{x\to0}(x+e^{2x})^{\frac{2}{x}} \]

Answer: e^6


Question: Evaluate the integral \( \int_{0}^{2\pi} \cos(\cos x)\,dx \).

Answer: 2\pi J_0(1)


Question: Evaluate the limit: \[ \lim_{x\rightarrow a}\frac{a^{a^x}-a^{x^a}}{a^x-x^a} \] without using l'Hospital's rule.

Answer: a^{a^a} \ln a


Question: Does $2r - \sigma(r)$ divide $\sigma(r)$ if $r$ is deficient-perfect? Recall that a positive integer $x$ is deficient-perfect if $D(x) \mid x$, where $D(x) = 2x - \sigma(x)$ is the deficiency of $x$ and $\sigma(x)$ is the sum of divisors of $x$. Provide a proof or counterexample to support your answer.

Answer: Yes


Question: Given that $x=1$ is a root of the polynomial $p(x)= x^4+ax^3+bx^2+ax+1$, find the range of values of $a$ for which $p(x)$ has a complex root that is not real.

Answer: (-4, 0)


Question: For how many different natural values of \(a\) does the Diophantine equation \(\frac{1}{x} = \frac{a}{x+y} - \frac{1}{y}\) have natural number solutions for \(x\) and \(y\)?

Answer: 1


Question: Evaluate the double integral \( \int_{0}^{\pi} \int_{x}^{\pi} \frac{\sin y}{y} \, dy \, dx \).

Answer: 2


Question: Is it necessary for \(M\) to be a finitely generated projective \(\mathbb{Z}\)-module for the functor composition \(\text{Hom}(-,N)\otimes_{\mathbb{Z}}M\) to be representable?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{m\to\infty} \dfrac{\int_{\cos^{-1}{(1/m)}}^{\pi/2} \dfrac{dt}{\sqrt{1-m^2 \cos^2 t}}}{(m-\sqrt{m^2-1})} \]

Answer: \pi


Question: Find the least positive integer $M$ such that $M^{49} \equiv 21 \pmod{209}$. \( \text{(Note: 209 = 11 * 19)} \)

Answer: 109


Question:
Consider the $2\pi$-periodic function $f(x)$ defined by:
\[
f(x) = \begin{cases}
1+x & \text{for } x \in [0, \pi) \\
-x-2 & \text{for } x \in [-\pi, 0)
\end{cases}
\]
Determine which of the following statements about the Fourier series of $f(x)$ is true:

(A) The Fourier series converges to $-\frac{1}{2}$ at $x=0$.
(B) The Fourier series does not converge pointwise at all points where $f$ is discontinuous.
(C) The Fourier series does not converge at $x=\frac{\pi}{2}$.
(D) The Fourier series converges to $1$ at $x=0$.

Select the correct option.

Answer: A


Question: Consider the series \( \sum^\infty_{n=1} a_n = \frac{1}{2}+\frac{1}{3}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\cdots \). Find \( \limsup_{n\to\infty} \frac{a_{n+1}}{a_n} \).

Answer: \infty


Question:
Verify whether the following evaluation of \( \partial_{t}F(t,y) \) is correct, given that \( F(t,y) = \int_{0}^{y} f(t,x)~dx \) and \( \partial_{t}f(t,x) = \partial_{x}g(t,x) \):
\[ \partial_{t}F(t,y) = \int_{0}^{y} \partial_{t}f(t,x)~dx = \int_{0}^{y} \partial_{x}g(t,x)~dx = g(t,y) - g(t,0) \]

Answer: g(t,y) - g(t,0)


Question: If \(x = \sqrt{2} + \sqrt{3} + \sqrt{6}\) is a root of the polynomial \(x^4 + ax^3 + bx^2 + cx + d = 0\), find the value of \(|a+b+c+d|\).

Answer: 93


Question: Consider the intersection of a parabola given by \( yc = x^2 \) and a hyperbola given by \( y^2 = x(x-h) \), where \( c^2 = a \) and \( c^2h = b \). Show that the x-coordinate of the intersection point in the first quadrant is a solution to the cubic equation \( x^3 + b = ax \).

Answer: x^3 + b = ax


Question: Find all positive integers $n$ such that $n$ divides $2^n + 1$. Provide a proof for your answer.

Answer: 3^k


Question: Determine the interval of convergence for the power series \( \sum_{n=0}^{\infty} \frac{(2x-3)^n}{n \ \ln n} \).

Answer: [1, 2)


Question: Determine the number of zeros of the polynomial \(4z^3 - 12z^2 + 2z + 10\) within the annulus defined by \(\frac{1}{2} < |z-1| < 2\).

Answer: 2


Question: Find the partial limit of the function \( f(x,y) = e^{\frac{2x}{\sqrt{x^2+y}}} \) at the point \((0,0)\) along the set \( A = \{(x,x): x > 0\} \).

Answer: 1


Question: Let $E$ be a finite set of $n \ge 2$ real numbers. Define the set $S = \{a + b \mid a, b \in E\}$. Determine the minimum possible number of elements in $S$. Provide your answer as a function of $n$. \( \boxed{} \)

Answer: 2n - 1


Question: Find a function \( f \) such that \( f(\log(x)) = x \cdot f(x) \).

Answer: 0


Question: Find the function \( f(x) \) given that \( f(x) \) is a differentiable function satisfying the equation \( f(x+y) = e^x f(y) + e^y f(x) \) for all \( x, y \in \mathbb{R} \) and \( f'(0) = 1 \).

Answer: x e^x


Question: Simplify the sum: $$\sum_{a=0}^{\left\lfloor \frac{p}{2} \right\rfloor}\frac{p!}{(a!)^2(p-2a)!}2^{p-2a},$$ where $p$ is a positive integer.

Answer: \dbinom{2p}{p}


Question: Let \( A \) be an \( n \times n \) matrix with all eigenvalues equal to 1 or 0. Does there exist a matrix \( B = XAX^{-1} \) for some invertible matrix \( X \) such that all elements of \( B \) are either 1 or 0?

Answer: Yes


Question:
Let \( X = C[-1,1] \) be an inner product space with the inner product defined by \( \langle f, g \rangle = \int_{-1}^{1} f \bar{g} \, dt \). Suppose \( M = \{ f \in X : f(t) = 0, \; -1 \leq t \leq 0 \} \). Determine which of the following statements is correct:

1. \( M = M^{\perp\perp} \)
2. \( M \oplus M^{\perp} = X \)

Answer: A


Question: Find the limit of the sequence \( \{x_n\} \) defined by the following recurrence relation: \[ x_1 = \sqrt[2022]{\frac{2022}{2021}} \] \[ x_{n+1} = \frac{n+2}{3n+11}(\sqrt{x_n} + \sqrt[3]{7+x_n}) \] for all \( n \in \mathbb{N}^* \).

Answer: 1


Question: Find the constant \( C \) such that \[ C \int_0^\infty \int_0^\infty \frac{e^{-\frac{x_1+x_2}{2}}}{x_1+x_2} \,dx_1 \,dx_2 = 1. \]

Answer: \dfrac{1}{2}


Question: Let \(a, b \in \mathbb{N}\) be two coprime numbers, so \(\gcd(a, b) = 1\). Find the number of ordered pairs \((a, b)\) such that \(\frac{a}{b} + \frac{201b}{10201a} \in \mathbb{N}\).

Answer: 2


Question: Let $G = U(32)$, the group of integers less than 32 that are relatively prime to 32, under multiplication. Let $H = \langle 1, 15 \rangle$. Determine the isomorphism type of the quotient group $G/H$ from the options: $\mathbf{Z}_8$, $\mathbf{Z}_2 \oplus \mathbf{Z}_4$, or $\mathbf{Z}_2 \oplus \mathbf{Z}_2 \oplus \mathbf{Z}_2$. Justify your answer by considering the orders of elements in $G/H$. What is the correct isomorphism type of $G/H$?

Answer: \mathbf{Z}_8


Question:
Let $E$ be a vector space over the field $\mathbb{K}$. Prove that the following statements are equivalent:

1. $V \subseteq E$ is absolutely convex.
2. $V \subseteq E$ is convex and $\lambda V \subseteq V$ for every $\lambda \in \mathbb{K}$ with $|\lambda| \leq 1$.
3. $V \subseteq E$ is convex and $\lambda V \subseteq V$ for every $\lambda \in \mathbb{K}$ with $\lambda = 0$ or $|\lambda| = 1$.

Demonstrate the equivalence of these conditions.

Answer: 1 \iff 2 \iff 3


Question: Evaluate the integral \( \int_0^\infty \frac{e^{-x^2}}{(x^2+1/2)^2} \, dx \).

Answer: \sqrt{\pi}


Question: Evaluate the sum of the series: \[ \sum_{m=1}^\infty \frac{\left(\frac{1}{2}\right)^\underline{m}}{m!} \]

Answer: \sqrt{2} - 1


Question: Determine the number of solutions for \( x \in \{1, 2, \ldots, 1000\} \) that satisfy the congruence \( x^2(x+1)^2 \equiv 0 \pmod{1000} \).

Answer: 40


Question: Evaluate the polynomial \( f(x) = x^{10} + 2x^9 - 2x^8 - 2x^7 + x^6 + 3x^2 + 6x + 2014 \) at \( x = \sqrt{2} - 1 \) and show that \( f(\sqrt{2} - 1) = 2017 \).

Answer: 2017