Question: Determine the values of \(x\) for which the series \(\sum_{n=1}^{\infty}\left(\frac{1}{\sqrt{n^2+4}}-\frac{x}{n+2}\right)\) is convergent.

Answer: 1


Question: An urn contains an unknown number of white balls, W. Five red balls are added to the urn. Balls are drawn one at a time without replacement until the last red ball is drawn on the 13th draw. Find the maximum likelihood estimate (MLE) for the number of white balls, W.

Answer: 8


Question: Determine the last two digits of the sum \( f(17) + f(18) + f(19) + f(20) \), where \( f(x) = x^{x^{x^x}} \).

Answer: 32


Question: Given a Hamiltonian action of a torus on a symplectic 4-manifold \( M \) with a moment map \( \mu: M \to \mathbb{R}^2 \), where \( \mu = (\mu_1, \mu_2) \), determine the expression for the moment map \( \phi \) for the circle action on \( M \) induced by the homomorphism \( S^1 \to T^2 \) given by \( \theta \mapsto (2\theta, 3\theta) \). Express \( \phi \) in terms of \( \mu_1 \) and \( \mu_2 \).

Answer: 2\mu_1 + 3\mu_2


Question: Given 0-1 sequences $a_n$, $b_n$, and $c_n$, with the limits \( a = \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N a_n \), \( c = \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N c_n \), \( d = \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N a_n c_n \), and \( b = \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N b_n \) existing, and given that \( a = b \) and \( d = ac \), determine if \( \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N b_n c_n = d \).

Answer: No


Question: A parabola given by the equation \( f(x) = x^2 \) is tilted by \( \pi/180 \) radians in the Cartesian plane. Determine the x-coordinate of the vertical line that intersects the tilted parabola exactly once.

Answer: \dfrac{\cos^2\left(\frac{\pi}{180}\right)}{4 \sin\left(\frac{\pi}{180}\right)}


Question: A grocery store has 10 apples, 13 pears, 8 oranges, and 2 bananas. Five customers enter the store one after another. Each customer can buy any combination of the remaining items, including buying nothing. How many different combinations of remaining items can exist in the store after all five customers have made their purchases?

Answer: 4158


Question:
Consider a Feller semigroup \((Q_t)_{t\geq 0}\) on a subset \(Y\subset\mathbb{R}^n\) and a continuous function \(f:X\times Y\to\mathbb{R}\) where \(X\subset \mathbb{R}^m\). For each \(x\in X\), the function \(f_x:=f(x,\cdot)\) belongs to \(C_0(Y)\), the space of continuous functions on \(Y\) vanishing at infinity. Assume there exists a bounded open neighborhood \(U(x)\) of \(x\) such that \(f\) is uniformly bounded on \(U(x)\times Y\). Define \(\varepsilon(t,x', y)\) by
\[\varepsilon(t,x', y)=\frac{1}{t}\int_0^t((Q_sf_{x'})(y)-f_{x'}(y))ds.\]
Does \(\varepsilon(t,x', y)\) converge to 0 as \(t\) approaches 0 uniformly for \(x'\in U(x)\), with \(y\) fixed?

Answer: Yes


Question: Determine the radius of convergence for the series \(\sum_{n=1}^{\infty}\frac{x^{n}}{n^{2}{(5+\cos(n\pi/3))^{n}}}\).

Answer: 4


Question: If $a$ is an odd perfect number, does there exist an even perfect number $b$ such that the product $ab$ is a perfect number?

Answer: No


Question: Calculate the area under the curve \( y = \sqrt{x} \) on the interval \([0, 1]\) using Darboux sums.

Answer: \dfrac{2}{3}


Question: Determine the abelianization of the fundamental group \(\pi_1(\mathbb{R}^3 \setminus k)\), where \(k\) is a knot in \(\mathbb{R}^3\).

Answer: \mathbb{Z}


Question: Let $M$ be a module over a commutative ring $R$. If $M^{\otimes n} = 0$ for some $n \geq 3$, is it possible that $M^{\otimes n-1}$ is nonzero? Justify your answer.

Answer: Yes


Question: Use the method of Lagrange Multipliers to find the maximum value of the function \( f(x,y,z) = x + 2y + 3z \) subject to the constraints given by the plane \( x - y + z = 1 \) and the cylinder \( x^2 + y^2 = 1 \).

Answer: 3 + \sqrt{29}


Question: Calculate the greatest common divisor of the polynomial \( p(x) = x^6 + x^5 + 2x^4 + x^3 + 2x^2 + x + 1 \) and its derivative \( p'(x) \).

Answer: x^2 + x + 1


Question: A point $A = (a,b)$ lies on the graph $y = x^2 + 1$. A point $B = (c,d)$ lies within the area defined by $(x+2)^2 + (y+2)^2 = r^2$. Define a matrix $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ such that $M$ always has an inverse (i.e., the determinant of $M$ is not zero). What is the maximum possible value of $50r^2$?

Answer: 40


Question: Determine the radius of convergence of the power series \( \sum_{n=0}^\infty a_n x^n \) given that the coefficients satisfy the recurrence relation \( a_{n-2} + (n^2 + \alpha^2)a_n = 0 \) for all \( n \geq 2 \).

Answer: \infty


Question: Let \( p \) be a prime number. Determine the total number of subgroups of the group \( G = \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z} \).

Answer: p + 3


Question: Evaluate the integral: \[ \int\limits_{0}^\infty \frac{\cos(\pi x)}{x}  \delta \left[ (x^2-1)(x-2) \right]  \mathrm{d}x \]

Answer: -\dfrac{1}{3}


Question: Given that \( T_1 \) is an \( \alpha \times 100\% \) lower confidence limit for \( \theta \) and \( T_2 \) is an \( \alpha \times 100\% \) upper confidence limit for \( \theta \), with the condition \( P(T_1 < T_2) = 1 \), find a \( (2\alpha - 1) \times 100\% \) confidence interval for \( \theta \).

Answer: [T_1, T_2]


Question: Calculate the surface area of the cylinder defined by \(x^2 + z^2 = 1\) that lies between the planes \(x + y - 4 = 0\) and \(y - z + 4 = 0\).

Answer: 16\pi


Question: Determine whether the inequality \( \sum\limits_{j=1}^n\frac{1}{j} > \int_1^{n+1} \frac{dt}{t} \) holds for all positive integers \( n \).

Answer: Yes


Question: Is it true that the Haar measure \(\mu\) on a compact group \(G\) is always completely positive, meaning every nonempty open set has positive measure? Provide a justification for your answer.

Answer: Yes


Question: Determine the exact decay rate of the sequence \( y_t \) defined by the recurrence relation \( y_{t+1} = y_t(1-y_t) \), where the initial value \( y_0 \) is in the interval \((0,1)\).

Answer: \dfrac{1}{t}


Question: What are the last three digits of the product of all odd numbers from 1 to 1000?

Answer: 625


Question: Determine the holonomy group of the 1-dimensional octonionic projective space, denoted as $\mathbb{O}P^1$. Provide your answer in terms of a well-known mathematical group.

Answer: SO(8)


Question: Let $p$ be a prime number. Determine if there exists a prime number $q$ such that $p$ is a primitive root of $q$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the contour integral \( \int_{|z|=1}\frac{z^n}{z-1}\,dz \) where the contour is the unit circle \(|z|=1\). Note that the pole is on the boundary of the contour.

Answer: \pi i


Question: Evaluate the limit: \[ y = \lim_{n \to \infty} \left( \sin \frac{\pi}{2n} \times \sin \frac{2\pi}{2n} \times \cdots \times \sin \frac{(n-1)\pi}{2n} \right) \] Find the value of \( y \).

Answer: 0


Question: Identify the well-known topological space that is homeomorphic to the quotient space of \(\mathbb{R}^2\) induced by the equivalence relation \((a_0, b_0) \sim (a_1, b_1)\) if and only if \(a_0 + b_0^2 = a_1 + b_1^2\).

Answer: \mathbb{R}


Question:
Let \( z_1 \) and \( z_2 \) be the solutions of the equation \( 1 - z + z^2 = 0 \). Evaluate the expression:
\[
E = (z_1^4 - z_1^3 + 2z_1^2 - 2z_1 + 1)^{2005} + (z_2^4 - z_2^3 + 2z_2^2 - 2z_2 + 1)^{2005}
\]

Answer: -1


Question: Determine the cardinality of the set \( S \) of all \( 3 \times 3 \) matrices with integer entries such that \( A A' = I \), where \( A' \) denotes the transpose of \( A \) and \( I \) is the identity matrix.

Answer: 48


Question:
Let $E$ be a set of edges on a graph $G$ with $n$ vertices, and let $\mathbf{x} \in \{-1,1\}^n$. Prove that if $G$ is not Eulerian, then the following sum is zero:
\[\sum_{\mathbf{x}} \prod_{ij\in E} x_i x_j=0\]

Answer: 0


Question: Find the imaginary part of the series \( \sum_{r=1}^\infty \frac{1}{r} \sin \frac{r\pi}{3} \).

Answer: \dfrac{\pi}{3}


Question: Find the maximum possible value of \(\left(\int_{0}^1\sqrt[3]{f(x)}\,dx\right)\left(\int_{0}^1\sqrt[5]{f(x)}\,dx\right)\left(\int_{0}^1\sqrt[7]{f(x)}\,dx\right)\) given that \(f:[0,1]\rightarrow (0,\infty)\) is a continuous function satisfying \(\int_{0}^1f(x)\,dx=1\).

Answer: 1


Question: Find the limit of the sequence \(a_n=\left(\frac{3n^2+7}{2n^2+4n}\right)^{3n}\) as \(n\) approaches infinity.

Answer: \infty


Question: Calculate the Euler Characteristic of the submanifold $M$ in $\mathbb{R}^4$ defined by the equations: $$M = \{(x,y,z,w) \mid 2x^2+2 = z^2+w^2, \ 3x^2+y^2 = z^2+w^2 \}.$$

Answer: 0


Question: Determine whether the expression \(x^{2k} - x^{2k-1} + x^{2k-2} + \ldots + x^2 - x + 1\) is non-negative for all real numbers \(x\) and positive integers \(k\).

Answer: Yes


Question: Calculate the expected value \( \mathbb{E}(e^{B(2)}) \), where \( B(t) \) is a standard Brownian motion with \( B(0) = 0 \).

Answer: e


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

Answer: -\infty


Question: Calculate the conditional expectation \( E[N(6) \mid N(4) = 4, N(10) = 12] \) for a Poisson process \( \{N(t), t \ge 0\} \) with rate \( \lambda = 2 \).

Answer: \dfrac{20}{3}


Question: Is there an infinite group that has only a finite number of subgroups? Answer yes or no.

Answer: No


Question: Let $A$ be an integral Noetherian ring of Krull dimension one and $K$ its field of fractions. Let $B$ be the set of elements of $K$ that are integral over $A$, i.e., $B$ is the normalization of $A$ in $K$. Is the morphism $A \to B$ finite?

Answer: Yes


Question: What is the Fourier transform of the Dirac delta function \( \delta(t) \)?

Answer: 1


Question: Determine the condition on \( \lambda \) for which the series \( \sum_{n=1}^{\infty} (a^{1/n}-1)^{\lambda} \) converges.

Answer: \lambda > 1


Question: Find the smallest integer \( m \in \mathbb{N} \) such that the polynomial \( f(x) = x^5 + 2x^4 + x^3 + 2x^2 + x + 1 \) divides \( x^{3^m} - x \) in \( \mathbb{F}_3[x] \).

Answer: 6


Question: Calculate the 100th derivative of the function \( f(x) = \frac{1}{1+x^2} \) at \( x = 0 \) without using complex numbers.

Answer: 100!


Question: Compute the integral \( \int_T G^* \ \eta_{S^2} \) where \( T \) is the torus obtained by rotating the circle \( \{ (x,0,z) \in \mathbb{R}^3 \ | \ (x-2)^2 + z^2 = 1 \} \) around the \( z \)-axis, \( G : T \to S^2 \) is the Gauss map, and \( \eta_{S^2} \) is the area form on \( S^2 \).

Answer: 0


Question: Minimize the expression \(|a + bw + cw^2|\) where \(w\) is a non-real cube root of unity and \(a, b, c\) are consecutive integers.

Answer: \sqrt{3}


Question:
In the inner product space $P([0,1])$ of all real polynomials on $[0,1]$ with inner product $\langle f,g\rangle=\int_0^1f(x)g(x)dx$, consider $V=\text{span}\{t^2\}$. Find $h(t) \in V$ such that $\|(2t-1)-h(t)\| \le \|(2t-1)-x(t)\|$ for any $x(t) \in V$. Determine $h(t)$ from the options below:

(a) $\frac{5t^2}{6}$

(b) $\frac{5t^2}{3}$

Answer: a


Question: Let \(a_n\) be a sequence of real numbers with a finite Cesaro limit, i.e., \(\frac{a_1+ \cdots + a_n}{n} \rightarrow c\) for some real number \(c\). If \(M_n= \max\{a_1, \cdots, a_n\}\), does \(\frac{M_n}{n} \rightarrow 0\)? Justify your answer.

Answer: No


Question: Find the greatest value of the expression \( P = x^3 + 2y^3 + 2x - y - 1 \) subject to the constraint \( x^2 + y^2 - x - y = 0 \).

Answer: 3


Question: Calculate the area of the planar figure enclosed by the loop of the curve given by \( y^2 = x^2 + x^3 \). Use the parametrization \( x(t) = t^2 - 1 \) and \( y(t) = t^3 - t \) to find the area. Express your answer as a positive number.

Answer: \dfrac{8}{15}


Question: Determine the values of \( p \in (0, \infty) \) for which the integral \( \int_{y=1}^{\infty}\int_{x=1}^{\sqrt{y}} \left| \cos x \sin y \right|^p \frac{1}{y^2} \, dx \, dy \) converges.

Answer: (0, \infty)


Question: Determine the number of ways to select 300 chocolate candies from 7 types, given that each type is selected in boxes of 20 candies, and for each type, between 1 and 5 boxes must be chosen. Use generating functions to find the solution.

Answer: 2415


Question: Let $z$ and $w$ be complex numbers such that $|2z - w| = 25$, $|z + 2w| = 5$, and $|z + w| = 2$. Find $|z|$.

Answer: 9


Question: In a game, Sheldon picks three non-zero real numbers, and Leonard arranges them as the coefficients of a quadratic equation \( ax^2 + bx + c = 0 \). Sheldon wins if the resulting equation has two distinct rational solutions; otherwise, Leonard wins. What is the maximum probability that Sheldon wins the game?

Answer: 1


Question: Determine whether the following statement is true for a function $f: \mathbb{R}^2 \to \mathbb{R}$, which is not necessarily well-behaved: \[ \frac{\partial f(t,u)}{\partial t}\Big|_{u=0} = \frac{\partial f(t,0)}{\partial t}. \] This means that the left-hand side is defined if and only if the right-hand side is defined, and their values are equal in that case.

Answer: True


Question: Let \( f:[a,b]\to\mathbb{R} \) be a smooth \( C^\infty \) function that attains a strict minimum at some point \( x_0 \in (a,b) \). Given that \( f'(x_0)=0 \) and \( f''(x_0) > 0 \), does there exist a neighborhood \( V \) of \( x_0 \) such that \( f(x) \leq f(y) \) for all \( x \in V \) and \( y \in [a,b] \setminus V \)?

Answer: No


Question: If a multivariate function $f: \mathbb{R}^m \rightarrow \mathbb{R}$ has a gradient at a point $x_0$, is it always differentiable at $x_0$? Assume that the gradient is defined such that for any unit vector $v$, the dot product $\nabla f \cdot v$ equals the directional derivative $D_v f(x_0)$. Provide a justification for your answer.

Answer: No


Question: Assume a problem $R \in \mathit{NP}$ can be solved with a verifier $M(x,y)$ that runs in $O(n^3)$ time, where $y$ is additional information no longer than $3\log n$ bits. Determine the function $f(n)$ such that $R \in \mathit{TIME}(f(n))$. Provide your answer as a function of $n$.

Answer: n^6


Question: Calculate \( \lim_{n\to\infty} ((a+b+c)^n+(a+\epsilon b+\epsilon^2c)^n+(a+\epsilon^2b+\epsilon c)^n) \) given that \( a, b, c \in \mathbb{R} \), \( \epsilon \in \mathbb{C} \setminus \mathbb{R} \) with \( \epsilon^3=1 \), \( |a+b+c|<1 \), and \( ab+bc+ac=0 \).

Answer: 0


Question: Evaluate the principal value of the integral: \[ \text{p.v.} \int_{-\infty}^{\infty} \frac{x \sin 4x}{x^2 - 1} \, dx \]

Answer: \pi \cos 4


Question: A student sits for 6 examination papers, each worth 100 marks. In how many ways can the student score a total of 240 marks across all papers?

Answer: 4188528351


Question: Evaluate the surface integral \(\iint _S F\cdot n \space dA\) using the divergence theorem, where \( F = [xy, yz, zx]\) and \(S\) is the surface of the cone defined by \(x^2 + y^2 \le 4z^2\) with \(0 \le z \le 2\).

Answer: 16\pi


Question: Let \( f: [0, \infty) \rightarrow \mathbb{R} \) be a continuous function. Prove that if \( \int_0^{\infty} f(x) \, dx \) converges, then \( \lim_{r \rightarrow \infty} \int_r^{\infty} f(x) \, dx = 0 \).

Answer: 0


Question: Determine the number of nine-digit odd numbers divisible by 375, where the digits from the second to the second-last are in non-decreasing order. If the number is represented as \( n = \overline{a_1a_2a_3a_4\ldots a_9} \), then \( a_2 \le a_3 \le a_4 \le a_5 \le \ldots \le a_8 \).

Answer: 186


Question: Calculate \( \lim\limits_{n \to \infty } \left(\frac{n+1}{n-1}\right)^{3n^{2}+1} \).

Answer: \infty


Question: Given the moment-generating function \( M_Y(s) = \alpha^6(0.1 + 2e^s + 0.1e^{4s} + 0.4e^{7s})^6 \), find the correct value of \( \alpha \) such that \( M_Y(0) = 1 \).

Answer: \dfrac{5}{13}


Question:
Let \( f(x) \) be a continuous function satisfying the differential equation:
\[ f(x) = (1+x^2)\left(1+\int_0^x \frac{f^2(t)\,dt}{1+t^2}\right) \]
Find \( f(1) \).

Answer: -6


Question: Find the area enclosed by the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) using Green's Theorem.

Answer: \pi ab


Question: Determine the number of roots of the polynomial \( f(z) = z^6 - 5z^4 + 3z^2 - 1 \) within the closed disk \( |z| \leq 1 \).

Answer: 4


Question: Evaluate the integral \( \int_{\phi_0}^1 \frac{D(\phi)}{-\alpha(1-\phi)} \,d\phi \) given that \( D \in C^1([0,1]) \), \( D(0)=0 \), \( D(u)>0 \,\,\forall\,\,u\in (0,1] \), and \( \phi_0 \in (0,1) \). Justify why the integral evaluates to \(-\infty\).

Answer: -\infty


Question: Let $f:S\rightarrow S$ be a holomorphic function defined on the unit open square centered at 0. Given that $f(0)=0$ and $f'(0)=1$, find $f''(0)$. Provide your answer as a single value.

Answer: 0


Question: Calculate the work done by the vector field \( F = (\sin(x), 0, 2z) \) along the path \( r = (\rho, \rho, \rho^2) \) for \( 0 \leq \rho \leq \pi \) by constructing a scalar potential function.

Answer: 2 + \pi^4


Question: Two players, A and B, take turns rolling a standard six-sided die. Player A rolls first. If a player rolls a 1 or 2, they win the game. If a player rolls a 3, 4, or 5, the other player gets a turn. If two consecutive 6's are rolled, the game ends in a tie. What is the probability that player A wins the game?

Answer: \dfrac{48}{85}


Question: If $A \subseteq \mathbb{R}^d$ has Lebesgue measure zero, is it always possible to find an $x$ such that $A \cap (A-x) = \emptyset$? Justify your answer.

Answer: Yes


Question: Compute the integral \( \int _0 ^{2\pi} \left(\sum_{n=1}^\infty \frac {\cos(nx)}{2^n}\right)^2 dx \).

Answer: \dfrac{\pi}{3}


Question: Determine which angle is greater: \(\alpha = 2\tan^{-1}(2\sqrt{2} - 1)\) or \(\beta = 3\sin^{-1}\left(\frac{1}{3}\right) + \sin^{-1}\left(\frac{3}{5}\right)\).

Answer: \alpha


Question: Let $T: \mathbb{C}^{11} \to \mathbb{C}^{11}$ be a linear transformation. Given that the dimension of the kernel of $T$ is 4, the dimension of the kernel of $T^3$ is 9, and the dimension of the kernel of $T^4$ is 11, find the dimension of the kernel of $T^2$. Use the rank-nullity theorem and the properties of nilpotent operators to determine your answer.

Answer: 7


Question:
If \(-a^{-b^{-c}}\) is a positive integer and \(a\), \(b\), and \(c\) are integers, which of the following must be true?

(a) \(a\) must be negative
(b) \(b\) must be negative
(c) \(c\) must be negative
(d) \(b\) must be an even positive integer
(e) None of the above

Answer: A


Question: Evaluate the contour integral \( \frac{1}{2 \pi i} \int_{|z| = 1} \frac{6z^{98}}{23z^{99}-2z^{81}+z^4-7} \, dz \) using the residue theorem.

Answer: \dfrac{6}{23}


Question: Determine the winding number of the function \( f(z) = z^3 + \frac{1}{(z-1)^2} \) around the origin as \( z \) moves counterclockwise along the circle \( |z| = 2 \).

Answer: 1


Question: Evaluate the limit: \[ \lim _{x\to 0^+}\left(\frac{\frac{e^{-\frac{1}{x}}}{x^3} +e^{-\frac{1}{\sqrt{x}}}}{e^{-\frac{1}{x}}\ln x}\right) \]

Answer: -\infty


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\sqrt[n]{\frac{1}{n!}\sum_{m=0}^n(m^m)}. \]

Answer: e


Question: Given integers $a, b, c, d$ such that $ad-bc \neq 0$, let $\Gamma = \langle (a,b), (c,d) \rangle$. Determine the index of $\Gamma$ in $\mathbb{Z}^2$. Express your answer in terms of $a, b, c,$ and $d$.

Answer: |ad - bc|


Question: Determine the values of $n$ for which the hypercube graph $Q_n$ has a Hamiltonian circuit.

Answer: n \geq 2


Question: Evaluate the infinite series \( \sum_{n=1}^{\infty} \frac{x_{n}}{n+2} \) where the sequence \( x_n \) is defined by the recurrence relation \( x_{n+2}=x_{n+1}-\frac{1}{2}x_{n} \) with initial conditions \( x_{0}=2 \) and \( x_{1}=1 \).

Answer: \pi - 3


Question: Determine the dimension of the eigenspace $E_1 = \mathrm{kernel}(I-R)$ for the Gaussian Markov operator \( R \) defined on \( L_1(\mathbb{R},\mathcal{B}(\mathbb{R}),Q) \), where \( Q = N(\nu,\tau^2) \). Given that the functions \( \mathbb{1}_{\mathbb{R}}: x \mapsto 1 \) and \( \mathrm{id}_{\mathbb{R}}: x \mapsto x \) are eigenfunctions corresponding to the eigenvalue 1, are there additional linearly independent eigenfunctions in \( E_1 \)?

Answer: 2


Question: Find the number of derangements of the multiset \( \{1, 1, 2, 2, 14\} \), where a derangement is defined as a permutation in which no element appears in its original position or the position of any of its identical elements.

Answer: 4


Question: Let $X_1, X_2, \ldots, X_n$ be independently and identically distributed random variables with probability density function $f(x) = e^{-x}$ for $0 < x < \infty$. Define $Y_n = \sqrt{n}(\bar{X}_n - 1)$, where $\bar{X}_n$ is the sample mean. Determine the limiting distribution of $Y_n$ as $n \to \infty$. Use the Central Limit Theorem in your solution.

Answer: \mathcal{N}(0, 1)


Question: Evaluate the integral \( \int_0^2 \int_{-4\sqrt{4-x^2}}^{4\sqrt{4-x^2}}(x^2-y^2)\,dy\,dx \) using polar coordinates.

Answer: -120\pi


Question: Consider the construction of the Cantor set, where at each step, the middle third of each segment is removed. The total length of the segments removed at each step forms a series. Verify if the series \( \sum_{n=0}^\infty \frac{1}{3}\left(\frac{2}{3}\right)^n \) correctly represents the total length of the segments removed in the construction of the Cantor set.

Answer: Yes


Question: Given that $f(x) + 3x^2 = 2f(1-x)$ and $\lim _{x\to 1}f(x) =7$, find $\lim _{x\to 0} f(x)$. \(\text{The correct answer is } \lim_{x\to 0} f(x)=14.\)

Answer: 14


Question: Can the function \( f(x) = x^{\frac{2}{1-x}} \) be defined at \( x = 1 \) so that it becomes continuous at that point?

Answer: \dfrac{1}{e^2}


Question: Given random variables $X$ and $Y$ with joint probability distribution $P[X = m, Y = k] = \frac{1}{25} \left(\frac{4}{5}\right)^{k-2}$ for $k > 1$ and $k > m \ge 1$, find the covariance $\text{Cov}(X, Y)$. \(\text{Cov}(X, Y) = ?\)

Answer: 20


Question: Suppose $f$ is a non-constant polynomial over a field $K$ and $a \neq b \in K$ such that $f(a) = f(b) = 0$. Let $n \in \mathbb{N}$ be the maximum $k$ such that $(X-b)^k \mid f$. Define $h$ as the polynomial such that $f = (X-b)^n h$. Is $a$ a zero of $h$?

Answer: Yes


Question: Compute the limit: \( \lim_{z\to i} \frac{\arctan(1+z^2)^2}{\sin^2(1+z^2)} \).

Answer: 1


Question: Given that $y_1(x) = e^x$ is one solution of the differential equation $(x - 1)y'' - xy' + y = 0$, find a fundamental set of solutions for the equation.

Answer: \{e^x, x\}


Question: Let \( f: [0, \infty) \rightarrow \mathbb{R} \) be a function such that \( \lim_{x\to \infty} f(x) = m \), where \( m \in \mathbb{R} \). Let \( l > 0 \) be fixed. Find \( \lim_{x\to \infty} \int_{x}^{x+l} f(s) \, ds. \)

Answer: ml


Question:
Find the coefficient of \( x^{12} \) in the expansion of the expression:
\[
\left(\frac{1 - x^{21}}{1 - x}\right)^2 \left(\frac{1 - x^{23}}{1 - x}\right)^3
\]

Answer: 1820


Question: Given independent random variables \(X_i\) for \(i=1, 2, \ldots, n\), each distributed as \(N(0,1)\), and a random variable \(Y = X_1 + 2X_2 + \cdots + nX_n\), find the distribution of the standardized random variable \(Y^* = \frac{Y-E(Y)}{\sqrt{D(Y)}}\).

Answer: N(0,1)


Question: Determine if there is a solution to the differential equation: $$\frac{d}{dx}\left(\frac{d}{dx}\left(f\left(x\right)\right)-f\left(x\right)\right)\cdot \,f\left(x\right)-\frac{d}{dx}\left(f\left(x\right)\right)\cdot \left(\frac{d}{dx}\left(f\left(x\right)\right)-f\left(x\right)\right)=0$$

Answer: Yes


Question: Determine whether the following statement is true or false: In every weighted graph with only positive edges and with at least 11 vertices, there exist at most \( 2n^{2} \) shortest paths between every two vertices, where \( n \) is the number of vertices.

Answer: False


Question: Determine the number of ring homomorphisms from \( \mathbb{Z}[x,y] \) to \( \mathbb{F}_2[x]/(1+x+x^2+x^3) \).

Answer: 64


Question: Given a function \( f(u) \le 2\sqrt{u}f(\sqrt{u}) + 1 \), determine the asymptotic growth of \( f(u) \) in terms of Big O notation.

Answer: O(u \log u)


Question: Determine the number of integers \( n \) for which the ratio \( R = \frac{p(n+2)}{p(n)} \) is an integer, where \( p(n) \) is the partition function of \( n \). For example, \( p(6) = 11 \) and \( p(8) = 22 \), so \( R = 2 \) is an integer in this case. Is this the only case?

Answer: 4


Question: Evaluate the integral \( \int_{-\infty} ^{\infty} \frac{x \sin{3x} }{x^4 +1}dx \) and express it in the form \( \pi^a e^{\frac{-b}{\sqrt{c}}}\sin \big({\frac {d}{\sqrt{e}}}\big) \), where \(a, b, c, d, e\) are positive integers and \(c\) and \(e\) are square-free numbers. Find the value of \(a+b+c+d+e\).

Answer: 11


Question: Calculate the volume enclosed by the surface \( z = \frac{100}{1+(x^2+4y^2)^2} \) and the \((x,y)\)-plane \( z = 0 \).

Answer: 25\pi^2


Question: A straight circular cone with a height of 2 meters and a radius of 1 meter is filled with a liquid. The cone is oriented with the pointy end downward. The density of the liquid at a distance of \( x \) meters from the pointy end is given by \( \rho = 10 - x^2 \) kg/m\(^3\). Determine the center of mass of the liquid in the cone.

Answer: \dfrac{55}{38}


Question:
Evaluate the limit \( \lim_{t \to 0} \frac{1}{t} \int_{-\infty}^{\infty} g\left(\frac{x}{t}\right) f(x) \, dx \), where \( g: \mathbb{R} \to \mathbb{R} \) is a continuous function satisfying the following conditions:

- \( g(x) \ge 0 \) for all \( x \in \mathbb{R} \),
- \( g(x) = 0 \) for all \( |x| \ge 1 \),
- \( \int_{-\infty}^{\infty} g(x) \, dx = 1 \).

Let \( f: \mathbb{R} \to \mathbb{R} \) be any continuous function.

Answer: f(0)


Question: Given an irreducible polynomial \( f(x) \in \mathbb{Q}[x] \) with \( \deg f = 8 \) and \( f(\alpha) = 0 \), let \( K \) be the splitting field of \( f \) over \( \mathbb{Q} \). The Galois group \( G(K/\mathbb{Q}) \) is abelian. If \( H = \{\sigma_1, \sigma_2, \sigma_3, \sigma_4\} \) is the set of automorphisms in \( G(K/\mathbb{Q}) \) that fix \( \alpha \), and \( F = \{x \in K \mid \sigma_1(x) = \sigma_2(x) = \sigma_3(x) = \sigma_4(x)\} \), find \([K : \mathbb{Q}]\).

Answer: 32


Question: Calculate the remainder when \(7^{1,000,000,000,000,000}\) is divided by 107.

Answer: 86


Question: Use the Chinese Remainder Theorem to compute \((1030 \times 989) \mod 3003\).

Answer: 653


Question: Determine whether the following claim is true: For any $n \in \mathbb{N}$, $x \in \mathbb{R}^{n}$, $r > 0$, and $z \in B(x,r)$, where $B(x,r)$ is a closed ball in $\mathbb{R}^{n}$ with center $x$ and radius $r$, there exists a positive constant $C$ such that \[ \int_{B(x,r)}|y-z|^{1-n}\,\mathrm{d}y \leq C r. \]

Answer: True


Question: Calculate the variance of the integral \( \int_0^1 X(t) \, dt \), where \( X(t) \) is a stationary random process with expected value \( E[X(t)] = m \) and covariance function \( r_X(\tau) = 2e^{-\lvert \tau \rvert} \).

Answer: \dfrac{4}{e}


Question: Find the limit \( \lim_{n \to \infty} \left( \frac{a_n}{3} + \frac{a_{n-1}}{3^2} + \frac{a_{n-2}}{3^3} + \cdots + \frac{a_1}{3^n} \right) \) given that \( \lim_{n \to \infty} a_n = a \).

Answer: \dfrac{a}{2}


Question: Compute \( \lim_{x \to 0} \frac{\sin(x)+\cos(x)-e^x}{\log(1+x^2)} \).

Answer: -1


Question: Solve the differential equation \( \frac{d}{dx}(2y y') = (y')^2 \) with the boundary conditions \( y(0) = 0 \) and \( y(1) = 0 \).

Answer: 0


Question: Evaluate the limit: \[ L = \lim_{x \to 0} \left( \frac{\ln((1+x)^{1+x})}{x^2} - \frac{1}{x} \right) \]

Answer: \dfrac{1}{2}


Question: Find the supremum of the function \( \frac{|x|^{2/3}-|y|^{2/3}}{|x-y|^{2/3}} \) within the unit ball centered at the origin, where \( x \neq y \) and \( x, y \in \mathbb{R}^n \).

Answer: 1


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{\sin^2(\pi Rx)}{R(\pi x)^2} \, dx \) for \( R > 0 \).

Answer: 1


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

Answer: 1


Question: Find the natural number \( n \) such that the equation \([\log_2 1] + [\log_2 2] + [\log_2 3] + \dots + [\log_2 n] = 1538\) holds, where \([\cdot]\) denotes the greatest integer function.

Answer: 255


Question: Find the constant $p$ such that the product of any positive number $N_0$ multiplied by successive random numbers between $0$ and $p$ will, on average, neither diverge to infinity nor converge to zero.

Answer: e


Question: Determine the value of \( \lim_{n\rightarrow \infty }\sum_{k=0}^{n} \frac{1}{{n \choose k}} \) if it converges.

Answer: 2


Question:
Determine the interval of convergence for the derivative of the function given by the power series:
\[ f(x) = \sum_{n=1}^{\infty}\frac{(-1)^{n+1}(x-5)^n}{n5^n} \]

Answer: (0, 10)


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

Answer: \dfrac{1}{6}


Question: Given the cyclic group \( \mathbb{Z}_7 \) with generator 3, determine the smallest non-negative integer \( n \leq 5 \) such that \( 3^n \equiv 6 \pmod{7} \).

Answer: 3


Question: Consider a topological space $X$ and a family of connected subsets $\{A_i\}_{i \in I} \subset X$ such that $A_i \cap A_{i+1} \neq \varnothing$ for every $i$ in an ordered index set $I$. Is the union $C = \bigcup_{i \in I} A_i$ necessarily a connected space if $I$ is any ordered set?

Answer: No


Question: Calculate the double integral \( \int\int_R xe^y \, dA \), where \( R \) is the region inside the circle \( x^2 + y^2 = 1 \) and outside the circle \( x^2 + y^2 = 2y \).

Answer: 0


Question: Compute the tensor product \( \mathbb{Z}[1/2] \otimes_{\mathbb{Z}} \mathbb{Z}/3 \).

Answer: \mathbb{Z}/3\mathbb{Z}


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

Answer: e^{ab}


Question: Determine the number of subgroups \( G < \mathbb{Z}^3 \) such that \( \mathbb{Z}^3/G \cong \mathbb{Z}/3\mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z} \).

Answer: 13


Question: Let \( I = [a, b] \) be a closed bounded interval and \( f: I \to \mathbb{R} \) a function such that for every \( x \in I \), there exist \( \delta_x, M_x > 0 \) such that for all \( u \in V_{\delta_x}(x) \), \( |f(u)| \le M_x \). Prove that there exists \( M > 0 \) such that for all \( x \in I \), \( |f(x)| \le M \).

Answer: M


Question: Is it true that in a finite simple group with abelian Sylow 2-subgroups, a Sylow 2-subgroup is not centralized by a nontrivial element of odd order?

Answer: True


Question: Let $p: Y \to X$ be a continuous open surjection between second countable, locally compact, Hausdorff spaces. For each $x \in X$, there is a positive Radon measure $\lambda^x$ on $Y$ such that $\text{supp}(\lambda^x) = p^{-1}(x)$ and for any $f \in C_c(Y)$, the function $x \mapsto \int_{Y} f \, d\lambda^x$ belongs to $C_c(X)$. Is it true that for any compact $K \subseteq Y$, we have $\sup_{x \in X} \lambda^x(K) < \infty$?

Answer: Yes


Question: Given a Markov chain \( \{X_n, n \geq 0\} \) with states 1, 2, 3 and the transition probability matrix \( P = \begin{bmatrix}0 & 0.4 & 0.6 \\ 1 & 0 & 0 \\ 0.3 & 0.3 & 0.4\end{bmatrix} \), and an initial distribution \( (0.5, 0, 0.5) \), calculate \( P(X_1=3|X_2=1) \).

Answer: 0.3


Question: Find the smallest positive number divisible by 10 such that the number of times the digit '1' appears in the sequence from 1 to that number is equal to the number itself.

Answer: 199990


Question: Let $G$ be an abelian group and $Aut(G)$ be the automorphism group of $G$. Does there always exist a nontrivial subgroup $H$ of $Aut(G)$ such that $\gcd(|H|, |G|) = 1$?

Answer: No


Question: Determine the values of \( k \) for which the quadratic form \( Q(x_1, x_2) = (4+k)x_1^2 + 2kx_1x_2 + 2x_2^2 \) is positive semi-definite.

Answer: [-2, 4]


Question: Show that \(-3\lambda\) is an eigenvalue of the operator \(L\) defined by \[ Lu = -\frac{1}{2}\frac{d^2u}{dx^2} + \lambda u - 3u_0^2 u \] where \(u_0\) is a positive radial symmetric nontrivial solution of \[ -\frac{1}{2}\frac{d^2u}{dx^2} + \lambda u - u^3 = 0 \] and that the corresponding eigenfunction is \(u_0^2\).

Answer: -3\lambda


Question: Simplify the combinatorial sum \(\frac{n!}{(n-k)!}-{1\over(n-k)!}{\sum _{m=1}^{k-1} (-1)^{m+1} (n-m)! S(k,k-m)}\) to show that it is equivalent to \((n-k+1)^k\).

Answer: (n - k + 1)^k


Question: Is the analytic map from the unit disk $B(0,1)$ to itself, satisfying $f(0)=\frac{1}{2}$ and $f'(0)=\frac{3}{4}$, unique?

Answer: Yes


Question:
Consider the polynomial \(f_r(x) = x(x-1)(x-2)(x-3) + r\), where \(r\) is a real number. Determine which of the following statements is true:

(A) \(f_r(x)\) has a real root only for finitely many values of \(r\);
(B) \(f_r(x)\) has a real root for every value of \(r\);
(C) \(f_r(x)\) can never have a repeated root;
(D) \(f_r(x)\) can have a repeated root only for finitely many values of \(r\).

Answer: D


Question: Determine the order of the group of torsion points $G_n = \{ x \in G \mid \exists n \in \mathbb{N}: nx = 0 \}$ for an elliptic curve over the complex numbers, where $G$ is an abelian group. Express your answer in terms of $n$. Assume you have already verified that for $n=2$, the order is $4$, and for $n=3$, the order is $9$. What is the general order of $G_n$?

Answer: n^2


Question: A finitely generated abelian group $G$ is generated by elements $a$ and $b$ with orders $|a|=16$ and $|b|=24$, respectively, and the relation $a^{2}=b^{3}$. Determine the order of the group $G$.

Answer: 48


Question: Given real random variables $X$ and $Y$, with $σ_{Y} = 2σ_{X}$ and $\text{corr}(X, Y ) = −\frac{3}{4}$, determine $\text{corr}(X + Y, X − Y )$. Provide your answer as a single value.

Answer: -\dfrac{3}{4}


Question: Let $s$ be a prime number of the form $6m+1$ for some positive integer $m$. Define $\psi(s) = 2s(s-1)$. Determine whether $(\psi(s), \psi(s+2)) = 12$ holds for all such primes $s$. If it does not hold, provide a counterexample.

Answer: 12


Question: Prove or disprove: If \( n \mid (2^n - 1) \), then \( n = 1 \).

Answer: 1


Question:
Given a sequence \((x_n)_{n\geq 0}\) with a fixed initial term \(x_0 > 0\) such that:
\[x_0 + x_1+\ldots+x_n=\frac{1}{\sqrt{x_{n+1}}}\]
Evaluate the limit:
\[\lim_{n\to \infty} n^2 x_{n}^3\]

Answer: \dfrac{1}{9}


Question: Determine the values of $a$ for which the function $f(x)=\frac{\arctan(x)}{\sqrt{x^a+x^{a+2}}}$ has a finite volume when rotated around the x-axis, where $f:(0, \infty) \to \mathbb{R}$.

Answer: (-1, 3)


Question: Calculate the work performed by the force field \( \mathbf{F}(x, y, z) = x^2 \mathbf{\hat{i}} + 4xy^3 \mathbf{\hat{j}} + y^2 x \mathbf{\hat{k}} \) on a particle that traverses the contour \( C \), where \( C \) is the boundary of the rectangular part of the plane \( z = y \) above the rectangle \( R = \{0 \leq x \leq 1, 0 \leq y \leq 3\} \). Use Stokes' theorem to find the work.

Answer: 90


Question:
Consider the one-dimensional wave equation given by:
\[ \frac{\partial^2u}{\partial t^2} - 4\frac{\partial^2u}{\partial x^2} = 0 \]
for \(-\infty < x < \infty\) and \(t > 0\), with the initial conditions:
\[ u(x,0) = \begin{cases} 16-x^2 & \text{if} \ |x| \leq 4 \\ 0 & \text{otherwise} \end{cases} \]
and
\[ \frac{\partial u}{\partial t}\bigg|_{(x,0)} = \begin{cases} 1 & \text{if} \ |x| \leq 2 \\ 0 & \text{otherwise} \end{cases} \]
For \(1 < t < 3\), determine the value of \(u(2,t)\) from the following options:
(a) \(\frac{1}{2}\bigg[16-(2-2t)^2\bigg]+\frac{1}{2}\bigg[1-\min \{1,t-1\}\bigg]\).
(b) \(\frac{1}{2}\bigg[32-(2-2t)^2-(2+2t)^2\bigg]+t\).
(c) \(\frac{1}{2}\bigg[32-(2-2t)^2-(2+2t)^2\bigg]+1\).
(d) \(\frac{1}{2}\bigg[16-(2-2t)^2\bigg]+\frac{1}{2}\bigg[1-\max \{1-t,-1\}\bigg]\).

Answer: D


Question: Determine the number of solutions to the polynomial congruence \( x^4 - 5x - 6 \equiv 0 \pmod{100^{100}} \).

Answer: 8


Question:
Given the sequences defined by:
\[ x_n = 1 + 2 + \dots + n \]
\[ y_n = x_1 + x_2 + \dots + x_n \]
\[ z_n = y_1 + y_2 + \dots + y_n \]
Find the value of \( z_{20} \).

Answer: 8855


Question: Suppose $\lim_{n\to\infty}P(X_{n}\leq x) = P(X\leq x)$ and $A_{n} \stackrel{p}{\longrightarrow} a$, where $a$ is a continuity point of $F_{X}(x) = P(X\leq x)$. Prove or disprove that $\lim_{n\to\infty}P(X_{n}\leq A_{n})=P(X\leq a)$. Justify your answer.

Answer: \lim_{n\to\infty}P(X_{n}\leq A_{n})=P(X\leq a)


Question:
In a metric space $(X, d)$, which of the following statements is true regarding an arbitrary open set $G$ in $X$?

A) $G$ is a countable union of closed sets.
B) $G$ cannot be a countable union of closed sets if $X$ is connected.
C) $G$ is a countable union of closed sets only if $X$ is countable.
D) $G$ is a countable union of closed sets only if $X$ is locally compact.

Select the correct option.

Answer: A


Question: Prove that if an \( n \times n \) matrix \( A \) satisfies \( AB = BA \) for any \( n \times n \) matrix \( B \), then \( A \) must be of the form \( cI \), where \( c \) is a scalar and \( I \) is the identity matrix.

Answer: A = cI


Question:
Determine the values of \( k \) for which the following equation holds:
\[ \frac{w - \overline{w}}{z - \overline{z}} = \frac{k}{|cz + d|^2} \]
where \( w = \frac{az + b}{cz + d} \) with \( a, b, c, d \in \mathbb{R} \) and \( z \in \mathbb{C} \).

Answer: ad - bc


Question: Determine the values of \( p \) for which the integral \( \int_1^2 \frac{\ln(x-1)}{(4-x^2)^p} \, \mathrm{d}x \) converges.

Answer: (-\infty, 2)


Question:
Given that $x, y, z > 0$ and $2(x^2 + y^2 + z^2) \leq 3(x + y + z - 1)$, find the minimum value of the expression:
\[ S = (x + y + z) \left( \frac{1}{\sqrt{2x^3 + x}} + \frac{1}{\sqrt{2y^3 + y}} + \frac{1}{\sqrt{2z^3 + z}} \right) \]

Answer: 3\sqrt{3}


Question: Consider the line $a+bt$ where $a,b>0$. Let $B(t)$ be Brownian motion and let $\tau=\inf\{t>0:B(t)=a+bt\}$ be the first hitting time of that line, with the understanding that $\tau=\infty$ if the line is never hit. Compute the probability that Brownian motion hits that line, i.e., $P(\tau<\infty)$. Express your answer in terms of $a$ and $b$.

Answer: e^{-2ab}


Question: Let \( f \) be a function analytic on the annulus \( \{ z \in \mathbb{C} : 1/2 < |z| < 2 \} \) except for a simple pole at \( z = 1 \). The residue of \( f \) at \( z = 1 \) is 1. Let \( \sum_n a_n z^n \) and \( \sum_n b_n z^n \) be the Laurent series expansions of \( f \) on the annuli \( \{ z : 1/2 < |z| < 1 \} \) and \( \{ z : 1 < |z| < 2 \} \), respectively. Compute \( b_n - a_n \) for every integer \( n \).

Answer: 1


Question:
Consider the sequence \(x_n = \{1, 1/2, 1/4, 1/8, \ldots\}\). Define two sequences based on \(x_n\):

1. \(y_n = x_n^a\)
2. \(z_n = x_n^b |\ln(x_n)|^c\)

Given that \(a > b\), determine which sequence, \(y_n\) or \(z_n\), approaches zero faster as \(n\) increases.

Answer: y_n


Question: Find the maximum value of the expression \( \frac{x^{100}}{1+x+x^2+\ldots+x^{200}} \) for positive \( x \).

Answer: \dfrac{1}{201}


Question: Let $x$ and $y$ be elements of a group $G$ such that $x \neq e$, $|y| = 2$, and $yxy^{-1} = x^2$. Determine the order of $x$, denoted as $|x|$.

Answer: 3


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_{0}^1 \frac{nx^{n-1}}{1+x} \, dx \]

Answer: \dfrac{1}{2}


Question: Find the inverse Laplace transform of the function $s$, i.e., determine $L^{-1}\{s\}$.

Answer: \delta'(t)


Question: Calculate the volume of the solid that lies outside the sphere defined by $r=2$ and inside the surface defined by $r=1+\cos\phi$ using a triple integral.

Answer: 0


Question: Determine whether the following statement is true or false: A set with a linear order where every initial segment is finite is necessarily countable. Provide a proof or a counterexample to support your answer.

Answer: True


Question: Find all invertible $n \times n$ matrices $A$ such that $A^2 + A = 0$. Provide a general form or description of such matrices.

Answer: -I


Question: In the group $G = AB$, where $A$ and $B$ are abelian groups, consider the commutator $[a,b]$ with $a \in A$ and $b \in B$. If $[a,b]$ commutes with $[x,y]$ for $x \in A$ and $y \in B$, does it necessarily follow that $[a,b]$ commutes with $[x^{-1},y^{-1}]$? Provide a proof or counterexample.

Answer: Yes


Question: Find all complex numbers \( z \) such that \(|z-1| = |z+3| = |z-i|\).

Answer: -1 - i


Question: For the curve \( y = x^3 \), a tangent at point \( A \) meets the curve again at point \( B \). The gradient at \( B \) is \( k \) times the gradient at \( A \). Determine the number of integral values of \( k \).

Answer: 1


Question: Evaluate the line integral \( \int _{C} y\,dx + (2x-z)\,dy + (z-x)\,dz \) where \( C \) is the intersection of the sphere \( x^{2}+y^{2}+z^{2}=4 \) and the plane \( z=1 \).

Answer: 3\pi


Question: Determine whether the following statement is true or false: If for any lower semicontinuous function $f : X \rightarrow (0,\infty)$, there exists $\varepsilon > 0$ such that $\varepsilon \leq f(x)$ for all $x \in X$, then $X$ is countably compact.

Answer: True


Question: Determine whether the set \(\{a \in \mathbb{R}^k : p(0) = 1, |p(t)| \le 1 \text{ for } 0 \le t \le 1 \},\) where \(p(t) = a_1 + a_2t + \ldots + a_kt^{k-1}\), is convex.

Answer: Yes


Question: Find the sum of all values of \( k \) such that the system of equations \( y = |x+23| + |x-5| + |x-48| \) and \( y = 2x + k \) has exactly one solution in real numbers.

Answer: 18


Question: Two airlines each operate a flight from country A to country B. A total of 1000 passengers choose between the two airlines, each with a probability of \( \frac{1}{2} \). Each airline offers a plane with 510 seats. What is the probability that at least one passenger does not get a seat on their chosen flight?

Answer: 0.507


Question: Given a population of x values following a known distribution, you draw 10 random samples to create a histogram. Define two random variables, $f_{i}$ and $f_{j}$, representing the counts of x values in the i-th and j-th bins, respectively. If the probabilities for an x value to belong to the i-th and j-th bins are $w_{i}$ and $w_{j}$, respectively, what is the value of $E(f_{i}f_{j})$?

Answer: 90 w_i w_j


Question: Let $C$ be a complex projective smooth hyperelliptic curve of genus $3$, and let $A_1, A_2, A_3$ be three distinct Weierstrass points on $C$. Consider the divisor $D = A_1 + A_2 + A_3$ and the line bundle $L$ associated with $D$. Determine if $h^0(C, L)$ is greater than one.

Answer: No


Question:
Find all functions \( f : \mathbb{N} \to \mathbb{N} \) that satisfy the equation:
\[ f \Big( f \big( f ( n ) \big) \Big) + 6 f ( n ) = 3 f \big( f ( n ) \big) + 4 n + 2001, \forall n \in \mathbb{N}. \]

Answer: f(n) = n + 667


Question: Calculate the surface integral of the vector field \( F = (-y, x, z) \) over the surface defined by the circle \( x^2 + y^2 = 9 \) in the plane \( z = 0 \) for \( x \ge 0, y \ge 0, z \ge 0 \). Use the parameterization \( \phi(r, \theta) = (r \cos \theta, r \sin \theta, 0) \) and verify if the surface integral evaluates to zero.

Answer: 0


Question: Determine the number of singular values of a linear map $T: V \to W$ given that $\dim V = n$ and $\dim W = m$. Provide the correct number of singular values for the linear map $T$. \( \text{(Assume $T$ is represented by a matrix in the standard basis.)} \)

Answer: \min(m, n)


Question: If $T$ is a linear transformation such that $T \circ T = I$, where $I$ is the identity map, is $T$ always diagonalizable?

Answer: Yes


Question:
Evaluate the double integral \( \int_{-\infty}^\infty\int_{-\infty}^\infty e^{-5x^2-5y^2+8xy} \, dx \, dy \). Use the hint that \( \int_{-\infty}^{\infty}e^{-x^2}dx = \sqrt{\pi} \). Choose the correct answer from the following options:

- \( \frac{\pi}{2} \)
- \( \frac{\pi}{3} \)
- \( \frac{\pi}{4} \)
- \( \frac{\pi}{5} \)
- \( \frac{\pi}{6} \)
- \( \frac{\pi}{8} \)

Answer: B


Question: In triangle $\triangle{ABC}$, $\angle{ABC}=90^{\circ}$, $AB=BC$, and $AC=\sqrt{3}-1$. Determine the minimum value of $AP+BP+CP$ for any point $P$ in the plane of $\triangle{ABC}$.

Answer: 1


Question: Let $\Omega$ be a circle of radius $r$ centered at the origin. Define $Z$ as the random variable representing the distance from the center of the circle to a point $(x, y)$ within the circle, and $X$ as the random variable representing the $x$-coordinate of that point. Calculate the covariance $\text{cov}(X, Z)$. Assume the points are uniformly distributed over the area of the circle.

Answer: 0


Question:
Compare the following two expressions to determine which one is smaller:
\[ A = \frac{\sqrt{2007}-\sqrt{2006}}{\sqrt{2008}-\sqrt{2007}} \]
and
\[ B = \frac{\sqrt{2010}-\sqrt{2009}}{\sqrt{2011}-\sqrt{2010}} \]

Answer: B


Question: Evaluate the limit: \[ \lim_{x\to0} \frac{1}{x}\int_0^{2x} (\sin t)^{t} \, dt \]

Answer: 2


Question: Let \(a, z \in \mathbb{C}\) and \(|z| = 1\). Given the function \(f(z) = \frac{z-a}{1-\bar{a}z}\), find the value of \(|z-a|^2 - |1-\bar{a}z|^2\).

Answer: 0


Question: Find the 2022nd derivative of the function \( f(x) = x^{2019} \cos(x) \) at \( x = 0 \). What is \( f^{(2022)}(0) \)?

Answer: 0


Question: In a unital $C^*$-algebra, suppose $x$ and $y$ are two positive elements such that $x \geq 0$ and $y \geq 0$. If $\|x\| \leq \|y\|$, can we conclude that $x \leq y$?

Answer: No


Question: Given the sequence \(x_1, x_2, x_3, \ldots\) defined by \(x_{n+1} = \frac{1}{2}\left(x_n + \frac{\alpha}{x_n}\right)\) with \(\sqrt{\alpha} < x_1\) and \(\alpha > 0\), prove that \(\lim_{n \to \infty} x_n = \sqrt{\alpha}\).

Answer: \sqrt{\alpha}


Question: Let $A$ be a non-negative square matrix. Normalize $A$ by its spectral radius $\sigma(A)$, and call it $A_2 = A/\sigma(A)$. Does this normalization preserve the ratio between the two largest eigenvalues of $A$? Specifically, is the ratio $\lambda_1/\lambda_2$ the same for $A$ and $A_2$, where $\lambda_1$ and $\lambda_2$ are the absolute values of the first and second largest eigenvalues of the two matrices?

Answer: Yes


Question: Assume $a_n > 0$ for all $n$ and that the series $\sum a_n$ converges. Determine $\lim_{n\to \infty} a_n\sqrt{n}$, if this limit exists.

Answer: 0


Question: Determine all entire functions \( f(z) \) such that \( 0 \) is a removable singularity of \( f\left(\frac{1}{z}\right) \).

Answer: f(z) = c


Question: Let \( \rho: G \to GL(V) \) be a finite-dimensional representation of a group \( G \). Show that if the tensor product representation \( \rho \otimes \rho: G \to GL(V \otimes V) \) is irreducible, then the vector space \( V \) is one-dimensional.

Answer: 1


Question: Given a sequence of independent random variables \( A_i \in \{0, 1\} \) for \( i \in N \), where \( P(A_i = 1) = \frac{2}{3i+2} \), simplify the expression \( P\left(\sum_{i \in N} A_i = 1\right) \).

Answer: 0


Question: If \( E \) is a splitting field for a division algebra \( D \), is it always true that \( E \) can be embedded in \( D \)? Provide a justification for your answer.

Answer: No


Question: Is the 0-error capacity of a 7-cycle known to be of the form \(7^q\) for some \(q \in \mathbb{Q}\)?

Answer: No


Question:
Determine the constant of integration \(C\) in the series expansion of the inverse hyperbolic sine function \(\sinh^{-1}(x)\) for \(|x| > 1\). The expansion is given by:
\[ f(x) = \ln(x) - \frac{1}{2} \frac{x^{-2}}{-2} + \frac{-1}{2} \frac{-3}{2} \frac{1}{2!} \frac{x^{-4}}{-4} - \ldots + C \]
Find the value of \(C\).

Answer: \ln 2


Question: Suppose $k$ is an algebraically closed field. According to Hilbert's weak Nullstellensatz, each maximal ideal in $k[x_1, \dots , x_n]$ has the form $\langle x_1 -a_1 , \dots , x_n - a_n\rangle $, where $a = (a_1 , \dots , a_n)$ is a point in affine space $k^n$. Is the point $a$ unique?

Answer: Yes


Question: Find the limit of the following expression as \( n \to \infty \): \[ \sum\limits_{k = 1}^{n - 1}\left(1 + \frac{k}{n}\right)\sin\frac{k\pi}{n^2} \]

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


Question: Find all real solutions to the equation: \[ x^3 - \sqrt[3]{6 + \sqrt[3]{x+6}} = 6 \]

Answer: 2


Question: Determine the number of elements in the quotient ring $\mathbb{Z}[i]/\langle 2+2i \rangle$.

Answer: 8


Question: Find the shortest path from point A to point B on a right-circular cone with vertex O, radius 20 units, and slant height 60 units. Point A is on the base of the cone, and point B is on the line joining OA such that AB = 10 units. The path must always go strictly uphill. Determine the shortest distance from A to B that satisfies this condition.

Answer: 10


Question: Find the power series expansion of the function \( f(x) = x \cos^2(2x) \) at the point \( x = 0 \). Then, determine the value of the 21st derivative \( f^{(21)}(0) \).

Answer: 21 \cdot 2^{39}


Question:
Let $G$ be a finite group of order 100, and let $H$ be a subgroup of $G$ with order 25. Suppose $a \in G$ has order 50. Determine which of the following statements is true:

1) $|\langle a \rangle H| = 50$
2) $a \in H$
3) $G \cong H \times \langle a^2 \rangle$
4) $\langle a \rangle H = G$

Consider the group $G = \mathbb{Z}_{25} \times \mathbb{Z}_{4}$, with $a = (1,2)$ and $H = \langle (1,0) \rangle$. Analyze the options based on this example.

Answer: 1


Question: Find all solutions in positive integers to the equation \(m^3 + 7p^2 = 2^n\), where \(p\) is a prime number.

Answer: (1, 3, 6)


Question: Find the smallest integer \( N \) such that all elements of the symmetric group \( S_{11} \) have orders dividing \( N \).

Answer: 27720


Question:
Given a double sequence \(a_{n,m}\) such that:

- For all \(n \in \mathbb{N}\), \(\lim_m a_{n,m}\) exists and \((a_{n,m})_{m \in \mathbb{N}}\) is non-decreasing.
- For all \(m \in \mathbb{N}\), \(\lim_n a_{n,m} = l\) exists and does not depend on \(m\).

Show that \(\lim_n\lim_m a_{n,m} = \lim_m\lim_n a_{n,m} = l\).

Answer: l


Question: Suppose that \( X \) is a separable and reflexive subspace of \( \ell_{\infty} \). Can \( X^{*} \) be identified with a subspace of \( \ell_{1} \)?

Answer: No


Question: Determine which sequence grows faster as \( n \) increases: \( f(n) = \sum_{i=0}^{\lfloor n/3 \rfloor} \binom{n}{3i+1} \) or \( g(n) = 2^n - 1 \).

Answer: g(n)


Question: Evaluate the integral \( \int_{-\pi/2}^{\pi/2}\ln(\sin x + \sqrt{\sin^2x+e^{\cos x}})\,dx. \)

Answer: 1


Question: Find the extreme values of the function given by the expression \(x^4 + y^4 - 2x^2 + 4xy - 2y^2\).

Answer: -8


Question: Let $k$ be an algebraically closed field and let $p(x) \in k[x]$ be a polynomial of degree $n \geq 1$ without repeated roots. Determine the dimension of the quotient vector space $k[x]/(p(x))$. Is it isomorphic to $k^n$?

Answer: n


Question: Consider a finite abelian group $G$ and let $H$ and $K$ be non-empty subsets of $G$ that are not disjoint. Does the equation $|HK| = \frac{|H| \cdot |K|}{|H \cap K|}$ hold for these subsets?

Answer: No


Question: Find the equation of a plane that is perpendicular to the xy-plane and contains the point (3, 2, 2) on the paraboloid \(36z = 4x^2 + 9y^2\). The cross-section of the paraboloid created by this plane has a slope of 0 at this point. What is the equation of the plane?

Answer: 2x + 3y = 12


Question: Evaluate the contour integral \( \int_{\mathcal{C}} \log(z) \, \mathrm{d}z \), where \( \log(z) \) is the branch of the complex logarithm with a branch cut along the negative real axis, and \( \mathcal{C}(t) = e^{it} \) for \( t \) from \(-\pi\) to \(\pi\).

Answer: -2\pi i


Question: Given that \((AA^t)^r = I\) for an \(m \times n\) real matrix \(A\) and a natural number \(r\), determine the possible values of \(\det(AA^t)\).

Answer: 1


Question:
Let \( \{x_k\}_{k=1}^{n} \) be a sequence whose terms come from the set \( \{2, 3, 6\} \). Given the equations:
\[ x_1 + x_2 + \ldots + x_n = 633 \]
and
\[ \frac{1}{x_1^2} + \frac{1}{x_2^2} + \ldots + \frac{1}{x_n^2} = \frac{2017}{36} \]
find the value of \( n \).

Answer: 262


Question:
Let \( R \) be a subring of \( \mathbb{Q} \) containing 1. Determine which one of the following statements is necessarily true:

A. \( R \) is a Principal Ideal Domain.
B. \( R \) contains infinitely many prime ideals.
C. \( R \) contains a prime ideal which is a maximal ideal.
D. For every maximal ideal \( m \) in \( R \), the residue field \( R/m \) is finite.

Answer: C


Question: Let \( f : \mathbb{C} \rightarrow \mathbb{C} \) be a nonconstant analytic function such that on \( \mathbb{R} \), the function is real-valued and equal to its own derivative. Compute \( \frac{f(i\frac{\pi}{2})}{f(i\pi)} \).

Answer: -i


Question:
Determine the possible set of eigenvalues for a $4 \times 4$ real skew-symmetric, orthogonal matrix from the following options:
1. \( \{\pm i\} \)
2. \( \{\pm i, \pm 1\} \)
3. \( \{\pm 1\} \)
4. \( \{\pm i, 0\} \)

Answer: 1


Question:
Given the linear spaces \( U \) and \( W \) in \( \mathbb{R}^4 \):
\[ U = \text{Span} \{ (2,2,1,-3), (-\lambda,1-\lambda,-1,4) \} \]
\[ W = \text{Span} \{ (1,-2,1,0), (-2,2,0,1) \} \]
Find the value of \( \lambda \) such that \( \dim(U \cap W) = 1 \).

Answer: 3


Question: Evaluate the integral \( \int_{|z| = 1}\frac{1}{(1-3z)(1-2z)^2}dz \) using the residue theorem, where \( z \) is a complex number.

Answer: 0


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

Answer: \dfrac{1}{3}


Question: Given the equation \( \int_{0}^{x} f(t) \, dt = x + \int_{x}^{1} t f(t) \, dt \), find the value of \( f(1) \).

Answer: \dfrac{1}{2}


Question:
Evaluate the integral \( I = \int_0^1 x^{\sin x + \cos x} \, dx \) and find the value of \([10I]\), where \([\cdot]\) denotes the greatest integer function. Choose the correct answer from the following options:

A) 3
B) 4
C) 5
D) 6

Answer: B


Question: Is a Turing machine with an arbitrary finite alphabet equivalent to a Turing machine with the alphabet \( \{0, 1\} \)? Specifically, given a Turing machine \( T \) on some finite alphabet \( \Sigma \supseteq \{0, 1\} \) that calculates a partial function \( \mathbb{N} \to \mathbb{N} \) using binary representation, does there exist a Turing machine \( T' \) using only the alphabet \( \{0, 1\} \) that calculates the same partial function?

Answer: Yes


Question: A box contains 14 balls: 7 black balls numbered from 1 to 7 and 7 white balls numbered from 1 to 7. A person picks 2 balls at a time and repeats the process until the box is empty. Determine the number of ways to draw the balls such that each time two balls are picked, one black and one white, the difference between their numbers is at most one.

Answer: 21


Question: Let $x_1, x_2, \ldots, x_{2014}$ be the roots of the polynomial $x^{2014} + x^{2013} + \cdots + x + 1 = 0$. Find the value of $\sum_{k=1}^{2014} \frac{1}{1-x_k}$.

Answer: 1007


Question: Let $G$ be a group with generators $a, b \in G$. Define $\mathrm{len}:G\to\mathbb{Z}_{\geq 0}$ by sending $g$ to the minimum length of a word in $a, b, a^{-1}, b^{-1}$ equal to $g$. Assume that for all $g \neq e \in G$, there are infinitely many $n \in \mathbb{Z}_{\geq 0}$ such that $\mathrm{len}(g^{n+1}) < \mathrm{len}(g^n)$. Must every element of $G$ be of finite order?

Answer: Yes


Question: Find the greatest value of \( \arg z \) achieved on the circle \( |z-6i|=3 \) in the complex plane.

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


Question: Use Stokes' Theorem to calculate the work done by the vector field \( \overrightarrow{F} = x^2 \hat{i} + 2x \hat{j} + z^2 \hat{k} \) along the anti-counterclockwise oriented boundary of the ellipse \( 4x^2 + y^2 = 4 \) in the xy-plane.

Answer: 4\pi


Question: Determine whether the following statement is true or false: Let \( f_{1}, f_{2}, f_{3}, f_{4} \in \mathbf{R}[x] \) be monic polynomials each of degree exactly two. Then there exists a real polynomial \( p \in \mathbf{R}[x] \) and a subset \( \{i,j\} \subseteq \{1,2,3,4\} \) with \( i \neq j \) such that \( f_{i} \circ p = c f_{j} \) for some \( c \in \mathbf{R} \).

Answer: True


Question: Let \( a \in \mathbb{C} \) be a root of the polynomial \( p(x) = x^5 + \sqrt{2}x^3 + \sqrt{5}x^2 + \sqrt{7}x + \sqrt{11} \). Show that \( a \) is algebraic over \( \mathbb{Q} \) with a degree at most 80.

Answer: 80


Question: Evaluate the limit \( \lim_{\epsilon \to 0} \frac{1}{\epsilon}\int_0^\infty f(x) e^{-x/\epsilon} \, dx \) where \( f \in L^1([0,\infty)) \cap C([0,\infty)) \).

Answer: f(0)


Question: Determine the Galois group of the polynomial \(x^5 + x - 1\) over \(\mathbb{Q}\).

Answer: C_2 \times S_3


Question: Evaluate the limit: \[ \lim_{x\to\infty}\left(\frac{x+n}{\ln(x+n)}-\frac{x}{\ln x}\right) \] for every integer \( n > 1 \).

Answer: 0


Question: Find all positive integer solutions \((x, y, z)\) to the equation \(7^x + 1 = 3^y + 5^z\).

Answer: (1, 1, 1)


Question: Given $n$ people and a list of pairs of people who must not be adjacent, determine the number of possible arrangements of these people. For example, if $n=4$ and the pairs are (1,2), (2,3), and (3,4), find the number of valid arrangements.

Answer: 2


Question: Find the limit \( \lim_{n \rightarrow \infty} \frac{\text{ex}(n; P)}{\binom{n}{2}} \), where \( \text{ex}(n; P) \) is the extremal number for the Petersen graph \( P \).

Answer: \dfrac{1}{2}


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

Answer: -2\pi i


Question: Determine the number of compositions of a positive integer \( n \) where all parts of the composition are odd numbers.

Answer: F_n


Question: Find the value of \( \left\lfloor (1^{-2/3}) + (2^{-2/3}) + (3^{-2/3}) + \cdots + (1000^{-2/3}) \right\rfloor \), where \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \).

Answer: 27


Question: Given that for coefficients $a_{n} \in \mathbb{R}$, the series \( \sum_{n=1}^{+ \infty} a_{n} \frac{1-e^{-n^{2}t}}{n} = 0 \) holds for every \( t > 0 \), does this imply that \( a_{n} = 0 \) for every \( n \in \mathbb{N} \)?

Answer: Yes


Question: Determine whether the set \( \{ (2n+1) \theta \bmod 1: n \in \mathbb{N} \} \) is dense in the interval \([0,1]\), given that \( \theta \in \mathbb{R} \setminus \mathbb{Q} \).

Answer: Yes


Question: Determine the number of integral values of \( p \) for which the roots of the quadratic equation \((4p−p^2  −5)x^2  −(2p−1)x+3p=0\) lie on either side of unity.

Answer: 2


Question: If six points are placed either along the boundary or inside of a unit square, such that the minimum distance of each of these points from the remaining five points is the same, what is the numerical value of this distance?

Answer: \dfrac{1}{2}


Question: Determine the number of non-ordered quadruples \((a, b, c, d)\) of non-negative integers that satisfy the equation \(a + b + c + d = 18\).

Answer: 84


Question:
Evaluate the integral \( \int_{0}^{ \infty} \int_{0}^{ \infty} \frac { e^{-(x+y)}}{x+y} \mathop{dx}\mathop{dy} \) and choose the correct answer from the following options:
(a) infinite
(b) finite but cannot be evaluated in closed form
(c) 1
(d) 2.

Answer: C


Question: Solve the congruence \( x^{11} + x^8 + 5 \equiv 0 \pmod{49} \).

Answer: 29


Question: If $A$ and $B$ are two normal operators such that $A + B = AB$, does it follow that $AB = BA$? Provide a justification for your answer.

Answer: AB = BA


Question: Find the minimum value of \( x^2 + y^2 + z^2 \) for a vector \((x, y, z)\) such that the linear mapping \( T : \mathbb{R}^3 \to \mathbb{R}^3 \) with eigenvalues 0 and 2, and eigenvectors \( v_1 = (0,1,1) \) for eigenvalue 2, and \( v_2 = (1,1,0) \), \( v_3 = (1,0,1) \) for eigenvalue 0, satisfies \( T((x,y,z)) = (0,2,2) \).

Answer: \dfrac{4}{3}


Question: Find the limit of the sequence \( a_k = \left(\dfrac{k^4 11^k + k^9 9^k}{7^{2k} +1}\right) \) as \( k \to \infty \).

Answer: 0


Question: Given a category \(\mathcal{C}\) and a functor \(F: \mathcal{C} \to \mathbf{Set}\) with the following properties: \(F\) is faithful, \(\mathcal{C}\) is complete and \(F\) is continuous, \(\mathcal{C}\) admits filtered colimits and \(F\) commutes with them, and \(F\) reflects isomorphisms. Let \(A, B, C\) be objects in \(\mathcal{C}\) and let \(f: A \to B\) and \(g: C \to B\) be morphisms with \(g\) being a monomorphism. Suppose there exists a map \(\tilde{h}: FA \to FC\) such that \(Ff = Fg \circ \tilde{h}\). Is there a morphism \(h: A \to C\) such that \(Fh = \tilde{h}\) and \(f = gh\)?

Answer: Yes


Question: Find all rational numbers \(\frac{p}{q}\) such that \(0 < p < q\), \(p\) and \(q\) are relatively prime, and \(pq = 25!\).

Answer: 256


Question: In a building with 1,800 light bulbs, the lifetime of each bulb (in days) is modeled as a Poisson process with an expected lifetime of 200 days. Calculate the probability that at most 10 bulbs fail on any given day.

Answer: 0.706


Question: Evaluate the integral $$\int_0^{2\pi}\sum_{n=1}^{N}a_n \sin (nt)\cos(n\xi) \sum_{n=1}^{N} a_n\sin(n\xi)\cos(n\tau) \, d\xi$$ and show that it equals zero.

Answer: 0


Question: Is it possible to color the plane \(\mathbb{R}^2\) using two colors such that any path that is entirely one color is a constant path?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\rightarrow\frac{\pi}{2}^\pm}{\log{\left({\frac{1}{\cos^4x}}\right)}-\tan^2{x}} \]

Answer: -\infty


Question: Evaluate the limit: \[ \lim_{x\to \infty} \left(\frac{f(x+1)}{f(x)}\right)^x \] where \( f(x) = (x - 2)(x - 3)(x - 4)(x - 5) \).

Answer: e^4


Question: Find the base with the smallest cardinality for the topological space (X, \(\tau\)), where X = \(\mathbb{R}^2\) and \(\tau = \{\emptyset, \mathbb{R}^2, \{(x, y) \in \mathbb{R}^2 \mid x^2+y^2<r \}, r \in \mathbb{R}, r>0\}\).

Answer: \aleph_0


Question: Determine the order of the best constant \(\lambda = \lambda(n)\) such that for any \(n \times n\) complex matrix \(A\) with trace zero, there exist \(n \times n\) matrices \(B\) and \(C\) satisfying \(A = BC - CB\) and \(\|B\| \cdot \|C\| \le \lambda \|A\|\).

Answer: O(n)


Question:
Determine which of the following cannot be the order of the quotient group \( \frac{G}{Z(G)} \), where \( Z(G) \) is the center of the group \( G \):

(A) 4
(B) 6
(C) 15
(D) 25

Note: If \( \frac{G}{Z(G)} \) is cyclic, then \( G \) is abelian.

Answer: C


Question: Compute the generic initial ideal (gin) of the ideal \(I = (x^2, y^2)\) in the polynomial ring \(K[x, y]\).

Answer: (x^2, xy)


Question: Compute the limit: \[ \lim_{x \to 0} \left(\frac{e^{x^2} -1}{x^2}\right)^\frac{1}{x^2} \]

Answer: \sqrt{e}


Question: Find the last two digits of \(2^{403}\).

Answer: 08


Question: Find the coefficient of $x^{80}$ in the power series expansion of \[ \frac{x^2(1+x^2+x^5)}{(1-x)^2(1+x)}. \]

Answer: 116


Question:
Given the functional equation \( f(x) - f(y) = f\left(\frac{x-y}{1-xy}\right) \) with domain \((-1,1)\), determine which of the following functions satisfies the equation:

(a) \( 2-\ln\left(\frac{1+x}{1-x}\right) \)
(b) \( \ln\left(\frac{1-x}{1+x}\right) \)
(c) \( \frac{2x}{1-x^2} \)
(d) \( \tan^{-1}\left(\frac{1+x}{1-x}\right) \)

Select the correct function from the options provided.

Answer: B


Question: Let $V$ be an $n$-dimensional vector space over the reals, where $n \ge 1$. Let $L(V)$ denote the vector space of all linear transformations $T:V \to V$, and let $A(V)$ be the set $A(V)=\{T \in L(V) \mid \text{dim}(T(V)) < n\}$. Determine all values of $n$ such that $A(V)$ is a subspace of $L(V)$. Provide your answer as a set of integers.

Answer: \{1\}


Question: Find the explicit form of the square-root of the nonnegative self-adjoint operator $U:L^2(\mu) \to L^2(\mu)$, defined by $f \mapsto \int f \,{\rm d}\mu$, where $(E, \mathcal{E}, \mu)$ is a probability space.

Answer: U


Question: Let \(f_n(x)\) be a sequence of functions such that \(f_n\rightarrow f\) weakly in \(L^2\), meaning for all \(g\in L^2\), \(|\int_{\mathbb{R}}(f_n(x)-f(x))g(x)\,dx|\rightarrow 0\) as \(n\rightarrow\infty\). Given a regular set \(A\subset\mathbb{R}\), such as an interval, does it follow that \(|\int_A(f_n(x)-f(x))g(x)\,dx|\rightarrow 0\)? Provide a justification for your answer.

Answer: Yes


Question: Given that $z_1, z_2, z_3, z_4$ are concyclic complex numbers and the conditions $a_1 + a_2 + a_3 + a_4 = 0$ and $a_1z_1 + a_2z_2 + a_3z_3 + a_4z_4 = 0$ hold, find the value of the expression $a_1|z_1|^2 + a_2|z_2|^2 + a_3|z_3|^2 + a_4|z_4|^2$, where $a_1, a_2, a_3, a_4 \in \mathbb{R}$. Use the fact that $|z|^2 = z\bar{z}$ in your solution.

Answer: 0


Question: Evaluate the integral \( \oint_{\gamma} \frac{\cos z}{z} \, dz \) using the residue theorem, where \( \gamma \) is a closed curve enclosing the singularity at \( z = 0 \).

Answer: 2\pi i


Question: Determine whether the line bundle described by the quotient \( E = \mathbb{C}^{n+1} \times_{\mathbb{C}^\times} \mathbb{C} \) with the action \( t \cdot ((z_0,\dots,z_n),\lambda)=(t(z_0,\dots,z_n),t^m\lambda) \) corresponds to \( \mathcal{O}(m) \) or \( \mathcal{O}(-m) \) on \( \mathbb{P}^n \).

Answer: \mathcal{O}(m)


Question: Consider the curve C, which is the intersection of the two cylinders given by the equations $e^z = x$ and $x^2 + y^2 = 2x$. Find the length of the portion of this curve where $x \geq \frac{3}{2}$.

Answer: 2 \ln 3


Question: Evaluate the surface integral \(\int_S (x^2+y^2+z^2)\,\mathrm{d}S\), where \(S\) is the unit sphere defined by \(x^2 + y^2 + z^2 = 1\).

Answer: 4\pi


Question: Given that \( y = \frac{\sin x}{x} \) is a solution to the differential equation \( xy'' + 2y' + xy = 0 \), find a particular solution to the differential equation \( xy'' + 2y' + xy = 1 \).

Answer: \dfrac{1}{x}


Question: Find the area of the region that lies between the polar curves \( r = 1 + \sin\theta \) and \( r = 1 + \cos\theta \).

Answer: 4\sqrt{2}


Question: Find the number of ways to distribute 22 identical objects to 9 distinct recipients, where 3 of the recipients can receive at most 3 objects each. Create the generating function for this distribution problem and determine the coefficient of \(x^{22}\) in the expansion.

Answer: 2081652


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

Answer: 1


Question:
Given the equations:

1. $a + b + c = 10$
2. $ab + bc + ca = 31$
3. $abc = 30$

Find the value of $2a - b + c$. Provide your answer as a single number.

Answer: 6


Question: Determine the number of solutions to the equation \( z = 4e^z \) that lie inside the unit circle in the complex plane.

Answer: 0


Question: Given the equation \( \frac{x^2 - y^2 + 2y - 1}{y^2 - x^2 + 2x - 1} = 2 \), find the value of \(x - y\).

Answer: \dfrac{1}{3}


Question: Find the maximum value of the function \( f(z) = |z^3 - z + 2| \) for \( |z| = 1 \).

Answer: \sqrt{13}


Question: In the generalized eigenvalue problem $SV = TVD$, where $S$ is a symmetric positive semidefinite matrix, $T$ is a symmetric positive definite matrix, $V$ is the eigenvector matrix, and $D$ is the eigenvalue matrix, can we guarantee that $V$ is orthogonal, i.e., $V^{H} V = I$?

Answer: No


Question: The fraction \( \frac{1}{97} \) has a decimal expansion with a repetend that is 96 digits long. If the last three digits of the repetend are \( A67 \), find the digit \( A \).

Answer: 5


Question: Let $f$ be a twice differentiable function on $(0,1)$ such that $\lim\limits_{x\to0+} f(x)=0$ and the estimates \(|f^{(k)}(x)|\le Cx^{-k},\ k=0,1,2;\ x\in(0,1)\) hold. Determine if it is true that \(\lim_{x\to0+} xf'(x)=0\).

Answer: 0


Question: Given the permutation \( \alpha \in S_{k+l} \) defined by \( \alpha = \left(\begin{array}{cccccc}1&\cdots&k&k+1&\cdots&k+l\\l+1&\cdots&l+k&1&\cdots&l\end{array}\right) \), where \( k, l \in \mathbb{Z}^+ \), determine the number of adjacent transpositions required to express \( \alpha \) as a composition of adjacent transpositions.

Answer: kl


Question: Determine the minimum value of \( A \) such that the inequality \( A(1.05^{n}) > \frac{10(1.05^{n+1}-(n+1)(1.05)+n)}{0.05^{2}} \) holds for all natural numbers \( n \).

Answer: 4200


Question: Determine the number of solutions to the equation \( \{x\}^2 = \{x^2\} \) within the interval \([1, 10]\), where \(\{\cdot\}\) denotes the fractional part function.

Answer: 91


Question:
Calculate the line integral of the tangential component of the vector field \( F(x,y,z) = \nabla f(x,y,z) \) along the curve given by the parametric equations:
\[ x = 8\sin(3\theta)\cos(\theta) \]
\[ y = 8\sin(3\theta)\sin(\theta) \]
\[ z = 8\cos(3\theta) \]
where \( f(x,y,z) = 7y^2 + 5e^{xz} + \ln(6(x^2+y^2+z^2)) \).

Answer: 0


Question: Determine whether the equation $-x + \frac{1}{\sqrt{x}} = c$ can be solved explicitly for $x > 0$, where $c \in \mathbb{R}$.

Answer: Yes


Question: Let \(x\) be an even integer with \(x \geq 2\). Determine the number of positive integers less than \(x^2\) that are divisible by \(x-1\) and have no even digits in their base \(x\) representation.

Answer: 2


Question: Use the divergence theorem to evaluate the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \) for the vector field \( \mathbf{F} = (xy^2, yz^2, x^2z) \), where \( S \) is the sphere of radius 1 centered at the origin.

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


Question: Given a finite morphism \(\pi:X\to Y\) of smooth projective curves over an algebraically closed field, both of genus greater than 1, consider the maps on the level of Jacobians: \(\pi_\ast : J_X \to J_Y\) and \(\pi^\ast : J_Y\to J_X\), satisfying \(\pi_\ast \pi^\ast = [\deg \pi]\). Let \(K\) be the kernel of \(\pi_\ast\), which is an abelian variety. Determine whether the degree of the isogeny from \(J_X\) to \(J_Y \oplus K\) is \(\deg \pi\) or \((\deg \pi)^2\).

Answer: (\deg \pi)^2


Question: Evaluate the triple integral: $$\int_{0}^{5}\int_{-z}^{z}\int_{0}^{\sqrt{z^2-y^2}}3xz~dxdydz$$

Answer: 1250


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

Answer: -1


Question: Determine the number of solutions to the equation \( \sin(x) = x^2 \).

Answer: 2


Question: Calculate the Galois group of the splitting field of the polynomial \(x^4 - 2x^2 - 1\) over \(\mathbb{Q}\).

Answer: D_4


Question: Find the derivative with respect to \( y \) of the integral \( \int_{-\infty}^{y^2} e^{-|x|} \, dx \).

Answer: 2y e^{-y^2}


Question: Suppose that \( f : W \rightarrow \mathbb{R}^n \) is a continuously differentiable mapping defined in an open neighborhood \( W \subset \mathbb{R}^n \) of the origin such that \( f(0)=0 \) and \( f'(0)=I \), the identity matrix. Is it always possible to choose a closed cell in \( W \) such that \( \det(f'(x)) \neq 0 \) for all \( x \) in the closed cell?

Answer: Yes


Question: Evaluate the limit of the series: \[ \lim_{n\to\infty} \sum_{k=1}^{n} \frac{1}{k(k+2)(k+4)} \]

Answer: \dfrac{11}{96}


Question: Determine if the function \( f(x) = \sin^2\left(x \sin \frac{1}{x}\right) \) is right differentiable at \( x = 0 \). If it is, find the right derivative of \( f \) at \( x = 0 \).

Answer: 0


Question:
Consider the function \( f(N_t, N_{t-1}) \) where \( N_t = N^* + n_t \) and \( N_{t-1} = N^* + n_{t-1} \). Determine the correct two-dimensional Taylor expansion of \( f(n_t+N^*, n_{t-1}+N^*) \) around the point \( (N^*, N^*) \). Choose the correct expression:

1. \( f(n_t+N^*, n_{t-1}+N^*) = f(N^*, N^*) + n_tf_{n_t}(N^*, N^*) + n_{t-1}f_{n_{n-1}}(N^*, N^*) + \mathcal{O}(2) \).

2. \( f(n_t+N^*, n_{t-1}+N^*) = f(N^*, N^*) + n_tf_{N_t}(N^*, N^*) + n_{t-1}f_{N_{n-1}}(N^*, N^*) + \mathcal{O}(2) \).

Answer: 2


Question: Determine the region of convergence for the Taylor series expansion of the function \( f(z) = \frac{z}{z-1} \) about \( z = i \).

Answer: |z - i| < \sqrt{2}


Question: Given the limit \( \lim_{x \to 0} \sqrt{x+1} = 1 \) and \( \epsilon = 0.1 \), find the value of \( \delta \) such that for all \( x \) satisfying \( 0 < |x| < \delta \), it follows that \( |\sqrt{x+1} - 1| < 0.1 \).

Answer: 0.19


Question:
Given the following premises:
- $E \land R \implies B$
- $E \implies R \lor P \lor L$
- $K \implies B$
- $\lnot(L \land B)$
- $P \implies \lnot K$

Determine which of the following statements cannot be deduced:

1. $E \land P$
2. $K \land E \implies R$
3. $L \land P \implies \lnot K$
4. $L \implies \lnot(K \land E)$

Answer: 1


Question: Let $X_1, X_2, \ldots$ be a sequence of independent random variables. For $k = 1, 2, \ldots$, suppose $P(X_{2k-1} = 1) = P(X_{2k-1} = -1) = \frac{1}{2}$, and the probability density function of $X_{2k}$ is $f(x) = \frac{1}{\sqrt{2}} e^{-\sqrt{2}|x|}$ for $-\infty < x < \infty$. Determine $\lim_{n \to \infty} P\left(\frac{X_1 + \cdots + X_{2n}}{\sqrt{2n}} \ge 1 \right)$.\n\n(a) $\Phi(1)$\n(b) $\frac{1}{2}$\n(c) $\Phi(-1)$\n(d) $1$\n\nwhere $\Phi(a)$ denotes the cumulative distribution function of the standard normal distribution at $a$. Choose the correct option.

Answer: C


Question: Let \( P_n \) be the vector space of all polynomials with real coefficients of degree at most \( n \). Define the linear transformation \( T: P_n \to P_{n-1} \) by \( T(p)(x) = p'(x) - \int_{0}^{x} p(t) \, dt \). Find the dimension of the null space of \( T \).

Answer: 0


Question: Evaluate the integral \( \int_{0}^{2\pi} \frac{1}{5-3\cos(x)} \,dx \).

Answer: \dfrac{\pi}{2}


Question: Find the smallest positive integer \( n \) such that \((113^{13})^n \equiv 113 \pmod{155}\).

Answer: 37


Question: Find the value of \( k \) such that the straight line \( x + ky = 1 \) intersects the curve \( 5x^2 + 12xy - 6y^2 + 4x - 2y + 3 = 0 \) at two points, and the lines joining these points to the origin are equally inclined.

Answer: -1


Question: Use the method of Lagrange multipliers to find the value of \( \lambda \) for the function \( F = a \log(x^2-y) + b\log(x^3-z) - \lambda (x^2-y+x^3-z -1) \), given the constraint \( x^2-y+x^3-z = 1 \).

Answer: a + b


Question: Determine the smallest integer $k$ such that $|x_k - \alpha| \le 10^{-16} |\alpha|$, where $\alpha$ is a solution of the equation $10x - \sin x = 3$. The iterative method is given by $x_{n+1} = \frac{\sin(x_n) + 3}{10}$ with an initial value of $x_0 = 0.33$. Use the iterative function $F(x) = \frac{\sin(x) + 3}{10}$ and the error term $e_k = |x_k - \alpha|$ to find $k$. Assume $e_{k+1} = |F'(\theta)| \cdot |x_k - \alpha|$ for some $\theta$ between $x_k$ and $\alpha$. Find $k$ such that $\frac{|x_{k+1} - \alpha|}{|F'(\theta) \alpha|} \le 10^{-16}$. Provide your answer as an integer $k$.

Answer: 14


Question: Determine the minimal degree of a smooth curve in $\mathbb{CP}^3$ that is not contained in any cubic surface.

Answer: 7


Question: Calculate the integral \[ W = \int_{{\textstyle\frac{\pi}{2}} + i \log(2)}^{{\textstyle\frac{\pi}{2}} + i \log (5)} \frac{\mathrm{d}z}{\cos(x) \cosh(y) + i \sin(x) \sinh(y)}. \]

Answer: \ln 2


Question: A line from the origin intersects the lines \(\frac{x-2}{1} = \frac{y-1}{-2} = \frac{z+1}{1}\) and \(\frac{x-\frac{8}{3}}{2} = \frac{y+3}{-1} = \frac{z-1}{1}\) at points P and Q, respectively. If the distance between P and Q is \(d\), find \(d^2\).

Answer: 6


Question: A coin-making machine produces pennies, each with a probability $P$ of turning up heads, where $P$ is drawn from a uniform distribution on $[0,1]$. You flip a coin from this machine once, and it comes up heads. Determine the conditional distribution function $F_{P|H}(p)$ for the probability of a head for that coin, given this outcome.

Answer: p^2


Question: Determine the tight big-O bound for the recurrence relation \( T(n) = T(n^{1/2}) + 1 \).

Answer: O(\log \log n)


Question:
Determine the set of points where the function \( f: \mathbb{R} \to \mathbb{R} \) is continuous, given by:
\[
f(x) = \begin{cases}
  x, & \text{if } x \notin \mathbb{Q} \\
  \frac{m}{2n+1}, & \text{if } x=\frac{m}{n}, \ m\in \mathbb{Z}, \ n \in \mathbb{N}, \ \text{and } m \text{ and } n \text{ are coprime}
\end{cases}.
\]

Answer: \{0\}


Question: Determine the number of invertible linear transformations in \( \mathbb{R}^3 \) that map a cube centered at the origin onto itself.

Answer: 48


Question: Determine the number of onto functions \( f : A \to A \) such that \((f \circ f)(1) = 2\), where \( A = \{1, 2, 3, \ldots, 9\} \).

Answer: 35280


Question: For the real vector space \( V \) of polynomials of degree less than or equal to \( d \), define \( \lVert p\rVert_k = \mathrm{max} \{\lvert p(0)\rvert,\lvert p^{(1)}(0)\rvert,\dots,\lvert p^{(k)}(0)\rvert \} \) for \( p \in V \), where \( p^{(i)}(0) \) is the \( i^{\mathrm{th}} \) derivative of \( p \) evaluated at 0. Determine the condition on \( k \) such that \( \lVert p\rVert_k \) defines a norm on \( V \). Choose the correct option: \( 1. \ k \ge d-1 \), \( 2. \ k < d \), \( 3. \ k \ge d \), \( 4. \ k < d-1 \).

Answer: C


Question: Evaluate the limit: \[ \lim_{z \to 0}\left(\frac{\sin z}{z}\right)^{1/z^2} \]

Answer: e^{-\frac{1}{6}}


Question: In how many ways can a random number of identical men be arranged in a line of 15 chairs such that no two men sit next to each other, and each empty chair has at least one man sitting next to it?

Answer: 65


Question: Determine whether the cardinality of the set of all real-valued functions is $2^{\mathfrak c}$, where $\mathfrak c$ is the cardinality of the real numbers.

Answer: 2^{\mathfrak{c}}


Question: Evaluate the integral: \[ \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{x^2 \tan x}{1+\cos^4{x}} \, dx \]

Answer: 0


Question: Calculate the limit: \[ \lim_{n \to \infty} \sin \left( 2\pi \sqrt{n^2 + n} \right) \]

Answer: 0


Question: Let $A$ and $B$ be $n \times n$ complex matrices such that $A$, $B$, and $[A,B] = AB - BA$ are all nilpotent matrices. Is it necessarily true that $\operatorname{trace}(AB) = 0$?

Answer: 0


Question: Compute the sum \( S_n = \sum_{k=0}^{n} \binom{n}{k}(-1)^k(n-k)^n \) and verify that \( S_n = n! \).

Answer: n!


Question: Let \(N_k\) be the number of pairs \((a, b)\) of non-negative integers such that \(ka + (k+1)b = n+1-k\). Find the value of \(N_1 + N_2 + \cdots + N_{n+1}\) for a given positive integer \(n\).

Answer: n+1


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f(3) = 3$ and $f'(3) = \frac{1}{2}$. Evaluate the limit: \[ \lim_{x \to 3} \int_{3}^{f(x)} \frac{2t^3}{x-3} \, dt. \]

Answer: 27


Question: In a semidirect product group where $N$ is a normal subgroup generated by $a$ and $b$, is it true that $ab = ba$? Justify your answer.

Answer: No


Question:
Let \(a, b, c\) be positive real numbers such that \(abc \leq 1\). Find the maximum value of the expression:
\[ P = \sqrt[3]{\frac{a^2 + a}{a^2 + a + 1}} + \sqrt[3]{\frac{b^2 + b}{b^2 + b + 1}} + \sqrt[3]{\frac{c^2 + c}{c^2 + c + 1}}. \]

Answer: \sqrt[3]{18}


Question: Find all complex numbers \( z \) such that \( \sin(z) = 0 \).

Answer: z = k\pi


Question: Find a model \( M \) such that \( M \models (\forall x)(\exists y)R(x,y) \land \neg(\exists y)(\forall x)R(x,y) \).

Answer: M


Question: Let \( f \) be a continuous function on \([0,\infty)\) such that \( \lim_{x\to \infty}(f(x)+\int_0^x f(t)dt) \) exists. Find \( \lim_{x\to \infty}f(x) \).

Answer: 0


Question: Find the value of \( a \) such that \( l = m \), where \( l = \lim_{x \rightarrow a}\frac{x^x-a^x}{x-a} \) and \( m = \lim_{x \rightarrow a}\frac{a^x-x^a}{x-a} \), given that \( a > 0 \).

Answer: e^2


Question: Determine whether the following statement is true for outer measure: For all sequences of sets \(A_j \subset \mathbb{R}^n\), \(j \in \mathbb{N}\), is it true that \[ m^*\left(\bigcup_{j\in \mathbb{N}}A_j\right) = \lim_{N\to\infty}m^*\left(\bigcup_{j=0}^{N}A_j\right) \]?

Answer: True


Question: Estimate the definite integral \( \int_{0}^{1} \frac{\sinh x}{x}\mathrm{d}x \) using the Maclaurin series expansion of \( \frac{\sinh x}{x} = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n+1)!} \). Determine the order \( n \) of the series required to ensure the error is at most \( 10^{-1} \).

Answer: 0


Question: Find an invariant of the recurrence relation \( S_{n+2} = 3S_{n+1} - S_n \) with initial conditions \( S_1 = S_2 = 1 \), in the form \( S^2_{n+1} + aS_nS_{n+1} + bS^2_n \).

Answer: S_{n+1}^2 - 3S_nS_{n+1} + S_n^2


Question:
Consider the series \(a_n=\sum_{k=0}^n 10^{-k^2}\). Determine which of the following sets the limit of this series belongs to:

A) doesn't exist
B) \([-1,0)\)
C) \(\{0,1,2\}\)
D) \(\Bbb Q\)
E) \(\Bbb R\setminus \Bbb Q\)

Answer: E


Question:
Let $f(x)$ be a differentiable function on the real line such that $\lim_{x\to \infty} f(x) = 1$ and $\lim_{x\to \infty} f'(x) = s$. Determine which of the following statements about $s$ is true:

- $s$ should be $0$
- $s$ need not be $0$ but $|s| < 1$
- $s > 1$
- $s < -1$

Answer: A


Question:
Determine the number of continuous functions \( f(x) \) defined on \([0,1]\) that satisfy the following conditions: \( I_1=\int_0^1 f(x)\,dx = 1 \), \( I_2=\int_0^1 x f(x)\,dx = a \), and \( I_3=\int_0^1 x^2 f(x)\,dx = a^2 \). Choose the correct answer from the options below:

(A) 1

(B) 2

(C) \(\infty\)

(D) 0

Answer: C


Question: Suppose $f \in L^1(\mathbb{R})$ is uniformly continuous on $\mathbb{R}$. Does the integrability of $f$ imply that $\lim\limits_{|x|\rightarrow \infty}f(x)=0$? Provide a justification for your answer.

Answer: Yes


Question: Given natural numbers \( u, v, m, n \) such that \( m \neq n \), \( vmn \neq 0 \), \( \gcd(u, m) = 1 \), and \( \gcd(v, n) = 1 \), determine if there exists a natural number \( L > 1 \) such that \( \gcd(u + Lm, v + Ln) = 1 \).

Answer: Yes


Question: Given the recursive sequence defined by \(a_{n+1} = \frac{1}{1-a_n}\) for \(n \geq 1\) with the condition \(a_1 = a_3\), find the value of \((a_{2001})^{2001}\). Choose the correct answer from the following options: \((A)\ 1,\ (B)\ -1,\ (C)\ 0,\ (D)\ \text{None of these}."

Answer: B


Question: Suppose a smooth surface $M$ has the property that for all spheres $S(p,R)$ (the set of all points at a distance $R \geq 0$ from $p \in M$, measured within the surface), there exists a different point $q(p,R) \in M$ and a distance $D(p,R) \geq 0$ such that $S(p,R) = S(q(p,R),D(p,R))$. Is $M$ isometric to a subset of a sphere?

Answer: Yes


Question: Determine whether there exist any positive uniformly integrable martingales that converge almost surely to 0. Justify your answer.

Answer: No


Question:
Determine the minimum number of Fourier coefficients needed to uniquely identify a sparse signal \( f = \sum_{n=1}^N c_n \delta_{t_n} \), where \( N \in \mathbb{N} \), \( c_n \in \mathbb{C} \), and \( t_n \in \mathbb{R} \) for \( n=1, \dots, N \). The \( k \)-th Fourier coefficient of \( f \) is given by
\[
\hat{f}(k) = \sum_{n=1}^N c_n e^{-2\pi i k t_n}.
\]
Find the smallest set \( K \subset \mathbb{Z} \) such that if \( \hat{f}(k) = \hat{h}(k) \) for all \( k \in K \), then \( f = h \), where \( h = \sum_{n=1}^N d_n \delta_{s_n} \) is another linear combination of Dirac delta functions.

Answer: 2N


Question:
Consider a group $G$ containing more than 12 elements of order 13. Determine which of the following statements is correct:

(A) $G$ is cyclic.
(B) $G$ has only one subgroup of order greater than 12.
(C) $G$ is never cyclic.
(D) None of these.

Answer: C


Question: Let $L\subset T^\ast S^n$ be a properly embedded Lagrangian submanifold homeomorphic to $\mathbb{R}^n$ with respect to the canonical symplectic structure on $T^\ast S^n$. Suppose $L$ intersects the zero section $S^n$ at exactly one point. Is $L$ necessarily Hamiltonian isotopic to a cotangent fiber?

Answer: Yes


Question:
Find all the eigenvalues of the differential equation:
\[-\frac{d^2y}{dx^2} + x^2y = \lambda y\]

Answer: 2n + 1


Question: Calculate the probability of drawing at least one repeated ball when drawing 10 times with replacement from a set of 100 balls numbered from 0 to 99.

Answer: 0.3718


Question: Evaluate the limit: $$\lim_{n \to \infty} \left( \lim_{x \to 0} \left(1+\sin^2(x)+\sin^2(2x)+\cdots+\sin^2(nx)\right)^\frac{1}{n^3\cdot x^2}\right)$$

Answer: e^{\frac{1}{3}}


Question: Determine the number of real solutions for the equation: \[ \cos^2 (x\sin(2x))+ \frac {1}{1+x^2}=\cos^2x+\sec^2x. \]

Answer: 1


Question: Calculate the limit: \[ \lim_{T\to\infty}\frac{1}{2T}\int_{-T}^T(\sin(t+1)-1)^4\,dt \]

Answer: \dfrac{35}{8}


Question: Evaluate the sum \( \sum_{k=0}^n \binom{2n + 1}{2k + 1} \).

Answer: 4^n


Question: Determine whether the field extension $[\Bbb Q(5^{1/2}, 5^{1/7}): \Bbb Q(5^{1/7})]$ is a normal extension.

Answer: Yes


Question: Find the area enclosed by the polar curve given by \( r(\theta) = 4 + \sin\theta + \cos\theta \) for \( 0 \le \theta \le 2\pi \).

Answer: 17\pi


Question: Determine the conditions on the constants \(a, b, c\) such that the function \(f(ax+by, cz)\) is harmonic in \(\mathbb{R}^3\).

Answer: a^2 + b^2 = c^2


Question: Let $X$ be a connected finite CW-complex. Suppose $k \geq 3$ is an integer. Can it occur that $\pi_n(X) = 0$ for all $n \geq k$ while $\pi_{k-1}(X) \neq 0$?

Answer: No


Question: Find the principal value of the complex exponential expression \((i^i)^i\).

Answer: -i


Question: Compute the expectation \( \mathbb{E}\left(W_s\int_s^t W_s \,\mathrm{d}s\right) \) where \((W_t)\) is a Brownian motion.

Answer: s(t - s)


Question: Does there exist a function \( f: \mathbb{R}^2 \to \mathbb{R} \) such that the set \( \{ (x, y) \in \mathbb{R}^2 \mid f(x, y) \le 1 \} \) has infinitely many connected components, while the set \( \{ (x, y) \in \mathbb{R}^2 \mid f(x, y) < 1 \} \) is connected?

Answer: Yes


Question: Let $f(n)$ be a non-negative real-valued function defined for each natural number $n$. If $f$ is convex and $\lim_{n\to\infty}f(n)$ exists as a finite number, is $f$ necessarily non-increasing? Provide a justification for your answer.

Answer: Yes


Question: Find all continuous functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the equation \( f(x+y) + f(x-y) = 2[f(x) + f(y)] \).

Answer: f(x) = ax^2


Question: Evaluate the limit: $$\lim_{x \to 0}\frac{\displaystyle\prod_{k=1}^{n}\left(1-\sqrt[k]{\cos x}\right)}{x^{2n-2}}$$

Answer: 0


Question: Given the joint density function of the vector \((X,Y)\) distributed uniformly over the region \(R = \{(x, y) : 0 < y < x < 1\}\), find \(\mathbb{P}(X+Y<1)\). The joint density function is given by \(f_{X,Y}(x,y)= 2\) if \((x,y) \in R\), and \(0\) elsewhere.

Answer: \dfrac{1}{2}


Question: Calculate the volume of the solid obtained by rotating the region bounded by the curve \( y = 2 \cos(x) \), the y-axis, and the x-axis about the y-axis.

Answer: 2\pi^2 - 4\pi


Question: Find all positive integer solutions \((k, n)\) to the equation \(1^n + 2^n + \ldots + n^n = k!\), where \(k!\) denotes the factorial of \(k\).

Answer: (1, 1)


Question: Determine which is larger: the number 3 or the expression \( \sqrt{2 + \sqrt{3 + \sqrt{4 + \sqrt{5 + \sqrt{\ldots + \sqrt{100}}}}}} \).

Answer: 3


Question: Find the limit: \[ \lim_{n \to +\infty}\left(\frac{1^p + 2^p + \cdots + n^p}{n^p} - \frac{n}{p + 1}\right), \] where \( p \in \mathbb{N} \).

Answer: \dfrac{1}{2}


Question: Find the limit of the probability $P(n)$ that $a^2 + b^2 \leq n^2$ as $n$ tends to infinity, where $a$ and $b$ are randomly chosen positive integers not exceeding $n$. Express your answer as a single value.

Answer: \dfrac{\pi}{4}


Question: Determine whether the statement "A submodule of a free module is torsion-free" is true or false. Provide a proof or a counterexample.

Answer: False


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

Answer: \dfrac{1}{3}


Question: Find the smallest positive integer \( N \) such that \( 13^N \equiv 1 \pmod{2013} \).

Answer: 30


Question: Let \( A \) be an \( n \times n \) matrix such that \( A^3 = A + I \). Prove that \( \det A > 0 \).

Answer: \det A > 0


Question: Let $(M,J, \omega)$ be a Kähler manifold, not necessarily compact. Given that the holonomy group is a subgroup of $U_n$, and there exists a constant ($\nabla \Omega = 0$) holomorphic non-vanishing $(n,0)$-form $\Omega$, determine if the holonomy group is contained in $SU_n$. Provide a justification for your answer.

Answer: Yes


Question: A collection of identical spheres can be arranged into a square pyramid, where the base layer consists of $n \times n$ spheres, the next layer has $(n-1) \times (n-1)$ spheres, and so on, up to the top layer of one sphere. The same collection can also be arranged into a single-layer $k \times k$ square, where $k < 100$. Determine the largest possible value of $k$.

Answer: 70


Question: Let $p$ be a prime number and $q = p^2$. Consider the ring $A = \mathbb{Z} / q \mathbb{Z}$ and a finite type module $M$ over $A$ with cardinality $q^N$, where $N$ is a positive integer. Suppose $S$ and $T$ are $A$-submodules of $M$ with cardinalities $q^s$ and $q^t$, respectively. If $s + t > N$, prove that $S \cap T \neq \{ 0 \}$. Additionally, determine a lower bound for the cardinality of $S \cap T$. Provide your answer and justification.

Answer: q^{s + t - N}


Question:
Calculate the limit of the sequence \(a_n\) defined by:
\[
a_n = \sum_{k=1}^{n-1} \frac{k}{n\sqrt{n^2+k^2}}
\]
as \(n\) approaches infinity, using the concept of Riemann integrals.

Answer: \sqrt{2} - 1


Question:
Given the surface parameterized by:
\[ x = u(3v^2 - u^2 - 1/3), \quad y = v(3u^2 - v^2 - 1/3), \quad z = 2uv \ . \]
Find the angle between the coordinate lines at a point on the surface.

Answer: \dfrac{\pi}{2}


Question: Given that \( \int^{2}_{1}x^x \, dx = k \), find the value of \( \int^{2}_{1}x^x \ln x \, dx \) in terms of \( k \).

Answer: 3 - k


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

Answer: 1


Question: Find all natural number pairs \((k, n)\) such that \((2n+1)^{n} = 8n^k + 2n^2 + 1\).

Answer: (1, 2)


Question: Evaluate the limit: \[ \lim_{n \to \infty} e^{-n} \sum_{k = 0}^n \frac{n^k}{k!} \]

Answer: \dfrac{1}{2}


Question: Determine if there exists a positive value of \( p \) such that the divergence of the vector field \( \mathbf{r} / (\| \mathbf{r} \|^p) \) is zero, where \( \mathbf{r} = (x, y, z) \) is the position vector in \( \mathbb{R}^3 \).

Answer: 3


Question: Let $G$ be the group of symmetries of an equilateral triangular lattice $L$. Find the index in $G$ of the subgroup $T \cap G$, where $T$ is the group of translations.

Answer: 12


Question: Find the smallest natural number \( n \) such that the number of divisors of \( \phi(n) \) is 2017, where \( \phi \) is the Euler's totient function.

Answer: 2^{2017}


Question: Let \(A\) be a subset of \(\mathbb{R}^n\), and let \(A^\circ\) and \(\overline{A}\) denote the interior and the closure of \(A\), respectively. Let \(m\) be the Lebesgue measure on \(\mathbb{R}^n\). Is it true that if \(m(A^\circ) = m(\overline{A})\), then \(m(\overline{A} \setminus A^\circ) = 0\)? Justify your answer.

Answer: Yes


Question:
Evaluate the limit of the following expression as \( x \to \infty \):
\[
\lim _{x\to\infty} \frac {3^x 2^0+3^{x-1} 2^1+3^{x-2} 2^2+\cdots+3^0 2^x}{2 (3^x+3^{x-1}+3^{x-2}+\cdots+3^0)}.
\]

Answer: 1


Question: Find the number of solutions to the equation \(a + b + c + d = 100\) where \(1 \leq a \leq 10\), \(b \geq 0\), \(c \geq 2\), and \(20 \leq d \leq 30\).

Answer: 7535


Question: Find the coefficient \( a_{-1} \) in the Laurent series expansion of the function \( f(z) = \frac{1}{2-e^z} \) around the singularity \( z_0 = \log(2) \).

Answer: -\dfrac{1}{2}


Question: Find the maximum Euclidean distance between two different vectors in the simplex \(\Delta_N = \{x \in \mathbb{R}^N : x_i \geq 0 \text{ and } \sum_{i=1}^N x_i = 1\}.\)

Answer: \sqrt{2}


Question: Determine the asymptotic relationship between the functions \( f(n) = \frac{\log n}{\log\log n} \) and \( g(n) = \log(n - \log n) \). Specifically, find whether \( f(n) = K(g(n)) \) for some asymptotic notation \( K \), such as \( \Omega, \Theta, \) or \( O \).

Answer: O


Question:
Let \(A\) be a \(3\times 3\) matrix with complex entries whose eigenvalues are \(1, \pm 2i\). Suppose that for some \(a, b, c \in \mathbb{C}\), the equation \(aA^{-1} = A^{2} + bA + cI\) holds, where \(I\) is the \(3\times 3\) identity matrix. Determine the correct tuple \((a, b, c)\) from the following options:

(A) \((-1, -4, 4)\),
(B) \((4, -1, 4)\),
(C) \((-1, 4, -2)\),
(D) \((-1, -2, 4)\).

Answer: B


Question: Determine the number of subgroups of the group $Z_{p^2} \oplus Z_{p^2}$ that are isomorphic to $Z_{p^2}$, where $p$ is a prime number.

Answer: p(p + 1)


Question: Evaluate the limit: \[ \lim_{(x,y)\to(0,0)} \frac{6(1-\cos(xy))}{x^2y\sin(2y)} \]

Answer: \dfrac{3}{2}


Question: Evaluate the limit: \[ \lim_{n\to \infty} \int_{0}^{n} \frac{1}{1+n^2\cos^2x} \, dx \]

Answer: 1


Question: Circle \(\omega\) is inscribed in a unit square \(PLUM\), and points \(I\) and \(E\) lie on \(\omega\) such that \(U, I,\) and \(E\) are collinear. Determine the greatest possible area of \(\triangle PIE\). Provide a proof for your answer.

Answer: \dfrac{1}{4}


Question: If a function $f$ is strictly convex, is its derivative $f'$ strictly increasing? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit \( \lim_{h\to 0}\frac{f(4h,1)-f(h,1)}{h} \) where \( f(x,y)=\arctan \frac{x}{\sqrt{x^2+y^2}} \).

Answer: 3


Question: Find the multiplicative inverse of $4x+3$ in the ring $\mathbb{Z}_{11}[x]/(x^2+1)$. Express your answer in the form $ax+b$ where $a, b \in \mathbb{Z}_{11}$.

Answer: 6x + 1


Question: Let $f$ be a function defined on the closed interval $[a,b]$. Does the Riemann-Stieltjes integrability of $f^3$ imply the Riemann-Stieltjes integrability of $f$? Provide a proof or a counterexample.

Answer: Yes


Question: Find the largest integer \( n \) such that \( 7^n \) divides \( \binom{2016}{1003} \).

Answer: 3


Question: Let \( B \) be a \( 4 \times 4 \) matrix where each entry is either \(-1\) or \(1\). Prove that \( \det(B) \) is divisible by 8.

Answer: 8


Question: If $G$ is a locally cyclic group, is it true that $\operatorname{Aut}(G)$ is abelian?

Answer: Yes


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

Answer: 2e


Question: Evaluate the limit: \[ \lim_{x\to 0}\dfrac{\cos\left(\frac{\pi}{2}\cos x\right)}{\sin(\sin x)} \] without using L'Hôpital's rule.

Answer: 0


Question: Compute the surface integral \( \iint_{S_1} \mathbf{F} \cdot \hat{\mathbf{n}} \ dS \) for the vector field \( \mathbf{F}(x,y, z) = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{[x^2+y^2+z^2]^{3/2}} \), where \( S_1 \) is the sphere given by \( x^2 + (y-2)^2 + z^2 = 9 \) oriented outwards. Note that \( \mathbf{F} \) is not defined at the origin.

Answer: 4\pi


Question: Evaluate the integral \( \int_{-\infty}^{\infty}\frac{x^4+1}{x^6+1}dx \) using partial fraction decomposition.

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


Question: Is it true that a connected, simply connected, nilpotent \(n\)-dimensional Lie group \(G\) is homeomorphic to \(\mathbb{R}^n\)?

Answer: Yes


Question: Let $P$ be a $7\times7$ matrix of rank 4, and let $\hat{a}$ be any vector in $\mathbb{R}^7$. Determine the minimum rank of the matrix $S = P + \hat{a}\hat{a}^t$.

Answer: 3


Question: A right circular cylindrical can has a volume of \(60\pi\) cubic inches. The metal for the top and bottom costs 4 cents per square inch, while the material for the side costs 2 cents per square inch. Determine the diameter of the can that minimizes the total cost.

Answer: 2\sqrt[3]{15}


Question:
Determine the degree of the polynomial expression:
\[ [x + (x^3 - 1)^{1/2}]^5 + [x - (x^3 - 1)^{1/2}]^5 \]

Answer: 7


Question: Evaluate the limit: \[ \lim _{n \rightarrow \infty} \sum_{k>n+\sqrt n}^\infty \frac{e^{-n}n^k}{k!} \] using properties of Poisson random variables and an appropriate limit theorem.

Answer: 1 - \Phi(1)


Question: A knight is placed on the bottom left corner of a $3 \times 3$ chessboard. Determine the number of ways to move the knight to the upper right corner in exactly 12 moves.

Answer: 992


Question: Consider $k$ Brownian particles diffusing in a three-dimensional solution, each with a diffusion coefficient $D$ (measured in $\mu^2/sec$). Determine the diffusion coefficient of a hypothetical particle located at the centroid of these $k$ Brownian particles.

Answer: \dfrac{D}{k}


Question: Determine whether the number of non-isomorphic posets of cardinality $n$ overwhelms the number of non-isomorphic groups of cardinality $n$. A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$, the inequality $f(n)\leq g(n+k)$ holds only for finitely many $n\in\mathbb{Z}_{\geq 1}$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: Every nontrivial subgroup $H$ of $S_9$ that contains an odd permutation also contains a transposition.

Answer: False


Question: Determine whether the following statement is true or false: Let $A \subseteq \mathbb{R}$ be an arbitrary set and $f:A\to \mathbb{R}$ be a continuous and one-to-one function. Then $f$ is strictly monotone.

Answer: B


Question: Find all pairs of natural numbers \((m, n)\) such that \(2^m m^2 = 9n^2 - 12n + 19\).

Answer: (2, 1)


Question: Let $T: V \to V$ be a linear transformation on a vector space $V$ with an inner product, and let $v \neq 0$ be a vector in $V$. If $T^3(v) = 0$, what can we conclude about $T(v)$? Choose from the following options: $0, v, 2v, 3v, 4v$. Provide a justification for your answer.

Answer: 0


Question:
Compute the determinant of the following 3x3 matrix:
\[A = \begin{pmatrix} a-b-c & 2a & 2a \\ 2b & b-c-a & 2b \\ 2c & 2c & c-a-b \end{pmatrix}.\]

Answer: (a + b + c)^3


Question: Given that $a, b, c, d$ are the real roots of the polynomial $x^4 + 2x^3 - 3x^2 - 3x + 2$, calculate $a^3 + b^3 + c^3 + d^3$.

Answer: -17


Question: Determine the values of \( t \) for which the expression \( \langle p, q \rangle = \int_{t}^{2} p(x)q(x) \, dx \) defines an inner product on the vector space \( V = P_2 \), the space of polynomials of degree at most 2.

Answer: (-\infty, 2)


Question: Determine the optimal constant \( C_{nr} \) such that for all matrices \( A, B \) in \( \mathbf{M}_n(\mathbb{C}) \), the inequality \( r([A,B]) \le C_{nr} r(A)r(B) \) holds, where \( r(A) \) is the numerical radius of \( A \).

Answer: 4


Question: A point is randomly chosen inside a circle with a radius of 1. Let $X$ be the distance from the point to the center of the circle, and $Y$ be the distance from the point to the boundary of the circle. Calculate the expected value of $\min(X, Y)$. Express $Y$ in terms of $X$ and find the expected value.

Answer: \dfrac{1}{4}


Question: Evaluate the integral \( \int_0^\pi \sin x \cdot P_n(\cos x ) \, dx \) where \( P_n \) is the \( n \)-th Legendre polynomial, and determine if it equals zero.

Answer: 0


Question: Determine the Galois group of the splitting field of the polynomial \( f(x) = x^3 - 3x + 1 \) over \( \mathbb{Q} \).

Answer: A_3


Question: Show that a 2-Sylow subgroup of the alternating group \(A_5\) has exactly 5 conjugates.

Answer: 5