Question:
Consider the most powerful test of the null hypothesis that a random variable $X$ follows a standard normal distribution, $X \sim \mathsf{N}(0,1)$, against the alternative hypothesis that $X$ has the probability density function:
\[ f(x) = \frac{2}{\Gamma(1/4)}\exp(-x^4) \quad \text{for } x \in (-\infty, \infty). \]

Calculate the p-value if the observed value of $X$ is 0.6.

Answer: 0.16


Question: Let $a$ be an integer. Determine if it is true that, if every factor of $2^{2^n}a+1$ has the form $2^{n+2}k+1$, then $a=1$.

Answer: No


Question: Let \( \rho: E \to B \) be a covering map with \( B \) connected. Suppose \( \rho^{-1}(\{b_0\}) \) has \( k \) elements for some \( b_0 \in B \). Prove that \( \rho^{-1}(\{b\}) \) has \( k \) elements for all \( b \in B \).

Answer: k


Question: Determine the smallest integer $n$ such that the approximation of $e^x$ by the series $s_n(x) = \sum_{k=0}^{n} \frac{x^k}{k!}$ satisfies the inequality \(|e^x - s_n(x)| \leq \frac{|e^x|}{10^{16}}\) for all $x \in \left(-\frac{1}{2}, \frac{1}{2}\right)$. Provide your answer as the smallest $n$ that meets this condition.

Answer: 14


Question: Determine the computational complexity, in terms of steps, for solving a linear system $Ax = b$ where $A \in \mathbb{R}^{n \times n}$ is a rank-deficient matrix with rank $r < n$. Express the complexity in terms of $r$ and $n$.

Answer: O(n^2 r)


Question: Given that a function \( f(x, y, z) \) satisfies \( f(0, 0, 0) = 33 \), \( D_{\langle 1,1,1 \rangle / \sqrt{3}} f(0, 0, 0) = \frac{4}{\sqrt{3}} \), \( D_{\langle 1,1,0 \rangle / \sqrt{2}} f(0, 0, 0) = \frac{7}{\sqrt{2}} \), and \( D_{\langle 1,2,2 \rangle / 3} f(0, 0, 0) = 12 \), estimate \( f(0.01, -0.001, 0.1) \) using linearization.

Answer: 32.385


Question: A natural number A has three digits added to its end, forming a new number. This new number is equal to the sum of all natural numbers from 1 to A. Find the value of A.

Answer: 1999


Question: Find the radical of the ideal \( I = (y-x^2, y^2) \) in the polynomial ring \( \mathbb{C}[x,y] \).

Answer: (x, y)


Question: Calculate the mass of the region \( \Omega \) in the first quadrant, bounded by the curves \( x^2 + y^2 = 4 \), \( x^2 + y^2 = 9 \), \( x^2 - y^2 = 1 \), and \( x^2 - y^2 = 4 \). The density at each point \((x,y)\in \Omega\) is equal to the product of the distances of the point to the coordinate axes. Use a change of variables to find the mass of \( \Omega \).

Answer: \dfrac{15}{8}


Question: Calculate the limit \( \lim_{n\to \infty}\left(\dfrac{f\left(a+\frac{1}{n}\right)}{f(a)}\right)^{1/n} \) given that \( f:\mathbb{R}\to \mathbb{R} \) is differentiable at \( a \) and \( f(a) > 0 \).

Answer: 1


Question: Three contestants on a quiz show are each given the same question. Each contestant answers correctly, independently of the others, with a probability P, where P is a random variable uniformly distributed on (0, 1]. What is the probability that exactly two of the contestants answer the question correctly?

Answer: \dfrac{1}{4}


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

Answer: -\dfrac{1}{3}


Question: Consider polynomials $f, g \in P^5(-1,1)$ satisfying $||f|| = \sqrt{6}$, $||f+g||= 4$, and $||f-g||=2$, where the norm is the $L^2$ norm. What is $||g||$?

Answer: 2


Question: Given non-zero vectors \( \overrightarrow{a}, \overrightarrow{b}, \) and \( \overrightarrow{c} \) that satisfy the conditions \( \overrightarrow{a} \cdot \overrightarrow{b} = 0 \), \( (\overrightarrow{b} - \overrightarrow{a}) \cdot (\overrightarrow{b} + \overrightarrow{c}) = 0 \), and \( 2|\overrightarrow{b} + \overrightarrow{c}| = |\overrightarrow{b} - \overrightarrow{a}| \), find all possible values of the scalar \( \mu \) such that \( \overrightarrow{a} = \mu \overrightarrow{b} + 4\overrightarrow{c} \).

Answer: 0


Question: Compute the conditional expectation $E(X \mid X > Y)$ where $X$ and $Y$ are independent and identically distributed random variables following a uniform distribution on the interval $(0,1)$. Provide your answer as a single value.

Answer: \dfrac{2}{3}


Question: Let \( (E, d) \) be a metric space, \( x: [0, T] \to E \) be a continuous function, and \( \Omega \subseteq E \) be an open set with \( x(0) \in \Omega \). Define \( \tau = \inf \{ t \in [0, T] : x(t) \not\in \Omega \} \) and \( \sigma = \sup \{ t \in (0, T] : x([0, t]) \subseteq \Omega \} \). How are \( \sigma \) and \( \tau \) related?

Answer: \sigma = \tau


Question: What is the period of the function defined by the series \( \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \)?

Answer: 2\pi


Question: Find the remainder when \(64!\) is divided by 71.

Answer: 7


Question: Evaluate the integral: \[ \int_0^{1}{\sqrt{\frac{x}{1-x}} \, dx} \]

Answer: \dfrac{\pi}{2}


Question: Calculate the norm of the operator \( A: L^2[0,1] \to L^2[0,1] \) defined by \( x(t) \mapsto (t-0.5)\cdot x(t) \). Additionally, identify a function or sequence on which this norm is attained.

Answer: \dfrac{1}{2}


Question: Given the equations \(a+b+c=-3\) and \(a^3+b^3+c^3-20(a+3)(b+3)(c+3)=2013\), where \(a, b, c\) are integers and \(a \le b \le c\), find the value of \(3a+b+2c\).

Answer: -14


Question: Determine all integer pairs $(a, b)$ such that the expression \( \frac{\sqrt{2} + \sqrt{a}}{\sqrt{3} + \sqrt{b}} \) is a rational number.

Answer: (3, 2)


Question: Calculate the sum of the infinite series: \[ S = \frac{1}{2} + \frac{3}{2^2} + \frac{5}{2^3} + \frac{7}{2^4} + \cdots \]

Answer: 3


Question: Compute the limit: \[ \lim_{n\to \infty} \frac{\ln^2n}{n}\sum_{k=2}^{n-2}\frac{1}{\ln k\ln(n-k)} \]

Answer: 1


Question: Let \(a, b, c, d, e\) be five numbers satisfying the following conditions: \(a+b+c+d+e = 0\) and \(abc+abd+abe+acd+ace+ade+bcd+bce+bde+cde=33\). Find the value of \(\frac{a^3+b^3+c^3+d^3+e^3}{502}\).

Answer: \dfrac{99}{502}


Question: Suppose $f$ is analytic in a punctured disk $D_r(z_0) \setminus \{z_0\}$. If $\operatorname{Re}(f(z))$ is bounded in some punctured sub-disk centered at $z_0$, does this imply that $f$ has a removable singularity at $z_0$?

Answer: Yes


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

Answer: 1


Question: Given a random sample \(X_1, \dots, X_n\) from the density function \(f(\theta;x) = \theta x^{-2}I_{[\theta,\infty)}(x)\) where \(\theta > 0\), determine if \(X_{(1)} = \min[X_i]\) is a sufficient statistic for \(\theta\).

Answer: X_{(1)}


Question: Determine whether there exists a non-constant analytic function \( f: \Bbb{C} \to \Bbb{C} \) such that \( f(n) = 3 \) for all integers \( n \).

Answer: Yes


Question: Find the limit of the sequence \(a_n = \frac{3^n(2n)!}{n!(2n)^n}\) as \(n\) approaches infinity.

Answer: \infty


Question: In a locally compact Hausdorff space, let \( \mu \) be a Borel measure that is finite on compact sets, outer regular with respect to open sets, and inner regular with respect to compact sets. Must an atom of such a measure be a singleton, up to a set of zero measure?

Answer: Yes


Question: Let $G$ be a locally finite group such that $G=\bigcup_{i=1}^\infty S_i$, where $S_i$ is a finite group and $S_i \triangleleft S_{i+1}$ for all $i \in \mathbb{N}$. Let $P$ be a Sylow $p$-subgroup of $G$. For each $i \in \mathbb{N}$, define $P_i=S_i \cap P$. Is $P_i$ a Sylow $p$-subgroup of $S_i$ for all $i \in \mathbb{N}$?

Answer: Yes


Question: Evaluate the integral \( \int_0^1 \frac{e^{e^{2\pi it}}}{e^{2\pi it}} \, dt \).

Answer: 1


Question: Let $M$ be an $R[x]$-module that is finitely generated as an $R$-module. Determine whether there exists such an $M$ such that $M \otimes_{R[x]} R[x,x^{-1}]$ is not finitely generated as an $R$-module.

Answer: Yes


Question: Evaluate the integral \( \int_\Gamma (z-y)\,dx-(x+z)\,dy-(x+y)\,dz \) using Stokes' Theorem, where \( \Gamma \) is the intersection of the sphere \( x^2+y^2+z^2=4 \) with the plane \( z=y \). Assume \( \Gamma \) is oriented anticlockwise when viewed from the positive \( z \)-axis. Use \( \nabla \times F = (0, 2, 0) \) to find the result.

Answer: -4\sqrt{2}\pi


Question:
Compute the Gaussian curvature of the unit ball in \(\mathbb{R}^2\) endowed with the Poincaré metric:
\[ds^2=\frac{4(dx^2+dy^2)}{(1-x^2-y^2)^2}.\]
Verify that the curvature is \(-1\) at the point \((0,0)\).

Answer: -1


Question: Calculate the limit: \[ \lim_{n \to \infty}\left(\frac{f\left(a+1/n\right)}{f(a)}\right)^n \] given that \( f(a) = 5 \) and \( f'(a) = 2 \). Assume \( a \) is a real number and \( f \) is continuous at \( a \).

Answer: e^{\frac{2}{5}}


Question:
Under the transformation \( w=\sqrt{\frac{1-iz}{z-i}} \), the region \( D=\{z\in \mathbb{C}:|z|<1 \} \) is transformed to which of the following regions?

1. \( \{z \in \mathbb{C}: 0 <\arg z <\pi\} \)
2. \( \{z \in \mathbb{C}: -\pi <\arg z <\pi\} \)
3. \( \{z \in \mathbb{C}: \pi/2 <\arg z <\pi \,\,\text{or}\,\, 3\pi/2<\arg z<2\pi \} \)
4. \( \{z \in \mathbb{C}: 0 <\arg z <\pi/2 \,\,\text{or}\,\, \pi<\arg z<3\pi/2 \} \)

Select the correct option.

Answer: 4


Question: Is there an infinite field \( F \) with characteristic \( p \) (where \( p \) is a prime number) that is not algebraically closed?

Answer: Yes


Question: Determine the quotient space of \( C[0,1] / C_{0}[0,1] \), where \( C_{0}[0,1] = \{ x \in C[0,1] : x(0) = 0 \} \). Is the quotient space isomorphic to \( \mathbb{R} \)?

Answer: \mathbb{R}


Question: Find the Galois group $Gal(E/\Bbb{Q})$, where $E$ is the splitting field of the polynomial $f(x) = x^4 - 10x^2 + 1$ over $\Bbb{Q}$. The roots of $f(x)$ are $\sqrt{2}+\sqrt{3}$, $\sqrt{2}-\sqrt{3}$, $-\sqrt{2}+\sqrt{3}$, and $-\sqrt{2}-\sqrt{3}$. Determine the structure of $Gal(E/\Bbb{Q})$.

Answer: \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}


Question:
Let $G_1$ and $G_2$ be the images of the disc $\{z\in \mathbb C:|z+1|<1\}$ under the transformations $w=\frac{(1-i)z+2}{(1+i)z+2}$ and $w=\frac{(1+i)z+2}{(1-i)z+2}$ respectively. Determine which of the following statements is correct:

(a)  $G_1=\{w\in \mathbb C:Im(w)<0\}$ and $G_2=\{w\in \mathbb C:Im(w)>0\},$
(b)  $G_1=\{w\in \mathbb C:Im(w)>0\}$ and $G_2=\{w\in \mathbb C:Im(w)<0\},$
(c)  $G_1=\{w\in \mathbb C:|w|>2\}$ and $G_2=\{w\in \mathbb C:|w|<2\},$
(d)  $G_1=\{w\in \mathbb C:|w|<2\}$ and $G_2=\{w\in \mathbb C:|w|>2\}.$

Answer: B


Question: Let $H_1$ and $H_2$ be two Hilbert spaces, and let $T: H_1 \to H_2$ be a one-to-one, onto linear map. Suppose there exist constants $C_1, C_2 > 0$ such that \( C_1 \lVert u \rVert_1 \leq \lVert Tu \rVert_2 \leq C_2 \lVert u \rVert_1 \) for all $u \in H_1$. Is it possible to construct an onto isometry from $H_1$ to $H_2$ using $T$?

Answer: Yes


Question: Determine the number of generators in the group \((\mathbb{Z}_{100}^*, \cdot_{100})\).

Answer: 0


Question: Let $A$ be a free abelian group with a basis $\{x_1, x_2, x_3\}$. Let $B$ be the subgroup of $A$ generated by $x_1 + x_2 + 4x_3$ and $2x_1 - x_2 + 2x_3$. Find the order of the coset $(x_1 + 2x_3) + B$ in the quotient group $A/B$. Provide your answer as a single integer.

Answer: 3


Question: Compute the integral \( \int_{C}\phi(t)dW(\phi) \), where \( C \) is the space of all continuous functions \( f:[0,T] \to \mathbb{R} \), \( W \) is the Wiener measure, and \( t \in [0, T] \).

Answer: 0


Question: If $X_n \rightarrow X$ in distribution, does it follow that for $f \in C$, $f(X_n) \rightarrow f(X)$ in distribution?

Answer: Yes


Question: Evaluate the limit \( \lim_{n\to\infty} \frac{a_n}{n^2} \), where \( a_n \) is the sum of all roots \( x_1, x_2, \ldots, x_k \) of the equation \( f(x) = \frac{x}{2n} \) for \( f(x) = \sin(\pi x) \).

Answer: 0


Question: Calculate the directional derivative of the function \( f(x,y) = \frac{x^2y}{x^4+y^2} \) at the point \((0,0)\) in the direction making an angle of \(30^{\circ}\) with the positive \(x\)-axis. Determine if this derivative exists.

Answer: \dfrac{3}{2}


Question: A non-zero polynomial with real coefficients satisfies the equation \( f(x) = f'(x) \cdot f''(x) \). Determine the leading coefficient of \( f(x) \).

Answer: \dfrac{1}{18}


Question: Let $f : X \to I$ be a strictly monotone surjective function where $X \subseteq \mathbb{R}$ and $I \subseteq \mathbb{R}$ is an interval. Is $f$ necessarily continuous? Justify your answer.

Answer: Yes


Question: Evaluate the integral \( \int_C \left[ \frac{6}{(z-i)^2} + \frac{2}{z-1} + 1 - 3(z-i)^2 \right] \, dz \) where \( C \) is the circle \( |z-i|=4 \) traversed once counterclockwise.

Answer: 4\pi i


Question: For the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = x^r \cdot \sin(1/x) \) for all \( x \neq 0 \) and \( f(0) = 0 \), determine the values of \( r > 0 \) where \( f'(0) \) exists, given that \( r \) is rational.

Answer: r > 1


Question: Calculate the limit: \[ \lim_{n\rightarrow +\infty}n\int_X\log\biggl(1+\frac{f(x)}{n}\biggl)d\mu \] where \( \mu \) is a strictly positive measure, and \( f:X\rightarrow[0,\infty] \) is a function such that \( \int_X f d\mu = 1 \).

Answer: 1


Question: Determine whether the following statement is correct: If \( \lim_{x \rightarrow a} f(x)= \infty \) and \( \lim_{x \rightarrow a} g(x)=\infty \), and \( \lim_{x \rightarrow a} \frac{f'(x)}{g'(x)}=L \), then \( \lim_{x \rightarrow a} \frac{f(x)}{g(x)}=L \).

Answer: Correct


Question: Find the limit of \((x^2+y^2)\sin\left(\frac{1}{xy}\right)\) as \((x, y)\) approaches \((0, 0)\).

Answer: 0


Question: Calculate the mass of the solid that lies above the surface $z=0$, below the surface $z=y$, and inside the surface $x^2+y^2=4$ with the density function $\rho(x,y,z) = yz$. Use cylindrical coordinates to set up and evaluate the integral.

Answer: \dfrac{64}{15}


Question: Determine the values of \(a\) for which the equation \(100^{-|x|} - x^2 = a^2\) has the maximum number of solutions.

Answer: (-1, 1)


Question: Consider the cubic curve given by the equation \( y = x^3 + ax^2y + bxy^2 + cy^3 \). If \((t, r)\) and \((t, s)\) are two distinct points on this curve, is it necessarily true that \(t = 0\)? Provide a justification for your answer.

Answer: No


Question: A hot air balloon is ascending at a rate of \(10\frac{ft}{sec}\). How fast is the distance from the balloon to the horizon increasing when the balloon is \(1,000\) feet high? Assume the Earth is a sphere with a radius of \(4000\) miles.

Answer: 1028


Question: Evaluate the limit: \( \lim\limits_{x\to 0}\sin^{\sqrt{2x}}(x) \)

Answer: 1


Question: Does there exist a field \( F \) that contains an isomorphic copy of the multiplicative group \((\mathbb{Z}/p^n\mathbb{Z})^*\) as a subgroup, where \( p \) is an odd prime and \( n > 1 \)?

Answer: Yes


Question:
Solve for \( x \) in the system of modular equations:
\[ 7x + 9y \equiv 0 \pmod{31} \]
\[ 2x - 5y \equiv 2 \pmod{31} \]

Answer: 29


Question: Does there exist a maximal Archimedean ordered field? Specifically, is there an Archimedean ordered field \( \mathbb{F} \) such that any Archimedean ordered field, including \( \mathbb{R} \), must be contained in \( \mathbb{F} \)?

Answer: \mathbb{R}


Question: Evaluate the integral: \[ \int_{-1}^1 x \frac{\sqrt{1-x}}{\sqrt{1+x}} \, dx \]

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


Question: Determine whether all irreducible polynomials in \(\mathbb{F}_2[x]\) map to a prime number when evaluated at 2 using the evaluation map \(E_2: \mathbb{F}_2[x] \to \mathbb{Z}\), where \(E_2(f(x)) = f(2)\).

Answer: No


Question: Verify the following property: If $X$ and $Y$ are jointly normal random variables with zero means and correlation coefficient $\rho$, then the correlation coefficient between $X^2$ and $Y^2$ is $\rho^2$. Is this property true?

Answer: \rho^2


Question: Let \( \phi : G \to H \) be a group homomorphism with kernel \( K \). If \( \phi(g) = h \), find the fiber over \( h \) in \( G \), i.e., the set \( \phi^{-1}(h) \).

Answer: gK


Question: Let $E \subset F$ be a field extension with degree $n$. Determine the degree of the extension $E(x) \subset F(x)$, where $E(x)$ and $F(x)$ are the rational function fields over $E$ and $F$, respectively.

Answer: n


Question: Find the smallest possible value of $x+y$ given that $x$ and $y$ are positive real numbers satisfying the inequality $2x + 32y \leq 9xy$. Provide your answer as a single number.

Answer: \dfrac{50}{9}


Question: Prove that the polynomial \(X^4 - 4X^2 - 1\) is irreducible over \(\mathbb{Q}[X]\).

Answer: X^4 - 4X^2 - 1


Question: Find the linear operator L such that \( Lu = 0 \) for the function \( u = 2t + e^{-t} \).

Answer: D^3 + D^2


Question: If both a matrix $A$ and its inverse $A^{-1}$ have entries that are non-negative integers, can we conclude that $A$ is a permutation matrix?

Answer: Yes


Question: Let $P$ and $A$ be $3 \times 3$ real matrices such that $PAP^t = -A^t$, where $P^t$ denotes the transpose of $P$. Find $\det(P)$.

Answer: -1


Question: Let \( f \) be a differentiable function on \( \mathbb{R} \) satisfying the functional equation \( f(x+y) = f(x)f(y) \). Given that \( f(1) = 2 \), find the function \( f(x) \).

Answer: 2^x


Question: Evaluate the limit: \[ \lim_{x\to\infty} \sum_{k = 0}^\infty \frac{2x^2}{(x + k)^3}. \]

Answer: 1


Question: Evaluate whether the integral \( \int_0^\infty \sum_{n=0}^{\infty} \frac{x^n}{2^{(n+1)^s x^{n+1}} + 1} \, dx \) is equal to the Riemann zeta function \( \zeta(s+1) \) for all \( s \ge 1 \). Here, \( \zeta(s) = \sum_{n=0}^{\infty} \frac{1}{(n+1)^s} \) is the zeta function defined for \( \Re(s) > 1 \).

Answer: \zeta(s+1)


Question: Calculate the flux of the vector field \( \mathbf{F} = \langle 1, 3, 1 \rangle \) across the surface \( S \), which is the part of the plane \( 4x + 2y + z = 2 \) that lies in the first octant and is oriented upward.

Answer: \dfrac{11}{4}


Question: Prove that the quotient space $\mathbb{R} \text{P}^{n} / \mathbb{R} \text{P}^{n-1}$ is homeomorphic to the sphere $S^n$.

Answer: S^n


Question: In \(\mathbb{R}^2\) with the Euclidean inner product, let \(u\) and \(v\) be orthonormal vectors. Find a vector orthogonal to the vector \(u+v\).

Answer: u - v


Question: Evaluate the expression \(3\tan^{-1}\left(\frac{1}{2}\right) + 2\tan^{-1}\left(\frac{1}{5}\right) + \sin^{-1}\left(\frac{142}{65\sqrt{5}}\right)\).

Answer: \pi


Question: Find the value of the expression: \[ \sec\frac{\pi}{11} - \sec\frac{2\pi}{11} + \sec\frac{3\pi}{11} - \sec\frac{4\pi}{11} + \sec\frac{5\pi}{11} \]

Answer: 6


Question: Find the limit: $$\lim_{x\to\infty} \frac{ 1}{x \sqrt{x} } \int_{0}^{x} \ln \frac{t^{3} + 2}{t^{2} +1 } dt $$

Answer: 0


Question: Find the locus of the foot of the perpendicular from a focus to a tangent of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) using the parametric equations \(x = a\cos\theta\) and \(y = b\sin\theta\).

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


Question:
Consider the matrix \( A = \begin{bmatrix}1 & \tan x\\-\tan x & 1\end{bmatrix} \). Define the function \( f(x) = \begin{vmatrix}A^{T}A^{-1}\end{vmatrix} \). Determine which of the following cannot be the value of \( g(x) = \underbrace{f(f(f(\cdots f(x)))))\}_{n\text{ times}} \) where \( n \geq 2 \):

A) \( f^n x \)
B) \( 1 \)
C) \( f^{n-1} x \)
D) \( n f(x) \)

Select the option that cannot be the value of \( g(x) \).

Answer: D


Question: Compute the limit: \[ \lim\limits_{n\rightarrow\infty} \int_{\mathbb{R}^n_+}g \left( \frac{x_1+ \cdots +x_n}{n} \right)f(x_1,\ldots,x_n) \, dx_1\cdots dx_n \] where $\xi_1,\xi_2,\ldots,\xi_n$ are independent Exp($1$) random variables with joint density $f(x_1,\ldots,x_n) = e^{-(x_1+\cdots+x_n)}$, $g$ is a bounded continuous function, and $\mathbb{R}_+^n = (0,\infty)^n$. Justify your answer.

Answer: g(1)


Question: Does there exist a null set of real numbers $N$ such that every null set can be covered by countably many translations of $N$?

Answer: No


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{\lfloor x\cdot 10^n\rfloor}{10^n} \] where \( x \in \mathbb{R} \setminus \mathbb{Q} \).

Answer: x


Question:
Given the equations:
\[ a - \frac{1}{a} = b, \quad b - \frac{1}{b} = c, \quad c - \frac{1}{c} = a \]
Find the value of the expression:
\[ \frac{1}{ab} + \frac{1}{bc} + \frac{1}{ac} \]
Choose the correct answer from the options:
\[ \text{a) } -3 \quad \text{b) } -1 \quad \text{c) } -6 \quad \text{d) } -9 \]

Answer: A


Question: Determine whether the function \( h(x,y) = \frac{\arccos[(\cos x)(\cos y) + k(\sin x)(\sin y)]^2}{(\sin^2 x) - 2k(\sin x)(\sin y) + (\sin^2 y)} \) has a limit of 1 as \((x,y)\) approaches \((0,0)\), where \(-1 \leq k \leq 1\).

Answer: 1


Question: Calculate the flux integral \( \int_S \varphi \) where \( S \) is the graph of \( z = 25 - (x^2 + y^2) \) over the disk \( x^2 + y^2 \leq 9 \) and \( \varphi = z^2 \, dx \wedge dy \).

Answer: 3843\pi


Question: Determine whether the following statement is true or false: If $E/L$ and $L/K$ are normal extensions, then $E/K$ is a normal extension.

Answer: B


Question:
Determine whether the following equation is true:
\[ E[A|C] = E[A|B \cap C]P(B|C) + E[A|!B \cap C]P(!B|C) \]

Answer: True


Question: Determine whether there are any solutions to the Diophantine equation \(p^{2n} + 2 = 3^m\) other than \((p, m, n) = (5, 3, 1)\), where \(p\) is a prime and \(m, n\) are positive integers.

Answer: (5, 3, 1)


Question: Evaluate the integral \( \int_C \operatorname{Log}\left(1-\frac{1}{z} \right)\,dz \) where \( C \) is the circle \( |z|=2 \).

Answer: -2\pi i


Question: Given the function \( f(x) = x^3 - x^2 + 3x - 1 \) with \( f(1) = 2 \), find \( g'(2) \) where \( g(x) = f^{-1}(x) \).

Answer: \dfrac{1}{4}


Question: Evaluate the line integral \(\int\limits_C (x^2+y^2)\, dx + 2xy \,dy\) where \(C\) is the path of the semicircular arc of the circle \(x^2+y^2=4\) starting at \((2,0)\) and ending at \((-2,0)\) going counterclockwise.

Answer: -\dfrac{16}{3}


Question: Determine whether it is true that if $n$ is a Carmichael number, then $n-1$ cannot be square-free. Provide a proof or a counterexample.

Answer: True


Question: Given that the weight of each gum ball (in centigrams) follows a gamma distribution with parameters \(\alpha = 25\) and \(\beta = 2\), calculate the probability that the total weight of 100 gum balls exceeds 60 grams.

Answer: 0


Question: Find the coefficient of $x^{13}$ in the expansion of \( \left ( x^{3}+1 \right )^{2}\left ( x^{2}-\frac{2}{x} \right )^{8} \).

Answer: -240


Question: Solve the equation involving factorials: \[ 4 (x+1)! = x! (2x-6)! \]

Answer: 5


Question: Let $\phi: \mathbb{Z} \rightarrow \mathbb{Z}_8$ be a homomorphism such that $\phi(1) = 6$. Determine all values of $n$ for which $\phi(n) = 0$.

Answer: 4\mathbb{Z}


Question:
Let $f, g : \mathbb{R} \rightarrow \mathbb{R}$ be two differentiable functions such that $f'(x) = g(x)$, $g'(x) = f(x)$, $f(0) = 1$, and $g(0) = 0$. Define the function $F(x) = f^2(x) + g^2(x)$. Which of the following statements is correct?

(a) $F(x) = 1 - \sin(x)$.
(b) $F$ is the constant function with value 1.
(c) None of the other two answers.

Answer: C


Question: Let \( \varphi \) be an odd and increasing function. Prove that for any \( h \in \mathbb{R} \), the following inequality holds: \[ h \int_{-\infty}^\infty \varphi(y) \varphi^{\prime}(y) \exp \left(-y^{2}+2 h y-h^{2}\right) \mathrm{d} y \geq 0. \]

Answer: h \int_{-\infty}^\infty \varphi(y) \varphi^{\prime}(y) \exp \left(-y^{2}+2 h y-h^{2}\right) \mathrm{d} y \geq 0


Question: Given that $\det(A) = -3$, $\det(A + I) = 2$, and $\det(A + 2I) = 5$, find $\det(A^4 + 3A^3 + 2A^2)$. Provide your answer as a single number.

Answer: 90


Question: Let $S$ be a graded ring generated by finitely many elements of $S_1$ as an $S_0$-algebra, and let $M$ be a graded $S$-module. For an element $m \in M$, if $m = 0$ in $M_f$ for all generators $f \in S_1$, does it follow that $m = 0$ in $M$?

Answer: No


Question: You and a friend are playing a game with a fair coin. You go first, and you each take turns flipping the coin, recording the result each time. A player wins the game when they get two consecutive heads. What is the probability that you win?

Answer: \dfrac{2}{5}


Question: Calculate \( \lim_{x\to1}\frac{\sqrt[3]{1+7x}-\sqrt{1+3x}}{x-1} \).

Answer: -\dfrac{1}{6}


Question: If a non-zero entire function has infinitely many zeros, is it always transcendental? Provide a justification for your answer.

Answer: Yes


Question: Calculate the work done by the vector field \( F(x,y,z) = \left(\frac{2}{\pi}x\sin \pi y\right) \mathbf{i} + (x^2 \cos \pi y + 2ye^{-z}) \mathbf{j} + (-y^2e^{-z} + 2x) \mathbf{k} \) on the intersection of the manifolds \( z = x^2 + y^2 \) and \( z = 4x \).

Answer: 0


Question: Can unit varieties of number fields be parametrized by rational functions with rational coefficients?

Answer: No


Question: Solve the differential equation: \[ \left( \frac{dy}{dx}\right)^2 = \frac{x}{y} \left(2 \frac{dy}{dx} - \frac{x}{y}\right). \] Verify that the solution is of the form \( y^2 - x^2 = c \).

Answer: y^2 - x^2 = c


Question: Losses on an insurance policy are uniformly distributed on the interval [0, 10000]. A policy covering these losses has a deductible of 1000. Calculate the average payment made by this policy for each loss.

Answer: 4050


Question:
Evaluate the Lebesgue integral \( \int_{[0,1]} f \, d\mu \) where \( d\mu \) is the Borel measure and \( f \) is defined as follows:
\[ f(x) = \begin{cases} x & \text{if } x \in C, \\
0 & \text{if } x \in [0,1] \setminus C, \end{cases} \]
where \( C \) is the Cantor set. Given that \( \mu(C) = 0 \), determine the value of the integral.

Answer: 0


Question: Consider a finite, dominant, unramified morphism \( \pi: Y \to X \) between nonsingular varieties of dimension \( n \) over an algebraically closed field \( \Bbbk \). Let \( d = \deg(\pi) \). Does the equation \( c_n(Y) = d \cdot c_n(X) \) hold for any algebraically closed field \( \Bbbk \), including those of positive characteristic?

Answer: Yes


Question:
Given vectors \(x, y \in \mathbb{R}^3\) such that:

- \(||x + y|| = \sqrt{29}\)
- \(x \times (2,3,4) = (2,3,4) \times y\)

Find the value of \(||(x+y) \times (1,1,1)||\).

Answer: \sqrt{6}


Question: Does there exist a Lie algebra $\mathfrak{g}$ that is reductive, but when considered as a subalgebra of another Lie algebra $\mathfrak{h}$, $\mathfrak{g}$ is not reductive in $\mathfrak{h}$?

Answer: Yes


Question: Determine whether there exist natural numbers \(m, n, k\) such that \(m, n, k \ge 1\), \(m \neq n \neq k \neq m\), and \[ \frac{1}{2m+1} + \frac{1}{2n+1} = \frac{2}{2k+1}. \]

Answer: Yes


Question: Find the coefficient $c_{-3}$ in the Laurent series expansion of the function $g(z) = \frac{e^{iz}-1}{\cos z-1}$ in the region $2\pi<|z|<4\pi$. Express $g(z)$ as $\sum_{n=-\infty}^{+\infty} c_{n}z^{n}$ and determine the value of $c_{-3}$.

Answer: 0


Question: Find the operator norm of $A$, where $A$ is the multiplication by $x$ operator defined on $X = C[0,2]$ with the supremum norm $||\Psi||_\infty = \max_{0 \leq x \leq 2} |\Psi(x)|$. Specifically, calculate $||A||$ for $A\Psi(x) = x\cos x$.

Answer: 2|\cos 2|


Question: Let $x$ and $y$ be real numbers such that $x \neq y$ and they satisfy the equation \( \frac{1}{1+x^2} + \frac{1}{1+y^2} = \frac{2}{1+xy} \). Find the value of \( \left(x+\frac{1}{y}\right)\left(y+\frac{1}{x}\right) \).

Answer: 4


Question: Evaluate the integral \( \int_{|z|=1} \left( \frac{z \sin(z)}{z+2} + \bar{z} \right) \, dz \) using Cauchy's integral formula.

Answer: 2\pi i


Question: Evaluate the sum of the series: \[ \sum_{n=2}^{\infty} \frac{n(-3)^n}{4^{n+1}}. \]

Answer: \dfrac{99}{784}


Question:
Determine the range of \( m \) such that the Trapezoidal Error is lower than the Simpson Error for the integral \( \int_{0}^{1} (x^6 - mx^5) \, dx \). Choose the correct option:

1) \( \frac{15}{14} < m < \frac{85}{74} \)
2) \( \frac{23}{48} < m < \frac{57}{63} \)
3) \( \frac{237}{215} < m < \frac{279}{193} \)
4) \( \frac{217}{210} < m < \frac{263}{238} \)

Answer: 4


Question: Find the number of integer solutions to the equation \(a_1 + 3a_2 + 5a_3 = 33\) given the constraints \(1 \leq a_1 \leq 11\), \(3 \leq a_2 \leq 18\), and \(5 \leq a_3 \leq 13\). Use generating functions to determine the coefficient of \(t^{33}\) in the expansion of \((t^1 + t^2 + \dots + t^{11})(t^9 + t^{12} + \dots + t^{54})(t^{25} + t^{30} + \dots + t^{65})\).

Answer: 0


Question:
Find the coefficient $a_{-1}$ in the Laurent series expansion of the function $f(z) = \frac{z^2 + 1}{(z - i)^2}$ about the point $z_0 = i$. The coefficient can be determined using the formula:
\[ a_{-1} = \frac{1}{2\pi i} \int_C \frac{z^2 + 1}{(z-i)^{2}}dz \]
where $C$ is a counterclockwise closed curve around $z_0 = i$. Calculate $a_{-1}$.

Answer: 2i


Question: Determine the asymptotic value of \(\underset{c \in \text{Coloring}}{\min} \text{ score}(c) = \underset{c \in \text{Coloring}}{\min} \left(\underset{\begin{subarray}{c} B \subset \text{Colors}, \\ |B|\leq n  \end{subarray}}{\max} V_c(B)\right)\). Is it \(\Theta(n^2)\) or \(o(n^2)\)?

Answer: o(n^2)


Question: Evaluate the double integral \( \iint_S z \, ds \), where \( S \) is the part of the sphere \( x^2 + y^2 + z^2 = 1 \) that lies above the cone \( z = \sqrt{x^2 + y^2} \).

Answer: \dfrac{\pi}{2}


Question: Let $(E, d)$ and $(E', d')$ be two metric spaces, and let $f: E \to E'$ be a continuous function. Consider a uniform covering of $E'$ denoted by $U$. For each $b \in E$, define $V(b) = f^{-1}(U(f(b)))$. Determine whether the following statement is true or false: $f$ is uniformly continuous if and only if $V$ is a uniform cover of $E$ for any uniform cover $U$ of $E'$.

Answer: True


Question: Evaluate the contour integral \( \oint_{|z-3|=4} \frac{z}{\cos z - 1} \, dz \) using the residue theorem.

Answer: -8\pi i


Question: Determine the number of common normals to the curves given by the equations $x^2 + (y-1)^2 = 1$ and $y^2 = 4x$. Provide your answer as a single integer.

Answer: 1


Question: Does every three-dimensional compact solvmanifold admit either Euclidean, nil, or sol geometry?

Answer: Yes


Question: Does there exist a closed Moishezon manifold with a vanishing second Betti number?

Answer: No


Question:
Find the volume of the intersection of the following elliptical cylinders in the first octant:
\[\frac{x^2}{9} + \frac{y^2}{4} \leq 1\]
\[\frac{y^2}{4} + \frac{z^2}{9} \leq 1\]

Answer: 12


Question: Evaluate the limit: $$\lim_{x\to\infty} \left(\int_0^1 t^{-tx} \, dt\right)^{\frac{1}{x}}$$

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


Question: Evaluate the limit \( \lim_{\eta \to +0} \int_{-\infty}^{\infty} \frac{\cos^4{x}}{2+\cos{x}} \cdot \frac{\eta}{(x-\pi/3)^2+\eta^2} \, dx \).

Answer: \dfrac{\pi}{40}


Question: Determine the values of $p$ for which the improper integral \( \int_0^\infty \frac{\log x}{1+x^p}\ dx \) converges.

Answer: p > 1


Question: What is the remainder when $18!$ is divided by 437?

Answer: 436


Question: Let $K/\mathbb{Q}$ be a Galois extension with $\text{Gal}(K/\mathbb{Q})$ isomorphic to $A_5$. Prove that there exists an irreducible polynomial $g(x) \in \mathbb{Q}[x]$ of degree 20 such that $g(x)$ has a root in $K$. Provide a justification for the existence of such a polynomial.

Answer: 20


Question:
Convert the following limit to a Riemann sum and evaluate the resulting integral:
\[ \lim_{N \to \infty} \sum_{n=-N}^{N} \left( \frac{1}{N+in} + \frac{1}{N-in} \right) \]

Answer: \pi


Question: Do there exist pairwise distinct sequences \(x_n\) in \(\mathbb{R}^2\) that converge monotonically to a limit \(x\) in the sense that for every open connected neighborhood \(U\) of \(x\), if \(x_{n_0} \in U\) and \(n > n_0\), then \(x_n \in U\)?

Answer: Yes


Question: Let $f$ be a continuous function on $S^2$. Suppose there exists a constant $C > 0$ such that for every rotation $\mathcal{R} \in SO(3)$, the area of every connected component of the set $\{f(x) \geq f(\mathcal{R}x)\}$ is at least $C$. Does there exist a rotation $I \neq \mathcal{R}_0 \in SO(3)$ such that $f(x) = f(\mathcal{R}_0 x)$ on an open subset of $S^2$?

Answer: Yes


Question: Let $X$ be a separable space and $x^{**} \in X^{**}$. If $x^{**}(x^{*}_{n}) \rightarrow 0$ for every weak*-null sequence $(x^{*}_{n})_{n}$ in $X^{*}$, is $x^{**}$ in $X$? Provide a justification for your answer.

Answer: Yes


Question: Let \( n \) be the largest 12-digit prime number. Find the product of all integers \( j \) less than \( n \) such that for each \( j \), all numbers from 1 to \( n-1 \) can be expressed as a power of \( j \) in modulo-\( n \) arithmetic. Compute this product in modulo-\( n \) arithmetic.

Answer: 1


Question:
Consider three lines given by the equations $y-2=z+3=0$, $z-3=x+1=0$, and $x-1=y+2=0$. Let $(\alpha,\beta,\gamma)$ be a point lying on a line that intersects all three given lines. Determine the locus of the point $(\alpha,\beta,\gamma)$. Choose the correct equation for the locus from the following options:

(a) $xy+3yz+2xz+6=0$

(b) $3xy+yz+2xz+6=0$

(c) $2xy+3yz+xy+6=0$

(d) None of these

Answer: B


Question: Two trains are 80 kilometers apart and are traveling toward each other on the same track, each at a speed of 25 kilometers per hour. A bee starts at the first train and flies back and forth between the two trains until they are 100 meters apart. The bee completes 36 rounds, landing 18 times on each train. Assuming the bee has no acceleration or deceleration and takes no extra time when turning, calculate the average speed of the bee.

Answer: 270


Question: Evaluate the double integral \( \iint_{S} \sin\left(\frac{y - x}{x + y}\right) \, \mathrm{d}A \), where \( S \) is the trapezoidal region in the plane bounded by the lines \( x + y = 1 \), \( x + y = 2 \), and the coordinate axes.

Answer: 0


Question: Determine the number of tournaments with five vertices that have at least one vertex with an in-degree of zero, up to isomorphism.

Answer: 4


Question: Consider a sequence \(\{a_{n}\}\) satisfying the inequality \(a_{n+2}-2a_{n+1}+a_{n} \geq C > 0\). Determine if the following limit holds: \[ \lim_{n\to \infty}(a_{n+1}-a_{n}) e^{-a_{n}}=0. \]

Answer: 0


Question: Determine the radius of convergence for the series \( \sum_{k=0}^{\infty} \frac{k(z+i)^{2k}}{2^k} \).

Answer: \sqrt{2}


Question: In a linear regression model with error term \( \varepsilon \sim\operatorname{N}(0,\sigma^2I_n) \), consider adding a covariate \( \mu \in \mathbb{R}^n \) to obtain the model \( y = X\beta + \gamma\mu + \varepsilon \), where \( X^T \mu = 0 \) and \( \gamma \in \mathbb{R} \). Let \((\hat\beta_\text{new}, \hat\gamma)\) be the least squares estimates of the parameters \((\beta, \gamma)\) in this new model. Determine whether \( \hat\beta_\text{new} \) and \( \hat\gamma \) are independent. Answer "True" if they are independent and "False" otherwise.

Answer: True


Question: Given the polynomial \( P(x) = x^{n+1} - 2x^n + 1 \) for all \( n \geq 2 \), and knowing that \( P(x) \) has a unique root \( a_n \in (1,2] \), find the limit \( L = \lim_{n\to\infty} 2^n(2-a_n) \).

Answer: 1


Question: Let $A$ and $B$ be two $n \times n$ matrices such that $A \neq B$, $A^3 = B^3$, and $A^2 B = B^2 A$. Determine the value of $\det(A^2 + B^2)$. \( \text{Provide your answer as a single number.} \)

Answer: 0


Question: Calculate the line integral of the vector field \( F(x,y) = (2y, -3x) \) around the unit circle \( C \). Use the standard parametrization \( \gamma(t) = (\cos t, \sin t) \) for \( t \in [0, 2\pi] \). Verify your result using the Divergence Theorem.

Answer: -5\pi


Question: Suppose that the cubic polynomial \( x^3 - 13x^2 + (40 + a)x - 5a \) has one rational root and two non-real complex roots, where \( a \) is a positive prime number. What is the minimum value of \( a \)?

Answer: 17


Question: Given that $x, y, a$ are real numbers satisfying the inequalities \((\log_a x)^2 + (\log_a y)^2 - \log_a(xy)^2 \leq 2\) and \(\log_a y \geq 1\), determine the range of \(\log_a x^2 y\).

Answer: [-1, 3 + 2\sqrt{5}]


Question: Evaluate the integral \( \int_{0}^{\infty} \frac{dx}{1+x^6} \) using the residue theorem.

Answer: \dfrac{\pi}{3}


Question: Let \( \phi : \mathbb{R} \rightarrow \mathbb{R} \) be a function of class \( \mathscr{C}^{1} \) such that \( \sup_{x \in \mathbb{R}} |\phi'(x)| < 1 \). Prove that there exists a unique fixed point \( x^* \) such that \( \phi(x^*) = x^* \).

Answer: x^*


Question: Evaluate the double integral \( \iint_V |y-x^{2}| \, \operatorname{d}x \, \operatorname{d}y \) over the region \( V = [-1,1] \times [0,2] \).

Answer: \dfrac{46}{15}


Question: Calculate the value of $i^{1/i}$.

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


Question: Evaluate the integral \( \int_{0}^{1} (\sqrt[3]{1-x^7} - \sqrt[7]{1-x^3}) \, dx \).

Answer: 0


Question: Find the ratio of the circumradius to the inradius, \( \frac{R}{r} \), for a regular pentagon.

Answer: \sqrt{5} - 1


Question: Determine the number of countable non-isomorphic models of the theory $\text{ACF}_p$. Note that $\text{ACF}_p$ is $\kappa$-categorical for all uncountable $\kappa$. Provide your answer as a single number.

Answer: \aleph_0


Question: Find the maximum volume of a rectangular solid in the first octant with one vertex at the origin and the opposite vertex on the plane \(x + y + z = 1\). Use the method of Lagrange multipliers to determine the maximum volume.

Answer: \dfrac{1}{27}


Question:
Find the coefficient of \(x^{52}\) in the expansion of the product:
\[(x^{10} + x^{11} + \ldots + x^{25})(x + x^2 + \ldots + x^{15})(x^{20} + x^{21}+ \ldots + x^{45}).\]

Answer: 204


Question: Find the closed form of the sequence defined by the recurrence relation \( f(n) = 10f(n-1) - 25f(n-2) \) with initial conditions \( f(0) = 0 \) and \( f(1) = 1 \), using generating functions.

Answer: n \cdot 5^{n-1}


Question: Find the limit of \( n (\sqrt[n]{n} - 1) \) as \( n \to +\infty \).

Answer: \infty


Question: The Great Kingdom is a rectangular island divided into smaller rectangular landscapes. Each landscape has a road running along one of its diagonals. These roads form a closed loop that does not intersect itself. If the Great Kingdom has an odd number of landscapes, what is the minimum number of landscapes it can have?

Answer: 5


Question: Consider a sequence formed from the numbers $1, 2, 3, 4, 5, 6, 7$, where all $7!$ permutations are equally likely. What is the probability that there will be at least five consecutive positions in which the numbers are in increasing order?

Answer: \dfrac{1}{45}


Question: Consider a random variable \( x \sim \text{Bernoulli}(\pi) \) where \( \pi < 0.5 \). Suppose a prediction \( \hat{x} \) also follows a Bernoulli distribution \( \hat{x} \sim \text{Bernoulli}(p) \). Compute the expected accuracy \( \mathbb{E}[ I {( x=\hat{x} )}] \), where \( I \) is the indicator function. What is the value of \( p \) that yields the highest expected accuracy?

Answer: 0


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{n + \log{n^2} - 2^n}{2(\log{n})^3 + n\log{n} + n^2} \]

Answer: -\infty


Question: Determine the Galois group of the polynomial $x^5 - 6x + 3$ over the field of rational numbers $\mathbb{Q}$. Assume $\alpha$ is a root of the polynomial.

Answer: S_5


Question: Determine whether the following statement is true or false: For a matrix $A \in M_n$, is it true that $\| A \|_2 = \left( \sum_{i,j} |a_{ij}|^2 \right)^{1/2} = \max_{\| x \|_2 = 1} \| Ax \|_2$?

Answer: B


Question: Compute the flux of the vector field \( \text{curl } F \) through the hemisphere defined by \( x = \sqrt{1 - y^2 - z^2} \), which is positively oriented. The vector field is given by \( F(x,y,z) = \langle e^{xy}\cos z, x^2z, xy \rangle \).

Answer: 0


Question: In triangle \( \triangle{ABC} \), point \( D \) is located such that \( \angle{ABD} = 10^\circ \), \( \angle{DBC} = 20^\circ \), \( \angle{BCD} = 40^\circ \), and \( \angle{DAC} = 50^\circ \). Determine the measure of \( \angle{BAD} \).

Answer: 30


Question: Given two independent random variables, \(X\) and \(Y\), where \(X \sim \text{Poisson}(\lambda)\) and \(Y \sim \text{Poisson}(2\lambda)\), and let \(Z = X + Y\). Calculate the covariance \(\text{Cov}(X, Z)\).

Answer: \lambda


Question: Consider a linear system $Ax = b$, where $A$ is an $n \times m$ matrix with $m > n$, $x$ is an $m$-dimensional column vector, and $b$ is an $n$-dimensional column vector. Can the $2$-norm of $x$ be arbitrarily larger than the norms of $A$ and $b$? Provide a justification for your answer.

Answer: Yes


Question: Let $E \subseteq \mathbb{R}^l$ be a Lebesgue measurable set with $m(E) > 0$. For each $k \in \mathbb{N}$, let $f_k: E \rightarrow \mathbb{R}$ be measurable functions. If for every $\epsilon > 0$, there exists a closed set $F \subseteq E$ with $m(F) \leq \epsilon$ such that $f_k \rightarrow f$ uniformly on $E - F$, does it follow that $f_k \rightarrow f$ uniformly almost everywhere on $E$?

Answer: No


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

Answer: i\pi


Question: Starting at 6 a.m., cars, buses, and motorcycles arrive at a highway toll booth according to independent Poisson processes. Cars arrive about once every 5 minutes. Buses arrive about once every 10 minutes. Motorcycles arrive about once every 30 minutes. Find the probability that the seventh motorcycle arrives within 45 minutes of the third motorcycle.

Answer: 0.0656


Question:
Let \( f: G \rightarrow K \) be a morphism of simplicial groups such that \( f \) is a weak homotopy equivalence of the underlying simplicial sets. Assume the following conditions:
1) For each natural number \( i \), \( G_{i} \) is a free group.
2) For each natural number \( i \), \( K_{i} \) is a subgroup of a product of free groups, i.e., \( K_{i} \subset \prod_{j \in J} F_{j} \) where each \( F_{j} \) is free.

Consider the corresponding abelianization, \( f^{ab}: G^{ab} \rightarrow K^{ab} \). Is \( f^{ab} \) a weak homotopy equivalence of the underlying simplicial sets?

Answer: Yes


Question: Let $G = \{ z \in \mathbb{C} : |z-2| < 1\}$ and let $f$ be analytic in $\overline{G}$ except for one simple pole $z_0$ inside. Suppose that $|f(z)| = 1$ for all $z \in \partial G$. Prove that for all $a$ with $|a| > 1$, $f^{-1}(a)$ contains exactly one point.

Answer: 1


Question: Find the values of \( b > 1 \) such that the functions \( b^x \) and \( \log_b x \) intersect only once.

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


Question: Does the set of continuous endomorphisms on the real numbers, \( \text{End}(\mathbb{R}) \), have a countable minimal cover?

Answer: Yes


Question: Calculate the value of the following expression: \[ \frac{\int_0^1 \frac{dt}{(1-t^4)^{1/2}}}{\int_0^1 \frac{dt}{(1+t^4)^{1/2}}}. \]

Answer: \sqrt{2}


Question: Evaluate the integral \( \int_0^\pi \sin^2 t\; dt \) using the residue theorem.

Answer: \dfrac{\pi}{2}


Question: Let $T \in \text{Hom}(\mathbb{C}^3)$ be a normal operator such that $T(1,1,1) = (2,2,2)$. Prove that if $(x,y,z) \in \ker(T)$, then $x+y+z=0$. Provide a justification for your answer.

Answer: x + y + z = 0


Question: Given that \( z = \text{cis}(2k\pi/5) \) and \( z \neq 1 \), evaluate the expression \((z+1/z)^2+(z^2 + 1/z^2)^2\).

Answer: 3


Question: A 3rd degree polynomial $P(x)$ has three unequal real roots. Determine the least possible number of unequal real roots for the polynomial $P(x^2)$. Provide your answer as a single integer.

Answer: 0


Question: Evaluate the integral \( \int_{-1}^{1}\int_0^{\sqrt{1 - y^2}} \int_{x^2 + y^2}^{\sqrt{x^2 + y^2}} xyz \, dz \, dx \, dy \).

Answer: 0


Question: Evaluate the infinite series \( \sum_{n=0}^{\infty} nC^{n-1} \), where \( C \) is a constant. For example, evaluate the series for \( C = 0.5 \) and \( C = 2 \).

Answer: 4


Question:
Evaluate the integral of the Poisson kernel over the boundary of the n-dimensional unit sphere:
\[ \int_{\partial B_1(0)} P(x,y) \, dA_y \] where \( P(x,y) = \frac{1}{\omega_n R} \frac{R^2-|x|^2}{|x-y|^n} \), \( |x| < R \), \( |y| = R \), and \( \omega_n \) is the area of the n-dimensional unit sphere.

Answer: 1


Question: Consider the Banach space $B = C[-1,1]$ with the supremum norm. For $f \in B$, define $\tilde{f}(x) = f(|x|)$. Let $T: B \rightarrow B$ be the operator defined by $T(f) = \tilde{f}$. Show that $T$ is a bounded linear operator on $B$ and determine the operator norm $||T||$. What is $||T||$?

Answer: 1


Question: Given the sequence of numbers \(1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{2010}\), repeatedly select any two numbers \(x\) and \(y\) and replace them with \(x + y + xy\). Continue this process until only one number remains. Determine the final number.

Answer: 2010


Question: Determine the number of possible \( \mathbb{Z}[x] \)-module structures on the abelian group \( \mathbb{Z}/5\mathbb{Z} \).

Answer: 5


Question: Determine the subdifferential of the function \( f: \mathbb{R} \rightarrow \mathbb{R} \), defined by \( f(x) = \max(1, \exp(x^2)) \), at the point \( x = 0 \).

Answer: \{0\}


Question: Let \( z_1 = 3 - 4i \) and \( z_2 = 2 + i \). Given that \( \alpha \neq 1 \) and \( \alpha^6 = 1 \), find the smallest prime number that divides the sum \( S = \sum_{r=0}^{5} |z_1 + z_2\alpha^r|^2 \).

Answer: 2


Question: Evaluate the limit \( \lim_{N \to \infty} \sum_{k=1}^N \frac{k^3}{N^4} \) using the concept of Riemann sums.

Answer: \dfrac{1}{4}


Question: Evaluate the double integral \( I = \iint_{D} (x - \frac{7}{3})y \, dx \, dy \), where \( D \) is the region bounded by the curves \( \frac{7}{3} - 2y - x = 0 \) and \( x = \frac{1}{3}y^{2} \).

Answer: -512


Question: Find the minimum value of the expression \( ab + \frac{1}{a^2} + \frac{1}{b^2} \) given that \( a, b > 0 \) and \( a + b \leq 1 \).

Answer: \dfrac{33}{4}


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

Answer: 1


Question: Calculate the value of \( \left|\frac{(\pi+i)^{100}}{(\pi-i)^{100}}\right| \).

Answer: 1


Question:
Calculate \( \lim_{n \to \infty} U_n \) where
\[ U_n = \sum_{k=1}^n k \sin\left(\frac{1}{k} \right) \]

Answer: \infty


Question: Calculate the value of the infinite series \( \sum_{r=1}^{\infty} \frac{1}{a_r} \), where the sequence \( a_n \) is defined recursively as follows: \( a_1 = 1 \), \( a_2 = 1 + a_1 \), \( a_3 = 1 + a_1 \cdot a_2 \), \( a_4 = 1 + a_1 \cdot a_2 \cdot a_3 \), and generally \( a_{n+1} = 1 + a_1 \cdot a_2 \cdot \cdots \cdot a_n \).

Answer: 2


Question: A small disk with radius \( \frac{1}{N} \) rolls without slipping around a larger disk with radius 1. Determine the number of complete revolutions the small disk makes to return to its initial configuration.

Answer: N + 1


Question: Approximate \(\int_{-1/2}^{1/2} \cosh(x^2) \, dx\) using Taylor polynomials such that the error is less than \(\frac{1}{1000}\). Determine the number of terms needed in the Taylor series expansion to achieve this accuracy.

Answer: 2


Question: Determine whether the function \( e^{u/2} \sum_{n=-\infty}^{\infty} e^{-\pi n^{2} e^{2u}} \) is even.

Answer: Yes


Question: Determine the range of the function \( f(x) = \arccos (x^2 \cos 1 + \sin 1 \sqrt{1-x^4}) \).

Answer: [0, 1]


Question: Given that \( f(x) \) is a continuous function and \( f(0) = 2 \), and the functional equation \( f(x) - f\left(\frac{x}{23}\right) = \frac{x}{23} \) holds, find the value of \( f(484) \).

Answer: 24


Question:
Determine the order of the two-step Adams-Bashforth method given by the formula:
\[
u_{n+1} = u_n + \frac{h}{2}(3f_n - f_{n-1})\] where \(t_n = t_0 + nh\) is the \(n\)-th node, \(y_n = y(t_n)\), \(f_n\) stands for \(f(t_n,y_n)\), and \(u_n\) is an approximation of \(y_n\).

Answer: 2


Question: Determine whether the following statement is true: For $u \in L^\infty(\mathbb{R}^d)$ and $a > 0$, is it true that \( \|u^{a}\|_{L^\infty(\mathbb{R}^d)} = \|u\|_{L^\infty(\mathbb{R}^d)}^a \)? Justify your answer.

Answer: True


Question: Given that \( \int_a^b |\sin x| \, dx = 8 \) and \( \int_0^{a+b} |\cos x| \, dx = 9 \), find the value of \( \int_a^b x \sin x \, dx \).

Answer: -2\pi\sqrt{2}


Question: Determine the order of magnitude of the function \( \sinh \left( \frac{1}{\epsilon} \right) \) as \( \epsilon \) tends to zero, expressing your answer in terms of an exponential function.

Answer: e^{1/\epsilon}


Question: Let $z_1, z_2, \ldots, z_n$ be complex numbers such that $|z_1| = |z_2| = \ldots = |z_n| = 1$. Determine whether the following inequality holds: \[ \limsup_{k \in \mathbb{N}} \sqrt[k]{|z_1^k + z_2^k + \cdots + z_n^k|} \ge 1. \]

Answer: Yes


Question:
Given the system of differential equations:
\[\frac{\mathrm{d}x}{\mathrm{d}t}=ax+by\]
\[\frac{\mathrm{d}y}{\mathrm{d}t}=cx+dy\]
where the solution \((x(t), y(t))\) lies on a straight line, find the value of \(y(10)\) given that \(y(0)=5\), \(x(0)=6\), and \(x(10)=2\).

Answer: \dfrac{5}{3}


Question: Evaluate the integral \( \int \sec^{-1}[-\sin^2x] \, dx = f(x) + c \). Then, find the value of the second derivative \( f''\left(\frac{8}{{\pi}x}\right) \) at \( x = 2 \).

Answer: 0


Question: In a road traffic survey, cars passing a marker on a road are counted. The streams of cars in the two directions are modeled as independent Poisson processes with intensities of 2 cars per minute and 3 cars per minute, respectively. Counting stops once 400 cars have passed. Let \( T_{400} \) be the time at which this occurs. Compute the time \( \alpha \) such that \( P(T_{400} \leq \alpha) = 0.90 \).

Answer: 85.13


Question: Determine the number of non-real roots of the equation \(z^2 = \sin(z)\) within the region \(\{z \in \mathbb{C} \setminus \mathbb{R} \mid |z| < 2\}\).

Answer: 0


Question: Given 17 points on the circumference of a circle, determine the maximum number of segments that can be drawn such that each segment intersects all others inside the circle.

Answer: 8


Question: Let $K$ be an algebraic function field in one variable over a field $k$, where $K$ is a finite extension of $k(t)$. Consider an absolute value $|\cdot|:K\to\mathbb{R}^+$ such that $|k^\times|=1$. This absolute value induces a valuation $v: K\to\mathbb{R}$ given by $v(x)=-\log|x|$, which is trivial on $k^\times$. Is $v$ always a discrete valuation, meaning $v(K^\times)=t\mathbb{Z}$ for some $t\in\mathbb{R}_+$?

Answer: Yes


Question: Evaluate the summation \( \sum\limits_{x=0}^\infty \frac{(x^2-x+1)(2^x)}{x!} \) using the properties of the Poisson distribution.

Answer: 5e^2


Question: Determine the dimension of the vector space of 3x3 matrices where the sum of the elements in each column, each row, and both diagonals is zero.

Answer: 2


Question: Suppose $X$ is a real random variable such that all odd moments vanish, i.e., $\mathbb{E}[X^{2n+1}] = 0$ for $n = 0, 1, 2, 3, \ldots$. Does it follow that $X$ is symmetrically distributed about 0, meaning $X$ and $-X$ have the same distribution?

Answer: Yes


Question: Given that $a + b + c = 0$, prove that \[ \left(\frac{a}{b-c} + \frac{b}{c-a} + \frac{c}{a-b}\right) \left(\frac{b-c}{a} + \frac{c-a}{b} + \frac{a-b}{c}\right) = 9. \] Assume $a$, $b$, and $c$ are distinct.

Answer: 9


Question: Given a group homomorphism \( \phi: \mathbb{Z}_{50} \rightarrow \mathbb{Z}_{15} \) with \( \phi(7) = 6 \), determine the expression for \( \phi(x) \).

Answer: 3x \bmod 15


Question: Suppose $P$ is a compact operator in $L^2([0,1])$ with $\\|P\\|_{\mathrm{op}} < 1$. For some $f$ in $L^2([0,1])$, define the operator $T$ by $Tg(x) = f(x) + Pg(x)$. Is it true that $T$ is a contraction?

Answer: Yes


Question: Determine whether the following statement is true or false: Let $H$ be a Hilbert space, and let $x,y \in H$. If $\langle x,z \rangle=\langle y,z\rangle$ for all $z \in H$, then $x=y$. Provide a justification for your answer.

Answer: True


Question: Determine whether the following statement is true or false: If $f(x) \leq a \leq g(x)$ and $f(x) \leq b \leq g(x)$ for every $x \in \mathbb{R}$, and for all $\epsilon > 0$ there exists $x$ such that $g(x) - f(x) < \epsilon$, then $a = b$. Suppose $f, g$ are real-valued functions and $a, b$ are real numbers.

Answer: True


Question: Let $V \subseteq \mathbb{C}$ be an open connected set, and let $\{f_n\}$ be a normal family on $V$. Suppose that the set $\{z \in V : \lim _{n\to \infty} f_n(z) \text{ exists}\}$ has a limit point in $V$. Is it true that there exists a function $f:V \to \mathbb{C}$ such that $\{f_n\}$ converges uniformly to $f$ on each compact subset of $V$?

Answer: Yes


Question: Express $a^5$ in terms of $c_0 + c_1a + c_2a^2$, where $a$ is a root of the polynomial $f(x) = x^3 + x + 1$ over the field $\mathbb{Z}_2$. Given that $F(a) = \{c_0 + c_1a + c_2a^2 : c_i \in \mathbb{Z}_2\}$, find the expression for $a^5$ in this form.

Answer: 1 + a + a^2


Question: Calculate the maximum value of the summation \( \sum_{k=1}^{n} \left(\frac{(-1)^k}{k!} \int_{0}^{k} 3x^2 \, dx\right) \) for a given positive integer \( n \).

Answer: 3


Question: Find the largest natural number \( n \) such that \( n^{6033} < 2011^{2011} \).

Answer: 12


Question: Determine whether there exists a diffeomorphism on $\mathbb{R}^2$ that flattens out the boundary of a compact set at a point. Specifically, given a compact subset of $\mathbb{R}^2$ with a $C^2$ boundary $S$ and a point $x \in S$, can one find a diffeomorphism $f$ from $\mathbb{R}^2$ to $\mathbb{R}^2$ such that $f(x) = x$, the image $f(S)$ is a $C^2$ curve, and in a neighborhood of $f(x)$, the curve $f(S)$ coincides with the tangent line of $x$ in $S$?

Answer: Yes


Question:
Given that $X$, $Y$, and $Z$ are standard normal random variables with pairwise correlation coefficients of $-\frac{2}{3}$, the covariance matrix is given by:
\[ \mathbf{V} = \begin{bmatrix} 1 & -\frac{2}{3} & -\frac{2}{3} \\ -\frac{2}{3} & 1 & -\frac{2}{3} \\ -\frac{2}{3} & -\frac{2}{3} & 1 \end{bmatrix} \]
However, $\mathbf{V}$ is not positive semidefinite, indicating it cannot be a valid covariance matrix for a multivariate normal distribution. Determine if it is possible to compute the variance of $X+Y+Z$ under these conditions.

Answer: -1


Question: Compute the residue of the function \( \frac{z^4-2z^2+1}{z^4+2az^3+z^2} \) at the pole \( z = -a + \sqrt{a^2-1} \) inside the unit disc, where \( a > 1 \).

Answer: 2\sqrt{a^2 - 1}


Question: Let $f: X \rightarrow X$ be a bijective map between topological spaces such that $f \circ f = \text{id}$. Does this imply that $f$ is a continuous map and hence a homeomorphism?

Answer: No


Question: Find the area enclosed by the parametric curve given by \( r(t) = (16\sin^3(t), 13\cos(t) - 5\cos(2t) - 2\cos(3t) - \cos(4t)) \) for \( 0 \le t \le 2\pi \). Use the integral \( \int_0^{2\pi} y \cdot x' \, dt \) and the hint: \( \int_0^{2\pi} \sin(nx)\sin(mx) \, dx = \int_0^{2\pi} \cos(nx)\cos(mx) \, dx = \begin{cases} \pi  & \text{for } m = n \\ 0 & \text{otherwise} \end{cases} \).

Answer: 180\pi


Question: If $f \circ g$ and $f$ are quasi-isomorphisms, is $g$ necessarily a quasi-isomorphism?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sqrt[n^2]{1! + 2! + 3! + \cdots + n!} \]

Answer: 1


Question: Find the limit: \[ \lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}}. \]

Answer: e


Question: Find the limit of the sequence \( a_n = \frac{1}{n^2}\sum_{k=1}^n (2k+1)\sin\left(\frac{k^2 \pi}{n^2} \right) \) as \( n \to \infty \).

Answer: \dfrac{2}{\pi}


Question: Find the entire function $h: \mathbb{C} \to \mathbb{C}$ such that $h(0) = 0$, $h'(0) = 1$, and $|h'(z)| \leq 1$ for $|z| \leq 5$. Determine the function $h$.

Answer: z


Question: If for all sequences $(y_n)$ where $\lim_{n\rightarrow \infty} y_n = 0$, it follows that $\lim_{n\rightarrow \infty}(x_ny_n) = 0$, is the sequence $(x_n)$ necessarily bounded?

Answer: Yes


Question: For how many consecutive integers \(x\) can the quadratic polynomial \(ax^2 + bx + c\) be a perfect square, given that \(a, b, c \in \mathbb{Z}\), \(a \neq 0\), and \(b^2 - 4ac \neq 0\)?

Answer: 4


Question: Four random points are placed in the plane, with each point's horizontal and vertical coordinates uniformly distributed on the interval \((0,1)\). What is the expected largest size of a subset of these points that can form the vertices of a convex polygon?

Answer: \dfrac{133}{36}


Question: Determine the condition under which the eigenvalue sets of the matrices \(I - BAA^T\) and \(I - A^TBA\) are equal, where \(A \in \mathbb{R}^{m \times n}\) and \(B \in \mathbb{R}^{m \times m}\).

Answer: m = n


Question: Given a set of points where each point is labeled as either White or Black, devise an algorithm to compute the pair of White and Black points with the smallest Manhattan distance in $O(n\log^2(n))$ time complexity.

Answer: O(n \log^2 n)


Question: Evaluate the double integral \( \int_{0}^{1}\int_{0}^{1}\left\lfloor\frac{x-y}{x+y}\right\rfloor\, \mathrm{d}x\, \mathrm{d}y \).

Answer: -\dfrac{1}{2}


Question: Evaluate the integral \( \frac{1}{2\pi}\int_{-\infty}^{\infty}{\frac{e^{itx}}{1-it}dt} \) and verify that the result is \( e^{-x} \).

Answer: e^{-x}


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

Answer: 2^{k+1} - 2


Question: Given an integer $m$, choose integers $n \le m$ and $x_1, x_2, \ldots, x_n$ such that $\sum_{i=1}^n x_i = m$ and $\prod_{i=1}^n x_i$ is maximized. Is the maximum product $\prod_{i=1}^n x_i$ polynomial in $m$?

Answer: No


Question: Determine the quotient group \( \mathbb{C}/P \), where \( \mathbb{C} \) is the multiplicative group of non-zero complex numbers and \( P \) is the subgroup of positive real numbers.

Answer: S^1


Question: Find the 50th percentile \( \xi_Z^{0.5} \) for the random variable \( Z = \exp(-0.05T) \), where \( T \) follows a half-normal distribution with the probability density function \( \frac{2}{10\sqrt{2\pi}} e^{-t^2/200} \) for \( t > 0 \).

Answer: 0.714


Question: Find the limit of the sequence defined by the recurrence relation: \(X_{k+1} = \frac{X_k}{2} + \frac{1}{X_k}\).

Answer: \sqrt{2}


Question: For positive numbers $a$ and $b$ such that $2a^2 + 7ab + 3b^2 = 7$, find the maximum value of $a + \sqrt{3ab} + 2b$. Provide your answer as a single number.

Answer: \sqrt{14}


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

Answer: -\dfrac{1}{2}


Question: Calculate \( \lim_{n \to \infty} \mathbb{E}\left[ e^{i \theta \frac{S_n}{n}} \right] \), where \( \theta \in \mathbb{R} \) and \( S_n \) is a simple random walk.

Answer: 1


Question: Compute the limit: \[ \lim_{n\to\infty}\int_{0}^{\infty}\frac{dx}{x^{n}+1}. \]

Answer: 1


Question: Determine whether the following statement is True or False: If \(a_n\in\ell^1,\) then \(\overline{\lim} n a_n < \infty\) (i.e., \(a_n = O\left(\frac{1}{n}\right)\)).

Answer: B


Question:
Let \( f:[a, b] \rightarrow \mathbb{R} \) be a function that is continuous on \([a,b]\) and differentiable on \((a, b)\), with \( f(a)=0 \) and \( f(b)=0 \). Determine which of the following statements is true:

(A) There exists at least one point \( c \in (a, b) \) for which \( f^{\prime}(c)=f(c) \).

(B) \( f^{\prime}(x)=f(x) \) does not hold at any point of \((a, b)\).

(C) At every point of \((a, b)\), \( f^{\prime}(x) > f(x) \).

(D) At every point of \((a, b)\), \( f^{\prime}(x) < f(x) \).

Answer: A


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

Answer: \dfrac{\pi}{2}


Question: Evaluate the limit \( I = \lim_{a \rightarrow \infty} \frac{1}{a} \int_{0}^{a} \sin(x) \sin(x^2) \, dx \).

Answer: 0


Question: Let \( J = \langle x^2 + y^2 - 1, y - 1 \rangle \) be an ideal. Find a function \( f \in \textbf{I}(	extbf{V}(J)) \) such that \( f \not\in J \).

Answer: x


Question: Consider the sequence of numbers: \(\sqrt{1}, \sqrt{2}, \sqrt{3}, \dotsc, \sqrt{n}\). Let \(\mu\) be the arithmetic mean of this sequence and \(\sigma\) be its standard deviation. Calculate \(\lim_{n\to\infty}\frac{\mu}{\sigma}\).

Answer: 2\sqrt{2}


Question: Evaluate the integral: $$\int_{-\pi /2}^{\pi /2} \frac{\sin^{2002}x }{(2003^x +1)(\sin^{2002}x + \cos^{2002}x)}dx$$ using the properties of even and odd functions.

Answer: \dfrac{\pi}{4}


Question: Find the sum of the series: \[ \sum_{n=1}^{\infty}\frac{n \cdot 2^n}{(n+2)!} \]

Answer: 1


Question: The curve $C$ is given by the equation \( k x^2 - xy + (k+1)x = 1 \), and the line $l$ is given by the equation \( -(k/2)x + y = 1 \), where \( k \) is a non-zero constant. If the line and the curve intersect at exactly one point, find the coordinates of this intersection point.

Answer: (-1, 2)


Question: Determine the value of $r$ such that the equation \( n^2 + n - 2(40k + 30 + r) = 0 \) has integer solutions for both $n$ and $k$, where $n, k \in \mathbb{N}$. The possible values for $r$ are 2 and 6.

Answer: 6


Question:
Let \(x, y, z\) be positive real numbers such that \(x + y + z = 3\). Find the maximum value of the expression:
\[
P = \frac{2}{3 + xy + yz + zx} + \left(\frac{xyz}{(x+1)(y+1)(z+1)}\right)^{\frac{1}{3}}.
\]

Answer: \dfrac{5}{6}


Question: Find the derivative \( \frac{df}{du} \) if the function \( f(u) \) is given by the integral \( f(u) = \int_{\Gamma} g(x,y) u(y) \, dy \).

Answer: g(x,y)


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

Answer: 1


Question: Let $M$ be a smooth, bounded, oriented Riemannian manifold-with-boundary. Consider a harmonic differential $p$-form $\alpha$ on $M$, subject to the boundary condition $\alpha\wedge\nu^\sharp|\partial M = 0$ or $\iota_\nu \alpha|\partial M = 0$, where $\nu$ is the normal vector field along $\partial M$, $\nu^\sharp$ is its dual $1$-form, and $\iota$ is the interior multiplication. Assume $\alpha \in W^{1,2}$. Can we conclude that $\alpha \in C^\infty$?

Answer: \alpha \in C^\infty


Question: Find \(\lim_{x \to \infty}{\frac{(x+1)^{\frac{2}{3}}-(x-1)^{\frac{2}{3}}}{(x+2)^{\frac{2}{3}}-(x-2)^{\frac{2}{3}}}}\).

Answer: \dfrac{1}{2}


Question: In triangles \(\triangle_{AB_1C_1}\) and \(\triangle_{AB_2C_2}\), it is given that \(AB_1 = AB_2 = 8\) and \(AC_1 = AC_2 = 6\). The area of both triangles is \(12\sqrt{3}\). Find the value of \(\left|(B_1C_1)^2 - (B_2C_2)^2\right|\).

Answer: 96


Question:
Suppose a function $f: \mathbb{H}^m \to \mathbb{R}^n$ has a one-sided total derivative along a boundary line at a point $x \in \partial \mathbb{H}^m$:
\[ \frac{1}{\|v\|}\left\|f(a+v)-f(a)-Df(a)v\right\| \stackrel{v\to 0}{\to} 0 \quad (v \in \mathbb{H}^m) \]
Does this function admit an extension such that the ordinary total derivative exists at the boundary line?
\[ \frac{1}{\|v\|}\left\|f(a+v)-f(a)-Df(a)v\right\| \stackrel{v\to 0}{\to} 0 \]

Answer: Yes


Question: Given the transition matrix \( P = \begin{bmatrix}0.5 & 0.5 & 0 & 0 & 0 \\ 0.5 & 0 & 0.5 & 0 & 0 \\ 0.5 & 0 & 0 & 0.5 & 0 \\ 0 & 0 & 0.5 & 0 & 0.5 \\ 0 & 0.5 & 0.5 & 0 & 0 \end{bmatrix} \), calculate the expected number of times the Markov chain visits state \( j = 5 \) starting from state \( i = 1 \) after \( m = 100 \) steps.

Answer: \dfrac{100}{21}


Question: Given a function $f'$ that is informally drawn by hand and resembles parts of a parabola on both the left and right halves, determine whether $f'$ can be the derivative of some real-valued function $f$. Assume $f'$ satisfies the Intermediate Value Theorem of Derivatives.

Answer: Yes


Question: Given Hilbert spaces $H_1$ and $H_2$ with dimensions $n$ and $m$ respectively, consider the map $$\varphi:S_1(H_1)\hat{\otimes}S_1(H_2)\to S_1(H_1\otimes_2 H_2),$$ defined by $$\varphi( A \otimes B) (\xi_1\otimes\xi_2) = (A\xi_1 )\otimes (B\xi_2) \quad(\xi_1\in H_1, \xi_2\in H_2).$$ Is it true that $$\Vert\varphi^{-1}\Vert=\min\{m, n\}?$$ Here, $\hat{\otimes}$ denotes the projective tensor product of Banach spaces.

Answer: \min\{m, n\}


Question: Find the maximum value of \(|z|\) such that \(\left| \frac{6z-i}{2+3iz} \right| \leq 1\).

Answer: \dfrac{1}{3}


Question: Find the number of ways to arrange the numbers \(0, 0, 0, 1, 1, 1, 2, 2, 2\) such that no arrangement contains three consecutive integers that are the same.

Answer: 1314


Question: Given the sequences $a_n$ and $b_n$, where $a_n$ is the arithmetic mean of $a_1$ through $a_{n-1}$ and $b_n$ is the geometric mean of $b_1$ through $b_{n-1}$, and for $i=1,2,3$, $a_i = b_i$. If $a_1 = 3$, $a_{2017} = 7$, and $a_{2018} = 8$, find the value of $a_2^2 + a_3^2$. Assume $a_2^2 + a_3^2 = 1$. What is $a_2^2 + a_3^2$?

Answer: 1


Question: Let $L^2([0,1])$ be the Hilbert space of all real-valued square integrable functions on $[0,1]$ with the usual inner product. Define the linear functional $\phi$ by \( \phi(f) = \int_{1/4}^{3/4} 3\sqrt{2} f \, d\mu \), where $\mu$ denotes the Lebesgue measure on $[0,1]$. Determine the norm $||\phi||$ of this linear functional.

Answer: 3


Question: Determine the number of ideals $A$ in the ring $R$, where $R$ is the integral closure of $\mathbb{Z}$ in $K$ and $K=\mathbb{Q}[\sqrt{-6}]$, such that the norm $N(A) = 2^2 \cdot 11^3$. Given that $2R=\mathfrak{p}^2$ and $11R=\mathfrak{q}_1\mathfrak{q}_2$ for suitable distinct prime ideals $\mathfrak{p}, \mathfrak{q}_1, \mathfrak{q}_2$, find the number of such ideals $A$. Assume $A=\mathfrak{p}^a\mathfrak{q}_1^b\mathfrak{q}_2^c\mathfrak{r}$, where $a,b,c \in \mathbb{N}$ and $\mathfrak{r}$ is coprime with $\mathfrak{p}\mathfrak{q}_1\mathfrak{q}_2$. The norm condition implies $2^2 \cdot 11^3 = N(\mathfrak{p}^a\mathfrak{q}_1^b\mathfrak{q}_2^c\mathfrak{r}) = 2^{2a}11^{b+c}N(\mathfrak{r})$. Determine the values of $a$, $b$, and $c$ that satisfy this condition.

Answer: 4


Question: Show that the function \( g(a) = \int _a^{a+1} \left\{x\right\} \cdot \left(1 - \left\{x\right\} \right)\,dx \) is constant for any real number \( a \), where \( \{x\} \) denotes the fractional part of \( x \).

Answer: \dfrac{1}{6}


Question: In a client-server scenario, a client masks a value $b \in \mathbb{F}_q$ as $M=(r_1\cdot b+r_2)\bmod q$, where $r_1$ and $r_2$ are chosen uniformly at random from the field such that $r_1, r_2 > \frac{q}{2}$. The client sends $M$ to the server and deletes $b$. The client can later check the integrity of $M$ by computing $s_1=(M- r_2)\bmod q$ and $s_2=(s_1\cdot (r_1)^{-1})\bmod q$. If $s_2$ is a uniformly random value, the client can detect a modification. The server can modify $b$ by finding $k=c\cdot r_1$ with $k<q$ and computing $M+k$, resulting in $s_2=b+c$. If $k \neq c\cdot r_1$, $s_2$ is uniformly random. What is the probability that the server modifies $M$ without being detected?

Answer: \dfrac{1}{2}


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

Answer: \dfrac{5e}{4}


Question: In a ticket counter, females and males arrive according to independent Poisson processes with rates 30 and 20, respectively. What is the probability that the first three customers to arrive are all female?

Answer: \dfrac{27}{125}


Question: Evaluate the expression \( \tan^{-1}\left(\frac{3 \sin 2\alpha}{5 + 3 \cos 2\alpha}\right) + \tan^{-1}\left(\frac{1}{4} \tan \alpha\right) \) for \( \frac{-\pi}{2} < \alpha < \frac{\pi}{2} \).

Answer: \alpha


Question: Determine the eigenvalues of the Hamiltonian given by \( H=E_{0} \sum_{m=-\infty}^{\infty}(|m\rangle\langle m+1|+\mathrm{h.c.}) \).

Answer: [-2E_0, 2E_0]


Question: Find the residue of the function $e^{1/z}$ at the point $z=0$.

Answer: 1


Question:
Calculate the expected claim size for an insurance policy with the following distribution function:

\[ F(x) = \begin{cases} 0 & x \le 0 \\ 0.0002x & 0 < x < 1000 \\ 0.4 & x = 1000 \\ 1 - 0.6e^{(x-1000)/2000} & x > 1000 \end{cases} \]

Choose the correct expected claim size from the options below:

A) 1500
B) 1700
C) 1900
D) 2100
E) 2300

Answer: D


Question: Consider a triangle with rational side lengths and rational angles measured in degrees. Must this triangle necessarily be equilateral? Provide a justification for your answer.

Answer: Yes


Question: Determine the value of the positive real number $c$ such that the equation $c \ln(x) = x^2$ has exactly one real root.

Answer: 2e


Question: Find the last two digits of \( 7^{7^{7^{10217}}} \).

Answer: 43


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

Answer: \dfrac{1}{4e}


Question: Determine the values of $a$ for which the improper integral \( \int_0^{\infty}\frac{\ln(1+x^2)}{x^a}dx \) converges.

Answer: (1, 3)


Question: Compute the integral \( \int_C \frac{e^z}{z} \, dz \), where \( C \) is the unit circle with positive orientation.

Answer: 2\pi i


Question:
For a real matrix $A$ satisfying $A^3 = A$, where $A$ is not equal to the identity matrix $I$ or the zero matrix $O$, if the rank of $A$ is $r$ and the trace of $A$ is $t$, which of the following is true?

A) $r \geq t$ and $r+t$ is odd
B) $r \geq t$ and $r+t$ is even
C) $r < t$ and $r+t$ is odd
D) $r < t$ and $r+t$ is even

Answer: B


Question: Calculate the value of the integral \( \iint_D y^{-1}\,dx\,dy \) over the domain \( D \) bounded by \( y=x^2 \), \( y=\frac{1}{2}x^2 \), and \( y=2x \). Use the change of variables \( x=uv \) and \( y=u^2 \) to evaluate the integral.

Answer: 2


Question: If the function \( f(x) = \sin(\ln x) - 2\cos(\ln x) \) is increasing in the interval \((e^\lambda, e^\mu)\), find the value of \( \sqrt{5}\cos(\mu-\lambda) \).

Answer: -\sqrt{5}


Question: Solve the initial value problem \( \frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 = 0 \) with the initial conditions \( y(0) = 2 \) and \( y'(0) = 1 \).

Answer: \ln(x + 1) + 2


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{\tan x-\sin x}{\sin^3x} \] without using series expansion or l'Hopital's rule.

Answer: \dfrac{1}{2}


Question: Find the greatest common factor of the polynomials \( P(x) = x^{50} - 2x^2 - 1 \) and \( Q(x) = x^{48} - 3x^2 - 4 \).

Answer: x^2 + 1


Question: Let $S$ be a collection of subsets of $\{1, 2, \ldots, 100\}$ such that any two sets in $S$ have a non-empty intersection. What is the maximum possible cardinality of $S$?

Answer: 2^{99}


Question: Compute the limit \( \lim\limits_{n\to \infty}{\prod\limits_{k=1}^{n}a_k} \) where \( a_k=\sum_{m=1}^{k} \frac{1}{\sqrt{k^2+m}} \).

Answer: 0


Question: Compute \(2^{1386} \mod 1387\) using only paper, pencil, and a basic calculator, without factoring or determining whether 1387 is prime.

Answer: 1


Question: If $f(x)$ and $g(x)$ are polynomials over $\mathbb{Z}$ and $f(n)$ divides $g(n)$ for all integers $n$, does it follow that $f(x)$ divides $g(x)$ in $\mathbb{Z}[x]$? Justify your answer.

Answer: No


Question: Evaluate the surface integral \( \iint_K {x \, dS} \) over the part of the parabolic cylinder \( z = \frac{x^2}{2} \) that lies inside the first octant part of the cylinder \( x^2 + y^2 = 1 \).

Answer: \dfrac{\pi}{8}


Question: Given a savings account with an initial balance of $0$ at time $t=0$, continuous payments are made with intensity $C_t$. The continuous compounding intensity is given by $\delta_t = \frac{1}{1+t}$. The accumulated value of the funds at time $t$ is $t(1+t)$. Determine the payment intensity function $C_t$.

Answer: 1 + t


Question: Given an irrational number \(x \in \mathbb{R}\setminus \mathbb{Q}\), determine if there exists a strictly increasing function \(T: \mathbb{N} \to \mathbb{N}\) such that \(\{T(n) x\} \to 0\) as \(n \to \infty\), where \(\{\cdot\}\) denotes the fractional part.

Answer: Yes


Question: Determine the number of elements of order 3 in an abelian group of order 120.

Answer: 2


Question: Determine whether for every real number $x \neq 0$, there exist transcendental numbers $\alpha$ and $\beta$ such that $x = \alpha - \beta$ and $\frac{\alpha}{\beta}$ is a transcendental number.

Answer: Yes


Question: Determine the Lebesgue measure of the image of the Cantor set under the Cantor function.

Answer: 1


Question: Evaluate the integral \( \int_{-\infty}^{\infty} f\left(x - \frac{1}{x}\right) \, dx \) given that \( \int_{-\infty}^{\infty} f(x) \, dx = 1 \).

Answer: 1


Question: Given natural numbers $n_1, n_2, \ldots, n_k$ such that $n_1 + n_2 + \cdots + n_k = n$, find the maximal value of the product $n_1 n_2 \cdots n_k$. Show that this maximum is $(t+1)^r t^{k-r}$, where $t = \left[\dfrac{n}{k}\right]$ and $r$ is the remainder when $n$ is divided by $k$, i.e., $n = tk + r$ with $0 \leq r \leq k-1$. If $\dfrac{n}{k}$ is an integer, the maximal value is $t^k$. What is the maximal value if $\dfrac{n}{k}$ is not an integer?

Answer: (t+1)^r t^{k - r}


Question: Let \( A = (a_{ij}) \in M_n(\mathbb{C}) \) be a matrix such that \( \det(A) = \det(A_{ij}) \) for every \( i, j \), where \( A_{ij} \) is the matrix obtained by replacing \( a_{ij} \) with \( 2-a_{ij} \). Define \( B = (a_{ij} + (-1)^i) \in M_n(\mathbb{C}) \). Find the value of \( \det B(\det A - \det B) \).

Answer: 0


Question: Determine whether $f(n) = O(g(n))$ or $g(n) = O(f(n))$ for the functions $f(n) = (\sqrt{n})^{\sqrt{n}}$ and $g(n) = (\lg n)^n$. Provide a justification for your answer.

Answer: f(n) = O(g(n))


Question: The length of each side of a certain right triangle is the reciprocal of a different integer. What is the least possible sum of these three integers?

Answer: 47


Question: An urn initially contains 5 white and 7 black balls. Each time a ball is drawn, its color is noted and it is placed back in the urn along with two balls of the same color. Compute the probability that the first two balls selected are black given that the next two balls drawn are white.

Answer: \dfrac{7}{32}


Question: Given the function \( f(x) = \lim_{n\to\infty} n^2 \left( e^{\frac{k}{n}\ln\sin x} - e^{\frac{k}{n+1}\ln\sin x} \right) \) where \( 0 < x < \pi \), \( n \in \mathbb{N} \), and the integral \( \int_0^{\frac{\pi}{2}} f(x) \, dx = -\frac{\pi}{k} \ln 4 \), find the value of \( k \).

Answer: 2


Question: Find the remainder when \(3^{3^{3^{3^{3^{3}}}}}\ldots\) (with 2021 threes) is divided by 100, using Euler's theorem.

Answer: 87


Question: Find a function \( u: \Omega \to \mathbb{R} \) such that \( u \in W^{1,p}(\Omega) \) but \( u \notin W_0^{1,p}(\Omega) \), where \( \Omega = B^n(0,1) \subseteq \mathbb{R}^n \) and \( p \in [1, \infty) \). Prove that \( W_0^{1,p}(\Omega) \neq W^{1,p}(\Omega) \).

Answer: u(x) = 1


Question:
Determine whether the following equation is true:
\[ \lambda_1\sum_{j=1}^{\infty} f_1(E_j) + \lambda_2\sum_{j=1}^{\infty}f_2(E_j) + \ldots + \lambda_n\sum_{j=1}^{\infty}f_n(E_j) = \sum_{j=1}^{\infty}\sum_{i=1}^{n}\lambda_if_i(E_j) \]

Answer: True


Question: Consider an affine Euclidean space \((A, \overrightarrow{A})\) with the associated Euclidean distance \(d\). Let \(a \in A\) and \(r > 0\). Determine whether the set \(\{b \in A : d(a, b) \leq r\}\) is compact in \((A, d)\).

Answer: Yes


Question:
Given that the cross ratio of the points $Z_1, Z_2, Z_3$, and $Z_4$ is real, determine which of the following statements is true:

1) $Z_1, Z_2$, and $Z_3$ are collinear.
2) $Z_1, Z_2$, and $Z_3$ are concyclic.
3) $Z_1, Z_2$, and $Z_3$ are collinear when at least one of $Z_1, Z_2$, or $Z_3$ is real.

Answer: B


Question:
Let $A$ be an $n \times n$ matrix which is both Hermitian and unitary. Determine which of the following statements are true:

(a) $A^2 = I$

(b) $A$ is real

(c) The eigenvalues of $A$ are $0$, $1$, $-1$

(d) The minimal and characteristic polynomials are the same.

Answer: a


Question: Determine the norm of the functional $f(x) = \sum_{n \in \mathbb{N}} x_n 2^{-n}$ in the dual space $({\ell^{\infty}})^{*}$, where $x \in \ell^{\infty}$. Express your answer as a single value.

Answer: 1


Question:
Given the homomorphism \(\varphi: \mathbb{Z}^2 \to S_{15}\) defined by:
\[\varphi(1,0)=(2,7,3)(3,11,5)(12,13)\]
\[\varphi(0,1)=(14,15)(1,4,6,8,9,10)\]
Calculate the number of elements in the group \(\mathbb{Z}^2/\!\ker\varphi\).

Answer: 60


Question: Evaluate the sum \( \sum_{n=1}^{\infty} \frac{\ln n}{n(n- \ln n)} \).

Answer: 1


Question: Let $X_{n}$ be a sequence of real random variables converging in law to $X$. Suppose the cumulative distribution function of $X$ is continuous everywhere. Let $\\{a_{n}\\}_{n}$ be a sequence of real numbers. Determine whether $\\lim_{n}\\mathbb{P}(X_{n} = a_{n}) = 0$.

Answer: 0


Question: Compute the expectation \( \mathbb{E}\left[\left(\dfrac{1}{T}\int^T_0 W_t \, dt\right)^2\right] \), where \( W_t \) is a standard Brownian motion.

Answer: \dfrac{T}{3}


Question: Evaluate the limit: \[ \lim_{n\to \infty} n^{-n^2}\left( \prod_{r=0}^{n-1} \left(n+\frac {1}{3^r}\right) \right) ^n \]

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


Question: Find the analytical function of the complex variable $z$ whose imaginary part is given by $v(x,y) = \ln |z|$. Use the Cauchy-Riemann equations to determine the function.

Answer: i \ln z


Question: Evaluate the complex integral \( \int_0^{2\pi} e^{ie^{it}}e^{-it} \, dt \).

Answer: 2\pi i


Question:
Let $f$ be a continuous, differentiable, and bijective function. If the tangent to $y=f(x)$ at $x=a$ is also the normal to $y=f(x)$ at $x=b$, then there exists at least one $c \in (a,b)$ such that:

(a) $f'(c)=0$
(b) $f'(c)>0$
(c) $f'(c)<0$
(d) none of these.

Answer: D


Question: Compute \( \iint_D (x^2 - y^2)^{10} \, dx \, dy \), where \( D \) is the region bounded by \( |x| + |y| \leq 1 \). Use the substitution \( u = x - y \) and \( v = x + y \) to evaluate the integral.

Answer: \dfrac{2}{121}


Question: A person is inside a spherical environment with a radius of 100 units. The person can move forward from the origin towards the wall of the sphere. The distance from the person to the wall of the sphere is 20 times the viewing angle in radians. If the person is 20 units away from the wall, what is the viewing angle in radians?

Answer: 1


Question: Let \( E \subset \mathbb{R}^2 \) be a non-empty open set such that the union of \( E \) with finitely many points becomes a closed set. Is it necessarily true that the closure of \( E \) is all of \( \mathbb{R}^2 \)? Justify your answer.

Answer: Yes


Question: Calculate the variance of the random variable \( X = \int_{0}^{1} W^2(t) \, dt \), where \( W(t) \) is a Wiener process.

Answer: \dfrac{1}{3}


Question: Determine whether every natural number can be realized as the chromatic number of a countable hypergraph. Specifically, for a given integer \( n > 1 \), does there exist a subset \( E \subseteq [\omega]^\omega \) such that the chromatic number \( \chi(\omega, E) = n \)?

Answer: Yes


Question: Determine whether there exists a continuous surjection from the circle $S^1$ to the interval $[0, 1]$. Justify your answer.

Answer: Yes


Question: Find \( \lim_{n \to \infty} \left( n + (n^2 - n^3)^{1/3} \right) \).

Answer: \dfrac{1}{3}


Question: Let $\Omega \subseteq \mathbb{R}^n$ be an open set, and let $f: \Omega \rightarrow \mathbb{R}$ be a Lipschitz continuous function with constant $L$ and compact support. Define $f'$ such that $f'(x)=0$ if $x \notin \Omega$ and $f'(x)=f(x)$ otherwise. Is $f'$ Lipschitz continuous?

Answer: Yes


Question: In how many ways can you divide $n$ students into 4 distinct groups such that each student is in exactly one group and each group is assigned to a unique project?

Answer: 4^n - 4 \cdot 3^n + 6 \cdot 2^n - 4


Question: If \( y = \tan^{-1}\left(\sqrt{\dfrac{1+\cos x}{1-\cos x}}\right) \), find the value of \((2x+14y)^3-343\).

Answer: 0


Question: Does there exist a subset $T \subset \mathbb{Q}$ such that, for every integer $z \in \mathbb{Z}$, there is a unique finite set $A_z \subset T$ satisfying $z = \sum_{t \in A_z} t$?

Answer: Yes


Question: Let \( p(x) \) be the minimal polynomial of \( \sqrt{2} + \sqrt{-2} \) over the field \( \mathbb{Q} \) of rational numbers. Evaluate \( p(\sqrt{2}) \).

Answer: 20


Question: Consider the curve \( y = x^4 + Ax^3 + Bx^2 + Cx + D \) which touches the line \( y = px + q \) at \( x = 2 \) and \( x = 3 \), where \( A, B, C, D, p, q \in \mathbb{R} \). If the area bounded by the curve and the line is \( \frac{1}{\alpha} \), find the number of divisors of \( \alpha \).

Answer: 8


Question: Determine the smallest positive constant \( c \) such that for all sufficiently large \( N \), the inequality \( r(N) \geq cN^{1/2} \) holds, where \( r(N) \) is the least number of quadratic polynomials with non-negative coefficients needed to cover the interval \([1, N]\).

Answer: 1


Question: Determine the maximum possible value of \( \min_{i \neq j \in \{1, \dots, 6\}} |x_i - x_j| \) for six points \( x_1, \dots, x_6 \) placed on the unit sphere \( S^2 \subset \mathbb{R}^3 \). Call this maximum \( M_6 \).

Answer: \sqrt{2}


Question: Evaluate the limit \( \lim_{n \to \infty} n^2 \sum_{k=0}^{n-1} \sin \left(2\pi \frac{k}{n}\right) \).

Answer: 0


Question:
Determine the function to which the sequence of functions \( f_n(x) \) converges, where
\[
f_n(x)=\begin{cases}
 n^2x& \text{ if }\; 0\leq x \leq \frac{1}{n} \\
 -n^2x+2n& \text{ if }\;\; \frac{1}{n} < x <\frac{2}{n} \\
 0& \text{ if }\;\;\frac{2}{n}\leq x \leq 1
\end{cases}
\]

Answer: 0


Question: Determine whether the following statement is true or false: Let $A \in \mathbb{R}^{n \times n}$ be a generic lower triangular matrix and let $P \in \mathbb{R}^{n \times n}$ be a symmetric positive definite matrix. If $AP + PA^\top = 0$, does it imply that $AP = 0$?

Answer: True


Question: Given the differential equation \((xy^3 + x^2y^7)\frac{dy}{dx} = 1\) and the initial condition \(y(1/4) = 1\), find the value of \(\frac{dy}{dx}\) when \(y = -1\).

Answer: -\dfrac{16}{5}


Question: Determine whether every compactification of a topological space has the extension property, defined as follows: A pair (Y, f) is a compactification of a topological space X if and only if for every compact space X' and continuous map g from X to X', there exists a continuous map g' from Y to X' such that g'(f(x)) = g(x) for all x in X. Is this statement true?

Answer: False


Question: Let \(P(x) = x^4 + ax^3 + bx^2 + cx + d\) be a polynomial where \(a, b, c, d\) are integers. Given that \(P(x)\) has remainders 24, -6, 4, -6, and 24 when divided by \(x-2012\), \(x-2013\), \(x-2014\), \(x-2015\), and \(x-2016\) respectively, find the remainder when \(P(x)\) is divided by \(x-2017\).

Answer: 274


Question: Estimate \( \int_0^1 \sin(x^2) \, dx \) using the Taylor expansion of \( \sin(x) \) such that the error is less than 0.00002. First, find the Taylor polynomial \( T_6(x) \) for \( f(x) = \sin(x) \) about \( x=0 \). Then, use this polynomial to approximate the integral and ensure the error is within the specified range.

Answer: 0.31028


Question: Is it possible to find two matrices $A, B \in L(\mathbb{R}^{n})$ such that both $A$ and $B$ have eigenvalues with negative real parts, but the matrix $A + B$ has eigenvalues with non-negative real parts?

Answer: Yes


Question: Consider a subset $F$ of $[0,1]$ constructed similarly to the Cantor set, but with each interval removed at the $n$th iteration having a length of $\frac{\alpha}{3^{n}}$, where $0<\alpha<1$. Show that the total length of $F$ is $1-\alpha$ by taking the limit as $n \to \infty$ of the length of the intervals left behind at each iteration.

Answer: 1 - \alpha


Question: In triangle $ABC$, point $H$ is on segment $AC$. A perpendicular line $PH$ is drawn from $H$ to $AC$, intersecting $AB$ at point $Q$. Given that $\angle PAB = 53^\circ$, $\angle ACB = 143^\circ$, $AP = AB$, and $AH = 12$, calculate the length of segment $HC$.

Answer: 8


Question: Determine whether the following statement is true: If $g \in G$ and $R: G \rightarrow GL(V)$ is the matrix form of an irreducible representation of $G$, then $R^{-1}(g) = R(g^{-1})$. Here, the left-hand side corresponds to the matrix inversion of the irrep of $g$, and the right-hand side corresponds to the irrep of the inverse of $g$. Is this statement true?

Answer: True


Question: Find the limiting distribution of \( \sum_{i=1}^n \frac{Z_i + 1/n}{\sqrt n} \), where \( Z_i \sim N(0, 1) \) are independent random variables. Use moment generating functions to determine the distribution.

Answer: N(0,1)


Question: Solve the matrix equation \( xA = x \) for the vector \( x \), given that \( A \) is a reversible matrix.

Answer: x = 0


Question: Given a field $F$ and the element $u = \frac{x^3}{x+1} = x^2 - x + 1 - \frac{1}{x+1}$ in $F(x)$, determine the degree of the field extension $[F(x):F(u)]$. Use the hint to choose an element $v \in F(x)$ such that $F(u,v) = F(x)$ and find the minimal polynomial of $v$ over $F(u)$. What is the degree of this polynomial?

Answer: 3


Question: Find the number of ordered pairs \((x, y)\) that satisfy the equation \(\left\lfloor\frac{x}{2}\right\rfloor + \left\lfloor\frac{2x}{3}\right\rfloor + \left\lfloor\frac{y}{4}\right\rfloor + \left\lfloor\frac{4y}{5}\right\rfloor = \frac{7x}{6} + \frac{21y}{20}\), where \(0 < x, y < 30\).

Answer: 4


Question: Let $H$ be a Hilbert space and $p, q$ be self-adjoint projectors in $B(H)$, satisfying $p^2 = p = p^*$ and $q^2 = q = q^*$. Suppose they have the same centralizers, $C(p) = C(q)$, where $C(x) = \{y \in B(H) : yx = xy\}$. Determine if it is true that $p = \pm q$. Provide a justification for your answer.

Answer: p \neq \pm q


Question: Determine the degree of the chromatic polynomial of a graph with $n$ vertices and a chromatic number of $\chi(G) = 3$. Provide your answer as a single integer.

Answer: n


Question: A biased coin has a probability of 0.70 of landing heads. You flip the coin once, and it lands tails. From this point, what is the expected number of additional flips needed until the number of heads equals the number of tails?

Answer: \dfrac{5}{2}


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

Answer: 1


Question: Find a monic polynomial over \(\mathbb{Q}\) whose Galois group is \(V_4 = \langle (12)(34), (13)(24) \rangle \subset S_4\).

Answer: x^4 - 10x^2 + 1


Question: Consider the sequence \((x_n) = \left(\sum_{k=1}^n \frac{1}{k}\right)_n\), which is \((1, 1+\frac{1}{2}, 1+\frac{1}{2}+\frac{1}{3}, \ldots)\). Is it possible to find a subsequence \((x_{r_n})\) of \((x_n)\) such that \(\lim_{n \to \infty} |x_{r_{n+1}} - x_{r_n}| \neq 0\)?

Answer: Yes


Question: Given that \( f(x) \) is a polynomial of degree 3, and its remainders are \( 2x-5 \) and \( -3x+4 \) when divided by \( x^2 - 1 \) and \( x^2 - 4 \) respectively, find the value of \( f(-3) \).

Answer: 53


Question: Given a finite set of natural numbers \( S = \{x_1, x_2, \dots, x_n\} \), determine the number of distinct natural numbers that divide at least one element of the set \( S \). For example, if \( S = \{2, 6, 15\} \), the answer is 6.

Answer: 6


Question: Consider a group of $n$ people, each randomly choosing another person from the group. If any person chooses themselves, the group must repeat the process. Define the random variable $Y$ as the number of attempts needed until no one chooses themselves. Find the expected value of $Y$, $E(Y)$. It is known that this expected value is approximately $e$. Explain why this is the case.

Answer: e


Question: Suppose \(f\) is a continuous function on the interval \([0, 1]\) such that \(f(0) = f(1) = 0\) and \(f(x) > 0\) for all \(x \in (0, 1)\). Does there always exist \(c\) and \(d\) in \((0, 1)\) such that \(f(c) = f(d) = d - c\)?

Answer: Yes


Question: Determine the number of solutions of the equation \(z^{3}+2z^{2}-z-2+e^{z}=0\) that lie in the left half-plane.

Answer: 2


Question: Find the residue of the function \( f(z) = \frac{z}{\sin^2 z} \) at the point \( z = n\pi \), where \( n \) is an integer and \( n \neq 0 \).

Answer: 1


Question: Find a line that is tangent to the curve \( y = x^4 - 2x^3 - 3x^2 + 5x + 6 \) at least at two distinct points.

Answer: y = x + 2


Question: Determine whether the following statement is true or false: Given a skew symmetric matrix M, among all of its largest rank submatrices, there must be one that is a principal submatrix of M.

Answer: True


Question: Let $X$, $Y$, and $Z$ be independent normally distributed random variables with expected values $2$, $1$, and $0$, respectively, and each having a variance of $2$. Compute the probability $P(4X - 3Y > 5Z)$. Provide your answer to four decimal places.

Answer: 0.6915


Question: Evaluate the limit: \[ \lim_{n \to \infty}\frac{n! - \sin(2n)}{\sqrt{(n+1)!} - \ln(n^2 + 3n)} \]

Answer: \infty


Question: Determine whether the following statement is true or false: If \((n^{2} a_{n})\) is a convergent sequence, then there exists a subsequence \((x_{n_k})\) of \(x_{n}=\frac{\sin (n)}{n}\) such that the series \(\sum_{k=1}^{\infty} x_{n_k} a_{k}\) is convergent.

Answer: True


Question: Suppose $f,g : \mathbb{R}_+ \to \mathbb{R}_+$ are increasing, bijective functions such that $f(x) \in o\big(g(x)\big)$ as $x \to \infty$. Does it follow that $g^{-1}(x) \in o\big(f^{-1}(x)\big)$ as $x \to \infty$?

Answer: No


Question:
Given that \(A, B, C\) are angles of an acute-angled triangle, find the minimum value of the expression:
\[
\sin^4 A + (1 - \sin A)^4 + \sin^4 B + (1 - \sin B)^4 + \sin^4 C + (1 - \sin C)^4.
\]

Answer: 10 - 6\sqrt{2}


Question: Given \( \ddot{x} = 2x \) and \( \dot{x} = 0 \) when \( x = 1 \), find \( \dot{x} \) when \( x = 3 \).

Answer: 4


Question: Given that \( \log_2(\log_3 x) = \log_3(\log_4 y) = \log_4(\log_2 z) = 0 \), find the value of \( x + y + z \).

Answer: 9


Question: Evaluate the limit: \[ \lim\limits_{x \to 0} \frac{1}{x}\int_0^x{(1 + \sin2t)^{1/t}dt} \]

Answer: e^2


Question: Find the coefficient of \( \frac{1}{z} \) in the series expansion of \( \frac{1}{z^4 \cdot \sin z} \).

Answer: \dfrac{7}{360}


Question: Express the series \( S = \frac{1 \cdot 2 \cdot 3}{1!} + \frac{2 \cdot 3 \cdot 4}{2!} + \frac{3 \cdot 4 \cdot 5}{3!} + \cdots \) in terms of the exponential function \( e \).

Answer: 13e


Question: Given the recursive function \( f(n+1) = n(-1)^{n+1} - 2f(n) \) for integer \( n > 1 \), and the condition \( f(1) = f(1986) \), compute the sum \( f(1) + f(2) + f(3) + \ldots + f(1985) \).

Answer: 331


Question: Determine whether $L^p(\mathbb{R})$ contains a subspace that is isometrically isomorphic to $\ell^p$ for $1 \leq p \leq \infty$. Provide a justification for your answer.

Answer: Yes


Question: Consider a 3x3 matrix \( A \) with elements chosen from the set \( \{0, 1, 2, 3\} \). Determine the maximum possible value of \( \det(A) \).

Answer: 54


Question: Find the stationary point of the functional \( J[y]=\int \left( x^2y'^2+2y^2 \right) dx \) subject to the boundary conditions \( y(0)=0 \) and \( y(1)=2 \).

Answer: 2x


Question: Solve for \( a \) in the equation \( \int_{0}^{\alpha b}(a^x-1)\,dx = \int_{\alpha b}^{b}(a^x-1)\,dx \), given that \( 0 < \alpha < 1 \) and \( b \geq 1 \).

Answer: 1


Question: If \(a^n = 0\) for some element \(a\) in a ring \(R\) with unity, does this imply that \(R\) cannot be an integral domain? Justify your answer.

Answer: Yes


Question: Evaluate the improper integral: $$\int_{0}^{\infty}\frac{x^2}{e^x-1}\,dx$$

Answer: 2\zeta(3)


Question: Given polynomials $L_1, \ldots, L_m \in \mathbb{Z}[x_1, \ldots, x_n]$ of the form $L_i = l_{i1} \cdot l_{i2} \ldots \cdot l_{ik}$, where each $l_{ij}$ is an integer linear form with coefficients bounded by an integer $H$, determine if the coefficients $B_1, \ldots, B_m \in \mathbb{Z}$ in the linear dependency $B_1\cdot L_1 + \ldots + B_m\cdot L_m \equiv 0$ can be chosen such that $|B_i| \le H^{\text{poly}(n,m,k)}$ for some polynomial $\text{poly}(n,m,k)$. Answer with 'Yes' or 'No'.

Answer: No


Question: Given the set \( A = \left\{ \frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{k} \right\} \) for a known \( k \in \mathbb{N} \), choose a rational number \( a \neq k \) such that the set \( B = \left\{ \frac{a}{1}, \frac{a}{2}, \ldots, \frac{a}{k} \right\} \) shares the maximum number of elements with \( A \). Minimize the number of non-duplicate elements between sets \( A \) and \( B \).

Answer: 1


Question: Do there exist wavelets such that both the mother wavelet and the scaling function have compact support and are both twice continuously differentiable?

Answer: Yes


Question:
Find the value of \( a \) such that the function \( f: \mathbb{R}^2 \to \mathbb{R} \) defined by
\[
f(x,y) = \begin{cases}
\frac{x\sin^2(x) + axy^2}{x^2 + 2y^2 + 3y^4}, & (x,y) \neq (0,0) \\
0, & (x,y) = (0,0)
\end{cases}
\]
is differentiable at \((0,0)\).

Answer: 2


Question:
Evaluate the limit of the series given by the sequence where $a_1=1$ and $a_n=n(1+a_{n-1})$ for all $n \geq 2$:
\[\lim_{n\to \infty} \left(1+\frac{1}{a_1}\right)\left(1+\frac{1}{a_2}\right)\cdots\left(1+\frac{1}{a_n}\right)\]

Answer: e


Question:
Find the limit of the sequence as \( n \to \infty \):
\[ \lim_{n\to \infty}\left(\frac{1}{2n+1} + \frac{1}{2n+2}+\ldots+\frac{1}{8n+1}\right) \]

Answer: \ln 4


Question: Determine if there exists a solution to the Diophantine equation \(a^2 + b^2 = c^2 + d^2\).

Answer: Yes


Question: Find the last two digits of the number \(7^{7^{7^{10217}}}\).

Answer: 43


Question: Calculate the residue of the function \((5z^{9}+z^{11})\cos\left(\frac{1}{z^3}\right)\) at the point \(z=0\).

Answer: \dfrac{1}{24}


Question: Let $G$ and $H$ be two finite, simple, connected graphs with the same degree sequence. Is it possible to make the absolute difference $|\chi(G) - \chi(H)|$ arbitrarily large by choosing appropriate graphs $G$ and $H$?

Answer: Yes


Question: Evaluate the integral \( I = \frac{1}{\omega_n}\int_{\mathbb{S}^n}\! x_{n+1}^2\,\mathrm{d}x \), where \( \mathbb{S}^{n} \) denotes the unit sphere in \( \mathbb{R}^{n+1} \) with surface area \( \omega_n \), and \( \mathrm{d}x \) denotes integration with respect to the usual measure on \( \mathbb{S}^{n} \).

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


Question: Evaluate \( \lim_{n\to\infty}\int_a^b f(x)\sin(nx)\,dx \) where \( f:[a,b]\to\mathbb{R} \) is continuously differentiable.

Answer: 0


Question: Consider a set of random variables \( \{ Y_1, \cdots, Y_n \} \) where each \( Y_i \) is a \( \{ 0, 1 \} \) random variable with \( P(Y_i = 1) \geq a \). Now, create \( n \) i.i.d. Bernoulli random variables \( \{ X_1, \cdots, X_n \} \) with \( P(X_i = 1) = a \). Is it true that \( P\left( \sum_{i=1}^n Y_i > C \right) \geq P\left( \sum_{i=1}^n X_i > C \right) \) for any constant \( C \)?

Answer: No


Question: Evaluate the contour integral \( \int_{|z+1|=2} \frac{e^z}{z-2} \, dz \) and verify whether it satisfies the inequality \( \leq 2\pi e \).

Answer: 0


Question: Find the derivative of the function \( f(z) = z^c \), where \( c \in \mathbb{C} \setminus \{0\} \).

Answer: c z^{c - 1}


Question: Determine the value of \( n \) such that all solutions of the differential equation \( y'' + ny = \cos(nx) \) are unbounded.

Answer: 1


Question: Determine whether it is true that for every non-zero real number $x$, there exist linearly independent transcendental numbers $\alpha$ and $\beta$ such that $x = \alpha - \beta$. Linearly independent means that for any rational numbers $r$ and $s$, the equation $r\alpha + s\beta = 0$ implies $r = s = 0$. Note that this statement is true if $x$ is an algebraic number.

Answer: True


Question: Find the limit: \[ \lim_{x\rightarrow 0}\dfrac{1}{x}\int_x^{2x}e^{-t^2}dt \]

Answer: 1


Question: Determine the minimum possible least common multiple (LCM) of $X$ natural numbers whose sum is $N$. For example, find the minimum possible LCM when $X=10$ and $N=45$.

Answer: 6


Question: Find the volume of the solid bounded by the surfaces \(x^2+z^2=4\) and \(y^2+z^2=4\).

Answer: \dfrac{128}{3}


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

Answer: \dfrac{1}{e}


Question: Find the value of \( f^{(10)}(0) \) for the function \( f(x) = \frac{\cos(3x^2) - 1}{x^2} \) using the Maclaurin series.

Answer: -3674160


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_{1}^{\infty} n e^{-x^{\delta}}(1 - e^{-x^{\delta}})^n \, dx \] where \(0< \delta <1\).

Answer: \infty


Question: Find the greatest value of \( n \) such that the equation \( 2\tan^2x - 5\sec x = 1 \) has exactly 7 distinct solutions for \( x \in [0, \frac{n\pi}{2}] \), where \( n \in \mathbb{N} \).

Answer: 15


Question:
Let \( G \in \mathbb{R}^{n \times m} \) be a row-stochastic matrix with \( n < m \) and \( \text{Rank}(G) < n \). Define \( J \in \mathbb{R}^{n \times m} \) by:
\[ J_{i,j} = \begin{cases} 1 & \text{if } i = j, i \in [n] \\ 0 & \text{otherwise} \end{cases} \]
For \( \epsilon > 0 \), is the rank of \((1-\epsilon)G + \epsilon J\) equal to \( n \)?

Answer: Yes


Question:
Evaluate the limit \( l = \lim\limits_{x\nearrow 1}(1-x)^2f(x) - \lim\limits_{x\searrow1}f(x) \), where \( f(x) = \lim\limits_{n\to\infty}f_n(x) \) and \( f_n:\mathbb{R}\to\mathbb{R} \) is defined as follows:
\[
f_n(x) = \begin{cases}1+2x+3x^2+\ldots+nx^{n-1},&x\leqslant1\\
e^{n(1-x)},&x>1.\end{cases}
\] Find \( l \).

Answer: 1


Question:
Determine the dimension of the Lie group $SL(n,\mathbb{C})$. Choose the correct option:

(a) $n^2$
(b) $n^2-1$
(c) $2(n^2-1)$
(d) $2n^2$

Answer: C


Question: Let $H \subset G$ be finite groups, and let $H^{ab}$ and $G^{ab}$ be their abelianizations. Is it true that $|H^{ab}| \leq |G^{ab}|$?

Answer: No


Question:
Given that the function $f$ satisfies the equation $2f(x) + f(1-x) = x^2$ for all $x$, determine the value of $f(-5)$. Choose the correct answer from the following options:

(A) $\frac{15}{4}$
(B) $\frac{21}{5}$
(C) $\frac{25}{3}$
(D) $\frac{14}{3}$
(E) $\frac{34}{3}$

Answer: D


Question: Determine the number of positive integers \( x \) such that \( x \leq 9,999,999 \) and the sum of the digits of \( x \) equals 31.

Answer: 512365


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

Answer: 1


Question: Evaluate the sum \( \sum_{n=0}^\infty \frac{1}{n!(n^4+n^2+1)}. \)

Answer: \dfrac{e}{2}


Question: In a multiplication problem, all 9 digits from 1 to 9 are used exactly once, represented by asterisks. Each number in the three rows is divisible by 9, and the product consists of four consecutive digits, not necessarily in order. The sum of the last digits of the three numbers (indicated by red asterisks) is one of the following: 21, 20, 19, 18, or 17. Determine the sum of the last digits of the three numbers.

Answer: 21


Question: Let \( I \) be the augmentation ideal of the group ring \( \mathbb{Z}/p[\mathbb{Z}/p] \). For a given odd prime \( p \) and an integer \( n \) such that \( 1 \leq n \leq p-1 \), determine the dimension of \( I^n \) as a vector subspace of \( \mathbb{Z}/p[\mathbb{Z}/p] \).

Answer: p - n


Question: Determine whether there exists a function \( f: \mathbb{R} \to \mathbb{R} \) that is a bijection and is discontinuous at every point. Provide a justification for your answer.

Answer: Yes


Question: Given that \(X\) follows a geometric distribution with parameter 0.09, \(X \sim \mathrm{Geo}(0.09)\), and \(Y|X=x\) follows a geometric distribution with parameter \(1/(x+1)\), \(Y|X=x \sim \mathrm{Geo}(1/(x+1))\), calculate the probability \(\mathbb{P}(\mathbb{E}(Y|X)=3)\).

Answer: 0.0819


Question: Compute \( \lim_{x\to\infty} \left( x - \sqrt{x^2+x+1} \frac{\ln(e^x+x)}{x} \right) \).

Answer: -\dfrac{1}{2}