Question: Let $R$ be a finite commutative ring and consider the full matrix ring $M_n(R)$ for $n > 1$. For a matrix $A \in M_n(R)$, is it true that the cardinality of the left annihilator of $A$ in $M_n(R)$ equals the cardinality of the right annihilator of $A$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to\infty}\sum_{k=1}^{\infty} \frac{(-1)^{k+1} (2^k-1)x^k}{k k!} \]

Answer: \ln 2


Question: Let $f$ be an entire function such that $|f(z)| \in \mathbb{Z}$ for all $z \in \mathbb{C}$. Is $f$ a constant function? Justify your answer.

Answer: Yes


Question: Find the product of all divisors of \(420^4\).

Answer: 420^{2250}


Question: Let \( f(x) = \int_0^x e^{x-y} f'(y) \, dy - (x^2 - x + 1) e^x \). Find the number of roots of the equation \( f(x) = 0 \).

Answer: 1


Question: Find a closed-form expression for the recursive formula \( f(n) = 1+\frac{1}{n}\sum_{i = 0}^{n - 1} f(i) \) with the base case \( f(0) = 0 \).

Answer: H_n


Question: Determine the number of equivalence classes in the set $S$ of all $2 \times 3$ real matrices, where each entry is $1, 0,$ or $−1$. Two matrices $M$ and $M'$ are equivalent if they have the same column space. How many equivalence classes are there in $S$?

Answer: 6


Question: Find the derivative of the function: \[ y = \bigg\lfloor{\arccos\left(\frac{1}{\tan\left(\sqrt{\arcsin x}\right)}\right)}\bigg\rfloor \]

Answer: 0


Question: Find the area enclosed by one loop of the parametric curve given by \( x(t) = t^3 - 3t \) and \( y(t) = t^2 + t + 1 \).

Answer: \dfrac{81}{20}


Question: Suppose the systems of linear equations $Ax = d$ and $A'x = d'$ are equivalent, where $A$ and $A'$ are full rank matrices. Additionally, consider the systems $(A+B)y = d+b$ and $(A'+B)y' = d'+b$, where $A+B$ and $A'+B$ also have full rank and have solutions $y$ and $y'$ respectively. Does the equivalency of $Ax = d$ and $A'x = d'$ imply that $y = y'$?

Answer: No


Question: Calculate the expected number of coin flips needed to obtain two consecutive tails (TT) when flipping a fair coin. Use a series approach to determine this expected value.

Answer: 6


Question:
Consider the set of integers \(\{1, 2, \ldots, 100\}\). Let \(\{x_1, x_2, \ldots, x_{100}\}\) be a permutation of these integers. Find the smallest possible value of the sum:
\[ S = |x_2 - x_1| + |x_3 - x_2| + \cdots + |x_{100} - x_{99}| + |x_1 - x_{100}|. \]

Answer: 198


Question: Find the slant asymptote of the function \( f(x) = xe^{\frac{1}{x-2}} \).

Answer: y = x + 1


Question: Evaluate the expression \( \frac{1^4 + 2007^4 + 2008^4}{1^2 + 2007^2 + 2008^2} \) without using a calculator. Provide your working and final answer.

Answer: 4030057


Question: Evaluate the integral \( \oint_{|z-1|=1} \frac{\exp(z)}{1-z} \, dz \) around a circle of radius 1 centered at \( z=1 \).

Answer: -2\pi i e


Question: In the context of the simplex method for linear programming, suppose the final iteration of the simplex tableau indicates nondegeneracy (no basic variable is at zero level) and the reduced cost of one of the non-basic variables is zero. Does this guarantee the existence of another optimal solution that is distinct from the current optimal solution?

Answer: Yes


Question: Determine whether the following statement is true or false: If \( \lim_{x\rightarrow x_{0}}f'(x)=a\in{\mathbb R} \), then \( f'(x_{0})=a \).

Answer: B


Question: Compute the variance of the geometric mean of two independent random variables, $Y_1$ and $Y_2$, each uniformly distributed on the interval $[0, 1]$. The geometric mean is given by $X = (Y_1 Y_2)^{1/2}$. What is the variance of $X$?

Answer: \dfrac{17}{324}


Question: Determine the number of solutions for the equation: \[2\sqrt{3} \arctan\sqrt{x} = \sqrt{(1 + x)}\]

Answer: 2


Question: Calculate the limit: \[ \lim_{x\to\infty} \left( \sqrt[100]{(x + 3 \cdot 1)(x + 3 \cdot 2)\cdots(x + 3 \cdot 100)} - x \right) \]

Answer: \dfrac{303}{2}


Question: Let $G$ be a Schmidt group and $Q = \langle a \rangle$ be a $q$-subgroup of $G$. Prove that $a^q \in Z(G)$.

Answer: a^q \in Z(G)


Question:
Determine the number of ordered pairs \((x, y)\) that satisfy the following system of equations:
\[\begin{align*}
\sin(x+y) &= \cos(x+y) \\
x^2 + y^2 &= \left(\frac{1995\pi}{4}\right)^2
\end{align*}\]

Answer: 2822


Question:
Find the coordinates of the point \(p = (3, x, y, 1)\) in \(\mathbb{R}^4\) such that the triangles \(\triangle acp\) and \(\triangle bcp\) are right-angled, where \(a = (3, 2, 5, 1)\), \(b = (3, 4, 7, 1)\), and \(c = (5, 8, 9, 3)\). Given that \(p\) lies on the same line as \(a\) and \(b\), and shares two coordinates with them, solve the following system of equations to find \(x\) and \(y\):

1. \(x^2 + y^2 - 10x - 14y + 61 = 0\)
2. \(x^2 + y^2 - 12x - 16y + 95 = 0\)

Additionally, use the equation \(x + y - 17 = 0\) derived from the conditions to find the exact coordinates of \(p\).

Answer: (3, 7, 10, 1)


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

Answer: +\infty


Question: In the context of Lebesgue measure on \(\mathbb{R}^n\), is a null set the same as a set of measure zero? Specifically, if a set has measure zero, can it be covered by a countable union of balls of arbitrarily small total length?

Answer: Yes


Question: Find the categorical sequential limit of the sequence \(... \to \mathbb{Q} \to \mathbb{Q} \to \mathbb{Q}\) in the category of abelian groups, where each arrow is multiplication by a fixed natural number \( n > 1 \).

Answer: \mathbb{Q}


Question: Consider iid uniform random variables $X_{1}, \dots, X_{n}$ on the interval $(0,1)$. Determine the limit, as $n$ approaches infinity, of $\min(nX_{1}, \dots, nX_{n}) = nM$, where $M= \min(X_{1}, \dots, X_{n})$. Prove that this limit is infinity.

Answer: \infty


Question: Let $p(x)$ be a function defined on $\mathbb{R}$ such that $p'(x) = p'(1-x)$ for all $x \in [0,1]$. Given that $p(0) = 1$ and $p(1) = 41$, find the value of $\int_{0}^{1} p(x) \, dx$. \( \text{Express your answer as a single number.} \)

Answer: 21


Question: Calculate the work done by the engine pistons when the pressure vector field is given by \( \mathbf{P} = (y - x^2) \mathbf{i} + (z - y^2) \mathbf{j} + (x - z^2) \mathbf{k} \) and the path \( C \) is the helix \( \mathbf{V}(t) = t\mathbf{i} + t^2\mathbf{j} + t^3\mathbf{k} \) from \((0,0,0)\) to \((1,1,1)\).

Answer: \dfrac{29}{60}


Question:
Determine which of the following expressions correctly represents the group U(2):

1. $U(2) = \{\exp(ix) \exp(i\theta\mathbf{\hat{n}}\cdot\mathbf{\sigma}) \,:\,\{x,\theta\}\in\mathbb{R}\wedge\mathbf{\hat{n}}\in S^2 \}$

2. $U(2) = \{\exp(ix)I+ \exp(i\theta\mathbf{\hat{n}}\cdot\mathbf{\sigma}) \,:\,\{x,\theta\}\in\mathbb{R}\wedge\mathbf{\hat{n}}\in S^2 \}$

Explain your reasoning based on the relationship $U(2)\simeq \langle SU(2)\cup U(1)\rangle$. (Here, $\langle A\rangle$ stands for the subgroup generated by the subset $A$.)

Answer: 1


Question:
Let \( \{x_n\} \) be a sequence of real numbers such that \( \lim_{n \to \infty} (x_{n+1} - x_n) = c \), where \( c \) is a positive real number. Determine the behavior of the sequence \( \left\{ \frac{x_n}{n} \right\} \):

A) is not bounded
B) is bounded but not convergent
C) converges to \( c \)
D) converges to 0

Choose the correct option.

Answer: C


Question: Let $X$ and $Y$ be independent standard normal random variables, i.e., $X, Y \sim \mathcal{N}(0,1)$. Find the conditional expectation $E(X \mid X^2 + Y^2)$. Provide your answer as a function of $X^2 + Y^2$.

Answer: 0


Question: If \(a\) is the largest root of the equation \(x^2(x-5) + 2 = 0\), find the value of \([a^4]\), where \([x]\) denotes the greatest integer less than or equal to \(x\).

Answer: 584


Question: Let $H$ be a Hilbert space, $M$ a closed subspace of $H$, and $u$ an operator in $B(H)$. Determine whether the following statement is true: If $(u(x),y)=0$ for all $x,y$ in $M$, then $u=0$ on $M$. Provide a justification for your answer.

Answer: False


Question: Calculate the work done by the vector field \( \vec{F} = (6zx^2e^{x^3} + yz)\vec{i} + (xz + y)\vec{j} + (xy + 2e^{x^3})\vec{k} \) to move a particle along the path \( \mathcal{C} \), which consists of the segments from \((0,0,0)\) to \((2,4,6)\), from \((2,4,6)\) to \((3,6,2)\), and from \((3,6,2)\) to \((0,0,1)\). Compute the line integral \( W = \int_{\mathcal{C}}{\vec{F}\cdot{d\vec{r}}} \).

Answer: 2


Question:
Given two bivariate polynomials \( p(x,y) \) and \( q(x,y) \) in \( \mathbb{C}[x,y] \) with total degrees \( m \) and \( n \) respectively, and assuming that \( p \) and \( q \) have no common polynomial factors, determine the number of solutions to the system of equations:
\[
\begin{cases}
p(x,y) = 0 \\
q(x,y) = 0
\end{cases}
\]
Assume that the number of solutions is finite. Express the number of solutions in terms of \( m \) and \( n \).

Answer: mn


Question:
Translate the following English sentence into predicate logic: "If some person is tall, then every person is tall." Choose the correct logical expression from the options below:

1. \( \exists x(P(x) \land T(x)) \to \forall x(P(x) \to T(x)) \)
2. \( \exists x(P(x) \land T(x)) \to \forall y(P(y) \to T(y)) \)

Answer: 2


Question: Evaluate the double integral \( \iint_D x(y+x^2)e^{y^2-x^4} \, dx \, dy \) where the domain \( D \) is defined by \( \{(x,y) \in \mathbb{R}^2: x^2 \leq y \leq x^2+1, 2-x^2 \leq y \leq 3-x^2 \} \).

Answer: 0


Question: Evaluate the constant \( \alpha \) such that \( \iiint_{g(S)}(2x+y-2z)\,dxdydz = \alpha \iiint_S z\,dxdydz \), where the transformation \( g: \mathbb{R}^3 \to \mathbb{R}^3 \) is defined by \( g(x,y,z) = (3x+4z, 2x-3z, x+3y) \) and \( S = \{(x,y,z) \in \mathbb{R}^3 : 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 1\} \).

Answer: 255


Question: In a path-connected subset of \(\mathbb{R}^2\) that is not simply connected, does there exist a simple loop that does not contract to a point?

Answer: Yes


Question: Given a complete graph $K_n$ with vertices labeled $1, \ldots, n$, each edge $ij$ (where $1 \leq i < j \leq n$) is assigned an ordered basis $(p^+_{ij}, p^-_{ij})$ of $\mathbb{C}^2$, representing complex polynomials of degree at most 1. An admissible choice involves selecting either $p^+_{ij}$ or $p^-_{ij}$ for each oriented edge $ij$, and assigning the opposite choice to the edge $ji$. For each vertex $i$, define the polynomial $p_i$ as the product of the chosen polynomials $p_{ij}$ for all $j \neq i$. Determine whether there always exists an admissible choice such that the polynomials $p_1, \ldots, p_n$ are linearly independent over $\mathbb{C}$. Provide a proof or counterexample.

Answer: Yes


Question: Find the value of \(9d^2\), where \(d\) is the minimum distance from the point \(\left(\frac{1}{3}, 2\right)\) to the curve \(y^3 = x^2\) in the first quadrant.

Answer: 13


Question: Given a vector space $V$ and a subspace $U$, if the quotient space $V/U$ is finite dimensional, does it imply that $V$ is finite dimensional? Provide a justification for your answer.

Answer: No


Question: Find the least square approximation of the function \( p(x) = x^{1/3} \) in the subspace \( W = \text{span}\{1, x, x^2 - \frac{1}{3}\} \) of the inner product space \( V = \mathcal{C}[-1, 1] \) with the inner product \( \langle f, g \rangle = \int_{-1}^{1} f(x)g(x) \, dx \).

Answer: \dfrac{9}{7}x


Question: Given the matrix \( A = \begin{bmatrix} z & z+1 & 1 \\ 1 & z & z+1 \\ z+1 & 1 & z \end{bmatrix} \), where \( z \) is a complex number, determine the number of different complex numbers \( z \) such that \( \det(A^4) = 16 \).

Answer: 10


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

Answer: \dfrac{1}{2}


Question: Determine whether the series \( \sum_{n=0}^\infty \frac{\sin(2n+1)}{2n+1} \) converges, and if so, find its sum.

Answer: \dfrac{\pi}{4}


Question: Find the limit of the sequence \(a_n = \sqrt[n!]{\frac{1}{2^{(n!)}}-\frac{1}{3^{(n!)}}}\).

Answer: \dfrac{1}{2}


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

Answer: -\pi


Question: Calculate the limit of the Lebesgue integrals: \[ \lim_{n \to \infty} \int_0^{n^2} e^{-x^2} \sin\left(\frac{x}{n}\right) \, dx. \]

Answer: 0


Question: Find the smallest positive integer that begins with the digit 1 and becomes three times larger when this digit is moved to the end of the number.

Answer: 142857


Question: Find the exact value of the solution to the equation \( \ln (x+1) = \frac{x}{4-x} \).

Answer: 0


Question: Determine the number of distinct solutions to the equation \( xyz + yvv = xxzy \), where each letter represents a different digit from 0 to 9. Each digit must be unique to a letter.

Answer: 1


Question: Find the maximum value of \((4a-3b)^2+(5b-4c)^2+(3c-5a)^2\) given that \(a^2+b^2+c^2=1\), where \(a, b,\) and \(c\) are real numbers.

Answer: 50


Question: Let $A$ be an orthogonal matrix of order $n$, where $n$ is odd, such that $A A^T = I$ and $\det A = 1$. Prove that $\det(A - I) = 0$.

Answer: 0


Question: Find the constant term in the expansion of \((3x^2 - \frac{2}{x})^{15}\).

Answer: 747242496


Question: Assume you have an uncountable set $M$. Is it true that there exists a subset $A \subseteq M$ such that neither $A$ nor $M \setminus A$ is countable? Provide a justification for your answer.

Answer: Yes


Question: Determine the dimension of the vector space of conics in the real projective plane with homogeneous coordinates \(\{x, y, z\}\) that are singular at the points \([1, i, 1]\) and \([i, -1, i]\).

Answer: 1


Question: Determine whether the following statement is true or false: Let \(x \in \ell^{p_{0}}\) for some \(1 \leq p_{0}<\infty\). Then \(\|x\|_{p} \rightarrow\|x\|_{\infty}\) as \(p \rightarrow \infty\). Justify your answer.

Answer: True


Question: Find the integer solution to the equation $x^{17} \equiv 17 \pmod{23}$. What is the value of $x$?

Answer: 10


Question: Let $X$ and $Y$ be two smooth projective varieties over a field $k$. Consider the Chow motives $M(X\times\mathbb{P}^n)\simeq M(X)\otimes M(\mathbb{P}^n)$ and $M(Y\times\mathbb{P}^n)\simeq M(Y)\otimes M(\mathbb{P}^n)$. Suppose that $M(X)\otimes M(\mathbb{P}^n)$ is a direct summand of $M(Y)\otimes M(\mathbb{P}^n)$, and that $M(X)$ is not a direct summand of $M(\mathbb{P}^n)$, nor is $M(\mathbb{P}^n)$ a direct summand of $M(Y)$. Is $M(X)$ a direct summand of $M(Y)$?

Answer: Yes


Question: Let $X, Y, Z$ be independent standard normal random variables. Compute the probability $P(3X + 2Y < 6Z - 7)$. Provide your answer as a numerical value.

Answer: 0.1587


Question: Evaluate the integral \( I = \int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t \) and verify that it equals \( 2\pi \).

Answer: 2\pi


Question: Evaluate the expression \( \frac{ \cos 6x + 6 \cos 4x + 15 \cos 2x + 10 }{ \cos 5x + 5 \cos 3x + 10 \cos x } \) and choose the correct answer from the following options: \( \text{A) } \cos 2x \), \( \text{B) } 2 \cos x \), \( \text{C) } \cos^2 x \), \( \text{D) } 1 + \cos x \).

Answer: B


Question: Given the joint density function \( f_{XY}(x,y) = 8xy \) for the random variables \( X \) and \( Y \) where \( 0 < y < x < 1 \), find the probability \( P[X < 2Y] \).

Answer: \dfrac{3}{4}


Question:
Find the sum of the series for $n > 3$ given by:
\[ S = \sum_{r=0}^{n} (-1)^r (x-r)(y-r)(z-r) \binom{n}{r} \]

Answer: 0


Question: Let $R$ be a commutative Noetherian ring such that for every $R$-module $M$ with finite projective dimension, every submodule of $M$ also has finite projective dimension. Given that $R$ is regular, does $R$ have finite global dimension?

Answer: Yes


Question: Calculate the integral \( \int_{0}^{3}\sqrt{4-x^2}\,dx \) using a Riemann sum.

Answer: \pi


Question: Let \( X \xrightarrow{f} Z \xleftarrow{g} Y \) be smooth manifolds and smooth maps, and let \( X \xrightarrow{K} Y \) be a continuous map such that \( f=g\circ K \). Let \( \omega\in\Omega^p(Z) \) be a smooth differential form such that the form \( g^*(\omega)\in\Omega^p(Y) \) is exact. Is \( f^*(\omega)\in\Omega^p(X) \) exact as well?

Answer: Yes


Question: Solve the equation for real numbers: $$\sqrt{x^{2}-x+2}+\sqrt{2x^{2}+2x+3}=\sqrt{2x^{2}-1}+\sqrt{x^{2}-3x-2}.$$

Answer: -2


Question: Evaluate \( \frac{\partial^6 f}{\partial x^4 \partial y^2}(0,0) \) for the function \( f(x,y) = \frac{1}{1-x^2y} \).

Answer: 48


Question: Calculate \( \liminf_{n\to\infty} \left( \frac{p_{n}}{p_{n}+p_{n+1}} \right) \), where \( p_{n} \) is the \( n \)-th prime number and \( p_{n+1} \) is the next prime number.

Answer: \dfrac{1}{2}


Question: Find the locus of the center of a circle that cuts the family of circles given by \(x^2+y^2+4x(\lambda-\frac{3}{2})+3y(\lambda-\frac{4}{3})-6(\lambda+2)=0\) orthogonally.

Answer: 4x + 3y = 6


Question: Determine the values of \( p \in \mathbb{R} \) for which the series \( \sum^{\infty}_{n=4}{\frac{1}{n\log (n)\log( \log(n))^p}} \) converges.

Answer: p > 1


Question: Determine the radius of convergence for the series \( \sum_{n \ge 1} \frac{(n!)^3}{(3n)!} z^{3n} \). Consider whether the ratio test should be applied to \( \frac{a_{k+3}}{a_k} \) instead of \( \frac{a_{k+1}}{a_k} \).

Answer: 3


Question: Construct a bijection between the set of all binary sequences and the power set of \( \mathbb{N} \). Define the function \( \phi : 2^{\mathbb{N}} \to \{0,1\}^{\mathbb{N}} \) by \( \phi(\{n_1,n_2,\ldots\}) = \) the binary sequence with 1s only at the positions \( n_k \) and 0s elsewhere. Is this function a bijection?

Answer: Yes


Question: Determine the value of \( t \) such that the test \( \delta = \chi[\bar{X}< a - t\sqrt{\frac{\sigma^2}{n}}] + \chi[\bar{X}> b + t\sqrt{\frac{\sigma^2}{n}}] \) for testing \( H_0: \theta \in [a, b] \) versus \( H_1: \theta \not\in [a, b] \) has a size of 0.05.

Answer: 1.96


Question: Find the last three digits of the number \(8^{8^8}\).

Answer: 656


Question: Evaluate the integral \( \int_{0}^{\infty} \frac{x^8 -1}{x^{10} + 1} \, dx \).

Answer: 0


Question: Given the transformation \( u = x^2 - y^2 \) and \( v = 2xy \), calculate \( \frac{\partial x}{\partial u} \) at the point \((0, 2)\), knowing that the transformation is bijective around the point \((1, 1)\).

Answer: \dfrac{1}{4}


Question: Find the coefficient of \(x^{12}y^{24}\) in the expansion of \((x^3 + 2xy^2 + y + 3)^{18}\).

Answer: 111890128896


Question: Do there exist families of groups $G_{s}$ such that for all $s \in [0,1]$, the cardinality of $G_{s}$ is $\mathfrak{c}$, where $\mathfrak{c}$ is the cardinality of the continuum, and for any $a, b \in [0,1]$ with $a \neq b$, the groups $G_{a}$ and $G_{b}$ are not isomorphic?

Answer: Yes


Question: Find the maximum area of a convex quadrilateral $ABCD$ where lines $AC$ and $BD$ are perpendicular and $AB + CD = 8$. Express your answer in terms of the maximum possible area.

Answer: 16


Question: Let \( S \) be a real symmetric \( n \times n \) matrix such that \( S^3 = I_n \). Does this imply that \( S = I_n \)? Justify your answer.

Answer: S = I_n


Question: Given a map \( f: T^2 \rightarrow T^2 \), where the induced homomorphism on the first homology group \( f^{**}: H_1(T^2) \rightarrow H_1(T^2) \) is given by \( f(x+y) = mx + ny \), calculate the degree of the map \( f \).

Answer: mn


Question: Determine the asymptotic complexity of the recurrence relation \( T(n) = T(\sqrt{n}) + 1 \).

Answer: O(\log \log n)


Question: Determine the interval to which the parameter $m$ belongs such that the polynomial equation \(x^4 - 2x^3 + mx^2 - 2x + 1 = 0\) has a double root \(x_1 = x_2\) in \(\mathbb{R} \setminus \{-1\}\).

Answer: 2


Question: Is there a finite group action on the 3-sphere, $S^3$, by smooth or analytic diffeomorphisms, that possesses an invariant knotted circle?

Answer: Yes


Question: Determine the number of values of \(x\) in the interval \((-90^\circ, 90^\circ)\) such that \(x \neq 36^\circ n\) for \(n = 0, \pm 1, \pm 2\) and both \(\tan x = \cot 5x\) and \(\sin 2x = \cos 4x\) are satisfied.

Answer: 3


Question: Calculate the line integral \( \int_\gamma (2x+1)dx+(3xy+1)dy \), where \( \gamma \) is the boundary of the triangle with vertices \( O(0,0) \), \( A(3,-3) \), and \( B(-6,-6) \).

Answer: 162


Question: Let $f$ be an entire function such that $f(0) = 1$ and $f\left(\frac{1}{n}\right) = 1 + \frac{i}{n} - \frac{1}{n^3}$ for each positive integer $n$. Find $f(1 - 2i)$. \( \text{(Express your answer in the form } a + bi \text{, where } a \text{ and } b \text{ are real numbers.)} \)

Answer: 14 - i


Question: In a Dedekind domain, let $R$ be a commutative, Noetherian ring with $1$. Consider a non-prime ideal $I$ of $R$, and elements $a, b \in R$ such that $a \not\in I$, $b \not\in I$, but $ab \in I$. Let $\mathcal{P}$ be the set of prime ideals appearing in the Dedekind factorization of $(I,a)$ or $(I,b)$. Is it always true that some ideal in $\mathcal{P}$ must appear in the Dedekind factorization of $I$? Provide a justification for your answer.

Answer: Yes


Question: Let $K$ be a nonarchimedean local field with residue characteristic $p \neq l$, and let $I_K$ be the inertia subgroup of its absolute Galois group. Consider an irreducible representation $V$ of $I_K$ over $\overline{\mathbb{F}}_l$ with finite image $I$. Suppose $V$ is primitive, meaning it is not induced from a proper subgroup of $I$. Can $I$ have an element of order $l$?

Answer: No


Question: Given a bivariate normal distribution with mean vector \( \mu = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \) and covariance matrix \( \Sigma = \begin{pmatrix} 2 & 0.4 \\ 0.4 & 1 \end{pmatrix} \), let \( Z_1 = X + Y \) and \( Z_2 = 2X - Y \). Calculate the covariance of \( Z_1 \) and \( Z_2 \).

Answer: 3.4


Question:
Consider the system of differential equations given by:
\[ \frac{dx}{dt} = \frac{3xy}{1+x^2+y^2} - \frac{1+x^2}{1+y^2} \\
\frac{dy}{dt} = x^2 - y^2. \]

Analyze the critical point at \( \begin{pmatrix} 1 \\ 1 \end{pmatrix} \) and determine its nature. Choose the correct classification from the following options:

A. an unstable node
B. a stable node
C. a saddle point
D. a stable spiral point
E. an unstable spiral point.

Answer: C


Question: Suppose $X_t \in \mathbb{R^+}$ for $t = 1, \ldots, T$, and for all $t$, $X_t = o_p(1)$, meaning $X_t$ goes to zero in probability as $T \to \infty$. As $T \to \infty$, does it follow that \( \frac{1}{T} \sum_{t=1}^{T} X_t = o_p(1) \)? Answer yes or no.

Answer: No


Question: Compute the integral \( \int_{-1}^1 \frac{e^{ax} \, dx}{\sqrt{1-x^2}} \), where \( a \in \mathbb{R} \).

Answer: \pi I_0(a)


Question: Is there a bounded real-valued sequence \( \{w_k\} \in l^\infty \) such that the Cesaro means diverge, i.e., \( \lim_{M\rightarrow\infty} \frac{\sum_{k=1}^M w_k}{M} \) does not exist?

Answer: Yes


Question: Given that \( a = s^3 \) and \( v = 3 \) when \( s = 4 \), find the value of \( v \).

Answer: 3


Question:
Let \(\Omega=\{(x,y)\in \mathbb{R}^2 \mid x^2+y^2<1\}\) be the open unit disk in \(\mathbb{R}^2\) with boundary \(\partial\Omega\). If \(u(x,y)\) is the solution of the Dirichlet Problem:
\[u_{xx}+u_{yy} = 0 \text{ in } \Omega\]
\[u(x,y) = 1-2y^2 \text{ on } \partial\Omega\]
Then find \(u\left(\frac{1}{2},0\right)\).

(a) \(-1\)  \(\quad\) (b) \(-\frac{1}{4}\)  \(\quad\) (c) \(\frac{1}{4}\)  \(\quad\) (d) \(1\)

Answer: C


Question: Find all prime numbers \( p \) such that \( p^2 \) can be expressed as the sum of the cubes of two positive integers.

Answer: 3


Question: Given the function $Z(x,y)$ defined by the equation $$yz + \sin(z) + xyz - y\cos (z) = -1$$ around the point $(1,1,0)$, calculate $\frac{\partial Z}{\partial y}(1,1)$. Provide your answer as a simplified fraction.

Answer: \dfrac{1}{3}


Question: Compute the integral \( \int_\gamma \frac{e^{iz}}{z^2}dz \), where \( \gamma \) is the unit circle. Verify if the result is \( -2\pi \).

Answer: -2\pi


Question: Evaluate \( \lim_{n\to\infty}\int_0^1\frac{nx}{1+n^2x^2}\,dx \) using the Dominated Convergence Theorem.

Answer: 0


Question: Evaluate the limit: \[ \lim_{z \to \infty} \left ( 1-\frac{4}{z+3} \right )^{z-2} \]

Answer: e^{-4}


Question: Evaluate the integral \( \oint_{|z|=1} \frac{\sin z}{z} \, dz \) using Cauchy's theorem.

Answer: 0


Question: Calculate the integral of the function \( e^{-\|x\|_\infty} \) over \( \mathbb{R}^d \). Use this result to determine the normalization constant \( z \) for the probability density function \( p(x) = \frac{1}{z} e^{-\|x\|_\infty} \).

Answer: 2^d d!


Question: Evaluate the definite integral: \[ \int_0^{\frac{\pi}{2}}{\sqrt{\sin(2\theta)} \cdot \sin(\theta)\,d\theta} \]

Answer: \dfrac{\pi}{4}


Question: If two categories $C$ and $D$ are equivalent, there is a bijection between the isomorphism classes of objects of $C$ and $D$. If, in addition, for each isomorphism class of objects of $C$, the corresponding isomorphism class of objects of $D$ has the same cardinality, does it follow that $C$ is isomorphic to $D$?

Answer: Yes


Question: Compute the integral \( \oint_{|z|=2} \frac{d z}{1+z+z^{2}+z^{3}} \) using the residue theorem.

Answer: 0


Question: Let $P(x) \in \mathbb{R}[x]$ be a polynomial with real coefficients such that for every natural number $n$, there exists a rational number $q$ such that $P(q) = n$. What is the largest possible value of $\deg P$?

Answer: 1


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

Answer: 64


Question: Evaluate the line integral \( \int_C (x^2 \, dx + y^2 \, dy + z^2 \, dz) \), where \( C \) is the curve consisting of line segments from \((0,1,0)\) to \((1,0,1)\) and then from \((1,0,1)\) to \((2,1,3)\).

Answer: \dfrac{35}{3}


Question: Determine the values of the positive constant $c$ for which the series \( \sum_{n=2}^\infty \frac{1}{n(\ln n)^c} \) is convergent.

Answer: c > 1


Question: Determine the values of \( \alpha \) for which there exists a nonconstant function \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(\alpha(x+y)) = f(x) + f(y) \) for all \( x, y \in \mathbb{R} \).

Answer: 1


Question: Calculate the integral \( \iint_R \frac{1}{x+y} \, dy \, dx \) where \( R \) is the region bounded by \( x=0, y=0, x+y=1, x+y=4 \) using the transformation \( T(u,v)=(u-uv,uv) \).

Answer: 3


Question:
Consider a triangle $\triangle ABC$ with centroid $G$, circumcenter $S$, and incenter $I$. Let $R$ and $r$ be the circumradius and inradius, respectively. Determine which of the following statements is incorrect:

(A) $SI^2 = R^2 (1 - \cos A \cos B \cos C)$, where $A, B, C$ are the angles of the triangle.

(B) $SI^2 = R^2 - 2Rr$.

(C) $SG^2 = R^2 - \frac{a^2 + b^2 + c^2}{9}$, where $a, b, c$ are the sides of the triangle.

(D) $SG \leq SI$.

Use the area $A$ of the triangle and the semi-perimeter $s$ to verify the statements:
$$R=\dfrac{abc}{4A} \quad \text{and} \quad r=\dfrac{A}{s}.$$

Answer: A


Question: If \( \alpha \) is the angle between the asymptotes of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) with eccentricity \( e \), find \( \sec \frac{\alpha}{2} \).

Answer: e


Question: Alice claims, "The probability that I'm lying is greater than \( p \)." Determine the probability that Alice is lying.

Answer: 1


Question: Find the maximum value of the expression \( p^3 + q^3 + r^3 + 4pqr \) given that \( p, q, r \) are non-negative real numbers such that \( p + q + r = 3 \) and each of \( p, q, r \) is less than or equal to 1.5.

Answer: \dfrac{243}{32}


Question: Consider the fixed circle $C_1$ with the equation $(x-1)^2 + y^2 = 1$ and a shrinking circle $C_2$ centered at $(0, 0)$ with radius $r$. Let $P$ be the point $(0, r)$, $Q$ the upper intersection point between $C_1$ and $C_2$, and $R$ the intersection of the line $PQ$ with the $x$-axis. Determine the position of $R$ as the radius $r$ of $C_2$ approaches zero ($r \to 0^+$).

Answer: (4, 0)


Question: Evaluate the integral \( \int_0^1 e^{-2 \pi jix} \sum_{k=-\infty}^\infty u_k e^{2 \pi kix} \, dx \), where \( j \) and \( k \) are integers, \( i \) is the imaginary unit, and \( u_k \) are complex numbers representing Fourier coefficients.

Answer: u_j


Question: Evaluate the surface integral \( \iint_{S} [ \nabla \phi \times \nabla \psi] \cdot \mathbf{n} \, dS \) where \( \phi=(x+y+z)^2 \) and \( \psi=x^2-y^2+z^2 \). The surface \( S \) is the curved surface of the hemisphere \( x^2+y^2+z^2=1 \) with \( z \ge 0 \).

Answer: -2\pi


Question: Find the value of \(x\) that maximizes the integral \(\int_{x}^{\infty} (y + c) \, dF(y)\), where \(F(y)\) is a cumulative distribution function and \(c\) is a constant.

Answer: -c


Question: Let $f$ be an entire function such that $f(n) = n$ for all integers $n$, and $\lim_{|z| \to \infty} |f(z)| = \infty$. Prove that $f(z) = z$ for all complex numbers $z$.

Answer: f(z) = z


Question:
Find an upper bound for the ratio of two Gaussian measures \( \frac{\gamma(U+m)}{\gamma(U)} \). Suppose that \(\gamma\) is the Gaussian measure on \(\mathbb{R}\), defined as
\[
\gamma(E) = \int_E \frac{1}{\sqrt{2\pi}} e^{-0.5|x|^2} dx.
\]
Let \(m \in \mathbb{R}\) be a fixed constant and \(U\) an arbitrary open set. Determine if this ratio is bounded by a constant that is independent of the set \(U\).

Answer: \infty


Question: Suppose \(f : [0,1] \to \mathbb{R}\) is a bounded function such that \(f\) is Riemann integrable on \([a,1]\) for every \(a \in (0,1)\). Is \(f\) Riemann integrable on \([0, 1]\)? Justify your answer.

Answer: Yes


Question:
For each positive real number $\lambda$, let $A_\lambda$ be the set of all natural numbers $n$ such that $|\sin\sqrt{n+1}-\sin\sqrt n|<\lambda$. Let $A_\lambda^c$ be the complement of $A_\lambda$ in the set of all natural numbers. Determine which of the following statements is true:

(A) $A_{1/2},A_{1/3},A_{2/5}$ are all finite sets
(B) $A_{1/3}$ is a finite set but $A_{1/2},A_{2/5}$ are infinite sets
(C) $A_{1/2}^c,A_{1/3}^c,A_{2/5}^c$ are all finite sets
(D) $A_{1/3},A_{2/5}$ are finite sets and $A_{1/2}$ is an infinite set

Choose the correct option.

Answer: C


Question:
Solve the ordinary differential equation $4y''\sqrt{y} = 9$ with the initial conditions $y(0) = 1$ and $y'(0) = 3$. Determine the value of $y\left(\frac{28}{9}\right)$. Choose the correct answer from the following options:

A) 14

B) 15

C) 15

D) 17

E) None of the above.

Answer: E


Question: Given the matrix \( A = \begin{bmatrix} 101 & 2 & 3 & 4 & 5 \\ 1 & 102 & 3 & 4 & 5 \\ 1 & 2 & 103 & 4 & 5 \\ 1 & 2 & 3 & 104 & 5 \\ 1 & 2 & 3 & 4 & 105 \end{bmatrix} \), find all the eigenvalues of \( A \). It is known that one of the eigenvalues is 115. Determine the remaining eigenvalues.

Answer: 100


Question:
Find the radius of convergence of the power series:
\[\sum_{k=0}^\infty a_k x^{a_k}\]
where \[a_k = \begin{cases} k & : k \text{ is prime}\\ 0 & : \text{otherwise} \end{cases}\]

Answer: 1


Question: Let $B$ be the unit ball in $\mathbb{R}^3$ and assume $f \in C^2(\mathbb{R}^3)$ is constant in a neighborhood of $\partial B$. Show that \( \int_B (f_{xx} + f_{yy} + f_{zz}) \, dv = 0 \).

Answer: 0


Question: Evaluate the expression \( \frac{(-1+i\sqrt{3})^{15}}{(1-i)^{20}} + \frac{(-1-i\sqrt{3})^{15}}{(1+i)^{20}} \) and determine the correct value. The given answer is \(-64\). Verify this result and identify any errors in the calculation.

Answer: -64


Question: Find the inverse of the polynomial $x^3 + x$ modulo $x^4 + x + 1$ in the field $F_2[x]$. Provide your answer as a polynomial in $F_2[x]$.

Answer: x^3 + x^2


Question: Given that \( \int_{0}^{1} f(x) \, dx = 1 \), \( \int_{0}^{1} x f(x) \, dx = 2 \), and \( \int_{0}^{1} x^2 f(x) \, dx = 3 \), determine the number of roots of \( f(x) = 0 \) in the interval \((0,1)\).

Answer: 2


Question: The line $lx + my = 1$ intersects the circle $x^2 + y^2 = a^2$ at points $A$ and $B$. If the chord $AB$ subtends an angle of $\frac{\pi}{4}$ at the origin, find the value of $a^2(l^2 + m^2)$. \(\text{Express your answer in terms of } a, l, \text{ and } m.\)

Answer: 4 - 2\sqrt{2}


Question:
Does there exist a function \( f\colon [0,1]\to\mathbb{R} \) such that:

- \( f \) is Lebesgue measurable; and
- For every continuous function \( g\colon [0,1]\to\mathbb{R} \), the set of points where \( f(x)\neq g(x) \) has positive measure?

Answer: Yes


Question: Calculate \( 121^{1002} \mod 100 \) using Euler's \( \phi \)-function.

Answer: 41


Question: Compute the dimension of the Zariski cotangent space of the variety $V(Y-X^2)$ at the origin in $k^2$, where $k$ is an algebraically closed field. Use the definition of the Zariski cotangent space as the quotient $\mathfrak{m}_{(0,0),V}/\mathfrak{m}_{(0,0),V}^2$, where $\mathfrak{m}_{(0,0),V}$ is the maximal ideal of the local ring $\mathcal{O}_{(0,0),V}$ at the origin.

Answer: 1


Question: Consider the vector space $V = \mathbb{R}^d$ and the set \( A = \{ (v_1, v_2) \in V \times V \mid v_1 \text{ and } v_2 \text{ are linearly independent} \} \). Let \( f: A \to \mathbb{R} \) be a map satisfying the following properties: \( f(a, b_1 + b_2) = f(a, b_1) + f(a, b_2) \) whenever \( (a, b_1), (a, b_2), (a, b_1 + b_2) \in A \); \( f(a_1 + a_2, b) = f(a_1, b) + f(a_2, b) \) whenever \( (a_1, b), (a_2, b), (a_1 + a_2, b) \in A \); and \( f(\lambda a, \mu b) = \lambda \mu f(a, b) \) whenever \( \lambda \mu \neq 0 \) and \( (a, b) \in A \). Can \( f \) always be extended to a bilinear map \( V \times V \to \mathbb{R} \)?

Answer: Yes


Question: Find all pairs of prime numbers \(p\) and \(q\) such that \(p \mid q+6\) and \(q \mid p+7\).

Answer: (19, 13)


Question: Determine the minimum side-length $s$ such that three equilateral triangles with side-length $s$ can completely cover a square with unit side length.

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


Question: Evaluate the double integral \( \iint_{R} |x+|y|| \, dx \, dy \) over the region \( R = [-1, 1] \times [-1, 1] \).

Answer: \dfrac{8}{3}


Question: Evaluate the integral: \[ \int_{0}^{2022} \left( x^{2} - \lfloor x \rfloor \lceil x \rceil \right) \, dx \]

Answer: 674


Question: Let $R$ be a regular local ring and $M$ be a faithfully flat $R$-module. Does there exist a surjective $R$-module homomorphism from $M$ to $R$?

Answer: Yes


Question: Determine whether the derivative of the function \( f(z) = z^2 \bar{z} \) exists at \( z = 0 \).

Answer: 0


Question: Let $G_1$ and $G_2$ be groups with subgroups $H_1$ and $H_2$, respectively. Suppose there is a bijection $f: G_1 \rightarrow G_2$ such that the restriction $f|_{H_1}$ is a bijection between $H_1$ and $H_2$. Must it be true that $|G_1 : H_1| = |G_2 : H_2|$?

Answer: No


Question: Find the sum of the infinite series: \[ \sum_{n=1}^{\infty} \frac{2n-1}{5^{2n-1}} \]

Answer: \dfrac{65}{288}


Question: Evaluate the double integral \(\iint_{R_{xy}} (x-y)^2\sin(x+y)\,dx\,dy\) over the parallelogram \(R_{xy}\) with vertices \((\pi,0),(2\pi,\pi),(\pi,2\pi),(0,\pi)\) using the transformation \(u = x-y\) and \(v = x+y\).

Answer: 0


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

Answer: 43


Question: Determine the linear minimum mean squared error (MMSE) estimator for \( y \) given \( x \) for a 2D random point \((x,y)\) uniformly distributed on a shaded region. Express the estimator in the form \( \hat{y} = wx \).

Answer: \hat{y} = \frac{1}{2}x


Question: Approximate to the nearest integer the value of the integral \( \int_0^4\int_0^4\int_0^4 \sqrt{x^2+y^2+z^2}\,dx\,dy\,dz.\)

Answer: 246


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

Answer: \dfrac{1}{2}


Question: Calculate \( \lim_{n \to \infty} a_{n} \) for \( a_{n} = \left((1+\frac{1}{n^{2}})^{n^{2}}(1-\frac{1}{n})^{n}(1+\frac{1}{n})\right)^{n} \).

Answer: \sqrt{e}


Question:
Find all twice continuously differentiable $2\pi$-periodic functions $f(x)$ that satisfy the differential equation:
\[ e^{ix} f''(x) + 5f'(x) + f(x) = 0. \]

Answer: 0


Question: Given that $a^2b = ba^2$ and $ab^2 = b^2a$ for elements $a$ and $b$ in a group $G$, determine whether it follows that $ab = ba$. Provide a justification for your answer.

Answer: No


Question: Evaluate the integral \( \iint_R y\, dx\, dy \) where \( R \) is the region bounded by the \( x \)-axis and the parabolas \( y^2 = 4-4x \) and \( y^2=4+4x \). Use the transformation \( x = u^2 - v^2 \) and \( y = 2uv \) to perform the evaluation.

Answer: 2


Question: Determine whether the series \(\sum_{k=1}^\infty \frac{1}{4^k(2k)}\binom{2k}{k}\) converges to \(\ln 2\).

Answer: \ln 2


Question: Are the one-point compactifications of homeomorphic locally compact Hausdorff spaces also homeomorphic?

Answer: Yes


Question:
Person A chooses a four-digit natural number \(a\). Person B chooses a natural number \(b\) and adds the square of it to \(a\). Person C chooses a natural number \(c\) and multiplies the square of it by \(a\). The results of B and C are then multiplied, resulting in 123456789. What number did A choose?

Given:
\[(b^2 + a)(c^2 \cdot a) = 123456789\]

Answer: 3607


Question: Find a basis for the ring $\mathbb{Z}_5[x]/\langle x^3-x^2-1 \rangle$. Treat the polynomial $x^3-x^2-1$ as the additive identity in this ring.

Answer: \{1, x, x^2\}


Question: Determine the optimal fraction \( q \) of your money to bet each time in order to grow your money by \( m \)-fold in the least amount of bets, given a biased coin with probability \( p \) of landing heads. Assume \( m \) is sufficiently large to ignore finite size effects. Is there an explicit formula for the optimal \( q \)?

Answer: 2p - 1


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{2x + x\cos(x) - 3\sin(x)}{x^4\sin(x)} \]

Answer: \dfrac{1}{60}


Question: Given the joint probability density function $f(x,y) = xe^{-x(1+y)}$ for $x, y \ge 0$, representing the useful lifetimes of components $X$ and $Y$ of a minicomputer, find the probability that the lifetime $X$ of the first component exceeds 3.

Answer: e^{-3}


Question: Find all prime numbers $p$ such that \( p \sum_{k=0}^{n}\frac{1}{2k+1} \) is an integer for a given positive integer $n$.

Answer: 3


Question: Suppose $X_0$ is a closed subset of a locally compact Hausdorff space $X$, and $f \in C_0(X_0)$. Is it always possible to extend $f$ to the whole of $X$?

Answer: Yes


Question: If $\mathbb{Z}^n$ is a quotient of $\mathbb{Z}^m$, must it be true that $n \le m$? Provide a proof or counterexample.

Answer: Yes


Question:
Find the maximum value of the function:
\[ f(y) = \max_{\mu>0} \left[ \exp\left(\frac{-n\mu^{2}}{\sigma^{2}}\right) \exp\left(\frac{2\mu}{\sigma^{2}}\sum_{k=1}^{n}y_{k}\right) \right] \]
where \( y_{k} \) is a random variable with a normal distribution.

Answer: \infty


Question: Determine whether there exists a Schauder basis for the space \( L^1(M) \), where \( M \) is a compact Lie group.

Answer: Yes


Question:
Calculate \( \limsup\limits_{j\to\infty} |a_j|^{1/j} \), where
\[ a_j = \sum_{j=1}^\infty \frac{(1+1/j)^{2j}}{e^j} \]

Answer: \dfrac{1}{e}


Question: Evaluate the limit: \[ \lim_{x \to \infty} \frac{1}{10^x} \frac{1-e^{it}}{1-e^{\frac{ti}{10}x}}. \]

Answer: 0


Question: Determine the values of \( \alpha \in \mathbb{R} \) for which the integral \( \int_0^{+\infty} \frac{1}{x^\alpha} \log\left(1+\frac{1}{x}\right)\arctan(x) \, dx \) is convergent.

Answer: (0, 2)


Question: Given two independent random variables $X$ and $Y$ with probability density functions $f(x) = 6x(1-x)$ and $g(y) = 2y$ respectively, where $x, y \\in [0,1]$, find the probability distribution of the random variable $Z = XY^2$. Provide the probability density function of $Z$.

Answer: 3(1 - z)^2


Question: Consider a matrix $A$ on $\mathbb{C}^n$ with one simple eigenvalue zero and all other eigenvalues having strictly negative real part. Let $v$ be the eigenvector corresponding to the eigenvalue $0$, and let $V$ be a subspace of $\mathbb{C}^n$ such that $v \notin V$ and $AV \subset V$. Does this imply that all eigenvalues of $A$ restricted to $V$ have strictly negative real part?

Answer: Yes


Question: Verify that the differential 1-form \( w = \frac{x\,dy-y\,dx}{x^2+y^2} \) is equal to the exterior derivative \( d\theta = d(\arctan\frac{y}{x}) \).

Answer: w = d\theta


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function such that \( f(2) = 2 \) and \( |f(x) - f(y)| \leq 5|x-y|^{\frac{3}{2}} \) for all \( x, y \in \mathbb{R} \). Define \( g(x) = x^3 f(x) \). Find the value of \( g'(2) \).

Answer: 24


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty }\sum_{1\leqslant k\leqslant n}\frac{1}{k\ln (n+k)} \]

Answer: 1


Question: Let \( b_n = \frac{1}{n} \sum_{i=1}^n \xi_{ni} \). If for each \( i = 1, \ldots, n \), we have \( \lim_{n \to \infty} \xi_{ni} = 0 \), can we conclude that \( \lim_{n \to \infty} b_n = 0 \)?

Answer: No


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

Answer: \ln 2


Question: For every natural number \( n > 1 \), does there exist a group \( G_n \) such that for every integer \( m \) with \( 1 < m \leq n \), there exists an element \( x \in G_n \setminus \{e\} \) where \( x^m = e \)? Here, \( e \) denotes the identity element of the group.

Answer: Yes


Question: Let $A$ be the $3 \times 3$ matrix such that for any $v \in \mathbb{R}^3$, $Av$ gives the projection of $v$ onto the plane $x+y+z=0$. Determine $A^{15}$. Express your answer in terms of $A$.

Answer: A


Question: Find the maximum possible value of the leading coefficient of a real polynomial \( P(x) \) of degree 6, given that \( 0 \leq P(x) \leq 1 \) for all \( -1 \leq x \leq 1 \).

Answer: 16


Question: Given a $5 \times 5$ matrix $B$ over the field $\mathbb{R}$, compute the dimension of the vector space of $5 \times 5$ matrices that commute with $B$.

Answer: 5


Question: Find the area of an infinitesimal elliptical ring between the ellipses given by \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=c\) and \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=c+dc\).

Answer: \pi a b \, dc


Question: Evaluate the limit: \[ \lim_{x \to 1} \frac{1 - x^{1/\pi}}{1 - x^{1/e}} \] without using de l'Hôpital's rule.

Answer: \dfrac{e}{\pi}


Question:
Let \( \omega \) be a complex number such that \( \omega^5 = 1 \) and \( \omega \neq 1 \). Evaluate the expression:
\[
\frac{\omega}{1 - \omega^2} + \frac{\omega^2}{1 - \omega^4} + \frac{\omega^3}{1 - \omega} + \frac{\omega^4}{1 - \omega^3}
\]

Answer: 0


Question: Evaluate the limit \( \lim_{x \to 0} x^{x^{x^{x^{x^{x\cdots}}}}} \).

Answer: 0


Question: Consider the finite field \( L \) with 121 elements. Determine how many roots the polynomial \( g(x) = x^9 - 1 \) has in \( L \), and find all these roots.

Answer: 3


Question: Determine whether the cardinal characteristic \( \mathfrak{ridiculous} \) is equal to \( \mathfrak{p} \), where \( \mathfrak{ridiculous} \) is defined as the minimal cardinality of a centered family of subsets of \( \mathbb{N} \) with no 2 to 1 image that has a pseudointersection.

Answer: \mathfrak{p}


Question: Calculate an approximation of the integral \( \int_0^1\int_0^1\frac{\log(xy)xy}{-1+\log(xy)}dxdy \).

Answer: 0.111


Question: Let \( A \subset \mathbb{R}^n \) be an open set and \( f: [0,1] \to A \) a continuous function. Prove that there exists a piecewise linear function \( g: [0,1] \to A \) such that \( g(0) = f(0) \) and \( g(1) = f(1) \).

Answer: g


Question: Given the function \( f(x) = x^5 + x^3 + x \), and assuming that \( f \) is invertible, find \( f^{-1}(3) \).

Answer: 1


Question: Calculate the work done by the vector field \( F = y^{2}\mathbf{i} + x^{3}\mathbf{j} \) over the curve \( C \), which is a counterclockwise path around the triangle with vertices \((0,0)\), \((1,0)\), and \((0,4)\).

Answer: -\dfrac{13}{3}


Question: Let $K$ be a non-archimedean discretely valued field, and let $f$ be a monic irreducible polynomial in $K[x]$. Consider an extended valuation $w$ to the splitting field of $f$, where the values of the roots of $f$ are equal. Given the values on the roots of $f$ and on $K$, is the valuation $w$ unique?

Answer: Yes


Question: Find the smallest positive integer \( n \) such that there exists a monomorphism from the cyclic group \( G = \langle g \rangle \) of order 30 to the symmetric group \( S_n \).

Answer: 10


Question: Solve the ordinary differential equation using the exact equation technique: \[(x^{2}-2xy)y'+y^{2}-2xy=0\]

Answer: xy = C(x + y)^3


Question: Evaluate the integral \( \int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ dx \) for \(a > 0\).

Answer: \pi a


Question: Determine the cohomology group $H^n(X \cup_{\Sigma \subset \partial X} C(\Sigma))$ for a compact manifold $X$ with boundary $\partial X$, where $\dim X \setminus \partial X = n$ and both $X$ and $\partial X$ are aspherical. Assume $H^n(X, \partial X) = \mathbb{Z}$ by Lefschetz duality.

Answer: \mathbb{Z}


Question: Let \( f(x) = x^3 - 3x + 1 \). Find the number of distinct real roots of the equation \( f(f(f(x))) = 3 \).

Answer: 11


Question: In a topological space $X$, let $A$ be an open set and $B$ be a set that is not open. Is it possible for $A$ and $B$ to be homeomorphic?

Answer: Yes


Question:
Let $f \in C_0^\infty(\mathbb{R}^n)$. Consider the Hilbert transform defined by
\[ Hf(x) = \operatorname{p.v.}\int_{\mathbb{R}}\frac{f(x-y)}{y} \, dy. \]
Determine if it is possible for $Hf = f$ almost everywhere, possibly after extending $H$ to an $L^p$ space.

Answer: No


Question: Determine the greatest interval around $x=0$ where the function $f(x) = x^5 - 5x + 3$ is invertible. Then, find the value of $(f^{-1})'(3)$. Provide your answer for $(f^{-1})'(3)$.

Answer: -\dfrac{1}{5}


Question: Solve the equation \((x+1)(2x^{3}-5)^{1/2}= 2x^{3}- 3x^{2}+x-2\) without squaring both sides.

Answer: 3


Question: A sphere with radius \( R \) has a cylindrical hole drilled through its center. If the height of the remaining solid is 6 cm, what is the volume of the remaining solid?

Answer: 36\pi


Question: If \( \tan{\frac{x}{2}} = \csc x - \sin x \), find the value of \( \tan^2{\frac{x}{2}} \).

Answer: \sqrt{5} - 2


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

Answer: 0


Question: Suppose that X has a Poisson distribution with rate \( \lambda \) and the conditional distribution of Y, given \( X=x \), is binomial with parameters \( x \) and \( p \). Find \( E(Y) \).

Answer: \lambda p


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

Answer: 1


Question: Find the area of the part of the plane \(2x + 2y - z = 10\) that lies within the cylinder \(x^2 + y^2 = 4\).

Answer: 12\pi


Question: Determine whether there exist non-empty sets $A$ and $B$ such that $A \times B \subseteq A$ in ZFC set theory.

Answer: Yes


Question: If $H$ and $K$ are subgroups of a group $G$, is the set $HK = \{hk \mid h \in H, k \in K\}$ always a subgroup of $G$? Justify your answer.

Answer: No


Question: Let $G$ and $H$ be two cyclic groups generated by $x$ and $y$ respectively. Determine the condition on the orders $m$ and $n$ of $x$ and $y$ so that the map $f(x^i) = y^i$ is a homomorphism.

Answer: n \mid m


Question: Calculate the numerical solution at \(x=1.2\) for the initial value problem \(y''=\frac{2}{x}y'-\frac{2}{x^2} y - \frac{1}{x^2}\), with \(y(1)=0\) and \(y'(1)=1\), using the Modified Euler's method. Use a step length of \(h=0.2\) and ensure the solution is accurate to 6 decimal places.

Answer: 0.220000


Question: Suppose $f \in L^2(\mathbb{R}^2)$ with compact support and $\frac{\partial ^{(k)}} {\partial x^{K}}f, \frac{\partial ^{(k)}} {\partial y^{K}}f \in L^2(\mathbb{R})$ for all $k \in \mathbb{N}$. Can we conclude that $f \in C^\infty$? Justify your answer based on the Nirenberg-Sobolev embedding.

Answer: Yes


Question: Set up the triple integral to find the volume of the region $Q$ bounded by the plane $x + 2y + 2z = 2$ and above the paraboloid $x = z^2 + y^2$. Specify the limits of integration and the order of integration.

Answer: \int_{-3}^{1} \int_{-1 - \sqrt{4 - (y + 1)^2}}^{-1 + \sqrt{4 - (y + 1)^2}} \int_{y^2 + z^2}^{2 - 2y - 2z} dx \, dz \, dy


Question:
Consider the Cauchy problem for the partial differential equation $u_t - uu_x = 0$ with the initial condition $u(x,0) = x$ for $x \in \mathbb{R}$. Determine which of the following statements is true about the solution $u(x,t)$:

- The solution $u$ exists for all $t > 0$.
- The solution $u$ exists for $t < \frac{1}{2}$ and breaks down at $t = \frac{1}{2}$.
- The solution $u$ exists for $t < 1$ and breaks down at $t = 1$.
- The solution $u$ exists for $t < 2$ and breaks down at $t = 2$.

Select the correct statement.

Answer: C


Question: Let $f: \mathbb{R}^m \to \mathbb{R}^n$ be a smooth function and let $y \in \mathbb{R}^m$ be a regular value. If $x \in f^{-1}(y)$, is there a neighborhood $U$ of $x$ such that $f$ has no critical points in $U$? Answer with "True" if such a neighborhood exists, or "False" if it does not.

Answer: A


Question:
Let \(H\) and \(K\) be affine subspaces of a real vector space, defined as follows:
\[ H = \{x \in X : f(x) = c\}, \text{ where } f \in X', f \neq 0, c \in \mathbb{R}, \]
\[ K = \{x \in X : g(x) = d\}, \text{ where } g \in X', g \neq 0, d \in \mathbb{R}. \]
Given that \(H \subset K\), determine if there exists a real number \(a\) such that \(g = af\) and \(d = ac\).

Answer: Yes


Question:
Find the function $u(x)$ that minimizes the functional $F[u(
cdot)] = \frac{1}{2}\int_0^1 f(u'(x)) \, dx$, where $f: \mathbb{R} \to \mathbb{R}$ is a strictly convex $C^2$ function. The function $u(x)$ must belong to the set $\mathscr{A} = \{ u: [0,1] \to \mathbb{R} \mid u \in C^1, u(0) = A, u(1) = B \}$. Determine the minimizer of $F$ on $\mathscr{A}$. Assume that $f'(u'(x)) = 2a$ for some constant $a \in \mathbb{R}$.

Answer: u(x) = A + (B - A)x


Question: Expand the function \( f(x) = \log(x + \sqrt{1 + x^2}) \) into a power series and determine its radius of convergence.

Answer: 1


Question: Determine the radius of convergence for the infinite series: \[ \sum_{n=0}^{\infty} \frac{2^n}{2^n+1} x^{n^2} \]

Answer: 1


Question: Determine whether the following implication is true or false: For a sequence \(\{a_n\}\) of positive operators in \(B(H)\), if \(\|a_n-a\| \to 0\), then \(\|a_n^{\frac{1}{2}}-a^{\frac{1}{2}}\| \to 0\).

Answer: True


Question: Find the minimum value of the sum \(r_1 + r_2\) for two distinct non-negative integers \(r_1\) and \(r_2\) such that the binomial coefficient \(\binom{1999}{r}\) is even for both \(r_1\) and \(r_2\).

Answer: 33


Question: Determine the smallest possible value of \( \epsilon \) and a sequence \( (a_n)_{n=1}^\infty \subset [0, 1] \) such that for all \( N \in \mathbb{N} \), the inequality \( d_{N} = \sup_{x \in [0,1]} \inf_{n=1}^N |x-a_n| \leq \frac{1+\epsilon}{N} \) holds. Note that a dyadic decomposition provides \( \epsilon = 1 \).

Answer: 1


Question: Evaluate the integral \( \int_C \frac{1}{z^2-1} \, dz \), where \( C \) is the circle \( |z| = 2 \), parametrized as a positively oriented simple closed curve.

Answer: 0


Question: Determine whether a Hamiltonian Path exists in a Maximal Non-Hamiltonian graph.

Answer: Yes


Question: Consider the Sierpinski triangle. Determine the number of triangular subsets that can be formed within this fractal structure.

Answer: \infty


Question: Determine the number of structurally different Latin squares of order 5, considering symmetries such as rotation, reflection, and permutation of symbols.

Answer: 12


Question: Find the least value of \(a\) for which the function \(f(x) = 2e^x - ae^{-x} + (2a+1)x - 3\) is increasing for all \(x \in \mathbb{R}\).

Answer: 0


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

Answer: \dfrac{\pi}{4}


Question: Find the coefficient of \(x^{10}\) in the expansion of \((1+x+x^2+\cdots+x^9)^6\).

Answer: 2997


Question: Determine the radius of convergence of the series \(\sum_{n=0}^{\infty}{\left(\frac{z+1}{3-i}\right)^{n^2}}.\)

Answer: \sqrt{10}


Question: On an island, there are 12 blue chameleons, 15 green chameleons, and 7 red chameleons. When two chameleons of different colors meet, they both change into the third color. What is the minimum number of meetings required to convert all the chameleons into the same color?

Answer: 15


Question:
Solve the following system of equations for \(x\) and \(y\):
\[\begin{cases}
2x^2+y^2=1,\\
x^2 + y \sqrt{1-x^2}=1+(1-y)\sqrt{x}.
\end{cases}\]

Answer: (0, 1)


Question: Find the value of \( A \) such that the double integral \( \int_{1}^{\infty}\int_{1}^{\infty} A(x+y)^2 e^{-(x+y)} \, dx \, dy = 1 \).

Answer: \dfrac{e^{2}}{18}


Question:
Find all pairs \((p, q)\) of prime numbers satisfying the equation:
\[ p^3 + 7q = q^9 + 5p^2 + 18p. \]

Answer: (29, 3)


Question: Does there exist a meromorphic function $f$, defined for all complex numbers except for possible poles, that has exactly two zeros $z_1$ and $z_2$ such that both $f'(z_1)$ and $f'(z_2)$ have a real part greater than 0?

Answer: Yes


Question: Determine the number of roots of the polynomial \( z^7 + 2z^3 + 1 \) in the region \( \frac{1}{2} \leq |z| < 1 \).

Answer: 3


Question: Evaluate the integral \( \int_0^\infty \frac{ \sin(\pi x)}{x(1-x^2)} \, \text{d}x \) using residue theory.

Answer: \pi


Question: Determine if there exist real numbers $x_0$ and integers $k$ and $q$ such that the recursive function \( x_{n+1} = \sqrt[k]{x_n^k + q} \) satisfies the conditions \( x_{Fn(1)} = x_{Fn(2)} = x_1 \) and \( x_{Fn(n)} - x_{Fn(n-1)} = q \), where \( Fn(n) \) is the $n^{th}$ Fibonacci number.

Answer: Yes


Question: Find the adjoint operator \( T^* \) of the linear operator \( T: V \to V \) defined by \( T(x) = x - \frac{2\langle v, x \rangle}{||v||^2}v \) for some fixed \( v \in V \), where \( V \) is a real inner product space.

Answer: T^*(x) = x - \frac{2\langle v, x \rangle}{\|v\|^2}v


Question: Prove that the infinite product \( \prod_{i=1}^n \frac{2i-1}{2i} \) approaches zero as \( n \to \infty \).

Answer: 0


Question: Evaluate the line integral \( \int 2x+y \,dx +xy\,dy \) along the curve \( y = x^2 + 1 \) from the point \((-1,2)\) to \((2,5)\).

Answer: \dfrac{141}{5}


Question: Determine whether there exists a triple of $n$-variable propositional formulas $\phi_1, \phi_2, \phi_3$ for some finite $n$ such that $\phi_1 \land \phi_2$, $\phi_1 \land \phi_3$, and $\phi_2 \land \phi_3$ are all not contradictions, but $\phi_1 \land \phi_2 \land \phi_3$ is a contradiction.

Answer: Yes


Question: Determine all real numbers \( \alpha \) such that the set \( \{(x,y)\in\mathbb{R}^2: \sin(x+y)=\alpha y\} \) represents the graph of a differentiable function \( g:\mathbb{R}\rightarrow\mathbb{R} \).

Answer: (-\infty, -1) \cup (1, \infty)


Question: Calculate the residue of the function \( \frac{\cos(z^3) - 4e^z}{z^8 - z} \) at the pole \( z = 0 \).

Answer: 3


Question: Do the strongly vanishing elements of $R[[x]]$ form an ideal?

Answer: Yes


Question: Find the exact value of the infinite series: $$\sum_{n=1}^{\infty}\frac{n^2+n+1}{3^n}$$

Answer: \dfrac{11}{4}


Question: Find the coefficient of \(x^{62}\) in the expansion of \((x^3 - \frac{1}{x^5})^{50}\).

Answer: -37353738800


Question: Does there exist a ring with a finitely generated ideal $I$ that has an infinite subset $M \subseteq I$ such that $M$ generates $I$, but no finite subset of $M$ generates $I$?

Answer: No


Question: Find the value of \(a \in \mathbb{R}\) such that the sequence \(x_n = \frac{1}{3} \cdot \frac{4}{6} \cdots \frac{3n-2}{3n} n^a\) converges to a positive real number.

Answer: \dfrac{2}{3}


Question: Evaluate the limit of the following integral: \[ \lim_{n \rightarrow \infty} \int_{[0,1]} \left(\frac{1 - \sin^2 (x)}{n}\right)^6 \, dm(x) \]

Answer: 0


Question: Evaluate the limit \( \lim_{x \to 0} [x - x^2 \ln(1+\frac{1}{x})] \) using the Taylor series expansion of the function \( f(x) = x^2 \ln(1+\frac{1}{x}) \) at \( x = 0 \).

Answer: 0


Question: Evaluate the limit \( L = \lim_{n \to \infty} \int_{0}^\infty \frac{1}{1+x^n} \, dx \).

Answer: 1


Question: Express the infinite product \((1+x)(1+x^2)(1+x^4)(1+x^8)\ldots\) in the form \(A+Bx+Cx^2\), given that \(x^3+x^2=1\) and \(x\in\mathbb{R}\).

Answer: 2 + 2x + x^2


Question: Consider a real power series \( f(x) = \sum_{n=0}^\infty a_n x^n \) with a positive radius of convergence \( R \). If \( f \) is non-constant within \((-R, R)\), is it true that all the zeroes of \( f \) in this interval are isolated?

Answer: Yes


Question: Determine the optimal threshold value, \( m \), for which you should spin a game board spinner a second time to maximize the expected value of your payoff. The spinner has 1000 equal sections numbered from 1 to 1000. Your payoff is the total score of your spins as long as this score does not exceed 1000; otherwise, your payoff is zero. What is the optimal value of \( m \)?

Answer: 414


Question: Evaluate the limit: \[ \lim _{t\to 0}\left(\int _t^{2t}\:\left(\frac{e^{2x}-1}{x^2}\right)dx\right) \]

Answer: 2 \ln 2


Question: Let \( x, y, z \) be non-zero real numbers such that \( \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 7 \) and \( \frac{y}{x} + \frac{z}{y} + \frac{x}{z} = 9 \). Find the value of \( \frac{x^3}{y^3} + \frac{y^3}{z^3} + \frac{z^3}{x^3} \).

Answer: 157


Question:
Consider the sets \( S = \{ x \in \mathbb{R} : x^6 - x^5 \le 100 \} \) and \( T = \{ x^2 - 2x : x \in (0, \infty) \} \). Determine whether the intersection \( S \cap T \) is:

- Closed and bounded in \( \mathbb{R} \)
- Closed but not bounded in \( \mathbb{R} \)
- Bounded but not closed in \( \mathbb{R} \)
- Neither closed nor bounded in \( \mathbb{R} \)

Answer: A


Question: Evaluate the improper integral: \[ \int_0^\infty \left(\frac{13x}{x^2+1} - \frac{65}{5x+1}\right) \, dx \] and determine its value.

Answer: -13 \ln 5


Question: Calculate the integral: $$\int_{\mathbb{R}^5}\frac{e^{-x^2-y^2-z^2}}{1+w^2+s^2}\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}w\,\mathrm{d}s$$

Answer: \infty


Question: Let \( X = [0,1] \). Consider a sequence \( \mu_n \) of regular signed Borel measures on \( X \) that converges to a measure \( \mu \) on \( X \) in the weak-star topology. This means that for any \( f \in C_0(X) \), we have \( \int_X f \, d\mu_n \to \int_X f \, d\mu \). Does this imply that the total variation norm \( |\mu_n|(X) \) converges to \( |\mu|(X) \)?

Answer: No


Question: Let \(x, y,\) and \(z\) be positive real numbers that satisfy \(2\log_{x}(2y) = 2\log_{2x}(4z) = \log_{2x^4}(8yz) \neq 0.\) The value of \(xy^5z\) can be expressed in the form \(\frac{1}{2^{p/q}},\) where \(p\) and \(q\) are relatively prime positive integers. Find \(p+q.\)

Answer: 49


Question: Let $R$ be a ring such that $\operatorname{char}(R) = 0$. Can there exist a subring $S$ of $R$ such that $\operatorname{char}(S) \neq 0$?

Answer: Yes


Question: Find the orthogonal trajectory of the family of curves given by the equation $y^2 = kx^3$. The final solution should be in the form $2x^2 + 3y^2 = d$, where $d$ is a constant.

Answer: 2x^2 + 3y^2 = d


Question: Let \( X \) and \( Y \) be topological spaces, with \( Y \) being contractible. Given continuous maps \( f, g: X \to Y \), for each fixed \( x \in X \), there exists a path \( P_x: [0,1] \to Y \) from \( f(x) \) to \( g(x) \) such that \( P_x(0) = f(x) \) and \( P_x(1) = g(x) \). Define \( F: X \times [0,1] \to Y \) by \( F(x,t) = P_x(t) \). Is \( F \) necessarily continuous for all \( t \in [0,1] \)?

Answer: No


Question: An urn contains an equal number of red and black balls. If you draw one ball at random, the probability of it being red or black is equal. If you draw two balls at random, the probability that they are the same color is not 50%. However, if the first pair of balls drawn are of the same color, and you draw another pair, the probability that the second pair is also of the same color is 50%. How many balls are in the urn?

Answer: 6


Question: Let $M$ be the set of continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $e^{f(x)} + f(x) \geq x + 1$ for all $x \in \mathbb{R}$. Find the minimum value of the integral $I(f) = \int_0^e f(x) \, dx$ for $f \in M$. \( \text{Provide your answer in the form of a single number.} \)

Answer: \dfrac{3}{2}


Question: Find \( \lim\limits_{n\to\infty}{e^n - e^{\frac{1}{n} + n}} \).

Answer: -\infty


Question: Determine whether the following statement is true or false: Let $X \subseteq \Bbb Q^2$. Suppose each continuous function $f:X \to \Bbb R^2$ is bounded. Then $X$ is finite.

Answer: B


Question: Let $X$, $Y_1$, and $Y_2$ be three different path-connected spaces. Prove that the amalgamated product of $\pi(X \star Y_1, x_o)$ $\star_{\pi(X, x_o)}$ $\pi(X \star Y_2, x_o)$ is the trivial fundamental group, where $X\star Y_1$ is the join, defined as the quotient space of $X \times Y_1 \times [0,1]$ under the identifications $(x,y_1,0) \sim (x,y_2,0)$ and $(x_1,y,1) \sim (x_2,y,1)$. Show that this fundamental group is trivial.

Answer: 0


Question: Given a fiber bundle \( F\hookrightarrow E\to B \) where \( F, E, \) and \( B \) are connected manifolds, \( F \) is simply-connected, and \( \pi_1(E)=\pi_1(B)= \mathbb{Z}_2 \), consider the universal covering manifolds \( \tilde{E} \) and \( \tilde{B} \) of \( E \) and \( B \) respectively. If the fiber bundle \( F\hookrightarrow \tilde{E}\to \tilde{B} \) is trivial, i.e., \( \tilde{E}=F\times \tilde{B} \), does it follow that \( E=F\times B \)?

Answer: No


Question: Find the Maclaurin series of \( f(x) = \frac{1}{1+x+x^2} \) and determine the radius of convergence of the series.

Answer: 1


Question:
Determine which of the following norms are equivalent on the vector space $C^1[0,1]$ of all real-valued continuously differentiable functions defined on $[0,1]$:

1. $||f||_{\infty} = \sup_{0 \le x \le 1} |f(x)|$
2. $||f||_1 = \int_0^1 |f(x)| \, dx$
3. $||f||_2 = ||f||_{\infty} + ||f'||_{\infty}$
4. $||f||_3 = \max\{||f||_{\infty}, ||f'||_{\infty}\}$

Select the correct statements:

(A) $||f||_{\infty}$ and $||f||_2$ are equivalent.
(B) $||f||_1$ and $||f||_2$ are equivalent.
(C) $||f||_1$ and $||f||_3$ are equivalent.
(D) $||f||_{\infty}$ and $||f||_3$ are equivalent.
(E) $||f||_2$ and $||f||_3$ are equivalent.

Answer: E


Question: Evaluate the integral \( \iiint\limits_T \mathrm{d}x\mathrm{d}y\mathrm{d}z \) where \( T = \{x > 0, y > 0, z > 0, x+y+z < 2\} \).

Answer: \dfrac{4}{3}


Question: What is the Krull dimension of the ring \( \mathbb{C}[x,y] / (xy) \)?

Answer: 1


Question: Given the function \( f(x) = e^x + x \), find \( \frac{d}{dx} f^{-1}(x) \) at \( x=1 \).

Answer: \dfrac{1}{2}


Question: Use the divergence theorem to calculate \( \iint_S F \cdot dS \) where \( F = \langle x^3, y^3, 4z^3 \rangle \) and \( S \) is the sphere \( x^2 + y^2 + z^2 = 25 \) oriented by the outward normal.

Answer: 15000\pi


Question: Evaluate the limit using l'Hospital's rule: \[ \lim_{x\to 0}\dfrac{1-(\cos x)^{\sin x}}{x^3} \]

Answer: \dfrac{1}{2}


Question: Find the limit: \( \lim_{(x,y) \to (0,0)} ({x^2+y^2})^{x^2y^2} \).

Answer: 1


Question: Calculate the volume of the region bounded by the intersection of the cylinders given by the equations $x^2 + y^2 = 1$ and $x^2 + z^2 = 1$. Provide your answer as a single number.

Answer: \dfrac{16}{3}


Question: Determine whether the following statement is true or false: For all primes \( p > 2 \), there are an infinite number of positive integers \( n \) for which \( p \) does not divide \( {2n \choose n} \).

Answer: True


Question: Determine the number of tangent lines to the curve \(y^2 - 2x^3 - 4y + 8 = 0\) that pass through the point \((1, 2)\).

Answer: 2


Question: Suppose $X$ is a non-empty set and $d_1, d_2$ are two metrics on $X$, forming the metric spaces $(X,d_1)$ and $(X,d_2)$. If there exists a surjective bi-Lipschitz map $\phi: (X,d_1) \to (X,d_2)$, are the metrics $d_1$ and $d_2$ equivalent?

Answer: Yes


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

Answer: 0


Question: Find the value of \( f(f(100)) \) given the function \( f(x) = ([\{x\}] + 3 - x^7)^{1/7} \).

Answer: 100


Question: Evaluate the integral \( \int_{C} (1 + z + z^2)(e^{\frac{1}{z}}+e^{\frac{1}{z-1}}+e^{\frac{1}{z-2}}) \, dz \), where \( C \) is the circle \(|z|=3\) in the complex plane.

Answer: 32\pi i


Question: Given the polynomial \( P(x) = 99(x^{101} - 1) - 101(x^{99} - 1) \), find the value of \( Q(1) \), where \( Q(x) \) is the quotient when \( P(x) \) is divided by \((x-1)^3\).

Answer: 656601


Question: Find the maximum value of \((xv-yu)^2\) subject to the constraints \(x^2+y^2=a^2\) and \(u^2+v^2=b^2\) using Lagrange Multipliers.

Answer: a^2b^2


Question:
Evaluate the line integral \( I = \int_{C}\mathbf{F}\cdot d\mathbf{R} \), where \( \mathbf{F} = \langle ye^{xy},xe^{xy}\rangle \) and \( C \) is the curve parameterized by \( \mathbf{R} = \langle \cos{t},\sin{t}\rangle \) for \( 0\leq t\leq 2\pi \). Choose the correct value of \( I \) from the following options:

(a) \( I = \pi \)
(b) \( I = 2\pi \)
(c) \( I = 0 \)
(d) \( I = 1 \)
(e) \( I = \frac{1}{2} \)

Answer: C


Question: Given a function \(f\in C[-2,1]\) such that \(\int_{-2}^1 f^2(x) \, dx = 84\) and \(\int_{-2}^1 f(x)(2-x) \, dx = 42\), find the minimum value of \(x^2 - f(x)\) for \(-2 \leq x \leq 1\).

Answer: -5


Question:
Determine which of the following statements is true regarding functions from the set \( A = \{x^2 : 0 < x < 1\} \) to the set \( B = \{x^3 : 1 < x < 2\} \):

1. There is a one-to-one, onto function from \( A \) to \( B \).
2. There is no one-to-one, onto function from \( A \) to \( B \) taking rationals to rationals.
3. There is no one-to-one function from \( A \) to \( B \) which is onto.
4. There is no onto function from \( A \) to \( B \) which is onto.

Answer: 1


Question: Determine whether the integral \( \int_{0}^{1}\frac{\ln(1+n^2x^2)}{n^2}~dx \) converges as \( n \to \infty \).

Answer: 0


Question: Find the 19th derivative of the function \( f(x) = \frac{x-1}{e^x} \) at \( x = 0 \).

Answer: 20


Question: Let $T$ be a transcendence basis of $\mathbb{R}$ over $\mathbb{Q}$, and let $F=\{A_n\subset T\colon n\in\mathbb{N}\}$ be a family of pairwise disjoint sets. Define the collection of sets $$C=\{pA+q\colon A\in F, \text{ and } p,q\in \mathbb{Q}\setminus\{0\}\}.$$ Are all the sets in $C$ pairwise disjoint algebraically independent sets?

Answer: Yes


Question: Find the minimum value of the function \( f(x, y, z) = \frac{1+x^2}{1+y+z^2} + \frac{1+y^2}{1+z+x^2} + \frac{1+z^2}{1+x+y^2} \) for \( x, y, z > -1 \).

Answer: 2


Question: Find the limit \( \lim_{n\to\infty}\sqrt[n]{4^n+9^n} \).

Answer: 9


Question:
Let \( f \) be a measurable function on \([0,1]\). Determine whether there exists a sequence of infinitely differentiable functions \( f_n \) such that at least one of the following conditions holds:

1. \( f_n \to f \) pointwise on \([0,1]\).
2. \( f_n \to f \) uniformly on \([0,1]\).
3. \( \int_0^1 |f_n - f| \to 0 \).

Provide a justification for your answer.

Answer: Yes


Question: A number $N$ consists of $2012$ digits. For any consecutive $m$ digits ($m \leq 2012$) taken from $N$ starting from any position, there exists another position in $N$ such that the $m$ consecutive digits starting from that position are in reverse order of the former sequence. The total number of possible values of $N$ can be expressed as $a \times 10^{b}$, where $a$ and $b$ are positive integers, and $a$ is not divisible by $10$. Find the value of $a+b$.

Answer: 1014


Question: Find the limit: \( \lim_{n \to \infty }\sqrt[n]{b^{2^{-n}}-1} \).

Answer: \dfrac{1}{2}


Question:
Translate the following regular grammar into a regular expression using concatenation, union, and star operations:

- Set of variables: \( V = \{S, T\} \)
- Set of terminals: \( \Sigma = \{a, b\} \)
- Set of relations:
  - \( S \to \varepsilon \)
  - \( S \to aS \)
  - \( S \to bT \)
  - \( T \to aT \)
  - \( T \to bS \)
- Start variable: \( S \)

Express the language generated by this grammar as a regular expression.

Answer: (a \cup b a^* b)^*


Question: Let $g(x)$ be a real-valued function defined on the entire real line $\mathbb{R}$. Suppose there are $k$ distinct real numbers $x_1, x_2, \ldots, x_k$ such that $g(x_i) = 0$ for each $i$. Let $G(x)$ be an antiderivative of $g(x)$. What is the largest number of real values $x$ for which $G(x) = 0$? Provide your answer as a single integer.

Answer: k+1


Question: Calculate the volume of the solid bounded by the surfaces $x^2 + y^2 - 2y = 0$, $z = x^2 + y^2$, and $z = 0$ using triple integrals.

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


Question: Evaluate the limit: \[ \lim_{x\to -\infty} \frac{x^4\sin\left( \frac{1}{x} \right) + x^2}{1+|x|^3} \]

Answer: -1


Question: Given the function \( f(x) = x^3 - \frac{4}{x} \), find the value of the inverse function \( f^{-1}(x) \) at \( x = 6 \).

Answer: 2


Question: Determine whether the sequence \( \sqrt[x!]{x} \) converges. If it converges, find its limit.

Answer: 1


Question: Evaluate the triple integral: \[ \int_0^{\pi}\int_x^{\pi}\int_0^2\frac{\sin y}{y}dzdydx. \]

Answer: 4


Question: Determine the number of zeroes (counting multiplicity) of the polynomial \( p(z) = z^6 + z^3 + 10z^2 + 4z + 3 \) that lie within the annulus defined by \( 1 < |z| < 2 \).

Answer: 4


Question: Is it true that every equivalence between derived categories preserves their distinguished triangles?

Answer: Yes


Question: Consider a random 2D walk starting at the origin. Take \( n \) independent steps of length 1, where each step is in the direction of \( \theta_i \), uniformly distributed on \([0, 2\pi]\). Let \( X, Y \) be the position after \( n \) steps, and define \( D = X^2 + Y^2 \). What is the expected value \( E[D] \)?

Answer: n


Question:
Evaluate the limit as \( h \to 0 \) of the expression:
\[ \frac{he^{-h}}{e^{-2h} + 3he^{-h} - 1} \]

Answer: 1


Question: Given the linear congruential generator (LCG) defined by the equation \( X_n = (X \times a + c) \mod m \), where \( a = 67 \), \( c = 0 \), \( m = 101 \), and \( X_n = 67 \), determine the value of \( X \). Assume \( X \) is a non-negative integer less than \( m \).

Answer: 1


Question: Evaluate the double integral \( \iint_{D} \frac{y^2}{x} \, dx \, dy \), where \( D \) is the region bounded by the curves \( y = x^2 \), \( y = 4x^2 \), \( xy = 1 \), and \( xy = 5 \).

Answer: 12


Question: Suppose $F: \mathbb{R}^n \to \mathbb{R}^n$ is a smooth vector field such that $F(0) = 0$ and $d_0F = A$, where $F(x) = Ax + O(|x|^2)$. Let $\phi_t: \mathbb{R}^n \to \mathbb{R}^n$ be the one-parameter family of diffeomorphisms generated by the integral curves of $F$. Determine if it is true that $d_0\phi_t = e^{tA}$. Justify your answer.

Answer: e^{tA}


Question: Find the maximum value of the function \( x^2 + y^2 \) subject to the constraint \( x^2 + xy + y^2 - 4 = 0 \) using the method of Lagrange multipliers.

Answer: 8


Question: Let $X$ be a nonempty closed convex subset of a finite-dimensional normed space, and let $Y$ be another subset of the same space. If the relative interior of $X$, denoted $\operatorname{ri} X$, is contained within the interior of $Y$, denoted $\operatorname{int} Y$, does it follow that $X \subset Y$? Answer yes or no.

Answer: No


Question: Evaluate the improper integral: \[ \int_{0}^{\infty} \frac{\log(1+x^2)}{1+x^2}~dx. \]

Answer: \pi \ln 2


Question: Given arbitrary $a, b \in (0, \infty)$, is it possible to find a function $f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$ such that $||f||_1 = a$ and $||f||_2 = b$?

Answer: Yes


Question: Calculate the variance of the unbiased estimator \( \widehat{\sigma_2^2} = \frac{1}{2}(Y_1-Y_2)^2 \) for a normal distribution.

Answer: 2\sigma^4


Question: Let $B$ and $W$ be independent Brownian motions, and let $\tau$ be a stopping time adapted to $\mathcal{F}^{W}$. Determine whether $E\left[\int_{0}^{\tau}B_{s}dW_{s}\right] = 0$ always holds.

Answer: 0


Question: Find the minimum value of \((\tan^2(A/2) + \tan^2(B/2) + \tan^2(C/2))\), where \(A\), \(B\), and \(C\) are the angles of a triangle.

Answer: 1


Question: Let $f:[0,1] \to \mathbb{R}$ be a function such that the set of its points of discontinuity has measure $0$ and the improper Riemann integral $\int_{0}^{1} f(x) \, dx$ exists. Is it true that $f$ is Lebesgue integrable on $[0,1]$? Provide a justification for your answer.

Answer: No


Question: Find the norm of the projection operator \( P: c \to c_0 \) defined by \( P(x_0,x_1,x_2,\ldots) = (x_0-\lim\limits_{n\to\infty}x_n,x_1-\lim\limits_{n\to\infty}x_n,x_2-\lim\limits_{n\to\infty}x_n,\ldots) \) in the space of convergent sequences with respect to the supremum norm.

Answer: 2


Question: Given that $x+y+z=1$, $x^2+y^2+z^2=2$, and $x^3+y^3+z^3=3$, find the value of $x^4+y^4+z^4$.

Answer: \dfrac{25}{6}


Question: Is it true that every involution \(\sigma\) (i.e., \(\sigma^2=\text{identity}\)) of an Enriques surface \(X\) acts trivially on \(K_X^{\otimes 2}\)? In other words, for any \(\omega\in K_X^{\otimes 2}\), do we have \(\sigma^* \omega=\omega\), where \(K_X^{\otimes 2}\) denotes the tensor square of the canonical bundle of \(X\)?

Answer: Yes


Question: In how many ways can you select an even number of green shirts from 50 drawers, each containing a blue shirt, a red shirt, a green shirt, and a yellow shirt, considering that the order of non-green shirts matters?

Answer: 2^{99} + 2^{49}


Question: Let \((M^n,g)\) be a compact Riemannian manifold with boundary \(\partial M=N\). Suppose \(|Rm_g| \le C_1\) on \(M\) and the second fundamental form of \(N\) is bounded by \(C_2\). Moreover, there exists a constant \(\epsilon\) such that \(\exp_x(t n_x) \in M\backslash N\) for any \(t \in (0,\epsilon]\) and \(x \in N\), where \(n_x\) is the inner unit normal vector. Determine if there exists a constant \(L=L(C_1,C_2,\epsilon)>0\) such that the map \(F(x,t)=\exp_x(t n_x)\) on \(N \times [0,L)\) is an embedding.

Answer: Yes


Question: Find the limit of the sequence \(x_n = \log\left(1+\frac{1}{n}\right)\sqrt[n]{n!}\) as \(n\) approaches infinity, given that \(n!\sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n\).

Answer: \dfrac{1}{e}


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0^+} \frac{x^{-x}-1}{x} \]

Answer: \infty


Question: Evaluate the limit \( \lim_{h \to 0}\frac{f(2h,h)}{h} \) given that \( f \in C^{\infty}(\mathbb{R}^2) \) satisfies \( f(\sin 2t, \sin t)=0 \) for all \( t\in \mathbb{R} \).

Answer: 0


Question:
Evaluate the integral \(I(a) = \int_{-1/a}^{1/a} (2x^6 - 2x^4 + 3) \cos^{-1}(ax) \, dx\) for \(a > 0\). Determine which of the following statements is true:

1. \(I(a) < \frac{129\pi}{35a}\).
2. \(I(a)\) is independent of \(a\).
3. \(I(a) = \frac{2}{7a^7} + \frac{2}{5a^5} + \frac{3}{a}\).
4. \(I(1) = 0\).
5. None of these.

Answer: 5


Question: Let $f:\{1,2,\ldots,n\}\to\{-1,1\}$ be a boolean function. Determine if $f$ can be extended to a polynomial $p:\mathbb{F}_q\to\mathbb{F}_q$ of degree at most $n$, where $q\in[n,2n]$ is a prime number, such that $p(x)=f(x)$ for all $x\in\{1,2,\ldots,n\}$. The polynomial $p(x)$ may take values other than $-1$ and $1$ for $x\not\in \{1,2,\ldots,n\}$. Provide a justification for your answer.

Answer: Yes


Question: Compute \( \lim_{n\to +\infty}n\left(\tan\left(\frac{\pi}{3}+\frac{1}{n} \right)-\sqrt{3}\right) \) without using L'Hôpital's rule.

Answer: 4


Question: Is every Lindelöf subspace of the ordinal space \(\omega_1\) countable?

Answer: Yes


Question: Compute the limit: \[ \lim_{n\to\infty} \int_0^{\infty}\frac{1}{\left(1+\frac{x}{n}\right)^nx^{\frac{1}{n}}}\ \mathrm{d}\lambda \]

Answer: 1


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

Answer: 1


Question: Prove that there exists a constant \( K > 0 \) such that for all \( x > K \), the interval \( (x, 2x] \) contains fewer prime numbers than the interval \( [0, x] \).

Answer: K


Question: Let $X$ be a smooth projective variety, and $D$ a Cartier divisor on $X$ inducing a surjective morphism $f: X \rightarrow C$, where $C$ is a curve. Can we conclude that the self-intersection number $D^2 = 0$?

Answer: 0


Question: Evaluate the integral \( \int_0^{\pi} \lfloor \pi^2 \cos^3 x \rfloor \sin x \, dx \), where \( \lfloor x \rfloor \) denotes the floor function.

Answer: -1


Question:
Solve the following system of congruences for \( x \):
\[ x \equiv 5 \pmod{7} \]
\[ x \equiv 5 \pmod{9} \]
\[ x^7 \equiv 5 \pmod{32} \]

Answer: 1517


Question: Consider a discontinuous real function \( f(x) \) with all nonnegative terms such that \( \int_a^b f(x) \, dx = 0 \). Does this necessarily imply that \( f(x) = 0 \) for all \( x \) in the interval \([a, b]\)? Provide a justification for your answer.

Answer: No


Question: Find the limit of the sequence \(x_n = \dfrac{\lfloor x \rfloor + \lfloor 3^2x \rfloor + \ldots + \lfloor (2n-1)^2x \rfloor}{n^3}\) as \(n\) approaches infinity.

Answer: \dfrac{4x}{3}


Question: Determine if the function \( g(n) = \frac{N\exp(N^3)}{N\lg N} \) satisfies \( T(N) = \Theta(g(N)) \) for \( T(N) = \frac{\exp(N^3)}{\lg N} \).

Answer: Yes


Question: Find the derivative of the function \( f(x) = \sqrt[5]{x^3 - \tan^3 x} \) at \( x = 0 \).

Answer: -1


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

Answer: 1


Question: A set $X$ contains $2^{56}$ distinct items. A set $A$ of $2^{24}$ items is randomly drawn from $X$ without replacement, and then all items are returned to $X$. Subsequently, a second set $B$ of $2^{24}$ items is drawn from $X$ without replacement. What is the expected number of items that appear in both sets $A$ and $B$?

Answer: \dfrac{1}{256}


Question: Determine the minimum number of symbols required to open a lock with 4 symbols (A, B, C, D) and a 3-symbol password, given that the lock does not "rest" after each attempt. Repeated symbols are allowed.

Answer: 66


Question: Let $V$ be a vector space and let $f$ and $g$ be linear maps from $V$ to $\mathbb{R}$. If $f(x) = 0$ whenever $g(x) = 0$, is $f(x)$ equal to some scalar times $g(x)$?

Answer: Yes


Question: Consider the function \( f = \sum_{k=1}^\infty a_k \chi_{A_k} \), where \( a_k \in \mathbb{R} \) and \( \chi_{A_k} \) is the indicator function on the set \( A_k \). Can this function \( f: \mathbb{R} \to \mathbb{R} \) produce uncountably many distinct values?

Answer: Yes


Question: Determine the total number of positive real values of $x$ such that $x$, $\lfloor x \rfloor$, and $\{x\}$ are in harmonic progression, where $\lfloor . \rfloor$ denotes the greatest integer function and $\{ . \}$ denotes the fractional part.

Answer: 1


Question: Determine the multiplicity of the root \(x = a\) for the polynomial \(Q(x) = \frac{1}{2}(x-a)(p'(x)+p'(a)) - p(x) + p(a)\), where \(p(x)\) is a polynomial of degree \(n\).

Answer: 3


Question: A rectangle can be divided into \( n \) equal squares. If the same rectangle can also be divided into \( n + 76 \) equal squares, find the value of \( n \).

Answer: 324


Question: Compute the integral of the differential form \( \omega = -x_1^2 x_2 \,dx_1 + x_1 x_2 \log(1+e^{2x_2})\,dx_2 \) along the unit circle \( \Gamma = \{x \in \mathbb{R}^2 \mid x_1^2 + x_2^2 = 1\} \) in the trigonometric direction. Use Green's theorem if necessary. What is the value of the integral?

Answer: \dfrac{\pi}{2}


Question: Evaluate the integral \( \int_0^\infty \frac{dx}{x^6 + 1} \) using complex integration techniques.

Answer: \dfrac{\pi}{3}


Question: Let $p$ be a prime and $x$ a fixed integer. Prove that there exists a unique $y \in \{0, 1, \dots, p^n-1\}$ such that $y \equiv x \pmod{p}$ and $y^p \equiv y \pmod{p^n}$.

Answer: y


Question: Evaluate the limit: \[ \lim_{n\to\infty} \frac{4^{n}+n!}{1-6^n} \]

Answer: -\infty


Question: Evaluate the limit: \[ \lim_{x\to \pi/2} (\tan x)^{\tan 2x} \]

Answer: 1


Question: Is it possible to fit pairwise disjoint open 3D balls with radii \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots\) inside a unit ball?

Answer: Yes


Question: Evaluate the double infinite summation: \[ \sum_{i=0}^\infty \sum_{j=0}^\infty (3 + i + j) \cdot \left( \frac{1}{3} \right)^{(1+i)} \cdot \left( \frac{2}{3} \right)^{(1+j)} \]

Answer: \dfrac{11}{2}


Question: Calculate the limit \( \lim_{x \to 0} \frac{e^{-x^2/2} - \cos x}{x^3 \sin x} \) without using L'Hôpital's rule or series expansion.

Answer: \dfrac{1}{12}


Question: Calculate the value of \(777^{1234} \mod 1111\).

Answer: 542


Question: Let \( f \) be a differentiable function on \( \mathbb{R} \). Given that \( \lim_{x\to 0} f\left(\frac{a}{x}+b\right) \neq 0 \) and \( \lim_{x\to 0} \left(f\left(\frac{a}{x}+b\right) - \frac{a}{x} f'\left(\frac{a}{x}+b\right)\right) = c \), find \( \lim_{x\to\infty} f(x) \).

Answer: c


Question: Evaluate the double integral: $$\int_0^2\left(\int_{\frac{y}{2}}^1 e^{x^2}\,dx\right)dy.$$

Answer: e - 1


Question:
Find the six-digit number $ABCDEF$ that satisfies the following conditions:

1. All digits $A, B, C, D, E, F$ are different.
2. The sum of the first two digits equals the sum of the last two digits, i.e., $A + B = E + F$.
3. The sum of all the digits equals the last two digits of the number, i.e., $A + B + C + D + E + F = 10E + F$.
4. The numbers $AB$, $CD$, and $EF$ are all prime numbers, i.e., $10A + B$, $10C + D$, and $10E + F$ are prime.
5. The sum of the last two digits is less than 10, i.e., $E + F < 10$.

What is the number $ABCDEF$?

Answer: 416723


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

Answer: (1, 3)


Question:
Solve the following system of equations for positive integers $x$, $y$, and $z$ and find the value of $x + y + z$:
\[ 9x + y - 8z = 0 \]
\[ 4x - 8y + 7z = 0 \]
\[ xy + yz + zx = 47 \]

Answer: 12


Question: For the function \(f(x) = x \cdot \ln(x)\), consider its Taylor polynomial of degree 4 centered at \(a = 1\). Determine the largest value of \(x > 1\) such that the error \(|E(x)|\) is guaranteed to be less than \(10^{-4}\). The error function is given by \(\frac{-6 \cdot (x-1)^5}{z^4 \cdot 5!}\), where \(z\) lies between \(x\) and \(a\). Find the maximum \(x\) satisfying this condition.

Answer: 1.2885


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

Answer: -7


Question: Determine the largest interval for the constant \( a \) such that the function \( f(x) = \sqrt{4e^{x} - e^{2x} - 1 - a} \) is defined for some real \( x \).

Answer: (-\infty, 3]


Question: Find the volume of the region contained within the cylinder \(x^2 + y^2 = 81\), bounded above by \(z = x\) and below by the \(xy\)-plane.

Answer: 486


Question: Determine the points of discontinuity of the function \( f(x) = \lim_{n\to \infty} \sum_{r=1}^n \frac{\lfloor 2rx \rfloor}{n^2} \).

Answer: \emptyset


Question: Suppose X, Y, and Z are independent normal random variables. X has a mean of 5 and variance of 16, Y has a mean of 7 and variance of 25, and Z has a mean of 10 and variance of 9. Find the probability that the average of X and Z is greater than Y.

Answer: 0.5357


Question: Given the matrix \( A = \begin{bmatrix} 3 & -3 \\ 3 & 3 \end{bmatrix} \), which has eigenvalues \(3 \pm 3i\) and eigenvectors \(\begin{bmatrix} 1 \\ \pm i \end{bmatrix}\), find a formula for the distance from the origin of the iterates \( x_k \) in the discrete dynamical system \( x_{k+1} = Ax_k \) starting with \( x_0 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \).

Answer: (3\sqrt{2})^k


Question:
Let \(a, b, c > 0\) such that \(abc = 8\). Find the minimum value of the expression:
\[
P = \sqrt{\log_2^2 a + 1} + \sqrt{\log_2^2 b + 1} + \sqrt{\log_2^2 c + 1}.
\]

Answer: 3\sqrt{2}


Question: Determine the interval of convergence for the series \( \sum_{n=1}^{\infty} \frac{n!x^n}{1\cdot 3\cdot 5\cdots (2n-1)} \).

Answer: (-2, 2)


Question: Calculate the length of the portion of the hypocycloid defined by \(x^{\frac{2}{3}}+y^{\frac{2}{3}}=1\) in the first quadrant, from the point \(\left(\frac{1}{8},\frac{3\sqrt{3}}{8}\right)\) to the point \((1, 0)\).

Answer: \dfrac{9}{8}


Question: Find the last two digits of the expression \(3^{3^{3^{\cdots}}}\) where there are a total of 2015 threes in the power tower.

Answer: 87


Question: Let $A_1 A_2 \cdots A_{11}$ be a regular $11$-gon inscribed in a circle with radius $2$. A point $P$ is located such that the distance from $P$ to the center of the circle is $3$. Calculate the sum of the squares of the distances from $P$ to each vertex of the $11$-gon: $$PA_1^2 + PA_2^2 + \cdots + PA_{11}^2.$$

Answer: 143


Question: Let \( f \in L^2(\mathbb{R}) \). Does this imply that \( f \to 0 \) as \( x \to \pm\infty \)?

Answer: No


Question: Consider the parabola given by the equation \( y^2 = 2px \). At a point \( A \) on the parabola, a tangent line \( L_1 \) is drawn, intersecting the x-axis at point \( B \). From point \( A \), a line \( L_2 \) is drawn perpendicular to the x-axis, intersecting the parabola again at point \( C \). A line \( L_3 \) passes through point \( B \) and is perpendicular to the x-axis, while a line \( L_4 \) passes through point \( C \) and is parallel to the x-axis. Determine the locus of intersection points \( F \) of lines \( L_3 \) and \( L_4 \).

Answer: y^2 = -2px


Question:
Find the limit of the sequence as \( n \to \infty \):
\[ \sum_{i=1}^n \frac{\cos(i^2)}{n^2+i^2} \]

Answer: 0


Question: Solve the equation \( \ln x - \sin 2x = 0 \) and round the solution for \( x \) to two decimal places.

Answer: 1.40


Question: Two independent points are uniformly distributed within a unit circle. What is the probability that the average of the distances from the points to the origin is less than \( \frac{1}{2} \)?

Answer: \dfrac{1}{6}


Question: Determine the condition under which the dihedral group \(D_n = \langle r,s\mid r^n = s^2 = (rs)^2 = 1\rangle\), of order \(2n\), can be expressed as a direct product \(G \times H\) of two nontrivial groups. Provide the value of \(n\) for which this is possible.

Answer: 2


Question: Provide an example of a quadratic integer in \(\mathbb{Q}[\sqrt{-1}]\) that is prime, but whose norm is not prime.

Answer: 3


Question: Determine the limit points of the set \( A = (0,1) \cup \{2\} \) in the lower limit topology \( \mathbb{R}_l \).

Answer: [0, 1)


Question: Consider groups $H$, $G$, and $G'$ such that $H$ injects into both $G$ and $G'$. Form the amalgamated product $G \ast_H G'$. Is the canonical map $G \rightarrow G \ast_H G'$ always injective?

Answer: Yes


Question: Determine whether the following statement is true or false: Let $\mathsf{V},\mathsf{W}$ be finite vector spaces and $\mathsf{T}: \mathsf{V} \to \mathsf{W}$ be a linear transformation. There exists a matrix $A$ such that $\mathsf{T} = \mathsf{L}_A$, where $\mathsf{L}_A$ denotes left-multiplication by $A$. Provide your answer as either "True" or "False".

Answer: False


Question: Determine the order of the poles of the function \( f(z) = \frac{z \sinh z}{(z^2-1)^2} \) at the points \( z_0 = 1 \) and \( z_0 = -1 \).

Answer: 2


Question: A manufacturer of car radios ships them to retailers in cartons of \( n \) radios. The profit per radio is \$59.50,\) less a shipping cost of \$25\) per carton, resulting in a profit of \( 59.5n - 25 \) per carton. To promote sales by assuring high quality, the manufacturer promises to pay the retailer \( 200X^2 \) if \( X \) radios in the carton are defective. Radios are produced independently, with a \( 5\% \) defect rate. Determine the number of radios that should be packed per carton to maximize the expected net profit per carton.

Answer: 50


Question: Given the joint density function \( f_{XY}(x, y) = kxy^2 \) for \( 0 \leq x, y \leq 1 \) and 0 otherwise, where \( k = 6 \), compute the probability \( \Pr(|X - Y| < 0.5) \).

Answer: \dfrac{7}{8}


Question: Solve the differential equation \( f(x)f''(x) - f(x)f'(x) - {f'(x)}^2 = 0 \) with the initial conditions \( f(0) = 0 \) and \( f'(0) = 0 \).

Answer: 0


Question: Calculate \( \int_{\partial S} -y\,dx + (x-xz)\,dy + e^x\,dz \) for the cylinder \( S = \{x^2 + y^2 = 1, 0 \leq z \leq 1\} \) with the orientation induced by the orientation of \( S \).

Answer: \pi


Question: Maximize $xy^2z^3$ subject to the constraint $x^3 + y^2 + z = 39$ and $x, y, z > 0$.

Answer: 3^{34/3}


Question: Evaluate the integral \( \int_0^\infty \frac{\sin t}{t} \, dt \) using Laplace transforms.

Answer: \dfrac{\pi}{2}


Question: Determine the residue of the function \( f(z) = \frac{\sin z}{1 - \cos z} \) at the point \( z_0 = 0 \).

Answer: 2


Question:
Find the complex conjugate roots of the auxiliary equation for the differential equation:
\[ \frac{d^4y}{dx^4}+2\frac{d^3y}{dx^3}+6\frac{d^2y}{dx^2}+2\frac{dy}{dx}+5y=0 \]
Given that $\sin x$ is a solution, determine the other pair of roots.

Answer: -1 \pm 2i


Question: Consider an oriented surface $F$ of finite type with Euler characteristic $\chi(F) < 0$. Let $\gamma_1$ be an oriented simple closed curve and $\gamma_2$ be an oriented closed curve on $F$, intersecting transversally at points $\{p_1, p_2, \ldots, p_n\}$. For a given hyperbolic metric on $F$, let $\tilde{\gamma}_1$ and $\tilde{\gamma}_2$ denote the geodesics in the free homotopy classes of $\gamma_1$ and $\gamma_2$, respectively. Let $\angle_{p_i}$ denote the angle between $\tilde{\gamma}_1$ and $\tilde{\gamma}_2$ at the intersection point $p_i$. Does there exist a hyperbolic metric on $F$ such that $\angle_{p_i}$ is an acute or obtuse angle for all $i \in \{1, 2, \ldots, n\}$?

Answer: Yes


Question: Evaluate the integral \( \int_{-\pi}^{\pi}\frac{1}{1+\sin^{2}t}dt \).

Answer: \pi \sqrt{2}


Question: Two players, A and B, alternately draw from the set \( S = \{0, 1\} \) with replacement. A player wins by drawing two consecutive zeros. Player A draws first. What is the probability that player A wins?

Answer: \dfrac{2}{5}


Question: Prove that for each integer \( n \geq 0 \), the following identity holds: \[ \sum_{k=0}^n \frac{c_k}{(n-k)!} = 1 \] where the coefficients \( c_n \) are defined by the generating function \( \sum_{n=0}^{\infty} c_n x^n = \frac{e^{-x}}{1-x} \).

Answer: 1


Question: Find a maximal ideal \(I\) in the ring \(\mathbb{Z}[i]\) such that \(\mathbb{Z}[i]/I\) is isomorphic to \(\mathbb{Z}/521\mathbb{Z}\).

Answer: (20 + 11i)


Question:
Let $X$ and $Y$ be random variables with the joint probability density function given by:
\[ f_{X,Y}(x,y) = \begin{cases} kye^{-y} & \text{if } 0 \leq |x| \leq y \\ 0 & \text{otherwise} \end{cases} \]
Find the value of the constant $k$.

Answer: \dfrac{1}{4}


Question: In a triangle, if \( a \cos A + b \cos B + c \cos C = \frac{a+b+c}{2} \), does this imply that the triangle is equilateral?

Answer: Yes


Question: Given 100 congruent triangles in the plane, is it always possible to translate 99 of them such that they completely cover the 100th triangle? A translation moves every point by the same amount in a given direction. Provide a yes or no answer with justification.

Answer: No


Question: Let $E$ be a compact subset of an open subset $U$ of the Euclidean plane. Is there a radius $r > 0$ such that for each point $z \in E$, the closed disc $\bar{D}(z, r)$ is contained in $U$?

Answer: Yes


Question: Consider a sequence of $n$ independent Bernoulli random variables, each with parameter $p_i$ such that $p_i \in \left[\frac{1}{2} - \frac{1}{n}, \frac{1}{2} + \frac{1}{n}\right]$. Determine if the central limit theorem can be applied to conclude that the sum of these random variables is normally distributed as $n \to \infty$. Justify your answer.

Answer: Yes


Question: Determine the common ring to which the quotient ring \( \mathbb{Z}_6[x]/\langle 2x-3 \rangle \) is isomorphic.

Answer: \mathbb{Z}_3


Question: Determine the area of the region bounded by the polar curve \( r = 3\sin(2\theta) \) for \( \theta \in [0, \pi/2] \) using Green's Theorem.

Answer: \dfrac{9\pi}{8}


Question: Let $G$ and $H$ be isomorphic groups. Is the number of isomorphisms between $G$ and $H$ equal to the order of $Aut(G,H)$?

Answer: Yes


Question: Find the maximum value of \(x^2 + 12xy + 2y^2\) subject to the constraint \(4x^2 + y^2 = 25\).

Answer: \dfrac{425}{4}


Question: Determine whether there exists a non-Noetherian ring $R$ such that every ascending chain of primary ideals in $R$ stabilizes.

Answer: Yes


Question:
Consider the spectral theorem for selfadjoint compact operators $L$ with infinite range, given by:
\[ Lx=\sum_{k=1}^{\infty} \alpha_k \langle x,f_k \rangle f_k, \]
where the $f_k$'s form an orthonormal system and the $\alpha_k$'s are real nonzero eigenvalues that tend to zero. If this sum is rearranged arbitrarily, will it still converge?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to 2^{-}} \left ( \frac{1}{\sqrt[3]{x^{2} -3x+2}} + \frac{1}{\sqrt[3]{x^{2} -5x+6}} \right ) \]

Answer: 0


Question: Determine whether there exists a sparse set $A \subseteq \mathbb{N}$ such that for every $n \in \mathbb{N}$, there exists an integer $z \in \mathbb{N}$ satisfying $|S_A(z)| \geq n$. A set $A$ is considered sparse if $\liminf_{n \to \infty} \frac{|A \cap \{1, \ldots, n\}|}{n} = 0$. Provide a proof or counterexample.

Answer: Yes


Question: Find the limit of the infinite series \( \lim_{n\rightarrow \infty} e^{-n}\displaystyle\sum_{k=0}^{n-1}\frac{n^{k-1}}{k!} \). What is the value of this limit?

Answer: 0


Question: Determine the smallest value of \( s \) such that the function \( f(x,y) = \frac{|x|^s|y|^{2s}}{x^2+y^2} \) for \((x,y) \neq (0,0)\) and \( f(x,y) = 0 \) for \((x,y) = (0,0)\) is continuous at \((0,0)\).

Answer: \dfrac{2}{3}


Question: Let $(\Omega,\mathcal{A},\mu)$ be a measurable space with $\mu(\Omega)<\infty$, and let $g$ be a nonnegative, integrable function. For any $\varepsilon > 0$, is it possible to choose a constant $K_{\varepsilon}>0$ such that \( \int_{\{g \geq K_{\varepsilon}\}} g \, d\mu < \varepsilon/2 \)?

Answer: Yes


Question: Let \((X, \le)\) be a finite lattice. Suppose \(f, g: (X, \le) \to (X, \le^{-1})\) are order-isomorphisms. Are \(f\) and \(g\) necessarily the same function?

Answer: No


Question: In a game of craps, if the first roll is neither a natural nor craps, the player can make an additional bet equal to their original one, betting that they will make their point before a seven appears. The payoff odds are as follows: 2:1 for a point of four or ten, 3:2 for a point of five or nine, and 6:5 for a point of six or eight. Calculate the expected winnings for the player if they make this additional bet when given the opportunity.

Answer: 0


Question: Find the limit \( \lim_{n \to \infty} \left( \sqrt[10]{n^{10} + 8n^9} - n \right) \) using the standard limits \( \lim_{n \to \infty} \frac{e^{\alpha_n}-1}{\alpha_n} = 1 \) and \( \lim_{n \to \infty} \frac{\ln(1+\beta_n)}{\beta_n} = 1 \) where \( \alpha_n, \beta_n \to 0 \).

Answer: \dfrac{4}{5}


Question: Compute the value of the Legendre symbol \( \left( \frac{11}{1729} \right) \) using the law of quadratic reciprocity.

Answer: -1


Question: Given the equations \( \frac{x(y+z-x)}{\log x} = \frac{y(z+x-y)}{\log y} = \frac{z(x+y-z)}{\log z} \) and \( ax^y y^x = by^z z^y = cz^x x^z \), determine the value of \( a + \frac{b}{c} \).

Answer: 2


Question: What is the cardinality of the number of digits (in decimal form) of an irrational number like $\pi$?

Answer: \aleph_0


Question: Find the minimum odd integer value of \(a > 1\) such that \(\int_{10}^{19} \frac{\sin x}{1+x^a}dx < \frac{1}{9}\).

Answer: 3


Question: Let \(x_1, x_2, \ldots, x_n\) be the roots of the equation \(x = \tan x\) such that \(x_1 < x_2 < \cdots < x_n\). Find the value of \(\lim_{n \to \infty} (x_{n+1} - x_n)\).

Answer: \pi


Question: Let A be a 3x3 matrix such that \( A^{2014} = 0 \). Must \( A^3 \) be the zero matrix? Justify your answer.

Answer: Yes


Question: Calculate the set $II = \{xy \mid x, y \in I\}$ for the ideal $I = (2, 1+\sqrt{-5})$ in the ring $\mathbb{Z}[\sqrt{-5}]$. Provide the process of calculation and the resulting set.

Answer: (2)


Question:
Find the function \( f(x) \) that satisfies the following conditions for \( x > 0 \):
1. \( xf(x^2) = f(x) \)
2. \( \int_{x}^{x^2} f(t) \, dt = \int_{1}^{x} f(t) \, dt \)
Assume that \( f(x) \in C(0, +\infty) \).

Answer: \dfrac{C}{x}


Question:
Let \( f \) be a function defined on \( \{(m,n): m \text{ and } n \text{ are positive integers} \} \) satisfying:

1. \( f(m,m+1)=\frac{1}{3} \), for all \( m \).
2. \( f(m,n)=f(m,k)+f(k,n)-2f(m,k) \cdot f(k,n) \) for all \( k \) such that \( m<k<n \).

Find the value of \( \frac{1}{3} f(1,98)-f(1,99) \).

Answer: -\dfrac{1}{3}


Question: In triangle $ABC$, $M$ is the midpoint of $BC$. Given that $\angle BAM = \angle C$ and $\angle MAC = 15^{\circ}$, find the measure of $\angle C$. \( \boxed{\text{?}} \)

Answer: 30


Question: Given an irreducible quartic polynomial $f(x) \in F[x]$ with roots $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ and Galois group $G = S_4$, determine the degree of the field extension $[F(\alpha_1+\alpha_2):F]$. Additionally, identify all subfields of this extension.

Answer: 6


Question: Compute the surface integral \( \iint\limits_S \vec{F} \cdot \vec{n} \, dS \) where \( \vec{F}(x,y,z) = (x,y,z) \) and the surface \( S \) is parameterized by \( \vec{r}(\alpha,\beta) = (\alpha \cos\beta, \alpha, \alpha \sin\beta) \) with \( \alpha \in [0,1] \) and \( \beta \in [0,2\pi] \).

Answer: 0


Question: Given three independent events $E_1, E_2, E_3$, the probability that only $E_1$ occurs is $\alpha$, only $E_2$ occurs is $\beta$, and only $E_3$ occurs is $\gamma$. The probability $p$ that none of the events occurs satisfies the equations $(\alpha-2\beta)p=\alpha\beta$ and $(\beta-3\gamma)p=2\beta\gamma$. If the probability of occurrence of $E_1$ is $X$ and the probability of occurrence of $E_3$ is $Y$, find the ratio $\frac{X}{Y}$.

Answer: 6


Question: Evaluate the limit: \[ \lim_{x \to 0^-} \frac{e^{\frac{1}{x}}}{x} \]

Answer: 0


Question: Calculate the limit: \[ \lim _{x \to 0} |x|^{\frac{1}{x^2}} \] without using series expansions or L'Hôpital's rule.

Answer: 0


Question: Evaluate the definite integral: $$\int_0^1 x\sqrt{1-x^2}\arcsin x \, dx$$

Answer: \dfrac{2}{9}


Question: Consider an \(n \times n\) square grid. Determine the number of ways to fill the grid with 0s and 1s such that the sum of the numbers in each row and each column is even.

Answer: 2^{(n-1)^2}


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

Answer: -\ln 2


Question: Determine the sum of the power series: \[ S = -\sum_{n=1}^{\infty}\frac{(1-x)^n}{n} \] where \( x = 1.74 \).

Answer: \ln(1.74)


Question: A drunk man is standing at the 17th meter of a 100-meter-long bridge. Each step, he has a 0.5 probability of moving forward or backward one meter. What is the expected number of steps he will take to reach either the beginning or the end of the bridge?

Answer: 1411