Question:
Determine whether there exists a positive integer $r \geq 3$ such that there is a sequence $b_1, b_2, \ldots, b_r$ of nonzero integers satisfying the equation:
\[(b_1)^3 + (b_2)^3 + \cdots + (b_{r-1})^3 = (b_r)^3.\]

Answer: Yes


Question: Determine the values of \( z \) for which the series \( \sum_{n=0}^{\infty} \frac{z^n}{1+z^n} \) converges using the ratio test.

Answer: |z| < 1


Question: Given the function \( f(x) = (60073 - x^{10})^{1/10} \), and the condition \( f'(2) = \frac{1}{f'(a)} \), where \( a \) is a positive integer, find the value of \( a \).

Answer: 3


Question: Calculate the surface area of the portion of the sphere defined by \(x^2 + y^2 + z^2 = 16z\) that lies within the paraboloid \(z = x^2 + y^2\).

Answer: 16\pi


Question:
Let $a$ and $b$ be the roots of the quadratic equation $x^2 - 6x + 4 = 0$. Define $P_n = a^n + b^n$. Evaluate the expression:
\[ \frac{P_{50}(P_{48} + P_{49}) - 6P_{49}^2 + 4P_{48}^2}{P_{48} \cdot P_{49}} \]
Choose the correct answer from the following options:
(A) 2
(B) 1
(C) 4
(D) 10

Answer: A


Question: Evaluate the limit: $$ \lim_{t\to 1^-} (1-t) \sum_{n=0}^\infty \frac{t^n}{1+t^n} $$

Answer: \ln 2


Question: Evaluate the limit \( \lim_{x \to \infty} F(x) \) where \( F(x) = \int_1^x \frac{\cos t + \cosh t}{e^t} \, dt \).

Answer: \infty


Question: In triangle \( \triangle ABC \), points \( A', B', C' \) lie on segments \( [BC], [AC], [BA] \) respectively, such that \( \vec{CB} = 3\vec{CA'} \), \( \vec{BA} = 3\vec{BC'} \), and \( \vec{AC} = 3\vec{AB'} \). The lines \( (CC') \) and \( (BB') \) intersect at \( R \), \( (AA') \) and \( (BB') \) intersect at \( T \), and \( (CC') \) and \( (AA') \) intersect at \( S \). If the area of triangle \( \triangle RST \) is \( S_{RST} \) and the area of triangle \( \triangle ABC \) is \( S_{ABC} \), find the value of \( k \) such that \( kS_{RST} = S_{ABC} \).

Answer: 7


Question:
Let $A$ be a bounded operator on a Hilbert space $E$. Define
$$N(A)=\sup\Big\{\|Ax\|^2-\big|{\langle Ax, x\rangle}\big|^2\;; \,\,x\in E, \,\|x\| = 1\Big\}.$$ If $N(A)=0$, does it follow that $A=0$?

Answer: No


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0} \frac{\ln(1-x)+\sin(x)}{x^2 e^x} \]

Answer: -\dfrac{1}{2}


Question: Find the values of the parameter \(a\) such that the limit \( \lim_{x \to 0} \frac{e^{x \ln(1-x)} - 1 + x^2 + ax^3}{x^4} \) is a non-zero real number.

Answer: \dfrac{1}{2}


Question:
Determine which of the following expressions for the posterior distribution in Gaussian Process Regression is correct:

1. \( p(\textbf{f} | \textbf{y}) \sim N\left(\sigma^{-2}\left( K^{-1} + \sigma^{-2}I\right)^{-1}\textbf{y}, (K^{-1} + \sigma^{-2}I)^{-1}\right) \)

2. \( p(\textbf{f} | \textbf{y}) \sim N(K(\sigma^2I + K)^{-1}\textbf{y}, \sigma^2 (\sigma^2I + K)^{-1}K) \)

Provide a justification for your choice.

Answer: 2


Question: Evaluate the definite integral: \[ \int_{\ln(0.5)}^{\ln(2)} \left( \frac{\displaystyle\sin x \frac{\sqrt{\sin^2(\cos x)+\pi e^{(x^4)}}}{1+(xe^{\cos x}\sin x)^2}+ 2\sin(x^2+2)\arctan\left(\frac{x^3}{3}\right) } {\displaystyle 1+e^{-\frac{x^2}{2}}+x^7 \sin(-\pi x)+\frac{12}{11}|x|^{2\pi+1}} \,d x\right) \] Determine if the integral evaluates to 0 by checking if the integrand is an odd function over the symmetric interval.

Answer: 0


Question: A family buys two insurance policies with independent losses uniformly distributed on the interval from $0$ to $10$. One policy has a deductible of $1$, and the other has a deductible of $2$. The family experiences exactly one loss under each policy. Calculate the probability that the total benefit paid to the family does not exceed $5$.

Answer: \dfrac{59}{200}


Question: Solve the differential equation \( y'(t) = \sin(t) + \int_0^t y(x)\cos(t-x)\,dx \) using the Laplace transform, given that \( y(0) = 0 \).

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


Question: Is there a probability measure \( p \) on the set \( C(\mathbb{R}) \) of continuous and bounded functions from \( \mathbb{R} \) to \( \mathbb{R} \) such that for all \( g \in C(\mathbb{R}) \) and for all \( \varepsilon > 0 \), \( p(\{f \in C(\mathbb{R}) : |f-g| < \varepsilon\}) > 0 \)?

Answer: No


Question: Evaluate the limit \( \lim_{x \to 0} \frac{(2x^{-t} - 1)^{-1/t}}{x} \) for \( t > 0 \).

Answer: 2^{-1/t}


Question: Find the intersection point of the tangent planes to the surface $S$ in $\Bbb R^3$ given by the graph of $f(x,y) = xy^2 − 2y^2 + e^x$ at the points $(x, y) = (0, 0)$, $(0, 1)$, and $(0, 2)$. Provide the coordinates of the intersection point.

Answer: (-2, 0, -1)


Question: Determine if the volume of an n-dimensional parallelepiped with edges \(v_1, v_2, \ldots, v_n\) is given by both \((\det[b_{i,j}])^{\frac{1}{2}}\), where \(b_{i,j} = v_i \cdot v_j\), and \(|\det A|\), where \(A\) is the matrix with columns \(v_1, v_2, \ldots, v_n\). Are these two expressions equivalent?

Answer: Yes


Question: Let $u$ be a unipotent element in the general linear group $\mathrm{GL}_n(\mathbb{F}_p)$, where $\mathbb{F}_p$ is a finite field with $p$ elements. Prove that the order of $u$ is $p^k$ for some positive integer $k$. A unipotent element $u$ satisfies $(u-I)^m = 0$ for some integer $m$. Show that the order of $u$ cannot divide $p^j - 1$ for any integer $j$, and thus must be a power of $p$. Provide a detailed proof of this result.

Answer: p^k


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

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


Question: Determine whether the function \( I(x,y) = \cos(xy) \) is an integrating factor for the differential equation \( [\tan(xy) + xy]dx + x^2dy = 0 \).

Answer: Yes


Question: Find the value of the series \( \sum_{n=0}^{\infty} \frac{\cos(nx)}{3^n} \) given that \( \cos x = \frac{1}{3} \) and \( x \in (0, \frac{\pi}{2}) \).

Answer: 1


Question:
Identify the missing conjugacy class in the list of conjugacy classes of the alternating group $A_7$: $
\mathrm{id}$, $(123)$, $(12345)$, $(12)(34)$, $(123)(456)$, $(1234)(56)$. Note that the conjugacy class of 7-cycles splits into two in $A_7$. Given that $A_7$ has 9 conjugacy classes, which one is missing?

Answer: (123)(45)(67)


Question: Does there exist a positive function $u$ on a domain $D \subset \mathbb{R}^n$ such that $\liminf_{y\to x}u(y)=0$ for all $x \in D$ but $u \neq 0$ almost everywhere?

Answer: Yes


Question: Find the number of permutations \( \pi \) of \([n]\) such that the numbers 1, 2, and 3 appear in the same cycle of \( \pi \), while the number 4 appears in a different cycle. Provide a general formula for any \( n \geq 4 \).

Answer: \dfrac{n!}{12}


Question: Find the flux of the vector field \( \overrightarrow{F} = -y \hat{i} + x \hat{j} \) through the surface that consists of the first octant of the sphere \( x^2 + y^2 + z^2 = a^2 \) where \( x, y, z \geq 0 \). Calculate the flux using both Cartesian and spherical coordinates.

Answer: 0


Question:
Find the limit of the following expression as \( j \to \infty \):
\[ \lim_{j\rightarrow \infty}\sin^2(2^{j-1}\pi f)\prod_{i=0}^{j-2} \cos^2(2^i\pi f) \]

Answer: 0


Question: Given that \(f\) is a continuous function satisfying the functional equation \(f(x) + f(2x+y) + 5xy = f(3x-y) + 2x^2 + 1\) for all \(x, y \in \mathbb{R}\), find \(|f(4)|\).

Answer: 7


Question: Compute \( \lim_{n \to \infty}\left(\frac{a_n+b_n}{2}\right)^n \) given that \( \lim_{n \to \infty} a_n^n = a > 0 \) and \( \lim_{n \to \infty} b_n^n = b > 0 \), where \( a_n, b_n > 0 \) for all \( n \in \mathbb{N} \).

Answer: \sqrt{ab}


Question: Evaluate the limit by expressing it as a definite integral: $$\lim_{n\to\infty}\frac{\pi}{2n}\left[\cos\left(\frac{\pi}{2n}\right)+\cos\left(\frac{\pi}{n}\right)+\cos\left(\frac{3\pi}{2n}\right)+\cdots+\cos\left(\frac{(n-1)\pi}{2n}\right)\right]$$

Answer: 1


Question: Determine whether there exists a continuous function \( f: [0,1] \to \mathbb{R} \) such that \( f(x) = 0 \) uncountably often and, for every \( x \) where \( f(x) = 0 \), in any neighborhood of \( x \) there exist points \( a \) and \( b \) such that \( f(a) > 0 \) and \( f(b) < 0 \).

Answer: Yes


Question: Let \( f(x) = x^{100} + a_{99}x^{99} + a_{98}x^{98} + \cdots + a_0 \) be a polynomial with real coefficients such that \( |f(0)| = f(1) \). Each root of \( f(x) = 0 \) is real and lies between 0 and 1. If the product of the roots does not exceed \( \frac{1}{m^{50}} \), find the value of \( m \).

Answer: 4


Question: Consider a probability space \((\Omega, \mathcal{F}, \mu)\) and an integrable function \(f\). Is it true that the measure \(\mu \{ \omega: |f(\omega)|=\alpha\}=0\) for all but countably many values of \(\alpha\)?

Answer: Yes


Question: How many known classifications and types of indifferent fixed-points are there in complex dynamics?

Answer: 3


Question: Determine whether the integral \( Y = \int_{0}^\infty Ae^{-B\sqrt{t}}\mathrm{d}t \) can be solved, where \( A \) and \( B \) are constants.

Answer: \dfrac{2A}{B^2}


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

Answer: 1


Question: If $1, \alpha_1, \alpha_2, \ldots, \alpha_{n-1}$ are the $n$th roots of unity, evaluate the sum: \[ \frac{1}{1-\alpha_1} + \frac{1}{1-\alpha_2} + \cdots + \frac{1}{1-\alpha_{n-1}}. \]

Answer: \dfrac{n-1}{2}


Question: Determine the homogeneous ideal generated by the empty set in the polynomial ring $k[x_0, \dots, x_n]$. Express your answer in terms of the ideal notation.

Answer: (0)


Question: Calculate the sum of the infinite series \( \sum_{n=1}^{\infty} n^2 \left( \frac{5}{6} \right)^{n-1} \).

Answer: 396


Question: Find the hundredth derivative at $x=0$ of the function $f(x) = \frac{x^2}{1+x^4}$ using its Taylor series expansion.

Answer: 0


Question: Given that $0 < A < 90^{\circ}$ and \( \sin A + \cos A + \tan A + \sec A + \operatorname{cosec} A + \cot A = 7 \), and that \( \sin A \) and \( \cos A \) are roots of the quadratic equation \( 4x^2 + 3x + k = 0 \), find the value of \( k \).

Answer: \dfrac{4}{31}


Question: Determine the number of homomorphisms from the group \( \Bbb{Z}_{10} \times \Bbb{Z}_{25} \) to the symmetric group \( S_4 \). Here, \( \Bbb{Z}_n \) denotes the integers from 0 to \( n-1 \) with addition modulo \( n \), and \( S_4 \) is the group of permutations of the set \( \{1, 2, 3, 4\} \).

Answer: 10


Question: Determine whether there exists a non-negative continuous function \( f: [0,1] \to \mathbb{R} \) such that \( \int_0^1 f^n \, dx \to 2 \) as \( n \to \infty \).

Answer: No


Question: Find the remainder when $3^{403} - 1$ is divided by $3^{2017} - 1$. Provide your answer as a single integer.

Answer: 3^{403} - 1


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{\tan( [-\pi^2]x^2)- \tan^2( [-\pi^2]x^2)}{\sin^2x} \] where \([\cdot]\) denotes the greatest integer function.

Answer: -10


Question: Calculate the value of \( \sup_{f \in \mathscr{H}} \inf_{z \in D} [f_x^2(z) + f_y^2(z)] \), where \( f: \mathbb{R}^2 \to \mathbb{R} \) is continuous, has partial derivatives in \( D = \{(x,y) : x^2 + y^2 \leq 1\} \), and satisfies \( \sup_D |f| \leq 1 \).

Answer: 1


Question:
Let $A$ be a finitely generated abelian group with $
u(A)$ representing the minimal number of elements needed to generate $A$. Determine $M(A) = \sup_{B \subseteq A} \nu(B)$, where the supremum is taken over all subgroups $B \subseteq A$. What is $M(A)$ in terms of $\nu(A)$?

Answer: \nu(A)


Question: Given that $X = \text{diag}(1,1,1,-1)$ and $C$ is an invertible matrix such that $C'XC = X$, determine if it follows that $CXC' = X$.

Answer: Yes


Question: Find all solutions to the equation $\exp(z) = z$ in the complex numbers $\Bbb{C}$. Provide your answer in terms of known functions or constants if possible.

Answer: z = -W_k(-1)


Question: Evaluate the Lebesgue integral \( \int_{0}^{\infty} 10^{-2[x]} \, dx \), where \([x]\) denotes the integer part of \( x \).

Answer: \dfrac{100}{99}


Question: If a sequence of sets \(A_n\) converges to a set \(A\) in the sense that \(\limsup A_n = \liminf A_n = A\), does it follow that the sequence of complements \(A_n^c\) converges to \(A^c\)?

Answer: Yes


Question: Let $G = \{a_1, a_2, \ldots, a_n\}$ be a finite abelian group such that there is no element $a \neq e$ with $a^2 = e$, where $e$ is the identity element. Evaluate the product $a_1 \cdot a_2 \cdot \ldots \cdot a_n$. \(\text{(Note: The group has no non-identity elements of order 2.)}\)

Answer: e


Question: A triangle is formed by the lattice points \((1, 1)\), \((9, 1)\), and \((9, n)\). For what integer value of \(n > 0\) are there exactly 560 lattice points strictly in the interior of the triangle?

Answer: 162


Question: Calculate the limit of the sequence: \[ \lim_{n\to\infty}\sum\limits_{i=1}^n\frac{2i^2-1}{2^i} = \lim\limits_{n\to\infty}\left(\frac{1}{2} + \frac{7}{2^2} + \ldots + \frac{2n^2 -1}{2^n}\right) \]

Answer: 11


Question: Given the polynomial \( \prod_{k=1}^{99} (x+k) = \sum_{k=0}^{99} a_k \cdot x^k \), and the expression \( M= \frac{\sum_{k=1}^{99} k\cdot a_k}{100!}= \sum_{i=1}^{99} \frac{1}{r_i} \) where \( r_i < r_j \) for all \( i < j \), find the value of \( S = \left\lfloor \sum_{k=1}^{99} \frac{1}{\sqrt{r_k-1}} \right\rfloor \).

Answer: 18


Question:
Determine if there exists a constant \( C \) such that the following inequality holds for all \( x \in \mathbb{R}^d \):
\[ \int_{|x-y|\leq |x|} |x|^{-s} |x-y|^{s-d} \, dy \leq C \]
where \( 0 < s < d \) and \( x, y \in \mathbb{R}^d \). Assume that \( |x-y| \leq \min \{|x|, |y|\} \) for all \( x, y \in \mathbb{R}^d \).

Answer: C


Question: Evaluate the asymptotic behavior of the sum \( \sum_{i=0}^{n-1} \left[i\frac{2^i}{2^n-1}\right] \) as \( n \) becomes large. What is the result of this sum for large values of \( n \)?

Answer: n - 2


Question: In a Hausdorff, second-countable, and locally compact space $E$, there exists a countable base consisting of relatively compact sets $\mathcal{B} = \{U_n : n \geq 1\}$. For any point $x \in E$, is there a sequence $(V_n)_n$ of base sets such that $V_n \to \{x\}$, meaning $V_1 \supseteq V_2 \supseteq \cdots$ and $\bigcap_{n=1}^{\infty} V_n = \{x\}$?

Answer: Yes


Question:
Suppose $\langle z_n \rangle_{n\ge0}$ is a sequence of complex numbers such that $\sum_{n=0}^\infty z_n$ converges. Given that $f$ is an entire function such that $f(z_n)=n$, determine which of the following statements is true:

(A) $f\equiv 0$.

(B) $f$ is unbounded.

(C) No such $f$ exists.

(D) $f$ has no zeroes.

Answer: C


Question: Determine if there exists a constant \(K\) such that for all positive numbers \(a\) and \(b\), the inequality \(|a-b|^p \leq K |a^p - b^p|\) holds, where \(p > 1\). If such a constant exists, find its explicit form.

Answer: 1


Question:
Given the step function \( h(x) = \begin{cases} 1, & x \geq 0 \\ 0, & x < 0 \end{cases} \), determine how the Heaviside distribution \( H \in \mathcal{D'}(\mathbb{R}) \) is defined. Choose the correct definition:

1. \( H(f) = h(x) \quad \forall f \in \mathcal{D}(\mathbb{R}) \)
2. \( H(f) = \int_{\mathbb{R}} h(x) f(x) \, dx \quad \forall f \in \mathcal{D}(\mathbb{R}) \)

Answer: 2


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0}\frac{x^{6000}-(\sin x)^{6000}}{x^{6002}} \]

Answer: 1000


Question: Determine the least value of \((\alpha - \beta)\) such that the area of the parallelogram formed by the lines \(x \cos \alpha + y \sin \alpha = p\), \(x \cos \alpha + y \sin \alpha = q\), \(x \cos \beta + y \sin \beta = r\), and \(x \cos \beta + y \sin \beta = s\) is minimized, given the values of \(p, q, r,\) and \(s\).

Answer: \dfrac{\pi}{2}


Question: Suppose that \( \int_1^\infty |f(x)|dx \) converges and \( \lim \limits_{x \to \infty} f(x) = L \). What is the value of \( L \)? Justify your answer.

Answer: 0


Question:
Given the joint density function of two random variables \(X\) and \(Y\):
\[
f_{X,Y}(x,y)=\begin{cases}0.0009x(1+y), & \text{if } (x,y)\in \Omega \\
0, & \text{else}\end{cases}
\]
and the marginal distributions:
\[
f_X(x)=0.0036(21x-2x^2-x\sqrt{x}), \quad \text{and} \quad f_Y(y)=0.0009(1+y)\frac{y^4}{512}
\]
where
\[
\Omega=\{(x,y):4\sqrt{x}<y \quad \text{and} \quad 0<y<12\}
\]
Calculate the conditional probability \(P(X<4|Y>4)\).

Answer: 0.42


Question: For a set $A = \{1, 2, 3, \ldots, n\}$, determine the maximum size of a transitive relation $T$ on $A$ such that $T \neq A \times A$. What is the largest possible size for $T$?

Answer: n^2 - n + 1


Question: Determine whether there exists a bijection between \( \mathbb{R}^m \) and \( \mathbb{R}^n \) for \( m, n \in \mathbb{N} \) with \( m \neq n \).

Answer: Yes


Question: Calculate the value of the infinite series: \[ \sum_{k=1}^{\infty}\operatorname{arccot}\frac{1-k^2+k^4}{2k} \]

Answer: \dfrac{\pi}{2}


Question: Calculate the limit \( \lim_{(x,y)\to(0,0)} \frac{xy(1-\cos(x^2+y^2))}{(x^2+y^2)^{\frac{5}{2}}} \).

Answer: 0


Question: Determine whether the following statement is true or false: If $\alpha \in K^{alg}$, where $K$ is a field, and $f \in K[X]$, then $\alpha$ is a zero of $f$ with multiplicity greater than one if and only if $f(\alpha) = f'(\alpha) = 0$. Provide a justification for your answer.

Answer: True


Question: Evaluate the limit: $$\lim_{n\to \infty}\ \frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot6\cdot\ldots\cdot2n}$$

Answer: 0


Question: A barrel of water is filled to a maximum height of 90 cm. After it stops raining, the water leaks out at a rate proportional to the square root of the water depth. If the water level drops from 90 cm to 85 cm in one hour, how long will it take for the barrel to become empty?

Answer: 18 + 3\sqrt{34}


Question: Suppose $f: \mathbb{R} \rightarrow \mathbb{C}$ is an $L^2$ function on the finite interval $(a,b)$, meaning that \( \int_{a}^b |f(x)|^2 \, dx < \infty \). Is it true that $f$ is also an $L^1$ function on the same interval, i.e., \( \int_{a}^b |f(x)| \, dx < \infty \)?

Answer: Yes


Question: Determine the values of $y$ for which the integral \( \int_{0}^{\infty}\frac{\arctan\left(x^{3}\right)\ln\left(1 + x^{2}\right)}{x^{y}} \,\mathrm{d}x \) converges.

Answer: 1 < y < 6


Question: Let $A$ and $B$ be $n \times n$ skew-symmetric matrices. Suppose there exist permutations $g, h: \{1, \ldots, n\} \to \{1, \ldots, n\}$ such that $A_{i,j} = B_{g(i),h(j)}$ for all $1 \leq i, j \leq n$. Must there exist a permutation $f: \{1, \ldots, n\} \to \{1, \ldots, n\}$ such that $A_{i,j} = B_{f(i),f(j)}$ for all $1 \leq i, j \leq n$?

Answer: Yes


Question: Given a curve in space with unit tangent vector \( T \), unit normal vector \( N \), and binormal vector \( B \), verify whether the following identity holds: \[ T \times B = -N \]

Answer: T \times B = -N


Question: Evaluate the limit \( \lim_{n\to\infty}n^2\left(\frac{1^k+2^k+\cdots+n^k}{n^{k+1}}-\frac{1}{k+1}-\frac{1}{2n}\right) \) for \( k \in (-1,1) \).

Answer: \dfrac{k}{12}


Question:
Determine which of the following polynomials divides the polynomial \( x^7 + x^2 + 1 \):

(A) \( x^5 - x^4 + x^2 - x + 1 \)

(B) \( x^5 + x^4 + 1 \)

(C) \( x^5 + x^4 + x^2 + x + 1 \)

(D) \( x^5 - x^4 + x^2 + x + 1 \)

Answer: A


Question: Find the norm of the bounded linear operator \( A: C[0,1] \to C[0,1] \) defined by \( Ax(t) = \int_0^1 e^{-ts} x(s) \, ds \), where \( x \in C[0,1] \) and \( t \in [0,1] \).

Answer: 1


Question: Calculate the conditional expectation \( E[B_t | B_t^2] \) for a continuous Brownian motion \( B_t \) at time \( t \geq 0 \).

Answer: 0


Question: Solve the differential equation: \( 2x + y^2 + 2xy y' = 0 \) and find the solution in the form \( x^2 + xy^2 = c \), where \( c \) is a constant.

Answer: x^2 + xy^2 = c


Question: Compute the limit: $$\lim_{n\to\infty} \sum_{k=0}^n \frac {\binom{n}{k}}{\binom{2n-1}{k}}$$

Answer: 2


Question: Consider a set of N points \( P = (x, y) \) that are independently and identically distributed uniformly inside the unit circle \( C_1 \) centered at the origin \((0,0)\). Each point \( P \) represents a point on the circumference of a circle \( C_{r_p} = C_{\sqrt{x^2+y^2}} \), where \( r_p \) is the radius of the circle that \( P \) lies on. Two points, \( P_1 \) and \( P_2 \), are randomly selected from the set. What is the probability that \( P_2 \) lies inside the circle \( C_{r_1} \) defined by \( P_1 \)?

Answer: \dfrac{1}{2}


Question: Given constants \(a_k\) for \(k = 0, 1, 2, \ldots, m\) such that \(\sum_{k=0}^m a_k = 0\) and \(\sum_{k=1}^m k a_k = -6\), find the value of \(\lim_{n \to \infty} \sum_{k=0}^m a_k \sqrt{n^2 + kn}\)."

Answer: -3


Question:
Consider the series whose nth term is defined as follows:
\[ a_n = \begin{cases} \frac{1}{n}, & \text{if } n \text{ is a perfect square} \\ \frac{1}{n^2}, & \text{if } n \text{ is not a perfect square} \end{cases} \]
Determine whether the series \( \sum a_n \) is:

(a) Convergent
(b) Divergent
(c) Divergent because \( \lim_{n \to \infty} n a_n \neq 0 \)

Answer: a


Question: Evaluate the surface integral \( \iint_S (x) \, dy \wedge dz + (x+y) \, dz \wedge dx + (x^2+2z) \, dx \wedge dy \) over the surface \( S \) of the solid bounded by the surfaces \( S_1: 2x^2 + y^2 = 4z \) and \( S_2: x^2 + 2z = 2 \). Use direct integration to find the result.

Answer: 4\pi


Question: In a triangle ABC, where A, B, and C are the angles and a, b, and c are the corresponding sides, find the real part of the expression \((a \cos B + b \cos A + i(a \sin B - b \sin A))^n\).

Answer: c^n


Question: Evaluate the limit: \[ \lim_{x \to \infty} \left( e^{-x^2} \int_{0}^{x} e^{t^2} \, dt \right) \]

Answer: 0


Question:
Given the lines:
\[ \frac{x+1}{4} = \frac{y-3}{1} = \frac{z}{k} \]
and
\[ \frac{x-1}{3} = \frac{y+2}{-2} = \frac{z}{1} \]
that lie on the same plane, find the parameter \(k\).

Answer: 2


Question:
Find the limit of the sequence \( x_n \) as \( n \to \infty \), where the sequence is defined by:
\[ x_n = \left(\lim_{p \to \infty}\left[\frac{1}{p}\sum_{k=1}^{p}\left(1+\frac{k}{p}\right)^{1/n}\right]\right)^{n} \]

Answer: \dfrac{4}{e}


Question: Determine the number of surjective ring homomorphisms from \(\mathbb{Z}[i]\) to \(\mathbb{F}_{11^2}\).

Answer: 2


Question: Let \( f_n \in L^1 \) such that for all \( g \in L^1 \), \( \|f_n * g - g\|_1 \rightarrow 0 \). Does this imply that \( \int f_n \rightarrow 1 \)? Provide a proof or a counterexample.

Answer: Yes


Question: Does there exist a continuous function $f$ such that $f$ is bounded with a codomain of $[0,1]$ and its second derivative $f''$ is unbounded with a codomain of $(-\infty,+\infty)$?

Answer: Yes


Question: Evaluate the surface integral \( \oint \oint_S (xy\bar{i} + z^2 \bar{k}) \bar{n} \, dS \) using the divergence theorem, where \( S \) is the surface enclosing the volume in the first octant bounded by the planes \( z= 0, y = x, y = 2x, x + y+ z = 6 \), and \( \bar{n} \) is the unit outer normal to \( S \).

Answer: \dfrac{57}{2}


Question: Consider a singular distribution \( F(x) \) on \([0,1]\), where the cumulative distribution function (CDF) \( F \) is continuous and its set of growth points has measure zero. An example of such a distribution is the Cantor function. Determine whether the characteristic function \( \varphi(t) = \int_0^1 \exp(itx) \, dF(x) \) always satisfies \( \varphi(t) \xrightarrow{t \rightarrow \infty} 0 \).

Answer: No


Question: Evaluate the integral \( \iiint_B (x^2+y^2+z^2)^2 \, dV \) where \( B = \{ (x,y,z) \in \mathbb{R}^3 : x^2+y^2+z^2 \leq 1, z \leq 0 \} \).

Answer: \dfrac{2\pi}{7}


Question: Compute the limit using L'Hôpital's rule: $$\lim_{x\to 0^{+}} \left(\frac{e^{x} - e^{-x}}{2}\right)^{\tan x}$$

Answer: 1


Question: Determine if the equations $a^2 + 5b^2 = 2$ and $a^2 + 5b^2 = 3$ have integer solutions. If either equation has solutions, it confirms that $(1 + \sqrt{-5})$ is a principal ideal in $\mathbf{Z}[\sqrt{-5}]$. Provide your answer as 'Yes' if there are solutions or 'No' if there are none.

Answer: No


Question: Let $H$ be a separable Hilbert space, and $T:{\frak Dom}(T) \to H$ a densely defined self-adjoint operator. Assume $T$ is diagonalisable, meaning $H$ has an orthonormal basis $\{b_k\}_{k \in \mathbb{N}_0}$ such that $T(b_k) = c_k b_k$ for all $k$, where $c_k \in \mathbb{R}$. For a bounded continuous function $f: \mathrm{spec}(T) \to \mathbb{C}$, is $f(T)$ the unique bounded operator on $H$ satisfying $f(T)(e_k) = f(c_k)e_k$ for all $k$?

Answer: Yes


Question: Let $X$ and $Y$ be real-valued random variables defined on the same probability space. Denote the characteristic function of $X$ by $\phi_X$. If $\phi_{X+Y} = \phi_X \phi_Y$, must $X$ and $Y$ be independent? Provide a justification for your answer.

Answer: No


Question: Let $Q \subset \mathbb{R}^2$ be the square with vertices $(0,0)$, $(4,0)$, $(4,4)$, and $(0,4)$. Consider a continuous function $f: Q \rightarrow Q$ that fixes the points $(0,0)$, $(4,0)$, $(4,4)$, and $(0,4)$. The diagonal $d$ of $Q$ is the line segment from $(0,0)$ to $(4,4)$, and $f(d)$ is its image under $f$, forming a continuous path in $Q$ from $(0,0)$ to $(4,4)$. This path divides $Q$ into regions. Let $Q'_0$ be the region containing the point $(4,0)$. Determine whether the point $f(2,1)$ is contained in $Q'_0$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{a^{\tan x} - a^{\sin x}}{\tan x - \sin x} \] without using L'Hospital's rule.

Answer: \ln a


Question: Prove that 26 is the only number that satisfies the equation \(x^2 + 1 = z = y^3 - 1\).

Answer: 26


Question: Determine whether the following statement is true: For a complete measure space $(\Omega, M, \mu)$, is it true that $L^1(\Omega, M, \mu) \cap L^\infty(\Omega, M, \mu)$ is continuously injected into $L^p(\Omega, M, \mu)$ for all $p \in [1,\infty]$?

Answer: True


Question:
Simplify the logical expression:
\[ \left[\{(\lnot p \lor q)\land(\lnot q \lor p)\}\rightarrow\{(q \lor \lnot p)\land p\}\right]\rightarrow\left[(p \leftrightarrow q)\lor(q \bigtriangleup p)\right] \]

Assume:
- \(p\) represents "Jenny goes to the movies"
- \(q\) represents "Jenny goes to the park"

Match the simplified expression to one of the following statements:

1. Jenny either goes to the park or goes to the movies
2. Jenny goes to the movies if and only if she goes to the park
3. Jenny goes to the movies or she doesn't go to the movies
4. Jenny doesn't go to the park but she goes to the movies

Use the provided truth table and identities for the strong disjunction \(\bigtriangleup\) to assist in the simplification.

Answer: 3


Question: Can two bicycles of different lengths leave the same set of tracks, aside from a straight line?

Answer: No


Question: Simplify the double summation: \( \sum_{i=1}^{n/2}\sum_{j=i}^{n-i}j \).

Answer: \dfrac{n^3}{8}


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} \frac{(2n)!}{n^{2n}} \]

Answer: 0


Question: Calculate the minimal polynomial of the matrix $E(i,j) = I_n + \alpha I_{(i,j)}$, where $I_n$ is the identity matrix of order $n$, $I_{(i,j)}$ is the matrix with 1 in the $(i,j)$ position and 0 elsewhere ($i \neq j$), and $\alpha \in \mathbb{K}$ with $\mathbb{K}$ a field, $\alpha \neq 0$. Provide your answer in terms of $n$, $i$, $j$, and $\alpha$.

Answer: (x - 1)^2


Question: Evaluate the limit: \[ \lim_{x\to 0}\left[1^{\frac{1}{\sin^2 x}}+2^{\frac{1}{\sin^2 x}}+3^{\frac{1}{\sin^2 x}}+\cdots + n^{\frac{1}{\sin^2 x}}\right]^{\sin^2x} \]

Answer: n


Question: Five people are passing a ball amongst themselves, starting with Alonzo. Each person who has the ball passes it to someone else. After the eighth pass, the ball must return to Alonzo. How many possible sequences of passes are there?

Answer: 13108


Question: Evaluate the integral \( \int _C e^z \, dz \) for any closed path \( C \) in the complex plane.

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\rightarrow +\infty}\sqrt{x}e^{-x}\left(\sum_{k=1}^{\infty}\frac{x^{k}}{k!\sqrt{k}}\right) \]

Answer: 1


Question: Calculate the limit: \[ \lim_{n \to \infty} \left(\frac{1}{3} + \frac{n^{1000}}{2^n}\right)^n \] without using the exponential function.

Answer: 0


Question: Let $X$ and $Y$ be independent standard Gaussian random variables. Calculate the conditional expectation $E[X|XY]$. Provide your answer in terms of $X$ and $Y$.

Answer: 0


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a 1-periodic function. For any real number \( c \), is it true that the variation of \( f \) on the interval \([0, 1]\) is equal to the variation on the interval \([c, c+1]\)?

Answer: Yes


Question: Given the integrals \( \frac{\int_{0}^{1}(1-x^3)^{49}dx}{\int_{0}^{1}(1-x^3)^{50}dx} = \frac{m}{n} \), where \( m \) and \( n \) are relatively prime integers, find the value of \( 2m-n \).

Answer: 152


Question: Find the equation of the line passing through the points (3, 4) and (1, 2).

Answer: y = x + 1


Question: Does there exist a non-nilpotent square matrix $A$ of order $n \geq 2$ with complex entries, where all real and imaginary parts are rational, such that there exists a power $k$ for which $A^k$ has integer real and imaginary parts?

Answer: Yes


Question: Determine whether there are any maximal ideals in the ring \( \mathbb{R}[x,y]/(xy-2) \). If so, find one maximal ideal.

Answer: (x - 2, y - 1)


Question: Does there exist an enumeration \(\{r_{n}\}_{n=1}^\infty\) of the rationals such that the complement of \(\bigcup_{n=1}^{\infty}{\left(r_{n}-\frac{1}{n},r_{n}+\frac{1}{n}\right)}\) in \(\mathbb{R}\) is non-empty? [Hint: Find an enumeration where the only rationals outside of a fixed bounded interval take the form \(r_n\), with \(n=m^2\) for some integer \(m\).]

Answer: Yes


Question: Two points are chosen uniformly and independently on the perimeter of a circle with radius 1. This divides the perimeter into two arcs. Determine the expected value of the length of the shorter arc.

Answer: \dfrac{\pi}{2}


Question: Find the limit as \( n \to \infty \) of the sequence defined by \( a_{n+1} = \frac{(a_n)^2}{6}(n+5)\int_{0}^{3/n}{e^{-2x^2}} \mathrm{d}x \), with the initial term \( a_1 = \frac{1}{4} \).

Answer: 0


Question: Let $G$ be a finite group. If $c_1, c_2, \ldots, c_h$ are the orders of the centralizers of elements from distinct conjugacy classes of $G$, prove that \( \frac{1}{c_1} + \frac{1}{c_2} + \cdots + \frac{1}{c_h} = 1. \)

Answer: 1


Question: Evaluate the limit \( \lim_{N \to \infty} \sum_{n=1}^{N} \frac{1}{(n+1)} \sum_{i=1}^{n} \frac{1}{i(n+1-i)} \).

Answer: 2\zeta(3)


Question: Consider the finite difference equation \((E):\ \sum_{k=0}^n P_k(z)g(z+k)=0\) where each \(P_k\) is a polynomial in \(\mathbb{C}[z]\). Is it true that every entire solution \(g\) of \((E)\) with exponential type less than \(\pi\) is an exponential polynomial?

Answer: Yes


Question: Given that \(a, b, c, d > 0\) and \(a + b = 3\), and \(\frac{a}{c} + \frac{b}{d} = 1\), find the minimum value of \(c + d\).

Answer: 6


Question: Consider the short exact sequence of divisible abelian groups: \(0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0\). Determine whether this sequence splits.

Answer: Yes


Question: Determine the last two digits of the sequence \((a_n)\) where \(a_1 = 3\) and \(a_{n+1} = 3^{a_n}\) for all \(n\).

Answer: 87


Question: Find \( \lim_{n\to\infty}\frac{1}{n}\left(n+\frac{n-1}{2}+\frac{n-2}{3}+\dots+\frac{2}{n-1}+\frac{1}{n}-\log(n!)\right) \).

Answer: \gamma


Question: Calculate the variance of the product XY, where X is a random variable following a Uniform(0,1) distribution and Y is a random variable following a Normal(0,1) distribution.

Answer: \dfrac{1}{3}


Question: Compute the conditional expectation \( E(W_t \,|\, |W_T|) \), where \( W_t \) is a one-dimensional Brownian motion and \( t < T \).

Answer: 0


Question: Determine the norm of the functional defined on $l_2$ by \( x = (x_n) \mapsto \sum \frac{x_n + x_{n+1}}{2^n} \).

Answer: 1


Question:
Determine whether there exists a point \(a < c < b\) such that the following equation holds:
\[ \int_a^b (b - x)^n f^{(n + 1)}(x)\,dx = \frac{f^{(n + 1)}(c)}{n + 1}(b - a)^{n + 1} \]
where \(\tilde{a} < a < b < \tilde{b}\), \(n\) is a nonnegative integer, and \(f(x)\) is a real-valued function that is \((n + 1)\)-times continuously differentiable on \((\tilde{a}, \tilde{b})\).

Answer: Yes


Question: Determine whether there exists an $n \times n$ matrix $A$ such that $A^2 = A$ and $\text{tr}(A) = n + 1$ for $n \geq 2$. Answer "True" if such a matrix exists and "False" otherwise.

Answer: False


Question: Given a non-negative integer $N$, a positive constant $C$, and a real number $x$, determine whether there exists a function $f \in C^{\infty}_c(\mathbb{R})$ with compact support such that $\partial^{N+1}f(x) > C\sum_{k=1}^{N}\sup|\partial^{k} f|$. Assume $x=0$ to simplify the problem.

Answer: Yes


Question: Determine the number of trailing zeros in the base 6 representation of 2015 factorial, denoted as \((2015!)_6\).

Answer: 1002


Question: Find the set of limit points of the sequences \(a_n = n \alpha - \lfloor n \alpha \rfloor\) and \(b_n = \sin(\pi n \alpha)\), where \(\alpha\) is an irrational number. Determine if the set of limit points for \((a_n)\) is the interval \([0, 1)\).

Answer: [0, 1)


Question: Use the method of Lagrange multipliers to find the maximum value of the function \( f(x, y, z) = xy^2z \) subject to the constraint \( x^2 + y^2 + z^2 = 36 \).

Answer: 162


Question: Is the kernel of an elliptic operator on sections of a vector bundle over a compact manifold a finite-dimensional space?

Answer: Yes


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

Answer: 0


Question: Suppose $(M,\omega)$ is a symplectic manifold and $L \subset M$ is a compact Lagrangian submanifold. Does the first Chern class $c_1(TM) \in H^2(M)$ vanish when restricted to the Lagrangian submanifold $L$?

Answer: Yes


Question:
Given that there exists an \( \epsilon > 0 \) such that \( \left\vert f(x) - g(x) \right\vert \leq \epsilon \) for every \( x \in (a,b) \), determine if the following inequality holds:
\[ \frac{1}{b-a} \left\vert \int_{a}^{b} f(x) \, dx - \int_{a}^{b} g(x) \, dx \right\vert \leq \epsilon \]
Assume that \( f(x) \) and \( g(x) \) are integrable and differentiable on \([a,b]\).

Answer: Yes


Question: Determine the value of the infinite tetration of \(-1\), denoted as \( {^{\infty}}(-1) \). Consider both the limit \( \lim_{n \to \infty} {^n}(-1) \) and the analytic continuation formula \( \frac{W(-\ln z)}{-\ln z} \). Which value is correct?

Answer: -1


Question: Let $A$ be a convex, absorbing subset of a real Banach space $X$ such that the closure $\mathrm{cl}(A)$ contains an open ball around $0 \in X$. Does this imply that $A$ itself contains an open ball around $0$?

Answer: Yes


Question:
Determine the number of solutions for the system of equations with $x, y \in (0,1)$:
\[ \sqrt{3} \cdot x = \cos(xy) \]
\[ e^{-xy} = 2y \]

Answer: 1


Question:
Determine the radius of convergence of the series:
\[ 1 + \frac{ab}{1c}z + \frac{a(a+1)b(b+1)}{1 \cdot 2 \cdot c(c+1)}z^2 + \cdots \]

Answer: 1


Question: Calculate the line integral \( \int_C \vec{F} \cdot d\vec{r} \) where \( C \) is the unit circle centered at the origin and oriented clockwise, and \( \vec{F} = \frac{-y\vec{i} + x\vec{j}}{x^2 + y^2} \).

Answer: -2\pi


Question: Let \((X_t)_{t \in \mathbb{N}}\) be a strictly stationary sequence of real random variables, and let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a measurable function. Is the sequence \((Y_t)\), where \(Y_t = f(X_t)\), also strictly stationary?

Answer: Yes


Question: Solve the equation \(x\lfloor x\rfloor\{x\}=58\) in rational numbers, where \(\lfloor x\rfloor\) is the greatest integer less than or equal to \(x\) and \(\{x\}\) is the fractional part of \(x\).

Answer: \dfrac{29}{3}


Question:
Consider the set \( P \) in \( \mathbb{R} \) defined as follows:

- Let \( E_0 = [0,1] = \{ 0.d_1d_2\ldots : 0 \leq d_j \leq 9 \text{ for all } j \} \).
- Let \( E_1 = \{ x \in E_0 : d_1 \neq 0 \} \).
- Let \( E_2 = \{ x \in E_1 : d_2 \neq 0 \} \).
- Continue this process to define \( E_n = \{ x \in E_{n-1} : d_n \neq 0 \} \) for all \( n \geq 1 \).
- Define the set \( P = \bigcap_{j=0}^{\infty} E_j \).

What is the length of the set \( P \)?

Answer: 0


Question:
Given that $f(x)$ is a function satisfying the differential equation $f(x) + f''(x) = 0$, and $g(x) = f^2(x) + f'^2(x)$ with $g(3) = 8$, find the value of $g(8)$.

Express your answer as a single number.

Answer: 8


Question: Is a complex symmetric matrix with a positive definite real part necessarily diagonalizable? Let \( M \in \mathbb{C}^{n \times n} \) be a complex-symmetric matrix. If the real part of \( M \) is positive-definite, determine whether \( M \) is diagonalizable.

Answer: Yes


Question: Determine the number of solutions to the equation $e^x - 2 = \cos x$.

Answer: 1


Question: Let $(x_n)$ be a sequence of real numbers. Suppose $e^{itx_n}$ and $e^{isx_n}$ are both convergent for some positive numbers $t$ and $s$ with $t/s$ irrational. Does it follow that $(x_n)$ is convergent? Provide a justification for your answer.

Answer: Yes


Question: Determine the remainder when \(7^{7^{2019}}\) is divided by 47.

Answer: 12


Question: Prove that \( \lim\limits_{x \to \infty} \frac{x!}{x^x} = 0 \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\to\infty} (\text{arccosh} \, x)^\frac{1}{x} \]

Answer: 1


Question: Find the minimal polynomial of the linear operator \( T \) on the vector space \( V = M_{n \times n}(R) \), where \( T(A) = A^t \) for each matrix \( A \).

Answer: x^2 - 1


Question: Compute the limit: \[ \lim_{n \to \infty}{\frac{\arcsin{\frac{1}{n}}-\arctan{\frac{1}{n}}}{\sin{\frac{1}{n}}-\tan{\frac{1}{n}}}} \]

Answer: -1


Question: Determine whether the following statement is true or false: If $|\int f_n - \int f | \to 0$, then $\int |f_n - f| \to 0$, given that $f_n$ and $f$ satisfy the conditions of the dominated convergence theorem.

Answer: A


Question: Let \( \Phi \) be a random variable with probability density function (p.d.f.) \( f(\phi) = \frac{1}{2}\sin\phi \) for \( \phi \in [0, \pi] \). Define \( h(\phi) = a\cos(\phi) + b \), where \( a \) and \( b \) are positive constants. Find the distribution of the random variable \( h(\Phi) \).

Answer: \frac{1}{2a}


Question: Simplify \((-2 + 2\sqrt{3}i)^{\frac{3}{2}}\) and express the result in rectangular form \(z = a + bi\).

Answer: -8


Question: Find the inverse polynomial \( f^{-1}(x) \) for \( f(x) = x^6 + x^5 + 1 \) in the quotient ring \( \frac{\mathbb{Q}[x]}{(x+2)} \).

Answer: \dfrac{1}{33}


Question:
Find all polynomials \( P(x) \in \mathbb{R}[x] \) that satisfy the equation:
\[ P(x+1) - 2P(x) + P(x-1) = 6x. \]

Answer: x^3 + ax + b


Question: Given the sequence defined by \(a_1 = 4\), \(a_2 = -2\), and \(a_{n} = 2a_{n-2} - 3a_{n-1}\), determine the smallest positive integer \(n\) for which \(|a_{n}| > 1,000,000\).

Answer: 12


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

Answer: \dfrac{\pi}{4}


Question: Determine the number of solutions in positive integers to the equation \(n_1 + n_2 + n_3 + n_4 + n_5 = 20\) under the condition that \(n_1 < n_2 < n_3 < n_4 < n_5\).

Answer: 7


Question: Evaluate the limit: $$\lim_{n\to \infty} \frac{1}{n^{2020}} \sum_{k=1}^n k^{2019}$$

Answer: \dfrac{1}{2020}


Question: Calculate the volume of the set \( \left\{ \begin{pmatrix} 2 & 33 & 444 \\ 0 & 3 & 4 \\ 0 & 0 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} : x^2 + y^2 + z^2 \leq 1 \right\} \) in \(\mathbb{R}^3\).

Answer: 32\pi


Question: Use Green's theorem to evaluate the line integral \( \int_{L} (x+2xy)dy + y^{2}dx \) where \( L \) is the circle defined by \( x^2 + y^2 = y \).

Answer: \dfrac{\pi}{4}


Question: Given that \( \int_0^\infty f(x)\, dx <\infty \), where \( f \) is a non-negative continuous function, determine whether \( \lim\limits_{n\rightarrow \infty} \int_n^{n+1}\sqrt{f(x)}\, dx = 0 \).

Answer: 0


Question: Find a maximal ideal in the polynomial ring $F[x]$, where $F$ is a field, other than the ideal $(x)$. Justify your answer.

Answer: (x - 1)


Question: Evaluate the integral: \[ \int_{0}^{\infty}\frac{\sin\left(x^{2}\right) + \cos\left(x^{2}\right) - 1}{x^{2}}\,\mathrm{d}x. \]

Answer: 0


Question: Let $B$ be a Banach algebra and $A$ be a bi-ideal of it. Suppose that for any $b \in B$, $Ab = \{0\}$ implies $b = 0$. Determine if for some $c \in B$, $cA = \{0\}$ implies $c = 0$. Provide a justification for your answer.

Answer: Yes


Question: Given the functional \( f(x) = \sum x_k a_k \), where \( x = (x_1, x_2, \ldots) \in l^p \) and \( a = (a_1, a_2, \ldots) \in l^q \), find the norm of \( f \).

Answer: \|a\|_q


Question:
Given a random variable $X$ with the cumulative distribution function (CDF):
\[ F(x) = \begin{cases} 1 - \left(\frac{2000}{2000 + x}\right), & x \ge 0 \\ 0, & \text{otherwise} \end{cases} \]
Let $Y = \max(0, X - 500)$. Calculate the 60th percentile of the positive values of $Y$, ignoring values of $Y$ where $0 \le X \le 500$. Provide your answer as a single number.

Answer: 3750


Question: Determine the number of distinct 7-permutations of the multiset \( \mathcal{M} = \{0^1,\ldots, 0^7,1^1,\ldots, 1^7, \cdots,9^1,\ldots, 9^7  \} \), where each digit from 0 to 9 appears 7 times. Two permutations are considered indistinguishable if they differ only by the exchange of identical digits. Calculate this number.

Answer: 10000000


Question: Determine whether there exists an entire function $f$, which is not a polynomial, such that there is a disk $D_\epsilon(0)$ for some $\epsilon > 0$ where the restriction of $f$ to $D_\epsilon(0)$ is injective.

Answer: Yes


Question: For a given prime number $p$, does there exist a prime number $q$ such that $p$ is a primitive root modulo $q$?

Answer: Yes


Question: Determine whether the Galois group of the polynomial \( f(x) = (x^2-ax+b)(x^2-cx+d) \), where \( f(x) \) is a separable polynomial with rational coefficients, is always a subgroup of the Klein four group \( C_2 \times C_2 \).

Answer: Yes


Question:
Solve the system of equations:
\[\begin{cases}
3-(y+1)^2 = \sqrt{x-y} \\
x+8y = \sqrt{x-y-9}
\end{cases}\]

Answer: (8, -1)


Question: Calculate the average projected area of a circle with radius 1 in 3D space onto the x-z plane, considering all possible orientations of the circle. Express your answer in terms of an integral if necessary.

Answer: \dfrac{\pi}{2}


Question:
Consider the paths \( c(t) = (\cos t, \sin t) \) for \( 0 \leq t \leq 2\pi \) and \( p(t) = (\cos t, \sin t) \) for \( 0 \leq t \leq 4\pi \). Determine whether \( \int_c F \cdot ds = \int_p F \cdot ds \) for every continuous vector field \( F \) on \( \mathbb{R}^2 \). Choose the correct option:

(a) True for all vector fields \( F \)
(b) True for some vector fields \( F \) and false for others
(c) False for all vector fields \( F \)
(d) It's complicated

Answer: B


Question: Find the smallest positive integer \( y \) such that \( a^y \equiv 1 \pmod{100} \) for all integers \( a \) with \( \gcd(a, 100) = 1 \).

Answer: 20


Question: Let \( u(x,y) = x^3 + ax^2y + bxy^2 + 2y^3 \) be a harmonic function, and \( v(x,y) \) be its harmonic conjugate. Given that \( v(0,0) = 1 \), find the value of \(|a+b+v(1,1)|\).

Answer: 10


Question: Given the matrix \(K_{3\times3} = \begin{bmatrix} 1&1&1 \\ 3&2&1 \\ 1&2&1 \end{bmatrix}\) with \(|K| = 2\), find the determinant of the matrix expression \(|2K^3 - 2K^4|\).

Answer: -384


Question: Consider a del Pezzo surface $X$ over $\mathbb{C}$, obtained by blowing up $\mathbb{P}^{2}$ at a collection of points, with $H$ as the hyperplane class of $\mathbb{P}^{2}$. Let $\Sigma$ be a smooth, irreducible curve on $X$ satisfying the inequality $-K_{X}\cdot \Sigma > \frac{1}{2}\Sigma \cdot \Sigma + \pi^{*}H \cdot \Sigma$. Is the genus of such curves bounded above?

Answer: Yes


Question: Let $f : S^{n-1} \to [0,b] \subset \mathbb{R}$ be a continuous function. Determine whether there exists a continuous extension $F : B^n \to [0,b]$ of $f$ that is strictly positive on the interior of $B^n$. Provide a justification for your answer.

Answer: Yes


Question: Determine the number of group homomorphisms from $A_5$ to $S_5$. Provide your answer as a single integer.

Answer: 121


Question: Let \( f(x) \) be a function that is thrice differentiable on \((a,b)\). Consider the function \( \phi(x) = f(b) - f(x) - (b-x)f'(x) - \frac{(b-x)^2}{2}f''(x) - (b-x)^3 \lambda \), which satisfies the conditions of Rolle's Theorem on \([a,b]\). If there exists some \( c \in (a,b) \) such that \( \phi'(c) = 0 \) and \( f(b) = f(a) + (b-a)f'(a) + \frac{(b-a)^2}{2}f''(a) + \mu (b-a)^3 f'''(c) \), find the value of \( \mu \).

Answer: \dfrac{1}{6}


Question:
Consider a real-valued continuous function \( f \) satisfying \( f(x+1) = f(x) \) for all \( x \in \mathbb{R} \). Define:
\[ g(t) = \int_{0}^{t} f(x) \, dx \]
\[ h(t) = \lim_{n \to \infty} \frac{g(t+n)}{n}, \quad t \in \mathbb{R} \]
Which of the following statements is true?
(A) \( h(t) \) is defined only for \( t = 0. \)
(B) \( h(t) \) is defined only when \( t \) is an integer.
(C) \( h(t) \) is defined for all \( t \in \mathbb{R} \) and is independent of \( t. \)
(D) None of the above is true.

Answer: C


Question: Given a light source at $L = (0, -2, 5)$ and a triangle $ABC$ with vertices $A = (0, 0, 2)$, $B = (3, 0, 2)$, and $C = (0, 0, 3)$, where point $C$ projects to $C' = (0, 3, 0)$ on the $xy$-plane, find the area of the shadow created by the triangle on the $xy$-plane.

Answer: \dfrac{25}{6}


Question: Determine the values of \( s > 0 \) for which the series \( \sum_{n=1}^{\infty} \frac{\sqrt{n+1} - \sqrt{n}}{(-1)^n n^s} \) converges.

Answer: s > 0


Question: In the given diagram, $ST$ is the perpendicular bisector of $PR$, and $SP$ is the angle bisector of $\angle QPR$. Given that $QS = 9\text{ cm}$ and $SR = 7\text{ cm}$, find the value of $x+y$ if $PR = \frac{x}{y}$, where $x$ and $y$ are co-prime integers.

Answer: 31


Question: Evaluate the triple integral \( \iiint_D (3+2xy)\,dV \) over the solid hemispherical dome \( D \) defined by \( x^2+y^2+z^2 \leq 4 \) and \( z \geq 0 \).

Answer: 16\pi


Question:
Compute the expectation of the stochastic process given by:
\[ S_t = 1 + \int_0^t \exp\left(\sigma B_s - \sigma^2\frac{s}{2}\right) dB_s, \]
where \( B_t \) is a standard Brownian motion.

Answer: 1


Question: A teacher is organizing 4 special lessons, each with exactly 3 participants. Any two students must attend at least one special lesson together. What is the maximum number of students who can participate in these special lessons?

Answer: 5


Question: Prove that the series \( \sum_{n=1}^\infty x_n \sin(nx) \) converges uniformly if and only if \( nx_n \to 0 \) as \( n \to \infty \), where \( x_n \) is a decreasing sequence with \( x_n > 0 \) for all \( n = 1, 2, \ldots \).

Answer: nx_n \to 0


Question: Let \( f \) be a function such that \( f \in L^1(0,1) \), \( f \ge 0 \), and \( f \) is continuous on \((0,1)\). Determine whether \( \lim_{x \to 0} x f(x) = 0 \).

Answer: 0


Question: Determine whether there exists a partial order relation on the group of integers \( \mathbb{Z} \) that makes it a partially ordered group or a totally ordered group, other than the usual \( \leq \).

Answer: Yes


Question: Calculate \( \int \int_X f(x, y) \, dx \, dy \) where \( f(x, y) = x \) and the region \( X \) is defined by the inequalities \( 2rx \leq x^2 + y^2 \leq R^2 \) with \( 0 < 2r < R \).

Answer: -\pi r^3


Question: Calculate \( 4239^4 \mod 19043 \) using the Chinese Remainder Theorem. First, find \( 4239^4 \mod 137 \) and \( 4239^4 \mod 139 \). Then, use these results to determine \( 4239^4 \mod 19043 \).

Answer: 808


Question: Find the integer \( n \) such that in a regular \( n \)-gon \( A_1A_2 \ldots A_n \), the following equation holds: \[ \frac{1}{A_1A_2} = \frac{1}{A_1A_3} + \frac{1}{A_1A_4}. \]

Answer: 7


Question: Is the number of irreducible components $m$ of an analytic hypersurface in a complex manifold always finite?

Answer: No


Question: A standard deck of 52 cards is shuffled and dealt. Let \(X_{1}\) be the number of cards appearing before the first ace and \(X_{2}\) be the number of cards between the first and second ace (not counting either ace). Calculate the probability \(P(X_{2} = 10 | X_{1} = 15)\).

Answer: \dfrac{5}{119}


Question: Find the last two digits of \( 14^{14^{14}} \).

Answer: 36


Question: Given the block triangular matrix \( M = \begin{pmatrix} A & 0 & 0 \\ B & A & 0 \\ C & D & A \end{pmatrix} \), where \( A, B, C, \) and \( D \) are diagonalizable square matrices, determine \( M^{\infty} := \lim_{n\rightarrow \infty} M^n \) in terms of \( A, B, C, \) and \( D \).

Answer: 0


Question: Evaluate the integral \( \int_\mathbb{R} \lfloor x \rfloor \, d\mu \), where \( \mu(A) = \#(\mathbb{N} \cap A) \) for any set \( A \subseteq \mathbb{R} \).

Answer: \infty


Question:
In a triangle \( \triangle ABC \), it is given that:
\[ \sin A = 2018 \cdot \sin B \cdot \sin C \]
\[ \cos A = 2018 \cdot \cos B \cdot \cos C \]
Determine the value of \( \tan A \).

Options:
- Not finite
- 1
- 0
- 2018
- 2019

Answer: 2019


Question: Evaluate the infinite product in closed form: \[ \prod_{n=1}^{\infty} \frac{1}{1+\pi^{1/2^n}} \]

Answer: 0


Question: Determine whether the expression \( \lim_{n\to\infty} \frac{1}{n}\left(\sin\frac{\pi}{n}+\sin\frac{2\pi}{n}+\cdots+\sin\frac{n\pi}{n}\right) \) converges or diverges.

Answer: \dfrac{2}{\pi}


Question: Find the volume of the solid obtained by rotating the region defined by \((x-1)^2+(y-1)^2 \le 1\) about the line \(y=\frac{1}{2}\).

Answer: \pi^2


Question: Let \( G \) be a group of order 42, and suppose \( G \) has a subgroup of order 6. Compute the number of conjugates of this subgroup in \( G \).

Answer: 7


Question: Provide an example of a non-principal ideal in the polynomial ring $K[x,y]$, where $K$ is a field.

Answer: (x, y)


Question: Determine whether the limit \( \lim_{x \to 0} \frac{f(x)}{|x|} \) exists, where \( f(x) = \langle A(x), x \rangle - \lambda \langle x, x \rangle \) and \( A \) is a linear operator on \( \mathbb{R}^3 \).

Answer: 0


Question: Suppose $T$ is a closable linear operator defined on a subspace of a Banach space, with the target space also being a Banach space. Let $G$ be a contraction operator defined on the target space of $T$. If the operator $GT$ is closable, can the domain of the closure of $GT$ be larger than the domain of the closure of $T$? Provide a justification for your answer.

Answer: Yes


Question: Let \( g \) be a monotonic strictly increasing, continuous, and positive function on \([0, \infty)\) such that \( \lim_{t \to \infty} g(t) = L \). Evaluate the limit: \[ \lim_{n \to \infty} \int_1^2 g(nx) \, \mathrm{d}x. \]

Answer: L


Question: Calculate the third derivative of the function \( f(x) = \frac{(x - 1)^3}{37x^6 - 61x^5 + 13x^2 - 74x + 25} \) at \( x = 1 \).

Answer: -\dfrac{1}{10}


Question: Evaluate the line integral \( \int_{(C)} xy \, dx + y^2 \, dy \) where \( (C) \) is the semicircular path defined by \( y \geq 0, x^2 + y^2 = 4 \), traversed in the clockwise direction.

Answer: 0


Question: Minimize the function \( xy + 49xz + 7yz \) subject to the constraint \( xyz = 1 \).

Answer: 21


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

Answer: e^2


Question: Calculate the dimension of the centralizer of the matrix \( A \), where \( A = \begin{bmatrix}8 & -9 & -2 \\ 6 & -7 & -2 \\ -6 & 9 & 4 \end{bmatrix} \). The centralizer \( V(A) \) is defined as \( \{X \in M_{3}(\mathbb{R}) : AX = XA \} \), where \( M_{3}(\mathbb{R}) \) is the vector space of \( 3 \times 3 \) real matrices. Determine the dimension of \( V(A) \).

Answer: 5


Question: Evaluate the integral \( I = \int_0^\infty \log\left(x + \frac{1}{x}\right) \frac{1}{1+x^2} \, dx \).

Answer: \pi \ln 2


Question: Let \((X_n)\) be a sequence of random variables such that \(X_n \to +\infty\) almost surely. Prove that for any \(M > 0\), \(P(X_n < M \text{ i.o.}) = 0\).

Answer: 0


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

Answer: (-1, \infty)


Question: If $T$ is a compact operator on a Hilbert space $H$, and $T$ has a polar decomposition $T = U|T|$, where $U$ is a partial isometry and $|T|$ is the positive part, does the partial isometry $U$ belong to the space of compact operators $\mathcal{K}(H)$?

Answer: No


Question: Does \(\mathbb{E}[(1-\varepsilon_n)^{X_n}]\rightarrow 1\) imply that \(\mathbb{P}(X_n\geq A_n)\rightarrow 0\)? Assume \(X_n\) is a sequence of random variables taking values on the non-negative integers, finite almost surely, \(\varepsilon_n\in(0,1)\), and \(\varepsilon_n A_n\geq O(1)\).

Answer: Yes


Question:
Let \( \vec{a}, \vec{b}, \vec{c}, \vec{d} \) be four unit vectors such that \((\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = 1\) and \(\vec{a} \cdot \vec{c} = \frac{1}{2}\). Determine which of the following statements is true:

(a) \(\vec{a}, \vec{b}, \vec{c}\) are non-coplanar.
(b) \(\vec{b}, \vec{c}, \vec{d}\) are non-coplanar.
(c) \(\vec{b}, \vec{d}\) are non-parallel.
(d) \(\vec{a}, \vec{d}\) are parallel and \(\vec{b}, \vec{c}\) are parallel.

Answer: C


Question: Find the integer part of the sum \( \sum_{n=1}^{1000000} \frac{1}{\sqrt{n}} \).

Answer: 1998


Question: Evaluate the expression \( 2^{2010}\frac{\int^{1}_{0}x^{1004}(1-x)^{1004}dx}{\int^{1}_{0}x^{1004}(1-x^{2010})^{1004}dx} \).

Answer: 4020


Question: Suppose $f(x)$ is such that \( \int_{-\infty}^\infty e^{tx} f(x) \, dx = \arcsin(t - \sqrt{1/2}) \) for all \( t \) where the right-side expression is defined. Compute \( \int_{-\infty}^\infty x f(x) \, dx \).

Answer: \sqrt{2}


Question: Given that the probability of success, \( p \), on an experiment is assigned a prior density \( f_p(s) = 2(1-s) \mathbb{1}_{[0,1]}(s) \), and the experiment is conducted independently three times resulting in two successes, find the posterior mean of \( p \).

Answer: \dfrac{1}{2}


Question: Evaluate the limit: $$ \lim_{n\to\infty}\left(\frac{(n!)^3}{(3n)!}\right)^{1/n} $$

Answer: \dfrac{1}{27}


Question: Let $k$ be a field and $P$ be a prime ideal of the polynomial ring $k[x_1, ..., x_n]$. Let $K$ be the field of fractions of the quotient ring $k[x_1, ..., x_n]/P$. If $K$ is a finite extension of $k$, does it follow that $P$ is a maximal ideal?

Answer: Yes


Question: Evaluate the triple integral \( \iiint_S z \sqrt{x^2 + y^2} \, dx \, dy \, dz \) over the region \( S \) bounded by \( x^2 + y^2 = 1 \), \( z = 0 \), and \( z = x \) with \( z \ge 0 \).

Answer: \dfrac{\pi}{20}


Question: Let $k$ be a field of characteristic 0, and let $X$ and $Y$ be smooth, projective, and geometrically integral $k$-schemes of finite type. Assume that both $X$ and $Y$ have 0-cycles of degree 1. Determine whether $X \times_{k} Y$ has a 0-cycle of degree 1.

Answer: Yes


Question: Let \( f: [0, 2\pi] \rightarrow \mathbb{R}^2 \) be a smooth function such that \( f([0, 2\pi]) \) is a smooth closed simple curve \( C \). Suppose the point \((0,0)\) lies inside the bounded open region enclosed by \( C \) and \( f(t) = (x(t), y(t)) \). Determine whether the function \( g(t) = x^2(t) + y^2(t) \) has at least 4 critical points in the interval \([0, 2\pi)\).

Answer: Yes


Question: Calculate the limit: $$\lim_{n\to\infty}(1+1/n^2)(1+2/n^2)\cdots(1+n/n^2).$$

Answer: \sqrt{e}


Question: Find the value of \( \alpha \) in degrees such that \( \alpha \csc\left(\frac{\alpha}{2}\right) = 120^\circ \).

Answer: 60


Question: Find the inverse Laplace transform of the function \( H(s) = \frac{1}{s^4 - s^2} \).

Answer: \sinh(t) - t


Question: Let \(X_1, X_2, \ldots, X_n\) be a random sample from a Poisson distribution with parameter \(\lambda = 1\). Define the statistic \(T = \bar{X}(n - \bar{X})\), where \(\bar{X}\) is the sample mean. Find an upper bound for the probability \(P(T = 0)\).

Answer: 2e^{-n}


Question: Is the torsion-free abelian group $G$, as described in the statement that for any positive integer $r$ there exists a countable torsion-free abelian group $G$ such that the direct sum of $m$ copies of $G$ is isomorphic to the direct sum of $n$ copies of $G$ if and only if $m \equiv n (\mod r)$, of finite rank?

Answer: No


Question: Find the value of \(a > 1\) at which the integral \(\int_{a}^{a^2} \frac{1}{x}\ln\Big(\frac{x-1}{32}\Big)dx\) reaches its minimum value.

Answer: 3


Question: Determine the number of abelian subgroups of order 15 in the symmetric group $S_8$. Provide your answer as a single integer.

Answer: 336


Question: Find the volume of the wedge cut from the first octant by the cylinder \(z = 12 - 3y^2\) and the plane \(x+y=2\).

Answer: 20


Question: Evaluate the line integral \( \int_c \frac{y^2}{2}dx + zdy + xdz \), where \( c \) is the curve of intersection of the plane \( x+z = 1 \) and the ellipsoid \( x^2+2y^2 + z^2 = 1 \).

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


Question: Evaluate the contour integral \( \oint_{|z - Ri| = R} \frac{1}{(z^2 + 1)^2} \, dz \) for the positively oriented circular contour \(|z - Ri| = R\), where \(R\) is a positive real number greater than \(\frac{1}{2}\).

Answer: \dfrac{\pi}{2}


Question: If \( a \) is sufficiently large compared with \( b \), and \( \sqrt{\frac{a}{a-b}} + \sqrt{\frac{a}{a+b}} = 2 + k\left(\frac{b}{a}\right)^2 \), find the value of \( k \).

Answer: \dfrac{3}{4}


Question: Let $f(x,y)$ be a differentiable function in $\mathbb{R}^2$ such that $f_x(x,y)y = f_y(x,y)x$ for all $(x,y) \in \mathbb{R}^2$. Find a differentiable function $g(r)$ on $[0,\infty)$ such that $g(\sqrt{x^2+y^2}) = f(x,y)$. Provide your answer as $g(r)$.

Answer: g(r)


Question: Let $A$ and $B$ be two complex matrices such that for all $s, t \in \mathbb{C}$, the matrix $sA + tB$ is nilpotent. Are $A$ and $B$ simultaneously triangularizable?

Answer: Yes


Question: Let $B_0\hookrightarrow B$ be a first order thickening, meaning it is a closed immersion with the ideal sheaf $I$ satisfying $I^2=0$. Let $X$ be a projective variety. Determine if every open neighborhood of $X\times B_0$ inside $X\times B$ is equal to $X\times B$. Consider all schemes over $\mathbb{C}$. Provide a justification for your answer.

Answer: Yes


Question:
Determine whether there exists an entire function $F$ such that:

1. $F(z) \neq 0$ for all $z \in \mathbb{C}$;
2. $1/F - 1 \in H^2(\mathbb{C}_+)$, where $H^2(\mathbb{C}_+)$ is the classical Hardy space in the upper half-plane;
3. $F$ is bounded in every horizontal half-plane $\{z: \text{Im}(z) > \delta\}$.

Provide a justification for your answer.

Answer: F(z) = 1


Question: Is the category Cat uniquely transportable as a concrete category over Set x Set via the functor U: Cat → Set x Set, where U maps a morphism A → B in Cat to (Ob(A) → Ob(B), Mor(A) → Mor(B))?

Answer: Yes


Question: Given the equations \(25(9x^2+y^2)+9z^2-15(5xy+yz+3zx)=0\) and \(x+y+z=18\), find all possible even integer values of \(y\) that are less than 15.

Answer: 6


Question: Let $f$ be a non-negative continuous function on the interval $[0,1]$. Suppose that the sequence $f^{1/n}$ converges uniformly. Determine the number of zeros of $f$. Provide your answer as a single integer.

Answer: 0


Question: Compute \( \lim_{x\to0} \ln^x(x) \).

Answer: 1


Question:
Consider a continuous distribution on \((0,1)\) with probability distribution function \(f\) and cumulative distribution function \(F\). Define the function
\[ g(r) = \frac{\int_r^1 x f(x) \, dx}{2 - F(r)} \]
where \(r \in (0,1)\). Let \(r_M\) be the value of \(r\) that maximizes \(g(r)\). Determine if there exists a positive constant \(c\), independent of the distribution, such that
\[ \int_{r_M}^1 x f(x) \, dx \geq c \int_0^1 x f(x) \, dx. \]

Answer: \dfrac{1}{2}


Question: Given that \(\lim_{n\to\infty}(a_na_{n+1})=1\), determine whether 0 can be a partial limit of the sequence \(a_n\).

Answer: Yes


Question: Consider the sequence of prime numbers. Define the first difference as the absolute difference between consecutive primes, the second difference as the absolute difference between consecutive first differences, and so on. Let \( a_{m,n} \) represent the \( n \)-th term in the \( m \)-th row of differences. Prove or disprove the conjecture that \( a_{m,1} = 1 \) for all \( m \geq 2 \).

Answer: 1


Question: Given the function \( g(x) = 1 - x^2 \) and \( f(g(x)) = \frac{x^4 + x^2}{1 + x^2} \), find the function \( f(x) \). Then, determine the value of \( f(1/2) \).

Answer: \dfrac{1}{2}


Question: Find the second degree Taylor polynomial of the function \( f(x,y) = e^{-x^2-y^2} \cos(xy) \) at the point \((x_0, y_0) = (0, 0)\).

Answer: 1 - x^2 - y^2


Question: In the binomial expansion of \((2\pi + 5)^n\), one of the terms is \(288000\pi^8\). Determine the value of \(n\).

Answer: 10


Question: Determine whether the following statement is true or false: If $f$ is continuous on $[a,b]$ and $\{ x_n\}$ is a sequence in $(a,b)$, then $\{f(x_n)\}$ has a convergent subsequence.

Answer: A


Question: 15 points $P_i$ are placed within the unit sphere. Find the maximum possible value of \(\sum_{1\leq i < j \leq 15} |P_i P_j|^2\) given that \(x_i^2+y_i^2+z_i^2 \leq 1\) for each point \(P_i = (x_i, y_i, z_i)\).

Answer: 225


Question:
Let $f, g \in C^2([0,1])$ such that $f'(0)g''(0) - f''(0)g'(0) \neq 0$ and $g'(x) \neq 0$ for all $x \in (0,1)$. Define $\theta(x)$ as a real number satisfying
\[ \frac{f(x) - f(0)}{g(x) - g(0)} = \frac{f'(\theta(x))}{g'(\theta(x))}. \]
Determine the value of \( \lim_{x \to 0^+} \frac{\theta(x)}{x} \).

Answer: \dfrac{1}{2}


Question:
Determine the number of integer values of $k$ from 1 to 200 for which the following limit holds:
\[ \lim\limits_{n \to \infty} \left( \cos^{2n} \left(\frac{k \pi}{3}\right) - \cos^{2n}\left(\frac{k \pi}{5}\right) \right) = 0. \]

Answer: 120


Question:
Evaluate the limit using the Stolz-Cesàro theorem:
\[ \lim_{n\to \infty} \frac{2021(1^{2020}+2^{2020}+3^{2020}+\ldots+n^{2020}) - n^{2021}}{2021(1^{2019}+2^{2019}+3^{2019}+\ldots+n^{2019})} = L \]

Answer: 1010


Question: Determine the minimum number of questions needed to find the positions of all numbers in a 64-cell grid, where each cell contains a unique number from 1 to 64. Each question allows you to find out the list of numbers situated in the cells you choose. What is the minimum number of questions required?

Answer: 6


Question: Determine whether the following statement is true or false: For a commutative and unitary ring $R$, is it true that $\bigcap_{P\in \mathrm{Spec}(R)} \mathrm{Id}(P)=\{0\}$, where $\mathrm{Id}(P)$ is the ideal of $R$ generated by idempotent elements of $P$?

Answer: True


Question: Given two square matrices $A$ and $B$ of the same order such that $AB = I$, is it true that $BA = I$? Justify your answer.

Answer: Yes


Question: Determine the number of abelian groups \( A \) of order \( p^4 \), where \( p \) is a prime number, such that there exists a subgroup \( N \) with \( A/N \cong \mathbb{Z}/p^3\mathbb{Z} \).

Answer: 2


Question: Evaluate the integral: \[ I = \int_{0}^{\frac{\pi}{2}} (\sqrt{\sin x} + \sqrt{\cos x})^{-4} \, dx \]

Answer: \dfrac{1}{3}


Question: Let $f,g: \mathbb{R}\rightarrow \mathbb{R}$ be two strictly convex functions, where $f$ is differentiable, $g$ is smooth, and $f \geq g$. Suppose that for some $x_0 \in \mathbb{R}$, the following conditions hold: $f(x_0) = g(x_0)$, $f'(x_0) = g'(x_0)$, and $g''(x_0) > 0$. Does there exist an interval $I$ containing $x_0$ such that $f$ is strongly convex on $I$?

Answer: Yes


Question: Let \(\alpha, \beta\) be the roots of the equation \(x^2 - x + p = 0\) and \(\gamma, \delta\) be the roots of the equation \(x^2 - 4x + q = 0\), where \(p\) and \(q\) are integers. If \(\alpha, \beta, \gamma, \delta\) form a geometric progression, find the value of \(p + q\).

Answer: -34


Question: Find the best approximation polynomial \( p_1(x) \in P_1 \) for \( f(x) = x^3 \) on the interval \([-1, 1]\) with respect to the infinity norm \( ||\cdot||_{\infty} \). Use Chebyshev polynomials in your solution.

Answer: \dfrac{3}{4}x


Question: Determine the necessary and sufficient condition on the integer \(m\) for the homomorphism \(\phi_m : C_n \rightarrow C_n\), defined by \(\phi_m(x^t) = x^{mt}\), to be an automorphism.

Answer: \gcd(m, n) = 1


Question:
Find a basis for the intersection of the subspaces \(L_1\) and \(L_2\) in \(\mathbb{R}^4\), where:
\[L_1 = \operatorname{span} \{ (1,2,0,3),(1,1,-1,2),(0,1,1,1) \}\]
\[L_2 = \operatorname{span} \{ (2,0,-2,1), (3,1,0,2),(4,2,2,3) \}\]


Question: Consider the real vector space \( V \) of polynomials of degree less than or equal to \( d \). For \( p \in V \), a norm is defined as \( \|p\|_k = \max \{|p(0)|,|p^{(1)}(0)|, \cdots, |p^{(k)}(0)|\} \), where \( p^{(i)}(0) \) is the \( i^{th} \) derivative of \( p \) evaluated at 0. Determine the condition under which \( \|p\|_k \) defines a norm. Choose the correct option: \( A. k \geq d-1 \), \( B. k < d \), \( C. k \geq d \), \( D. k < d-1 \).

Answer: C


Question:
Let $F_1$ and $F_2$ be the fractional parts of $(44-\sqrt{2017})^{2017}$ and $(44+\sqrt{2017})^{2017}$, respectively. Determine the range in which $F_1 + F_2$ lies.

(a) $0$ and $0.45$
(b) $0.45$ and $0.9$
(c) $0.9$ and $1.35$
(d) $1.35$ and $1.8$

Answer: C


Question: Suppose $f$ is a vector field and $g$ is an integral curve of $f$ whose domain includes $[0, \infty)$. If $\lim_{t \to \infty} g(t) = p$, is it necessarily true that $f(p) = 0$? Provide a justification for your answer.

Answer: Yes


Question: Given a sequence of independent real-valued random variables \((X_n)\) such that \(E[X_n] = 0\), \(E[X_n^2] = \frac{1}{n^2}\), and \(\sum_{n = 1}^\infty X_n < \infty\) almost surely, find \(E\left[ \sum_{n = 1}^\infty X_n \right]\).

Answer: 0


Question: Compute the limit: \[ \lim_{T \rightarrow \infty} \frac{1}{T} \int_{0}^{T} \sin(\alpha x)\cos(\beta x) \,dx \] given that \(|\alpha| \neq |\beta|\).

Answer: 0


Question: What is the name of the equivalence relation defined by $A \simeq B$ if $A = P^t BP$ for some invertible matrix $P$?

Answer: congruence


Question: Determine the least possible radius of a graph with 100 vertices and a maximum degree of 4.

Answer: 4


Question: Evaluate the integral \( \int_0^{2\pi} |a\cos(x)+b\sin(x)|dx \) given that \( a^2+b^2=1 \).

Answer: 4


Question: Evaluate the integral \( \int_1^2 \frac{1}{x^2} \, dx \) using a Riemann sum.

Answer: \dfrac{1}{2}


Question: Given real numbers \(x, y, z \neq 0\) satisfying the equations \(x^2 - xy + yz = y^2 - yz + zx = z^2 - zx + xy\), calculate the value of \(\frac{(x + y + z)^3}{xyz}\).

Answer: 27


Question: A fly moves through the point \((\sqrt{3}, 1, 2)\) at time \(t = 0\) along the curve of intersection of the surfaces \(z = x^2 - y^2\) and \(z^2 = x^2 + y^2\). The temperature in the room is given by \(T(x,y,z) = \frac{1}{2}(4x^2 + y^2 + z^2)\). If the fly's vertical velocity at this time is 5, what is the rate of change of temperature \(\frac{dT}{dt}\) experienced by the fly at \(t = 0\)?

Answer: \dfrac{155}{4}


Question: Evaluate the limit: \[ \lim_{x\to +\infty} x\left(\frac{\pi}{4} - \arctan\left(\frac{x}{x+1}\right)\right) \]

Answer: \dfrac{1}{2}


Question: In a 4x4 magic square where all rows, columns, and the two main diagonals must sum to the same given value, what is the minimum number of squares that need to be filled to ensure a unique solution for the entire square?

Answer: 7


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

Answer: 0


Question: Solve the equation \( \bar{z} - iz^2 = -\sqrt{3} - 3i \) for the complex number \( z \), without using the identity method. Provide the solution as a complex number.

Answer: -\sqrt{3}


Question: Given the interval \(\mathcal{I} = \{[a,b) \mid a,b \in \mathbb{R}, a < b\}\), define the length function \(\ell(I) = b-a\) for \(I \in \mathcal{I}\). Deduce that the length of an open interval \((a,b)\) is \(b-a\) using the properties of the length function.

Answer: b - a


Question: Is the power set of the positive integers uncountable when its members are expressed as sequences of zeros and ones using modulo two arithmetic?

Answer: Yes


Question: Let $X$ be a smooth projective variety of Picard number one, and let $f:X\dashrightarrow X$ be a birational automorphism which is not an automorphism. Must $f$ necessarily contract a divisor? Provide a justification for your answer.

Answer: Yes


Question: Find the sum of the infinite series: \[ \sum_{n=0}^\infty \frac{2n^7 + n^6 + n^5 + 2n^2}{n!} \]

Answer: 2013e


Question: Determine the range of values for $r$ such that the function $f$ is continuous on the interval $[0, \infty)$. Provide your answer as an interval or union of intervals.

Answer: (0, \infty)


Question: Let $V$ be a finite-dimensional vector space, and let $U_1, U_2, \ldots, U_n$ be subspaces of $V$. Consider the lattice $L$ generated by these subspaces, which is the smallest collection of subspaces containing the $U_i$ and closed under intersections and sums. Is the lattice $L$ finite?

Answer: Yes


Question:
Solve the recurrence relation for $S(n, 2)$ given by:
\[ S(n, 2) = 2S(n-1, 2) + S(n-1, 1) \]
Find $S(n, 2)$ in terms of $n$. The answer is $2^{n-1} - 1$. Show how to obtain this result.

Answer: 2^{n-1} - 1


Question: Let $\{A_n\}_{n=1}^\infty$ be a sequence of finite sets such that no two sets in the sequence are pairwise disjoint. Must there exist a finite set $F$ such that no two sets in the sequence $\{ A_n \cap F\}_{n=1}^\infty$ are pairwise disjoint?

Answer: Yes


Question: Evaluate the limit: $$\psi=\lim_{x\to0}{\frac{\sqrt{1-\cos x+\sqrt{1-\cos x+\sqrt{1-\cos x+\cdots}}}-1}{x^2}}$$

Answer: \dfrac{1}{2}


Question: Differentiate the function \( f(x) = \frac{1}{2}||A x - b||_2^2 \) with respect to \( x \) and simplify the expression. Show that the derivative is \( A^T A x - A^T b \).

Answer: A^T A x - A^T b


Question: Let \(\lambda_k\), \(k = 1, \dots, N\) be distinct real numbers. Consider the function \(g(x) = \sum_{k =1}^N \beta_k e^{i\lambda_k x}\), where \(\beta_k\) are complex numbers not all zero. Does there exist a constant \(c > 0\) and a sequence \(x_n \to \infty\) such that \(|g(x_n)| \ge c\) for all \(n\)?

Answer: Yes


Question: Compute the Lefschetz number \( L(f) \) for the map \( f : S^n \to S^n \) defined by \( f(x_0, x_1, \dots, x_n) = (x_0, -x_1, \dots, -x_n) \), where \( S^n \) is the unit sphere in \( \mathbb{R}^{n+1} \).

Answer: 2


Question: Find the exact value of $k$ such that the cubic equation $x^3 + kx + 4 = 0$ has exactly 2 distinct real roots.

Answer: -3\sqrt[3]{4}


Question: Determine the 2-dimensional Lebesgue measure of the set \( \Gamma^2 := \{ (\{at\},\{bt\}) \; : \; t \in \mathbb{R} \} \subset \mathbb{R}^2 \), where \( \{x\} \) denotes the fractional part of \( x \), and \( a, b \) are two irrational numbers such that \( a/b \) is irrational.

Answer: 0


Question:
Determine which of the following Taylor expansions is correct for a function $f: \mathbb{R}^n \to \mathbb{R}$, when expanding around $x$ for $y = x + \delta v$ with $\delta > 0$ small:

Expansion 1:
\[
f(y) = f(x) + \delta v^\top \nabla f(x) + \frac{\delta^2}{2}v^\top\nabla^2 f(x) v + \mathcal{O}(\delta^3)
\]

Expansion 2:
\[
f(y) = f(x) + \delta v^\top \nabla f(x) + \frac{\delta^2}{2}v^\top \nabla^2 f(x^-) v
\]
where $x^-$ lies on the line between $y$ and $x$. Which expansion is correct?

Answer: 1


Question:
Determine the number of unit preserving ring homomorphisms from the ring \( \mathbb{Z}[\sqrt{2}] \) to \( \mathbb{Z}_2 \times \mathbb{Z}_7 \times \mathbb{Z}_{17} \). Choose the correct answer from the following options:

(a) 0
(b) 2
(c) 4
(d) 8

Answer: C


Question: A point on a rifle bullet travels according to the position function \( r(t) = \langle 5t^2, \cos(t^2), \sin(t^2) \rangle \). Find the value of \( t \) at which the tangential component of acceleration is equal to the normal component.

Answer: \sqrt[4]{\dfrac{13}{2}}


Question: Find the degree of the field extension \( \mathbb{Q}(\zeta_{169},\sqrt[169]{34}) \) over \( \mathbb{Q} \).

Answer: 26364


Question: Suppose \( \lim\limits_{x \to \infty} \frac{f(x)}{x \cdot g(x) } =0 \) for every function \( g(x) \) such that there exists \( x_0 \) with \( g(x) > 1 \) for all \( x \ge x_0 \). Does this imply that \( f(x) = O(x) \)?

Answer: Yes


Question: Determine whether $P(\liminf X_n \geq 1) = 1$ and $P(\limsup X_n \leq 1) = 1$ implies $P(\lim X_n = 1) = 1$ for a sequence of random variables $(X_n)$. Provide a justification for your answer.

Answer: Yes


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

Answer: 0


Question:
Find the matrix derivative of the following minimization problem with respect to \( \Theta \):
\[ \min_{\Theta} \frac{C_1}{2} \sum_{j=1}^N ||\vec{\theta_j}||^2 \]
where \( \Theta = (\vec{\theta_1}, \vec{\theta_2}, ..., \vec{\theta_N}) \).

Answer: C_1 \Theta


Question: Let $\{a_n\}_{n\geqslant 1}$ be a sequence such that $a_n > 0$ for all $n$ and $\lim_{n\to\infty} a_n = 0$. Determine the set $\bigcup_{n=1}^\infty [a_n,1)$. Is it $(0,1)$ or $[0,1)$?

Answer: (0,1)


Question: Calculate the area between the curves defined by the equation \((x^2+y^2)^2 = x^2 - y^2\) for \(x \ge 0\) using double integrals.

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_0^{2\pi} \frac{\cos(t)}{4+\sin(t)} \, dt \) using complex analysis techniques.

Answer: 0


Question: Determine the center of the division ring \(D\), where \(D\) consists of all Laurent series \(\sum_{m=-k}^{\infty} f_iy^i\) in \(y\) over the field \(F\) of rational functions in \(x\) over \(\mathbb{C}\), satisfying \(yr(x)y^{-1}=r(2x)\) for any \(r(x) \in F\).

Answer: \mathbb{C}


Question: Is it true that for every IA automorphism \(\alpha \in \text{Aut}(F(X_n))\), the map \(x_i \mapsto \varphi(\alpha(x_i))\) for \(1 \le i \le n-1\) defines an element in \(\text{Aut}(F(X_{n-1}))\)?

Answer: Yes


Question: Solve the equation \(x^{\log_3 2} = \sqrt{x} + 1\) for \(x\).

Answer: 9


Question: Consider the first order language \(\mathcal{L}=\{\in,\in'\}\) with two binary relational symbols \(\in , \in'\) and \(ZFC\) as a \(\{\in\}\)-theory. Suppose \(\in'\) is defined using a \(\{\in\}\)-formula \(\varphi(x,y)\). If all \(\in'\)-elementary embeddings of the universe preserve \(\in\), is it sufficient to conclude that there exists a \(\{\in'\}\)-formula \(\psi(x,y)\) such that \(ZFC\cup\{\forall x\forall y~(x\in' y\leftrightarrow \varphi (x,y))\}\vdash \forall x\forall y~(x\in y\leftrightarrow \psi (x,y))\)? Answer yes or no.

Answer: Yes


Question: Consider all the natural numbers from $000000$ to $999999$. A number is called "lucky" if the sum of its first three digits equals the sum of its last three digits. A number is called "medium" if the sum of all its digits equals $27$. Find the difference between the number of lucky numbers and medium numbers.

Answer: 0


Question: Determine the missing digit $y$ in the 41-digit number $N = \overline{\underbrace{8\ldots8}_{20} \;y \; \underbrace{9\ldots9}_{20}}$ such that $N$ is divisible by 7.

Answer: 5


Question: Find the increasing sequence of positive integers \( \{a_n\} \) such that for every pair of positive integers \( i \) and \( j \), the number of divisors of \( i+j \) is equal to the number of divisors of \( a_i + a_j \).

Answer: a_n = n


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\ln(x+1) + 1 - e^x}{x^2} \] without using L'Hospital's rule or series expansion.

Answer: -1


Question: Calculate the value of the 25th derivative of the function \( f(x) = \cos(x^3) \) at \( x = 0 \). That is, find \( f^{(25)}(0) \).

Answer: 0


Question:
Let \( z_1 \) and \( z_2 \) be two complex numbers that satisfy the following conditions:
\[ \frac{z_2}{\bar{z_1}} = \frac{3}{8} \left(\cos(75^{\circ}) + i\sin(75^{\circ})\right) \]
\[ z_1 z_2^2 = \frac{1}{3} \left(\cos(120^{\circ}) + i\sin(120^{\circ})\right) \]
Determine which of the following can be a possible value for \( \sqrt{z_1} \):
(a) \( \frac{2}{\sqrt{3}} \text{cis}(135^{\circ}) \)
(b) \( \frac{2}{3} \text{cis}(155^{\circ}) \)
(c) \( \frac{2}{\sqrt{3}} \text{cis}(195^{\circ}) \)
(d) \( \frac{2}{\sqrt{3}} \text{cis}(175^{\circ}) \)
(e) \( \frac{2}{3} \text{cis}(215^{\circ}) \)

Answer: C


Question: Determine the cardinality of the set \( A \) of all continuous functions \( f : [0,1] \to [0,\infty) \) that satisfy the condition \( \int_{0}^{x} f(t) \, dt \geq f(x) \) for all \( x \in [0,1] \).

Answer: 1


Question: Find the maximum value of $x^2 + y^2$ given that $(x, y)$ satisfy the equations $2x^2 + 5xy + 3y^2 = 2$ and $6x^2 + 8xy + 4y^2 = 3$. Calculus methods are not allowed.

Answer: \dfrac{5}{2}


Question: Given that the Net Income at time $n$ is $N_n = N_0(1+g)^n$ and the Book Value of Equity at time $n$ is $B_n = B_0 + \sum_{t=1}^{n} N_0(1+g)^t$, determine the time $n$ at which the fraction \( \frac{N_n}{B_{n-1}} \) converges to the constant growth rate $g$. Assume $N_0$, $B_0$, and $g$ are given constants. Does this convergence occur as $n \to \infty$?

Answer: \infty


Question: Evaluate the limit: $$\lim_{x \to 0^+} \left((\sin(x))^{\frac{1}{x}} + \left(\frac{1}{x}\right)^{\sin(x)}\right).$$

Answer: 1


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{\ln(n^{15}+n+\ln n)}{\ln(\sqrt{n^{22}+1772}+666n^6+13)} \]

Answer: \dfrac{15}{11}


Question: Evaluate the limit: \[ L = \lim_{{n\to\infty}}\sqrt[n]{\int_0^1{(\arctan(1 + x^n))^n \, dx}} \]

Answer: \arctan 2


Question: Let $X \subset \mathbb{R}^n$ be a bounded and connected set, and let $f: X \to \mathbb{R}$ be a polynomial function. Consider a point $x \in X$ and the nearest global maximum $m \in X$ to $x$. Suppose that in every neighborhood of $x$, there exists a point $y$ and a path $p_t$ such that $p_0 = y$, $p_1 = m$, and for every $t < t'$, $f(p_t) < f(p_{t'})$ and $d(x, p_t) < d(x, p_{t'})$. Does this imply that $x$ is not a local maximum of $f$?

Answer: Yes


Question: Consider a family of topological spaces \(\{ X_{\alpha} \}\) and let \(X = \prod_{\alpha} X_\alpha\) be their product with the product topology. Is \(X\) uniquely determined up to homeomorphism by the following property: There exist continuous maps \(\pi_{\alpha} : X \to X_\alpha\) such that for any topological space \(Y\) and any family of continuous maps \(f_\alpha \in C(Y, X_\alpha)\), there exists a unique continuous map \(F \in C(Y,X)\) satisfying \(f_{\alpha} = \pi_{\alpha} \circ F\) for each \(\alpha\)? Provide a justification for your answer.

Answer: Yes


Question: Calculate the flux of the vector field \( \mathbf{F}=(x-y+xy,-2x+y,xz) \) through the flat triangle with vertices at \((1,0,0), (0,1,0), (0,0,1)\).

Answer: -\dfrac{1}{12}


Question:
Use the Neyman-Pearson lemma to find the most powerful statistical test for testing the hypothesis \( H_0 : \theta = \theta_1 \) versus \( H_1 : \theta = \theta_2 \) at a significance level of \( \alpha = 0.05 \). Given the following probability distributions for a discrete random variable \( X \) with possible values \( x = 1, 2, 3, 4, 5 \):

\[
\begin{array}{c|ccccc}
  x & 1 & 2 & 3 & 4 & 5 \\
  \hline
  p(x; \theta_1) & 0.05 & 0.1 & 0.6 & 0.02 & 0.23 \\
  p(x; \theta_2) & 0.45 & 0.01 & 0.19 & 0.05 & 0.3 \\
  p(x; \theta_3) & 0.15 & 0.4 & 0.05 & 0.3 & 0.1 \\
\end{array}
\]

Determine the rejection region for the test.

Answer: \{1\}


Question: Calculate the limit \( \lim\limits_{n \to \infty} \int_0^2 \frac{nx^{\frac{n}{2}}}{(1+x^n)} \, dx \) using the convergence theorem.

Answer: \pi


Question: Find the degree of the splitting field of the polynomial \( P(X) = X^4 + 2 \) over \( \mathbb{Q} \).

Answer: 8


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

Answer: \dfrac{\pi}{2}


Question: For what value of \(a\) are the graphs of \(y = a^x\) and \(y = \log_a x\) tangent to each other at exactly one point?

Answer: e^{1/e}


Question: Evaluate the integral \( I = \int_{-\infty}^\infty e^{-\frac{y^2}{2}} \, dy \).

Answer: \sqrt{2\pi}


Question: True or False: If \((f_n):[0, 1] \to [0, 1]\) is a sequence of continuous functions that converges to \(f\) pointwise, then \(f\) is Riemann integrable and \(\int_0^1{f_n} \to \int_0^1f.\)

Answer: B


Question: Determine the degree of the field extension \([\mathbb{Q}(i,\sqrt{-3}):\mathbb{Q}]\).

Answer: 4


Question: Find the last two digits of \(3^{2024}\).

Answer: 81


Question: Let $I_0 = [a,b]$ with $b > a$. Consider a measurable set $A \subset I_0$ such that for all distinct rational numbers $p$ and $q$, the sets $(\{p\} + A)$ and $(\{q\} + A)$ are disjoint. Determine the Lebesgue measure $\mu(A)$. \(\mu(A)\) should be zero if the sets are disjoint for all distinct $p$ and $q$. What is $\mu(A)$?

Answer: 0


Question: Given that \(\cos \alpha, \cos \beta, \cos \gamma\) are the roots of the polynomial equation \(9x^3 - 9x^2 - x + 1 = 0\), where \(\alpha, \beta, \gamma \in [0, \pi]\), find the value of \(\alpha + \beta + \gamma\).

Answer: \pi


Question: Calculate the area of the ellipse given by the equation \(x^2 - 2xy + 2y^2 + 4y = 8\) using the change of variables \(u = x - y\) and \(v = y + 2\).

Answer: 12\pi


Question: Let $X$ be a Hausdorff space such that the irrationals $\mathbb{P}$ form a dense subspace of $X$. Consider the Cantor set $C$, where the set of non-endpoints of $C$ is homeomorphic to $\mathbb{P}$. If $f: C \to X$ is a continuous surjection such that $f|_{\mathbb{P}}$ is the identity map, is $X$ necessarily disconnected?

Answer: Yes


Question:
Find all rational numbers \(\frac{m}{n}\) such that:
1. \(0 < \frac{m}{n} < 1\)
2. \(\text{H.C.F.}(m, n) = 1\)
3. \(m \cdot n = 30!\)

Answer: 512


Question: Solve the equation $x$&$2$ = $2$&$x$ for $x \in \mathbb{R}$, where the operator & is defined such that $x$&$n$ = $x^{x^{x^{...^{x}}}}$ ($n$ times).

Answer: 2


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty} \frac{(n+2)^{\log(n+2)}}{(n+1)^{\log(n+1)}} \]

Answer: 1


Question:
Consider a smooth, simple closed curve $C \subset \mathbb{R}^2$. A thief is inside $C$, and the police can place countably infinite officers on $C$. The thief and officers move simultaneously and continuously with a maximum speed of 1. Officers are restricted to move on $C$ and can pass through each other. The thief is caught if his coordinates coincide with an officer's. Determine which of the following is true:

1. The thief always has a plan to escape $C$.
2. The officers always have a plan to prevent the thief from escaping.
3. It depends on the shape of $C$.

Which option is correct?

Answer: 2


Question: Determine the fundamental period of the function \( f(x) = \sin x + \tan x + \tan\frac{x}{2} + \tan\frac{x}{4} + \tan\frac{x}{8} + \tan\frac{x}{16} \).

Answer: 16\pi


Question: Let \(H^*(M)\) be the cohomology algebra of the oriented manifold \(M\) with rational coefficients. Consider \(M = \mathbb{C}P^m\), where \(H^*(M) = \mathbb{Q}[x]/(x^{m+1})\) and \(|x| = 2\). Choose the basis \(b_i = x^i\) for \(i = 1, \ldots, m\). Compute the dual basis elements \(d_i\) such that \(b_i \smile d_j = \delta_{i,j}w\), where \(\smile\) is the cup product and \(w\) is the orientation class of \(M\).

Answer: d_j = x^{m - j}


Question:
Find the radius of curvature of the polar curve \(r^2 = a^2 \cos 2\theta\) at any point \((r, \theta)\) using the formula for the radius of curvature of polar curves:
\[ R = \frac{(r^2 + r_1^2)^{3/2}}{r^2 + 2r_1^2 - rr_2} \]

Answer: \dfrac{a^2}{3r}


Question: Determine whether the limit \( \lim_{n \to +\infty} f(n) \approx 4.018 \) exists, where \( f(n) = \frac{\sum A}{\sum_{b \in B} b^3 - b} b_n \), with \( A \) being the set of primes of the form \( 3j^2 + 2 \) and \( B \) being the set where \( 3b_i^2 + 2 \) is prime. For example, when \( n=3 \), \( f(3) = 4.5 \), and for larger \( n \), \( f(n) \approx 4.018 \).

Answer: 4.018


Question: Suppose that $f$ and $g$ are everywhere continuous but nowhere differentiable functions. Can the composition $f \circ g$ be differentiable?

Answer: Yes


Question: Find all natural number solutions \((m, n, p)\) to the equation \(4^m - 3^n = p^2\).

Answer: (1, 1, 1)


Question: Consider a sequence \( \{a_0, a_1, a_2, \ldots\} \) that has no limit point. Does this imply that \( |a_n| \to \infty \) as \( n \to \infty \)? Provide a justification for your answer.

Answer: Yes


Question:
Given two simple graphs on the same vertex set, one red and one blue, and a subset of vertices $V_k = \{v_1, \ldots, v_k\}$ such that for $i=1, \ldots, k$, $\deg(v_i) \ge i$ in both graphs, determine if it is always possible to find a family of vertex-disjoint paths such that:

- For $i=1, \ldots, k$, every $v_i$ is contained in a path.
- Each path consists of vertices only from $V_k$ except for exactly one of its endpoints, which must be outside of $V_k$.
- In each path, the red and blue edges are alternating.

Answer: Yes


Question: Find the maximum value of the function \( f(x,y) = x^2 - y^2 + 2xy - 2x - 2y + 1 \) subject to the constraint \( x^2 + y^2 \leq 2x \).

Answer: \sqrt{2}


Question: Find the complex number \(a\) such that the polynomial \(x^2-x+a\) divides \(x^{13}+x+90\).

Answer: 2


Question: If $\theta$ is a rational number, is $e^{i\pi\theta}$ an algebraic number?

Answer: Yes


Question: Let $A$ be a $k$-algebra and $l/k$ be a finite field extension of $k$. Suppose $A \otimes_k l$ is an integral domain. Does it follow that the map $A \rightarrow A \otimes_k l$ defined by $a \mapsto a \otimes 1$ is injective?

Answer: Yes


Question: Show that using the Poisson distribution, \( \lim_{n \to +\infty} e^{-n} \sum_{k=1}^{n}\frac{n^k}{k!} = \frac {1}{2} \).

Answer: \dfrac{1}{2}


Question: Find a maximal ideal in the ring \( F_7[x]/(x^2+5) \), where \( F_7 \) is the finite field with 7 elements.

Answer: (x - 3)


Question: Determine the number of distinct abelian groups of order 3240. Use the prime factorization of 3240, which is \(2^3 \cdot 3^4 \cdot 5\), and apply the structure theorem for finitely generated abelian groups to find the number of such groups.

Answer: 15


Question: Evaluate the double limit \( \lim_{x\to 1^+} \lim_{n\to\infty}\frac{\ln(n!)}{n^x} \). Does the order of limits impact the result?

Answer: 0


Question: Evaluate the limit using L'Hopital's Rule: \[ \lim_{x\to 1/2} \left(\frac{\tan(\pi x)}{2x-1}+\frac{2}{\pi(2x-1)^2}\right) \]

Answer: \dfrac{\pi}{6}


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

Answer: -\dfrac{1}{6}


Question: Let $p > 2$ and $c > 0$. Prove that the set \[ \left \{ x \in [0,1] : \left | x-\frac aq\right | \leq \frac c {q^p} \text{ for infinitely many $a,q \in \mathbb N^*$ } \right \} \] has measure zero, using the Borel-Cantelli Lemma.

Answer: 0


Question: Determine the dimension of the subspace $W$ of $\mathcal{L}(\mathbb{R}[x])$, where $W$ is generated by linear operators on $\mathbb{R}[x]$ that have no adjoint. The vector space $\mathbb{R}[x]$ consists of all polynomials with real coefficients, and the inner product is defined by $\langle f,g\rangle=\int_{0}^{1}f(t)g(t)dt$. What is the dimension of $W$?

Answer: \infty


Question: Let \( X_1, X_2, \ldots, X_n \) be a random sample from a normal distribution with mean 0 and variance \( \sigma^2 \). Define the random variable \( T = \frac{\bar{X}\sqrt{n}}{S} \), where \( \bar{X} \) is the sample mean and \( S^2 \) is the sample variance. Determine the distribution of \( T^2 = \left(\frac{\bar{X}\sqrt{n}}{S}\right)^2 \).

Answer: F(1, n-1)


Question: Let $I = [0,1] \times [0,1]$ and $E \subset \mathbb{R}^2$ be a set of zero Lebesgue measure. Determine whether the following statement is true: $$\overline{I \setminus E} = I.$$

Answer: True


Question: Evaluate the integral \( \int_k \frac{\cos(z)}{z^4}dz \), where \( k \) is the rectangle with corners \(-2-2i, 2-2i, 2+i, -2+i\).

Answer: 0


Question: A random vector \((X,Y)\) has a continuous distribution with a density function \[f(x,y)=\begin{cases}c\cdot x & \text{when }0 \leq x \leq 2, \max\{0,1-x\} \leq y \leq 2-x\\ 0& \text{otherwise}\end{cases}\] where \(c > 0\) is a constant. Find the variance of \(Y\) conditioned on \(X = 1.5\), denoted as \(\text{Var}(Y | X = 1.5)\).

Answer: \dfrac{1}{48}


Question: Determine for which values of $n$ the ring $\mathbb{C}[x_1,\dots,x_n]/(\sum_{i=1}^n x_i^2)$ contains no nilpotent elements.

Answer: n \geq 2


Question: Evaluate the integral \( \int_{\gamma} \frac{1}{z} \, dz \) where \( \gamma \) is a circle centered at \( z_0 \neq 0 \) with radius \( r < |z_0| \).

Answer: 0


Question: Evaluate the integral: $$\int_1^3 \sqrt{x^4+1} \, dx$$

Answer: 8.980


Question: Given a function \( f(x, y) \) with continuous partial derivatives everywhere, and \( \nabla f(0, -18) = -2i + 3j \), define a new function \( g(x, y) = f(xy + x^2, xy - y^2) \). Calculate \( \nabla g(3, -3) \).

Answer: -15\mathbf{i} + 21\mathbf{j}


Question: Consider two sets of real numbers, \(A = \{a_1, a_2, \ldots, a_{100}\}\) and \(B = \{b_1, b_2, \ldots, b_{50}\}\). Define a mapping \(f: A \to B\) such that every element in \(B\) has a pre-image and \(f(a_1) \leq f(a_2) \leq \ldots \leq f(a_{100})\). Determine the number of such mappings.

Answer: \dbinom{99}{49}


Question: If \(M\) is a linear operator on \(\mathbb{R}^3\) with unique and real eigenvalues \(\lambda_1 < \lambda_2 < \lambda_3\), and there exists a vector \(x \in \mathbb{R}^3 \setminus \{0\}\) such that \(\lim_{n \to \infty} ||M^n x|| = 0\), what are the possible values of \(\lambda_1\)?

Answer: (-\infty, 1)


Question:
Consider a matrix $E_0$ with non-negative entries. Define a sequence of matrices $(E_n)$ where $E_{n+1}$ is obtained from $E_n$ by performing the following operations:

1. Divide each entry by the sum of all entries in its column to make the matrix column-stochastic.
2. Divide each entry by the sum of all entries in its row to make the matrix row-stochastic.

Given that $E_0$ is a $c \times 5$ matrix, where $c \in \{1, 2, \ldots, 5\}$, determine the limit of $E_n$ as $n \to \infty$. Assume that the entries of $E_0$ are such that the process is well-defined and converges.

Answer: \frac{1}{5}


Question: Determine whether the expression \( \sin\left(\frac{tx_2}{2}\right)\cos(f(t)-tx_2)-\sin\left(\frac{tx_1}{2}\right)\cos(f(t)-tx_1) \) can be bounded by \( (tx_2-tx_1) \) given that \( x_2 > x_1 \ge 0 \) and \( t \ge 0 \).

Answer: Yes


Question: Determine whether the set of discontinuities of the Dirichlet function is countable or uncountable.

Answer: uncountable


Question: Determine whether the following statement is true or false: If \( \lim_{\{f(x),g(x)\}\to\infty}\frac{f(x)}{g(x)}=1 \), then \( \lim_{\{f(x),g(x)\}\to\infty}f(x)-g(x)=0 \).

Answer: B


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{n^{\ln(n)}}{2^n} \]

Answer: 0


Question: If \( \frac{3-\tan^2\left(\frac{\pi}{7}\right)}{1-\tan^2\left(\frac{\pi}{7}\right)} = k \cos\left(\frac{\pi}{7}\right) \), find the value of \( k \).

Answer: 4


Question: Find the real number \( p \) such that the substitution \( z = x^p \) reduces the differential equation \( x^2y'' + xy' + (4x^4 - 1)y = 0 \) to a Bessel equation for some function in \( z \).

Answer: 2


Question: Evaluate the integral \( I = \int_{-1}^0 \int_{0}^1 \delta(x-y) \, dx \, dy \), where \( \delta(t) \) is the Dirac delta function defined as the limit of a symmetric Gaussian probability density function.

Answer: 0


Question: Evaluate the integral: \[ \int_0^{2 \pi} \sin^4 \theta \, \mathrm{d} \theta \]

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


Question: Determine whether the function $g(x) = (\mathbb{P}(Z>f(x)))^2$, where $Z$ is a standard normal random variable, is maximized at the point $x_0 = \operatorname{argmin} f(x)$. Is this statement true?

Answer: True


Question: A variable $X$ is uniformly distributed over the interval (1,5) with probability 0.9 and over the interval (2,44) with probability 0.1. Calculate the variance of $X$. \( \text{Provide your answer in the form of a single number.} \)

Answer: 51.9


Question: Find the radius of convergence of the series \( S = \sum_{n=1}^{\infty}\frac{n+7\sqrt{n+1}}{(n+1)2^n}z^{2n-1} \).

Answer: \sqrt{2}


Question: Calculate the expected value of the expression \(x_{(1)} + x_{(4)} - x_{(2)} - x_{(3)}\), where \(x_{(1)}, x_{(2)}, x_{(3)}, x_{(4)}\) are the order statistics of four i.i.d. random variables representing dice rolls.

Answer: 0


Question: Determine if there exists a solution for $f(a)$ in terms of the known function $g(x)$ in the integral equation: $$f(a)=\int_0^a g(x)f(x)\,dx.$$

Answer: f(a) = 0


Question: Suppose \( \lim_{x\to\infty}f(x)g(x)=0 \) and \( \lim_{x\to\infty}g(x)\ne0 \). Must it be true that \( \lim_{x\to\infty}f(x)=0 \)?

Answer: Yes


Question:
Given complex numbers \(a, b, c\) such that:
\[ a+b+c \in \mathbb{Z}, \]
\[ a^2+b^2+c^2 = -3, \]
\[ a^3+b^3+c^3 = -46, \]
\[ a^4+b^4+c^4 = -123, \]
find \((a^{10}+b^{10}+c^{10}) \pmod{1000}.\)

Answer: 162


Question: Evaluate the integral \( \int_C \frac{e^{i\pi z}}{2z^2-5z+2} \, dz \), where \( C \) is the curve \( \cos t + i \sin t \) for \( t \in [0, 2\pi] \). Find the greatest integer less than or equal to the value of this integral.

Answer: 2


Question: Find the maximum possible length of the smallest side of a triangle with vertices in or on the unit cube. What is this length?

Answer: \sqrt{2}


Question: Is it possible to isometrically embed the unit sphere $S^m$ into the unit sphere $S^n$ such that the tangent space at a point $p \in S^m$ matches a given $m$-dimensional subspace $V \subset T_pS^n$, where $m < n$?

Answer: Yes


Question: Evaluate the expression: \[ \frac{(2020^2 - 20100)(20100^2 - 100^2)(2000^2 + 20100)}{10(2010^6 - 10^6)} \]

Answer: 10


Question: Calculate the line integral \( \oint_C \vec{F} \cdot \mathrm{d} \vec{r} \) where \( \vec{F} = \langle y^2, x \rangle \) and \( C: x^2 + 4y^2 = 4 \). Use Green's theorem to evaluate the integral.

Answer: 2\pi


Question: Is every Hilbert space isomorphic to an $L^2$ space? Specifically, for any Hilbert space $H$, does there exist a measure space $(X, \mathcal{M}, \mu)$ such that $H \cong L^2(\mu)$?

Answer: Yes


Question: Determine the values of \(a\) for which the matrix \(\begin{bmatrix} 1 & a & 1 \\ a & 0 & 0 \\ 1 & 0 & 1 \end{bmatrix}\) is positive semidefinite or indefinite using the principal minor methodology.

Answer: 0


Question: Given the function \(F(x)\) defined by the identity \(F((F(x)+x)^k) = (F(x)+x)^2 - x\), determine the maximum value of \(k\) such that \(F'(1)\) is an integer. Express this maximum value of \(k\) as \(\frac{a}{b}\), where \(a\) and \(b\) are coprime integers. What is the value of \(a+b\)?

Answer: 5


Question: For a Noetherian scheme $X$ and an ample line bundle $\mathcal{L}$ on $X$, does there exist an integer $n \geq 1$ such that the canonical map $X \to \mathbb{P}(\Gamma(\mathcal{L}^{\otimes n}))$ is a closed immersion?

Answer: Yes


Question: Compute \( \sum_{k=0}^n \binom{2n}{k} \binom{n}{k} \) using generating functions.

Answer: \dbinom{3n}{n}


Question: Is the derivative of a continuous function Riemann integrable on a closed interval [a, b]? Specifically, if f' is the derivative of a continuous function f defined on [a, b], is f' Riemann integrable on [a, b]?

Answer: No


Question: Evaluate the integral \( \int_{\frac{1}{2}}^{2} \frac{1}{x} \tan\left(x - \frac{1}{x}\right) \, dx \).

Answer: 0


Question: Find all values of \( c \in \mathbb{R} \) such that for the map \( F: \mathbb{R} \to \mathbb{R}^2 \) defined by the differential equation \( D[F] = XF \), where \( X = \begin{bmatrix} -1 & 1 \\ c & -1 \end{bmatrix} \), the limit \( \lim_{t \to \infty} F(t) = 0 \).

Answer: (-\infty, 1)


Question:
Let $A$ be the set of all continuous functions $f:[0,1] \to \mathbb{R}^+ \cup \{0\}$ that satisfy the condition:
\[ \int_{0}^{x} f(t) \, dt \ge f(x), \forall x \in [0,1]. \]
Which of the following statements is true about the set $A$?
(A) $A$ has cardinality 1.
(B) $A$ has cardinality 2.
(C) $A$ is infinite.
(D) $A = \emptyset$.

Answer: A


Question: Find the integer value of \(x\) such that \(7^{95} \equiv x^3 \pmod{10}\).

Answer: 7


Question:
Determine whether the function \( f(x) \) defined on \([0,1]\) is integrable, where:

- For every irrational number \( x \) in \([0,1]\), \( f(x) = 1 \).
- For every rational number \( x \) in \([0,1]\), \( f(x) = \frac{a-2}{a} \), where \( a \) is the smallest natural number for which \( ax \) is an integer.

If the integral \( \int_0^1 f(x) \, dx \) exists, find its value.

Answer: 1


Question:
Assume a real sequence \(1=a_1\leq a_2\le \cdots \leq a_n\), and \(a_{i+1}-a_i\leq \sqrt{a_i}\). Prove or disprove that the following holds:
\[\sum_{i=1}^{n-1} \frac{a_{i+1}-a_i}{a_i} \in O(\log n)\]

Answer: O(\log n)


Question: Evaluate the limit: \[ \lim_{x \to 0} \left(\frac{2f(x)-x}{x^2}\right) \] where \( f(x) = \lim_{n \to \infty} \frac{x}{n} \left(\frac{1}{1+e^{-x/n}}+\frac{1}{1+e^{-2x/n}}+\dots+\frac{1}{1+e^{-x}} \right) \).

Answer: \dfrac{1}{4}


Question: Compute the coefficients $a_n$ for $n<0$ in the Laurent expansion of $\cot(\pi z)$ on the annulus $1<|z|<2$, where $\cot(\pi z)=\sum_{n=-\infty}^{\infty} a_n z^n$. Specifically, find $a_{-1}$.

Answer: \dfrac{3}{\pi}


Question: Determine whether there exists a real-valued positive function \( f : \mathbb{R} \rightarrow \mathbb{R} \) such that \( f(x) > 0 \) for all \( x \) and \( \int_{-\infty}^\infty f(x) \, dx = \int_{-\infty}^\infty f(x)^2 \, dx = 1 \).

Answer: Yes


Question: Can there be a division ring containing an algebraically closed field $K$ such that the dimension of the division ring over $K$ is finite?

Answer: No


Question: Determine all positive values of \( p \) such that the function \( f(x) = \left| \frac{1}{\ln(x)} \right| \) is in the space \( L^p(0,1) \).

Answer: (0, 1)


Question: Calculate the limit: $$ \lim_{x\to 0}\frac{2-x}{x^3}e^{(x-1)/x^2} $$

Answer: 0


Question: Evaluate the improper integral: \[ \int_{-\infty}^{+\infty} \frac{x^2}{(x^2-8x+20)^2} \, dx \]

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


Question: Let \(\delta = \frac{d}{dx} + q(x)\) be a differential operator defined on a dense subspace of \(L^2((0,1),d\mu)\), where \(\mu\) is a finite measure, and \(q: \mathbb{R} \rightarrow \mathbb{R}\). Denote by \(\delta^*\) the formal adjoint of \(\delta\) with respect to \(L^2((0,1),d\mu)\). Assume \(f \in L^2((0,1),d\mu)\) is an eigenfunction of the operator \(\delta^* \delta\) with eigenvalue \(\lambda\), i.e., \(\delta^* \delta f = \lambda f\). Prove or disprove that \(\lambda \ge 0\).

Answer: \lambda \ge 0


Question: Find the smallest integer $n$ such that $n$ is divisible by a perfect square, $n+1$ is divisible by a perfect cube, and $n+2$ is divisible by a perfect fourth power.

Answer: 350


Question: Let \( n = p_1^{k_1} + p_2^{k_2} + \ldots + p_m^{k_m} \), where \( p_1, p_2, \ldots, p_m \) are distinct prime numbers and \( k_1, k_2, \ldots, k_m \) are positive integers. Determine the number of ways \( n \) can be expressed as a product of two positive integers that have no common factors, considering that the order of the factors matters (i.e., \( a \cdot b \) and \( b \cdot a \) are regarded as different).

Answer: 2^m