Question: Let $f:X \rightarrow Y$ be a finite covering map between compact oriented surfaces, and let $K$ be the kernel of the induced map $f_*: H_1(X) \rightarrow H_1(Y)$, where homology has rational coefficients. Is $K$ nondegenerate with respect to the symplectic algebraic intersection pairing $\omega$ on $H_1(X)$? In other words, for all nonzero $k \in K$, does there exist some $k' \in K$ such that $\omega(k,k') \neq 0$?

Answer: Yes


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

Answer: \dfrac{1}{2}


Question: Let \(X_1, X_2, X_3\) be three independent random variables, each uniformly distributed over the interval \([0, 1]\). Determine the greatest value of the probability \(P(X_1 + X_2 \leq X_3)\).

Answer: \dfrac{1}{6}


Question: Determine the smallest positive integer \( m \) for which \( \underbrace{100^{100^{\ldots^{100}}}}_m > \underbrace{3^{3^{\ldots^3}}}_{100} \).

Answer: 99


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

Answer: \infty


Question: Let $R$ be a commutative Noetherian ring with unity, and let $I$ be an ideal of $R$ such that $R/I \cong R$. Is it true that $I = (0)$?

Answer: (0)


Question: Consider n balls, each labeled with a number from 1 to 365, placed in a row. A ball is selected at random. Let P be the probability that the number on the selected ball is the same as the number on any one of the balls placed to its left, with all balls to the left having distinct numbers. Determine the position K, counting from left to right, that maximizes P.

Answer: 20


Question: Given a function \( f(t) = \sum_{k=1}^N a_k e^{\mathrm{i}\phi_k t} \) with the condition \( 0 \leq f(t) \leq C \) for all \( t \in \mathbb{R} \), where \( C > 0 \) is a constant, \( \phi_k \) are real numbers, and \( a_k \) are complex numbers, find a bound for \( \sum_{k=1}^N |a_k| \) in terms of \( C \) and \( N \). Assume that \( f \) is a real function.

Answer: C


Question: Find the value of the infinite product series: \(2^{\frac{1}{4}} \cdot 4^{\frac{1}{8}} \cdot 8^{\frac{1}{16}} \cdot \ldots \).

Answer: 2


Question: Evaluate the limit: \[ \lim\limits_{x\to0}\frac{1}{x}\int_{0}^{x}[1-\tan{2t}]^{1/t}dt = e^{-2}. \]

Answer: e^{-2}


Question:
Evaluate the determinant:
\[\begin{vmatrix} \cos 2019^{\circ} & \cos 1131^{\circ}  & \cos 1869^{\circ}  & \cos  1101^{\circ}\\ -\cos 1131^{\circ} & \cos 2019^{\circ}  & -\cos 1101^{\circ}  & \cos  1869^{\circ}\\  -\cos 1869^{\circ} & \cos 1101^{\circ}  & \cos 2019^{\circ}  & -\cos  1131^{\circ} \\  -\cos 1101^{\circ} & -\cos 1869^{\circ}  & \cos 1131^{\circ}  & \cos  2019^{\circ}\end{vmatrix}.\]

Answer: 4


Question: Provide an epsilon-delta proof to show that \( \lim\limits_{x\to 0} e^{-1/x^2}=0 \).

Answer: 0


Question: What is the remainder when \(2^{1990}\) is divided by \(1990\)?

Answer: 1024


Question: Use Green's Theorem to compute the area enclosed by the parametric curve given by \( x = 2\cos(t) \) and \( y = \sin(2t) \) for \( 0 \le t \le 2\pi \).

Answer: \dfrac{16}{3}


Question: Calculate the residue of the function \( \frac{(z^6+1)^2}{(z^5)(z^2-2)(z^2-\frac{1}{2})} \) at \( z = 0 \).

Answer: \dfrac{21}{4}


Question: Determine the number of natural numbers 'a' such that there exists a natural number 'b' for which the equation \((a,b) + [a,b] = ab\) holds, where \((a,b)\) denotes the greatest common divisor and \([a,b]\) denotes the least common multiple of 'a' and 'b'.

Answer: 1


Question: Let $X_i$ be copies of the standard real Gaussian random variable $X$. Let $b$ be the expectation of $\log |X|$. Assume that the correlations $EX_iX_j$ are bounded by $\delta_{|i-j|}$ in absolute value, where $\delta_k$ is a fast decreasing sequence. Is there a constant $C$, depending only on the sequence $\delta_k$, such that for all $t_i > 0$, the inequality \[ E\prod_i|X_i|^{t_i} \le \exp\left( b\sum_i t_i + C\sum_i t_i^2 \right) \] holds true?

Answer: Yes


Question: Let $A$ and $B$ be subsets of $\mathbb{R}$. Determine whether it is true that $m(A+B) = m(A) + m(B)$, given that the sum $A+B$ is measurable.

Answer: False


Question: Find the sum of the infinite series: \[ \sum_{i=1}^{\infty} i \cdot 3 \left(\frac{1}{3}\right)^{i+1} \]

Answer: \dfrac{3}{4}


Question: A die is initially placed with the number 6 on top. It is then turned randomly to one of the four adjacent faces. Let $P_n$ be the probability that after $n$ such turns, the die will again show the number 6 on top. Determine the limit $\lim_{n \to \infty} P_n$. \( \text{Express your answer as a single value.} \)

Answer: \dfrac{1}{6}


Question: Evaluate the integral \( \int_{-\infty}^{\infty} A x^2 e^{-x^2/(2\sigma^2)} \, dx \), given that \( \int_{-\infty}^{\infty} A e^{-x^2/(2\sigma^2)} \, dx = 1 \). Here, \( A \) is a constant, and the expression inside the second integral represents the Gaussian or Normal distribution.

Answer: \sigma^2


Question: For a non-degenerate bilinear form on a vector space $V$, is the right orthogonal complement of the left orthogonal complement of a subspace $W$ equal to $W$?

Answer: Yes


Question: In triangle $ABC$, points $E$ and $F$ divide sides $AC$ and $AB$ respectively such that $\frac{AE}{EC} = 4$ and $\frac{AF}{FB} = 1$. Point $D$ is on side $BC$, and $G$ is the intersection of $EF$ and $AD$. If $\frac{AG}{GD} = \frac{3}{2}$, find the ratio $\frac{BD}{DC}$.

Answer: \dfrac{4}{5}


Question: Find \( \lim_{x\to+\infty} (3^x-x)^{1/(4x)} \).

Answer: \sqrt[4]{3}


Question: Given that $g(x)$ is a polynomial function satisfying the equation $g(x)g(y) = g(x) + g(y) + g(xy) - 2$ for all real numbers $x$ and $y$, and that $g(2) = 5$, find the value of $g(3)$.

Answer: 10


Question: Evaluate the triple integral \( \iiint\limits_{V}(x-2)\,dV \) where the region \( V \) is defined by the inequality \( \frac{(x-2)^2}{9} + \frac{(y+3)^2}{25} + \frac{(z+1)^2}{16} < 1 \).

Answer: 0


Question: Does the forgetful functor \(\mathbf{Comp} \rightarrow \mathbf{Top}\) from compact spaces to topological spaces have a left-adjoint?

Answer: No


Question: Is it possible to partition the real numbers \( \mathbb{R} \) into two disjoint subsets \( X \) and \( Y \) such that both are dense in \( \mathbb{R} \) and both are locally uncountable? A set \( X \subset \mathbb{R} \) is locally uncountable if for any nonempty open subset \( U \) of \( \mathbb{R} \), the intersection \( U \cap X \) has cardinality strictly larger than the natural numbers.

Answer: Yes


Question: Determine whether the following statement is true or false: If a function has only one point of discontinuity, then it is Riemann integrable and its indefinite integral is a differentiable function. Provide a justification for your answer.

Answer: B


Question:
Is the following version of the fundamental lemma of the calculus of variations valid? Let $U$ be an open subset of $\mathbb{R}^n$ with smooth boundary $\partial U$. Consider a function $f$ in $L^2(U)$. Suppose that for every $h$ in the Sobolev space $H^2_0(U)$ it holds that
$$\int_U f \Delta h=0.$$ Where $\Delta$ is the Laplacian operator. Can we conclude that $f=0$ almost everywhere on $U$?

Answer: No


Question: Consider a process where you roll an n-sided die, and the result is m. You then roll an m-sided die, and continue this process until you roll a 1. What is the expected number of rolls needed to reach a result of 1, starting with an n-sided die?

Answer: 1 + \sum_{k=1}^{n-1} \frac{1}{k}


Question: Find the value of \(a \in \mathbb{R}\) such that the image of the function \(f(x) = \frac{x^2 + ax + 1}{x^2 + x + 1}\) is contained within the interval \([0, 2]\).

Answer: [0, 2]


Question: Let \( f(n) \) be a function defined for \( n \ge 2 \) and \( n \in \mathbb{N} \) which follows the recurrence relation for \( n \ge 3 \): \[ f(n) = f(n-1) + \frac{4 \cdot (-1)^{(n-1)} \cdot \left( \sum_{d \vert (n-1)} \chi(d) \right)}{n-1} \] where \( d \vert (n-1) \) means \( d \) divides \( n-1 \). Assume that \( f(2) = -4 \). The function \( \chi(d) \) is defined as: \[ \chi(d) = \begin{cases} 1, & \text{if } d = 4k+1 \text{ where } k \text{ is a whole number} \\ -1, & \text{if } d = 4k+3 \text{ where } k \text{ is a whole number} \\ 0, & \text{if } d \text{ is an even natural number} \end{cases} \] Find \( \lim_{n \to \infty} f(n) \).

Answer: -\pi \ln 2


Question: Find the largest positive integer \( n \) such that for all positive real numbers \( a, b, \) and \( c \), the inequality \[ n \cdot \left(\frac{abc}{a+b+c}\right) \leq (a+b)^2 + (a+b+4c)^2 \] holds.

Answer: 100


Question: Consider a circle of arbitrary radius, surrounded by 6 circles of equal radius, all touching the central circle and each other. If the central circle is replaced by 6 smaller circles of equal radius that fit inside it, by what constant factor do the radii of these smaller circles change relative to the original surrounding circles?

Answer: \dfrac{1}{3}


Question: Suppose G and H are finite groups of the same order such that the function \( F_G = F_H \), where \( F_G(d_1,d_2) \) is defined as the number of elements in G that have order equal to \( d_1 \) and can be expressed in the form \( x^{d_2} \) for some \( x \in G \). Does there necessarily exist a 1-isomorphism between G and H?

Answer: Yes


Question: Determine whether the following statement is true: For each $T > 0$, let $B^T$ be a Brownian bridge on $[0, T]$, conditioned to start and end at $0$. Then, $\mathbb{E}[|\exp(\sup_{0 \leq t \leq T} B^T_t) - 1|] \to 0$ as $T \to 0^+$. Provide a justification for your answer.

Answer: True


Question: In \(\mathbb{R}^4\) with the Euclidean inner product, consider the hyperplane \(H\) given by the equation \(3x - 5y + 2z + w = 3\). Within \(H\), there is a 2-plane \(P\) defined by the equations \(4x - 4y - z = 1\) and \(x - 3z + y - w = -2\). Also within \(H\), there is a line \(L\) described by the parametric equations \(x = t+2\), \(y = t+1\), \(z = -t-2\), and \(w = 4t+6\). Determine the distance from the point \(Q = (-1, -1, 3, -12)\) to the intersection point \(S\) of the 2-plane \(P\) and the line \(L\).

Answer: \sqrt{17}


Question: Determine whether the zero vector is contained in the convex hull of the set \( \{(\underbrace{\alpha_1, \cdots, \alpha_1}_{\lambda_1}, \cdots , \underbrace{\alpha_d , \cdots, \alpha_d}_{\lambda_d}) \in \mathbb{Z}^k: \, (\alpha_1, \cdots , \alpha_d) \in \mathbb{Z}^d,\, \sum_{i=1}^d \alpha_i = 0\} \), where \((\lambda_1 , \cdots , \lambda_d) \vdash k\) is a partition of \(k\) of length \(d\) and not all \(\alpha_i\)s are zero.

Answer: Yes


Question: Given the equations \((a+b)(c+d) = 143\), \((a+c)(b+d) = 150\), and \((a+d)(b+c) = 169\), where \(a, b, c, d\) are real numbers, find the smallest possible value of \(a^2 + b^2 + c^2 + d^2\).

Answer: 214


Question: For which values of integers \(r\) and \(k\) does the polynomial \(x^{p^k} - x\) divide \(x^{p^r} - x\) in \(\mathbb{Z}[x]\), where \(p\) is a prime number?

Answer: k \mid r


Question: Determine whether the following statement is true or false: If every linear operator on a normed space $X$ is bounded, then $X$ is finite dimensional.

Answer: A


Question: Evaluate \( \lim_{x \to 0}(f(x) + g(x)) \) given that \( \lim_{x\to 0}f(x)=\infty \) and \( g(x)=\sin\left(\frac{1}{x}\right) \).

Answer: \infty


Question: Evaluate the limit \( \lim_{x \to 1} x^{\frac{1}{1-x}} \) using L'Hospital's rule and logarithms.

Answer: \dfrac{1}{e}


Question: Evaluate the integral \( \int_{|z|=3} \frac{e^{\frac{1}{1-z}}}{z} \, dz \) using the residue theorem, considering the residue at infinity.

Answer: 2\pi i


Question:
Let \( a, b \in \mathbb{R} \) with \( a < b \). Show that the following limit holds:
\[ \lim_{n \to \infty} \int_a^b \frac{((b-a)e^{2x})}{(b-a)n^2e^{2x}+b^2} \, dx = 0. \]

Answer: 0


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

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


Question: Evaluate the limit: \( \lim_{x \rightarrow 0} \frac{e^x+e^{-x}-2}{x^2+2x} \) without using L'Hopital's rule.

Answer: 0


Question: Does there exist a measurable function \( f: X \to Y \) such that \( f^{-1}(B) \) is not a measurable set for some subset \( B \subset Y \)?

Answer: Yes


Question: A cone-shaped drinking cup is made from a circular piece of paper with radius \( r \) by cutting out a sector and joining the edges. If the cut is made such that the volume of the resulting cone is maximized, what is the ratio between the radius and height of the cone?

Answer: \sqrt{2}


Question: Evaluate the limit: $$\lim_{x \to 0^+} \frac{x-\sqrt{x^2+1}\cdot \ln \big(\sqrt{x^2+1} + x \big)}{2x^2\sqrt{x^2+1}\cdot \ln\big(\sqrt{x^2+1} + x \big)}$$

Answer: -\dfrac{1}{6}


Question: Evaluate the volume integral \( \int\int\int_{V}(x-y)\,dV \) where \( V \) is the volume enclosed by the surface \( S = \{(x,y,z) : (x^{2}+y^{2})^{2}+z^{4}=16; z\geq0\} \).

Answer: 0


Question: Evaluate the integral \( \int_C \frac{f'(z)}{f(z)} \, dz \) where \( f(z) = z^5 - 3iz^2 + 2z - 1 + i \) and the contour \( C \) encloses all the zeros of \( f(z) \).

Answer: 10\pi i


Question:
Determine whether the statement $T(n) \in \Theta(n^2)$ is true given that:
\[ \frac{n^2}{\log{(n)}} \leq T(n) \leq \frac{n^2}{\log{(n)}} + n \]

Answer: False


Question: Determine whether there exists a subfield $F \subseteq K$ such that $[K:F]=2$, where $F \simeq K \simeq \mathbb{Q}(x)$. Provide a justification for your answer.

Answer: Yes


Question: Let $f:[c,d] \rightarrow \mathbb{R}$ be a continuous function that is piecewise analytic. There exists $N \in \mathbb{N}$ and an increasing sequence $(i_{n})_{1 \le n \le N}$, with $i_{1}=c$ and $i_{N}=d$, such that for all $x \in (c, d)$, \[ f(x)=\sum_{n=1}^{N-1} \chi_{[i_n, i_{n+1})}(x) f_{n}(x), \] where $f_n$ are analytic functions on $[i_{n}, i_{n+1}]$ and $\chi_{[i_{n}, i_{n+1})}$ are characteristic functions for $n=1, \ldots, N-1$. Determine if the set $Z(f)$ of zeros of $f$ in $(c,d)$ has a finite number of connected components.

Answer: Yes


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a function such that \((x-2)f(x)-(x+1)f(x-1) = 3\) for all real numbers \(x\). Given that \(f(2) = 5\), find the value of \(f(2013)\).

Answer: 8157014183


Question: In how many different ways can 16 identical objects be distributed among 7 different people such that 3 of them can receive a maximum of 2 objects each, 3 of them must receive at least 2 objects each, and the remaining person has no restrictions?

Answer: 3483


Question: Determine the number of elements in the cyclic subgroup \(\left\langle -\frac{1}{2}+\frac{\sqrt{3}}{2} i \right\rangle\) of \(\mathbb{C}^*\).

Answer: 3


Question: Determine whether the following statement is true or false: "Every non Lebesgue measurable set in $\Bbb{R}^N$ has empty interior."

Answer: B


Question: Evaluate the limit: \[ \lim_{x\to0}\frac{1}{x^3}\int_{0}^{x}\sin(\sin(t^2))dt \]

Answer: \dfrac{1}{3}


Question:
Determine whether the following two statements are equivalent:

Statement 1: $p(x) \in P$ for all $x \in X$ if and only if $q$.

Statement 2: For all $x \in X$, $p(x) \in P$ if $q$. Furthermore, if $\neg q$, then there exists $x \in X$ such that $p(x) \notin P$.

Are these two statements equivalent?

Answer: Yes


Question: Assume that $X$ is a path-wise connected Hausdorff space with a non-trivial fundamental group. Does there always exist a simple closed curve in $X$ which is non-null-homotopic?

Answer: Yes


Question: For which prime numbers $p$ is $(p-2)! - 1$ a power of $p$?

Answer: 5


Question:
Consider a discrete-time Markov chain with state space \( \{0, 1, 2\} \) and transition matrix:
\[
\begin{bmatrix}
1/3 & 1/3 & 1/3 \\
1/3 & 1/3 & 1/3 \\
1/3 & 1/3 & 1/3
\end{bmatrix}
\]
Find the expected value of the time it takes for the chain to move from state 0 to state 2.

Answer: 3


Question: Estimate the number of iterations of Newton's method needed to find a root of \( f(x) = \cos(x) - x \) to within \( 10^{-100} \).

Answer: 7


Question:
Determine the number of nontrivial families of increasing substrings of the set \( \{1, 2, \ldots, k\} \). The rules for forming these families are as follows:

1. Do not consider the trivial case of picking the entire set \( \{1, 2, \ldots, k\} \) as a whole.
2. Pick at least one increasing substring, but multiple different ones can be picked.
3. Substrings may not overlap with one another.
4. Increasing substrings must consist only of consecutive numbers.
5. A substring cannot be a single number.

For example, for \( k = 4 \), the possible families are:
\[ \Big\{ \{1,2\}, \{1,2,3\}, \{2,3\}, \{3,4\}, \{2,3,4\}, \{\{1,2\}, \{3,4\}\} \Big\} \]
Thus, the number of families in this case is 6.

Find the number of such families for a general \( k \).

Answer: 2^{k-1} - 2


Question: Consider a polynomial $P(x) = g(x) \cdot (x - \beta)$, where $g(x)$ is a polynomial and $\beta$ is an element of the finite field $\mathbb{F}_p$. Evaluate $P$ at points $\textbf{x} = (x_1, x_2, \ldots, x_n)$ to obtain $\textbf{y} = (y_1, y_2, \ldots, y_n)$. Suppose some of the $y_i$ values are accidentally changed to random values $y'_i$. Interpolate the points $(x_1, y_1), (x_2, y_2), \ldots, (x_i, y'_i), \ldots, (x_j, y'_j), \ldots, (x_n, y_n)$ to obtain a new polynomial $P'$. What is the probability that $P'$ has $\beta$ as a root? Assume $x_i \neq 0$, $x_i \neq x_j$, and all values are in the finite field $\mathbb{F}_p$ for a large prime $p$. Provide your answer as a probability.

Answer: \dfrac{1}{p}


Question: Given a set of numbers \(a_1, a_2, \ldots, a_k\) where \(1 \leq k \leq n\), find the number of distinct ways to sum to a number \(N\) using these numbers. Each number \(a_i\) can be used more than once, and different orders of the same numbers are considered distinct. For example, if \(N = 19\) and the available numbers are 4 and 5, the number of ways is 4, corresponding to the sequences [4,5,5,5], [5,4,5,5], [5,5,4,5], and [5,5,5,4]. Determine the number of such distinct sequences for a given \(N\) and set of numbers.

Answer: 4


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

Answer: \dfrac{1}{2}


Question: Evaluate the line integral \( \oint_C (4x \, dx + 9y \, dy + 3(x^2 + y^2) \, dz) \) where \( C \) is the boundary of the surface \( z = 4 - x^2 - y^2 \) in the first octant (\( x, y, z \ge 0 \)), oriented counterclockwise as viewed from above, using Stokes' Theorem.

Answer: 0


Question: Evaluate the surface integral \( \int_S \frac{\partial \phi}{\partial n} \, ds \), where \( S \) is the surface of the unit sphere centered at the origin. The function \( \phi: \mathbb{R}^3 \to \mathbb{R} \) is strictly positive and satisfies \( |\nabla \phi|^2 = 4\phi \) and \( \Delta(\phi^2) = 20\phi \). Here, \( \frac{\partial \phi}{\partial n} \) is the directional derivative in the direction of the unit outward normal to \( S \), and \( ds \) is the surface measure of \( S \).

Answer: 8\pi


Question: Find the intersection of the fields \(\mathbb{Q}(\pi^2)\) and \(\mathbb{Q}(\pi^2 - \pi)\).

Answer: \mathbb{Q}


Question: For how many different real values of \(k\) do there exist real numbers \(x\), \(y\), and \(z\) such that \(\frac{x+y}{z}=\frac{y+z}{x}=\frac{z+x}{y}=k\)? Determine the number of such values of \(k\).

Answer: 2


Question: Find all functions \( f: [0, \infty) \to [0, \infty) \) that are differentiable and convex, satisfying the conditions \( f(0) = 0 \) and \( f'(x) \cdot f(f(x)) = x \) for all \( x \geq 0 \).

Answer: f(x) = x


Question: Determine whether the series \( \sin(\theta) + \frac{1}{3}\sin(3\theta) + \frac{1}{5}\sin(5\theta) + \ldots \) converges to \( \frac{\pi}{4} \) for \( 0 < \theta < \pi \).

Answer: \frac{\pi}{4}


Question: Evaluate the limit: \[ \lim\limits_{n \to \infty}\frac{[x]1!+2![2x]+\dots+[nx]n!}{(n+1)!} \] where \([x], [2x], \ldots, [nx]\) are the floor functions.

Answer: x


Question: Let $G = (V, E)$ be an infinite, simple, undirected graph where for every vertex $v \in V$, the degree $\text{deg}(v) \geq \aleph_0$. Given an integer $k \geq 1$, determine if there exists a subset $E^{(k)} \subseteq E$ such that the graph $(V, E^{(k)})$ is $k$-regular, meaning every vertex has exactly $k$ neighbors.

Answer: Yes


Question: Find the integer $n$ such that $\mathbb{Z}[\sqrt{2}]/(3-\sqrt{2}) \simeq \mathbb{Z}_n$. Use the first ring isomorphism theorem to determine the appropriate homomorphism and its kernel.

Answer: 7


Question: For which values of the constant \( c \) is the zero-set of the function \( f(x,y,z) = \cos^2(x) \cdot \sinh^2(y) - z^2 + c \) a regular surface?

Answer: c \neq 0


Question: Let \(f: \mathbb{R} \rightarrow \mathbb{C}\) be a continuous and differentiable function on \([a, b]\). Determine if there exists a \(c \in (a,b)\) such that \[\frac{|f(b) -f(a)|}{b - a} \leq |f'(c)|.\]

Answer: Yes


Question: Determine the smallest integer $m$ such that every subset of size $m$ from the set $M = \{1, 2, \ldots, 2013\}$ contains at least two numbers $a$ and $b$ where $a$ is divisible by $b$. What is the value of $m$?

Answer: 1008


Question: Find the orthogonal trajectories of the family of curves given by the equation \( y^2 = x^2(1 - cx) \).

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


Question: Identify a group that is isomorphic to the additive group $\mathbb{R} / \mathbb{Z}$.

Answer: S^1


Question: A talk show host has 10 new jokes. Each night, he tells some of these jokes. What is the maximum number of nights you can listen to the show such that on no night do you hear all the jokes you heard on any other night? For example, hearing jokes 1, 2, and 3 on one night, jokes 3 and 4 on another, and jokes 1, 2, and 4 on a third is acceptable, but hearing jokes 1 and 2 on one night and joke 2 on another is not.

Answer: 252


Question: Find all positive integers $n$ such that if $1 = a_1 < a_2 < a_3 < \ldots < a_k = n$ are the divisors of $n$, then $n = a_2^2 + a_3^3$. Determine all such $n$.

Answer: 68


Question: The number of new customers in a mall each day follows a Poisson distribution with \( \lambda = 50 \). Find the approximate probability that after one year (200 working days), the total number of customers that visited was between 950 and 1100.

Answer: 0


Question:
Determine whether it is possible to solve for $x$ in the equation:
\[ B = 0.16Y^{0.36}\cdot Z^{0.57} \cdot x^{0.032} \cdot \ln(1+17.5x) \]

Answer: Yes


Question:
Given the vector field \( \bar{F}(x,y,z) = (x,y,z) \) in Cartesian coordinates, its divergence is known to be 3. Express \( \bar{F} \) in cylindrical coordinates as \( \bar{F}(r,\theta,z) = (r\cos\theta, r\sin\theta, z) \). Calculate the divergence of \( \bar{F} \) in cylindrical coordinates using the formula:
\[ \nabla \cdot \bar{F} = \frac{1}{r}\frac{\partial(r\bar{F}_r)}{\partial r} + \frac{1}{r}\frac{\partial(\bar{F}_\theta)}{\partial \theta} + \frac{\partial(\bar{F}_z)}{\partial z} \]
Verify that the divergence in cylindrical coordinates is also 3.

Answer: 3


Question: In the "magnificent" graph game, players take turns choosing edges on a complete graph \( K_n \). A player loses if they cannot make a move, and a player wins if, at the end of their turn, the current graph contains a magnificent graph as one of its connected components. A graph is magnificent if it is connected, has at least two vertices of degree greater than 1, and for every vertex of degree greater than 1, the degree is unique among the vertices. For a set of vertices of size 1, 2, 3, or 4, no magnificent graphs are possible. Player 1 wins if and only if \( {n \choose 2} \) is odd, i.e., \( n \equiv_4 2 \) or \( n \equiv_4 3 \). Does the player associated with the parity of \( {n \choose 2} \) always win the game?

Answer: Yes


Question: Evaluate the double integral \( \iint_D \sqrt{x} \ dx \ dy \) where \( D = \{(x,y) \mid x^2+y^2 < x\} \) using the change of variables to polar coordinates.

Answer: \dfrac{8}{15}


Question: Determine whether the condition \( \sum_{a \in A} \frac{1}{a} < \infty \) implies that \( \lim\sup_{n \to \infty} \frac{|A \cap \{1, \ldots, n\}|}{n} = 0 \) for a subset \( A \) of the positive integers \( \mathbb{N} \).

Answer: Yes


Question: In a directed complete graph with $n$ vertices, where each edge has a direction, determine the maximum number $k$ such that there always exists a directed path of $k$ distinct vertices $a_1, a_2, \ldots, a_k$ with $a_1 \rightarrow a_2 \rightarrow \ldots \rightarrow a_k$. What is the value of $k$?

Answer: n


Question: Solve the equation \(x^2 = \log^2(x)\) for \(x\).

Answer: W(1)


Question: Let $M$ be an $R$-module and $I \subset R$ an ideal. If $M$ is finitely generated, is $M/IM$ finitely generated? Justify your answer.

Answer: Yes


Question:
Evaluate the limit using the dominated convergence theorem:
\[ \lim_{n\rightarrow \infty} n^2 \int_0^1 (1-x)^n \sin(\pi x)\, dx \]

Answer: \pi


Question: Determine all positive divisors \(d\) of \(3n^2\) such that \(n^2 + d\) is a perfect square for some positive integer \(n\).

Answer: 3k^2


Question: Determine the plane curves for which the length of the segment of the normal lines between the curve and the x-axis is constant. If $\alpha(t)$ is such a curve, find the length of $\alpha(t) - Q$, where $Q$ is the point where the normal line intersects the x-axis.

Answer: c


Question: Given $w = f(z) = u + iv$ and $u - v = e^x(\cos y - \sin y)$, express $w$ in terms of the complex variable $z = x + iy$. Assume $f(z)$ is analytic.

Answer: e^z


Question: Determine whether 2021 can be the hypotenuse of a Pythagorean triangle. In other words, does there exist a pair of integers \(x\) and \(y\) such that \(2021^2 = x^2 + y^2\)? Provide a proof for your answer.

Answer: No


Question: Consider the sequence \( \{P_n\} \) defined as follows: \( P_1 = 2 \), and for any integer \( n \geq 2 \), \( P_n \) is the maximum prime factor of \( 1 + P_1P_2P_3\ldots P_{n-1} \). Determine whether there exists a term in this sequence that is equal to 11.

Answer: No


Question: How many $2 \times 2$ orthogonal matrices $A$ satisfy both $A^3 = I_2$ and $A^2 = A^T$, where $I_2$ denotes the $2 \times 2$ identity matrix and $A^T$ denotes the transpose of $A$?

Answer: 3


Question: Determine a region in the xy-plane where the differential equation \( \frac{dy}{dx} = y^{4/5} \) has a unique solution.

Answer: y \neq 0


Question: Evaluate the infinite series \( f(x) = \sum_{n=1}^{\infty}\frac{(n-1)!}{\prod_{r=1}^{n}(x+r)} \) for \( x \in \mathbb{R}^{+} \).

Answer: \dfrac{1}{x}


Question: Consider a convex polytope $P \subseteq \mathbb{R}^n$ defined by finitely many linear inequalities, and an open set $O \subseteq \mathbb{R}^n$ such that $O \cap P$ contains the relative interior of $P$. Let $f: O \cup P \to \mathbb{R}$ be a function that is continuously differentiable on $O$, convex on $P$, and satisfies the condition that for all $y \in P \setminus O$ and sequences $x_n \in P \cap O$ approaching $y$, we have $\lim_{n} \|\nabla f(x_n)\| = \infty$. Is it guaranteed that if $x^* \in \mathrm{argmin}_{x \in P} f(x)$, then $x^* \in O \cap P$?

Answer: Yes


Question: Determine whether $4^{1536} - 9^{4824}$ is divisible by 35.

Answer: Yes


Question: Evaluate \( \lim_{n\rightarrow \infty}\int_0^\infty \frac{x^{n-2}\cos(n\pi x)}{1+x^n} \, dx \).

Answer: 0


Question:
Calculate the expected value \( \langle m \rangle \) given by the formula:
\[ \langle m \rangle = \sum_{-j\le m\le j} m \ P(m) \]
where \( P(m) = \frac{e^{xm/j}}{Z} \) and \( Z = \sum_{-j\le m\le j} e^{xm/j} \).

Verify that the result simplifies to:
\[ \langle m \rangle = j \cdot B_j (x) \]
where \( B_j(x) = \frac{2j+1}{2j} \coth \left( \frac{2j+1}{2j} x \right) - \frac{1}{2j} \coth \left( \frac{1}{2j} x \right) \).

Answer: \langle m \rangle = j \cdot B_j (x)


Question: Solve the equation: \(x^3 - 3x^2 + 2x - 5 = 0\).

Answer: 1 + \sqrt[3]{\frac{45 + \sqrt{2013}}{18}} + \sqrt[3]{\frac{45 - \sqrt{2013}}{18}}


Question:
Evaluate the limit without using L'Hospital's Rule:
\[ \lim_{\Delta z\to 0}\frac{e^{\Delta z^2+2z\Delta z}-1}{\Delta z} \]
where \(\Delta z=\Delta x+i\Delta y\) and \(z=x+iy\).

Answer: 2z


Question: Consider the smooth compact manifolds $X$ and $Y$. Let $C^{-\infty}(X \times Y)$ denote the space of generalized functions on the product manifold $X \times Y$. Define the canonical linear map \( T: C^{-\infty}(X \times Y) \to \text{Bil}(C^\infty(X), C^\infty(Y)) \), where the target is the space of continuous bilinear functionals from \( C^\infty(X) \times C^\infty(Y) \to \mathbb{C} \). The map is given by \( (T\Phi)(f,g) = \Phi(f \otimes g) \). It is known that \( T \) is an isomorphism of vector spaces. Determine the standard topologies on the source and target for which \( T \) is an isomorphism of topological vector spaces. Specifically, consider the strong topology on \( C^{-\infty}(X \times Y) \) and the topology on the target given by the seminorms \( ||B||_{K,L} = \sup_{k \in K, l \in L} |B(k,l)| \), where \( K \subset C^\infty(X) \) and \( L \subset C^\infty(Y) \) are arbitrary bounded subsets. Is \( T \) a topological isomorphism under these conditions?

Answer: Yes


Question: Let \( L: \mathbb{R}^n \to \mathbb{R}^m \) be a linear function. Prove that if \( \lim_{\hat{h} \to \hat{0}} \frac{\|L(\hat{h})\|}{\|\hat{h}\|} = 0 \), then \( L \equiv 0 \).

Answer: L \equiv 0


Question: Consider the integral \( \int_\gamma \frac{dz}{z-a} \) where \( \gamma = C(r,z_0) \) is a circle oriented in the positive direction. If \( a \) is any point inside the disk \( D(r,z_0) \), does the integral evaluate to \( 2 \pi i \)?

Answer: 2\pi i


Question: Find the value of \(|a+b+c|\) for the circle given by the equation \(x^2+y^2+ax+by+c=0\), where every tangent to the circle is perpendicular to exactly one member of the family of lines \(x+y-2+\lambda(7x-3y-4)=0\) at the point of contact, and the circle touches only one member of the family \((2x-3y)+\mu(x-y-1)=0\).

Answer: 7


Question: Calculate the line integral \( \int_C (x-y) \, dz \) using Stokes' theorem, where \( C \) is the curve formed by the intersection of the paraboloid \( z = x^2 + y^2 \) and the plane \( z - 2x - 2y = 1 \). Orient \( C \) counterclockwise as viewed from the point \( (0,0,10) \).

Answer: 12\pi


Question: For a real-valued random variable $X$, the $k$th raw moment is defined as $\mathbb{E}[X^k]$, and the $k$th central moment is defined as $\mathbb{E}[(X-\mathbb{E}[X])^k]$. Determine if the $k$th central moment is always less than or equal to the $k$th raw moment for even $k$. Provide a justification for your answer.

Answer: Yes


Question: Determine the order of the Galois group of the polynomial \( f(x) = x^2 + 2ix + 1 \). The roots of the polynomial are \( x_1 = -i + i\sqrt{2} \) and \( x_2 = -i - i\sqrt{2} \). Let \( L = \mathbb{Q}(x_1, x_2) \).

Answer: 4


Question: Determine the number of natural numbers \( n \leq 1000 \) that cannot be expressed in the form \( n = a^2 - b^2 - c^2 \), where \( a, b, \) and \( c \) are non-negative integers satisfying the condition \( a \geq b + c \).

Answer: 0


Question: Let $x_1, x_2, x_3, x_4$ be the roots of the polynomial equation $x^4 + kx^2 + 90x - 2009 = 0$. Given that $x_1x_2 = 49$, determine the value of $k$.

Answer: 7


Question: Let \( f \) and \( g \) be holomorphic functions within a neighborhood of \( z_0 \). Suppose \( z_0 \) is a zero of order \( k \) of \( f \), and a zero of order \( l \) of \( g \) with \( l > k \). Find \( \lim _ { z \rightarrow z _ { 0 } } \frac { f ( z ) } { g ( z ) } \).

Answer: \infty


Question: Let $X$ be a locally compact Hausdorff space, and let $K \subset X$ be a compact subspace. Given an open set $U \subset X$ such that $K \subset U$, determine whether there exists an open set $V \subset X$ such that $K \subset V \subset U$ and the closure $\overline{V}$ is compact.

Answer: Yes


Question: Let $N$ be a $12 \times 12$ matrix such that $N^4 = 0$. Prove that the rank of $N$ is at most 9.

Answer: 9


Question: Let $f: [0,\infty) \to \mathbb{R}$ be a continuous function such that for every $a > 0$, \( \lim_{n \to +\infty} f(na) = 0 \), where $n$ is an integer. Does this imply that \( \lim_{t \to +\infty} f(t) = 0 \), where $t$ is real?

Answer: Yes


Question: For a Hermitian nonnegative-definite matrix $A$ with complex entries, if $Ax$ is always real for any real vector $x$, is $A$ necessarily a real matrix? Provide a justification for your answer.

Answer: Yes


Question: Determine whether the sequence \(a_n = \frac{n^n}{n!}\) converges or diverges as \(n\) approaches infinity. If it converges, find its limit.

Answer: \infty


Question: Compute the integral \( \int_{|z|=1} |z^5 - 1|^2 |dz| \).

Answer: 4\pi


Question: Consider sequences of probability measures $P_n$ and $Q_n$ on $\mathbb{R}^n$. Let $f_n(t) = \int e^{itx} \, dP_n(x)$ and $g_n(t) = \int e^{itx} \, dQ_n(x)$ be their characteristic functions. If for all $t$, $f_n(t) - g_n(t)$ converges to zero as $n$ approaches infinity, do the sequences $P_n$ and $Q_n$ merge with respect to bounded uniformly continuous functions? (That is, for every bounded uniformly continuous function $f$, does $\int f \, dP_n - \int f \, dQ_n$ converge to zero as $n$ approaches infinity?)

Answer: Yes


Question: Determine whether the limit \( \lim_{n\rightarrow\infty}\int_0^\infty \frac{\sin(x^n)}{x^n}\,dx \) exists, and if it does, find its value.

Answer: 1


Question: In a complete countably seminormed space $X$, consider a sequence of linear continuous mappings $T_k: X \rightarrow \mathbb{R}$ such that for each $x \in X$, the sequence $(T_k(x))_{k \in \mathbb{N}}$ is bounded. By the Banach-Steinhaus theorem for Fréchet spaces, the family $(T_k)_{k \in \mathbb{N}}$ is equi-continuous. For each $k \in \mathbb{N}$, there exists $N_k$ such that $T_k$ is continuous with respect to the seminorm $p_{N_k}$, i.e., $|T_k(x)| \leq M p_{N_k}(x)$ for all $x \in X$, where $M > 0$ depends on $k$. Does there exist an $N \in \mathbb{N}$ and $M > 0$, independent of $k \in \mathbb{N}$, such that $|T_k(x)| \leq M p_N(x)$ for all $x \in X$ and $k \in \mathbb{N}$?

Answer: Yes


Question:
Let \( f \) be a differentiable function on \( \mathbb{R} \) such that \( f(x+y), f(x)f(y), f(x-y) \) (in that order) form an arithmetic progression for all \( x, y \in \mathbb{R} \). Given that \( f(0) \neq 0 \), determine which of the following is true:

\( (A) \ f'(0) = -1 \)

\( (B) \ f'(0) = 1 \)

\( (C) \ f'(1) - f'(-1) = 0 \)

\( (D) \ f'(1) + f'(-1) = 0 \)

Answer: D


Question: Consider a smooth and continuous real-valued vector function \( \mathbf{r} \) such that \( \mathbf{r} \cdot \mathbf{r} = \| \mathbf{r} \|^2 \). Given that \( (\| \mathbf{r} \|^2)' = 2\mathbf{r} \cdot \mathbf{r}' = 0 \), does it follow that \( \|\mathbf{r}\| \) is a constant? Justify your answer.

Answer: Yes


Question: Find the value of \( f^{(3)}(0) \) for the function \( f(x) = \sin^3(\ln(1+x)) \) using Taylor expansion.

Answer: 6


Question: Compute the operator norm of the matrix \( \begin{bmatrix}3 & 1 \\ 1 & 1\end{bmatrix} \).

Answer: 2 + \sqrt{2}


Question:
Evaluate the limit of the following expression as \( N \) approaches infinity:
\[ \lim_{N \to \infty} \frac{N(N-1)(N-2)\ldots 1}{(N+ \frac{1}{M})(N-1 + \frac{1}{M})(N-2 + \frac{1}{M})\ldots(1 + \frac{1}{M})} \]
where \( M \) is a constant.

Answer: 0


Question: Evaluate \( \lim_{x\to 0} \frac {\arcsin(x) \sqrt{\sin(x)}}{\sqrt{2x-x^2}} \).

Answer: 0


Question: Prove that \( \lim_{n\to\infty}\frac{Li(n)}{\pi(n)} =1 \), where \( Li(n) = \int_{2}^{n} \frac{dx}{\ln(x)} \) and \( \pi(n) \) is the prime counting function. Use the Prime Number Theorem and L'Hopital's rule in your proof.

Answer: 1


Question: Is there a shape such that the surface area-to-volume ratio approaches 1 as a characteristic length increases?

Answer: Yes


Question: Given a value $A$ and a range $[(1-\epsilon)A, (1+\epsilon)A]$, consider a series of $m$ random variables $\chi_i$ with $E[\chi_i] = A$. The probability $Pr[\chi_i \geq (1-\epsilon)A] > p_1 = 1-\gamma$, where $0 < \gamma < \frac{1}{2}$, and $Pr[\chi_i \geq (1+\epsilon)A] \leq p_2$, calculated using Markov's inequality. What is the probability that the minimum value of $\chi_i$ from $m$ values is within the range $[(1-\epsilon)A, (1+\epsilon)A]$? Express your answer in terms of $p_1$, $p_2$, and $m$. Verify if the expression $p_1^m(1-p_2^m)$ is correct.

Answer: p_1^m - p_2^m


Question: Evaluate the line integral \( \oint\limits_C {(x\sin ({y^2}) - {y^2})dx + ({x^2}y\cos ({y^2}) +3x)dy} \) where \( C \) is the counterclockwise boundary of the trapezoid with vertices \((0,-2),(1,-1),(1,1),(0,2)\).

Answer: 9


Question:
Consider the function \( f \) which is continuous on the interval \([0,1]\). Determine the existence and value of the limit:
\[ \lim_{n \to \infty}\sum_{i=0}^{[n/2]} \frac{1}{n} f \left(\frac{i}{n} \right) \]
where \([y]\) denotes the greatest integer less than or equal to \(y\). Choose the correct option:

(A) The limit does not exist.
(B) The limit exists and is equal to \( \frac{1}{2} \int_{0}^{1} f(x)\, dx \).
(C) The limit exists and is equal to \( \int_{0}^{1} f(x)\, dx \).
(D) The limit exists and is equal to \( \int_{0}^{1/2} f(x)\, dx \).

Answer: D


Question:
Given the partial differential equation \( \frac{\partial^2u}{\partial x\partial t}=e^{-t}\cos x \) with the conditions \( \frac{\partial u}{\partial t}=0 \) at \( x=0 \) and \( u=0 \) at \( t=0 \), determine the correct solution from the following options:

(a) \((1-e^{-t})\sin x\)
(b) \((-e^{-t})\sin x+g(x)\)
(c) \((1+e^{-t})\cos x\)
(d) \((-e^{-t})\sin x\)

Answer: a


Question: Evaluate the integral \( \int_0^\infty \frac{1}{(1+x^2)(1+x^{2018})}\,dx \).

Answer: \dfrac{\pi}{4}


Question: Evaluate the limit: $$\lim_{x \rightarrow \infty} \frac{x^5 \sin \frac{1}{x} + \cos x}{x^2 + x \sin x}$$

Answer: \infty


Question: Calculate the limit: \( \lim\limits_{z \to 0} {\exp(-1/z^4)} \)

Answer: 0


Question: Determine whether it is true that in a primitive Pythagorean triple $(a, b, c)$, the hypotenuse $c$ is never divisible by 7.

Answer: True


Question: Does there exist a sequence of linear maps \((M_n)\) in \(\mathcal{L}(X,Y)\), where \(X\) and \(Y\) are Banach spaces, that converges pointwise but not in norm?

Answer: Yes


Question: Let $V$ be a finite-dimensional real linear space, and let $K$ be a compact subgroup of $GL(V)$ with the usual topology. Is there a basis of $V$ such that every $f \in K$ is an orthogonal matrix with respect to this basis?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{{n \choose n/2}}{2^n} \]

Answer: 0


Question: Evaluate the double sum: \[ \sum_{j=2}^{\infty} \left( \sum_{k=2}^{\infty} k^{-j} \right) \]

Answer: 1


Question: Evaluate the limit: $$\lim_{n\to\infty}\frac{n^{n}}{e^{n^{3/2}}}.$$

Answer: 0


Question: Let $E = \left\{ x \mid x = \sum_{k=1}^{\infty} \frac{a_k}{10^k}, a_k \in \{3,7\} \right\}$. Find the set $E'$, which is the set of all accumulation points of $E$. Provide your answer as a set.

Answer: E


Question: Consider the functions \( f(x) = x^2 + 2bx + 9 \) and \( g(x) = 3a(x+b) \), where \( a \) and \( b \) are real numbers. Each pair \((a, b)\) can be considered as coordinates of a point in the xy-plane. Let \( S \) be the set of all such points \((a, b)\) for which the graphs of \( y = f(x) \) and \( y = g(x) \) do not intersect. Find the area of the region defined by \( S \).

Answer: 6\pi


Question:
Determine the integer value of \( n \) such that the following limit is a finite non-zero number:
\[ \lim_{x\to 0} \frac{\cos^2 x - \cos x - e^x \cos x + e^x - \frac{x^3}{2}}{x^n} \]

Answer: 4


Question: Determine the values of \( b \in \mathbb{R} \) for which the series \( \sum_{n=1}^{\infty}\frac{b^{2n^2}}{n^n} \) converges.

Answer: [-1, 1]


Question: Calculate the expectation \( E[W_s e^{W_s - W_t}] \) for a Brownian process \( W_s \) where \( 0 \leq s < t \) and \( W_0 = 0 \).

Answer: 0


Question: Determine whether it is possible to find random variables $X, Y, Z$ such that $\Bbb P(X>Y) > \frac{1}{2}$, $\Bbb P(Y>Z) > \frac{1}{2}$, and $\Bbb P(Z>X) > \frac{1}{2}$. Provide a justification for your answer.

Answer: Yes


Question: Let $X$ be a Banach space, $U \subset X$ an open subset, and $(\Omega, \mathcal A, \mu)$ a measure space. Consider a function $f: U \times \Omega \to \mathcal R$ that is continuously Fréchet-differentiable in the first argument and measurable in the second argument. Fix $u \in U$, and assume $f(u,\omega)$ and $Df(u,\omega)(x)$ for all $x \in U$ are Lebesgue-integrable over $\omega$. Determine if $f(\tilde u,\omega)$ is integrable for all $\tilde u$ in some neighborhood of $u$. Provide a justification for your answer.

Answer: Yes


Question: Your favorite baseball team is playing against your uncle's favorite team in the World Series. At the beginning of each game, you and your uncle bet on the game's outcome. If you bet \( b \) dollars and your team wins the game, your uncle gives you an IOU for \( b \) dollars. But if they lose the game, you give him an IOU for \( b \) dollars. When the series is over, all outstanding IOUs are settled in cash. You would like to walk away with \( \$100 \) in cash if your team wins the series, and lose \( \$100 \) if your team loses the series. How much should you bet on the opening game to achieve this outcome? (The first team to win a total of four games wins the series.)

Answer: 31.25


Question: Consider a nondiagonalizable matrix $A \in \mathbb{R}^{n \times n}$. Let $\hat{A} = A + E$, where $E$ is a small random diagonal matrix with diagonal entries independently sampled from $N(0, \epsilon^2)$ for an arbitrarily small $\epsilon > 0$. Is it true that $\hat{A}$ is diagonalizable with probability 1, given that $A$ is fixed and the randomness is only from $E$?

Answer: Yes


Question: Evaluate the contour integral \( \oint_{|z|=1} \sqrt{z} \, dz \) around the unit circle, where \( z = e^{2\pi it} \) for \( t \) from 0 to 1. Assume a branch cut along the positive real axis.

Answer: -\dfrac{4}{3}


Question: If $E$ is an elliptic curve over the finite field $\mathbb{F}_p$ with $p \geq 5$ and the number of points $\#E(\mathbb{F}_p)$ is even, does $E$ have a non-trivial 2-torsion point defined over $\mathbb{F}_p$? In other words, if $E$ is given by the equation $y^2 = f(x)$, is the polynomial $f(x)$ reducible over $\mathbb{F}_p$?

Answer: Yes


Question: Evaluate the integral \( \int_1^{1000} \frac{dx}{x+\lfloor \log_{10}(x) \rfloor} \).

Answer: \ln \left( \dfrac{168670}{187} \right)


Question: Evaluate the following limit: \[ \lim_{\delta\rightarrow0^+}\int_{-\infty}^\infty(g(k)-g(k+\delta))\ \text{dk} \] where \( g(x) = \sin{(c)} + \cos{(c)}(x-c) \) is the first order Taylor series approximation of \( y = \sin{x} \) about \( c \).

Answer: 0


Question: Let the radius of convergence of the power series \( \sum_{n=0}^\infty a_n z^n \) be \( R_1 \) and that of \( \sum_{n=0}^\infty b_n z^n \) be \( R_2 \). Consider the power series \( \sum_{n=0}^\infty a_n b_n z^n \). If \[ \lim\sup \left| \frac{a_{n+1}b_{n+1}}{a_n b_n} \right| = \lim\sup \left| \frac{a_{n+1}}{a_n} \right| \lim\sup \left| \frac{b_{n+1}}{b_n} \right|, \] prove that the radius of convergence \( R \) of the series \( \sum_{n=0}^\infty a_n b_n z^n \) satisfies \( R_1 R_2 \leq R \).

Answer: R_1 R_2 \leq R


Question: Evaluate the integral \( \oint\limits_{C} u \, dy + v \, dx \) where \( C \) is the circle \( |z|=3 \), and \( u \) and \( v \) are the real and imaginary parts of the function \( f(z) = \frac{1}{z^2-6z+8} \). The curve \( C \) is oriented counterclockwise. Determine the value of the integral.

Answer: -\pi


Question: Evaluate the sum \( S = \sum_{n=1}^{\infty} n \) using the concept of the Riemann zeta function, where \( S = \zeta(-1) \).

Answer: -\dfrac{1}{12}


Question: Calculate the value of the following summation: \[ \sum_{k=0}^{13} \binom{26}{2k}\frac{(2k)!}{k!2^k} \]

Answer: 532985208200576


Question: Find the limit of \( \frac{e^{-1/x^2}}{x^{100}} \) as \( x \to 0 \).

Answer: 0


Question: Simplify the expression \((V' A V)^{-1}\) in terms of the matrices \(A^{-1}, V,\) and \(V'\), given that \(V \in \Re^{n \times k}\) with \(n > k\), \(V'V = I\), and \(\mathrm{rank}(V) = \text{rank}(V' A V) = k\). The matrix \(A\) is \(n \times n\) and its inverse \(A^{-1}\) is known.

Answer: (V' A V)^{-1} = V' A^{-1} V


Question: Find the multiplicative inverse of \(n+1\) modulo \(n^2\), where \(n\) is an integer greater than 1. That is, determine the integer \(z\) such that \((n+1)z \equiv 1 \pmod{n^2}\).

Answer: n^2 - n + 1


Question: Does there exist a topological space $X$ and a subspace $A \subseteq X$ such that $X$ is connected, $A$ is connected, and for all $a \in A$, the subspace $A \setminus \{a\}$ is connected while the subspace $X \setminus \{a\}$ is not connected? Additionally, can $A$ have a non-empty interior?

Answer: Yes


Question: Calculate the limit: \[ \lim_{x \to 0} \frac{\sqrt{1+ \tan x} - \sqrt{1+ \sin x}}{x^3} \] without using L'Hospital's Rule.

Answer: \dfrac{1}{4}


Question: Find the length of the graph of the polar equation given by \( r = 8\sin^2 (\theta/2) \) for \( 0 \leq \theta \leq \pi \).

Answer: 16


Question: Suppose \( f: (0,1) \to \mathbb{R}_+ \) is a measurable function and \( f \in L((0,1)) \). Is it true that if \( \int_{(0,1)} f(x) \, d\mu = 0 \), then \( f(x) = 0 \) almost everywhere on \((0,1)\)?

Answer: Yes


Question: Evaluate the integral \( \int_0^{+\infty} \frac{\log(t)}{1+t^2} \, dt \).

Answer: 0


Question: Let $A$ be a countable dense subset of $[0,1]$. If $f(n)=0$ for all $n \in A$, does it follow that $f(x)=0$ almost everywhere in $[0,1]$? Justify your answer.

Answer: No


Question:
Solve the equation for $t$:
\[0.6e^{-\dfrac{t}{20}}\left(1-\dfrac{t}{20}\right)+0.5e^{\dfrac{-30+t}{10}}\left(-1+\dfrac{30-t}{10}\right)=0\]

Answer: 20


Question: Let $(X,\mathcal{A},\mu)$ be a measurable space and $\{u_n\}_{n=1}^\infty$ be a sequence in $\mathcal{L}^1(\mu)$ that converges pointwise to $u \in \mathcal{L}^1(\mu)$. If $\int u_n \, d\mu \to 0$, prove that $\int u \, d\mu = 0$. Provide a justification for your answer.

Answer: 0


Question: Let \( f: [0,1]^2 \rightarrow \mathbb{R} \) be a twice continuously differentiable function such that for every \( x \in [0,1] \), there exists an interval \( I_x \subset [0,1] \) where \( f(x,y) = 0 \) for all \( y \in I_x \). Does there exist an open ball in \( [0, 1]^2 \) where \( f \) is identically zero?

Answer: Yes


Question: Determine whether there exists a positive decreasing sequence \( \{a_n\}_{n \geq 1} \) such that \( a_n \searrow 0 \), \( \sum a_n < +\infty \), and \( \frac{1}{n^{1+\epsilon}} \in o(a_n) \) for all \( \epsilon > 0 \).

Answer: Yes


Question: Is it true that in a finite field $K$, the equation $x^2 = a$ has at most two roots for a given $a \in K$?

Answer: True


Question: A transport company operates 40 trams and 80 buses. Each day, independently, each tram breaks down with a probability of 0.01, and each bus breaks down with a probability of 0.02. Let $X$ be the number of trams that break down in a day, and $Y$ be the number of buses that break down on the same day. Calculate the covariance of $X$ and $X \cdot Y$. Provide your answer as a single numerical value.

Answer: 0.6336


Question: Calculate the limit: \[ \lim_{x \to 0} \frac{\ln(\cos(2x))}{x \sin x} \]

Answer: -2


Question: Determine the base system \(x\) such that the repeating decimal \(\overline{.17}_x\) is equal to \(\frac{1}{5}\).

Answer: 9


Question: Apples are being packed in a box, where each apple has an expected weight of 200 g and a variance of 20 g. Packing stops as soon as the total weight reaches 10 kg or more. Calculate the probability that the number of apples, $N$, in the box is less than or equal to 49, i.e., find $P(N \leq 49)$. Use the central limit theorem to solve this problem.

Answer: 0


Question:
Compute the limit:
\[
\lim_{x \to 0}\ \dfrac{x^2e^{x^2}+\dfrac{1}{2}\log(1+2x^2)-2x^2}{1+\arctan(x^6)-\cos(x^3)}
\]
using Taylor series expansion around the origin. Do not use L'Hopital's rule.

Answer: \dfrac{11}{9}


Question: Evaluate the limit: \[ \lim_{n \to \infty} n^3 \left(\int_0^{\pi}\cosh(2\cos(x)) \, dx - \pi\sum_{k=0}^n \frac{1}{(k!)^2} \right) \]

Answer: 0


Question: Alice and Bob play a game with the set $A = \{1, 2, \ldots, n\}$, where $n \ge 2$. Bob starts by choosing any number from $A$. Subsequently, each player alternates turns, choosing a number that is distinct from those already chosen and differs by 1 from an already chosen number. Alice wins if the sum of the numbers she chooses is composite; otherwise, Bob wins. Determine which player has a winning strategy.

Answer: Alice


Question: Find the irreducible polynomial $f \in \mathbb{Z}_p[x]$ such that $\mathbb{F}_q \cong \mathbb{Z}_p[x]/(f)$, where $q = p^n$ and $\mathbb{F}_q$ is the finite field with $q$ elements.

Answer: f(x)


Question: Find the remainder when \(x^{1000000}\) is divided by \(x^3 + x + 1\) in \(\mathbb{Z}_2[x]\).

Answer: x


Question: In a zero-dimensional topology, where the space has a base of sets that are both open and closed, is every open set also a closed set?

Answer: No


Question: Evaluate the limit: \[ \lim \limits_{n \to \infty} \int_{-\infty}^\infty e^{-x^2}\cos(nx)\, dx \]

Answer: 0


Question: Find the volume of the solid bounded by the cylinder \(x^2 + y^2 = 1\), the xy-plane, and the plane \(x + z = 1\).

Answer: \pi


Question: Find the remainder when \((x-1)^{2013}\) is divided by \(x^2-2x+2\).

Answer: x - 1


Question: Let $X_1, X_2, X_3, X_4, X_5$ be independent random variables, each uniformly distributed over the interval $(0,1)$. Calculate the probability that $X_1 + X_2 + X_3 + X_4 + X_5 < 1$. Provide your answer as a single probability value.

Answer: \dfrac{1}{120}


Question: Determine the number of ring homomorphisms from \(\mathbb{Z}[x]\) to \(\mathbb{Z}_{12}\).

Answer: 12


Question: Calculate \( \lim_{x \to 0} (\cot^3(x) \cdot x \cdot (\ln(1+3x)-3x)) \).

Answer: -\dfrac{9}{2}


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

Answer: 1


Question: Evaluate the triple integral \( \iiint\limits_{D} (-4z + y^2 + x^2) \, \mathrm{d}V \) over the region \( D = \{(x,y,z) \in \mathbb{R}^{3} \mid x \ge 0, 0 \le y \le x, x^2 + y^2 \le 16, 0 \le z \le 5\} \).

Answer: -20\pi


Question: Let $A$ be a non-empty bounded closed set in $\Bbb C$ that does not have accumulation points in $\Bbb C$. Define $d = \inf \{|a-b| : a, b \in A, a \neq b\}$. Prove that $d > 0$.

Answer: d > 0


Question: Determine the minimum value of $d - n$ such that the probability that the images of $n$ $2n$-dimensional linearly independent vectors over $\mathbb{F}_2$ remain linearly independent after projection onto a random $d$-dimensional subspace is at least $1 - \frac{1}{\mathrm{poly}(n)}$. Provide your answer in terms of $n$.

Answer: \log n


Question: Find the smallest positive argument \( \theta \) for the complex number \( z = re^{i\theta} \) such that \( \sum_{k=0}^9 z^k = 0 \).

Answer: \dfrac{\pi}{5}


Question: If $U$ is a Lie group with a closed subgroup $K$ such that both $U$ and $U/K$ are simply-connected, is $K$ necessarily connected?

Answer: Yes


Question: Find the probability \( P(X_1 < X_2 \mid X_1 < 2X_2) \) given that the random variables \( X_1 \) and \( X_2 \) are independent and identically distributed with the probability density function \( f(x) = e^{-x} \) for \( 0 < x < \infty \).

Answer: \dfrac{3}{4}


Question: Let \( p(x) = 2 + 4x + 3x^2 + 5x^3 + 3x^4 + 4x^5 + 2x^6 \). For which \( k \) with \( 0 < k < 5 \) is the integral \( I_k = \int_0^\infty \frac{x^k}{p(x)} \, dx \) smallest?

Answer: 2


Question:
Determine whether the following equality holds:
\[ \frac{1}{(1+a\cos t)^2} = \frac{1}{(1+a)^2} + a\cdot\frac{t^2}{(1+a)^3} + a\cdot\frac{t^4(8a-1)}{12(1+a)^4} + \dots \]

Answer: Yes


Question: Determine whether \( \lim \sup_{n\to\infty}\frac{m_n}{n} = 1 \), where \( m_n \) is the largest integer such that there exists a subset \( T \subseteq \{0,1\}^n \) with \(|T|=n\) and for distinct \( x, y \in T \), the Hamming distance \( d^H_n(x,y) \geq m_n \).

Answer: 1


Question: Consider an infinite set A that satisfies the T1 axiom. Does A have no limit points? Justify your answer.

Answer: No


Question: Determine whether there exists an ordered 4-tuple of rational numbers \((a, b, c, d)\) such that \((b, d) \neq (0, 0)\) and \(2a^2 + 3b^2 + 30c^2 + 45d^2 = 2\).

Answer: Yes


Question: Compute the expected value of the product of the coordinates of a point randomly selected on a circle of radius 1 centered at the point (1,1).

Answer: 1


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{4^n-C(2n+1,n)}{2^n}. \]

Answer: \infty


Question: Determine the number of onto group homomorphisms from \((\mathbb{Q}, +)\) to \((\mathbb{Q} \setminus \{0\}, \cdot)\).

Answer: 0


Question:
Does there exist a non-zero linear operator \( P : L^1(\mu) \to L^1(\mu) \) on a \( \sigma \)-additive measure space \((\Omega,\Sigma,\mu)\) such that:

1. \( \|Pf\| \leq \|f\| \) for all \( f \in L^1(\mu) \),
2. \( f \geq 0 \Rightarrow Pf \geq 0 \),

and there is no \( \alpha \in (0,1] \) satisfying \( \alpha \|f\| \leq \|Pf\| \) for all \( f \in L^1(\mu) \)?

Answer: Yes


Question: Determine the rank of the matrix $I_n + M$, where $M$ is a skew-symmetric matrix.

Answer: n


Question: Suppose 100 points in the plane are colored using two colors, red and white. Each red point is the center of a circle passing through at least three white points. What is the least possible number of white points?

Answer: 10


Question: Find the volume of the region bounded by the surfaces \( z = x^2 + y^2 \) and \( z = 1 - x^2 - y^2 \).

Answer: \dfrac{\pi}{4}


Question: Let \( \kappa \) be an infinite cardinal. Is there a cardinal \( \alpha \geq \kappa \) such that the intersection number \( i(\alpha) < \alpha \)?

Answer: Yes


Question: Determine the Galois group of the field extension \(\mathbb{F}_3(x^4) \subset \mathbb{F}_{3^2}(x)\).

Answer: D_4


Question: Evaluate the integral \( \int_0^\infty \frac{\sin x}{x} \, dx \).

Answer: \dfrac{\pi}{2}


Question:
Determine whether the limit \( \lim\limits_{x \to +\infty} f(x) \) exists for the function \( f \) defined by:
\[
f(x)=\begin{cases}\frac{\sin(\pi x^2)}{x^2\sin(\pi x)} & \text{if } x \in \mathbb{R} \setminus \mathbb{Z} \\
\frac{2}{n} & \text{if } x=n \in \mathbb{Z} \setminus \{0\} \end{cases}
\]

Answer: 0


Question: Determine the number of equivalence relations on the set \(\{1, 2, 3, 4, 5, 6, 7\}\) that must include the pairs \(\{(6,4), (4,7), (3,3), (5,1)\}\) and must not include the pairs \(\{(1,2), (6,1), (2,4)\}\).

Answer: 4


Question: Evaluate the limit: \[ \lim_{x\to0}\frac{\int_x^{x^2}\sinh(t)\sin(t)\,dt}{\int_0^x t^3\csc(t)\,dt}. \]

Answer: -1


Question: Consider the sequences generated by the polynomials \( P(x) = x + 1 \) and \( Q(x) = x^2 + 1 \). Starting with the pair \((x_1, y_1) = (1, 3)\), each subsequent pair \((x_{k+1}, y_{k+1})\) is generated by either \((P(x_k), Q(y_k))\) or \((Q(x_k), P(y_k))\). A positive integer \( n \) is called cute if there exists at least one sequence such that \( x_n = y_n \). Determine all cute integers \( n \).

Answer: 3


Question: Compute the integral of the 1-form \( \omega = \frac{x\,\mathrm{d}y - y\,\mathrm{d}x}{x^2+y^2} \) over the unit circle parameterized by \( \gamma(t) = (\cos(t), \sin(t)) \) for \( t \in [0, 2\pi] \).

Answer: 2\pi


Question: Find the number of solutions to the equation \(u_1 + u_2 + \ldots + u_5 = 24\) where each \(u_i\) satisfies \(1 \le u_i \le 7\). Use generating functions to determine the coefficient of \(x^{24}\) in the expansion of \((x + x^2 + \ldots + x^7)^5\).

Answer: 1015


Question: Evaluate the limit: \[ \lim_{h \to 0} \int_{-1}^{1}\frac{h}{h^2+x^2}~dx \]

Answer: \pi


Question: Evaluate the integral \( \int_0^{\infty} \frac{\sqrt{x}}{(x+1)^2} \, \mathrm{d}x \).

Answer: \dfrac{\pi}{2}


Question: Given the functions \( f(t) = t-1 \), \( g(t) = t+1 \), and \( h(t) = t^2+1 \), does the function \( t \to t^2 \) belong to the set \( \mathbb{Q}\{f,g,h\} \)?

Answer: Yes


Question: Given a triangle with vertices at \((1,1), (5,3), (7,1)\), find the shear factor \(i\) in the matrix \(\begin{pmatrix} 1 & i \\ 0 & 1 \end{pmatrix}\) such that the transformed triangle becomes a right triangle with the right angle at \((1,1)\).

Answer: -2


Question: Evaluate the limit: $$\lim_{x\to\infty} \left(\sqrt{x^2+5x-2}-\sqrt{4x^2-3x+7}+\sqrt{x^2+7x+5}\right)$$

Answer: \dfrac{27}{4}


Question: Given that $f(a)$ is the area of a region $S$ defined by points $A(a,a^2)$ and $B(b,b^2)$ with $\overline{AB}=1$, find the limit: \[ \lim_{a\to \infty} a^3 f(a). \]

Answer: \dfrac{1}{48}


Question: Let \(f(x)\) be a twice differentiable function on \([1,3]\) such that \(f(1) = f(3)\) and \(|f''(x)| < 2\) for all \(x\) in \([1,3]\). Determine the range of \(|f'(x)|\) on the interval \([1,3]\).

Answer: [0, 2)


Question: Set up and evaluate a triple integral to find the volume of the region bounded by the paraboloid \( z=1-\frac{x^2}{9}-\frac{y^2}{100} \) and the \( xy \)-plane.

Answer: 15\pi


Question: Given a natural number \( n \), evaluate the limit: \[ \lim_{x \to 0} \left( \lfloor n \cdot \frac{\sin x}{x}\rfloor + \lfloor n \cdot \frac{\tan x}{x} \rfloor \right) \]

Answer: 2n - 1


Question: What is the expected number of points in the interval \([0,1]\) for the sine point process, which is a determinantal point process with kernel \(K(x,y)=\frac{\sin \pi(x-y)}{\pi(x-y)}\) when \(x\neq y\) and \(1\) when \(x=y\)?

Answer: 1


Question: Find a function \( M(x) \) such that the differential equation \( x^2u_{xx}-2xu_x+2u=\lambda x^2u \) for \( 0<x<1 \), with boundary conditions \( u_x(0)=0 \) and \( u(1)=u_x(1) \), can be transformed into the form \( w_{xx}=-\lambda w \) using the substitution \( u(x)=M(x)w(x) \).

Answer: x


Question: Given a function $f: \Bbb Z \to \Bbb Z$ such that $f(11) = 1$ and $f(a)f(b) = f(a+b) + f(a-b)$ for all integers $a$ and $b$, find the value of $f(550)$. \( \boxed{-1} \)

Answer: -1


Question: Determine the number of equivalence relations \(S\) over the power set of natural numbers \(\mathcal P(\mathbb N)\) such that the equivalence class of \(\{8\}\) under \(S\) is \(\{A \in \mathcal P(\mathbb N) \mid A \neq \{1\} \wedge A \neq \{2\}\}\).

Answer: 2


Question: Given the joint density function \( f_{X,Y}(x,y) = \frac{2}{3} (x+2y) \) for \( 0 < x < 1, 0 < y < 1 \), find the probability \( P(X > Y) \).

Answer: \dfrac{4}{9}


Question: Find an upper bound for the expression \( \frac{x \phi(x)}{2 \Phi(x) - 1} \) for \( x \geq 0 \), where \( \phi(x) \) and \( \Phi(x) \) are the probability density function and cumulative distribution function of a standard normal distribution, respectively. Show that this expression is upper bounded by \( \frac{1}{2} \), with the bound achieved at \( x = 0 \).

Answer: \dfrac{1}{2}


Question: Find a primitive root of 338.

Answer: 7


Question: Determine the cardinality of the set of formulas in a first-order language.

Answer: \aleph_0


Question: Evaluate the integral \( \int_0^{2\pi} \frac{1}{1+8\cos^2(\theta)}d\theta \) using Cauchy's residue theorem.

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


Question: Evaluate the limit: \[ \lim_{n\to\infty}\int_{-\pi/4}^{\pi/4}\frac{n\cos(x)}{n^2x^2+1}\,dx \]

Answer: \pi


Question: Evaluate the improper integral \( \int_{-\infty}^{\infty} \frac{x^2}{x^6+9} \, dx \).

Answer: \dfrac{\pi}{9}


Question: Find the variance of the sum of 51 randomly selected numbers from the set \( \{1, 2, \ldots, 159\} \).

Answer: 73440


Question: Compute the limit: \[ \lim_{x\to\frac{2}{3}}\frac{x^2 \cos(\pi/x)}{x - \frac{2}{3}} \] without using l'Hôpital's rule.

Answer: -\pi


Question: Let $R \subseteq S$ be two integral domains where $S$ is finitely generated over $R$. Given that $Frac(S)$ is finitely generated over $Frac(R)$ and hence finite by Zariski's lemma, determine if $S$ is finite as a module over some localization $R[1/f]$ where $0 \neq f \in R$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the line integral \( \int_\Gamma \vec{F} \cdot \vec{dr} \) where \( \vec{F}(x,y,z) = (6xy + 4xz)\vec{i} + (3x^2 + 2yz)\vec{j} + (2x^2 + y^2)\vec{k} \) and \( \Gamma \) is the path \( y = x^2, z = 0 \) from \((0,0,0)\) to \((2,4,0)\) followed by the line segment from \((2,4,0)\) to \((1,1,2)\).

Answer: 9


Question: Let $S$ be an uncountable set. Let $u(S)$ denote the set of all uncountable subsets of $S$, and let $P(S)$ denote the power set of $S$, which is the set of all subsets of $S$. Does there exist an injection $f: P(S) \to u(S)$?

Answer: Yes


Question: Determine the radius of convergence for the series \( \sum_{n=0}^\infty 2^{-n}z^{n^2} \).

Answer: 1


Question: Determine whether the Weyl group $W(D_n)$ is a quotient of the Weyl group $W(B_n)$. Justify your answer.

Answer: Yes


Question: Find the volume of a truncated pyramid with a height of 10 units. The larger base is a triangle with sides measuring 27 units, 29 units, and 52 units. The smaller, truncated base has a perimeter of 72 units. Calculate the volume of the truncated pyramid.

Answer: 1900


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

Answer: \dfrac{5}{36}


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

Answer: \infty


Question: Determine the algebraic degree of \( \tan(\pi/180) \).

Answer: 48


Question: Calculate the curl of the vector field \((\mathbf{b} \cdot \mathbf{r}) \mathbf{b}\), where \(\mathbf{b}\) is a constant vector and \(\mathbf{r}\) is the position vector.

Answer: \mathbf{0}


Question: Suppose $A$ and $B$ are nonempty sets. If there are sets $C$ and $D$ such that $(A \times B) \cup (B \times A) = C \times D$, prove that $A = B = C = D$. Provide a justification for your answer.

Answer: A = B = C = D


Question: Find the sum \( \displaystyle \sum_{k = 1}^{2004}\dfrac1{1+\tan^2 \left(\dfrac{k\pi}{2\cdot 2005}\right)}. \)

Answer: 1002


Question: How many points with integer coordinates lie either wholly inside or on the perimeter of triangle $PQR$ with vertices $P=(-24,17)$, $Q=(15,-35)$, and $R=(100,169)$?

Answer: 6206


Question: Consider the set \( S = \left\{ \sum_{i=1}^k (-1)^{f(i)}i \mid k \in \mathbf{N}, f: \mathbf{N} \to \{1,2\} \right\} \). Determine whether \( S \) is equal to the set of all integers \( \mathbf{Z} \).

Answer: \mathbf{Z}


Question: Evaluate the integral \( \int_C e^z \, dz \) where \( C \) is any curve joining \( -i\pi/2 \) and \( i\pi/2 \).

Answer: 2i


Question: Consider the harmonic series defined by \( S_n = \sum_{k=1}^{n} \frac{1}{k} \). For another sequence \( \{x_k\} \), define \( S'_n = \sum_{k=1}^{n} x_k \). Is it possible for \( S'_n \) to diverge while satisfying \( \lim_{n \to \infty} \frac{S'_n}{S_n} = 0 \)?

Answer: Yes


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

Answer: -J_1(2)


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty }(n+1)\int_{0}^{1} x^n f(x) \, dx \] where \( f(x) \) is a positive real-valued continuous function on \([0,1]\).

Answer: f(1)


Question: Calculate the volume of the region between the surfaces \( z = x^2 + 9y^2 \) and \( z = 18 - x^2 - 9y^2 \) using Fubini's Theorem.

Answer: 27\pi


Question:
Prove by induction that for all integers \( n \geq 2 \), the following identity holds:
\[ \sum_{k=2}^n (k-1)(k)\binom{n}{k} = n(n-1)2^{n-2} \]

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


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

Answer: 0


Question: Consider a set of $N$ points $X = (x_i)_{i=1}^N$. Define the score function $S: X \rightarrow \mathbb{N}$ as follows: \[ S(X) = \sum_{i=1}^N S(x_i) \] where \[ S(x_i) = 2 \cdot \left| \{x_j; \ |x_i-x_j| \in [1,2]\} \right| + \left| \{x_j; \ |x_i-x_j| \in [2,3]\} \right| \] and $|\bullet|$ denotes the cardinality of the set. Additionally, it is required that for all $i \neq j$, $|x_i-x_j| \ge \frac{1}{2}$. Determine whether it is true that any configuration of $N$ points with the maximal possible score is contained within a domain of diameter $c\sqrt{N}$ for some fixed constant $c$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the limit of \((1-\sin x)^{(\tan \frac{x}{2} -1)}\) as \(x\to \frac{\pi}{2}\).

Answer: 1


Question: Determine whether the map $f: X \setminus \{0\} \to S$, defined by $f(x) = \frac{x}{\|x\|}$, is an open map for any dimension of the normed linear space $X$. Here, $S = \{x : \|x\| = 1\}$ is the unit sphere of $X$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_{-1}^{4} ||x^2 + x - 6| - 6| \, dx \).

Answer: \dfrac{71}{6}


Question: Given independent random variables $X$ and $Y$, where $X$ has a density function $p_X(t) = \frac{t}{2} \cdot I_{[0,2]}(t)$ and $Y$ is uniformly distributed on $[0,3]$, find the probability that segments of lengths $X$, $Y$, and 1 can form a triangle.

Answer: \dfrac{11}{18}


Question:
Find the natural numbers \((n, m)\) such that the following system of equations is satisfied:
\[\begin{cases} 13n - 9m = 110 \\ \text{lcm}(n, m) - \text{gcd}(n, m) = 3n + 12 \end{cases}\]

Answer: (14, 8)


Question: Calculate the value of the sum: \[ \sum_{k=0}^{19} (-1)^k \frac{1}{k+1} \binom{19}{k} \]

Answer: \dfrac{1}{20}


Question: Determine the number of solutions to the congruence equation \(x_1 + 2x_2 + 3x_3 + 4x_4 + 5x_5 + 6x_6 + 7x_7 + 8x_8 + 9x_9 + 10x_{10} \equiv 0 \pmod{11}\), where \(x_1, x_2, \ldots, x_9 \in \{0, 1, 2, \ldots, 9\}\) and \(x_{10} \in \{0, 1, 2, \ldots, 10\}\).

Answer: 1000000000


Question: For positive integers \( n \) and \( k \) (excluding \( n = 1 \)), does \( \frac{(n+k)!}{n!} \) ever equal \( n! \)?

Answer: No


Question:
Consider the polynomial function \(f(x) = x^4 - 4x^3 + 4x^2 + c\), where \(c\) is a real number. Determine which of the following statements is true:

(A) \(f(x)\) has infinitely many zeros in \((1, 2)\) for all \(c\).
(B) \(f(x)\) has exactly one zero in \((1, 2)\) if \(-1 < c < 0\).
(C) \(f(x)\) has double zeros in \((1, 2)\) if \(-1 < c < 0\).
(D) Whatever the value of \(c\), \(f(x)\) has no zero in \((1, 2)\).

Answer: B


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

Answer: e - 1


Question:
Determine the value of \( c \in \mathbb{R} \) such that \( \lim_{(x,y) \to (0,0)} f(x,y) \) exists, where
\[
f(x,y) = \begin{cases}
\frac{1-\cos(2xy)}{x^2y\sin(\pi y)} & \text{if } x \neq 0, y \neq 0, \\
c & \text{if } x = 0 \text{ or } y = 0.
\end{cases}
\]

Answer: \dfrac{2}{\pi}


Question: Calculate the Minkowski bound for the number field $\mathbb{Z}[\sqrt{8}]$. Use the formula $$M=\left(\frac{4}{\pi}\right)^{s}\frac{n!}{n^n}\left| \Delta \right|^{\frac{1}{2}}$$ where $s$ is the number of pairs of complex embeddings, $n$ is the degree of the field, and $\Delta$ is the discriminant. For $\mathbb{Z}[\sqrt{8}]$, $s=0$, $n=2$, and $\Delta=32$. What is the correct value of $M$?

Answer: 2\sqrt{2}


Question: Given two random variables $X$ and $Y$ with the joint density function \( f(x,y) = axy \mathbb{1}_{\{X>0, Y>0, X+Y \leq 1\}} \), determine the value of the parameter \( a \).

Answer: 24


Question:
Determine the smallest real value \( C \) such that for all functions \( f: [0,1] \to \mathbb{R} \) in the set \( M \), the inequality \( f(x) \le Cx \) holds for all \( x \in [0,1] \). The set \( M \) consists of functions satisfying the following properties:

- \( f(x) \ge 0 \) for all \( x \in [0,1] \),
- \( f(1) = 1 \),
- \( f(x+y) \ge f(x) + f(y) \) for all \( x, y \) such that \( x+y \in [0,1] \).

Answer: 1


Question: Evaluate the integral: $$ \int \cos x \cdot \cos^2(2x) \cdot \cos^3(3x) \cdot \cos^4(4x) \cdot \ldots \cdot \cos^{2002}(2002x) \, dx $$

Answer: 0


Question: Let \( f: [0, a] \to \mathbb{R} \) be a continuous and one-to-one function with \( f(0) = 0 \). Verify that the following equation holds: \[ \int_0^{f(a)} f^{-1}(t)\,dt + \int_0^a f(t)\,dt = af(a). \]

Answer: af(a)


Question: Calculate \(67^{26^{42^{23}}} \mod 990\).

Answer: 661


Question: Consider a random subset \( C \) of \( \mathbb{N} \setminus \{1, 2\} \), where for each \( n \in \mathbb{N} \setminus \{1, 2\} \), the probability that \( n \in C \) is given by \( P(n \in C) = \frac{1}{\ln(n)} \). The events of different numbers belonging to \( C \) are independent. This model is known as Cramer's Prime Model. Determine the probability that there exists an integer \( N \in \mathbb{N} \) such that for all \( n > N \), \( 2n \in C + C \), where \( + \) denotes pointwise summation.

Answer: 1


Question: Use the method of Lagrange multipliers to find the maximum value of the function \( f(x,y,z) = 25x^{\frac{2}{5}}y^{\frac{1}{5}}z^{\frac{1}{5}} \) subject to the constraint \( 24000 = 80x + 12y + 10z \).

Answer: 2310


Question: A piece of paper with a 1x1 cm square grid is folded into a hollow cylinder with a height of 50 cm and a base circumference of 50 cm. Can the surface of this cylinder be completely covered using a shape made of four 1x1 cm squares? Provide a yes or no answer.

Answer: Yes


Question: Simplify the expression \( \arctan \left(\frac{1}{2}\tan (2A)\right) +  \arctan (\cot (A)) + \arctan (\cot ^{3}(A)) \) for \(0< A< \frac{\pi}{4}\).

Answer: \pi


Question: In the field of surreal numbers $No$, does the infinite sum $\underbrace{\frac{1}{\omega} + \frac{1}{\omega} + \ldots}_{\omega\text{ times}}$ equal 1?

Answer: 1


Question:
Determine the correct solution to the integral equation \( y(x) - \lambda \int_{0}^{2\pi} |\pi - t| \sin(x) y(t) \, dt = x \). Choose from the following options:

1. \( y(x) = x - \lambda \pi^3 \sin(x) \)
2. \( y(x) = x + \lambda \pi^3 \sin(x) \)
3. \( y(x) = x + \lambda \pi^3 \cot(x) \)
4. \( y(x) = x + \lambda \pi^2 \sin(x) \)

Answer: 2


Question: Evaluate the integral \( \int\int_R y^{-N(x)} \, dA \), where \( R \) is the unit square \([0, 1] \times [0, 1]\) in the plane, \( \mu \) is the usual Lebesgue measure on the real Cartesian plane, and \( N(x) \) is the function that assigns to each real number \( x \) in the unit interval the positive integer that indicates the first place in the decimal expansion of \( x \) after the decimal point where the first \( 0 \) occurs. If there are two expansions, use the expansion that ends in a string of zeroes. If \( 0 \) does not occur, let \( N(x) = \infty \). For example, \( N(0.0) = 1, N(0.5) = 2, N(1/9) = \infty \), and \( N(0.4763014 \dots) = 5 \).

Answer: \infty


Question:
Consider the group $G = \{ z \in \mathbb{C} \mid z^n = 1 \text{ for some positive integer } n \}$, with group structure given by multiplication of complex numbers. Determine the nature of the group $G$ by choosing the correct option:

1. $G$ is a group of finite order.
2. $G$ is a group of infinite order, but every element of $G$ has finite order.
3. $G$ is a cyclic group.
4. None of these.

Select the correct option.

Answer: B


Question: Evaluate the integral \( \int_{|z| = 2} \frac{1}{(z^2 +1)^n} \, dz \) for \( n \in \mathbb{N} \).

Answer: 0


Question: Calculate the average square distance from the origin to a point in the region \( D = \{(x,y) \mid y^2 + 1 \le x \le 3\} \).

Answer: \dfrac{193}{35}


Question: Evaluate the integral \( \oint_{|z| = 2} \frac{e^z}{z-1} \, dz \) using the Cauchy Integral Theorem.

Answer: 2\pi i e


Question: Does a unicursal octagram exist, where all vertices of a regular octagon are connected with a single continuous trace?

Answer: Yes


Question: Let $a$ and $b$ be two integers such that $(a+b)$ is a solution to the equation $x^2 + ax + b = 0$. Find the minimal value of the product $ab$.

Answer: -54


Question: Let \( T : \mathbb{R}^4 \to \mathbb{R}^4 \) be a linear transformation such that there exists a vector \( v \in \mathbb{R}^4 \) for which \( v, T(v), T(T(v)) \) are linearly independent and \( T(T(T(v))) = 0 \). Determine the rank of the transformation \( T \).

Answer: 2


Question: Consider the sequence of random variables $X_n$ which assumes the values $1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{n}$, each with equal probability. Determine whether $X_n$ converges in probability to a constant as $n \to \infty$.

Answer: 0


Question: A tank with a diameter of $40\,\rm m$ and a height of $15\,\rm m$ is initially $50\%$ full with $92$-Octane fuel. The fuel is being drained at a rate of $500\,\rm m^3$ per hour, while $87$-Octane fuel is being added at a rate of $3000\,\rm m^3$ per hour. Assuming the fuel mixes evenly, determine how many hours it will take for the fuel in the tank to become $91$-Octane.

Answer: 0.77


Question: Adele and Bryce play a game with 50 cards numbered from 1 to 50. They take turns picking a card, with Adele going first. Adele wins if the sum of the numbers on her cards is divisible by 25. Can Adele guarantee a win?

Answer: Yes


Question: Calculate the limit: \[ \lim_{x \to -1^+}\left(\ln(x+1)+\frac{1}{x+1}\right) \]

Answer: +\infty


Question: Solve the differential equation \( y''(1+\ln(x)) + \frac{1}{x}y' = 2+\ln(x) \) given the initial conditions \( y(1) = \frac{1}{2} \) and \( y'(1) = 1 \).

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


Question: Find the correlation between the random variables $|D|$ and $U$, where $D \sim N(0,1)$ and $U$ is defined as $U = 1$ if $D \ge 0$ and $U = 0$ if $D < 0$. Calculate the correlation coefficient between $|D|$ and $U$.

Answer: 0


Question: Determine whether every sufficiently large integer can be expressed as \( a^2 + b^3 + c^6 \) for some non-negative integers \( a, b, \) and \( c \).

Answer: Yes


Question: Given that \( \sum\limits_{n=2}^\infty \frac{1}{(n^2-n)^3} = 10 - \pi^2 \), find the closed form of \( \sum\limits_{n=1}^\infty \frac{1}{n^3} \).

Answer: \zeta(3)


Question: Alice has 2010 buttons and 2010 bulbs. Each button is connected to exactly one bulb, but some bulbs may be connected to multiple buttons, and some bulbs may not be connected to any buttons. A move consists of pressing several buttons simultaneously and observing which bulbs light up (a bulb lights up if at least one of its connected buttons is pressed). What is the minimum number of moves Alice must make to determine the connection between each button and bulb?

Answer: 11


Question: Find all triples \((m, p, q)\) where \(m\) is a positive integer and \(p, q\) are primes such that \(2^m p^2 + 1 = q^5\).

Answer: (1, 11, 3)


Question: Evaluate the limit: $$\lim_{x\rightarrow 0}\frac{e^{x^{2}}-\cos(x)}{\sin^2x}$$ without using Maclaurin series or L'Hôpital's Rule.

Answer: \dfrac{3}{2}


Question: In a Poisson process with a rate of three calls per day, the fire department receives calls, one-third of which are false alarms. Firemen are paid based on the number of fires they actually fight, with an expected pay of 60 Dollars per fire. Determine the expected reward for the firemen until time t.

Answer: 120t


Question: Find the closed form solution for the integral \( \frac{1}{2\pi}\int_0^{2\pi} e^{ik(x-\sin x)} \, dx \), where \( k \in \mathbb{Z} \). The solution is related to the modified Bessel function of the first kind.

Answer: J_k(k)


Question: The digits $1, 2, \cdots, 9$ are arranged in random order to form a nine-digit number. What is the probability that this number is divisible by $11$?

Answer: \dfrac{11}{126}


Question: Evaluate the complex integral \( \int_\gamma \left( \frac{1}{z} + e^{z^2} \right) dz \), where \( \gamma(t) = e^{-it} \) for \( t \in [0, 4\pi] \). Verify if the approach of solving the integral by parametrization and geometric series is correct, and find the value of the integral.

Answer: -4\pi i


Question: Find the limit of \(\left(1+\frac{1}{a_n}\right)\left(1+\frac{1}{a_{n-1}}\right)\cdots\left(1+\frac{1}{a_1}\right)\) as \(n\) tends to infinity, where \(a_n = n(1+a_{n-1})\) and \(a_1 = 1\).

Answer: e


Question: A and B play a game where each has a 50% chance of winning any given game. The game continues until one player wins n consecutive games. What is the expected total number of games played until this occurs?

Answer: 2^n - 1


Question: Find the closure of the set \( M = \left\{ \frac{1}{n} \::\: n \in \mathbb{N} \right\} \) in the Zariski topology on the affine line \( \mathbb{A}^1 \).

Answer: \mathbb{A}^1


Question: Evaluate the double integral \( \int_0^2\int_0^{\sqrt{2x}}y^2|x-y|\,dydx \).

Answer: \dfrac{24}{35}


Question: Determine the values of \( p \in (0, \infty) \) for which the function \( f(t) = \tan(t) \) on the interval \( (0, \frac{\pi}{2}) \) belongs to the space \( \mathcal{L}^p(\lambda) \). Specifically, evaluate the integral \( \int_0^{\pi/2} \tan(t)^p \, dt \) and find the range of \( p \) for which this integral is finite.

Answer: (0, 1)


Question:
Let $f$ and $g$ be two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Determine which of the following statements is true:

(a) If $g$ is continuous, then $f \circ g$ is continuous.
(b) If $f$ is continuous, then $f \circ g$ is continuous.
(c) If $f$ and $f \circ g$ are continuous, then $g$ is continuous.
(d) If $g$ and $f \circ g$ are continuous, then $f$ is continuous.

Answer: C


Question: Given two probability measures \(\mathbb{G}^{H}\) and \(\mathbb{G}^{L}\) with the same support, and an integrable function \(\gamma\) such that \(\mathbb{G}^{H}(B)=\int_{B}\gamma(r)\mathbb{G}^{L}(dr)\) for every subset \(B\) in the Borel \(\sigma\)-algebra of \([0,1]\), where \(\gamma\) is almost surely positive, finite, different than 1, and increasing, can we assert that \(|\mathbb{G}^{H}(B)-\mathbb{G}^{L}(B)|>0\) for any half-closed interval \(B=(a,b]\) such that \(0<\mathbb{G}^{L}(B)<1\)?

Answer: No


Question: Evaluate the limit: \[ \lim_{x \to \infty} x^{\frac{x^{\ln x}}{\ln x^x}} \]

Answer: \infty


Question: Find the distance between the point \( p = (1,0,0,1) \) and the subspace \( L = (\operatorname{span}{(1,0,2,1),(2,1,2,3),(0,1,-2,1)})^\perp \) in \( \mathbb{R}^4 \).

Answer: \dfrac{\sqrt{14}}{3}


Question: Evaluate the double integral \( \iint_B x \, dx \, dy \) where \( B = \{ (x, y) : x^2 + y^2 \le x \} \).

Answer: \dfrac{\pi}{8}


Question: Consider a singular matrix $A$ of size $n \times n$ with complex entries. Is it possible to find two different unitary matrices $U_1$ and $U_2$ such that $A = U_1P = U_2P$, where $P = \sqrt{A^*A}$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim\limits_{h \to 0}\frac{\int_{0}^{\frac{\pi}{3}+h^4e^{\frac{1}{h^2}}}\cos^{3}x\,dx -\int_{0}^{\frac\pi3}\cos^{3}x\,dx}{h^4e^{\frac{1}{h^2}}}.$$

Answer: 0


Question: Let \( f(x) \) be a function such that \( f : \mathbb{R} \to \mathbb{R} \) and \( f(x) + f\left(1-\frac{1}{x}\right) = \tan^{-1}(x) \) for all real \( x \) except \( 0 \). Find the value of \( \int_0^1 f(x) \, \mathrm{d}x \).

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


Question: Let $f: [a, \infty) \to \mathbb{R}$ be a nonincreasing function. Suppose for all $t > s \geq a$ and all $c > 0$, the inequality \( f(s+c) + f(t) \leq f(t+c) + f(s) \) holds. Does this condition imply that $f$ is convex?

Answer: Yes


Question: Given the functional equation \( f(1) + f(2) + \cdots + f(n) = n^2 f(n) \) for all \( n > 1 \) and the initial condition \( f(1) = 2005 \), determine the value of \( f(2004) \).

Answer: \dfrac{1}{1002}


Question: Let $R$ be a commutative ring with unity, and consider the $R$-module $R^n$. Let $S$ be a maximal linearly independent subset of $R^n$ over $R$. Is it true that $|S| = n$? Justify your answer.

Answer: Yes


Question: Determine whether the logical statement \( \forall x\,P(x) \to \exists x\,P(x) \) is true.

Answer: true


Question: Let $X$ be a topological vector space and $f:[a,b]\to X$ be a function. We say that $f$ is of bounded semi-variation on $[a,b]$ if the set $SV(f,[a,b])$, consisting of all elements of the form \(\sum_{i=1}^n a_i[f(t_i)-f(t_{i-1})]\) where $a=t_0<t_1<\dots<t_n=b$ and $|a_i|\le 1$, is a bounded subset in $X$. Is the set $f[a,b]=\{f(x):x\in [a,b]\}$ a bounded set in $X$ whenever $f$ is of bounded semi-variation?

Answer: Yes


Question: Find the minimum value of the function \( f(x, y, z) = (x-1)^2 + \left(\frac{y}{x} -1\right)^2 + \left(\frac{z}{y} -1\right)^2 + \left(\frac{4}{z} -1\right)^2 \) subject to the constraints \( 1 \le x \le y \le z \le 4 \).

Answer: 12 - 8\sqrt{2}


Question: Evaluate the integral: \[ \int_{0}^1 \binom{207}{7} x^{200} (1-x)^7 \, dx \]

Answer: \dfrac{1}{208}


Question: Evaluate the complex integral \( \oint_\gamma \frac{\bar{z}}{z-2} \, dz \), where \( \gamma \) is the circle \( |z-2| = 2 \) with positive orientation.

Answer: 4\pi i


Question: Evaluate the contour integral \( \int_{|z|=10} \frac{e^z}{4z^2+9\pi^2}\ dz \).

Answer: -\dfrac{i}{3}


Question: Find the remainder when $652^{679}$ is divided by 851.

Answer: 717


Question: Let $V$ be a dense subset of a real Hilbert space $H$, and let $(ullet, ullet)$ denote the inner product in $H$. Prove that if $(u, v) = 0$ for all $v \in V$, then $u = 0$. Provide a detailed explanation of your reasoning.

Answer: 0


Question:
Solve for \( t \) in the equation:
\[ 5000 = 21t - 111.37e^{-\frac{12t}{6.5625}} + 1207.12 \]

Answer: 180.61


Question: Find the degree of the field extension \( \mathbb{Q}(2^{1/2}, 2^{1/4}, 2^{1/8}) \) over \( \mathbb{Q} \).

Answer: 8


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{x - \int_{0}^{x} \cos u^2 \, du}{6\sin^{-1}x - 6x - x^3} \]

Answer: \dfrac{2}{9}


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

Answer: 0


Question: In triangle ABC, angle BAC is 22 degrees. A circle with center O has tangents AB produced, AC produced, and BC. Find the measure of angle BOC in degrees.

Answer: 79


Question: Determine the value of \( \lambda \) for which the improper integral \[ I = \int_{0}^{\infty} \left(\frac{x+1}{3x^2 + \lambda} - \frac{\lambda}{2x+1}\right)dx \] converges to a finite number.

Answer: \dfrac{2}{3}


Question: Determine whether there exists a structuring element $B$ such that $X \oplus B \supseteq X \ominus B$ is not true when $B$ does not contain the origin of $\mathbb{E}$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the triple integral \( \iiint_Q \frac{1}{x^2 + y^2 + z^2} \, dV \), where \( Q \) is the solid region above the xy-plane. Use spherical coordinates to perform the integration.

Answer: \infty


Question: Consider the optimization problem: minimize \(f(x)=||Ax-b||_2^2\), where \(A\) is an \(m\times n\) matrix with \(m\geq n\), and \(b\) is a vector of length \(m\). Assume that the rank of \(A\) is equal to \(n\). If \(x_*\) is a local minimizer, then \(\nabla f(x_*)=0\). Is this condition also sufficient for optimality?

Answer: Yes


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

Answer: \dfrac{5\pi}{3}


Question:
Calculate the volume of the region defined by the inequality:
\[(2x+y+z)^2 + (x+2y+z)^2 + (x+y+2z)^2 \leq 1.\]

Answer: \dfrac{\pi}{3}


Question: Let $G$ be an undirected graph with a global bound on the degree of vertices and on the length of paths. Define an equivalence relation $v \sim u$ if there exists an automorphism $f$ of $G$ such that $f(u) = v$. Is the quotient of $G$ with respect to $\sim$ finite?

Answer: Yes


Question: Calculate the limit \( \lim\limits_{x \to 0^+} \frac{\sqrt{\sin x}-\sin\sqrt{ x}}{x\sqrt{x}} \) without using Taylor series or L'Hôpital's rule.

Answer: \dfrac{1}{6}


Question: Let $T$ be a tree with exactly two vertices of degree 7 and exactly 20 vertices of degree 100. What is the minimum possible number of vertices in the tree $T$?

Answer: 1994


Question: Find the minimum value of the functional \( J(y) = \int_a^b \sqrt{1+y'(x)^2} \, dx \) for \( y \in \mathcal{A} \), where \( \mathcal{A} \) is the set of continuously differentiable functions on the interval \([a,b]\).

Answer: b - a


Question: Given 3 types of coins, determine the number of ways to select 20 coins such that no type of coin is selected more than 8 times.

Answer: 15


Question: Does there exist a $2 \times 2$ matrix $A$ with real entries such that $\text{Det}(A) = \text{Tr}(A) = 0$ and $A$ has a square root?

Answer: Yes


Question: Given a symmetric $n \times n$ Gaussian random matrix $A$ where the entries above the diagonal are independent $N(0,1)$ random variables and the diagonal entries are independent $N(0,2)$ random variables, use Sudakov-Fernique's inequality or Gordon’s inequality to derive a bound on the operator norm of $A$. Specifically, construct a Gaussian process $Y_u$ such that $\mathbb{E}(X_u-X_v)^2 \leq \mathbb{E}(Y_u-Y_v)^2$ and $\mathbb{E}(\sup_{u\in S^{n-1}}Y_u) \leq 2\sqrt{n}$, where $X_u = \langle A u, u \rangle$ for $u \in S^{n-1}$. Provide the bound on $\|A\|$. \[ \boxed{} \]

Answer: 2\sqrt{n}


Question: Determine the density of the set \( S_1 = \{ a \mid \exists m \in \mathbb{N} ; \sigma_1(m) = a \} \) in \( \mathbb{N} \), where \( \sigma_1(n) \) is the sum of divisors function for the first power.

Answer: 0


Question: Given two fixed coprime integers $d_1$ and $d_2$ with $\frac{d_1}{d_2} \neq \pm 1$, determine if for any positive integer $n$, there exists a prime number $p$ such that the order of $d_1d_2^{-1}$ in the multiplicative group of the field $\mathbb{Z}/p\mathbb{Z}$ is greater than $n$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: If \( \lim\limits_{n\to\infty}a(n)=0 \) and \( \lim\limits_{n\to\infty}a(n)b(n)=1 \), then \( \lim\limits_{n\to\infty}|b(n)|=\infty \).

Answer: True


Question: Given the joint density function of random variables \(X\) and \(Y\), \(f_{X,Y}(x,y) = \frac{(xy-2x-2y+4)}{32}\) for \(2 \le x \le y \le 6\), find the conditional probability \(P(X > 3 \mid Y = 5)\).

Answer: \dfrac{8}{9}


Question: Evaluate the integral \( \iiint_A x^2 y^2 z \, dx \, dy \, dz \) where the region \( A \) is defined by \( A = \{ (x, y, z) \in \mathbb{R}^3 \mid 0 \leq z \leq x^2 + y^2, \ |x| + |y| = 1 \} \).

Answer: \dfrac{1}{450}


Question: Evaluate the limit: \[ \lim_{(x,y) \to (0,0)} \frac{\sin(x - y)}{\sqrt{x} + \sqrt{y}} \]

Answer: 0


Question:
Compute the limit of the determinant of the following \( n \times n \) matrix as \( n \to \infty \):
\[
\lim_{n \rightarrow \infty} \begin{vmatrix}
1+x & -x & 0 & 0 & \cdots & 0 & 0 \\
-\frac{1}{2} & 1+\frac{x}{2} & -x & 0 & \cdots & 0 & 0 \\
0 & -\frac{1}{3} & 1+\frac{x}{3} & -x & \cdots & 0 & 0 \\
0 & 0 & -\frac{1}{4} & 1+\frac{x}{4} & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\
0 & 0 & 0 & 0 & \cdots & 1+\frac{x}{n-1} & -x \\
0 & 0 & 0 & 0 & \cdots & -\frac{1}{n} & 1+\frac{x}{n} \\
\end{vmatrix}
\]

Answer: e^x


Question: Determine the number of ring homomorphisms from the ring \( \mathbb{Z}[x,y]/(x^3+y^2-1) \) to the ring \( \mathbb{Z}_7 \), where \( \mathbb{Z}_7 \) is the ring of integers modulo 7.

Answer: 11


Question: Given the differential equation \( \frac{dy}{dx} = 1-y^2 \) and the condition that the function \( y \) is symmetric about the point \((\ln(9),0)\), find the value of \( y(\ln(4)) \).

Answer: -\dfrac{65}{97}


Question: Evaluate the integral \( \int_0^{\infty}\frac{\cos{x}}{x^2+1} \, dx \) using Feynman's technique.

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


Question: Calculate the volume enclosed by the paraboloid \(4z = x^2 + y^2\) and the cone \((z+1)^2 = x^2 + y^2\) above the xy-plane using a triple integral.

Answer: \dfrac{\pi}{3}


Question: Find the formula to calculate the centroid of the solid of revolution formed by rotating the curve \( y = \sin(x) \) from \( x = 0 \) to \( x = \pi \) around the \( x \)-axis.

Answer: \left( \frac{\pi}{2}, 0, 0 \right)


Question: Compute the limit: $$\lim_{n\to\infty}\frac{a_n}{n},$$ where $a_n$ is the product of the digits of $n$ in base 10.

Answer: 0


Question: Find the shortest distance from the origin to the surface defined by the equation \( x = yz + 10 \). Use the method of Lagrange multipliers with the function \( f(x, y, z) = x^2 + y^2 + z^2 \) and the constraint \( \phi(x, y, z) = x - yz = 10 \). Calculate the shortest distance and provide your answer as a numerical value.

Answer: \sqrt{19}


Question: Determine the smallest degree of the Taylor polynomial needed to approximate \( \cos(43) \) with an error less than \( \frac{8}{60000} \). Use the Taylor series error formula: \( |R_n(x)| \leq \frac{1}{(n+1)!}|x-a|^{n+1} \). Assume the angle is in degrees.

Answer: 6


Question: Find the digit at the unit place of the integral part of the number \(\left(15+\sqrt{220}\right)^{19}+\left(15+\sqrt{220}\right)^{82}\).

Answer: 9


Question: Determine the number of distinct ways to fill a 3 × 3 grid with 0s and 1s, given that the center cell must contain a 0 and two grids are considered identical if one can be transformed into the other through a symmetry operation.

Answer: 51


Question: Evaluate the trigonometric series: \[ \tan^{2}\left(\frac{\pi}{16}\right) + \tan^{2}\left(\frac{2\pi}{16}\right) + \tan^{2}\left(\frac{3\pi}{16}\right) + \cdots + \tan^{2}\left(\frac{7\pi}{16}\right) \]

Answer: 35


Question: Given the probability density function $f(x) = 3x^2$ for a continuous random variable $X$ on the interval $[0,1]$, find the moment generating function (MGF) of $X$. Using the MGF, determine $E[X^3]$. If there is an issue with the MGF, identify and correct it to find the correct $E[X^3]$. Provide your final answer for $E[X^3]$.

Answer: \dfrac{1}{2}


Question: Show that the integral of \( h(x) = \nabla^2 g(x) \) over \( \mathbb{R}^3 \) is 1, where \( g(x) = \frac{1}{4 \pi |x|} \) and \( x \in \mathbb{R}^3 \).

Answer: 1


Question: Find all functions \( f : (0, \infty) \to (0, \infty) \) such that for all \( x, y, z > 0 \) with \( xyz = 1 \), the equation \[ f\left(x + \frac{1}{y}\right) + f\left(y + \frac{1}{z}\right) + f\left(z + \frac{1}{x}\right) = 1 \] holds.

Answer: \dfrac{1}{3}


Question: In a game, you start with two coins. Each round, you bet one coin. You have a 1/4 chance of losing the coin and a 3/4 chance of winning an additional coin. You continue playing until you either go bankrupt or play indefinitely. Define \( f(i) \) as the probability of going bankrupt with \( i \) coins. Given that \( f(0) = 1 \) and \( f(i) = \frac{1}{4}f(i-1) + \frac{3}{4}f(i+1) \), determine the probability that you will go bankrupt starting with two coins.

Answer: \dfrac{1}{9}


Question: Calculate the total work done by the force field \( F = (y^2-x^2)i+(2-x+y)j \) in moving a particle along the curve \( y = x^3 \) from \((-1,-1)\) to \((1,1)\).

Answer: \dfrac{76}{21}


Question: Determine if there is a closed form solution to the equation $ae^a = 1$.

Answer: W(1)


Question:
Find the value of \( k \) such that the length of the parabola \( p(x) = kx^2 + d \) is equal to the length of the catenary, both passing through the points \((-2, \cosh(2))\) and \((2, \cosh(2))\). The equation for the equality of lengths is given by:
\[ 2\sinh(2) = \frac{1}{2} \cdot (4k\cdot \sqrt{1+(4k)^2} + \arsinh(4k)) \]
Solve this equation for \( k \).

Answer: 0.868


Question:
Find the smallest positive integer \( x \) such that:
- \( x \equiv 1 \pmod{7} \)
- \( x \equiv 1 \pmod{9} \)
- \( x \equiv 3 \pmod{64} \)

Answer: 3907


Question:
Does a circle with circumference 1 minimize the mean squared curvature among all non-intersecting closed curves in the plane of fixed length 1 with a continuous second derivative? The mean squared curvature is given by:
\[ \langle \kappa \rangle = \int_0^1 \left|\frac{d^2 \mathbf{x}(s)}{ds^2}\right|^2 ds, \]
where the curve is parameterized by arclength.

Answer: Yes


Question: Determine if the derivative of the function \( F(x) = \int_0^x |\cos(1/u)| \, du \) exists at \( x = 0 \) and find \( F'(0) \).

Answer: \dfrac{2}{\pi}


Question: Determine whether there exists a graph homomorphism from the 6-hypercube to the 3-cube.

Answer: Yes


Question: Calculate the limit using Taylor's series: \[ \lim_{x \to \infty} \left( x - x^2 \ln\left(1+\frac{1}{x}\right) \right) \] without using L'Hospital's rule.

Answer: \dfrac{1}{2}


Question: Evaluate the definite integral \( I = \int_0^\pi \frac{\sin^2(\theta) \, d\theta}{10-6\cos(\theta)}. \)

Answer: \dfrac{\pi}{18}


Question:
Given two non-zero numbers $x$ and $y$ such that $x^{2} + xy + y^{2} = 0$, find the value of the expression:
\[ \left(\frac{x}{x + y}\right)^{2013} + \left(\frac{y}{x + y}\right)^{2013} \]

Answer: -2


Question:
Given the system of equations:
\[\begin{cases} x^2 + xy - y^2 = u \\ 2xy + y^2 = v \end{cases}\]
find the value of \( \left(\frac{\partial x}{\partial u}\right)_{v} \) at the point where \( x = 2 \) and \( y = -1 \).

Answer: \dfrac{1}{7}


Question: On a 5×5 chessboard, one square contains -1 and the other 24 squares contain +1. In one move, you may reverse the signs of all numbers in an a×a subsquare where a>1. Determine the square on which -1 should be placed so that it is possible to make all squares +1.

Answer: (3,3)


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

Answer: e^2


Question: Let \( \Omega \) be an open subset of \( \mathbb{R}^N \). Does there exist a continuous function \( g: \mathbb{R}^N \to \mathbb{R} \) such that \( \Omega = \{ x \in \mathbb{R}^N : g(x) > 0 \} \)?

Answer: Yes


Question: Determine whether the following statement is true or false: If $E \subset \mathbb{R}^n$ is a Lebesgue measurable set with finite measure, then $E$ is bounded.

Answer: B


Question: Find the derivative of the trace of the product of a matrix by its transpose, where \( A \) is a \( p \times n \) matrix. Specifically, compute \( \frac{d}{dA} \text{tr}(AA^T) \).

Answer: 2A


Question: Find the least positive integer $M$ such that $M^{77} \equiv 14 \pmod{31}$. Use Fermat's Little Theorem to assist in your solution.

Answer: 18


Question: Evaluate the integral: \[ \int_{0}^{\frac{\sqrt{2}-1}{2}} \frac{dx}{(2x+1)\sqrt{x^2+x}} \]

Answer: \dfrac{\pi}{4}


Question: Point C moves along the top arc of a circle of radius 1 centered at the origin O(0, 0) from point A(-1, 0) to point B(1, 0) such that the angle BOC decreases at a constant rate of 1 radian per minute. Determine the rate at which the area of triangle ABC changes at the moment when |AC| = 1.

Answer: \dfrac{1}{2}


Question: Is it true that any prime ideal of a finite direct product ring \(R=\prod_{i=1}^nR_i\) is of the form \(P=\prod_{i=1}^nI_i\), where \(I_j\) is a prime ideal of \(R_j\) for some \(j\) and \(I_i=R_i\), for \(i\neq j\)?

Answer: Yes


Question: Let $A$ be an algebra over a commutative ring $R$, and suppose $A$ is projective as a module over $R$. Let $M$ be a right $A$-module. Is $M$ automatically projective as a right module over $R$?

Answer: No


Question: Find the condition on positive integers \(m\) and \(n\) such that the quotient rings \(\mathbb{Z}[x,y]/(x^2-y^n)\) and \(\mathbb{Z}[x,y]/(x^2-y^m)\) are isomorphic.

Answer: m = n


Question: Let \(n\geq 1\) be an integer and suppose \(S\subseteq \mathbb{R}^n\) is a countable and dense subset. Is \(S\) homeomorphic to \(\mathbb{Q}^n\) when both sets carry the topology inherited from the Euclidean topology on \(\mathbb{R}^n\)?

Answer: Yes


Question: Find the inverse Fourier transform of the function \( \frac{1}{(1+iw)^2} \).

Answer: t e^{-t} u(t)


Question:
Determine the value of \( \mu \) that minimizes the expected cost of fines and cleanup for a bottle filling process. A bottle is considered underfilled if it contains less than 500 ml and overfilled if it contains more than 560 ml. The purchaser fines the supplier \( \$1 \) per underfilled bottle, and it costs the manufacturer \( \$2 \) to clean up an overfilled bottle. Given that the variance \( \sigma^2 = 20^2 \), the expected cost \( E[X] \) is given by:
\[
E[X] = \phi\left(\frac{500-\mu}{20}\right) + 2\left[1-\phi\left(\frac{560-\mu}{20}\right)\right]
\]
Find \( \mu \) such that the derivative of \( E[X] \) with respect to \( \mu \) is zero:
\[
0 = -\frac{1}{20} \phi\left(\frac{500-\mu}{20}\right) + \frac{2}{20}\phi\left(\frac{560-\mu}{20}\right)
\]

Answer: 525.38


Question: Given that \(z_1\) and \(z_2\) are the roots of the equation \(3z^2 + 3z + b = 0\), and the points \(O(0)\), \(A(z_1)\), and \(B(z_2)\) form an equilateral triangle, determine the value of \(b\).

Answer: 1


Question: True or False: If $f$ is a Riemann-integrable function on $[a,b]$, then the set of points where $f$ is continuous is dense in $[a,b]$.

Answer: A


Question: Calculate the value of the sum \( f(1/2009) + f(2/2009) + \ldots + f(2008/2009) \) where \( f(x) = \frac{e^{2x-1}}{1+e^{2x-1}} \).

Answer: 1004


Question: Determine the values of \( \alpha > 0 \) for which the improper integral \( \int_0^1 \frac{x^\alpha}{x+x^2} \, dx \) converges.

Answer: \alpha > 0


Question: Calculate the probability \( P(N(4) \ge 6 \mid S(4) = 3) \), where \( N(t) \) is the number of events that have occurred between 0 and \( t \), and \( S(n) \) is the arrival time of the \( n \)th event in a Poisson process.

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


Question: Given a simple graph on $n$ vertices, determine the maximum number of graphs that are isomorphic to the given graph. Is it $\Theta(n^2!)$ or $\Theta(n!)$?

Answer: \Theta(n!)


Question: Verify Green's Theorem for the region D bounded by the lines $x=2$, $y=0$, and $y=2x$ using the functions $f(x,y)=(2x^2)y$ and $g(x,y)=2x^3$. Determine if the line integral around the boundary of D equals the double integral over D.

Answer: 32


Question: Point \( P \) is inside \( \triangle ABC \). Line segments \( APD \), \( BPE \), and \( CPF \) are drawn with \( D \) on \( BC \), \( E \) on \( AC \), and \( F \) on \( AB \). Given that \( AP=6 \), \( BP=9 \), \( PD=6 \), \( PE=3 \), and \( CF=20 \), find the area of \( \triangle ABC \).

Answer: 108


Question:
Let $A$ be a $4 \times 4$ matrix such that $\text{rank}(A+I) = 3$ and $\text{rank}(A-3I) = 2$. Determine which of the following statements is true about $A$:

(a) $A$ has $0$ as an eigenvalue.
(b) $A$ is not diagonalizable.
(c) $A$ has at least $3$ different eigenvalues.
(d) $A$ has at least $3$ linearly independent eigenvectors.
(e) None of the above statements (a-d) can be inferred about $A$.

Choose the correct option.

Answer: D


Question: Determine whether the following statement is true or false: If a differentiable function $f(x)$ satisfies $(f(x))^{n} = f(nx)$ for all $x \in \mathbb{R}$, then $f'(x)f(nx) = f(x)f'(nx)$. Justify your answer.

Answer: True


Question: Let \( f: V \to W \) be a linear map and an epimorphism, meaning \( f(V) = W \). Given that \( \dim V = n \) and \( \dim W = m \), and \( E \) is a \( k \)-dimensional subspace of \( W \), find the dimension of the preimage \( f^{-1}(E) \).

Answer: n - m + k


Question: Let \( f: \mathbb{R}^n \rightarrow \mathbb{R} \) be a convex function where \( n > 1 \). Suppose for every \( x \in \mathbb{R}^n \), there exists a \( t_0 \) such that \( t_0 = \arg\min_{t \in \mathbb{R}} f(tx) \). Does there exist a point \( x_0 \in \mathbb{R}^n \) that is a global minimum of \( f \)? Note: \( t_0 \) does not need to be unique.

Answer: Yes


Question: Given a linear map $Z \in M_{7\times7}(\mathbb{R})$ such that $Z^2 - 8Z^{-1} = 0$, find the determinant of $Z$. Express your answer as a single number.

Answer: 128