Question: Let $X$ be a Hausdorff topological space, and let $f: X \to X$ be a continuous map such that $f \circ f = f$. Is it true that $f(X)$ is closed in $X$? Justify your answer.

Answer: Yes


Question: Determine the number of subfields of the splitting field of the polynomial \(x^{42} - 1\) over \(\mathbb{Q}\).

Answer: 10


Question: Find the sum of the convergent series \( \sum_{n=2}^\infty \frac{n+5}{(n^2-1)(n+2)} \).

Answer: \dfrac{7}{6}


Question: If \( p \) is a prime number and \( k \) is an even integer, what is the probability that \( p + k \) is also a prime number?

Answer: 0


Question: Let $X$ be a simply-connected $\mathbb{Z}_p$-homology $n$-sphere for some prime $p$. Is $X$ homotopy equivalent to the $n$-sphere?

Answer: No


Question: Determine the number of solutions to the equation \(x_1 + x_2 + \cdots + x_{10} = 15\) where \(x_1 \geq 1\) and \(0 \leq x_i \leq 9\) for \(i = 1\) to \(10\).

Answer: 808753


Question: Find the coefficient of $x^{15}$ in the expansion of the expression $$(x + x^2 + x^3 + x^4 + x^5) \cdot (x^2 + x^3 + x^4 + \cdots)^5.$$

Answer: 126


Question: Given the function $f: \mathbb{R} \to \mathbb{R}$ satisfying the functional equation \( f(x)f(y) = f(x+y) + f(x-y) \) and the condition \( f(1) = 3 \), find the value of \( f(7) \).

Answer: 843


Question:
Determine the maximum order of persistence of excitation for the periodic input signal $u(t)$ given by:
\[
u(t) = \begin{cases} 0, & t = 0,3 \\ 1, & t = 1,4 \\ -1, & t = 2,5 \end{cases}\]

Answer: 2


Question: Evaluate the limit: \[ \lim_{x\to0^-}{\frac{\ln(\cot(-3x))}{\ln(\cot(-2x))}} \]

Answer: 1


Question: How many sequences can be formed using the letters $E, H, I, R, S, W$ such that none of the subsequences $WIR$, $IHR$, or $SIE$ appear in the sequence?

Answer: 426


Question: Let $a$ and $b$ be real numbers such that $a^3 - 3a^2 + 5a + 11 = 0$ and $b^3 - 3b^2 + 5b - 17 = 0$. Find the value of $a + b$. \( \boxed{} \)

Answer: 2


Question: Suppose \((1+x)^{2y}\sim 1\) as \(y\to\infty\). Does this imply \(x\sim 1/y\) as \(y\to\infty\)?

Answer: No


Question:
Assume that positive random variables \(X\) and \(Y\) are identically distributed with \(E[X] = E[Y] = \mu < \infty\), and are not necessarily independent. Compute the covariance:
\[\text{Cov}\left(X+Y, \frac{X}{X+Y}\right)\]

Answer: 0


Question:
Let \(f: \mathbb{R} \to \mathbb{R}\) be a differentiable function such that \((f(x))^7 = x - f(x)\). Find the area bounded by the curve \(y = f(x)\) between the ordinates \(x = 0\) and \(x = \sqrt{3}\) and the x-axis. Choose the correct option:

(A) \(\frac{f(\sqrt{3})}{8}[8 \sqrt{3} - (f(\sqrt{3}))^7 - 4 f(\sqrt{3})]\)

(B) \(\frac{f(\sqrt{3})}{8}[8 \sqrt{3} - (f(\sqrt{3}))^7]\)

(C) \(\sqrt{3} f(\sqrt{3}) - \frac{93}{8}\)

(D) None of these.

Answer: A


Question: Let \( y = f(x) \) be an infinitely differentiable function on the real numbers such that \( f(0) \neq 0 \) and \( \frac{d^n y}{dx^n} \neq 0 \) at \( x = 0 \) for \( n = 1, 2, 3, 4 \). If \[ \lim_{x \to 0} \frac{f(4x) + af(3x) + bf(2x) + cf(x) + df(0)}{x^4} \] exists, find the value of \( 25a + 50b + 100c + 500d \).

Answer: 300


Question: Evaluate the integral \( \int_{-1}^{1} u^2(1-u^2)^{3/2} \, du \).

Answer: \dfrac{\pi}{16}


Question: Determine the value of \(a\) for which the function \(f(x) = x^3 + 3x^2 + ax + 3\) is an open map from \(\mathbb{R}\) to \(\mathbb{R}\).

Answer: 3


Question: Emails arrive from two professors at rates of 4 emails per hour and 6 emails per hour, respectively. What is the expected time you will have to wait until you receive one email from each professor?

Answer: \dfrac{19}{60}


Question: Consider a decreasing sequence of bounded open sets \(V_{n}\) in \(\mathbb{R}^{m}\) with \(m \geq 1\). Suppose \(\cap V_{n} = \emptyset\) and \(F := \cap \overline{V_{n}}\) is connected. Is there an \(N\) such that each \(x \in F\) belongs to the boundary of \(V_{n}\) for all \(n \geq N\)?

Answer: Yes


Question: Evaluate the integral: \[ \int_{-1}^1 \! \int_0^{\sqrt{1-x^2}} \! \int_0^{\sqrt{1-y^2-x^2}} \! 1 \, dz  \, dy  \, dx \]

Answer: \dfrac{\pi}{3}


Question: Solve for \(x\) in the equation \(a^x + b^x = c\), where \(a, b \in (0,1)\) and \(c \in (0,2)\), with \(x \in \mathbb{R}_{>0}\).

Answer: x


Question: Calculate the line integral \( \oint\limits_{C} 2x + y^2 \, ds \) using polar coordinates, where \( C = \{(x,y) \in \mathbb{R}^2 \mid x^2 + 3y^2 = 8\} \). Verify that the result is \( \frac{8\pi}{3} \).

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


Question: Minimize the function $2x + y$ subject to the constraint $x^2 + y^2 = 1$. Use the Karush-Kuhn-Tucker (KKT) conditions to find the solution.

Answer: -\sqrt{5}


Question: Evaluate the line integral \( \oint\limits_C \ \frac{x-y}{x^2+y^2}dx+\frac{x+y}{x^2+y^2}dy \) over the closed path \( C \), where \( C \) is formed by the line \( x=2 \) and the parabola \( y^2=2(x+2) \).

Answer: 2\pi


Question: Determine the number of zeros (counting multiplicities) of the polynomial \( f(z) = z^{100} + 8z^{10} - 3z^3 + z^2 + z + 1 \) inside the closed unit disc \( |z| < 1 \).

Answer: 10


Question: Assume ZFC. Let $B \subseteq \mathbb{R}$ be a set that is not Borel-measurable. Can one conclude that $B$ necessarily has the cardinality of the continuum without assuming either the continuum hypothesis or its negation?

Answer: No


Question: Two merchants share a telephone package such that one keeps \( \frac{2}{7} \) of the total and the other \( \frac{5}{7} \). The first resells them in batches of 51 and the second in batches of 31. When they finish, the first has 46 telephones left and the second 19. What is the lowest initial number of telephones that makes this possible?

Answer: 1589


Question: Given that the quotient of the division of the polynomial \( P(2x) \) by \( P(x) \) is 16, find the quotient of the division of \( P(3x) \) by \( P(x) \). Assume \( P(x) = \sum_{1\leq i\leq n}\alpha_i x^i \in \mathbb{R}[x] \).

Answer: 81


Question: Given the recurrence relation \( a_{r+1} = \sqrt{\frac{1}{2}(a_r + 1)} \) for \( r \in \{0\} \cup \mathbb{N} \), find the value of \( \chi = \cos\left(\frac{\sqrt{1-a_0^2}}{\prod_{k=1}^{\infty}a_k}\right) \).

Answer: a_0


Question:
Suppose we have three random variables $X, Y,$ and $Z$ such that $P(X > Y) = P(Y > Z) = \frac{2}{3}$. Determine which of the following statements is correct:

1. $P(Z > X) > 0.5$.
2. $X-Z$ has a positive mean.
3. The median of $X$ is larger than the median of $Z$.
4. The median of $Y$ is smaller than that of $X$.
5. None of the above is correct.

Answer: 5


Question: Find the asymptotic bound \( \theta \) for the recurrence relation \( T(N) = N + T(N-3) \) given that \( T(1) = \theta(1) \).

Answer: \Theta(N^2)


Question: Let \( n \ge 2 \) and \( f: S^{n} \to \mathbb{R} \) be a continuous function, where \( S^n \) is the n-dimensional sphere. Define the set \( A = \{t \in f(S^{n}) \mid f^{-1}(t) \text{ is finite}\} \). Prove that the set \( A \) has cardinality at most 2.

Answer: 2


Question: Consider a collection $(X_a)_{a\in[0,1]}$ of i.i.d. random variables following the uniform distribution on [0,1]. For every real number $a \in [0,1]$, there is a random variable $X_a$. Determine the probability of the event that at least one of the variables $X_a$ equals $1/2$. What is this probability?

Answer: 0


Question: Calculate the integral \( \int_{[0,1]\times [0,1]} xy \, dx \, dy \) using the definition of the Riemann integral. Use a partition of \([0,1]\times[0,1]\) and compute the Riemann sum to find the value of the integral.

Answer: \dfrac{1}{4}


Question: A general population follows an exponential distribution with parameter \( \lambda \). A sample of six elements was drawn: \( 3.2, 2.5, 0.1, 5.1, 4.7, 4.1 \). Using the most-powerful test, verify the null hypothesis that \( \lambda = \frac{1}{2} \) against the alternative hypothesis that \( \lambda = \frac{1}{3} \). Assume a significance level of \( \alpha = 0.05 \). Determine the critical value \( c_2 \) for the test statistic \( \sum_{i=1}^{n} x_{i} \).

Answer: 21.026


Question:
Given that \(\alpha\) and \(\beta\) are continuous and differentiable functions satisfying the following conditions:
\[\alpha'(x) = \beta(x),\]
\[\beta(x)^3 = (\alpha(x) - 1)^2,\]
\[\alpha(a) = 0,\]
\[\alpha(b) = 2,\]
compute the definite integral:
\[\int_a^b{\alpha(t)^2 \beta(t)^4 \,dt}.\]

Answer: \dfrac{16}{15}


Question: Determine if the function \( f(x) = \begin{cases}\ln |x|+1 & x<0\\\ln |x| & x>0\end{cases} \) is a valid counterexample to the claim that the antiderivative of an odd function is even. The derivative of \( f(x) \) is \( \frac{1}{x} \), which is an odd function. Is \( f(x) \) an even function?

Answer: No


Question: Given a sequence of complex numbers $(a_n)_{n\ge 0}$ such that $\lim_{n\to \infty} a_n=L\neq 0$, evaluate the limit: $$\lim_{x\to 1^-} (1-x)\sum_{n\ge 0} a_n x^n.$$

Answer: L


Question: Let $f(x,y)$ be a function analytic with respect to both $x$ and $y$. Assume $U$ is some open set in $\mathbb{R}$. If $f: U \times U \rightarrow \mathbb{R}$ is such that for a fixed $y$, the function $x \mapsto f(x,y)$ is analytic, and for a fixed $x$, the function $y \mapsto f(x,y)$ is analytic, is it true that for a fixed $y$, the function $x \mapsto \frac{\partial}{\partial y} f(x,y)$ is also analytic?

Answer: Yes


Question: In the ring \( \mathbb{Z}_n = \{0, 1, \ldots, n-1\} \) under addition and multiplication modulo \( n \), suppose \( a, b, c, x \in \mathbb{Z}_n \) are nonzero and the cyclic order \( R(a, b, c) \) holds. Under what conditions does \( R(ax, bx, cx) \) hold?

Answer: \gcd(x, n) = 1


Question: Determine whether the expected value of the product of three dependent normal variables, $X_1, X_2, X_3 \sim N(0, \sigma_i)$, is zero. Specifically, is it true that $E[X_1X_2X_3] = 0$?

Answer: 0


Question: Suppose that $(f_n)_{n=1}^\infty$ and $(g_n)_{n=1}^\infty$ are two sequences of functions that converge uniformly to some limits. If the supremum norm $\|f_n - g_m\| = \sup_y |f_n(y) - g_m(y)| \to 0$ as $n, m \to \infty$, do $(f_n)_{n=1}^\infty$ and $(g_n)_{n=1}^\infty$ converge uniformly to the same limit? Justify your answer.

Answer: Yes


Question: Suppose $f:[0,T]\to X$ is a measurable map where $X$ is a Hilbert space. Let $R(t):X \to X^*$ be an isometric isomorphism such that \(\lVert R(t)f(t)\rVert_{X^*} = \lVert f(t) \rVert_X\) is also measurable. Is the function \(t \mapsto R(t)f(t)\) measurable?

Answer: No


Question:
Determine whether the following proposition is true or false:

Proposition: If a continuous function $f(x)$ defined on $[c,b)$ is differentiable on $(c,b)$ and if $\lim_{x\to c+0} f'(x)=+\infty$ or $\lim_{x\to c+0} f'(x)=-\infty$, then $f(x)$ is not differentiable at $c$. Provide a justification for your answer.

Answer: True


Question: Evaluate the limit: \[ \lim_{x \to 0} (\sqrt{4+x}-1)^{1/(e^x-1)} \] without using l'Hôpital's rule, utilizing notable limits instead.

Answer: e^{\frac{1}{4}}


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

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_{0}^{1}\frac{x-1}{\ln x}dx \).

Answer: \ln 2


Question: Is a projection on a closed subspace of a Banach space always bounded?

Answer: No


Question:
Convert the following triple integral from rectangular coordinates to both cylindrical and spherical coordinates and evaluate the simplest iterated integral:
\[\int_{-1}^1\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\int_1^{1+\sqrt{1-x^2-y^2}}x\,dz\,dy\,dx\]

Answer: 0


Question: Suppose $f: [0, \infty) \to \mathbb{R}$ is a continuous function that is nearly eventually almost periodic with periods $1$ and $a$, where $a$ is irrational and $0 < a < 1$. Define $F: [0, 1) \to \mathbb{R}$ by $F(x) = \lim_{n \to \infty} f(x + n)$. Is $F$ necessarily constant almost everywhere?

Answer: Yes


Question: Given that \( f(x) = \int_{0}^{x} \sqrt{f(t)} \, dt \), where \( f: [0, \infty) \to [0, \infty) \) is continuous on \([0, \infty)\) and differentiable on \((0, \infty)\), find the value of \( f(6) \).

Answer: 9


Question: Consider the polynomial \( p(z) = z^7 + z(z-3)^3 + 1 \). Determine the number of zeros of \( p(z) \), including multiplicities, within the region \( B_1(3) \), where \( B_1(3) \) denotes the open ball of radius 1 centered at 3. Use Rouché's Theorem to solve the problem without the aid of a calculator.

Answer: 0


Question: Let $G$ be a finite group and $N \triangleleft G$. Assume that $xN$ is an element of prime power order in the quotient group $G/N$. Does there exist an element of prime power order in the coset $xN$?

Answer: Yes


Question: Determine a non-trivial lower bound for the 'de Polignac constant' \( K_{Pol} = \sum_{i>0}2^{-k_{i}} \), where \( k_{i} \) is the \( i \)-th positive integer such that \( 2k_{i} \) is a Polignac number. A Polignac number is a number that is the difference of two consecutive primes in infinitely many ways. Is there a known non-trivial lower bound for \( K_{Pol} \)?

Answer: 2^{-123}


Question: Let \( f, g \in L^2 \). Find the limit \( \lim_{t \to \infty} \frac{\|f+tg\|_2-\|f\|_2}{t} \).

Answer: \|g\|_2


Question: Determine whether there exist integers \(a, b, c\) such that \(a^7 + b^7 + c^7 = 45\).

Answer: No


Question: Evaluate the limit: $$\lim_{x \to 0}\left( \frac{1}{x^2} - \cot^2x\right)$$

Answer: \dfrac{2}{3}


Question: Evaluate the integral \( I = \int_0^{\pi} e^{\cos\theta}\cos(\sin\theta)\ d\theta \).

Answer: \pi


Question: Determine whether there exists a map \( F: S^{1}\times\cdots\times S^{1} \rightarrow S^{n} \) (where there are \( n \) copies of \( S^{1} \)) that has a nonzero degree.

Answer: Yes


Question: Let $G$ be a finite abelian group and $C(G,\mathbb{C})$ the group ring on $G$ with multiplication defined as convolution $\ast$ and unit $\delta_0$. Consider a $|G|$-element subset $B \subseteq C(G,\mathbb{C})$ consisting of orthogonal idempotents with respect to $\ast$, such that $\sum_{b \in B} b = \delta_0$. Is it necessarily true that $B = \widehat{G}$, where $\widehat{G}$ denotes the character group of $G$?

Answer: No


Question: Does every zero-dimensional commutative ring have a bounded index of nilpotency? A commutative ring is called zero-dimensional if all its prime ideals are maximal, and a ring is said to have a bounded index of nilpotency if there is a positive integer $n$ such that $x^n = 0$ for every nilpotent $x$ in the ring.

Answer: No


Question: If $B \subseteq A \subseteq X$, is the uniform subspace $B$ of $X$ identical to the uniform subspace $B$ of the uniform subspace $A$ of $X$?

Answer: Yes


Question: Given the permutation \( t = (1\; 2)(3\; 4)(5\; 6)(7\; 8)(9\; 10) \), determine whether there exists an \( n \)-cycle \( s \) (where \( n \geq 10 \)) such that \( t = s^k \) for some integer \( k \).

Answer: Yes


Question: Compute the determinant of the linear transformation $T: V \to V$ defined by $T(X) = AX + XA$, where $A = \begin{pmatrix}1 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1 \end{pmatrix}$. What is $\det T$?

Answer: 0


Question: Find the Lebesgue measure of the Borel set \( A = \bigcup_{n=1}^{\infty} [n, n + \frac{1}{n}] \).

Answer: \infty


Question: Given that the real roots of the polynomial \(x^3 - 3x + 1\) are \(\alpha, \beta,\) and \(\gamma\), find the value of the cyclic expression \(\alpha^2 - \gamma + \beta^2 - \alpha + \gamma^2 - \beta\).

Answer: 6


Question: Let $M_{1}$ and $M_{2}$ be 2-dimensional, connected, smooth, orientable manifolds of finite topological type. Consider a smooth, surjective map $F: M_{1} \rightarrow M_{2}$ such that the pre-image of each point in $M_{2}$ is finite. Additionally, assume there exists a constant $K > 0$ such that $|F^{-1}(p)| \leq K$ for all $p \in M_{2}$. Must $F$ locally be a branched covering?

Answer: No


Question: Describe the quotient group \( \mathbb{Z} \times \mathbb{Z} / \langle (1, 2) \rangle \).

Answer: \mathbb{Z}


Question: Evaluate whether the expression \( \int xe^x \, dx = \sum\limits_{n=2}^{+\infty}(-1)^n\frac{x^n}{n!}e^x + C \) is a correct representation of the integral of \( xe^x \).

Answer: Correct


Question: If $X$ admits a universal covering space and $\alpha$ and $\gamma$ are homotopic paths between $x$ and $p(y)$, is the loop $\alpha*\gamma^{-1}$ nullhomotopic?

Answer: Yes


Question:
Evaluate the expression:
\[ \frac{\sum_{i=0}^{100}{{k}\choose{i}}{{m-k}\choose{100-i}}\frac{k-i}{m-100}}{{m}\choose{100}} \]
where:
- \(m-k > 100\)
- \(k > 100\)

Choose the correct answer:
- \(\frac{k}{m}\)
- \(\frac{m}{k}\)
- \(\frac{k}{m^2}\)
- \(\frac{m}{k}\)

Answer: \frac{k}{m}


Question: Given the implicit function defined by \( f(x, y, z) = \exp(xyz) \) and the condition \( f(x, y, g(x, y)) = 2 \) with \( g(1, 1) = \ln 2 \), calculate the partial derivatives \( \frac{\partial g}{\partial x}(1,1) \) and \( \frac{\partial g}{\partial y}(1,1) \).

Answer: -\ln 2


Question: Let $c=2^{\aleph_0}$, $x=2^c$, $y=2^{2^c}$, and $z=2^{2^{2^c}}$. Arrange the expressions $x^{y^z}$, $x^{z^y}$, and $y^{z^x}$ in ascending order.

Answer: x^{z^y} = y^{z^x} < x^{y^z}


Question: Evaluate the limit: \[ \lim_{r\to 0}\frac{1}{r}\int_{1}^{\exp(4r)}\sqrt{1 + \frac{3}{x}}\mathrm{d}x \]

Answer: 8


Question: Evaluate the line integral \( \int\limits_c dz \) from \((1,2)\) to \((2,1)\), where the exact differential is given by \( dz = 3x^2(x^2+y^2)dx + 2y(x^3+y^4)dy \).

Answer: \dfrac{8}{5}


Question: Determine the minimum number of colors needed to color an infinite grid such that no square has all four of its vertices the same color. Assume the grid consists of all integer coordinate points \((p, q)\) where \(p, q \in \mathbb{Z}\).

Answer: 4


Question:
Let $T_1$ and $T_2$ be non-zero linear transformations from $V$ to $W$. Consider the following statements:

P: If $R(T_1) \cap R(T_2) = \{0\}$, then $T_1$ and $T_2$ are linearly independent.

Q: If $N(T_1) \cap N(T_2) = \{0\}$, then $T_1$ and $T_2$ are linearly independent.

Determine which of the following is true:

1. P and Q both are false.
2. P and Q both are true.
3. P is true and Q is false.
4. P is false and Q is true.

Answer: 3


Question: Determine the value of the constant \( k \) such that the function \( f(x) = \frac{k|x|}{(1+|x|)^4} \) is a probability density function. This requires that \( \int_{-\infty}^{\infty} f(x) \, dx = 1 \).

Answer: 3


Question: Evaluate the limit: \[ \lim_{z \to 1+i} \frac{z^{2} -4z+4+2i}{z^{2}-2z+2} \]

Answer: 1+i


Question:
Determine the number of vectors \((x, x_1, x_2)\) with \(x, x_1, x_2 > 0\) that satisfy the equation:
\[
\left | \log(xx_{1}) \right |+\left | \log(xx_{2}) \right |+\left | \log\frac{x}{x_{1}} \right |+\left | \log\frac{x}{x_{2}} \right |= \left | \log x_{1}+\log x_{2} \right |
\]

Answer: 1


Question: Calculate the flux of the curl of the vector field \( F(x, y, z) = (x - yz, xz, y) \) across the surface \( S \), which is obtained by rotating the graph of \( x = 2^z + 3^z \) for \( z \in [0, 1] \) around the \( z \)-axis, with normal vectors oriented outward.

Answer: 50\pi


Question:
Evaluate the contour integral \( \oint_{|z|=1}\frac{2\Re(z)}{z+1}dz \) over the positively oriented unit circle. Choose the correct answer from the following options:

(a) 0
(b) \( \pi i \)
(c) \( 2\pi i \)
(d) \( 4\pi i \)

Answer: a


Question: Find the infimum of the function \( f(x) = (|\log x| - 1)x^{\frac{1}{2}} \) for \( x \in (0, \infty) \) using derivatives. For \( x > 1 \), the derivative is given by \( \frac{d}{dx} = \frac{\log x + 1}{2x^{\frac{1}{2}}} \), and for \( x < 1 \), the derivative is \( \frac{d}{dx} = \frac{-\log x - 3}{2x^{\frac{1}{2}}} \).

Answer: -1


Question: Given the spherical metric \(ds^2 = dr^2 + r^2(d\theta^2 + \sin^2(\theta) d\phi^2)\), determine if the covariant derivative \(\nabla_\theta \theta\) is equal to 1.

Answer: 0


Question: Evaluate the line integral \( \int_C (x + xy + y) \, ds \) where \( C \) is the path of the arc along the circle given by \( x^2 + y^2 = 4 \), starting at the point \((2,0)\) and going counterclockwise, making an inscribed angle of \( \frac{7}{6}\pi \).

Answer: 3 + 2\sqrt{3}


Question: Compute the limit \( \lim_{n \to \infty} n\left(\frac{{\pi}^2}{6} - \sum_{k=1}^{n} \frac{1}{k^2} \right) \).

Answer: 1


Question: Evaluate the integral \( F(x) = \int \frac{(1+x) \lfloor (1-x+x^2)(1+x+x^2)+x^2 \rfloor}{1+2x+3x^2+4x^3+3x^4+2x^5+x^6} \, dx \). Find the value of \( \lfloor F(99) - F(3) \rfloor \).

Answer: 3


Question: Determine whether the space of continuous functions $C[a,b]$ with the inner product $\langle x(t),y(t)\rangle=\int_a^b x(t)y(t)dt$ is separable.

Answer: Yes


Question:
Determine whether the infimum of the Riemann Upper Sum is equal to the limit of the Upper Sum as the partition size approaches zero. Specifically, consider the Riemann upper sum \( U_\Gamma = \sum_{k=1}^N[\sup_{x\in I_k}f(x)]v(I_k) \), where \( \Gamma = \{I_k\} \) is a partition of \( I \) into finite nonoverlapping intervals, and \( |\Gamma| = \max_k(\text{diam } I_k) \). Are the following two expressions equal?

1) \( \inf_\Gamma U_\Gamma \)
2) \( \lim_{|\Gamma|\to 0} U_\Gamma \)

Answer: Yes


Question: Given a continuous function $f$ such that \( \int f(x)\phi''(x) \, dx = 0 \) for all test functions \( \phi \in D(\mathbb{R}) \), determine the type of function $f$. Provide a proof for your answer.

Answer: f(x) = ax + b


Question: In a game with successive levels, the probability of a player passing the first level is \( \frac{1}{2} \), the probability of passing the second level given they passed the first is \( \frac{1}{3} \), and the probability of passing the third level given they passed the second is \( \frac{1}{4} \). What is the expected level at which a random player stops?

Answer: \dfrac{5}{3}


Question: A liquid drug is marketed in phials containing a nominal 1.5 ml, but the amounts can vary slightly. The volume in each phial is modeled by a normal distribution with a mean of 1.55 ml and a standard deviation of \(\sigma\) ml. The phials are sold in packs of 5 randomly chosen phials. It is required that in less than 0.5% of the packs, the total volume of the drug is less than 7.5 ml. Find the greatest possible value of \(\sigma\).

Answer: 0.0434


Question: Let $\frac{1}{a_1}$, $\frac{1}{a_2}$, $\frac{1}{a_3}$, ... be a sequence of positive numbers defined by: \(a_1=1, a_{n+1}=a_n+\frac{1}{a_n}\). Find the integer part of $a_{100}$.

Answer: 14


Question: Let $(a_{i,j})_{i,j\in\mathbb{N}}$ be a double sequence such that for each $k < \infty$, \(\sup_{1\leq i\leq k}|a_{i,j}| \to 0\) as \(j \to \infty\), and \(a_{i,i} \to 0\) as \(i \to \infty\). Does it follow that \(\sup_{1\leq i\leq k}|a_{i,k}|\to 0\) as \(k \to \infty\)?

Answer: No


Question: Evaluate the definite integral: \[ \int_0^4\sqrt{x^2+4}\,dx + \int_2^{2\sqrt{5}}\sqrt{x^2-4}\,dx \]

Answer: 8\sqrt{5}


Question: Find the maximum value of \( \operatorname{Re} \left\{ \frac{x^* Ax}{x^* x}:0 \ne x \in \mathbb{C}^n\right\} \) for the matrix \( A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \).

Answer: 2


Question: Determine the radius of convergence of the series \( \sum_{n=0}^\infty a_n z^n \), where \( a_n \) is the number of pairs of integers \((x,y)\) such that \( x^2 + y^2 \leq n^2 \).

Answer: 1


Question: Find the maximum value of the expression \( P = \frac{4}{\sqrt{a^2+b^2+c^2+4}} - \frac{9}{(a+b)\sqrt{(a+2c)(b+2c)}} \) where \( a, b, c > 0 \).

Answer: \dfrac{5}{8}


Question: If $a(x)$ and $b(x)$ are rational functions in $x$ with complex coefficients, and $b(x)^2 = a(x)^3 + 1$, does it necessarily follow that $a(x)$ and $b(x)$ are constant? Justify your answer.

Answer: Yes


Question: Let $G$ be a group and $x$ be the only element in $G$ such that $x^2 = e$, where $e$ is the identity element of $G$. Prove that $x \in Z(G)$, where $Z(G) = \{ u \in G \mid ua = au \text{ for all } a \in G \}$.

Answer: x \in Z(G)


Question: Determine the values of the real parameter \( \alpha > 0 \) for which the function \( \frac{1}{|x|^{\alpha} + |y|^{\alpha} +1} \) is in \( L^2(\mathbb{R}^2) \).

Answer: \alpha > 1


Question: Determine whether there exists a left-Ore ring that does not have the invariant basis number (IBN) property.

Answer: Yes


Question: Let \( a \in (0,1] \) satisfy the equation \( a^{2008} - 2a + 1 = 0 \). Define \( S \) as \( S = 1 + a + a^2 + a^3 + \cdots + a^{2007} \). Find the sum of all possible values of \( S \).

Answer: 2010


Question: Determine the range of the function \( f(z) = |1-z| + |1+z^2| \) where \( z \) is a complex number such that \( |z| = 1 \).

Answer: [\sqrt{2}, 4]


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

Answer: \dfrac{\pi}{8}


Question:
Evaluate the limit \( \lim\limits_{x\to 0} f(x) \) where
\[
 f(x) = \frac{e^{x^{2}} \sin x - x \left(1+\frac{5 x^2}{6}\right)}{x^{5}}
\]

Answer: \dfrac{41}{120}


Question: Suppose $f \in L(E)$ and $f(x) > 0$ for all $x \in E$. Show that \( \lim_{k \to \infty} \int_{E}(f(x))^{\frac{1}{k}}dx = m(E) \).

Answer: m(E)


Question: Consider a convex polytope \(P\) in \(\mathbb{R}^d\) with \(f\) facets. Let \(\text{Proj}(P)\) be the projection of \(P\) into \(\mathbb{R}^2\). Can \(\text{Proj}(P)\) have more than \(f\) facets?

Answer: Yes


Question: Determine the Galois group of the polynomial \( f(x) = x^5 - x + 1 \) over the finite field \( \mathbb{F}_7 \).

Answer: \mathbb{Z}/6\mathbb{Z}


Question: Determine the asymptotic behavior of the function \( CL(n) \), where \( CL(n) = \max \{len_c(G) \mid |G| \leq n \} \) and \( len_c(G) \) is defined as the minimum sum of the sizes of supports of permutations in a generating set of a finite group \( G \) isomorphic to a subgroup of \( S_\infty \). Given the bounds \( CL(n) = O(n \log(n)) \) and \( CL(n) = \Omega(n) \), is either bound tight?

Answer: CL(n) = \Theta(n)


Question: Given the sequence defined by $x_1 = 18$ and $x_{n+1} = \frac{10}{3}x_n - x_{n-1}$, find the value of $x_2$ if the sequence $(x_n)$ converges.

Answer: 6


Question: Identify the ring structure of $\mathbb{Z}[x]/(x^2+3, 5)$. What is this ring isomorphic to?

Answer: \mathbb{F}_{25}


Question: Under the assumptions that $Z \to X$ is a closed immersion of schemes and both $\mathcal{O}_Z$ and $\mathcal{O}_X$ are coherent sheaves of $\mathcal{O}_Z$, resp. $\mathcal{O}_X$-modules, is it true that if $\mathcal{F}$ is a finitely presented quasi-coherent $\mathcal{O}_Z$-module, then $i_*\mathcal{F}$ is also finitely presented?

Answer: Yes


Question: Let \(\alpha\) and \(\beta\) be cycles in \(S_n\). If there exists an element \(i_1\) that is moved by both \(\alpha\) and \(\beta\), and if \(\alpha^k(i_1) = \beta^k(i_1)\) for all positive integers \(k\), prove that \(\alpha = \beta\).

Answer: \alpha = \beta


Question: Does the triple $(j_\ast A_X^\ast,j_\ast F^\bullet,\tau^{\leq\bullet} j_\ast A_X^\ast)$ induce the canonical mixed Hodge structure on $H^\ast(X)\simeq H^\ast(\bar{X},j_\ast A_X^\ast)$?

Answer: No


Question: Consider a line $y = kx + b$ where $k < 0$ and $b > 0$, which is tangent to the parabola $y = x^2 - 4x + 4$. Find the maximal area of the region bounded by the lines $y = kx + b$, $y = 0$, and $x = 0$. Provide your answer as a single numerical value.

Answer: \dfrac{64}{27}


Question: Given the function \( f(x) = \int_{0}^{x} \frac{1}{\sqrt{1+t^3}} \, dt \), let \( g(x) \) be the inverse of \( f(x) \). Find the value of \( \lambda \) such that \( \lambda g''(x) = g(x)^2 \).

Answer: \dfrac{2}{3}


Question:
Determine the period of the limit cycle for the system of differential equations given by:
\[ \frac{d}{dt}v(t) = -x(t) - \frac{1}{2}v(t)(x(t)^2 + v(t)^2 - 1) \]
\[ \frac{d}{dt}x(t) = v(t) \]
where the limit cycle is described by the equation \(x^2 + v^2 = 1\).

Answer: 2\pi


Question: Given $n$ points $A_1, A_2, \ldots, A_n$ on the unit circle in $\mathbb{R}^2$, determine whether there exists a point $M$ on the unit circle such that the product of the distances from $M$ to each point $A_i$ is equal to 1, i.e., $\prod_{i=1}^n MA_i = 1$. Provide a justification for your answer.

Answer: Yes


Question:
Given the partial differential equation:
\[ \frac{\partial}{\partial t} f(x,y,t) = x \frac{\partial f}{\partial y} - y \frac{\partial f}{\partial x} \]
and the initial condition:
\[ f(x,y,0) = x^2 + y^2 \]
determine the function \( f(x,y,t) \) for all time \( t \).

Answer: x^2 + y^2


Question: Let $\mathbb{F}$ be an ordered field, and let $a, b, q, r \in \mathbb{F}$ with $a < b$. Consider the polynomial $f(x) = x^3 + qx + r$. Given that $f(a) > 0$, $f(b) > 0$, and the discriminant $\Delta = -4p^3 - 27q^2$ of $f$ is negative, determine if $f(x) > 0$ for all $x \in [a, b]$. Is this statement true?

Answer: True


Question: Evaluate the integral \( \frac{1}{2\pi i}\int_{C} |1+z+z^2|^2 \, dz \), where \( C \) is the unit circle centered at the origin in \( \mathbb{C} \).

Answer: 2


Question: Determine the norm of the linear functional \( f(x) = \int_0^1 x(t) \, dt - x(1/2) \) on the space \( X = C[0,1] \) of continuous functions with the max norm.

Answer: 2


Question: Is it possible to construct a free abelian group $G$ of finite rank $n$ such that $G$ has a proper subgroup $H$ of rank $n$? Justify your answer.

Answer: Yes


Question: Given a multiset of symbols \((A, A, B, B, B, C, D, D)\), find the nth permutation of the symbols when sorted in alphabetical order.

Answer: BDCADABB


Question: Find the locus of the third vertex of a triangle formed by three tangents to the parabola \(y^2=4ax\), given that two of the vertices lie on the parabola \(x^2=4by\).

Answer: x^2 = 4by


Question: Prove that for any integer \( n \), the expression \( n^{9^9} + 4 \) is not divisible by 37.

Answer: n^{9^9} + 4


Question: Given that \( \lim_{n \rightarrow \infty} a_n(1 - c_n) = L > 0 \) and \( \lim_{n \rightarrow \infty} c_n = 1 \), calculate the limit \( \lim_{n \rightarrow \infty} a_n(1 - c_n^{\frac{1}{m}}) \) where \( m > 1 \) is an integer.

Answer: \dfrac{L}{m}


Question: Suppose $0 \leq f(x)$ on $[a,b]$. If $f^2(x)$ is Riemann integrable on $[a,b]$, can we deduce that $f(x)$ is also Riemann integrable on $[a,b]$?

Answer: Yes


Question: Determine the number of ways to choose three subsets \(A, B,\) and \(C\) of \(\{1, \ldots, n\}\) such that \(A \subseteq B \subseteq C\).

Answer: 4^n


Question: Let $\Gamma$ be a discrete group acting continuously on a metric space $X$. Let $a_1, a_2 \in X$, and suppose $a_2$ is a limit point of the orbit $\Gamma a_1$. Suppose $a_3$ is a limit point of the orbit $\Gamma a_2$. Must $a_3$ also be a limit point of $\Gamma a_1$? Assume that the metric on $X$ is nonarchimedean, and that the closure of any orbit of $\Gamma$ is compact.

Answer: Yes


Question: Let $A \in \mathbb{R}^{n \times n}$ be a matrix where the column vectors $k_i \in \mathbb{R}^n$ for $i = 1, 2, \ldots, n$ satisfy the condition $k_i = (i+2)k_{i+2}$ for $i = 1, 2, \ldots, n-2$. Given that $n > 3$, choose one eigenvalue of $A$ and determine the smallest possible dimension of the eigenspace corresponding to the chosen eigenvalue.

Answer: 1


Question: Let \( \{a_n\}_{n\geq1} \) be an absolutely convergent sequence. Determine whether the sequence \( K_n = \frac{1}{\ln(n^2+1)}\sum_{k=1}^{n}a_k\frac{3k^3-2k}{7-k^3}\sin k \) is convergent.

Answer: 0


Question: Find the coefficient of $x^{11}$ in the expansion of $(1+x^2)^4 (1+x^3)^7 (1+x^4)^{12}$.

Answer: 1113


Question: Given a continuous map $f: D^2 \rightarrow S^2$ such that $f(-x) = -f(x)$ for $x \in S^1$, determine if the degree of the restriction $f|_{S^1}$ is zero. Justify your answer.

Answer: 0


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

Answer: \dfrac{\pi}{6}


Question: In triangle \(ABC\), where \(AB = 3\), \(BC = 4\), and \(CA = 5\), the area of \(\triangle ABC\) is 6 square units. If point \(A\) is reflected across point \(B\) to become \(A'\), point \(B\) is reflected across point \(C\) to become \(B'\), and point \(C\) is reflected across point \(A\) to become \(C'\), what is the area of triangle \(A'B'C'\)?

Answer: 42


Question: Given that $A$ and $B$ are independent random variables uniformly distributed over $(0,1)$, consider the quadratic equation $x^2 + 2Ax + B = 0$. If both roots of this equation are real, what is the probability that they lie within the unit disc?

Answer: \dfrac{1}{4}


Question: Evaluate the limit: \[ \lim_{y \to 0^+} (\cosh (3/y))^y \]

Answer: e^3


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

Answer: (\ln 2)^2


Question: Evaluate the limit: $$ \lim_{x\to\infty} \frac { (1+2 \sqrt[6]{x})^{2016} } {1+(2 + (3+4x^6)^7)^8} $$

Answer: 2^{1904}


Question: Is it possible to approximate a function $f \in L^p([a,b])$ arbitrarily well with simple functions of the form $\sum_{k=1}^m \alpha_k \chi_{A_k}$, where each $A_k$ is an interval, given that $X=[a,b]\subset \mathbb{R}$ is equipped with the Borel $\sigma$-algebra and the Lebesgue measure?

Answer: Yes


Question: Find the positive root of the equation \(x = 5 - 5e^{-x}\) given that one root is \(x_1 = 0\).

Answer: 4.965


Question: In an isosceles triangle $ABC$ with $AB = BC$ and $	riangle ABC$ having an angle $B$ measuring $48^\circ$, points $P$ and $Q$ are marked on the median $BM$. The extension of $AQ$ intersects $BC$ at point $F$. Given that $B, P, Q,$ and $M$ form a harmonic range, calculate the measure of angle $PFB$.

Answer: 66^\circ


Question: Consider a group $G$ of order $35$ acting on a set $S$ with $16$ elements. Determine whether the action must have a fixed point, and justify your answer.

Answer: Yes


Question: Determine the greatest common divisor (gcd) of the polynomials $X$ and $Y$ in the ring $k[X,Y]$, where $k$ is a unique factorization domain (UFD).

Answer: 1


Question: Determine whether there exists a finite non-commutative local ring $R$ with identity such that the Jacobson radical $J(R)$ is simple as a left module.

Answer: Yes


Question: Calculate the line integral \( \int_{\partial F} 2xy\, dx + x^2 \, dy + (1+x-z) \, dz \) for the intersection of the surfaces \( z=x^2+y^2 \) and \( 2x+2y+z=7 \). The curve should be traversed clockwise with respect to the origin.

Answer: 18\pi


Question: Determine the range of \(a\) such that the function \(f(x) = \int_{0}^{x} |\log_{2}(\log_3(\log_4(\cos t + a)))|\, dt\) is increasing for all real values of \(x\).

Answer: (5, \infty)


Question: Given an $n$-bit binary number with the property \((a2^{n-1}+b2^{n-2}+k)\mod 3 = p\), where the two most significant bits $a$ and $b$ cannot both be 1 concurrently, determine if it is possible to uniquely calculate $a$ and $b$ given $k$, $n$, and $p$. Provide a 'yes' or 'no' answer.

Answer: Yes


Question: Let $A$ and $B$ be $n \times n$ matrices. If for every $x \in \mathbb{R}^n$, $Ax = 0$ implies $Bx = 0$, which of the following statements is true?\n\n(a) There exists an $n \times n$ matrix $C$ such that $A = BC$.\n(b) There exists an $n \times n$ matrix $C$ such that $A = CB$.\n(c) There exists an $n \times n$ matrix $C$ such that $B = CA$.\n(d) There exists an $n \times n$ matrix $C$ such that $B = AC$.\n(e) There exists an invertible $n \times n$ matrix $C$ such that $B = C^{-1}AC$.\n\nChoose the correct option.

Answer: C


Question: Evaluate the integral \( \int_{|z|=1} \frac{1}{z^2 -\frac{3}{2}z + 1} \, dz \) using the residue method, given that the roots of the denominator are \( z=\frac{3}{4} \pm i \frac{\sqrt{7}}{4} \). Note that these roots lie on the boundary \(|z|=1\).

Answer: 0


Question:
Given two random variables $X$ and $Y$, consider the following conditions:

1. There exists a (not necessarily measurable) function $f: \mathbb{R} \to \mathbb{R}$ such that $Y = f(X)$.
2. There exists a Borel-measurable function $f: \mathbb{R} \to \mathbb{R}$ such that $Y = f(X)$ almost surely.

Does the first condition imply the second condition?

Answer: Yes


Question: Is there a (1,1) bridge knot in $S^3$ or a lens space such that its complement is hyperbolic?

Answer: Yes


Question: Verify that \( \lim_{h \to 0} \frac{f(x + 2h) - 2f(x+h) + f(x)}{h^{2}} = f''(x) \).

Answer: f''(x)


Question: Does there exist an irreducible unitary representation \( \pi \) of the universal covering group \( G \) of \( \mathrm{SL}_2({\mathbb R}) \) such that its central character is injective?

Answer: Yes


Question: Find the minimum value of \( \frac{a}{b + 1} + \frac{b}{a + 1} + \frac{1}{a + b} \) where \( a, b > 0 \) and \( a + b \le 1 \).

Answer: \dfrac{5}{3}


Question: Calculate the determinant of the matrix \([d_{i,j}]_{n \times n}\), where for every \(i,j \in \{1,2,...,n\}\), \(d_{i,j} = \min\{i,j\}\).

Answer: 1


Question: Calculate the limit: \[ \lim_{n \to \infty} \left(\frac{n^2 + 8n - 1}{n^2 - 4n - 5}\right)^n \] without using logarithm rules.

Answer: e^{12}


Question: Find the radius of convergence for the power series \( \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1} \) using the formula \( R = \frac{1}{\limsup |a_n|^{1/n}} \).

Answer: 1


Question: Given the function \( g(x) \) such that \( g(x+1) + g(x-1) = g(x) \) for all \( x \in \mathbb{R} \), determine the value of \( p \) for which \( g(x+p) = g(x) \).

Answer: 6


Question: Given two independent random variables, $X \sim \text{Normal}(1,1)$ and $Y \sim \text{Gamma}(\lambda,p)$ with $E(Y) = 1$ and $\text{Var}(Y) = 1/2$, find $E((X+Y)^3)$. Assume $X$ and $Y$ are independent.

Answer: \dfrac{35}{2}


Question: Evaluate the double integral \( \iint_E x \, dx \, dy \) over the region \( E = \{(x,y) \mid 0 \le x, 0 \le y \le 1, 1 \le x^2 + y^2 \le 4\} \).

Answer: \dfrac{3}{2}


Question: Let $X$ be a local complete intersection subvariety of codimension two in $\mathbb{P}^n$. By Bertini's theorem, there exists an irreducible hypersurface $Y$ containing $X$. Can $Y$ be chosen such that $X$ is a Cartier divisor on $Y$?

Answer: Yes


Question: Determine whether the inequality \( \frac{n^2}{x}-\left\lfloor\frac{n^2}{x}\right\rfloor+\frac{2n+1}{x}-\left\lfloor\frac{2n+1}{x}\right\rfloor < 1 \) holds for all integer values of \( x > 2n+1 \), given that \( n > 0 \) and \( 0 < x < n^2 \).

Answer: No


Question: Evaluate the integral \( I = \int_{0}^{\infty} \delta(\sin(\pi x)) \cdot 2^{-x} \, dx \), where \( \delta(b(x)) \) is the Dirac delta function with \( b(x) = \sin(\pi x) \).

Answer: \dfrac{2}{\pi}


Question: Let $\xi_{1},\xi_{2},\ldots$ be a sequence of independent random variables such that $P(\xi_{i} = 2^i) = \frac{1}{2^i}$ and $P(\xi_{i} = 0) = 1 - \frac{1}{2^i}$ for $i \geq 1$. Find the almost sure value of the limit $\lim_{n\to \infty}(\xi_{1}+\cdots+\xi_{n})/n$. Apply the Law of Large Numbers to determine the limiting value.

Answer: 0


Question: True or False: If \( \sum a_n \) is a conditionally convergent series, then there exists an increasing sequence of integers \( k_1, k_2, k_3, \ldots \) such that the sequence \( \left( \sum_{n=1}^{k_1} a_n, \sum_{n=k_1+1}^{k_2} a_n, \sum_{n=k_2+1}^{k_3} a_n, \ldots \right) \) is an alternating sequence of decreasing terms.

Answer: True


Question: Find the minimum value of \( n \) such that for any prime number \( p > 3 \), the expression \( 105 \mid (9^{p^2} - 29^p + n) \) holds true.

Answer: 20


Question: Evaluate the expression \(\frac{(5+6)(5^2+6^2)(5^4+6^4)\cdots(5^{1024}+6^{1024})+5^{2048}}{3^{1024}}\).

Answer: 12^{1024}


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

Answer: \infty


Question: Determine whether the statement is true or false: A matrix $A = QQ^{T}$, where $Q$ is a square matrix, is invertible if and only if $A$ is positive definite.

Answer: A


Question: Does every primitive digraph have a directed cycle?

Answer: Yes


Question: Let \( f(x) \) be a non-constant rational polynomial such that \( f(f(x)) = 3f(x)^4 - 1 \). Determine the polynomial \( f(x) \).

Answer: 3x^4 - 1


Question:
Determine the set of real values of \( p \) for which the equation \( | 2x + 3 | + | 2x - 3 | = px + 6 \) has more than two solutions. Choose the correct option:
A) \( (4, 0) \)
B) \( \mathbb{R} \setminus \{4, 0, -4\} \)
C) \( \{0\} \)
D) None

Answer: C


Question: Calculate the limit: \[ \lim_{n\to \infty} \frac{1}{n^{n}}\int_{0}^{n}x^n\arctan(\sqrt[n]{x})dx \]

Answer: \dfrac{\pi}{4}


Question: Find the Fourier series representation of \( \cos(x) \) over the interval \([-\pi, \pi]\).

Answer: \cos(x)


Question: Determine the radius of convergence of the series \( \sum_{n=0}^{\infty}\frac{(-1)^n z^{(n)}}{n!} \), where \( z^{(n)} \) denotes the Pochhammer symbol.

Answer: 1


Question: Determine whether the function \( f(x) = e^{-x} \) is Lebesgue-integrable over the interval \([0, \infty)\).

Answer: Yes


Question: Does the set of irrational numbers, \( \mathbb{R} \setminus \mathbb{Q} \), contain a connected proper subset that is not a single point?

Answer: No


Question: Evaluate the limit: \[ \lim_{x \to 0} \int_{0}^{x} \frac{\cos(t^3)}{t+x} \, dt \]

Answer: \ln 2


Question: If a function $f(x)$ is absolutely continuous on the interval $[a,b]$, is the function $e^{f(x)}$ also absolutely continuous on $[a,b]$?

Answer: Yes


Question: Calculate the total curvature of the logarithmic spiral given by the parameterization \( \exp(t) \cdot (\cos(t), \sin(t)) \) for \( t \in [0, 2\pi] \).

Answer: 2\pi


Question: Determine whether the following statement is true: For distinct odd primes \(p\) and \(q\), \((pq,(p-1)(q-1)) =1\) if and only if \((pq,\operatorname{lcm}(p-1,q-1))=1\).

Answer: True


Question: Given the functions $f: \mathbb{R}^2 \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}^2$, with $g(0) = (1,2)$, $g'(0) = (-1,4)$, and $(f \circ g)'(0) = -3$, and knowing that $\frac{df}{dx}(1,2) = \frac{df}{dy}(1,2)$, compute the value of $\frac{df}{dx}(1,2)$. Use the property that $(f \circ g)'(0) = f'(g(0)) \cdot g'(0)$. What is $\frac{df}{dx}(1,2)$?

Answer: -1


Question: Determine whether the following statement is true: For distinct prime numbers $p_1, p_2, \ldots, p_n$ and positive integers $r_1, r_2, \ldots, r_n$, is it true that $C_{p_1^{r_1} p_2^{r_2} \ldots p_n^{r_n}} \cong C_{p_1^{r_1}} \times C_{p_2^{r_2}} \times \ldots \times C_{p_n^{r_n}}$?

Answer: True


Question: Find the derivative of \( \log(\sec(x^2)) \) from first principles.

Answer: 2x \tan(x^2)


Question: Given the velocity field of a fluid defined by \( u = kx, \, v = -ky, \, w = 0 \) where \( k \) is a constant, and the density \( \rho = \rho_0 + Aye^{kt} \), find the rate of change of density for each individual fluid particle. Here, \( \rho_0 \) and \( A \) are constants.

Answer: 0


Question: Given that \( f(x + y) = f(x) + f(y) + 5xy \) and \( \lim_{h\to0}\frac {f(h)}{h}=3 \), find \( f'(x) \).

Answer: 5x + 3


Question: Given vectors $\alpha_1, \dots , \alpha_{m}$ in an $n$-dimensional Euclidean space, such that $(\alpha_i, \alpha_j) \leq 0$ for $i\neq j$ and $(\alpha_i,\alpha_i)\neq 0$, find the maximum value of $m$. Provide a justification for your answer.

Answer: 2n


Question: Prove that for any prime number $p \geq 5$, the sum $1^2 + 2^2 + 3^2 + \ldots + (p-1)^2$ is divisible by $p$.

Answer: 0


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

Answer: \ln 2 - 1


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} \frac{3}{n} \sum_{k=1}^n \left(\frac{2n+3k}{n} \right)^2 \]

Answer: 39


Question: Determine whether there exists a unique group homomorphism from the additive group \((\mathbb{Q}, +)\) to the additive group \((\mathbb{Z}, +)\).

Answer: Yes


Question: Given two Banach spaces $X$ and $Y$, and a bounded surjective linear operator $T: X \to Y$, can we always find a linear map $g: Y \to X$ such that $T \circ g = \text{Id}_Y$?

Answer: Yes


Question: Let $F$ be a closed subset of the real line without isolated points. Can $F$ be expressed as a union of closed intervals with positive length?

Answer: No


Question: Given the joint density function \( f(x,y) = \frac{e^{-x/y}e^{-y}}{y} \) for \( x, y > 0 \), calculate the conditional probability \( P(X > 1 \mid Y = y) \).

Answer: e^{-1/y}


Question: Evaluate the integral \( \int_0^1 \int_0^1 \cdots \int_0^1 \lfloor x_1 + x_2 + \cdots + x_n \rfloor \, dx_1 \, dx_2 \cdots dx_n \), where the integrand is the floor function of the sum of the variables.

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


Question:
Consider the series \( \sum_{n=1}^\infty \frac{\cos(n\pi)}{\ln(6n)} \). Select the first correct reason from the list below why the given series converges:

A. Geometric series
B. Comparison with a convergent p-series
C. Integral test
D. Ratio test
E. Alternating series test

Answer: E


Question: Determine the number of solutions to the equation \(X^\top X = B^\top B\), where \(B \in \mathbb{R}^{d \times n}\) with \(d \geq n\) and \(B^\top B\) is non-singular.

Answer: \infty


Question: Evaluate the integral \( \dfrac{1-|a|^{2}}{\pi}\int_{|z|=1} \dfrac{|dz|}{|z+a|^{2}} \) where \( a \in \mathbb{C} \) and \( |a| < 1 \).

Answer: 2


Question: Find the 39th derivative of the function \( f(x) = e^{-x^2} \) evaluated at \( x = 0 \).

Answer: 0


Question: Let $(A,d_1)$ and $(B,d_2)$ be two metric spaces where $A$ is compact. Suppose $f: A \to B$ is a continuous bijective function. Is the inverse function $f^{-1}$ continuous?

Answer: Yes


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a continuous function such that \( f(f(x)) = 1 + x \). Find the value of \( f'(0) \).

Answer: 1


Question:
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the functional equation:
\[ f(x + f(x + y)) = f(2x) + y, \quad \forall x, y \in \mathbb{R} \]

Answer: f(x) = x


Question: Let \( \{x_n\} \) be a bounded sequence of real numbers such that \( 2x_n \le x_{n-1} + x_{n+1} \) for all \( n \in \mathbb{N} \). Prove that \( \lim_{n \to \infty} (x_{n+1} - x_n) = 0 \).

Answer: 0


Question: Does a function \( f \) that is uniformly continuous on the interval \((a, b)\) necessarily have a limit \( \lim_{x \to a^+} f(x) \)? Provide a justification for your answer.

Answer: Yes


Question:
Let $S_n$ be the group of $n$-permutations, and denote the number of inversions of $\sigma \in S_n$ by $\ell(\sigma)$. Assume $n > 2$. Determine whether the following cancellation property holds:
\[ \sum_{\sigma \in S_n} (-1)^{\ell(\sigma)} \sum_{i=1}^n i(i-\sigma(i)) = 0. \]

Answer: 0


Question: Find the function \( f(x) \) that satisfies the q-difference equation \( a(x)f(x) = f(qx) \), where \( q \) is a real, positive number less than 1, and \( a(x) \) is a known, continuous function with \( a(0) = 0 \). Additionally, \( f(0) = 0 \) and the solution for \( x < 0 \) does not depend on the solution for \( x > 0 \). For example, consider the equation \( xf(x) = f(qx) \). Determine \( f(x) \).

Answer: 0


Question: Given $g(x) = 2x^2 - 2x - 1$, find a function $f(x)$ such that $(f \circ g)(x) = -8x^6 + 24x^5 - 8x^4 - 24x^3 + 6x^2 + 10x + 6$. Determine $f(x)$. $f(x) =$

Answer: -x^3 + x^2 + 4


Question: Let $\Sigma$ be the splitting field for $x^5 - 7$ over $\mathbb{Q}$. Determine how many roots of the polynomial $x^4 + 7x^3 + 6x^2 + 3x + 3$ lie in $\Sigma$. Provide your answer as a single integer.

Answer: 1


Question: Find a group $G$ such that the map $\varphi: G \to \mathrm{Aut}(G)$, defined by $g \mapsto f_g$ where $f_g$ is conjugation by $g$, is an isomorphism.

Answer: S_3


Question: Determine the number of solutions to the equation \(f(x) = 0\), where \(f(x) = \sum\limits_{i=1}^{2020}\frac{i^2}{x-i}\).

Answer: 2019


Question: Is it possible for a non-deterministic Turing machine in state $q_1$ reading symbol $a$ to have two different transitions: one to $(q_{accept}, <space>, L)$ and another to $(q_{reject}, <space>, L)$?

Answer: Yes


Question: Assume a remailer reorders messages and maintains a pool of \( n-1 \) messages at all times. When the \( n \)th message arrives, one of the \( n \) messages is selected at random and forwarded. An attacker sends enough messages to force the \( n-1 \) messages in the original pool to be sent. Assuming the message to be sent is chosen according to a uniform random distribution, what is the expected number of messages the attacker would have to send to achieve this goal?

Answer: n H_{n-1}


Question: Use Parseval's identity to evaluate the integral \( \int_{-\pi}^{\pi}\left|\sum_{n=1}^{\infty}\frac{1}{2^n}e^{inx}\right|^2 dx \).

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


Question: Evaluate the integral \( \int_{|z+1|=2} \frac{z^2}{4-z^2}dz \) using the residue theorem.

Answer: 2\pi i


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

Answer: 0


Question: Evaluate the integral \( \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{\cos^2 x}{1+a^x} \, dx \), where \( a \) is a constant.

Answer: \dfrac{\pi}{4}


Question:
Calculate the limit of the function as \(x\to\infty\):
\[ \lim_{x\to\infty}\left(\frac{c+\sqrt{x}}{-c+\sqrt{x}}\right)^x\exp(-2c\sqrt{x}) \] where \(c\) is a constant.

Answer: 1


Question:
Determine the value of \( p \) such that the following limit converges:
\[ \lim _ { x \to \infty } \left( \left( n + \left( \frac { x ^ 2 + 2 } { x ^ 2 - x + 1 } \right) ^ { 2 x + 1 } \right) \cdot \sin \left( \frac { x \pi } 2 \right) \right) \]

Answer: -e^2


Question: Evaluate the integral \( \int_{0}^{1} (1-x^{\frac{1}{a}})^{-\frac{1}{2}} \, dx \) as \( a \to 0 \).

Answer: 1


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

Answer: e^{-1/3}


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{1 - \cos(1 - \cos x)}{x^4} \]

Answer: \dfrac{1}{8}


Question:
Identify the function whose Taylor series is given by \( \sum^\infty_{n=0}\frac {x^{4n}}{(2n)!} \). Choose from the following options:

(a) \( e^{x^4} \)
(b) \( 1+ x^2\frac {\sin(x^2)}{2} \)
(c) \( \cos^2(x^2) \)
(d) \( (1+x^2)\ln(1+x^2) \)
(e) \( \frac {1}{2} (e^{x^2} + e^{-x^2}) \)

Answer: E


Question: Compute the Zariski tangent space of the parabola $X = V(y - x^2) \subset \mathbb{A}^2_k$ at the origin using the definition $T_0X = (\mathfrak{m}_{X,0}/\mathfrak{m}_{X,0}^2)^{\ast}$. Determine the quotient $\mathfrak{m}_{X,0}/\mathfrak{m}_{X,0}^2$ and its dual.

Answer: k


Question: Evaluate the limit: \[ \lim_{x \to 0^+} x \int_x^1 \frac{\cos t}{t^2} \, dt \]

Answer: 1


Question: Evaluate the integral \( \int_{|z|=1} \frac{\sin(z)}{z} \mathrm{d}z \) using Cauchy's integral formula.

Answer: 0


Question: If two vector spaces $V$ and $W$ are isomorphic, do they necessarily have the same dimension? Provide a justification for your answer.

Answer: Yes


Question: Determine whether the function \( \varphi(x) = \int_2^x \frac{1}{\log(t)}e^{-\frac{1}{\sqrt{t}}}~dt \) is an asymptotic lower bound for the prime counting function \( \pi(x) \).

Answer: Yes


Question: For a given integer $n \in \mathbb{N}$, determine the smallest integer $m \in \mathbb{N}$ such that the inequality \[ 3^{3^{3^{3^{\cdots^{3}}}}} \text{ (with $m$ layers of 3) } > 9^{9^{9^{9^{\cdots^{9}}}}} \text{ (with $n$ layers of 9) } \] holds true.

Answer: n + 1


Question: Determine whether the cardinality of the set of sequences of functions from \( \mathbb{R} \) to \( \mathbb{R} \) is \( \beth_1 \), where \( \beth_0 \) is the cardinality of the natural numbers, \( \beth_1 \) is the cardinality of the continuum, and \( \beth_2 \) is the cardinality of the set of all functions from \( \mathbb{R} \) to \( \mathbb{R} \).

Answer: \beth_2


Question:
Given two linear transformations $S : \mathbb{R}^3 \to \mathbb{R}^4$ and $T : \mathbb{R}^4 \to \mathbb{R}^3$ such that $T \circ S$ is the identity mapping on $\mathbb{R}^3$, determine the nature of the mapping $S \circ T$ on $\mathbb{R}^4$. Choose the correct option:

- It is the identity mapping.
- It is injective but not surjective.
- It is surjective but not injective.
- It is neither injective nor surjective.

Answer: D


Question: Evaluate the integral \( \int_{C}\frac{1}{1+\sqrt{z+2}} \) where \( C \) is the positively oriented unit circle. Determine if Cauchy's Theorem can be applied to conclude the value of the integral.

Answer: 0


Question: How many ways are there to distribute 16 identical balloons and 11 different toys to 9 different children, ensuring that each child receives at least one toy?

Answer: 308255312534400


Question:
Is the identity map the only function \( f: \mathbb{Z}_{+} \to \mathbb{Z}_{+} \) that satisfies the following properties?

1. \( f(mn) = f(m)f(n) \) for all positive integers \( m \) and \( n \).
2. \( f(m) < f(n) \) whenever \( m < n \).
3. \( f(p) \) is prime for any prime number \( p \).

Determine whether the identity map is the unique function that meets these criteria.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{(x,y) \to (0,0)} \frac{e^{-\frac{1}{x^2+y^2}}}{x^4+y^4} \]

Answer: 0


Question: Calculate the gradient of the function \( f(X) = X^{T}X \) using both the directional derivative and the product rule. Verify if the expressions obtained from both methods are equivalent. If not, identify the discrepancy.

Answer: 2X


Question: Determine the group \( I := \langle a, b, c \ | \ a^2, b^2, c^2, (ab)^2, (ac)^3, (bc)^7, (abc)^{19}, (abcbc)^{25} \rangle \).

Answer: 1


Question: Evaluate the infinite series: \[ \sum_{n=2}^{\infty}\frac{n^4+3n^2+10n+10}{2^n(n^4+4)} \]

Answer: \dfrac{11}{10}


Question: Find the value of the following summation: \( \sum\limits_{k=1}^{10}\left(\sin{\frac{2k\pi}{11}}-i\cos\frac{2k\pi}{11}\right) \).

Answer: i


Question: Calculate the limit of the series: \[ \sum_{n=0}^{\infty} \sum_{k=0}^{n} \frac{1}{3^k} \cdot \frac{1}{2^{n-k}} \]

Answer: 3


Question: Determine the value of the infinite power tower \( \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdots}}} \).

Answer: 2


Question: Find the second derivative of the function \( f(x) = \frac{\sin x}{x - \pi} \) at \( x = \pi \).

Answer: \dfrac{1}{3}


Question: Find the values of \( \lambda \) where the functions \( f(\lambda) = e^{\frac{\ln(\frac{-\lambda\theta^k}{3\lambda-2T})}{k}} \) and \( g(\lambda) = \frac{\ln(\frac{-\lambda}{3\lambda-2T})}{k}+\theta \) intersect, given that \( \theta, k, \) and \( T \) are positive real numbers.

Answer: \dfrac{T}{2}


Question: Evaluate the limit using Taylor expansion: \[ \lim_{x\to 0}\frac{\sinh x^4-x^4}{(x-\sin x)^4} \]

Answer: 216


Question:
Determine the correct predicate formula to express that a set \( p \) is equal to the set \( \{a, b\} \). Choose between the following options:

1. \( p = \{a, b\} ::=  \forall z \Big((z \ne a \land z \ne b) \implies z \not \in p\Big) \)
2. \( p = \{a, b\} ::=  \forall z \Big((z \ne a \land z \ne b) \iff z \not \in p\Big) \)

Which one is correct?

Answer: 2


Question: Let $U$ be an open subset of $\mathbb{C}$ that is bounded, and let $z_1,z_2 \in U$ be distinct points. Determine if there exists an analytic function $f: U \rightarrow \mathbb{D}$ such that $|f(z_1)-f(z_2)| = \sup \{|g(z_1)- g(z_2)| : g: U \rightarrow \mathbb{D} \text{ analytic} \}$.

Answer: Yes


Question: Determine the range of values for \( p \) such that the series \( \sum_{n=2}^{\infty} \frac{1}{n (\ln n)^p} \) converges.

Answer: p > 1


Question: Determine the prime numbers \(p\) for which the set \(\mathfrak{B} = \{(1,2,0)^t, (2,1,2)^t, (3,1,1)^t\}\) forms a basis of \(\mathbb{F}_p^3\).

Answer: p \neq 7


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\int^{1}_{0}x^{2019}\cdot \sin (nx)\,dx \]

Answer: 0


Question: Let \( \{X_n, n=1,2,\ldots\} \) and \( \{Y_n, n=1,2,\ldots\} \) be two sequences of random variables, where elements from \( X_n \) are independent of elements from \( Y_n \). Suppose \( X_n \) converges in probability to \( X \), and \( Y_n \) converges in probability to \( Y \). Does \( (X_n - Y_n)^2 \) converge in probability to \( (X - Y)^2 \)?

Answer: Yes


Question: Compute \( \Delta^*\Delta_*\mathcal{O}_X \) for the diagonal embedding \( \Delta:X\to X\times X \) of a variety \( X \).

Answer: \mathcal{O}_X


Question: Is the symmetric group $S_n$ a normal subgroup of the general linear group $GL(n,\mathbb{R})$ when $S_n$ acts on $\mathbb{R}^n$ by permuting the coordinates?

Answer: No


Question: Let $G$ be a nonabelian finite simple group of Lie type over a finite field $F$, and let $s \in G$ be a semisimple element, i.e., an element with order coprime to $\text{Char}(F)$. Suppose $T$ is a maximal torus of $G$ containing $s$. Is it true that $T \subseteq C_G(s)$, where $C_G(s) = \{g \in G : sg = gs\}$?

Answer: Yes


Question: Find the radius of convergence of the power series \( \sum_{n=0}^{\infty} p(n) a_n z^n \), where \( p(x) \) is a polynomial of degree \( d \) and \( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = R > 0 \).

Answer: \dfrac{1}{R}


Question: What is the covariance of $B_t$ and $B_{t^2}$, where $B_t$ is the standard Brownian Motion?

Answer: \min(t, t^2)


Question: Evaluate the surface integral \( \iint_D \langle x,y,-2\rangle \cdot \mathbf{n} \, dS \), where \( D \) is the surface defined by \( z = 1 - x^2 - y^2 \) with \( x^2 + y^2 \leq 1 \), oriented upwards. Use an appropriate parameterization to find the flux over this surface.

Answer: -\pi


Question: Evaluate the definite integral \( \int_0^2 x^2 \,d(x^2) \). Determine whether the limits of integration should be applied to \( x \) or \( x^2 \).

Answer: 8


Question: Find the minimum value of \( P = \dfrac{1}{(a-b)^2} + \dfrac{1}{(b-c)^2} + \dfrac{1}{(c-a)^2} \) given that \( a, b, c \in \mathbb{R}^+ \) and \( a^2 + b^2 + c^2 = 3 \).

Answer: \dfrac{15}{4}


Question: Find the maximum volume of a box with dimensions \(x\), \(y\), and \(z\) such that the sum of its sides \(x + y + z = k\) using Lagrange multipliers.

Answer: \dfrac{k^3}{27}


Question: In an isosceles right triangle $ABC$ with $BC$ as the hypotenuse and $AB = AC = 6$, point $E$ is on $BC$ and point $F$ is on $AB$. Find the length of $EF$ such that the sum $AE + EF + FC$ is minimized.

Answer: \sqrt{5}


Question: Given a number $m$, such as $9874$, determine the number of natural numbers less than $m$ that do not contain the digit $3$. Provide your answer for $m = 9874$. Note that $1 \leq m < 10^{16}$.

Answer: 6455


Question: Determine the number of integers \( n \) with \( 1 \leq n \leq 2017 \) such that \( 323 \) divides the expression \( 20^n + 16^n - 3^n - 1 \).

Answer: 1008


Question: Given that the radius of convergence of the power series \( \sum_{n=1}^{\infty}c_{n}z^{n} \) is finite and nonzero, find the radius of convergence of the series \( \sum_{n=1}^{\infty}c_{n}z^{n^{2}} \).

Answer: 1


Question:
Given three unit vectors \( \vec{a}, \vec{b}, \vec{c} \) such that \( |\vec{a} + \vec{b} + \vec{c}| = \sqrt{3} \), find the maximum value of the expression:
\[
(\vec{a} \times \vec{b}) \cdot (\vec{b} \times \vec{c}) + (\vec{b} \times \vec{c}) \cdot (\vec{c} \times \vec{a}) + (\vec{c} \times \vec{a}) \cdot (\vec{a} \times \vec{b}).
\]

Answer: 0


Question: Let $X$ be a set. Define mappings $F, G : \text{Map}(X, X)\times\text{Map}(X, X)\to\text{Map}(X, X)$ such that $F(f, g) = f\circ g$ and $G(f, g) = g\circ f$. What is the necessary and sufficient condition on $X$ that ensures $F=G$?

Answer: |X| \leq 1


Question: Let \( A \) and \( B \) be \( L \)-algebras, where \( L \) is an arbitrary set of operation symbols, and let \( h: A \to B \) be a homomorphism. Is the set \( h(A) = \{h(a) : a \in A\} \) the universe of a subalgebra of \( B \)? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sum_{k=0}^n \frac{\binom{n}{k}}{n^k(k+1)} \]

Answer: e - 1


Question:
Evaluate the determinant:
\[
\left|
\begin{array}{ccc}
  b^2c^2 & bc & b+c \\
  c^2a^2 & ca & c+a \\
  a^2b^2 & ab & a+b
\end{array}
\right|
\]

Answer: 0


Question:
Solve the equation for real numbers \(x\), given that \(a\) and \(b\) are real:
\[
\left| a^2-2a-b^2-4b-x \right| + \left| a^2-2a-b^2-4b-3x+2 \right| + \left| a^2-2a+b^2+4b+2x \right| + a^2-2a+b^2+4b+18 \left| x-2 \right| + 11x = 20.
\]

Answer: 2


Question:
Compute the difference between the integrals:
\[ I_{2n}\left(\sum_{i=0}^{2n+1} ix^i\right) - I\left(\sum_{i=0}^{2n+1} ix^i\right). \]
Here, \( I(f(x)) = \int_{-1}^{1} f(x) \, dx \) and \( I_{2n}(f(x)) = \int_{-1}^{1} p_{2n}(x) \, dx \), where \( p_{2n}(x) \) is a polynomial of degree at most \( 2n \) interpolating the function \( f(x) \) at the points \( x_i = -1 + ih \) for \( i = 0, 1, \ldots, 2n \) and \( h = \frac{1}{n} \).

Answer: 0


Question: Determine the degree of the splitting field of the polynomial \( f(x) = x^3 + 18x + 3 \) over \( \mathbb{Q} \). What is \( [T: \mathbb{Q}] \), where \( T \) is the splitting field of \( f(x) \)?

Answer: 6


Question: Let $n$ be a positive integer. Determine whether the following statement is true or false: If $a$ and $b$ are integers with $b-a > \sqrt{n}$, then there exists an integer $x$ coprime to $n$ in the interval $a \le x \le b$. Provide a justification for your answer.

Answer: True


Question: Evaluate the definite integral: \[ \int_{-3\pi}^{5\pi}\cot^{-1}(\tan x)\,dx \]

Answer: 4\pi^2


Question: What is the Gromov-Witten invariant \(\langle p, p, \ell\rangle_{0, 1}\) that counts degree 1, genus 0 curves in \(\mathbb{CP}^2\), where \(p\) and \(\ell\) are the cohomology classes Poincaré dual to a point and a line, respectively?

Answer: 1


Question: If the product of the factors of $30^{12}$ that are congruent to 1 mod 7 can be expressed as $2^{a} \cdot 3^{b} \cdot 5^{c},$ find $a+b+c$. Compute the value of $a+b+c$. \( \text{(Express your answer as a single integer.)} \)

Answer: 6642


Question: Find the radius of convergence of the Maclaurin series for the function \( f(x) = \int_{0}^{x} \ln{(t+\sqrt{t^2+1})} \, dt. \)

Answer: 1


Question: Let $Q_n$ and $Q$ be compact positive and symmetric operators. Define $A_n = {Q_n}^{\frac12}$ and $A=Q^{\frac12}$. If $Q_n$ converges to $Q$ with respect to the operator norm, does $A_n$ converge to $A$?

Answer: Yes


Question:
Find the value of \( y \) that maximizes the function:
\[ f(y) = \frac{\left(\frac{3}{4}\right)^{25} 25!}{3^y (25-y)! y!} \]

Answer: 6


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

Answer: 2


Question: Find all possible values of \(|z^z|\) using the polar form of the complex number \(z\).

Answer: (0, \infty)


Question: Evaluate the limit: $$ \lim_{x \to \infty} \frac{e^{x}}{(1+\frac{1}{x})^{x^{2}}}. $$

Answer: \sqrt{e}


Question: Does the result that normalizing Gamma random variables with shape parameters $(\alpha_1, \alpha_2, \dots, \alpha_K)$ and scale parameter $\theta = 1$ to obtain a Dirichlet distribution hold for any scale parameter $\theta > 0$?

Answer: Yes


Question:
Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that for all real numbers \( x \) and \( y \), the following equation holds:
\[ f \left( y^2 - f(x) \right) = y f(x)^2 + f \left( x^2 y + y \right). \]

Answer: 0


Question:
The quadratic equation \(p(x)=0\) with real coefficients has purely imaginary roots. Determine the nature of the roots of the equation \(p(p(x))=0\). Choose one of the following options:

(a) Only purely imaginary roots
(b) All real roots
(c) Two real and two imaginary roots
(d) Neither real nor purely imaginary roots

Answer: D


Question: Find the integer $d$ such that $17^d \equiv 8 \pmod{55}$. This is part of an RSA encryption problem.

Answer: 7


Question: Determine the number of points on the elliptic curve given by \(x^3 + x + 1 = y^2\) over the finite field \(\mathbb{F}_{125}\).

Answer: 108


Question: Evaluate \( \alpha^2(\alpha^2+\alpha+1) \) in the Galois field \( GF(27) \), where \( \alpha \) is a root of the irreducible polynomial \( x^3+2x+1 \) over \( \mathbb{Z}_3 \).

Answer: 2\alpha^2 + 2


Question: Determine the radius of convergence for the function \( f(z) = \frac{\sin(z)}{e^z} \) centered at \( z_0 = 1 + 7i \).

Answer: \infty


Question: Let $A$ be a finite-dimensional C*-algebra. Determine whether every quotient of $A$ is of the form $PAP$ for some projection $P \in A$. Provide a justification for your answer.

Answer: Yes


Question: Determine the value to which the sequence \( \frac{X_i}{2^i} \) converges in probability, given that \( X_1, X_2, \ldots, X_i \) are independent, identically distributed random variables with \( E[X_i] = 2 \) and \( \text{Var}(X_i) = 9 \).

Answer: 0


Question: Find the maximum value of the function \( f(x) = \frac{x^4 - x^2}{x^6 + 2x^3 - 1} \) for \( x > 1 \).

Answer: \dfrac{1}{6}


Question: Determine whether the following statement is true or false: For a smooth $S^2$-bundle $\pi:E\to B$ where $B$ is a Riemann surface, is $E$ a ruled surface?

Answer: True


Question: Determine if the Diophantine equation \( y^2 = x^2 - x^3 + x^4 \) has integer solutions other than \( x = 1, y = 1 \).

Answer: Yes


Question: Consider the sequence of real numbers \((a_n)_{n \in \mathbb{N}}\) such that \(a_n > 0\) and the series \(\sum_{n=1}^\infty \frac{a_n}{n} < \infty\). Is the sequence \((a_n)_{n \in \mathbb{N}}\) necessarily bounded?

Answer: No


Question: Let $S_n$ be the set of the first $n$ prime numbers, and let $s_n$ be the $n$-th prime number. Define $K_{n}$ as the set of numbers $\{k\ |\ s_n+1\leq k\leq2s_n^2\}$. What is the probability that an element of $K_n$ has no prime factors in $S_n$? Calculate this probability for $n=3$.

Answer: \dfrac{13}{45}


Question: In the 2D system of differential equations given by \(x' = Ax + b\), verify whether the term \(A^{-1}b\) is correct in the solution \(x = c_1e^{\lambda_1t}\eta_1 + c_2e^{\lambda_2t}\eta_2 + A^{-1}b\), where \(\lambda_i\) and \(\eta_i\) are the eigenvalues and eigenvectors of \(A\), respectively.

Answer: -A^{-1}b


Question: In how many distinct ways can you arrange visits to 5 places, labeled A, B, C, D, and E, such that each place is visited exactly twice, and place E is never visited consecutively?

Answer: 90720


Question: Find a prime factor of \( \frac{3^{41} - 1}{2} \) without using prime factorization.

Answer: 83


Question: Find the maximum value of the expression \(4\sin^2 x + 3\cos^2 x + \sin(x/2) + \cos(x/2)\).

Answer: 4 + \sqrt{2}


Question: Evaluate the limit: \[ \lim_{x \to 0} \int_{0}^{x^{2}} \frac{\cos(t^{2})}{x \sin x} \, dt \]

Answer: 1


Question: Let $X_1, X_2, X_3$ be independent and identically distributed random variables, each following a uniform distribution on the interval (0, 1). Calculate the expected value of the expression $E\left(\frac{X_1 + X_2}{X_1 + X_2 + X_3}\right)$. What is the value?

Answer: \dfrac{2}{3}


Question: Determine whether the multiplicative group of the finite field $GF(2^{2016})$ contains a subgroup of order 2016.

Answer: No


Question: Let \( \Omega \subseteq \mathbb{R}^n \) be a bounded set with \( n \ge 2 \). Show that for all \( x \in \mathbb{R}^n \), the integral \( I = \int_\Omega |x-y|^{1-n} \, d\lambda(y) \) is finite, where \( |\cdot| \) denotes the Euclidean norm and \( \lambda \) is the Lebesgue measure on the Borel \( \sigma \)-algebra of \( \mathbb{R}^n \).

Answer: I < \infty


Question: Calculate the flux of the vector field \( \bar{F} = \langle x, y, x \rangle \) over the surface \( z = 1 - x - y \) in the first octant using the upward unit normal vector. The flux is given by the integral \( \int\int_S \bar{F} \cdot \bar{n} \, dA \), where \( \bar{n} = \frac{1}{\sqrt{3}} \langle -1, -1, 1 \rangle \) and \( dS = \sqrt{3} \, dA \). Evaluate the integral \( \int_0^1 \int_0^{-x+1} (-y) \, dy \, dx \) and determine the correct flux value.

Answer: -\dfrac{1}{6}


Question:
Let \( f: \mathbb{R} \to \mathbb{R} \) be an infinitely differentiable function that vanishes at 10 distinct points in \( \mathbb{R} \). Suppose \( f^{(n)} \) denotes the \( n \)-th derivative of \( f \), for \( n \ge 1 \). Determine which of the following statements is true:

A. \( f^{(n)} \) has at least 10 zeros, for \( 1 \le n \le 8 \)
B. \( f^{(n)} \) has at least one zero, for \( 1 \le n \le 9 \)
C. \( f^{(n)} \) has at least 10 zeros, for \( n \ge 10 \)
D. \( f^{(n)} \) has at least 10 zeros, for \( n \ge 9 \)

Answer: B


Question: Determine whether the following statement is true: Given a continuous injection \( f : U \to \mathbb{R}^m \), where \( U \) is a nonempty open subset of \( \mathbb{R}^n \), it holds that \( n \leq m \).

Answer: True


Question: Evaluate the limit: \( \displaystyle\lim_{z\to3i}\frac{-3z}{z^2-9-6iz} \). Determine whether the limit is \( \infty \) or does not exist.

Answer: \infty


Question: Calculate the flux of the vector field \( F(x,y,z) = (3z^2y, 2x^2y, 2y^2z) \) through the surface \( S = \{ (x,y,z) \in \mathbb{R}^3 : 2\sqrt{x^2+y^2} \leq z \leq 1+x^2+y^2 \} \). Use the divergence theorem to evaluate the flux.

Answer: \dfrac{\pi}{15}


Question:
If \(\lim\limits_{n\to \infty}a_n=a\), determine the value of \(\lim\limits_{n\to \infty}\frac{1}{\ln(n)}\sum_{r=1}^{n}\frac{a_r}{r}\). Choose the correct option:

(A) 0
(B) \(a\)
(C) \(\frac{a}{2}\)
(D) None of these

Answer: B


Question: Determine the dimension of a maximal commutative vector subspace of $M_3(\mathbb{C})$. A vector subspace $V$ of $M_3(\mathbb{C})$ is commutative if for every $A, B \in V$, $AB = BA$. Additionally, $V$ is maximal if any larger subspace $W$ containing $V$ has some $A, B \in W$ such that $AB \neq BA$. What is $\dim V$?

Answer: 3


Question: Let $R$ be a commutative ring and $P_1, P_2, \dots, P_n$ be prime ideals of $R$ such that $P_i \not\subseteq \bigcup _{j \not = i} P_j$ for all $1\le i\le n$. Define $S:=R\setminus(P_1 \cup \cdots \cup P_n)$. Prove that the localization $S^{-1}R$ has exactly $n$ maximal ideals.

Answer: n


Question: Find the limit of \( \frac{\log(1+x^2 + y^2)}{x^2+y^2+yx^3} \) as \((x,y)\) approaches \((0,0)\).

Answer: 1


Question: Determine the number of subgroups in a non-cyclic group of order 25.

Answer: 8


Question: Is the product of $k$ pairwise distinct Galois radii of an integer $n$ always a $k$-quasi-Galois radius of $n$?

Answer: No


Question: How many simple graphs with 10 vertices and 15 edges contain a complete subgraph with 5 vertices?

Answer: 81709614


Question: Evaluate the limit \( \lim_{k \to \infty} \int_0^1 \frac{(1-x)^k \cos(k/x)}{\sqrt{x}} \, dx \).

Answer: 0


Question: Evaluate the limit: $$\lim _{n\to \infty }\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots+\left(-1\right)^{n-1}\cdot \frac{1}{2n-1}\right).$$

Answer: \dfrac{\pi}{4}


Question: Determine whether there exist two different bilinear forms on a vector space over $\mathbb{F}_2$ that have the same associated quadratic form.

Answer: Yes


Question: Evaluate the integral \( \int_1^4 \frac{1}{x^3} \, dx \) using Riemann sums with \( \Delta x = \frac{3}{n} \) and \( x_i = 1 + \frac{3i}{n} \).

Answer: \dfrac{15}{32}


Question: Evaluate the limit \( \lim_{x\to a}\frac{a^x-x^a}{x-a} \) without using L'Hopital's rule or Taylor series, where \( a > 0 \).

Answer: a^a (\ln a - 1)


Question: A person wishes to visit 6 different cities, each exactly twice, such that no city is visited consecutively. In how many distinct ways can this be done?

Answer: 2631600


Question: Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that \( f(ax+b)+c \le x \le f(x+c)+b \) for all \( x \in \mathbb{R} \), where \( a, b, c \) are real constants.

Answer: f(x) = x - b - c


Question:
Consider a sequence of embedded minimal disks $\Sigma_n$ in $\mathbb{R}^3$ satisfying the following conditions:
1. $0 \in \Sigma_n \subset B(0, r_n)$ with $r_n \to \infty$ as $n \to \infty$.
2. $\partial \Sigma_n \subset \partial B(0, r_n)$.
3. $|K_{\Sigma_n}(p)| \le 1$ for all $p \in \Sigma_n$, where $K_{\Sigma_n}$ is the Gauss curvature of $\Sigma_n$.

If the surfaces $\Sigma_n$ do not have a local uniform area bound, does there exist a subsequence of $\Sigma_n$ that converges to a complete minimal surface in $\mathbb{R}^3$?

Answer: Yes


Question: In triangle ABC, let P be a point inside the triangle. The cevians AP, BP, and CP intersect the opposite sides at points A', B', and C', respectively. Determine the minimum value of the expression \((1 + 2 \cdot \frac{PA'}{PA})(1 + 2 \cdot \frac{PB'}{PB})(1 + 2 \cdot \frac{PC'}{PC})\).

Answer: 8


Question: Consider the function \( f : \mathbb{C} \to \mathbb{C} \) with a single pole at \( z = 0 \) and analytic elsewhere. The function is even, meaning \( f(z) = f(-z) \) for all \( z \in \mathbb{C} \). Determine \( \text{Res}(f, 0) \).

Answer: 0


Question: Let $g$ be a strongly convex function, meaning that for all $x, y$ in the domain and $t \in [0,1]$, the inequality \( g(t x + (1-t)y) \le tg(x)+(1-t)g(y) - \frac{1}{2}mt(1-t)||x-y||_2^2 \) holds for some $m > 0$. Let $A$ be a full-rank linear transformation, and define $f = g \circ A$. Is the function $f$ strongly convex?

Answer: Yes


Question: In \(\triangle CDF\), the incircle touches sides \(CD\), \(DF\), and \(FC\) at points \(B\), \(E\), and \(G\) respectively. The cevians \(CE\), \(FB\), and \(DG\) intersect at point \(H\). Given that \(CD\) is divided in a \(5:3\) ratio at \(B\) and \(CF\) is divided in a \(3:2\) ratio at \(G\), find the ratio \(\frac{CH}{HE}\).

Answer: \dfrac{19}{6}


Question: Determine the result of the tensor product \( \mathbb{C}(x) \otimes_{\mathcal{O}_{X,x}} \mathbb{C}(x) \), where \( X \) is a complex manifold with structure sheaf \( \mathcal{O}_X \), and \( \mathbb{C}(x) \) is the residue field at a point \( x \in X \). Is this tensor product still \( \mathbb{C}(x) \)?

Answer: \mathbb{C}(x)


Question: In a normed vector space \((V, ||\cdot||)\), let \(A\) be a compact and convex subset of \(V\), and let \(x \in V \setminus A\). Does there exist an open, convex subset \(U\) of \(V\) such that \(A \subseteq U\) and \(x \notin U\)?

Answer: Yes


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

Answer: \ln 2


Question: Evaluate the mass of the region \( V = \{ (x,y,z) \mid x^2+y^2+z^2 \le 2, z \ge 0, x^2+y^2 \ge 1 \} \) given that the density of mass is \( \phi(x,y,z) = z \).

Answer: \dfrac{\pi}{4}


Question: Determine the values of \( p \) for which the Lebesgue integral \( \int_0 ^ {\infty} x ^ x e ^ {- x^{p} } dx \) is finite.

Answer: p > 1


Question: Determine the number of solutions for the equation \( \cot^{-1}{\sqrt{4-x^2} + \cos^{-1}{(x^2-5)}} = \frac{3\pi}{2} \).

Answer: 2


Question: A tangent is drawn to the ellipse \(x^2 + 2y^2 = 6\) at the point \((2, 1)\). Let \(A\) and \(B\) be the feet of the perpendiculars from the two foci of the ellipse onto the tangent. Find the length of \(AB\).

Answer: \sqrt{6}


Question: Given a connected graph with 10 vertices and 20 edges, where 3 edges have a weight of 3, 4 edges have a weight of 4, and the remaining edges have a weight of 9, determine the minimum weight of a spanning tree of the graph.

Answer: 43


Question: Is the space \(C([0,T] ; L^1(\mathbb{R}^{N}) \cap L^{\infty}(\mathbb{R}^{N}))\) a complete metric space under the norm \(||u|| = \sup_{0 < t < T} (||u(t)||_{L^{1}(\mathbb{R^{N}})} + ||u(t)||_{L^{\infty}(\mathbb{R^{N})}})\)? Consider \(T > 0\) and \(N \in \mathbb{N}\).

Answer: Yes


Question: Given that $E[X^n] = 3n$ for a random variable $X$, find the expected value $E[e^X]$. Use the series expansion $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$ to assist in your calculation.

Answer: 1 + 3e


Question: Calculate the sum of the telescoping series: $$\sum_{n=1}^{\infty} \ln \left( \frac{\left(\frac{n+3}{n+1}\right)^{n+1}}{\left(\frac{n+2}{n}\right)^n} \right).$$

Answer: 2 - \ln 3


Question: Consider a random vector \((X_1,X_2,X_3,Y)\) with \(Y\in L^1\), and let \(f\) be the Borelian function such that \(E[Y\mid X_1, X_2, X_3]=f(X_1, X_2, X_3)\). Define \[I = E\left[{\bigl(Y-f(X'_1,X_2,X_3)\bigr)}^2\right] - E\left[{\bigl(Y-f(X_1,X_2,X_3)\bigr)}^2\right]\] where \(X'_1\) is a random variable having the same distribution as \(X_1\) but is independent of all other random variables \(X_2,X_3,Y\). Is it possible that \(I < 0\)?

Answer: No


Question: Does there exist a function $f(x)$ such that its antiderivative can be expressed in terms of elementary functions, but its derivative cannot be expressed in terms of elementary functions?

Answer: No


Question: Evaluate the integral \( f(x,y) = \int_{-\infty}^\infty p_X(x-z)p_Y(y-z)dz \), where \( X \) and \( Y \) are normal random variables with zero variance and zero mean, and \( x \) and \( y \) are real numbers.

Answer: \delta(x - y)


Question:
Show that for any non-negative measurable function \( r(x) \) with \( \{r=0\} \) having zero Lebesgue measure, the following limit holds:
\[
\lim_{\theta \to +\infty} \frac{\int_0^\infty r(x)\exp\left(-\frac{1}{2}(x-\theta)^2\right)\, dx}{\int_{-\infty}^0 \exp\left(-\frac{1}{2}(x-\theta)^2\right)\, dx} = \infty.
\]

Answer: \infty


Question: Let \( f: [-1, 1] \to \mathbb{R} \) be a continuous function satisfying the functional equation \( f(2x^2 - 1) = (x^3 + x)f(x) \). Determine the value of \( \lim_{x \to 0} \frac{f(\cos x)}{\sin x} \).

Answer: 0


Question: Find the coefficient of $\frac{x^3}{3!}$ in the series expansion of the exponential generating function $e^{e^x-1}$ for Bell numbers. Verify that this coefficient corresponds to the Bell number $B_3$.

Answer: 5


Question: Find the smallest integer $n > 1$ such that the last three digits of $n^2$ are the same as the last three digits of $n$. What is this integer?

Answer: 376


Question: A simple bipartite graph \( G \) has 20 vertices, where 18 vertices have degree 7 and the remaining 2 vertices have degree 6. Determine the chromatic number of the complement of \( G \).

Answer: 10


Question: Evaluate the definite integral \( \int_0^3 x \, d[x] \), where \([x]\) is the greatest integer less than or equal to \(x\).

Answer: 6


Question: Calculate the limit: $$\lim \limits_{x \to 1 }\frac{\sqrt[3]{7+x^3}-\sqrt{3+x^2}}{x-1}$$ without using l'Hôpital's rule.

Answer: -\dfrac{1}{4}


Question: Given the function \( f(x) = x^2 \ln(x) \), find the derivative of its inverse function \( (f^{-1})'(y_0) \) at the point \( y_0 = e^2 \).

Answer: \dfrac{1}{3e}


Question: Is the pullback of a covering map $p:E\rightarrow B$ along a continuous map $f:C\rightarrow B$ a covering map?

Answer: Yes


Question: Let \( f \in \mathbb{Q}[x] \) such that \( \gcd(f, f') = 1 \). Does this imply that \( f \) has no multiple irreducible factors in \( \mathbb{C}[x] \)? Justify your answer.

Answer: Yes


Question:
Evaluate the sum: \( \sum_{l=2}^{100} \frac{1}{\log_{l} 100!} \). Choose the correct answer from the following options:

A) 0.01
B) 0.1
C) 1
D) 10

Answer: C


Question: Determine the maximum number of distinct positive integers $k_1, k_2, \ldots, k_n$ such that $k_1^2 + k_2^2 + \cdots + k_n^2 = 2002$. What is the maximum value of $n$?

Answer: 17


Question: Determine the number of independent components of an antisymmetric rank five tensor \( T_{ijklm} \) in 3-D space.

Answer: 0


Question: Let \( \pi:X\rightarrow \mathbb{A}^{n} \) be a finite surjective morphism, where \( X \) is an irreducible affine variety with \( \text{dim}X=n \). Let \( K \) be a proper closed subset of \( X \). Is the dimension of \( \pi(K) \) strictly less than \( n \)? Justify your answer.

Answer: Yes


Question:
Given the ordinary differential equation \( dy + (y\cot(x) - 2\cos(x))dx = 0 \), find which of the following functions is a solution:

1. \( y = \cos(x) + C\sin(x) \)
2. \( y = \cos(x) + C\csc(x) \)
3. \( y = -\sin(x) + C\cos(x) \)
4. \( y = \sin(x) + C\csc(x) \)
5. \( y = \sin(x) + C\sec(x) \)

where \( C \in \mathbb{R} \).

Answer: D


Question: A nonzero polynomial with rational coefficients has all of the numbers \(1+\sqrt{2}, \; 2+\sqrt{3}, \;3+\sqrt{4},\; \dots, \;1000+\sqrt{1001}\) as roots. What is the smallest possible degree of such a polynomial?

Answer: 1970


Question: Compute the sum of the series \( \sum_{k=1}^\infty \frac{k^2}{k!} \).

Answer: 2e


Question: Evaluate the double integral \( \int_0^{1/2} \int_{1/4}^{y^2} y \cos(16 \pi x^2) \, dx \, dy \) by first changing the order of integration.

Answer: 0


Question: Given that $X$ and $Y$ are uniformly distributed on the region $0 < |x| + |y| < 1$, calculate the conditional probability $P(Y > 1/4 \mid X = 1/2)$. Provide your answer as a probability value.

Answer: \dfrac{1}{4}


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

Answer: 0


Question: Consider a sequence of residual subsets $E_k \subset [0,1]$ such that $E_{k+1} \subset E_k$ and $\dim_H E_k = 1$ for all $k$. Determine whether the Hausdorff dimension of the intersection $\bigcap_k E_k$ is 1.

Answer: 0


Question:
Find the value of \( k \) that satisfies the equation:
\[
(1+\frac{e^k}{e^k+1})^{25} = \frac{3000(\frac{e^k}{e^k+1})-300}{2500(\frac{e^k}{e^k+1})-300}
\]

Answer: -1.98


Question: Evaluate the series \(\sum_{n=0}^{\infty}\frac{nx^n}{(1+x)^{n+1}}\) and verify that it equals \(x\).

Answer: x


Question: Four ants are placed on the vertices of a square with side length 1. Each ant starts moving towards its left neighbor. What is the length of the path each ant travels until they meet at the center of the square?

Answer: 1


Question: Determine the number of real solutions to the equation \( \sin x \cdot \sin 2x \cdot \sin 3x = 1 \) for \( x \in \mathbb{R} \).

Answer: 0


Question: Let $A$ and $B$ be two $n \times n$ primitive row-stochastic matrices, meaning all entries are non-negative, each row sums to 1, and there exists an integer $p \geq 1$ such that all entries of $A^p$ and $B^p$ are strictly positive. Assume $A_{i,j} = 0$ if and only if $B_{i,j} = 0$ for all $i, j$. Suppose there exists $k_0 \geq 1$ such that $A^k = B^k$ for all $k \geq k_0$. Does this imply that $A = B$?

Answer: A = B


Question: Find all positive integers \( n \) such that \( \frac{2^{n-1} + 1}{n} \) is an integer.

Answer: 1


Question:
Determine whether there exist three non-constant functions \( f, g, h \) such that the following equalities hold for all \( x \):
\[ f(g(h(x))) = f(h(g(x))) = g(f(h(x))) = g(h(f(x))) = h(f(g(x))) = h(g(f(x))) \]

Answer: Yes


Question: Solve the differential equation \((1-x)y'(x) + y(x) = 0\) using a power series, given the initial condition \(y(\frac{1}{2}) = 17\).

Answer: 34(1 - x)


Question: Determine whether the function \( f(x,y) = \frac{\sin(x^4+y^4)}{x^2+y^2} \) for \((x,y) \neq (0,0)\) and \( f(0,0) = 0 \) is differentiable at the point \((0,0)\).

Answer: Yes


Question: How many terms are in the simplified form of \((a+b+c)^{2006}+(a-b-c)^{2006}\)?

Answer: 1008016


Question: Find the coefficient of \(r^2w^3b^4\) in the expansion of \(4(r+b+w)^3(b^2+r^2+w^2)^3\).

Answer: 120


Question: Determine whether the following statement is true or false: Every linear transformation on \(\mathbb{R}^5\) has an invariant 3-dimensional subspace.

Answer: True


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

Answer: \dfrac{\pi}{2}


Question: Evaluate the iterated integral: \[ \int_0^8\int_{y^{1/3}}^2 4e^{x^4} \, dx\,dy \]

Answer: e^{16} - 1


Question: Evaluate the integral \( \int_{-\infty}^\infty \frac{1}{x^6+1} \, dx \) using the identity \( \frac{1}{x^6+1} = \Im\left[\frac{1}{x^3-i}\right] \).

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


Question: Consider a function \( f: [0, \infty) \to [0, \infty) \) such that \( |f(x) - f(y)| \le \frac{1}{2}|x-y| \) for all \( x, y \in [0, \infty) \). Does this function always have a fixed point?

Answer: Yes


Question: Find the integer $x$ such that $x^{45} \equiv 7 \pmod{113}$. Provide your answer as a single integer $x$.

Answer: 83


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

Answer: 1


Question: In a game with 2015 marbles, two players, Alex and Brad, take turns removing marbles from the jar. The number of marbles removed on each turn must be a power of two. Alex goes first, and the player who removes the last marble wins. Is there a winning strategy for either player?

Answer: Alex


Question: Determine the values of \( p \) for which the sequence \( \langle a_n \rangle \) converges, given that \( 0 < a_{n+1} \leq a_n + \frac{1}{n^p} \) for \( n \geq 1 \).

Answer: p > 1


Question: Let $G$, $H_1$, and $H_2$ be finite abelian groups. Given that $G \times H_1 \cong G \times H_2$, prove that $H_1 \cong H_2$. Provide a detailed proof of this statement.

Answer: H_1 \cong H_2


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

Answer: \ln 3


Question: Let \( \mu:X'\rightarrow X \) be a birational morphism of normal complex projective varieties. Consider the ideal sheaves \( I_1= \mu_*\mathcal{O}_{X'}(-\sum d(E)E) \) and \( I_2=\mu_*\mathcal{O}_{X'}(-\sum(d(E)+1)E) \), where the \( d(E) \)'s are non-negative integers and the \( E \)'s are prime divisors. Suppose \( x\in X \) is such that \( x\in \mu(E_0) \) for a prime Cartier divisor \( E_0 \) with \( d(E_0)>0 \). Is the stalk \( (I_2)_x \) strictly contained in \( (I_1)_x \)?

Answer: Yes


Question: Find all values of $a$ such that \( \lim_{x \to \infty} \left( \frac{x+a}{x-a} \right)^x = e. \)

Answer: \dfrac{1}{2}


Question: Let $X$ be an arbitrary infinite set. Determine whether there always exists a surjective function $f: X \rightarrow X$ that is two-to-one.

Answer: Yes


Question: Is the representation sphere $S(V)$ dualizable in the monoidal category of naive $G$-spectra, where $G$ is a finite group acting on a vector space $V$?

Answer: Yes


Question: Determine the domain of convergence for the series \( \sum_{n=0}^{\infty} (-1)^n x^{(n+1)^2} \).

Answer: (-1, 1)


Question: Let \( A \) be a symmetric \( n \times n \) matrix, and let \( D \) be the diagonal matrix with eigenvalues \( (\lambda_1, \ldots, \lambda_n) \). Suppose \( O \) is the orthogonal matrix such that \( A = O^TDO \). Is it true that \( \max_{i=1, \ldots, n} |\lambda_i| \leq ||A||_\infty \)?

Answer: Yes


Question: Evaluate the integral \( \int_0^5 (x^2+1) \, d \lfloor x \rfloor \), where \( \lfloor \cdot \rfloor \) is the greatest integer function.

Answer: 60


Question: Let $A_1$ and $A_2$ be the vertices of the conic $C_1 : 4(x - 3)^2 + 9(y - 2)^2 - 36 = 0$. A point $P$ moves in the plane such that $|PA_1 - PA_2| = 3\sqrt{2}$. The locus of $P$ is another conic $C_2$. Let $D_1$ be the distance between the foci of conic $C_2$, $D_2$ be the product of the perpendiculars from the points $A_1$ and $A_2$ upon any tangent drawn to conic $C_2$, and $D_3$ be the length of the tangent drawn from any point on the auxiliary circle of conic $C_1$ to the auxiliary circle of conic $C_2$. Find the value of $\left(\frac{D_1 \cdot D_2}{D_3^2}\right)^2$. \(\boxed{\text{Answer}}\)

Answer: 36


Question: Determine whether the radius of convergence of the series \( \sum \frac{f(n)}{g(n)}r^n \), where \( f(x) \) and \( g(x) \) are polynomials and \( g(n) \neq 0 \) for each \( n \in \mathbb{N} \), is \( R = 1 \).

Answer: 1


Question: Does every countable subset of the set of all countable limit ordinals have its least upper bound within the set of countable limit ordinals?

Answer: Yes


Question: How many powers of 2 have only the digits 0, 1, 2, 4, and 8 in their decimal representation?

Answer: 7


Question: Evaluate the Big-O notation of the expression \( \log_2 \binom{n}{n/2} \).

Answer: O(n)


Question: Evaluate the surface integral \( \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{N} \, dS \) for the vector field \( \mathbf{F} = (x, y, z) \) using Stokes' Theorem.

Answer: 0


Question: Evaluate the limit \( \lim_{x \to 1} \Theta_2(0,x) \) where \( \Theta_2(0,x) = 2x^{1/4}\sum_{k=0}^\infty x^{k(k+1)} \).

Answer: \infty


Question: Develop a mathematical model to determine the number of triples of integers that sum to zero, where each integer is randomly chosen from a uniform distribution between \(-M\) and \(M\), and \(M\) is not small.

Answer: 3M^2 + 3M + 1


Question: Find the remainder when \(3!^{{{5!}^{...}}^{2013!}}\) is divided by 11.

Answer: 1


Question: Determine whether there exists a non-constant entire function \( f : \mathbb{C} \to \mathbb{C} \) such that \( f(n+\frac{1}{n}) = 0 \) for all \( n \in \mathbb{N} \).

Answer: Yes


Question: Solve the equation \((x + x^{\ln x})^{10} = 2^{10}\) for \(x\).

Answer: 1


Question: Reduce \(1819^5 \pmod{2669}\) using the Chinese Remainder Theorem. Given that \(2669 = 17 \cdot 157\) and \(1819 = 17 \cdot 107\), first solve \((17 \cdot 107)^5 \pmod{17}\) and \((17 \cdot 107)^5 \pmod{157}\).

Answer: 1547


Question: Calculate the expected value of the absolute difference between two independent random variables X and Y, each uniformly distributed over the interval (0, 1). That is, find \( E[|X-Y|] \) where \( X, Y \sim \text{Unif}(0, 1) \).

Answer: \dfrac{1}{3}


Question: Decide whether there exists a polynomial of two variables that is bounded below and has its set of values as an open subset of \(\mathbb{R}\).

Answer: Yes


Question: Determine the value to which the infinite product \( C_\infty(n) = \prod_{0 < q < \infty \land q \neq n} \frac{1}{e^{\frac{i\pi q}{t}}-e^{\frac{i\pi n}{t}}} \) converges, where \( t, n \in \mathbb{N} \).

Answer: 0


Question: Suppose $(X_t)$ and $(Y_t)$ are real-valued stochastic processes defined on probability spaces $(\Omega_1, \mathcal{F}_1, P_1)$ and $(\Omega_2, \mathcal{F}_2, P_2)$, respectively, with $(Y_t)$ being a Markov process. If $P_1\circ X_t^{-1}=P_2\circ Y_t^{-1}$ as measures on $\mathbb{R}$, must $(X_t)$ be a Markov process?

Answer: No


Question: Consider a surface diffeomorphic to a cylinder, where the boundary consists of two curves that are diffeomorphic to circles. If these boundary curves are geodesics, can the surface have non-vanishing Gaussian curvature everywhere (i.e., $K \neq 0$ everywhere)? Provide a justification for your answer.

Answer: No


Question: Suppose \((X, d)\) is a metric space such that every contraction map \(f: X \to X\) has a unique fixed point. Is \(X\) necessarily complete?

Answer: Yes


Question: Determine the values of the parameter \(a\) for which the function \(f(x,y) = \frac{x^a \sin(y)}{x^2 + 4y^2}\) for all \((x,y) \neq (0,0)\) and \(f(0,0) = 0\) is continuous at \((0,0)\).

Answer: a > 1


Question: Determine whether the function \( f: (C_c(\mathbb{R}), \|\cdot\|_{\infty}) \rightarrow \mathbb{R} \) defined by \( f(x) = \int_{-\infty}^{\infty} x(t) \, dt \) for all \( x \in C_c(\mathbb{R}) \) is continuous. Answer True or False.

Answer: False


Question: Determine whether the following statement is true: \( f(y) = \inf_{y \in [0,1]} \{ y - y \ln(y) \ge a \} \neq a^2 \).

Answer: True


Question: Suppose $g$ is the inverse of a one-to-one differentiable function $f$ and $H = g \circ g$. Given that $f(4) = 3$, $g(4) = 5$, $f'(4) = \frac{1}{2}$, and $f'(5) = 2$, find $H'(3)$. Provide your answer as a single number.

Answer: 1


Question: Let \( f: \overline{\mathbb{D}} \to \mathbb{C} \) be a continuous function that is holomorphic on \( \mathbb{D} \). If \( f(\mathbb{T}) \subset \mathbb{R} \), is \( f \) constant? Justify your answer.

Answer: Yes


Question: Let $\mathbb{V}$ be a finite-dimensional vector space over a field $F$. Suppose $\mathcal{B}^* = \{f_1, \ldots, f_n\}$ is a basis for the dual space $\mathbb{V}^*$. Does there exist a basis $\mathcal{B} = \{\alpha_1, \ldots, \alpha_n\}$ for the vector space $\mathbb{V}$ such that $f_i(\alpha_j) = \delta_{ij}$ for all $i, j$? Justify your answer.

Answer: Yes


Question: Find the roots of the polynomial $Y^3 - Y + 1$ in the ring $\mathbb{F}_3[Y]/(Y^3 - Y^2 + 1)$. Provide your answer as a set of elements in this ring.

Answer: \{Y + 2Y^2, 1 + Y + 2Y^2, 2 + Y + 2Y^2\}


Question: Assume \( \int_{[0,1]} x^n \, d\mu = 0 \) for all non-negative integers \( n \). Is it true that the measure \( \mu \) is zero on the interval \([0,1]\)?

Answer: Yes


Question: Let \( f(z) \) be a function with a continuous second derivative, and define \( u(x,t) = f(x+ct) \). It is known that \( \frac{\partial^2 u}{\partial t^2} - 12 \frac{\partial^2 u}{\partial x \partial t} + 36 \frac{\partial^2 u}{\partial x^2} = 0 \). Find the value of \( c \).

Answer: 6


Question: Determine the group that is isomorphic to \( G = \left(\mathbb{Z}_{2^\infty} \oplus \frac{\mathbb{Q}}{\mathbb{Z}} \oplus \mathbb{Q}\right) \otimes_{\mathbb{Z}} \mathbb{Q} \). Is it \( 0 \), \( \mathbb{Q} \), \( \mathbb{Q}/\mathbb{Z} \), or \( \mathbb{Z}_{2^\infty} \)?

Answer: \mathbb{Q}


Question: Let $q(z)$ be a rational function where the degree of the denominator exceeds that of the numerator by $l$, with $l \geq 2$. Prove that \( \sum_{c \in \mathbb{C}} \operatorname{Res}_{c} q = 0 \).

Answer: 0


Question: Evaluate the integral \( \int_0^\infty \frac{1}{(1+x^{\varphi})^{\varphi}}\,dx \), where \( \varphi = \frac{\sqrt{5}+1}{2} \) is the Golden Ratio.

Answer: 1


Question: Find a real sequence \((u_j)_{j \in \mathbb{N}}\) such that the series \(\sum_{j=1}^\infty u_j\) converges and \(\sum_{j=0}^\infty (2^j+1)\exp(-u_j2^j) < \infty.\)

Answer: u_j = \dfrac{j}{2^j}


Question: Determine the subgroup of the symmetric group $S_n$ that has the maximal proportion of derangements. Can this proportion exceed $\frac{n-1}{n}$?

Answer: \frac{n-1}{n}


Question: Determine the order of the group $G$ defined by the generators $a$ and $b$ with the relations $a^5 = e$, $b^4 = e$, and $ab = ba^{-1}$, where $e$ is the identity element.

Answer: 20


Question: Determine the number of solutions to the equation \( \underbrace{f(f(f( \ldots f}_{2016~\text{times}}(x) \ldots) =x \) on the interval \(0 \leq x \leq 1\), where \(f(x)\) is a differentiable function satisfying \(0 < f(x) < 1\) and \(0 < f'(x) < 1\) for all \(x\) in the interval \(0 \le x \le 1\).

Answer: 1


Question: Suppose we have maps $f:Z \leftrightarrows X:g$ that are both $\pi_*$ isomorphisms and satisfy $f \circ g \simeq \operatorname{id}_X$. Additionally, assume that $Z$ is a CW complex. Do these conditions imply that the maps form a homotopy equivalence?

Answer: Yes


Question: Is the first hitting time of a left-continuous process into an open set a stopping time when working with a right-continuous filtration?

Answer: Yes


Question:
Find the sum of all possible values of the real number \(a\) for which there exist non-negative real numbers \(x_1, x_2, \ldots, x_5\) satisfying the following system of equations:
\[
\sum_{k=1}^5 kx_k = a, \\
\sum_{k=1}^5 k^3x_k = a^2, \\
\sum_{k=1}^5 k^5x_k = a^3.
\]

Answer: 55


Question: Determine whether the limit \( \lim\limits_{x\to 1^{-}}\sum_{n=1}^{\infty}\frac{x^n}{n(n+1)} \) exists, and if it does, find its value.

Answer: 1


Question: Let $X$ and $Y$ be absolutely continuous random variables where $X$ is uniformly distributed over $[0,12]$ and the conditional density of $Y$ given $X$ is $f_{Y|X}(y|x) = \frac{1}{x}$ for $y \in [0,x]$ and $0$ otherwise. Compute the covariance $\text{Cov}(X,Y)$. Use the formula $\text{Cov}(X,Y) = E(XY) - E(X)E(Y)$. Assume $f_X(x) = \frac{1}{12}$ for $0 \leq x \leq 12$. Determine the joint density $f_{XY}(x,y)$ and find the appropriate range for integration to compute $f_Y(y)$. Finally, calculate $E(XY)$, $E(X)$, and $E(Y)$ to find $\text{Cov}(X,Y)$. \( \text{Cov}(X,Y) = \boxed{?} \)

Answer: 6


Question: Given that \( f(x) \in L^2 [-\pi, \pi] \) and \( \int_{-\pi} ^ {\pi }f(x)e^{-inx}dx =0 \) for all integers \( n \), determine the value of \( \int_{-\pi} ^ {\pi }|f(x) |^2 dx \).

Answer: 0


Question:
Consider a linear transformation \(T: V \to V\) where \(\dim(V) = n\). It is known that there exists an integer \(m\), with \(0 \le m \le n\), such that:
\[\{\mathbf{0}\} = K(T^0) \subsetneq K(T^1) \subsetneq K(T^2) \subsetneq \cdots \subsetneq K(T^m) = K(T^{m+1}) = K(T^{m+2}) = \cdots\]
Let the nullity of \(T^i\) be \(n_i\), forming the sequence:
\[0, n_1, n_2, \cdots, n_{m-1}, n_m, n_m, n_m, \cdots\]
Define \(s_i = n_i - n_{i-1}\), representing the increase in the dimension of the kernel from \(T^{i-1}\) to \(T^i\). It is clear that \(s_i > 0\) for \(0 \le i \le m\).

Prove that \(s_{i-1} \ge s_i\) for \(1 \le i \le m\).

Answer: s_{i-1} \ge s_i


Question: In a triangle with an inscribed circle of radius \( r = 1 \) and one of its sides \( a = 3 \), find the minimum possible area of the triangle.

Answer: \dfrac{27}{5}


Question: Find the five-digit integer \(ABCDE\) such that \(4 \times ABCDE = EDCBA\), where \(A, B, C, D, E\) are distinct digits and \(A, E \neq 0\).

Answer: 21978


Question: Calculate the volume of the solid obtained by rotating the region \( C = \{(x,y,z) \in \mathbb{R}^3 : (x-R)^2+z^2\le r^2, y=0\} \) around the \( z \)-axis, where \( 0<r<R \).

Answer: 2\pi^2 R r^2


Question: Consider the sequence \((a_n)\) where each term satisfies \(a_n \in \left(\frac{1}{2}, 1\right)\) for all \(n \geq 0\). Define the sequence \((x_n)_{n=0}^{\infty}\) by \(x_0 = a_0\) and \(x_{n+1} = \frac{a_{n+1} + x_n}{1 + a_{n+1}x_n}\). Determine the limit \(\lim_{n \to \infty} x_n\), if it exists. If the sequence diverges, state so.

Answer: 1


Question: Find the discrete logarithm of -1 to the base of a primitive root \( g \) in \( \mathbb{Z}_n \), where \( n \) is an odd prime. Specifically, determine \( \log_g (-1) \).

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


Question:
Find the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfies the following conditions for all \( x \in \mathbb{R} \) and \( x \neq 0 \):
\[ f(-x) = -f(x), \quad f(x+1) = f(x) + 1, \quad \text{and} \quad f\left(\frac{1}{x}\right) = \frac{f(x)}{x^2}. \]

Answer: f(x) = x


Question: Given a function \(f:(a,b)\to\mathbb{R}\) with \(f'\) continuous, satisfying \(f'(x)+(f(x))^2+1\geq 0\) for all \(x\in(a,b)\), and the limits \(\lim_{x\to a}f(x)=\infty\) and \(\lim_{x\to b}f(x)=-\infty\), find the value of \(b-a\).

Answer: \pi


Question: Evaluate the complex line integral \( \int_C z \, dz \), where \( C \) is the path consisting of the line segment from \( 0 \) to \( 1+i \), followed by the line segment from \( 1+i \) to \( 2 \).

Answer: 2


Question:
Determine the radius of convergence of the hypergeometric series:
\[ F(\alpha,\beta,\gamma,z)=1+\sum_{n=1}^\infty\frac{\alpha(\alpha+1)\cdots(\alpha+n-1)\beta(\beta+1)\cdots(\beta+n-1)}{n!\gamma(\gamma+1)\cdots(\gamma+n-1)}z^n \]
where \(\alpha,\beta\in\mathbb{C}\) and \(\gamma \neq 0,-1,-2,\ldots\).

Answer: 1


Question: Compute the rank of the $\mathbb{Q}$ vector space $\text{Hom}_R(M,R/(x^2))$, where $R=\mathbb{Q}[x]$ and $M$ is the cokernel of the map $R^2 \rightarrow R^2$ given by the matrix \(\begin{bmatrix} x-1 & 0 \\ 1-x & x^2 \end{bmatrix}\).

Answer: 2


Question:
Solve the following system of equations for the variables $x$, $y$, $z$, $t$, and $u$:
\[ x+y+z=10 \]
\[ xy=4 \]
\[ 2x=t+uy \]
\[ 2y=t+ux \]
\[ 2z=t \]
Then, find the minimum value of the function $f(x,y,z) = x^2 + y^2 + z^2$. Provide your answer as a single value.

Answer: 42


Question: Determine whether the implication $p \multimap p$ holds in linear logic.

Answer: p \multimap p


Question: Find \(\lim_{x \to 0} x\left[\frac{1}{x}\right]\). Provide a formal \(\epsilon\text{-}\delta\) proof for your answer.

Answer: 1