Question: Determine whether the function \( f: \mathbb{R}^m \to \mathbb{R} \) defined by \[ f(x) = \begin{cases} \dfrac{\sin||x||}{e^{||x||}-1} & \text{if } x \ne 0 \\ 1 & \text{if } x = 0 \end{cases} \] is continuous at \( x = 0 \).

Answer: Yes


Question: Find all pairs of prime numbers $(p, q)$ such that $p^3 - q^5 = (p+q)^2$.

Answer: (7, 3)


Question: Find the limit: \[ \lim_{x \to \infty} \left(1 + \frac{2}{x} + \frac{3}{x^2}\right)^{7x} \]

Answer: e^{14}


Question: Given the function \( f(x) = x + \frac{2}{1 \cdot 3}x^3 + \frac{2 \cdot 4}{1 \cdot 3 \cdot 5}x^5 + \frac{2 \cdot 4 \cdot 6}{1 \cdot 3 \cdot 5 \cdot 7}x^7 + \cdots \) for all \( x \in (0,1) \), if the value of \( f\left(\frac{1}{2}\right) \) is \( \frac{\pi}{a\sqrt{b}} \) where \( a, b \in \mathbb{R} \), find \( |a+b| \).

Answer: 6


Question: For the 3-sphere, consider the set of primes $P_{n}$ that divide the order of the homotopy group $\\pi_{i}S^{3}$ for some $i \leq n$. Does the cardinality of $P_{n}$ tend to infinity as $n$ increases?

Answer: Yes


Question: Compute the integral \( \int_{|z|=1} \overline{f(z)} f'(z) \, dz \) for the function \( f(z) = \frac{z-a}{1-\overline{a}z} \) where \(|a|<1\).

Answer: 2\pi i


Question: Consider an $n$-dimensional Brownian motion $B = (B^1, 	imes, B^n)$, where $B_t - B_s \sim N(0, (t - s) E_n)$ for $s < t$. Are the components $B^k$ of $B$ independent?

Answer: Yes


Question: Determine the image of the unit disk \(|z| \leq 1\) under the mapping \(w = \prod_{k=1}^{n} \frac{z_k - z}{1-\bar{z_k}z}\), where \(0<|z_k|<1\).

Answer: |w| \leq 1


Question: What is the largest positive integer $r$ such that there are infinitely many elliptic curves over the rationals with rank at least $r$?

Answer: 19


Question: Find the Taylor series expansion of the function \( f(z) = \frac{1}{(1+z)^3} \) using the binomial expansion. Also, determine the radius of convergence of the series.

Answer: 1


Question: Two circles intersect and form three regions A, B, and C, each with equal area. The center of circle A is at (2, 2). Determine the coordinates of the center of circle B.

Answer: (4, 2)


Question: Determine the number of ways to parenthesize the tensor product of $n$ elements $a_1, a_2, \ldots, a_n$ in numerical order, where $n \in \mathbb{N}$. Assume that the tensor product is associative but not strictly so. For example, for $n=3$, there are two ways: $(a_1 \otimes a_2) \otimes a_3$ and $a_1 \otimes (a_2 \otimes a_3)$. Find the number of ways for a general $n$. Express your answer in terms of the $n$th Catalan number.

Answer: C_{n-1}


Question: What is the smallest field containing both the rational numbers \( \mathbb{Q} \) and the transcendental number \( e \)? Is it the field of real numbers \( \mathbb{R} \) or a proper subfield of \( \mathbb{R} \)?

Answer: \mathbb{Q}(e)


Question: Find the maximum value of the expression \(p^2r^3 + q^2s^3\) given that \(p, q, r, s\) are non-negative real numbers satisfying \(p^5 + q^5 \leq 1\) and \(r^5 + s^5 \leq 1\).

Answer: 1


Question: Determine whether the following statement is true or false: If $W$ and $V$ are open sets and $Z$ is dense, then $\overline{W}\cap Z\subseteq V$ implies $\overline{W}\subseteq\overline{V}$. Provide a proof or counterexample.

Answer: True


Question: Is the Hermitian logarithm \(P^*{D'}P\) the only Hermitian logarithm of a positive definite matrix \(A\)?

Answer: Yes


Question: Determine the values of $C$ for which the integral \( \int_0^{\infty}\left(\frac{7x}{x^2+1}-\frac{7C}{3x+1}\right)dx \) converges.

Answer: 3


Question: Find the sum of the digits of the square root of the number formed by 2018 digits of 4, followed by 2017 digits of 8, and ending with a single digit of 9.

Answer: 12109


Question: Let \( f : \mathbb{R} \to \mathbb{R} \) be defined by \( f(x) = x^r \sin(1/x) \) for \( x \neq 0 \), and \( f(0) = 0 \), where \( r > 0 \) is a rational number. Determine the values of \( r \) for which \( f'(0) \) exists.

Answer: r > 1


Question: Given a differentiable function \( f(x) \) that satisfies the functional equation \( f(x+y) = f(x)e^y + f(y)e^x + 2xye^x e^y \) for all \( x, y \in \mathbb{R} \), and the condition \( f'(0) = 0 \), determine the number of solutions to the equation \( f(x) = 0 \).

Answer: 1


Question: Let $f:X\rightarrow Y$ be a projective map of schemes, where $Y$ is of finite type over $\mathbb{Z}[1/N]$ for some large $N$, and $R^1 f_* \mathcal{O}_X=0$. Given that $\mathrm{Pic}_{X/Y}$ is a scheme of dimension 0 over $Y$, is it true that the $p$-torsion $\mathrm{Pic}_{X/Y}[p]$ is zero for sufficiently large prime $p$?

Answer: Yes


Question: An urn initially contains 6 black balls and 0 white balls. In each turn, one ball is drawn at random, removed, and replaced with a ball of the opposite color. Calculate the expected number of turns until all 6 balls in the urn are white.

Answer: \dfrac{416}{5}


Question: Do there exist two non-isomorphic finite groups $G$ and $H$ such that $|G| = |H|$ and $Pv(G) = Pv(H)$?

Answer: Yes


Question: Evaluate the integral \( \int_{\mathbb{R}} x \, d\mu(x) \) where the measure \( \mu = (1-p)\delta_0 + p\delta_1 \) is defined on the Borel sets of \( \mathbb{R} \). Here, \( \delta_x(B) = 1 \) if \( x \in B \) and \( 0 \) otherwise, for a Borel set \( B \).

Answer: p


Question: Find the ratio of the area of the region defined by \( \log_{10}(2 + x^2 + y^2) \leq 2 + \log_{10}(x + y) \) to the area of the region defined by \( \log_{10}(1 + x^2 + y^2) \leq 1 + \log_{10}(x + y) \).

Answer: 102


Question: Compute the norm of the inverse operator \((iI + \frac{1}{2}U)^{-1}\), where \(U\) is the bilateral shift operator on \(\ell^2(\mathbb{Z})\) defined by \(U\delta_k = \delta_{k+1}\) for \(k \in \mathbb{Z}\), and \(I\) is the identity operator on \(\ell^2(\mathbb{Z})\).

Answer: 2


Question: Determine the number of solutions to the equation \(x^y - y^x = x + y\) where both \(x\) and \(y\) are prime numbers.

Answer: 1


Question: Find the value of \( z \) such that \( \frac{\partial x}{\partial y}\bigg|_{y=0}=0 \) for the polynomial \( P(x,y,z)=x^3+[Q+f(y)]zx^2+cy^2f(y)x+cy^2f(y)Qz=0 \). Given that \( x=F(z)=-[Q+f(0)]z \) satisfies \( P(F(z),0,z)=0 \) for all \( z \).

Answer: 0


Question: In a predator-prey system, the populations of prey (R) and predators (W) are modeled by the following differential equations: \( \frac{dR}{dt} = 6R - 2RW \) and \( \frac{dW}{dt} = -4W + 5RW \). Find the value of \( RW \) when the system is in equilibrium with \( W \neq 0 \) and \( R \neq 0 \).

Answer: \dfrac{12}{5}


Question:
Given a function \( f(x) = \begin{cases} f_1(x) & \text{if } x \leq 0 \\ f_2(x) & \text{if } x > 0 \end{cases} \) with the following properties:

- \( f \) is continuous over its domain.
- \( f_1 \) is differentiable over \( (-\infty, 0) \).
- \( f_2 \) is differentiable over \( (0, +\infty) \).
- \( \lim_{x \to 0^-} f_1'(x) = \lim_{x \to 0^+} f_2'(x) = L \).

Determine if \( f'(0) \) exists and equals \( L \).

Answer: L


Question:
Given polynomial functions \( P, Q, R: \mathbb{C} \rightarrow \mathbb{C} \) with maximum degree 2, and complex numbers \( a, b, c \) such that:
\[
\begin{vmatrix}
P(a) & Q(a) & R(a) \\
P(b) & Q(b) & R(b) \\
P(c) & Q(c) & R(c) \\
\end{vmatrix} = 1,
\]
calculate the following sum of determinants:
\[
\begin{vmatrix}
P(1) & Q(1) & R(1) \\
P(b) & Q(b) & R(b) \\
P(c) & Q(c) & R(c) \\
\end{vmatrix} +
\begin{vmatrix}
P(a) & Q(a) & R(a) \\
P(1) & Q(1) & R(1) \\
P(c) & Q(c) & R(c) \\
\end{vmatrix} +
\begin{vmatrix}
P(a) & Q(a) & R(a) \\
P(b) & Q(b) & R(b) \\
P(1) & Q(1) & R(1) \\
\end{vmatrix}.
\]

Answer: 1


Question: Consider a sequence of independent random variables $X_1, X_2, \dots, X_n$ that are uniformly distributed on the interval $[0,1]$. Let $X_{\lfloor n/3 \rfloor}^{(n)}$ denote the $\lfloor n/3 \rfloor$-th smallest value in this sample of size $n$. Determine the limit of $X_{\lfloor n/3 \rfloor}^{(n)}$ as $n$ approaches infinity.

Answer: \dfrac{1}{3}


Question: Determine whether the magnitudes of the short-time Fourier transform expressions \( V_{g}f(x,w) \) and \( V_{g}f(-w,x) \) are equal. Specifically, is it true that \( |V_{g}f(x,w)| = |V_{g}f(-w,x)| \)?

Answer: No


Question: Is it possible to sort 10 coins using exactly 20 single weightings, where each weighting can determine which of two coins is heavier, given that any two coins have different weights?

Answer: No


Question: Evaluate the surface integral \( \iint_S \vec{F} \cdot d\vec{s} \) where \( \vec{F} = 3x\hat{i} + 2y\hat{j} -5z\hat{k} \) and \( S \) is the portion of the surface \( y = x^2 + z^2 \) that lies behind \( y = 1 \), oriented in the direction of the positive \( y \)-axis.

Answer: 2\pi


Question: Evaluate the line integral \( I = \int_{C} \overrightarrow{E} \cdot d\overrightarrow{r} \) for the vector field \( \overrightarrow{E} = 7x^2\hat{e}_x + 3y\hat{e}_y - 2xz\hat{e}_z \), where the contour \( C \) is the straight line from the point \((0,0,0)\) to \((1,2,0)\).

Answer: \dfrac{25}{3}


Question: Calculate the flow of the vector field \( \mathbf{F} = (z, y, x) \) through the sphere centered at \((0, 0, 0)\) with radius 1, using a suitable surface integral. Assume the flow is from the center of the sphere outwards.

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


Question:
Evaluate the line integral of the vector field \( \mathbf{F} = [xy, x^2y^2] \) along the curve \( C \), which is a quarter-circle from \((2,0)\) to \((0,2)\) with center at \((0,0)\). Use the parametric representation \( \mathbf{r}(t) = [2\cos(t), 2\sin(t)] \) for the curve. Compute the integral:
\[ \int_C \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) \, dt \]
Verify your result with the known answer of \( \frac{8}{5} \).

Answer: \dfrac{8}{5}


Question: Determine whether \( \overline{L^1(\mathbb{R}) \cap L^p(\mathbb{R})} = L^p(\mathbb{R}) \) holds for \( 1 < p < 2 \).

Answer: Yes


Question: Calculate \( f(5) \) where \( f(n) = \sum_{A \subseteq N} \sum_{B \subseteq N} |A \cap B| \) and \( N = \{1, 2, \ldots, n\} \).

Answer: 1280


Question: Calculate the Cauchy Principal Value of the integral \( P.V. \int_{-\infty}^{\infty} \frac{dx}{x(x^2+1)} \).

Answer: 0


Question:
Let $V_t$ and $W_t$ be independent Brownian motions defined on $[0,T]$. Let $X_t$ and $Y_t$ be stochastic processes adapted to the filtrations generated by $V_t$ and $W_t$, respectively. Prove or disprove the following statement:
\[ \mathbb{E} \left[ \left(\int_0^T X_t \, dV_t \right) \left(\int_0^T Y_t \, dW_t \right) \right] = 0. \]

Answer: 0


Question: Determine the values of \(a\) for which the integral \(\int_{0}^{1}(-\ln x)^a dx\) converges.

Answer: a > -1


Question: Determine whether the inequality \( \sum_{\genfrac{}{}{0pt}{1}{p>N}{p\text{ is prime}}}\frac{1}{p^2} < \frac{1}{N} \) holds for every positive integer \( N \).

Answer: Yes


Question: An urn contains a large number of balls with 50% black, 30% white, and 20% red. Balls are extracted randomly, one at a time, recorded, and returned to the urn. What is the average number of balls one should extract until all three colors have appeared?

Answer: \dfrac{559}{84}


Question: Determine the number of natural numbers \( n \) that satisfy the equation \( \varphi(n) + \tau(n^2) = n \), where \( \varphi \) is the Euler's totient function and \( \tau \) is the divisor function.

Answer: 2


Question: Let $G$ and $G'$ be two finite simple groups, and let $p$ be a prime divisor of both $|G|$ and $|G'|$. Suppose every Sylow $p$-subgroup of $G$ and $G'$ is a cyclic group of prime order $C_p$. If the number of Sylow $p$-subgroups of $G$ is equal to the number of Sylow $p$-subgroups of $G'$, is it true that $G \cong G'$?

Answer: No


Question:
Show that the composition of boundary maps in the Koszul complex is zero. Specifically, prove that for the boundary map \( d_p \) defined by:
\[
d_{p}(g_{i_1} \wedge \dots \wedge g_{i_p}) = \sum_{j=1}^{p} (-1)^{j-1}x_{i_j}g_{i_1} \wedge \dots \wedge \widehat{g_{i_j}} \wedge \dots \wedge g_{i_{p}},
\]
the following holds:
\[
d_{p-1}\left( \sum_{j=1}^{p} (-1)^{j-1} x_{i_j}g_{i_1} \wedge \dots \wedge \widehat{g_{i_j}} \wedge \dots \wedge g_{i_{p}} \right) = 0.\]

Answer: d_{p-1} \circ d_p = 0


Question: Determine the domain of convergence for the integral \( \int_{1}^{+\infty}\frac{dt}{1+t^x} \) where \( x \in \mathbb{R} \).

Answer: (1, +\infty)


Question: Determine whether all singular functions are of bounded variation. Provide a justification for your answer.

Answer: No


Question:
Evaluate the double integral over the unit disk $R$ in the $xy$-plane:
\[ \iint_R \left(5-x^2\sin x+y^3\cos y\right) \, dA \]

Answer: 5\pi


Question: Find the best order 2 least squares approximation to the data points \((0,0), (0.25,1), (0.5,0), (0.75,-1)\) using the basis functions \(1\) and \(\cos(2\pi t)\).

Answer: 0


Question: Find the minimum integer \( m \) such that \( \lim_{(x,y)\to(0,0)} \frac{x^{\frac{m}{3}}|x-y|}{\sqrt{x^2+y^2}} \in \mathbb{R} \).

Answer: 1


Question:
Let \(a, b, c\) be variables that can take any real or complex values. Given that \(ab + bc + ca = \frac{1}{2}\), \(a + b + c = 2\), and \(abc = 4\), find the value of the expression:
\[
\frac{1}{ab + c - 1} + \frac{1}{bc + a - 1} + \frac{1}{ac + b - 1}
\]

Answer: -\dfrac{2}{9}


Question: Given a Riemannian metric $g$ on $\mathbb{R}^n$ and a vector $v_1 \in \mathbb{R}^n$ such that $g(v_1,v_1)=1$, determine if it is possible to find a basis $\{v_1,v_2,...,v_n\}$ of $\mathbb{R}^n$ such that $g(v_i,v_j)=\delta_{ij}$ for all $i,j=1,...,n$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the sequence \(b_n = \frac{(n+1)^3 - n^3 - cn^2}{n}\) converges, and if so, find its limit. Assume \(c\in\mathbb{C}\).

Answer: 3


Question: Determine the number of bijections \(f: \mathbb{N} \to \mathbb{N}\) such that \(\sum_{n=1}^{\infty} \frac{f(n)}{n^2} < \epsilon\).

Answer: 0


Question: Evaluate the double integral \( \iint_S (\nabla \times \vec{F}) \cdot \hat{n} \, dS \) over the surface \( S \) defined by \( x^2 + y^2 + z = 1 \) with \( z \geq 0 \), where \( \vec{F} = (x+2y)e^z \mathbf{i} + (ye^z + x^2) \mathbf{j} + y^2z \mathbf{k} \). If \( \left| \iint_S (\nabla \times \vec{F}) \cdot \hat{n} \, dS \right| = \alpha \pi \), find \( \alpha \).

Answer: 2


Question: Find the constant \( k \) such that \( \int_{6}^{16} \frac{dx}{\sqrt{x^3 + 7x^2 + 8x - 16}} = \frac{\pi}{k} \).

Answer: 6\sqrt{5}


Question: Let $B$ be a Banach space. Determine whether there exists a Hilbert space $H$ that is homeomorphic to $B$. Provide a justification for your answer.

Answer: Yes


Question: Suppose $M$ and $N$ are simple modules over a commutative ring $R$, and $P_N$ is the projective cover of $N$. If there is a nonzero morphism $P_N \to M$, does this imply $N \cong M$?

Answer: Yes


Question: Let $N$ be the product of all divisors of $2013^{2013}$. Determine the last two digits of $N$ in its decimal notation.

Answer: 41


Question: Determine the positive integer values of $n$ for which the fraction \( \frac{4n+9}{2n^2+7n+6} \) is a terminating decimal.

Answer: 6


Question:
Suppose that $<$ is a total order on $\mathbb{Z}^{n}_{\geq 0}$ satisfying the following conditions:
1. $\alpha < \beta$ if and only if $\alpha + \gamma < \beta + \gamma$ for all $\alpha, \beta, \gamma$.
2. The vector $\vec{0}$ is the smallest element in $\mathbb{Z}^{n}_{\geq 0}$ under this order.

Does it follow that $\mathbb{Z}^{n}_{\geq 0}$ is well-ordered under this ordering, meaning that every non-empty subset has a least element?

Answer: Yes


Question: Determine whether the following statement is true or false: If \( f \in L^1(\mathbb{R}) \) is continuous but not uniformly continuous, then \( \lim_{|x|\to\infty}|f(x)|=0 \).

Answer: B


Question: Compute the limit: \[ \lim_{n \to \infty} \int_0^{n^2} e^{-x^2}n\sin\frac{x}{n} \, dx. \]

Answer: \dfrac{1}{2}


Question: Let $R$ be a finite commutative ring, $J$ a maximal ideal of $R$, and $n$ a positive integer with $n \geq 2$. Determine whether every ideal of the quotient ring $R/J^n$ is principal.

Answer: No


Question: Solve the equation $x^{2x} + 27^2 = 54x^x$ for all possible values of $x$.

Answer: 3


Question: Given a monic polynomial \( f(X) \) with integer coefficients and an odd prime number \( p \), suppose that for every positive integer \( m \), there exists an integer \( a \) such that \( f(a) \equiv 0 \pmod{p^m} \). Does there exist an integer \( b \) such that \( f(b) = 0 \)? Justify your answer.

Answer: No


Question: Evaluate the limit: \( \lim_{n\rightarrow\infty}\left(\lim_{x\rightarrow\infty}x-n\right) \).

Answer: \infty


Question: Evaluate the limit: \[ \lim\limits_{n\to\infty} \frac{3+\sqrt{3}+\sqrt[3]{3}+\dots+\sqrt[n]{3}-n}{\ln n}. \]

Answer: \ln 3


Question: Evaluate the sum \( \sum_{k=2}^{\infty}\log{(1-1/k^2)} \).

Answer: -\ln 2


Question: Let $G$ be an infinite group with a finite generating set $S$. For $n \geq 1$, let $p_n$ be the probability that a random word in $S \cup S^{-1}$ of length at most $n$ represents the identity. Determine if it is possible for $p_n$ to not approach $0$ as $n$ approaches infinity.

Answer: No


Question: If a distribution $f$ has all weak derivatives of order $k$ equal to zero on $\mathbb{R}^n$, must $f$ be a polynomial of degree at most $k-1$? Prove or disprove this statement.

Answer: Yes


Question: Let $H$ be an infinite-dimensional separable Hilbert space with an orthonormal basis $(e_n)_{n \in \mathbb{N}}$. Define $f_N = \frac{1}{N} \sum_{n=1}^{N^2} e_n$ for all $N \geq 1$. Prove that $f_N \rightarrow 0$ weakly as $N \rightarrow \infty$.

Answer: 0


Question: A rectangle with sides $(x+y+z)$ and $(a+b+c)$ is divided into 9 smaller rectangles. Given that the areas of 4 of the smaller rectangles are $ax=8$, $ay=10$, $by=5$, and $cz=12$, find the minimum possible area of the rectangle.

Answer: 75


Question: Evaluate the integral \( \int_0^{\infty} \frac{2 \sin x \cos^2 x}{x e^{x \sqrt{3}}} \, dx \).

Answer: \dfrac{\pi}{4}


Question: If \( t = \sum_{i=1}^{\infty} \tan^{-1}\left(\frac{1}{2i^2}\right) \), find the value of \( \tan t \).

Answer: 1


Question: Determine whether \( \biggl\|\frac{vv^T}{v^Tv}\biggr\| = 1 \) for any vector \( v \in \mathbb{R}^n \), where \( \|.\| \) denotes a matrix norm.

Answer: 1


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

Answer: 1


Question: Find the spectrum of the operator \( A : L_2[-1,1] \to L_2[-1,1] \) defined by \((Ax)(t) = \int_{-1}^{1} t^2s x(s) \, ds\).

Answer: \{0\}


Question: Find an integer solution to the equation \(2x^2 - y^2 = 1\) other than \((1, 1)\).

Answer: (5, 7)


Question: Determine the number of homomorphisms from the group \( \mathbb{Z} \times (\mathbb{Z} / 6\mathbb{Z}) \) to the group \( (\mathbb{Z} /2\mathbb{Z} ) \times (\mathbb{Z} /60\mathbb{Z}) \).

Answer: 1440


Question: Let $z \in \mathbb{C}$ be a root of the equation $z^2 - 2z + 3 = 0$. Find the modulus of the complex number $f(z) = z^{17} - z^{15} + 6z^{14} + 3z^2 - 5z + 9$. Express your answer as a single number.

Answer: \sqrt{3}


Question: Compute the residue of the function \( \frac{1}{e^{\frac{1}{z}}-1} \) at the point \( z=0 \).

Answer: \dfrac{1}{12}


Question: Consider the numbers 1 through 1000 arranged in a line. Each number, except the rightmost, must differ by 1 from at least one number to its right. Determine the total number of such possible arrangements.

Answer: 2^{999}


Question: Let $A$ be a set with two relations $R$ and $S$ such that $(A,R)$ and $(A,S)$ are well-orders with the same order type, i.e., $(A,R) \cong (A,S) \cong (\alpha, \in)$ for some ordinal $\alpha$. Is it true that $R = S$?

Answer: No


Question: Evaluate the surface integral \( \iint_S (x^2+y^2) \, dS \), where \( S \) is the surface of the cone defined by \( z^2 = 3(x^2 + y^2) \) bounded between \( z = 0 \) and \( z = 3 \).

Answer: 9\pi


Question: Determine the relationship between \(\alpha\) and \(\beta\) such that the function \(f_n(x) = \frac{2n^\alpha x}{1+n^\beta x^2}\) is uniformly convergent over the interval \([0,1]\), given that \(\beta > 2\alpha \ge 0\).

Answer: \beta > 2\alpha


Question: Let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Consider a smooth bijective map $f: D \to D$ with an everywhere invertible differential. Is $f$ a diffeomorphism of the closed disk?

Answer: Yes


Question: Determine the modulus of continuity of the function \( f(x) = \sin\left(\frac{1}{x}\right) \) on the interval \( I = (0, 1) \). The modulus of continuity is defined as \( w(f,r) = \sup\{|f(x)-f(y)| : x,y \in I, |x-y| \leq r\} \).

Answer: 2


Question: Find the function \( f: \mathbb{R} \to \mathbb{R} \) that satisfies the equation \( f\bigl(yf(x+y)+f(x)\bigr)=4x+2yf(x+y) \).

Answer: 2x


Question: Find a generator of the multiplicative group of the quotient ring \( \mathbb{Z}_3[x]/(x^2+x+2) \).

Answer: x


Question: Given an embedding $h: D^n \to S^n$, is the closure of the complement $S^n - h(D^n)$ homeomorphic to $D^n$?

Answer: No


Question: Given that \( f(x) = x^2 f(2x) \) and the values \( f(4) = -2 \), \( f'(4) = 1 \), and \( f''(4) = -1 \), find \( f''(2) \).

Answer: -4


Question: Let \( f:[0,1]\rightarrow[0,1] \) be a twice differentiable function satisfying \( f(ax+(1-a)y)\leq af(x)+(1-a)f(y) \) for all \( x,y\in[0,1] \) and any \( a\in[0,1] \). Prove that for all \( x\in[0,1] \), \( f''(x)\geq0 \).

Answer: f''(x) \geq 0


Question: Determine if the function \( f(z) = \frac{2}{\pi}\log\left(\frac{1-z}{1+z}\right) - 1 \) is a valid conformal map from the unit disk \( \mathbb{D} \) to the infinite strip \( |\Im(z)| < 1 \).

Answer: Yes


Question: Find the polynomial \( P(x) \) with integer coefficients such that for all integer \( x \), the equation \( P(P(x)) = P(x) + P(x^2) \) holds.

Answer: 0


Question:
Let $p$ and $q$ be distinct primes. There is a proper subgroup $J$ of the additive group of integers which contains exactly three elements of the set $\{p,p+q,pq,p^q,q^p\}$. Which three elements are in $J$?

$A. ~pq,~p^q,~q^p$
$B. ~p+q,~pq,~p^q$
$C. ~ p,~p+q,~pq$
$D. ~p,~p^q,~q^p$
$E. ~p,~pq,~p^q$

Answer: E


Question: Given a set \( A = \{1, 3, 6\} \), calculate the total sum of the expression \((\text{sum of all numbers in the piece}) \times (\text{length of the piece})\) over all possible contiguous partitions of the set. For example, the partitions are \( \{\{1\},\{3\},\{6\}\} \), \( \{\{1,3\},\{6\}\} \), \( \{\{1\},\{3,6\}\} \), and \( \{\{1,3,6\}\} \). Compute the total sum of values for these partitions.

Answer: 73


Question: Determine the domain of the function \( f(x) = \int_0^1 \frac{t^x - 1}{\ln(t)} \, dt \), where \( x \) is a real number.

Answer: (-1, \infty)


Question:
Compute the genus of the graph with vertex set \( V(G) = \{u_1, \cdots, u_7, v_1, \cdots, v_9, w_1, \cdots, w_5\} \) and edge set \( E(G) = \bigcup_{i=1}^{11} E_i \), where:

\[ E_1 = \{u_1u_j \mid 2 \leq j \leq 7\} \cup \{u_1v_j \mid j=1,3,5,6,8,9\} \cup \{u_1w_3, u_1w_4\} \; ; \]
\[ E_2 = \{u_2u_j \mid j=3,4,6,7\} \cup \{u_2v_j \mid j=4,5,6,7\} \cup \{u_2w_5\} \; ; \]
\[ E_3 = \{u_3u_j \mid j=4,5,7\} \cup \{u_3v_j \mid j=1,7,8,9\} \cup \{u_3w_2\} \; ; \]
\[ E_4 = \{u_4u_j \mid j=5,6\} \cup \{u_4v_j \mid j=1,2,3,4\} \cup \{u_4w_1\} \; ; \]
\[ E_5 = \{u_5u_j \mid j=6,7\} \cup \{u_5v_j \mid j=4,5,6,7\} \cup \{u_5w_5\} \; ; \]
\[ E_6 = \{u_6u_7\} \cup \{u_6v_j \mid j=1,7,8,9\} \cup \{u_6w_2\} \; ; \]
\[ E_7 = \{u_7v_j \mid j=1,2,3,4\} \cup \{u_7w_1\} \; ; \]
\[ E_8 = \{v_1v_5, v_1v_6\} \; ; \]
\[ E_9 = \{v_2v_7\} \; ; \]
\[ E_{10} = \{v_3v_7\} \; ; \]
\[ E_{11} = \{v_4v_8, v_4v_9\} \; . \]

Answer: 1


Question: Let $M$ be the set of natural numbers such that there exists a group of this order which is not solvable. Determine the minimal distance $D$ between two numbers in $M$. Given examples show $D \leq 12$, and the Feit-Thompson theorem implies $D > 1$. What is the value of $D$?

Answer: 12


Question: Evaluate the integral: $$\int_0^{2\pi}\cot(e^{it})ie^{it}dt$$

Answer: 2\pi i


Question: Determine the number of roots of the polynomial \( f(x) = x^8 - 1 \) in the ring \( \mathbb{Z}/91\mathbb{Z} \).

Answer: 8


Question: In how many ways can 7 dwarfs be divided into 4 non-empty teams, where the order of the teams and the order of the dwarfs within the teams do not matter?

Answer: 350


Question: Let $Q$ be a finite quiver with stationary parts denoted by $e_{i}$. Suppose there are two arrows $f$ and $g$ such that their composition $f \circ g$ equals $e_{i}$. Does this imply that $f = g = e_{i}$?

Answer: Yes


Question: Determine the number of points where the functions \( f(x) = x^2 \) and \( g(x) = x \sin x + \cos x \) intersect.

Answer: 2


Question: Calculate the sum of the trigonometric series: \[ \cos \frac{2\pi}{2013} + \cos \frac{4\pi}{2013} + \cdots + \cos \frac{2010\pi}{2013} + \cos \frac{2012\pi}{2013} \] without using a calculator.

Answer: -\dfrac{1}{2}


Question: Evaluate the integral \( \int d^3 \mathbf{r} \frac{1}{1+ \mathbf{r} \cdot \mathbf{r}} \delta(\mathbf{r} - \mathbf{r}_0) \) where \( \mathbf{r}_0 = (2,-1,3) \).

Answer: \dfrac{1}{15}


Question: Calculate the contour integral \( \oint_{|z|=3} \frac{\sin z}{z(z-2i)} \, dz \) using the Residue Theorem in both the trigonometric and inverse trigonometric senses. Determine the value of the integral in each case.

Answer: \pi i \sinh 2


Question: What is the minimal number of years such that it is possible to assign people in a Secret Santa, where each person gives and receives exactly one present, and no one gives a present to someone they have given a present to in any of the previous years? Assume there are \( n \) people involved. Provide your answer in terms of \( n \).

Answer: n-1


Question: Find the mass of the surface of the sphere defined by \( x^2 + y^2 + z^2 = 16 \), where the density at any point on the surface is equal to the distance from the xy-plane.

Answer: 128\pi


Question: Find the largest positive real number \( \delta \) such that, for all real numbers \( x \) and \( y \), the inequality \( | \cos x - \cos y| < \sqrt{2} \) holds whenever \( |x - y| < \delta. \)

Answer: \dfrac{\pi}{2}


Question: Determine whether the algebra \( \mathbb{C}^3 \), with pointwise addition and multiplication, has only one simple module up to isomorphism. Answer true or false.

Answer: False


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

Answer: \dfrac{1}{2}


Question: Does every continuous action of the circle group $S^1$ on $\mathbb{R}^n$ have a fixed point?

Answer: Yes


Question:
Given positive real numbers \(x_1, x_2, \ldots, x_{24}\) such that \( \sum_{i=1}^{24} x_i = 1 \), determine the maximum value of the expression:
\[
\left( \sum_{i=1}^{24} \sqrt{x_i}\right) \left(\sum_{i=1}^{24} \frac{1}{\sqrt{1+x_i} } \right).
\]

Answer: \dfrac{576}{5}


Question: Let $U=\{ (x,y,z) \: : \: x-z=0\}$ and $V=\{(x,y,z) \: :\: z=0\}$ be two subspaces of $\mathbb{R}^3$. Define $W=\{f:\mathbb{R}^3 \rightarrow \mathbb{R}^3 \: : f(U) \subseteq V \: \: f(V) \subseteq U \}$. Find the dimension of $W$ using its matrix representation.

Answer: 5


Question: If $X_n\overset{d}{\to}c$ for some constant $c$, does it imply that $X_n\overset{\text{a.s.}}{\to}c$? Provide a justification for your answer.

Answer: No


Question: Determine the maximum number of pairwise non-homotopic and pairwise disjoint simple closed curves that can be placed on an orientable genus $g$ surface $\Sigma_g$. Provide an upper bound for this number.

Answer: 3g - 3


Question: Calculate the area of the triangle formed by the points $A:(1,2,-3,3)$, $B:(3,-6,-4,2)$, and $C:(-3,-16,-4,0)$ in $\mathbb{R}^4$. Express your answer as an absolute value.

Answer: 35


Question: Let $A$ be an $n \times n$ non-singular complex matrix and define $B = (\bar{A})' A$, where $(\bar{A})'$ is the conjugate transpose of $A$. Determine whether the statement "If $x$ is an eigenvalue of $B$, then $x$ is real and positive" is true or false.

Answer: True


Question: Is it true that any element \( \gamma \in H_1(\Sigma, \partial \Sigma; \mathbb{Z}) \) can be represented by a disjoint union of simple closed curves and properly embedded arcs on a compact oriented surface \( \Sigma \) with non-empty boundary?

Answer: Yes


Question: Let $X$ be a normal integral variety over $\mathbb{C}$ and $D \subset X$ be a Cartier divisor in $X$. Is the associated reduced scheme $D_{\mathrm{red}}$ also necessarily a Cartier divisor in $X$? Answer yes or no.

Answer: No


Question: Let \( f \) be a relation on \( \mathbb{R} \) defined by \( xfy \Leftrightarrow (y(2x-3)-3x = y(x^2-2x)-5x^3) \). Let \( g \) be the restriction of \( f \) to \( \mathbb{Z}^+ \), such that \( g(n) = f(n) \) for \( n \in \mathbb{Z}^+ \). Determine the value of \( a \in \mathbb{R} \) such that \( g \in \Theta(n^a) \).

Answer: 1


Question: Determine the values of $p > 0$ for which the series \( \sum_{n=2}^{\infty} \frac{1}{(\log 1)^p+(\log 2)^p+\cdots +(\log n)^p} \) converges.

Answer: p > 1


Question: Consider a set of 220 cars. What is the probability that a specific car, say car i, is selected x times before another specific car, say car j, is ever selected? Assume the selection process is random. Calculate this probability for x = 6.

Answer: \dfrac{1}{64}


Question:
Given the system of equations for complex numbers $x, y, z$:
\[ x^2 - yz = i, \]
\[ y^2 - zx = i, \]
\[ z^2 - xy = i, \]
find the value of $x^2 + y^2 + z^2$. You do not need to find the individual values of $x, y, z$. Express your answer in terms of $i$. \( \boxed{} \)

Answer: 2i


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

Answer: (\ln 2)^2


Question: Given the sequence \(a_n = \sqrt[n]{(\frac{2012}{2013})^n - (\frac{2011}{2012})^n}\) and the sequence \(A_n\) defined by \(A_1 = 17\) and \(A_{n+1} = A_n \cdot a_n\), determine the limit of \(A_n\) as \(n\) approaches infinity.

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\ln(\cos(2x))}{\ln(\cos(3x))} \]

Answer: \dfrac{4}{9}


Question: Compute the determinant of the matrix \( M = \text{diag}(1, 2, \dots, n) - n I_n + n 1_n 1_n^T \), where \( I_n \) is the \( n \times n \) identity matrix and \( 1_n \) is the \( n \times 1 \) vector of all ones.

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


Question: Suppose $f : \Omega \to \mathbb{C}$ is a holomorphic function on some open set $\Omega$. If $f'(z) \neq 0$ for some $z \in \Omega$, does there necessarily exist a neighborhood $U$ of $z$ where $f$ is injective? Answer 'yes' or 'no' and justify your answer.

Answer: yes


Question: A mouse is trapped in a maze and can initially choose to go right or left. If it goes left, it will walk for 3 minutes and return to the initial position. If it goes right, with probability \( \frac{1}{3} \) it will exit the maze after 2 minutes, and with probability \( \frac{2}{3} \) it will return to the initial position after 5 minutes. Assuming the mouse randomly chooses left or right, calculate the variance of the time it remains in the maze.

Answer: 438


Question:
Calculate the Jacobian determinant for the transformation given by \( u = x \cdot y \) and \( v = \frac{y}{x} \). Express the Jacobian determinant as:
\[ J = \begin{vmatrix} \frac{dx}{du} & \frac{dx}{dv} \\ \frac{dy}{du} & \frac{dy}{dv} \end{vmatrix} \]

Answer: \dfrac{1}{2v}


Question:
Evaluate the definite integral for $a > 0$:
\[ I = \int_{0}^{1} \left( \left( 1-x^a \right)^{\frac{1}{a}} - x \right)^2 \, dx \]

Answer: \dfrac{1}{3}


Question: If a sequence of uniformly continuous functions on a compact set converges uniformly, is the sequence necessarily bounded?

Answer: Yes


Question: Find the real number \( a \) such that \( \lim_{n\to\infty}{(n^3+an^2)^{\frac{1}{3}}-(n^2-an)^{\frac{1}{2}}}=1 \).

Answer: \dfrac{6}{5}


Question: Does there exist a continuous and bounded function \( f(x) \) on \( \mathbb{R} \) such that \( \lim\limits_{x\to +\infty} f(x) \) does not exist, but the limit of the sequence \( \lim\limits_{n\to \infty} f(n) \) exists for \( n \in \mathbb{N} \)?

Answer: Yes


Question: Evaluate the limit: \(\lim \limits_{x \to 0} \frac{\sin x^2}{\sqrt{1+x\sin x} - \sqrt{\cos x}}\)

Answer: \dfrac{4}{3}


Question: Evaluate the integral \( \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\exp\left(n x-\frac{x^2}{2}\right) \sin(2 \pi x)\ dx \) for \( n \in \mathbb{N}\setminus \{0\} \).

Answer: 0


Question: What is the smallest number of subsets needed to partition the set of all integers such that none of the subsets contain an infinite arithmetic progression?

Answer: 2


Question: Find the residue of the function \( f(z) = \frac{z^3+5}{z(z-1)^3} \) at the pole \( z_0 = 1 \).

Answer: 6


Question: A computer component has a lifetime (in years) that follows a Gamma distribution with a mean of 6 years and a variance of 18 years squared. Calculate the probability that the component will last at least 9 years.

Answer: 4e^{-3}


Question: Find the curvature of the ellipse given by the equation \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) at the point \(P=(a,0)\).

Answer: \dfrac{a}{b^2}


Question: Find all real roots of the equation \((1-4x)^4 + 32x^4 = \frac{1}{27}\).

Answer: \dfrac{1}{6}


Question: Evaluate \( \frac{dy}{dx} \bigg\vert_{x=0} \) for the function \( y = \frac{x}{1+\frac{x}{1+\frac{x}{1+\ldots}}} \).

Answer: 1


Question: Compute \(2^{111455} \pmod{2012}\).

Answer: 36


Question: Find the maximum value of the function \( f(x,y,z) = y \) subject to the constraints \( y^2 + x^2 + z^2 = 3 \) and \( y + x + z = 1 \).

Answer: \dfrac{5}{3}


Question:
Given that $x_1, x_2, \ldots, x_n$ are non-negative real numbers such that $x_1 + x_2 + \cdots + x_n = 1$, find the maximum value of the expression:
\[ x_1^2 + x_2^2 + \cdots + x_n^2 + \sqrt{x_1 x_2 \cdots x_n} \] for a positive integer $n \ge 2$. Provide your answer as a positive integer.

Answer: 1


Question: Consider the function \( f(x) \) which counts the number of prime powers less than \( x \). Evaluate the limit \( \lim_{x \to \infty} \frac{f(x)}{x} \). What is the value of this limit?

Answer: 0


Question: Calculate the expected value \( E[X_{n+1} | X_0 = i_0, \ldots, X_{n-1} = x_{n-1}, X_n = i] \) for a Markov chain, given that \( \sum_{j}^n j p_{i,j} = i \).

Answer: i


Question: Calculate the remainder when \( 14^{20192019} \) is divided by 60.

Answer: 44


Question: Find an asymptotic upper bound for the recurrence relation \( T(n) = 4T\left(\frac{n}{2}\right) + n^2 \log_2 n \).

Answer: O(n^2 \log^2 n)


Question: Multiply the polynomials \((x^2 + x + 1)\) and \((x^2 + x + 1)\) in the Galois field \(\text{GF}(8) = \mathbb{Z}_2[x]/\langle x^3 + x + 1 \rangle\). Simplify the result to find the equivalent polynomial in \(\text{GF}(8)\).

Answer: x + 1


Question: Determine the number of 11-digit strings that use each of the digits 0 through 9 at least once and do not have any consecutive identical digits.

Answer: 163296000


Question:
Find the sum of the series:
\[ 1 + \frac{1\cdot3}{6} + \frac{1\cdot3\cdot5}{6\cdot8} + \cdots \]

Answer: 4


Question: In triangle ABC, given that AB = CD and the angles are marked in the diagram, find the measure of angle $x$ in degrees.

Answer: 30


Question: Evaluate the definite integral \( \int_{0}^{\pi}\lfloor\cot x\rfloor \, dx \), where \( \lfloor a \rfloor \) denotes the greatest integer less than or equal to \( a \).

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


Question: Given that the complex number \(4 - 5i\) is a root of the quadratic equation \(z^2 + (a + 8i)z + (-39 + bi) = 0\), where \(a\) and \(b\) are real numbers, find the other root of the equation.

Answer: -6 - 3i


Question: Let \( A_1, A_2, \ldots, A_n \) be convex sets in a vector space. Suppose \( x \in \operatorname{co}(A_1 \cup A_2 \cup \dotsb \cup A_n) \). Is it true that \( x = t_1 a_1 + \dotsb + t_n a_n \) such that \( \sum_{i=1}^n t_i = 1 \), \( t_i \ge 0 \), and \( a_i \in A_i \) for each \( i \)?

Answer: Yes


Question:
Evaluate the limit \( \lim_{a \to -\infty} b \) for a point \( P(a,b) \) lying on the curve satisfying the differential equation:
\[
2xy^2dx + 2x^2 y dy - \tan(x^2y^2) dx = 0.
\]
Choose the correct option:
(a) 0
(b) -1
(c) 1
(d) does not exist.

Answer: A


Question: Find the smallest possible value of \( n \) such that there are exactly 77,000 ordered quadruples \((a, b, c, d)\) with \( \gcd(a, b, c, d) = 77 \) and \( \mathrm{lcm}(a, b, c, d) = n \).

Answer: 27720


Question: Let $M^n$ be an $n$-dimensional topological closed manifold. Suppose there exists an embedding $i:M \to M \times [0,1]$ such that $i(M)$ is contained in the interior of $M \times [0,1]$ and separates $M \times \{0\}$ and $M \times \{1\}$. Show that the region bounded by $M \times \{0\}$ and $i(M)$ is homeomorphic to $M \times [0,1]$.

Answer: M \times [0,1]


Question: Three fair dice are cast. In 10 independent casts, let X be the number of times all three faces are alike, and let Y be the number of times only two faces are alike. Find the joint probability distribution function of X and Y, and compute the expected value of 6XY.

Answer: \dfrac{25}{4}


Question: Is the function \( f(x,y) = d(x,y) \) continuous from a metric space \( X \times X \) to \( \mathbb{R} \), where \((X, d)\) is a metric space?

Answer: Yes


Question: Is the ring $\mathbb{Z}^2$ a quotient of the polynomial ring $\mathbb{Z}[x_1, x_2, \dots, x_n]$ for some integer $n$?

Answer: Yes


Question: Let $k \in (0,1)$ be fixed and $L \in \mathbb{R}$ be finite. If $\limsup_{x \to \infty} f(kx) = L$ and $\liminf_{x \to \infty} f\left(\frac{x}{k}\right) = L$, determine whether $\lim_{x \to \infty} f(x) = L$. Provide a justification for your answer.

Answer: L


Question: Given a group of $n$ bandits, where $1 \leq n \leq 30$, they want to ensure that at least $k$ bandits, where $1 \leq k \leq n$, must agree to retrieve their hidden treasure. The treasure is secured with multiple locks, and each lock can be opened if at least one bandit in a group has a key to it. Determine the minimum number of locks needed such that any group of at least $k$ bandits can open all the locks, but no group of fewer than $k$ bandits can do so. For example, if $n=3$ and $k=2$, the minimum number of locks needed is 3. Find the number of locks needed for given values of $n$ and $k$.

Answer: \dbinom{n}{k-1}


Question:
Let \((X,Y)\) be a uniformly chosen point from the region \(A \subset \mathbb{R}^2\), where \(A = \{(x,y) \mid |y| \leq x \cdot e^{-x}\}\). The joint probability density function is given by:
\[
 f(x,y) = \begin{cases} \dfrac{a}{\text{area of}~ A} & (x,y)\in A \\ 0 & \text{else} \end{cases}
\]
Compute the expected value \(E[X]\) given that the area of \(A\) is 2.

Answer: 2


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sqrt[n]{\frac{1}{n}-\frac{1}{2^n}} \] without using L'Hôpital's rule.

Answer: 1


Question: Let \( f(x) = e^x + x^3 + x - 3 \). Given that this function is invertible, find the value of \( f^{-1}(-2) \).

Answer: 0


Question: Determine the number of elements in the group of invertible $2 \times 2$ matrices over the field of seven elements.

Answer: 2016


Question:
Determine whether there exist matrices $A, B, C, D$ in $M_n(K)$ for some $n$ and field $K$ such that the following conditions are satisfied:

1. $AB = BA$
2. $CD = DC$
3. $AD - BC = I_n$
4. $DA - CB \neq I_n$

Answer: Yes


Question: Find all monic polynomials \( f \in \mathbb{Z}[x] \) such that there exists a positive integer \( N \) for which every prime \( p > N \) divides \( 2(f(p)!)+1 \), where \( f(p) \) is positive.

Answer: x - 3


Question: In triangle $ABC$, $AB = 7$, $BC = 9$, and $\angle B = 74^\circ$. The angle bisector of $\angle B$ intersects the perpendicular bisector of $AC$ at point $P$. If $AC \cap BP = D$, calculate the length of segment $PD$.

Answer: 3.7


Question: Determine whether the expression \( \frac{\left(1+ \sqrt{k^4-k^2+1}\right)^k + \left(1-\sqrt{k^4-k^2+1}\right)^k}{2^k} \) is an integer for every non-negative integer \( k \).

Answer: Yes


Question: Determine the cardinality of the set of all obtuse triangles on the Cartesian plane.

Answer: \mathfrak{c}


Question: Consider two finitely presented groups \( \langle \Gamma | R \rangle \) and \( \langle \Gamma | S \rangle \) where \( \langle R \rangle \subsetneq \langle S \rangle \). Could these groups be isomorphic?

Answer: Yes


Question: Let $(G,+)$ be an Abelian topological group, and let $V$ be an open connected neighborhood of zero in $G$. A subset $A$ of $G$ is called convex if for each $x \in G$, the condition $2x \in A + A$ implies that $x \in A$. Does there exist an open convex neighborhood of zero $W$ such that $W \subset V$?

Answer: Yes


Question: Let $a, b, c, d, e, f$ be nonnegative real numbers such that $a^2 + b^2 + c^2 + d^2 + e^2 + f^2 = 6$ and $ab + cd + ef = 3$. Determine the maximum value of $a+b+c+d+e+f$. Provide your answer as a single number.

Answer: 6


Question: Determine whether the statement \( \log(i^2) \neq 2\log(i) \) is true or false, given that \( \log(z) = \ln|z| + i\arg(z) \).

Answer: B


Question: Given the recursive sequence defined by $x_{n+1} = \sqrt{\frac{1}{2}(1+x_n)}$ with $-1 < x_0 < 1$, find the limit \( \lim_{n \rightarrow \infty} \cos\left(\frac{\sqrt{1-(x_0)^2}}{x_1 x_2 \cdots x_n}\right) \).

Answer: x_0


Question: Let $X$ and $Y$ be compact metric spaces, and let $\pi: X \to Y$ be a continuous onto map. Suppose $\nu$ is a Borel probability measure on $Y$ such that $\nu$-almost every $y \in Y$ has a unique $\pi$-preimage. Does there exist a measurable map $s: Y \to X$ such that for $\nu$-almost every $y \in Y$, $(\pi \circ s)(y) = y$?

Answer: Yes


Question: Evaluate the double integral \( \iint_R |xy| \, dx \, dy \), where \( R \) is the region enclosed by \( x^2 + 4y^2 \ge 1 \) and \( x^2 + y^2 \le 1 \).

Answer: \dfrac{3}{8}


Question: Find the smallest positive integer consisting only of the digit '1' repeated multiple times (e.g., 111, 1111) that is divisible by 2019.

Answer: 672


Question: Calculate the limit: \[ L = \lim_{n \to \infty} -(n - n^{n/(1+n)}) \]

Answer: -\infty


Question: Show that \( \lim_{a \rightarrow \infty} \int_0^1 f(x)x\sin(ax^2) \, dx = 0 \) given that \( f \) is integrable on \((0,1)\).

Answer: 0


Question: Determine whether every continuous automorphism of \( \mathbb{C}_p \) can be expressed as \( \sigma(x) = \lim_{n\to \infty}\bar{\sigma}(x_n) \), where \( x = \lim_{n\to \infty}x_n \), \( \bar{\sigma} \in \text{Gal}(\bar{\mathbb{Q}}_p/ \mathbb{Q}_p) \), and \( x_n \in \bar{\mathbb{Q}}_p \).

Answer: Yes


Question: Determine whether there exists a function \( f(x, y): \mathbb{R}^2 \to \mathbb{R} \) such that \( f(x, y) \) has directional derivatives at the point \((0,0)\) in every direction, but \((0,0)\) is a point of discontinuity for \( f \).

Answer: Yes


Question: Evaluate the line integral \( \oint_C \frac{1}{y} \, dx + \frac{1}{x} \, dy \) using Green's theorem, where \( C \) is the boundary of the region defined by \( x=1, x=4, y=1, y^2=x. \)

Answer: \dfrac{3}{4}


Question: Find the minimum distance between the circle given by the equation \(x^2 + y^2 = 9\) and the curve given by the equation \(2x^2 + 10y^2 + 6xy = 1\).

Answer: 2


Question: Evaluate the limit: $$\lim\limits_{(x,y)\to(1, 8)} \frac{\tan(y-8) \sin^2(y-8x)}{(x - 1)^2 + (y - 8)^2}$$

Answer: 0


Question: Find the number of integer values of \(n\) such that the order of the tensor product \(\mathbb{Z}_n \otimes \mathbb{Z}_{12}\) is equal to \(\frac{n}{2}\).

Answer: 2


Question: In triangle \(\triangle ABC\), let \(D\), \(E\), and \(F\) be the midpoints of \(BC\), \(CA\), and \(AB\) respectively, and let \(G\) be the intersection of \(AD\) and \(BE\). If \(AG = 15\), \(BG = 13\), and \(FG = 7\), find the area of triangle \(\triangle ABC\).

Answer: 252


Question: Find a closed expression for the series \( \sum_{n=0}^\infty \frac{n^2a^n}{n!} \), where \( a \in (0,\infty) \).

Answer: a(a + 1)e^a


Question: Determine whether there exists a finite constant $c$ such that the inequality \[ \left|\mathbb{P}\left(\sqrt{12n}\left(\bar{X} - \frac{1}{2}\right)\le x\right) - \Phi(x)\right| \le \frac{c}{n} \] holds uniformly in $x$, where $X_i \sim U[0,1]$. Provide a justification for your answer.

Answer: Yes


Question:
Given a set of positive numbers $X = \{x_1, x_2, x_3, \dots, x_n \}$ such that $\sum x_i = m$, determine the value of $x_i$ that maximizes the summation:
\[ S = \sum_{x_i \in X, x_j \in X} (x_ix_j)^3 \]

Answer: m


Question: Let \( \{a_n\} \) be a sequence such that \( \{ a_{2n}\} \rightarrow L \) and \( \{a_{2n+1}\} \rightarrow L \). Prove that \( \{a_n\} \rightarrow L \).

Answer: L


Question: Evaluate the limit: \[ \lim _{x \rightarrow 0} \dfrac{2f(x)-f(2x)}{x-f(x)} \] where \( f(x) = \int _0^x \dfrac{\sin t}{t} dt \).

Answer: 6


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

Answer: \dfrac{1}{e}


Question: Determine the order of the pole at \( z_0 = 1 \) for the function \( \frac{\sin^3(z-1)}{\log^4(z)[1-\cos(z-1)]^2} \).

Answer: 5


Question: Determine whether the quotient ring \( A = \mathbb{R}[x]/\langle x^2-x+1\rangle \) is isomorphic to the complex numbers \( \mathbb{C} \).

Answer: \mathbb{C}


Question: Find the limit of the function \( f(x,y) = (x^2+y^2)^{x^2y^2} \) as \((x, y)\) approaches \((0, 0)\).

Answer: 1


Question: Consider a sequence of functions $f_n$ defined on a compact interval that uniformly converges to a function $f$. If $f$ is Riemann integrable, must each $f_n$ also be Riemann integrable? Answer 'True' or 'False' and justify your answer.

Answer: False


Question: Let \( \gamma: [a,b] \to \mathbb{C} \) be a piecewise continuously differentiable path, and let its image be \( \gamma^* \). Suppose \( f: \gamma^* \to \mathbb{C} \) is continuous. Does \( \max_{z \in \gamma^*} |f(z)| \) always exist?

Answer: Yes


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

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


Question: Suppose that $X_1, \ldots, X_{100}$ are random variables with $\operatorname{E} X_i = 100$ and $\operatorname{E} X_i^2 = 10100$. If $\operatorname{Cov}(X_i, X_j) = -1$ for $i \neq j$, what is $\operatorname{Var}(S)$, where $S = \sum_{i=1}^{100} X_i$?

Answer: 100


Question: Evaluate the surface integral \( \iint_S \textbf{F} \cdot \textbf{n} \, d\alpha \) where \( S \) is the truncated half-cylinder defined by \( z = \sqrt{4-y^2} \) for \( 0 \le x \le 1 \), with \( \textbf{n} \) as the outward-pointing normal. The vector field is given by \( \textbf{F}(x,y,z) = (x^2+y^2, yz, z^2) \). Calculate the integral and verify that the result is 16.

Answer: 16


Question: Two players, $A$ and $B$, play a game where $A$ starts with $n_A \geq 0$ dollars and $B$ starts with $n_B \geq 0$ dollars. A fair coin is tossed repeatedly. If it lands heads, $B$ gives a dollar to $A$. If it lands tails, $A$ gives a dollar to $B$. The game ends when one player loses all their money. Determine the average number of steps until the game ends.

Answer: n_A n_B


Question: Let $G$ be a finite $p$-group and $H$ a subgroup of $G$. For subgroups $K_1$ and $K_2$ of $H$ that are conjugate in $G$, does it necessarily follow that $r(K_1,H) = r(K_2,H)$, where $r(K,H)$ is the smallest $r$ such that there exists a chain of subgroups $K=L_0 \lhd L_1 \lhd \cdots \lhd L_r = H$, with each $L_i/L_{i-1}$ cyclic?

Answer: Yes


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

Answer: 1


Question: Evaluate the line integral \( \int_{C}{\bf{F}}\cdot d{\bf{s}} \) where \( F=\left[\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}\right] \) and \( C \) is the closed curve consisting of the line segments from \((-1,0)\) to \((0,-2)\), \((0,-2)\) to \((2,0)\), \((2,0)\) to \((3,4)\), \((3,4)\) to \((0,5)\), and \((0,5)\) to \((-1,0)\).

Answer: 0


Question: Determine whether the following logical implication is true: If $P(x) = P$ is a constant for all $x \in E$, does $\forall x \in E, (P \vee Q(x))$ imply $(P) \vee (\forall x \in E, Q(x))$?

Answer: True


Question: Is there a Hausdorff space \((X, \tau)\) with \(|X| > 1\) such that the box product \(\Box_{n \in \omega} X\) is connected?

Answer: No


Question: Let \( A \) be a \( 2 \times 2 \) complex matrix with \( \det(A) = -6 \) and \( \text{trace}(A) = 1 \). Find the trace of \( A^4 - A^3 \).

Answer: 78


Question: Let $\Omega$ be a domain in $\mathbb{C}^n$. Consider the algebra $A(\Omega) = \mathcal{C}(\overline{\Omega}) \cap \mathcal{O}(\Omega)$. Let $\{F_\nu\}$ be a sequence in $A(\Omega)$ such that $F_\nu(\overline{\Omega}) \subseteq \overline{\mathbb{D}}$ for every $\nu \in \mathbb{Z}_+$. By Montel's Theorem, there is a subsequence of this sequence which converges uniformly on compact sets of $\Omega$. Let $F$ be the limit function of this subsequence. Can $F$ be extended continuously to the boundary of $\Omega$, i.e., does $F$ belong to $A(\Omega)$?

Answer: No


Question:
Find the limit of the sequence \((x_n)\) defined by the recursive relation:
\[ x_1 = a > 0, \quad x_{n+1} = \frac{2x_n \cdot \cos\left(\frac{\pi}{2^n+1}\right)}{x_n+1} \]
Determine \( \lim_{n \to \infty} x_n \).

Answer: 1


Question:
Determine the points of differentiability of the function \( f(z) = |z|^2 + i \bar{z} + 1 \). Choose the correct option:

1) \( i \)
2) \( 1 \)
3) \( -i \)
4) No point in \( \mathbb{C} \)

Answer: C


Question: Evaluate the integral: \[ \int_{1/4}^4 \frac{1}{x}\sin\left(x-\frac{1}{x}\right)\,dx \]

Answer: 0


Question: Find the sixth derivative of the function \(x^4e^{x^2}\) using the Maclaurin series.

Answer: 720


Question: Calculate the value of the series: \( \cos^2\theta + \cos^2(\theta+1^\circ) + \cos^2(\theta+2^\circ) + \ldots + \cos^2(\theta+179^\circ) \).

Answer: 90


Question: Evaluate the limit \( \lim_{x\to 0} (e^x + x)^{1/x} \) using L'Hopital's rule.

Answer: e^2


Question: Find all solutions to the equation \(|z+\sqrt{z^2-1}|=1\).

Answer: [-1, 1]


Question: Determine the radius of convergence of the power series \( \sum_{n=1}^\infty \frac{n^n}{(n!)^2}z^n \) using the root test.

Answer: \infty


Question:
Given the joint probability density function:
\[
f(x,y) =
\begin{cases}
e^{-x-y},  & \text{for } x>0 \text{ and } y>0 \\
0, & \text{elsewhere}
\end{cases}
\]
Find the probability \( P(X+Y>3) \).

Answer: 4e^{-3}


Question: Determine the fundamental group \( \pi_1(W_n, w_0) \) of the space \( W_n \), which is formed by taking \( n \) copies of the circle \( S^1 \) and identifying all \( n \) base points to form a new base point \( w_0 \).

Answer: F_n


Question: Calculate the work done when a unit particle is moved anticlockwise around a circle with center \((0,0,4)\) and radius 2 in the plane \(z=4\) within the force field defined by \(\mathbf{F}=(xy+z)\boldsymbol{\hat{i}} + (2x+y)\boldsymbol{\hat{j}}+(x+y+z)\boldsymbol{\hat{k}}\).

Answer: 8\pi


Question: Calculate \( L = \lim\limits_{n \to \infty} \frac{1}{n}\cdot\log\left(3^\frac{n}{1} + 3^\frac{n}{2} + \dots + 3^\frac{n}{n}\right) \).

Answer: \log 3


Question: Is every smooth automorphism of a Lie Group $G$ a diffeomorphism?

Answer: Yes


Question: If \( G \) is an abelian group with generators \( x \) and \( y \) of order 16 and 24 respectively, and \( x^{2} = y^{3} \), what is the order of \( G \)?

Answer: 48


Question: Determine the order of the Galois group of the polynomial \( f(x) = x^5 - 7 \) over \( \mathbb{Q} \).

Answer: 20


Question: Find the area enclosed by one loop of the lemniscate given by the equation \( r^2 = 81\cos(2\theta) \).

Answer: \dfrac{81}{2}


Question: What is the computational complexity of eigenvalue decomposition for a unitary matrix? Is it O(n^3)?

Answer: O(n^3)


Question: Find the norm of the operator \( T_f: (C([a,b],\mathbb{C}), \lVert \cdot \rVert_1) \to \mathbb{C} \) defined by \( T_f(g) = \int_a^b f(x)g(x) \, dx \) for a given function \( f \in C([a,b],\mathbb{C}) \).

Answer: \|f\|_\infty


Question: Let \( f \) be a real-valued function of a variable \( x \) such that \( f'(x) \) takes both positive and negative values and \( f''(x) > 0 \) for all \( x \). Prove that there exists a real number \( p \) such that \( f \) is increasing for all \( x \ge p \).

Answer: p


Question: Determine whether every sane hypergraph is summable. A hypergraph $H=(V,E)$ is defined as sane if $V$ is finite, $E \neq \emptyset$, and $\emptyset \notin E$, and $e\not\subseteq e'$ whenever $e\neq e' \in E$. It is summable if there exists a map $f:V\to \mathbb{Z}_{\geq 0}$ such that for all $e, e' \in E$, $$\sum_{v\in e}f(v) = \sum_{w\in e'}f(w) > 0.$$ Is every sane hypergraph summable?

Answer: Yes


Question: Find the minimum distance between the point \([6,6]\) and the circle defined by \(x^2 + y^2 = 25\) in the metric space where the distance \(p_1(x,y)\) is given by the sum of the absolute differences of their coordinates, \(p_1(x,y) = \sum |x_k - y_k|\).

Answer: 12 - 5\sqrt{2}


Question: Find the closed form of the integral \( \int_0^\infty e^{-x}\ln(kx) \, dx \).

Answer: \ln(k) - \gamma


Question: Find the number of ordered pairs \((\alpha, \beta)\) that satisfy the equations \(\cos(\alpha-\beta)=1\) and \(\cos(\alpha+\beta)=\frac{1}{e}\), where \((\alpha, \beta)\) are within the interval \([\-\pi, \pi]\).

Answer: 4


Question: Compute the area of the square defined by $0 \leq u,v \leq 1$ on a surface $M$ with the metric given by \( ds^2 = f^2 \, du^2 + f^{-2} \, dv^2 \), where $f$ is a positive function of $u$ and $v$. Is the area obtained by integrating the metric over the square?

Answer: 1


Question: Find the radius of convergence for the power series \( \sum_{n=2}^\infty a_nx^n \) given that \( n^3 < |a_n| < n^4 \).

Answer: 1


Question: Determine whether the following statement is true: For a closed oriented surface $S$ and a finite set of points $X$ on $S$, is it true that $$ H^1(S \setminus X, \mathbb{C}) \simeq H^1(S,X,\mathbb{C}) $$?

Answer: True


Question: If \(\Theta\) is a random variable uniformly distributed over \([-\pi, \pi]\), determine the probability density function \(f_\Theta(\theta)\) when \(\Theta\) is restricted to the interval \([-\pi/4, \pi/4]\).

Answer: \dfrac{2}{\pi}


Question: Determine the group that \( \langle (135)(246), (12)(34)(56) \rangle \subset S_6 \) is isomorphic to, using only the groups \( D_n, S_n, \mathbb{Z}_n \), and their direct sums, where \( S_n \) is the permutation group of \( n \) elements, and \( D_n \) is the dihedral group of order \( 2n \).

Answer: \mathbb{Z}_6


Question: Calculate the probability that the distance of a randomly chosen point A from the origin in the unit disk is greater than twice the distance of another randomly chosen point B from the origin in the same disk. Express this probability as \( P(R_1 > 2R_2) \), where \( R_1 \) and \( R_2 \) are the distances of points A and B from the origin, respectively.

Answer: \dfrac{1}{8}


Question: Determine the limit \( \lim_{n \to \infty} (1+a_n)^{c_n} e^{-a_n c_n} \), given that \( a_n \to 0 \) and \( c_n a_n^2 \to 0 \) as \( n \to \infty \).

Answer: 1


Question: Determine whether there exists a matrix $X \in \mathbb{R}^{m \times n}$ such that $ABX = A$, where $A \in \mathbb{R}^{n \times n}$ with $\text{rank}(A) = n$ and $B \in \mathbb{R}^{n \times m}$. If such a matrix $X$ exists, provide an example.

Answer: X = B^\top (B B^\top)^{-1}


Question: Find the area bounded by one branch of the trochoid given by \(x(t)=at-b\sin t,\ y(t)=a-b\cos t,\ 0<b<a\) and its tangent line at the lowest point. Express your answer in terms of \(a\) and \(b\).

Answer: \pi b (2a + b)


Question: Determine whether the number \( \alpha = 3 - \sqrt[5]{5} - \sqrt[5]{25} \) is algebraic over \( \mathbb{Q} \). If it is algebraic, find the degree of the extension \( [\mathbb{Q}(\alpha):\mathbb{Q}] \).

Answer: 5


Question: Suppose $u$ is a harmonic function on the exterior of the unit disk, $u \in \text{har}(\mathbb{C} \setminus \mathbb{D})$, that vanishes on the boundary of the unit disk and at infinity. Prove that $u$ is identically zero in $\mathbb{C} \setminus \mathbb{D}$.

Answer: 0


Question: Let $X$ and $Y$ be irreducible projective schemes over $\mathbb{C}$ with $X \subset Y$. Let $x \in X$ be a closed point. Assume that for any positive integer $n$ and any morphism from $\mathrm{Spec} (\mathbb{C}[t]/(t^n))$ to $Y$ such that its composition with the natural morphism from $\mathrm{Spec}(\mathbb{C})$ to $\mathrm{Spec}(\mathbb{C}[t]/(t^n))$ corresponds to the closed point $x$, this morphism factors through $X$. Does this imply that there exists an open neighbourhood $U$ of $x$ in $Y$ such that $U$ is contained in $X$?

Answer: Yes


Question: Find a polynomial \( f(x) \) with integer coefficients such that \( f(\sqrt{2} + i) = 0 \).

Answer: x^4 - 2x^2 + 9


Question: Determine the set of positive numbers \( b \) such that \( b^x \geq 1+x \) for all real numbers \( x \).

Answer: e


Question: Evaluate the integral \( I = \int_0^{1000} \frac{e^{-10x} \sin(x)}{x} \, dx \) to within \( \pm 10^{-5} \).

Answer: 0.09967


Question: Consider a discrete random walk starting at position $x=0$ with equal probabilities of moving left or right. What is the probability that the particle will never leave the interval $[-n, n]$, where $n$ is a positive integer?

Answer: 0


Question: Find the volume of the region bounded by the inequalities $x^2 + y^2 \leq 4$, $x \geq 0$, and $0 < z < x^2 - y^2$. Set up and evaluate the appropriate triple integral to determine this volume.

Answer: 4


Question: Given a group \( G \) with order \( |G| = 231 \), determine the maximum number of subgroups of order 3 that \( G \) can have.

Answer: 7


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

Answer: 1


Question: Find the value of $x$ such that $x^{\sqrt{x}} = \sqrt[\frac{1}{\sqrt{2}}]{\dfrac{1}{\sqrt{2}}}$.

Answer: \dfrac{1}{2}


Question: Compute the dimension of the variety \( Z(x_1x_2, x_2x_3, x_1x_3) \) in \( \mathbb{A}_k^3 \).

Answer: 1


Question: Determine whether the following statement is true or false: For every integer \( n > 0 \), there exists an integer \( N \) such that for any two-coloring of the subsets of a set with \( N \) elements, there is always an \( n \)-element subset whose subsets are all the same color.

Answer: False


Question: Determine the upper bound time complexity (Big-O notation) for the recurrence relation: \[ T(n) = T\left(\frac{n}{3}\right) + T\left(\frac{2n}{3} + 4\right) + a \times n \]

Answer: O(n \log n)


Question: Given the expansion \((1+x+x^2)^n = \sum_{r=0}^{2n} a_r x^r\), find the value of the expression \(a_r - {n \choose 1}a_{r-1} + {n \choose 2}a_{r-2} - {n \choose 3}a_{r-3} + \cdots + (-1)^r{n \choose r}a_0\) when \(r\) is not a multiple of 3.

Answer: 0


Question:
Let $f(x)$ be a continuous real-valued function on $[a,b]$ and let $M=\max\{|f(x)| : x \in [a,b]\}$. Determine whether the following statement is true:
\[
M= \lim_{n\to\infty}\left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n}.
\]

Answer: True


Question: Let \( f:[0,2)\to \mathbb{R} \) be defined by \( f(x) = \begin{cases} x^2 & \text{if } 0 \le x \le 1 \\ 3-x & \text{if } 1 < x < 2 \end{cases} \). Find the Lebesgue outer measure \( m^*(A) \), where \( A = f^{-1}\left(\frac{9}{16},\frac{5}{4}\right) \).

Answer: \dfrac{1}{2}


Question: Compute the limit \( \lim_{x \to \infty} x e^{-x^2} \int_0^x e^{t^2} \, dt \).

Answer: \dfrac{1}{2}


Question: Find the volume of the region inside both the cylinder given by the equation \(x^2 + y^2 = 1\) and the ellipsoid given by the equation \(4x^2 + 4y^2 + z^2 = 64\) in \(\mathbb{R}^3\). Use cylindrical coordinates to set up the integral for the volume, considering the boundaries \(-2\sqrt{15} \leq z \leq 2\sqrt{15}\), \(0 \leq \theta \leq 2\pi\), and \(r = 1\). Verify if these boundaries are correct for calculating the volume of the intersection.

Answer: \dfrac{8\pi}{3} \left(64 - 15\sqrt{15}\right)


Question: What is the probability that, when drawing cards one at a time from a shuffled standard deck of 52 cards, the number of red cards drawn never exceeds the number of black cards drawn at any point during the process?

Answer: \dfrac{1}{27}


Question: Use the fourth-order Runge-Kutta method to approximate \( x(2) \) for the differential equation \( \frac{dx}{dt} = \frac{1}{2x} \) with the initial condition \( x(1) = 1 \) and step size \( h = 1 \). Provide your answer rounded to five decimal places.

Answer: 1.41438


Question:
Determine the order of accuracy of the difference formula:
$$Q(h) = \frac{f(x-2h) - 8f(x-h) + 8f(x+h) -f(x+2h)}{12h}$$
using the Taylor expansions around $x$ for $f(x-2h)$, $f(x-h)$, $f(x+h)$, and $f(x+2h)$. Given that the truncation error is zero, what is the order of accuracy of this formula?

Answer: 4


Question: Calculate the 32nd derivative of the function \( f(x) = \exp{[\sin{(x^{10}+x^{11})}]} \) at \( x = 0 \).

Answer: 0


Question: Let \( A \in \mathbb{R}^{n\times n} \) be a matrix with real eigenvalues. If \( B = X^TAX \) for an invertible matrix \( X \in \mathbb{R}^{n\times n} \), does \( B \) also have real eigenvalues? Assume \( X \) is symmetric and positive definite.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{a\to 0}\frac{\delta_{a}+\delta_{-a}-2\delta}{a^{2}} \] where \( \delta_{a} \) is the Dirac distribution at the point \( a \).

Answer: \delta''


Question: Find all functions \( f \in C^1(\mathbb{R}, \mathbb{R}) \) such that \( f^2 + (1 + f')^2 \leq 1 \).

Answer: 0


Question: Find all continuous functions $f$ on the interval $[0,1]$ such that for every $x$ in $(0,1)$, the following holds: \[ \int_0^x f(t)\ dt = \int_x^1 f(t)\ dt. \]

Answer: 0


Question: An insurer finds that the time in hours it takes to process a loss amount $x$ has a uniform distribution on the interval $(\sqrt{x}, 2\sqrt{x})$. Given that the loss distribution has a probability density function $f(x) = \frac{1}{x^2}$ for $x > 1$, find the expected time to process a claim of a random amount.

Answer: 3


Question: Determine whether the infinite product \( \frac{1}{2}\prod_{n=1}^\infty\frac{\operatorname{prime}(n)-1}{\operatorname{prime}(n)} \) converges, and if so, find its value.

Answer: 0


Question: Given \(x, y, z, t \in \mathbb{R}\) such that \(x^2 + y^2 = 9\), \(z^2 + t^2 = 4\), and \(xt - yz = 6\), find the greatest value of \(xz\).

Answer: 3


Question: Is it true that any element of the completion of a valued ring $R$ that is algebraic over the field of fractions of $R$ is in the henselization of $R$?

Answer: Yes


Question: Let $X$ and $Y$ be independent and identically distributed random variables, each following a uniform distribution on the interval (0, 1). Calculate the expected value of $X$ given that $X > Y$, denoted as $E(X|X>Y)$. What is this expected value?

Answer: \dfrac{2}{3}


Question: Given that $f$ is a function with a continuous second derivative, $f(\pi) = 1$, and \( \int_0^\pi{(f(x)+f''(x))\sin{x}dx}=2 \), find the value of $f(0)$. \( \boxed{} \)

Answer: 1


Question: Find the limit \( \lim _{n\to \infty }\left(\sum _{k=1}^n\left(\frac{k}{3^k}\right)\right) \).

Answer: \dfrac{3}{4}


Question: Evaluate the integral \( \int_{C(0,1)} |z-1||dz| \), where \( C(0,1) \) is the circle of radius 1 centered at the origin.

Answer: 8


Question: Let $f\colon [0,1]\rightarrow [0,1]$ be a continuous function such that $f(0)=f(1)$. Define the set \( A = \{ (t,s) \in [0,1] \times [0,1] \mid t \neq s, f(t) = f(s) \} \). Determine the cardinality of the set $A$. \( \text{card}(A) = ? \)

Answer: \mathfrak{c}


Question: Derive the critical region for the likelihood ratio test (LRT) statistic for the following hypothesis test: Let $X$ be a random variable with pdf $f(x; \theta) = \theta x^{\theta - 1}$ where $0 < x < 1$ and $\theta > 0$. Consider the null hypothesis $H_0: \theta = 1/2$ versus the alternative hypothesis $H_A: \theta = 1/4$. Determine the most powerful test using a significance level of $\alpha = 0.05$. Specifically, identify whether the critical region is $R = \{ X \in (0, 1) \ | \ X > c_2 \}$ or $R = \{ X \in (0, 1) \ | \ X < c_2 \}$, and calculate the value of $c_2$.

Answer: 0.0025


Question:
Find the surface area of the part of the cylinder \(y^2+z^2=2z\) that is cut off by the cone \(x^2=y^2+z^2\) using a double integral. Use the formula for the surface area of the surface \(z=f(x,y)\) over the region \(R\):
\[ S=\iint_R\sqrt{f_x^2+f_y^2+1}\,dx\,dy \]
Determine the appropriate region \(R\) and calculate the surface area. The answer is \(16\).

Answer: 16


Question: Determine the values of \( \alpha \) for which the double integral \( \iint_D \frac{1}{(x+y)^{\alpha}}\ dxdy \) converges, where \( D = \{0 \leq y \leq 1-x, \ 0 \leq x \leq 1\} \).

Answer: \alpha < 2


Question: Given that \(X_i \sim \text{Poisson}(\theta)\), find the maximum likelihood estimator (MLE) of \(\pi(\theta) = \exp(-\theta)\). Is \(\exp(-\bar{X})\) the MLE of \(\pi\)?

Answer: e^{-\bar{X}}


Question: Given that for any positive integer \(n\), the equation \(ax^n + by^n = 1 + 2^{n+1}\) holds, find the value of \(x^a + y^b\).

Answer: 5


Question: Determine the order of convergence of the secant method when applied to a function $f$ that has a double root at $r$, i.e., $f(r) = f'(r) = 0$.

Answer: 1


Question: Let \( G \) be a graph with vertex set \( V = \{1, 2, \ldots, 20\} \). Vertices \( u, v \in V \) are adjacent if and only if \( \gcd(u, v) = 1 \). Find the chromatic number \( \chi(G) \) of the graph.

Answer: 9


Question: Find the largest integer \( n \) such that \( 10^{17} < n! \).

Answer: 19


Question: Determine whether the sequence \( (\cos (\pi \sqrt{ n^2 + n}))_{n=1}^\infty \) converges.

Answer: 0


Question: Find the maximum possible value of \(|[a]|\) such that the equation \(3\sqrt{-2(x+3)}-1=|x+3|+a\) has exactly two real roots, where \([\cdot]\) denotes the greatest integer function.

Answer: 3


Question: Given that we have \( n \) variables, each with a pairwise correlation of 1, and the variance of their sum is 1, find the lower bound on the sum of the individual variances.

Answer: \dfrac{1}{n}


Question: Evaluate the integral \( \int_0^{\infty}\frac{\mathrm{d}z}{z^6+1} \) using the method of residues.

Answer: \dfrac{\pi}{3}


Question:
Solve for $x \in \mathbb{R}$:
\[ \dfrac{\sqrt{x^2-x+2}}{1+\sqrt{-x^2+x+2}} - \dfrac{\sqrt{x^2+x}}{1+\sqrt{-x^2-x+4}} = x^2-1 \]

Answer: 1


Question:
Determine the radius of convergence of the power series \( \sum_{n=0}^\infty a_n x^n \), where the coefficients \( a_n \) are defined by the recurrence relation:
\[
\begin{aligned}
a_{n+2} &= \frac{n(n+1) a_{n+1} - a_n}{(n+2)(n+1)}, \quad n \ge 1, \\
a_2 &= -\frac{a_0}{2}
\end{aligned}
\]
with arbitrary initial values \( a_0 \) and \( a_1 \).

Answer: 1


Question: Evaluate the double integral \( \iint_D 2x \cos(2(x-y)x) \, dx \, dy \), where \( D \) is the rectangle with corners at \((\pi/6, 2+\pi/6), (\pi/6, 4+\pi/6), (\pi/4, 4+\pi/4), \text{ and } (\pi/4, 2+\pi/4)\).

Answer: -\dfrac{5}{16}


Question: Determine the number of integral solutions \((x, y)\) to the equation \(xy(x + y + 1) = 2019^{2020} + 1\).

Answer: 0


Question: Given that \( \Im[f'(z)]=6x(2y-1) \) and \( f(0)=3-2i \), \( f(1)=6-5i \), find the value of \( f(1+i) \).

Answer: 6 + 3i


Question: Evaluate the limit: \[ \lim_{x \to +\infty} \left((x+7)^\frac{1}{17} - x^\frac{1}{17}\right) \]

Answer: 0


Question: Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed random variables. Calculate $E[X_1 \mid X_1 + X_2 + \cdots + X_n = x]$. Provide your answer in terms of $x$ and any relevant parameters of the distribution.

Answer: \dfrac{x}{n}


Question: How many unique bracelets can be made with 6 stones in a ring using three different colors, if each bracelet must contain at least one stone of each color and two bracelets are considered identical if one is a rotation or a flip of the other?

Answer: 56


Question: Determine if a number can be part of two different Pythagorean triplets. Specifically, given a Pythagorean triplet \((a, b, c)\) such that \(a^2 + b^2 = c^2\), is it possible to have another triplet \((a, d, e)\) such that \(a^2 + d^2 = e^2\) and \(b \neq d\)?

Answer: Yes


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

Answer: f(x) = x


Question: Let $A$ and $B$ be $n \times n$ Hermitian matrices. Denote by $(\alpha_i)$, $(\beta_i)$, and $(\gamma_i)$ the eigenvalues of $A$, $B$, and $A+B$, respectively. Assume there exist permutations $\sigma, \tau \in \mathfrak{S}_n$ such that \( \gamma_i = \alpha_{\sigma(i)} + \beta_{\tau(i)} \) for all $1 \le i \le n$. Does this imply that $A$ and $B$ commute?

Answer: Yes


Question: Find the value of \( f'(1) \) for the function \( f(x) = \sum_{i=1}^{\infty} (-1)^{i-1}\frac{x^i}{i} \).

Answer: \dfrac{1}{2}


Question:
Given a Markov chain with the following transition matrix and time weights for each state:

\[ \begin{bmatrix} \frac{3}{4} & \frac{1}{4} & 0 & 0 \\
\frac{4}{5} & 0 & \frac{1}{5} & 0 \\
\frac{6}{7} & 0 & 0 & \frac{1}{7} \\
0 & 0 & 0 & 1 \end{bmatrix} \]

and time weights:
- \( t_0 = 5 \)
- \( t_1 = 10 \)
- \( t_2 = 15 \)
- \( t_3 = 20 \)

Calculate the expected time to reach the absorbing state \( t_3 \) starting from \( t_0 \).

Answer: 1155


Question:
Consider the limiting joint distribution of the rescaled eigenvalues for Gaussian ensembles, given by:
\[
\lim_{N\to\infty}P(N^{2/3}(\lambda_N-2)\le s_1,\dots, N^{2/3}(\lambda_{N-k+1}-2)\le s_k)=F_{\beta, k}(s_1,\dots, s_k)
\]
where $\lambda_N, \lambda_{N-1}, \ldots, \lambda_{N-k+1}$ are the $k$ largest eigenvalues. Using the continuous mapping theorem, determine if the gap $\lambda_N-\lambda_{N-k+1}=O_p(N^{-2/3})$.

Answer: Yes


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

Answer: 1


Question: Find an integer $n$ such that $n$, $n+1$, $n+2$, $n+3$, and $n+4$ are all composite numbers.

Answer: 24


Question: Which plane figure, having the same area as others, minimizes the mean distance between two randomly selected points on its perimeter? Assume the points are selected independently and uniformly along the perimeter. Provide a justification for your answer.

Answer: circle


Question: Evaluate the integral \( \int_{-4}^4 \{nx\} \, dx \), where \( n \in \mathbb{R} \) and \( \{\cdot\} \) denotes the fractional part function.

Answer: 4


Question: Evaluate the limit: $$\lim_{t\to 0} \frac{\left(\sqrt{t+9}-3 \sqrt[3]{2 t+1}\right)}{\sqrt[3]{t+8}-2 \sqrt[3]{3 t+1}}$$

Answer: \dfrac{22}{23}


Question: Evaluate the integral \( \int x^2 \, d\mu \) where \( C \) is the Cantor set and \( \mu \) is the uniform measure on \( C \).

Answer: \dfrac{3}{8}


Question: Given that the variables $x$, $y$, and $z$ follow Gaussian distributions with means $\mu_x = \mu_y = \mu_z = 1,000,000$ and standard deviations $\sigma_x = \sigma_y = \sigma_z = 200,000$, calculate the probability that the function \[ f(x, y, z) = \frac{x}{1.1} + \frac{y}{1.1^2} + \frac{z}{1.1^3} - 2,000,000 > 0. \]

Answer: 0.9545


Question: Consider a $4 \times 4 \times 4$ cube where each $1 \times 1 \times 4$ segment (row, column, or depth) contains exactly one red cube. There are 16 red cubes in total. Determine the number of unique configurations of this cube, where two configurations are considered the same if one can be transformed into the other by rotating the cube along one or more of its three central axes.

Answer: 24


Question: Compute the limit: \[ \lim_{x\to0^+}\frac{\pi/2- \arctan(1/x^2)-\sin(x^3)-1+\cos(x)}{x\tan(x)+e^{x^3}-1} \]

Answer: \dfrac{1}{2}


Question: Find the exact sum of the series \( \sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1} \) using the Fourier series representation of the function \( f(x) = x \) on the interval \([-\pi, \pi]\).

Answer: \dfrac{\pi}{4}


Question: Find a sequence of polynomials \((f_n)\) such that \(f_n \rightarrow 0\) pointwise on \([0,1]\) and \(\int_0^1 f_n(x) \, dx \rightarrow 3\). Calculate \(\int_0^1 \sup_n |f_n(x)| \, dx\) for this sequence of polynomials.

Answer: \infty


Question: Consider the polynomial \( p(x) = x^4 + 4 \) in the ring \( \mathbb{Q}[x] \). Does the set of zeroes of \( p(x) \) in \( \mathbb{C} \) form a group under multiplication? Answer Yes or No.

Answer: No


Question: Evaluate the limit: \[ \lim_{x \to 0} \int_0^x \frac{\cos(t^3)}{t+x} \, dt \]

Answer: \ln 2


Question: Find the smallest integer \(x\) such that \(\gcd(50 + x^2, 4x + 2) = 6\).

Answer: 4


Question: Find the value of \( \alpha \in \mathbb{R} \) such that the limit \( \lim_{x \to 1^+} \frac{(\sqrt{x}-1)^\sqrt{2}}{(x-1)^\alpha} \) exists and is a real number.

Answer: \sqrt{2}


Question: Find the monic real polynomial \( f(x) \) such that \( f(1/x) = f''(x) = c \) as \( x \to \infty \), and \( f(x) \ge f(1) \) for all real \( x \). What is \( f(x) \)?

Answer: x^2 - 2x + 2


Question: Let $R$ be a ring with 1 and $N$ be a submodule of an $R$-module $M$. If $M$ is free of finite rank, is $M/N$ necessarily free of finite rank? Justify your answer.

Answer: No


Question: Find all integer values of \( n \) such that the equation \( \frac{1}{a} + \frac{1}{b} = \frac{n}{a+b} \) holds for some non-zero integers \( a \) and \( b \) with \( a + b \neq 0 \).

Answer: 4


Question: Determine the cardinality of the language \( \mathcal{L} \) constructed from a finite alphabet \( \mathcal{A} \) using the Kleene closure, where \( \mathcal{L} = \{ \text{string} \mid \text{string} = x^* \land x \in \mathcal{A} \} \).

Answer: \aleph_0


Question: Find the function \( f(x) \) such that \( f(x) = f'(x)\int{f(x)}\,dx \).

Answer: 0


Question: Given the elliptic curve $E: y^2 = x^3 + 2x + 2 \pmod{17}$ and the point $P = (5, 1)$ on this curve, compute the coordinates of $2P = P + P$. Use the formula for the slope $m = \frac{3x_1^2 + a}{2y_1}$ to find the slope and then determine the coordinates $(x_3, y_3)$ of the point $2P$. Express your final answer as $(x_3, y_3) \pmod{17}$.

Answer: (6, 3)


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

Answer: \dfrac{1}{4}


Question: For the sequence \( a_n = (5 + 5^n)^{-4} \), find the number \( r \) such that \( \frac{a_n}{r^n} \) has a finite non-zero limit.

Answer: \dfrac{1}{625}


Question: Given that \( \lim_{x\rightarrow \infty} f(x)=0 \), where \( f'(x)=\dfrac{-f(x)}{\sqrt{f^2(x)+g^2(x)}} \) and \( g'(x)=1-\dfrac{g(x)}{\sqrt{f^2(x)+g^2(x)}} \) for all \( x \in \mathbb{R} \), with initial conditions \( g(0)=0 \) and \( f(0)=10 \), find the value of \( \lim_{x\rightarrow \infty} g(x) \).

Answer: 5


Question: Find the length of the curve of intersection between the parabolic cylinder $x^2 = 2y$ and the surface $3z = xy$ from the origin to the point $(6, 18, 36)$. Express your answer as a single numerical value.

Answer: 42


Question: Let $K_1$ and $K_2$ be parallel lines with line $K_3$ passing through both of them. Points $A$ and $H$ are not located on these lines. Points $C$ and $F$ are on line $K_3$. Draw lines $AG$, $AH$, and $AF$. Point $E$ is on line $AF$ such that $EC$ is perpendicular to $DC$. Find the measurement of the angle at point $H$ that is labeled as $x$. Assume $AXDC$ is a cyclic quadrilateral when $CE$ and $AG$ intersect at point $X$. What is the measure of angle $x$?

Answer: 90^\circ


Question: Determine the maximum number of points $P$ in a plane such that all triangles formed by any three points in $P$ are not obtuse. Consider degenerate triangles as well.

Answer: 5


Question: Let \( H = \langle (1,1), (1,-1) \rangle \) be a subgroup of \( G = \mathbb{Z}^2 \). Find the index \( |G:H| \).

Answer: 2


Question: Determine the radius of convergence of the series \( \sum \frac{(-1)^n z^{2n+1}}{\log n} \).

Answer: 1


Question: Let $G$ be a group of order $pq$, where $p$ and $q$ are distinct prime numbers. Determine the order of the center $Z(G)$. Choose the correct option: (i) 1 or $p$, (ii) 1 or $q$, (iii) 1 or $pq$, (iv) None of these.

Answer: iii


Question: Determine the characteristic of the ring \( \mathbb{Z}_3[x]/\langle x^3+2x+1 \rangle \).

Answer: 3


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a continuous function with \( f(1) = 5 \) and \( f(3) = 11 \). Define \( g(x) = \int_1^3 f(x+t) \, dt \). Find \( g'(0) \).

Answer: 6


Question: Calculate the expected value \( E(\ln(X)) \) where \( X \sim \text{Beta}(1/y, 1) \).

Answer: -y


Question:
Let $\| \cdot \|$ be the Euclidean norm on $\mathbb{R}^n$, and let $0<p<1$. Define the function $d: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ by
$$d(x,y)=||x-y||^p, \quad x,y \in \mathbb{R}^n. $$
Does the function $d(x,y)$ induce the usual topology on $\mathbb{R}^n$?

Answer: Yes


Question:
Determine the interval in which the parameter \(a\) must lie if the point of minima of the function \(f(x) = 1 + a^2x - x^3\) satisfies the inequality \(\frac{x^2 + x + 1}{x^2 + 5x + 6} < 0\). Choose the correct interval from the options below:

(A) \((-3\sqrt{3}, 3\sqrt{3})\)
(B) \((-2\sqrt{3}, -3\sqrt{3})\)
(C) \((2\sqrt{3}, 3\sqrt{3})\)
(D) \((-3\sqrt{3}, -2\sqrt{3}) \cup (2\sqrt{3}, 3\sqrt{3})\)

Answer: D


Question: Evaluate the integral \( \int_\Gamma \frac{1}{z} \, dz \), where \( \Gamma \) is the semicircular contour in the right half-plane, traversed from \( -i \) to \( i \).

Answer: i\pi


Question: Given the sequence defined by $a(1) = 2$ and $a(n+1) = a(n)^2 - a(n) + 1$ for $n \geq 1$, find the value of the infinite series $$\sum_{n=1}^{\infty} \frac{1}{a(n)}.$$

Answer: 1


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

Answer: \infty


Question: Evaluate the limit \( \lim_{x\to \infty} e^{-x^2} \int_x^{x+1/x} e^{t^2} \, dt \) using L'Hôpital's rule. Verify whether the differentiation of the numerator should be \( \frac{e^{(x+1/x)^2} - e^{x^2}}{2xe^{x^2}} \) or \( \frac{(1 - 1/x^2) e^{(x+1/x)^2} - e^{x^2}}{2xe^{x^2}} \).

Answer: 0


Question: Given two positive-valued functions $f, g: [0,1]\to(0,1)$, a pair $(f,g)$ is defined as good if, for every real number $r$, the set $Z_r := \{x ~\big|~ {g(x) / f(x)} = r \}$ has a measure of zero in $\mathbb{R}$. For example, if $f(x)=1/2$ and $g(x)=(x+1)/2$, the pair $(f,g)$ is good. Conversely, if $f(x)=1/2$ and $g(x)=3/4$, the pair $(f,g)$ is not good. Given a pair $(f,g)$ and a constant $\epsilon>0$, does there always exist a function $h: [0,1]\to [0,\epsilon]$ such that the pair $(f, g+h)$ is good? In other words, is it always possible to make a pair "good" by a slight perturbation?

Answer: Yes


Question: Let $X$ be a space with a free and properly discontinuous $\mathbb{Z}/2$-action, and consider the $2$-sheeted covering map $p: X \to X/(\mathbb{Z}/2)$. We have an associated vector bundle \( \xi: \mathbb{R}^2 \to X \times_{\mathbb{Z}/2} \mathbb{R}^2 \to X/(\mathbb{Z}/2) \), where $\mathbb{Z}/2$ acts on $\mathbb{R}^2$ by reversing the order of coordinates $(x,y) \mapsto (y,x)$. If $\xi$ is a trivial bundle, can we conclude that the covering map $p$ is a trivial $2$-sheeted covering map, i.e., $X = (X/(\mathbb{Z}/2)) \times \mathbb{Z}/2$?

Answer: Yes


Question:
Given the transition matrix of a Markov chain:
\[
\begin{bmatrix}
1&0&0&0\\
0.1&0.6&0.1&0.2\\
0.2&0.3&0.4&0.1\\
0&0&0&1
\end{bmatrix}
\]
with states \( \{0, 1, 2, 3\} \), calculate the probability that the Markov chain, starting from state 1, will end in state 0.

Answer: \dfrac{8}{21}


Question: Let \(f(x) = x^2 + ax + b\) be a quadratic polynomial with integer roots. If for all non-zero real \(x\), the equation \(f\left(x + \frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right)\) holds, find the value of \(a^2 + b^2\).

Answer: 13


Question: Evaluate the integral \( \int_C \frac{dz}{z(2z+1)} \) where \( C \) is the unit circle contour.

Answer: 0


Question: Find the last two digits of \(12^{12^{12^{12}}}\) using Euler's theorem.

Answer: 16


Question: On a manifold with a metric tensor, let \( \Omega \) be the volume form. Is it true that the Hodge star operator applied to \( \Omega \) results in 1, i.e., \( *\Omega = 1 \)?

Answer: 1


Question: Evaluate the contour integral \( \frac{1}{2i\pi} \oint_C \frac{z^2}{z^2+4} \, dz \) where \( C \) is the square with vertices at \( \pm 2 \) and \( \pm 2 + 4i \). Determine if the evaluation using the Cauchy Integral Theorem, resulting in the integrals being zero, is correct.

Answer: i


Question: Find the volume of the solid bounded by the paraboloid $z = x^2 + y^2$ and the plane $z = 0$, above the region enclosed by the circle $x^2 + y^2 = 2y$ in the xy-plane.

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


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty} \sqrt{2^n}\int_a^b \cos^n \left(x+\frac{\pi}{4}\right)\,dx \] given that the interval \([a,b]\) is contained within \((0,\frac{\pi}{2})\).

Answer: 0


Question:
Calculate the value of \( f(2017) + f(2013) \) where the function \( f(n) \) is defined as:
\[
f(n) = \left(\frac{1+i}{\sqrt{2}}\right)^n + \left(\frac{1-i}{\sqrt{2}}\right)^n
\] and \( i^2 = -1 \).

Answer: 0


Question: Consider a smooth closed simply-connected 4-manifold $M$ with $w_1 = w_2 = 0$. Can the tangent bundle $TM$ be trivialized in the complement of a point?

Answer: Yes


Question:
Consider the function $f: \mathbb{C} \to \mathbb{C}$ defined by \( f(x+iy) = x^3 + 3xy^2 + i(y^3 + 3x^2y) \). Determine which of the following statements is true:

1) $f'(1+i)$ exists and $|f'(1+i)| = 3\sqrt{5}$.
2) $f$ is analytic at the origin.
3) $f$ is not differentiable at $i$.
4) $f$ is differentiable at $1$.

Use the Cauchy-Riemann equations to justify your answer.

Answer: 4


Question: Consider the functions \( \varphi, \psi : \mathbb{R} \rightarrow \mathbb{R} \) defined by \( \varphi(x) = x^3 \) and \( \psi(x) = x \) for \( x < 0 \) and \( \psi(x) = 2x \) for \( x \geq 0 \). Do the atlases \( A_1 = \{ \text{id}_\mathbb{R} \} \), \( A_2 = \{ \varphi \} \), and \( A_3 = \{ \psi \} \) induce different differentiable structures on \( \mathbb{R} \)?

Answer: Yes


Question: Determine whether the infinite series \( \sum_{k=1}^\infty \frac{1}{n}\cos\left(\frac{k\pi}{2n}\right) \) equals \( \frac{2}{\pi} \).

Answer: \frac{2}{\pi}


Question: Let \( f \) be a function satisfying \( f(x+y) = f(x) + f(y) \) and \( f(x) = x^2 g(x) \) for all \( x \), where \( g(x) \) is a continuous function. Find \( f''(x) \).

Answer: 0


Question: Given a prime number \(p\) of the form \(4k+1\), determine if the integer right-angled triangle with hypotenuse \(p\) is unique.

Answer: Yes


Question: Find the coefficient $a_{-1}$ in the Laurent series expansion of the function \( f(z) = z^3 \cos\left(\frac{1}{z}\right) e^{\frac{1}{z^2}} \).

Answer: \dfrac{1}{24}


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

Answer: \infty


Question: Given the joint mass function \( P(k, n) = \frac{2^{-k}C}{n} \) for \( k = 1, 2, \ldots \) and \( n = 1, 2, \ldots, k \), where \( C \) is a suitable constant, compute the conditional expectation \( E(X|Y = y) \).

Answer: y + 1


Question: Find the flux of the vector field \( \mathbf{F} = 3\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \) across the part of the plane \( 1x + 3y + z = 3 \) that lies in the first octant, oriented upwards.

Answer: 18


Question: What is the smallest integer \( n \geq 5 \) such that the edge set of the complete graph \( K_n \) can be partitioned into edge-disjoint copies of \( K_4 \)?

Answer: 13


Question: Evaluate the limit: \[ \lim_{n\to\infty} \int_{0}^{\pi/2}\frac{1}{1+x \tan^{n} x }dx \]

Answer: \dfrac{\pi}{4}


Question: Find the number of ways to rearrange the sequence "1234123567" such that no two identical numbers are adjacent to each other.

Answer: 236880


Question: Determine the number of non-trivial ring homomorphisms from \( \mathbb{Z}_{12} \) to \( \mathbb{Z}_{28} \).

Answer: 1


Question: Given an integer lattice \(\mathcal{L}(B) = \sum_{i=1}^n x_i b_i : x_i \in \mathbb{Z}\), and a point \(t\) that lies in the space spanned by the lattice, determine if it is possible to check in polynomial time whether \(t\) lies inside the fundamental parallelepiped generated by the lattice.

Answer: Yes


Question: Given the function \(f(x) = \frac{1}{2}(x + \sqrt{x^2 + 4})\), find the value of \(f^{-1}(x) + f^{-1}(\frac{1}{x})\) for \(x \neq 0\).

Answer: 0


Question: Evaluate \( \lim\limits_{n\to\infty} \left(1+\frac{1}{n}\right)^{n^2}\cdot\left(1+\frac{1}{n+1}\right)^{-(n+1)^2} \) using the definition of \( e \).

Answer: \dfrac{1}{e}


Question: Solve the differential equation \( y(t)[y''(t)+2\lambda y'(t)]=(y'(t))^2 \) with the initial condition \( y(0)=0 \). Find the solution for \( \lambda \neq 0 \).

Answer: 0


Question: In triangle $ABC$, $BC = 8$ and the length of median $AD$ is 5. Determine the difference between the largest possible value and the smallest possible value of $AB^2 + AC^2$. Find $M - m$. \( \)

Answer: 0


Question: In an experiment where a fair coin is tossed 10 times, let X be the number of heads observed. Define Y as \( Y = X^2 \). Calculate the covariance between X and Y.

Answer: 25


Question: Let \( f(z) \) be an analytic function satisfying the functional equation \( f(z) = z + f(z^2) \). Define the coefficients \( a_n = \frac{f^{(n)}(0)}{n!} \). Determine the radius of convergence of the power series \( \sum_{n=1}^{\infty} a_n z^n \).

Answer: 1


Question: Let $f:X\rightarrow [0,1]$ be a continuous function. Determine whether the statement $a<b\implies \overline{f^{-1}[0,a)}\subset f^{-1}[0,b)$ is true.

Answer: True


Question: Determine whether the function \( f: \mathbb{R}^2 \setminus \{(0,0)\} \to \mathbb{R} \) defined by \( f(x,y) = e^{xy\log(x^2+y^2)} \) can be continuously extended to the point \((0,0)\).

Answer: 1


Question: Determine the values of \( n \) for which the integral \( \int_0^\infty \frac{\sin(x+x^2)}{x^n} \, \mathrm{d}x \) converges.

Answer: (-1, 2)


Question: In an equilateral triangle $\triangle ABC$ inscribed in the unit circle $C(0,1)$, find the maximum value of $\overline{PA}\cdot\overline{PB}\cdot\overline{PC}$, where $P$ is a variable point in the closed disk $\bar{D}(0,2)$. Provide your answer as a single numerical value.

Answer: 9


Question: Compute the limit: $$\lim_{n \rightarrow \infty} \left[n \; \arccos \left( \left(\frac{n^2-1}{n^2+1}\right)^{\cos \frac{1}{n}} \right)\right]$$

Answer: 2


Question: Let $G$ be an infinite group, and let $X$ and $Y$ be two sets on which $G$ acts freely. Assume $X$ and $Y$ have the same cardinality, and $|X| > |G|$. Is there a bijection $f: X \to Y$ such that $f(gx) = gf(x)$ for all $g \in G$ and all $x \in X$?

Answer: Yes


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

Answer: \sqrt{e}


Question: Suppose $f: V \to V$ is a linear transformation such that for every basis $E = \{ e_1, \ldots, e_n \}$ of $V$, the matrix representation $[f]_E^E$ of $f$ with respect to $E$ is equal to its transpose, i.e., $[f]_E^E = ([f]_E^E)^T$. Does this imply that $f$ is a multiple of the identity function, $f = \alpha I$ for some $\alpha \in \mathbb{R}$?

Answer: Yes


Question: For a game of paintball tag with $N$ players standing in a circle, each player takes turns clockwise starting from player 1. On their turn, a player can either randomly tag another player or shoot in the air. If no player is tagged in a complete round, the referee randomly selects a winner. Players aim to maximize their winning probabilities and prefer shooting in the air if it is equally beneficial as tagging. For sufficiently large $N$, will player 1 always choose to tag at the start of the game?

Answer: Yes


Question: Prove by induction that \(133\) divides \(11^{n+1} + 12^{2n-1}\) for all positive integers \(n\).

Answer: 133


Question: Evaluate the double integral \( \int_0^1 \int_0^1 e^{\max(x^2, y^2)} \, dx \, dy \).

Answer: e - 1


Question: Compute the limit \( \lim_{n\rightarrow\infty}\left(\frac{1+\sqrt[n]{a}}{2}\right)^n \) without using L'Hopital's rule or Taylor series. Provide your answer as a simplified expression.

Answer: \sqrt{a}


Question: Evaluate the integral \( \int_\gamma \frac{dz}{(z^2+1)^2} \), where \( \gamma \) is the semicircle \([-R,R]\cup\{z\in\mathbb{C}:|z|=R\ \text{and}\ \text{Im}{z}>0\}\) traced in the positive direction, and \( R>1 \).

Answer: \dfrac{\pi}{2}


Question: Solve the non-linear congruence equation: \( x^{11} \equiv 7 \pmod{61} \). Find the value of \( x \) that satisfies this equation.

Answer: 31


Question:
Find all continuous and bounded functions $g$ such that for all $x \in \mathbb{R}$, the following equation holds:
\[ 4g(x) = g(x+1) + g(x-1) + g(x+\pi) + g(x-\pi) \]

Answer: g(x) = C


Question: Given \( z = \frac{\sqrt{3} - i}{2} \), find the smallest positive integer \( n \) such that \((z^{95} + i^{67})^{94} = z^n\), where \( i = \sqrt{-1} \).

Answer: 10


Question:
Given that \(\lim_{x\to +\infty}\frac{f(x+1)}{f(x)}=2\), calculate the following limit:
\[\lim_{x\to +\infty}\frac{f(x+14)-3147f(x+1)}{f(x+2)+f(x)}\]

Answer: 2018


Question: Let $X$ be a scheme locally of finite type over a field $k$. Assume that every closed point in $X$ is also open. Is $X$ a discrete scheme?

Answer: Yes


Question: Let $G$ be a group that is not weakly nilpotent, and let $Z^*(G)$ be its hypercenter. Suppose $x, y \in G$ such that $\langle x, y \rangle$ is not nilpotent. Is it true that for all $u, v \in Z^*(G)$, the subgroup $\langle xu, yv \rangle$ is also not nilpotent? Justify your answer.

Answer: Yes


Question: Find the 66th derivative of the function \( F(x) = \int_0^x \cos(t^3) \, dt \) at \( x = 0 \). The function can be expressed as an infinite series: \( \sum_{n \ge 1} \left( \frac{(-1)^n}{(2n)!} \cdot \frac{x^{6n+1}}{6n+1} \right) \). Determine \( F^{(66)}(0) \).

Answer: 0


Question:
Let $S = \{0\} \cup \{\frac{1}{4n+7} : n =1,2,\ldots\}$. Determine the number of analytic functions that vanish only on the set $S$. Choose the correct answer from the following options:

(a) $\infty$
(b) $0$
(c) $1$
(d) $2$

Answer: B


Question: Find a cube root of 97 modulo 101.

Answer: 31


Question: A round cake is cut with 4 vertical slices, resulting in the maximum number of pieces. Determine the number of ways to distribute these pieces among three people such that each person receives at least one piece.

Answer: 171006


Question: Determine whether there exists a distributive lattice \( L \) such that for no set \( X \), there is a surjective lattice homomorphism \( s: \text{Part}(X) \to L \).

Answer: Yes


Question: Given three integers \(a, b,\) and \(c\) such that \(a, b, c, a + b - c, a + c - b, b + c - a,\) and \(a + b + c\) are seven distinct prime numbers, and \(800\) is an element in the set \(\{a + b, b + c, c + a\}\). Determine the maximum possible value of \(d\), where \(d\) is the difference between the largest and smallest of these seven primes.

Answer: 1594


Question: Let $M$ and $N$ be smooth manifolds, and let $f: M \rightarrow N$ be a local diffeomorphism. Is it true that $f(M)$ is open in $N$? Justify your answer.

Answer: Yes


Question: Determine the cardinality of the set \( \bigcup_{i \in I} A_i \) where each set \( A_i \) and the index set \( I \) have cardinality \( 2^{\aleph_0} \). Is \( |\bigcup_{i \in I} A_i| = 2^{\aleph_0} \)?

Answer: 2^{\aleph_0}


Question: Determine whether the set \( \{ u \in H^1(\Omega) : 0 < a \leq u(x) \leq b\quad \text{a.e.}\} \) is closed, where \( \Omega \subset \mathbb{R}^1 \) is an interval.

Answer: Yes


Question: Find \( \lim_{n\to\infty} \lvert x_n \rvert ^\frac{1}{n} \) given that the sequence \( x_{n+1}=3x_n + \frac{2}{x_n^2} \) with \( x_1=1 \).

Answer: 3


Question: Determine whether the following statement is true or false: \( \forall x, y \in \mathbb{Q}: (\sin(x)=\sin(y))\Rightarrow (x=y) \). Justify your answer.

Answer: True


Question: Use Green's Theorem to evaluate the line integral \( \int_C P\,dx + Q\,dy \), where \( P = xy \), \( Q = x^2 \), and \( C \) is the first quadrant loop of the graph \( r = \sin 2\theta \).

Answer: \dfrac{16}{105}


Question:
Starting with the number 1, determine the minimum number of operations needed to reach \(2^{2018}\) if you can either multiply the current number by 2 or square it in each step. Choose the correct answer from the following options:

A) 15
B) 16
C) 17
D) 18
E) 12

Answer: C


Question: Determine a formula or method to calculate the number of distinct cycle type forms in the symmetric group \( S_n \) for any positive integer \( n \).

Answer: p(n)


Question: Determine the values of \( r \) for which the function \( f(x,y,z) = \begin{cases} \frac{(x + y + z)^r}{x^2+y^2+z^2}, & (x,y,z) \neq (0,0,0) \\ 0, & (x,y,z) = (0,0,0) \end{cases} \) is continuous on \( \mathbb{R}^3 \).

Answer: r > 2


Question: Does the equation \( f(s) = \int_0^s K(s,t)f(t)dt \) imply that \( f = 0 \) for \( f \in C[0,1] \) and \( K(s,t) \in C([0,1]\times[0,1]) \)?

Answer: 0


Question: Given the functional equation \( e^x f(x) = 2 + \int_0^x \sqrt{1+x^4} \, dx \), find the value of \( \frac{d}{dx} \left(f^{-1}(x)\right) \bigg|_{x=2} \).

Answer: -1


Question: In a right triangle ABC with a right angle at vertex C, a semicircle is constructed with center P on leg AC such that the semicircle is tangent to leg BC at C, tangent to the hypotenuse AB, and intersects leg AC at Q between A and C. Given that the ratio of AQ to QC is 2:3 and BC = 12, find the length of AC.

Answer: 8\sqrt{10}


Question: An investor invests $900 in a stock. Each trading day, the investment has a probability of 0.5 of increasing by $10 and a probability of 0.5 of decreasing by $8. The changes in price on different trading days are mutually independent. Calculate the probability that the investment has a value greater than $969 at the end of 252 trading days.

Answer: 0.907


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{2\sin x-\sin 2x}{x-\sin x}. \]

Answer: 6


Question: Determine the size of the set generated by an unrestricted number of operations on the elements of the set \( F = \{c, x, \exp(x), \ln(x) \} \), where \( c \) is a constant term. The operations allowed are addition \( f+g \), multiplication \( f\cdot g \), division \( f/g \), and composition \( f\circ g \).

Answer: \aleph_0


Question: In the context of the commutative C*-algebra $C_0(\Omega)$, let $\mu$ be a complex Radon measure on $\Omega$. By the Riesz representation theorem, $\mu$ can be viewed as a bounded linear functional on $C_0(\Omega)$. According to the polar decomposition, there exists a partial isometry $u$ in the W*-algebra $C_0(\Omega)^{**}$ such that $\mu = u|\mu|$. Is the absolute value $|\mu|$ equivalent to the total variation of $\mu$?

Answer: Yes


Question: Find the remainder when \((12371^{56} + 34)^{28}\) is divided by 111.

Answer: 70


Question: Evaluate the summation: \[ \sum_{r=0}^{50} (-1)^r \frac{\binom{50}{r}}{r+2} \]

Answer: \dfrac{1}{2652}


Question: Determine the interval for the parameter \( c \) such that the expression \( \frac{x^2-x+c}{x^2+x+2c} \) can take all real values for any real number \( x \).

Answer: (-6, 0)


Question: Consider a normalized dataset $X$ of size $m \times n$, where $m$ is the number of records and $n$ is the number of attributes, with values between 0 and 1. Let $Y = XR$, where $R$ is an $n \times p$ matrix with $p < n$. If $R$ is drawn from a uniform distribution $U(0,1)$, does the transformation preserve the Euclidean distances between instances in $X$?

Answer: No


Question: Evaluate the integral \( \int_{\partial B(\frac{1}{2}, \frac{3}{4})} \frac{dz}{z(z^2-1)} \) using partial fraction decomposition and appropriate theorems.

Answer: -\pi i


Question:
Let $f(x,y)$ be a real-valued function in two variables. Suppose $x_n \to x_0$ and $y_n \to y_0$ as $n \to \infty$. Assume $f(x,y)$ is differentiable in $x$ at $(x_0, y_0)$ with derivative $f'_x(x_0,y_0)$, and $f'_x(x,y)$ is continuous at $(x_0,y_0)$. Consider the Taylor expansion about $(x_0,y_n)$:
\[f(x_n,y_n) = f(x_0,y_n) + f'_x(x_0,y_n)(x_n - x_0) + R_n(x_n,y_n)\]
Is it necessarily true that $R_n(x_n,y_n) = o(|x_n - x_0|)$?

Answer: Yes


Question: Find all zeros of the equation $e^z = 1+2z$ within the unit disk $|z| < 1$. Provide your answer in terms of complex numbers.

Answer: 0


Question: Evaluate the sum \( \sum_{b=0}^{N-1}\left(\frac{b}{p}\right)\zeta_M^{-kb} \) for an odd prime \( p \) such that \( p \mid N \), \( M \mid N \), \( (k,M)=1 \), and \( p \not\mid M \). Show that the sum is zero.

Answer: 0


Question: In how many ways can 15 different books be distributed among 5 children such that each child receives at least one book?

Answer: 25292030400


Question: Let $f:(-1,1) \rightarrow \mathbb{R}$ be a smooth function such that $f(0) = 1$, $|f^{(n)}(x)| \le n!$ for every $x \in (-1,1)$ and for every $n \in \mathbb{N}$, and $f'(\frac{1}{m+1}) = 0$ for every $m \in \mathbb{N}$. Determine the function $f(x)$ for every $x \in (-1,1)$. What is $f(x)$?

Answer: 1


Question: Compute the integral \( \int_{0}^{2\pi}\frac{Re^{it}}{z_0+Re^{it}}dt \) for \( z_0 \in \mathbb{C} \) such that \( z_0 \neq 0 \) and \( R < |z_0| \), without using the Cauchy Theorem.

Answer: 0


Question: Given that $X$ and $Y$ have a joint density function \( f_{X, Y}(x, y) = \frac{1}{x^2y^2} \) for \( x \geq 1 \) and \( y \geq 1 \), and the transformations \( U = 4XY \) and \( V = \frac{X}{Y} \), determine the correct domain for the variable \( V \).

Answer: (0, \infty)


Question: Determine the smallest integer $n$ such that the complex projective space $\mathbb{CP}^8$ can be embedded in the Euclidean space $\mathbb{R}^n$. It is known that $\mathbb{CP}^8$ cannot be embedded in $\mathbb{R}^{23}$ or less, and it can be immersed in $\mathbb{R}^{31}$. What is the smallest such $n$?

Answer: 30


Question: Given that $E(X) = 0$, $E(Y) = 0$, and $Cov(X,Y) = \rho$, find $E(X^2Y^2)$ using the relationship $Var(XY) = E(X^2Y^2) - E(XY)^2$. Assume $Cov(X,Y) = \rho \sigma_x \sigma_y$ and $E(XY) = \rho$. Provide your answer in terms of $\rho$. \( \boxed{} \)

Answer: 1 + 2\rho^2


Question: Determine the value of \( x \) such that the sequence \( a_0, a_1, \ldots \) defined by \( a_n = 4a_{n-1} - 4a_{n-2} \) with initial conditions \( a_0 = 1 \) and \( a_1 = x \) has a limit of \(-\infty\) as \( n \to \infty \).

Answer: x < 2


Question: Let $A \in M_n$ be a doubly stochastic matrix. Is the largest singular value of $A$, denoted as $\sigma_1(A)$, equal to 1?

Answer: 1


Question:
Let $f$ be a function from $GR(p^2,m)$ to $GR(p^2,1)=Z_{p^2}$, where $p$ is an odd prime and $m > 1$ is a positive integer. For $f$ to be a bent function, verify if the following condition holds:

$$\left|\sum_{x \in GR(p^2,m)}  w^{Tr(f(x)-ax)}\right|=|GR(p^2,1)|^{m/2}$$

where $w=e^{2\pi i/p^2}$, $a \in GR(p^2,m)$, Tr is the trace function from $GR(p^2,m)$ to $GR(p^2,1)$, and $ax$ is the dot product of $a$ with $x$. Determine if this condition is true.

Answer: True


Question:
Given four groups, each containing four unique people, determine the total number of ways to form new groups by selecting one person from each group. For example, if the groups are:

Group 1: \(A, B, C, D\)
Group 2: \(H, I, J, K\)
Group 3: \(M, N, O, P\)
Group 4: \(X, Y, Z, T\)

How many different new groups of four people can be formed by selecting one person from each group?

Answer: 256