Question: Determine the almost sure limit of the sequence \( W_n = \frac{X_1^3 + \cdots + X_n^3}{X_1^2 + \cdots + X_n^2} \), where \( X_1, X_2, \ldots \) are independent random variables each following a Uniform(0,1) distribution.

Answer: \dfrac{3}{4}


Question: Determine the number of solutions to the equation \( \lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 8x \rfloor + \lfloor 16x \rfloor + \lfloor 32x \rfloor = 12345 \), where \( \lfloor \cdot \rfloor \) denotes the floor function.

Answer: 0


Question: Find the conditions on the parameter $a$ such that the equation \((x-2)e^x + a(x-1)^2 = 0\) has more than one real root.

Answer: a > 0


Question: Given that $x, y > 1$ and $(\ln x)^2 + (\ln y)^2 = \ln x^2 + \ln y^2$, determine the maximum value of $x^{\ln y}$.

Answer: e^4


Question: In triangle $ABC$, where $\angle BAC = 60^\circ$, $BC = 1$, and $AC \geq AB$, let $H$, $I$, and $O$ denote the orthocenter, incenter, and circumcenter, respectively. If the area of pentagon $BCOIH$ is maximized, find the measure of $\angle CBA$. \(\text{(AMC 12A, 2011)}\)

Answer: 90^\circ


Question:
Let $A \in \mathbb{R}^{n \times n}$ be a symmetric positive semi-definite matrix such that $\operatorname{Tr}(A) \le n$. Determine a lower bound for the quantity:
\[ \operatorname{Tr} \left( \left( (A+2I )^{-1} \right)^3\right) \]
where $I$ is the identity matrix.

Answer: \dfrac{n}{27}


Question: Evaluate the limit: $$ \lim_{x\to 0} x\left(\left\lfloor\frac{1}{x}\right\rfloor +\left\lfloor\frac{2}{x}\right\rfloor +\cdots +\left\lfloor\frac{10}{x}\right\rfloor\right) $$

Answer: 55


Question: Evaluate the limit \( \lim_{x\to 0^+}\int_x^{2x}\frac{\sin t}{t^2}dt \).

Answer: \ln 2


Question: Determine the number of zeros of the function \( f(z) = z^2 + \cosh(iz) \) within the disk \( |z| < \frac{1}{2} \).

Answer: 0


Question: Given a matrix $A \in M_5(\mathbb{R})$ that satisfies the equation $A^2 - 4A - I = 0$, where $I$ is the identity matrix, let $a_1, a_2, a_3, a_4, a_5$ be the eigenvalues of $A$. Calculate the sum \( (a_1 - \frac{1}{a_1}) + (a_2 - \frac{1}{a_2}) + \dots + (a_5 - \frac{1}{a_5}) \).

Answer: 20


Question: Evaluate the limit: $$\lim _{n\to \infty }\frac{\sum_{k=2}^{n} \frac{1}{k\ln k}}{\ln (\ln n)}$$ without using Taylor series.

Answer: 1


Question: Given that \( p + q + r = 1 \) where \( p, q, \) and \( r \) are positive numbers, find the minimum value of the expression \( \left(\frac{1-p}{p}\right) \left(\frac{1-q}{q}\right) \left(\frac{1-r}{r}\right) \).

Answer: 8


Question: Let $G$ be a finite group and $H$ be an abelian subgroup of $G$. If the order of $H$ is a power of a prime $p$, must $H$ be isomorphic to a direct product of copies of the cyclic group of order $p$? Answer yes or no.

Answer: No


Question: Is Proposition 1.17 in John Lee's "Introduction to Smooth Manifolds" still valid when $M$ is a smooth manifold with boundary?

Answer: Yes


Question: Find the unique primitive Pythagorean triple (a, b, c) such that the area of the triangle is equal to twice its perimeter. Here, a and b are the legs, and c is the hypotenuse of the triangle.

Answer: (9, 40, 41)


Question: Calculate the probability \( \mathbb{P}(Z_1 > 0, Z_1 + Z_2 < 0) \) where \( Z_1 \) and \( Z_2 \) are independent standard normal random variables.

Answer: \dfrac{1}{8}


Question: Find the maximum possible value of \( \sum_{i=1}^{20}{\cos^2 2x_i} \) given that the real numbers \( x_1, x_2, \dots, x_{20} \) satisfy \( \sum_{i=1}^{20}{\sin x_i} = 0 \).

Answer: 20


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

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


Question: Given the elliptic curve \(E\) over \(\mathbb{F}_{11}\) defined by \(y^2 = x^3 + 5\) and the function \(f(x,y)=\frac{2x+y-3}{3x-y+2}\), calculate \(f(\mathcal{O})\) using projective coordinates.

Answer: 10


Question: Find the coefficient of \(x^{30}\) in the expression \(\left(\frac{1-x^{11}}{1-x}\right)^3\left(\frac{1-x^{21}}{1-x}\right)\).

Answer: 1111


Question: Let \( f: \Bbb{R} \to \Bbb{R} \) be a surjective function restricted to positive values. Consider the function \( g(x, y) = \frac{\ln f(x) + \ln f(y)}{\ln 2} - 2 \). Determine if \( g(x, y) \) is surjective when \( y \) is fixed to a value \( y_0 \) such that \( f(y_0) = 1 \).

Answer: Yes


Question: Determine whether the equality $|A^X|=|B^X|$ implies $|A|=|B|$ for arbitrary sets $A$ and $B$, given that $X$ is a nonempty set.

Answer: No


Question: Evaluate the double integral: $$\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}e^{-5x^2-5y^2+8xy}dxdy$$

Answer: \dfrac{\pi}{3}


Question: Let $p$ be a prime other than $2$ or $5$. Let $n$ be the least positive integer such that $10^n \equiv 1 \pmod{p}$. By Fermat's Little Theorem, $n \leq p - 1$. Determine whether $10^n - 1$ divides $10^{p-1} - 1$. Answer "yes" or "no."

Answer: yes


Question: Calculate the limit: $$\lim_{n \to \infty} \Bigg(\tan\Bigg(\frac{n\pi}{2n+1}\Bigg)\Bigg)^{\frac{1}{n}}$$

Answer: 1


Question: Is $L_E^p(G)$, the set of $E$-spectral functions in $L^p(G)$ for an infinite compact abelian group $G$, with $E \subseteq \Gamma$ where $\Gamma$ is the dual group, and $1<p<\infty$, a Banach subspace of $L^p(G)$?

Answer: Yes


Question:
Let $f(x)$ be a twice differentiable function such that $f(0)=0$ and $f(1)=9$ and $f(x)+f''(x)>2f'(x)$ for all $x \in [0,1]$. Determine which of the following statements is true about $f(x)$:

A) $f(x) \geq 9xe^{x-1}$

B) $f(x) \leq 9xe^{x-1}$

C) $f(x) \geq 27xe^{x-1}$

D) None of the above

Answer: B


Question: Evaluate the integral \( \iint_D |x-y| \, dx \, dy \) where \( D = \{(x, y) \in \mathbb{R}^2 : x \ge 0, 0 \le y \le 3-2x \} \).

Answer: \dfrac{15}{8}


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

Answer: 1


Question: Compute the limit: $$\lim_{n \to \infty} \frac1{2\sqrt n\log n} \frac{\sin(2\pi b\sqrt n\log n+\pi b)}{\sin(\pi b)}-\frac1{2\sqrt n\log n}\,,$$ where $b\in \mathbb{R}\setminus \mathbb{Z}$.

Answer: 0


Question: Evaluate the integral: $$\int_{1}^{5} \frac{\log(7-x)}{\log((7-x)(x+1))} \, dx.$$

Answer: 2


Question: Solve the functional equation $f(2x) = f(x) \cdot \sin(x)$. Provide the function $f(x)$ that satisfies this equation.

Answer: 0


Question: Determine if every group of order 35 is cyclic.

Answer: Yes


Question: Find the chromatic number of a connected graph that contains exactly two cycles, where the cycles share exactly one edge.

Answer: 3


Question: Solve the equation \( \arcsin(2x^2 - 1) + 2\arcsin x = -\frac{\pi}{2} \).

Answer: [-1, 0]


Question: Evaluate the limit: $$\lim_{n\to\infty}\sqrt[n]{5^n-20n^2}$$

Answer: 5


Question: Find the function \( f(x) \) that satisfies the functional equation \( f(x+y) = f(x)f(y) \) for all real numbers \( x \) and \( y \), given that \( f'(0) = 2 \).

Answer: e^{2x}


Question: In the product topology, determine whether the closure of the box of two points is equal to the box of the closures of the points. Provide a true or false answer.

Answer: True


Question: Do two isomorphic finite groups have identical character tables? Provide a justification for your answer.

Answer: Yes


Question: Determine whether the statement is true or false: An integral is path-independent if and only if the differential form being integrated is exact.

Answer: A


Question: Given the function \( f(x) = \frac{ax + b}{cx + d} \), where \( a, b, c, d \in \mathbb{R} \) and \( abcd \neq 0 \), if \( f(f(x)) = x \) for all \( x \) in the domain of \( f \), determine the value of \( a + d \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{(1+x)^{1/x} - e + \frac{1}{2}ex}{x^2} \] using a method other than Taylor series expansion.

Answer: \dfrac{11e}{24}


Question:
Let $X_1, X_2, \ldots, X_{10}$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. Calculate the probability that the sample mean $\bar{X}$ satisfies the inequality:

$$\frac{\sum_{i=1}^{10} (X_i-\mu)^2}{\sigma^2} < \frac{\sum_{i=1}^{10} (X_i-\bar{X})^2}{\sigma^2}.$$

Answer: 0


Question: Determine an upper bound for the Euclidean norm of a time-varying, real, square \( n \times n \) matrix \( A \), given that all the variables in the matrix are bounded.

Answer: nM


Question: Find the sum of the series \( \sum_{i = 0}^{\infty}\frac{F_i}{7^i} \), where \( F_i \) is the \( i \)-th Fibonacci number.

Answer: \dfrac{7}{41}


Question: Find a subspace \( Y \subseteq l^2 \) such that \( Y \) is not closed. Define \( Y = \{ x = (x_n) \in l^2 \mid x_n = 0 \text{ except for a finite number of indices} \} \). Prove that \( Y \) is not closed in \( l^2 \).

Answer: Y


Question: Evaluate the limit: $$\lim_{n\to\infty} \frac{1+\frac{n}{1!}+\cdots+\frac{n^n}{n!}}{e^n}$$

Answer: \dfrac{1}{2}


Question: Prove that the quotient space \( \frac{D}{\delta(D)\setminus [0,1]^{n-1}\times\{0\}} \) is homeomorphic to the \( n \)-dimensional disk \( D \), where \( D=[0,1]^n \) and \( n \ge 2 \).

Answer: D


Question: Compute the closed-form expression for the integral \( R^* = \int\int \max_i\{\theta_i+\epsilon_i-1,0\}dF(\theta_i)dG(\epsilon_i) \), where \( \theta_i \) is uniformly distributed over \([1,2]\) and \( \epsilon_i \) is uniformly distributed over \([-1/2,1/2]\), with both random variables being independent.

Answer: \dfrac{25}{48}


Question: For a $k$-digit number that does not contain the digit '0', determine the minimum number of prime numbers that can be found in its decimal expansion.

Answer: 0


Question: Given the function $y = f(x)$ with a minimum value of $-3$ at $x = 0$ and a maximum value of $3$ at $x = -2$, where the graph is a conic curve, determine the number of critical points for the function $y = g(x) = 3^{2f(x) + 4x - 3} - 2^{-f(x) - 2x + 3}$. Find the derivative $g'(x)$ and solve $g'(x) = 0$ to find the number of critical points.

Answer: 2


Question: Evaluate the greatest integer function \([10I]\), where \(I\) is the integral \(\int_0^1 x^{e^{i\theta}} \, dx\) with \(\theta = \frac{\pi}{4}\).

Answer: 5


Question: Maximize \( \int_{0}^{1} f(x)^5 \, dx \) over all functions \( f: [0,1] \to [-1,1] \) such that \( \int_{0}^{1} f(x)^3 \, dx = 0 \) and \( \int_{0}^{1} f(x) \, dx = 0 \).

Answer: 0


Question: Determine the radius of convergence of the power series \( \sum_{n=1}^{\infty}\left(\sqrt{n+a}-\sqrt{n}\right) x^n \), where \( a > 0 \).

Answer: 1


Question: Suppose that $E \subseteq [0, 1]$ has the property that for each $x \in E$, there exists $r_x > 0$ such that the intersection $B(x, r_x) \cap E = (x-r_x, x+r_x) \cap E$ has Lebesgue measure zero. Does it follow that $E$ itself has Lebesgue measure zero?

Answer: Yes


Question: Determine the function $f(x)$ that satisfies the equation \( f(x)f(y) + 2 = f(x) + f(y) + f(xy) \) for all real numbers $x$ and $y$, given that $f'(0) = 0$ and $f'(1) = 2$. Provide your answer as a function $f(x)$.

Answer: x^2 + 1


Question: Determine whether the operator $4I - T$, where $T : C_\mathbb{C}([0,1]) \to C_\mathbb{C}([0,1])$ is given by \((Tf)(t) = \int_0^t 2sf(s)ds + f(1)\), has an inverse in the space of bounded linear operators on $C_\mathbb{C}([0,1])$. Use the condition that $\lVert 3I - T \rVert < 1$ to justify your answer.

Answer: Yes


Question: Evaluate the surface integral \( \iint_S (z + x^2y) \, dS \), where \( S \) is the portion of the cylinder \( y^2 + z^2 = 1 \) that lies between the planes \( x = 0 \) and \( x = 3 \), with \( x, y, z \geq 0 \).

Answer: 12


Question: Determine whether the expression \(2^{2n+2} + 2^{m+2} + 1\) is a perfect square for natural numbers \(m\) and \(n\) satisfying \(m \leq 2n\) if and only if \(m = n\).

Answer: m = n


Question: Prove that if a sequence of ordinals \( \alpha_i \) increases to a limit ordinal \( \alpha \), then the sequence of cardinal numbers \( \aleph_{\alpha_i} \) also increases to \( \aleph_{\alpha} \).

Answer: \aleph_{\alpha}


Question: Calculate the trace of the operator $T$ defined by \[ T = \frac{1}{\#G} \sum_{g \in G} \phi(g^{-1}) \rho_g \] where $\phi$ is a class function from a finite group $G$ to a field $k$, and $\rho$ is an irreducible representation of $G$. Assume that the characteristic of $k$ is zero and $(\phi, \chi_\rho) = 0$. What is the trace of $T$?

Answer: 0


Question: Evaluate the limit: $$\lim_{k\to\infty}\left(1+\frac{2}{3k}\right)^{9k^2/4}$$

Answer: \infty


Question: Determine whether there are any polynomial solutions to the functional equation \( f(x)f(1/x) = f(x+1/x) + 1 \) with \( f(1) = 2 \), other than \( f(x) = x + 1 \).

Answer: f(x) = x + 1


Question: Evaluate the limit: $$\lim_{(x,y)\to(1,1)} \dfrac{(x-y)(x+y-2)}{\sqrt{(x-1)^2+(y-1)^2}}$$

Answer: 0


Question: Use Stokes' theorem to evaluate the integral \( I = \int\limits_C \textbf{F} \centerdot \textbf{ds} \) where \( \textbf{F} = 3zx\textbf{i} + 3xy\textbf{j} + yz\textbf{k} \) and \( C \) is the path consisting of the three edges of the triangle \( \Delta ABC \) formed by the portion of the plane \( x + y + z = 1 \) in the first octant of 3-space, oriented as shown.

Answer: \dfrac{7}{6}


Question:
Construct a second-degree polynomial function \( f(x,y) \) such that:
1. \( f(x,y) \) is a polynomial of degree 2.
2. The gradient \( \nabla f(3,-2) \) is non-zero.
3. The maximum value of \( f \) constrained by \( x+y=1 \) occurs at the point \( (3,-2) \).
4. \( f \) is non-constant along the line \( x+y=1 \).

Find such a polynomial \( f(x,y) \).

Answer: xy + 5x


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{x-\int_{0}^{x}\cos u^2\,du}{6\sin^{-1}x-6x-x^3} \] using L'Hôpital's Rule.

Answer: \dfrac{2}{9}


Question:
Given an orthonormal basis $\{\varphi_n\}$ of the space $L^2(E)$, for any function $f\in L^2(E)$, the series representation is
\[ f=\sum_{n=1}^{\infty}\langle f,\varphi_n\rangle\varphi_n, \]
where the series converges absolutely. Consequently, as $n$ approaches infinity, the absolute value of the inner product $\langle f,\varphi_n\rangle$ tends to zero, leading to
\[ \lim_{n\to\infty}\int_E f(x)\overline{\varphi_n(x)}dx=0. \]
Does this imply that
\[ \lim_{n\to\infty}\int_E f(x)\varphi_n(x)dx=0? \]

Answer: Yes


Question: Evaluate the surface integral \( \iint_S (\nabla \times \vec{F}) \cdot \vec{n} \, dS \) where \( \vec{F} = y\hat{i} + (x - 2xz)\hat{j} - xy\hat{k} \) and \( S \) is the surface of the hemisphere defined by \( x^2 + y^2 + z^2 = a^2 \) for \( z \ge 0 \).

Answer: 0


Question: Determine whether the following statement is true or false: If $p$ is an odd prime and $p$ divides the sum of two integers raised to the power of $p$, then $p^2$ also divides that sum.

Answer: True


Question:
Solve for $x$ in real numbers, where $a$ and $b$ are real, in the equation:

$$\left| a^2 - 2a - b^2 - 4b - x \right| + \left| a^2 - 2a - b^2 - 4b - 3x + 2 \right| + \left| a^2 - 2a + b^2 + 4b + 2x \right| + (a^2 - 2a + b^2 + 4b + 18|x - 2|) + 11x = 20.$$

Answer: 2


Question: Evaluate the limit: $$\lim_{n \to \infty}\left( \frac{1}{\sqrt{n^3+1}} + \frac{1}{\sqrt{n^3+4}} + \cdots + \frac{1}{\sqrt{n^3+n^2}}\right)$$

Answer: 0


Question: If a group $G$ is isomorphic to its quotient group $G/H$, does this imply that the subgroup $H$ is trivial?

Answer: No


Question: Determine whether the following statement is true or false: Any open set in a metric space is an uncountable set.

Answer: B


Question: Evaluate the alternating series \( \sum_{n=1}^{\infty}\frac{(-1)^n (n+1)}{n!} \).

Answer: -1


Question: Determine whether adding an $F_\sigma$-subset to a metrizable topology can result in a non-Hausdorff topology. Answer 'yes' or 'no'.

Answer: no


Question: Determine the largest constant \( c > 0 \) such that the inequality \[ c \leq \frac{\sup_{t\in [0,1]} |x+ty|}{|x|+|y|} \] holds for all real numbers \( x \) and \( y \).

Answer: \dfrac{1}{3}


Question: Solve the equation \(\sqrt{\frac{\sqrt{x!}\times \sqrt{x!!}\times \sqrt{x!!!}}{\sqrt{\left( x-2 \right)!+x}}}=12\) by simplifying the radicals and finding the value of \(x\).

Answer: 6


Question: Let $(G,*)$ be a group with an equivalence relation $\sim$ such that for all $a,a',b,b' \in G$, if $a \sim a'$ and $b \sim b'$, then $a*b \sim a'*b'$. Does this condition necessarily imply that for all $a,b \in G$, if $a \sim b$, then $a^{-1} \sim b^{-1}$?

Answer: Yes


Question: Given the function \( f(x) = x^3 + x + 1 \), find the value of \((f^{-1})'(1)\).

Answer: 1


Question: Calculate the degree of the map \( f_k: S^1 \to S^1 \) defined by \( f_k(\cos(2\pi t), \sin(2\pi t)) = (\cos(2k\pi t), \sin(2k\pi t)) \).

Answer: k


Question: Evaluate the integral \( \int_0^{\pi/2} (\cos^4 \theta + \sin^4 \theta) \, d\theta \) without using polar or rectangular coordinates.

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


Question: Evaluate the limit: $$ \lim_{x \to 0} \frac{\sin^2(\ln(1+x^2)) \arccos(\ln(1+x^2))}{\tan(\ln(1+x^2)^{1/3}) (1+x^2)^{1/2}} $$

Answer: 0


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_{\epsilon}^1\frac {n+1}{2^{n+1}}\frac{(1+t)^{n+1}-(1-t)^{n+1}}{t}\,dt \] where \(\epsilon \in (0, 1)\).

Answer: 2


Question: Determine the number of non-trivial homomorphisms from the symmetric group $S_5$ to the alternating group $A_6$.

Answer: 45


Question: Consider a convex and compact set $K \subset \mathbb{R}^n$, with $0$ lying in the interior of $K$. Let $f: K \rightarrow \mathbb{R}^n$ be a continuous function such that $f(\partial K) \subset K$. Does it necessarily follow that $f(K) \subset K$? Answer yes or no.

Answer: No


Question: Evaluate the infinite series: $$\sum_{n=1}^{\infty} \dfrac{1}{(3n-2)(3n+4)}.$$

Answer: \dfrac{5}{24}


Question: Find the residue of the function \( f(z) = \frac{\sin z}{\cos^2 z} \) at its second-order pole \( z = \frac{\pi}{2} \).

Answer: 0


Question: Calculate the expected time until a bank balance, modeled by a Brownian motion with positive drift, first becomes negative.

Answer: \infty


Question: Determine the order of growth for a positive, increasing, and concave function \( u: \mathbb{R}_+ \to \mathbb{R} \) as \( x \to \infty \). Assuming the function is differentiable, express \( u(x) \) in the form \( u(x) = O(...) \) and find the upper bound.

Answer: O(x)


Question: Find the positive integer $x$ that satisfies the equation $x(x+1)(x+2)(x+3) + 1 = 379^2$. Use any method you find appropriate.

Answer: 18


Question: Find the limit \( \lim\limits_{n \to \infty} S^n(x) \), where \( S^n(x) = \sin(\sin(\ldots\sin(x)\ldots)) \) is the composition of the sine function with itself \( n \) times, starting with \( S(x) = \sin(x) \).

Answer: 0


Question: Determine the rank of the elliptic curve given by the equation \( y^2 = x^3 + p^3x \) where \( p \) is a prime number such that \( p \equiv 7 \pmod{16} \).

Answer: 0


Question: Find the solution to the recurrence relation \(T(n)=1+2\sum_{i=1}^{n-1}T(i)\) for \(n > 1\), with the initial condition \(T(1)=1\).

Answer: 3^{n-1}


Question: Determine the minimum number of lines required to touch every square of an \(n \times n\) grid, given that the lines must pass inside the squares and not on their edges or corners.

Answer: n


Question: Evaluate the series \( \sum_{n=0}^{\infty}\binom{2n}{n}\left(-\dfrac{6}{25}\right)^{n} \) using the Taylor expansion of \((1+x)^{-\frac{1}{2}}\).

Answer: \dfrac{5}{7}


Question: Determine whether there exists a surjective function from the set of finite length binary strings \(\{0,1\}^*\) to the set of infinite length binary strings \(\{0,1\}^\omega\) that contain a finite number of 1's.

Answer: Yes


Question: Determine the cardinality of the set $M$, where $M$ consists of all subsets of the natural numbers $\mathbb{N}$ such that both the subset and its complement are infinite.

Answer: 2^{\aleph_0}


Question: Determine whether the inequality \(2^{x+y}\cdot x^x\cdot y^y\geq(x+y)^{x+y}\) holds for all positive values of \(x\) and \(y\).

Answer: Yes


Question: Determine which constant functions are in the range of the linear map $T: C[0,1] \rightarrow C[0,1]$ defined by $T(f) = f - \int_{0}^{1}f$.

Answer: 0


Question: Calculate the length of the polar curve given by \( r = 2\theta^2 \) for \( 0 \leq \theta \leq \sqrt{5} \). Use the formula for the length of a polar curve: \[ \mathscr{L} = \int_0^\sqrt{5} \sqrt{(r(\theta))^2 + (r'(\theta))^2} \, d\theta = \int_0^\sqrt{5} \sqrt{(2\theta^2)^2 + (4\theta)^2} \, d\theta. \] Determine the correct length of the curve.

Answer: \dfrac{38}{3}


Question: Given a function $H(w)$, determine an upper bound on the Lipschitz constant of this function, assuming that $\|x_i\| \leq 1$ for all $i$.

Answer: 1


Question: Given that $f$ is a continuous function on the interval $[0,1]$, determine the value of the limit: $$ \lim_{n\rightarrow \infty} \int_{0}^{1} f(x^n) \, dx. $$ Choices: (A) $f(1)$, (B) $f(0)$, (C) $1$, (D) $0$. Is the conclusion that the limit is $0$ correct?

Answer: B


Question: Find the general solution of the functional equation \( f(x+h)=e^{x} f(h)+e^{h} f(x) \) given that \( \lim _{x \rightarrow 0} \frac{f(x)}{x}=0 \).

Answer: 0


Question: Find the radius of convergence for the complex series \(\sum_{n=1}^{\infty} \frac{z^{2n}n!}{(1+n^2)^n}\).

Answer: \infty


Question: Consider two polynomials, $P_1(x)$ and $P_2(x)$, with integer coefficients, and integers $n$ and $a$. Determine the number of distinct ordered pairs $(P_1(x), P_2(x))$ such that $(P_1(x))^2 + (P_2(x))^2 = (x^n - a)^2$, excluding the trivial solution $(x^n - a, 0)$. Assume that $x^n - a$ is irreducible over the Gaussian integers $\mathbb{Z}[i][x]$. Provide your answer as a single integer.

Answer: 3


Question: Evaluate the limit \( \lim_{x\to{0^+}}\frac{\log_{\cos{x}}{\sin{x}}}{\log_{\cos{\frac{x}{2}}}\sin{\frac{x}{2}}} \).

Answer: \dfrac{1}{4}


Question: Consider the separable Hilbert space $H$ with an orthonormal basis $(e_i)_i$. The weak operator topology (WOT) on the bounded linear operators $B(H)$ is defined using seminorms $|(Tu, v)|$ for all vectors $u, v \in H$. Let $\mathcal{T}$ be the topology generated by seminorms $|(Te_i, e_j)|$. Do $\mathcal{T}$ and WOT agree on norm-bounded subsets of $B(H)$?

Answer: Yes


Question: Verify the third isomorphism theorem \( \frac{G/N}{H/N} \cong G/H \) for \(G = \mathbb{Z} \oplus \mathbb{Z}\), \(N = \langle(1,0)\rangle\), and \(H = \langle(1,0), (0,5)\rangle\). Determine the structure of the quotient group \(G/N\).

Answer: \mathbb{Z}


Question: Determine whether the function \( f(x) = \sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}}e^{\frac{x}{n}} \) is differentiable on the entire real line \( \mathbb{R} \).

Answer: Yes


Question: Evaluate the limit: $$ \lim_{x \to a} \frac{x}{x-a} \int_{a}^{x} f(t) \, dt $$ where $f(x)$ is continuous in a neighborhood of $a$.

Answer: a f(a)


Question: Determine whether the following claim is true or false: Let $\boldsymbol{f}:U\to\mathbb{R}^n$, where $U\subset\mathbb{R}^m$ is open, and $\boldsymbol{a}\in U$, be a function such that there exists a matrix $A\in\mathbb{R}^{n\times m}$ for which, for every smooth curve $\boldsymbol{\gamma}: I\to U$, where $I$ is an open interval and $0\in I$, with $\boldsymbol{\gamma}(0)=\boldsymbol{a}$ and $\boldsymbol{\gamma}'(0)\ne \boldsymbol{0}$, the composition $h(t)=\boldsymbol{f}(\boldsymbol{\gamma}(t))$ is differentiable at $t=0$ and $h'(0)=A\boldsymbol{\gamma}'(0)$. Then $\boldsymbol{f}$ is differentiable at $\boldsymbol{x}=\boldsymbol{a}$. Is this claim true or false?

Answer: True


Question: Determine whether the following statement is true or false: If an operator $A$ maps a Banach space $X$ to a Hilbert space $Y$ and is compact, then the closure of the image of the unit ball under $A$ is infinite-dimensional.

Answer: B


Question: Let $R$ be a commutative ring and $J_1, J_2$ be two non-zero proper ideals of $R$. Suppose there exist $r_1, r_2 \in R$ such that for any $a \in R$, the conditions $r - r_1 \in J_1$ and $r - r_2 \in J_2$ are equivalent to $r \in J_2$ and $r - a \in J_1$. Does this imply that $J_1 + J_2 = R$?

Answer: J_1 + J_2 = R


Question: Determine the constant $c$ in the distribution function $F(x) = c\int_{-\infty}^x e^{-|t|}\, dt$ using the limit $\lim_{{x \to \infty}} F(x) = 1$.

Answer: \dfrac{1}{2}


Question: Determine the maximum number of linearly independent real $n \times n$ matrices $A_1, A_2, \ldots, A_m$ such that $A_i A_j = 0$ for all $i \neq j$. The matrices are not necessarily diagonalizable.

Answer: n


Question: Given the equation \(\left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right)=1\), find the expression for \(x+y\).

Answer: 0


Question: Calculate the limit: $$ \lim\limits_{n \to +\infty} \dfrac{\sqrt[n] {n^3} + \sqrt[n] {7}}{3\sqrt[n]{n^2} + \sqrt[n]{3n}} $$

Answer: \dfrac{1}{2}


Question: Let $X_1, X_2, \dots, X_n$ be independent, uniformly distributed random variables on the interval [0, 1]. Calculate the probability that the maximum of $X_3$ and $X_5$ is less than the minimum of $X_1, X_2, X_4, X_6, X_7$. \( \text{Express your answer as a single probability value.} \)

Answer: \dfrac{1}{21}


Question: Let \( g(z) = z^4 + iz^3 + 1 \). Determine the number of zeros that \( g \) has in the region \( \{z \in \mathbb{C}: \text{Re}(z), \text{Im}(z) > 0\} \).

Answer: 1


Question: In a unitary ring $R$, if $a, b \in R$ satisfy $ab = 1$, is $a$ necessarily invertible with $a^{-1} = b$?

Answer: No


Question: Is the set \( K = \{f(x) \in R \mid f(2) = 0\} \) a maximal ideal in the ring of functions \( R = \{f:\mathbb{R}\rightarrow \mathbb{R}\} \) with operations defined as \((f + g)(x) = f(x) + g(x)\) and \((fg)(x) = f(x)g(x)\)?

Answer: Yes


Question: Evaluate the series for $n \geq 2$: $$\sum_{k=2}^{n}\frac{n!}{(n-k)!(k-2)!}$$

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


Question: Find the values of $x$ for which the series $\sum_{n=1}^\infty \frac{(x-2)^n}{n^2}$ converges.

Answer: [1, 3]


Question:
Find all continuous functions \( f: (a, b) \to \mathbb{R} \) that satisfy the Jensen equation:

\[ f\left(\frac{x+y}{2}\right) = \frac{f(x) + f(y)}{2} \]

Answer: f(x) = mx + c


Question: Evaluate the series: $$\sum_{n=0}^{\infty} (-1)^n\frac{\pi^{4n}}{n!}$$

Answer: e^{-\pi^4}


Question: Determine the dimension of the variety \( Y = V(x_1 x_2 - x_3 x_4) \subset \mathbb{A}^4 \) using the relationship between Krull dimension and the height of prime ideals.

Answer: 3


Question: Calculate the probability of rolling exactly one triple with the other 9 numbers different from each other and from the triple when rolling 12 dice.

Answer: 0


Question: Given a vector-valued function $f : \mathbb{R} \to \mathbb{R}^n$ such that $|f(t)|=1$ for all $t \in \mathbb{R}$, determine the value of the inner product $\langle f(t), f'(t) \rangle$.

Answer: 0


Question: Evaluate the integral \( \int_0^\infty \frac{x^{1/2}}{x^2+4} \, dx \) using contour integration.

Answer: \dfrac{\pi}{2}


Question: Determine whether the statement is true or false: Every subspace of a connected space is connected.

Answer: B


Question: Determine the rank of the matrix $W_{ij} = v_iv_j^T - v_jv_i^T$, where $v_i$ and $v_j$ are linearly independent vectors in $\mathbb{R}^n$ and $i \neq j$. Provide your answer as a single integer.

Answer: 2


Question: Let $A$ be a binary matrix with $\textrm{rank}(A)=r$. If $C$ is a matrix such that $A+C$ is a matrix with all entries being 1, find the minimum possible value of $\textrm{rank}(C)$. \(\text{Express your answer as a single integer.}\)

Answer: r - 1


Question:
Determine if the following representation of the derivative $f'(x)$ is correct for all real numbers $x$:
$$f'(x)=\sum_{k=0}^{2009}\prod_{l=0, l \neq k}^{2009}(x+l)$$
where $f(x)=\prod_{k=0}^{2009}(x+k)-c$.

Answer: Correct


Question: Consider a sequence of matrices defined recursively by \(A_{k+1} = \frac{3A_k + I}{2}\), where \(A_0\) is a given initial matrix, and \(I\) is the identity matrix. Find the limit of the sequence as \(k\) approaches infinity.

Answer: -I


Question: Find the area of the region enclosed by the sphere $x^2 + y^2 + z^2 = 1$ and the plane $x + y + z = m$ for $|m| > \sqrt{3}$.

Answer: 0


Question: Find the value of $f(3)$ for all functions $f$ that satisfy the functional equation $f(xy) = f(x)f(y)$ and the condition $f(9) = 2$. Provide your answer as a single number.

Answer: \sqrt{2}


Question: Compute the volume of the solid bounded by the surface \( z = 3\sqrt{x^2 + y^2} \), the plane \( z = 0 \), and the cylinder \( x^2 + (y - 1)^2 = 1 \).

Answer: \dfrac{32}{3}


Question:
Determine the functional derivative \( \frac{\delta \mathfrak{F}}{\delta f} \) of the Fourier transform \( \mathfrak{F} \) given by:
\[ f \mapsto \int_{-\infty}^{\infty} f(x)e^{ix}dx. \]

Answer: e^{ix}


Question: Let $(X,d)$ be a complete metric space, and let $f:X \to X$ be a function such that $d(f(x),f(y)) \geq L d(x,y)$ for all $x,y \in X$ with $x \neq y$, where $L > 1$. Does $f$ necessarily have a unique fixed point? Provide a justification for your answer.

Answer: No


Question: Suppose \((x_n)_{n\in\mathbb{N}}\) is a sequence in \(\mathbb{R}\) that converges to \(x\in\mathbb{R}\). If \(f:\mathbb{R}\longrightarrow \mathbb{R}\) is a bounded function, does it follow that \((f(x_n))_{n\in\mathbb{N}}\) has a convergent subsequence? Answer yes or no.

Answer: Yes


Question: Calculate the kurtosis of the random variable $W$, where $W$ is defined as follows: Let $X$ be a standard normal random variable and $Y$ be an independent uniform random variable on $[0,1]$. Define $W = X$ if $Y \leq \frac{1}{2}$ and $W = -X$ if $Y > \frac{1}{2}$. Provide your answer as a single numerical value.

Answer: 3


Question: Solve for \( x \) in the congruence equation \( 35 \equiv 7^x \pmod{41} \) without using brute force.

Answer: 19


Question: Evaluate the contour integral \( \oint_{|z|=1} \frac{\cos z}{z^3} \, dz \).

Answer: -\pi i


Question: Calculate the integral \( \int_C e^z\,dz \) where \( C \) is the circle \( |z - j\pi/2| = \pi/2 \) from the point \( z = 0 \) to the point \( z = j\pi \).

Answer: -2


Question: Determine whether there are any solutions in positive integers to the equation \( a^n + b^{n-i} = c^{n-2i} \) where \( n-2i > 1 \).

Answer: Yes


Question: Determine whether an odd multi-perfect number can be a perfect square.

Answer: No


Question:
Find the limit of the function as \(x\) approaches 0:

\[ \lim_{x\rightarrow0}\frac{\sin 2x - 2\sin x}{x^3} \]

Answer: -1


Question: Determine if the function $g: [0,1] \rightarrow \mathbb{S}^1$, defined by $g(t) = (\cos(2\pi t), \sin(2\pi t))$, is a homeomorphism when $[0,1]$ is equipped with the equivalence relation that identifies points $0$ and $1$. Use the theorem that states: If $g: X \rightarrow Z$ is continuous and surjective, and the equivalence relation on $X$ is defined as $x \sim y \Leftrightarrow g(x) = g(y)$, then the quotient topology on $X/\sim$ induces a bijective continuous function $f: \frac{X}{\sim} \rightarrow Z$. This function is a homeomorphism if and only if $g$ is a quotient map.

Answer: Yes


Question: Evaluate the limit of the sequence: \[ \lim_{n\to\infty}\frac{\sqrt{n!}}{\prod_{k=1}^{n+1}(1+\sqrt{k})} \]

Answer: 0


Question: Calculate the limit: $$ \lim_{(x,y) \to (0,0)}\frac{\sin(x^2+y^2)}{\ln(3x^2y^2+1)} $$

Answer: \infty


Question: Calculate the limit: $$ \lim _{(x,y,z)\to (0,0,0) } \frac{x^3+y^3+z^3}{x^2+y^2+z^2} $$ using only rectangular coordinates.

Answer: 0


Question: Evaluate the triple integral \(\int\int\int_V (x^2 + y^2 + z^2)\, \mathrm{d}x\, \mathrm{d}y\, \mathrm{d}z\) over the region \(V\) bounded by the planes \(x = 0\), \(y = 0\), \(z = 0\), and \(x + y + z = b\) where \(b > 0\).

Answer: \dfrac{b^5}{20}


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{g(x)}{x-a} \delta(x-a) \, dx \), where \( \delta(x) \) is the Dirac delta distribution and \( g(x) \) is a function nonsingular everywhere.

Answer: g'(a)


Question: Let $V$ be a vector space over $\mathbb{R}$ with dimension $n$. Given a basis $v = \{ v_1, \dots, v_n\}$ for $V$, define the set $$S_{n,k} = \{ v_1 + v_2 + \dots + v_k, v_2 + \dots + v_{k+1}, \dots, v_n + v_1 + \dots + v_{k-1}\}.$$ Determine the values of $n$ and $k$ for which $S_{n,k}$ is a basis for $V$. Provide your answer as a set of conditions or specific values for $n$ and $k$.

Answer: \gcd(n, k) = 1


Question: Verify that \(|\int_\gamma e^z - \bar{z}| \leq 57\) using the ML estimate, where \(\gamma\) is the boundary of the triangle with vertices at \(0, 3i, -4\). Given that the length of the triangle \(L = 12\), determine the maximum value \(M\) such that \(ML = 57\).

Answer: \dfrac{19}{4}


Question: Find the remainder when the sum \(1 + b + b^2 + \dots + b^{\varphi(m)-1}\) is divided by \(m\), given that \(b\) and \(m\) are relatively prime.

Answer: 0


Question: Determine whether there exists a first-order logic formula, without using the identity operator, that can only be satisfied by infinite structures, assuming the structures are non-empty.

Answer: Yes


Question: Evaluate the limit \(\lim_{n \to \infty} \frac{(3n)!}{3^n (n!)^3}\) using the ratio test.

Answer: \infty


Question: Determine whether the equation \( \frac{a^n-b^n}{a-b}=c^n \) can have a solution for \(a, b, c, n \in \mathbb{N}\), given that \(a^n-b^n \neq a-b\) and \(\gcd(a-b, c) = \gcd(a, b) = 1\).

Answer: Yes


Question: Given $A = \dfrac{(p^2+q^2+r^2)^2}{p^2q^2+q^2r^2+r^2p^2}$ and $B=\dfrac{q^2-pr}{p^2+q^2+r^2}$, where $p+q+r=0$, find the value of $A^2-4B$. Express your answer as a single number.

Answer: 14


Question: What is the largest factorial that has only three distinct digits in its decimal representation?

Answer: 5040


Question: Find the radical of the ideal \( I = (xy) \) in the polynomial ring \( R = \mathbb{Z}[x,y] \).

Answer: (xy)


Question: Consider an $n$-dimensional sphere of radius $\sqrt{n}$. A point $(x_1, x_2, \ldots, x_n)$ is chosen randomly on this sphere. Determine the probability that the first coordinate $x_1$ is less than or equal to a given value $a$ as $n$ approaches infinity.

Answer: \Phi(a)


Question: Find the limit of \( \frac{b_n}{a_n} \) as \( n \to \infty \), given that \( \lim_{n\to\infty}a_n=0 \) and \( \lim_{n\to\infty}b_n=\infty \).

Answer: \infty


Question: Given that \(2f(x) + f(-x) = \frac{1}{x}\sin\left(x-\frac{1}{x}\right)\), find the value of \(\int_{\frac{1}{e}}^e f(x)\,dx\).

Answer: 0


Question: Evaluate the integral: \[ \int_{\frac{\pi}{2}}^{\frac{5\pi}{2}} \frac{e^{\cot^{-1}\cos(x)}}{e^{\cot^{-1}\sin(x)}+e^{\cot^{-1}\cos(x)}} \, dx \]

Answer: \pi


Question: Calculate \(5^{3^{1000}} \mod 101\). Use Fermat's Little Theorem or Euler's Theorem to simplify the calculation.

Answer: 5


Question: Determine whether every automorphism of a group that fixes all maximal subgroups is the identity automorphism.

Answer: No


Question: Find the minimal polynomial over the field $F(y)$ for the polynomial $g(s) = (s^8+1)-ys^4$, where $y=\frac{x^8+1}{x^4} \in F(x)$. Provide your answer as a polynomial in terms of $s$.

Answer: s^8 - y s^4 + 1


Question: Evaluate the limit: $$\lim_{n\rightarrow\infty}\int_{0}^{\frac{\pi}{4}}\frac{n\sin x \cos^n x}{1+x^2}dx$$

Answer: 1


Question: Evaluate the limit: $$ \lim_{x\rightarrow 0} \frac{\sin x^2}{ \ln ( \cos x^2 \cos x + \sin x^2 \sin x)} $$ without using L'Hôpital's rule.

Answer: -2


Question: Compute the residue of the function \( \frac{(\sin(z))^2}{(\sinh(z))^5} \) at \( z = 0 \).

Answer: -\dfrac{7}{6}


Question: Calculate the work done by the force field \( F(x,y) = -xi + 6yj \) along the path \( C: y = x^3 \) from the point \((0,0)\) to \((6,216)\).

Answer: 139950


Question: Determine the index and normality of the subgroup of direct motions within the group of motions in \(\mathbb{R}^2\). Use the properties of reflections about lines to justify your answer.

Answer: 2


Question: Given the matrix $B$ with the first row as $(t, 3, 1)$, the second row as $(3, t, 0)$, and the third row as $(1, 0, t)$, find the values of $t$ such that the matrix $B$ is positive definite.

Answer: t > \sqrt{10}


Question: Evaluate the integral \( \int_{\frac{1}{e}}^e f(x) \, dx \), given that \( 2f(x) + f(-x) = \frac{1}{x}\sin\left(x-\frac{1}{x}\right) \) for all \( x \).

Answer: 0


Question: Students arrive at a help centre according to a Poisson process with rate $r$. When there are $n \geq 2$ students in the centre, the first two students to leave do so at random $Exp (4r)$ times. Currently, there is only one student in the centre. What is the expected time until there are three students in the centre?

Answer: \dfrac{10}{r}


Question: Consider two trees $T_1=(V,E_1)$ and $T_2=(V,E_2)$ with the same vertex set $V$. Let $G=(V,E_1 \cup E_2)$ be the graph formed by the union of the edge sets of $T_1$ and $T_2$. Prove that there exists a vertex in $G$ with degree at most 3.

Answer: 3


Question: Determine whether the sequence \(d_n = a_n^3 - \sqrt{b_n} + \cos(a_n - b_n)\), where \(a_1,a_2,a_3,...\) and \(b_1,b_2,b_3,...\) are Cauchy sequences in \([0,\infty)\), is a Cauchy sequence.

Answer: Yes


Question: Let $U$ be an open subset of $\mathbb{R}^n$ that is $C^k$ ($0 \leq k \leq \infty$). Assume that $h : \mathbb{R}^n \rightarrow \mathbb{R}^n$ is a $C^k$ diffeomorphism. Does $h(U)$ also inherit the $C^k$ property of $U$?

Answer: Yes


Question: Consider a sequence of 58 terms defined as $P+n$, where $P$ is the product of all prime numbers less than or equal to 61, and $n$ takes values from 2 to 59 in succession. Determine the number $N$ of primes in this sequence.

Answer: 0


Question: Let $X$ be a set and $T_1, T_2$ be topologies on $X$. Assume that for any topological space $Y$, the following conditions hold: (1) A function $f:Y\rightarrow X$ is continuous with respect to $T_1$ if and only if it is continuous with respect to $T_2$. (2) A function $f:X\rightarrow Y$ is continuous with respect to $T_2$ if and only if it is continuous with respect to $T_1$. Must $T_1$ and $T_2$ be the same topologies?

Answer: T_1 = T_2


Question: Let $A, B, C$ be three collinear points with $B$ between $A$ and $C$. Consider all circles passing through $B$ and $C$. Determine the type of curve on which the points of contact of the tangents from $A$ to these circles lie.

Answer: circle


Question: In a metric space $X$, consider two complete metric spaces $X'$ and $Y$, with isometries $i : X \rightarrow X'$ and $f : X \rightarrow Y$. Let $(i(x_n))_n$ be a sequence in $i[X]$ converging to $p \in \overline{i[X]} = X'$. Since $(i(x_n))_n$ is Cauchy, $(x_n)_n$ is Cauchy in $X$, implying $(f(x_n))_n$ is Cauchy in $Y$ and converges to $q \in Y$. If $(i(y_n))_n$ is another sequence converging to $p$, does $(f(y_n))_n$ also converge to $q$ in $Y$?

Answer: Yes


Question: Determine the smallest integer \( n \) such that the probability \( \Bbb P\left(\bar X_n - 0.1 \leq p \leq \bar X_n + 0.1\right) \geq 0.9 \), where \( \bar X_n = \frac{\text{number of heads}}{n} \) is the relative frequency of heads in \( n \) tosses of a biased coin with probability \( \Bbb P(\text{head}) = p \).

Answer: 68


Question:
Given that \(a \neq 0\), \(b \neq 0\), \(c \neq 0\), solve the following system of equations to find the value of \(a + b + c\):\[\begin{cases}
\frac{1}{a} + \frac{1}{b} + \frac{1}{a+x} = 0\\
\frac{1}{a} + \frac{1}{c} + \frac{1}{a+y} = 0\\
\frac{1}{a} + \frac{1}{x} + \frac{1}{y} = 0
\end{cases}\]

Answer: 0


Question:
Calculate the arc length of the curve defined by the parametric equations:

\[ X(t)=\int_{1}^{t}\frac{\cos z}{z^2}dz, \quad Y(t)=\int_{1}^{t}\frac{\sin z}{z^2}dz \]

from the point where the tangent is perpendicular to the x-axis to the point where \(t=\infty\).

Answer: \dfrac{2}{\pi}


Question: Evaluate the limit \( \lim_{n \to \infty}\int_{0}^{1}nx^ne^{x^2}dx \).

Answer: e


Question:
Consider three circles given by the equations \(x^2+y^2+2\lambda_ix-c^2=0\) for \(i=1,2,3\), where their centers are located on the x-axis. If the distances from the origin to the centers of these circles form a geometric progression (GP), determine the relationship between the lengths of tangents drawn to these circles from any point on the circle \(x^2+y^2=c^2\):

(a) GP
(b) Arithmetic progression (AP)
(c) Harmonic progression (HP)

Given that the centers of the circles are at \((-\lambda_1, 0)\), \((-\lambda_2, 0)\), and \((-\lambda_3, 0)\), with distances \(\lambda_1, \lambda_2, \lambda_3\) from the origin satisfying the GP condition: \(\lambda_2^2 = \lambda_1\lambda_3\). Determine the correct relationship.

Answer: a


Question: Solve the equation: $$x^{(x+1)} = (x+1)^x$$

Answer: 2.293


Question: Determine the chromatic number of a graph $G$ where the vertices represent integers $1, 2, \dots, 99$, and two vertices are connected if their greatest common divisor is at least $3$. \( \text{(Provide your answer as a single integer.)} \)

Answer: 33


Question: Suppose $u \in H^1(\mathbb{R}^2) \cap C(\mathbb{R}^2)$. Determine whether $\limsup_{|x| \to \infty} u(x) = 0$.

Answer: 0


Question: If the arrow category \( \mathcal{C}^{\rightarrow} \) of a category \( \mathcal{C} \) has an initial object, does \( \mathcal{C} \) also have an initial object?

Answer: Yes


Question: Given a polynomial $P(x) \in \mathbb{Z}[x]$ that does not have a root with modulus 1, determine whether there exists a polynomial $Q(x) \in \mathbb{Z}[x]$ such that the product $P(x)Q(x)$ has a big coefficient.

Answer: Yes


Question: Evaluate the integral \( \int_{-\infty}^{\infty} dk\left[\Im\left( \frac{1}{(\omega+i\eta+k)^2+(\omega+i\eta+k+q)^2}\right) \right] \) for an infinitesimal \( \eta \).

Answer: 0


Question:
Solve the inequality for $n$ variables $x_1, x_2, \ldots, x_n$ where $a > 2$ and $x_i \in [a, 2a]$ for all $i$, given that $3x_i > 2a$ for all $i$:
\[ \log_{x_1} \sqrt{3x_2 - 2a} + \log_{x_2} \sqrt{3x_3 - 2a} + \cdots + \log_{x_n} \sqrt{3x_1 - 2a} \geq \frac{n}{2}. \]

Answer: x_i = a


Question: Evaluate the limit: $$\lim_{n\to\infty}\frac{n^n}{(n+k)!}, \ n,k\in\Bbb N$$

Answer: \infty


Question: Determine whether the limit of a uniformly convergent sequence of convex functions is also convex. Specifically, if a sequence of functions \((f_{n})\) converges uniformly to a function \(f\) on a set \(E\), and each \(f_{n}\) is convex, is \(f\) necessarily convex?

Answer: Yes


Question: Calculate the expected number of attempts required to find a quadratic non-residue modulo a prime number \( p \) using a random algorithm.

Answer: 2


Question: Calculate the integral \( \int_{C} f(z) \, dz \), where \( f(z) = \frac{\bar{z}}{z+i} \) and \( C \) is the circle \( |z+i|=3 \).

Answer: -2\pi


Question: Determine whether the polynomials \( p_n(z) = \left(1 + \frac{z}{n}\right)^n \) converge uniformly on compact subsets of \( \mathbb{C} \) to \( e^z \).

Answer: Yes


Question: Calculate the last two digits of \( f(17) + f(18) + f(19) + f(20) \) where \( f(x) = x^{x^{x^x}} \).

Answer: 32


Question: Determine the total number of square matrices of order \( n \) with entries 1 or -1, such that the product of each row and each column is equal to -1.

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


Question: Calculate the maximum value of the real part of the Rayleigh quotient for the matrix \( A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \). Express your answer as \( \max\left(\operatorname{Re} \left\{ \frac{x^* A x}{x^* x} : 0 \neq x \in \mathbb{C}^2 \right\} \right) \), where \( x \) is a complex 2-dimensional vector.

Answer: 2


Question: Calculate the limit: $$\lim _{x\to 0}\left(1-\cos^{2}\left(\frac{\pi}{4(1-\tan x)}\right)\right)^{\frac{1}{\ln(\cos x)}}$$

Answer: \infty


Question: Given a Lebesgue measurable function $f rom [0,1]^2 	o \mathbb{R}$, where the iterated integrals over any measurable subsets of $[0,1]$ exist and are equal, determine if it is true that $f$ must be integrable over $[0,1]^2$ if it is bounded.

Answer: Yes


Question: Evaluate the surface integral \(\iint_S F\cdot n \,ds\) for the vector field \(F(x,y,z) = (z^2 - x, -xy, 3z)\) over the solid \(S\) bounded by the equations \(z = 4 - y^2, x = 0, x = 3\), and \(z = 0\), where the normal vector is pointing outward. Do not use the Divergence Theorem.

Answer: 16


Question: Determine whether Grandi's Sum, given by the series \(1 - 1 + 1 - 1 + 1 - 1 + \ldots + (-1)^n + \ldots\), is equal to \(\frac{1}{2}\).

Answer: \frac{1}{2}


Question: For finite groups $G$ and $H$, let $G^{ab}$ and $H^{ab}$ denote their respective abelianizations. Determine if it is possible for $|G^{ab}|$ to be greater than $|H^{ab}|$ even when $|G| < |H|$. Provide a justification for your answer.

Answer: Yes


Question: Determine the real values of \(\alpha\) for which the function \(g_{\alpha}(x) = x^{\alpha}\log(x)\) is uniformly continuous on the interval \([1, \infty)\), given that \(0 < \alpha < 1\).

Answer: (0, 1)


Question: Evaluate the integral \( \int_0^1\int_{3x}^3\int_0^{\sqrt{9-y^2}} z^2 \, dz \, dy \, dx \) using the double-angle formula for sine.

Answer: \dfrac{27}{5}


Question: Determine the minimum achievable accuracy for an algorithm $\psi$ that approximates an elementary function $f$ within a floating-point system characterized by $e$ bits for the exponent and $m$ bits for the mantissa, given that the domain $D(f) = [1,2)$ and the range $f(D(f)) = [1,2)$. Provide your answer in terms of $e$ and $m$.

Answer: 2^{-(m+1)}


Question: Using the chain rule, find $g'(1)$ if $f(x) = x^2 + x + e^x$ and $g(x) = f^{-1}(x)$. Note that $f(x)$ is not a one-to-one function and does not have an inverse function expressible in terms of elementary functions. However, find $g'(1)$ by other means.

Answer: \dfrac{1}{2}


Question: Let $A$ and $B$ be Hausdorff topological spaces, where $A$ is compactly generated. A function $f: A \times B \rightarrow \mathbb{R}$ is continuous on $K \times B$ for every compact subset $K$ of $A$. Is $f$ necessarily continuous on $A \times B$ when the product sets are given the product topology? Answer yes or no.

Answer: No


Question: For any natural number $n$, determine the greatest common divisor of $n! + 1$ and $(n+1)! + 1$. Express your answer as a single number.

Answer: 1


Question: Evaluate the limit \( \lim_{n \to \infty} \int_{0}^{n}\dfrac{\cos \left(\frac{x}{n}\right)}{1+n^{1/2}x^{3/2}}\, dx. \)

Answer: 0


Question: Consider the open hemisphere $S_+^{n+1}$ of the standard Euclidean sphere centered at the north pole, and let $M^n$ be a compact, connected, and oriented hypersurface of $S_+^{n+1}$. If $M$ has nowhere zero Gauss-Kronecker curvature, do all its principal curvatures have the same sign?

Answer: Yes


Question: Determine whether the following statement is true or false: If \(P(A \mid C) > P(A \mid C^c)\) and \(P(B \mid C) > P(B \mid C^c)\), then \(P(AB \mid C) > P(AB \mid C^c)\).

Answer: False


Question: In $\Delta ABC$, where $AB = 2017$ units, point $O$ is the midpoint of $AB$. Point $D$ lies inside the triangle such that $\angle DAP = \angle DBQ$. If $\Delta APD$ is similar to $\Delta BQD$, find the difference $OP - OQ$. Express your answer as a single numerical value.

Answer: 0


Question: Let $Y$ and $Z$ be independent random variables, each with the exponential density $f(t)=0.25e^{-0.25t}$ for $t > 0$. Find the probability that the inequality $Y \geq 16Z$ holds.

Answer: \dfrac{1}{17}


Question: Given 8 independent runners with finish times distributed as $X_1, X_2, \ldots, X_8 \sim \text{Exp}(1)$, find the expected value of the difference between the time of the first and second runners to finish, $E(T_2 - T_1)$, where $T_1$ is the time of the first runner to finish and $T_2$ is the time of the second runner to finish.

Answer: \dfrac{1}{7}


Question: Determine the smallest integer $d^*$ such that the inequality \( \frac{\left(\sum_{i=1}^d i^{-p}\right)^2}{\sum_{i=1}^d i^{-2p}} \ge \frac{1}{2}\frac{\zeta(p)^2}{\zeta(2p)} \) holds, where $\zeta(s)$ is the Riemann Zeta function and $H_d^{(p)}$ are the Harmonic numbers.

Answer: 2


Question: Evaluate the infinite series \(\sum_{i=2}^\infty \frac{i^2}{i!}\) and express your answer in the form \(2e - 1\).

Answer: 2e - 1


Question: Let $\{X_n\}_{n\ge 1}$ be a sequence of random variables. If the expectation and variance of $X_n$ both tend to 0, does $X_n$ tend to 0 weakly? If not, provide a counter-example.

Answer: Yes


Question: Determine the number of real roots of the equation \( 1+\frac{x}{1}+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\frac{x^5}{5}+\frac{x^6}{6}+\frac{x^7}{7} = 0 \).

Answer: 1


Question: Determine the minimal polynomial of the linear transformation \(T: M_3(F) \rightarrow M_3(F)\) defined by \(T(B) = B^T\), where \(B^T\) is the transpose of the matrix \(B\).

Answer: x^2 - 1


Question: Determine whether there exists a nonempty compact subset $C_\infty \subset C([0,\infty),\mathbb{R}^n)$ such that $C_{\infty\mid T} \subset C_T$ for all $T>0$, given that for each $T>0$, $C_T \subset C([0,T],\mathbb{R}^n)$ is a nonempty compact subset and if $T_1 < T_2$, then $C_{T_2\mid T_1} \subset C_{T_1}$, where $C_{T_2\mid T_1} = \{f(\,)\in C([0,T_1],\mathbb{R}^n)\mid \exists g(\,)\in C_{T_2},\, f(t) = g(t),\, t \in [0,T_1]\}$.

Answer: Yes


Question: Let $X$ and $Y$ be independent random variables with the probability distribution function $P(n) = 3^{-n}$ for $n = 1, 2, 3, \ldots$. Calculate $P(X \geq 3Y)$. \( \boxed{?} \)

Answer: \dfrac{3}{160}


Question: Given the Markov chain with state space \(\{1, 2, 3\}\) and transition matrix \(P = \begin{pmatrix} 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{3} & \frac{1}{2} & \frac{1}{6} \\ 0 & 1 & 0 \end{pmatrix}\), where the initial state \(X_0 = 1\) almost surely, find the probability that the chain returns to state 1 before reaching state 3 for the first time. Let \(u(i)\) denote the probability of visiting state 1 before state 3 when starting from state \(i\).

Answer: \dfrac{1}{3}


Question: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f'(x)$ exists for each $x \in \mathbb{R}$ and $f'$ is bounded on $\mathbb{R}$. Does this imply that $f$ is uniformly continuous on $\mathbb{R}$? Answer yes or no and justify your answer.

Answer: Yes


Question: Find the smallest irreducible algebraic set in \(\mathbb{R}^3\) that contains all points with integer coordinates.

Answer: \mathbb{R}^3


Question: Find a closed-form expression for the 19th derivative of the function \( f(x) = e^{-x}(x-1) \).

Answer: (20 - x)e^{-x}


Question: Find the number of elements of order 1008 in the abelian group $G = \mathbb{Z}_{20} \times \mathbb{Z}_{28} \times \mathbb{Z}_{36}$.

Answer: 0


Question: Evaluate the limit using the Gamma function: $$\lim_{c\to \infty} \int_{0}^c xe^{-x}dx$$

Answer: 1


Question: Determine whether the function \( f(x) = 3\sin^4(x) - 2\sin^6(x) + 3\cos^4(x) - 2\cos^6(x) - 1 \) is a constant function.

Answer: 0


Question: Evaluate the integral \( \int_0^{\pi/2}\int_{x}^{\pi/2} \frac{\cos{y}}{y}\, dy \,dx \).

Answer: 1


Question: Calculate the contour integral of \( \frac{1}{z+2} \) over the circle of radius 2 centered at \( z=-3 \).

Answer: 2\pi i


Question: Determine whether it is possible to construct disjoint sets $A$ and $B$ in $\mathcal{P}(\mathbb{R})$ such that the outer measure of their union is strictly less than the sum of their individual outer measures, using the Axiom of Choice.

Answer: Yes


Question: Find the minimum value of \( P = \dfrac{1}{1 + x^2} + \dfrac{1}{1 + y^2} + 2xy \) subject to the constraint \( xy \geq 1 \).

Answer: 3


Question: Evaluate whether the rising factorial $0^{\bar{0}}$ is equal to 1 using the relationship $a^{\bar{n}}=\frac{\Gamma(a+n)}{\Gamma(a)}$. Consider the case when $a=n=0$.

Answer: 1


Question: Evaluate the limit: $$ \lim_{x \rightarrow \infty} \left(1+\dfrac{1}{x} \right)^{\sqrt{2x^2 +3x+1}} $$ without using L'Hopital's rule.

Answer: e^{\sqrt{2}}


Question: Evaluate the limit: $$\lim\limits_{n\to \infty} \sin \bigl( (2 + \sqrt 3 )^n\pi \bigr) \quad \text{ for } n \in \mathbb{N}.$$

Answer: 0


Question: Can the Gaussian integral be evaluated using complex analysis techniques, such as contour integrals?

Answer: \sqrt{\pi}


Question: Evaluate the limit: $$ \lim_{x\to 3}\left(\frac{x}{x-3}\int_{3}^{x}\frac{\sin t}{t} dt\right) $$ without using L'Hopital's rule.

Answer: \sin 3


Question: Find the analytic extension of the function $f(x) = |x|$ from the real numbers to the complex plane, excluding zero.

Answer: \sqrt{z^2}


Question: Evaluate the sum of the infinite series: $$\sum_{n=2}^{\infty} \frac{(n-1)^2+2(n-1)+3}{3^{n-1}}$$

Answer: \dfrac{9}{2}


Question: Determine whether the following statement is true or false: For any class of rings $\{R_1, R_2, \dots, R_n \}$, we have $(\prod_{i=1}^{n}R_i)[x] \cong \prod_{i=1}^{n} R_i[x]$.

Answer: True


Question: Evaluate the integral \( \int_{0}^{4}\sqrt{9t^2+t^4}\,dt \).

Answer: \dfrac{98}{3}


Question: Determine whether the limit \( \lim_{x\rightarrow \infty} \frac{u(x)}{\pi(x)} \) exists, where \( u(x) \) is the number of primes below \( x \) such that the sum of their digits is also prime, and \( \pi(x) \) is the prime counting function.

Answer: 0


Question: Evaluate the integral \( \int_\Gamma \frac{z}{\bar{z}^2}\; dz \), where \( \Gamma \) is the circle of radius 3 centered at the origin, traversed once in the clockwise direction from the point \((3, 0)\).

Answer: 0


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

Answer: 2 - \pi


Question: Consider the ellipse \(\frac{x^2}{25}+\frac{y^2}{9}=1\). Four lines, \(L_1\), \(L_2\), \(L_3\), and \(L_4\), are parallel to each other with slopes \(9\) and \(-\frac{1}{25}\), respectively. All these lines are tangent to the ellipse. Determine the area of the parallelogram formed by these tangent lines.

Answer: 60


Question: How many positive divisors of $36^{20}$ have a remainder of $1$ when divided by $7$ and a remainder of $2$ when divided by $13$?

Answer: 21


Question: Determine whether the following statement is true or false: If $X$ is compactly embedded in $Y$ and $B$ is a closed subspace of $Y$, then $B$ is compactly embedded in $X$. Provide a justification for your answer.

Answer: False


Question: Determine whether the following statement is true or false: If $V$ is a vector space over $\mathbb{C}$ and $T: V \to V$ is a linear transformation with $0$ as its only eigenvalue, then $T$ must be nilpotent.

Answer: False


Question: Calculate the group of homomorphisms $\text{Hom}(C_n, B)$, where $C_n$ is the cyclic group of order $n$ and $B$ is an abelian group. The group composition is defined by $(g_1*g_2)(y) = g_1(y)+g_2(y)$. Provide your answer in terms of $B$.

Answer: B[n]


Question: Let $H$ be an infinite dimensional Hilbert space and $K:H\rightarrow H$ a compact self-adjoint operator. If $|\mathrm{spec}(K)|<\infty$, does it follow that $0$ is an eigenvalue of $K$? Provide a justification for your answer.

Answer: Yes


Question: Consider a sequence of random variables $(Y_n)$ where each $Y_n$ takes the same value as a fixed random variable $Y_1$. If $\lim_{n\rightarrow\infty}\mathbb{E}[Y_n]=\nu$, does this imply that $\lim_{n\rightarrow\infty}\mathbb{E}[|Y_n-\nu|]=0$? Provide a justification for your answer.

Answer: No


Question: Compute the line integral \( \int_C \textbf{F} \cdot ds \) for the vector field \( \textbf{F}(x,y) = \frac{1}{x+y}\textbf{i}+\frac{1}{x+y}\textbf{j} \), where \( C \) is the segment of the unit circle from \((1,0)\) to \((0,1)\).

Answer: 0


Question: Evaluate the limit \( \lim_{n\to \infty} n(\sqrt[n^2]{nI_n}-1) \), where \( I_n=\int_0^{\pi/4}e^{nx}(\tan^{n-1}x+\tan^nx+\tan^{n+1}x)dx \).

Answer: \dfrac{\pi}{4}


Question: Determine the radius of convergence of the power series \( \sum_{k=0}^{\infty} a_k x^k \), given that the series \( \sum_{k=0}^\infty (-1)^k a_k \) converges and \( \sum_{k=0}^\infty a_k \) diverges.

Answer: 1


Question: For which values of $a$ is the $3 \times 3$ matrix $A$ with characteristic polynomial $p(x) = x^3 - 3x + a$ diagonalizable over the reals? Consider that a matrix is diagonalizable if it has distinct real eigenvalues.

Answer: (-2, 2)


Question: Determine whether there exists a positive integer $s > 10$ such that the Riemann zeta function $\zeta(s)$, when divided by a rational constant $c = \frac{1}{d}$ with $d$ being another positive integer, yields a power of $\pi$. Specifically, is there an $s$ for which $\pi = (d\zeta(s))^{1/s}$?

Answer: No


Question: Evaluate the limit: $$\lim_{h \to 0} \, \left(\cos2h\right)^{\frac{1}{2h}}$$

Answer: 1


Question: Let $f: A \rightarrow B$ be a ring homomorphism between two commutative rings with identity. Given that the induced map $f' : \text{Spec}(B) \rightarrow \text{Spec}(A)$ is surjective, every prime ideal of $A$ is a contracted ideal. Determine if the converse holds: If every prime ideal of $A$ is contracted from $B$, does it imply that $f'$ is surjective?

Answer: Yes


Question: Evaluate the triple integral \(\iiint\frac{xz}{1+x^2+y^2}\,dz\,dy\,dx\), where the region of integration is defined by \(1 \leq x^2 + y^2 \leq 3\) and \(0 \leq z \leq 3\).

Answer: 0


Question: Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be a continuously differentiable map satisfying $\|f(x)-f(y)\|\geq\|x-y\|$ for all $x,y\in\mathbb{R}^{n}.$ Determine which of the following statements is true: \(A.\) $f$ is onto. \(B.\) $f(\mathbb{R}^{n})$ is a closed subset of $\mathbb{R}^{n}.$ \(C.\) $f(\mathbb{R}^{n})$ is an open subset of $\mathbb{R}^{n}.$ \(D.\) $f(0)=0.$

Answer: ABC


Question: Use the Fourier sine series expansion to prove that \( \sum_{k=1}^{\infty} \frac{(-1)^{k-1}}{2k-1} = \frac{\pi}{4} \). Consider the range \([0, \pi]\).

Answer: \dfrac{\pi}{4}


Question: Determine the number of elements in the symmetric group S6 that commute with the permutation (12)(34)(56).

Answer: 48


Question: Find the limit of the sequence \(x_n\) defined by \(x_1=1\) and \(x_{n+1}=2+\frac{3}{x_n}+\frac{1}{n}\).

Answer: 3


Question: How many $3 \times 3$ matrices with real entries satisfy the condition that their transpose is equal to their inverse?

Answer: \infty


Question: Find the range of values of $k$ such that the equation $x^3 - hx^2 + kx - 9 = 0$ has only one real root, which is $1$.

Answer: (3, 15)


Question: Calculate the limit: $$\lim_{n\to\infty}\frac{[x] + [2x]+ ... +[nx]}{n^2}$$ where \([x]\) denotes the floor function.

Answer: \dfrac{x}{2}


Question: Calculate the expected order of an element chosen uniformly at random from the Alternating group $A_5$. Provide your answer as a single numerical value.

Answer: \dfrac{211}{60}


Question:
Consider a map $f: \Bbb Z \rightarrow \Bbb Q$. Determine which of the following properties the map can have:

A. Bijective and increasing
B. Onto and decreasing
C. Bijective and satisfies $f(n) \ge 0$ if $n \le 0$
D. Has uncountable images

Select the correct option.

Answer: C


Question: Solve the partial differential equation \( \frac{\partial u}{\partial x}=4\frac{\partial u}{\partial y} \) with the initial condition \( u(0, y)=8e^{-3y} \).

Answer: 8e^{-12x - 3y}


Question: Evaluate the limit \( \lim_{n\to \infty} \int_{[0,\infty]} \frac{n\sin\left(\frac{x}{n}\right)}{x(1+x^2)}\,dm(x) \), where \( m \) denotes the Lebesgue measure on the real line \( \mathbb{R} \). Consider using the dominated convergence theorem to analyze the integrand's behavior.

Answer: \dfrac{\pi}{2}


Question: Determine whether it is possible to define an inner product on $\ell^1(\mathbb{N})$ that induces a norm not equivalent to $\|\cdot\|_1$. Provide a justification for your answer.

Answer: Yes


Question:
Determine the number of distinct square root matrices $B$ for the matrix $A=\begin{pmatrix} 5&0\\0&9\end{pmatrix}$, where $B=\begin{pmatrix} a&b\\c&d\end{pmatrix}$ satisfies $BB=A$. Use the following equations derived from $BB=A$ to find your answer:
1. $a^2 + bc = 5$
2. $ab + bd = 0$
3. $ca + cd = 0$
4. $cb + d^2 = 9$

What can be deduced about $b$ and $c$ from equations 2 and 3?

Answer: 4


Question: Compute the expected value $E(N_1N_2)$ for a Poisson process $N_t$ with rate $\lambda = 2$.

Answer: 10


Question: Determine the degree of the field extension \(\mathbb{Q}(2^{1/2},2^{1/4})\supset\mathbb{Q}\) and find a basis for this extension.

Answer: 4


Question: Evaluate the limit of \( \prod\limits_{k=3}^n\frac{k^3+8}{k^3-8} \) as \( n \to \infty \).

Answer: \dfrac{7}{2}


Question:
Find the value of $x$ that satisfies the equation:
\[ \arccsc(\sqrt{37}) + \arcsin\left(\frac{x}{\sqrt{4+x^2}}\right) = \frac{1}{2} \arcsin\left(\frac{3}{5}\right) \]

Note: Convert all inverse trigonometric functions to $\arctan$ to solve this problem.

Answer: \dfrac{6}{19}


Question:
Suppose there exists a continuous function $f: (0, \infty) \rightarrow \mathbb{R}$ such that for all $x, y > 0$, the following property holds:
$$\int_{1}^{xy}f(t)dt = y\int_{1}^{x}f(t)dt + x\int_{1}^{y}f(t)dt$$
If $f(1) = 3$, find the expression for $f(x)$ for any $x > 0$. \( \boxed{} \)

Answer: 3 \ln x + 3


Question: Evaluate the integral \( \int_0^{+\infty} \frac{x^2}{(x^2+1)^2} \, dx \) using the method of contour integration.

Answer: \dfrac{\pi}{4}


Question: Let $f$ be a real and continuous function such that $f(x+1) = f(x)$ for all $x \in \mathbb{R}$. Define $g(t) = \int_0^t f(x) \, dx$ for $t \in \mathbb{R}$. Determine for which values of $t$ the function $h(t) = \lim_{n \to \infty} \frac{g(t+n)}{n}$ is defined. Choose the correct option: (a) $t=0$ only (b) integer $t$ only (c) $\forall t \in \mathbb{R}$ and $h(t)$ is independent of $t$ (d) none of the above is true.

Answer: C


Question: Evaluate the improper integral \(I=\int_{-\infty}^\infty\frac{x\sin(\pi x)}{x^2+2x+5}dx\) using complex analysis techniques.

Answer: -\pi e^{-2\pi}


Question: Evaluate the limit \(\lim_{x \to 0} \frac {\cos(xe^x)-\cos(xe^{-x})}{\arcsin^3x}\) using asymptotic simplifications and Taylor series expansions, without using L'Hopital's Rule.

Answer: -2


Question:
Let $A$ be an $n \times n$ matrix with real entries such that $A^2 = -I$. Determine which of the following statements is true:

(a) $n$ is even.
(b) $A = \pm I$.
(c) All eigenvalues of $A$ are real.
(d) $A$ is a diagonal matrix.

Answer: a


Question: Evaluate the integral \( \dfrac{1}{2-\pi} \int_{|z|=2} \dfrac{dz}{z \sin z} \).

Answer: 0


Question: Calculate the volume enclosed by the planes $x=0$, $z=0$, and the surfaces $x=-2z^2+5$, $y=x^2z$. Provide your answer as a single numerical value.

Answer: \dfrac{625}{48}


Question: Given the parabola $y^2 = 3x$, two perpendicular straight lines are drawn from the origin, intersecting the parabola at points P and Q. Determine the equation of the locus of the centers of mass of all triangles OPQ, where O is the origin.

Answer: y^2 = x - 2


Question: Given that \((x^2-5x+4)(y^2+y+1)<2y\) holds for all real numbers \(y\), determine the interval \((2, b)\) for \(x\). What is the value of \(b\)?

Answer: 3


Question: In the context of an algebraically closed field of characteristic 0 and finitely dimensional semisimple Lie algebras, is the decomposition of a representation into a direct sum of irreducible subrepresentations unique up to ordering?

Answer: Yes


Question: Consider the free Lie algebra $L$ generated by the countably infinite set $X = \{x_i\}_{i \in \mathbb{N}}$ over a field $k$. A Lie monomial is a bracketed expression consisting of elements from $X$. Define a filtration of $L$ by ideals $L^{(n)}$ using weights $w(x_i) = s$ for $i \in (s-1)p + 1, \ldots, sp$, where $p$ is a fixed prime. The weight of a Lie monomial $[x_{i_1}, \ldots, x_{i_m}]$ is the sum of the weights of its generators. Determine if this filtration can be well-defined such that $L^{(n)}$ is spanned by Lie monomials with weight $\geqslant n$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: If a space $X$ is open in its Stone-Cech compactification $\beta X$, then any cover of $X$ by $\beta X$-open sets also covers $\beta X$. Provide a justification for your answer.

Answer: B


Question: Evaluate the limit of the function as $j$ approaches infinity: $$\lim_{j\rightarrow \infty}\sin^2(2^{j-1}\pi f)\prod_{i=0}^{j-2} \cos^2(2^i\pi f)$$ where $f$ represents frequency.

Answer: 0


Question: Evaluate the line integral \( \int_{\gamma} -y^3 \, dx + x^3 \, dy \), where \( \gamma \) is the positively oriented boundary of the unit circle.

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


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:
Let $\pi_1$ and $\pi_2$ be two distinct irreducible representations of a finite group $G$, both having the same degree. Determine whether the following sum equals the zero matrix:
\[\sum_{g \in G}  \pi_1(g)   \pi_2(g^{-1})^\top = 0.\]

Answer: 0


Question:
Find the value of $n$ for which the following integral is equal to $\frac{1}{\sqrt{n}}$:

$$\int_{-\infty}^{\infty}(-1)^xe^{\frac{-\pi x}{2}(nx+1)}dx=\frac{1}{\sqrt{n}}$$

Assume that $n > 0$.

Answer: \dfrac{3\pi}{4\ln 2}


Question: Find the function $g(x)$ for each $x > 0$ if $g$ is continuous on the positive real axis and satisfies the equation $$\int_1^{xy} g(t) \, dt = y \int_1^x g(t) \, dt + x \int_1^y g(t) \, dt$$ for all $x > 0$ and $y > 0$, given that $g(1) = 5$. Provide your answer as an explicit expression for $g(x)$. \( \boxed{\text{your answer here}} \)

Answer: 5(\ln x + 1)


Question: Determine whether there exist continuously differentiable functions $f$ and $g$ with $f(y) = g(y) = 0$ for some $y \in \mathbb{R}^n$ (where $n \neq 1$) and $f'(y) = a g'(y) \neq 0$ for some $a \in \mathbb{R}$, such that $\lim_{x \to y} \frac{f(x)}{g(x)}$ does not exist.

Answer: Yes


Question: Determine the set of points where the function \( f(x,y) = \begin{cases} x^{4/3} \sin(y/x) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \) is differentiable.

Answer: \mathbb{R}^2


Question: Evaluate the definite integral: \[ \int _{-1}^{3} \left[\arctan \left(\dfrac{x}{x^2+1}\right) + \arctan \left(\dfrac{x^2 + 1}{x}\right)\right]dx. \]

Answer: \pi


Question: Find all real solutions to the equation \((x+a^3)^3 = a-x\).

Answer: a - a^3


Question: Find the remainder when $3^{2023}$ is divided by 2023.

Answer: 1116


Question: Is it possible to find a countable family of open sets in $\mathbb{R}$, denoted by $G$, such that each set in $G$ contains $\mathbb{Z}$, but there exists an open set $B \subset \mathbb{R}$ with $\mathbb{Z} \subset B$ for which $G_n \not\subset B$ for any $G_n \in G$? Answer 'yes' or 'no'.

Answer: yes


Question: Determine the values of $k$ for which the substitution $u = \frac{y}{x}$ makes the differential equation $\frac{dy}{dx} = f(x,y)$ separable, given that $f(x,y)$ is homogeneous of degree $k$.

Answer: 0


Question: Calculate the volume of the region bounded between the surfaces given by the equations $x^2 + y^2 + z = 1$ and $z = x^2 + (y-1)^2$.

Answer: \dfrac{\pi}{16}


Question: Does every facet of a cyclic $d$-polytope with $N$ facets share a $(d-2)$-dimensional face with at least one other facet? Answer yes or no.

Answer: Yes


Question: Prove that in a field \( F \), the equation \( \frac{x}{y} = \frac{w}{z} \) holds if and only if \( xz = yw \).

Answer: xz = yw


Question: Evaluate the improper integral \( \int_{0}^{\infty} e^{-x}\sin(x) \, dx \) using integration by parts.

Answer: \dfrac{1}{2}


Question: Determine whether a bounded function defined on a compact interval of \(\mathbb{R}\) can have an infinite Riemann integral. Provide a justification for your answer.

Answer: No


Question: Determine whether the limit \( \lim_{m \to \infty, n \to \infty} \frac{1}{m} \text{Trace}[W_m(n)\Omega_m(n)] \) is always equal to 1, given that \( \Omega_m(n) \rightarrow \Omega_m \) and \( W_m(n) \rightarrow \Omega_m^{-1} \) as \( n \rightarrow \infty \), where \( \Omega_m^{-1} \) exists for any natural number \( m \).

Answer: 1


Question:
Determine the minimum number of states in a DFA that accepts the language \(L_n \subseteq \{0,1\}^*\) defined by:

\[L_n=\{s\in \{0,1\}^*\, |\,\, \text{for every substring } t \text{ of } s, \text{abs}(N_0(t)-N_1(t)) \leq n \}\]

where \(\text{abs}\) denotes the absolute value and \(N_0(t),N_1(t)\) denote the number of \(0\)'s and \(1\)'s in \(t\) respectively.

Answer: \frac{(n+1)(n+2)}{2} + 1


Question: Determine the number of ordered triplets of positive integers \((x, y, z)\) such that \(z! = x! + y!\) and \(w! = x! + y! + z!\).

Answer: 0


Question: Calculate the value of \( \sum\limits_{n=0}^{\infty} \sum\limits_{k=0}^{n} \binom{n}{k} \left(\frac{1}{2}\right)^k \).

Answer: \infty


Question: Given the matrix \(A = \begin{pmatrix} 0 & a & b & c \\ 0 & 0 & d & e \\ 0 & 0 & 1 & f \\ 0 & 0 & 0 & 2 \end{pmatrix}\), find the values of \(a, b, c, d, e,\) and \(f\) such that \(A\) is diagonalizable. The characteristic polynomial of \(A\) is \(\lambda^2(\lambda - 1)(\lambda - 2)\), implying \(\lambda \in \{0, 1, 2\}\). Determine the conditions on \(a\) and the relationships between \(b, c, d, e,\) and \(f\) that ensure \(A\) is diagonalizable.

Answer: a = 0


Question: Evaluate the limit \( \lim_{n \to \infty} n^2 \int_{n}^{5n}\frac{x^3}{1+x^6}dx \).

Answer: \dfrac{12}{25}


Question: Determine whether the value of the integral \( \int\limits_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)} \), where \( a \) is a non-zero real number, remains constant regardless of \( a \).

Answer: \dfrac{\pi}{4}


Question:
Evaluate the sum:

$$G=\left(\sum_{k=0}^{8} 4^{2 k}({ }^{16} C_{2 k})(-1)^k\right)^{2}+\left(\sum_{k=0}^{8} 4^{2 k+1}\left({ }^{16} C_{2 k+1}\right)(-1)^k\right)^{2}$$

Answer: 17^{16}


Question: Evaluate the limit: \[ \lim_{(x, y, z) \to (0, 0, 0)} \frac{x^2 y^2 z^2}{(x^4 + y^4 + z^4)^{a/4}}. \]

Answer: 0


Question: Determine whether there exist any nonzero $n \times n$ matrices, where $n \geq 3$, that satisfy the condition: for any invertible matrix $X$ with determinant 1, it holds that $$ XJX^{-1} = XX^TJ. $$

Answer: No


Question: Calculate the volume of the solid formed by revolving the curve \( y = \frac{\sin x}{x} \) around the y-axis from \( x = \frac{\pi}{3} \) to \( x = \pi \).

Answer: 3\pi


Question: Determine whether the Hardy-Littlewood maximal function \( Mf(x) \) defined as \[ Mf(x) = \sup_{r>0}\frac{1}{m(B_r(x))}\int_{B_r(x)} f(y) \, dy \] is equal to \[ Mf(x) = \sup_{R>0}\sup_{r \in (0,R)}\frac{1}{m(B_r(x))}\int_{B_r(x)} f(y) \, dy \]? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{\infty} e^{-x^2} x^3 \, \mathrm{d}x \).

Answer: \dfrac{1}{2}


Question: Evaluate the limit: $$ \lim_{x\to+\infty}\left(\frac{2}{\pi}\text{arccot}\;x\right)^{x} $$

Answer: 0


Question: Evaluate the integral: $$\int_{0}^{\pi} \cos(x) \log(\sin^2(x) + 1) \, dx$$

Answer: 0


Question: Determine whether there exists a module over the group ring $\mathbb{Q}G$ that is projective but not free, where $G$ is a nontrivial finite group.

Answer: Yes


Question: Calculate the limit \( \lim _{n\to \infty }\:\left(\int _0^{\infty \:}\left(\cos\left(x\right)\right)^n e^{-2x}\:dx\right) \).

Answer: 0


Question: Determine whether there exists a nonconstant entire function that is bounded on the real axis. Provide a yes or no answer.

Answer: Yes


Question: Find the minimum value of the expression \(\dfrac{x}{2-x}+\dfrac{y}{2-y}+\dfrac{z}{2-z}\), given that \(x, y, z > 0\) and \(x + y + z = 1\).

Answer: \dfrac{3}{5}


Question:
Calculate the volume of the solid in \(\mathbb{R}^3\) defined by the inequality:
\[(2x + y + z)^2 + (x + 2y + z)^2 + (x + y + 2z)^2 \leq 1.\]
After simplification, the inequality becomes:
\[6(x^2 + y^2 + z^2) + 10(xy + yz + xz) \leq 1.\]
Find the volume of this solid.

Answer: \dfrac{\pi}{3}


Question: Find the value of $x$ such that for two independent standard normal random variables $X$ and $Y$, the probability $Pr(X > Y, X \geq x) = \frac{3}{8}$.

Answer: 0


Question: Consider an algebraically closed field $K$. If $G$ is an algebraic subgroup of $(K,+)^3$ with dimension 1, determine whether there always exists a line $L$ in $K^3$ that intersects $G$ only at the origin.

Answer: Yes


Question: Let $g: \mathbb{R} \to \mathbb{R}$ be an infinitely differentiable function with 12 distinct zeros. Determine the values of $n \ge 1$ for which the $n$-th derivative $g^{(n)}$ has at least 2 zeros.

Answer: 1 \leq n \leq 10


Question: Determine the number of distinct ways to color the faces of a regular icosahedron using two colors.

Answer: 17824


Question: Evaluate the limit: $$\lim_{x\to 0}\left(\frac{1-\cos \left(x\right)\sqrt{\cos \left(2x\right)}\cdot \sqrt[3]{\cos \left(3x\right)}}{x^2}\right)$$

Answer: 3


Question: If $N$ is a von Neumann algebra factor of type $III$, is $\Bbb M_n(N)$ also a von Neumann algebra factor of type $III$?

Answer: Yes


Question: If all the zeros of a monic polynomial \( f: \mathbb{C} \rightarrow \mathbb{C} \) are real, does this imply that all the zeros of its derivative \( f'(z) \) are also real?

Answer: Yes


Question: Evaluate the contour integral \( \oint_C |z|^2 \, dz \) for the square contour \( C \) with vertices at \((0,0), (1,0), (1,1),\) and \((0,1)\).

Answer: -1 + i


Question: Consider \(\mathbb{R}^3\) with the Cartesian coordinates \(x=(x_1,x_2,x_3)\). Let \(K\) be a compact subset of \(\mathbb{R}^3\) and consider the function \(f:\mathbb{R}^3\times K\to \mathbb{R}\) defined by: \[f(x,y)=\frac{x_1-y_1}{\|x-y\|}\] Determine whether \(|f|\) is bounded in its entire domain \(\mathbb{R}^3\times K\).

Answer: Yes


Question: Evaluate \( \frac{dy(0)}{dx} \) for the function \( Q(\sqrt{x}) \), where \( Q(x) = \frac{1}{\sqrt{2\pi}} \int_x^\infty e^{-\frac{u^2}{2}} du \) and \( \frac{dy}{dx} = - \frac{e^{-\frac{x}{4}}}{4\sqrt{\pi x}} \) for \( x \ge 0 \).

Answer: -\infty


Question: Find the integer value of \( x \) such that \( x^{11} \equiv 10 \pmod{41} \).

Answer: 10


Question: Calculate the dimension $h^0$ of the normal bundle $N_{C/\mathbb{P}^3}$ for the twisted cubic $C \subset \mathbb{P}^3$ defined by the ideal $I = (xz - y^2, yw - z^2, xw - yz)$. Use a method different from the standard approach.

Answer: 12


Question: Evaluate the integral \( \int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz \), where \( \varGamma \) is the contour \( |z|=4\pi/3 \).

Answer: -6\pi i


Question: Suppose $A$ and $B$ are matrices with dimensions $n \times K$ and $n \times M$ respectively, where $n > K$ and $\text{rank}(A) = K$. If $\text{rank}(A + B) = K - s$ for some $s > 0$, determine the minimum possible dimension of the column space of $B$. \(\text{Express your answer as an integer.}\)

Answer: s


Question: Calculate the sum \( \sum_{k \in S(m,n)} \varphi(k) \) where \( S(a,b) = \{ k \in \mathbb{Z} \mid a \equiv b \pmod{k} \} \) and \( \varphi \) is Euler's totient function.

Answer: |m - n|


Question: Determine whether the following statement is true or false: If a function $f(z)$ is holomorphic, then its conjugate, $\bar{f}(z)$, is equal to $f$ evaluated at the conjugate of $z$, i.e., $\bar{f}(z)=f(\bar{z})$. Provide your answer as either "True" or "False".

Answer: False


Question: Determine whether a simple algebraic extension $\mathbb{F}(a)$ of degree $n$ over a field $\mathbb{F}$ can be embedded in the matrix ring $M_n(\mathbb{F})$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$ \lim_{x \to 0}\left(-\frac{1}{3 !}+\frac{x^{2}}{5 !}-\frac{x^{4}}{7 !}+\frac{x^{6}}{9!}+\cdots\right) $$

Answer: -\dfrac{1}{6}


Question: For a bounded set \(\Omega \subset \mathbb{R}^d\), determine the maximum value of \(\epsilon\) such that there exists an infinite subset \(\mathcal{A} \subset \mathcal{B}(\Omega)\) with the property that \(|A \triangle B| \geq \epsilon\) for all distinct sets \(A, B \in \mathcal{A}\).

Answer: \dfrac{|\Omega|}{2}


Question: Suppose $A$ is an $n \times m$ matrix and $B = (A^T A)^{-1} A^T$. If a matrix $C$ satisfies the conditions $CAC = C$, $ACA = A$, and $(CA)^T = CA$, determine whether $C = B$. Justify your answer.

Answer: C \neq B


Question: Evaluate the infinite series: $$\sum_{n=0}^{\infty} \frac{(2n+1)!}{2^{3n} \; (n!)^2}$$

Answer: 2\sqrt{2}


Question: Determine whether the following statement is true or false: If the span of linearly independent vectors $\{u_1, u_2, \dots, u_n\}$ is contained in the span of another set of linearly independent vectors $\{v_1, v_2, \dots, v_n\}$ in a vector space $V$, then both spans are equal.

Answer: A


Question: Compute the covariance of two random variables, $X$ and $Y$, where $Y$ follows an exponential distribution with parameter $3$ ($Y \sim \text{Exp}(3)$). The variable $X$ is chosen uniformly at random from the interval $[Y, 3Y]$ given the value of $Y$. Use the covariance formula $\text{Cov}(X, Y) = E[XY] - E[X] \cdot E[Y]$ to find the covariance.

Answer: \dfrac{2}{9}


Question: Calculate the class number of a field using the Minkowski bound of 6 and the ideal norms: 1, 3, 4, and 6.

Answer: 2


Question: Find all continuous functions \( f \) defined over \( \mathbb{R}^{+*} \) such that \( f(x) = \sqrt{2 + f(x^2)}.\)

Answer: 2


Question: Evaluate the limit: $$\lim_{k \to \infty} \frac{\int_{0}^{1} x^{2k+1} e^x \, dx}{\int_{0}^{1} x^{2k} e^x \, dx}$$

Answer: 1


Question: Determine if there exists a non-constant entire function $f$ such that $|f|$ is subharmonic.

Answer: Yes


Question: Is there a function $f:[a,b]\rightarrow\mathbb{R}$, where $a < b$ are real numbers, that is Riemann integrable, has an antiderivative on $[a,b]$, but is not continuous?

Answer: Yes


Question: Determine the base of the logarithm in the inequality: $$\omega(n) < \frac{\log n}{\log \log n} + 1.4573 \frac{\log n}{(\log \log n)^{2}}.$$

Answer: e


Question: A particle is placed at point A in a given figure. It moves to each of its adjacent points with equal probability. Determine the probability that the particle will have visited all points after an infinite amount of time.

Answer: 1


Question: Find the limit of the expression as $n$ approaches infinity: $$\lim_{n \to \infty}\frac{((n+1)!)^{\frac{1}{n+1}}}{(n!)^{\frac{1}{n}}}.$$

Answer: 1


Question: Let $X$ be a random variable uniformly distributed on the interval $(-1,1)$, and let $Y = X^2$. Compute the covariance $\operatorname{Cov}(X,Y)$. \(\boxed{\text{Your answer here}}\)

Answer: 0


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

Answer: 0


Question: Evaluate the limit \( \lim\limits_{x\to 0}\frac{\int_0^x(x-t)t^2\ dt}{x\sin^3x} \) using Taylor series expansion.

Answer: \dfrac{1}{12}


Question: Evaluate the limit \( \lim_{(x,y)\to(0,0)} g \left(\frac{x^4 + y^4}{x^2 + y^2}\right) \) given that \( \lim_{z\to 0}\frac{g(z)}{z}=2. \)

Answer: 0


Question: Find the covering radius of the code $C = \{0000, 1111, 2222\}$ in the vector space $F_3^4$. The covering radius is the smallest non-negative integer $y$ such that the union of spheres of radius $y$ centered at each codeword in $C$ covers the entire space $F_3^4$. Determine this covering radius.

Answer: 2


Question: Solve the equation: \[ \sqrt{x-\sqrt{x-\sqrt{x-\sqrt{x-5}}}}=5 \]

Answer: 30


Question: Find the maximum order of any element in the alternating group $A_{12}$.

Answer: 35


Question: Find the possible values for the trace of a matrix with characteristic polynomial \((x-2)^2(x-3)^3\) and minimal polynomial \((x-2)(x-3)^2\).

Answer: 13


Question: Evaluate the limit of the sequence \(\{a_n\}\) as \(n\) approaches infinity, where \(a_n = \frac{n\cos(n\pi+\frac{\pi}{3})+n(-1)^n}{n^2+1}\).

Answer: 0


Question: Determine the natural numbers \( n \) for which the fraction \( \frac{2n^{7}+1}{3n^{3}+2} \) is reducible.

Answer: n \equiv 435 \pmod{1163}


Question: Evaluate the limit: $$\lim_{n\to\infty} \int_0^\infty \left(1+ \frac{x}{n}\right)^{-n} \sin\left(\frac{x}{n}\right) dx$$

Answer: 0


Question: What is the probability that an event with probability "p" occurs exactly twice in an infinite sequence of independent trials?

Answer: 0


Question: Calculate the surface integral of the vector field \( F = (-y, x, z) \) over the region defined by \( x^2 + y^2 = 9 \) for \( x \geq 0, y \geq 0, \) and \( z \geq 0 \) using the parameterization \( \phi(r,\theta) = (r\cos\theta, r\sin\theta, 0) \). Verify if the result is \( 0 \).

Answer: 0


Question: Evaluate the integral \( \int_{0}^{\pi}\dfrac{1}{1+e^{-\cos x}}dx \) using the properties of definite integrals.

Answer: \dfrac{\pi}{2}


Question:
Consider the set $E = \left\{\frac{1}{n} \mid n \in \mathbb{N}\right\}$. For each $m \in \mathbb{N}$, define the function $f_m : E \to \mathbb{R}$ by:

\[
f_m(x) =
\begin{cases}
\cos(m x) & \text{if } x \geq \frac{1}{m} \\
0 & \text{if } \frac{1}{m+10} < x < \frac{1}{m} \\
x & \text{if } x \leq \frac{1}{m+10}
\end{cases}
\]

Determine which of the following statements is true:

1. No subsequence of $(f_m)_{m \geq 1}$ converges at every point of $E$.
2. Every subsequence of $(f_m)_{m \geq 1}$ converges at every point of $E$.
3. There exist infinitely many subsequences of $(f_m)_{m \geq 1}$ which converge at every point of $E$.
4. There exists a subsequence of $(f_m)_{m \geq 1}$ which converges to $0$ at every point of $E$.

Answer: 3


Question: Consider two sequences defined using Knuth's up-arrow notation. Let $G_1 = 3\uparrow\uparrow\uparrow\uparrow3$, and define $G_n = \underbrace{3 \uparrow \ldots \uparrow 3}_{G_{n-1} \ \text{times}}$ for $n \geq 2$. Now, let $H_1 = 4\uparrow\uparrow\uparrow\uparrow4$, and define $H_n = \underbrace{4 \uparrow \ldots \uparrow 4}_{H_{n-1} \ \text{times}}$ for $n \geq 2$. Determine the smallest $n$ such that $H_n > G_n$. What is this value of $n$?

Answer: 1


Question: Determine whether the language $L = \{0^{2^n} \mid n \geq 0\}$ can be decided in polynomial time.

Answer: Yes


Question: Determine whether there exists a Galois extension $K/\mathbb{Q}(i)$ such that $\mathrm{Gal}(K/\mathbb{Q}(i)) \cong D_8$, where $D_8$ is the dihedral group of order 8.

Answer: Yes


Question: Let \(f_n:[0,1]\to\mathbb{R}\) be infinitely differentiable functions such that \(\lim_{n\to\infty}\int_0^1\!f_n^2\,dx=0\), and for all \(\varepsilon>0\) there exists \(N\in\mathbb{N}\) such that for all \(n,m\geq N\), \(\int_0^1\!(f'_n-f'_m)^2\,dx<\varepsilon\). Determine whether it must be true that \(\lim_{n\to\infty}\int_0^1\!{f'_n}^2\,dx=0\).

Answer: Yes


Question: Is there a time $t$ at which the sum of the series of cosine functions $A_1\cos(2 \pi f_1 t) + A_2\cos(2\pi f_2t) + \cdots + A_n\cos(2 \pi f_n t)$ equals zero, given that the frequencies $f_i$ may be multiples of a common fundamental frequency or unrelated?

Answer: Yes


Question: Find the limit of the sequence \(\left(\left( 1 + \frac{1}{n} \right) \left( 1 + \frac{2}{n} \right) \cdots \left( 1 + \frac{n}{n} \right) \right)^{\frac{1}{n}}\).

Answer: \dfrac{4}{e}


Question: Evaluate the infinite product \( \prod_{r=1}^{\infty} \left(1-\frac{1}{\sqrt{r+1}}\right) \).

Answer: 0


Question: Evaluate the double integral in polar coordinates: $$\int_{0}^{2\pi} \int_{0}^{\infty}\frac{\sin(r\cos \theta)\sin(r \sin(\theta))}{2\pi}e^{-\left( \frac{r^{2}}{2}\right)}rdr d \theta$$

Answer: 0


Question: Calculate the expected value of the continuous random variable $X_n$ with the probability density function $f_n(x) = \frac{x^n}{n!} e^{-x}$ for $x > 0$.

Answer: n + 1


Question: Consider the set of digits \( \{1, 2, 3, \ldots, 8\} \). How many permutations of these digits have the property that the product of any five consecutive digits is divisible by 5?

Answer: 10080


Question: Evaluate the limit: $$\lim_{x \to 0^+}\frac{1}{x}\int_{0}^{5x}\sin^2\left(\frac{1}{t}\right)\,dt$$

Answer: \dfrac{5}{2}


Question: Determine whether the terms of a Laurent series expansion of an analytic function $f(z)$ can be differentiated term by term within an annulus $r < |z| < R$ to obtain the Laurent series for $f'(z)$. Provide a justification for your answer.

Answer: Yes


Question: For a non-constant holomorphic function $f$ in the annulus $A(a,b)=\{z:a<|z|< b\}$, equality holds in Hadamard's three-circle theorem if and only if $f$ is of the form $f(z) = cz^k$ for some constant $c$ and integer $k$. Determine the form of such functions $f$.

Answer: f(z) = cz^k


Question: Given the equation \( (3+4)(3^2+4^2)(3^4+4^4)(3^8+4^8)(3^{16}+4^{16})(3^{32}+4^{32}) = 4^x - 3^x \), find the value of \( x \).

Answer: 64


Question: Determine if 25 is an mth power residue modulo 43 using Euler's criterion, where p is a prime and m is an integer.

Answer: Yes


Question:
Determine the pointwise limit \( f(x) = \lim_{n \to \infty} f_n(x) \) for the function sequence \( f_n : [0, 2] \to \mathbb{R} \) defined by
\[f_n(x) = \begin{cases} n^3 x^2, & 0 < x < \frac{1}{n}; \\ n^3\left(x - \frac{2}{n}\right)^2, & \frac{1}{n} \le x < \frac{2}{n}; \\ 0, & \text{otherwise.}\end{cases}\]

Answer: 0


Question: Find the limit of the sequence \( y_n=\sum_{k=1}^n\frac{k \cos^2k}{n^2+k \sin^2k} \) as \( n \to \infty \).

Answer: \dfrac{1}{4}


Question: Evaluate the integral \( \int_0^{\frac{\pi}{2}} \frac{\sin^{1395}x}{\sin^{1395}x + \cos^{1395}x} \, dx \).

Answer: \dfrac{\pi}{4}


Question: Evaluate the limit \(\lim_{(x,y) \rightarrow(0,0)} \frac{1-\cos(xy)}{\sqrt{x^2+y^2}(x^2+y^2)}\).

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to 0} \frac{ e^x - e^{-x}-2\ln(1+x)}{x \sin x}$$ using a method different from L'Hospital's Rule.

Answer: 1


Question: Let $G$ be a finite group, and let $d(G)$ denote the minimal size of the generating set of $G$. If $Q$ is a quotient group of $G$, is it true that $d(Q) \le d(G)$? Answer true or false.

Answer: A


Question: Determine the number of unique deal combinations in the Klondike solitaire card game, taking into account both the order of the cards and the game's rules.

Answer: 52!


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

Answer: -\dfrac{1}{2}


Question: Determine whether the expression \( c = \sqrt{\frac{p^2 + 2a^2q^2 - a^2}{a^2q^2}} \) can represent a rational number, given that \( p \) and \( q \) are integers, and \( a \) and \( c \) are rational numbers.

Answer: Yes


Question: Determine the number of four-digit numbers with distinct digits that are divisible by 99.

Answer: 72


Question: Given positive numbers $a<1$ and $	ext{finite } C>0$, consider a real nonnegative, bounded, and nonincreasing function $f(i)$ that satisfies $f(i) \leq Ca^i$ for all positive integers $i$. Determine whether there exist finite positive numbers $C_0$ and $b$ such that $f(i) \leq C_0i^{-b}$ for any positive integer $i$. Provide a justification for your answer.

Answer: Yes


Question: Find the limit of \( \frac{\log(n!)}{n\log(n)} \) as \( n \to \infty \).

Answer: 1


Question: Evaluate the limit \( \lim_{n \to \infty} n \int_{0}^{1} x^n p(x) \, dx \), where \( p(x)=a_0+a_1x+a_2x^2+\cdots+a_kx^k \) is a polynomial of degree \( k \).

Answer: p(1)


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

Answer: \dfrac{1}{2}


Question: Calculate the correlation between a standard normal random variable $X \sim \mathcal{N}(0, 1)$ and $X^2$. Provide your answer as a numerical value.

Answer: 0


Question: Find the limit of the integral \( \int_0^1 \frac{ne^x\cos x}{1 + n^2x^2} \, dx \) as \( n \rightarrow \infty \).

Answer: \dfrac{\pi}{2}


Question: Calculate the derivative of the cross-entropy loss function \( J = -\sum_i p_i \log{q_i} \) with respect to \( z_j \), where \( q_j \) is defined using the softmax function.

Answer: q_j - p_j


Question:
Consider the exponential distribution with probability density function (pdf) \( f(x;\lambda) = \lambda \exp(-\lambda x) \). Let \( Y \) be a statistic defined as:

\[
Y(X)= \left\{ \begin{array}{lcc} 0 & X < 1 \\\\\ 1 &\mbox{ otherwise} \\\\\ \end{array} \right.
\]

Is \( Y(X) \) a sufficient statistic for \( \lambda \)?

Answer: No


Question: Given a sequence of real numbers, if the set of all subsequential limits is bounded, does it necessarily imply that the original sequence is also bounded? Answer with 'Yes' or 'No' and justify your answer.

Answer: No


Question: Determine whether the set of all true statements provable by induction has the same cardinality as the set of natural numbers.

Answer: Yes


Question: Determine the interval for the parameter $m$ such that the expression \( \frac{mx^2+3x-4}{-4x^2+3x+m} \) can take all real values for real numbers \( x \).

Answer: [1, 7]


Question: Consider a compact Hausdorff space $X$ with cardinality $2^\kappa$, where $\kappa$ is an infinite cardinal. Let $U$ be a dense open subset of $X$. Determine the smallest possible cardinality of $U$. Can it be $2^\kappa$ or $\kappa$?

Answer: 2^\kappa


Question: Determine the largest possible rank of a non-zero $5 \times 5$ complex matrix $B$ such that $B^2 = 0$. \( \text{Provide your answer as a single integer.} \)

Answer: 2


Question: Determine if the ideal \((x_1-a_1,x_2-a_2,\dots,x_n-a_n)\) is maximal in the polynomial ring \(K[x_1,x_2,\dots,x_n]\), where \(K\) is a field and \(a_1,a_2,\dots,a_n\) are elements of \(K\).

Answer: Yes


Question: Given a matrix $B$ with a known trace, determine the minimum number of distinct pairs $(\vec y_i, \vec z_j)$ required such that the linear measurements $B\vec y_i \cdot \vec z_j$ uniquely determine $B$. Assume $\vec y_i$ and $\vec z_j$ are known vectors.

Answer: n^2 - 1


Question:
Let $S=\{(a,b,c):a,b,c\in\mathbb{R}\}$ such that $f(z)=x+ay+i(bx+cy)$ is analytic. Determine the isomorphic structure of the linear span of $S$ from the following options:

(a) $\mathbb{C}(\mathbb{R})$
(b) $\mathbb{R}^2(\mathbb{R})$
(c) $\mathbb{R}(\mathbb{R})$
(d) $\mathbb{R}^3(\mathbb{R})$

Using the Cauchy-Riemann equations, we find $a+b=0$ and $c=1$. Therefore, $S=\{(a,-a,1):a\in\mathbb{R}\}$. What is $L(S)$ isomorphic to?

Answer: B


Question: Suppose \(\{f_n\}\) is a uniformly bounded sequence of holomorphic functions in a plane open set \(\Omega\), and \(\{f_n(z)\}\) converges for every \(z \in \Omega\). Does it necessarily follow that the convergence is uniform on every compact subset of \(\Omega\)? Answer "Yes" or "No".

Answer: Yes


Question: Determine the number of solutions of the polynomial equation \(z^4 - 7z^3 - 2z^2 + z - 3 = 0\) that lie inside the unit disc.

Answer: 3


Question: Let $X \sim N(0,1)$ and $Y \sim N(X, 1)$, where $Y-X$ is independent of $X$. Compute the probability $P(Y<0 \mid X>0)$. This is related to pricing a Slalom option in a Normal model without drift.

Answer: \dfrac{1}{4}


Question: Suppose a group $G$ acts simplicially on a $\Delta$-complex $X$, where each element of $G$ maps each simplex of $X$ onto another simplex by a linear homeomorphism. If the action is free, is it necessarily a covering space action? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim\limits_{n\rightarrow \infty} \frac{1}{n^3}\ln \left( \frac{(n!)^n}{(1!2!...n!)^3} \right)$$

Answer: 0


Question: In a right-angled triangle $\Delta ABC$ with $\angle C = 90^\circ$, point $E$ is on side $BC$ such that $AC = BE$. A point $D$ is on side $AB$ with $DE$ perpendicular to $BC$. Given that $DE + BC = 1$ and $BD = \frac{1}{2}$, find the measure of $\angle B$. \( \text{(Express your answer in degrees.)} \)

Answer: 30


Question: Find the number of non-isomorphic graphs with 25 vertices and 303 edges by considering their complements.

Answer: 0


Question: In triangle ABC, vertex C forms a 30-degree angle, and the triangle is inscribed in a circle with center O and a radius of 9 cm. Determine the radius R of a circle that is tangent to segments AO and BO, as well as the arc AB.

Answer: 3


Question: Determine whether there exist two subgroups of SYM($\aleph_0$) with infinite support that are not isomorphic but have the same order.

Answer: Yes


Question:
Let \( f(x) \) be defined as follows:
\[
f(x) = \begin{cases} e^{-1/x^2}, & x \neq 0 \\ 0, & x = 0 \end{cases}
\]
Calculate \( f'''(0) \).

Answer: 0


Question: Find the smallest positive integer $k$ such that the limit \( \lim_{n \to \infty} \frac{\sin (n^2)}{n^k} \) exists.

Answer: 1


Question:
Find the six-digit number $ABCDEF$ such that:
1. All digits $A, B, C, D, E, F$ are distinct.
2. $A + B = E + F$.
3. $A + B + C + D + E + F = 10E + F$.
4. $10A + B$, $10C + D$, and $10E + F$ are prime numbers.
5. $E + F < 10$.

Determine the number $ABCDEF$.

Answer: 416723


Question: Determine the denominator of the coefficient of the integral of the polynomial \((x-\alpha)^5(x-\beta)^4\) with respect to \(x\), given that \(\beta > \alpha\).

Answer: 2520


Question: If \( \varphi:R^{m}\to R^{n} \) is an epimorphism of free modules over a nontrivial commutative ring, prove or disprove that \( m \geq n \).

Answer: m \geq n


Question: If $E[X^2]$ exists, does it imply that $E[X]$ also exists? Answer yes or no and justify your answer.

Answer: Yes


Question: Determine whether the sequence \( \{b_n\} = \frac{n!}{2^{n^{2}}} \) converges or diverges as \( n \to \infty \).

Answer: 0


Question: Find the radius of convergence of the series: $$ \sum_{n=0}^\infty \frac{x^{n^2}}{2^n} $$

Answer: 1


Question: Let $U_{\theta} \sim U[0,\theta]$ be independent for each $\theta \in [0,1]$. Define $X = \sup_{\theta \in [0,1]} U_{\theta}$. Find $\mathbb{P}(X \leq x)$.

Answer: x e^{1 - x}


Question: Given that A and B are invertible square matrices of the same order, solve for the matrix X in the equation $BAX=(BA)^2$. Express your answer in terms of A and B.

Answer: BA


Question: Let $p$ be a prime number greater than 2, and let $e_1, e_2, \ldots, e_p$ be $p$-th roots of unity in $\mathbb{C}$, which are not necessarily distinct. If \( e_1 + e_2 + \cdots + e_p = 0 \), determine whether all $e_i$'s must be distinct.

Answer: Yes


Question: Find the exact value of the limit: $$\lim_{x \to 0}\frac{e^{x\cos{x^2}}-e^{x}}{x^5}$$

Answer: -\dfrac{1}{2}


Question: Evaluate the limit: $$\lim_{n \to \infty} \int_1^2 \frac{1}{1+x^{-n}}dx$$

Answer: 1


Question: Evaluate the integral \( \int_1^\infty \frac{dx}{x^2\sqrt{x^2-1}} \).

Answer: 1


Question: Find the maximum value of the function \( f(x) = \sum_{n=0}^\infty x^n(1-x)^n \) on the interval \((0, 1)\).

Answer: \dfrac{4}{3}


Question: Determine the accumulation points of the sequence \(a_n\) defined by \(a_1 = 0\) and \(a_{n+1} = \sqrt{2a_n + 3} + 2\).

Answer: 3 + 2\sqrt{2}


Question: Evaluate the integral: $$\int_{(0,1)\times \mathbb{R}} \frac{|x|}{2\pi(1+x^2y^2)} \, d(x,y)$$ given that \(\int \frac{1}{1+y^2} \, dy = \pi\). Find the value of the integral.

Answer: \dfrac{1}{2}


Question: Determine the property of the coefficients in the quadratic equation $x^2 + 2xy + y^2 = 1$ that causes the graph to transform from an ellipse into two parallel lines when the coefficient of the $xy$ term changes from 1 to 2. Explain how this transformation relates to the discriminant $D = b^2 - 4ac$.

Answer: D = 0


Question: Find the sharp lower bound of the expression $e^x + e^y + e^z$ given that $x, y, z$ are positive real numbers satisfying $x + y + z = 3k$, where $k$ is a positive integer.

Answer: 3e^k


Question: Find the function represented by the power series \( \sum_{k=0}^{\infty}\frac{2(-1)^k x^{k+2}}{k!} \).

Answer: 2x^2 e^{-x}


Question: Evaluate the limit: \[ L = \lim_{n \to \infty} \int_{0}^{\pi} x^{1/n} \sin x \, dx \]

Answer: 2


Question: Calculate the minimum number of check bits required to create a correctable code with a minimum distance of 4 for a list of 10 key words.

Answer: 4


Question: Determine whether the function \( f(z) = \frac{z^2}{(e^z-1)^2} \) has a bounded neighborhood around the point \( z = 0 \) without using L'Hopital's rule.

Answer: Yes


Question: Find the last digit of the power tower of consecutive integers from $22$ to $67$ in base-$10$ notation. In other words, determine the last digit of $22^{23^{24^{25^{26^{27^{28^{\cdot^{\cdot^{\cdot^{67}}}}}}}}}}$. Use number theory concepts if necessary.

Answer: 2


Question:
Determine the value of \(\phi\) that maximizes the function:

\[f(\phi) = \frac{\sum_{u=0}^t u\theta^u /u!(t-u)!}{\sum_{u=0}^t \theta^u /u!(t-u)!}\]

Answer: \infty


Question: Determine the Hausdorff dimension of a Cantor set where, at each step, a larger percentage than the previous middle-$\alpha$-th is removed from the remaining intervals. For example, starting with the interval $[0,1]$, after the first step, remove the middle-$(\frac{1}{3}+\varepsilon\cdot\frac{1}{3})$ from $\left[0,\frac{1}{3}\right]$. What is the Hausdorff dimension of this "skinny" Cantor set?

Answer: \dfrac{\log 2}{\log \left( \dfrac{6}{2 - \varepsilon} \right)}


Question: Evaluate the limit \( I = \lim_{x \to 1} \Gamma(1 - x) \cos\left(\frac{\pi}{2}x\right) \), where \( \Gamma(x) \) denotes the Gamma function.

Answer: \dfrac{\pi}{2}


Question: Evaluate the limit: \[ \lim_{{n \to \infty}} \left( \frac{10^n+5^n}{2^n+5^n} - \frac{8^n-4^n+\sin(4^n)}{(-1)^n+4^n} \right) \] without using L'Hôpital's rule.

Answer: 2


Question: Determine the least possible degree of a polynomial with real coefficients that has the roots $2\omega, 2+3\omega, 2+3\omega^2$, and $2-\omega-\omega^2$, where $\omega$ is a primitive cube root of unity.

Answer: 5


Question: Calculate the line integral of the vector field \( F(x,y) = \left(\frac{x}{x^2+y^2}, \frac{-y}{x^2+y^2}\right) \) along the closed curve formed by the union of the line segments from \((1,0)\) to \((0,1)\) and from \((0,1)\) to \((-1,0)\), and the upper half-circle centered at the origin from \((-1,0)\) to \((1,0)\).

Answer: 0


Question: Let \(H\) be any nontrivial subgroup of the alternating group \(\mathfrak{A}_n\), where \(n \geq 5\). Determine whether \(H\) contains an element \(\sigma\) whose order is a prime \(p\), such that \(\sigma\) can be expressed as \(\sigma = c_1c_2\ldots c_r\), where \(c_i\) are disjoint \(p\)-cycles and \(o(\sigma) = \text{lcm}(l_1, l_2, \ldots, l_r) = p\).

Answer: Yes


Question: Given that X and Y are independently distributed Poisson random variables with means 1 and 3 respectively, find the conditional probability P(X ≤ Y | X + Y = 1).

Answer: \dfrac{3}{4}


Question: Calculate the expected value of \( \Phi\left(\frac{W_{t+a}}{\sqrt{2+t}}\right) \), where \( W_t \) is a standard Brownian motion and \( \Phi \) is the cumulative distribution function of the standard normal distribution.

Answer: \dfrac{1}{2}


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

Answer: (-e, e)


Question: Given two sets of linearly independent basis vectors, $V = \{v_1, \ldots, v_n\}$ and $U = \{u_1, \ldots, u_n\}$, both spanning $\mathbb{R}^n$, determine if it is possible to construct a new set $T$ by combining a subset of $V$ and a subset of $U$ such that $T = \{v_1, \ldots, v_m\} \cup \{u_1, \ldots, u_o\}$ with $m + o = n$, and $T$ also spans $\mathbb{R}^n$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the volume of the solid region bounded by the surfaces $z=10-r^2$ and $z=2+r^2$ in cylindrical coordinates. Verify whether the volume is $\frac{64\pi}{3}$ or $16\pi$. Provide the correct volume.

Answer: 16\pi


Question: Calculate the value of \((319!)!\).

Answer: (319!)!


Question: Determine whether any set of distinct rational numbers can always be arranged to form an arithmetic progression. Answer with 'Yes' or 'No' and justify your response.

Answer: No


Question: If $f(x)$ is a polynomial satisfying $$f(x)f(y) = f(x) + f(y) + f(xy) - 2$$ for all real $x$ and $y$ and $f(3) = 10$, find the value of $f(4)$. \( \boxed{?} \)

Answer: 17