Question: Evaluate the triple integral \( \iiint_D \frac{dV}{\sqrt{x^{2}+y^{2}+z^{2}}} \) over the region \( D \), which is the sphere defined by the equation \( x^2+(y-1)^2+z^2=1 \). Use spherical coordinates to set up and compute the integral.

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


Question: Given two operators or non-zero matrices $A$ and $B$, where $A \neq B$, $\text{tr}(A) = 1$, $\text{tr}(B) = 1$, and $\text{tr}(A - B) = 0$, find a lower bound for the Schatten 1-norm $\|A - B\|_1$. Provide your answer as a numerical value or expression.

Answer: 0


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{n \log_2( \log_2 n)}{3^{\log_2 n^2}}. \]

Answer: 0


Question:
Determine the integral surface of the partial differential equation \(xu_x + yu_y = 0\) that satisfies the condition \(u(1,y) = y\). Choose the correct option:

(a) \(u(x,y) = \frac{y}{x}\)

(b) \(u(x,y) = \frac{2y}{x+1}\)

(c) \(u(x,y) = \frac{y}{2-x}\)

(d) \(u(x,y) = y + x - 1\)

Answer: a


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_1^n x^{-n} \, dx. \]

Answer: 0


Question: Evaluate the integral: \[ \int_0^{\pi/2} \frac {\sqrt{\sin x}}{\sqrt {\sin x} + \sqrt {\cos x}}\, \mathrm{d}x. \]

Answer: \dfrac{\pi}{4}


Question: Let $A$ be a $3 \times 3$ matrix given by $A = [a_{ij}]$ and $B$ be a column vector such that $B^T A B$ is a null matrix for every column vector $B$. If $C = A - A^T$ and $a_{13} = 1$, $a_{23} = -5$, $a_{21} = 15$, find the value of $\det(\operatorname{adj} A) + \det(\operatorname{adj} C)$. \(A\) is a null matrix, and \(C\) is skew-symmetric.

Answer: 0


Question: Determine the size of the group of linear functions of the form \( f(x) = ax + b \) under composition, where \( a \neq 0 \) and \( a, b \in \mathbb{F}_q \). Here, \( q = p^n \) for some prime \( p \).

Answer: q(q - 1)


Question: Determine all values of $m$ such that the inequality \( \log(60x^2+120x+10m-10) > 1+3\log(x+1) \) has exactly 4 integer solutions for $x$.

Answer: (-9, 2]


Question: Consider the set $X = \mathbb{R}^2 - \{(0,0),(1,0),(2,0),(3,0)\}$. Let $V$ be the vector space of irrotational vector fields over $X$, and let $W$ be the vector space of conservative vector fields over $X$. Determine the dimension of the quotient vector space $V/W$. Provide your answer as a single integer.

Answer: 4


Question: Let \(\{X_i\}\) be a sequence of i.i.d. random variables with mean 0 and variance \(\sigma^2 > 0\). Compute the limit: \[ \lim_{n \to \infty}E\left(\frac{|\overline{X}_n|}{|\overline{X}_n| + 1}\right)^2 \] where \(\overline{X}_n\) is the sample mean of the first \(n\) random variables.

Answer: 0


Question: Find the number of integer solutions to the equation \( x + y + z = 100 \) such that \( 3 \leq x \leq y \leq z \leq 60 \).

Answer: 430


Question: Suppose $u$ is a real-valued continuous function on $\overline{\mathbb{D}}$, harmonic on $\mathbb{D} \setminus \{0\}$, and $u = 0$ on $\partial \mathbb{D}$. Show that $u$ is constant in $\mathbb{D}$.

Answer: 0


Question: Find the infimum of the expression \( \inf\limits_{k \ge n} k^{2}\chi_{[0,\frac{1}{k}]} \), where \( \chi \) is the indicator function.

Answer: 0


Question:
Consider the equation \(|x^2 + bx + c| = k\). Determine which of the following conditions must be true for the equation to have four real roots:

(a) \(b^2 - 4c > 0\) and \(0 < k < \frac{4c - b^2}{4}\)

(b) \(b^2 - 4c < 0\) and \(0 < k < \frac{4c - b^2}{4}\)

(c) \(b^2 - 4c > 0\) and \(0 < k > \frac{4c - b^2}{4}\)

Answer: a


Question:
Let $f_1(x)$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$, and let $f_2(x)$ be a constant function. Suppose that for all $a \in \mathbb{R}$, the following limit exists:
\[ \lim_{h\to 0} \frac{ f_1(a+ h) - 2f_1(a) + f_1(a-h) }{h^2} = f_2(a). \]
Is it true that $f_1(x)$ must be a polynomial function of degree at most 2?

Answer: Yes


Question: Evaluate the line integral of the vector field \( \mathbf{E} = \mathbf{a_x}y + \mathbf{a_y}x \) from the point \( P_3(3,4,-1) \) to \( P_4(4,-3,-1) \) by converting both \( \mathbf{E} \) and the points \( P_3 \) and \( P_4 \) into cylindrical coordinates. Calculate the integral \( \int \mathbf{E} \cdot \text{d}l \).

Answer: -24


Question:
Consider the functional inequality \( \frac{f(x)}{f(y)} \le 2^{(x-y)^2} \) for all \(x, y\) in the domain of \(f\). Determine which of the following functions \(f\) satisfies this inequality:

(A) \(\sqrt{x}+x^3\)

(B) \(\int_0^{\sin^2 x} \sin^{-1}\sqrt{t}\,dt + \int_0^{\cos^2 x} \cos^{-1}\sqrt{t}\,dt\)

(C) \(\int_0^x 2t^3\,dt\)

(D) \(\frac{5}{2} x^2 + x^3 - 32\).

Answer: B


Question: Suppose a power series \(\sum a_n z^n\) with \(a_n,z \in \mathbb{C}\) converges for some single point \(z_0\) in the complex plane. Can we conclude that it converges on any disk with radius \(r\leq |z_0|\)? Justify your answer.

Answer: Yes


Question: Let $X$ be an infinite set. Is it possible to construct a sequence $X = X_0 \supset X_1 \supset X_2 \supset \cdots$ such that for each $i \geq 0$, $\operatorname{card}(X_i) = \operatorname{card}(X)$ and $\bigcap_{i \geq 0} X_i = \emptyset$?

Answer: Yes


Question: Compute the volume of the set \( A = \{ (x,y,z) \in \mathbb{R}^3 \colon (x,y) \in \Omega, 0 \le z \le f(x,y) \} \), where \( f(x,y) = 1 - \tfrac{x^2}{4} - y^2 \) and \( \Omega = \{ (x,y) \in \mathbb{R}^2 \colon f(x,y) \ge 0 \} \).

Answer: \pi


Question: Let \( \Gamma \) be a discrete subgroup of \( GL_n \), and let \( G \) be its Zariski closure. Suppose \( H \) is an algebraic cocompact normal subgroup of \( G \). Is \( H \cap \Gamma \) of finite index in \( \Gamma \)? Justify your answer.

Answer: Yes


Question: Find all positive integer solutions $(a, b, c)$ such that $a! + b! = c!$.

Answer: (1, 1, 2)


Question: Determine the number of non-isomorphic graphs with 30 vertices and 432 edges.

Answer: 5


Question: Determine the number of pairs \((p, q)\) where both quadratic equations \(x^2 - px + q = 0\) and \(x^2 - qx + p = 0\) have unequal integral roots, given that \(p\) and \(q\) are natural numbers.

Answer: 2


Question: Evaluate the line integral \( \int_{C}{x\,\mathrm{d}z - z\,\mathrm{d}y + y\,\mathrm{d}z} \) where the curve \( C \) is parameterized by \( \gamma(t) = (\cos t, \sin t, 2t) \) for \( t \in [0, 2\pi] \).

Answer: 0


Question: Find the constant term \(c\) in the large \(x\) expansion of the expression \(\ln \frac{1+e^{ax}}{1+e^{-bx}}\), where \(a > 0\) and \(b > 0\). The leading term is \(ax\).

Answer: 0


Question: Determine all points $z_0$ for which the complex derivative of the function $f(z) = z \operatorname{Re}(z)$ exists.

Answer: 0


Question: Determine whether an injective function from \(\mathbb{R}\) to \(\mathbb{R}\) can have countably many discontinuities. Provide a justification for your answer.

Answer: Yes


Question: Find the maximum value of $k$ for which the sum \( \sum_{i=0}^k \binom {10}{i} \binom{15}{k-i} +\sum_{i=0}^{k+1} \binom {12}{i} \binom {13}{k+1-i} \) exists.

Answer: 25


Question: Determine the number of integral solutions for the polynomial equation \( f(x) = x^4 - ax^3 - bx^2 - cx - d = 0 \), where \( a, b, c, d \in \mathbb{Z}^+ \) and \( a \geq b \geq c \geq d \).

Answer: 0


Question: In a round of the DES cryptosystem, the input is the 64-bit string $1010101010101010101010101010101010101010101010101010101010101010$, and the round key is the 48-bit string $111111111111111111111111111111111111111111111111$. Compute the 53rd bit of the output for this round.

Answer: 1


Question: Determine whether the sequence \( x_n = \sum_{k=1}^{n^2} \frac{1}{\sqrt{n^4+k}} \) is convergent. If it is convergent, find its limit.

Answer: 1


Question: Does the space $L^1(\mathbb{X})$, where $\mathbb{X}$ is a measure space, have Schur's property, meaning that weak convergence implies strong convergence?

Answer: No


Question:
Given the current ELO score, the opponent's ELO score, the result of the match (win or loss), and the constant K, determine the previous ELO score. Use the following equations based on the match result:

For a win:
\[ ELO_{Current} = ELO_{Previous} + K \cdot \left(1 - \frac{1}{1 + 10^{\frac{(ELO_{Opponent} - ELO_{Previous})}{400}}}\right) \]

For a loss:
\[ ELO_{Current} = ELO_{Previous} + K \cdot \left(0 - \frac{1}{1 + 10^{\frac{(ELO_{Opponent} - ELO_{Previous})}{400}}}\right) \]

Calculate the previous ELO score, \( ELO_{Previous} \).

Answer: ELO_{Previous}


Question: Determine the signature of the quadratic form \( Q(x_1, \ldots, x_n) = \sum_{i,j=1}^{n} a_i a_j x_i x_j \) in \( \mathbb{R}^n \), where \( a = (a_1, \ldots, a_n) \neq 0 \) is a vector in \( \mathbb{R}^n \).

Answer: (1, 0)


Question: Calculate the limit: \[ \lim_{x \to 0} \frac{f(x) - f(\ln(1+x))}{x^3} \] given that \( f \) is a differentiable function on \([-1,1]\), doubly differentiable at \(x=0\), with \( f'(0) = 0 \) and \( f''(0) = 4 \).

Answer: 2


Question: Determine whether a function $f$ with the property that for all $\phi \in L^2$, $f\phi \in L^1$, is necessarily in $L^2$. Provide a justification for your answer.

Answer: Yes


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

Answer: \infty


Question: Suppose $S^1$ acts freely on a manifold $M$. Determine the Euler characteristic of $M$.

Answer: 0


Question: Find the minimum value of the expression \(2\csc(2x) + \sec(x) + \csc(x)\) for \(x \in (0, \frac{\pi}{2})\).

Answer: 2 + 2\sqrt{2}


Question: Determine the radius of convergence of the series \( \sum_{n=1}^{\infty}x^{n!} \).

Answer: 1


Question: Compute the pullback of the 1-form \( \omega = -\frac{y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy \) on \( \mathbf{R}^2 \) under the map \( c_{R,n}: [0,1] \to \mathbf{R}^2 \) given by \( t \mapsto (R\cos 2\pi n t, R\sin 2\pi n t) \). Show that the pullback is \( 2\pi n \, dt \).

Answer: 2\pi n \, dt


Question: Determine the measure of the set of all 1-generic sets in Cantor space $2^\omega$. A set $A$ is 1-generic if it forces its jump, meaning for any $e \in \omega$, there exists $\sigma \preceq A$ such that $\Phi^{\sigma}_{e}(e) \downarrow \vee (\forall \tau \succeq \sigma)(\Phi^{\tau}_{e}(e) \uparrow)$. What is the measure of $G = \{f \in 2^{\omega} : f \text{ is 1-generic}\}$?

Answer: 1


Question:
Suppose $f$ is holomorphic on the strip $\{ z \in \mathbb{C}: |\text{Im} \, z| < \delta \}$. Consider the expansion $f(z) = \sum_{k=0}^\infty a_k z^k$. Determine if there exist positive constants $M$ and $b$ such that
$$
|a_k| \leq M b^k,
$$
where $M$ and $b$ depend on $f$. Provide a justification for your answer.

Answer: Yes


Question:
Consider an automorphism \( f \) of the octonions algebra. It is given that \( f(x) = x \) for \( x \in \mathbb{R} \) and that \( f \) restricted to \( \text{Im} \mathbb{O} \) is in \( SO(7) \). There exists an orthonormal basis \( e_1, e_2, \ldots, e_7 \) in \( \mathbb{O} \) and real numbers \( \phi_1, \phi_2, \phi_3 \) such that:
\[
f(e_1) = e_1, \\
f(e_2) = \cos \phi_1 e_2 - \sin \phi_1 e_3, \\
f(e_3) = \sin \phi_1 e_2 + \cos \phi_1 e_3, \\
f(e_4) = \cos \phi_2 e_4 - \sin \phi_2 e_5, \\
f(e_5) = \sin \phi_2 e_4 + \cos \phi_2 e_5, \\
f(e_6) = \cos \phi_3 e_6 - \sin \phi_3 e_7, \\
f(e_7) = \sin \phi_3 e_6 + \cos \phi_3 e_7.
\]
Does there exist a Cayley's triple \((i, j, l)\) in \( \text{Im} \mathbb{O} \) (i.e., unit elements of \( \text{Im} \mathbb{O} \) satisfying \( l \perp i, j, ij \)) such that:
\[
e_1 = i, \\
e_2 = j, \\
e_3 = k, \\
e_4 = l, \\
e_5 = il, \\
e_6 = jl, \\
e_7 = (ij)l?
\]

Answer: Yes


Question:
Determine the smallest integer \( n \) such that the error term in the Taylor series expansion of \( \int_{1}^{2} \frac{e^x}{x} \, dx \) is less than \( 5.0 \times 10^{-6} \). Use the Taylor series expansion:
\[
\frac{e^x}{x} = \frac{1}{x} + 1 + \frac{x}{2!} + \frac{x^2}{3!} + \cdots + \frac{x^{n-1}}{n!} + \frac{x^n e^{\xi}}{(n+1)!}
\]
Integrate term by term and find \( n \) such that:
\[
\int_{1}^{2} \frac{x^n e^{\xi}}{(n+1)!} \, dx < 5.0 \times 10^{-6}
\]

Answer: 12


Question: If $A$ and $B$ are non-singular square matrices of the same order and $(A+B)^{-1}=A^{-1}+B^{-1}$, which of the following statements is true for every $A$ and $B$?\n\nA) $AB^{-1}=BA^{-1}$\nB) $AB=BA$\nC) $AB^{-1}A=BA^{-1}B$\nD) $A=A^{-1}$ and $B=B^{-1}$

Answer: C


Question: Evaluate the definite integral: \[ \int_{0}^{2\pi} \frac{1}{5+4\cos(x)} \, dx \]

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


Question: Determine the intervals on which the series \( \sum_{n=1}^\infty ne^{-nx} \) converges uniformly.

Answer: [a, \infty)


Question: For which values of \(a\) and \(b\) are there three distinct real numbers \(x_1, x_2, x_3\) such that for the function \(f(x) = \frac{1}{ax+b}\), the conditions \(f(x_1) = x_2\), \(f(x_2) = x_3\), and \(f(x_3) = x_1\) are satisfied?

Answer: a = -b^2


Question:
Given a network $(G,
u, s, t)$ with capacities $\nu = \mu + \mu'$, where $f$ is an $s$-$t$ flow in $(G, \mu, s, t)$ and $f'$ is an $s$-$t$ flow in $(G, \mu', s, t)$, prove that there exists an $s$-$t$ flow in $(G, \nu, s, t)$ with a value of $|f| + |f'|$.

Answer: |f| + |f'|


Question:
Let \( A \) be a subset of \( \mathbb{R}^n \) and let \( x \in \mathbb{R}^n \). Determine under which condition there exists a point \( y_0 \in A \) such that the distance \( d(x, y_0) = d(x, A) \), where \( d(x, A) = \inf\{d(x, y) : y \in A\} \). Choose the correct option:

1. \( A \) is a non-empty subset of \( \mathbb{R}^n \).
2. \( A \) is a non-empty closed subset of \( \mathbb{R}^n \).
3. \( A \) is a non-empty compact subset of \( \mathbb{R}^n \).
4. \( A \) is a non-empty bounded subset of \( \mathbb{R}^n \).

Answer: 2


Question:
Verify the correctness of the following calculation using L'Hospital's rule to confirm that \( \lim_{x \to \infty} h_t(x) = 0 \) for the function \( h_t(x) = t \cdot x \cdot (1-t)^x \) with \( t \in (0,1) \):
\[
\begin{align}
\lim\limits_{x\to \infty}h_t(x)&=\lim\limits_{x\to \infty}t\cdot x\cdot (1-t)^x=t\cdot \lim\limits_{x\to \infty}x\cdot (1-t)^x=t\cdot \lim\limits_{x\to \infty}\frac{x}{\frac{1}{(1-t)^x}}\\
&=t\cdot \lim\limits_{x\to \infty}\frac{x}{(1-t)^{-x}}=t\cdot \lim\limits_{x\to \infty}\frac{x}{e^{-x\cdot \ln(1-t)}}\stackrel{L'H}{=}t\cdot \lim\limits_{x\to \infty}\frac{1}{e^{-x\cdot \ln(1-t)}\cdot (-\ln(1-t))}\\
&=-\frac{t}{\ln(1-t)}\cdot \lim\limits_{x\to \infty}\frac{1}{e^{-x\cdot \ln(1-t)}}\stackrel{L'H}{=}-\frac{t}{\ln(1-t)}\cdot \lim\limits_{x\to \infty}\frac{0}{e^{-x\cdot \ln(1-t)}\cdot (-\ln(1-t))}=0.
\end{align}
\]

Answer: 0


Question: Calculate the flux of the vector field \( \boldsymbol{\mathrm{F}} = x\boldsymbol{\mathrm{i}} + 4y \boldsymbol{\mathrm{j}} \) outwards across the triangle with vertices at \((0,0), (2,0), (0,2)\).

Answer: 10


Question: Suppose that \(a = \binom{7}{0} + \binom{7}{3} + \binom{7}{6}\), \(b = \binom{7}{1} + \binom{7}{4} + \binom{7}{7}\), and \(c = \binom{7}{2} + \binom{7}{5}\). Compute the value of \(a^3 + b^3 + c^3 - 3abc\).

Answer: 128


Question: Evaluate the integral by expressing the integrand as a power series: \[ \int_0^1 \ln\frac{1}{1 - x} \, dx \]

Answer: 1


Question: Find the integer \( y \) such that \( y^3 \equiv 427 \pmod{3901} \).

Answer: 2932


Question: Let $X$ be an inner product space. Suppose there exists an element $x \in X$ such that $\{x\}^{\perp} = \{0\}$. Is $X$ necessarily one-dimensional?

Answer: Yes


Question: Evaluate the limit: $$\lim_{n \to \infty} \int_0^1 \frac{nx^{n-1}}{1+x} \, dx$$ assuming that the interchange of limit and integration is permissible.

Answer: \dfrac{1}{2}


Question:
Determine the tightest upper bound \( C \) such that for all vectors \( w, v \in \mathbb{R}^n \) with \( w_j \in (0,1) \) and \( \sum_{j=1}^n w_j = 1 \), the following inequality holds:
\[ \sum_{j=1}^n v_j^2 w_j \leq C \sum_{j=1}^n v_j^2. \]

Answer: 1


Question: A server processes events from clients, where each client sends an event every second. The server processes an event in t seconds, where t < 1 second (e.g., 0.01 seconds). We want to serve 70,000 clients. The server can process up to n simultaneous requests, and a timeout occurs if this limit is exceeded. What should be the value of n so that a timeout occurs on less than 1% of requests?

Answer: 762


Question: Find \( \lim_{n \rightarrow \infty} \int_{0}^{1} \frac{n^3 x^{3/4}}{ 1 + n^4 x^2} \, dx. \)

Answer: 0


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

Answer: 2 \ln 2 - 1


Question: Determine whether there exists a function \( f \) such that the sequence of functions \( f_n(x) = \sum_{k=1}^n \frac{1}{2^{n-k}} \mathcal{X}_{[a_{k-1},a_k)}(x) \) converges to \( f \) in measure, where \( a_n = \sum_{k=1}^n \frac{1}{k} \) and \( a_0 = 0 \).

Answer: 0


Question: Let $R$ be a local (not Noetherian) domain satisfying the ascending chain condition (a.c.c.) on radical ideals and the descending chain condition (d.c.c.) on prime ideals. Is the Krull dimension of $R$ finite?

Answer: Yes


Question: Compute the Krull dimension of the ring \( k[x,y,z]/(xz, yz) \).

Answer: 2


Question: If a finite group $G$ is solvable, is the derived subgroup $[G,G]$ nilpotent?

Answer: No


Question: Find the radius of the circle $C_2$ that touches the parabola $y^2 = 4x$ at the point $R = (9,6)$ and passes through the focus of the parabola, which is $(1,0)$. The point $R$ is determined by the tangent drawn from the point $P = (6,5)$ to the parabola.

Answer: 5\sqrt{10}


Question: Evaluate the limit: \[ \lim_{x\to0} \frac{\tan x-\sin x}{x-\sin x} \] without using L'Hospital's rule or series expansions.

Answer: 3


Question: Given that \(\lim_{x\to 0} \frac{ae^x - b\cos x + ce^{-x}}{x\sin x} = 2\), find the value of \(a + b + c\).

Answer: 4


Question: Consider a monic polynomial $f(x)$ of degree 5 such that the graphs of $f(|x|)$ and $|f(x)|$ are the same. Given that $4$ is a root of $f(x)$, find the value of $f(1)$. Provide your answer as a single number.

Answer: 225


Question: Evaluate the integral \( \int _0^{+\infty }\frac{1}{\left(1+x^2\right)\left(1+x^{\alpha }\right)}dx \) where \( \alpha \) is a constant.

Answer: \dfrac{\pi}{4}


Question:
Determine whether the following statement is true or false: For every nonempty set $A$ of real numbers having an upper bound, and for every $d \in \mathbb{R}$, the statement
$$[(\sup A = d, d \notin A) \implies (\exists_{N \in \mathbb{N}}{\forall_{n \ge N}{\exists_{a \in A}{\;d - \tfrac{1}{n} < a \le d - \tfrac{1}{n -1}}}})]$$ holds. Justify your answer with an example or counterexample.

Answer: False


Question: Determine whether the Grassmannian of lines in $\mathbb{P}_k^3$, denoted as $\mathbb{G}(1,3)$, can be embedded into $\mathbb{P}_k^5$ when $k$ is an algebraically closed field of characteristic 2.

Answer: Yes


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

Answer: \infty


Question:
Given the finite difference approximations:
- \( \frac{\partial}{\partial x} \rightarrow O(\Delta^4) \)
- \( \frac{\partial^2}{\partial x^2} \rightarrow O(\Delta^2) \)

Determine the resulting order of accuracy for the approximation of \( \frac{\partial^3 f}{\partial x^3} \) using the approach:
\[ \frac{\partial}{\partial x} \left(\frac{\partial^2 f}{\partial x^2}\right) \rightarrow O(\Delta^?) \]

Answer: 2


Question: Evaluate the definite integral: \[ \int_0^1 \left(\sqrt{1-x^3}-\sqrt[3]{1-x^2}\right) \, dx. \]

Answer: 0


Question: Determine whether there exist projections $P$ and $Q$ in $M_2(\mathbb{F}_3)$ such that $Q-P$ is a positive matrix and $PQ \neq P$. Provide a justification for your answer.

Answer: Yes


Question: Given 5 observations: 4, 3, 5, 2, 6 from a \(\text{binomial}(10, p)\) random variable, find an unbiased estimate of \((1+p)^{10}\).

Answer: \dfrac{124}{5}


Question: Evaluate the integral \( \int_{0}^1 \frac{x^a - 1}{\log x} \, dx \) for \( a \geq 0 \).

Answer: \ln(a + 1)


Question: Given the joint probability density function \(f(x,y) = 4 e^{-2(x+y)}\) for \(x, y > 0\), find the conditional expectation \(E(X \mid X+Y=4)\).

Answer: 2


Question: Consider the immersion $f: \mathbb{R}^2 \to \mathbb{R}^4$ defined by \[ f(\theta, \phi) = \frac{1}{\sqrt{2}}(\cos\,\theta, \sin\,\theta, \cos\,\phi, \sin\,\phi). \] The image of $f$ is a torus $\mathbb{T}^2$. Show that the sectional curvature of the image of $f$ is zero.

Answer: 0


Question:
Let $G$ be a subcubic graph with vertices of degree 1 or 3. Suppose $G$ has an edge coloring $\varphi$ using colors from $\{1,2,3,4,5\}$ such that:

1. Each edge is colored with a set of two elements from $\{1,2,3,4,5\}$.
2. For any vertex $v$ of degree 3 with incident edges $e_1, e_2, e_3$, there exists $\{a,b,c\} \subseteq \{1,2,3,4,5\}$ such that $\varphi(e_1)=\{a,b\}$, $\varphi(e_2)=\{a,c\}$, and $\varphi(e_3)=\{b,c\}$.
3. If the distance between two edges $e_1$ and $e_2$ is 2, then $\varphi(e_1) \neq \varphi(e_2)$.

Does $G$ have a 2-distance vertex 4-coloring, meaning a proper vertex 4-coloring such that every two vertices at distance 2 receive distinct colors?

Answer: Yes


Question:
In the Oldenburger-Kolakoski sequence, does the number of 1s equal the number of 2s infinitely many times? The Oldenburger-Kolakoski sequence, $OK$, is the unique sequence of $1$s and $2$s that starts with $1$ and is its own runlength sequence:
$$OK = (1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,\ldots).$$
For $n \geq 1$, let $a(n)$ be the number of indices $h \leq n$ such that $OK(h) =1$, and let $b(n)$ be the number of $h \leq n$ such that $OK(h) = 2.$

Determine if $a(n) = b(n)$ for infinitely many $n$. For example, the first ten runs are $1,22,11,2,1,22,1,22,11,2$, and the lengths of these runs are $1,2,2,1,1,2,1,2,2,1$. In the OEIS, the sequence is A000002.

Answer: Yes


Question: Find the value of $x$ such that $[809x] = [214]$ in $\mathbb{Z}_{5124}$, where $0 \le x < 5124$. Use the inverse $[5105]$ of $[809]$ in $\mathbb{Z}_{5124}$ to help solve the problem.

Answer: 1058


Question:
Consider two random vectors $u=(u_1,...,u_n)$ and $v=(v_1,...,v_n)$, independently and uniformly distributed on the unit sphere in $\mathbb{R}^n$. Define the random variables $X=\sum_{i=1}^n u_i^2 v_i^2$ and $Y=u_1^2 v_1^2$. For the ratio of expectations given by
\[ r_n(\alpha)=\frac{\mathbb{E}\{\exp[-\alpha^2\frac{1-X}{2}]\}}{\mathbb{E}\{\exp[-\alpha^2(1-Y)]\}} \]
determine if there exists a finite upper bound for $r_n(\alpha)$ that is independent of $\alpha$, for fixed $n$ and large $\alpha$. Provide a justification for your answer.

Answer: Yes


Question: Compute the line integral \( \oint_C \frac{\text{d}x}{y} -\frac{\text{d}y}{x} \) where \( C \) is the circle \( x^2+y^2=1 \).

Answer: -4\pi


Question: Find the degree of the field extension \(\mathbb{Q}(\sqrt[4]{5},\sqrt[6]{7})\) over \(\mathbb{Q}\).

Answer: 24


Question: Let $A$ be a square, invertible matrix such that $A^4 = A$. Determine all real eigenvalues of $A$.

Answer: 1


Question: Determine whether \(1^n + 2^n + \cdots + m^n\) divides \((1+2+ \cdots +m)^n\) for any even integers \(m, n \geq 2\).

Answer: No


Question: Given the sequence of sets \(A_n\) defined as follows: \(A_n = (-1/n, 1]\) if \(n\) is odd, and \(A_n = (-1, 1/n]\) if \(n\) is even, find \(\lim\inf_{n \to \infty} A_n\).

Answer: \{0\}


Question: Suppose that $g$ is a Lebesgue integrable function. Show that \( \int |g(x + \epsilon) - g(x)|\,dx \to 0 \) as \( \epsilon \to 0 \).

Answer: 0


Question: Given a mechanical device with a lifetime distribution characterized by the hazard rate function \( \lambda(t) = 3.9 \cdot t^2 \) for \( t > 0 \), calculate the probability that the device's lifetime is between 1 and 2.

Answer: 0.2725


Question: Evaluate the limit: \[ \lim_{n\to\infty}\int_{0}^{2\pi}\left(\frac{\sin{(nx)}}{x^2+n^2}+\frac{\cos{(nx)}}{1+x}\right)dx \]

Answer: 0


Question: Determine the radius of convergence of the series \( \sum\limits_{n=0}^\infty \frac{5^{n+1}}{\sqrt[n]{(2n)!}}z^{n} \).

Answer: \dfrac{1}{5}


Question: Determine the degree of the irreducible polynomial over \( \mathbb{Q} \) for \( z + z^{-1} \), where \( z \) is a primitive 7th root of unity.

Answer: 3


Question: Given the sequence defined by \( a_{n+1} = (n+1)a_n + n \) with \( a_0 = 0 \), find the remainder of \( a_{101} \) when divided by 102.

Answer: 101


Question: Suppose $X$ is a normal space such that for every topological space $Y$, if $f: X \to Y$ is an embedding, then $f(X)$ is closed in $Y$. Does it follow that $X$ is compact?

Answer: Yes


Question: Calculate the degrees of freedom (DOF) for a natural cubic spline with knots at \{10, 20, 30, 50\}, given that the curve is constrained to be linear before the first knot and after the last knot.

Answer: 4


Question: Solve the equation \( \sum_{n=1}^{\infty }\sum_{k=1}^{\infty }\frac{1}{n}\left(\frac{x}{n}\right)^k=0 \).

Answer: 0


Question: Prove that the sum \( \sum_{i=0}^{5x - 1} 2^{4i} \equiv 0 \pmod{11} \) for any integer \( x \).

Answer: 0


Question: Compute the dimension of the Dolbeault cohomology group \(H^{(0,1)}\left(B(2)-B(1)\right)\), where \(B(r)\) is the ball of radius \(r\) in \(\mathbb{C}^2\).

Answer: 0


Question: Let $\alpha$ be a root of $x^3+x^2+1 \in \mathbb{F}_2$ and $K=\mathbb{F}_2(\alpha)$. Suppose that $f$ is an irreducible polynomial in $K[x]$ of degree 4. Let $L$ be the splitting field of $f$ over $K$. Determine the number of elements in $L$. \( \boxed{4096} \)

Answer: 4096


Question: Evaluate the integral \( \iint_{\mathbb{R}^2} e^{x^2+y^2} \, dx \, dy \).

Answer: \infty


Question: Calculate the degree of the field extension \([\mathbb{Q}(\sqrt[n]{m}):\mathbb{Q}]\), where \(m\) is an integer with a prime factorization of the form \(m = p \prod p_i^{n_i}\), and \(p \neq p_i\). Assume \(n\) is a natural number.

Answer: n


Question:
Determine the number of real solutions for the system of equations:
\[ x + \lfloor y \rfloor - \{z\} = 2.98, \]
\[ \lfloor x \rfloor + \{y\} - z = 4.05, \]
\[ -\{x\} + y + \lfloor z \rfloor = 5.01. \]

Answer: 0


Question: Does there exist a continuous function \( f: [0,1] \to \mathbb{C} \) such that \( f(0) = 0 \), \( f(1) = 1 \), and there is no pair of points \( p, q \in f([0,1]) \) with \( q - p = 0.75 \)?

Answer: Yes


Question: Find the limit of the sequence \( a_n = \left(1 - \frac{1}{\sqrt{2}}\right) \cdots \left(1 - \frac{1}{\sqrt{n+1}}\right) \) as \( n \to \infty \).

Answer: 0


Question:
Given the sets:

- \( A = \{ x : x \text{ is a 3-digit natural number} \} \)
- \( B = \{ x : x = 9k + 2, \; k \in \mathbb{N} \} \)
- \( C = \{ x : x = 9k + I, \; k \in \mathbb{Z} \} \) with \( 0 \leq I < 9 \)

The sum of the elements of \( A \cap (B \cup C) \) is given by \( 400 \times 274 \). Find the value of \( I \).

Answer: 5


Question: Let $V$ be an inner-product space and $T:V \to V$ be a linear operator with $T^\ast$ as the adjoint of $T$. Prove that if $v \in \ker(T^\ast T)$, then $T(v) = 0$.

Answer: T(v) = 0


Question: Chris flips a coin 16 times. Given that exactly 12 of the flips land heads, what is the probability that Chris never flips five heads in a row? Express your answer as a common fraction.

Answer: \dfrac{16}{91}


Question: Evaluate the line integral \(\oint_C {\bf B}\cdot d{\bf x}\) for the vector field \({\bf B}({\bf x})=\rho^{-1}e_{\phi}\) in cylindrical coordinates, where \(\rho \neq 0\). The curve \(C\) is the circle defined by \(z=0, \rho=1\), and \(0\le \phi \le 2\pi\).

Answer: 2\pi


Question: Find the limit \( \lim_{n \to \infty} \frac{n!}{n^n} \) using the Squeeze Theorem.

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{1+1/2+1/3+\ldots+1/n}{(\pi^{n}+e^{n})^{1/n}\ln n} \]

Answer: \dfrac{1}{\pi}


Question: Find the fourth derivative of the function \( f(x) \) at \( x=0 \) given that \( f(2x^2-1) = 2x \times f(x) \).

Answer: 0


Question: Consider the differential equation \( f(t,x) + g(t,x) \dot{x} = 0 \) with \((t,x) \in U \subset \mathbb{R}^2\) and \(U\) open. This equation is called exact if there exists a continuous and differentiable function \( F: U \to \mathbb{R} \) such that \( \frac{\partial F}{\partial t}(t,x) = f(t,x) \) and \( \frac{\partial F}{\partial x}(t,x) = g(t,x) \) for \((t,x) \in U\). Prove that if the equation is exact, then the solutions are differentiable functions \( x = x(t) \) with graphs in \( U \) such that \( F(t, x(t)) \) is constant.

Answer: F(t, x(t)) = C


Question: Determine the values of $p$ for which the improper integral \( \int_1^{\infty}\frac{dx}{ x^2\sqrt{x^p -1}} \) converges.

Answer: p > 0


Question: Determine the cardinality of the set \( M = \{ A \in P(\mathbb{N}) \mid A \text{ and } A' \text{ are infinite} \} \).

Answer: 2^{\aleph_0}


Question: Find the constant term in the antiderivative of the function \( f(x) = \log(\log x) + \log^{-2}x \) given that the antiderivative passes through the point \((e, 1998-e)\).

Answer: 1998


Question:
Let $a$, $b$, and $c$ be three positive real numbers such that $a^2 + b^2 + c^2 = 3$. Find the minimum value of the expression:
\[ P = \frac{a^2}{b + 2c} + \frac{b^2}{c + 2a} + \frac{c^2}{a + 2b}. \]

Answer: 1


Question: Use Rouché’s Theorem to determine the number of roots of the polynomial \( z^5 + z^4 + 3 \) in the annulus \( 1 < |z| < 2 \).

Answer: 5


Question: Determine the values of \( r \) for which the integral \( \int_{1}^{\infty} \frac{\cos(x)}{x^r} \, dx \) converges.

Answer: (0, \infty)


Question: Evaluate the triple integral: \[ \int_{-5}^5\int_0^{\sqrt{25-x^2}}\int_0^6\frac{1}{1+x^2+y^2}\,dz\,dy\,dx \]

Answer: 3\pi \ln 26


Question: Let $f(x)$ be a real function such that $\lim\limits_{x\to+\infty} f(x)=c\in\mathbb{R}$. Suppose $\lim\limits_{x\to+\infty} f'(x)$ exists. Is it true that $\lim\limits_{x\to+\infty} f'(x)=0$? Additionally, if $f\in C^1(\mathbb{R})$ and the same hypothesis holds, does $\lim\limits_{x\to+\infty} f'(x)=0$?

Answer: 0


Question: Determine whether 50 is a quadratic residue modulo 71. If it is, find the principal square root of 50 in \( \mathbb{Z}_{71}^x \).

Answer: 60


Question: Compute the outer measure of the set \( \left\{ \left(1+\frac{1}{n}\right)^n : n \in \mathbb{N} \right\} \).

Answer: 0


Question: Calculate the remainder when $10^{5^{101}}$ is divided by 21.

Answer: 19


Question: Calculate the complex integral \( \oint_{D(0,1)} \overline{z} e^z \, \mathrm{d}z \), where \( D(0,1) \) is the unit circle.

Answer: 2\pi i


Question: Determine the number of elements of order 196 in the abelian group $A_2 = C_4 \times C_4 \times C_{49} \times C_7$. Provide a method to find this number.

Answer: 3528


Question: Two unbiased coins are tossed repeatedly. Determine the expected number of tosses required until the last five outcomes of both coins are identical. What is this expected number of tosses?

Answer: 62


Question: Find the derivative \( \frac{dy}{dx} \) if \( y = \frac{1}{1 + x^{n-m} + x^{p-m}} + \frac{1}{1 + x^{m-n} + x^{p-n}} + \frac{1}{1 + x^{m-p} + x^{n-p}} \).

Answer: 0


Question: How many terms of the Maclaurin expansion of \((1+x)^{82}\) are needed to approximate \(1.1^{82}\) to an accuracy of \(10^{-6}\)?

Answer: 28


Question: Calculate the limit: \[ \lim _{x\to \infty} x^2\int _0 ^x e^{t^3-x^3}dt \]

Answer: \dfrac{1}{3}


Question: Evaluate the integral \( \int_0^{2 \pi} \arcsin(\sin(2x)) \, dx \).

Answer: 0


Question: Let $A=\{a_{1}, a_{2}, \ldots, a_{p}\}$ be a complete system of residues modulo a prime number $p \geq 5$. Define $N_p = \sum_{1\leq i < j \leq p}a_ia_j$. Prove that $N_p \equiv 0 \pmod{p}$.

Answer: 0


Question: Given that \( \log_{10}2 \approx 0.30103 \), find the smallest positive integer \( n \) such that the decimal representation of \( 2^{10n} \) does not begin with the digit 1.

Answer: 30


Question: Evaluate the integral \( \int_{0}^{1} \left( (1-x^7)^{\frac{1}{3}} - (1-x^3)^{\frac{1}{7}} \right) \, dx \).

Answer: 0


Question: Does there exist a complex series with a radius of convergence of 1 that diverges at exactly three points on the unit circle \(|z|=1\)?

Answer: Yes


Question: Calculate the volume of the region bounded by the surfaces $z = x^2$, $y = x^2$, $z = 0$, and $y = 4$ using double integrals.

Answer: \dfrac{128}{15}


Question: Evaluate the limit \( F(0) \) where \( F(x) = \lim_{y \to 0} \frac{\cos{x} - \cos{y}}{2(x^2 + y^2)} \).

Answer: \dfrac{1}{4}


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

Answer: e


Question: Consider the sequence defined by \( a_1 = \sqrt{2}\sqrt{3}i \) and \( a_{n+1} = \frac{i a_n}{n+1} \). Determine whether the sequence \( a_n \) converges.

Answer: 0


Question: Calculate the flux of the vector field \( \nabla \times A \) through the surface \( S = \{ z = 2(1-x^2-y^2),\ z \geq 0 \} \), where \( A = (y, z, x) \).

Answer: -\pi


Question: Expand \( \sqrt{z+\mathrm{i}} \) as a power series in terms of \( z-1 \). Determine the radius of convergence of this series and identify the largest disk centered at \( z=1 \) where the function \( \sqrt{z+\mathrm{i}} \) and the power series coincide.

Answer: \sqrt{2}


Question: Determine whether the sentence \( \sigma : \forall x \exists y \exists z \ (x \neq y \wedge y \neq z \wedge x \neq z \wedge x R y \wedge z R x) \) has a finite model.

Answer: Yes


Question: Let \( D \) be a domain in \( \mathbb{C} \), which is a connected open subset of the complex numbers. If one point is removed from \( D \), is the resulting set still connected?

Answer: Yes


Question: Let $(X,d)$ be a connected compact metric space. Given a radius $R > 0$, does there exist a finite family $B_1, B_2, \ldots, B_n$ of open balls in $X$ such that $B_i \cap B_{i+1} \neq \emptyset$ for $i = 1, \ldots, n-1$ and $X = \bigcup_{k=1}^n B_k$?

Answer: Yes


Question: Evaluate the integral: \[ \int_0^1 \sqrt{\frac{\left\{\frac{1}{x}\right\}}{1-\left\{\frac{1}{x}\right\}}}\frac{\mathrm{d}x}{1-x} \]

Answer: \pi


Question: Let $A$ be a $4 \times 4$ real matrix with eigenvalues $1, -1, 2, -2$. If $B = 2A + A^{-1} - I$, find the determinant of $B$. Use the fact that $A$ satisfies its characteristic equation $(x-1)(x+1)(x-2)(x+2)=0$, which implies $A^4 - 5A^2 + 4I = 0$. Determine the determinant of $B$. \( \text{det}(B) = ? \)

Answer: 154


Question: Given that the series \( \sum_{n=3}^{\infty}a_n x^{n} \) has a radius of convergence \( R \), determine the radius of convergence for the series \( \sum_{n=3}^{\infty}a_n x^{n^2} \).

Answer: 1


Question: Determine the number of elements \( \alpha \in F_{83} \) such that the polynomial \( t^2 + 5t + \alpha \) is irreducible over the field \( F_{83} \).

Answer: 41


Question:
What is the expected magnitude of the sum of two unit vectors in the Cartesian plane if the directions of both vectors are randomly chosen from the interval \([-\pi, \pi]\)?

A.) \(\dfrac{4}{3}\)

B.) \(\sqrt{2}\)

C.) \(\dfrac{4}{\pi}\)

D.) \(\dfrac{9}{2\pi}\)

Answer: C


Question: Suppose \( A \) is a 3x3 non-singular matrix such that \((A - 3I)(A - 5I) = O\), where \(I\) is the 3x3 identity matrix and \(O\) is the 3x3 zero matrix. If \(\alpha A + \beta A^{-1} = 4I\), find the value of \(\alpha + \beta\).

Answer: 8


Question: Find a noncyclic group \( G \) such that the order of \( G \) is equal to the order of its automorphism group, \( |\mathrm{Aut}(G)| \).

Answer: S_3


Question: Suppose $X$ is a topological space and $\mathcal{F}$ is a discrete family of closed subsets of $X$. Is it true that any two members of $\mathcal{F}$ are disjoint? Justify your answer.

Answer: Yes


Question: Find \( \lim_{n\to\infty} P_n \) where \( P_n = \left(1+\frac{1}{a_1}\right)\left(1+\frac{1}{a_2}\right)\left(1+\frac{1}{a_3}\right)\cdots\left(1+\frac{1}{a_n}\right) \) and \( a_1=1 \) with \( a_n=n(a_{n-1}+1) \) for \( n=2,3,\ldots \).

Answer: e


Question: Find the limit of \( \prod_{i=2}^n \frac{2i^2-i-1}{2i^2+i-1} \) as \( n \) approaches infinity.

Answer: 0


Question: Calculate the value of \(5^{{9}{^{13}}^{17}} \mod 11\).

Answer: 9


Question:
In a triangle \(\Delta ABC\) inscribed in a circle of radius one unit, the internal angle bisectors of angles \(\angle A, \angle B, \angle C\) meet the circle at the points \(A_1, B_1, C_1\) respectively. Find the value of the expression:
\[ S = \frac{\sum AA_1 \cos\left(\frac{A}{2}\right)}{\sum \sin A} \]

Answer: 2


Question: Determine all values of $a$ for which the integral \( I = \int_{0}^{+\infty}\frac{dx}{x^2+x^a} \) converges.

Answer: (-\infty, 1)


Question: Calculate the limit: \[ \lim_{x\rightarrow 0}\frac{1}{x^n}e^{-\frac{1}{x^2}} \]

Answer: 0


Question: Let $A$ be a unital $C^*$-algebra and $a, b \in A_+$ be positive elements with $a \in \text{Her}(b) := \overline{bAb}$. Determine if there exists a positive, continuous function $f \in C(\sigma(b) \cup \{0\})$ such that $f(0) = 0$ and $a \le f(b)$. Provide a justification for your answer.

Answer: Yes


Question:
Calculate the Fourier coefficient \( \hat{f}(2) \) for the function \( f(x) = (2\cos(\pi x))^{10} \), where \( f \in L^{1} \) is a one-period function. The Fourier coefficient is given by:
\[ \hat{f}(n) = \int_{0}^{1} e^{-2\pi inx} (2\cos(\pi x))^{10} \, dx, \quad n \in \mathbb{Z} \]
Use the binomial expansion \( \cos(\pi x)^{10} = \left(\frac{e^{i\pi x} + e^{-i\pi x}}{2}\right)^{10} \) to express the integrand and compute \( \hat{f}(2) \).

Answer: 120


Question: Given an $n$-dimensional manifold $M$ that is Hausdorff and satisfies the second countable axiom, for each point $p \in M$, there exists an open neighborhood $U_{r(p)}(p)$ that is diffeomorphic to the open ball $B_{r(p)} = \{ x \in \mathbb{R}^n \mid |x| < r(p) \}$ via a differential mapping $\varphi_p$ such that $\varphi_p(p) = 0$. Can we find at most countably many open sets $U_i = U_{r'(p_i)}(p_i) \subset U_{r(p_i)}(p_i)$ such that the collection $\{U_i\}$ forms a locally finite open covering of $M$?

Answer: Yes


Question: Find the number of ordered integer solutions \((A, B, C, D, E, F)\) such that \( A \times B \times C \times D \times E \times F = 7 \times 10^7 \).

Answer: 120434688


Question: Jim rolls a fair six-sided die repeatedly and records the sum of the outcomes until the die shows a 6 for the third time. Let \( S \) be the sum of the outcomes. For example, if he rolls \(1, 4, 6, 3, 4, 6, 6\), he stops after the last 6, and \( S = 30 \). Find \( E[S] \), the expected value of \( S \).

Answer: 63


Question: Determine the minimum number of times you need to toss a coin to estimate the probability of getting heads within 0.1 of its true value with at least 90% confidence, using Chebyshev's Inequality.

Answer: 250


Question: Let \( S = \sum_{k=1}^{100} X_k \) where each \( X_k \sim \text{Poisson}(1) \). Use the Central Limit Theorem to approximate \( P(S \ge 200) \).

Answer: 0


Question: Calculate the limit: \[ \lim_{x \to 1} \big(4^x - 3^x\big)^{\frac{1}{x - 1}} \] without using L'Hôpital's rule or Taylor series.

Answer: \dfrac{256}{27}


Question: Let \( a \) be a natural number such that \( a^4 \not\equiv 1 \pmod{29} \), \( a^7 \not\equiv 1 \pmod{29} \), and \( a^{14} \not\equiv 1 \pmod{29} \). Calculate the order of \( a \) modulo 29.

Answer: 28


Question: Determine all possible orders of elements in the quotient group $\mathbb{Z}^4 / H$, where $H$ is the subgroup of $\mathbb{Z}^4$ consisting of 4-tuples $(a,b,c,d)$ satisfying $8 \mid (a-c)$ and $a+2b+3c+4d=0$. Provide your answer as a set of integers.

Answer: \{1, 2, 4\}


Question:
Find all real numbers \(x\) such that:
\[
\left( \sqrt{2-\sqrt{2} }\right)^x+\left( \sqrt{2+\sqrt{2} }\right)^x=2^x.
\]

Answer: 2


Question: Compute the integral \( \int_{\gamma} e^z \, dz \), where \( \gamma \) is the semicircular arc defined by \( \gamma = 3e^{i\theta} \) for \( 0 \leq \theta \leq \pi \).

Answer: e^{-3} - e^{3}


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

Answer: [-1, 0]


Question: Prove that \(3^m + 3^n + 1\) cannot be a perfect square for any positive integers \(m\) and \(n\).

Answer: 3^m + 3^n + 1


Question: Given a linearly recurrent sequence \(a_k\) such that \(a_k \geq 0\) and \(\sum_{k=1}^{\infty} a_k = 1\), determine if there always exists a constant \(x\) and a constant \(r < 1\) such that \(a_k \leq x r^k\) for all \(k\).

Answer: Yes


Question: Find all natural numbers \(m\) such that \(998^m - 1\) divides \(1994^m\).

Answer: 1


Question: Consider a convex Euclidean polytope $P$ with $n$ facets defined by $\langle f_i , x\rangle = 1$ and $m$ vertices $v_j$, where $1\leq i\leq n$ and $1\leq j\leq m$. The origin is not contained in any bounding hyperplane containing a facet of $P$. Is the polytope $P’$, with $m$ facets given by $\langle v_j , x\rangle = 1$ and $n$ vertices $f_i$, combinatorially equivalent to the polar dual $P^*$ of $P$?

Answer: Yes


Question: Determine whether the region \( D = \bigcup_{n=1}^{\infty} \left\{ (x , y) \in \mathbf {R}^2 :\;\; x^2 + y^2 \,\leq\, 1 - \frac {1}{n} \right\} \) is open or closed.

Answer: open


Question: Evaluate the integral: \[ \int_0^{\frac{\pi}{2}}\left(\frac{x}{\sin x}\right)^2\text{d}x \]

Answer: \pi \ln 2


Question: Determine the values of \( \alpha \) for which the series \( \sum_{n = 1}^{\infty}\left(2^{n^{-\alpha}} - 1\right) \) converges.

Answer: \alpha > 1


Question: Compute the integral \( \oint_{\partial D(0,8)}\frac{1+\cos^2(z)}{z-\pi}dz \) using Cauchy's integral formula.

Answer: 4\pi i


Question: Determine whether there exists a natural number \( n \) such that \( 3^n \equiv 7 \pmod{10^{2011}} \).

Answer: No


Question: The function \( y = f(x) \) is represented parametrically by \( x = t^5 - 5t^3 - 20t + 7 \) and \( y = 4t^3 - 3t^2 - 18t + 3 \) for \( -2 < t < 2 \). Determine the value of \( t \) at which the minimum of \( y = f(x) \) occurs.

Answer: \dfrac{3}{2}


Question:
Let \( g:(\alpha,\beta)\rightarrow\mathbb{R} \) be a twice differentiable function such that
\[
g(x)=\frac{x^2g''(x)+xg'(x)}{x^2+1},
\]
and \(0<\alpha<\beta\). Given that \(g(\alpha)=g(\beta)=0\), determine if \(g(x)=0\) for all \(x\in(\alpha,\beta)\).

Answer: g(x) = 0


Question: Evaluate the limit: \[ \lim_{x \to 1^+} \frac{\log_{10}{(x-1)}}{\sqrt{x-1}} \]

Answer: -\infty


Question: Consider the function \(f(x) = \frac{cx}{2x+3}\), where \(x \neq -\frac{3}{2}\). Find all values of \(c\) for which \(f(f(x)) = x\).

Answer: -3


Question: Determine the values of $m$ and $n$ for which the affine spaces $\mathbb{A}^n(k)$ and $\mathbb{A}^m(k)$, equipped with the Zariski topology, are homeomorphic, where $k$ is an algebraically closed field.

Answer: m = n


Question: Determine whether every finite dimensional $C^*$-algebra is an $AW^*$-algebra.

Answer: Yes


Question: For which integer value of $m$ does the sum \( \sum_{n=1}^m \sin n \) achieve its maximum value?

Answer: 9


Question: Given that the number of defects per yard in a certain fabric, Y, follows a Poisson distribution with parameter \( \lambda \), where \( \lambda \) is a random variable with a density function \( f(\lambda) = e^{-\lambda} \) for \( \lambda \geq 0 \), find the variance of Y.

Answer: 2


Question: Does the statement "If a characteristic function \( \varphi \) is integrable \( (\varphi \in L^{1}) \), then there is a continuous bounded density function \( f(x) = \frac{1}{2\pi}\int_{\mathbb{R}}\exp(-itx)\varphi(t)dt \; \forall x \in \mathbb{R} \)" imply that if \( \mathbf{Q} \) is a discrete probability distribution, then \( \varphi \) is not integrable?

Answer: Yes


Question: Determine the number of ordered pairs of positive integers \((a, b)\) such that \(a! + \frac{b!}{a!}\) is a perfect square.

Answer: 6


Question: Determine the quotient space of \(\mathbb{C}^n\) under the action of the symmetric group \(S_n\), where the action is defined by permuting the coordinates: \(\sigma(x_1, x_2, \ldots, x_n) = (x_{\sigma(1)}, x_{\sigma(2)}, \ldots, x_{\sigma(n)})\).

Answer: \mathbb{C}^n


Question: Evaluate the integral: $$\int_0^\infty \frac{\log x}{1+x^2} \, dx$$

Answer: 0


Question:
Compute the lower Dini derivative of the function \( f(x) = \begin{cases} x^2 \sin\left(\frac{1}{x}\right), & x < 0 \\ 0, & x=0 \\ 2 x^2 \sin\left(\frac{1}{x}\right), & x > 0 \end{cases} \) at \( x=0 \) using the definition:
\[Df(x) = \lim_{\varepsilon \rightarrow 0} \inf_{0 < \delta \le \varepsilon} \frac{f(x+\delta) - f(x)}{\delta}\]

Answer: 0


Question: Given a positive integer \( n \), determine if there exists a positive integer \( p \) such that the unique solution \( x_p \) of the equation \((1-x)^p = x\) satisfies \(0 < x_p \le \frac{1}{n}\).

Answer: Yes


Question: Determine the ideal $I(V)$ for the variety $V(x+y) = \{(x,y) \in \mathbb{R}^2 : x = -y\}$. The ideal $I(V)$ is defined as $I(V) = \{f \in \mathbb{R}[x,y] : f(a,b) = 0 \text{ for all } (a,b) \in V\}$. Find the pattern for functions $f$ that belong to this ideal.

Answer: (x + y)


Question: Solve the differential equation \((\cos x - x \sin x + y^2)\,dx + (2xy)\,dy = 0\) with the initial condition \(y(\pi) = 1\). Find the function \(y(x)\).

Answer: \sqrt{-\cos x}


Question:
Consider the region \( \Omega : x^2 + y^2 < 4 \) in \( \mathbb{R}^2 \) with boundary \( \partial \Omega \). Let \( u(x,y) \) be the solution to the Dirichlet problem given by the Laplace equation:
\[
\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \quad (x,y) \in \Omega
\]
with the boundary condition \( u(x,y) = 1 + 2x^2 \) for all \( (x,y) \in \partial \Omega \). Find the value of \( u(0,1) \).

Answer: 4


Question: Does there exist a probability distribution defined on \(x \in (0, \infty)\) such that \(p(x) = p(1/x)\) for all \(x\), and \(\int_0^\infty p(x) \, dx = 1\)?

Answer: Yes


Question:
Given a Markov chain with state space \( \{0, 1, 2\} \), where state 0 represents making no money, state 1 represents making a small amount of money, and state 2 represents making a large amount of money, the transition matrix is:
\[
\begin{bmatrix}
0.1 & 0.6 & 0.3 \\
0.4 & 0.4 & 0.2 \\
0.7 & 0.3 & 0
\end{bmatrix}
\]
Calculate \( P(X_5 = 2 \mid X_3 = 0, X_1 = 1) \).

Answer: 0.15


Question: Determine the characteristics of the partial differential equation \( \frac{\partial\rho}{\partial t} + 4 \frac{\partial\rho}{\partial x}= 0 \).

Answer: x - 4t = C


Question: Determine the area of the figure formed by the implicit function \(x^{n} + y^{n} = c\) as \(n\), an even positive integer, approaches infinity. Express the area as a function of \(c\).

Answer: 4


Question: Find the minimum value of the function \( f(x,y,z) = x^2+y^2+z^2 \) subject to the constraints \( x+2y+z=8 \) and \( x-y=5 \).

Answer: \dfrac{358}{11}


Question: Calculate the value of the integral \( \int_{(-1, 1)} |x| d\mu \), where the measure \( \mu \) on the measurable space \((\mathbb{R}, \mathcal{B})\) is defined by \( \mu\left( (a, b] \right) = \int_a^b \frac{1}{1+x^2}dx \) for all intervals \((a,b]\).

Answer: \ln 2


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f(2) = 6 \) and \( f'(2) = \frac{1}{48} \). Find \( \lim_{x \to 2} \int_{6}^{f(x)} \frac{4t^3}{x-2} \, dt \).

Answer: 18


Question: Find the equation of a plane that is invariant under the transformation given by the matrix \( \begin{pmatrix}4&-23&17\\ 11&-43&30\\ 15&-54&37\end{pmatrix} \).

Answer: 3x - 3y + z = 0


Question:
Let $A$ and $B$ be square matrices such that $ABA = B^2$. Determine which of the following statements is true:

1. If $A$ is invertible, then $B$ is invertible.
2. If $B$ is invertible, then $A$ is invertible.

Answer: 2


Question: Evaluate the limit: \[ A = \lim_{n \to \infty} \left(\binom{n}{0} \binom{n}{1} \binom{n}{2}\cdots \binom{n}{n}\right)^{1/((n)(n+1))} \]

Answer: \sqrt{e}


Question: Compute the limit: $$\lim_{x\rightarrow 0}\frac{3x^2-3x\sin x}{x^2+x\cos\frac{1}{x}}$$

Answer: 0


Question: Find the minimal polynomial of the element $\sqrt{2} + \sqrt{2}i$ over the field $\mathbb{Q}$. Provide your answer as a polynomial with rational coefficients.

Answer: x^4 + 16


Question: Evaluate the integral \( \int_m^{n}\frac{1}{\frac{b}{x -\frac{1}{2b}}-\lfloor{b/x}\rfloor} \, dx \) where \( b, m, n, \) and \( x \) are natural numbers such that \( m < n < b \).

Answer: n - m


Question: Calculate the limit of the function \( g(m) = \frac{\left[\Gamma\left(1+\frac{1}{m}\right)\right]^2}{\Gamma\left(1+\frac{2}{m}\right)} \) as \( m \to 0^+ \), if the limit exists.

Answer: 0


Question: Given the solution to the differential equation \( \frac{\partial c}{\partial t} = D\frac{\partial^2c}{\partial x^2} \) with boundary conditions \( c(x,0) = C^\infty \), \( c(0,t)=0 \), and \( c(\infty,t)=C^\infty \), the concentration is given by \( c(x,t) = C^\infty \operatorname{erfc}\Big(\frac{x}{2\sqrt{Dt}}\Big) \). The flux at \( x=0 \) is \( J_{(x=0)} = \frac{C^\infty\sqrt{D}}{\sqrt{\pi t}} \). Determine the limit of \( J \) as \( t \to 0 \).

Answer: \infty


Question: What is the expected number of randomly chosen real numbers between 0 and 1 needed to reach a sum of 1?

Answer: e


Question: Evaluate the limit: \[ \lim_{x\to\infty}\left(\frac {x^5+\pi x^4+e}{x^5+ex^4+\pi}\right)^x \]

Answer: e^{\pi - e}


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

Answer: \dfrac{1}{2}


Question: Evaluate the integral: \[ \int_{-2}^{2} \frac{x^2}{1+5^x} \, dx \]

Answer: \dfrac{8}{3}


Question: Find the coefficient of \(x^{-1}\) in the expansion of \(\left(\frac{1}{2x}+3x\right)^5 (x+1)^4\).

Answer: \dfrac{541}{32}


Question: Evaluate the integral \( \iint\limits_D \frac{\cos x}{\cos x + \cos y} \, dA \), where \( D \) is the region defined by \( x^2 + y^2 \leq 1 \).

Answer: \dfrac{\pi}{2}


Question: Find the last three digits of the number $7^{100} - 3^{100}$. Express your answer as a three-digit number.

Answer: 000


Question:
Let $k$ be a field, $n$ be an integer $
\geq 2$, and $x, y_1, \dots, y_n$ be indeterminates. Consider the ideal $I = \langle x^2y_1, x^2y_2 - xy_1, \dots, x^ny_n - xy_1 \rangle$ in the polynomial ring $k[x, y_1, \dots, y_n]$. Determine whether $xy_1 \in I$.

Answer: xy_1 \notin I


Question: Find all polynomials $Q(x)$ such that \( Q(x)(x^2-6x+8) = Q(x-2)(x^2-6x) \) for all real numbers \( x \).

Answer: 0


Question: Determine whether there exists a surjective homomorphism from the group $U(15)$ to the cyclic group $\mathbb{Z}_4$. $U(15)$ is the multiplicative group of positive integers less than 15 that are coprime with 15, and it is given by $U(15) = \{1, 2, 4, 7, 8, 11, 13, 14\}$. $\mathbb{Z}_4$ is the cyclic group of integers modulo 4.

Answer: Yes


Question: Calculate the limit \( \lim_{x\to\infty}\intop_{x}^{2x}\frac{1}{t}\cos\left(\frac{1}{t^{2}}\right)dt \).

Answer: \ln 2


Question: Is any $F$-stable maximal torus contained in some $F$-stable Borel subgroup of an affine algebraic group $G$ over the algebraic closure of a finite field $\mathbb{F}_q$? Assume $F$ is a Frobenius morphism defining an $\mathbb{F}_q$ structure on $G$. It is known that each $F$-stable Borel subgroup of $G$ contains an $F$-stable maximal torus. Determine if the converse is true.

Answer: No


Question: Let \( X_1, X_2, \ldots, X_{40} \) be independent random variables, each uniformly distributed on \([-1/2, 1/2]\). What is the probability that the sum \( Y = X_1 + X_2 + \cdots + X_{40} \) is greater than 3?

Answer: 0.050


Question: Given a quadratic function $f(x) = ax^2 + bx + c$ that is bijective from $[0,2]$ to $[0,2]$, find the value of $f(2)$. Provide your answer as a single number.

Answer: 2


Question: Consider the polynomial \( f(x) = x^4 - 2x^3 - x^2 + 17x + 6 \) with roots \( x_1, x_2, x_3, x_4 \). Let \( g(x) = x^5 - 3x^4 + 22x^2 - 18x - 2 \). Find the value of \( \sum_{k=1}^{4} g(x_k) \).

Answer: 63


Question: Given a matrix $B$ with dimensions $k \times p$ where $k \leq p$ and rank$(B) = k$, and a positive definite matrix $A$ with dimensions $p \times p$, determine if the matrix $BAB'$ is of full rank $k$. Justify your answer.

Answer: Yes


Question: Calculate the energy of the signal given by \( s(t) = \int_{-4}^{3} e^{\pi i t u} \, du \). The energy \( E \) is defined as \( E = \int_{-\infty}^{\infty} |s(t)|^2 \, dt \).

Answer: 14


Question:
Given a function \( f \) with a continuous derivative on \([0,1]\) such that \(0 < f'(x) \le 1\) and \(f(0) = 0\), determine the minimum value of the expression:
\[ I = \frac{\left(\int_0^1 f(x) \, dx\right)^2}{\int_0^1 f^3(x) \, dx} \]
Choose the correct answer from the following options:
(A) \( I \ge \frac{2}{3} \)
(B) \( I \ge \frac{1}{2} \)
(C) \( I \ge \frac{1}{3} \)
(D) \( I \ge 1 \)

Answer: D


Question: If $A$ is a skew symmetric matrix such that $A^TA=I$, find $A^{4n-1}$.

Answer: -A


Question: Find the Jordan measure of the set \( M = \left\{ \begin{pmatrix} -r +2 t -t\\ 3r+t \\2r+2t \end{pmatrix} : 1\leq r \leq 3, 0 < s < 1, -1 < t \leq 2 \right\} \).

Answer: 0


Question: Use the Central Limit Theorem to approximate the 0.95-quantile \( q_{0.05}(\mathcal{A}_n) \) for the sample size \( n=100 \), given that \( q_{0.05}(\mathcal{A}_1)=3.84 \). Assume \( \mathcal{A}_1 \sim \chi^2_1 \) and \( n \cdot \mathcal{A}_n \sim \chi^2_n \).

Answer: 1.23


Question: For an even positive integer \( n \), numbers \( 1, 2, \ldots, n^2 \) are placed on an \( n \times n \) chessboard such that each number appears exactly once. Let \( S_1 \) be the sum of the numbers on the black squares and \( S_2 \) be the sum of the numbers on the white squares. Determine all values of \( n \) for which \( \frac{S_1}{S_2} = \frac{39}{64} \).

Answer: 206


Question: Suppose $R$ is a ring and $S \subseteq R$ is a subring. Moreover, suppose $R$ is integral over $S$ and $R$ is finitely generated as an $S$-module. Is it true that for any maximal ideal of $S$, there are at most $t$ maximal ideals of $R$?

Answer: Yes


Question: Find the maximum value of the function \( \phi(r) = a_1 J_0(\alpha r) + a_2 J_0(\beta r) \) on the interval \([0, l]\), where \( a_1 + a_2 = 1 \), \( \alpha \) and \( \beta \) are positive parameters, and \( \phi(l) = 0 \). The function \( \phi \) is non-negative on \([0, l]\). Here, \( J_0 \) denotes the Bessel function of the first kind.

Answer: 1


Question: Consider the integral domain \(R=\mathbb{Z}[a_1,\ldots,a_n]\), which is finitely generated over \(\mathbb{Z}\). Determine whether the quotient group \((R,+)/(\mathbb{Z},+)\) can contain a divisible element, where a "divisible element" is defined as an element \(e \neq 0\) such that for every positive integer \(n\), there exists an element \(f\) satisfying \(e = nf\).

Answer: No


Question: Find the probability that a random variable \( \xi \sim \text{Beta}(1,2) \) is less than another independent random variable \( \eta \sim \text{Exp}(1) \). Calculate \( P(\xi < \eta) \).

Answer: \dfrac{2}{e}


Question: Determine the maximum number of subsets that can be chosen from a set of 20 elements such that no two subsets have more than 2 elements in common.

Answer: 1351


Question: Compute the \( \mathbb{C} \)-dimension of the vector space \( \mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \), where \( \mathbb{R} \) is the field of real numbers and \( \mathbb{C} \) is the field of complex numbers. Consider \( \mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \) as a \( \mathbb{C} \)-vector space via \( a(b\otimes c) := ab \otimes c, \) for \( a, b, c \in \mathbb{C} \).

Answer: 2


Question: Consider a unit circle on a plane with a frog initially at its center. The frog makes an infinite series of jumps. For the $n^{th}$ jump, the frog chooses a random point $x_n$ on the circle and jumps to the midpoint of the segment connecting its current position $r_n$ with the point $x_n$. Determine the expected value of the square of the frog's distance to the center in the limit of an infinite number of jumps: \( \lim\limits_{n\to\infty}E[|r_n|^2] \).

Answer: \dfrac{1}{3}


Question: Let \( f \) be a holomorphic function such that \( a \) is a zero of order 2 in its domain \( D_f \). Evaluate \( \int _\gamma \frac{f'(z)}{f(z)}dz \), where \( \gamma \) is a circle centered at \( a \) contained in \( D_f \), and all other zeroes of \( f \) lie outside the circle bounded by \( \gamma \).

Answer: 4\pi i


Question: Using the Newton-Raphson method, find the value of $f(0)$ for the implicit function $3x^7 + 2y^5 - x^3 + y^3 - 3 = 0$. Provide your answer to an appropriate degree of accuracy.

Answer: 1


Question: Let $Y$ and $Z$ be homotopy equivalent subspaces of a topological space $X$. Does it follow that the relative homology groups $H_n(X,Y)$ and $H_n(X,Z)$ are isomorphic for all $n$?

Answer: No


Question:
Evaluate the line integral of the differential form from the point \((0,0,0)\) to \((1,1,1)\):
\[ \int^{(1,1,1)}_{(0,0,0)} 3x^2y^{10}z^{10} \, dx + 10x^3y^9z^{10} \, dy + 10x^3y^{10}z^9 \, dz \]

Answer: 1


Question: Given a set of points \(\{x_i\}\) on a circle, determine if it is always possible to position an arc with length equal to \(\frac{1}{c}\) of the circle's circumference such that at most \(\frac{1}{c}\) of the points \(\{x_i\}\) lie on the arc. Assume \(c\) is an integer.

Answer: Yes


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

Answer: 1


Question: Calculate the sum of the series: \[ \sum_{k=1}^{\infty} \frac{k^2}{2^k} \]

Answer: 6


Question: Evaluate the limit: \[ \lim_{x\to \infty} \int_x^{x+1} \frac{t^2+1}{t^2+20t+8}\, dt \]

Answer: 1


Question: Given the equation \(f(x-1) + f(x+3) = f(x+1) + f(x+5)\), determine the period of the function \(f(x)\).

Answer: 8


Question: Find the smallest natural number \( k \) such that the polynomial \( x^{2k} + 1 + (x+1)^{2k} \) is not divisible by \( x^2 + x + 1 \).

Answer: 3


Question: Determine the number of differentiable functions on $(0, \infty)$ that satisfy the equation $f^{-1} = f'$. Provide your answer as a single integer.

Answer: 1


Question: Evaluate the limit \( \lim_{n \to \infty} \frac{n! + n^n}{1 + \ln(n)} \).

Answer: \infty


Question: Evaluate the limit: $$ \lim_{n \to \infty} \frac{1^{1/3} + 2^{1/3} + \cdots + n^{1/3}}{n \cdot n^{1/3}} $$

Answer: \dfrac{3}{4}


Question: Find the remainder when \(100!\) is divided by \(97^2\).

Answer: 8827


Question: Compute the volume of the region \( R \) defined by \( R = \{(x,y,z) : 3x^2 \leq z \leq 3, \ -z \leq y \leq z\} \).

Answer: \dfrac{72}{5}


Question: If \( \sum a_n \) converges and for almost all \( n \), \( a_n > 0 \), does it imply that the series \( \sum a_n \) converges absolutely?

Answer: Yes


Question: Solve the equation \(\sqrt[3]{7x+19}+\sqrt[3]{7x-19}=\sqrt[3]{2}\) using algebraic methods.

Answer: \dfrac{7}{4}


Question: Find the remainder when the 81-digit number consisting entirely of ones is divided by 729.

Answer: 81


Question: Determine the values of $c$ for which the function $f(x) = x^3 + 3x^2 + cx + 3$ is an open map from $\mathbb{R}$ to $\mathbb{R}$. Provide your answer in terms of $c$.

Answer: c \geq 3


Question: Calculate the flux of the vector field \( \mathbf{F} = (x^2+y^2+2+z^2)\mathbf{i} + (e^{x^2} + y^2)\mathbf{j} + (3+x)\mathbf{k} \) outward across the part of the spherical surface \( x^2 + y^2 + z^2 = 2az + 3a^2 \) that is above the \( xy \)-plane, where \( a > 0 \).

Answer: 9\pi a^2


Question: Determine the number of ways to select 13 numbers from the set \( \{1, 2, 3, 4, 5\} \) such that their arithmetic mean is 4.6.

Answer: 0


Question:
Let $M$ be a nonempty manifold with $\dim M = n$, where $n > 0$ and $0 < k < n$. Determine the dimension of $\Omega^{k}M$. Choose the correct answer from the following options:

1. $\infty$
2. $\displaystyle\binom{n}{k}$
3. $k(k-1)/2$

Answer: 1


Question: What is the length of the shortest sequence that contains all possible 4-digit sequences (from 0000 to 9999) as substrings?

Answer: 10003


Question: Determine the value of \( \alpha \) such that the function \( f : [0, 1] \rightarrow \mathbb{R} \) defined by \[ f(x) = \begin{cases} x^\alpha \sin{\frac{1}{x}}, & x \neq 0 \\ 0, & x = 0 \end{cases} \] is twice differentiable at the point \( x = 0 \).

Answer: \alpha > 3


Question: Calculate the sum \( u_0 u_1 + u_1 u_2 + \ldots + u_{n-2} u_{n-1} + u_{n-1} u_0 \), where \( u_k = \cos\frac{2k\pi}{n} + i \sin\frac{2k\pi}{n} \).

Answer: 0


Question: Given a differentiable function \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(1) = 2 \) and \( f(x+y) = 2^x f(y) + 4^y f(x) \) for all \( x, y \in \mathbb{R} \), find the minimum value of \( f(x) \).

Answer: -\dfrac{1}{4}


Question: Solve the equation \(\sqrt{x^{2}+ 8}- \sqrt{x^{2}+ 3}+ 2x^{3}- x- 2= 0\).

Answer: 1


Question: Evaluate the line integral \( \int_C (y+e^\sqrt{x}) \, dx + (xe^{y^2}) \, dy \) where \( C \) is the boundary of the triangle with vertices \((0,0), (4,2), (0,2)\), oriented counterclockwise.

Answer: e^4 - 5


Question: Does there exist a family \( A \) of subsets of \( \mathbb{N} \) such that \( A \) has the cardinality of the continuum and for any two elements of \( A \), one is a subset of the other?

Answer: Yes


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

Answer: 2e - 5


Question: Is a Borel subset of an analytic subset of a Polish space still analytic? Assume $A$ is a Borel subset of $B$, where $B$ is an analytic subset of a Polish space $C$. Determine whether $A$ is an analytic set in $C$. Provide a justification for your answer.

Answer: Yes


Question: Find the derivative of \( \ln(\sec(x)) \) using first principles.

Answer: \tan(x)


Question: Does there exist a unitary matrix $U$ in $M_{n}(\mathbb{C})$ but not in $D_{n}$ such that $UD_{n}U^{*}= D_{n}$?

Answer: Yes


Question: Compute \( \sqrt{-4} \times \sqrt{-9} \). Assume the principal square root for negative numbers.

Answer: -6


Question: Determine all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the following conditions: \( f(x + y) + f(x - y) = 2 f(x) f(y) \) for all \( x, y \in \mathbb{R} \) and \( \lim\limits_{x \to \infty} f(x) = 0 \).

Answer: 0


Question: Let \( p(z) = \prod_{k=1}^{n}(z-\lambda_k)^{m_k} \), where each \( \lambda_k \) is in the set \( H = \{ z \in \mathbb{C} \mid \operatorname{Im}(\frac{z-a}{b}) \geq 0 \} \), with \( a \in \mathbb{C} \) and \( b \neq 0 \) in \( \mathbb{C} \). Prove that the roots of the derivative \( p'(z) \) are also in \( H \).

Answer: H


Question: Given a vector \(x = (x_1, \dotsc, x_n) \in \mathbb{R}^n\) and an arbitrary norm \(||\cdot||\), determine whether \(||x|| < c\) implies \(|x_i| < c\) for all components \(x_i\) and some constant \(c > 0\).

Answer: No


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

Answer: f(x) = c


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

Answer: [-1, 1)


Question: Consider the operator $T: L^\infty((0,1)\times (a,b)) \to L^\infty((0,1)\times (a,b))$ defined by \[ Tf(x,y) = \begin{cases} f\left(x+\frac{y}{b},y\right), & 0 < x < 1-\frac{y}{b}, \\ \frac{1}{2}f\left(x+\frac{y}{b}-1,y\right), & 1-\frac{y}{b} < x < 1, \end{cases} \] where $0 < a < b$. Determine whether the following statement is true: \[ \lim_{n \to \infty} \frac{\log \|T^n\|}{n} < 0. \]

Answer: True


Question: Evaluate the integral \( \int_{0}^{\infty} \frac{1}{x^6 + 1} \,\mathrm{d}x \) using methods from real analysis, without employing complex analysis techniques.

Answer: \dfrac{\pi}{3}


Question: The reflection of the parabola \(y^2 = 4(x - 1)\) in the line \(x + y = 2\) is the curve \(Ax + By = x^2\). Find the value of \(A + B\).

Answer: 0


Question: Determine the maximum number of perfect matchings in a genus $g$ balanced bipartite graph, where the number of vertices for each color is within a difference of 1. Specifically, is the maximum number of perfect matchings $2^{O(n\log g)}$ for $k=2$?

Answer: 2^{O(n \log g)}


Question: Show that the expectation of the Ito integral with respect to a standard Wiener process is zero: \[ \mathbb{E}\left[\int^t_0 g(W_s,s)\,dW_s\right] = 0. \] Here, \( g(W_s,s) \) is a simple process, and the integral is defined as the limit \( \lim_{n \to \infty} \sum_{k=0}^{n-1} f(W_{t_i},t_i)(W_{t_{i+1}}-W_{t_i}) \), where \( t_i = \frac{it}{n} \), \( 0=t_0 < t_1 < \ldots < t_{n-1} < t_n = t \), and \( n \in \mathbb{N} \).

Answer: 0


Question: Evaluate the integral \( \int_{|z|=2} \frac{\operatorname{Im} z}{z-\alpha} \, dz \) where \( \alpha \) is a complex number satisfying \( 0<|\alpha|<1 \), and \( \operatorname{Im} z \) denotes the imaginary part of \( z \).

Answer: \pi \alpha


Question:
Verify the uniform convergence of the Poisson kernel. Show that $P(r, \theta) \to 0$ uniformly on the set $\{ \theta : |\theta| \geq \theta_* \}$ as $r \to a^-$, where
\[ P(r,\theta) = \frac{a^2-r^2}{a^2-2r\cos(\theta)+r^2} \]
for $0 \leq r < a$ and $-\pi \leq \theta \leq \pi$. Use the hint: $a^2 - 2ar \cos(\theta) + r^2 = (a - r)^2 + 2ar(1 - \cos \theta)$. Determine if the following limit is correct:
\[ \lim_{r \to a^-} \max_{|\theta| \geq \theta_*} \left| \frac{a^2-r^2}{(a - r)^2 + 2ar(1 - \cos \theta)} \right| = 0. \]

Answer: 0


Question: Consider the set \( A = \{1, 2, \ldots, 10\} \). Define a function \( f: A \rightarrow A \) such that \( f^{30}(x) = x \) for all \( x \in A \), where \( f^{30} \) denotes the 30th composition of \( f \) with itself. Given that 30 is the smallest number for which \( f^{30} \) becomes the identity function, how many such functions \( f \) are there?

Answer: 120960


Question: Evaluate the infimum \( I = \inf_{f \in \mathcal{F}} \left( \sup_{0 \leq t \leq 1} \left| 1 - f(t) \right| + \int_{0}^{1} \left| 1 - f(t) \right| dt \right) \), where \( \mathcal{F} \) is the vector space of all continuous functions \( f: [0,1] \rightarrow \mathbb{R} \) such that \( f(0) = 0 \).

Answer: 1


Question: Evaluate the integral \( \int_0^\pi \sin^4\left(x+\sin 3x\right)dx \).

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


Question: Verify the expression for the derivative of the integral of the survival function: \( \frac{\mathsf d}{\mathsf dx}\int_x^\infty \bar F(u)\ \mathsf du = -\bar F(x) \).

Answer: -\bar{F}(x)


Question: Find the limit: \[ \lim_{n \to \infty} \sum_{m=1}^{\infty} \frac{1}{m^2 + n^2} \]

Answer: 0


Question: Calculate the limit: $$\lim_{x\to \pm\infty}\sqrt[3]{(x-1)^2}-\sqrt[3]{(x+1)^2}$$

Answer: 0


Question: Evaluate the expression \([-x^{n+1} e^{-x} ]_{x=0}^{x=\infty}\), where \(n\) is any integer greater than 1.

Answer: 0


Question: Let \( n \geq 1 \) and \( p \) be a prime number. Determine whether there always exists an integer \( m \geq 1 \) such that \( p^m \equiv m \pmod{n} \).

Answer: Yes


Question: Let \(X \sim N(1,4)\) be a Gaussian random variable. Find the value of the moment generating function \(M_X(t) = E(e^{tX})\) at the point \(t=1\).

Answer: e^3


Question: An object with an initial temperature $T_0$ is placed in a room with a constant temperature of 30°C. After 10 minutes, the object's temperature is -10°C, and after another 10 minutes, it is -5°C. Using Newton's law of cooling, determine the initial temperature $T_0$ of the object.

Answer: -\dfrac{110}{7}


Question: In triangle ABC, cevians AD, BE, and CF intersect at point P. Given that \( \frac{|BP|}{|PE|} = \frac{3}{2} \) and \( \frac{|FP|}{|PC|} = \frac{3}{4} \), find the ratio \( \frac{|AP|}{|PD|} \).

Answer: \dfrac{29}{6}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sum_{k=0}^n \frac{\binom{n}{k}}{(2n)^k} \]

Answer: \sqrt{e}


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

Answer: e


Question: Let $\bar{k}$ be an algebraic closure of a field $k$. Suppose $R$ is an integral domain such that $\bar{k} \supseteq R \supseteq k$. Is $R$ necessarily a field? Justify your answer.

Answer: Yes


Question: Determine the interval of convergence for the power series \( \sum_{k=1}^\infty\frac{(x+2)^k}{k(k+1)^{\frac{1}{2}}} \).

Answer: [-3, -1]


Question: Find the distribution of the random variable $Z = XY$, where $X$ and $Y$ are independent and uniformly distributed on the interval $(0,1)$. Use the method of conditional expectation to derive the distribution.

Answer: -\ln z


Question: Given a discrete random variable $X$ that can take integer values from $0$ to $20$, and the mean of $X$ is $0.005$, determine the maximum possible variance of $X$. Provide your answer as a single number.

Answer: 0.099975


Question: Determine whether the solution to the recurrence relation \( T(n) = T\left(\frac{1}{8n}\right) + 8 \) is infinite.

Answer: \infty


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_{\frac{1}{n+1}}^{\frac{1}{n}} \frac{\sin(x)}{x^3} \, dx. \]

Answer: 1


Question: Evaluate the line integral of the vector field \( F(x,y,z) = (zy)i + (y^3 + xz)j + \left(\frac{1}{z} + xy\right)k \) along the path \( C \), where \( C \) is the quarter circle with center \((1,1,1)\) going from \((1,0,1)\) to \((0,1,1)\).

Answer: \dfrac{1}{4}


Question: Determine the smallest dimension a non-commutative C*-algebra can have.

Answer: 4


Question: Evaluate the limit: \[ \lim_{x\rightarrow1}{\frac{1+\log{x}-e^{x-1}}{(x-1)^2}} \] using L'Hospital's rule.

Answer: -1


Question:
Determine the functions \( f: \mathbb{R} \to \mathbb{R} \) that have a primitive \( F: \mathbb{R} \to \mathbb{R} \) satisfying the following properties:

1. \( F(x) \geq f(x) \geq 0 \) for all \( x \in \mathbb{R} \).
2. There exists \( x_0 \in \mathbb{R} \) such that \( F(x_0) = 0 \).

Answer: 0


Question: Let \((x_1, x_2, \ldots, x_n)\) be \(n\) independent random variables, each uniformly distributed as \(x_i \sim U(\theta, 3\theta)\) with an unknown parameter \(0.5 \le \theta \le 1\). Determine the domain of the minimal number of samples \(\min_n\) such that \(n \ge \min_n\) ensures \[ P(2\theta - 0.05 \le \hat{X} \le 2\theta + 0.05) \ge 0.97 \] where \(\hat{X} = \frac{1}{n} \sum_{i=1}^n x_i\).

Answer: 628


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0} \frac{\arcsin\frac{x}{\sqrt{1-x^{2}}} }{\ln (1-x)} \] without using L'Hôpital's rule.

Answer: -1


Question: Let $G$ be a finite group, $N$ a normal subgroup of $G$, and $K$ a characteristic subgroup of $N$. Given that $Z(G/N) = 1$ and $Z(G) = 1$, determine if it follows that $Z(G/K) = 1$.

Answer: No


Question: Determine the value of \( a \in \mathbb{C} \) such that the function \( f(z) = z^2 + a\overline{z}^2 + 4z\overline{z} + 2z - 8\overline{z} + 1 + 2i \) is not holomorphic at any point in \( \mathbb{C} \).

Answer: -2


Question: Consider a $C^*$-algebra $A$ and elements $a, b \in A$ such that $0 \leq a \leq 1$ and $0 \leq b \leq 1$. Is it true that $0 \leq ab \leq 1$ if $A$ is not abelian?

Answer: No


Question: Evaluate the integral: $$\int \limits _{0}^{10} \int \limits _{0}^{10} \frac{x^2y^2}{(x^2+y^2)^{5/2}}\ \delta(x)\ \mathrm{d}x\mathrm{d}y$$

Answer: 0


Question: Determine an upper bound on the chromatic number of the square of a tree with $n$ vertices. Consider the maximum degree $\Delta$ of the tree in your analysis.

Answer: \Delta + 1


Question: Describe the topological space obtained by forming the adjunction space $X \sqcup_{f} Y$, where $X = D^2$ is the closed unit disk in the plane, $A = S^1$ is the boundary of $D^2$, $Y$ is the singleton set $\{1\}$, and $f: S^1 \to Y$ is the constant map defined by $f(x) = 1$. What well-known topological space does this adjunction space resemble?

Answer: S^2


Question: A die is rolled repeatedly until every number from 1 to 6 appears at least once. What is the probability that the number 1 appears exactly once during this process?

Answer: \dfrac{49}{120}


Question:
Let \( A \) be a \( 4 \times 4 \) matrix with eigenvalues \(-5, -2, 1, 4\). Determine which of the following is an eigenvalue of the block matrix \( \begin{pmatrix} A & I \\ I & A \end{pmatrix} \), where \( I \) is the \( 4 \times 4 \) identity matrix.

(A) \(-5\)
(B) \(-7\)
(C) \(2\)
(D) \(1\)

Answer: C


Question: Let $G = \left( \mathbb{Z}/32\mathbb{Z} \right)^{\times}$ and $H = \{1, 31\}$. The group $G/H$ is isomorphic to one of $\mathbb{Z}_8$, $\mathbb{Z}_4 \oplus \mathbb{Z}_2$, or $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$. Determine which group $G/H$ is isomorphic to by elimination.

Answer: \mathbb{Z}_8


Question: Find the limit of \(x^{-4}\left(1-e^{x^2/2}\cos \left(x\right)\right)\) as \(x\to0\) using the expansions of \(\exp\) and \(\cos\).

Answer: \dfrac{1}{12}


Question: Given that \(X\) and \(Y\) are independent and identically distributed standard normal random variables, calculate the probability \( P(X > 0 \mid X + Y > 0) \).

Answer: \dfrac{3}{4}


Question: Find \( \lim\limits_{n \to \infty} \frac{\ln(n+1)}{\ln(n)} \frac{\ln(n!)}{\ln((n+1)!)} \).

Answer: 1


Question: Given a holomorphic embedding \( f: \mathbb{C} \to \mathbb{C}^2 \), determine if it is true that for any disjoint union of sets \( U = \bigcup_{j=1}^N U_j \), the preimage \( f^{-1}(U) \) is also a disjoint union of the same number of sets.

Answer: True


Question: Find the solution set of the equation \(3^x = 2^x + 1\) using algebraic methods.

Answer: 1


Question: Determine the number of real solutions to the equation \( \cos(\cos(\cos x)) = \sin(\sin(\sin x)) \).

Answer: \infty


Question: Verify whether $p=9631$ is the only prime satisfying the equation $23005\cdot (2^n-1)\cdot 2^n +1=p^2$ for $n=6$. Note that $9631$ is the inverse of $37^2$.

Answer: 9631


Question:
Let $b$ be a real number such that $b > 1$. Compute the limit:
\[ \lim_{n\to\infty}n\cdot\left(\frac1{b^n}+\frac1{b^{2n}}+\frac1{b^{3n}}+\dots+\frac1{b^{(n-1)n}}+\frac1{b^{n^2}}\right). \]

Answer: 0


Question: Let $G$ be a Hausdorff group. Suppose $A$ is a closed subgroup of $G$ and $B$ is a discrete subgroup of $G$. Is the product set $AB$ necessarily closed in $G$?

Answer: No


Question: Evaluate \( \lim_{n \rightarrow \infty} \int_{0}^{1} x \cos\left(\frac{x}{n}\right) \, dx \).

Answer: \dfrac{1}{2}


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

Answer: \dfrac{e}{2}


Question: Evaluate the integral \( \int_{-1}^{1} \exp(x+e^{x})\,dx \), where \( \exp(x)=e^x \).

Answer: e^e - e^{1/e}


Question: A piggy bank contains 45 loonies and 25 toonies. Determine the number of ways to distribute the coins such that Jamie receives no loonies, Julie receives no toonies but at least 10 loonies, and Brenda receives an odd number of toonies. Use generating functions to find the solution.

Answer: 468


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

Answer: 0


Question: Let $f:(c,+\infty)\to \mathbb{R}$ be a differentiable function. Given that $\lim_{x\to+\infty}f(x)=a$ and $\lim_{x\to+\infty}f'(x)=b$ exist with $a\in \mathbb{R}$, prove that $b=0$. Use the hint: $f(n+1)-f(n)=f'(x_n)$, where $x_n\to \infty$. Provide a detailed explanation of your reasoning.

Answer: 0


Question: Given that \( a^{100} \equiv 2 \pmod{73} \) and \( a^{101} \equiv 69 \pmod{73} \), find \( a \equiv x \pmod{73} \).

Answer: 71


Question: What is the Kleisli category of the monad \( \mathbb{T} : \mathbf{Set} \to \mathbf{Set} \) associated with the Lawvere theory for the category of sets?

Answer: \mathbf{Set}


Question: Evaluate the limit: $$\lim_{x\to {\pi \over 4}}\frac{\sqrt[3]{\tan(x)} -1}{2\sin^2x - 1} $$

Answer: \dfrac{1}{3}


Question: Evaluate the double integral \( \int\int_A x\, dxdy \), where \( A \) is the region bounded by the parabolas \( y = x^2, y = x^2 + 4, y = (x-1)^2, y = (x-1)^2 + 4 \). Use an appropriate change of variables to simplify the integration.

Answer: 4


Question: Determine the value of $r$ for which the sequence $a_n = \dfrac{4^n}{n^3} + (r^2)^n$ and the series $S = \sum_{n=0}^\infty \frac{4^n}{n^3 + (r^2)^n}$ converge.

Answer: |r| > 2


Question: Determine whether the limit of the recursive sequence \(a_{n+1}=\frac{n+1}{\frac{1}{a_{n}}-(n+1)}\) converges to the same value regardless of the initial value \(a_0\). If it does, find the limit of the sequence.

Answer: -1


Question: Given distinct proper rigid transformations \(A, B \in \operatorname{SE}(n)\), determine the maximum dimension of the nullspace of \(A - B\). Specifically, find the maximum dimension of \(\operatorname{Eq}(A, B) = \{x \in \mathbb{R}^n\ |\ A(x) = B(x)\}\) for all integers \(n \geq 2\).

Answer: n - 2


Question: Evaluate the integral \( \int_{1/2}^{2} \frac{1}{x} \sin \left(x - \frac{1}{x}\right) \, dx \). Choose the correct value from the following options: \((A)\ 0\), \((B)\ \frac{3}{4}\), \((C)\ \frac{3}{4}\), \((D)\ 2\).

Answer: A


Question: Consider the sequence defined by \( a_n = \sqrt{n} + \sqrt[3]{n} + \cdots + \sqrt[n]{n} \). Find the limit of the sequence \( a_{n+1} - a_n \).

Answer: 1


Question: Let \( a, b \) be elements of a unital C*-algebra \( A \) with \( 0 \leq a, b \leq 1 \). Determine whether there exists a state \( \tau \) on \( A \) such that \( |\tau(ab)| = \|ab\| \). Consider the case when \( a, b \) are projections and \( \|ab\| = 1 \).

Answer: Yes


Question: Is every continuous function defined on the sphere $B$ given by $x^2 + y^2 + z^2 \le 1$ Riemann-integrable?

Answer: Yes


Question: Determine the number of distinct ways to color a 3x3 grid with exactly five red squares and four blue squares, considering two colorings identical if one can be obtained from the other by a rotation or flip of the grid.

Answer: 22


Question: Let $X$ be the ordinal space $[0,\Omega)$ with the order topology, where $\Omega$ is the first uncountable ordinal. Let $f: X \rightarrow \mathbb{R}$ be a continuous real-valued function on $X$. Is $f$ constant on some interval $(\alpha, \Omega)$, where $\alpha$ is a countable ordinal?

Answer: Yes


Question: Let \( f:G \to G \) be a function such that \( f(a)f(b)f(c)=f(x)f(y)f(z) \) for all \( a,b,c,x,y,z \in G \) with \( abc=xyz=e \). Is there an element \( g \in G \) such that the function \( h:G\to G \) defined by \( h(x):=gf(x) \) is a homomorphism?

Answer: Yes


Question: Evaluate the limit of the integral: \[ \lim \limits_{x \to \infty} \int_0^\infty \sin \left(e^{xt}\right)\,dt \]

Answer: 0


Question: Find the infimum of \( \int_0^1 (u'(t))^2 \, dt \) for continuously differentiable functions \( u \) on \([0,1]\) such that \( u(0) = 0 \) and \( \max_{t \in [0,1]} |u(t)| = 1 \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{1^k + 2^k + 3^k + \dots + n^k - \frac{1}{k+1} n^{k+1}}{n^k} \]

Answer: \dfrac{1}{2}


Question: Let $X$ be a connected topological space, and $Y$ a space whose connected components are not all open. Can there exist an open continuous surjective map $Y \rightarrow X$ with finite fibers?

Answer: Yes


Question: Determine the number of equivalence relations on the set \( S = \{1, 2, 3, 4, 5, 6, 7, 8\} \) that have exactly 3 equivalence classes.

Answer: 966


Question: Find the function \( f \) that has a power series representation at \( 0 \) converging on \((-1, 1)\), given that there exists a sequence of distinct elements \((x_n)\) such that \(x_n \to 0\) and \(f(x_n) = 0\) for all \(x_n\).

Answer: 0


Question:
Consider the sequence defined by \(x_n=\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\left(1-\frac{1}{10}\right)^2\left(1-\frac{1}{15}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2\) for \(n>1\). Determine \(\lim_{n\to\infty} x_n\). Choose the correct answer from the following options:

A) \(\frac{1}{3}\)
B) \(\frac{1}{9}\)
C) \(\frac{1}{81}\)
D) 0

Answer: B


Question: In \(\triangle ABC\), points \(L\) and \(M\) are on \(AB\) and \(AC\) such that \(AL = \frac{2}{5}AB\) and \(AM = \frac{3}{4}AC\). Lines \(BM\) and \(CL\) intersect at point \(P\), and the extension of line \(AP\) intersects \(BC\) at point \(N\). Find the value of \(\frac{BN}{BC}\).

Answer: \dfrac{9}{11}


Question: Determine the degree of the field extension \([\mathbb{R}(\theta):\mathbb{R}]\) given that \(\theta\) is a root of the polynomial \(\theta^3 - 4\theta^2 + 6\theta + 6\) and \(\theta \notin \mathbb{R}\).

Answer: 2


Question: Evaluate the contour integral \( \oint_C \frac{dz}{z-2} \) around the circle \( |z-2| = 4 \).

Answer: 2\pi i


Question: Determine the number of linear functions from \(\{0,1\}^n\) to \(\{0,1\}\), where a function \(F\) is linear if for all \(x, y \in \{0,1\}^n\), \(F(x \oplus y) = F(x) \oplus F(y)\).

Answer: 2^n


Question: If $f$ is a measurable function on a measure space $X$ and $fg \in L^1$ for all $g \in L^q$, must $f$ be in $L^p$, where $p$ is the conjugate of $q$? Assume that $\|fg\|_1 \leq C\|g\|_q$ for some constant $C$. Does this follow from the Riesz Representation theorem even if such a $C$ is not given?

Answer: f \in L^p


Question: Find the derivative of the function \( \phi(t) = \int_0^1 e^{\sqrt{x^2 + t^2}} \, \mathrm{d}x \) at \( t = 0 \).

Answer: 0


Question: Determine whether the sequence \(y_n = \left(1 - \frac{1}{2^n}, \frac{1}{2^{n+1}}, \frac{1}{2^{n+2}},\dots,\frac{1}{2^{2n}},0,0,0,\dots\right)\) converges in \(\ell^1\) to \((1,0,0,0,\dots)\).

Answer: Yes


Question: For any prime $p$, does there exist a constant $C$ such that if $x \ge C$, then at most one integer among $x+1, x+2, \dots, x+p$ has its greatest prime factor less than or equal to $p$?

Answer: Yes


Question: Determine whether there exists a natural number \( p \) such that the sequence \( \frac{1}{\left| n^p \sin(n/2) \right|} \) is bounded for \( n \in \mathbb{N} \).

Answer: 2


Question: Compute the value of the series \( \sum_{n=0}^\infty \frac{\sin((2n+1)\phi)}{2n+1} \) for \( 0 < \phi < \pi \).

Answer: \dfrac{\pi}{4}


Question: Evaluate the limit of the sequence: \[ \lim_{n \to \infty} \sin\left(2\pi(n^2+n^{1/2})^{1/2}\right). \]

Answer: 0


Question:
Determine the upper-bound on the computational complexity of the function $a_n$, which represents the number of directed acyclic graphs on $n$ vertices, given by the recurrence:
$$a_n = \sum_{k=1}^{n}(-1)^{k-1}\binom{n}{k}2^{k(n-k)}a_{n-k}$$
Assume the cost of multiplication and binomial coefficient evaluation is constant. Let $T(n)$ be the time required for computing $a_n$. Analyze the recurrence:
$$T(n) = \sum_{k=1}^{n}T(n-k)$$
with $T(0) = c$. What is the upper-bound on $T(n)$ in big O notation?

Answer: O(2^n)


Question: Determine whether the equation \(A^TAx = A^Tb\) is always consistent for any \(m \times n\) matrix \(A\) and vector \(b \in \mathbb{R}^n\). Answer True or False.

Answer: True


Question: Let $p$ be a prime number greater than 2, and let $e_1, e_2, \ldots, e_p$ be some $p$-th roots of unity in $\mathbb{C}$, not necessarily distinct. If \( e_1 + e_2 + \cdots + e_p = 0 \), can we conclude that all $e_i$ are distinct? Justify your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{z \to 0} \frac{\sin(wz)}{\sin(z)} \] where \( w \in \mathbb{C} \).

Answer: w


Question: Evaluate the limit: \[ \lim_{x\to \infty} (2\arctan x -\pi)\ln x \]

Answer: 0


Question: Calculate the residue of the function \( f(z) = \frac{e^z + 1}{\sin^2(iz)} \) at the point \( z = i\pi \).

Answer: 1


Question: Evaluate the integral \( \int_{-1}^1 P_5(x) + P_4^2(x) \, dx \), where \( P_5(x) \) and \( P_4(x) \) are Legendre polynomials.

Answer: \dfrac{2}{9}


Question: Determine the maximal Hausdorff dimension of the set on which a Lipschitz continuous function $f: \mathbb{R}^n \to \mathbb{R}$ is differentiable and $|Df| = L$, where $L$ is the strict Lipschitz constant of $f$.

Answer: n


Question: Given the function \( f(x,y) = \frac{y\ln(1+x^2 + ay^2) } {x^2 + 2y^2} \) for \((x,y)\neq (0,0)\), and \(f(0,0)=0\), with the additional information that \( f_y (0,0) = 2 \), find the value of \(a\).

Answer: 4


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0^+} \frac{e^{-a/x}}{x}, \quad a>0 \]

Answer: 0


Question: What is the probability that the greatest common divisor of two natural numbers, $a$ and $b$, is twice a prime number? Denote the set of prime numbers by $P$. Express your answer as a numerical probability.

Answer: 0.0687


Question: Determine whether there exists a real analytic diffeomorphism of the two-dimensional disk onto itself, other than the identity, such that its restriction to the boundary of the disk is the identity.

Answer: Yes


Question:
Let $G$ be a non-abelian group of order $20$. Determine the order of $\operatorname{Aut}(G)$. Choose the correct option:
(a) $1$
(b) $10$
(c) $30$
(d) $40$

Answer: D


Question: Evaluate the contour integral \( \int_C \frac{dz}{\sin^3 z} \), where \( C \) is the unit circle in the complex plane with positive orientation.

Answer: \pi i


Question: Find the sum of the infinite series: \[ \sum_{n=1} ^{\infty} \frac{(-1)^n}{(n+1)(n+3)(n+5)}. \]

Answer: -\dfrac{7}{480}


Question:
Find all positive integer pairs \((x, y)\) that satisfy the system of equations:
\[ 2^x + \log_3 x = y^2 \]
and
\[ 2^y + \log_3 y = x^2 \]

Answer: (3, 3)


Question: Calculate the surface integral \( \int_S dx \wedge dy + dy \wedge dz \), where \( S \) is the surface defined by \( S = \{(x,y,z) : x = z^2 + y^2 - 1, x < 0\} \).

Answer: \pi


Question: Determine whether the sequence \(a_n=\sqrt[n]{\sum_{k=1}^{n}(2-\frac{1}{k})^k}\) is convergent.

Answer: 2


Question: Find the value of $x$ such that \(1! + 2! + \ldots + 100! \equiv x \pmod{19}\).

Answer: 8


Question: Determine if the infinite sum \( \frac{1}{20}\sum_{n=1}^{\infty}\left(n^2+n\right)\left(\frac{4}{5}\right)^{n-1} \) can be simplified further.

Answer: \dfrac{25}{2}


Question: Determine the asymptotically slowest growing function $f(x)$ such that there exist constants $a$ and $b$ for which, for all $x > a$, there is always a prime number between $x$ and $x + bf(x)$. Consider whether $f(x) = x$, $f(x) = \sqrt{x}$, $f(x) = \log(x)$, or $f(x) = \log\log(x)$ could be such a function.

Answer: x


Question: Evaluate the line integral \( \int_{\gamma} \frac{1}{z^2-1} \, dz \), where \( \gamma(t) = 1 + e^{it} \) for \( 0 \leq t \leq 2\pi \).

Answer: \pi i


Question: Find the radius of convergence of the power series \(\sum_{n=1}^{\infty} a_n z^{n^2}\), where \(a_0 = 1\) and \(a_n = \frac{a_{n-1}}{3^n}\) for \(n \geq 1\).

Answer: \sqrt{3}


Question: Determine the number of solutions to the equation \(|\ln|x|\;|=\sin(\pi x)\).

Answer: 6


Question: Find the sum of the series \( \sum_{n=0}^{\infty} \frac{n^3}{n!} \).

Answer: 5e


Question: Let $V$ be the vector space of all continuous functions from $[0,1]$ to $\mathbb{R}$ over the field $\mathbb{R}$. Determine the rank of the linear transformation $T: V \rightarrow V$ defined by \[ T(f(x)) = \int_{0}^{1} (3x^3y - 5x^4y^2) f(y) \, dy. \]

Answer: 2


Question: Find the order of the group generated by all cycles of length $n$. For example, when $n=4$, the set of cycles is: \( G = \{(1234), (1243), (1342), (1324), (1432), (1423)\} \). Determine the order of the group generated by this set.

Answer: 24


Question: Find the residue of the function \( f(z) = \frac{1}{z - \sin z} \) at the pole \( z = 0 \).

Answer: \dfrac{3}{10}


Question: Is there any norm on \( \mathbb{R}^n \) such that for every vector \( \mathbf{x} = (x_1, \ldots, x_n) \in \mathbb{R}^n \), the equality \( ||\mathbf{x}'|| = ||\mathbf{x}|| \) holds, where \( \mathbf{x}' = (|x_1|, \ldots, |x_n|) \)?

Answer: Yes


Question: Let \( f \in \mathbb{R}[x] \) be a polynomial of degree \( d \). Given real numbers \( x_1, \dots, x_n \), consider the matrix \( A \) defined by \( A_{ij} = f(x_i + x_j) \). Prove that the rank of \( A \) is at most \( d+1 \).

Answer: d+1


Question: Determine the number of zeros of the function \( z^4 + 3iz^2 + z - 2 + i \) in the upper half of the complex plane.

Answer: 2


Question: Determine the order of the alternating group $A_6$. Show that the order is given by \( \frac{6!}{2} = 360 \).

Answer: 360


Question: Evaluate the expression \[ P = \dfrac{a^2}{2a^2+bc} + \dfrac{b^2}{2b^2+ca} + \dfrac{c^2}{2c^2+ab} \] given that \( a + b + c = 0 \).

Answer: 1


Question: Determine the positive integer values of \( k \) for which the series \( \sum_{n=1}^{\infty} \frac{(n!)^6}{(kn)!} \) is convergent.

Answer: k \geq 6


Question: Let $G$ be a compact, simply connected Lie group, and let $H$ be a Lie subgroup of $G$ that is also compact, simply connected, and has the same dimension as $G$. Prove or disprove that $H = G$.

Answer: H = G


Question: Let $f: (0, \infty) \to \mathbb{R}$ be a continuous function. Determine whether $f$ maps any Cauchy sequence to a Cauchy sequence. Answer with 'True' or 'False'.

Answer: False


Question: Find the limit of the sequence: \[ \lim_{n \to \infty} \left( \frac{1}{n^k} \sum_{j=1}^n j^k - \frac{n}{k+1} \right) \] where \( k \in \mathbb{N} \setminus \{ 0 \} \).

Answer: \dfrac{1}{2}


Question: Let $A$ be an $n \times n$ matrix with real entries such that the numbers in each column sum to $0$, $a_{ij} \in \{0,1\}$ for all $i \neq j$, and $a_{ij} = 0 \leftrightarrow a_{ji} = 1$ for all $i \neq j$. Given that $A$ is non-invertible, is the linear combination of columns that results in the zero vector necessarily unique (up to a constant factor)?

Answer: Yes


Question: Solve the functional equation \( f(x+y) = \frac{f(x) + f(y)}{1 - f(x)f(y)} \). Given that \( f(0) = 0 \), \( f(n) = 0 \) for all \( n \in \mathbb{N} \), \( f(\frac{1}{2}) = 0 \), and \( f(-x) = -f(x) \), find the function \( f(x) \).

Answer: 0


Question:
Find the distributional limit \( \lim_{\varepsilon\rightarrow 0^+}f_{\varepsilon} \) in \( D'(\mathbb{R}) \), where \( f_\varepsilon(x) \) is defined as follows:

\[
f_\varepsilon(x) = \begin{cases}
\frac{1}{\varepsilon^3}, & x \in (0,\varepsilon) \cup (2\varepsilon,3\varepsilon), \\
\frac{-2}{\varepsilon^3}, & x \in (\varepsilon,2\varepsilon), \\
0, & x \notin (0,3\varepsilon).
\end{cases}
\]

Answer: \delta''


Question:
Consider two real polynomials $f(x)$ and $g(x)$ of degrees $m \geq 2$ and $n \geq 1$, respectively, that satisfy the equation:
\[ f(x^2+1) = f(x)g(x) \]
for every real number $x$. Determine the nature of the real roots of $f(x)$ from the following options:

(A) $f$ has exactly one real root $x_0$ such that $f'(x_0) \neq 0$.
(B) $f$ has exactly one real root $x_0$ such that $f'(x_0) = 0$.
(C) $f$ has $m$ distinct real roots.
(D) $f$ has no real root.

Select the correct option.

Answer: D


Question: Prove that the following binomial sum equals \(4^n\): \[ \sum_{k=0}^n \binom{2k}{k} \binom{2n-2k}{n-k} = 4^n. \]

Answer: 4^n


Question: For what values of \(a\) is a \(3 \times 3\) matrix \(A\) with characteristic polynomial \(x^3 - 3x + a\) guaranteed to be diagonalizable over the real numbers?

Answer: (-2, 2)


Question: Let \( f(x) \) be a function defined by \( f(x) = \sin|x| + 5\sin(2013x) \) for \( -\pi < x < \pi \) and \( f(x + 2\pi) = f(x) \) for all \( x \in \mathbb{R} \). The Fourier series for \( f(x) \) is given by \( a_0 + \sum_{n=1}^{\infty}(a_n\cos(nx) + b_n\sin(nx)) \). Find the exact value of \( \sum_{n=1}^{\infty}b_n \).

Answer: 5


Question: Determine if the integral \( \int_0^\infty \frac{xe^{-bx}}{\sqrt{x^2 +a^2}}dx \) can be solved analytically, where \(a\) and \(b\) are constants.

Answer: a K_1(a b)


Question: Given that \(\alpha\) is a real root of the equation \(\alpha^3 - 3\alpha^2 + 5\alpha - 17 = 0\) and \(\beta\) is a real root of the equation \(\beta^3 - 3\beta^2 + 5\beta + 11 = 0\), find the value of \(\alpha + \beta\). The curve \(y = x^3 - 3x^2 + 5x\) is strictly increasing, and the equation \(y = x^3 - 3x^2 + 5x - 17 = 0\) intersects the x-axis between 3 and 4, while \(y = x^3 - 3x^2 + 5x + 11 = 0\) intersects the x-axis between -2 and -1. Determine if it is possible to find the exact value of \(\alpha + \beta\).

Answer: 2


Question: In a graded ring $B = \bigoplus_{d \ge 0} B_d$, can there exist a non-zero element $b \in B$ such that $b \in B_e \cap B_f$ with $e \neq f$? Justify your answer.

Answer: No


Question: Given a number N, find the sum of all possible subsequences of its digits, where the order of digits in each subsequence is preserved. For example, if N = 389, the subsequences are 3, 8, 9, 38, 39, 89, and 389. Calculate the total sum of these subsequences.

Answer: 575


Question: Evaluate the limit of the integral \( \int_{0}^{1} \frac{n\cos^2x}{1 + n^2x^2} \, dx \) as \( n \to \infty \).

Answer: \dfrac{\pi}{2}


Question: Find the expression for a holomorphic function \( f \) that maps the annulus \( a < |z| < b \) onto itself, permuting the inner boundary \( |z| = a \) and the outer boundary \( |z| = b \), where \( 0 < a < b \).

Answer: \dfrac{ab}{z}


Question: Let N couples be randomly seated at a rectangular table, with men on one side and women on the other. Define the random variable X as the number of couples seated directly across from each other. Calculate the variance of X.

Answer: 1


Question: Given a balanced bipartite graph with $2n$ vertices, determine the upper bound for the maximum number of edges such that the graph is planar. Is there a bound stronger than $6n-6$?

Answer: 4n - 4


Question: Does every $p$-group have a faithful unipotent representation with coefficients in $\mathbb{F}_p$ or some finite extension thereof?

Answer: Yes


Question: Evaluate the integral of the differential form \( \varphi = e^x dx \wedge dy + y dz \wedge dx + x dy \wedge dz \) over the surface \( S \) of the cylinder defined by \( x^2 + y^2 = 1 \) between \( z = 1 \) and \( z = 3 \).

Answer: 4\pi


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

Answer: 4\sqrt{2} - 5


Question: Compute the contour integral \( \oint_{|z-\frac{\pi}{2}|=\pi+1} z \cdot \tan(z) \, dz \).

Answer: -3\pi^2 i


Question: Find the radius of convergence of the power series of the function \( \frac{1}{z^2+z+1} \) around \( z=1 \).

Answer: \sqrt{3}


Question: Compute \( \int_{\gamma} \frac{z}{z^3-1} \, dz \) where \( \gamma(t) = 2e^{it} \), \( t \in [0, 2\pi] \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{(x,y)\rightarrow (0,0)} \frac{\tan(x^2+y^2)}{\arctan\left(\frac{1}{x^2+y^2}\right)} \]

Answer: 0


Question: Let $\ell_1$ and $\ell_2$ be two lines passing through the point $M_0 = (1,1,0)$ that lie on the hyperboloid $x^2 + y^2 - 3z^2 = 2$. Calculate the cosine of the angle between the two lines.

Answer: \dfrac{1}{2}


Question: Compute $10^{1001} \pmod{1001}$. Use Euler's theorem and any other necessary techniques to find the result.

Answer: 901


Question: Determine the type and order of the singularity at \( z=0 \) for the function \( \frac{1}{z(e^z-1)} \).

Answer: 2


Question: Evaluate the integral \( \frac{1}{\sqrt{2\pi t}} \int_{\mathbb{R}} e^{-\frac{(x-iut)^2}{2t}} \, dx \). Is the result equal to 1?

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \to 0}\frac{(e^{2\tan(x)}-1) \cdot \ln(2-\cos^2(x))}{\sqrt{1+x^3}-(\cos x)} \] using notable limits.

Answer: 0


Question: Given the sequence \((a_{n})_{n \in \mathbb{N}}\) with \(a_{0} > 0\) and the recurrence relation \(a_{n+1} = a_{n} + \frac{1}{a_{n}^2 + a_{n} + 1}\), find the limit \(\lim_{n \to \infty} \frac{a_{n}^3}{n}\).

Answer: 3


Question: Evaluate the limit \( \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{3n} \left(\frac{k}{n}\right)^2 \) by expressing it as a definite integral.

Answer: 9


Question: Given the joint density function \( f(x, y)=\begin{cases} \dfrac{x^2+y^2}{50} & 0<x<2, 1<y<4, \\\ 0 & \text{elsewhere.} \end{cases}\), calculate the probability \( P(X+Y>4) \).

Answer: \dfrac{8}{15}


Question: Let \( f(x) = \begin{cases} 0 & \text{if } -1 \leq x \leq 0 \\ x^4 & \text{if } 0 < x \leq 1 \end{cases} \). If \( f(x) = \sum_{k=0}^{n} \frac{f^{(k)}(0)}{k!}x^{k} + \frac{f^{(n+1)}(\xi)}{(n+1)!}x^{n+1} \) is the Taylor's formula for \( f \) about \( x=0 \) with the maximum possible value of \( n \), find the value of \( \xi \) for \( 0 < x \leq 1 \).

Answer: \xi \in (0, x)


Question: Let $X$ be a totally ordered set and $A \subset X$. Define $B$ as the upper closure of $A$. Prove that $\inf B$ exists if and only if $\inf A$ exists, and in that case, $\inf A = \inf B$. Provide a detailed proof of this statement.

Answer: \inf A = \inf B


Question: In a circle of 100 balls, 3 are red and 97 are blue. The balls are randomly placed in the circle. Determine the most probable minimum interval between the red balls.

Answer: 0


Question: Prove that \( \dfrac{1}{2\pi i} \oint_{C}\dfrac{f'(z)}{f(z)-f_{0}}dz=N \), where \( N \) is the number of points \( z \) such that \( f(z)=f_{0} \) inside the contour \( C \). Assume that \( f'(z) \) and \( f(z) \) are analytic inside and on \( C \), and that \( f(z) \neq f_{0} \) on the boundary of \( C \).

Answer: N


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

Answer: 0


Question: Evaluate the rational map \( \hat{\phi} : \mathbb{P}^2 \setminus \{(1:0:0),(0:1:0),(0:0:1)\} \to \mathbb{P}^2 \) given by \( \phi:(t_0:t_1:t_2) \mapsto (\frac{1}{t_0}:\frac{1}{t_1}:\frac{1}{t_2}) \) at the point \((1:1:0)\).

Answer: (0:0:1)


Question: Consider a directed graph $G$ where for every pair of vertices $u$ and $v$, there is either a directed edge from $u$ to $v$ or from $v$ to $u$, but not both. Starting from the vertex $v_1$ with the most outgoing edges, is it possible to find a path that visits all vertices exactly once, such that the path is $v_1, v_2, \ldots, v_n$ and there is a directed edge $(v_i, v_{i+1})$ for $1 \leq i < n$? (Here, $n$ is the total number of vertices in the graph.)

Answer: Yes


Question: Using Green's Theorem, evaluate the line integral \( I = \int_C \vec{F} \cdot d\vec{r} \) where \( \vec{F}(x,y) = \langle y, xy \rangle \) and \( C \) is the unit circle centered at the origin, oriented counterclockwise. Determine the appropriate bounds for the double integral \( \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA \) and evaluate the integral.

Answer: -\pi


Question: Find the inverse Laplace transform of the function given by the integral: \[ f(t) = \int_t^\infty \frac{e^{-u}}{u} \, du \]

Answer: \dfrac{u(t - 1)}{t}


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

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


Question:
Determine the logical relationship between the following two statements for an orthonormal system \((e_i)_{i \in I}\) in an inner product space \((X, \langle \cdot , \cdot \rangle)\):

(i) \(\text{span}\{e_i:i \in I\}\) is dense in \(X\);
(ii) If \(\langle x,e_i \rangle=0\) for some \(x \in X\) and all \(i \in I\), then \(x=0\).

Replace \(\star\) with either \(\Rightarrow\), \(\Leftarrow\), or \(\iff\) to make the statement "(i) \(\star\) (ii)" true. If your choice is \(\Rightarrow\) (or \(\Leftarrow\)), provide an example where \(\Leftarrow\) (or \(\Rightarrow\)) fails. Consider the span of \((1/j)\) and \(e_n\), \(n\geq 2\) in \(l^2\) as a hint.

Answer: \Rightarrow


Question: Given a round table with $n$ seats, define $s(n)$ as the smallest number of people such that the probability of at least two people sitting next to each other is at least 0.5 when they choose their seats randomly. Determine whether $\lim_{n \to \infty} \frac{s(n)}{n}$ exists, and if so, find its value.

Answer: 0


Question: Suppose $S\subset{\mathbb R}^n$ is an infinite subset in general position, meaning the intersection of $S$ with every affine subspace of dimension $d<n$ contains at most $(d+1)$ points. Determine if there exist two distinct points $a,b\in S$ such that the point $(a+b)/2$ lies in the interior of the convex hull of $S$. Provide a justification for your answer.

Answer: Yes