Question: Alice sends emails according to a Poisson process with rate \( \lambda_A \) per hour. Let \( Y_1 \) be the time at which Alice's first email is sent. You arrive at time 1 and are informed that Alice has sent exactly one email by this time. What is the conditional expectation of \( Y_1 \) given this information?

Answer: \dfrac{1}{2}


Question: Compute the limit: \[ \lim_{n\to\infty}\int_0^{\frac{n}{2}}\left(1-\frac{2x}{n}\right)^n \, d\lambda(x) \] where \( \lambda \) is the Lebesgue measure on \( \mathbb{R} \).

Answer: \dfrac{1}{2}


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

Answer: \emptyset


Question: Solve the inequality: \[ 2\log_3(x^2-4) + 3\sqrt{\log_3(x+2)^2} - \log_3(x-2)^2 \leq 4 \]

Answer: [-2 - \sqrt{3}, -3]


Question: Find all integer solutions \((x, y, z)\) to the equation \(\overline{xyz} = \frac{3}{2} x! y! z!\), where \(\overline{xyz}\) represents the three-digit number formed by the digits \(x, y, z\).

Answer: (4, 3, 2)


Question: If you have two infinite product measures, $P$ and $Q$, and a single $B^{\infty }$-measurable random variable $X$ (where $B^{\infty}$ is the tail $\sigma$-field) such that $P(X=c)=1$ and $Q(X=d)=1$, must $c=d$?

Answer: No


Question:
Solve the following system of congruences:
\[\begin{align*}
x &\equiv 1 \pmod{2} \\
x &\equiv 2 \pmod{3} \\
x &\equiv 3 \pmod{4} \\
x &\equiv 4 \pmod{5} \\
x &\equiv 5 \pmod{6} \\
x &\equiv 0 \pmod{7}
\end{align*}\]
Find the smallest non-negative solution for \(x\).

Answer: 119


Question: Compute the integral \( \int_{\gamma} \frac{f(z)}{z^3}dz \), where \( f(z)=az^3+bz^2+cz+d \) and \( \gamma(t)=e^{it} \) for \( t \in [0, 4\pi] \).

Answer: 4\pi i b


Question: Suppose $X_n$ and $Y_n$ are sequences of random variables defined on a common probability space such that $X_n \leq Y_n$ almost surely, and both $X_n$ and $Y_n$ converge weakly to the same distribution $L$ as $n \to \infty$. Is it true that $Y_n - X_n$ converges weakly to 0?

Answer: Yes


Question: Given a set $X$ and $n$ subsets $S_1, S_2, \ldots, S_n$ of $X$, each of cardinality $n$, can you always totally order each $S_i$ such that each $k$-th transverse set $T_k$ contains exactly $n$ distinct elements of $X$ for all $1 \leq k \leq n$? Here, $T_k$ is defined as the set containing the $k$-th smallest element from each ordered $S_i$. Assume some $S_i$ may be equal.

Answer: Yes


Question: Determine the number of integer values of \( y \) such that there are no more than 2021 integer values of \( x \) satisfying the inequality \( \log_2(x + y^2 + 1) - 3^{y^2 + y - 3x} < 0 \).

Answer: 78


Question: Let \( f \) be a function that is continuous on \([0, +\infty)\) and differentiable on \((0, +\infty)\). Given that \( f(0) = 0 \) and \(|f'(x)| < |f(x)|\) for all \( x > 0 \), prove that \( f(x) = 0 \) for all \( x \geq 0 \).

Answer: 0


Question: Let \( n \geq 2 \). Consider a sequence \( p_i = 2^{-k_i} \) where \( k_i \in \mathbb{N} \) such that \( p_1 \geq p_2 \geq \ldots \geq p_n \) and \( \sum_{i=1}^{n} p_i = 1 \). Prove that \( p_{n-1} = p_n \).

Answer: p_{n-1} = p_n


Question: Let \( \Omega \subset \mathbb{R}^N \) be a bounded domain and suppose that \( \mu_n \) is a sequence of non-negative Radon measures that converges to a Radon measure \( \mu \) in the weak star sense, i.e., \( \int_\Omega \phi \, d\mu_n \to \int_\Omega \phi \, d\mu \) for all \( \phi \in C_0(\Omega) \), where \( C_0(\Omega) \) denotes the set of continuous functions with compact support. Can you conclude that \( \mu \geq 0 \)?

Answer: \mu \geq 0


Question:
Given that $\sin q \ne \cos q$, and $x, y, z$ satisfy the following equations:
\[ x\cos p - y \sin p + z = \cos q + 1, \]
\[ x\sin p + y\cos p + z = 1 - \sin q, \]
\[ x\cos(p+q) - y\sin(p+q) + z = 2, \]
find the value of $x^2 + y^2 + z^2$.

Answer: 2


Question: Determine the number of times the prime number 5 appears in the prime factorization of the product \(1! \times 2! \times \cdots \times 100!\).

Answer: 1124


Question: Determine whether the following statement is true or false: Every subgroup of a simple group is itself simple. Provide a justification for your answer.

Answer: B


Question:
Determine if the following equality holds for a given set of parameters:
\[ \max_{i\in I}\max_{a_j \in P_j}\left\{\sum_j a_{ij}x_j - b_i\right\} = \max_{a_j \in P_j}\max_{i\in I}\left\{\sum_j a_{ij}x_j - b_i\right\} \]
where $P_j$ is a closed convex set, $I$ is a finite index set, $b_i$ is a known parameter, and $x_j$ is a nonnegative variable. The vector $a_j=(a_{1j},a_{2j},...,a_{mj})$ represents the column vectors of $A$. Provide a justification for your conclusion.

Answer: Yes


Question:
Determine whether the series given by the expression
\[
x_{k+1} = x_1 \sum^{\lfloor \frac{k}{2}\rfloor}_{m=0} \left(\frac{(-1)^m \alpha^{k-2m}}{m!} \prod^{2m-1}_{i=m}(k-i)\right) - x_0 \sum^{\lfloor \frac{k-1}{2}\rfloor}_{m=0} \left(\frac{(-1)^m \alpha^{k-1-2m}}{m!} \prod^{2m-1}_{i=m}(k-1-i)\right)
\]
converges or diverges for large \( k \) depending on the value of \( \alpha = 2 + h^2 \lambda \). Consider the stability of the system described by the matrix
\[
A = \begin{bmatrix}\alpha & -1 \\ 1 & 0\end{bmatrix}
\]
and its eigenvalues. Analyze the behavior of the system for different values of \( \alpha \) and determine the conditions under which the series converges.

Answer: |\alpha| < 2


Question: Given that \( y_1(t) = t \) is a solution to the differential equation \((t-1)y'' - ty' + y = 0\), find a second independent solution using the reduction of order method. Express the second solution in the form \( y_2(t) = v(t)t \) for some function \( v(t) \).

Answer: e^t


Question:
Calculate the volume of the solid bounded above by the paraboloid \( z = 16 - x^2 - y^2 \) and below by the plane \( z = 6 \). Use the fact that the volume \( V \) can be expressed as:
\[ V = \frac{1}{3} \int_{S} (x, y, z) \cdot \frac{\vec{n}}{||\vec{n}||} \, dS, \]
where \( S \) is the surface of the solid. Verify that the correct volume is \( 50\pi \).

Answer: 50\pi


Question: Evaluate the double integral \( \int\int xy \, dx\,dy \) over the square with corners \((0,0), (1,1), (2,0),\) and \((1,−1)\) using the change of variables \(x = \frac{u+v}{2}\) and \(y = \frac{u-v}{2}\). Compute the integral using the correct limits and Jacobian, and verify if the solution is zero.

Answer: 0


Question: Suppose you have a null set \( S \) in \( \mathbb{R}^n \). Determine whether there always exists an immersion \( i: \mathbb{R} \hookrightarrow \mathbb{R}^n \) such that for almost all \( x \in \mathbb{R} \), \( i(x) \) does not intersect \( S \) (i.e., \( i(\mathbb{R}) \cap S \) is a null set in \( i(\mathbb{R}) \)).

Answer: Yes


Question:
Given that \( \{e_i\}_{n=1}^{\infty} \) is an orthonormal sequence in a Hilbert space \( H \), and \( x \neq 0 \in H \), determine the value of \( \lim_{n\to\infty}\langle x,e_n\rangle \). Choose the correct option:

1. \( \lim_{n\to\infty}\langle x,e_n\rangle \) does not exist
2. \( \lim_{n\to\infty}\langle x,e_n\rangle = \|x\| \)
3. \( \lim_{n\to\infty}\langle x,e_n\rangle = 1 \)
4. \( \lim_{n\to\infty}\langle x,e_n\rangle = 0 \)

Answer: 4


Question:
Given three hypotheses about the defectiveness of a box:

- \( A \equiv \text{the box has } \frac{1}{3} \text{ defective items} \)
- \( B \equiv \text{the box has } \frac{1}{6} \text{ defective items} \)
- \( C \equiv \text{the box has } \frac{99}{100} \text{ defective items} \)

With prior probabilities given by:

\[ P(A|X) = \frac{1}{11}(1-10^{-6}) \]
\[ P(B|X) = \frac{10}{11}(1-10^{-6}) \]
\[ P(C|X) = 10^{-6} \]

The threshold fraction \( f_t \) is defined such that as the number of tests \( m \) goes to infinity with \( f = \frac{m_b}{m} \rightarrow \text{constant} \), the evidence \( e(C|DX) \) tends to \(+\infty\) if \( f > f_t \) and \(-\infty\) if \( f < f_t \). Here, \( m_b \) is the number of defective draws, \( D \) is the data, and \( e(A|B) \) is the evidence for \( A \) given \( B \) defined as:

\[ e(A|B) = 10 \log_{10} \frac{P(A|B)}{P(\overline{A}|B)} \]

Using the evidence formula for two hypotheses at a time:

\[ e(C|DX) = e(C|X) + 10 \log_{10} \frac{P(D|CX)}{P(D|\overline{C}X)} \]

where in a two-hypothesis case between \( C \) and \( A \), \( \overline{C} = A \). The evidence terms are \( b = 4.73 \) for a bad draw and \( g = -18.24 \) for a good draw. The total evidence added to the prior evidence is:

\[ bm_b + g(m - m_b) = (b-g)m_b + gm = m((b-g)f + g) \]

For this to be zero, \( f \) must be:

\[ f_t = -\frac{g}{b-g} \]

Calculate the threshold fraction \( f_t \).

Answer: 0.794


Question: Given the joint probability density function \( f(x, y) = e^{-(x+y)} \) for \( 0 \le x, y < \infty \), calculate the probability \( P(X < Y) \).

Answer: \dfrac{1}{2}


Question: Find the minimum value of \[ P = \frac{(x+1)^2(y+1)^2}{z^2+1} + \frac{(y+1)^2(z+1)^2}{x^2+1} + \frac{(z+1)^2(x+1)^2}{y^2+1} \] given that \( x > 0, y > 0, z > 0 \) and \( x + y + z = 3 \).

Answer: 24


Question: Let $M$ be a complete Riemannian manifold with sectional curvature $K$. Let $\tilde{M}$ be the simply connected complete Riemannian manifold of constant curvature $\kappa$ with $K \geq \kappa$ and $\dim M = \dim \tilde{M} = n$. Let $p, q \in M$ and $\gamma$ be a shortest geodesic from $p$ to $q$ with $L(\gamma) = r$, i.e., $\text{dist}(p, q) = r$. Determine if for an arbitrary $\tilde{p} \in \tilde{M}$, there exists $\tilde{q} \in \tilde{M}$ such that $\text{dist}(\tilde{p}, \tilde{q}) \geq r$. Provide a justification for your answer.

Answer: Yes


Question: Solve the differential equation \((x+y)\frac{dy}{dx} -(4x+y)=0\) using the substitution \(z=\frac{y}{x}\). Given the initial condition \(y(1)=2\), find the value of \(y(3)\).

Answer: 6


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a function such that \( |f(x) - f(y)| \le 6|x-y|^2 \) for all \( x, y \in \mathbb{R} \). Given that \( f(3) = 6 \), find the value of \( f(6) \).

Answer: 6


Question:
Determine the number of terms needed in the series expansion to approximate π to within $10^{-3}$ using the formula:

$$\frac{\pi}{4} = 4\arctan\left(\frac{1}{5}\right) - \arctan\left(\frac{1}{239}\right)$$

and the Maclaurin series for the arctangent function:

$$\arctan(x)=\sum_{i=1}^n\frac{(-1)^{i+1}x^{2i-1}}{2i-1}$$

Find the smallest integer $n$ such that the approximation error is less than $10^{-3}$. Use the inequality:

$$\frac{4}{2n+1} \left(\frac{4}{5^{2n+1}}-\frac{1}{239^{2n+1}}\right)<10^{-3}$$

Answer: 3


Question: Find the value of \( \sum_{n=1}^{2020} n^{2019} \pmod{2019} \).

Answer: 1


Question: Evaluate the integral \( \int_{0}^{2\pi} f(e^{it})\cos t \,\mathrm dt \) for \( f \) an analytic function.

Answer: \pi f'(0)


Question: Let $a$ be the remainder when $1124^{2017}$ is divided by $2017$. Find the value of $a^{2048}$ modulo $45$. Express your answer as a single integer.

Answer: 1


Question: Find the limit superior: \[ \limsup_{|z|\to\infty}\frac{\log|e^{-iz}|}{|z|} \] where \(z = x + iy\), and \(x, y \in \mathbb{R}\).

Answer: 1


Question: If a bounded operator between Banach spaces has a closed image, does its adjoint necessarily have a weak$^*$-closed image? Provide a justification for your answer.

Answer: Yes


Question: Find \( \lim_{n\to\infty} \cos\left(\frac{\pi}{4}\right) \cos\left(\frac{\pi}{8}\right)\ldots \cos\left(\frac{\pi}{2^n}\right) \).

Answer: \dfrac{2}{\pi}


Question: Let \( P(x) = x^5 + x^2 + 1 \) and suppose the roots of this polynomial are \( x_1, x_2, x_3, x_4, x_5 \). Define \( g(x) = x^2 - 2 \). Find the value of \( g(x_1)g(x_2)g(x_3)g(x_4)g(x_5) \).

Answer: -23


Question: Find the supremum of \( \int_0^1 f(t^a) \, dt \) subject to the condition \( \int_0^1 |f(t)| \, dt \leq 1 \), where \( 0 < a < 1 \) is fixed.

Answer: \dfrac{1}{a}


Question: Compute the limit: \[ \lim_{n \to \infty} \left( \frac{1}{2\sqrt{n}\log n} \frac{\sin(2\pi b\sqrt{n}\log n+\pi b)}{\sin(\pi b)} - \frac{1}{2\sqrt{n}\log n} \right) \] where \( b \in \mathbb{R} \).

Answer: 0


Question: Determine the range of values for $a$ such that the function \( f(x) = ax + \cos(2x) + \sin(x) + \cos(x) \) is strictly increasing for all real numbers $x$.

Answer: \left( \dfrac{17}{8}, \infty \right)


Question: Determine the last three digits of the number $2003^{2009} + 2009^{2017} + 2015^{2025}$ when expressed in base 3.

Answer: 110


Question: Determine whether the limit \( \lim_{n\to\infty}\underset{n}{\underbrace{\cos(\cos(...\cos x))}} \) exists. If it exists, find the limit.

Answer: 0.739085


Question: Determine the set of all real numbers $a$ such that the sequence $\{n^2 a^n\}_{n \in \mathbb{N}}$ is convergent.

Answer: (-1, 1)


Question: Calculate the area of the region that lies inside the cardioid given by \( r = 1 + \cos(\theta) \) and outside the circle given by \( r = \cos(\theta) \).

Answer: \dfrac{5\pi}{4}


Question: Determine the smallest integer \(d\) such that every simple finite planar graph can be oriented so that the out-degree of each vertex is at most \(d\).

Answer: 3


Question: Find \( \lim_{x \to 0^-} x^x \).

Answer: 1


Question: Determine the stalk of the structure sheaf at a point $x$ in the projective line $X = \mathbb{P}_k^1$, where $k$ is an algebraically closed field. Specifically, if $x = (t-a) \in \text{Spec } k[t]$, what is $\mathcal{O}_{X,x}$?

Answer: k[t]_{(t - a)}


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

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


Question: Find the volume of the region bounded by the hemisphere \( y=\sqrt{4-x^2-z^2} \) and the planes \( y=x \) and \( y=\sqrt{3}x \) using spherical polar coordinates.

Answer: \dfrac{4\pi}{9}


Question: Let $w$ be a linear transformation on an $n$-dimensional space $E$ such that $w^2 = i$. Prove that $\det(w) = (-1)^r$, where $r$ is the rank of the map $i - w$.

Answer: (-1)^r


Question: Determine the number of turns the curve described by the function \( f(z) = z^{40} + 4 \) makes about the origin in the complex plane when \( z \) traverses the circle \( |z| = 2 \) once.

Answer: 40


Question: Determine the probability that the sum \( S_n = \sum_{i=1}^n X_i \) equals zero for infinitely many \( n \). Express this probability as \( \mathbb{P} [S_n=0 \text{ for infinitely many } n] \).

Answer: 1


Question: Find the limit of the expression \( x\ln\left[\exp(a/x)+\exp(b/x)\right] \) as \( x \) approaches zero.

Answer: \max(a, b)


Question: Given the joint density function \(f_{X,Y}(x,y)=6e^{-x}e^{-2y}\) for \(0<x<y< \infty\), where \(X\) and \(Y\) are the fail times for two components of a machine, find the expected time of failure of the machine. The machine fails when component \(Y\) fails.

Answer: \dfrac{5}{6}


Question: Let $g(x)$ be a polynomial satisfying the relation \( g(x)g(y) = g(x) + g(y) + g(xy) - 2 \) for all real values of $x$ and $y$. Given that \( g(2) = 5 \), find the value of \( g(3) \).

Answer: 10


Question: Let \( A \in M^{\Bbb C}_{n \times n} \) be a square matrix with a minimal polynomial of degree \( k \). Determine the dimension of \( W = \operatorname{span} \{ I, A, A^2, A^3, \ldots, A^m, \ldots \} \).

Answer: k


Question: Given a Poisson process with a rate of 16.6 meteorites per minute, and knowing that 7 meteorites were observed during the first minute, calculate the expected time until the 10th meteorite is observed.

Answer: \dfrac{98}{83}


Question: Is every function that is continuously differentiable on a compact subset \( K \subseteq \mathbb{R}^n \) also Lipschitz on that compact set?

Answer: Yes


Question:
Evaluate the line integral \( \int_{\gamma_1 \cup \gamma_2} x \, dx + x^2 y \, dy \) using Green's theorem, where:

- \( \gamma_1(t) = (2\cos t, 2\sin t) \), \( t \in \left[-\frac{\pi}{2}, \pi\right] \)
- \( \gamma_2(t) = (\cos t, \sin t) \), \( t \in \left[-\frac{\pi}{2}, \pi\right] \)

The region enclosed by these curves forms three-quarters of a disc. Calculate the integral and verify the result.

Answer: -\dfrac{7}{4}


Question: Determine the dimension of the splitting field of the polynomial \( f(x) = x^3 - x + 1 \) over \( \mathbb{Q} \).

Answer: 6


Question: Given that \(3 > b > 0\) and \(a + b = 3\), find the minimum value of the expression \((3^a + 3^{2b} + 3^{2-b})^3\).

Answer: 8748


Question: Evaluate the complex integral \( \int_{|z|=3} \frac{1}{z+2} \, dz \).

Answer: 2\pi i


Question: Consider the set \( M = \{x \in \mathbb{R}^n : x^{⊤} A x = r\} \subset \mathbb{R}^n \), where \( r > 0 \) and \( A \) is a symmetric, positive semidefinite matrix \( A \in \mathbb{R}^{n \times n} \). Determine if \( M \) is a \( C^l \)-submanifold of \( \mathbb{R}^n \). If it is, find its dimension.

Answer: n - 1


Question: Given the joint density function \( f_{XY}(x,y) = 6xy \) for \( 0 \leq x \leq 1 \) and \( 0 \leq y \leq \sqrt{x} \), calculate the expectation \( \mathbb{E}(Y) \).

Answer: \dfrac{4}{7}


Question: Let $k$ be a field, and $R$ a $k$-algebra. Suppose that for every prime ideal $\mathfrak{p}$ of $R$, the localization $R_{\mathfrak{p}}$ is a finitely generated $k$-algebra. Does this imply that $R$ is a finitely generated $k$-algebra?

Answer: Yes


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

Answer: 1


Question: Given a real-valued function $f$ such that $f(x+y) = f(x)f(y)$ for all $x, y \in \mathbb{R}$ and $f(2) = 5$, find the value of $f(5)$. \( \boxed{} \)

Answer: 25\sqrt{5}


Question: Determine the limits of integration for the volume enclosed by the paraboloid $z = x^2 + y^2$ and the plane $z = 3 - 2y$. Evaluate the triple integral to show that the volume is $8\pi$. Use the hint that $\cos^4\theta = \frac{1}{8}\cos4\theta + \frac{1}{2}\cos2\theta + \frac{3}{8}$ if necessary.

Answer: 8\pi


Question:
Verify the following identity for any uniformly bounded function $f$:

Let $\mathcal{F}$ be the Fourier transform, $P$ the law of a random variable $X$, and $\varphi$ the characteristic function of $X$. Show that:
\[ \int_{\mathbb{R}} f \ast \mathcal{F}^{-1}\left[\frac{1}{\varphi}(-\bullet)\right](x) P(dx) = f(0) \]

Answer: f(0)


Question: Provide an example of a locally nilpotent group that is not nilpotent.

Answer: UT(\infty, \mathbb{Z})


Question: Suppose $G$ is a group with exactly 8 elements of order 10. How many cyclic subgroups of order 10 does $G$ have?

Answer: 2


Question:
Given the equations:

1. $a^2x^3 + b^2y^3 + c^2z^3 = p^5$
2. $ax^2 = by^2 = cz^2$
3. $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{p}$

Express $\sqrt{a} + \sqrt{b} + \sqrt{c}$ in terms of $p$.

Answer: \sqrt{p}


Question: Calculate the expectation \( \tilde{E}\left(B_t - \int_0^t B_s \,ds\right) \) where \( B \) is a standard Brownian motion with \( B_0 = 0 \) under the probability measure \( P \), and \( \tilde{B} = B_t - \int_0^t B_s ds \) for \( t \in [0,T] \), \( T>0 \). The expectation \( \tilde{E} \) is taken under the probability measure given by Girsanov's Theorem. Is this expectation zero? If not, calculate it explicitly.

Answer: 0


Question: Find the derivative of the function \( f(x) = \sqrt[3]{(x-1)(x-2021)}|\sin(\pi x)| \) at \( x = 1 \).

Answer: 0


Question: Consider a category $\mathcal{C}$ where every fiber product exists. Let $S' \rightarrow S$ be a morphism in $\mathcal{C}$. Define the slice categories $(\mathcal{C}\downarrow S)$ and $(\mathcal{C}\downarrow S')$. Let $F\colon (\mathcal{C}\downarrow S) \rightarrow (\mathcal{C}\downarrow S')$ be the functor given by $F(X) = X\times_S S'$. Does the functor $F$ preserve limits?

Answer: Yes


Question: Compute the limit: \[ \lim _{ n\to \infty } \sum _{ r=1 }^{ n }{ \frac { \lfloor 2rx \rfloor }{ n^{ 2 } } } \] where \( \lfloor \cdot \rfloor \) denotes the greatest integer function.

Answer: x


Question: Find the total number of right-angled triangles with integer sides whose inradius is 2013.

Answer: 27


Question: Consider a group $G$ with a finite symmetric generating set $S$. Let $g \in G$ be an element of infinite order. The asymptotic translation length of $g$ on the Cayley graph $Cay(G,S)$ is defined as \( \tau(g) = \lim_{n \to \infty} \frac{||g^n||}{n} \), where \(||.||\) denotes the $S$-word length. Is it possible for $g$ to have infinite order while \( \tau(g) = 0 \)?

Answer: Yes


Question: Given the area \(D\) bounded by \(y^2-x^2=1\), \(y^2-x^2=4\), \(x=-\frac{y}{2}\), and \(x=\frac{y}{2}\), find the value of the integral \(\iint_D \frac{y^2-x^2}{y^2} \, dA\) using the change of variables \(u = y^2 - x^2\) and \(v = \frac{x}{y}\). Calculate the Jacobian determinant and express \(du \, dv\) in terms of \(dx \, dy\). Verify that \(|2(v^2 - 1)| = 2(1-v^2)\) in this context.

Answer: \dfrac{3}{2}


Question: Given a Markov matrix \(A=\begin{bmatrix} \lambda_1 & 1-\lambda_1\\ 1-\lambda_2 & \lambda_2 \end{bmatrix}\), calculate the sum of the series \(I + A + A^2 + A^3 + \ldots\). Assume that \(|A| < 1\) and both \(\lambda_1\) and \(\lambda_2\) are probabilities.

Answer: \frac{1}{2 - \lambda_1 - \lambda_2} \begin{bmatrix} 1 - \lambda_2 & 1 - \lambda_1 \\ 1 - \lambda_2 & 1 - \lambda_1 \end{bmatrix}


Question: Given that \( P(1) + P(2) + \ldots + P(n) = n^7 + \lambda \) for all positive integers \( n \), where \( \lambda \in \mathbb{R} \) and \( P \) is a polynomial in \( \mathbb{R}[X] \), find the value of \( P(\lambda) \).

Answer: 1


Question: Determine the domain of \( z \) such that the Möbius transformation \( w = T(z) = \frac{1+z}{1-z} \) maps to the region \( \{ \Re(w) > 0 \} \).

Answer: |z| < 1


Question: Find the number of all seven-digit numbers that start with the digit 1 and end with the digit 9, such that the difference between any two adjacent digits is either 1 or 2.

Answer: 44


Question: Find the limit of the sequence \(\left(\frac{1}{2},\frac{1}{2}\right),\left(\frac{2}{3},\frac{2}{3}\right),\ldots,\left(\frac{n}{n+1},\frac{n}{n+1}\right),\ldots\).

Answer: (1, 1)


Question: Let $x_i$ be positive integers for $i=1,2,...,11$. If $x_i + x_{i+1} \geq 100$ and $|x_i - x_{i+1}| \geq 20$ for $i=1,2,...,10$, and $x_{11} + x_1 \geq 100$ and $|x_{11} - x_1| \geq 20$, what is the minimum possible value of $\sum_{i=1}^{11} x_i$?

Answer: 580


Question: Determine if the surface defined by the equation \( xy - y \log(z) + \sin(xz) = 0 \) can be expressed in the form \( z = f(x,y) \) near the point \((0,2,1)\).

Answer: Yes


Question: Consider a sequence \((X_i)_{1 \leq i \leq n}\) of independent and identically distributed random variables such that \(P(X_i = 1) = P(X_i = -1) = 1/2\). Is it necessary for the sample space \(\Omega\) to have at least \(2^n\) elements to support such a sequence of random variables?

Answer: Yes


Question: Determine the set of values for \( k \) such that the inequality \( 9^x - k \cdot 3^x - k + 3 \leq 0 \) is satisfied for at least one real \( x \) in the interval \((0,1)\).

Answer: (2, \infty)


Question:
Determine whether there exists a random variable $X$ such that:

1. $X \geq 0$ almost surely,
2. $E[X] \leq \epsilon$ for some $\epsilon > 0$,
3. $E[X^2] \geq 1$,
4. $E[X^3] \leq 10^{100}$.

Provide a justification for your answer.

Answer: Yes


Question: Given the equation \( 1a_1 + 2a_2 + \cdots + 20a_{20} = 1 \), where the \( a_j \) are real numbers, find the value of \( a_{12} \) that minimizes the expression \( 1a_1^2 + 2a_2^2 + \cdots + 20a_{20}^2 \).

Answer: \dfrac{1}{210}


Question: Determine the range of \( \alpha \) for which the improper double integral \[ I = \iint_{x^2+y^2>0} \frac{\sin(x^2+y^2)}{(x^2+4y^2)^{\alpha}}\, dx\, dy \] is convergent.

Answer: 0 < \alpha < 2


Question: Calculate the limit of the multivariate function \( f(x_1, x_2, \ldots, x_m) = \frac{1}{(1+\sum_{i=1}^{m}x_i)^{\sum_{i=1}^{m+1}a_i}}\Big(\prod_{i=1}^{m}b_i^{x_i}\Big)\Big(\prod_{i=1}^mx_i^{a_i}\Big) \) as \( x_1, x_2, \ldots, x_m \to \infty \), where \( a_1, \ldots, a_{m+1} \) are positive real numbers and \( 0 < b_i < 1 \) for \( i=1, 2, \ldots, m \).

Answer: 0


Question: Suppose $x_i \geq 0$ for $i = 1, \ldots, n$ and $\frac{1}{n}\sum_{i=1}^n x_i \to s > 0$ as $n \to \infty$. Does this imply that $\frac{\max_{i \le n} x_i}{\sum_{i=1}^n x_i} \to 0$ as $n \to \infty$? Determine whether this statement is true or false.

Answer: False


Question: Count the number of ways to choose distinct subsets $A_0, A_1, \ldots, A_n$ of the set $\{1, 2, \ldots, n\}$ such that $A_0 \subset A_1 \subset \ldots \subset A_n$. Express your answer in terms of $n$.

Answer: (n+2)^n


Question: Let $\Pi$ be the plane through the point $P$ with position vector $\mathbf{p} = (1,2,3)$ parallel to both $\mathbf{a} = (3,0,2)$ and $\mathbf{b} = (1,-2,1)$. Determine which one of the following vector equations is true for every point $V$ (with position vector $\mathbf{v}$) in the plane $\Pi$: $(A) ~ \mathbf{v} \times \mathbf{p} = \lambda \mathbf{a}+\mu\mathbf{b}; ~(B) ~\mathbf{v} = \mathbf{p}+\mu (\mathbf{a}+\mathbf{b});~ (C) ~ \mathbf{v} \cdot \mathbf{p} = \mathbf{p}\cdot(\mathbf{a}+\mathbf{b});~(D)~ (\mathbf{v} - \mathbf{p})\cdot (\mathbf{a} \times \mathbf{b}) = 0~(E) ~ \mathbf{v} = \lambda \mathbf{p}+\mu (\mathbf{a}+\mathbf{b}); ~ (F) ~ \text{none of these}.$

Answer: D


Question:
Does there exist a function \(f:\mathbb{R}^n\rightarrow\mathbb{R}\) such that all the \(n!\) iterated limits
\[\lim_{x_{\sigma(1)} \to 0} \lim_{x_{\sigma(2)}\to 0} \dots \lim_{x_{\sigma(n)} \to 0} f(x_{\sigma(1)},x_{\sigma(2)},\dots,x_{\sigma(n)})\]
(where \(\sigma\) is a permutation of \(1,2, \dots, n \)) are finite and take different values?

Answer: Yes


Question: Evaluate the line integral \( \int _C F \cdot dr \), where \( F(x,y) = xi + e^{y^2}j \) and \( C \) is the curve given by \( r(t) = \frac{1}{t^2+1}i + \left(3+10t^2\cos\left(\frac{\pi t}{2}\right)\right)j \) for \( 0 \le t \le 1 \).

Answer: -\dfrac{3}{8}


Question:
Verify which of the following expressions for \( \frac{\partial B}{\partial T} \) is correct when applied to the Black-Body radiation law by Planck:

1. \( \frac{\partial B}{\partial T} = \frac{c}{(e^\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}e^\frac{hf}{kT} \)

2. \( \frac{\partial B}{\partial T} = \frac{2h^2f^4}{kc^2}\frac{1}{T^2}\frac{e^\frac{hf}{kT}}{\left(e^\frac{hf}{kT}-1\right)^2} \)

The Black-Body radiation law by Planck is given by:
\[ B(f,T)=\frac{2hf^3}{c^2\left(e^\frac{hf}{kT}-1\right)} \]

Answer: 2


Question: Evaluate the integral: \[ \int_{0}^{\pi} e^{\cos t}\,dt \]

Answer: \pi I_0(1)


Question: Determine the number of Polish group topologies that can be placed on \( \text{PSL}_2(\mathbb{C}) \).

Answer: 1


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

Answer: 1


Question: Determine the number of connected components of the complement of a closed curve in the plane. Let \( \gamma:[0,1] \rightarrow \mathbb{R}^2 \) be a continuous, closed curve such that \( \gamma(0) = \gamma(1) \). Find the number of connected components of the set \( \mathbb{R}^2 \setminus \gamma \), where \( \gamma \) denotes the image of the curve.

Answer: 2


Question: Does there exist an infinite non-abelian group such that all its irreducible characters are linear?

Answer: No


Question: Evaluate the double summation \( \sum_{j=0}^{n}\sum_{i=j}^{n} {n \choose i}{i \choose j} \).

Answer: 3^n


Question: Prove that the fraction field of the integral domain $F[X,Y]/(f)$ is isomorphic to $F(X)[Y]/(f)$, where $F$ is a field and $f$ is an irreducible polynomial in $F[X,Y]$ involving the variable $Y$. Assume $f$ is also irreducible in $F(X)[Y]$.

Answer: F(X)[Y]/(f)


Question: Determine if the ratio of the volume of a standard simplex in \(\mathbb{R}^n\) to the volume of its dual, where the dual's vertices are the Euclidean barycenters of the faces of the simplex, is \(\frac{1}{n^n}\).

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


Question: Calculate the sum of the series \( \sum^{\infty}_{n=2}\ln\left(1- \frac{1}{n^2}\right) \).

Answer: -\ln 2


Question: How many ordered triples \((x, y, z)\) of integer solutions are there to the following system of equations? \[ \begin{align} x^2+y^2+z^2&=194 \\ x^2z^2+y^2z^2&=4225 \end{align} \]

Answer: 48


Question: Find all solutions to the congruence equation \(x^7 \equiv -2 \pmod{101}\), given that \(2\) is a primitive root modulo \(101\).

Answer: 15


Question: Evaluate the limit: $$\lim_{x\rightarrow 0, y\rightarrow 0}(x+y) \sin\frac{1}{x} \sin\frac{1}{y}$$

Answer: 0


Question:
Let \( \Gamma \) be a set of predicate formulas, and let \( A, B \) be two predicate formulas. If the theory \( \Gamma \cup \{ \sim (A \wedge B) \} \) is consistent, determine which of the following is true:

1) At most one of \( \Gamma \cup \{ \sim (A \wedge B) \} \) or \( \Gamma \cup \{ \sim (A \vee B) \} \) is consistent.
2) At least one of \( \Gamma \cup \{ \sim A \} \) or \( \Gamma \cup \{ \sim B \} \) is consistent.

Answer: 2


Question: Evaluate the integral \( \int_{-1}^{1} \frac{e^{-1/x} \, dx}{x^2(1+e^{-2/x})} \).

Answer: \pi - 2\arctan(e)


Question: Find the expectation of the random variable \( \frac{X_i}{\bar{X}} \), where \( X_i \) are IID random variables and \( \bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i \). Express \( E\left[\frac{X_i}{\bar{X}}\right] \) in terms of \( n \) and the distribution of \( X_i \).

Answer: 1


Question: Compute the sum of the infinite series: \[ \sum_{n=1}^\infty \frac{5 + 4n - 1}{3^{2n+1}}. \]

Answer: \dfrac{17}{48}


Question: Determine whether there exists an integer \( N > 0 \) such that the equation \( \varphi(n) = N \) has infinitely many solutions for \( n > 0 \), where \( \varphi \) is the Euler's totient function.

Answer: No


Question: Evaluate the integral \( \int_{\gamma} \frac{e^{iz}}{z^2} \, dz \) where \( \gamma(t) = e^{it} \) for \( 0 \le t \le 2\pi \).

Answer: -2\pi


Question: If $a_n$ converges in probability to $0$ but $b_n$ does not have a probability limit, does $a_n + b_n$ converge in probability to $0$?

Answer: No


Question: Evaluate the integral \(\int _{|z|=1} \left(\frac{e^z}{z+2} - 3 \bar{z}\right) dz\).

Answer: -6\pi i


Question: Given a vector space $X$ with a norm induced by an inner product, where $x, y \in X$, and the conditions $||3x+y|| = 2$ and $||x+3y|| = 2$, calculate the value of $4||x+y||^2 + ||x-y||^2$. Use the identity $||x+y||^2 + ||x-y||^2 = 2(||x||^2 + ||y||^2)$ to assist in your calculation.

Answer: 4


Question: Let $K$ be a field, and let $a_1, a_2, \ldots, a_n$ be algebraic over $K$. Consider the extension $K[a_1, \ldots, a_n]$. If $a_1, a_2, \ldots, a_n$ are linearly independent, non-zero, and $c_1, c_2, \ldots, c_n \in K$ with $c_i \neq 0$ for all $i$, is it possible for the extension $K[c_1a_1 + c_2a_2 + \ldots + c_na_n]$ to be different from $K[a_1, \ldots, a_n]$?

Answer: Yes


Question: Find the limit: \[ \lim_{n \to \infty} n \sin(2\pi en!) \] where \( e \) is the base of the natural logarithm.

Answer: 2\pi


Question: Consider a Jordan-Hölder filtration of a finite $R$-module $M$: $0 = M_0 \subset M_1 \subset \cdots \subset M_n = M$. Is it true that for every $i$, there exists a submodule $N_i \subset M$ such that $N_i \cong M_i/M_{i-1}$?

Answer: No


Question: Verify Stokes' theorem for the vector field \( \vec{a}=\vec{r}\times\hat{k} \), where \( \vec{r}=x\hat{i}+y\hat{j}+z\hat{k} \), and \( \lbrace\hat{i},\hat{j},\hat{k}\rbrace \) is the standard basis for \( \mathbb{R}^3 \). Consider the hemisphere \( C \) of radius \( c \) centered at \( (0,0,0) \) in the upper half region of \( \mathbb{R}^3 \) (i.e., \( z\geq0 \)).

Answer: -2\pi c^2


Question: Let $p, \ell_1, \ell_2$ be distinct prime numbers, and $x_1, x_2 \in \overline{\mathbf{Q}}^\times$. If \( \frac{\log_p x_1}{\log_p \ell_1} = \frac{\log_p x_2}{\log_p \ell_2} \), does it follow that both ratios \( \frac{\log_p x_i}{\log_p \ell_i} \) must be in \( \mathbf{Q} \)?

Answer: Yes


Question: Let \( U \subseteq \mathbb{R}^2 \) be an open, bounded, connected subset, and let \( f: U \to U \) be a smooth map such that \( Jf = 1 \) (i.e., \( f \) is area-preserving). Suppose that \( h = f \circ f \) is affine. Must \( f \) be affine?

Answer: Yes


Question: Solve the difference equation \( y_k = \frac{2}{7} y_{k+1} + \frac{2}{7} y_{k-1} + \frac{3}{7} y_{k-2} \) for \( k \geq -1 \), given the initial conditions \( y_0 = 1 \), \( y_{-1} = \frac{1}{3} \), and \( y_{-2} = \frac{1}{9} \).

Answer: 3^k


Question: Let $M$ be a $C^1$-embedded $n$-submanifold (without boundary) of $\mathbb{R}^{n+k}$. Determine whether it is true that for every compact set $K$ in $\mathbb{R}^{n+k}$, the $n$-dimensional Hausdorff measure of $M \cap K$ is finite.

Answer: True


Question: Evaluate the sum \( a_k = \sum_{j=\lceil k/2\rceil}^k {j \choose k-j} \) in closed form using a generating function.

Answer: F_{k+1}


Question: Is it true that for any integer \( N \geq 1 \), there exists a function \( f(N) \) such that for any integer \( x \geq f(N) \), the integers \( x+1, x+2, \ldots, x+N \) are multiplicatively independent? Provide a justification for your answer.

Answer: Yes


Question: Determine the number of zeros of the equation \( ze^{3-z} - 5^2 = 0 \) inside the unit disk.

Answer: 0


Question: If a closed plane curve has a total signed curvature of either $2\pi$ or $-2\pi$, must the curve be a simple closed plane curve? Answer yes or no and justify your answer.

Answer: No


Question: Determine the limit of the sequence \( \{x_n\} \) where \( x_n = \frac{M_n + a}{M_n(M_n + b)} \), given that \( M_n = o_p(\sqrt{n}) \), \( M_n > 0 \), and \( \mathbb{P}(\liminf\{M_n > K\}) = 1 \) for any integer \( K \). Here, \( a \) and \( b \) are positive constants.

Answer: 0


Question: Is it possible for a matrix $A$ with eigenvalue $\lambda$ that for some positive integer $n$, the inequality \( \text{dim}(N((A-\lambda I)^{n+1})) - \text{dim}(N((A-\lambda I)^{n})) > 1 \) holds? Justify your answer.

Answer: Yes


Question:
Let $a_1=1$, and $a_n=n(a_{n-1}+1)$ for all $n=2,3,\ldots$
Define \[P(n)=\left(1+\frac{1}{a_1}\right)\left(1+\frac{1}{a_2}\right)\cdots\left(1+\frac{1}{a_n}\right)\]
Compute \[\lim_{n\to\infty}P(n)\]

Answer: e


Question: Solve the equation \( \sqrt[5]{x^3-6x^2+9x} = \sqrt[3]{x^5+6x^2-9x} \) for \( x \in \mathbb{R} \).

Answer: 0


Question: Given a package of 10 potatoes weighing exactly 500 grams, let $X_1, \dots, X_{10}$ represent the weights of each potato. Are the random variables $X_1, \dots, X_{10}$ independent? Compute the correlation coefficient $\rho(X, Y)$ where $X = X_1$ and $Y = \sum_{i=2}^{10} X_i$. Use the formula $\rho = \frac{\text{cov}(X,Y)}{\sigma_X \sigma_Y}$, where $\text{cov}(X,Y) = E[XY] - E[X]E[Y]$. Assume the total weight constraint and independence conditions as necessary.

Answer: -1


Question: Find a lower bound for the expression \(||a-b|-|a+b||\) where \(a, b \in \mathbb{C}\) are complex numbers.

Answer: 0


Question: In an irreducible Markov diagram, you are stepping from vertex to vertex. At each step, you choose one of the four adjacent vertices to move to, each with equal likelihood and independent of history. If you start at vertex A, what is the expected number of steps until you reach vertex B?

Answer: 4


Question:
Let $A$ be a C*-algebra with no one-dimensional irreducible representations, meaning there is no closed two-sided ideal $I \subseteq A$ such that $A/I \cong \mathbb{C}$. Define $J$ as the two-sided ideal generated by additive commutators in $A$:
\[
J := \left\{ \sum_{k=1}^n a_k[b_k,c_k]d_k : a_k, b_k, c_k, d_k \in A \right\}.
\]
Is it true that $A = J$? Note that $J$ is a dense, two-sided ideal and $A/J$ is a commutative algebra.

Answer: A = J


Question:
Evaluate the limit:
\[ \lim \limits_{n \to \infty} \left(\sqrt[3]{\frac{1}{n^4}} + \sqrt[3]{\frac{2}{n^4}} + \sqrt[3]{\frac{3}{n^4}} + \dots + \sqrt[3]{\frac{n}{n^4}} \right) \]

Answer: \dfrac{3}{4}


Question: Is the exponential of a Jordan block always an upper triangular matrix? Provide a proof or counterexample.

Answer: Yes


Question: Find all values of the constant $k$ for which the function $u(x,y) = x^{3} - kxy^{2} + 12xy - 12x$ is the real part of a holomorphic function.

Answer: 3


Question: Find the solution set of the equation \(200x = \sum_{k=1}^{100}{\left( \left| x-k \right|+\left| x+k \right| \right)}\).

Answer: [100, \infty)


Question: Find the positive integer $N$ such that $N(N - 101)$ is a perfect square.

Answer: 2601


Question:
Determine the range of \(a\) such that the function \(f(x) = \left\lfloor \frac{(x-2)^3}{a} \right\rfloor \sin(x-2) + a \cos(x-2)\) is continuous and differentiable on the interval \([4,6]\). Choose the correct option:
(a) \(a \in [8,64]\)
(b) \(a \in (0,8]\)
(c) \(a \in [64, \infty)\)

Answer: C


Question: Evaluate the limit: $$\lim_{x\to 0^+}\cfrac{x^{x^x}}{x}.$$

Answer: 1


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

Answer: \dfrac{4}{e}


Question: Find the function \( f(x) \) that satisfies the equation \( x^2 f(x) + f(1-x) = 2x - x^4 \).

Answer: 1 - x^2


Question: Find all subgroups of the multiplicative group of non-zero real numbers, \(\mathbb{R}^*\), that have an index of 2.

Answer: \mathbb{R}^+


Question: Determine the smallest natural number $k$ such that the series \( \sum\limits_{n=1}^\infty\frac{(n!)^{k-1}(27n)^n}{(kn)!e^n} \) converges.

Answer: 4


Question: Find the value of the expression \( \frac{x^2+y^2+x+y-1}{xy-1} \) for positive integers \( x \) and \( y \) such that the expression is an integer.

Answer: 7


Question: Determine the number of integer values of $k$ for which the equation $x^3 - 27x + k = 0$ has at least two distinct integer roots.

Answer: 2


Question: Find the maximum natural number \( n \) such that the product of two consecutive positive integers is equal to \( n^2 + 10n + 3 \).

Answer: 17


Question: Find the x-coordinate of the intercept, other than $x=1$, between the circle $x^2 + y^2 = 1$ and the function $y = \log(x)$. Solve the equation $x^2 + (\log x)^2 = 1$.

Answer: 0.4


Question: Evaluate the integral \( \int_{-\infty}^{+\infty} \text{sinc}(x) \, dx \).

Answer: \pi


Question: Consider a sequence \( \{x_n\}_{n=0}^\infty \) such that \( \frac{\vert x_n\vert}{\vert x_{n-1}\vert^2}\leq C \), where \( C>0 \) is a constant independent of \( n \). If \( \vert x_0\vert<1 \), is it guaranteed that \( \lim_{n\to\infty}x_n=0 \)?

Answer: No


Question: Let $A$ and $B$ be matrices in $\mathbb{R}^{n\times n}$. If the matrix $A+tB$ is nilpotent for different real values $t_0, t_1, \ldots, t_n$, are the matrices $A$ and $B$ necessarily nilpotent?

Answer: Yes


Question: Let \(x_1, x_2, \ldots, x_{2560}\) be the roots of the polynomial \(x^{2560} + 2x^{2559} + 3x^{2558} + \cdots + 2561\). Find the value of the product \((x_1^2 + 1)(x_2^2 + 1) \cdots (x_{2560}^2 + 1)\).

Answer: 3279361


Question: Given the differential equation $y'(t) = \frac{t}{4} - \frac{1}{y^2 + 1}$ with the initial condition $y(0) = 1$, and the function $x(t) = 1 - \frac{t}{2}$, approximate $|x(t) - y(t)|$ for $|t| < \frac{1}{2}$ using a corollary of Gronwall's theorem.

Answer: \dfrac{t^3}{48}


Question: Find the minimum value of \(a + b + \frac{1}{ab}\) given that \(a\) and \(b\) are positive real numbers satisfying \(a^2 + b^2 = 1\).

Answer: 2 + \sqrt{2}


Question: Let \( f = (1\ 2\ 3\ 4\ 5\ 6) \circ (7\ 8\ 9\ 10) \in S_{10} \). Does there exist a positive integer \( n \) such that, when \( f^n \) is decomposed into disjoint cycles, one of the cycles has length 5? Justify your answer.

Answer: No


Question: Find the surface area of the region above the plane \(x + 2z + 1 = 0\) and below the surface defined by \(\sqrt{x^2 + y^2} + z = 1\) using a double integral. The region of integration in the x-y plane is an ellipse. Express the surface area in terms of a double integral and evaluate it.

Answer: 2\sqrt{6}\pi


Question: Determine whether the following statement is true: For a bounded function $F: \mathbb{R}_{\ge 0} \to \mathbb{R}$, the condition \( \int_0^\infty \frac{x^k s}{(s^2 + x^2)^{(k+3)/2}} F(x) \, dx = 0 \) for all \( s > 0 \) implies \( F \equiv 0 \), where \( k \in \mathbb{N} \) is a positive constant.

Answer: True


Question: Let $G_n$ be the subgroup of $S_{n^2}$ generated by the red arrow permutations as shown in the picture for $n = 5$. Is $G_n$ a strict subgroup of $S_{n^2}$?

Answer: Yes


Question: Evaluate the limit \( \lim_{n\rightarrow \infty}\binom{2n}{n}^{\frac{1}{n}} \) without using the limit as a sum or Stirling's approximation.

Answer: 4


Question: Find the volume of the solid obtained by rotating the region between the circles defined by $x^2 + y^2 = 16$ and $(x-4)^2 + y^2 = 16$ about the x-axis.

Answer: 32\pi


Question:
Given that $abc = (1-a)(1-b)(1-c)$ and $0 \le a,b,c \le 1$, find the minimum value of the expression:
\[ a(1-c) + b(1-a) + c(1-b) \]

Answer: \dfrac{3}{4}


Question: Find all non-negative integer solutions \((x, y)\) to the Diophantine equation \(2^x - 3^y = 2021\).

Answer: (11, 3)


Question: If a function \( f: [a,b] \to \mathbb{C} \) is Lebesgue measurable, does it necessarily imply that the function is integrable, i.e., a member of \( L^1([a,b]) \)?

Answer: No


Question: Let \( w = e^{\frac{4\pi i}{7}} \). Evaluate the expression \((2+w)(2+w^2)(2+w^3)(2+w^4)(2+w^5)(2+w^6)\).

Answer: 43


Question:
Solve the system of equations:
\[\begin{cases}
\sqrt{x^2+y^2}+\sqrt{x-y}=6 \\
x^2-y^2-x+y=12
\end{cases}\]

Answer: (4, 0)


Question: Suppose $n$ is the least integer such that a space $X$ may be embedded into $\mathbb{R}^n$. Is it true that for all $m > n$, $H_m(X) = 0$?

Answer: Yes


Question: Is a Jordan curve encircling a finite-perimeter set rectifiable? Let \(\gamma:[0,1]\rightarrow \mathbb{R}^2\) be a continuous simple closed curve (Jordan curve). Assume that the internal bounded set \(E\) has finite perimeter \(\text{Per}(E)\) in the sense of Caccioppoli sets. Is it true that the curve \(\gamma\) is rectifiable, meaning \(\text{Len}(\gamma)<+\infty\)?

Answer: Yes


Question:
Find the minimum length of cable required to connect an antenna to the points A, B, C, and D. The antenna must be placed on the same axis as point A, and its position can be between A and B, or behind either A or B. Use the relationship:

\[ D_1 + D_2 = \sqrt{2^2 + X^2} + \sqrt{(12-x)^2+1^2} \]

where \(D_1\) and \(D_2\) are the distances from the antenna to points A and B, respectively. Determine the optimal position of the antenna to minimize the total cable length.

Answer: 3\sqrt{17}


Question:
Solve the inequality for \(x\):
\[ \frac{x+c\sqrt{x}}{2x+c\sqrt{x+c^2}+c^2} \leq u \]
where \(x \in \mathbb{R}^{\geq 0}\), \(c \in \mathbb{R}^{\gt 1}\), and \(u \in \mathbb{R}^{\gt 0.5}\).

Answer: x \geq 0


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

Answer: -\dfrac{1}{4}


Question:
A crystal moves on a horizontal plane according to the law \( r(t) = (12t\hat{i} + ct^2\hat{j}) \, \text{m} \), where \( t \) is time in seconds and \( c \) is a positive constant with acceleration units. Given that the radius of curvature is \( 4 \, \text{m} \) at \( t = 0 \), find the tangential acceleration at \( t = 2 \, \text{s} \).

Choose the correct answer:
\[
\begin{array}{ll}
1.&\frac{216}{37}\sqrt{37}\,\frac{\text{m}}{\text{s}^2}\\
2.&216\sqrt{37}\,\frac{\text{m}}{\text{s}^2}\\
3.&\frac{\sqrt{37}}{37}\,\frac{\text{m}}{\text{s}^2}\\
4.&\frac{72}{37}\sqrt{37}\,\frac{\text{m}}{\text{s}^2}\\
5.&\frac{144}{37}\sqrt{37}\,\frac{\text{m}}{\text{s}^2}\\
\end{array}
\]

Answer: 1


Question: Find \( B \in \mathbb{R} \) such that \[ \lim_{n \rightarrow \infty} \left( \frac{2}{\sqrt{n^2 + 4n} - n} \right) ^{B(n+2)} \in \left( \frac{1}{2}, 2 \right). \]

Answer: (-\ln 2, \ln 2)


Question:
Consider a Hilbert space $H$ with an orthonormal basis $\{e_n : n \ge 1\}$. Let $T: H \to H$ be a bounded linear operator. Determine which of the following statements CANNOT be true:

(A) $T(e_n) = e_1$ for all $n$.

(B) $T(e_n) = e_{n+1}$ for all $n$.

(C) $T(e_n) = \sqrt{\frac{n+1}{n}} e_n$ for all $n$.

(D) $T(e_n) = e_{n-1}$ for all $n \ge 2$ and $T(e_1) = 0$.

Select the option that is not possible.

Answer: A


Question: Let $H = H_\mathbb{C}$ be a complex inner product space. Suppose $x, y \in H$ and for every $c \in \mathbb{C}$, the inequality $\|x\| \leq \|x + cy\|$ holds. Prove that $(x, y) = 0$, where the norm is defined as $\|x\|^2 := (x, x) \geq 0$.

Answer: (x, y) = 0


Question: Let $G = \{a_1, a_2, \ldots, a_n\}$ be an abelian group of order $n$. Show that $(a_1 a_2 \cdots a_n)^2 = e$, where $e$ is the identity element of the group.

Answer: e


Question: Evaluate the limit: $$\lim_{n\to\infty}n\sin(2\pi \mathrm{e}n!)$$

Answer: 2\pi


Question: Find \( \alpha \in \mathbb{Z}[i] \) such that the ideal \((3+5i,1+3i)\) is equal to the ideal \((\alpha)\).

Answer: 1 - i


Question: Determine the residue field of the valuation ring \( V = \{ r \in \mathbb{F}_q(t) : v(r) \ge 0 \} \) with respect to the valuation \(-\deg\), where \( \mathbb{F}_q(t) \) is the field of rational functions over the finite field \( \mathbb{F}_q \). Express the residue field \( V/M \) in a more explicit or familiar form.

Answer: \mathbb{F}_q


Question: Evaluate the integral \( \frac{1}{3} \int_{0}^{\pi} (8 + 8\cos\theta)^{\frac{3}{2}} \, d\theta \) to find the flux of the vector field \( F = (x\sqrt{y^2+z^2}, -z, y) \) through the closed surface defined by \( R = \{z^2 - 4z + y^2 \le 0, 0 \le x \le 1\} \).

Answer: \dfrac{256}{9}


Question: Let \(\mathcal{A}\) be an almost disjoint (AD) family of infinite subsets of \(\omega\), and let \(\mathcal{M}\) be a maximal almost disjoint (MAD) family such that \(\mathcal{A} \subseteq \mathcal{M}\). If \(\mathcal{A}\) is of true cardinality \(\mathfrak{c}\), is \(\mathcal{M}\) necessarily of true cardinality \(\mathfrak{c}\)? Provide a justification for your answer.

Answer: Yes


Question: Find the area of the circle described by the polar equation \( r = \cos(\phi) - \sin(\phi) \).

Answer: \dfrac{\pi}{2}


Question: Let $X$ be an affine scheme and $U \subset X$ be an affine open subset. Consider the inclusion map $i: U \to X$ and the induced morphism of rings $\phi: \mathcal{O}_X(X) \to \mathcal{O}_X(U)$. Is $\phi$ the restriction map?

Answer: Yes


Question: Let \( f: \mathbb{R}^n \to \mathbb{R}^m \) be a function with a continuous Jacobian \( J(f)(x) \) of rank \( m \) for all \( x \in \mathbb{R}^n \), where \( n > m \). Suppose \( E \subset \mathbb{R}^m \) is a set of Lebesgue measure 0. Is it true that \( f^{-1}(E) \) also has Lebesgue measure 0?

Answer: Yes


Question: In triangle $ABC$, point $P$ is inside the triangle such that lines $AP$, $BP$, and $CP$ intersect sides $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Given that $AP = a$, $BP = b$, $CP = c$, $PD = PE = PF = 3$, and $a + b + c = 43$, find the value of $abc$.

Answer: 441


Question:
Evaluate the limit:
\[ \lim_{n\to \infty}\sqrt[n]{\int_0^1\int_0^1\int_0^1\frac{x^n(1-x)^ny^n(1-y)^nz^n(1-z)^n}{[1-(1-xy)z]^{n+1}}d x dydz} \]

Answer: 17 - 12\sqrt{2}


Question:
Calculate the absolute value of the line integral:
\[ \oint_C (4z+2xy)dx + (x^2+z^2)dy+(2yz+x)dz \]
where \(C\) is the intersection of the surfaces \(z=\sqrt{x^2+y^2}\) and \(x^2+y^2 = 2y\). Find a suitable parameterization for the curve \(C\) and evaluate the integral.

Answer: 8


Question: Given the joint probability density function \( f_{X,Y}(x,y) = \begin{cases} 8xy, & \text{for } 0 < y < x < 1, \\ 0, & \text{elsewhere} \end{cases} \), find \( E(XY) \) using the conditional expectation \( E(Y|X) = \frac{2x}{3} \).

Answer: \dfrac{4}{9}


Question: Construct a line segment with a length of \( \sqrt[3]{2} \) using geometric methods.

Answer: \sqrt[3]{2}


Question: Given four independent random variables $X_1, X_2, X_3, X_4$ each uniformly distributed on the interval $(0,1)$, let $(X_{(1)}, X_{(2)}, X_{(3)}, X_{(4)})$ denote the order statistics. Compute the probability $P(X_{(2)} \leq 3X_{(1)})$. Use integration techniques to find your answer.

Answer: \dfrac{2}{3}


Question: Find the value of $u(1, 2)$, where $u(x, y)$ is the solution of the first order partial differential equation $$xu_x+(x^2+y)u_y=u$$ for all $x, y \in \mathbb{R}$, satisfying the initial condition $u(2, y) = y - 4$. Use the Lagrange method to solve the PDE.

Answer: 1


Question: Evaluate the expression \( \sum_{n=1}^{10}\sum_{m=1}^{10} \tan^{-1}\left(\frac{m}{n}\right) = k\pi \). Determine the total number of positive divisors of \( k \).

Answer: 3


Question: Evaluate the surface integral \( \int_{S} (x + y + z) \, dS \) where \( S \) is the boundary of the unit ball \( x^2 + y^2 + z^2 = 1 \).

Answer: 0


Question: Does $SL(2,\mathbb{Z})$ have a finite-dimensional faithful unitary representation?

Answer: No


Question:
Determine the value of \( a \) such that the following integral is equivalent to the given triple integral:

\[ \iiint \sqrt{x^2+y^2+z^2} \,dx \,dy \,dz \]

where the region is bounded by \( z=\sqrt{x^2+y^2} \) and \( z=4 \). After converting to spherical coordinates, the integral becomes:

\[ a\int_{0}^{\pi/4} \frac{(\sin \phi)d\phi}{\cos^4\phi} \]

Answer: 128\pi


Question: What is the probability that none of the urns is empty when distributing n balls into n urns? Express your answer in terms of n.

Answer: \dfrac{n!}{n^n}


Question: Determine the last two digits of the number \(17^{362}\).

Answer: 89


Question: Solve the ordinary differential equation \( y'' + ty' - y = 0 \) using the Laplace transform, given the initial conditions \( y(0) = 0 \) and \( y'(0) = 5 \).

Answer: 5t


Question: Find the minimum distance between the curves $y = e^x$ and $y = \ln x$.

Answer: \sqrt{2}


Question: Evaluate the limit: $$\lim_{x\rightarrow\infty} \frac{1}{x}\left(\frac{\Gamma(2x-1)}{\Gamma(x-1)}\right)^{1/x}$$

Answer: \dfrac{4}{e}


Question: Evaluate the definite integral: \[ \int _{-\pi}^{\pi}\frac{\sin(3x)}{\sin(x)}\frac{1}{1+2^x}  \, dx \]

Answer: \pi


Question: Let $E$ be an algebraic extension of a field $F$, and let $\sigma: F \to L$ be an embedding of $F$ into an algebraically closed field $L$, where $L$ is the algebraic closure of $\sigma F$. Does there exist an extension of $\sigma$ to an embedding of $E$ into $L$?

Answer: Yes


Question: Let \( X \) be a Banach space. Suppose there exists a sequence \( \{x_n\} \) in \( X \) such that for all finite \( A \subseteq \mathbb{N} \) and for all functions \( \alpha:A\to\{-1,1\} \), we have \( \|\sum_{n\in A}\alpha_nx_n\| = |A| \). Does this imply that the closure of the subspace spanned by \( \{x_n\} \) is isomorphic to \( \ell^1 \)?

Answer: Yes


Question: Find the convergence value of the series \( \sum_{i=1}^{\infty} i^2 \times (0.4)^i \).

Answer: \dfrac{70}{27}


Question: Find the multiplicative inverse of the coset \((1 + x) + (x^2 + x + 1)\mathbb{Q}[x]\) in the field \(\mathbb{Q}[x]/(x^2 + x + 1)\mathbb{Q}[x]\).

Answer: -x


Question: Given two complex numbers $z$ and $a$, where $z = (-1)a^{3\pi i}$ and the argument of $a$ is $\frac{\pi}{3}$, find the argument of $z$. Express your answer in terms of $\pi$. \( \text{arg}(z) = \boxed{?} \)

Answer: \pi


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

Answer: \dfrac{1}{3}


Question: Determine all values of \( p > 0 \) such that \( \log x \in L^p((0,1)) \).

Answer: (0, \infty)


Question: Verify the conjecture that \( \sum_{k=-\infty}^\infty \exp(-(x-k)^2/2) = \sqrt{2\pi} \) for any real value of \( x \).

Answer: \sqrt{2\pi}


Question: Find the minimum value of the function \( f(x, y, z) = x^z + y^z - (xy)^{\frac{z}{4}} \) for all positive real numbers \( x, y, \) and \( z \).

Answer: -\dfrac{1}{8}


Question:
Prove that for distinct real numbers \(a\), \(b\), and \(c\), the following identity holds:
\[
\frac{a^3}{(a-b)(a-c)} + \frac{b^3}{(b-c)(b-a)} + \frac{c^3}{(c-a)(c-b)} = a + b + c.
\]

Answer: a + b + c


Question: Let \( f(x) = ax^3 + bx^2 + cx + d \) and \( g(x) = x^2 + x - 2 \). Given that \( \lim_{x \to 1} \frac{f(x)}{g(x)} = 1 \) and \( \lim_{x \to -2} \frac{f(x)}{g(x)} = 4 \), find the value of \( \frac{c^2 + d^2}{a^2 + b^2} \).

Answer: 16


Question:
Calculate the limit of the sequence defined by:
\[ x_n = \frac{1}{n}\left(n + \frac{n-1}{2} + \frac{n-2}{3} + \ldots + \frac{2}{n-1} + \frac{1}{n} - \log(n!)\right) \]
Use asymptotic equivalences to find the limit, considering that \( \log(n!) \sim n\log(n) \).

Answer: \gamma


Question: Determine whether the sequence \(a_n = \frac{1}{n} \cdot \frac{1\cdot3\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot\ldots\cdot(2n)}\) converges, and if so, find its limit.

Answer: 0


Question: Consider a multilinear map \( T: V \times W \rightarrow Y \) where \( V \) and \( W \) are infinite-dimensional normed vector spaces. Given that \( \|T(x,y)\| \le C(x) \|y\| \) and \( \|T(x,y)\| \le D(y)\|x\| \), determine if there exists a constant \( K \) such that \( \|T(x,y)\| \le K \|x\| \|y\| \) for all \( x \in V \) and \( y \in W \).

Answer: No


Question: Solve the equation $e^x = x^e$ for $x$ in the domain $(0, \infty)$. Provide the solution as a single value.

Answer: e


Question:
Let $E$ be a rank $r$ vector bundle on a quasiprojective scheme. Consider the natural map:
\[ \Lambda^r H^0(E) \to H^0(\Lambda^r E) \]
Is it true that $E$ can always be twisted by an ample line bundle so that this map becomes surjective?

Answer: Yes


Question: Let $\vec p = (p_k)_{k \in \omega}$ and $\vec q = (q_k)_{k \in \omega}$ be two increasing sequences of prime numbers. Consider the Polish groups $F_{\vec p} = \prod_{k \in \omega} F_{p_k}$ and $F_{\vec q} = \prod_{k \in \omega} F_{q_k}$, where $F_n$ is the free group with $n$ generators endowed with the discrete topology. If the groups $F_{\vec p}$ and $F_{\vec q}$ are topologically isomorphic, must $\vec p = \vec q$? Answer yes or no.

Answer: Yes


Question: Let \(G = (V, E)\) be a simple graph that can be embedded on a torus such that every region is bordered by exactly 3 edges. Determine all possible values for the expression \(|V| - |E| + r\), where \(r\) is the number of regions into which the graph \(G\) divides the torus.

Answer: 0


Question: Can a separable topological space have a cardinality greater than that of \( \mathbb{R} \)?

Answer: Yes


Question: Given the joint probability distribution function \( f(x,y) = 10xy^2 \) for \( 0 < x < y < 1 \), find the probability \( P(X+Y \leq 1) \).

Answer: \dfrac{11}{96}


Question: Is it possible to obtain two concentric generalized circles (parallel straight lines) through the inversion of two circles with respect to a third circle?

Answer: Yes


Question: Find the rational number \( \frac{p}{q} \) that is closest to but not equal to \( \frac{22}{7} \) among all rational numbers with a denominator less than 100. What is the value of \( p - 3q \)?

Answer: 14


Question: Find the point on the paraboloid \(z = \frac{x^2}{4} + \frac{y^2}{25}\) that is closest to the point \((3, 0, 0)\).

Answer: (2, 0, 1)


Question: Determine whether the function \( f(x) = \begin{cases} x^2 \cos\left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases} \) is differentiable at \( x = 0 \). Justify your answer.

Answer: 0


Question: Consider the function \( f(x) = \max \{-11x - 37, x - 1, 9x + 3\} \) defined for all real \( x. \) Let \( p(x) \) be a quadratic polynomial tangent to the graph of \( f \) at three distinct points with \( x \)-coordinates \( x_1, x_2, x_3. \) Find \( x_1 + x_2 + x_3. \)

Answer: -\dfrac{11}{2}


Question: Evaluate the definite integral \(\int_0^{2016} x (x-1)(x-2)\cdots (x-2016)\,dx\).

Answer: 0


Question: Evaluate the triple integral \( \int_{-2}^{-1} \int_{-2}^{-1} \int_{-2}^{-1} \frac{x^2}{x^2+y^2+z^2} \, dx \, dy \, dz \).

Answer: \dfrac{1}{3}


Question: Determine the set of values of $k$ for which the polynomial equation \(3x^{4}+4x^3-12x^2+k=0\) has four real roots.

Answer: [0, 5]


Question: Determine the infimum of the set \( A = \left\{ \frac{m^2-n}{m^2+n^2} : m, n \in \mathbb{N}, m > 2n > 1 \right\} \).

Answer: \dfrac{23}{29}


Question: Let \( f(x) = \sin(4x - f(x)) \). Find \( f'(0) \).

Answer: 2


Question: Evaluate the integral \( \int_C \frac{5z-2}{z(z-1)} \, dz \) where \( C \) is the circle \( |z| = 2 \).

Answer: 10\pi i


Question: Find the smallest positive integer \( k \) such that for any positive integer \( n \), the inequality \( \frac{1}{k} \leq \frac{3-a_{n+1}}{7-a_n} \) holds, where \( a_1 = \sqrt{2} \) and \( a_{n+1} = \sqrt{2 + a_n} \) for all \( n \in \mathbb{N} \).

Answer: 5


Question: Let $X \sim N(0, 1)$. Calculate $E(X^4)$. Provide your answer as a single number.

Answer: 3


Question: Determine the Krull dimension of the algebra \( \mathbb{Q}[X,Y,Z]/(X^{2}-Y,Z^{2}) \).

Answer: 1


Question: Given the equation \(y^{-\frac{1}{4}} + y^{\frac{1}{4}} = 2x\), and the differential equation \((x^2-1)\frac{d^2y}{dx^2} + ax\frac{dy}{dx} + by = 0\), find the value of \(|a-b|\).

Answer: 17


Question: Compute \( \lim_{\theta\rightarrow 0}\frac{\sin{(\tan{\theta})}-\sin{(\sin{\theta})}}{\tan{(\tan{\theta})}-\tan{(\sin{\theta})}} \).

Answer: 1


Question: Given a continuous function \( f: \mathbb{R}_+^2 \to \mathbb{R}_+ \) and a positive integer \( k \), the equation \( g = (1+f(g,x))^k \) has a unique root for \( g \) when \( x \) is in the interval \([a,b]\) with \( 0 < a < b < 1 \). Does this imply that the function \( g(x) \) is continuous on \([a,b]\)?

Answer: Yes


Question: Determine the value of \( m \) that maximizes the sum: \[ \sum_{i=0}^m \binom{10}{i}\binom{20}{m-i} \] where \( \binom{p}{q} = 0 \) if \( p < q \).

Answer: 15


Question: Does there exist an uncountable set $S$ of sets such that it is closed under finite union and uncountable union, but not always closed under countably infinite union?

Answer: Yes


Question:
Solve the system of exponential equations:
\[\begin{cases}
x\cdot2^{x-y}+3y\cdot2^{2x+y-1}=1 \\
x\cdot2^{2x+y+1}+3y\cdot8^{x+y}=1
\end{cases}\]

Answer: (1, -1)


Question: Compute \( F_{1000} \mod F_{11} \), where \( F_n \) denotes the Fibonacci numbers.

Answer: 34


Question: If each point \((x,y)\) in the plane, where both coordinates are rational numbers, represents a tree, how far can you see if you are standing at the point \((0,0)\)?

Answer: \infty


Question: Does there exist a finite basis for the polynomial identities of a finite-dimensional complex Lie algebra \( \mathfrak{g} \) that satisfies the identity \( [[X,Y],[U,V]] = 0 \) for all \( X,Y,U,V \in \mathfrak{g} \)?

Answer: Yes


Question: Given that the polynomial \(x^4 - 2x^3 - 3x^2 + 4x - 1 = 0\) has four distinct real roots \(x_1, x_2, x_3, x_4\) such that \(x_1 < x_2 < x_3 < x_4\) and the product of two roots is unity, find the value of \(x_1x_2 + x_1x_3 + x_2x_4 + x_3x_4\) without explicitly finding the roots.

Answer: 1


Question: Find an imprimitive permutation group acting on a set of 12 elements such that the group has the maximum possible order. Provide the order of this group.

Answer: 1036800


Question: Determine the values of \( a > 0 \) for which the improper integral \( \int_{0}^{\infty}\frac{\sin x}{x^{a}} \, dx \) converges.

Answer: (0, 2)


Question: Find the torque around the x-axis of the triangle with vertices (0, 0), (1, 4), and (1, 0), assuming the density is 1.

Answer: \dfrac{8}{3}


Question:
Evaluate the limit using Taylor series expansions:
\[ \lim_{x \to 0}{\frac{x^2-\frac{x^6}{2}-x^2 \cos (x^2)}{\sin (x^{10})}} \]

Answer: -\dfrac{1}{24}


Question: Evaluate the degree of the field extension \([L:\mathbb{Q}]\), where \(L\) is the splitting field of the polynomial \(x^4 - 2\) over \(\mathbb{Q}\).

Answer: 8


Question: Determine the number of 5-Sylow subgroups in the symmetric group \( S_5 \).

Answer: 6


Question: Determine whether the following statement is true: A bounded positive function $v \in C^{2}(\mathbb{R}^N)$ that decays to zero and is $s$-harmonic ($s \in (0, 1)$) outside a ball behaves like $|x|^{2s-N}$ near infinity. Specifically, if $N > 2s$, then $|x|^{N-2s} v(x) \to l$ as $|x| \to \infty$ for some $l > 0$. Is this statement true?

Answer: True


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

Answer: \dfrac{k}{12}


Question: Find the upper bound for the expression \(E = |z_1-z_2|^2 + |z_2-z_3|^2 + |z_3-z_1|^2\), where \(z_1, z_2, z_3\) are unimodular complex numbers (i.e., \(|z_i| = 1\) for each \(i\)).

Answer: 9


Question: Given the system of differential equations \( \dot y = Ay \), where \( y = \begin{bmatrix} R(t) & F(t) \end{bmatrix}^{T} \) represents the populations of rabbits \( R(t) \) and foxes \( F(t) \), and \( A = \begin{bmatrix} 98 & -1170 \\ 7 & -83 \end{bmatrix} \), determine the number of foxes needed at time \( t = 0 \) if the rabbit population begins at 53,300 and grows as \( R(t) = R_0 e^{8t} \). The eigenvalues of \( A \) are \( \lambda = 8 \) and \( 7 \).

Answer: 4100


Question: Let $U$ be an open set of the Riemann sphere, and let $z_i$ be $n$ distinct points in $U$. Consider $E$, the vector space of meromorphic functions on $U$ with poles of order no more than 2, and $F$, the subspace of $E$ consisting of functions that are holomorphic in a neighborhood of each $z_i$. Determine if the quotient space $E/F$ has finite dimension, and if so, find its dimension.

Answer: 2n


Question: Identify the well-known ring that is isomorphic to the ring of homomorphisms from \( \mathbb{Z} \) to \( \mathbb{Z} \).

Answer: \mathbb{Z}


Question: Find the multiplicative inverse of the formal power series \( f(T) = \sum_{n\geq 0}(n+1)T^n \) in \( \mathbb{C} [T] \).

Answer: 1 - 2T + T^2


Question: Given that \(a, b, c,\) and \(d\) are positive integers such that \(a + b + c + d = 63\), find the maximum value of the expression \(ab + bc + cd\).

Answer: 991


Question: Solve the inequality $x + 3^x < 4$ analytically and find the range of values for $x$ that satisfy this inequality.

Answer: (-\infty, 1)


Question: Let \( f \) be a continuous real function on \( \mathbb{R}^1 \), such that \( f'(x) \) exists for all \( x \neq 0 \) and \( f'(x) \to 3 \) as \( x \to 0 \). Does it follow that \( f'(0) \) exists? Justify your answer.

Answer: Yes


Question: Compute the double integral \( I = \int_{-\infty}^{\infty}\int_{0}^{\infty} \cos k\xi \cdot u(\xi ) \, dkd\xi \), where \( u(\xi) = a_{0} \operatorname{sech}^{2} \left( \sqrt{\frac{a_{0}}{6+4a_{0}}} \, \xi \right) \) and \( a_0 \) is a constant.

Answer: \pi a_0


Question: Calculate the contour integral \( \int_{|z-1| = 2} \frac{1}{z} \, dz \) directly by using the substitution \( z = 1 + 2e^{i\theta} \).

Answer: 2\pi i


Question: Find an ideal \( I \) of \( \mathbb{Z}[\sqrt{3}] \) generated by an integer prime such that \( \mathbb{Z}[\sqrt{3}]/I \) is not an integral domain.

Answer: (2)


Question:
Determine whether there exists a function \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x \in \mathbb{R} \), the following condition holds:
\[ \prod_{n=0}^{\infty} (f(x)^2-(2n+1)^2)=0 \]
In other words, for all \( x \), \( f(x) \) must be an odd integer.

Answer: Yes


Question: Find the domain \( D \) of the function \( f(z) = \sum (-1)^n(z-1)^n - \sum i^{n-1} (z-i)^n \). Then determine \( f(z) \) for all \( z \in D \).

Answer: 0


Question: Evaluate the integral \( I_1 = \int_{-1}^{1} \{x+1\}\{x^2+2\} + \{x^2+2\}\{x^3+4\} \, dx \), where \( \{\} \) denotes the fractional part. Find the value of \( I_1 \).

Answer: \dfrac{2}{3}


Question: Determine the spectrum of the linear operator $T: L^2(0,1) \to L^2(0,1)$ defined by $T(f) = tf(t)$. Consider the properties of $T$ being self-adjoint and having a norm of 1, and find the approximate spectrum by identifying functions $f_n$ with unit norm such that $\int_0^1 |\lambda - t|^2 |f_n(t)|^2 \, dt \to 0$ for $\lambda \in [-1,1]$. What is the spectrum of $T$?

Answer: [0,1]


Question:
Let \( X_1 \) and \( X_2 \) have the joint density function \( f(x_1, x_2) \) given by:
\[
f(x_1, x_2) = \begin{cases} 2, & 0 < x_1 < 1, \ 0 < x_2 < 1, \ 0 < x_1 + x_2 < 1 \\ 0, & \text{otherwise} \end{cases}
\]
Find the probability \( P \left( \frac{1}{6} < X_1 < X_2 \right) \).

Answer: \dfrac{2}{9}


Question: Determine the range of $p \in [1, +\infty]$ for which the sequence \( f_h(x) = \frac{h^2 x}{h^4 + x^4} \) converges in \( L^p(]0, +\infty[) \). Specifically, find the values of $p$ such that \( \lim_{h \to 0} ||f_h||_p = 0 \).

Answer: \emptyset


Question: If $x, y, z$ are positive real numbers such that $x + y + z = 1$, determine whether the inequality \( \left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right)\left(1+\frac{1}{z}\right) \ge 64 \) holds true.

Answer: 64


Question: Compute the first Chern class, \( c_1(\mathcal{O}_C(1)) \), where \( C \) is the twisted cubic defined by the map \( \nu: \mathbb{P}^1_{\mathbb{C}} \to \mathbb{P}^3_{\mathbb{C}} \) given by \( \nu(s,t)=(s^3,s^2t,st^2,t^3) \). Assume \( c_1(\mathcal{O}_{\mathbb{P}^3}(1))=1 \).

Answer: 3


Question: Let $U = \text{Spec}(A)$ and $V = \text{Spec}(B)$ be affine schemes, and let $X$ be a separated scheme. Suppose there exist morphisms $U \to X$ and $V \to X$. Is the natural map \( A \otimes_{\Gamma(X, \mathcal{O}_X)} B \to \Gamma(U \times_X V, \mathcal{O}_{U \times_X V}) \) surjective?

Answer: Yes


Question: Given a list of $2^n$ nonzero vectors in $GF(2^n)$, where $n > 1$, and the sum of all vectors is zero, determine if there exists a subset of $2^{n-1}$ vectors whose sum is also zero. Assume the vectors are elements of $(\mathbb{Z}/2\mathbb{Z})^n$ and are not necessarily distinct.

Answer: Yes


Question: In rectangle $ABCD$, where $AB = 6$ and $BC = 3$, a point $P$ is chosen on $AB$ such that $\angle APD = 2 \angle CPB$. Determine the length of $AP$. \( \text{Express your answer as a decimal.} \)

Answer: 1.354


Question: Let \( p \) and \( q \) be distinct primes. Determine the maximum number of distinct solutions to the congruence \( x^2 - a \equiv 0 \pmod{pq} \).

Answer: 4


Question: Evaluate the infinite series \(\sum_{n=1}^{\infty}{\frac{\prod_{r=0}^{n-1}{(\pi-r})}{2^nn!}}\) and express it in the form \(\left(\frac{A}{B}\right)^{\pi}-C\), where \(A\), \(B\), and \(C\) are coprime integers. Find the value of \(A + B - C\).

Answer: 4


Question: Determine whether there exists a hereditary $\sigma$-space $X$ that is not a $Q$-space. A topological space $X$ is a $\sigma$-space if every $F_{\sigma}$-subset of $X$ is $G_{\delta}$. A $Q$-space is a topological space where any subset is $F_{\sigma}$. A hereditary $\sigma$-space is one where every subset is a $\sigma$-space.

Answer: Yes


Question: Let \( f : \mathbb{R} \to \mathbb{C} \) be a \( 2\pi \)-periodic function determined by its values on \([-\pi, \pi]\). Define \( \sigma_n(f)(x) = \frac{1}{n}(S_0(f)(x) + \ldots + S_n(f)(x)) \) for \( x \in [-\pi, \pi] \) and \( n \in \mathbb{N} \), where \( S_n(f)(x) = \sum_{k=-n}^n \hat{f}(k) e^{ikx} \) and \( \hat{f}(k) = \frac{1}{2\pi} \int_{-\pi}^\pi f(t)e^{-ikt}dt \). It can be shown that \( \sigma_n(f) = f * F_n \), where \( F_n(x) = \frac{1}{2n\pi} \frac{\sin^2(nx/2)}{\sin^2(x/2)} \). Given that \( F_n \) are approximate identities, for \( f \in L^1[-\pi, \pi] \) and \( f \) continuous at some \( x \in [-\pi, \pi] \), is it true that \( \lim_{n \to \infty} \sigma_n(f)(x) = f(x) \)?

Answer: Yes


Question: Evaluate the complex integral \( \int_{|z|=10} \frac{z^2-2z}{(z+1)^2(z^2+4)} \, dz \) using the Cauchy Integral Formula.

Answer: 0


Question: Determine the density character of the subspace $\ell_\infty(\kappa, S)$ of $\ell_\infty(S)$, where $S$ is an uncountable set and $\kappa < |S|$. Is the density character $2^\omega \cdot \kappa$?

Answer: 2^\omega \cdot \kappa


Question: Determine if there exists an integer $n > 2$ such that the numbers $n$, $n^2$, and $n^3$ all start with the same digit, which is not 1.

Answer: Yes


Question: Is the determinant map \( \det: M \rightarrow S^1 \) surjective, where \( M \) is the set of \( n \times n \) unitary matrices (\( n > 1 \)) and \( S^1 \) is the subset of \( \mathbb{C} \) such that \( |z| = 1 \)?

Answer: Yes


Question:
Suppose 8 real numbers $a, b, c, d$ and $x, y, z, w$ satisfy the following conditions:
\[ a^2 + b^2 + c^2 + d^2 = x^2 + y^2 + z^2 + w^2 = 1, \]
\[ ax + by + cz + dw = 0. \]
Determine if the following expression is equal to zero:
\[ (ay-bx)(cw-dz)-(az-cx)(bw-dy)+(aw-dx)(bz-cy) = 0. \]

Answer: 0


Question: Let $S$ be a set with 5 elements. Determine the number of relations on $S$ that are reflexive and symmetric but not transitive.

Answer: 972


Question: Find the generator of the kernel of the evaluation homomorphism \( \phi: \mathbb{R}[x] \to \mathbb{R} \) defined by \( f(x) \to f(1+\sqrt{2}) \).

Answer: x - 1 - \sqrt{2}


Question: Evaluate the limit: \[ \lim_{x \to \infty} \int_0^x e^{t^2 - x^2} \, dt = 0. \]

Answer: 0


Question: Is it possible to partition the set of edges and diagonals of a 100-gon into subsets of three elements, all originating from the same vertex?

Answer: Yes


Question: Compute the Lebesgue measure of the set \(A = \{(x,y) \in \mathbb{R}^2 : xy = 1\}\).

Answer: 0


Question: Calculate the 13th derivative of the function \( g(x) = \frac{x^2 - 1}{x^2 + 2} \) at \( x = 0 \).

Answer: 0


Question: Given the recursive sequence defined by $a_0 = 0$, $a_1 = 1$, and $a_{n+1} = a_n + 2a_{n-1}$ for $n \ge 1$, find the limit \( \lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n} \).

Answer: 2


Question: Determine the order of the group \( \text{Aut}(E/\mathbb{Q}) \), where \( E \) is the splitting field of the polynomial \( x^4 + x^2 + 1 \) over \( \mathbb{Q} \).

Answer: 2


Question: Let $H=(V,E)$ be a hypergraph where every member of $E$ has at most $n$ elements, with $n$ being a positive integer. Is there necessarily a maximal matchable set $S \subseteq V$?

Answer: Yes


Question: Let $G$ be a direct product of nonabelian simple groups $T_1, T_2, \dots, T_d$ with $d > 1$. Determine whether $G$ can be generated by two elements.

Answer: Yes


Question: Given that each coordinate of a k-tuple can take one of 3 values, determine the maximum number of 2-different k-tuples that can be formed.

Answer: 3^{k-1}


Question: Determine the degree of the splitting field of the polynomial \( f(x) = x^3 + 5x + 5 \) over the finite field \( \mathbb{Z}_3 \). What is \([K : \mathbb{Z}_3]\), where \(K\) is the splitting field?

Answer: 3


Question: Let \( Z \sim G(p) \) and \( W \sim G(2p) \) be independent random variables. Given that \( P(W > Z - 1) = \frac{3}{7} \), calculate the value of \( p \).

Answer: \dfrac{1}{3}


Question: Find the value of the infinite series: \[ \sum_{n=1}^{\infty}\frac{n^2-n+1}{n!} \]

Answer: 2e - 1


Question: Find the length of the cardioid given by the polar equation \( r = 1 - \cos(\theta) \).

Answer: 8


Question: In a unital Banach algebra $A$, if $x \in A$ satisfies $\|x\| \geq 1$, determine whether the element $1-x$ can be invertible.

Answer: Yes


Question: Does the OEIS sequence A059046 contain any odd squares \(u^2\) such that \(\omega(u) \geq 2\), where \(\omega(x)\) is the number of distinct prime factors of \(x\)?

Answer: No


Question: In a $101 \times 101$ square grid, each unit square contains one chocolate, and there are doorways between adjacent squares. It costs $\$1$ to pass through a doorway. Jake can start from any square and must collect all $5101$ chocolates. What is the minimum amount of money Jake needs to ensure he can collect all the chocolates?

Answer: 10200


Question: Determine the number of roots of the function \( f(x) = \sum_{r=1}^{2009} \frac{r}{x-r} \).

Answer: 2008


Question: Determine the value of \( \lim\sup_{n\to\infty} E_n \), where \( E_n \) is the expected number of adjacent indices in a bijection \( \varphi: \{1,\ldots, 2n\} \to \big(\{1,\dots,n\}\times\{0,1\}\big) \) such that \( k \in \{1,\ldots, 2n-1\} \) is an adjacent index if \( \pi_0(\varphi(k)) = \pi_0(\varphi(k+1)) \).

Answer: 1


Question: Find the slope of the tangent line to the curve given by the equation \(2(x^{2}+y^{2})^{2} = 25(x^{2}-y^{2})\) at the point \((-3,1)\).

Answer: \dfrac{9}{13}


Question: Find the value of the infinite series \( \sum_{k=1}^{\infty}\frac{k^n}{k!} \) for a given integer \( n \). For example, when \( n = 6 \), the sum is \( 203e \).

Answer: B_n e


Question: Evaluate the expression \( F(m,n) = \sum\limits_{i=0}^n\frac{\binom{m+i}{i}}{2^{m+i+1}} + \sum\limits_{i=0}^m\frac{\binom{n+i}{i}}{2^{n+i+1}} \) for positive integers \( m \) and \( n \). Verify the hypothesis that \( F(m,n) = 1 \) for all positive integers \( m \) and \( n \).

Answer: 1


Question: Evaluate the limit: $$\lim _{ x\rightarrow 1 }{ \frac { x\log { \left( x \right) -x+1 }  }{ \left( x-1 \right) \log { \left( x \right)  }  }  } $$

Answer: \dfrac{1}{2}


Question: If $a$ and $b$ are integers such that for every prime $p$, there exists an integer $n_p$ satisfying $b \equiv a^{n_p} \pmod{p}$, can we conclude that there exists an integer $n$ such that $b = a^n$?

Answer: Yes


Question: Evaluate the contour integral \( \oint_{|z|=3} \frac{1}{2-z} e^{\frac{1}{z}} \,dz \) by finding the residue at \( z=0 \).

Answer: -2\pi i


Question: Solve the differential equation \( T'' + T' + n^2 T = 0 \) with initial conditions \( T(0) = 0 \) and \( T'(0) = 0 \), where \( n \in \mathbb{N} \). Express the solution in the form \( e^{-\frac{t}{2}}[A \sin(\frac{\sqrt{4n^2-1}}{2} t) + B \cos(\frac{\sqrt{4n^2-1}}{2} t)] \).

Answer: 0


Question:
Determine the condition on \( \alpha \) such that the expression \( \langle (x_1,y_1), (x_2,y_2) \rangle = x_1x_2 - \alpha(x_1y_2 + x_2y_1) + y_1y_2 \) is an inner product in \( \mathbb{R}^2 \). Choose the correct option:

(a) \( \forall \alpha \in \mathbb{R} \)
(b) iff \( \alpha = 0 \)
(c) iff \( \alpha < 1 \)
(d) iff \( |\alpha| < 1 \)

Answer: D


Question: Evaluate the limit: \[ \lim _{n \rightarrow \infty} \sum_{r=1}^{n}\binom{2r}{r}\left(\frac{1}{6}\right)^{r} \]

Answer: \sqrt{3} - 1


Question: Evaluate the limit: $$\lim_{x\to 0} \int_0^1 \cos\left(\frac{1}{xt}\right)\, dt$$

Answer: 0


Question: Solve the congruence $x^3 \equiv 27 \pmod{41}$.

Answer: 3


Question: Use complex analysis to evaluate the integral \( \int_{-\infty}^\infty \frac{dx}{(1+x^2)^3} \).

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


Question: In a game between Alice and Bob, they alternately select distinct nonzero digits from 1 to 9 until they have chosen seven such digits. The digits are joined in the order selected to form a seven-digit number \( \overline{A_1B_2A_3B_4A_5B_6A_7} \). Alice wins if this number is the last seven decimal digits of a perfect seventh power. Determine which player has a winning strategy.

Answer: Alice


Question: Let \( \{f_n(x)\} \) be a sequence in \( L^2[0,1] \) such that \( \lim f_n = f \) almost everywhere, where \( f(x) \in L^2[0,1] \). Compute the value of \( \lim_{n\to \infty} \int_0^1 f_n(x) \sin (nx) \, dx. \)

Answer: 0


Question: Given the linear transformation \( f: M_{2 \times 3}(\mathbb{R}) \rightarrow \mathbb{R}^4 \) defined by \( f\begin{pmatrix} a & b & c \\ i & j & k \end{pmatrix} = (a+b+c, a+i+k, b+j-k, c-i-j) \), and the subspace \( F \subseteq \mathbb{R}^4 \) spanned by the vectors \( (1,2,1,0), (2,1,0,-1), (0,1,2,1) \), calculate the dimension and find a basis for the preimage \( f^{-1}(F) \).

Answer: 5


Question: Given the joint probability density function \( f(x,y) = \frac{1}{4}(x-y)e^{-x} \) for \( 0 < x < \infty \) and \( -x < y < x \), compute the expected value of the random variable \( Z = \frac{Y}{X} \).

Answer: -\dfrac{1}{3}


Question: Evaluate the line integral \(\int_C (x+2y)dx + x^2dy\), where \(C\) consists of line segments from \((0,0)\) to \((2,1)\) and from \((2,1)\) to \((3,0)\). Use the parametric equations \(\langle t, t/2\rangle\) for \(0 \leq t \leq 2\) and \(\langle t, 3-t\rangle\) for \(2 \leq t \leq 3\) to solve the integral.

Answer: \dfrac{5}{2}


Question: Given that $a^k \mid b^{k+100}$ for every integer $k$, prove that $a \mid b$.

Answer: a \mid b


Question: Let \((\Omega, \mathcal{F}, P)\) be a probability space, and let \(X_1, X_2, \dots\) be a sequence of identically distributed random variables. Define \(S_n = X_1 + \cdots + X_n\) and let \(\mathcal{G} = \sigma(S_n)\), the \(\sigma\)-field generated by \(S_n\). Compute the conditional expectation \(E(X_1 \vert S_n)\).

Answer: \dfrac{S_n}{n}


Question:
Find the maximum likelihood estimator (MLE) of \( \Theta \) for the probability mass function given by:
\[ f(x, \Theta) = \left(\frac{x}{\Theta}\right)^{\Theta A'(\Theta)} \exp(A(\Theta) + C(x)) \]
using the random sample \{8, 4, 1/2, 1\}.

Answer: 2


Question: Calculate \(113^{17} \mod 91\) without using a calculator that supports large numbers.

Answer: 29


Question: Find the length of the arc of the curve given by the equation \(8y = x^4 + 2x^{-2}\) from \(x = 1\) to \(x = 2\).

Answer: \dfrac{33}{16}


Question: Let $G$ be a finite 2-group and $x$ a non-central element of $G$ such that $C_G(x) \leq \text{cl}(x) \cup Z(G)$, where $\text{cl}(x)$ denotes the conjugacy class of $x$ in $G$. Determine if $|C_G(x) : Z(G)| = 2$.

Answer: 2


Question: Suppose $K$ is a Killing vector satisfying Killing's equation, $\nabla_bK_a+\nabla_aK_b=0$. Prove that the following identity holds: \[ \nabla_a\nabla_bK_c + \nabla_b\nabla_cK_a + \nabla_c\nabla_aK_b = 0. \]

Answer: \nabla_a\nabla_bK_c + \nabla_b\nabla_cK_a + \nabla_c\nabla_aK_b = 0


Question: Given the numbers $1, 2, 3, \ldots, 101$ arranged in any order, is it always possible to remove 90 numbers such that the remaining 11 numbers are either in strictly increasing or strictly decreasing order?

Answer: Yes


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

Evaluate the integral \( \int \frac{dx}{x^2 \sqrt{2x} + 1}dx \). Is it equal to \( \ln \left| x^2 \sqrt{2x} +1 \right| + C \)?

Answer: B


Question: Suppose $f$ is analytic on $\bar{B}(0; 1)$ and satisfies $|f(z)| < 1$ for $|z| = 1$. Find the number of solutions (counting multiplicities) of the equation $f(z) = z^n$, where $n$ is an integer greater than or equal to 1.

Answer: n


Question: In a party, each person shakes hands with exactly 20 other people. For each pair of people who shake hands with each other, there is exactly 1 other person who shakes hands with both of them. For each pair of people who do not shake hands with each other, there are exactly 6 other people who shake hands with both of them. Determine the total number of people at the party.

Answer: 81


Question: A function \( f \) satisfies the condition \( f[f(x) - e^x] = e + 1 \) for all \( x \in \Bbb R \). Find \( f(\ln 2) \).

Answer: 3


Question: Determine the total number of different game situations possible in a standard Connect Four game played on a 7-column by 6-row grid.

Answer: 4531985219092


Question: Solve the equation \(5^{x} + 2^{y} = 2^{x} + 5^{y} = \frac{7}{10}\) and find the value of \(\frac{1}{x+y}\).

Answer: -\dfrac{1}{2}


Question: Determine the values of $p$ for which the sequence $\{f_n\}$, defined by \[ f_n(x) = \begin{cases} 1 - nx, & \text{if } 0 \leq x \leq \frac{1}{n}, \\ 0, & \text{otherwise}, \end{cases} \] is a Cauchy sequence in the space $(C[0,1], \lVert\cdot\rVert_p)$ for $1 \leq p \leq \infty$. Additionally, if the sequence is Cauchy, determine whether it converges.

Answer: [1, \infty)


Question:
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \), the following equation holds:
\[ (x^2 - y^2) \cdot f(xy) = x \cdot f(x^2y) - y \cdot f(xy^2) \]

Answer: f(x) = kx


Question: Given a circle with radius \( r \), a point \( P \) on the circle, and an angle \( a \) at point \( P \) subtending an arc of the circle, find the angle \( b \) at the center of the circle that subtends the same arc. Assume you know the distance from \( P \) to the center of the circle. Determine the angle \( b \).

Answer: 2a


Question: Let $R$ be a unital ring. Suppose $f, g \in R$ define the same function on points of $\text{Spec } R$, meaning $f(p) = g(p)$ for all $p \in \text{Spec } R$. Does it follow that $f = g$ in $R$? Justify your answer.

Answer: No


Question: Given three points $A = (\text{latitude}, \text{longitude})$, $B = (\text{latitude}, \text{longitude})$, and $C = (\text{latitude}, \text{longitude})$, calculate the angle between the road connecting $A$ and $B$ and the road connecting $B$ and $C$.

Answer: \theta


Question: Show that the function \( z e^{\lambda-z} - 1 \) has exactly one real root within the unit disk for all real \( \lambda > 1 \).

Answer: 1


Question: Evaluate the triple integral: \[ \int_0^1 \int_1^2 \int_0^{x+y} 12(4x+y+3z)^2 \, dz \, dy \, dx \]

Answer: 1272


Question:
Find the minimum value of the expression \(x^5 + y^5 + z^5 - 5xyz\) where \(x, y,\) and \(z\) are positive numbers. You may use the inequality:
\[ (t_1\cdot t_2\cdot t_3\cdots t_n)^{\frac{1}{n}} \leq \frac{t_1+t_2+t_3+\cdots +t_n}{n} \]

Answer: -2


Question: Calculate the genus of the projective curve given by the equation \(3x^{3}+4y^{3}+5z^{3} = 0\).

Answer: 1


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{\sin x}{x+i} \, dx \). Use the hint that \( \sin x = \frac{e^{ix} - e^{-ix}}{2i} \).

Answer: \dfrac{\pi}{e}


Question: Find the exponential generating function for the sequence defined by \( a_n = (-1)^n \sum_{k=0}^{n} k \binom{n}{k} \).

Answer: -x e^{-2x}


Question:
If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2-4x+1=0\), find the value of
\[f(\alpha, \beta) =\frac {\beta^3}{2} \csc^2\left (\frac {1}{2}\arctan \frac {\beta}{\alpha}\right) +\frac {\alpha^3}{2} \sec^2\left (\frac {1}{2}\arctan \frac {\alpha}{\beta}\right)\]

Answer: 56


Question: Compute the value of the double geometric series \(\sum_{(i,j)\in L}\frac{1}{18}\left(\frac{5}{6}\right)^i\left(\frac{2}{4}\right)^j\), where \(L=\{(i,j)\in\mathbb{N}^2; 0\le i\le j\}\).

Answer: \dfrac{4}{21}


Question: Let $M$ be a compact manifold and $f$ a smooth function on $M$. Determine whether the $1$-form $df$ always has zeros on $M$. Provide a justification for your answer.

Answer: Yes


Question: Prove that there exists a constant \( C > 0 \) such that \( a^{c_1\ln T} \leq T^{-C} \) for all sufficiently large \( T \).

Answer: C = -c_1 \ln a


Question: Evaluate the integral of the function \( f(x, y) = 1 + xy \) over the region \( D = \{(x, y) \in \mathbb{R}^2 : 1 \leq x^2 + y^2 \leq 2 \text{ and } y \geq 0\} \). Use polar coordinates to perform the integration.

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


Question: Determine the values of \( \alpha \geq 0 \) for which the integral \( \int_e^4 \frac{dt}{\ln^\alpha[\ln(t)]} \) converges.

Answer: [0, 1)


Question: Is it true that if a function X is F-measurable, or a Borel function, then it is constant on the atoms of F in all cases?

Answer: Yes


Question: Given a compact Riemann surface $S$ and two distinct points $p$ and $q$ on $S$, determine whether there exists a meromorphic function on $S$ that is zero at $p$ but not at $q$. Provide a justification for your answer.

Answer: Yes


Question: Determine the number of ring isomorphisms from \( \mathbb{Z}^n \) to \( \mathbb{Z}^n \).

Answer: n!


Question: Consider a group $G$ of order $105$. Determine the number of Sylow 3-subgroups of $G$.

Answer: 7


Question: Consider an algebraic cone $X$ in $\mathbb{C}^n$ with $\dim_0 X = p$. A linear projection $\pi: \mathbb{C}^n \to \mathbb{C}^p$ is given. Are there coordinates $(z_1, \ldots, z_n)$ in $\mathbb{C}^n$ and a linear projection $\pi: \mathbb{C}^n \to \mathbb{C}^p$ such that the branch locus $br(\pi|_X)$ equals the singular set $Sing(X)$, meaning $br(\pi|_{Reg(X)}) = \emptyset$, and $\pi^{-1}(0) \cap X = \{0\}$?

Answer: Yes


Question: Are the numbers \(2^{2^1}+1, 2^{2^2}+1, \dots\) relatively prime to each other?

Answer: Yes


Question: Calculate the flux of the vector field \( \mathbf{F} = (3x^3y^2 + 3)\mathbf{i} + \left(\frac{y^2 + 2x}{3}\right)\mathbf{j} + (3yz^3 + 3)\mathbf{k} \) through the box with opposite corners at \((0, 1, 0)\) and \((2, 2, 1)\), where the faces are parallel to the coordinate planes. Use the divergence theorem to find the correct flux value.

Answer: 67


Question:
Which of the following is correct for the partial derivative of the modulus squared of a complex number with respect to the complex number itself, considering $z = a+ib$?

1) Assuming $z$ and $z^*$ are independent, \( \frac{\partial |z|^2}{\partial z} = \frac{\partial (zz^*)}{\partial z} = z^* \)

2) \( \frac{\partial |z|^2}{\partial z} = \frac{\partial (a^2+b^2)}{\partial (a+ib)} = \frac{\partial (a^2+b^2)}{\partial a}\frac{\partial a}{\partial (a+ib)} + \frac{\partial (a^2+b^2)}{\partial b}\frac{\partial b}{\partial (a+ib)}  = 2a +2b\frac{1}{i} = 2 (a-ib) = 2 z^* \)

Answer: 1


Question: Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ be a $C^2$ function, and suppose the origin is a non-degenerate critical point. If $f(x, mx)$ is a local minimum at the origin for all $m$, does $f$ have a local minimum at the origin? Provide a justification for your answer.

Answer: Yes


Question: Consider the function defined as follows: \[ h(x) = \begin{cases} 0, & x \in \mathbb{R} \setminus \mathbb{Q} \\ x^2, & x \in \mathbb{Q} \end{cases} \] Determine if this function is differentiable at the point where it is continuous.

Answer: 0


Question: Determine the number of integer roots of the equation \(x^8 - 24x^7 - 18x^5 + 39x^2 + 1155 = 0\).

Answer: 0


Question: Determine whether there exist two topologies on the same underlying set such that exactly one of them is metrizable, they share the same convergent sequences with the same limits, but they are not the same topology.

Answer: Yes


Question: Determine the radius of convergence of the power series \( \sum_{n=1}^{\infty}\frac{n}{n+1}\left(\frac{2x+1}{x}\right)^n \).

Answer: \dfrac{1}{3}


Question: Solve the equation \( \sqrt{3\sqrt[3]{x}+1}=\sqrt[3]{2\sqrt{x+1}-1} \).

Answer: 0


Question: Determine whether the winding number \(n(\gamma, z)\) is zero for all points \(z\) in the complex plane that are exterior to the closed curve \(\gamma\).

Answer: Yes


Question: Determine the value of the Haar measure \( \lambda \) on the set \( \{(0,0,\ldots)\} \) in \( \{0,1\}^{\mathbb{N}} \).

Answer: 0


Question: Evaluate the line integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \) where \( \mathbf{F} = \langle -y^2, x, z^2 \rangle \) and \( C \) is the curve of intersection of the plane \( y+z=2 \) and the cylinder \( x^2+y^2=1 \).

Answer: \pi


Question: Find the last four digits of \(7^{65}\).

Answer: 8807


Question: Is the Markov inequality a special case of the Chebyshev inequality when \( p = 1 \)?

Answer: No


Question: Does the wedge sum of two circles have the fixed point property? Provide a justification for your answer.

Answer: No


Question: Given the polynomial \( P(x) = x^4 + ax^3 + bx^2 + cx + d \) with the conditions \( P(1) = 10 \), \( P(2) = 20 \), and \( P(3) = 30 \), find the value of \( P(12) + P(-8) \).

Answer: 19840


Question:
In the context of $K$-theory, consider a CW pair $(X,A)$ and the associated long exact sequence:
\[ \cdots \rightarrow K(SX) \rightarrow K(SA) \rightarrow K(X/A) \rightarrow K(X) \rightarrow K(A). \]
If the inclusion map $A \rightarrow X$ has a splitting $X \rightarrow A$, making $K(X) \rightarrow K(A)$ surjective, does this imply that $K(X/A) \rightarrow K(X)$ is injective? Provide a justification for your answer.

Answer: Yes


Question: Does the function \( f(z) = e^{z^2} \) have an antiderivative over the entire complex plane? Provide a justification for your answer.

Answer: Yes


Question: Given three discrete random variables A, B, and C, if \( P(A=a)P(C=c) = P(B=b)P(C=c) \) for all 3-tuples \((a,b,c)\in \Omega_A \times \Omega_B \times \Omega_C\), does this imply that \( P(A=a) = P(B=b) \) for all 2-tuples \((a,b)\in \Omega_A \times \Omega_B\)? Assume that the entire sample space has some values of C with nonzero probability.

Answer: Yes


Question: Given a $202 \times 202$ matrix $A$ with $\|A\|_2 = 100$ and $\|A\|_f = 101$, determine the sharpest lower bound on the 2-norm condition number $k(A)$. Assume $k(A) = \|A\|_2 \cdot \|A^{-1}\|_2$. Provide your answer as a single numerical value.

Answer: 100


Question: Let $G$ be a group acting as a covering space action on a space $X$ such that for each $x \in X$, there exists a neighborhood $U_x$ where the images $g(U_x)$ for different $g \in G$ are disjoint. In other words, $g_1(U_x) \cap g_2(U_x) \neq \emptyset$ implies $g_1 = g_2$. If $H$ is a subgroup of $G$, is $H$ also a covering space action that satisfies this condition?

Answer: Yes


Question: Find the remainder when \( \sum_{r=0}^{1008}{2016 \choose 2r}{3^{2016-2r}8^r} \) is divided by 2017.

Answer: 1


Question: Let $X$, $Y$, and $Z$ be linear spaces, and let $T: X \times Y \rightarrow Z$ be a bilinear mapping. Suppose $(e_i)_{i \in I}$ is a basis for $X$ and $(f_j)_{j \in J}$ is a basis for $Y$, such that $(T(e_i, f_j))_{i \in I, j \in J}$ forms a basis for $Z$. If $(e'_i)_{i \in I}$ and $(f'_j)_{j \in J}$ are other bases for $X$ and $Y$, respectively, is $(T(e'_i, f'_j))_{i \in I, j \in J}$ a basis for $Z$?

Answer: Yes


Question: Evaluate the limit: \[ \lim\limits_{x\to 0}\frac{1}{x}\int_{0}^{x}\sin^{2}\left(\frac{1}{u}\right)du \]

Answer: \dfrac{1}{2}


Question: Let $p$ be a prime integer and $f$ a function from $A = \mathbb{F}_p^n$ to $\mathbb{F}_p$, with $n \geq p + 1$. Is it true that for all $x \in A$, the following holds: \[ \sum_{\sigma \in S_n} s(\sigma) \times f(x_\sigma) = 0 \] where $s(\sigma)$ is the signature of the permutation $\sigma$, and $S_n$ is the group of all bijections of $U_n = \{1, \ldots, n\}$ to $U_n$? Assume $x = (x_1, \ldots, x_n)$ and $x_\sigma = (x_{\sigma(1)}, \ldots, x_{\sigma(n)})$. The answer is known to be yes for $p = 2$. What about other cases?

Answer: Yes


Question: Calculate \( \mathbb{E}[B^4(t)] \) where \( B(t) \) is a Brownian motion.

Answer: 3t^2


Question: Consider independent random variables $X_1, X_2, \ldots, X_n, \ldots$ such that $E[X_i] = 1 + \frac{1}{1+i^2}$ and $Var[X_i] = \sqrt{i}$ for $i = 1, 2, \ldots$. Determine the variance of $\sum\limits_{i=1}^n \frac{X_i}{i^{1/4}}$. Express your answer in terms of $n$.

Answer: n


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

Suppose $N$ and $L$ are submodules of $M$ with $N \subseteq L$. If $M/L \cong M/N$, then $L = N$. Consider both the general case and the specific case where $L = \ker \varphi$ for some $\varphi \in \text{Hom}(M,P)$. Provide a justification for your answer.

Answer: False


Question: Determine the values of \(a \in \mathbb{R}\) for which there exists a non-constant function \(f: (0, 1] \rightarrow \mathbb{R}\) such that \(a + f(x+y-xy) + f(x)f(y) \leq f(x) + f(y)\) holds for all \(x, y \in (0, 1]\).

Answer: (-\infty, 0]


Question: Given that \(\alpha, \beta, \gamma\) are the roots of the equation \(x^3 - 4x + 2 = 0\), find the value of \([(\alpha - \beta)(\beta - \gamma)(\gamma - \alpha)]^2\).

Answer: 148


Question:
A merry mouse moves in a maze. If it is at time $n$ in a room with $k$ horizontal or vertical adjacent rooms, it will be at time $n+1$ in one of the $k$ adjacent rooms, choosing one at random, each with probability $1 / k$. A fat lazy cat remains all the time in room $3,$ and a piece of cheese waits for the mouse in room $5$. The mouse starts in room $1$. The cat is not completely lazy: If the mouse enters the room inhabited by the cat, the cat will eat it. Also, if the mouse eats the cheese, it rests forever. Let $X_{n}$ be the position of the mouse at time $n$. What is the probability that the mouse ever gets to eat the cheese?

The transition matrix is as follows:
$$P=\begin{pmatrix}0 & 1/2 & 0 & 1/2 & 0 \\
1/2 & 0 & 1/2 & 0 & 0 \\
0 & 1/2 & 0 & 1/2 & 0 \\
1/3 & 0 & 1/3 & 0 & 1/3 \\
0 & 0 & 0 & 1 & 0 \\
\end{pmatrix}$$

Answer: \dfrac{2}{7}


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{\sin(\sin(\sin x)) - x}{x^3} \]

Answer: -\dfrac{1}{2}


Question: Let \( P_n = \{ p : [0,1] \rightarrow \mathbb{R} \mid \deg(p) \le n \} \) be the space of polynomials of degree at most \( n \). Define the norm \( \lVert p(t) \rVert = \max_{0 \le k \le n} |a_k| \) where \( p(t) = a_n t^n + \ldots + a_1 t + a_0 \). Consider the linear operator \( T : P_n \rightarrow P_n \) defined by \( T p(t) = \frac{d}{dt} p(t) \). Find the norm \( \lVert T \rVert \).

Answer: n


Question: Does the space $H(\operatorname{div};\Omega)$ have a Schauder basis, where $\Omega$ is an open bounded subset of $\mathbb{R}^n$ for $n \in \{2,3\}$ and $H(\operatorname{div};\Omega) = \{v \in L^2(\Omega) : \operatorname{div} v \in L^2(\Omega)\}$?

Answer: Yes


Question: Determine the number of integer values of \(a\) for which there exists at least one complex number \(z\) satisfying both the inequality \(|z-ai|=a+4\) and the inequality \(|z-2|<1\).

Answer: 2


Question: Calculate \( \lim_{n\to\infty}\int_{0}^{1}\frac{nx}{1+nx^2} \, dx \).

Answer: \infty


Question: How many integers $n$ have the property that there exist positive integers $a, b, c$ such that $a^n + b^n = c^n$? Provide your answer as a single integer.

Answer: 3


Question: Determine whether the following statement is true or false: Any continuous function from the open unit interval \((0, 1)\) to itself has a fixed point.

Answer: B


Question: Given the equation of a rectangular hyperbola: \(2x^2+3xy-2y^2-6x+13y-36=0\) and one of its asymptotes \(x+2y-5=0\), find the other asymptote. The second asymptote is of the form \(2x-y+\lambda=0\). Determine the value of \(\lambda\).

Answer: 4


Question: Let \( \varphi \) be a continuous complex-valued function on a closed contour \( \gamma \). Define the function \( F(z) = \int_{\gamma} \frac{\varphi(w)}{w-z}~dw \). Prove that \( \lim_{z \to \infty} F(z) = 0 \).

Answer: 0


Question: Calculate the expected number of moves required to get a cat, a fish, a dog, and your lunch across a river, following these rules: A troll rolls a die showing one of the four items, and you must move that item across the river. If the cat and fish are left alone on one side, the cat will eat the fish. If the dog and lunch are left alone, the dog will eat the lunch. If necessary, the troll will re-roll the die to prevent any item from being eaten. Determine the expected number of moves needed to get all items safely across the river.

Answer: 18


Question: If $A$ is a square matrix of order 3 such that $|A|=2$, find the value of $|(	ext{adj}A^{-1})^{-1}|$. Given that $A^{-1}=\frac{\text{adj}(A)}{|A|}$, $\text{adj}(A^{-1})=\frac{1}{2} \text{adj}(\text{adj}(A))$, and $|\text{adj}(A^{-1})|=\frac{1}{2} |A|^2$, determine the correct value of $|\text{adj}(A^{-1})^{-1}|$. The provided answer is 4.

Answer: 4


Question: Determine whether the number \( x = r \alpha e^{2i\pi \theta} \), where \( r = \sqrt{\frac{\sqrt{2}}{3-\sqrt{3}}} \), \( \alpha = 1-\frac{\sqrt{3}}{2} + \frac{i}{2} \), and \( \theta = \frac{1}{48} \), has a degree greater than 2 over \( \mathbb{Q}(i) \).

Answer: Yes


Question: Compute the integral \( \int_{2-i\infty}^{2+i\infty} \frac{1}{t} \, \mathrm{d}t \).

Answer: i\pi


Question: Evaluate the double integral \( \iint_R xy \, dx \, dy \) over the parallelogram defined by the lines \(2x-y=1\), \(2x-y=3\), \(x+y=-2\), and \(x+y=0\). Use the variable substitution \(u=2x-y\) and \(v=x+y\) to transform the integral.

Answer: -\dfrac{44}{81}


Question:
Determine the number of values of \(\alpha\) in the interval \([0, 2\pi)\) that satisfy the equation:
\[ \alpha = \left(12\left(12\alpha \bmod 2\pi\right) \bmod 2\pi\right) \]

Answer: 143


Question: Determine the minimum degree of the Taylor polynomial for \( \sin(x) \) needed to approximate \( \sin(100) \) such that the error is less than \( 10^{-4} \).

Answer: 5


Question: Given that a natural number $n$ satisfies the congruences $n \equiv 1 \pmod{3}$, $n \equiv 1 \pmod{4}$, and $n \equiv 4 \pmod{5}$, calculate $5^n \pmod{13}$.

Answer: 5


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}e^{-n}\sum_{j=n}^{4n}\frac{n^j}{j!} \]

Answer: \dfrac{1}{2}


Question: Find the integer $n$ such that $n$ divides $2^n + 2$ and $100 < n < 1000$. What is the value of $n$?

Answer: 946


Question:
Let \( a_k = \frac{k^4 - 17k^2 + 16}{k^4 - 8k^2 + 16} \) for integers \( k \geq 5 \). Find the limit:
\[ \lim_{n \to +\infty} a_5 a_6 a_7 \cdots a_n \]

Answer: \dfrac{1}{14}


Question: Find the limit: \[ \lim_{n \to \infty } \left(a_0 \sqrt{n} + a_1 \sqrt{n + 1} + \cdots + a_k \sqrt{n + k}\right) \] given that \( a_0 + a_1 + \cdots + a_k = 0 \) and \( a_0, a_1, \ldots, a_k \) are real numbers.

Answer: 0


Question: Find the greatest integer less than or equal to \( \sum\limits_{n=1}^{9999} \frac {1} {n^{\frac 1 4}}. \)

Answer: 1332


Question: Evaluate the limit: $$\lim_{x \rightarrow \infty} \frac{(x - 2)^{3/4} \sqrt{x^x}}{\sqrt{{(x - 2)^x}{(x - 1)}{\sqrt{x}}}}$$ and show that it converges to \(e\).

Answer: e


Question: Calculate the limit: \[ \lim_{x \to \infty} \frac{x^{x-1}}{(x-1)^x} \]

Answer: 0


Question: Is any differential graded (dg) coalgebra weakly equivalent to its cohomology as an $A_\infty$ coalgebra?

Answer: Yes


Question: Let $X$ be a random variable following a Poisson distribution with parameter $\lambda$. Calculate $E[X(X-1)(X-2)(X-3)]$ using the definition of expectation.

Answer: \lambda^4


Question: Calculate the derivative \( \frac{ \partial { \mathrm{tr}(XX^TXX^T)}}{\partial X} \).

Answer: 4 X X^\top X


Question: Consider a simple Lie algebra \( \frak{g} \) over \( \mathbb{R} \) or \( \mathbb{C} \) and a fundamental representation \( V_{\lambda_i} \). Determine if the sum of the weights of all the non-trivial weight spaces of \( V_{\lambda_i} \) in the dual of a Cartan subalgebra \( \frak{h} \subseteq \frak{g} \) is always zero.

Answer: Yes


Question: Determine whether the expression $x^n - (x-1)^n$ can be a prime number if $n$ is not a prime number.

Answer: No


Question: Evaluate the integral \( \int_C \frac{z \, dz}{(z+2)(z-1)} \) where \( C \) is the circle \( |z| = 4 \).

Answer: 2\pi i


Question: Prove that for a finite measure \( \mu \) on \([0,1]\), the norm of the bounded linear functional \( T(f) = \int f \, d\mu \) on \( C[0,1] \) is given by \( \|T\| = \mu([0,1]) \).

Answer: \mu([0,1])


Question: Evaluate the limit: $$\lim_{n\to\infty}\int_{1}^{n}\dfrac{1}{x^{n}}dx.$$

Answer: 0


Question: Consider the torus defined by \( \mathbb{R}^2 \mod((4,1),(1,-4)) \) with an area of 17. Determine if it is possible to cover this torus with seventeen labeled tiles in two distinct ways such that any pair of tiles is adjacent (either at an edge or a vertex) in at least one of the two patterns.

Answer: Yes


Question: Given that the series \( \sum_{n=1}^{\infty} a_n \) converges and each \( a_n > 0 \), find the value of \( \lim_{n \to \infty} n \sqrt[n]{a_1 a_2 \cdots a_n} \).

Answer: 0


Question: Determine the degree of the field extension $[F_p(X,Y) : F_p(X^p,Y^p)]$, where $F_p$ is a finite field of characteristic $p$. Provide your answer as a single integer.

Answer: p^2


Question:
Calculate the value of $E = f(1) + f(3) + f(5) + \dots + f(999)$, where the function $f$ is defined as follows:
\[ f(x) = \frac{1}{\sqrt[3]{x^2+2x+1} + \sqrt[3]{x^2-1} + \sqrt[3]{x^2-2x+1}}. \]

Answer: 5


Question: Evaluate the triple integral \( \iiint_S x^2 \, dx\, dy\, dz \), where \( S \) is the region bounded by \( 4x^2 + y^2 = 4 \), \( z + x = 2 \), and \( z = 0 \).

Answer: \pi