Question: In triangle ABC, angle B measures 56 degrees and angle C measures 36 degrees. Point P is on side AB and point Q is on side AC such that BP = CQ = 1. M is the midpoint of side BC, with BM = MC = 5. N is the midpoint of segment PQ. Find the measure of angle BMN.

Answer: 80


Question: Compare the two numbers \(\log_3 4\) and \(\sqrt[4]{2}\) without using a calculator. Determine which is greater, or if they are equal.

Answer: \log_3 4 > \sqrt[4]{2}


Question: Evaluate the integral \( \int_{0}^{2\pi}\frac{1}{1+\cos^2(\theta)}d\theta \) using contour integration techniques.

Answer: \pi \sqrt{2}


Question: Find the largest integer \( n \) such that \( 3^n \) divides every product \( abc \), where \( a, b, c \) are positive integers satisfying \( a^2 + b^2 = c^2 \) and \( 3 \mid c \).

Answer: 4


Question: Evaluate the integral \( \int_{|\vec{r}| \le R} \frac{y}{|\vec{r} - \vec{r}~'|} d^3V \) over a sphere of radius \( R \), where \( \vec{r} \) is a vector in three-dimensional space. Use cylindrical coordinates \( (s, \theta, z) \) to express the integral and consider the symmetry of the problem to simplify the evaluation.

Answer: 0


Question: Calculate the exact value of the sum: \( \tan^2(5^\circ) + \tan^2(10^\circ) + \tan^2(15^\circ) + \cdots + \tan^2(85^\circ) \).

Answer: \dfrac{595}{3}


Question: Given a set where \(|I_{r}| \leq O(r^{2})\) and \(|I_{r}| \geq \rho^{r}\) with \(\rho = 1 + \epsilon\), prove that there exists a threshold \(r^* = f(\rho)\) such that the inequality \(\rho^r \leq O(r^2)\) holds for all \(r = 0, 1, 2, \ldots, r^*-1\), but not for any \(r \geq r^*\). Assume \(\epsilon = 0.01\) and \(O(r^2) = r^2\).

Answer: 1466


Question: Let $A$ be a finite subset of an additive abelian group $G$ with $|A| = n > 3$. Can $A$ be permuted as $\{a_1, a_2, \ldots, a_n\}$ such that all the sums $a_1 + a_2 + a_3, a_2 + a_3 + a_4, \ldots, a_{n-2} + a_{n-1} + a_n, a_{n-1} + a_n + a_1, a_n + a_1 + a_2$ are pairwise distinct?

Answer: Yes


Question:
Calculate the norm of the operator \( \varphi : l_{1} \rightarrow \mathbb{C} \), where
\[ \varphi( (x_n)_{n=1}^{\infty} ) = \sum_{n=1}^{\infty} (-4)^{-n} x_{2n}.\]

Answer: \dfrac{1}{4}


Question: Evaluate the triple integral \( \int \int \int (x^2 + y^2 + z^2) \, dx \, dy \, dz \) over the region \( R \) bounded by the planes \( x+y+z = 1 \), \( y = 0 \), and \( z = 0 \).

Answer: \dfrac{1}{20}


Question: Let \( \alpha: \mathbb{R}_+ \rightarrow \mathbb{R}_+ \) be a \( \mathcal{K} \) function such that \( \alpha(0) = 0 \) and \( \alpha(a) > \alpha(b) \) for \( a > b \). Suppose \( \alpha < \mathrm{id} \), meaning \( \alpha(s) < s \) for all \( s > 0 \). Given \( s \in \mathbb{R}_+ \), determine whether \( \lim_{k \rightarrow \infty} \underbrace{\alpha\circ\alpha\circ \cdots \circ \alpha}_{k}(s) = 0 \).

Answer: 0


Question: Determine whether the sequence \( x_n = \frac{ \sqrt[n]{n!} }{ n } \) converges. If it converges, find the limit of the sequence.

Answer: \dfrac{1}{e}


Question: Evaluate the integral \( \int_S (x \, dy \, dz + y \, dz \, dx + z \, dx \, dy) \) over the surface \( S \) of a unit cube using the Gauss divergence theorem.

Answer: 3


Question: Let $K$ be an algebraically closed field, and $G \subset (K,+)^3$ be an algebraic subgroup of dimension 1. Determine if there exists a line $L \subset K^3$ such that $L \cap G = \{0\}$.

Answer: Yes


Question: Given that the rate of decrease of a reactant, represented by $Q(t)$, is proportional to $Q^3(t)$, such that $Q' = -kQ^3$, and knowing that $Q(0) = \frac{1}{8}$, determine the time it will take for the reactant to be reduced to one half of its original amount.

Answer: \dfrac{96}{k}


Question: Find the fundamental group \( \pi_1(X) \) of the topological space \( X = (S^1 \times S^2) \setminus (A \cup B) \), where \( A = \{(0,1)\} \times S^2 \) and \( B = S^1 \times \{(0,0,1)\} \).

Answer: 0


Question: Let \( f(x) \) be a real-valued function such that \( f(x) > 0 \) for all \( x \in \mathbb{R} \). The function \( f(x) \) is symmetrical about \( x = 2 \) and \( x = 4 \). Given that \( \prod_{r=1}^{50} f(r) = 2^{50} \), find the minimum value of the sum \( S = \sum_{r=1}^{50} f(r+4) \).

Answer: 100


Question: Two towers, each of height $N$, consist of $N$ coins. A machine moves a coin from one tower to the other with equal probability. On average, how many moves will it take before one of the towers is empty?

Answer: N^2


Question:
Consider a \(\mathrm{C}^*\)-algebra \(A\) that is infinite dimensional, separable, and generated by projections, with finitely many characters. Is there a projection \(p_C\) such that for all characters (non-zero *-homomorphisms \(\varphi:A\rightarrow \mathbb{C}\)) and all \(f\in A\), the following conditions hold?

1. \(\varphi(f)=\varphi(fp_C)=\varphi(p_C f)=\varphi(p_C fp_C)\)
2. \(\varphi(p_C)=1\)
3. For \(q_C:=1_{A}-p_C\), \(\varphi(q_C)=\varphi(fq_C)=\nu(q_C f)=0\)

Answer: Yes


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

Answer: \dfrac{8}{3}


Question: Evaluate the integral: \[ \int_0^{\frac{\pi}{2}} \frac{\sin^3(t)}{\sin^3(t)+\cos^3(t)} \, dt \]

Answer: \dfrac{\pi}{4}


Question:
Determine whether it is possible for integers $a, b, c$ to be coprime and satisfy the following equations:
\[ a+b = x^n \]
\[ b+c = y^n \]
\[ c+a = z^n \]
where $x, y, z \in \mathbb{Z}$ and $n$ is a positive integer. Assume $n$ is an odd prime.

Answer: Yes


Question: Find the orthogonal complement of the set \( \{f \in V: f(0) = 0\} \) in the vector space \( V \) of continuous functions on the interval \([0, 2\pi]\), with the scalar product defined by \( \langle f, g \rangle = \int_{0}^{2\pi} f(x)g(x) \, dx \).

Answer: \{0\}


Question: Let \( f: V \to V \) be a linear map where \( \dim \operatorname{Im}(f) < \infty \). Is there a polynomial \( P \) such that \( P(f) = 0 \)?

Answer: Yes


Question: Given that $x, y, z > 0$ and $x^2 + y^2 + z^2 = 1$, find the minimum value of the expression \( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \).

Answer: 3\sqrt{3}


Question: Consider the sequence \((q_n)_{n \in \mathbb{N}}\) which is an enumeration of the set \(\mathbb{Q} \cap [0,1]\). Does this sequence have convergent subsequences? If so, what are the possible limits of these subsequences?

Answer: [0,1]


Question:
Let \( F(x) \) be the primitive function of \( f(x) = |\sin x| \). Calculate the limit:
\[ \lim_{x \to \infty} \frac{F(x)}{x} \]

Answer: \dfrac{2}{\pi}


Question: Find the infinite sum of the probabilities p(n), where p(n) is the probability that the product of the faces is prime when rolling n fair dice. For example, p(1) = \( \frac{1}{2} \), p(2) = \( \frac{1}{6} \), and p(3) = \( \frac{1}{24} \). Use the hint: Consider differentiating both sides of the infinite geometric series \( \sum_{n=0}^{\infty} r^n = \frac{1}{1-r} \) for \( |r| < 1 \).

Answer: \dfrac{18}{25}


Question: Determine the values of \( \alpha > 0 \) for which the series \( \sum_{k=1}^{\infty} \frac{1}{(k+1)\ln(k+1)^\alpha} \) converges.

Answer: \alpha > 1


Question: Evaluate the integral \( \int_S x\,dy\,dz+y\,dz\,dx+z\,dx\,dy \), where \( S=\{(x,y,z)\in \mathbb{R}^{3}: x^2+y^2+z^2=1\} \).

Answer: 4\pi


Question: Evaluate the double integral \( \iint_R (x+y) \, dA \) where \( R \) is the trapezoidal region with vertices at \((0,0), (5,0), (5/2, 5/2), (5/2, -5/2)\). Use the transformation \( x = 2u + 3v \) and \( y = 2u - 3v \).

Answer: \dfrac{125}{8}


Question: Ten non-negative numbers satisfy the conditions \( \sum_{i=1}^{10} x_i = 5 \) and \( \sum_{i=1}^{10} x_i^2 = 6.1 \). Determine the greatest value among these numbers.

Answer: 2.3


Question: Is $A_5$ the only non-abelian finite simple group such that the number of distinct sizes of orbits under the action of its automorphism group is 4?

Answer: A_5


Question: Determine the range of the expression \( \tan^2\frac{A}{2} + \tan^2\frac{B}{2} + \tan^2\frac{C}{2} \) where \( A, B, C \) are the angles of a triangle. Choose the correct option: \( >1 \), \( <1 \), \( \ge1 \), or \( \le1 \).

Answer: \ge 1


Question:
Evaluate the expression \((\sin^{-1}a)^2-(\cos^{-1}b)^2+(\sec^{-1}c)^2-(\csc^{-1}d)^2\) given that:
\[(\sin^{-1}a)^2+(\cos^{-1}b)^2+(\sec^{-1}c)^2+(\csc^{-1}d)^2=\frac{5\pi^2}{2}\]

Answer: 0


Question: Let $G = \{ 1 + a_1x + a_2x^2 + a_3x^3 : a_i \in \mathbb{Z}/2\mathbb{Z} \}$, and define a binary operation on $G$ by $p(x) * q(x) = p(x)q(x) \bmod{x^4}$. This makes $G$ a $\mathbb{Z}$-module. Find the invariant factors of $G$. \( \text{(Hint: Consider the structure of } G \text{ as a finite abelian group.)} \)

Answer: (2, 4)


Question: Does a convergent and bounded sequence of positive semidefinite matrices always converge to a positive semidefinite matrix?

Answer: Yes


Question: In a parallelogram \(ABCD\), let \(K\) and \(L\) be the midpoints of \(BC\) and \(CD\), respectively. Given that \(\angle BAD = \frac{\pi}{3}\) and the points \(A\), \(B\), \(K\), and \(L\) lie on a common circle, find \(\angle ADB\).

Answer: \dfrac{\pi}{4}


Question: For what values of \( q > 1 \) is the function \( f_q(x) = \sum\limits_{k=1}^\infty q^{-k} a_k \) of bounded variation, where \( x = 0.a_1a_2\dots \) is the decimal expansion of a number \( x \), \( 0 < x < 1 \), and if two decimal expansions of \( x \) exist, the one that ends with 0's is taken?

Answer: q > 10


Question: Determine the value of $h(0,y)$ such that the function $h:\mathbb{R}^2 \to \mathbb{R}$, defined by \[ h(x,y) = \begin{cases} \displaystyle \int_0^x \frac{\pi}{x} \sin \left (\frac{\pi}{x} y \right) \, dy & \text{if } x \neq 0, \\ h(0,y) & \text{if } x = 0 \end{cases} \] is continuous at $x=0$.

Answer: 2


Question: Consider a Poisson process $N_t$ and a continuous process of finite variation $X_t$. Is the quadratic covariation $\langle X, N \rangle_t$ equal to zero?

Answer: 0


Question: Evaluate the limit: \[ \lim _{n\to \infty }n\int _1^2 \frac{dx}{x^2(1+x^n)} \] without using Taylor expansion.

Answer: \ln 2


Question: Let \( f(x) \) be a function defined for all \( x \in \mathbb{R} \) such that \( f(x+y) = f(x) \cdot f(y) \) for all \( x, y \in \mathbb{R} \). Given that \( f'(0) = 3 \) and \( f(5) = 2 \), find the value of \( f'(5) \).

Answer: 6


Question: Find the minimum value of \( a \) such that the function \( f(x) = \frac{3x}{7+x^2} \) has an inverse, given that the domain is defined as \( x \geq a \).

Answer: \sqrt{7}


Question: Let \(\alpha: \mathbb{S}^1 \to \mathbb{R}^2\) be a \(C^2\) curve satisfying \(|\dot \alpha|=1\). Define \(b(\theta) = \langle \alpha(\theta), \dot \alpha(\theta) \rangle\). Suppose \(b(\theta+c_1) - b(\theta) = c_2\) holds for every \(\theta\), where \(c_1\) and \(c_2\) are constants. Determine whether \(c_2 = 0\).

Answer: c_2 = 0


Question:
A planar curve is given by the parametric equations \(x=\cos(t)+t\) and \(y=t^2+2t+1\). Determine the curvature of the curve at the point \(P=(1,1)\). Choose the correct answer from the following options:

(a) \(\frac{1}{\sqrt{5}}\),
(b) \(\frac{4}{5\sqrt{5}}\),
(c) \(\frac{4}{5}\),
(d) \(\frac{3}{5\sqrt{5}}\),
(e) \(\frac{5}{3\sqrt{5}}\).

Answer: B


Question: Let \((X_n)_{n \in \mathbb{N} }\) be a sequence of random variables such that \(X_n \overset {\mathbb{P}}{\rightarrow} \mu\). Let \(g:\mathbb{R} \rightarrow \mathbb{R}\) be a function that is continuous everywhere except at \(\mu\), with left and right limits at \(\mu\) equal to zero. Define \(h\) as the continuous extension of \(g\) with \(h(\mu)=0\). By the continuous mapping theorem, \(h(X_n) \overset {\mathbb{P}}{\rightarrow} h(\mu)\). Does it follow that \(g(X_n) \overset {\mathbb{P}}{\rightarrow} h(\mu)\)?

Answer: No


Question: Evaluate the limit \( \lim_{n\rightarrow \infty}\sqrt[n]{f\left(\frac{1}{n}\right)g\left(\frac{n}{n}\right)+f\left(\frac{2}{n}\right)g\left(\frac{n-1}{n}\right)+\cdots+f\left(\frac{n}{n}\right)g\left(\frac{1}{n}\right)} \) where \( f:[0,1]\rightarrow [0,\infty) \) and \( g:[0,1]\rightarrow [0,\infty) \) are continuous functions.

Answer: 1


Question: Given a matrix $A \in M_4(\mathbb{C})$ with the characteristic polynomial $p_A(x)=(x^2+3)(x^2+\sqrt{3}x+3)$, find $A^{12}$.

Answer: 729I


Question:
Does there exist a homotopy \( H: \Bbb S^1 \times [1,2] \times [0,1] \to \Bbb R^2 \) such that:
- \( H(r\cos\theta, r\sin\theta, 1) = f(\cos\theta, \sin\theta) \) for all \( 1 \leq r \leq 2 \), and
- \( H(\cos\theta, \sin\theta, t) = f(\cos\theta, \sin\theta) = f(2\cos\theta, 2\sin\theta) = H(2\cos\theta, 2\sin\theta, t) \) for all \( t \in [0,1] \)?

Here, \( f: \Bbb S^1 \times [1,2] \to \Bbb R^2 \) is a map such that \( f(\cos\theta, \sin\theta) = f(2\cos\theta, 2\sin\theta) \) for all \( \theta \in [0,2\pi] \).

Answer: Yes


Question: Find all pairs of prime numbers \(p\) and \(q\) such that \(p^{q-1} + q^{p-1} = m^2\) for some integer \(m\).

Answer: (2, 2)


Question: Find the remainder of \(2^{3^{5^{7^{11}}}}\) when divided by 2016 using Euler's totient function.

Answer: 512


Question: Consider a population that, in each generation, has the following probabilities: it dies out with probability 0.2, stays constant with probability 0.5, and doubles with probability 0.3. What is the expected long-term behavior of this population?

Answer: \infty


Question: Given the probability density function \( f(x,y) = 3x \) for \( 0 \le y \le x \le 1 \) and \( f(x,y) = 0 \) elsewhere, find the probability \( P(Y \le X/2) \).

Answer: \dfrac{1}{2}


Question: Evaluate the contour integral \( \oint_{|z|=2} \frac{f'(z)}{f(z)} \, dz \) for the function \( f(z) = \frac{2z^4 - 7z^3 - 2z - i}{(z-i)^2(3z^2 - z + 9)^2} \).

Answer: -6\pi i


Question: Find the maximum distance between two parallel normals of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), where \(a > b\). Express your answer in terms of \(a\) and \(b\).

Answer: 2a


Question: Solve the differential equation \( u''(x) + \frac{u(x)}{4x^2} = 0 \) with the boundary conditions \( u(1) = 0 \) and \( u(2) = 0 \).

Answer: 0


Question: Calculate the value of the infinite series: \[ \sum_{n=1}^{\infty}\ln\left(1+x^{2^{n}}\right) \]

Answer: -\ln(1 - x^2)


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

Answer: e^2


Question: Evaluate \( L = \lim_{n \to \infty} \sum_{r=1}^{n} \left( \frac{1}{r!} \prod_{i=1}^{r} \left( \frac{i}{2} + \frac{1}{3} \right) \right) \). Find \( \lfloor L \rfloor \).

Answer: 2


Question: Show that \(\frac{d^n}{dx^n}(x^2-1)^n\bigg|_{x=1} = 2^n n!\).

Answer: 2^n n!


Question: Evaluate the limit: \[ \lim_{x \to \infty} \left[(x^3+x^2+1)^{1/3}-(x^2+x)^{1/2}\right] \]

Answer: -\dfrac{1}{6}


Question: Determine the maximum number of solutions to the equation \( f(z) = az + b \) for a holomorphic function \( f : \mathbb{C} \to \mathbb{C} \) such that the real part of \( f''(z) \) is strictly positive for all \( z \in \mathbb{C} \).

Answer: 2


Question: How many distinct necklaces can be made using exactly three red beads and three black beads?

Answer: 3


Question: Two players, A and B, toss coins simultaneously. Player A has n coins, and player B has n+1 coins. What is the probability that player B will have more heads than player A? Express your answer as a decimal or a fraction.

Answer: \dfrac{1}{2}


Question: Solve the differential equation \((x^2 + y^2)\,dx - 2xy\,dy = 0\). Find the solution in the form \(x^2 - y^2 = Cx\).

Answer: x^2 - y^2 = Cx


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

Answer: \dfrac{\pi}{2}


Question: Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $f(3x) - f(x) = x$. Given that $f(8) = 7$, find the value of $f(14)$. \( \boxed{?} \)

Answer: 10


Question: Let \( f:\mathbb{R} \to \mathbb{R} \) be a differentiable function defined by \( f(x) = x + \int_0^1 (xy + x^2)f(y) \, dy \). Find the value of \( \int_0^1 f(x) \, dx \).

Answer: \dfrac{36}{23}


Question: Calculate the surface area of the region of the plane given by \( z = 1 + x + 2y \) that lies inside the cone defined by \( z = \sqrt{2x^2 + 10y^2} \).

Answer: 10\sqrt{3}\pi


Question: Find the value of \( \lim_{n\to\infty}\sum_{k=1}^{n}\frac{1}{\sqrt{n^{2}+kn}} \).

Answer: 2\sqrt{2} - 2


Question: Determine whether the following statement is true: For all integers \( m = \overline{a_k a_{k-1} \ldots a_1 a_0} \), if \( f(m \times m) = f(m) \times f(m) \), then \( a_i \in \{0, 1, 2, 3\} \) for all digits \( a_i \) of \( m \). Here, \( f(n) \) denotes the integer obtained by reversing the digits of \( n \).

Answer: True


Question:
Let $x = 20062007$. Evaluate the expression:
\[ A = \sqrt{x^2 + \sqrt{4x^2 + \sqrt{16x^2 + \sqrt{100x^2 + 39x + \sqrt{3}}}}}. \]
Find the greatest integer not exceeding $A$. Provide your answer as an integer.

Answer: 20062008


Question: Evaluate the integral for the variance of the integral of a stationary stochastic process when the process is uncorrelated (white noise process): $$\text{Var}[I_{uncorr}] = \sigma^2 \int_0^L \int_0^L \delta(t_1-t_2)\,\mathrm{dt_1\,dt_2}$$ where $\delta$ is the Dirac delta function.

Answer: \sigma^2 L


Question: Given the polynomial function \( f(x) = x^3 - 12x^2 + Ax + B \) where \( f(x) > 0 \), and the conditions \( f(f(f(3))) = 3 \) and \( f(f(f(f(4)))) = 4 \), determine the value of \( f(7) \).

Answer: 31


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

Answer: \dfrac{27}{4e}


Question: Evaluate the limit \( \lim_{x\to0^+}\frac{x^2}{e^{-\frac{1}{x^2}}\cos(\frac{1}{x^2})^2} \). Determine whether the limit is infinite or does not exist.

Answer: \infty


Question: Let \( f(x) = x + \tan^3 x \) and \( g(x) = f^{-1}(x) \). Find the value of \( 56g'\left(\frac{\pi}{4}+1\right) \).

Answer: 8


Question: Determine the value of \( p \) that minimizes the expression \( S = \frac{p-1}{p^n-1}\sum_{i=1}^n(n-i) p^{i-1} \) for the maximum possible value of \( n \).

Answer: 2


Question:
Given the solutions $\phi_{1}$ and $\phi_{2}$ to the initial value problem:
\[ x_{1}'=3x_{1}+x_{2}, \quad x_{1}(0)=1 \]
\[ x_{2}'=-x_{1}+x_{2}, \quad x_{2}(0)=-1 \]
find a second order differential equation that $\phi_{1}$ satisfies. Additionally, compute $\phi_{1}'(0)$. Provide your answer for $\phi_{1}'(0)$ in the form \( \boxed{} \).

Answer: 2


Question: Find the value of the product \( \sin 1^\circ \sin 3^\circ \sin 5^\circ \cdots \sin 179^\circ \).

Answer: \dfrac{1}{2^{89}}


Question: Given that \(E(X^2) = 3\) and \(E(X) = 1\), find the value of \(\sum\limits_{i=1}^{\infty} i P(X \geq i)\). Use the hint: \(\sum\limits_{i=1}^{\infty} i P(X \geq i) = \frac{1}{2}[E(X^2) - E(X)]\).

Answer: 1


Question: Given a sequence \((z_n)\) such that \(\lim_{n \to \infty} z_n = 0\), and a function \(f(n)\) such that \(f(n) \in \mathcal{O}(g(n))\) as \(n \to \infty\), where \(g(n) = \frac{n \cdot z_n^2}{n-1}\), does it follow that \(\lim_{n \to \infty} f(n) = 0\)?

Answer: 0


Question: Find the residue of the function \( f(z) = (z^2+4iz)\sin\left(\frac{3}{z+2i}\right) \) at \( z_0 = -2i \).

Answer: \dfrac{15}{2}


Question: Determine if there is a computable bound on the size of solutions for the set \( \mathcal{S}(z) \) when it is finite, where \( \mathcal{S}(z) \) is defined as the set of all solutions \( ab+cd=z \) satisfying \( \min(\vert a\vert,\vert b\vert)>\max(\vert c\vert,\vert d\vert) \) with \( a,b,c,d \) in the set \( \mathbb{Z}[i] \) of Gaussian integers.

Answer: Yes


Question:
Let \( f(x) \) be a \(2\pi\)-periodic function defined on \([0, 2\pi)\) as follows:
\[
f(x) = \begin{cases}
1, & x \in [0, \frac{7\pi}{16}) \\
5, & x \in [\frac{7\pi}{16}, \frac{11\pi}{16}) \\
4, & x \in [\frac{11\pi}{16}, 2\pi)
\end{cases}
\]
Find the Fourier series coefficients \(a_0\), \(a_n\), and \(b_n\) for \(f(x)\). Then, determine the value of the Fourier series at \(x = \frac{7\pi}{16}\).

Answer: 3


Question: Let $Y$ be an affine variety in $k^n$, defined by the polynomials $f_1, \dots, f_s \in A = k[y_1, \dots, y_n]$, where $k$ is an algebraically closed field. Consider $g \in A$ such that $f_1, \dots, f_s, g$ are algebraically dependent over $k$. Is it possible for the complement of the variety $W = V(f_1, \dots, f_s, g)$ in $Y$ to have infinite cardinality, i.e., can the set $Y \setminus W$ be infinite?

Answer: Yes


Question: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f'(x) = [f(x)]^2$ for all $x \in \mathbb{R}$ and $f(0) = 0$. Prove that $f(x) = 0$ for all $x \in \mathbb{R}$.

Answer: 0


Question: Calculate the value of the derivative of the Riemann xi function, \( \xi(s) \), at \( s = \frac{1}{2} \). The Riemann xi function is defined as: \[ \xi(s) = \frac{s(s-1)}{2}\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)\zeta(s) \]

Answer: 0


Question:
Consider the sequence of functions \(f_n: \mathbb{R} \to \mathbb{R}\) defined by \(f_n(x) = e^{-nx^2}\). This sequence converges pointwise to the function \(f(x)\) given by:
\[
f(x) = \begin{cases} 1 & \text{if } x = 0 \\ 0 & \text{if } x \neq 0 \end{cases}
\]
Determine \(\|f_n - f\|_\infty = \sup_{x \in \mathbb{R}} |e^{-nx^2} - f(x)|\). Explain why this supremum equals 1.

Answer: 1


Question:
Let \(f(x)\) be a positive, continuous, decreasing function such that \(\int_1^\infty f(x) \, dx = 5\). Which of the following statements must be true about the series \(\sum_{n=1}^\infty f(n)\)?

(a) \(\sum_{n=1}^\infty f(n) = 0\)

(b) \(\sum_{n=1}^\infty f(n)\) converges, and \(\sum_{n=1}^\infty f(n) < 5\)

(c) \(\sum_{n=1}^\infty f(n) = 5\)

(d) \(\sum_{n=1}^\infty f(n)\) converges, and \(\sum_{n=1}^\infty f(n) > 5\)

(e) \(\sum_{n=1}^\infty f(n)\) diverges.

Answer: D


Question: Determine the number of elements of order 10 in the symmetric group \( S_7 \).

Answer: 504


Question: Find the number of ways to distribute 30 marks among four parts of a test, where the first three parts can each receive up to 10 marks and the fourth part can receive up to 20 marks. Assume all marks are integers.

Answer: 1111


Question: Determine whether the inequality \(|D(u)(p)| \leq \left|\frac{\partial u}{\partial \nu}(p)\right|\) holds, where \(\nu\) is the unit normal of \(\partial U\), \(p \in \partial U\), and \(u=0\) on \(\partial U\).

Answer: Yes


Question: Determine the number of solutions to the equation \(\sqrt{3} \cdot x = \cos(xy)\) for \((x, y) \in (0,1)^2\).

Answer: \infty


Question: What space is homeomorphic to the space obtained by identifying antipodal points on the unit circle $S^1$ in $\Bbb{R}^2$?

Answer: S^1


Question:
Let $S: \mathbb{R}^3 \to \mathbb{R}^4$ and $T: \mathbb{R}^4 \to \mathbb{R}^3$ be linear transformations such that $T \circ S$ is the identity map on $\mathbb{R}^3$. Determine which of the following statements about $S \circ T$ is correct:

- $S \circ T$ is the identity map on $\mathbb{R}^4$.
- $S \circ T$ is one-to-one, but not onto.
- $S \circ T$ is not one-to-one, but onto.
- $S \circ T$ is neither one-to-one nor onto.

Answer: D


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

Answer: \ln 2


Question: Evaluate the integral \( \int_{|z|=2} \frac{1}{z^2+z+1}\,dz \) without using partial fraction decomposition.

Answer: 0


Question: Consider a sequence of bounded functions that converges. Can the limiting function be unbounded?

Answer: Yes


Question: How many seven-digit numbers are divisible by 11 and have a sum of their digits equal to 59?

Answer: 40


Question: Evaluate the limit using L'Hopital's rule: $$\lim_{K\to\infty} K^s\left(1-\frac{x}{2r}\right)^{K-s}$$ where $s \in \mathbb{Z}^+$, $r \in \mathbb{Z}^+$, and $0 < x < 1$ are constants.

Answer: 0


Question: Calculate the average value of the function \( f(\pi) = \sum_{i=1}^7 |\pi_i - i| \) over all permutations \( \pi \) of the numbers 1 to 7.

Answer: 16


Question: Compute the value of the infinite series \( \sum_{n=3}^{\infty}\frac{(n-3)!}{(n+2)!} \).

Answer: \dfrac{1}{96}


Question: Given the differential equation \( \ddot{x}(t) + w^2 x(t) = 0 \) with initial conditions \( x(0) = 1 \) and \( \dot{x}(0) = 0 \), find the numerical value of \( w \) such that \( x(2\pi) = \cosh(2\pi) \).

Answer: i


Question: Express the probability \( P(X_6 = 7, X_5 = 3 | X_4 = 1, X_9 = 3) \) in terms of the transition probabilities \( p(i,j) \) for a discrete time Markov chain, given that \( X_0 = 3 \).

Answer: \dfrac{p(1,3) \cdot p(3,7) \cdot p^{(3)}(7,3)}{p^{(5)}(1,3)}


Question: Let \( u: \mathbb{R} \to \mathbb{R} \) be a function that is continuous and piecewise linear with a finite number of discontinuities of the derivative. Determine whether \( u \) is absolutely continuous.

Answer: Yes


Question: Calculate the expected value of the stochastic process \( E(U(t)) = E(e^{t+W(t)/2}) \), where \( W(t) \) is a Wiener process. Verify that \( E(U(t)) = e^{9t/8} \).

Answer: e^{\frac{9}{8}t}


Question:
Given the plane curve \((x(t),y(t)) = (2 \cos t, \sin t)\), find the radius of curvature at the point \((0,1)\). The point corresponds to time \(t=\pi/2\). The radius \(R\) is given by \(1/\kappa\) where \(\kappa\) is the curvature, defined as:
\[ \kappa = \frac{|v \times a |}{ |v|^3}, \]
with \(v\) and \(a\) being the velocity and acceleration vectors, respectively. Calculate \(R\) at the given point.

Answer: 4


Question: Find the smallest integer $n$ such that the symmetric group $S_n$ contains an element of order 1000.

Answer: 133


Question: Determine the number of elements in the Lie algebra $sl(5, F)$, where $F$ is the finite field with 5 elements.

Answer: 5^{24}


Question: Does there exist a function that is discontinuous everywhere but has a range that is an interval?

Answer: Yes


Question: Determine whether the following statement is true: If two plane curves $C$ and $C'$ of degree $n$ intersect at $n^2$ points, and $np$ of these points lie on a curve of degree $p < n$, then the remaining $n(n - p)$ points will lie on a curve of degree $n - p$.

Answer: True


Question: Evaluate the limit: \[ \lim_{n\to\infty} n\left|e^\frac{i2\pi}{n}-1\right| \]

Answer: 2\pi


Question: A wire is divided into two parts. One part is shaped into a square, and the other part is shaped into a circle. Let the ratio of the circumference of the circle to the perimeter of the square be \( r \) when the sum of the areas of the square and circle is minimized. Find \( r \).

Answer: \dfrac{\pi}{4}


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

Answer: e^{-\frac{1}{6}}


Question: Solve the equation $x^x = \frac{1}{256}$ for $x$ without using the Lambert W function.

Answer: -4


Question: Let \( \alpha = (178)(2593)(46) \) be a known permutation in \( S_9 \). Find a permutation \( \beta \) such that \( \alpha = \beta\alpha\beta^{-1} \).

Answer: (178)


Question: Evaluate the integral \( \frac{3}{4\pi}\int\int\int\limits_{\mathbb{R}^3} e^{-\sqrt{x^2+y^2+z^2}^3} \, dV \).

Answer: 1


Question: Find the greatest integer value of \( k \) such that the inequality \( 2k^3 + 5k^2 + 3k - 132000 < 0 \) holds true.

Answer: 39


Question: Determine whether the following statement is true or false: If a sequence of functions in $C_c(\mathbb{R})$ converges pointwise to $g(x) = \exp(-x^2)$, then it must converge uniformly to $g$. Provide a justification for your answer.

Answer: False


Question: Evaluate the double integral \( \int \int_{x^2 + y^2 \leq 1} \frac{(x-y)^2}{x^2 + y^2} \, dx \, dy \).

Answer: \pi


Question: Let $f:X\rightarrow Y$ be a proper and surjective morphism of projective $k$-varieties, where $k$ is a field of characteristic $0$. Let $L$ be a line bundle over $X$. Assume that $L$ is nef and generically ample. Does there exist a line bundle $A$ on $Y$ such that $L\otimes f^*A$ is big?

Answer: Yes


Question: Find the coordinates of the point $P$ on the parametric curve $(x,y,z)=(2−t,−1−t^2,−2t−3t^3)$ such that the tangent line at $P$ passes through the point $(2,8,−162)$. Provide your answer as $(x, y, z)$.

Answer: (5, -10, 87)


Question: Let $A \in GL(n, \mathbb{C})$ be a symmetric matrix. Determine whether there always exists a matrix $B \in GL(n, \mathbb{C})$ such that $A = B^T B$. Provide a proof or counterexample.

Answer: Yes


Question: Determine the distribution of the random variable $X$, given that $Y \sim N(29.5, 6.49)$ and $X \sim N(y, 0.16)$, where $y$ is sampled from $Y$. What is the resulting distribution of $X$?

Answer: N(29.5, 6.65)


Question: Let $L \in GL_n(\mathbb{Q})$ be a linear invertible map on $\mathbb{Q}^n$. If $L^{-1} = L + L^2$, prove that $3 \mid n$.

Answer: 3


Question: Let $p > 3$ be a prime number, and let $G \leq \mathrm{PGL}_2(\mathbb{F}_p)$ be a solvable subgroup. Determine if the action of $G$ on $\mathbb{P}^1(\mathbb{F}_p)$ can be transitive.

Answer: Yes


Question: Let \( f(x) = x \sin(x^N) \). Determine the smallest integer \( k \) such that the \( k \)-th derivative of \( f(x) \) evaluated at 0, \( f^{(k)}(0) \), is not equal to zero. Express your answer in terms of \( N \).

Answer: N + 1


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\sum_{k=0}^{n}{n \choose k}|x^{n-k}(1-x)^{k}-y^{n-k}(1-y)^{k}| \] where \( x \in (0,\frac{1}{2}) \) and \( y \in (0,\frac{1}{2}] \). Specifically, find the value of the limit when \( y = \frac{1}{2} \): \[ \lim_{n\rightarrow \infty}\sum_{k=0}^{n}{n \choose k}|x^{n-k}(1-x)^{k}-\frac{1}{2^{n}}| \]

Answer: 2


Question: Does there exist an extreme vector in the positive cone $A^{+}$ of a unital $C^*$-algebra $A$?

Answer: Yes


Question: Given that $x - \frac{1}{x} = \sqrt{2}i$, find the value of $x^{2187} - \frac{1}{x^{2187}}$. Express your answer in terms of $i$. \( \boxed{} \)

Answer: \sqrt{2}i


Question:
Determine if there exists a constant \(C \geq 0\) such that the following inequality holds for functions \(f(x)\) and \(g(x)\) from \(\mathbb{R}\) to \(\mathbb{R}\):
\[ \left|X-Y\right| \leq C\left|\max (-f(x)) - \max (-g(x))\right| \leq C\max \left|f(x) - g(x)\right| \]
where \(X = \max\left[0, \max(-f(x))\right]\) and \(Y = \max\left[0, \max(-g(x))\right]\).

Answer: 1


Question: Determine the number of zeros of the polynomial \( h(z) = z^8 + 5z^7 - 20 \) inside the unit disk using Rouché's theorem.

Answer: 0


Question: Given a lower triangular matrix $L$ of dimension $n \times n$ with distinct elements along the diagonal, is the number of unitary similarities that transform $L$ into another lower triangular matrix exactly $n!$? Justify your answer.

Answer: No


Question: Evaluate the limit: \[ \lim_{n \to \infty} n \int_0^1 (\cos x - \sin x)^n \, dx \]

Answer: 1


Question: Find a transformation that maps a rectangular region $D$ in the $uv$-plane onto the region $R = \{(x, y) \in \mathbb{R}^2 : x \geq 0, y \geq 0, 1 \leq x+y \leq 2\}$. Use this transformation to evaluate the integral \( \iint_R \frac{y}{x + y} \, dA \).

Answer: \dfrac{3}{4}


Question: Let $n,m>0$ be positive integers and consider the canonical projection $\pi: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^n$ defined by $\pi(x,y) = x$, where $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^m$. Does $\pi$ have a section, i.e., a continuous right-inverse?

Answer: Yes


Question: There are 11 rings arranged in a circle, numbered from 1 to 11. Exactly 9 of these rings are fake, and exactly 2 are real. You can choose any 5 consecutive rings and ask how many real rings are in that selection. What is the minimum number of questions needed to identify at least one real ring?

Answer: 4


Question:
Determine the set of values for $p \in \mathbb{R}$ such that the function $f(x) = \sqrt{\log_7\left(\frac{2x^2 + px + 5}{x^2 + 1}\right)}$ is defined for every real number $x$. Choose from the following options:

A: $(-4,4)$

B: $(-\infty,-4)$

C: $(-\infty,4)$

D: $[-4,4]$

E: Empty

Answer: D


Question: Let \((r_n)\) and \((s_n)\) be sequences of positive real numbers. Suppose that for every \(\epsilon>0\), the inequality \(r_{n}^{1+\epsilon}<s_n^{\epsilon}\) holds for all \(n\) sufficiently large. If for every \(\epsilon>0\), \(\lim_{n\to\infty}s_{n}^{\epsilon}=0\), determine whether \(\lim_{n\to\infty}r_{n}=0\).

Answer: 0


Question: Let $(E, \mathcal{A}, \mu)$ be a finite measure space and $\{f_n\}$ be a sequence of integrable functions such that for all $\epsilon \in (0, \frac{\sqrt{2}}{2}]$, \[ \lim_{n \to \infty} \mu(\{t : \sup_{k \geq n} |f_k(t) - f(t)| > \epsilon\}) = 0. \] Determine if it follows that \[ \mu(\{t : \lim_{n \to \infty} f_n(t) \neq f(t) \text{ or } \lim_{n \to \infty} f_n(t) \text{ does not exist}\}) = 0. \]

Answer: 0


Question: Evaluate the integral \( I = \int^{\infty}_{0} \frac{x \cdot \ln x}{(1+x^2)^2} \, dx \) using integration by parts.

Answer: 0


Question: Let $G$ be a $2$-edge-connected graph drawn in the plane, where edges intersect only at their endpoints. Determine whether it is always possible to orient the edges of $G$ such that for each vertex $v$, there are no three consecutive edges (in the clockwise direction) all oriented towards $v$ or all oriented away from $v$.

Answer: Yes


Question: Given that \( \frac{a_n}{a_{n-1}} = \frac{3n-1}{3n} \) for \( n = 1, 2, \ldots \) and \( a_0 = 1 \), find \( \lim_{n \to \infty} a_n \).

Answer: 0


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

Answer: 0


Question: Determine whether it is possible for the equation \( f(x) = g(x) + h(x) \) to hold if \( f(x) \) is an even function, \( g(x) \) is an odd function, and \( h(x) \) is neither even nor odd.

Answer: Yes


Question: Determine whether 1331 is a porous number. A porous number is defined as a number $k$ which is not a multiple of 10 such that every $m$ with a sum of digits equal to $k$ and $k$ being a divisor of both $m$ and rev($m$) has a zero in its digits. Here, rev($m$) is the digit reversal of $m$. Provide your answer with justification.

Answer: 1331


Question: Find the minimum value of the function \( f(x,y) = \frac{4}{x^2+y^2+1} + 2xy \) within the domain \( \frac{1}{5} \leq x^2+y^2 \leq 4 \).

Answer: -\dfrac{16}{5}


Question: Let \( f(x) = \int_2^x \frac{1}{1+t^4} \, dt \) and \( g(x) = f^{-1}(x) \). Find \( \frac{d}{dx}(g(x)) \) at \( x = 0 \).

Answer: 17


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{x - \sin x}{x^3} \] without using L'Hôpital's rule.

Answer: \dfrac{1}{6}


Question:
Determine the nature of the value of the expression:
\[ \log_{10}(\tan 6^\circ) + \log_{10} (\tan 12^\circ) + \log_{10} (\tan 18^\circ)+\cdots+\log_{10}(\tan 84^\circ) \]
where the angles are in degrees. Choose the correct option:
(A) Whole number
(B) An irrational number
(C) A negative number

Answer: A


Question:
Suppose \(X\) and \(Y\) have a joint density \(f_{X, Y}(x, y) = c\) for all \(x, y\) satisfying:
\(0<y<x<1\) or \(0<y<1-x\) and \(0<x<1\). Find the value of \(c\) such that the total probability is 1.

Answer: \dfrac{4}{3}


Question: Evaluate the integral: $$\int_{-\infty}^\infty z^2 e^{-\frac{z^2}{2}}\ dz$$

Answer: \sqrt{2\pi}


Question: Let $G$ be a finite non-abelian group, and let $A$ and $B$ be subgroups of $G$ with order $p$, where $p$ is a prime number greater than 2. Given that $A$ and $B$ are isomorphic to $\mathbb{Z}_p$, determine whether the product $ab = ba$ holds for any elements $a \in A$ and $b \in B$. Provide a justification for your answer.

Answer: No


Question: Determine the number of solutions to the equation \(x^p \equiv x \pmod{p^n}\) in \(\mathbb{Z}_{p^n}\), where \(p\) is a prime number and \(n\) is a positive integer.

Answer: p


Question: Let $f$ be a measurable function on a measure space $(X, \mathcal{F}, \mu)$ and $p > 0$ such that $||f||_p := \left(\int_X |f|^p \, d\mu\right)^{1/p} < \infty$. Prove or disprove that there exists a sequence of simple functions $\{g_n\}$ such that $||g_n - f||_p \to 0$ as $n \to \infty$. \( \text{(True or False)} \)

Answer: A


Question:
Calculate the value of $S-T$, where $S$ and $T$ are defined as follows:
\[ S = \sum_{k=0}^{n} (-1)^k \frac{1}{k+m+1} \binom{n}{k} \]
\[ T = \sum_{k=0}^{m} (-1)^k \frac{1}{k+n+1} \binom{m}{k} \]

Answer: 0


Question: Let $X$ and $Y$ be random variables with joint density \(f(x,y) = x + y\) for \(0 \leq x \leq 1\) and \(0 \leq y \leq 1\). Define \(T = \max(X,Y)\). Calculate \(\mathbb{E}[XT]\).

Answer: \dfrac{7}{15}


Question: Determine the cardinality of the equivalence class of \( \frac{1}{\sqrt{2}} \) under the relation defined on the interval \( I = [0, 1] \) by \( x \sim y \) if and only if \( x - y \in \mathbb{Q} \).

Answer: \aleph_0


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{\ln(\tan(x)+1)-\sin(x)}{x\sin(x)} \] without using L'Hôpital's rule.

Answer: -\dfrac{1}{2}


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

Answer: 0


Question: There are N cards numbered from 1 to N in a box. Two cards are drawn randomly, and the smaller number is divided by the larger number to form a ratio Z. Calculate the expected value of Z, denoted as \( \mathbb{E}(Z) \).

Answer: \dfrac{1}{2}


Question:
Given a point $D$, a line $l$, and a circle $K$ with center $C$ and radius $r$ in the plane $\mathbb{R}^2$, find a point $G$ on the line $l$ such that the following conditions are satisfied:

1. The line $GC$ intersects the circle $K$ at a point $H$.
2. The distance from $G$ to $D$ is equal to the distance from $G$ to $H$, i.e., $\left\lVert\vec{GD}\right\rVert = \left\lVert\vec{GH}\right\rVert$.

Assume that $D$ and $C$ lie on the same side of $l$, $D$ is exterior to $K$, and $l$ intersects $K$ at two different points without passing through $C$. Find the point $G$ that satisfies these conditions.

Answer: G


Question: What is the average length of a cycle in a permutation of the set \( \{1, 2, 3, \dots, n\} \)?

Answer: \dfrac{n}{H_n}


Question: Evaluate the series \(\sum_{n=1}^{\infty} \frac{2^{[\sqrt{n}]}+2^{-[\sqrt{n}]}}{2^n}\), where \([\ \ ]\) denotes the Nearest Integer Function.

Answer: 3


Question: Tangents are drawn from the point \((-2\sqrt{3}, 2)\) to the hyperbola \(y^2 - x^2 = 4\). The chord of contact of these tangents subtends an angle \(\theta\) at the center of the hyperbola. Calculate the value of \(12 \tan^2 \theta\).

Answer: 9


Question: Evaluate the limit \( \lim \limits_{r \to \infty} \frac {r^C \int_0^{\frac{\pi}{2}} x^r \sin(x)\, dx}{\int_0^{\frac{\pi}{2}} x^r \cos(x)\, dx} = L \). Given that \(C\in\mathbb{R}\) and \(L>0\), find the value of \(\pi L - C\).

Answer: 3


Question: Let \( A \) be a real \( 2 \times 2 \) matrix such that \( A^6 = I \), where \( I \) denotes the identity \( 2 \times 2 \) matrix. Determine the total number of possibilities for the characteristic polynomial of \( A \).

Answer: 5


Question: Given a topological space \( E \) with the topology \( \theta = \{ G \subseteq E \mid \text{card}(E \setminus G) < \infty \} \cup \{ \emptyset \} \), where \( \text{card}(E) = \infty \), find the interior \( \mathring{A} \) of a set \( A \) such that \( \text{card}(A) = \infty \) and \( \text{card}(E \setminus A) = \infty \).

Answer: \emptyset


Question: Find the expectation of $X^4$ for an exponential random variable $X$ with parameter $a$.

Answer: \dfrac{24}{a^4}


Question:
Determine whether there exists a bivariate function \( f: [0,1]^2 \to \mathbb{R} \) that satisfies the following conditions:

1. \( 0 \leq f(x,y) \leq 1 \) for all \( x, y \in [0,1] \).
2. \( f(x,y) + f(y,x) = 1 \) for all \( x, y \in [0,1] \).
3. \( \int_0^1 f(x,y) \, dy = \frac{1}{2} \) for all \( x \in [0,1] \).

Provide a justification for your answer.

Answer: Yes


Question: Given a set $B$ of colored balls with $|B| = 4k$, where $k \in \mathbb{N}$, and a set $C$ of colors with $|C| = |B|$, each ball is painted in $k$ different colors from $C$. If any two balls have a common color, determine if it is possible to guarantee that at least $0.251 \cdot |B|$ balls share a common color for arbitrarily large $k$. Provide a lower estimate better than $|B|/4$.

Answer: 0.251 \cdot |B|


Question: Evaluate the integral \( \int_{-\infty}^{\infty} e^{-\pi(x + i\xi)^2} \, dx \), where \( \xi \) is a real constant.

Answer: 1


Question: Evaluate the limit \( \lim_{n \to \infty} \int_{0}^1 x^{n+1} f(x) \, dx \), where \( f : [0, 1] \to \mathbb{R} \) is Riemann integrable.

Answer: 0


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

Answer: 1


Question: Find the value of $a$ where the functions $\sin(x)$ and $\ln(x)$ intersect, i.e., solve $\sin(x) = \ln(x)$ for $x$. This value of $a$ is used in setting up the integral for the volume of the solid created by revolving the region bounded by $\sin(x)$, $\ln(x)$, and the $x$-axis about the line $y=5$. Provide the value of $a$.

Answer: 2.219


Question: Evaluate the limit: $$\lim_{x\to\infty}\left(\cos\frac{3}{x}\right)^{x^2}$$

Answer: e^{-\frac{9}{2}}


Question: Find the integer $b$ such that \((2+\sqrt{3})^n = 5042 + b\sqrt{3}\).

Answer: 2911


Question: Compute the degree of the line bundle \( \mathcal{O}_X(nX_1+mX_3)|_{X_2} \) on the component \( X_2 \) of the fiber \( X_t \) over \( t=0 \), where \( X_1, X_2, X_3 \) are smooth irreducible components intersecting transversally in a chain, with \( X_1 \) intersecting \( X_2 \) at point \( P \) and \( X_2 \) intersecting \( X_3 \) at point \( Q \), and \( P \neq Q \). Here, \( m \) and \( n \) are integers.

Answer: n + m


Question: Given that \(a, b, c\) are the roots of the equation \(7x^3 - 25x + 42 = 0\), find the value of the expression \((a+b)^3 + (b+c)^3 + (c+a)^3\).

Answer: 18


Question: Evaluate the line integral \( \int_C (x+y) \, ds \) where \( C \) is the triangle with vertices \((0,0)\), \((1,0)\), and \((0,1)\), traversed in a counterclockwise direction.

Answer: 1 + \sqrt{2}


Question: Find the real value(s) of \(a\) (where \(a \neq -1\)) for which the limit \( \lim_{ n \to \infty} \frac{ 1^a+2^a+\cdots+n^a}{(n+1)^{a-1}[ (na+1)+(na+2)+\cdots+(na+n)]} = \frac{1}{60} \).

Answer: 7


Question: Consider the random experiment in which three fair dice are rolled simultaneously. Let $X$ be the sum of the values of these three dice. Let $Y_1$ be the maximum of the three values, $Y_2$ be their product, and $Y = Y_1 + Y_2$. Define $Z = E[X | Y]$, the conditional expectation of $X$ given $Y$. Find the expected value of the random variable $Z$. Enter your answer as a fraction, such as $\frac{3}{2}$.

Answer: \dfrac{21}{2}


Question: Find all odd integers \( n \) greater than 1 such that for any relatively prime divisors \( a, b \) of \( n \), the number \( a+b-1 \) is also a divisor of \( n \).

Answer: p^k


Question: Find the supremum of the set \( S = \{ x \in \mathbb{R} : x \ge 0, \sum_{n=1}^\infty x^{\sqrt{n}} < \infty \} \).

Answer: 1


Question: Suppose the scores of an exam follow a normal distribution, and the correlation coefficient \( \rho = 0.8 \) for exam1 and exam2. If a student's score percentile rank (PR) is 90% in exam1, estimate his/her PR in exam2.

Answer: 85


Question: Let $X_t$ be a bounded continuous martingale adapted to the filtration $\mathcal{F}_t$ such that \(\Bbb{E} \bigg[\int_0^t  e^{X_s} \, d\langle X\rangle_s\bigg] = 0\) and $X_0=0$. Prove that $X_t$ is identically zero.

Answer: X_t \equiv 0


Question:
Determine whether the following equation is true for a 3-dimensional Riemannian manifold $(\Sigma,h_{ij})$ with a 2-dimensional submanifold $S$:
\[2(\nabla_i R^{ij})n_j=(\nabla_kR)n^k\] where $\nabla$ is the Levi-Civita Connection with respect to $h_{ij}$, $R_{ij}$ and $R$ are the Ricci and scalar curvatures, respectively, and $n_i$ is the unit normal to $S$. Provide a justification for your answer.

Answer: True


Question: Calculate the double integral \( \iint_D xy \, dx \, dy \), where \( D \) is the region bounded by a hexagon inscribed in the circle \( x^2 + y^2 = 1 \) with one vertex at the point (1,0).

Answer: 0


Question: Find the volume of the solid that lies above the X-Y plane and below the elliptic paraboloid given by \( x^2 + \frac{y^2}{4} = z \), and is cut off by the plane \( z = 9 \).

Answer: 81\pi


Question: Find the $a_5$ Fourier coefficient for the function $f(t) = t^3 - 2t$ over the interval $-\pi < t < \pi$. Provide your answer to two decimal places.

Answer: 0.00


Question: Find the one's digit of the remainder when \( |5555^{2222} + 2222^{5555}| \) is divided by 7.

Answer: 0


Question: Determine whether the following statement is true or false: In a Lie algebra $\mathcal{L}$, the product of two ideals $[I, J]$ is equal to the intersection $I \cap J$. Justify your answer.

Answer: False


Question: Find the number of positive integer triples \((x, y, z)\) such that \([x, y] = 72\), \([x, z] = 600\), and \([y, z] = 900\), where \([a, b]\) denotes the least common multiple of \(a\) and \(b\).

Answer: 15


Question: Calculate \(\mathbb{E}(X + Y)^2\) where \(X \sim P(\lambda_{1})\) and \(Y \sim P(\lambda_{2})\) are independent random variables. Given that \(\sum_{n=0}^{\infty} P(X+Y>n) = 2\), find the value of \(\mathbb{E}(X + Y)^2\).

Answer: 6


Question: Evaluate the contour integral \( \oint_C \frac{2}{z(z+2)} \, dz \) where \( C \) is the circle \( |z| = 1 \) in the complex plane.

Answer: 2\pi i


Question: Determine the maximal interval of existence for solutions of the differential equation \( x'(t) = -x(t) + \sin x(t) + t^3 \) in \( \mathbb{R} \).

Answer: (-\infty, \infty)


Question:
Determine the range of the constant $k$ for which all the roots of the equation \(x^4 + 4x^3 - 8x^2 + k = 0\) are real. Choose the correct option:

(a) \((1,3)\)
(b) \([1,3]\)
(c) \((-1,3)\)
(d) \((-1,3]\)

Answer: B


Question: Determine the number of elements in the quotient ring $\mathbb{Z}[i]/(2+i)$, where $\mathbb{Z}[i]$ is the ring of Gaussian integers.

Answer: 5


Question: Use the Laurent series of \( e^{1/z} \) on the punctured plane \( \mathbb{C} - \{0\} \) to compute \( \frac{1}{\pi} \int_0^\pi e^{\cos\theta}\cos(\sin\theta-n\theta) \, d\theta \) for \( n = 0, 1, 2, \ldots \).

Answer: \dfrac{1}{n!}


Question: Evaluate the limit: \[ \lim_{n \to \infty}\left(\sqrt[3]{n^3+3n}-\sqrt{n^2-2n}\right) \]

Answer: 1


Question: Find the smallest positive integer value of \( c \) such that \( \frac{1}{2015} = \frac{a}{5} + \frac{b}{13} - \frac{c}{31} \), where \( a \), \( b \), and \( c \) are positive integers.

Answer: 41


Question: Find the smallest integer \( n \geq 2 \) such that there exists a matrix \( B \in M_{4}(\mathbb{C}) \) with \( B^n = \begin{bmatrix}0 & 0 & 0 & 1\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\end{bmatrix} \).

Answer: 2


Question: Find the upper bound of \( \left|\frac{1}{z^4 + 3z^2 + 2}\right| \) where \(|z| = 2\) and \(z \in \mathbb{C}\).

Answer: \dfrac{1}{6}


Question: Find the integer value of \( x \) such that \( 0 \leq x \leq 143 \) and \( x^{103} \equiv 2 \pmod{143} \).

Answer: 128


Question: Find a positive integer $n$ such that there exists a subfield of $\mathbb{Q}_n$, where $\mathbb{Q}_n = \mathbb{Q}(\omega)$ and $\omega$ is a primitive $n$th root of unity, that is not a cyclotomic extension of $\mathbb{Q}$. Provide the smallest such $n$. \(\text{Note: A cyclotomic extension is a field extension obtained by adjoining a root of unity to } \mathbb{Q}.\)

Answer: 5


Question: Determine the smallest integer \( N \) such that the inequality \( \sum_{k=0}^{17} \binom{N}{k} \times 0.1^k \times 0.9^{N-k} < 0.004 \) holds.

Answer: 317


Question: Solve the inequality: \[ x + \log(5^x - 1) < x \cdot \log(2) + \log(20) \]

Answer: (0, 1)


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

If $1<p<\infty$, $|f_n|\leq 1$, $f_n\rightarrow f$ in measure, and $g_n\rightarrow g$ in $L_p$, then $f_ng_n\rightarrow fg$ in $L_p$. Provide a proof or a counterexample to justify your answer.

Answer: True


Question:
Calculate the volume of the region in \( \mathbb{R}^3 \) bounded by the planes \( y = 1 \), \( y = -x \), \( z = -x \), and the coordinate planes \( x = 0 \) and \( z = 0 \). Evaluate the integral:
\[ \int_{0}^{1} \int_{0}^{-y}\int_{0}^{-x} \;dz \;dx\;dy \]

Answer: \dfrac{1}{6}


Question: Suppose we have a $C^k$ map $f: \mathbb{R}^{n+1} \rightarrow \mathbb{R}^n$. Define a map $g: \mathbb{R}^n \rightarrow \mathbb{R}^n$ by \( g(x_1, \ldots, x_n) = f(x_1, \ldots, x_n, 0) \). Must $g$ be a $C^k$ map? Provide a justification for your answer.

Answer: Yes


Question: Given that \( X \sim N(\mu, \sigma^2) \) and \( P(X \leq 15) = 0.80 \), find the smallest value of \( \alpha \) such that \( P(X \leq 10) = \alpha \).

Answer: 0


Question:
Given the continuous-time state-space representation:
\[ \frac{d}{dt}x(t)=Ax(t)+Bu(t), \quad y(t)=Cx(t), \quad t \in \mathbb{R}^+, \]
with matrices:
\[ A =\begin{bmatrix}-4&-5&0&5\\-4&-4&0&4\\0&0&-2&0\\-4&-5&0&5\end{bmatrix}, \quad B=\begin{bmatrix}2\\0\\0\\1\end{bmatrix}, \quad C=\begin{bmatrix}1&-1&0&0 \end{bmatrix}. \]
The matrix \( A \) satisfies \( AV=V\Lambda \) with:
\[ \Lambda=\begin{bmatrix}-4&0&0&0\\0&-2&0&0\\0&0&0&0\\0&0&0&1\end{bmatrix}, \quad V=\begin{bmatrix}1&0&0&1\\1&0&-1&0\\0&1&0&0\\1&0&-1&1\end{bmatrix}, \quad V^{-1}=\begin{bmatrix}1&1&0&-1\\0&0&1&0\\1&0&0&-1\\0&-1&0&1\end{bmatrix}. \]
Determine the transfer function \( H(s) \) of the system, with all pole/zero cancellations removed, from the following options:

A) \( H(s)=\frac{2s^2+6s-8}{s(s+4)(s-1)} \)

B) \( H(s)=\frac{2s-2}{s(s-1)} \)

C) \( H(s)=\frac{2s^3+10s^2+4s-16}{s(s+4)(s+2)(s-1)} \)

D) \( H(s)=\frac{2}{s} \)

E) None of the above

Answer: E


Question: Determine the probability distribution of the x-coordinate for a random point on the triangle with vertices at (1,0,0), (0,1,0), and (0,0,1).

Answer: 2(1 - x)


Question: Determine the radius of convergence of the power series \( \sum_{n=-\infty}^0 n \, 3^{-n} z^n. \)

Answer: 3


Question: Evaluate the limit: $$\lim_{x \to 0} \frac{a^x \sin(bx) - b^x \sin(ax)}{\tan(bx) - \tan(ax)}$$ where $a \neq b$. Determine the value of the limit.

Answer: 1


Question: What is the distribution of the test statistic when the partial likelihood of the Cox model is used in place of the full likelihood in a likelihood ratio test? Is it still a chi-squared distribution with q degrees of freedom?

Answer: \chi^2_q


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\arctan x}{e^{2x} - 1} \] without using L'Hôpital's rule.

Answer: \dfrac{1}{2}


Question: What is the completion of the metric space \((\mathbb{Q}, d)\), where \(d(x, y) = |x - y|\)?

Answer: \mathbb{R}


Question: Let $f(x) = x^2 + x$ for all real $x$. There exist positive integers $m$ and $n$, and distinct nonzero real numbers $y$ and $z$, such that $f(y) = f(z) = m + \sqrt{n}$ and $f(1/y) + f(1/z) = 1/10$. Compute $100m + n$. \( \text{(Express your answer as an integer.)} \)

Answer: 1735


Question:
The solutions to the equation \((z+6)^8=81\) form a convex regular polygon in the complex plane. Three of the vertices of this polygon are labeled \(A, B,\) and \(C\). What is the least possible area of \(\triangle ABC\)?

(A) \(\frac{1}{6}\sqrt{6}\)
(B) \(\frac{3}{2}\sqrt{2}-\frac{3}{2}\)
(C) \(2\sqrt{3}-3\sqrt{2}\)
(D) \(\frac{1}{2}\sqrt{2}\)
(E) \(\sqrt{3}-1\)

Answer: B


Question: Determine the number of terms needed in the series expansion of the integral \( \int_{0}^{1} e^{-x^2} \, dx \) to approximate it within an error of 0.01.

Answer: 4


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

Answer: \sqrt{2\pi}


Question: Determine the smallest integer $k$ such that a $k$-vector with binary entries can be used to communicate any integer between 1 and 1000 to a friend, even after one of its components is altered by a function $f_i$, which sets the $i$-th component to 1.

Answer: 14


Question: For a certain value of \(c\), the limit \(\lim_{x \to \infty}\left[\left(x^{5} + 7x^{4} + 2\right)^c - x\right]\) is finite and non-zero. Let \(l\) be this limit. Calculate \(l - 2c\).

Answer: 1


Question: Given two independent random samples $X_1, \ldots, X_{100}$ and $Y_1, \ldots, Y_{50}$ from a normal distribution $N(\mu, 1)$, calculate the probability that the 80% confidence intervals for $\mu$ constructed from these samples are disjoint.

Answer: 0.074


Question: Determine the dimension of the space of \( n \times n \) matrices with non-zero trace, given that the dimension of the space of all \( n \times n \) matrices is \( n^2 \) and the dimension of the space of \( n \times n \) matrices with trace zero is \( n^2 - 1 \).

Answer: n^2


Question: If an infinite sum of positive terms \(\sum x_n\) diverges, must there exist a sequence \(y_n\) such that \(\frac{y_n}{x_n} \to 0\) as \(n \to \infty\) and \(\sum y_n\) also diverges? Provide a proof or counterexample.

Answer: Yes


Question: Determine the number of non-constant polynomial factors with leading coefficient 1, where the other coefficients may be complex, for the polynomial \(x^{2015} + 18\).

Answer: 2^{2015} - 1


Question: Does a ring homomorphism $f: R \to S$ induce a ring homomorphism $H^*(X,A;R) \to H^*(X,A;S)$ on cohomology rings?

Answer: Yes


Question: Show that the centralisers \(C_{S_6}(s)\) and \(C_{S_6}(t)\) are isomorphic to \(S_3 \times C_3\), where \(s = (123)\) and \(t = (123)(456)\) are elements of \(S_6\).

Answer: S_3 \times C_3


Question:
Evaluate the sum:
\[ \sum_{x=0}^\infty x(x-1) {2+x \choose x}(0.008)(0.8)^x \]

Answer: 192


Question: Let A be a positive definite matrix. If the inverse of A is expressed as \( \text{inverse}(A) = \frac{1}{\det(A)} B \), is the matrix B still positive definite?

Answer: Yes


Question: Express \( g(1+\alpha) \) in the basis \( \{ 1, \alpha \} \) of the vector space \( E = \mathbb{Z}/3[x]/(x^2+1) \), where \( g(x) = 2x^4 + x^2 + 1 \) and \( \alpha \) is a root of \( x^2 + 1 \).

Answer: 2 + 2\alpha


Question: Determine the highest weight of the trivial representation of a Lie algebra $g$. Express this weight using fundamental weights.

Answer: 0


Question: Evaluate the integral \( \int_{\Bbb R ^2} e^{-x^2-y^2} \, dx \, dy \).

Answer: \pi


Question: In square $ABCD$, points $E$ and $F$ are located on sides $BC$ and $CD$, respectively, such that $\angle EAF = 45^\circ$. The lines $AE$ and $AF$ intersect the diagonal $BD$ at points $P$ and $Q$, respectively. Determine the ratio $\frac{[\Delta AEF]}{[\Delta APQ]}$. \(\text{Express your answer as a simplified fraction.}\)

Answer: 2


Question: Determine the total number of automorphisms of the field extension \( \mathbb{Q}(\sqrt{2},\sqrt{3}) \).

Answer: 4


Question: Let $a$, $b$, and $c$ be three complex numbers such that $|a| = |b| = |c| = 1$ and $2(a+b+c) - 3abc$ is a real number. Given that the arguments of $a$, $b$, and $c$ are all positive, find the minimum value of $\max(\arg(a), \arg(b), \arg(c))$. Provide your answer in radians.

Answer: \dfrac{\pi}{6}


Question: Evaluate the limit: \[ \lim_{n \to \infty}\sum_{k=1}^n\frac{1}{\sqrt[3]{n^3+k}} \]

Answer: 1


Question: Suppose \(X\) and \(Y\) are independent and identically distributed random variables with an exponential distribution of parameter \(\lambda\). Let \(Z = X + Y\). Determine the conditional density function \(f_X(x \mid Z = z)\).

Answer: \dfrac{1}{z}


Question: Find the residue of the function \( f(z) = \frac{\sin z}{(z^2+1)^2} \) at \( z = \infty \).

Answer: \dfrac{1}{2e}


Question: Find the polynomial $p$ such that $p(x-1) = x+1$ in the ring $\mathbb{C}[x]/(x^3-x^2)$. Provide your answer as a polynomial in terms of $x$.

Answer: x + 2


Question: Evaluate the integral \( \int_{\mathcal{C}} \frac{\sin z}{z-\pi/2}\,\mathrm{d}z \), where \( \mathcal{C} \) is the contour defined by \( |z| = \pi/2 \).

Answer: \pi i


Question: Suppose $H$ is a separable Hilbert space. If $P$ and $Q$ are two finite-dimensional orthonormal projections, is there a finite-dimensional orthonormal projection $L$ such that $L(H) \supset P(H)$ and $L(H) \supset Q(H)$?

Answer: Yes


Question: Evaluate the integral: $$\int^{\pi/2}_0 \frac{\cos^4x + \sin x \cos^3 x + \sin^2x\cos^2x + \sin^3x\cos x}{\sin^4x + \cos^4x + 2\sin x\cos^3x + 2\sin^2x\cos^2x + 2\sin^3x\cos x} \, dx$$

Answer: \dfrac{\pi}{4}


Question: Consider a topological space $X$ with countable tightness, denoted by $t(X) = \omega$. If $X$ has a dense subset $A$ with $|A| < 2^\omega$, does it follow that $|X| \le 2^\omega$? Provide a justification for your answer.

Answer: No


Question: Find the length of the curve given by the polar equation \( r = 1 + \sin{\theta} \) for \( 0 \leq \theta \leq 2\pi \).

Answer: 8


Question: Determine if there exists a prime number of the form \(n^m - 1\) for any integer \(n > 2\) and even integer \(m > 2\).

Answer: No


Question: A person can be in one of three states: employed, unemployed, or early retirement. The transition rates per year are as follows: an employed person becomes unemployed with a rate of 0.2, an unemployed person becomes employed with a rate of 4, and an unemployed person retires early with a rate of 0.5. A working person never retires directly, and a retired person stays retired forever. A person starts employed, earning $280,000 per year while employed and $224,000 per year while unemployed, with no earnings in retirement. Calculate the expected total earnings before the person retires early.

Answer: 13048000


Question:
Find the pointwise limit of the sequence of functions \( f_n(x) \) and determine whether the convergence is uniform. The sequence is defined as follows:
\[
f_n(x) = \begin{cases}
 nx^2, & \text{if } x \in \left[0, \frac{1}{\sqrt{n}}\right], \\
 0, & \text{if } x \in \left(\frac{1}{\sqrt{n}}, 1\right].
\end{cases}
\]

Answer: 0


Question: Evaluate the complex integral \( \int \limits_{\beta} f(z) \, dz \) where \( \beta \) is the path consisting of the line segment from \(-i\) to \(2+5i\), followed by the line segment from \(2+5i\) to \(5i\), and \( f(z) = iy + x^2 \) for \( z = x + yi \).

Answer: -12 + 2i


Question: Find the volume of the solid defined by the inequalities: \(x^2+y^2\leq4\), \(z=2+x^2+y^2\), and \(z\geq-1\).

Answer: 20\pi


Question: Determine the number of imaginary roots of the equation \( \sum_{n=1}^{100} \frac {n^2}{x-x_{n} }= 101 \), where each \( x_n \) is a real number.

Answer: 0


Question: Find the value of \( L = \limsup_{n \to \infty} \sum_{k=0}^n (-1)^k |\sin k| \).

Answer: \infty


Question: Determine the ideal \( I(X) \) for the subset \( X \subseteq \mathbb{A}^2 \) defined by the equations \( x^2 + y^2 = 1 \) and \( x = 1 \), where \( k \) is an algebraically closed field and \( \mathbb{A}^2 \) is affine.

Answer: (x - 1, y)


Question: Suppose for $p \geq 1$ that \( \left( \sum^\infty_{k=0} |a_k|^p \right)^{1/p} < + \infty \). Determine whether it is necessarily true that \( \sum^\infty_{k=0} |a_k|^p \log|a_k| \leq 0 \). Justify your answer.

Answer: No


Question: Find the quadratic equation \( y = ax^2 + bx + c \) that best fits the points \((-1, -3)\), \((0, 0)\), \((1, -1)\), and \((2, 1)\) using the least squares method. Use the normal equation \( A^T A X = A^T B \) to determine the coefficients \(a\), \(b\), and \(c\). Then, calculate the exact value of \( y(1) \) on the quadratic regression curve.

Answer: \dfrac{1}{20}


Question: Determine the condition on $\beta$ and $p$ such that the function $f: \mathbb{R}^n \to \mathbb{R}$ defined by \[ f(x) = \frac{1}{|x|^\beta} e^{-\frac{|x|^2}{2}} \] belongs to $L^p(\mathbb{R}^n)$. Express your answer as an inequality involving $n$, $\beta$, and $p$.

Answer: \beta p < n


Question: Calculate the flux of the vector field \( F = \langle x, y, z^2 \rangle \) across the unit sphere centered at the origin. Express your answer in terms of \( \pi \).

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


Question: Let $\mathcal{O}$ be the ring of integers of a number field, and let $\{\mathfrak{p}_i,\,i \in \mathbb{N}\}$ be a sequence of pairwise distinct prime ideals. Determine whether the intersection $\bigcap_i \mathfrak{p}_i$ is equal to $\{0\}$. Provide a justification for your answer.

Answer: \{0\}


Question: Find the local minimum value of the function \( h(x) = \frac{x^2 + \frac{1}{x^2}}{x - \frac{1}{x}} \) for \( x \in \mathbb{R} \setminus \{-1, 0, 1\} \).

Answer: 2\sqrt{2}


Question: Evaluate the integral \( I = \frac{1 - |a|^2}{\pi} \int_{|z|=1} \frac{|dz|}{|z + a|^2} \), where \( a \in \mathbb{C} \) and \( |a| < 1 \), over the simple closed contour \(|z| = 1\) with positive orientation.

Answer: 2


Question: Let $(X, \mu)$ be a measure space and $E_n$ be a sequence of measurable sets such that $\mu(X \setminus E_n) \to 0$. Define $G$ as the set of points $x \in X$ that belong to only finitely many of the sets $E_n$. Prove that $G$ is measurable and that $\mu(G) = 0$. \( \text{(Hint: Consider the complement of } G \text{ and use properties of } \sigma\text{-algebras.)} \)

Answer: 0


Question:
Given two games with the same number of players, each having a set of equilibria, consider the following probabilities:

1. \( Pr(\text{players pick } Y_1 \text{ in game 1}| \text{the set of equilibria of game 1 is } \mathcal{A}) \)
2. \( Pr(\text{players pick } X_1 \text{ in game 2}| \text{the set of equilibria of game 2 is } \mathcal{B}) \)

Assume that the selection rule of game 1 is independent of the selection rule of game 2. Does this imply that:

\[
Pr(\text{players pick } Y_1 \text{ in game 1}| \text{the set of equilibria of game 1 is } \mathcal{A}) \times Pr(\text{players pick } X_1 \text{ in game 2}| \text{the set of equilibria of game 2 is } \mathcal{B}) = Pr(\text{players pick } Y_1 \text{ in game 1}, \text{players pick } X_1 \text{ in game 2}| \text{the set of equilibria of game 1 is } \mathcal{A}, \text{the set of equilibria of game 2 is } \mathcal{B})
\]

Or are additional conditions required?

Answer: Yes


Question: Determine whether there exists a plane figure, other than the entire plane, that has three centers of symmetry which are not collinear.

Answer: Yes


Question: A matrix $A$ of size $n \times n$ is chosen at random, where each element is chosen uniformly and independently from among the numbers $1, \dots , \sqrt{n}$. Consider submatrices (not necessarily continuous) of size $2 \times 2$, defined by choosing two different rows $r,s$ and two different columns $c,d$. A submatrix is called "good" if all four elements in it are equal. Let $X$ be the number of good submatrices. Express $E[X]$ in big $O$ notation.

Answer: O(n^{5/2})


Question: Find the smallest positive integer \( x \) such that \( 10^x \equiv 1 \pmod{1997} \).

Answer: 998


Question: Evaluate the limit \( \lim_{n\to\infty}\frac{[x]+[2^2x]+\dots +[n^2x]}{n^3} \), where \([.]\) denotes the integral part of \(.\).

Answer: \dfrac{x}{3}


Question:
Let $f : (S, d_S) \to (T, d_T )$ be a bijective continuous function of metric spaces, and $f^{-1}: (T, d_T ) \to (S, d_S)$ be the inverse function. Determine which of the following statements is true:

(a) $f^{-1}$ is always a continuous function.
(b) $f^{-1}$ is continuous if $(S, d_S)$ is compact.
(c) $f^{-1}$ is continuous if $(T, d_T )$ is compact.
(d) $f^{-1}$ is continuous if $(S, d_S)$ is connected.

Select the correct option(s).

Answer: B


Question: Let \( X \sim N(0,1) \) be a standard normal random variable, and define \( U = 1 \) if \( X \ge 0 \) and \( U = 0 \) if \( X < 0 \). Find the covariance \( \text{cov}(\text{abs}(X), U) \), where \( \text{abs}() \) denotes the absolute value function.

Answer: 0


Question: Determine whether the function \( f(x) = x^2 \sin\left(\frac{1}{x^2}\right) \) for \( x \neq 0 \) and \( f(x) = 0 \) for \( x = 0 \) is differentiable on the entire real line.

Answer: Yes


Question: Find the radius of convergence for the power series \( \sum_{n=0}^{\infty} a_n z^n \) where \( a_n = \frac{n^2-5}{4^n+3n} \).

Answer: 4


Question: Given that $x, y, z > 0$ and $3x(x+y+z) = yz$, find the minimum value of $P = \frac{y+z}{x}$. Provide your answer in the form of a single number.

Answer: 6 + 4\sqrt{3}


Question: How many times must you roll a die with an unknown number of sides to determine the number of sides it has, given that you can only see the results of the rolls?

Answer: \infty


Question: Consider a quadratic Galois extension of \( \mathbb{Q} \), denoted as \( k \), with Galois group \( G \). Let \( \mathcal{O}_k \) be the ring of integers in \( k \). Is \( \mathcal{O}_k \) stable under the action of \( G \)? In other words, is it true that for every \( x \in \mathcal{O}_k \) and every \( \sigma \in G \), we have \( \sigma x \in \mathcal{O}_k \)?

Answer: Yes


Question: How many 3rd order determinants, formed using 0 and 1 as elements, have a determinant value of 0? The possible determinant values are -2, -1, 0, 1, and 2, with the number of determinants with value 2 equal to those with value -2, and the number with value 1 equal to those with value -1.

Answer: 338


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

Answer: (0, 1)


Question: Evaluate the integral \( \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} dx \, dy \, \exp(-5x^2+8xy-5y^2) \).

Answer: \dfrac{\pi}{3}


Question: Given the functional equation \( f(xy) = f(x) + f(y) + \frac{x+y-1}{xy} \) and the condition \( f'(1) = 2 \), find \( \lfloor f(e^{100}) \rfloor \).

Answer: 99


Question: Given a matrix with eigenvalues \(0, 0, 1, 2, 3\), determine the rank of the matrix.

Answer: 3


Question: Evaluate the limit: \[ \lim_{y\rightarrow \infty}y \ln\left(\frac{\sin(x+1/y)}{\sin x}\right), \quad x\in \left(0,\frac{\pi}{2}\right) \] without using L'Hospital's rule.

Answer: \cot x


Question: For an $n \times n$ matrix $A$ and $\lambda \in \mathbb{R}$, suppose $(A - \lambda I)^{k} v = 0$ for some non-zero $v \in \mathbb{R}^{n}$ where $k > 1$. Is $\lambda$ necessarily an eigenvalue of $A$? Provide a justification for your answer.

Answer: Yes


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

Answer: 0


Question: Find the value of $a$ such that the equation $(x^2 + x)^2 + a(x^2 + x) + 4 = 0$ has all four real roots, with two of them being equal.

Answer: -4


Question: Determine the condition on \(k\) for which the integral \( \int_{\mathbb{R}^n} \frac{dx}{(1+|x|^2)^{k/2}} \) converges, where \(n\) is a positive integer.

Answer: k > n


Question: Suppose \((S_t)_{t \geq 0}\) is a right continuous, uniformly integrable submartingale. Is the family \((S_T)_{T \equiv \text{stopping time}}\) uniformly integrable?

Answer: Yes


Question: Evaluate the integral \( \int\limits_{-0.5 \pi}^{0.5 \pi} \sin(x) F(\cos(x)) \, dx \), where \( F(x) = \int\limits_{0}^{x} e^{\sin(t)} \, dt \).

Answer: 0


Question: Find the smallest cardinality multiset \( x \) such that every element of the set \( z = \{ 1, 5, 6, 10, 11, 35, 36 \} \) can be expressed as a sum of elements from \( x \). For example, one possible multiset is \( x = \{1, 5, 5, 30\} \).

Answer: 4


Question: Suppose that $A \subseteq B$ and $A$ is isomorphic to $P$ with isomorphism $\phi$, while $B$ is isomorphic to $Q$ with isomorphism $\psi$. Does it follow that $P \subseteq Q$? Justify your answer.

Answer: No


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

Answer: 2m


Question: Given a deck of 52 cards, how many times must it be shuffled to make it "random"? Assume a riffle shuffle is used, where the deck is cut into two stacks and interleaved. Provide your answer as a whole number of shuffles.

Answer: 7


Question: Evaluate the limit: \[ \lim_{x\to 0}(\cos x +\sin x)^{1/\tan x} \]

Answer: e


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

Answer: 0


Question: Suppose you have \( n \) balls in a bag. Each time, you pick a random positive integer \( k \leq n \) and remove \( k \) balls from the bag. This process is repeated until the bag is empty. What is the expected number of times required to empty the bag?

Answer: H_n


Question: Determine the cardinality of the set \( \bigcup_n \mathcal{P} (B-A_n) \), where \( n \in \mathbb{N} - \{0,1\} \), \( A_n=[0,1/n] \), and \( B=\{1/k | k \in \mathbb{N} - \{0\}\} \).

Answer: \aleph_0


Question: Let $R$ be a ring with identity such that the category of left $R$-modules has a simple generator $T$. Is $T$ isomorphic to any simple left $R$-module $M$?

Answer: Yes


Question: Find the outer Lebesgue measure of the set \( \bigcup_{n \geq 1} [n-2^{-n}, n+2^{-n}] \).

Answer: 2


Question: Given the joint probability density function \( f_{1,2}(x_1, x_2) = \frac{1}{4}(x_1 - x_2)e^{-x_1} \) for \( 0 < x_1 < \infty \) and \( -x_1 < x_2 < x_1 \), compute the expected value \( E[X_2] \).

Answer: -1


Question: Find the multiplicative inverse of the polynomial $x^2 + 1$ in the finite field $GF(2^4)$, given the irreducible polynomial $m(x) = x^4 + x + 1$. Express your answer as a polynomial in $GF(2^4)$. \( \boxed{} \)

Answer: x^3 + x + 1


Question: Let $F$ be a symmetric cumulative distribution function around $\frac{1}{2}$ with support $[0,1]$, such that $F(x) = 1 - F(1-x)$ for all $x \in [0,1]$. Prove or disprove the following identity: \[ \int_0^1 \left(x - \frac{1}{2}\right) F(x) (1 - F(x)) \, dx = 0. \]

Answer: 0


Question: Evaluate the limit \( \lim_{x \rightarrow +\infty} \sum_{n=0}^{\infty} \frac{(-x)^n}{1+n!} \).

Answer: 0


Question: Given a probability space $(\Omega,\mathcal{F},P)$ and a Young function $\varphi:[0,\infty)\rightarrow[0,\infty)$ with its conjugate Young function $\psi$, consider a sequence $(x_n)\subset L^\varphi$ of non-negative and disjoint random variables such that $\Vert x_n\Vert_\varphi=1$ for all $n\in\mathbb{N}$. Can a norm-bounded sequence $(y_n)\subset L^\psi$ of non-negative random variables be found such that $\mathbb{E}[x_n y_n]=1$ and $\text{supp}(y_n)\subset\text{supp}(x_n)$ for all $n\in\mathbb{N}$?

Answer: Yes


Question: Let \( C \) be a standard normal random variable, and let \( G \) be another random variable defined as \( G = \Phi(C) \), where \( \Phi \) is the cumulative distribution function (CDF) of the standard normal distribution. Determine the probability density function (PDF) of the random variable \( G \).

Answer: 1


Question: Evaluate the infinite series \( \sum_{k=0}^\infty \frac{k(vt)^k e^{-vt}}{k!} \) and verify that the result is equal to \( vt \), where \( v \) is a constant.

Answer: vt


Question: Let $H_1$, $H_2$, and $H_3$ be normal subgroups of a group $G$. Define the subgroups $H^1 = \langle H_2 \cup H_3 \rangle$, $H^2 = \langle H_1 \cup H_3 \rangle$, and $H^3 = \langle H_1 \cup H_2 \rangle$. Suppose the homomorphism $p: G \rightarrow G/H^1 \times G/H^2 \times G/H^3$ given by $p(g) = (gH^1, gH^2, gH^3)$ is an isomorphism. Does this imply that $G$ is the internal direct product of $H_1$, $H_2$, and $H_3$?

Answer: Yes


Question: Find the minimum value of the expression \( \sqrt{\frac{1}{3}x^4+1}+\sqrt{\frac{1}{3}y^4+1}+\sqrt{\frac{1}{3}z^4+1} \) given that \( x, y, z \) are positive real numbers such that \( x + y + z = xyz \).

Answer: 6


Question:
Given the moment generating function (MGF) of a discrete random variable $X$:
\[ M_X(t) = \left(\frac{1}{2}e^{2t} + \frac{1}{2}e^{4t}\right)^7 \]
find the probability $p_X(15)$. Assume $X$ is the sum of 7 independent identically distributed random variables, each taking values 2 or 4 with equal probability.

Answer: 0


Question: Given the function defined implicitly by the equation \(3\sin(x^2yz) - 3x + 2y^2 + e^{yz} = 0\) near the point \((1,1,0)\), find the maximum value of the directional derivative of \(z\) at the point \((1,1)\).

Answer: \dfrac{5}{4}


Question: Compute the limit: \[ \lim_{n \rightarrow \infty}\sqrt{n}(A_{n+1} − A_n) \] where \( A_n = \frac{1}{n}(a_1 + a_2 + \cdots + a_n) \) and \( \{a_n\} \) is a sequence of real numbers satisfying \( |a_n| \leq 1 \) for all \( n \geq 1 \).

Answer: 0


Question: Find the limit of the sequence \( \{a_{n}\} \) defined by the following recurrence relations: \( a_{1}=0, a_{2}=\frac{1}{2}, a_{n+1}=1+a_{n}+a^{3}_{n-1} \) for \( n > 1 \).

Answer: \infty


Question: A grasshopper starts on island $A$ and hops to one of the two other islands, $B$ or $C$, every minute. In how many different ways can the grasshopper end up on island $C$ after seven minutes, given that it cannot hop to the same island consecutively?

Answer: 43


Question: Simplify the expression \( \sum_{k=1}^n (n-k)2^{k-1} \) using combinatorial methods.

Answer: 2^n - n - 1


Question:
Let \(\alpha:(-1,1)\to O(n,\mathbb{R})\) be a smooth map such that \(\alpha(0)=I\), the identity matrix. Determine which of the following statements is true about \(\alpha'(0)\):

(a) \(\alpha'(0)\) is symmetric.
(b) \(\alpha'(0)\) is skew-symmetric.
(c) \(\alpha'(0)\) is singular.
(d) \(\alpha'(0)\) is non-singular.

Here, \(O(n,\mathbb{R})\) denotes the set of all \(n\times n\) real orthogonal matrices.

Answer: B


Question:
Determine whether the following claim is true:

Let $A$ be a Noetherian domain, $\mathfrak{m}$ a maximal ideal, $s$ a non-zero element of $\mathfrak{m}$, and $d = \dim A_\mathfrak{m}$. For any $\epsilon > 0$, there exists a positive integer $n$ such that for any ideal $I$ satisfying:

1) $I \subset \mathfrak{m}^n$
2) $\sqrt{I} = \mathfrak{m}$
3) $I$ can be generated by $d$ elements,

the following inequality holds:

$$ \frac{\text{length}(A/(I+As))}{\text{length}(A/I)} < \epsilon $$

Note: The claim may be false if the requirement that the number of generators of $I$ be bounded is dropped. For example, consider $A = k[x,s]$ with $\mathfrak{m} = (x,s)$ and the ideal $I_{n,m} = s\mathfrak{m}^{n-1} + \mathfrak{m}^m$. It can be shown that for any $n$, $$\lim_{m \to \infty} \frac{\text{length}(A/(I_{n,m}+As))}{\text{length}(A/I_{n,m})} = 1.$$

Answer: True


Question:
The persons $A$, $B$, and $C$ are playing a game where the person with the highest points wins. The points are distributed as follows:

- $A$'s points are a Poisson distributed random variable with parameter 2.
- $B$'s points are a Poisson distributed random variable with parameter 3.
- $C$'s points are a Poisson distributed random variable with parameter 4.

The points of each player are independent of each other. The outcome of the points was 1, 2, and 3, but it is unknown who received which score. What is the conditional probability that person $C$ won the game?

Answer: \dfrac{40}{81}


Question: Determine the number of points of discontinuity for the function \( f(\theta) = \int_{-1}^{1} \frac{\sin\theta \, dx}{1-2x\cos\theta +x^2} \) where \( \theta \in [0,2\pi] \).

Answer: 3


Question: Consider two sets $A$ and $B$ defined on bounded intervals. If $A$ and $B$ have the same cardinality, $A \cap B$ is non-empty, and the Lebesgue outer measure of $A$ is greater than zero, is it true that the Lebesgue outer measure of $B$ is also greater than zero?

Answer: No


Question: Determine the number of integer solutions to the equation \(x^2 + 4xy = 10^{100}\).

Answer: 19594


Question: Let $f$ be an entire function that takes every value no more than three times. Determine the possible form of $f$.

Answer: f(z) = az^3 + bz^2 + cz + d


Question: Calculate the conditional expectation $E[X \mid Y]$ where $Y = |X|$ and $X \sim N(0,1)$. Provide your answer as a function of $Y$.

Answer: 0


Question: Calculate the surface integral \( \int_S \mathbf{u} \cdot \mathbf{dS} \) to determine how many cubic meters of fluid cross the upper hemisphere \( x^2+y^2+z^2=1 \), \( z\ge0 \) per second, given that the velocity of the flow is \( \mathbf{u}=\mathbf{i}+x\mathbf{j}+z\mathbf{k} \) meters per second.

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


Question: Let $\{v_1, \ldots, v_n\}$ be an orthonormal basis of an inner product space $V$, and let $F:V\to V$ be a linear operator such that for all $j$, \(\langle Fv_j, Fv_j \rangle =  \langle F^*v_j, F^*v_j \rangle.\) Does this imply that $F$ is a normal operator?

Answer: No


Question: Evaluate the limit: $$ \lim_{x \to \infty} \frac{(2x)!}{(x! \cdot 2^x)^2} $$

Answer: 0


Question: Determine the number of distinct (up to isomorphism) groups of order 21.

Answer: 2


Question: Evaluate the integral of the conjugate of a complex number $z$, denoted by $\overline{z}$, around the closed contour consisting of the portion of the parabola $y = x^2$ from $(0,0)$ to $(1,1)$ followed by the line segment from $z = 1 + i$ to $z = 0$. Use the definition of the complex integral to find the result.

Answer: \dfrac{i}{3}


Question: Find the intersection of the sets $A = \{z : \arg\left(\frac{z-1}{z}\right) = \frac{\pi}{2}\}$ and $B = \{z : \arg(z-1) = \pi\}$, where $A$ and $B$ are sets of complex numbers. Determine if $A \cap B = \varnothing$.

Answer: A \cap B = \varnothing


Question:
Determine whether the function \( f: \mathbb{R}_l \to \mathbb{R} \) defined by
\[
f(x) =
\begin{cases}
1,  & \text{if } x \geq 0 \\
-1, & \text{if } x < 0  \\
\end{cases}
\]
is continuous, where \( \mathbb{R}_l \) is the set of real numbers with the lower limit topology and \( \mathbb{R} \) is the set of real numbers with the standard topology.

Answer: Yes


Question:
Determine the limit of the sequence given by:
\[ \lim_{n \to \infty} \left( \prod_{i=1}^n X_i \right)^{1/n} \]
where \( X_n \) are independent random variables uniformly distributed over the interval \([0,1]\).

Answer: \dfrac{1}{e}


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

Answer: (-1, 1]


Question: Given that the columns of matrix $U$ are orthonormal, determine which of the following representations is the correct projection matrix onto the column space of $U$: $UU^T$ or $U(U^TU)^{-1}U^T$. Justify your answer.

Answer: UU^T


Question: Let $F$ be an infinite field such that $F^*$, the multiplicative group of $F$, is a torsion group. Determine whether $F^*$ has a proper subgroup with finite index.

Answer: No


Question: Evaluate the definite integral: \[ \int_0^{\pi/2} (\sin x)^{\cos x} \left( \cos x \cot x - \log (\sin x)^{\sin x} \right)\, dx \]

Answer: 1


Question:
Find the maximum integer \( n \) such that the Taylor series expansion of \( \sinh(x^4) \) is given by:
\[ \sinh(x^4) = x^4 + \frac{1}{6}x^{12} + \mathcal{o}(x^n) \quad \text{as} \ x \to 0. \]

Answer: 19


Question: Calculate the integral \( \int_{0}^{\pi} e^{a\cos(\theta)} \cos(a\sin(\theta)) \, d\theta \) using complex analysis.

Answer: \pi


Question: Is there an absolute constant $C_0$ such that the norm $C_{p,n}$ of the Hardy-Littlewood maximal operator $\mathcal{M}$ in $L^p(\mathbb{R}^n)$ satisfies $C_{p,n} < C_0$ for all $p \geq 2$ and all $n$? Consider $C_0 = 10$ as a potential candidate.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to \infty} \left(\frac{x+\sqrt[3]{x^4+1}}{x+\sqrt[6]{9x^8+3}}\right)^{5x+1} \]

Answer: 0


Question:
Determine the smallest integer \( N \) such that there exist two \( N \times N \) matrices \( X \) and \( Y \) satisfying the following conditions:

1. \( \diag(X) = \diag(XYY) = \diag(YXY) = \diag(YYX) \neq \mathbf{0} \)
2. \( \diag(Y) = \diag(XXY) = \diag(XYX) = \diag(YXX) \neq \mathbf{0} \)
3. \( \diag(XY) = \diag(YX) \neq \mathbf{0} \)
4. \( \diag(XXYY) = \diag(XYYX) = \diag(YYXX) = \diag(XYXY) = \diag(YXYX) \neq \mathbf{0} \)
5. The values of \( \diag(X) \), \( \diag(Y) \), \( \diag(XY) \), and \( \diag(XXYY) \) are all different.

In particular, determine if there exists a solution for \( N \leq 3 \). Here, \( \diag(X) \) denotes the diagonal of matrix \( X \), and \( \mathbf{0} \) is the \( N \times 1 \) zero vector.

Answer: 2


Question: Determine the values of \(p\) and \(q\), both greater than 0, for which the series \(\sum_{n=1}^{\infty} \frac{(p+1)(p+2)\cdots(p+n)}{(q+1)(q+2)\cdots(q+n)}\) converges.

Answer: q > p + 1


Question: Representatives from three research institutes are to form a commission of 9 researchers. Determine the number of ways to form this commission such that no institute has an absolute majority.

Answer: 11130


Question:
Determine the angle \( \theta \) that eliminates the mixed term in the quadratic equation \( 3x^2 - 2xy - 10x + 3y^2 - 2y + 8 = 0 \) by using the coordinate transformation:
\[ x = \cos \theta X - \sin \theta Y \]
\[ y = \cos \theta Y + \sin \theta X \]

Answer: \dfrac{\pi}{4}


Question: Determine whether the double integral \( \iint_Q f(x,y) \, dx \, dy \) exists, where \( f(x,y) = \begin{cases} 1, & x = y \\ 0, & x \neq y \end{cases} \) and \( Q = [0,1] \times [0,1] \). If it exists, find its value.

Answer: 0


Question: Consider the curve of intersection of the cylinder \(\frac{x^2}{25} + \frac{y^2}{9} = 1\) with the plane \(3z = 4y\). Let \(L\) be the line tangent to this curve at the point \((0,-3,-4)\). Determine the x-coordinate of the point where \(L\) intersects the plane \(2x - 3y - 4z = 27\).

Answer: 1


Question: Let $p$ be a positive integer. For each positive integer $N$, define the function $F: \{1, \ldots , pN\} \rightarrow \{1, \ldots, N\}$ by $F(n) = \lceil n/p \rceil$. Let $r \in [0,1]$. Determine the limit of the expected value $\mathbb{E}(\#F(A)/N)$, where $A \subseteq \{1, \ldots, pN\}$ is a subset of cardinality $\lfloor rpN \rfloor$ selected randomly.

Answer: 1 - (1 - r)^p


Question:
Given the functions:
\[ \begin{cases} f(x) = g(x+1) \\ f(y) = 2018y + 2016y + \cdots + 2y \\ g(x) = h(2x) - x \end{cases} \]
If \( h(2018) = a^3 \) and \( a \in \mathbb{Z} \), find the value of \( a \).

Answer: 1009


Question: Let \( c:[0,2] \to \mathbb{R}^3 \) be the curve given by \( c(t) = \left( \frac{t^3}{3}, t^2, 2t \right) \). Find the value of \( m > 0 \) such that there exists a \( C^{\infty} \) bijection \( f:[0,m] \to [0,2] \) with \( f'(s) > 0 \) for every \( s \in [0,m] \), and the curve \( b:[0,m] \to \mathbb{R}^3 \) defined by \( b(s) = c(f(s)) \) satisfies \( |b'(s)| = 1 \) for every \( s \in [0,m] \).

Answer: \dfrac{20}{3}


Question: Suppose $X$ and $Y$ are independent random variables, each following a standard normal distribution $N(0,1)$ with the probability density function given by $$f(x)=\frac{1}{\sqrt{2\pi}}\exp{\left(\frac{-x^2}{2}\right)}.$$ Compute the cumulative distribution function of the variable $$R=\sqrt{X^2+Y^2}.$$

Answer: 1 - e^{-r^2/2}


Question: Determine all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the inequality \( f(c)-b^2 \leq f(a+b)-f(a)+f(c) \leq b^2+f(c) \) for all real numbers \( a, b, \) and \( c. \)

Answer: f(x) = k


Question:
Given a function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ that is continuous and has a limit at $p=\infty$, such that $\lim_{x\rightarrow p}f(x,y)=b(y)$, and a function $g:\mathbb{R} \rightarrow \mathbb{R}$ that is continuous and has a limit at $p$, $\lim_{x\rightarrow p}g(x)=c$, determine if the following statement is true:
$$\lim_{x\rightarrow p}f(x,g(x))=\lim_{x\rightarrow p}f(x,c)=b(c)$$

Answer: False


Question:
Prove that for an odd prime number \( p \) and an integer \( k \) such that \( p-1 \nmid k \), the following congruence holds:
\[ 1^k + 2^k + \cdots + (p-1)^k \equiv 0 \pmod{p}. \]

Answer: 0


Question: Find the maximum value of \(|z^2 - 2iz + 1|\) given that \(|z| = 3\). Use the triangle inequality to determine this maximum value.

Answer: 14


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

Answer: 2\pi i (1 - e)


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

Answer: 1


Question: Determine whether the following statement is true: For a Hankel matrix \([I_{i+j}]_{i,j=0}^n\), where \(I_k\) denotes the enumeration of involutions among permutations in \(\mathfrak{S}_k\), is it true that \(\det\left[I_{i+j}\right]_{i,j=0}^n = n!!\), where \(n!! = 1!2!\cdots n!\)?

Answer: True


Question: Use Green's Theorem to find the counter-clockwise circulation for the vector field \( \mathbf{F} = (x + 3y)\mathbf{i} + (2x - y)\mathbf{j} \) around the curve parameterized by \( x = \cos(t) \), \( y = \sin(t) \) for \( 0 \le t \le 2\pi \).

Answer: -\pi


Question: Find the maximum value of the infinite power tower \(x^{x^{x^{\cdots}}}\).

Answer: e


Question: An urn contains \( r > 0 \) red balls and \( b > 0 \) black balls. A ball is drawn at random from the urn, its color noted, and returned to the urn. Additionally, \( d > 0 \) balls of the same color are added to the urn. This process of drawing a ball and adding \( d \) balls of the same color is continued indefinitely. Define \( X_i = 1 \) if the \( i \)-th draw results in a red ball, and \( 0 \) otherwise. Compute the expected value \( E\left(\sum_{i=1}^{\infty} X_i\right) \).

Answer: \infty


Question: Calculate the area of the part of the surface \( z = x^2 + y^2 + 2 \) that lies above the disc \( x^2 + y^2 \le 1 \), correct to four decimal places, by expressing the area in terms of a single integral.

Answer: 5.3304


Question: Suppose that $x_1, x_2, \ldots, x_n$ (where $n > 2$) are real numbers such that $x_i = -x_{n-i+1}$ for $1 \leq i \leq n$. Consider the sum \( S = \sum_{1 \leq i < j < k \leq n} x_i x_j x_k \), where the summation is taken over all distinct indices $i, j, k$. What is the value of $S$?

Answer: 0


Question: Find the convolution of the functions \( f(t) = t \) and \( g(t) = e^{t} \). Express your answer as \((f*g)(t) = ?\).

Answer: e^t - t - 1


Question: Evaluate the limit: \[ \lim_{x \to 0}\left(\frac{\tan\left(\pi\cos^2x\right)}{x^2}\right) \] without using L'Hospital's rule.

Answer: -\pi


Question: Determine the dimension of the vector space \( \mathcal{L}_F(K) \) over \( K \), where \( K/F \) is a finite-dimensional field extension with \([K:F]=n\).

Answer: n


Question: In the polynomial \((x-1)(x^2-2)(x^3-3) \ldots (x^{11}-11)\), find the coefficient of \(x^{60}\).

Answer: 1


Question: Evaluate the integral: \[ \int_{0}^{\frac{\pi}{4}} (\tan^{9}x + 5\tan^{7}x + 5\tan^{5}x + \tan^{3}x) \, dx \]

Answer: \dfrac{25}{24}


Question: Given vectors $v_1, \ldots, v_k \in V$ and linear functionals $\phi_1, \ldots, \phi_k \in V^*$, where $k$ is the dimension of $V$, prove that if $v_1, \ldots, v_k$ are linearly dependent, then $\det[\phi_i(v_j)] = 0$. Assume $v_r$ is a linear combination of the others and consider the matrix $[\phi_i(v_j)]$ whose $i$th row and $j$th column is $\phi_i(v_j)$. Show that the determinant of this matrix is zero.

Answer: 0


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

Answer: \dfrac{4}{e}


Question: Consider the set $I = \bigcup_{n=1}^{\infty} R_n$, where each $R_n$ is a closed set. Is it possible for $I$ to contain a non-empty open interval?

Answer: Yes


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

Answer: 0


Question: A circle is drawn with its center at (8,0) and radius \( r \) such that the circle intersects the ellipse \( x^2 + 4y^2 = 16 \) at right angles. Find the radius \( r \) of the circle.

Answer: \sqrt{39}


Question: Evaluate the limit: \[ \lim_{x \rightarrow 0^+} \frac{\int_{0}^{x^2} (1 + \sin(t))^{1/t} dt}{x \sin(x)} \]

Answer: e


Question: Evaluate the line integral \( \oint_C -x^2 y \, dx + xy^2 \, dy \) using Green's Theorem, where \( C \) is the circle with radius 2 centered at the origin.

Answer: 8\pi


Question: Evaluate the limit: \[ \lim _{x\to 0 }\frac{\sin^2x - x^2 \cos^2x}{x^2 \sin^2x} \] without using Taylor expansion.

Answer: \dfrac{2}{3}


Question: Consider a bounded linear operator $A$ from a subspace $X_1$ of a Banach space to a subspace $Y_1$ of another Banach space, where $\mathrm{codim}X_1+\mathrm{codim}Y_1<\infty$. If $A$ maps one open subset $X$ of $X_1$ onto one open subset $Y$ of $Y_1$, is it true that $A$ maps $X_1$ onto $Y_1$, i.e., is $A$ an open map?

Answer: Yes


Question:
Consider the polynomial equation \(x^6+2x^5+4x^4+8x^3+16x^2+32x+64=0\). Determine the number of roots \(x_i\) for which \(|x_i|=2\). Choose the correct option:

(A) \(|x_i|=2\) for exactly one value of \(i\)
(B) \(|x_i|=2\) for exactly two values of \(i\)
(C) \(|x_i|=2\) for all values of \(i\)
(D) \(|x_i|=2\) for no value of \(i\)

Answer: C


Question: Let \(a, b, c \in \mathbb{Z}\) with \(1 < a < 10\), and \(c\) is a prime number. Define the function \(f(x) = ax^2 + bx + c\). Given that \(f(f(1)) = f(f(2)) = f(f(3))\), find the value of \(f'(f(1)) + f(f'(2)) + f'(f(3))\).

Answer: 95


Question: Determine the Galois group of the polynomial \( f(x) = x^5 - 5x^4 + 10x^3 - 10x^2 - 135x + 131 \) over \( \mathbb{Q} \).

Answer: S_5


Question: Is it true that an endomorphism whose minimal polynomial has the same degree as the dimension of the vector space is similar to a companion matrix?

Answer: True


Question:
Evaluate the contour integral \( \oint (z^* - 3) \, dz \) over the following path in the complex plane, traversed anticlockwise:

1. From \((0,0)\) to \((2,0)\) along the line \(y = 0\).
2. From \((2,0)\) to \((0,2i)\) along the arc of the circle \(|z| = 2\) in the first quadrant.
3. From \((0,2i)\) to \((0,0)\) along the line \(x = 0\).

Provide the value of the integral.

Answer: 2\pi i


Question: Evaluate the limit: $$\lim \limits_{N\to \infty}\frac{1}{N}\left(\sum_{m=1}^\infty\frac{1}{m^2}\left|\frac{\sin(mN)}{\sin(m)}\right|\right)$$

Answer: 0


Question: Find \( \lim_{n \to \infty} \sqrt[n]{n!} \).

Answer: \infty


Question: Suppose $f$ and $g$ are polynomials in $n$ variables, where $n \ge 2$, over a field $E$. If $f$ and $g$ are relatively prime over $E$, are they still relatively prime over a field extension $F$ of $E$? Provide a justification for your answer.

Answer: Yes


Question: Determine the largest constant \(c > 1\) such that every triangle-free graph with chromatic number \(n\) has at least \(\Theta(c^n)\) vertices.

Answer: 2


Question: Given two fuzzy topological spaces $(X, \tau_1)$ and $(Y, \tau_2)$, where $X=\{a\}$, $Y=\{x, y\}$, $\tau_1 =\{0_X, 1_X, \{(a, 0.3)\}\}$, $\tau_2 =\{0_Y, 1_Y, \{(x, 0.2), (y, 0.2)\}\}$, and a fuzzy function $f:X \rightarrow Y$ such that $f(a)=x$, find the inverse of the fuzzy set $\{(x, 0.5), (y, 0.3)\}$ in $X$. What is the inverse fuzzy set in $X$?

Answer: \{(a, 0.5)\}


Question: Simplify the expression \( \epsilon_{ij \ell} \, \epsilon_{km \ell} \, \epsilon_{ijm} \, a_k \) using the properties of the Levi-Civita symbol and the Kronecker delta.

Answer: 0


Question: Suppose $g(x)$ is a differentiable, real-valued, periodic function with period $a$ such that for all $u$, $\int_u^{u+a} g(x)\,dx=0$. Determine whether it is true that $\int_u^{u+a}g(x+g(x))\,dx=0$ for all $u$.

Answer: 0


Question: Evaluate the limit \( \lim_{x \to 1^-} \prod_{n=0}^{\infty} \left[\frac{1+x^{n+1}}{1+x^n}\right]^{x^n} = L. \) Find \( \left\lceil \frac{1}{L} \right\rceil. \)

Answer: 2


Question: Find the equation of a plane that is perpendicular to the $xy$-plane and intersects the paraboloid $z = x^2 + 4y^2$ at the point $(2, 1, 8)$. The intersection between the plane and the paraboloid is a parabola with a slope of 0 at the given point.

Answer: x + 2y = 4


Question:
Combine the following two integrals into a single integral and choose the correct option:

\[ \int\frac{\csc^2x}{\sqrt{\cot^2x-1}}\,dx - 2\int\frac{\sin x}{\sqrt{1-2\sin^2x}}\,dx \]

Options:

A. \( \int\sqrt{\csc^2x-2}\,dx \)

B. \( \int\frac{\sqrt{\cos 2x}}{2\sin x}\,dx \)

C. \( \int\frac{\sqrt{\cos 2x}}{\sin x}\,dx \)

D. \( \int\sqrt{\csc^2x-1}\,dx \)

Answer: C


Question:
Determine the domain of the function:
\[ f(x) = \frac{ \ln(1-\sqrt{x^2+1}) }{\sqrt{(\arctan(x)-\frac{ \pi }{ 4 })(\arctan(x)-\frac{ \pi }{ 3 })} } \]

Answer: \emptyset


Question: Let $R$ be a ring, $A$ be an $R$-module, and $B, C$ be submodules such that $A = B \oplus C$. We know that $\frac{A}{B} \cong C$ using the second isomorphism theorem. Is the converse true? If $\frac{A}{B} \cong C$, can we deduce that $A \cong B \oplus C$?

Answer: No


Question: Evaluate the integral \( \int_{|z|=1} \frac{\sin(z)}{z^3} \, dz \) over the unit circle \(|z|=1\) in the complex plane without using the residue theorem.

Answer: 0


Question: Let \( f(x+y) = f(x)f(y) \) for all \( x, y \in \mathbb{R} \), with \( f(5) = 2 \) and \( f'(0) = 3 \). Find \( f'(5) \).

Answer: 6


Question: Given the function \(f(x,y) = (x + \sin y)(\cos(\pi x^2 + y))\), consider the change of variables \(u = f(x, y)\) and \(v = xy\). What is the Jacobian determinant of the transformation from \((x, y)\) to \((u, v)\) at the point \((x, y) = (2, 0)\)?

Answer: 2


Question: Given that $f(x,y)$ is a homogeneous function of order 3, with $f_{x}(2,1)=7$ and $f_{y}(8,4)=5$, find the value of $f(12,6)$. The expected answer is $1030.5$. Use the properties of homogeneous functions and their derivatives to solve the problem.

Answer: 1030.5


Question: Given that \( \sum_{k=0}^{2n}(-1)^k\cdot \binom{2n}{k}^2 = A \), find the value of \( \sum_{k=0}^{2n}(-1)^k\cdot (k-2n)\cdot \binom{2n}{k}^2 \) in terms of \( n \) and \( A \).

Answer: -nA


Question: Let $u_1=(1,1,1)$, $u_2=(2,0,0)$, and $u_3=(0,3,0)$ be a basis of $\mathbb{R}^3$. Calculate the coordinates of $w=2e^{*2}+3e^{*3}$ in the dual basis $\{u^{*1}, u^{*2}, u^{*3}\}$.

Answer: (5, 0, 6)


Question: Calculate the integral \( \int_{-1}^1 |z| \, dz \) where the integration contour is the upper semi-circle with unit radius.

Answer: 2


Question:
Determine whether the following statement is true for representations \(\pi\) and \(\mu\) of a subgroup \(H \leq G\), where \(G\) is a finite group:
\[
Ind_H^G(\pi \oplus \mu) \simeq Ind_H^G(\pi) \oplus Ind_H^G(\mu)
\]
Is the induction of the direct sum of two representations equal to the direct sum of the induced representations?

Answer: True


Question: Determine whether the following statement is true: $E_{X|Y}[X|Y]=E_{Z|Y}[E_{X|Y,Z}[X|Y,Z]]$. Justify your answer.

Answer: True


Question: Compute the limit \( \lim_{n\to\infty } n\left [\int_0^{\frac {\pi}{4}}\tan^n \left ( \frac{x}{n} \right )\mathrm {d}x\right]^{\frac{1}{n}} \).

Answer: \dfrac{\pi}{4}


Question: Determine the number of positive roots of the equation \(x^{x+1} = (x+1)^x\).

Answer: 1


Question: Evaluate the triple integral of \( f(x,y,z) = (x^8 + y^6 + z^4)\sin(z^3) + 2y - 3 \) over the unit ball.

Answer: -4\pi


Question: Given that $f(x+y) = f(x)f(y)$ for all $x$ and $y$, $f(5) = -2$, and $f'(0) = 3$, find the value of $f'(5)$.

Answer: -6


Question: Is the identity \((D(n+2)-(D(n+3)+1)) (D(n+1)-(D(n+2)+1))(a_1 D(n+1) + a_2 D(n+2)+a_3 D(n+3)+a_4) \equiv 0 \pmod{p}\) true for constants \(a_i\) that depend only on the prime \(p\)?

Answer: Yes


Question: Determine whether the matrix \( A = \begin{pmatrix} 3 & 2 & 0 \\ -4 & -3 & 0 \\ 0 & 0 & 8 \end{pmatrix} \) has a cube root. Justify your answer based on the matrix's diagonalizability and eigenvectors.

Answer: Yes


Question: Calculate \( \lfloor 1000S \rfloor \) where \( S = \sum_{n=1}^{\infty} \frac{1}{2^{n^2}} = \frac{1}{2^1}+\frac{1}{2^4}+\frac{1}{2^9}+\cdots. \)

Answer: 564


Question: Let $G$ be a non-abelian group and let $a, b \in G$ such that $\operatorname{ord}(a) = 4$ and $ba = ab^3$. Given that $\operatorname{ord}(b)$ is an odd prime number, determine the order of $b$.

Answer: 5


Question: Find the angle between the polynomials $2x^3 - 3x^2$ and $1$ in the space of polynomials of degree less than or equal to three, using the inner product defined by \( \langle p,q\rangle = p(0)q(0) + p'(0)q'(0) + p(1)q(1) + p'(1)q'(1) \).

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


Question:
Solve the partial differential equation using the method of characteristics:
\[ xu_x + yu_y + zu_z = 1 \]
with the initial condition:
\[ u = 0 \text{ on } x + y + z = 1. \]

Answer: \ln(x + y + z)


Question: If the quadratic equations \(3x^2+ax+1=0\) and \(2x^2+bx+1=0\) have a common root, find the value of \(5ab-2a^2-3b^2\).

Answer: 1


Question: Construct a polynomial of degree $m$ over a field $K$ such that its Galois group is the full symmetric group $S_m$. Provide a method or example of such a polynomial.

Answer: x^m - x - 1


Question:
Find the number of integral solutions \((x,y)\) to the equation:
\[x^3+3x^2y+3xy^2+2y^3=50653\]

Answer: 2


Question: Given the equation $x^y \cdot y^x = 16$, find the value of $\frac{dy}{dx}$ at the point $(2, 2)$. What is the derivative at this point?

Answer: -1


Question: Solve the differential equation \( \frac{dy}{dx}=\frac{x^2+2xy-y^2}{x^2-2xy-y^2} \) with the initial condition \( y(1)=-1 \).

Answer: y = -x


Question: Find the shortest distance between the surface defined by \( H = \{(x,y,z) \in \mathbb{R}^{3}: x^2+y^2 - z^2 + 4 = 0 \} \) and the point \( p = (2,4,0) \).

Answer: \sqrt{14}


Question: Consider a $5 \times 5$ invertible matrix with entries from \(0,1\). What is the minimum difference between the number of \(1\)'s and \(0\)'s?

Answer: 1


Question: Evaluate the integral \( \int_C \frac{dz}{z-3i} \) where \( C \) is the circle \( |z| = \pi \) oriented counterclockwise. Determine if the Cauchy Integral Theorem applies.

Answer: 2\pi i


Question: Evaluate the limit: \[ \lim_{n\to\infty}\int_{0}^{\infty}e^{-x}\sin\left(\frac{n}{x}\right)~\text{d}x = 0. \]

Answer: 0


Question: Let $f: [0,1] \rightarrow [0,1]$ be a continuous function. Choose any point $x_0 \in [0,1]$ and define a sequence recursively by $x_{n+1} = f(x_n)$. Suppose $\lim_{n \rightarrow \infty} (x_{n+1} - x_n) = 0$. Does the sequence $\{x_n\}$ converge?

Answer: Yes


Question: Is there an infinite sequence of natural numbers such that any finite partial sum of this sequence is not a perfect power?

Answer: Yes


Question: Calculate the constant radius of curvature of the curve \( r(u) = (\cos(u), \sin(u), u) \). Given that the velocity vector has length \( \sqrt{2} \), determine the radius of curvature.

Answer: 2


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

Answer: 1


Question: Find the equation of the tangent line to the parametric curve given by \( x(t) = 2\cos(t)\cos(2t) \) and \( y(t) = 2\sin(t) - \sin(2t) \) at \( t = 0 \).

Answer: y = 0


Question: Suppose $a$ and $b$ are positive integers such that $\gcd(a,b)$ is divisible by exactly 7 distinct primes and $\mathop{\text{lcm}}[a,b]$ is divisible by exactly 28 distinct primes. If $a$ has fewer distinct prime factors than $b$, what is the maximum number of distinct prime factors that $a$ can have?

Answer: 17


Question: Let $X$ be a smooth projective toric variety over $\mathbb{C}$, acted upon by the compact torus $T=(S^1)^n$. The $T$-equivariant cohomology $H^*_T(X)$ is an algebra over the ring $H^*_T(pt)$. Consider the ideal $H^{>0}_T(pt)\cdot H^*_T(X)$. Is the quotient algebra $H^*_T(X)/H^{>0}_T(pt)\cdot H^*_T(X)$ isomorphic to $H^*(X)$ as a graded algebra?

Answer: Yes


Question: Use Green's Theorem to evaluate the line integral \( \int_C y^4\, dx + 2xy^3\, dy \), where \( C \) is the ellipse \( x^2 + 2y^2 = 2 \).

Answer: 0


Question: Determine whether the measure of the set \( A = \{x \in \mathbb{R} \mid \exists\,\text{infinitely many pairs of integers } p,q \text{ such that } |x-p/q| \leq 1/q^3\} \) is equal to 0.

Answer: 0


Question: Determine the value of \( a \) such that the curves \( ax^2+4xy+2y^2+x+y+5=0 \) and \( ax^2+6xy+5y^2+2x+3y+8=0 \) intersect at four concyclic points. Choose the correct option: \( a \) = 4, \( a \) = -4, \( a \) = 6, \( a \) = -6.

Answer: -4