Question: Let \( V \) be a vector space and \( \phi \in \text{End}_\mathbb{C}(V) \) a linear map with the characteristic polynomial \( P_\phi(X) = (x-2)^3(x-5)^2 \). How many possible Jordan normal forms are there for \( \phi \)?

Answer: 6


Question:
Consider the sequence \(x_n = \sqrt[n]{2^{n\sin 1} + 2^{n\sin 2} + \cdots + 2^{n\sin n}}\) for \(n \ge 2\). Determine the behavior of the sequence \((x_n)_{n \ge 2}\) by choosing one of the following options:

A. convergent
B. bounded and divergent
C. unbounded and divergent
D. has negative terms
E. has infinite limit

Answer: A


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

Answer: 1


Question: Evaluate the limit: $$\lim_{x\to0^+}\frac{(-3\sqrt{x}+x^3+\sin(x^6))(\text{e}^{4\sqrt{x}}-1)}{4\ln (1+x)+x^4-x^6}$$

Answer: -3


Question:
Determine the radius of convergence of the power series \( \sum_{n=0}^{\infty} a_n x^n \), where \( a_n = \frac{\sin(n!)}{n!} \) and \( a_0 = 0 \). Choose the correct inequality for the radius of convergence \( R \):

- \( R \geq 1 \)
- \( R \geq 2\pi \)
- \( R \leq 4\pi \)
- \( R \leq \pi \)

Answer: R \geq 1


Question: Is the center of the Heisenberg group also its commutator subgroup?

Answer: Yes


Question: Find the volume of the solid obtained by rotating the curve \( y = \cosh x \) around the \( y \)-axis from \( y = 0 \) to \( y = 1 \).

Answer: 2\pi\left(1 - \frac{1}{e}\right)


Question: Find the total mass of a solid located between the surface \(z=4-x^2-y^2\) and the cone \(z=3\sqrt{x^2+y^2}\) in the first octant, given that its density is \(\rho(x, y, z)=2\) at all points.

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


Question: Given a gamma distribution with shape \(\alpha=2\) and rate \(\lambda=10\), find an expression for \(\mathbb{E}[X^{-2}]\).

Answer: \infty


Question: Evaluate the principal value of the integral: \[ \int_0^{\infty} \frac{\cos x - 1}{x^2} \, dx \]

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


Question: Evaluate the improper integral: \[ \int_{0}^{1} \frac{\log x}{x^2} \, dx \]

Answer: -\infty


Question: Determine whether there exists a function \( f: \Bbb R \to \Bbb R \) that is discontinuous at some point and satisfies the functional equation \( f(x+y) = f(x) + f(y) \) for all \( x, y \in \Bbb R \).

Answer: Yes


Question: Determine the infimum of the set \( A = \{ |\sin(n)| : n \in \mathbb{N} \} \).

Answer: 0


Question: A die is rolled repeatedly. Let $X$ be the random variable representing the number of rolls needed to get a 4, and $Y$ be the random variable representing the number of rolls needed to get a 1. Calculate the expected value $E[X|Y=7]$. Provide your answer as a single number.

Answer: \dfrac{86317}{15625}


Question: Assume $f(x) \in L_1([a,b])$ and $x_0 \in [a,b]$ is a point such that $f(x) \to +\infty$ as $x \to x_0$. Does there always exist a function $g(x) \in L_1([a,b])$ such that $f(x) = o(g(x))$ as $x \to x_0$?

Answer: Yes


Question: Find the remainder when the number \( S = \sum_{i=0}^{99} (n+i)^6 + 2^{2^{2558}} + 1 \) is divided by 100, where \( n \) is a positive integer.

Answer: 67


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0}\frac{\int_{0}^{\sin{x^2}}e^{t^2}dt}{x^2} \]

Answer: 1


Question: Let $G$ be a hyperbolic group, and let $N$ be the subgroup of $G$ generated by its non-torsion elements. Is the quotient group $G/N$ necessarily finite?

Answer: Yes


Question: Let \( a(x):\mathbb{R}_+\to\mathbb{R} \) be a locally integrable function in the Lebesgue sense, and define \( A(x) = \int_{0}^{x} a(t)\,dt. \) If \( A(x) \) is differentiable almost everywhere, is it always true that \( A'(x) = a(x) \)? Provide a justification for your answer.

Answer: Yes


Question: Given that \( \lim_{n \rightarrow\infty} ~ x_{n+1} - x_n= c \) where \( c > 0 \), determine whether the sequence \( \{x_n/n\} \) is convergent or divergent.

Answer: c


Question: Determine whether the limit \( \lim_{(x,y) \to (0,0)} \frac{xy}{\sqrt{x^2+y^2}} \) exists and, if so, find its value.

Answer: 0


Question: Let \( f(x) \) be a non-negative continuous and bounded function for all \( x \geq 0 \). If \((\cos x)f'(x)\leq (\sin x-\cos x)f(x)\) for all \(x\geq 0\), can it be concluded that \(f(x)\) is a constant function?

Answer: Yes


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

Answer: \dfrac{3}{4}


Question: Determine if there exists a natural number \( n \geq 2 \) such that the integral \( \int_{0}^{+ \infty} \frac{\ln r}{(1 + r^2)^{n}} r \, dr \) converges.

Answer: Yes


Question: A small freight elevator has a maximum capacity, $C$, which is normally distributed with a mean of $400\text{ lbs}$ and a standard deviation of $4\text{ lbs}$. The weight of each box being loaded into the elevator is a random variable with a mean of $30\text{ lbs}$ and a standard deviation of $0.3\text{ lbs}$. Assuming the random variable $C$ is independent of the weight of the boxes, determine the maximum number of boxes that can be loaded into the elevator such that the probability of exceeding the elevator's capacity is less than or equal to 0.2.

Answer: 13


Question:
Prove that for every integer $n > m$ and $m > 1$, the following holds:
\[ \sum_{k=1}^n (-1)^k k^m {n \choose k} = 0. \]

Answer: 0


Question: In an equilateral triangle \(\triangle ABC\), a point \(P\) is located inside the triangle such that \(\angle PAC = x\), \(\angle PCA = 3x\), and \(\angle PBC = 2x\). Determine the value of \(x\) in degrees.

Answer: 6


Question: Given a matrix $A \in \mathbb{R}^{m \times n}$, find a matrix $X \in \mathbb{R}^{m \times r}$ with orthonormal columns such that the Frobenius norm $\|X^T A\|$ is maximized. Express the solution in terms of the singular value decomposition of $A$.

Answer: U_r


Question: Given a sequence \(\delta_n\) of positive numbers decreasing to 0 and a fixed real number \(t > 0\), determine whether there exists a sequence of integers \(N_n\) such that \(N_n \delta_n \rightarrow t\).

Answer: Yes


Question: Calculate the sum \( \sum\limits_{s \subset S} [\max(s) - \min(s)] \) for the set \( S = \{1, 5, 2\} \).

Answer: 12


Question:
Let $K$ be a complete discrete valued field, and $f \in \mathcal{O}_K[X]$ be a monic irreducible polynomial. Let $L$ be the splitting field of $f$, and $a \in \mathcal{O}_K$ with $|a| < 1$. Define
\[ \mathfrak{X} = \operatorname{Sp}( L\langle X,Y\rangle /(aY - f(X))) = \operatorname{Sp} L \langle X \rangle\langle f/a \rangle \]
as a rational subdomain of $\mathbb{D}_L^1$. Is it true that each connected component of $\mathfrak{X}$ contains at least one zero of $f$? Provide a justification for your answer.

Answer: Yes


Question:
Find the asymptotic bounds, preferably in \(\Theta\) notation, for the recurrence relation:
\[ T(n) = T\left(\frac{n}{\log n}\right) + \log \log n \]

Answer: \Theta(\log n)


Question:
Find the supremum of the function \( \frac{f(x)}{g(x)} \), where
\[
f(x) = \dfrac{\beta \exp(-\beta x) \int\limits_{-\infty}^{x} \exp\left(-\frac{(x-\mu)^2}{2 \sigma^2}\right) \,\mathrm{d}x}{\int\limits_{-\infty}^{\infty}\beta \exp(-\beta x) \,\mathrm{d}x \int\limits_{-\infty}^{x} \exp\left(-\frac{(x-\mu)^2}{2 \sigma^2}\right) \,\mathrm{d}x}
\]
and
\[
g(x)=e^{-x} \;.
\]

Answer: \infty


Question: Let $p$, $q$, and $r$ be prime numbers such that $p < q < r$ and $pq + qr + rp - p - q - r = 357$. Find the sum of all possible values of $r$.

Answer: 196


Question: Determine the type of conic section represented by the equation \(\sqrt{x} + \sqrt{y} = \sqrt{a}\) after squaring it twice. Is it a parabola, hyperbola, ellipse, or circle?

Answer: parabola


Question: Find the smallest integer \( n \) such that the product \((p_1^2+1)(p_2^2+1) \cdots (p_n^2+1)\) is divisible by \(10^6\), where \(p_k\) is the \(k^{th}\) prime number.

Answer: 7


Question: Consider a finite group $G$ defined by generators and relations. Suppose we create a new group $G'$ by imposing one additional relation on $G$ that is not implied by any of the original relations. Is it always true that the order of $G'$ is strictly less than the order of $G$?

Answer: Yes


Question: Find the sum of the infinite series \( \sum_{k=1}^{\infty} \frac {1}{(k)(k+2)(k+4)} \).

Answer: \dfrac{11}{96}


Question: A circular disc is divided into three sectors marked with the letters O, A, and B. The disc is spun six times, and the letter that comes in front of a fixed marker is recorded each time. Assuming the boundaries of the sectors do not come in front of the marker, determine the maximum probability of forming the sequence 'BAOBAA'. Let x, y, and z represent the probabilities of the letters O, A, and B, respectively. Maximize the expression zyxzyy.

Answer: \dfrac{1}{432}


Question: Consider a linearly ordered set $L$ with the property that every order-preserving injection $f : L \rightarrow L$ is expansive, meaning $f(x) \geq x$ for all $x \in L$. Is $L$ necessarily well-ordered?

Answer: Yes


Question: Find the value of the normalizing constant \(a\) for the probability density function given by \(f(x) = ax^2 e^{-kx}\) for \(k > 0\) and \(0 \le x \le \infty\).

Answer: \dfrac{k^3}{2}


Question: Calculate the discriminant of the number field $K = \mathbb{Q}(\sqrt{2}, \sqrt{3})$. Given that $K = \mathbb{Q}(\sqrt{2} + \sqrt{3})$ and $[K: \mathbb{Q}] = 4$, determine $d(K)$. The conjugates of $\sqrt{2} + \sqrt{3}$ are $x_1 = \sqrt{2} + \sqrt{3}$, $x_2 = \sqrt{2} - \sqrt{3}$, $x_3 = -\sqrt{2} + \sqrt{3}$, and $x_4 = -\sqrt{2} - \sqrt{3}$. An integral basis of $K$ is $S = \{1, \sqrt{2} + \sqrt{3}, 5 + 2\sqrt{6}, (\sqrt{2} + \sqrt{3})^3\}$. Compute $D(S)$, the square of the determinant of the matrix formed by these conjugates, to find $d(K)$. The correct value of $D(S)$ is $147456$. What is $d(K)$?

Answer: 147456


Question: Determine the interval of convergence of the power series \( \sum_{n=0}^\infty (\frac{\pi}{2} - \arctan(n))(x)^{n} \).

Answer: [-1, 1)


Question: A trough is 3 feet long and 1 foot high, with a vertical cross-section parallel to an end shaped like the graph of $x^2$ from -1 to 1. The trough is full of water. Calculate the amount of work required to pump all the water over the top of the trough. Assume the weight of water is 62 pounds per cubic foot.

Answer: \dfrac{496}{5}


Question: Evaluate the integral \( \int_{S^2} \frac{dS}{((x-a)^2 +y^2+z^2)^{1/2}} \), where \( a > 1 \) and \( S \) is the unit sphere.

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


Question: Let \( f \) be a continuous function such that \( f(f(x)) = 1-x \) for all \( x \in [0,1] \). Evaluate the integral \( J = \int_{0}^{1} f(x) \, dx \).

Answer: \dfrac{1}{2}


Question: Evaluate the infinite series: $$\sum_{n=1}^\infty\frac{n}{1\cdot3\cdot5\cdots(2n+1)}$$ What is its value?

Answer: \dfrac{1}{2}


Question: Calculate \( \lim\limits_{x\to 0}\bigg(x^2\bigg(1+2+3+\ldots+\bigg\lfloor\dfrac{1}{\mid x\mid}\bigg\rfloor\bigg)\bigg) \).

Answer: \dfrac{1}{2}


Question: Determine the values of $z$ for which the sequence \( \frac{2^n z^{2^n}}{1-z^{2^n}} \) converges.

Answer: |z| < 1


Question: Given that \( F(n) = \sum_{d|n} f(d) \) is a multiplicative function with \( F(1) = 1 \), and \( F(p^e) = e + 1 \) for all primes \( p \), find the formula for the function \( f(n) \).

Answer: 1


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

Answer: \dfrac{1}{2}


Question: Consider the number of fixed necklaces of length \( n \) composed of \( a \) types of beads, denoted as \( N(n,a) \). Determine the limit of the ratio \( \frac{N(n+1,a)}{N(n,a)} \) as \( n \to \infty \).

Answer: a


Question: Let $E$ be an arbitrary closed subset of $\mathbb{R}$. Is it possible to construct a function $f: \mathbb{R} \to \mathbb{R}$ such that the set of all points where $f$ is not bounded is precisely $E$?

Answer: Yes


Question: Find all holomorphic functions on the unit disk that satisfy the condition \( f\left(\frac{1}{n^2+1}\right) = 0 \) for all \( n \in \mathbb{N} \).

Answer: 0


Question: Suppose \( f: \mathbb{R} \rightarrow \mathbb{R} \) is a continuous function such that the restricted function \( f\upharpoonright:(-\infty,a] \rightarrow \text{range}(f) \) is invertible, and the shifted function \( g(x) := f(x+a) \) is even. Given \( a > 0 \) and \( c > a \), determine the value of \( f^{-1}(f(c)) \).

Answer: 2a - c


Question: Determine the Galois group of the polynomial \(x^{12} + x^{11} + \dots + x^2 + x + 1\) and state whether it is solvable.

Answer: C_{12}


Question: Is it correct to say that if \( \frac{d^2L(\hat{\theta})}{d\theta^2} > 0 \), then \( \hat{\theta} \) cannot be the maximum likelihood estimator (MLE) of \( L(\theta) \)?

Answer: Yes


Question:
Evaluate the integral \(\int_{\mathbb{R}} \left(\int_{\mathbb{R}} f(x,y)\,dx \right)dy\) for the function \(f\colon \mathbb{R}^2 \rightarrow \mathbb{R}\) defined as:
\[
f(x,y) =  \begin{cases} 1, & \text{for} \ x \ge 0, \ x \le y < x+1 \\ -1, & \text{for} \ x \ge 0, \ x+1 \le y<x+2 \\ 0, & \text{elsewhere}\end{cases}
\]

Answer: 1


Question: Evaluate the contour integral \( \oint_{|z|=2} \frac{dz}{z^4 - 1} \) using the Cauchy integral formula.

Answer: 0


Question: Determine the Lebesgue measure of the set of numbers in the interval (0,1) whose decimal expansion contains one hundred consecutive 4s.

Answer: 1


Question: Let $A$ be a finite dimensional symmetric algebra over a field, assumed to be connected. A non-projective indecomposable module $M$ is called strange if $\text{Ext}^i(M,M)=0$ for all but finitely many $i$. Is it true that strange $M$ have complexity equal to one, meaning that the terms in a minimal projective resolution have bounded dimension?

Answer: Yes


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

Answer: 1


Question: Find all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the equation \( f(x) + f \left( \frac{x - 1}{x} \right) = \frac{5x^2 - x - 5}{x} \) for all nonzero \( x \).

Answer: 5x - 3


Question: Determine whether for any two sets of points \( \{p_1, \dots, p_k\} \) and \( \{q_1, \dots, q_k\} \) in \( \mathbb{R}^n \), the spaces \( \mathbb{R}^n - \{p_1, \dots, p_k\} \) and \( \mathbb{R}^n - \{q_1, \dots, q_k\} \) are homeomorphic.

Answer: Yes


Question: Find the closed-form expression for the series: \[ \frac{1}{1} + \frac{1}{1+2^2} + \frac{1}{1+2^2+3^2} + \cdots \]

Answer: 18 - 24 \ln 2


Question: Evaluate the limit: \[ \lim_{N\to\infty}\frac{1}{{(\log N)}^2}\,\sum_{i=1}^N \,\frac{\log i}{i} \] Choose the correct answer from the following options: (A) 0, (B) 1, (C) 1/2, (D) None of these.

Answer: C


Question: Evaluate the limit \( \lim_{x \to 1} \frac{f(x)}{\ln x} \), where \( f(x) = \int_1^{x^2} e^{-\sin t} \, dt \).

Answer: 2e^{-\sin 1}


Question: Let $X$ be a finite metric space. Determine whether there exists a natural number $n$ such that there is an isometry from $X$ into $(\mathbb{R}^n, ||\cdot||_\infty)$, where $\mathbb{R}^n$ is equipped with the supremum metric.

Answer: Yes


Question: Determine the number of solutions \(x(t)\) to the differential equation \(x^{\prime}(t) = x(t) - e^{-t^2}\) that satisfy \(\lim_{t \to \pm \infty}x(t) = 0\).

Answer: 1


Question: Suppose $M^2$ is a closed Riemannian manifold and $u$ is a $C^2$ function on $M\setminus S$, where $S$ is a closed set of measure zero consisting possibly of a countable number of points. Determine whether $\int_M \Delta u = 0$. Provide a justification for your answer.

Answer: 0


Question:
Find a simple function \( g(n) \) such that \( f(n) = \Theta(g(n)) \), where
\[ f(n) = \sum_{i=1}^n 3(4^i) + 3(3^i) - i^{19} + 20. \]

Answer: 4^n


Question: Evaluate the line integral \( \int_L \vec{F} \cdot \vec{dr} \), where \( L \) is the path given by \( x = 3\cos t, y = 4\sin t, z = t^2 \) from \((3,0,0)\) to \((0,4,\frac{\pi^2}{4})\), and \( \vec{F} = yz(2x+y+z)\hat{i} + xz(x+2y+z)\hat{j} + xy(x+y+2z)\hat{k} \).

Answer: 0


Question:
Consider a Markov chain \( \{X_n : n \ge 0 \} \) with state space \( \{0, 1, 2, 3\} \) and transition matrix
\[P=\begin{pmatrix}
\frac{1}{4} & 0 & \frac{1}{2} & \frac{1}{4}\\
0 & \frac{1}{5} & 0 & \frac{4}{5}\\
0 & 1 & 0 & 0\\
\frac{1}{3} & \frac{1}{3} & 0 & \frac{1}{3}\\
\end{pmatrix}\]
A new process \( \{Z_n : n \ge 0\} \) is defined by \(Z_n = 0\) if \(X_n = 0\) or \(1\), and \(Z_n = X_n\) if \(X_n = 2\) or \(3\). Find \(P(Z_{n+1} = 2 \mid Z_n = 0, Z_{n−1} = 2)\).

Answer: 0


Question: Compute \( \lim_{(x,y)\to(\alpha,0)} \left(1+\frac{x}{y}\right)^y. \)

Answer: 1


Question:
Given the matrix \( M = \begin{bmatrix}2&1\\1&2\end{bmatrix} \), find the value of \( \cos(\pi M/6) \). Choose the correct option from the following:

1. \( \begin{bmatrix}1/2&1\\1&1/2\end{bmatrix} \)
2. \( \begin{bmatrix}\sqrt{3}/4&-\sqrt{3}/4\\-\sqrt{3}/4&\sqrt{3}/4\end{bmatrix} \)
3. \( \begin{bmatrix}\sqrt{3}/4&\sqrt{3}/4\\\sqrt{3}/4&\sqrt{3}/4\end{bmatrix} \)
4. \( \begin{bmatrix}1/2&\sqrt{3}/2\\\sqrt{3}/2&1/2\end{bmatrix} \)

Answer: 2


Question: Evaluate the integral \( \int_{-\infty}^\infty \frac{x^2}{x^6 + 1} \, dx \) using the residue theorem.

Answer: \dfrac{\pi}{3}


Question: Determine the smallest dimension \( n \geq 2 \) such that every coloring with 2 colors of the points in \( \mathbb{R}^n \) contains a monochromatic equilateral triangle with side length 1.

Answer: 3


Question: Determine which expression is larger: \(99^{50} + 100^{50}\) or \(101^{50}\). Use the Binomial Theorem to justify your answer.

Answer: 101^{50}


Question: You have a standard six-sided die and you roll it repeatedly, writing down the numbers that come up. You win when two of your rolled numbers add up to 7. How many turns do you expect to play before you win?

Answer: \dfrac{21}{5}


Question: When $n$ standard six-sided dice are rolled, the probability of obtaining a sum of $1994$ is greater than zero and is the same as the probability of obtaining a sum of $S$. What is the smallest possible value of $S$?

Answer: 337


Question: Consider a strictly convex surface in \(\mathbb{R}^3\) that bounds a strictly convex closed region. Determine whether the Gauss map on this surface has an invertible derivative everywhere.

Answer: Yes


Question:
Determine the limit of the sequence \( a_n = \sqrt[n]{ f\left(\frac{1}{n}\right)f\left(\frac{2}{n}\right) \cdots f\left(\frac{n}{n}\right) } \), where \( f(x) = 1 - \sin x \). Which of the following statements is correct?

(a) \( \{a_n\} \) converges to 0
(b) \( \{a_n\} \) diverges to \( \infty \)
(c) \( \{a_n\} \) converges and \( \lim_{n \rightarrow \infty} a_n > 0 \)
(d) None of the above options is correct

Answer: C


Question: In a normed vector space $V$, if a subspace $U$ contains an open ball of $V$, prove or disprove that $U = V$.

Answer: U = V


Question: Given the modular equation \(c = A^e \pmod{n}\), where \(e = 17\), \(n = 3233\), and \(c = 2790\), find the value of \(A\).

Answer: 65


Question: Does a non-trivial totally ordered group have a maximum element?

Answer: No


Question: Let $R$ be a commutative ring with unity. If the Krull dimensions of all the localizations $S^{-1}R$ are zero, where $S$ runs among multiplicative subsets of $R$, is it true that the Krull dimension of $R$ is zero?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to 0}\dfrac{e^{-x^2/2}-\cos(x)}{x^3\sin(x)} \]

Answer: \dfrac{1}{12}


Question: Determine whether a non-linear map that preserves the inner product in a prehilbertian space necessarily preserves distance. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_0^{2\pi} \log |1-w e^{i\theta}| \, d\theta \) where \( w \) is a complex number such that \( |w| < 1 \).

Answer: 0


Question: Find the residue of the function \( f(z) = \sin(1/z) \) at the singularity point \( z = 0 \).

Answer: 1


Question: Solve the congruence equation: \( x^{37} \equiv 54 \mod 97 \).

Answer: 43


Question: Compute the expected value \( E[e^{(i-1)Z}] \) where \( Z \) is a standard normal random variable and \( i \) is the imaginary unit.

Answer: e^{-i}


Question: Evaluate the integral \( I = \int_0^1 \frac{dx}{\sqrt{(1-x^2)(1-x^2k^2)}} \) for \( 0 < k < 1 \).

Answer: K(k)


Question:
Let $p_1, p_2, \ldots, p_k$ be distinct prime numbers. Define $f(n, l)$ as the number of elements in the set $\{n+1, n+2, \ldots, n+l\}$ that are divisible by at least one of the primes $p_1, p_2, \ldots, p_k$. Determine if the following statement is true:
\[ \sup_{n, m, l \in \mathbb{N}} |f(n, l) - f(m, l)| = O(k). \]

Answer: True


Question: Let \( A \in \text{Mat}_{n,n}(\mathbb{C}) \) be a diagonalizable matrix with distinct eigenvalues \( u_1, \ldots, u_k \). Prove that \((A-u_1I_n) \cdots (A-u_k I_n) = 0\).

Answer: 0


Question: Find the total number of distinct values of \( x \in [0,1] \) for which \( \int\limits_0^x \frac{t^2}{1 + t^4} \, dt = 2x - 1 \).

Answer: 1


Question: Find the volume of the segment of the cylinder defined by the equation \(x^2 + y^2 = 1\), which is bounded above by the plane \(z = 12 + x + y\) and below by \(z = 0\). Express your answer in terms of \(\pi\).

Answer: 12\pi


Question: Determine the values of \( \alpha \) for which the improper integral \( \int_{B}\frac {1}{|x|^\alpha}dV \) exists, where \( B \) is the ball defined by \( |x| \le 1 \) in \( \mathbb{R}^n \).

Answer: \alpha < n


Question: Evaluate the limit of the complex function as \( z \to 0 \): \[ \lim_{z \to 0} z \log(\sin \pi z) \] where \( z \in \mathbb{C} \).

Answer: 0


Question: Determine whether the following statement is true or false: Let $M$ be a module over a commutative unital ring. If every linear form assigns zero to a vector in $M$, then the vector is the zero vector.

Answer: False


Question: You and infinitely many friends each have $1 in a casino. Slot machines cost $1 to play and have three outcomes: Win $2 with probability $\frac{1}{5}$, Lose with probability $\frac{1}{5}$, or Null (get your $1 back) with probability $\frac{3}{5}$. If you win, you pass the extra dollar to a friend, and both of you continue playing on separate machines. If a player loses, they stop playing. If a player gets a Null, they play again. Assuming infinite plays and machines, what is the probability that all players eventually go broke? Use the hint: $P(\text{Broke}|2 \text{ dollars}) = (P(\text{Broke}|1 \text{ dollar}))^2$. Determine $P(\text{Broke}|1 \text{ dollar})$. \(\text{Hint: } P(\text{Broke}|2 \text{ dollars}) = (P(\text{Broke}|1 \text{ dollar}))^2\)

Answer: 1


Question: If \( O \in M_3(\mathbb{R}) \) is an orthogonal matrix with \( \det O = -1 \), prove that \( \lambda = -1 \) is an eigenvalue of \( O \).

Answer: -1


Question: Find the number of strongly monotonic increasing functions \(f:[n]\to \mathcal{P}([k])\) with respect to the subset relation, where \(n > k+1\). Use the pigeonhole principle to determine the value of \(a_{n,k}\).

Answer: 0


Question: Find all real numbers \(x\) such that \(x^{x^5} = 5\).

Answer: \sqrt[5]{5}


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

Answer: \ln 2


Question: Let $p$ be a prime number. Find the degree of the extension $[E:\mathbb{Q}]$, where $E$ is a splitting field of the polynomial $x^p - 2$ over $\mathbb{Q}$.

Answer: p(p-1)


Question: Find the length of the curve defined by $5y^3 = x^2$ that lies inside the circle $x^2 + y^2 = 6$. Provide your answer in exact form.

Answer: \dfrac{134}{27}


Question: Given the equations \(a^3+b^3+2(a^2+b^2)=b^3+c^3+2(b^2+c^2)=c^3+a^3+2(c^2+a^2)\) with the condition \(a \neq b \neq c\), find the value of \(ab+bc+ca\).

Answer: 0


Question: Find \(\lim_{x \to \infty} \left(\frac{x^2+2}{x^2+1}\right)^{3x^2+\frac{1}{x}}\) using the fact that \(\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e.\)

Answer: e^3


Question: Find all continuous functions \( f : (0, \infty) \to \mathbb{R} \) such that \( f(x+y) = xf(y) + yf(x) \) for all \( x, y > 0 \), given that \( f \) is continuous at \( x = 1 \).

Answer: 0


Question: Suppose \( \lim_{x\to b^-} \int_a^x f(u) \, du = 0 \). Does this imply that \( \lim_{x\to b^-} \int_a^x g(u)f(u) \, du = 0 \)? Answer yes or no and justify your answer.

Answer: No


Question: Solve for $x$ in the equation: \[ \log_a(3x-4a) + \log_a(3x) = \frac{2}{\log_2 a} + \log_a(1-2a) \] where $0 < a < \frac{1}{2}$. What is the value of $x$?

Answer: \dfrac{2}{3}


Question:
Suppose that \(|A(n)| < B\) and \(\lim_{s \to 0^{+}}\sum_{n=1}^{\infty}a_{n}n^{-s}=a\) where \(A(x)=\sum_{n \leq x}a_{n}\). Prove that:
\[\lim_{x \to \infty}\sum_{n \leq x} a_{n}(1-\log{n}/\log{x})=a\]

Answer: a


Question: If $A$ is a non-square matrix with orthonormal columns, what is the Moore-Penrose pseudoinverse $A^+$ of $A$?

Answer: A^\top


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

Answer: 0


Question: Evaluate the double integral \( \int_{0}^{\infty} \int_{0}^{x} \frac{h(x)}{x} \mathop{dy} \mathop{dx} \), where \( h(x) \) is a non-negative function satisfying \( \int_{0}^{\infty} h(x) \mathop{dx} = 1 \).

Answer: 1


Question: Find the smallest positive integer \( n \) such that \( \frac{3^n}{n!} \leq 10^{-6} \).

Answer: 17


Question:
Consider a convex function \( \phi : \mathbb{R}^n \to \mathbb{R} \). For a point \( x \in \mathbb{R}^n \), the subdifferential is defined as:
\[ \partial \phi(x) = \{ y \in \mathbb{R}^n \mid \phi(z) \geq \phi(x) + y \cdot (z-x) \quad \forall z \in \mathbb{R}^n \} \]
If \( \partial \phi(x) = \{y_0\} \), does it imply that \( \phi \) is differentiable at \( x \) and \( \nabla \phi(x) = y_0 \)?

Answer: Yes


Question:
Suppose $X$ is a random variable and $c$ is a constant. Let $a_n = E((X-c)^n)$ be finite for all $n \geq 1$. Determine for how many values of $n$ the condition $P(X = c) = 1$ holds if $a_n = 0$. Choose the correct option:

1) At least one $n \geq 1$
2) At least one odd $n$
3) At least one even $n$
4) At least two values of $n$

Answer: C


Question:
Determine the greatest value of \( \lim_{x \to \infty} \frac{f'(x)}{f(x)} \) where \( f(x) \) is a solution to one of the following differential equations:

1. \( y'' + 5y' - 6 = 0, \; y(0) = -1, \; y'(0) = 1 \)
2. \( 4y'' - 8y' + 3y = 0, \; y(0) = 2, \; y'(0) = 1 \)
3. \( 4y'' - 4y' + y = 0, \; y(0) = 2, \; y'(0) = 1 \)
4. \( 2x^2y'' + 3xy' - y = 0, \; x > 0, \; y(1) = 0, \; y(2) = 1 \)

Answer: \dfrac{1}{2}


Question: Find the sum of the series: \[ \sum\limits_{k=0}^\infty\frac{3k}{k!}x^{3k-1} \]

Answer: 3x^2 e^{x^3}


Question: Let \( f(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \frac{x^5}{5} \) and \( g(x) = f^{-1}(x) \). Find \( g'''(0) \).

Answer: 1


Question: Evaluate the sum: \[ \sum_{n=1}^{2017}\left(\left((n+2)^4\bmod{(n+1)^4}\right)\bmod{4}\right) \]

Answer: 4030


Question: Find the coefficient of the \(x^{40}\) term in the generating function \((x^2 + x^3 + x^4 + x^5)^8(x^5+x^{10})^2\).

Answer: 17706


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

Answer: 2


Question: Evaluate the limit: \[ \lim_{n\to \infty} \int_{[0, \pi/4]} \sin(nx)\log(1+x^n)\,d\lambda(x) \]

Answer: 0


Question: Find the integer \(x\) such that \((9^4)^x \equiv 12 \pmod{23}\).

Answer: 6


Question: Determine whether the alternating group $A_5$ can act transitively on a set with 6 elements.

Answer: Yes


Question:
Determine which condition among the following ensures that the bilinear transformation \( w(z) = \frac{az+b}{cz+d} \) maps a straight line in the z-plane to the circle \(|w|=1\) in the w-plane:

1. \(|b|=|d|\)
2. \(|a|=|c|\)
3. \(|a|=|d|\)
4. \(|b|=|c|\)

Answer: 2


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty}\frac{(5n^3-3n^2+7)(n+1)^n}{n^{n+1}(n+1)^2} \]

Answer: 5e


Question: Find the coefficient of \(x^{14}\) in the expansion of \((x^0 + x^1 + x^2 + x^3 + x^4)^6\).

Answer: 1506


Question:
Given a fourth degree polynomial \( p(x) \) with the following conditions:
\[ \Delta^4P(0) = 24 \]
\[ \Delta^3P(0) = 6 \]
\[ \Delta^2P(0) = 0 \]
where \( \Delta P(x) = P(x+1) - P(x) \), compute \( \Delta^2P(10) \).

Answer: 1140


Question: Determine the dimension of the vector space \( \operatorname{End}_{\mathbb{C}} \mathbb{H} \) as a real vector space.

Answer: 8


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

Answer: -\ln 2


Question: Find all positive integers $n$ such that the sum of the digits of $2^n$ is equal to the sum of the digits of $5^n$. Let $s(k)$ denote the sum of the digits of the number $k$. Determine all $n$ for which $s(2^n) = s(5^n)$. Provide a complete solution.

Answer: 3


Question: Given a set of values \( \{a_1, a_2, a_3, \ldots, a_i\} \), calculate the number of possible decreasing sequences of points \( A_i(x_i, y_i) \) such that each point satisfies the condition \( x_i + y_i = a_i \) with non-negative integer coordinates. A sequence is considered decreasing if for each pair of consecutive points \( A_i(x_i, y_i) \) and \( A_{i+1}(x_{i+1}, y_{i+1}) \), the conditions \( x_i \leq x_{i+1} \) and \( y_i \geq y_{i+1} \) hold. For example, for the values \( a_1 = 4, a_2 = 5, a_3 = 3 \), determine the number of such sequences.

Answer: 10


Question: Given that the number of digits in \( n! \) is 6, find the value of \( n \).

Answer: 9


Question: Let $A$ be a complex unital algebra and $M$ be a proper left ideal in $A$. Suppose $\\{L_i\colon i \in I\\}$ is an uncountable family of left ideals such that for each $i$, $L_i \subseteq M$ and $\text{dim }M/L_i = 2$, and for $s \neq t$, neither $L_s \subseteq L_t$ nor $L_t \subseteq L_s$. Can $M$ be finitely generated?

Answer: Yes


Question: Suppose \( f : \mathbb{R}^4 \rightarrow \mathbb{R}^4 \) is an invertible function that maps every straight line to another straight line, preserving the constancy of velocity. Is \( f \) necessarily of the form \( f(x) = Ax + b \) for some matrix \( A \) and vector \( b \)?

Answer: f(x) = Ax + b


Question: Determine the number of integer values of $k$ such that the polynomial \(x^3 - 27x + k\) has at least two distinct integer roots.

Answer: 2


Question: If $H$ is a Lie subgroup of $G$, is $T_{e}H$ a subspace of $T_{e}G$?

Answer: Yes


Question: Consider the entire function $f(z) = z(z-i)$. Define the set \( S = \left\{ \frac{1}{|f(z)|} \mid |z| \geq 2 \right\} \). Determine the value(s) of $z$ at which the maximum of the set $S$ is attained.

Answer: 2i


Question: Determine the number of points inside a triangle \( \triangle ABC \) such that exactly 27 rays can be drawn from each point, intersecting the sides of \( \triangle ABC \) to divide it into 27 smaller triangles of equal area. Provide a proof for your answer.

Answer: 1


Question: Evaluate the limit: \[ \lambda = \lim_{z \to \infty} \left[ 2^z - \left(\frac{4}{3}\right)^z - \zeta(\zeta(z)) \right] \]

Answer: -\infty


Question: In a population of 10,000 birds consisting of 8 equally sized species, determine the minimum number of birds you need to catch to have an 80% probability of capturing at least one bird from each species.

Answer: 28


Question: Given an analytic function \( f: \mathbb{C} \to \mathbb{C} \) defined by \( f(z) = u(x) + iv(y) \) for complex numbers \( z = x + iy \), and knowing that \( f(20) = 17 \) and \( f(17) = 20 \), determine the value of \( f(2017) \).

Answer: -1980


Question: Evaluate the line integral \( \oint_C x \, dx \) over the closed curve \( C \) defined by the path consisting of the segments \( x=0 \), \( y=0 \), and \( y=-x+1 \).

Answer: 0


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

Answer: \infty


Question: Determine \( \lim_{T \to 0} \frac{1}{T} \int_0^T S_0 e^{(\mu - \frac{1}{2}\sigma^2)t + \sigma W_t} \, dt \), where \( dS_t = \mu S_t \, dt + \sigma S_t \, dW_t \) with initial condition \( S_0 > 0 \).

Answer: S_0


Question: Determine whether the following statement is true or false: If \( A \) and \( B \) are orthogonal projection matrices, then the product \( AB \) is also a projection matrix.

Answer: False


Question: Solve the logarithmic equation: \[ \log_{\sqrt{5}}x \cdot \sqrt{\log_x{5\sqrt{5}}+\log_{\sqrt{5}}{5\sqrt{5}}} = -\sqrt{6} \]

Answer: \dfrac{1}{5}


Question: If $A^+$ is the pseudoinverse of matrix $A$, is it always true that $(A^+)^+ = A$? Provide a justification for your answer.

Answer: Yes


Question: Determine whether $\aleph_0^{\aleph_0} = 2^{\aleph_0}$ or $\aleph_0^{\aleph_0} > 2^{\aleph_0}$. Justify your answer.

Answer: \aleph_0^{\aleph_0} = 2^{\aleph_0}


Question: Find a closed-form expression for the sum \( \binom{n}{1} + 3\binom{n}{3} + 5\binom{n}{5} + \cdots \), where \( n > 1 \). You may find the identity \( k\binom{n}{k} = n\binom{n-1}{k-1} \) helpful.

Answer: n \cdot 2^{n-2}


Question: Determine whether the series \( \lim_{n\to \infty}\frac{1}{n^{1/2}}\sum_{k=0}^n\left(\frac{1}{2^{1/2}}(1+i)\right)^k \) converges or diverges.

Answer: 0


Question: Verify Stokes' theorem for the vector field \( \vec{F} = xz \vec{i} - y \vec{j} + x^2y \vec{k} \) where \( S \) is the surface of the region bounded by \( x = 0, y = 0, 2x + y + 2z = 8 \), excluding the part in the xz-plane. Ensure that the curve direction is chosen such that the surface is on the left-hand side when moving along the curve. Calculate both the surface integral of the curl of \( \vec{F} \) over \( S \) and the line integral of \( \vec{F} \) around the boundary of \( S \), and verify if they are equal.

Answer: \dfrac{32}{3}


Question: Calculate the radius of the symmetric random walk on \( \mathbb{Z} \), given by \( \limsup_k (p^{(k)}(0,0))^\frac{1}{k} \), where \( p^{(k)}(0,0) \) denotes the probability of returning to 0 in \( k \) steps, and the transition probabilities are \( p(n,n+1)=p(n,n-1)=\frac{1}{2} \).

Answer: 1


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

Given that $f(n) = o(g(n))$ and $f, g, h: \mathbb{N} \rightarrow \mathbb{N}$ are increasing sequences that approach infinity, is it true that $h(f(n)) = o(h(g(n)))$?

Answer: False


Question: Find all real numbers \(x\) such that \(\int _0^1 \frac{dt}{1+\{x+t\}} = \ln 2\).

Answer: \mathbb{R}


Question: Determine the number of solutions to the equation \(3xe^x + 1 = 0\).

Answer: 2


Question: Given a point \( x \) in \( \mathbb{R}^2 \) that lies within the unit circle of the metric space with the \( \infty \)-norm but not within the unit circle of the metric space with the 1-norm, determine if there exists a \( p \) such that \( \|x\|_p = 1 \) for some \( 1 \leq p \leq \infty \).

Answer: Yes


Question: Suppose \( \phi:H^{n} \cap V \rightarrow M \) is a diffeomorphism between a relatively open subspace of the half-space \( H^{n} \) and a set \( M \), making \( M \) an \( n \)-manifold with boundary. If \( C \subseteq M \) is such that \( \phi^{-1}(C) \subseteq H^{n} \cap V \) is an \( m \)-manifold with boundary, does it follow that \( C \) is also an \( m \)-manifold with boundary? In other words, do diffeomorphisms preserve the dimension of subsets?

Answer: Yes


Question: Determine the number of ordered triplets \((x, y, z)\) such that \(x, y, z\) are prime numbers and \(x^y + 1 = z\). Choose from the following options: \(0, 1, 2, \infty\).

Answer: 1


Question: Evaluate the integral \( \int_0^{+\infty} \frac{x^2}{(x^2+9)(x^2+4)} \, dx \).

Answer: \dfrac{\pi}{10}


Question: Let $W$ be the Lambert W function, and let $Y$ be a real-valued function such that $Y e^Y = x$ for $x \in \mathbb{R}$. Prove or disprove that $Y = kW\left(\frac{1}{k}x\right)$ for a non-zero $k \in \mathbb{R}$. \[ \text{Given: } Ye^Y = x \iff Y = W(x) \]

Answer: Y = W(x)


Question: Find the absolute conditional minimum of the function \( F(x_1, x_2, \ldots, x_n) = \sum_{i=1}^n x_i \) subject to the constraint \( G(x_1, x_2, \ldots, x_n) = \prod_{i=1}^n x_i - 1 = 0 \).

Answer: n


Question: Find the constant \( C \) such that the continuous function \( f(x) = 2x^5 \) satisfies the equation \[ \int_{0}^{x}f(t)dt=\int_{x}^{1}t^2f(t)dt+\frac{x^{6}}{3}+\frac{x^8}{4} + C \] for all \( x \in \mathbb{R} \).

Answer: -\dfrac{1}{4}


Question: Determine whether the sequence \(x_n = \frac{\sin 1}{2} + \frac{\sin 2}{2^2} + \frac{\sin 3}{2^3} + \dots + \frac{\sin n}{2^n}\) is a Cauchy sequence.

Answer: Yes


Question: Find the value of $k$ such that \( \arg\left(\frac{z_1^k z_2}{2i}\right) = \pi \), given that \(|z| = |z - 3i|\) and \(z_1\) and \(z_2\) are two possible values of \(z\) with \(|z| = 3\) and \(\arg(z_1) = \frac{\pi}{6}\).

Answer: 4


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

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to 0} \left( \left[ 1^{\csc^2(x)} + 2^{\csc^2(x)} + 3^{\csc^2(x)} + \cdots + 100^{\csc^2(x)} \right]^{\sin^2(x)} \right) \]

Answer: 100


Question: Evaluate the integral \( \int_0^{2\pi} e^{\cos(x)}\cos(\sin x) \, \mathrm{d}x \).

Answer: 2\pi


Question: Determine the number of elements of order 4 in the symmetric group $S_7$. Consider the possible cycle structures that result in permutations of order 4.

Answer: 840


Question: Let matrix $A$ be symmetric and positive semidefinite. Prove that if there exists a diagonal element $a_{ii} = 0$, then the determinant of $A$ is zero, i.e., $$\det (A) = 0.$$

Answer: 0


Question: Let \( f(x) = x^6 - 2x^3 - 8 \) and \( g(x) = x^2 + 2x + 4 \). If \( a_1, a_2, \ldots, a_6 \) are the roots of \( f(x) \), find the value of \( \prod_{n=1}^{6} g(a_n) \).

Answer: 1600


Question:
Let \( f, g: \mathbf{R} \to \mathbf{R} \) be two strictly increasing continuous functions such that, for all \( a, b, c, d \in \mathbf{R} \), it holds
\[
f(a) + f(b) = f(c) + f(d)
\]
if and only if
\[
\forall h \in \mathbf{R},\,\, g(a+h) + g(b+h) = g(c+h) + g(d+h).
\]
Assuming \( f(0) = g(0) = 0 \) and \( f(1) = g(1) = 1 \), determine if \( f(t) = g(t) = t \) for all \( t \).

Answer: f(t) = g(t) = t


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

Answer: \infty


Question: Given that \((a_{n+1} - a_n) \rightarrow g\), prove that \(\frac{a_n}{n} \rightarrow g\).

Answer: g


Question: Find a prime number \( p \) such that \( p^4 \equiv N^2 \pmod{4p^2} \), where \( N \) is a given integer that is not prime and \( p < N \). For example, verify that \( 7^4 \equiv 21^2 \pmod{4 \times 7^2} \).

Answer: 7


Question: A laser is stationed at coordinates \((1,2,4)\), and it is pointing towards the mirror \(\pi:2x-y-z=0\). After being reflected by the mirror, the laser hits the point \((3,2,-2)\). What are the coordinates of the reflection point \(P\) on the mirror?

Answer: (5, -2, 12)


Question:
Find all functions \( f: \mathbb{Z}^2 \to [0,1] \) such that for any \( x, y \in \mathbb{Z} \), the following equation holds:
\[ f(x,y) = \frac{f(x-1,y) + f(x,y-1)}{2}. \]

Answer: f(x,y) = c


Question: What is the maximum value of the sine function when the domain is the set of complex numbers, \(\mathbb{C}\)?

Answer: \infty


Question: Determine the number of real solutions to the equation \( \sin^5 x - \cos^5 x = \frac{1}{\cos x} - \frac{1}{\sin x} \) given that \( \cos x \neq \sin x \).

Answer: 0


Question: Find the coefficient of \(x^{100}\) in the expression \(\frac{1}{(1-x^5)(1-x^{10})}\).

Answer: 11


Question: Calculate the limit \( \lim_{(\Delta x,\Delta y)\to(0,0)} \frac{\Delta x(\sin (\Delta y) -\Delta y)}{\sqrt{((\Delta x)^2+(\Delta y)^2})^3} \).

Answer: 0


Question: Let \( f:(a,b] \rightarrow \mathbb{R} \) be a function such that the Riemann integral \( F(c):= \int_c^b f(x) \, dx \) exists for all \( c \in (a,b) \), and the improper integral \( F(a)=\int_a^b f(x) \, dx \) also exists. Is the function \( F \) continuous on \([a,b]\)? Justify your answer.

Answer: Yes


Question:
Given a positive area $S$, find the maximum upper bound $\tau$ of the definite integral of a continuous, differentiable, one-variable function $f(t)$ such that:
\[ \int_{0}^\tau f(t)\,dt=S.\]
The function $f(t)$ satisfies the following properties:
1) $f(0)=a$, where $a>0$ is a given constant;
2) $f(t)>0$ for $t \in [0,\tau)$;
3) $f(\tau)=0$. Determine the maximum value of $\tau$.

Answer: \dfrac{2S}{a}


Question: Determine the number of consecutive zeroes at the end of the number \((2146!)_{10}\) when expressed in base 26.

Answer: 177


Question: Calculate the area bounded by the curve given by the equation \(x^3 + y^3 = 3xy\) using Green's theorem. Use the parametrization \(y(t) = \frac{3t^2}{1+t^3}\) and \(x(t) = \frac{3t}{1+t^3}\). Evaluate the integral \(\frac{1}{2} \int_0^\infty \frac{9t^2}{(1+t^3)^2} \, dt\) to find the area.

Answer: \dfrac{3}{2}


Question: Let $a$, $b$, $c$, and $d$ be four complex numbers on the unit circle. Given that the line joining $a$ and $b$ is perpendicular to the line joining $c$ and $d$, find a simple expression for $d$ in terms of $a$, $b$, and $c$.

Answer: -\dfrac{ab}{c}


Question: Given a sequence of i.i.d. random variables $(X_i)_{i\geq 1}$ with $\mathbb{E}(X)=0$ and $\mathbb{E}(X^2)=1$, consider the sum $S_n=\sum_{k=1}^n{X_k}$. Is it true that $|S_n - S_{n-1}| = 1$ for all $n \geq 2$, regardless of the distribution of the sequence?

Answer: No


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

Answer: 3 - e


Question: Calculate the time-averaged power of the continuous signal given by: \[ z(t) = \sqrt{E} \sum_{k=-\infty}^{+\infty} A_k g(t-kT) \] where \( A_k = 1 \) if \(|k|\) is a prime number and \( A_k = -1 \) otherwise, and \( g(t) = 1 \) for \( 0 \leq t \leq T \) and \( g(t) = 0 \) otherwise. \( T \) is the fundamental period. The power of a continuous signal is defined by: \[ \lim_{T\to\infty} \frac{1}{2T} \cdot \int_{-T}^{T} |x(t)|^2 dt \]

Answer: E


Question: If the adjugate matrix \( \text{adj}(A) \) of an \( n \times n \) real matrix \( A \) is symmetric, must \( A \) also be symmetric? Justify your answer.

Answer: No


Question: Evaluate the surface integral \( \int \mathbf{r} \cdot d\mathbf{S} \) over the surface of a sphere with radius \( a \) centered at the origin, where \( \mathbf{r} \) is the position vector.

Answer: 4\pi a^3


Question: Consider the set $L$ of countable limit ordinals, where $0$ is regarded as a limit ordinal. For any $\alpha \in \omega_1$, there exists a unique $\beta_0 \in L$ and $n_0 \in \omega$ such that $\alpha = \beta_0 + n_0$. Given that $L$ is an uncountable well-ordered set with each element having at most countably many predecessors, there exists an order isomorphism $\phi: \omega_1 \to L$. Consequently, $\beta_0 = \phi(\beta_1 + n_1)$ for some unique $\beta_1 \in L$ and $n_1 \in \omega$, and this process can be continued. Define a map $n: \omega_1 \to \omega^\omega$ by $n(\alpha) = (n_0, n_1, \dots)$. Is the map $n: \omega_1 \to \omega^\omega$ injective?

Answer: Yes


Question: Find the remainder when \(2^{2^{17}} + 1\) is divided by 19.

Answer: 7


Question: Does there exist a function on \(\mathbb{R}^n\) whose graph does not have content zero?

Answer: Yes


Question: Evaluate the triple integral \( \iiint z(x^2+y^2+z^2)^{-3/2} \, dx \, dy \, dz \) over the region defined by the part of the ball \( x^2+y^2+z^2 \le 81 \) where \( z \ge 4.5 \).

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


Question: Evaluate the limit: \[ \lim _{x\to 1}\left(\frac{\ln \left(\left|x-2\right|\right)}{\ln \left(x\right)}\right) \] without using L'Hopital's rule or graphing.

Answer: -1


Question: Determine whether the series \( \sum_{k=1}^{n} \left(\frac{1}{k \cdot \log(n)}\right) \) converges to 1 as \( n \to \infty \).

Answer: 1


Question: Determine the maximum period of the sequence defined by the recurrence relation \(s_{n+1} = s_n + s_{n-2}\) with the initial seed \((s_0, s_1, s_2) = (a, b, c)\) in \(\mathbb{Z}_2\), where \((a, b, c)\) is not equal to \((0, 0, 0)\).

Answer: 7


Question: Evaluate the line integral \( \int (x+y) \, dx + dy \) along the path \( g(t) = (t, t^2) \) from \( t = 0 \) to \( t = 1 \).

Answer: \dfrac{11}{6}


Question: Let $X$ be a random variable with $X \sim \text{Geom}(1/3)$. Let $Y$ be another random variable which has a $\text{Binomial}(N,1/4)$ distribution where $N$ is the value taken by the random variable $X$. Compute the expectation value of the product $XY$. Use the law of total probability or any other method to find $E(XY)$. Express your answer as a sum: $E(XY) = \sum (N E(Y|X=N)P(X=N))$. What is the value of this sum?

Answer: \dfrac{15}{4}


Question: Solve the congruence equation $x^5 \equiv 4 \pmod{7}$. Determine the integer value(s) of $x$ that satisfy this equation.

Answer: 2


Question:
Given two probability measures \(\mu\) and \(\mu'\) over the sigma algebra \(\Sigma\) of Lebesgue measurable subsets of \([0,1]\), both assigning measure 0 to all and only null sets, consider the collection of functions:
\[\{f_X:X\rightarrow \overline{X} \mid X\in \Sigma, 0<\lambda(X)<1\}\]
such that for every partition of \([0,1]\) into two sets \(X\) and \(\overline{X}\), the function \(f_X\) preserves the measure between \(\mu_X\) and \(\mu_{\overline{X}}\) and between \(\mu'_X\) and \(\mu'_{\overline{X}}\). That is, \(\mu_X(f^{-1}(Y)) = \mu_{\overline{X}}(Y)\) and similarly for \(\mu'\). Does it follow that \(\mu = \mu'\)?

Answer: \mu = \mu'


Question: Determine the accumulation points of the sequence \( \{x_n\} \) where \( x_n = n\sin(n) \) for \( n \in \mathbb{N} \).

Answer: \mathbb{R}


Question:
Evaluate \( n^4 \lim_{n \to \infty} \prod_{r=1}^{n} a_r \) given that the terms \( a_1, a_2, a_3, \ldots, a_n \) of a series satisfy the relation:
\[
\frac{n+1}{a_{n+1}} - \frac{n-1}{a_n} = \frac{2(n+2)}{n}, \quad n \geq 1, \; n \in \mathbb{N}.
\]

Answer: 24


Question: Let $A \in \mathbb{R}^{n \times n}$ be a normal matrix, meaning $AA^t = A^tA$. If $A^+ \in \mathbb{R}^{n \times n}$ is the Moore-Penrose pseudoinverse of $A$, does the equation $AA^+ = A^+A$ hold?

Answer: Yes


Question:
Find the dimension of the subspace \(W\) of \(\operatorname{Hom}(\mathbb{R}^3, \mathbb{R}^3)\), the space of all linear transformations from \(\mathbb{R}^3\) to \(\mathbb{R}^3\), defined by:
\[
W = \{L \in \operatorname{Hom}(\mathbb{R}^3, \mathbb{R}^3) : \operatorname{Im}(L) \subset S\}\]
where \(S = \{(x^1, x^2, x^3) \in \mathbb{R}^3 : x^1 + x^2 + x^3 - 1 \leq 0, \ x^1 + x^2 + x^3 + 1 \geq 0\}\).

Answer: 6


Question: Evaluate the limit of the expression \( x^2\left(1+\frac{1}{x}\right)^{x}-e\,x^{3}\ln\left(1+\frac{1}{x}\right) \) as \( x \to \infty \).

Answer: \dfrac{e}{8}


Question: Let $M$ be a real analytic manifold and $E \subset M$ be a subanalytic set. Is $E$ measurable?

Answer: Yes


Question: Compute the expected value \( E[HXY] \) where \( H \) is distributed binomially with parameters \( XY \) and \( \frac{1}{2} \), and \( X \) and \( Y \) are independent variables both distributed geometrically with parameter \( \frac{1}{2} \).

Answer: 18


Question:
Given complex numbers $x, y, z$ satisfying the equations:
\[ x+y+z = 1 \]
\[ x^2+y^2+z^2 = 2 \]
\[ x^3+y^3+z^3 = 3 \]
Find the value of $x^4+y^4+z^4$ expressed as $\frac{a}{b}$, where $a$ and $b$ are positive coprime integers. What is $a+b$?

Answer: 31


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

Answer: 3


Question: For which values of \(a\) is the function \(f_{a}(x)\) bounded, where \(f_{a}(x) = \begin{cases} x^{a}\cos\left(\frac{1}{x}\right), & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases}\)?

Answer: 0


Question: Consider a symmetric random walk \(S_n\) where each step \(X_i\) satisfies \(P(X_i=1)=P(X_i=-1)=\frac{1}{2}\). Define \(\tau_k=\inf\{n : |S_n|=k\}\). Calculate the expected value \(E[\tau_k]\).

Answer: k^2


Question: Let $A \subset \mathbb{R}$ be a set such that $A \times \mathbb{R}$ is measurable in $\mathrm{B}(\mathbb{R}) \times \mathrm{B}(\mathbb{R})$, the product $\sigma$-algebra of the Borel $\sigma$-algebra of $\mathbb{R}$. Must $A$ be measurable in $\mathrm{B}(\mathbb{R})$?

Answer: Yes


Question: Evaluate the contour integral \( \frac{1}{2\pi j} \oint_C \text{Re}\{ z\} \, dz \) where \( C \) is the counterclockwise unit circle.

Answer: \dfrac{1}{2}


Question:
Find the condition on $t$ such that the limit \( h(t) = \lim\limits_{n\to\infty}\frac{g(t+n)}{n} \) is defined, where $f$ is a real and continuous function satisfying $f(x+1)=f(x)$ for all $x \in \mathbb{R}$, and $g(t)=\int_0^t f(x) \, dx$. Choose the correct option:

(a) $t=0$ only
(b) integer $t$ only
(c) $\forall t \in \mathbb{R}$ and $h(t)$ is independent of $t$
(d) none of the above is true

Answer: C


Question:
Show that for a function \( g \in C^2(\mathbb{R}^2) \) with compact support, the following integral equality holds:
\[
-\frac{1}{2 \pi} \iint_{\mathbb{R}^2} \Delta g(z) \log \frac{1}{|z-\xi|} \, dx \, dy = g(\xi)
\]

Answer: g(\xi)


Question: Let $X_1$ and $X_2$ be two independent uniformly distributed random variables on $[0,1]$. Calculate the expectation of $\min(X_1,X_2)$ given that $\max(X_1,X_2) > 0.5$. Provide your answer as a fraction.

Answer: \dfrac{7}{18}


Question:
Find the density function of the random variable $V = XY$, where $X$ and $Y$ are continuous random variables with the joint density function given by $f(x,y) = x + y$ for $0 < x, y < 1$. Use the formula for the density of a product of two random variables:
\[ f_V(v) = \int_{0}^{1} f\left(x, \frac{v}{x}\right) \frac{1}{x} \, dx \]

Answer: 2(1 - v)


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

Answer: 0


Question: Find the largest integer that divides $p^4 - 1$ for all prime numbers $p$ greater than 5.

Answer: 240


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{\mathcal{P}(n)}{n} \] where \( \mathcal{P}(n) \) is the number of prime numbers that divide \( n \).

Answer: 0


Question: Determine whether the following statement is true or false: If a bounded function is Riemann integrable over an interval, then it has a finite number of discontinuities.

Answer: B


Question: Find the minimal polynomial of the matrix \( A = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} \).

Answer: x^4 - 1


Question: Calculate the total number of transitive relations on the set \( A = \{a, b, c\} \).

Answer: 171


Question: Consider $2n$ sets, each containing an even number of integers. Each integer is independently colored red with probability $\frac{1}{2}$. A set is defined as good if at least half of its elements are red. A coloring is considered good if at least $n$ out of the $2n$ sets are good. Given that $p$ is the probability that a random coloring is good, is it always true that $p > \frac{1}{2}$?

Answer: Yes


Question: Let $M$ be a countable model of $ZFC$ and $M[G]$ be a (set) generic extension of $M$. Suppose $N$ is a countable model of $ZF$ such that \(M \subseteq N \subseteq M[G]\) and that $N = M(x)$ for some $x \in N$. Is $N$ a symmetric extension of $M$?

Answer: Yes


Question: Let $n$ be a positive integer. Find the minimum value of \( \sum_{k=1}^n \frac{x^k_k}{k} \), where $x_1, x_2, \ldots, x_n$ are positive real numbers such that \( \sum_{k=1}^n \frac{1}{x_k} = n \).

Answer: \sum_{k=1}^n \frac{1}{k}


Question: Let \(A\) be a real square matrix with a simple eigenvalue \(1\) and corresponding eigenvector \(v\). If \(v\) spans the kernel of \(I-A\), does \(v\) also span the kernel of \((I-A)^2\)? Justify your answer.

Answer: Yes


Question: Evaluate the triple integral \( \iiint xyz \, dx \, dy \, dz \) over the volume of the tetrahedron with vertices \((0,0,0)\), \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\).

Answer: \dfrac{1}{720}


Question: Let $G = \mathbb{Z}/30\mathbb{Z} \oplus \mathbb{Z}$ and $H = \mathbb{Z}/15\mathbb{Z} \oplus \mathbb{Z}/7\mathbb{Z}$ be abelian groups. Determine the number of group homomorphisms from $G$ to $H$, i.e., find the number of elements in $\text{Hom}_{\mathbb{Z}}(G, H)$. \(\text{Hom}_{\mathbb{Z}}(G, H)\)

Answer: 1575


Question: Find the arc length of the curve given by the equation \(6xy = x^4 + 3\) from \(x = 1\) to \(x = 2\).

Answer: \dfrac{17}{12}


Question: Determine the order of differentiability of the tangent bundle of a $C^2$-manifold. Is the tangent bundle $C^1$, or is it $C^2$?

Answer: C^1


Question: Find all pairs of positive integers \((a, b)\) such that \(a > b\) and \((a-b)^{ab} = a^b \cdot b^a\).

Answer: (4, 2)


Question: Determine whether the following statement is true or false: Let \( X \) be a random variable and \( \{X_{n}\}_{n \in \mathbb{N}} \) a sequence of random variables. The sequence \( X_{n} \rightarrow X \) in probability if and only if \( \lim_{n\to\infty} \mathbb{E}\left(\frac{|X_{n}-X|}{1+|X_{n}-X|}\right)=0 \).

Answer: True


Question: Evaluate the integral \( \int_0^{\infty} \frac{1-e^{-ax}}{x e^x} \, dx \) and verify that the result is \( \log(a+1) \) for \( a > -1 \).

Answer: \log(a+1)


Question: Find the line integral of the vector field \( F = \langle yz + y, xz+5x, xy+2y \rangle \) around the intersection of the unit sphere and the plane \( x + y = 14 \). Use the surface integral \( \iint \nabla \times F \cdot \mathbf{n} \, dA \), where \( \mathbf{n} \) is the unit normal vector to the plane.

Answer: 0


Question: Calculate the line integral of the vector field \( \vec{F} = \left(x \frac{e^{x^2+y^2} - e}{x^2+y^2}, y \frac{e^{x^2+y^2} - e}{x^2+y^2}\right) \) along the curve \( \gamma(t) = (4 \cos t, 4 \sin^2 t) \) for \( t \in [0, \pi/2] \).

Answer: 0


Question: Suppose $A, B \subset B(\mathcal{H})$ are $C^*$-algebras, and let $\{p_n\} \subset B(\mathcal{H})$ be a monotone sequence of projections such that $p_n \rightarrow 1$ in the strong operator topology, $\forall n$, $p_n A p_n \simeq p_n B p_n$, and each $p_n A p_n$ is finite dimensional. Is it true that $A \simeq B$?

Answer: No


Question: Find all positive real numbers \(x\) that satisfy the inequality \(x^{n+1} - x^n - 2(-1)^n x \geq 0\) for all natural numbers \(n\).

Answer: [2, \infty)


Question: Find the minimal irreducible polynomial of $a = e^{\frac{2 \pi i}{8}}$ over $\mathbb{Q}$. Note that $a$ is a root of the polynomial $f(x) = x^8 - 1 = (x^4 - 1)(x^4 + 1)$. Determine if $x^4 + 1$ is irreducible over $\mathbb{Q}$ and find the minimal polynomial $Irr(a, \mathbb{Q})$. Provide your answer as a polynomial.

Answer: x^4 + 1


Question: Determine whether the following statement is true or false: Let $T:H \rightarrow H$ be a continuous linear operator, where $H$ is a pre-Hilbert space. Then $H=\ker(T) \oplus\ker(T)^{\perp}$, where $\ker(T)^{\perp}$ is isomorphic to $H/\ker(T)$. Furthermore, $\ker(T)^{\perp}$ is isomorphic to $\text{ran}(T)$ as a vector space.

Answer: False


Question: Compute the line integral \( I = \oint_{C} F \cdot dr \) for the vector field \( F = (x, z, 2y) \in \mathbb{R}^{3} \), where \( C \) is the curve obtained from the intersection of the plane \( z = x \) and the cylinder \( x^{2} + y^{2} = 1 \), oriented counterclockwise. Use Stoke's Theorem to evaluate the integral.

Answer: -\pi


Question: Suppose \((\Omega,\mathscr{F},\mu)\) is a measure space and \(f,g\) are \((\mathscr{F},\mathscr{B})\)-measurable functions where \(\mathscr{B}\) is the Borel algebra on \(\mathbb{R}\). If \(\mu(\{f<g\})>0\), determine whether there exists a constant \(\xi \in \mathbb{R}\) such that the set \(\{f\leq \xi < g\}\) has nonzero measure.

Answer: Yes


Question: A positive integer is written on each vertex of a pentagon, with a different integer on each vertex. On each side of the pentagon, the least common multiple (LCM) of the numbers at the vertices forming that side is written. If the same integer $n$ is written on each side, determine the minimum possible value of $n$.

Answer: 30


Question: Find the minimal polynomial of the element $a = \bar{x}$ in the field extension $K = \mathbb{Q}[x]/(x^2 - 5)$ over $\mathbb{Q}$.

Answer: x^2 - 5


Question: Does there exist an uncountable dense subset of \( \mathbb{R} \) whose complement is also uncountable and dense?

Answer: Yes


Question:
Compute the limit:
\[ \lim_{n\to\infty} e^{-itn^{1/4}}\left(1-\frac{1}{\sqrt{n}}+\frac{1}{\sqrt{n}}e^{i\frac{t}{n^{1/4}}} \right)^{n} \] where \( t \) is a real number.

Answer: e^{-t^2/2}


Question: Let \( f(x) \) be defined as follows: \( f(x) = \sin x \) for \( x \in [0, \frac{\pi}{2}] \), \( f(x) + f(\pi-x) = 2 \) for \( x \in (\frac{\pi}{2}, \pi] \), and \( f(x) = f(2\pi-x) \) for \( x \in (\pi, 2\pi] \). Calculate the area enclosed by \( y = f(x) \) and the \( x \)-axis over the interval \([0, 2\pi]\). If this area is expressed as \( a\pi + b \), find \( a^2 + b^2 \).

Answer: 4


Question: Determine if $k=1$ is the only real number for which the integral \( \int_0^\infty \log ( k+\exp (-x^2)) \, dx \) converges.

Answer: 1


Question: Calculate the sum of the series: \((2^2-1)(6^2-1) + (4^2-1)(8^2-1) + \cdots + (100^2-1)(104^2-1)\).

Answer: 1157089290


Question: Let $f: A \to \mathbb{R}$ be a differentiable function on an open convex set $A \subset \mathbb{R}^{n}$. If $\| \nabla f \| \leq M$ on $A$ for some $M > 0$, determine whether it is possible to have \(|f(x) - f(y)| \leq M \| x - y \|\) for all $x, y \in A$. Justify your answer.

Answer: Yes


Question:
Determine which of the following properties the subspace $\mathbb{Q} \times [0,1]$ of $\mathbb{R}^2$ (with the usual topology) possesses:

1. Dense in $\mathbb{R}^2$
2. Connected
3. Separable
4. Compact

Answer: 3


Question: Evaluate \( \lim_{n \to \infty} \sum_{k=1}^n \frac{k^2}{n^2 + k^2} \).

Answer: \infty


Question: Find the value of \( r \) for which a non-trivial fixed point appears in the equation \( x = r\sin\pi x \), given that \( 0 \leq x \leq 1 \) and \( 0 \leq r \leq 1 \). The trivial fixed point is \( x = 0 \).

Answer: \dfrac{1}{\pi}


Question: Evaluate the integral \( \iint_{D}\left(x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}\right)\mathrm{d}x\mathrm{d}y \) where \( f(x,y) \) is defined on \( D=\{(x,y)\mid x^2+y^2\le1\} \) and has continuous second-order partial derivatives in \( D \). Given that \( \frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}=\mathrm{e}^{-(x^2+y^2)} \), find the value of the integral.

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


Question:
Consider the function defined by $f(z) = 2$ for $\Re(z) = 2$ and $-2 < \Im(z) < 2$. Determine if this function can be uniquely extended to the entire complex plane by analytic continuation. Choose the correct answer:

(a) This cannot be done - period!
(b) Yes, it can be done, the function is $f(z) = 2$ for $z \in \mathbb{C}$.
(c) Yes, it can be done, the function is $f(z) = \Re(z)$ for $z \in \mathbb{C}$.
(d) It can’t be done because we can’t choose between (b) and (c) above.
(e) Other - please elaborate.

Answer: B


Question: Calculate the value of \(4^{4^4} \mod 10\).

Answer: 6


Question: Evaluate the limit: \[ \lim_{x \to 0} 2 \frac{(\cosh x - 1)^{1/x^2}}{x^2} \]

Answer: 0


Question: Determine the limit: \[ \lim_{n\to \infty} \sum_{k=n}^\infty \frac{1}{k} \]

Answer: \infty


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

Answer: 2


Question: Given that $f(f(x)) = x^2 + 2$ and $f$ is a strictly increasing function, find the value of $f(11)$. Assume $x, f(x)$ are positive integers.

Answer: 38


Question: Let \( A \) and \( B \) be real matrices of order \( n \geq 6 \). Suppose \( A + \alpha B \) is a projection operator for any \( \alpha \in \mathbb{R} \). Determine whether the following statement is true or false: If \( A \) is an orthogonal projection, then \( A \neq B \).

Answer: False


Question: Evaluate the integral \( \frac{2}{\pi \sigma_k} \int_{-\infty}^{\infty} \frac{\sin^2(\frac{\sigma_k}{2}(v_gt-x))}{(v_gt-x)^2} \, dx \) and verify if it equals 1.

Answer: 1


Question: Given an arithmetic sequence $a_1, a_2, a_3, \dots$, where $b_i = \lfloor a_i \rfloor$ (the greatest integer less than or equal to $a_i$), is it possible to determine the common difference $d = a_{i+1} - a_i$ using only the sequence $\{b_i\}$?

Answer: Yes


Question: Compute the gradient of the function \( f(x) = \|x\|^p \), where \( f(x) = (x_1^2 + x_2^2 + \ldots + x_N^2)^{p/2} \).

Answer: p \|x\|^{p-2} x


Question: Let $f$ be a $C^{\infty}$ function such that $f(0)=0$ and $f'(x)>0$ for $x \in (-\varepsilon, \varepsilon)$. Determine whether it is true that $\frac{1}{\left\lvert f(x) \right\rvert} \le \frac{C}{\left\lvert x \right\rvert}$ for some $C>0$ in a neighborhood of zero.

Answer: Yes


Question: Calculate the value of the sum: \[ \sum_{k=1}^{10} k \binom{10}{k} \binom{20}{10-k} \]

Answer: 100150050


Question: Find the minimal polynomial of \(\alpha^{21}\) over \(\mathbb{Z}_2\), given that \(\alpha\) satisfies the equation \(\alpha^6 + \alpha + 1 = 0\) over \(\mathbb{Z}_2\).

Answer: x^2 + x + 1


Question: Find the derivative of the matrix expression $ABB^T$ with respect to the matrix $A$. Express your answer in terms of $B$. Is the derivative simply $BB^T$?

Answer: BB^T


Question: Determine the values of \( p \in \mathbb{R} \) for which the integral \( \int_0^\infty \frac{\sin x}{x^p} \, dx \) converges.

Answer: (0, 2)


Question: Calculate the limit: \[ \lim_{x \to \infty}{\frac{1}{x}\int^{3x}_{x/3}} g(t)\, dt \] where the function \( g: (0, +\infty) \rightarrow \mathbb{R} \) is unbounded, continuous, and has a limit at infinity equal to \( \pi \).

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


Question: Compute the limit: \[ \lim_{n \to \infty} \int_{0}^{\pi} \frac{\sin x}{1 + 3\cos^{2}(nx)} \, \mathrm{d}x. \]

Answer: 1


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

Answer: \mathbb{R}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{\int_{0}^1\left(x^2-x-2\right)^n \, dx}{\int_{0}^1\left(4x^2-2x-2\right)^n \, dx} \]

Answer: 2


Question: Let $G$ be a finite abelian group and let $m = \operatorname{Max}\{\text{the order of } a : a \in G\}$. For every $g \in G$, is $m$ divisible by the order of $g$?

Answer: Yes


Question: Given a $4 \times 4$ matrix $A$ with a determinant of $3$, find the determinant of the cofactor matrix $\text{cof}(A)$. Express your answer in terms of the determinant of $A$. \( \text{det}(A) = 3 \)

Answer: 27


Question: Let $X_n$ be a sequence of random variables such that $X_n > 0$ for all $n$ and $X_n \to_P 0$ as $n \to \infty$. Let $Y_n$ be another sequence of random variables such that $Y_n > 0$ and $Y_n \leq X_n$ with probability $p_n$. If $p_n \to 1$, prove or disprove that $Y_n \to_P 0$.

Answer: Y_n \to_P 0


Question: Find the length of the longest line segment contained in the region defined by the following conditions in the $x-y$ plane: $y = v$, $x = u + v$, and $u^2 + v^2 \leq 1$, where $u$ and $v$ are parameters. What is the length of this longest segment?

Answer: 1 + \sqrt{5}


Question:
Given a real sequence $(x_n)$ that satisfies the following conditions:
\[ \lim_{n \to \infty} (x_{2n} + x_{2n+1}) = 315 \]
and
\[ \lim_{n \to \infty} (x_{2n} + x_{2n-1}) = 2016, \]
determine \[ \lim_{n \to \infty} \frac{x_{2n}}{x_{2n+1}}. \]

Answer: -1


Question: Let \( F \subseteq \mathbb{C} \) be the splitting field of \( x^7 - 2 \) over \( \mathbb{Q} \), and let \( z = e^{2\pi i/7} \) be a primitive seventh root of unity. Determine the degree of the extension \([F:\mathbb{Q}(z)]\].

Answer: 7


Question: Given \( U_n=\int_0^\frac{\pi}{2} x\sin^n x \, dx \), find the value of \( \frac{100U_{10}-1}{U_8} \).

Answer: 90


Question: Let $X_n \sim \operatorname{Bin}(n,p_n)$ where $p_n \to 0$ and $np_n \to \infty$ as $n \to \infty$. Show that the standardized random variable \( \frac{X_n - np_n}{\sqrt{np_n}} \) converges in distribution to a standard normal distribution, \( N(0,1) \), as \( n \to \infty \).

Answer: N(0,1)


Question: Evaluate the double integral \( \iint_R xy^2 \sqrt{x^2+y^2} \, dx \, dy \), where the region \( R \) is defined as \( \{ (x, y) \in \mathbb{R}^2 : 1 \leq x^2 + y^2 \leq 4, \, y \geq 0 \} \).

Answer: 0


Question: Evaluate the contour integral \( \int_{\gamma(0,1)} \frac{\cos(z)}{z^2}dz \), where \( \gamma(0,1) \) is the circular contour defined by \( z = e^{it} \) for \( 0 \leq t \leq 2\pi \).

Answer: 0


Question:
Determine the value of \( k \) for which the function \( f(x,y) \) is continuous at \((0,0)\):
\[
f(x,y)=\begin{cases}\frac{xy}{(x^2+y^2)^{5/2}}\left[1-\cos (x^2+y^2)\right] & (x,y)\ne(0,0)\\ k & (x,y)=(0,0)
\end{cases}
\]

Answer: 0


Question: Evaluate the sum \( \sum_{k_1+\cdots+k_n=m}\frac{1}{k_1!\cdots k_n!} \) for fixed integers \( m \) and \( n \), where the sum is taken over all non-negative integer solutions \( k_1, k_2, \ldots, k_n \) such that \( k_1 + k_2 + \cdots + k_n = m \).

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


Question: In a finite semigroup $S$, two elements $s, t \in S$ are called $\mathcal{R}$-equivalent if $s = t$ or there exist $x, y \in S$ such that $s = tx$ and $t = sy$. Equivalently, they generate the same right ideal, i.e., $sS \cup \{s\} = tS \cup \{t\}$. If two idempotents $e, f \in S$ are $\mathcal{R}$-equivalent, does this imply that $e = f$? Provide a proof or counterexample.

Answer: No


Question: If \(x = \omega - \omega^2 - 2\), where \(\omega\) is a cube root of unity not equal to 1, find the value of \(x^4 + 3x^3 + 2x^2 - 11x - 6\).

Answer: 1


Question: Compute the line integral of the vector field \( \vec{A}(x,y,z) = (yz)\hat{i} + (xz)\hat{j} + (xy)\hat{k} \) along any path connecting the points \((0,0,0)\) to \((1,1,1)\). Given that \( \nabla \times \vec{A} = 0 \), find the potential function \( V \) such that \( \vec{A} = \nabla V \), and evaluate \( V(1,1,1) - V(0,0,0) \).

Answer: 1


Question: Let \( f(x) = \int_0^x \sum_{i=0}^{n-1} \frac{(x-t)^i}{i!} \, dt. \) Find the nth derivative \( f^{(n)}(x). \)

Answer: 1


Question: Suppose you throw a fair die 1995 times. What is the probability that the total sum of the outcomes is divisible by 3?

Answer: \dfrac{1}{3}


Question:
Consider the function:
\[ f_n(x) = x + 3^3x^3 + 5^3x^5 + \ldots + (2n-1)^3x^{2n-1}, \quad x \in (0, 1). \]
As \( n \) becomes larger, which of the following functions does \( f_n(x) \) approach?

(a) \( \frac{x(x+1)(x^4+22x^2+1)}{(x-1)^4(x+1)^4} \)

(b) \( \frac{x(x^2+1)(x^4+22x^2+1)}{(x-1)^4(x+1)^4} \)

(c) \( \frac{x^2(x+1)(x^4+22x^2+1)}{(x-1)^4(x+1)^4} \)

(d) \( \frac{x^2(x^2+1)(x^4+22x^2+1)}{(x-1)^4(x+1)^4} \)

Answer: B


Question: Given an invertible matrix $M \in GL_n(\mathbb{F})$ and a number $1 \leq m \leq n-1$, is it always possible to permute the rows of $M$ to make it $m$-good, where $M$ is partitioned as a block matrix $\begin{bmatrix} A & B \\ C & D \end{bmatrix}$ with $A$ as an $m \times m$ matrix and $D$ as an $(n-m) \times (n-m)$ matrix, and both $A$ and $D$ are invertible? Assume $\mathbb{F}=\mathbb{Z}_p$. Provide a justification for your answer.

Answer: Yes


Question:
Find the minimum value of the expression:
\[ \sum_{i=1}^n \frac{1}{\prod_{j \ne i} |x_j - x_i|} \]
where \(x_1, x_2, \ldots, x_n\) are real numbers in the interval \([-1, 1]\).

Answer: 2^{n-2}


Question:
Consider the half-space problem for the fractional Laplacian:
\[
  \left\{\begin{aligned}
(-\Delta)^s v &= f(v)  &&\text{in } \mathbb{R}^N_+\\
 v  & =0 &&\text{in  }  \mathbb{R}^N \setminus \mathbb{R}^N_+
\end{aligned}
  \right.
\]
where $s \in (0, 1)$ and $v$ changes sign. If $v$ is bounded and $f(v) = 0$, does it imply that $v \equiv 0$? Provide a justification for your answer.

Answer: v \equiv 0


Question: Evaluate the expression \( n^{\aleph_0} \) for \( n \in \mathbb{N} \) and determine if it can be expressed as \( \aleph_1 \).

Answer: \aleph_1


Question: Determine the number of functions \( f: \mathbb{Z} \to \mathbb{Z} \) that satisfy the equation \( 3f(f(x)) - 7f(x) + 2x = 0 \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \to 0, y \to 0} \frac{(x \tan y + 4)^{1/2} - 2}{y \sin x} \]

Answer: \dfrac{1}{4}


Question: Evaluate the ratio \( \frac{I_1}{I_2} \), where \( I_1 = \int_{0}^{1} \frac{\tan^{-1}x}{x} \, dx \) and \( I_2 = \int_{0}^{\pi/2} \frac{x}{\sin x} \, dx \).

Answer: \dfrac{1}{2}


Question: Find the locus of the point of intersection of lines given by the equation \( y + 2at = t(x - at^2) \), where \( t \) is a parameter, such that the lines are at right angles.

Answer: y^2 = a(x - 3a)


Question: Consider the modular surface $M = \mathbb{H}/SL(2,\mathbb{Z})$. Determine whether every bounded geodesic on $M$ is closed.

Answer: No


Question: Determine the radius of convergence for the series \( \sum x^{n!} \).

Answer: 1


Question: Determine the values of $p$ for which the series \( \sum_{n=1}^\infty \frac{\ln(n)}{n^p} \) is convergent.

Answer: p > 1


Question: Calculate the complex line integral of \( f(z) = z^{-1} \) around the square with sides of length 2, centered at the origin, oriented counterclockwise. What is the value of this integral?

Answer: 2\pi i


Question: Does the flow of a smooth vector field on a closed smooth manifold exist for all time $t \in \mathbb{R}$?

Answer: Yes


Question: Calculate the expected value \( \mathbb{E}\left(\frac{X_1}{X_1 + X_2 + X_3}\right) \) where \( X_1, X_2, X_3 \) are independent and identically distributed exponential random variables with parameter \( \lambda \).

Answer: \dfrac{1}{3}


Question: Given the differential equation \( y'(x) = y(x) + \int_0^1 y \, dx \) with the initial condition \( y(0) = 1 \), find the value of \( y(\ln\frac{11-3e}{2}) \).

Answer: 4


Question: Find the smallest integer \( n \) such that \( n^n \) does not divide \( 2016! \).

Answer: 47


Question:
Let $x_0, x_1, \dots, x_n$ be distinct real numbers and define $\omega(x) = (x-x_0)(x-x_1)\dots(x-x_n)$. Determine the value of the sum:
\[\sum_{k=0}^{n}\frac{\omega''(x_k)}{\omega'(x_k)}\]

Answer: 0


Question: Given two non-zero square matrices \( A \) and \( B \) such that \( A^2 B = BA \), find the value of \( k \) if \( (AB)^{10} = A^k B^{10} \).

Answer: 1023


Question: Let \( \M \) and \( \N \) be smooth, compact, connected, oriented manifolds of the same dimension. Consider a sequence of smooth orientation-preserving embeddings \( f_n: \M \to \N \) that converges uniformly to a smooth immersion \( f: \M \to \N \). Is it true that for every point \( q \in \N \), the inverse image \( f^{-1}(q) \) is finite?

Answer: Yes


Question: Consider a number field $K/\mathbb{Q}$ and the embedding of $K^* \hookrightarrow GL_n(\mathbb{Q})$. This is the set of rational points of a $\mathbb{Q}$-algebraic group $G \subseteq GL_n(\mathbb{C})$. Is it true that any $\mathbb{Q}$-character of $G$ is of the form $g \mapsto \det(g)^k$ for some $k \in \mathbb{Z}$? In other words, on $G_\mathbb{Q} = K^*$, does the character look like $x \mapsto N_\mathbb{Q}^K(x^k)$?

Answer: Yes


Question: Find all integer values of $a$ such that the polynomial $x^2 - x + a$ divides $x^{13} + x + 90$. Provide your answer as a list of integers.

Answer: 2


Question: Calculate the volume of the region that lies outside the cone defined by \( \varphi = \frac{\pi}{4} \) and inside the sphere defined by \( \rho = 4\cos(\varphi) \).

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


Question: Evaluate the integral \( \iiint_E z \,{\rm d}v \), where \( E \) is the region between the spheres \( x^2 + y^2 + z^2 = 1 \) and \( x^2 + y^2 + z^2 = 3 \) in the first octant.

Answer: \dfrac{\pi}{2}


Question: Determine whether there exists a nonzero smooth function \( x: \mathbb{R} \to \mathbb{R} \) such that for all \( k \in \mathbb{N}_0 \), there exists a constant \( M > 0 \) with \( |\rD^k x(t)| \le M \) for all \( t \in \mathbb{R} \) and \( \lim_{s \to \pm\infty} |\rD^k x(s)| = 0 \).

Answer: Yes


Question: Determine whether there exists an infinite graph $G = (V,E)$ such that the chromatic number $\chi(G)$ is finite, but the coloring number $\text{Col}(G)$ is infinite.

Answer: Yes


Question: Find a lower bound for the probability \( P(25 \le \sum_{i=1}^{30}X_{i} \le 35) \), where \( X_{1},...,X_{30} \) are independent Poisson random variables with mean 1, using Chebyshev's inequality.

Answer: 0


Question:
Determine the nature of the set of all continuous functions \( f:[0,1]\rightarrow \mathbb{R} \) satisfying \( \int_0^1 t^n f(t) \, dt=0 \) for all \( n=1,2,\ldots \). Is this set:

(a) empty,
(b) containing a single element,
(c) countably infinite, or
(d) uncountably infinite?

Answer: b


Question: In a 60-minute match, goals are scored according to a Poisson process with an average time between goals of 15 minutes (i.e., \(\lambda = \frac{1}{15}\)). Assuming that at least three goals are scored in the game, what is the mean time at which the third goal is scored?

Answer: 33.46


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

Answer: -2i


Question: Find all bijections \( f: [0,1] \rightarrow [0,1] \) that satisfy the functional equation \( f(2x - f(x)) = x \) for all \( x \in [0,1] \).

Answer: f(x) = x


Question: Evaluate the limit of the integral \( \int_{0}^{\pi}\frac {\sin x}{1+\cos^2(nx)}\mathrm{d}x \) as \( n \to +\infty \).

Answer: \sqrt{2}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_{\mathbb{R}} \frac{1}{n} \, \mathrm{d}m \] where the integral is a Lebesgue integral over the real numbers.

Answer: \infty


Question: Compute the degree \( d(f_k) \) of the map \( f_k: S^1 \rightarrow S^1 \) defined by \( f_k(z) = z^k \) for any integer \( k \).

Answer: k


Question: In the Zariski topology on the $n$-dimensional affine space, can a closed set other than the entire space contain a non-empty open set? Provide a justification for your answer.

Answer: No


Question:
Simplify the expression involving inverse trigonometric functions:
\[ \left( \frac{1}{y^2} \left( \frac{\cos(\tan^{-1} y) + y \sin(\tan^{-1} y)}{\cot(\sin^{-1} y) + \tan(\sin^{-1} y)} \right)^2 + y^4 \right)^{\frac{1}{2}} \]
What is the simplified value of this expression?

Answer: 1


Question: Find the trace of the matrix \( x(x^\top x)^{-1} x^\top \), where \( x_i^\top = (1, x_{1,i}, \ldots, x_{d,i}) \in \mathbb{R}^{1 \times (d+1)} \) for \( i \in \{1, \ldots, n\} \) are linearly independent, and \( x \in \mathbb{R}^{n \times (d+1)} \) with the \(i\)-th row of \(x\) equal to \(x_i^\top\).

Answer: d+1


Question: Find the value of $k$ if the angle between the planes given by the equation $6x^2+4y^2-10z^2+3yz+4zx-11xy=0$ is $\cos^{-1}(k)$. What is $k$?

Answer: 0


Question: Let $X$ be a separable bounded metric space, and let $A$ and $B$ be subsets of $X$ such that $d(A, B) > 0$. Does there exist sets $A'$ and $B'$ such that each is a finite union of open balls, $A' \cap B' = \emptyset$, $A \subset A'$, and $B \subset B'$?

Answer: No


Question: Determine whether Alice has a winning strategy for odd values of \( n \) in the following game: There are \( n \ge 2 \) points around a circle. Alice and Bob take turns coloring the points, with Alice starting. Alice colors a point red, and Bob colors a point blue. After all points are colored, Alice finds the maximum number of consecutive red points, \( R \), and Bob finds the maximum number of consecutive blue points, \( B \). Alice wins if \( R > B \), Bob wins if \( B > R \), and it's a draw if \( R = B \). Does Alice have a winning strategy for odd \( n \)?

Answer: Yes


Question: Find the limit of the sequence \( \left[ \bigg(1+\frac{1}{n}\bigg)\bigg(1+\frac{2}{n}\bigg)\bigg(1+\frac{3}{n}\bigg)\cdots\bigg(1+\frac{n}{n}\bigg) \right]^{1/n} \) as \( n \to \infty \).

Answer: \dfrac{4}{e}


Question: Determine the number of ordered pairs \((F_1, F_2)\) such that \(F_1 \not\subseteq F_2'\), where \(F_1, F_2 \subseteq \{a, b, c, d\}\) and \(F_2' = F_2 \cup \{c \mid d \in F_2\}\).

Answer: 157


Question: Let $S$ be the set of all continuous real functions $f(x)$ on the interval $[0,1]$ such that \(\lim_{x\rightarrow 0}f(x)/x^k=0\) for \(k=1,2,...\). Prove or disprove: 0 is the only common root of the elements of $S$. Provide a justification for your answer.

Answer: 0


Question: If for every $y\in\mathbb{R}^N$,  $$\int_{y+B_R}|w_n|^2dx \rightarrow 0 ,$$ as $n\rightarrow \infty$, where $R>0$ is some number and $B_R$ denotes the ball of radius $R$ at the origin, does it follow that  $\displaystyle\sup_{y\in \mathbb{R}^N}\int_{y+B_R}|w_n|^2dx\rightarrow 0$ as $n\rightarrow \infty$?

Answer: No


Question: Evaluate the limit: \[ \lim_{n\to\infty} \left( \frac {c_n}{n} - \frac{c_n}{p_n} - \frac {c_n}{n^2} \right) \] where \( c_n \) is the nth composite number (excluding 1) and \( p_n \) is the nth prime number.

Answer: 1


Question:
Suppose $f : \mathbb{R} \to \mathbb{R}$ is continuous. Fix $a \in \mathbb{R}$ and define
$$
F(x) := \int_a^x f(t) \, \mathrm{d}t.
$$
Is it true that $F'(x) = f(x)$ for all $x < a$? Justify your answer.

Answer: Yes


Question: Find the digit at the hundred's place of \(33^{33}\).

Answer: 5


Question: Evaluate the line integral \( \int_{\alpha} f \, ds \) where \( f(x, y) = x^2 y^2 \) and the path \( \alpha(t) = (\cos t, \sin t) \) for \( 0 \leq t \leq 4\pi \).

Answer: \dfrac{\pi}{2}


Question: Suppose $Q$ is an $m \times n$ matrix where $m > n$ with orthonormal columns. Determine whether $QQ^T = I$ is true or false.

Answer: B


Question:
Consider a function \( f: \mathbb{R}^7 \to \mathbb{R} \) that is four times continuously differentiable (\( f \in C^4 \)). Determine the number of distinct elements in the set of fourth-order partial derivatives of \( f \) at the origin, given by:
\[ \left\{ \frac{\partial^4 f}{\partial x_{i_4} \partial x_{i_3} \partial x_{i_2} \partial x_{i_1}}(0) : i_1, i_2, i_3, i_4 \in \{1, 2, 3, 4, 5, 6, 7\} \right\} \]
where the order of differentiation does not matter. Express your answer as a binomial coefficient.

Answer: \dbinom{10}{4}


Question: Consider a non-oriented Riemannian manifold $M$ of dimension $m$. Using the Nash embedding theorem, there exists an isometric embedding $\phi: M \longrightarrow \mathbb{R}^n$ for some sufficiently large $n$. The Lebesgue measure on $\mathbb{R}^n$ induces a measure on the $m$-submanifold $\phi(M)$, and hence on $M$. If $\phi$ and $\psi$ are two such isometric embeddings of $M$ into $\mathbb{R}^n$ and $\mathbb{R}^N$ respectively, the induced measures from $\mathbb{R}^n$ and $\mathbb{R}^N$ are the same due to the isometries. Does this argument correctly establish a well-defined measure on the non-oriented manifold $M$?

Answer: Yes


Question: Determine whether there exists a non-trivial group homomorphism \( f: \mathbb{R} \rightarrow \mathbb{Z} \). Specifically, is there an \( a \in \mathbb{R} \) such that \( f(a) \neq 0 \)?

Answer: No


Question: Determine whether the supremum of the expression \( \min_{n\ge 1} \frac{1-x}{(1-xy)x^{n-1}(1-y^n)} \) over all \( x, y \in (0,1) \) is finite or infinite.

Answer: \infty


Question: Given two polynomials \( f_a(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_0 \) and \( f_b(x) = b_{n-1} x^{n-1} + \cdots + b_0 \) with coefficients in \( \mathbb{R} \), where \( f_a \) and \( f_b \) are coprime in \( \mathbb{C} \), determine if there exists a scalar \( s \in \mathbb{R} \) such that the polynomial \( f_a(x) + s f_b(x) \) has distinct roots in \( \mathbb{C} \).

Answer: Yes


Question: Evaluate the integral \( \int_{C} \frac{\cosh(4z)}{(z-4)^3} \, dz \), where \( C \) consists of the contour \( |z|=6 \) oriented counterclockwise and the contour \( |z-3|=2 \) oriented clockwise. Note that \( z=4 \) is inside \( |z-3|=2 \) and not inside \( C \).

Answer: 0


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

Answer: \dfrac{\pi}{6}


Question: Given the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(f(x)) = 3 + 2x$ and the condition $f(0) = 3$, find the value of $f(3)$. \( \boxed{} \)

Answer: 9


Question: Evaluate the limit \( \lim_{r\rightarrow +\infty} \int_{B(0,r)} \frac{y}{1+\sqrt{(x^2+y^2)^5}} \, dx \, dy \). Use polar coordinates to simplify the integral and determine if the limit exists.

Answer: 0


Question: Find a function \( f(x) \) such that \( \int_{1}^{\infty} f(x) \, dx \) converges, but \( \lim_{x \to \infty} f(x) \) does not exist.

Answer: \sin(x^2)


Question: If $E$ and $F$ are two normed spaces with $E \neq \{0\}$, and $L(E,F)$, the set of all bounded linear maps from $E$ to $F$, is a complete normed space under the operator norm, is $F$ necessarily complete?

Answer: Yes


Question: Classify the singularity of the function \( f(z) = \left(\frac{\sin 3z}{z^2}-\frac{3}{z}\right) \) at \( z=0 \) and calculate its residue.

Answer: 0


Question: Let $(G, \ast)$ be a group with identity $e$ and order $2n$, where $n$ is a positive integer. Does there exist an element $x \in G$ such that $x \ast x = e$ and $x \neq e$?

Answer: Yes


Question: Determine the set of values for which the series \( \sum_{n=1}^\infty \left(\frac{n^2+1}{3n^2+4}\right)^n(x-2)^n \) converges.

Answer: (-1, 5)


Question: Determine whether \( \lfloor \sqrt{n!} \rfloor \) can be a prime number for any integer \( n > 1 \).

Answer: Yes


Question: In how many ways can you assign integer grades to 5 students such that their average grade is 60, with each grade being between 0 and 100 inclusive?

Answer: 47952376


Question: Evaluate the limit: \[ \lim_{n \to \infty} n \int_1^e x^a (\log (x))^n \,dx. \]

Answer: e^{a+1}


Question:
Determine the correct bound for the norm of the operator \( T: l_2 \to l_2 \) defined by \( T((x_1,x_2,\ldots,x_n,\ldots))=(x_2-x_1,x_3-x_2,\ldots,x_{n+1}-x_n,\ldots) \). Which of the following is true?

- \( \|T\|=1 \)
- \( \|T\|\geq2 \)
- \( 1<\|T\|\leq2 \)
- None of the above.

Answer: D


Question: Given a positive number $d \in \mathbb{R}$ and a sequence of numbers $a_n \in [0,d]$ for $n \in \mathbb{N}$ with the properties that \( \sum_{i=1}^{\infty} a_i^r \in \mathbb{Z} \) for all \( r \in \mathbb{N} \) and \( \sum_{i=1}^{\infty} a_i \leq d \), determine whether it follows that only finitely many of the $a_i$ are non-zero.

Answer: Yes


Question: Find the smallest value of \( T \) such that \( |y(x)| \leq 0.1 \) for all \( x > T \), where \( y(x) = e^{-\frac{x}{5}} \left[2\cos\left(\frac{\sqrt{34}}{5} x\right) + \frac{7}{\sqrt{34}}\sin\left(\frac{\sqrt{34}}{5}x\right)\right] \).

Answer: 15.75


Question:
Let $A$ be a $10 \times 10$ matrix with complex entries, where all eigenvalues are non-negative real numbers and at least one eigenvalue is strictly positive. Determine which of the following conditions can be satisfied by some matrix $B$:

A. $AB - BA = B$
B. $AB - BA = A$
C. $AB + BA = A$
D. $AB + BA = B$

Answer: ACD


Question: Express the sum \( \text{cis}75^\circ + \text{cis}83^\circ + \text{cis}91^\circ + \dots + \text{cis}147^\circ \) in the form \( r \, \text{cis } \theta \), where \( 0 \le \theta < 360^\circ \). Find the value of \( \theta \) in degrees.

Answer: 111


Question: Find the degree of the field extension \( \mathbb{Q}(\zeta_9 + \zeta^{-1}_9) \) over \( \mathbb{Q} \). Use the concepts of the tower of extensions and minimal polynomials, without employing Galois theory.

Answer: 3


Question: Let \( T \) be a compact selfadjoint operator on a separable Hilbert space. Given that the diagonal of \( T \) with respect to some fixed orthonormal basis is in \( \ell^1(\mathbb{N}) \), determine whether \( T \) is a trace class operator.

Answer: Yes


Question: Determine the number of elements in the set \( \{z \in \mathbb{C} \mid z^{60} = -1, z^k \neq -1 \text{ for } 0 < k < 60\} \).

Answer: 32


Question: Calculate the expected value of a continuous random variable $X \sim \mathcal{N}(3.85, 1.87^2)$ over the interval from 1.1 to 6.6.

Answer: 3.85


Question: Find the value of \(x\) that minimizes the function \(f(x) = \int_{0}^{1} |t-x|^3 dt\).

Answer: \dfrac{1}{2}


Question: Evaluate the line integral \( \int_\gamma \frac{-y^2 \, dx + 2xy \, dy}{x^2 + y^4} \) along the path \( \gamma: r(t) = (t, 2t^2 - 2) \) for \( -1 \leq t \leq 1 \).

Answer: -\pi


Question: Let $\sim$ be an equivalence relation on $[0,1)$ defined by $x \sim y$ if $x - y$ is rational. Let $N$ be a set containing exactly one element from each equivalence class of this relation. If $E$ is a Lebesgue measurable subset of $N$, prove that $E$ has measure 0.

Answer: 0


Question:
Determine the value of \( p \) such that the function \( f(x) \) is differentiable at \( x = 0 \), where
\[
f(x) = \begin{cases}
e^{−\frac{1}{|x|}} & \text{if } x \ne 0\\
p & \text{if } x = 0
\end{cases}.
\]

Answer: 0


Question: Given a bounded and measurable set \(K \subset \mathbb{R}^N\), does there exist a smooth cutoff function \(\eta \in C^\infty(\mathbb{R}^N)\) such that \(\eta = 1\) on \(K\) and zero outside a neighborhood of \(K\)?

Answer: Yes


Question: Evaluate the integral: \[ \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, \mathrm{d}x. \]

Answer: \dfrac{\pi}{4}


Question: Let \(f\) and \(g\) be two co-prime polynomials in \(\Bbb{C}[x,y]\). Can the expression \(af + bg = 1\) always be written, where \(a, b \in \Bbb{C}[x,y]\)? Justify your answer.

Answer: No


Question: Determine the class equation of the tetrahedral group, which consists of 1 element of order 1, 8 elements of order 3, and 3 elements of order 2. Provide the class equation without using brute force on $A_4$.

Answer: 1 + 3 + 4 + 4


Question: Evaluate the limit \( \lim_{t\rightarrow 0}\frac{1}{t}\int_{0}^{1}x(f(x+t)-f(x))\, dx \) given that \( f:[0,1]\rightarrow \mathbb{R} \) is a continuous function with \( \int_{0}^{1}f(x)dx=0 \).

Answer: f(1)


Question: Determine whether the sequence \( x_n = \frac{1}{n^2} \sum_{k=1}^n \frac{k^2}{(k+1)\log (k+1)} \) converges to 0 as \( n \to \infty \).

Answer: 0


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

Answer: e


Question: Evaluate the limit: $$\lim_{x\to\pi/4} \frac{\ln(\tan(x))}{x-\pi/4}$$

Answer: 2


Question: Is the matrix logarithm defined for all matrices that can be expressed in the form $e^X$?

Answer: Yes


Question: Given that $33335^2 \equiv 670705093^2 \pmod{670726081}$, find a nontrivial factor of $670726081$ using the information that $\gcd(33335 + 670705093, 670726081)$ is a nontrivial factor.

Answer: 12347


Question: Determine whether the following statement is true for all $n > 1$: For the symmetric group $S_n$, the sum of the signatures of all elements is zero, i.e., \( \sum_{g \in S_n} \varepsilon(g) = 0 \), where \( \varepsilon \) is the signature function.

Answer: True


Question: Integrate \( f(x) = \operatorname{Im} z \) counterclockwise around the unit circle.

Answer: -\pi


Question: Is every monoid isomorphic to a submonoid of a full transformation monoid? A full transformation monoid is defined as the set of all functions from a set $X$ to $X$, with function composition as the operation and the identity function as the identity element.

Answer: Yes


Question: Find a tight upper bound for the function \( f(m,n) \) given the following conditions: \( f(m,n) \leq \frac{f(m-1,n-1) + f(m,n-2)}{2} \), \( f(m,m-r) = 0 \) for integer \( r \geq 1 \), and \( f(0,n) = 1 \).

Answer: 2^{-m}


Question:
Let $Z$ be a random variable with a continuous distribution having density
$$
f_{Z}(z)=\begin{cases}
2z & \text{if } 0 \leq z \leq 1 \\
0 & \text{otherwise}
\end{cases}
$$
Let $X$ and $Y$ be two random variables, independent given $Z=z$, satisfying
$$
\mathbf{P}(X=1 \mid Z=z)=z=1-\mathbf{P}(X=0 \mid Z=z)
$$
and
$$
\mathbf{P}(Y=1 \mid Z=z)=z=1-\mathbf{P}(Y=0 \mid Z=z)
$$
Find the conditional probability
$$
\mathbf{P}(Z \leq 1/2 \mid X=1 \text{ and } Y=1).
$$

Answer: \dfrac{1}{16}


Question: The plane \(P_1 : 4x + 7y + 4z + 81 = 0\) is rotated through a right angle about its line of intersection with the plane \(P_2 : 5x + 3y + 10z = 25\). If the plane in its new position is denoted by \(P\), and the distance of this plane from the origin is \(\sqrt{d}\), find \(d\).

Answer: 212


Question: The tangent at the point \( P(x_0, y_0) \) to the parabola \( y^2 = 4ax \) meets the parabola \( y^2 = 4a(x+b) \) at points \( Q \) and \( R \). Find the coordinates of the midpoint of \( QR \).

Answer: (x_0, y_0)


Question: Find the value of \( t_0 \) such that the tangent line to the curve \( c(t) = (2t, t^2, -\frac{t^3}{3}) \) at \( c(t_0) \) is parallel to the line given by the equations \( x = -8z \) and \( x = 2y \).

Answer: \dfrac{1}{2}


Question: Given an $n$-dimensional vector space $V$ with basis $\{e_i\}$, consider the Young diagram $\lambda$ for the partition $1+1+\ldots+1=n$ and the corresponding Young symmetrizer $c_{\lambda}$. Let $\hat{c}_{\lambda}$ be the scalar multiple of $c_{\lambda}$ such that $\hat{c}_{\lambda}$ is idempotent. Show that the space $V^{\otimes n}.\hat{c}_{\lambda}$ is 1-dimensional.

Answer: 1


Question: Suppose $(X_1,X_2)$ is independent of $(Y_1,Y_2)$. Does it follow that $X_1$ is independent of $Y_1$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{n\to\infty}\frac {1-\frac {1}{2} + \frac {1}{3} -\frac {1}{4}+ ... + \frac {1}{2n-1}-\frac{1}{2n}}{\frac {1}{n+1} + \frac {1}{n+2} + \frac {1}{n+3} + ... + \frac {1}{2n}}$$

Answer: 1


Question: Let \( V = L(\mathbb{R}^5, \mathbb{R}^5) \) be the real vector space of linear operators on \( \mathbb{R}^5 \). Given a nonzero vector \( x \in \mathbb{R}^5 \), define the subspace \( W = \{ T \in V \mid Tx = 0 \} \). Determine the dimension of \( W \).

Answer: 20


Question: Given an angle \( \angle AOB \) less than \( 180^\circ \) and a point \( P \) inside the angular region, construct a line segment \( CD \) passing through \( P \) such that \( C \) lies on ray \( OA \), \( D \) lies on ray \( OB \), and the segment \( CP : PD = 1 : 2 \). Provide a proof for your construction using only a ruler and compass.

Answer: CD


Question:
Consider the function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $|f(x)|\leq\frac{1}{1+2x^{2}}$ for all $x\in\mathbb{R}$. Define the functions $f_{n}(x)=f(x+n)$ for $n\in\mathbb{N}$. Determine whether the series $\sum_{n=1}^{\infty}f_{n}(x)$ converges uniformly on the intervals $[0,1]$ and $[-1,0]$. Choose the correct option:

A. The series converges uniformly on $[0,1]$ but not on $[-1,0]$.
B. The series converges uniformly on $[-1,0]$ but not on $[0,1]$.
C. The series converges uniformly on both $[0,1]$ and $[-1,0]$.
D. The series converges uniformly on neither $[0,1]$ nor $[-1,0]$.

Provide a justification for your choice.

Answer: C


Question: Find the interval of values for the parameter \( a \) such that the expression \( \sqrt{\log_4(\log_3(\log_2(x^2-2x+a)))} \) is defined for all real numbers \( x \).

Answer: [9, \infty)


Question: Given the equation \( f(x+\frac{y}{2})-f(x-\frac{y}{2})=2x^2y+5y^2 \), find the derivative \( f'(3) \). Assume \( y \) is a constant.

Answer: 18


Question: Evaluate the limit: \( \lim_{x\to 0^+}(x^{x}-1)\ln(x) \) using L'Hospital's rule.

Answer: 0


Question: Solve the equation $x^{1/4} = -1$ for $x$.

Answer: 1


Question: Calculate the mass of the surface defined by \( z = \sqrt{2xy} \) over the region \( 1 \leq x \leq 5 \) and \( 1 \leq y \leq 4 \), given that the density function is \( \sigma(x,y,z) = 6z \).

Answer: 396


Question: Evaluate the infinite series: \[ \sum_{i=2}^\infty \frac{i^2}{i!} \] and show that it equals \( 2e - 1 \).

Answer: 2e - 1


Question: Compute the element of the fundamental group of a Möbius strip represented by the loop \((\cos 10\pi t, \sin 10\pi t)\).

Answer: 5


Question: Find the minimum integer value of \( n \) such that if \( a^n \mid b^3 \), then \( a^3 \mid b^2 \).

Answer: 5


Question: Determine whether the 2nd Eisenstein Series, defined by \(E_{2}(\tau)=1-24\sum_{n=0}^{\infty} \frac{nq^{2n}}{1-q^{2n}}\), where \(q=e^{i\pi \tau}\) and \(\tau\) is in the upper-half plane, can output complex numbers.

Answer: Yes


Question: Determine which is greater: $100^{101}$ or $101^{100}$, without using a calculator or logarithms.

Answer: 100^{101}


Question: Prove that for every real number \(x\) and every integer \(n\), the following identity holds: \[ \sum_{k=0}^{2n} \binom{2n}{k} \sin((n-k)x) = 0. \]

Answer: 0


Question: Given a circle of radius 1, starting from a fixed point, points are placed along the circumference at curved distances of 0, 1, 2, ..., respectively. Determine the minimum number of these points needed to ensure that at least two of them are less than a distance of \( \frac{1}{5} \) apart.

Answer: 20


Question: Let $X$ be a real Banach space and $f: \mathbb{R} \rightarrow X$ be a function that is continuous on $\mathbb{R}$, of class $C^1$ on $\mathbb{R} \setminus \{0\}$, and satisfies $\lim_{x \rightarrow 0} f'(x) = g$ for some limit $g$. Is $f$ of class $C^1$ on $\mathbb{R}$?

Answer: Yes


Question: Suppose $F$ is a field and $\overline{F}$ is its algebraic closure. Let $I \subseteq \overline{F}[x_0, \ldots, x_d]$ be a homogeneous ideal generated by homogeneous polynomials. If $I$ can be generated by elements of $F[x_0, \ldots, x_d]$, must it also be possible to generate $I$ by homogeneous elements of $F[x_0, \ldots, x_d]$?

Answer: Yes


Question:
The tangent line to the curve of intersection of the surface \(x^2+y^2=z\) and the plane \(x+z=3\) at the point \((1,1,2)\) passes through which of the following points?

(A) \((-1,-2,4)\)

(B) \((-1,4,4)\)

(C) \((3,4,4)\)

(D) \((-1,4,0)\)

Answer: B


Question:
Given a positive semi-definite matrix \( X \), with its singular value decomposition (SVD) expressed as:
\[
X = U \begin{bmatrix}\Omega & 0 \\ 0 & 0 \end{bmatrix} U^T,
\]
where \( U \) is an orthogonal matrix and \( \Omega \) is positive definite, determine if the matrix \( Y \) given by:
\[
Y = U \begin{bmatrix}\Omega^{-1} & 0 \\ 0 & 0 \end{bmatrix} U^T,
\]
is the unique Moore-Penrose pseudo-inverse of \( X \).

Answer: Yes


Question: Prove that \( \lim_{n\to\infty}n\left(a^{\frac{1}{n}}-1\right)=\ln a \) using only sequence limits, without involving the limit of a function.

Answer: \ln a


Question: Determine whether the equation $x \cdot \tanh(x) = C$ admits a real solution for a given constant $C$.

Answer: C \geq 0


Question: In triangle $PAT$, where $\angle P = 36^\circ$, $\angle A = 56^\circ$, and $PA = 10$, points $U$ and $G$ lie on sides $TP$ and $TA$ such that $PU = AG = 1$. Let $M$ and $N$ be the midpoints of segments $PA$ and $UG$, respectively. Calculate the measure in degrees of the acute angle $\angle NMA$. \( \text{Provide your answer in degrees.} \)

Answer: 80


Question: Evaluate the limit: \[ \lim_{x \to 0} \left( \cos(xe^x) - \ln(1-x) - x \right)^{1/x^3} \]

Answer: e^{-\frac{2}{3}}


Question: Let $M$ be a finite module over a Noetherian ring $A$. Consider a minimal injective resolution $0 \rightarrow M \rightarrow I^1 \rightarrow I^2 \rightarrow \cdots$. Is each $I^k$ a finite $A$-module?

Answer: No


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

Answer: 0


Question: Let \( \beta = zdx \wedge dy + xzdx \wedge dz \) be a differential 2-form on \( \mathbb{R}^3 \), and let \( \mathbb{X} = (0, -x, -1) \) be a vector field on \( \mathbb{R}^3 \). Compute the contraction \( i_\mathbb{X}\beta \) and simplify the expression by combining terms where possible.

Answer: 2xz\,dx


Question:
Consider the discrete-time state-space system:
\[x(t+1) = Ax(t)+Bu(t), \quad y(t) = Cx(t)\]
with
\[
A =\begin{bmatrix}
0 & 1 & 0 \\
1 & 0 & 1 \\
1 & 1 & 0 \\
\end{bmatrix},
\quad B =\begin{bmatrix}
0 \\
0 \\
1 \\
\end{bmatrix},
\quad C =\begin{bmatrix}
1 & 0 & 0\\
\end{bmatrix}
\]
Determine the minimal time \(T \in \mathbb{N}\) such that there exists an input signal \(u : \mathbb{N} \rightarrow \mathbb{R}\) for which the state \(x(T,0,u) = x_{f}\) with \(x_{f} = \begin{bmatrix} 0 & 1 & 1\end{bmatrix} ^{T}\).

Answer: 2


Question:
Let $f(x) = x^4 + ax^3 + bx^2 + cx + d$ be a polynomial with real coefficients. If $f'(x)$ has three distinct real roots, determine which of the following statements is true:

(a) There always exists a real constant $k$ such that $f(x) = k$ has exactly one real root.
(b) There always exists a real constant $k$ such that $f(x) = k$ has three distinct real roots.
(c) $f(x) = k$ has four distinct real roots for any real number $k$.
(d) $f(x) = k$ has two real and two complex roots for any real number $k$.

Choose the correct option.

Answer: B


Question: Evaluate the limit: \[ \lim_{n\to\infty}\dfrac{\ln(n+1) \sin(n)}{n \tan(n)} \]

Answer: 0


Question: Verify that the discriminant of the cubic equation \(x^3 + px + q = 0\) is given by \(\Delta = -4p^3 - 27q^2\).

Answer: -4p^3 - 27q^2


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

Let \(f_n\) be a sequence of non-negative functions in \(L^1(X, m)\), where \((X, \Sigma, \mu)\) is a probability space, such that \(\|f_n - f\|_{L^1(X)} \to 0\) for some \(f \in L^1(X)\). Then there exists a subsequence \(f_{n_k}\) and a function \(g \in L^1(X, \mu)\) such that
\[f_{n_k}(x) \le g(x) \quad \text{for \(\mu\)-a.e. } x \in X.\]

Answer: True


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{\cos x}{ (x^2 +1)^2}\, dx \) using contour integration.

Answer: \dfrac{\pi}{e}


Question: Determine whether there exists an infinite Hausdorff space \((X, \tau)\) such that for every proper equivalence relation \(\vartheta\) on \(X\), the space \(X\) is not homeomorphic to the quotient space \(X/\vartheta\).

Answer: Yes


Question: Let \( \phi \) be an automorphism of the quaternion group \( \mathcal{Q}_8 \) such that \( \phi(i) = -i \) and \( \phi(j) = -k \). Determine the order of \( \phi \).

Answer: 2


Question: Let \( F \) be a metrizable locally convex space, assumed to be a Banach space, and let \( E \) be a complete locally convex space, assumed to be a Fréchet space. Consider a continuous linear map \( T: F \to E \) and a subspace \( H \subset E^{*} \) such that \( \overline{T^*H}^{F^*} = \overline{T^*E^*}^{F^*} \). Let \( B \subset F \) be an absolutely convex bounded closed set, assumed to be the unit ball. Does the \( \sigma(E,H) \)-compactness of \( \overline{TB}^E \) imply that \( \overline{TB}^E \) is \( \sigma(E,E^{*}) \)-compact?

Answer: Yes


Question: Does every graph arise as the commutativity graph of some group? A graph is defined as a set $G$ with a relation $\bot$ that is reflexive and symmetric. For a group, the commutativity graph is defined by $x \,\bot\, y \iff xy=yx$. Determine if every graph can be represented in this way.

Answer: No


Question: Let \(X_1, X_2, \dots\) be independent and identically distributed real random variables with \(\operatorname{Var}(X_1) = 1\) and \(E(X_1) = 0\). Define the random walk \(S_n = X_1 + X_2 + \cdots + X_n\) for \(n \in \mathbb{N}\). Prove that \(\lim\limits_{n \to \infty} E(|S_n|) = \infty\).

Answer: \infty


Question: Evaluate the integral \( \oint \frac{z-7}{z^{23}-1} \, \mathrm{d}z \) over the circle of radius 3 centered at the origin.

Answer: 0


Question: Evaluate the integral \(2\int_{-2}^2\int_0^{\sqrt{4-x^2}}(4-y)\,dx\,dy\) to find the volume of the cylinder defined by \(x^2+y^2=4\) and cut by the plane \(z=4-y\). Verify if the integral evaluates to \(16\pi\).

Answer: 16\pi