Question: Evaluate the line integral of the vector field \( F(x,y) = \langle \cos x, \sin x \rangle \) along the path \( C \), which is the triangle with vertices \((0,0)\), \((\pi/3,0)\), and \((0,\pi)\). Use a direct parameterization method without applying Green's Theorem.

Answer: \dfrac{3}{2}


Question: Solve the recurrence relation for the time complexity: \[ T(n) = T(\sqrt{n}) + n^2 \]

Answer: O(n^2)


Question:
Estimate \( \ln(3) \) using the Taylor expansion of \( \ln(x) \) about \( a = 2 \) up to the third order. Use the derivatives:

\[ f(x) = \ln(x) \]
\[ f'(x) = \frac{1}{x} \]
\[ f''(x) = -\frac{1}{x^2} \]
\[ f'''(x) = \frac{2}{x^3} \]

Apply the Taylor series formula:

\[ f(x) = f(a) + (x-a)f'(a) + \frac{(x-a)^2}{2!}f''(a) + \frac{(x-a)^3}{3!}f'''(a) + \ldots \]

Calculate the estimate for \( \ln(3) \) without using a calculator.

Answer: \ln(2) + \dfrac{5}{12}


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

Answer: \dfrac{1}{2}


Question: Given 4 distinct positive integers, how many ways are there to sum them, considering the order of addition?

Answer: 120


Question: Show that the sequence \(a_n = \left(\frac{n^2+8n-1}{n^2-4n-5}\right)^n\) converges to \(e^{12}\).

Answer: e^{12}


Question: Determine whether the element $x \in K(x)$ is algebraic over the field $K(x)^p$, where $K$ is a field with characteristic $p > 0$ and $K(x)$ is the field of rational functions over $K$. Provide a justification for your answer.

Answer: Yes


Question:
Determine the total variation of the two-variable step function given by:
\[ f(x,y) = \begin{cases} 1 & \text{when } 0<x<a,\ 0<y<b; \\\ 0 & \text{otherwise} \end{cases} \]

Answer: 2(a + b)


Question: Evaluate the surface integral \( \iint_S \vec{F} \cdot d\vec{S} \), where \( \vec{F} = y\hat{i} + x\hat{j} + zy\hat{k} \) and \( S \) is the part of the surface \( z = x^2 + y^2 \) that lies above \( z = 1 \). Assume \( S \) has an upwards orientation.

Answer: 0


Question: Evaluate the integral \( \int_a^b \delta(t-a)e^{-iwt}dt \) and determine its result in terms of the sifting property of the Dirac delta function.

Answer: e^{-i\omega a}


Question: There are 16 people seated around a round table for a meeting. Every hour, a new session begins. In each session, individuals whose neighbors in the previous session were both sitting or both standing will sit, while those whose neighbors were in opposite states (one sitting, one standing) will stand. Determine the minimum number of sessions required to ensure that everyone is sitting at the table.

Answer: 4


Question: Find the smallest integer $k \geq 1$ such that the inequality \[ 1 - 3 \cdot 2^{1-4k^2} + 3 \cdot 2^{3-(2k+1)^2} > 0.5927 \] holds.

Answer: 1


Question: Use Stokes' theorem to evaluate the line integral \( \int_C{ \vec{F} \cdot \vec{dr}} \), where \( \vec{F}(x, y, z) = \langle 2 x^2 y , \frac{2 x^3}{3}, 2xy \rangle \) and \( C \) is the curve of intersection of the hyperbolic paraboloid \( z = y^2 - x^2 \) and the cylinder \( x^2 + y^2 = 1 \), oriented counterclockwise as viewed from above.

Answer: 2\pi


Question: Calculate the residue of the function \( z^2 e^{\frac{1}{z - 2}} \) at \( z = 2 \).

Answer: \dfrac{37}{6}


Question: Consider a 3x3 matrix with real entries. What is the probability that this matrix is non-singular?

Answer: 1


Question: Let \(a_n\) be a sequence of real numbers such that \[\lim_{n\to\infty}|a_n+3((n-2)/n)^n|^{1/n}=\frac{3}{5}.\] Determine \(\lim_{n\to\infty}a_n\).

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


Question: Determine the asymptotic number of non-equivalent weight functions on a set of \( n \) elements. Specifically, find the optimal degree of the polynomial in the exponent for the number of such functions, given that there are at most \( 2^{O(n^4)} \) distinct weight functions according to a result by Frank and Tardos.

Answer: 4


Question: Determine whether the nilpotency class of the group $G_d$, defined by the presentation \( \langle x,y \mid x^{2^{d+1}}=1, x^4=y^2, [x,y,x]=x^{2^{d}}, [x,y,y]=1\rangle \) for integer \( d > 2 \), is exactly 3.

Answer: 3


Question: Let $f$ be a non-invertible bounded outer function on the unit disk. Does $f$ have a radial limit of $0$ somewhere?

Answer: No


Question: Evaluate the expectation \( E[B(u)B(u+v)B(u+v+w)B(u+v+w+x)] \), where \( \{B(t); t \ge 0\} \) is a standard Brownian motion, and \( 0 < u < u+v < u+v+w < u+v+w+x \) with \( u, v, w, x > 0 \).

Answer: 3u^2 + 3uv + uw


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

Answer: 1


Question: Find a 4-tuple \((a, b, n, k) \in \mathbb{N}^4\) with \(a, b, n, k > 1\) and \(a \neq b\) such that the reverse of \(a^n\) is equal to \(b^k\).

Answer: (12, 21, 2, 2)


Question: Compute the path integral \( \int_\gamma \frac{1}{z+1} \, \mathrm{d}z \) where \( \gamma \) is a path from \(-i\) to \(i\) such that \( \gamma(t) \in \mathbb{C}\setminus(-\infty,-1] \) for all \( t \in [0,1] \).

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


Question: Evaluate the integral \( \iiint_{B} (3x^2 + y^2 + z^2 + 2) \, \mathrm{d}x \, \mathrm{d}y \, \mathrm{d}z \), where \( B \) is the unit ball defined by \( \{ (x, y, z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 < 1 \} \).

Answer: 4\pi


Question: Find all functions \( f: [0, 1] \rightarrow [0, 1] \) that satisfy the following conditions: \( f(0) = 0 \), \( f(1) = 1 \), and \( f(x+y) - f(x) = f(x) - f(x-y) \) for any \( x, y \geq 0 \) such that \( (x-y), (x+y) \in [0, 1] \).

Answer: f(x) = x


Question: Is the discrete topology admissible?

Answer: Yes


Question: Evaluate the integral \( \int_{-\pi}^{\pi} \cos(e^{it}) \, dt \).

Answer: 2\pi


Question: Find the number of integral solutions to the equation \(x^2 + y^2 = 10^{10}\).

Answer: 44


Question:
Given that $A, B, C$ are the roots of the equation $x^3 - 5x^2 + x + 1 = 0$, find the value of the expression:
\[ \frac{A}{B+C} + \frac{B}{A+C} + \frac{C}{A+B} \]

Answer: \dfrac{56}{3}


Question: Does there exist a left group that is not a right group?

Answer: Yes


Question: Evaluate the integral \( \oint_{|z-1|=1} \overline{z}^2 \, dz \), where the contour is a circle of radius 1 centered at \((1,0)\).

Answer: 4\pi i


Question: Determine whether the function \( l(z) = \text{Re}(z)^2 \) is differentiable at a point \( z_0 \in \mathbb{C} \). If it is differentiable, find \( l'(z_0) \). Additionally, identify the set of points in \( \mathbb{C} \) where \( l \) is differentiable.

Answer: 0


Question: For a 3x3 matrix \( A \) with the characteristic polynomial \( c^3 - 3c^2 + 4 = 0 \), find the trace of the adjoint of \( A \).

Answer: 0


Question: Find the minimum value of \(a^2+b^2+c^2+2abc\) given that \(a+b+c=3\) and \(a,b,c\geq0\).

Answer: \dfrac{9}{2}


Question: A point inside square ABCD is at distances 3 meters, 4 meters, and 5 meters from vertices A, B, and C, respectively. Determine the area of square ABCD.

Answer: 17 + 4\sqrt{14}


Question: Evaluate the infinite series: \[ \sum_{n=0}^{\infty}\frac{1}{(n^4+n^2+1)n!}. \]

Answer: \dfrac{e}{2}


Question: Evaluate the integral \( \iiint_R z \sqrt{x^2+y^2+z^2} \, dV \) where \( R \) is the solid hemisphere that lies above the \( xy \)-plane with center at the origin and radius of 1.

Answer: \dfrac{\pi}{5}


Question: Let \( f: D(0,1) \to \mathbb{C} \) be a holomorphic function, where \( D(0,1) = \{ z \in \mathbb{C} \mid |z| < 1 \} \), and assume that \( |f(z)| \leq 2 \) for all \( z \) in \( D(0,1) \). Estimate \( |f^{(3)}(i/3)| \).

Answer: \dfrac{81}{2}


Question:
Find the most general solution of the partial differential equation:
\[ 6\frac{\partial^2u}{\partial x^2} - 5\frac{\partial^2u}{\partial x\partial y} + \frac{\partial^2u}{\partial y^2} = 1 \]
by making the change of variables \( \xi = x + 2y \) and \( \eta = x + 3y \). Then, determine the specific solution that satisfies the initial conditions \( u = 0 \) and \( \frac{\partial u}{\partial y} = x \) when \( y = 0 \).

Answer: xy + 3y^2


Question: Consider a box function $f$ from $[0, 1]$ to itself. Define its oscillation function $Of$ as \(Of(x) = \lim _{d \to 0} \sup _{y, z \in B_d (x)} |f(y) - f(z)|\). Prove or disprove that for every box function $f$, the equality $OOOf = OOf$ holds.

Answer: OOOf = OOf


Question: Let $X$ be a topological $n$-manifold and $C$ a finite partition of $X$ into topological manifolds with subspace topologies. Is there at least one $M \in C$ such that $\dim(M) = n$?

Answer: Yes


Question: Solve the differential equation \( \frac{dy}{dx} = \frac{xy + 3x - y - 3}{xy - 4x + 6y - 24} \) and find the value of \( \left(\frac{x+6}{y+3}\right)^7 \).

Answer: 1


Question: A hexagon is inscribed in a circle of radius \( r \). Four of the edges of the hexagon are 10 feet long, and two of the edges are 20 feet long, but the exact arrangement of the edges is unknown. Using either Ptolemy's Theorem or the Law of Cosines, determine the value of \( r \) to three decimal places.

Answer: 13.660


Question: Let $A$ be a $227 \times 227$ matrix with entries in $\mathbb{Z}_{227}$ such that all of its eigenvalues are distinct. What is the trace of the matrix $A$?

Answer: 0


Question: How many distinct ways can the letters of the word COMBINATION be arranged such that none of the patterns CAN, BIN, or NIB appear?

Answer: 4541040


Question: Given a vector \(x\) of size \(N \times 1\) and a symmetric sparse matrix \(K\) of size \(N \times N\), is it possible to decompose \(x^T K x\) as \(y^T y\) such that \(y = K^{1/2} x\)? Assume \(K\) is a correlation matrix with local correlations, making it a band matrix.

Answer: Yes


Question: Let $G$ be a finite group with $|G| = pq^m r$, where $p < q$, $p^2$ does not divide $|G|$, and $(r, pq^m) = 1$. Suppose $H_p$ and $H_q$ are Sylow subgroups of $G$ with $H_p \lhd G$. Prove that if $H_p H_q \lhd G$, then $H_q \lhd G$.

Answer: H_q \lhd G


Question: Find the maximum value of the function \( f(x) = \sqrt{(x-3)^2 + (x^2-2)^2} - \sqrt{x^2 + (x^2-1)^2} \), where \( P(x, x^2) \) is a point on the parabola \( y = x^2 \).

Answer: \sqrt{10}


Question:
Determine which known group the quotient group $S_4/N$ is isomorphic to, where $N = \{1, (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)\}$ is a normal subgroup of $S_4$. The set of cosets is $G/N = \{a, b, c\}$, where:

$$a = \{(1), (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)\},$$
$$b = \{(1, 3)(1, 2), (2, 3)(2, 4), (1, 2)(1, 4), (1, 4)(1, 3)\},$$
$$c = \{(1, 2)(1, 3), (2, 4)(2, 3), (1, 4)(1, 2), (1, 3)(1, 4)\}.$$

Answer: S_3


Question: Let $n \geq 2$ be an integer and consider the vector space $V=F^{2n}$ with the standard basis $e_1,\ldots,e_{2n}$. The second exterior power $\Lambda^2(V)$ contains the element $\omega=e_1 \wedge e_2 + e_3 \wedge e_4 + \cdots + e_{2n-1} \wedge e_{2n}$. Since $\Lambda^{2n}(V)$ is spanned by $x := e_1 \wedge \cdots \wedge e_{2n}$, the $n$-fold wedge product $\omega \wedge \cdots \wedge \omega$ is a scalar multiple of $x$. Compute the scalar $c$ such that $c x = \omega \wedge \cdots \wedge \omega$. What is the value of $c$?

Answer: n!


Question: Evaluate the limit: \[ L = \lim_{n \to \infty} \frac{\left(2\sqrt[n]{n}-1\right)^n}{n^2} \]

Answer: 1


Question: Suppose $X$ is a compact space and $f_n: X \to \mathbb{R}$ is a sequence of continuous functions for all $n \in \mathbb{N}$. The sequence $f_n$ converges pointwise to a continuous function $f$ on $X$, and the sequence $\{f_n\}$ is uniformly bounded on $X$. Does it follow that the convergence is uniform?

Answer: No


Question: Determine the largest number \( R \) such that the Laurent series of the function \( f(z) = \frac{2\sin(z)}{z^2-4} + \frac{\cos(z)}{z-3i} \) about \( z = -2 \) converges for \( 0 < |z+2| < R \).

Answer: \sqrt{13}


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

Answer: 0


Question: Find the limit of the sequence \( s_n = \int_0^1 \frac{nx^{n-1}}{1+x} \, dx \) as \( n \to \infty \).

Answer: \dfrac{1}{2}


Question:
Given that \(a, b, c\) are positive real numbers such that \(a+b+c=3\), find the minimum value of the expression:
\[ 7(a^4+b^4+c^4)+\frac{ab+bc+ca}{a^2b+b^2c+c^2a} \]

Answer: 22


Question: Calculate the norm \( \|\Lambda\| \) of the linear functional \( \Lambda(f) = \int_{0}^{1} (9t^4 - 18t^3 + 11t^2 - 2t) f(t) \, dt \) in the Banach space \( C([0, 1]) \).

Answer: \dfrac{49}{810}


Question: Let \((a_n)_n\) be a convergent sequence. Determine the value of \( \lim_{n \to \infty} n(a_{n+1} - a_n) \) if it exists.

Answer: 0


Question: Find the sum of the series \( \cos \frac{\pi}{3} + \frac{1}{2}\cos\frac{2\pi}{3} + \frac{1}{3}\cos \frac{3\pi}{3} + \cdots \).

Answer: 0


Question: Is it true that if a random variable $X$ has a finite mean, then $E[|X|] < \infty$? In other words, does $E[X]<\infty$ imply $X \in L^1$?

Answer: Yes


Question: Given the equation \(2a + 3b + 4c = 100\) where \(a, b, c\) are natural numbers, find the maximum value of the expression \(2a^2 + 5b^2 + 8c^2\).

Answer: 4534


Question: Determine whether the limit \( \lim_{N \to \infty} \int_{N}^{e^N} xe^{-x^{2016}} \, dx \) exists. Justify your answer.

Answer: 0


Question: Let $G$ be the Galois group of a field with nine elements over its subfield with three elements. Determine the number of orbits for the action of $G$ on the fields with nine elements.

Answer: 6


Question: Determine the dimension over \(\mathbb{R}\) of the space of \(n \times n\) Hermitian matrices.

Answer: n^2


Question: A particle of mass \(m\) starts from rest at time \(t=0\) and is moved along the \(x\)-axis with constant acceleration \(a\) from \(x=0\) to \(x=h\) against a variable force of magnitude \(F(t)=t^2\). Find the work done in moving the particle from \(x=0\) to \(x=h\).

Answer: \dfrac{h^2}{a}


Question: Find the measure of the external angle bisector of angle $A$ in triangle $ABC$ given that $b - c = 20$ and $b \cdot c (p - b)(p - c) = 10c$, where $p$ is the semiperimeter of the triangle.

Answer: 1


Question: Find the leading order asymptotic approximation of the integral \( I = \int_{-1}^{1} e^{xt^3} \, dt \) as \( x \to \infty \).

Answer: \dfrac{e^{x}}{3x}


Question: Evaluate the limit \( \lim_{x \to\infty} \frac{\log\log f(x)}{\log x} \) given that the function \( f\colon [1,\infty) \to (1,\infty) \) is monotone increasing and satisfies \( f(x)^2 \le f(4x) \le 2013^{\sqrt x} \).

Answer: \dfrac{1}{2}


Question: Evaluate the expression \( \left\lfloor \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{1000}} \right\rfloor \), where \( \lfloor \cdot \rfloor \) denotes the greatest integer function.

Answer: 60


Question: Let \( A \) be a non-zero \( 3 \times 3 \) matrix with real entries such that \( A^3 + A = 0 \). Prove that \( \mathrm{rank}(A) = 2 \).

Answer: 2


Question:
Consider the initial value problem given by the partial differential equation \((x - y) u_x + (y - x - u) u_y = u\) with the initial condition \(u(x, 0) = 1\). Determine which of the following expressions the solution satisfies:

1) \(u^2(x + y + u) + (y - x - u) = 0\)
2) \(u^2(x - y + u) + (y - x - u) = 0\)
3) \(u^2(x - y + u) - (y + x + u) = 0\)
4) \(u^2(y - x + u) + (y + x - u) = 0\)

Answer: 1


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{1}{\ln (n)} \sum_{k=1}^{n} \frac{a_k}{k} \] given that \( \lim_{n \to \infty} a_n = a \).

Answer: a


Question: Compute \( \int_\gamma \overline{\zeta} \, d\zeta \) where \( \gamma \) is the circle of radius 1 centered at 0, oriented counterclockwise, using Cauchy's Integral Formula.

Answer: 2\pi i


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

Answer: \sqrt{2}


Question: Let \(X_i \sim G_{0.5}\) for \(i \geq 1\). Define \(Z_n = \frac{1}{\sqrt{n}}\sum_{i=1}^{n}(X_i-2)\). Approximate \(P(-1 \leq Z_n \leq 2)\) for large \(n\).

Answer: 0.6816


Question: Let $f(z): \mathbb{C} \rightarrow \mathbb{C}$ be an analytic function such that $|f(z) - 1| < 1$ in a domain $\Omega \subset \mathbb{C}$. Prove that for any closed curve $\gamma$ in $\Omega$, the integral \( \int_{\gamma} \frac{f'(z)}{f(z)} \, dz = 0 \).

Answer: 0


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a continuous function such that \( \int_{-\infty}^{+\infty} f(x) \sin(nx) \, dx = 0 \) and \( \int_{-\infty}^{+\infty} f(x) \cos(nx) \, dx = 0 \) for all \( n \in \mathbb{Z} \). Additionally, both integrals \( \int_{-\infty}^{+\infty} |f(x) \sin(nx)| \, dx \) and \( \int_{-\infty}^{+\infty} |f(x) \cos(nx)| \, dx \) are finite. Does this imply that \( f \equiv 0 \)?

Answer: No


Question: Let $X$ be a scheme and $\mathcal{L}$ an invertible sheaf in $Qcoh(X)$. Consider the functor $F: Qcoh(X) \to Qcoh(X)$ defined by $F(\mathcal{M}) = \mathcal{M} \otimes \mathcal{L}$. Is this functor exact?

Answer: Yes


Question: Let $G$ be a group with a normal subgroup $N$ such that $[G:N] = 4$. Prove that there exists a subgroup of $G$ with index 2.

Answer: 2


Question:
Let $A$ be a $5 \times 5$ matrix such that $\det(A) \neq 0$ and $4A = 5B$. Calculate the value of the following expression:
\[ \frac{1}{25} \cdot \det(4(A^t)^2B^{-1}A^{-1}) \]

Answer: 125


Question: Find the remainder when $40!$ is divided by $5^{10}$. Express your answer as $40! \mod 5^{10}$.

Answer: 7812500


Question: In a finite group $G$, is it always possible to express any element $z \in G$ as $z = gh^2$, where $g$ is an element of order $2^k$ for some non-negative integer $k$, and $h$ is an element of odd order?

Answer: Yes


Question: Let \(X_i\) be i.i.d. nonnegative random variables such that \(E[X_1]<\infty\). Prove or disprove that the sequence \(m_n:=\frac{1}{n} \sum_{i=1}^n X_i1_{\{X_i \geq \sqrt{n}\}}\) converges almost surely to 0.

Answer: 0


Question: Determine whether the following conjecture is true: Let $X$ be a noncompact complex manifold that contains no positive dimensional compact analytic sets. There must be strictly plurisubharmonic functions on $X$. Is this conjecture true?

Answer: True


Question:
Determine whether the following inequality holds for any two positive definite matrices \(\mathbf{X}\) and \(\mathbf{Y}\), and any \(\lambda \in (0,1)\):
\[
(1-\lambda)\mathbf{X}^{-1}+\lambda\mathbf{Y}^{-1} \succeq ((1-\lambda)\mathbf{X}+\lambda\mathbf{Y})^{-1}
\]
where the notation \(\mathbf{A} \succeq \mathbf{B}\) indicates that \(\mathbf{A-B}\) is positive semi-definite.

Answer: Yes


Question: Given that $y_1$ and $y_2$ form a fundamental set of solutions for the differential equation $t^2y'' - 3ty' + t^3y = 0$ with $W[y_1, y_2](1) = 4$, find the value of $W[y_1, y_2](7)$.

Answer: 1372


Question: Show that \( E\left[\int_0^\infty e^{-A_t} \, dA_t\right] \) is bounded above by a constant, where \( A_t \) is an increasing stochastic process of locally integrable variation starting from 0. Note that \( A_t \) may be purely discontinuous.

Answer: 1


Question: Given the boundary value problem \(\begin{cases}-\Delta u=0 & \text{in}\  B(0,2)\\u(x,y)=3xy+2 & \text{on} \ \partial B(0,2)\end{cases}\), compute \(u(0)\).

Answer: 2


Question: Determine whether the number 7 is prime, irreducible, or neither in the ring \( \mathbb{Z}_{21} \).

Answer: prime


Question: Evaluate the limit: \[ \lim_{n\to \infty} \left[n+n^2\log\left(\frac{n}{n+1}\right)\right] \] and determine if it equals \( \frac{1}{2} \).

Answer: \dfrac{1}{2}


Question: Determine if the sequence $b_r$, defined by the condition $\gcd(C_n+1, C_n-1) > 1$ where $C_n = \frac{\binom{2n}{n}}{n+1}$ is the Catalan sequence, is equal to $2^n - 1$ for $n \leq 4200$. The starting terms for $b_r$ are: 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095.

Answer: 2^n - 1


Question: Determine the number of 4x4 matrices \( A = [a_{ij}] \) where \( 1 \le i,j \le 4 \), each element \( a_{ij} = \pm 1 \), and the sum of the elements in each row and each column is zero.

Answer: 90


Question: Let \( X = \{ f \in C[0,1] : f(0) = 0 \} \) be a subspace of continuous functions on \([0,1]\) with the property that \( f(0) = 0 \). Define the linear operator \( T: X \to \mathbb{R} \) by \( T_f = \int_0^1 f(t) \, dt \). Compute the operator norm \( \|T\| \) when \( X \) is endowed with the \( \|\cdot\|_{\infty} \) norm.

Answer: 1


Question: Express the sum of coefficients \( \sum_{n=0}^{\infty}C_n \) from the formal power series \( \sum_{k=1}^{\infty} (-1)^{k+1}a_k \sum_{j=0}^k (-1)^j\binom{k}{j} X^j = \sum_{n=0}^{\infty}C_n X^n \) in terms of the sequence \( \{a_k\} \).

Answer: 0


Question: Find the non-zero value of $k$ such that the average area between the curves $x^2$ and $ax$ for $x \ge 0$, as $a$ varies from $0$ to $k$, equals $x^2$. What is this value of $k$?

Answer: 24


Question: Is the genus of $CP^N$ equal to 0 for all $N$?

Answer: 0


Question: Determine whether the codeword (1,1,1,1,1,1,1) is part of every binary cyclic code of length 7. Justify your answer.

Answer: No


Question: Let \(x = (9 + 4\sqrt{5})^{48}\). If \(x = [x] + f\), where \([x]\) is the integral part of \(x\) and \(f\) is its fractional part, find the value of \(x(1-f)\).

Answer: 1


Question: Determine whether the radical of the ideal $4\mathbb{Z}$ is equal to $2\mathbb{Z}$. Justify your answer.

Answer: Yes


Question: Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the circle given by the equation \(x^2+(y-R)^2=r^2\), where \(R > r > 0\).

Answer: 2\pi^2 R r^2


Question: Given that $Z \sim N(0,1)$, find the variance of $Y = a + bZ + cZ^2$.

Answer: b^2 + 2c^2


Question:
Given \( a = \sqrt[3]{2-\sqrt{3}} + \sqrt[3]{2+\sqrt{3}} \), find the value of the expression:
\[ a^5 + a^4 - 2a^3 - 7a^2 - 7a + 16. \]

Answer: 20


Question: Find the quotient group of the normal subgroup \( N = \{ \pm 1, \pm i \} \) of the quaternion group \( H = \{ \pm 1, \pm i, \pm j, \pm k \} \). Determine the structure of the quotient group \( H/N \).

Answer: \mathbb{Z}_2


Question: In the right triangle $ABC$, which is right-angled at $B$, the height $BH$ is drawn. Let $P$ and $Q$ be the intersections of triangles $AHB$ and $BHC$. The line $PQ$ intersects $BH$ at point $E$, where the ratio $\frac{BE}{EH} = 5\sqrt{2}$. The inradius of triangle $ABC$ is $10$. Calculate the inradius of triangle $PHQ$.

Answer: 1


Question:
Find the limit of the following sum as \( n \to \infty \):
\[ \lim_{n \to \infty} \frac{2}{n^2} \sum_{j=0}^{n-1} \sum_{k=j+1}^{n-1} \frac{k}{n} \]

Answer: \dfrac{2}{3}


Question: Determine whether there exists an n-cycle $s$ (where $n \ge 10$) such that $\tau = s^k$ for some integer $k$, given $\tau = (1\ 2)(3\ 4)(5\ 6)(7\ 8)(9\ 10)$. Provide a justification for your answer.

Answer: Yes


Question: Determine the value of \(a > 0\) such that the integral \(I = \int_1^\infty \int_1^\infty (x+y)^{-a} \, dx \, dy\) converges.

Answer: 2


Question: Evaluate the summation: \[ \sum_{r=0}^{n} (-1)^r \binom{n}{r}\frac{1+r\ln 10}{(1+\ln 10^n)^r} \]

Answer: 0


Question: Evaluate \( x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} \) for the function \( z = x\tan^{-1} \frac{y}{x} + y\sin^{-1} \frac{x}{y} + 2 \) at the point (1,1).

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


Question: Consider a closed subgroup of a Lie group $G$ that is a connected Lie group with Lie algebra $g$, where $l$ is an abelian ideal of $g$. Let $K$ be the connected Lie subgroup of $G$ with Lie algebra $l$. Is $K$ necessarily closed in $G$?

Answer: No


Question: If a complex number \( z \) (where \( z \neq 2 \)) satisfies the equation \( z^2 = 4z + \lvert z \rvert^2 + \frac{16}{\lvert z \rvert^3} \), find the value of \( \lvert z \rvert^4 \).

Answer: 4


Question:
Calculate the determinant of the following matrix, given that the polynomial function $P \in \mathbb{R}[X]$ has a remainder of the form $ax^3 + bx^2 + cx$ when divided by $(x-1)(x-2)(x-3)(x-4)$:

$$\begin{vmatrix}
P(1) & 1 & 1 & 1 \\
P(2) & 2 & 4 & 8 \\
P(3) & 3 & 9 & 27 \\
P(4) & 4 & 16 & 64
\end{vmatrix}$$

Answer: 0


Question: For a matrix $M \in \mathbb{R}^{n \times n}_{\geq 0}$ with nonnegative entries, define $m$ as the smallest positive integer such that all the entries of $M^m$ are strictly positive, if such an $m$ exists. Determine, as a function of $n$, the maximum possible finite value of $m$ over all possible choices of $M$. Express your answer in terms of $n$.

Answer: (n-1)^2 + 1


Question: Determine the maximum step size for the forward Euler method such that the solution of the system of ODEs \( a' = -a - 2b \) and \( b' = 2a - b \) with initial conditions \( a(0)=1 \) and \( b(0)=-1 \) goes to zero. Use the eigenvalues of the matrix \( \begin{pmatrix} -1-h & -2-2h \\ 2+2h & -1-h \end{pmatrix} \) to find the solution stability.

Answer: \dfrac{2}{5}


Question: Find the value of $k$ such that the matrix $H = E - k \alpha \alpha^T$ is orthogonal, given that $\alpha$ is an $n$-dimensional vector and $\alpha^T \alpha = 1$. Express your answer as a single number.

Answer: 2


Question: Find the value of \(x\) such that for every integer \(a\), the equation \(a^{31x} \equiv a \pmod{271}\) holds true.

Answer: 61


Question: Calculate the line integral \( \int_{C} xy\,dx + 2y^2\,dy \), where \( C \) is composed of two parts: the arc of the circle from \( (2,0) \) to \( (0,2) \) and the line segment from \( (0,2) \) to \( (0,0) \).

Answer: -\dfrac{8}{3}


Question:
Given real numbers \(a\), \(b\), and \(c\) such that:
\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 4 \left( \frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+a} \right) = \frac{c}{a+b} + \frac{a}{b+c} + \frac{b}{c+a} = 4 \]
Determine the value of \(abc\).

Answer: \dfrac{49}{23}


Question: Evaluate the integral \( \int_{\mathbb{R}^1} f(x) \, dx \), where \( f(x) = |(A-x) \cap B| \) and \( A, B \) are measurable subsets of \( \mathbb{R}^1 \). Here, \( |\cdot| \) denotes the measure.

Answer: |A| \cdot |B|


Question: Find the limit: \[ \lim_{x\rightarrow0}\frac{\sqrt{\cos x}-\sqrt[3]{\cos x}}{\sin^{2}x} \]

Answer: -\dfrac{1}{12}


Question: Show that the Riemannian metric \( g_r \) on a sphere of radius \( r \) in \( \mathbb{R}^n \), induced by its inclusion in \( \mathbb{R}^n \), is related to the metric \( g_1 \) on the unit sphere by the equation \( g_r = r^2 g_1 \).

Answer: g_r = r^2 g_1


Question: Let \( A \) and \( B \) be \( 4 \times 4 \) matrices with real entries such that they satisfy the equations \( A^2 - 2B + I = 0 \) and \( B^2 - 2A + I = 0 \). Given that \( |A-B| \neq 0 \), find the value of \( \det(A+B) \).

Answer: 16


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty }\left(\sum_{k=0}^{n-1}{\frac{e^{\frac{k}{n}}}{n}}\right). \]

Answer: e - 1


Question: Evaluate the integral \( \int_0^{\infty} e^{-x} \sum_{k=0}^{\infty} \frac{(-1)^k (2015x)^{2k}}{((2k)!)^2} \, dx \).

Answer: \cos(2015)


Question: Let \((X, \delta)\) be a metric space and \((Y, \tau)\) be a topological space. Assume that \((X, \delta)\) is homeomorphic to \((Y, \tau)\). Is \((Y, \tau)\) metrizable, i.e., is there a metric that induces the topology on \(Y\)? If not, provide an example.

Answer: Yes


Question: Determine if the following is an equivalent definition to the epsilon-delta definition of a limit: \( \forall\mbox{ Natural Numbers }N>0 \,\,\exists\delta>0\mbox{ s.t. }0<|x−x_0|<\delta \implies|f(x)−f(x_0)|<1/N \).

Answer: Yes


Question: Let $S$ be the disk $|z|<3$ in the complex plane, and let $f: S \rightarrow \mathbb{C}$ be an analytic function such that $f\left(1+\frac{\sqrt{2}}{n}i\right) = -\frac{2}{n^2}$ for each natural number $n$. Determine the value of $f(\sqrt{2})$. \( \boxed{?} \)

Answer: 3 - 2\sqrt{2}


Question:
Prove that \( k \geq r \) given the following conditions:
\[ r = \frac{\lambda - \mu + \sqrt{(\lambda - \mu)^2 + 4(k - \mu)}}{2} \]
where:
- \( v, k, \lambda, \mu \geq 0 \)
- \( \mu > 0 \)
- \( k > \lambda, \mu \)
- \( r > 0 > s + 1 \)
- \( \lambda - \mu \geq -\frac{2k}{v-1} \)

Answer: k \geq r


Question: Calculate the probability \( P(Z(3) > Z(2), Z(2) > 0) \) for a Brownian motion process \( Z(t) \), where \( Z(3) \) and \( Z(2) \) are on the same sample path.

Answer: \dfrac{1}{4}


Question: Determine whether the function \( f(x,y) = \begin{cases}\big|1+xy^2\big|^\dfrac{1}{x^2+y^2} & \quad\hfill (x,y)\neq(0,0)\\ 1 &\quad\hfill (x,y)=(0,0)\end{cases} \) is continuous at the point \((0,0)\).

Answer: Yes


Question: A 6-sided die is biased such that rolling a 1 is twice as likely as rolling any other number, and the probabilities for rolling 2, 3, 4, 5, and 6 are equal. Specifically, the probabilities are given by: \( p(1) = \frac{2}{7} \) and \( p(2) = p(3) = p(4) = p(5) = p(6) = \frac{1}{7} \). What is the expected number of rolls needed to obtain each value at least once?

Answer: \dfrac{323}{20}


Question: Let $X$ be an infinite dimensional Banach space and $T: X \rightarrow Y$ be a compact operator between Banach spaces. Does there always exist a sequence $x_n$ of norm $1$ such that $T(x_n)$ converges to zero?

Answer: Yes


Question: Evaluate the contour integral \( \int_{C(0,1)^+} \sin(z) \, dz \), where \( C(0,1)^+ \) is the positively oriented unit circle centered at the origin.

Answer: 0


Question: Determine the conditions on the numbers \(a, b,\) and \(c\) such that the linear operator \(T: \mathbb{R}^{4} \to \mathbb{R}^{4}\), defined by \(T(x_{1},x_{2},x_{3},x_{4})=(0,ax_{1},bx_{2},cx_{3})\), is diagonalizable.

Answer: a = b = c = 0


Question: Evaluate the limit \( \lim\limits_{x\to \infty}\left(\frac{20^x-1}{19x}\right)^{\frac{1}{x}} \).

Answer: 20


Question: Calculate the expected value \( \mathbb{E}[\cosh(B_t)] \) for a Brownian motion \( B_t \) at time \( t \), where \( B_t \sim N(0, t) \).

Answer: e^{t/2}


Question: Compute the limit \( \lim_{x \to 0} \left( \frac{1}{x^4} - \frac{\int_0^{x^2} e^{-u^2} \, du}{x^6} \right) \) without using L'Hôpital's rule or Taylor series.

Answer: \dfrac{1}{3}


Question: Let $f \in L^{1}([0,1],m)$. Assume that for every interval $I \subset [0,1]$ with $m(I) = c$, we have \( \int_{I} f \, dm = 0 \). Does it follow that $f = 0$ almost everywhere? Here, $0 < c < 1$ is a fixed constant.

Answer: No


Question: Determine the values of \( n \) for which the following integral equation holds: \[ \int_0^1 \ln(x^n+1)\ln(2-x^n)\, dx = 1. \]

Answer: 1


Question: Find the expectation of the inverse of the sum of squares of 10 independent standard normal variables, \(X_1, X_2, \ldots, X_{10}\). That is, compute \(E\left(\frac{1}{X_1^2 + X_2^2 + \cdots + X_{10}^2}\right)\).

Answer: \dfrac{1}{8}


Question: What is the algorithm with the lowest time complexity for finding the nullspace of a singular $n \times n$ square matrix $A$? For example, using Singular Value Decomposition (SVD) has a time complexity of $O(n^3)$. Provide the algorithm and its time complexity.

Answer: O(n^\omega)


Question: Determine which is greater: $(\sqrt{5})^e$ or $e^{\sqrt{5}}$. Provide a formal proof for your conclusion.

Answer: e^{\sqrt{5}}


Question:
Find all functions \( f : \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the following conditions for all \( x, y \in \mathbb{R} \):
1. \( |f(x)| = |x| \)
2. \( f(f(-y)) = -f(y) \)
3. \( f(f(x)) = f(x) \)

Answer: f(x) = x


Question: Evaluate the line integral \( \int \limits_C (y + z)\,dx + (z + x)\,dy + (x + y)\,dz \) using Stokes' theorem, where \( C \) is the intersection of the cylinder \( x^2 + y^2 = 2y \) and the plane \( y = z \). Assume the curve is oriented counterclockwise when viewed from above.

Answer: 0


Question: Evaluate the complex contour integral \( \int_\gamma \frac{\cos(z)}{z^3} \mathrm{d}z \), where \( \gamma(t) = e^{it} \) for \( t \in [0, 2\pi] \). Use the series expansion for \( \cos(z) \) to compute the integral.

Answer: -\pi i


Question: Find an element in the group \((\mathbb{Z}/100\mathbb{Z})^\ast\) that has order 20.

Answer: 3


Question: Compute the value of the exterior 2-form \( \theta = \sum_{i=1}^{n-1} x_i \wedge x_{i+1} \) on the vectors \( A = (1, 1, 1, \dots, 1) \) and \( B = (-1, 1, -1, \dots, (-1)^n) \) in \( \mathbb{R}^n \). What is \( \theta(A, B) \)?

Answer: 1 + (-1)^n


Question: Find the radius of convergence of the series \( \sum_{n=0}^{\infty} \left(\frac{z}{1+z}\right)^n. \)

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_{-\infty}^\infty \frac{e^{ix}}{x} \, dx \) using contour integration techniques. Verify if the result \( I_0 = i\pi \) is correct.

Answer: i\pi


Question: Is it possible to construct the quadratrix of Hippias using only one degree of freedom?

Answer: Yes


Question: Find the remainder when \(2(26!)\) is divided by 29.

Answer: 28


Question: Does there exist a knot $K \subset \mathbb{S}^3$ such that $K$ is not slice in $B^4$, but there exists a rational homology ball $W^4$ with $\partial W = \mathbb{S}^3$ and a properly embedded smooth disk $(D, \partial D) \to (W, K)$?

Answer: Yes


Question: Evaluate the limit of the integral: \[ \lim_{n\rightarrow \infty} \int_{\mathbb{R}^2}e^{-(x^2+y^2)^n}dxdy \]

Answer: \pi


Question: Determine the limit of the sequence defined by successive compositions of the natural logarithm function, starting with a positive number greater than 1. Does this sequence converge to zero as the number of compositions approaches infinity?

Answer: 0


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

Answer: 1


Question: Calculate the integral \( \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{x^{2}}{(\sqrt{x^2+y^2}+1)^5} \, dx \, dy \) by using a suitable transformation of variables.

Answer: \dfrac{\pi}{4}


Question: Determine the values of $m$ for which the quadratic equation \((m+1)x^2 -2(m-1)x + m=0\) has roots $\alpha$ and $\beta$ such that $-1 < \alpha < 1 < \beta < 2$.

Answer: (-\infty, -8)


Question: Given that \( f(x+1) = \lim_{n\rightarrow \infty} \left(\frac{n+x}{n-2}\right)^{n} \), find the expression for \( f(x) \).

Answer: e^{x + 1}


Question: Given that the equation \( \cos(ax^2 + bx + c) = -1 \) has two distinct real roots at \( x=1 \) and \( x=-1 \), where \( a, b, c \in (2,5) \), find the maximum value of \( \frac{ac}{b^2} \).

Answer: 1


Question: Let \(f: \mathbb{R} \to \mathbb{R}\) be a continuous, differentiable function such that \(f(x) = 0\) for any \(x \le 0\) and \(f(x) > 0\) for \(x > 0\). Determine if there exists a positive real number \(a\) such that for any \(x \in (0, a]\), \(f'(x) > 0\).

Answer: No


Question: Evaluate the limit: $$\lim_{x \to 0}\frac{\cos(x) - 8x\sin(x/2) - \cos(3x)}{x^4}$$

Answer: -\dfrac{19}{6}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\left(\sqrt{\frac{9^n}{n^2}+\frac{3^n}{5n}+2}-\frac{3^n}{n}\right) \]

Answer: \dfrac{1}{10}


Question: Find the coefficient of \(x^{22}\) in the expansion of \(\frac{1+3x}{(1-x)^8}\) using generating functions.

Answer: 5112900


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{\int_{0}^{x}f(t)dt}{x} \] given that \( f(0) = 3 \) and \( f \) is continuous.

Answer: 3


Question: Evaluate the limit: \[ \lim_{x \to \infty} \left(\frac{x+a}{x-a}\right)^x \] without using L'Hôpital's rule.

Answer: e^{2a}


Question: Given the function \( f(x) = x^3 - x + 1 \), determine the number of distinct real values of \( x \) for which \( f(f(x)) = 0 \).

Answer: 1


Question: Find the smallest value of $k$ such that for all real numbers $a_1, a_2, \ldots, a_n$, the inequality \( \sum_{i=1}^n a_i^3 \leq k \sqrt{\sum_{i=1}^n a_i^6} \) holds, where $n$ is an integer and $n \geq 3$.

Answer: \sqrt{n}


Question: Consider the space \( X = (B(\mathbb{R},\mathbb{R}), \|.\|) \) of all bounded linear operators from \( \mathbb{R} \) to \( \mathbb{R} \). Define the operator \( T: X \rightarrow X \) by \( Tx(t) = x(t - \Delta) \), where \( \Delta > 0 \) is fixed. Determine whether the operator \( T \) is linear and bounded. If \( T \) is bounded, find its norm.

Answer: 1


Question: Find the total number of ordered triplets \( \langle \mathbf{u, v, w} \rangle \) of \( n \)-dimensional vectors, where each vector is of the form \( \{-1, 1\}^n \), such that the \( L_{\infty} \)-norm of the vector sum \( \mathbf{u+v+w} \) is less than or equal to 1.

Answer: 6^n


Question: Let $k$ be an algebraically closed field and $V$ a smooth projective variety over $k$. Consider a map $\phi: V \to V$. Do the coefficients of the characteristic polynomial of the induced map $\phi^*: H^i_{dR}(V/k) \to H^i_{dR}(V/k)$ lie in the prime subfield of $k$?

Answer: Yes


Question: Evaluate \( \lim\limits_{n \to \infty} n\int ^{2\pi }_{0}x\sin x (e^{\frac {x}{n}}-1){\rm d}x \).

Answer: -4\pi^2


Question: In the given figure, $O$ and $O_1$ are centers of circles, and the arcs $\overset{\LARGE{\frown}}{AO_1}$ and $\overset{\LARGE{\frown}}{O_1B}$ are equal. If $AD = 4\sqrt{2}$, calculate the area of the shaded region $CEOD$. Express your answer in terms of $\sqrt{2}$.

Answer: 2(4 - \sqrt{2})


Question: Find the value of \( b \) such that the process \( M_t = e^{5B_t - bt} \) is a martingale, where \( B_t \) is a standard Brownian motion.

Answer: \dfrac{25}{2}


Question: Evaluate the limit \( \lim_{x\to 0^+} x \int^1_x \frac{f(t)}{\sin^2t}dt \), where \( f \) is a differentiable function.

Answer: f(0)


Question: Determine whether \( \mathbb{R} \) is isomorphic to \( \mathbb{R} \oplus \mathbb{R} \) as vector spaces over \( \mathbb{Q} \). Answer with 'Yes' or 'No'.

Answer: Yes


Question: Evaluate \( \lim_{n \to \infty} \int_{0}^{n} \frac{\cos(x/n)}{\sqrt{x+\cos(x/n)}}dx \).

Answer: \infty


Question: Let $A$ be a $2 \times 2$ matrix with eigenvalues $1$ and $-1$. Find the determinant of the matrix $S = 2A + A^{-1} - I$. \( \text{(Note: } I \text{ is the identity matrix.)} \)

Answer: -8


Question: Determine the image set of the function \( f(x) = \frac{\sin(\pi x)}{x(1-x)} \) for \( x \in (0, 1) \).

Answer: (\pi, 4]


Question: For a given positive integer \( N \), determine if there exists an integer \( n > N \) such that the first \( n \) prime numbers can be divided into two sets with equal sum.

Answer: Yes


Question: Given \( \frac{3a+2b}{a}=\frac{2a+5b}{c}=\frac{5c-2b}{b} \), find the value of \( \frac{2a-3b}{a+b} \).

Answer: -\dfrac{1}{2}


Question: Let \( \alpha \) and \( \beta \) be two complex numbers with \( \beta \neq 0 \), and let \( f(z) \) be a polynomial function on \( \mathbb{C} \) such that \( f(z) = \alpha \) whenever \( z^5 = \beta \). Determine the minimum possible degree of the polynomial \( f(z) \).

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to\infty} \left(\frac{5x-2}{5x+4}\right)^{5x+1}$$

Answer: e^{-6}


Question: Let $G$ and $H$ be groups. Suppose the category $P(G)$, defined as a preorder with objects as elements of $G$ and morphisms $	ext{Hom}(a, b) = \{ab^{-1}\}$, is isomorphic to the category $P(H)$. Is $G$ isomorphic to $H$?

Answer: Yes


Question: Given the set \( S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \) and the function \( f: S \rightarrow S \) defined by \( f(x) = 6x^2 + 3x + 8 \mod 11 \), find the set \( f^{-1}(\{0, 5\}) \).

Answer: \{2, 3, 6, 10\}


Question: Evaluate the improper integral \( \int_{-\infty}^0 e^x \sin x \, dx \). Justify any non-trivial steps in your solution.

Answer: -\dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} \int_0^n \frac{1}{n+n^2\sin(xn^{-2})} \, dx \]

Answer: \ln 2


Question: Find a single generator for the kernel of the homomorphism \( \varphi : \mathbb{Z}[i] \to \mathbb{F}_{13} \) defined by \( \varphi(a+bi) = a + 5b \pmod{13} \). Given that \( \ker(\varphi) \) is the ideal generated by \( 13 \) and \( i-5 \), determine a single generator for \( \ker(\varphi) \).

Answer: 3 + 2i


Question: Evaluate the integral \( \int_{-\infty}^\infty \frac{1}{(4+x^2)^2} \mathrm{d}x \).

Answer: \dfrac{\pi}{16}


Question: Find all polynomials \( p(x) \) with real coefficients such that \( p(x+1) = p(x) + 2x + 1 \).

Answer: x^2 + c


Question: Provide an example of an injective but not surjective function from an infinite set to itself.

Answer: f(n) = n + 1


Question: Consider i.i.d. random variables $X_1, X_2, \ldots, X_n$ distributed according to a Weibull distribution with shape parameter $0 < \epsilon < 1$, such that $\mathbf{Pr}[X_i \geq t] = e^{-\Theta(t^{\epsilon})}$. Define the random variable $S_n = X_1 + X_2 + \ldots + X_n$. As $n$ tends to infinity, is it true that there exists a constant $C = C(\epsilon)$ such that $\mathbf{Pr}[S_n \geq C n] \leq e^{-\Omega_{\epsilon}(n^{\alpha})}$ for some $\alpha = \alpha(\epsilon) > 0$? If so, determine the largest possible value of $\alpha$. Assume that standard MGF-based methods are not applicable due to the heavy-tailed nature of $X_i$. What is the largest $\alpha$ one can achieve?

Answer: \epsilon


Question: Solve the equation \( \sqrt{x^2-9} = \frac{2(x+3)}{(x-3)^2} - x \). Find the value of \( x \) that satisfies this equation, considering any necessary restrictions.

Answer: 8 - \sqrt{13}


Question: Suppose a function \( f:\mathbb{R}\rightarrow\mathbb{R} \) is analytic at 0, meaning it is represented by its Taylor series in a neighborhood \( U \) of 0. Is it necessarily true that \( f \) is analytic at every point in \( U \)? Answer yes or no.

Answer: Yes


Question:
There are 2n people seated around a circular table, and m cookies are distributed among them. The cookies may be passed around under the following rules:
- Each person may only pass cookies to his or her neighbours.
- Each time someone passes a cookie, he or she must also eat a cookie.

Let A be one of these people. Find the least m such that no matter how m cookies are distributed to begin with, there is a strategy to pass cookies so that A receives at least one cookie.

Answer: 2^n


Question: Compute the norm of the bounded linear functional \( f \) defined by \( f(x) = x'(t_0) \) for all \( x \) in the space \( C'[a,b] \), where \( C'[a,b] \) is the space of all continuously differentiable functions on \([a,b]\) with the norm \( ||x|| = \max_{a \leq t \leq b} |x(t)| + \max_{a \leq t \leq b} |x'(t)| \). Here, \( t_0 \) is a fixed point in \([a,b]\). What is the norm of \( f \)?

Answer: 1


Question: Evaluate the triple summation \( S = \sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}\frac{1}{3^{i+j+k}} \) where \( i \neq j \neq k \).

Answer: \dfrac{81}{208}


Question: Find the limit of the sequence \( u_n = \int_{0}^{1} \frac{1}{1+t+t^2+\ldots+t^n} \, \mathrm{d}t \) as \( n \to \infty \).

Answer: \dfrac{1}{2}


Question: Determine whether the following statement is true or false: For a finitely generated module $M$ over a principal ideal domain (PID) $R$, the annihilator of the product of two prime elements $sr$ is the direct sum of the annihilators of $s$ and $r$, i.e., $$\operatorname{Ann}_{M}(sr)=\operatorname{Ann}_M(s) \oplus \operatorname{Ann}_M(r),$$ where $r,s\in R$ are prime elements.

Answer: False


Question:
Given that \(a + b + c = 0\), find the value of the expression:
\[
\frac{a^2}{2a^2+bc} + \frac{b^2}{2b^2+ac} + \frac{c^2}{2c^2+ab}
\]

Answer: 1


Question: Calculate the volume of the region bounded by the surfaces given by the equations \(xy=1\), \(xy=9\), \(xz=4\), \(xz=36\), \(yz=25\), and \(yz=49\). Use the transformation \(u=xy\), \(v=xz\), \(w=yz\) and the Jacobian determinant to set up the integral in the \(uvw\) coordinate system.

Answer: 64


Question:
Determine the number of real roots of the equation:
\[ \frac{x^{2000}}{2001} + 2\sqrt{3}x^2 - 2\sqrt{5}x + \sqrt{3} = 0 \]
Choose the correct answer:
A) 0  B) 11 C) 12 D) 1 E) None of these

Answer: A


Question: Let $X$ be an irreducible smooth projective variety over $\mathbb{C}$, and let $G$ be an affine algebraic group over $\mathbb{C}$. Consider a holomorphic principal $G$-bundle $p : E_G \longrightarrow X$ on $X$. The adjoint vector bundle of $E_G$, denoted $ad(E_G) = E_G \times^G \mathfrak{g}$, is associated with the adjoint representation $ad : G \longrightarrow \text{End}(\mathfrak{g})$ of $G$ on its Lie algebra $\mathfrak{g}$. The fibers of $ad(E_G)$ are $\mathbb{C}$-linearly isomorphic to $\mathfrak{g}$. Consider $ad(E_G)$ as a sheaf of $\mathcal{O}_X$-modules on $X$. Is there an $\mathcal{O}_X$-bilinear homomorphism $[,] : ad(E_G) \times ad(E_G) \to ad(E_G)$ that gives a Lie algebra structure on the sheaf $ad(E_G)$?

Answer: Yes


Question: Determine the profinite completion of the group $S^1$, where $S^1 = \{ z \in \mathbb{C} : |z| = 1 \}$ is a compact and abelian group.

Answer: \{1\}


Question: Does a non-measurable set always contain a bounded non-measurable subset? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim \limits_{(x,y) \to (0,0)} \frac{\ln(1+2x^2+4y^2)}{\arctan(x^2+2y^2)}. \]

Answer: 2


Question: If a set of random variables \(\{X_i\}_{i=1}^n\) are each distributed as a chi-squared distribution with one degree of freedom (\(\chi_1^2\)), and they are dependent, will the expectation of their sum \(\mathcal{G} = \sum_{i=1}^n X_i\) still be \(n\)? Assume that the expectation of each \(X_i\) is \(E[X_i]=1\).

Answer: n


Question:
Evaluate the integral of the quasi-binomial distribution score $U$ for one sample, defined as:
\[ Q(\mu,y)= \int_{y}^{\mu} \frac{y-t}{t(1-t)} \, dt, \quad y\in\{0,1\}. \]

Answer: y \ln \mu + (1 - y) \ln(1 - \mu)


Question: Let \( p \) be a prime number greater than 6. Show that 1920 divides \( p^4 - 10p^2 + 9 \).

Answer: 1920


Question: If $V$ is an irreducible representation of a semisimple Lie algebra with highest weight $\lambda$, what is the highest weight of the dual representation $V^*$?

Answer: -w_0(\lambda)


Question: Calculate the probability of obtaining 50 trillion or more successes when an experiment with a 10% success rate is repeated 100 trillion times.

Answer: 0


Question: Does there exist an uncountable set of sets such that each set has a distinct cardinality?

Answer: Yes


Question: Determine which of the following sets is recursively enumerable: \( L_1 = \{ M \mid \text{Turing Machine } M \text{ terminates for at least 637 inputs} \} \) and \( L_2 = \{ M \mid \text{Turing Machine } M \text{ terminates for at most 636 inputs} \} \).

Answer: L_1


Question: How many ways can you arrange 15 pens, consisting of 5 red, 5 green, and 5 blue pens, such that no 5 pens of the same color are consecutive?

Answer: 748560


Question: Given the joint density function \( f_{x,y} = \begin{cases} 2 , & 0 < y < x < 1 \\ 0 , & \text{elsewhere} \end{cases} \), find the probability \( P(X > 2Y) \).

Answer: \dfrac{1}{2}


Question: Find the minimum value of \[ P = \frac{1}{(1+x)^2} + \frac{1}{(1+y)^2} + \frac{1}{(1+z)^2} + \frac{4}{(1+x)(1+y)(1+z)} \] given that \( x, y, z > 0 \) and \( y + z = x(y^2 + z^2) \).

Answer: \dfrac{91}{108}


Question: What is the least number of square roots needed to express \(\sqrt{1}+\sqrt{2}+\cdots+\sqrt{100}\) in the form \(a+b\sqrt{c}+d\sqrt{e}+\cdots\) where \(a,b,c,d,e,\ldots\) are all integers?

Answer: 60


Question: Suppose a function $f$ satisfies the differential equation $y'' + by = 0$. Can you conclude that $f$ is of class $C^{\infty}$, meaning it has derivatives of all orders?

Answer: Yes


Question: Is the space of continuous functions on a compact space $C(X)$ dense in the space of continuous functions on the space of probability measures $C(\mathcal{M}(X))$?

Answer: No


Question: A particle moves in a straight line with acceleration given by \( a = -kv^n \), where \( n \neq 1, 2 \) and \( k \) is a constant. Determine the value of \( n \) such that the average velocity of the particle, averaged over the time until it stops, is one third of its initial velocity.

Answer: \dfrac{1}{2}


Question: Determine whether the following statement is true or false: If a language $L$ is context-sensitive, then every grammar $G$ that generates $L$ is also context-sensitive.

Answer: B


Question: Let $(R, \mathfrak{m})$ be a Noetherian local ring, and consider the associated graded ring $\mathrm{gr}_{\mathfrak{m}}(R) = \bigoplus_{n \ge 0} \mathfrak{m}^n/\mathfrak{m}^{n+1}$. If $x \in \mathfrak{m}/\mathfrak{m}^2$ is not a zero divisor on $\mathrm{gr}_{\mathfrak{m}}(R)$, is $x$ also not a zero divisor on $R$? Provide a justification for your answer.

Answer: Yes


Question: Find the condition on the parameter $k$ such that the function \( g(x) = kx - \ln(ex + 1 - x) \) is positive for all \( x \in (0, 1] \).

Answer: k \geq e - 1


Question: Compute the line integral of the vector field \( \vec{F} = \langle x^2, xy \rangle \) over the curve defined by the part of the circle \( x^2 + y^2 = 9 \) with \( x \le 0, y \ge 0 \), oriented clockwise.

Answer: 0


Question: Suppose $K$ is a finite connected simplicial complex that is a closed combinatorial surface, and $C$ is a subcomplex of $K$ such that $C$ is a simplicial circle. Is there a vertex $v \in K$ such that the link of $v$ is $C$?

Answer: No


Question: Determine whether the embedding of \( \mathcal{l}^1 \) into \( \mathcal{l}^p \) is continuous for any \( p \geq 1 \).

Answer: Yes


Question: In a $2 \times 3 \times 3$ array of letters, how many five-letter strings can be formed if you are allowed to repeat letters but cannot stall and repeat? An example of an acceptable string is "DETER", while an unacceptable string is "ADDER". Disregard the requirement that the string be an actual English word.

Answer: 1503378


Question: Evaluate the limit: \[ \lim_{n\to \infty}\sqrt[n^2 +n]{\prod_{i=0}^n \frac{n!}{i!(n-i)!}} \]

Answer: \sqrt{e}


Question: Suppose $f : \{ z \in \mathbb{C} : \mathrm{Re}(z) > 0 \} \to \mathbb{C}$ is a holomorphic function in the right half-plane with the property that $\lim_{z \to 0} f(z)$ exists in $\mathbb{C}$. If $(z_n)$ is a sequence converging to a point in $\{ z \in \mathbb{C} : \mathrm{Re}(z) > 0 \} \cup \{ 0 \}$ such that $f(z_n) = 0$ for all $n$, does $f$ vanish identically? Justify your answer.

Answer: Yes


Question: Compute the limit: \[ \lim_{x \rightarrow \frac{\pi}{4}^+} \left[ \tan\left(\frac{\pi}{8} + x\right) \right]^{\tan(2x)} \]

Answer: 0


Question: Determine whether the expression $4\cdot n! - 4n + 1$ can be a perfect square for any integer $n > 4$. Provide a justification for your answer.

Answer: No


Question: Find the locus of the circumcenter of triangle $\Delta AQB$, where $A$ and $B$ are points of tangency from a point $Q(\alpha, \beta)$ on the line $x - 2y + 8 = 0$ to the circle $x^2 + y^2 = 4$. The point $P(8, 8)$ lies on the parabola $y^2 = 8x$, and a tangent is drawn at $P$. Determine the locus of the circumcenter of $\Delta AQB$.

Answer: x - 2y + 4 = 0


Question: If the direct sum \( \bigoplus_{i \in I} P_i \) of a family of \( R \)-modules is projective, does it follow that each \( P_i \) is a projective module for all \( i \in I \)?

Answer: Yes


Question: Compute the degree of the field extension \([\mathbb{Q}(\sqrt{3}, \sqrt[3]{2}): \mathbb{Q}]\).

Answer: 6


Question: Determine whether the integral \( \int_0^\infty \frac{x^\theta}{1+x^2} \, \mathrm{d}x \) converges for different values of \( \theta \).

Answer: (-1, 1)


Question: Consider the function \( f(a,b) = h(a) + g(b) \) with the constraint \( a + b = 1 \). Determine whether \( f(a,b) \) is increasing in \( a \) by analyzing the expression \( \frac{dh(a)}{da} - \frac{dg(b)}{db} \). Is the function \( f(a,b) \) increasing in \( a \) if this expression is positive?

Answer: Yes


Question: In a complete metric space, does the condition \( \lim_{t\rightarrow \infty}||x(t)-x_e||=0 \) imply that there exists a \( T \in \mathbb{R} \) such that for all \( \epsilon > 0 \) and \( t \geq T \), \( ||x(t)-x_e||<\epsilon \)?

Answer: No


Question: Let \( V \) be a finite-dimensional real inner product space, and let \( T: V \rightarrow V \) be a Hermitian operator. Suppose \( n \) is an odd positive integer. Prove that there exists an operator \( S: V \rightarrow V \) such that \( T = S^n \).

Answer: S^n = T


Question: Given a positive integer \(n\) and an integer \(m\), find the number of ways to choose integers \(i, j, k\) such that \(1 < i, j, k < m\) and the expression \((((n \mod i) \mod j) \mod k) \mod n\) achieves its maximum value \(L\). Assume \(L = n - 1\).

Answer: 0


Question: A box contains 5 balls marked with the number 1, 3 balls marked with the number 3, and 2 balls marked with the number 9. We randomly select a ball 100 times with replacement. What is the probability that the product of the numbers on the selected 100 balls is between \(3^{60}\) and \(3^{100}\)?

Answer: 0.91


Question: Evaluate the integral \( \int_2^4 \left( \log_x 2 - \frac{(\log_x 2)^2}{\ln 2} \right) \, dx \).

Answer: 0


Question: Evaluate the limit using Taylor's Theorem: \[ \lim_{x \to 0} \left( \frac{1}{\sin^2{(x)}} - \frac{1}{\sin{(x^2)}} \right) \]

Answer: \dfrac{1}{3}


Question: A graph with 40 vertices is given. It is known that among any 5 vertices, there is one connected to the other 4. What is the minimum possible number of edges in this graph?

Answer: 774


Question: Find the limit of the sequence as \( n \to \infty \): \[ a_n = \frac{\sqrt[3n]{4} - 6 \sqrt[3n]{2} + 9}{\sqrt[2n]{9} - 4 \sqrt[2n]{3} + 4} \]

Answer: 4


Question: Let \( \Pi_1, \Pi_2, \) and \( \Pi_3 \) be the planes with Cartesian equations \( x + 2y + 3z = 5 \), \( x - y + 2z = 7 \), and \( 3x - 3y + kz = 10 \) respectively, where \( k \) is a constant. Find the value of \( k \) given that \( \Pi_1, \Pi_2, \) and \( \Pi_3 \) do not have a unique point of intersection.

Answer: 6


Question: Calculate the line integral \( \int_{CR} \vec{G} \cdot d\vec{r} \) for the vector field \( \vec{G} = \left( \frac{x}{\sqrt{x^2+y^2-1}}, \frac{y}{\sqrt{x^2+y^2-1}} \right) \), where \( CR \) is a circle centered at the origin with radius \( R > 1 \). Assume \( x^2 + y^2 > 1 \).

Answer: 0


Question:
Find the constant \( c \in (0, 2\pi) \) such that the Fourier transform \( \hat{f}(m) = 0 \) for all \( m \in 7\mathbb{Z} \backslash \{0\} \), where the function \( f_c(x) \) is defined as:
\[
f_c(x) = \begin{cases} \frac{x}{c}, & 0 \leq x \leq c \\ \frac{2\pi - x}{2\pi - c}, & c < x \leq 2\pi \end{cases}
\]
The Fourier transform is given by:
\[
\hat{f}(m) = \frac{1}{2\pi} \left[ \int_0^c \frac{x}{c} e^{-imx} \, dx + \int_c^{2\pi} \frac{2\pi - x}{2\pi - c} e^{-imx} \, dx \right]
\]

Answer: \dfrac{2\pi}{7}


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

Answer: \dfrac{\pi}{2}


Question: Determine whether the ideal \( \langle x^2+y^2+z^2-1,3x^2+y^2-z^2-1,x+z\rangle \) in \( \mathbb{C}[x,y,z] \) is a radical ideal.

Answer: Yes


Question: Verify that the product of the Jacobian determinant and its inverse is 1, given the transformations: \( x = e^u \cos v \) and \( y = e^u \sin v \).

Answer: 1


Question: Is the one-dimensional Hausdorff measure of the graph of a continuous function defined on \([0,1]\) at least 1? Let \(f:[0,1]\to\mathbb{R}\) be continuous. We define the graph of \(f\) to be the set \(\text{graph}\,f=\{(x,f(x)):x\in [0,1]\}\). Is \(\mathcal{H}^1(\text{graph}\,f)\ge 1\), where \(\mathcal{H}^1\) is the one-dimensional Hausdorff measure on \(\mathbb{R}^2\)?

Answer: Yes


Question: Show that \(\sum\limits_{i=1}^n \log\left(\frac{n}{i}\right) = \Theta(n)\).

Answer: \Theta(n)


Question: Given a function $f$ with domain $[-1,1]$ such that the coordinates of each point $(x,y)$ on its graph satisfy $x^2 + y^2 = 1$, find the total number of points at which the function $f$ is necessarily continuous.

Answer: 2


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

Answer: \sqrt{2}


Question: Suppose \( f(x,y) \) is defined on \([0,1]\times[0,1]\) and is continuous in each dimension. Prove that \[ \lim_{m \to \infty ,n \to \infty } f\left(\frac{{\left\lfloor {mx} \right\rfloor }}{m},\frac{{\left\lfloor {ny} \right\rfloor }}{n}\right) = f(x,y) \] for any \(x, y \in [0,1]\).

Answer: f(x,y)


Question: Determine the signature of the symmetric bilinear form on \( \mathbb{R}^n \) defined by \( e_i \cdot e_j = 1 - \delta_{ij} \), where \( e_1 \cdot e_1 = e_2 \cdot e_2 = e_3 \cdot e_3 = \cdots = 0 \) and \( e_1 \cdot e_2 = e_1 \cdot e_3 = e_2 \cdot e_3 = \cdots = 1 \). The signature is the number of vectors in an orthogonal basis squaring to \(+1, -1, 0\). What is the signature for \( n > 1 \)?

Answer: (1, n-1, 0)


Question: Determine whether the function \( f: \mathbb{N} \rightarrow \mathbb{N} \) defined by \( f(x) = ((x \bmod 9) + 1) \cdot 10^{\lfloor \frac{x}{9} \rfloor} \) has an inverse that maps elements from its image back to its domain.

Answer: Yes


Question:
A machine has two components and fails when both components fail. The number of years from now until the first component fails, \(X\), and the number of years from now until the machine fails, \(Y\), are random variables with joint density function
\[f(x,y)=\begin{cases}\frac{1}{18}e^{-(x+y)/6}&\text{if }0<x<y\\0&\text{otherwise}\end{cases}\]
Calculate \(\operatorname{Var}(Y|X=2)\).

Answer: 36


Question: Calculate the expectation of the expression \( n(\bar{X}-\theta)^2/\sigma_{n}^2 \) given i.i.d. data points \( X_{1},...,X_{n} \) from an unknown smooth distribution \( f(x) \) with \( EX=\theta \). Asymptotically, this expectation is expected to be 1 as the test statistic converges in distribution to \( \chi_{1}^{2} \).

Answer: 1


Question: Given that 2011 is a prime number, calculate \(2009! \mod 2011\).

Answer: 1


Question: Evaluate the definite integral \( \int_{-2}^{2}\int_{0}^{\sqrt{1-x^2}}\sqrt[3]{x^2+y^2} \, dx \, dy \).

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


Question: Find all functions \( f: \mathbb{N} \cup \{0\} \to \mathbb{N} \cup \{0\} \) such that for all \( x \in \mathbb{N} \cup \{0\} \), the equation \( f(x+1) + 1 = f(f(x) + 1) \) holds.

Answer: f(x) = x + 1


Question: For \(\alpha=\cos\left(\frac{2\pi}{7}\right)+i \sin\left(\frac{2\pi}{7}\right)\), find the value of \(n\) such that \(\left|\sum_{r=0}^{3n-1}  (\alpha^{2^{r}})\right|^2=32.\)

Answer: 4


Question: The surface defined by the equation \( z = x^2 - y^2 \) is intersected by the plane \( y = 3x \), resulting in a curve. Determine the slope of this curve at the point \( (1, 3, -8) \).

Answer: -16


Question: Determine the exact number of real roots of the equation \(x^{2016} - x^{2011} + x^{2006} - x^{2001} + 1 = 0\).

Answer: 0


Question: If the limit of a density function exists, does it necessarily follow that the limit is zero? Formally, if \( \exists a \in \mathbb{R} \lim_{t \rightarrow \infty} f(t) = a \), must \( a = 0 \)? Provide a justification for your answer.

Answer: 0


Question: Determine the number of distinct subgroups of the automorphism group of the field \( \mathbf{F}_{3^{100}} \).

Answer: 9


Question: Given that $x + y + z + w = 29$ where $x$, $y$, and $z$ are real numbers greater than 2, find the maximum possible value of the expression $(x-1)(y+3)(z-1)(w-2)$. Provide your answer as a single number.

Answer: 2401


Question: Does there exist a meromorphic function on \(\mathbb{C}\) such that all of its Taylor coefficients are prime numbers? Specifically, is there an unbounded sequence \(a_0, a_1, \ldots \in \mathbb{N}\) of primes such that the function \(O(z) = \sum_{n \ge 0} a_n z^n\) is meromorphic?

Answer: No


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

Answer: 0


Question: Suppose $D$ is a simple closed curve, and $z_0$ is a point not on the boundary $\partial D$. Determine whether the integral \( \oint_{\partial D} \log|z_0 - z| \, dz = 0 \) holds true by applying the Cauchy–Goursat theorem.

Answer: 0


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

Answer: -3


Question: Compute the value of the product: \[ \prod_{x=1}^{44}\left(1-\frac{1}{\tan(x^\circ)}\right) \]

Answer: 2^{22}


Question: Determine whether the function \( F(x,y) = \sin(x)\arctan(y) \) is Lipschitz continuous with respect to the \( \|\cdot\|_\infty \) norm. If it is, find the Lipschitz constant.

Answer: \dfrac{\pi}{2}


Question: Evaluate the triple integral \( \int\int\int_D (x+y+z) \, dx\,dy\,dz \) where the region \( D \) is defined by \( 0 \leq x \leq y \leq z \leq 1 \).

Answer: \dfrac{1}{4}


Question: Let $X$ be a set and $\sim$ the equivalence relation generated by $(x,0)\sim (y,0)$ for $(x,0),(y,0)\in X\times [0,1]$. Determine whether the statement $(x,t_x)\sim (y,t_y) \implies (x,0)\sim(y,0)$ is true.

Answer: True


Question: Suppose that $u$ is a harmonic nonconstant function on a disk $D(z_0,R)$ with $u(z_0)=0$. Is it true that on each circle $C(z_0,r)$, where $0<r<R$, the function $u$ will take both positive and negative values? Justify your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \to \infty} n \cos x \cos(\cos x) \cdots \underbrace{\cos(\cos(\cdots(\cos x)))}_{n \text{ times of } \cos}. \]

Answer: 0


Question: Find the limit \( \lim_{n \to \infty}{\frac{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n-1)}{2\cdot 4 \cdot 6 \cdot \ldots \cdot 2n}} \) using the squeeze theorem.

Answer: 0


Question:
Given the function defined by the following conditions:
\[
\begin{align}
f'(x) &= \frac{1}{2} f(x)\quad\text{for all } x,\\
f(0) &= 1,\\
|f(x)| &\le 6\quad\text{for all } x \in [-3, 3],
\end{align}
\]
find the higher error bound of the approximation of \(f(2)\) using the 6th degree MacLaurin polynomial \(T_6(2)\).

Answer: \dfrac{1}{840}


Question: Determine whether the limit \( \lim_{n\to\infty}(n!-2)^2 \) exists with respect to the topology on \( \mathbb{Z} \) defined by the basis sets \( B(c, r) = \{c+kr \mid k \in \mathbb{Z}\} \), where \( c \) and \( r \) are integers and \( r \) is positive.

Answer: 4


Question: There are an equal number of red, yellow, green, and blue cards. One card is placed in a box, and the rest are randomly selected. It is observed that red cards are most frequently selected, followed by yellow, green, and blue. What is the most probable color of the card in the box?

Answer: blue


Question: Determine the function that has the longest sequence of derivatives before returning to the original function. What is the length of this sequence?

Answer: 4


Question:
Given the joint probability density function of random variables $X$ and $Y$:
\[
f(x,y)=\begin{cases}
24xy, & 0<x<1, 0<y<1, x+y<1 \\
0, & \text{elsewhere}
\end{cases}
\]
Calculate the probability $P(X+Y<\frac{1}{2})$. Determine the appropriate limits for the double integral to find this probability.

Answer: \dfrac{1}{16}


Question: Find the remainder when \( {}^{119}C_{33} \) is divided by 5.

Answer: 3


Question: Solve the equation \( \frac{4}{\pi}\arctan{(1 + x)}\cosh{x} = 1 \) for \( x \).

Answer: 0


Question:
Let \( \{X_n, n \geq 1\} \) be a sequence of random variables with finite means satisfying the condition:
\[ E(X_{n+1} | X_0, X_1, \ldots, X_n) = aX_n + bX_{n-1}, \quad n \geq 1 \]
where \( a > 0, b < 1 \), and \( a + b = 1 \). Determine the value of \( \alpha \) such that \( S_n = \alpha X_n + X_{n-1}, \ n \geq 1 \) is a martingale with respect to the natural filtration generated by the sequence \( X_n \).

Answer: \dfrac{1}{b}


Question: A large university has a 13-day registration period for students. The number of elapsed days before a randomly selected student registers follows a continuous distribution with a density function that is symmetric about day 6.5 and proportional to \( \frac{1}{t+1} \) between days 0 and 6.5. A student registers at the 60th percentile of this distribution. Calculate the number of elapsed days in the registration period for this student.

Answer: 8.99


Question:
Evaluate the limit \( \lim_{(x,y)\rightarrow (0,0)}\frac{x^5-\sin(y^3)}{4x^4+3y^2} \) and determine if the following computations are correct:
\[
0\leq \left|\frac{x^5-\sin(y^3)}{4x^4+3y^2}\right|
\leq\frac{|x^5|}{4x^4}+\frac{|y^3|}{3y^2}
=\frac{|x|}{4}+\frac{|y|}{3}
\]
Then, by the squeeze theorem, the limit is zero.

Answer: 0


Question: Find the minimum value of \( p \) such that \( 5^p \equiv 1 \pmod{p} \).

Answer: 2


Question:
Determine whether the inclusion map \(i:C^{1}_{2\pi}\rightarrow C_{2\pi}\) is a compact operator, where:

- \(C^{1}_{2\pi}=\{u\in C^{1}((0,2\pi),\mathbf{R}^n):u(0)=u(2\pi)\}\)
- \(C_{2\pi}=\{u\in C^{0}((0,2\pi),\mathbf{R}^n):u(0)=u(2\pi)\}\)

The space \(C_{2\pi}\) is equipped with the norm \(\|u\|_0=\max|u(s)|\), and \(C^{1}_{2\pi}\) is equipped with the norm \(\|u\|_1=\|u\|_0+\|\frac{du}{ds}\|_0\).

Answer: Yes


Question: Evaluate the integral: $$\int_0^1 \frac{\sin(\ln(x))}{\ln(x)}\,dx$$

Answer: \dfrac{\pi}{4}


Question: What is the characteristic function of the Dirac delta function?

Answer: 1


Question: Does there always exist a smooth non-vanishing normal vector field on a smooth regular simple closed curve in \(\mathbb{R}^3\)? That is, can we find a smooth function \(n: C \to \mathbb{R}^3 \setminus \{0\}\) such that \(n(s)\) is perpendicular to the tangent vector at \(s\) for every \(s \in C\)?

Answer: Yes


Question: Compute the curvature of the helix \( \gamma(t) = (\cos(t), \sin(t), t) \) using the definition \( \kappa(t) = \frac{|a^{\perp}(t)|}{|v(t)|^2} \), where \( a^{\perp} \) is the perpendicular component of the acceleration and \( v(t) = (-\sin(t), \cos(t), 1) \).

Answer: \dfrac{1}{2}


Question: Find the value of the following expectation involving Brownian motion: \[ E\left[\int_{0}^{T} W_t^{n} dW_t\right] \] where $n$ is a constant power of $W_t$.

Answer: 0


Question: Let $X$ be a projective variety with two morphisms $f:X\rightarrow Y$ and $g:X\rightarrow Z$ with irreducible fibers of positive dimension. Assume that $Pic(X) = f^{*}Pic(Y)\oplus g^{*}Pic(Z)$. If $D$ is a divisor on $X$, we can write $D = f^{*}D_Y + g^{*}D_Z$, where $D_Y$ and $D_Z$ are divisors on $Y$ and $Z$ respectively. If $D$ is effective, are $D_Y$ and $D_Z$ effective as well?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left( \frac{n!}{n^n} \right)^{\frac{1}{n}} \] using basic tools of calculus.

Answer: \dfrac{1}{e}


Question: Evaluate the integral \( \int_{C} \frac{z^2+1}{(z+1)(z+2)} \, dz \) where \( C \) is the circle \( |z|=\frac{3}{2} \). Choose the correct answer from the following options: \( a) \, 0 \), \( b) \, \pi i \), \( c) \, 2\pi i \), \( d) \, 4\pi i \).

Answer: D


Question: Given a set \( X = \{x_1, x_2, \ldots, x_n\} \) where \( x_i = 2^i \), a subset \( S \) is formed by selecting each \( x_i \) independently with probability \( p_i = \frac{1}{2^i} \). What is the expected value of the smallest number in the subset \( S \)?

Answer: 2


Question: Evaluate the limit: $$ \lim_{x\to0}\left(\frac{1+x2^x}{1+x3^x}\right)^\frac{1}{x^2} $$

Answer: \dfrac{2}{3}


Question: Determine whether the infinite product \(\prod_{n=1}^{\infty}\left(1-\frac{1}{p_n}\right)\), where \(p_n\) is the nth prime number, converges to a nonzero value.

Answer: 0


Question:
Solve the periodic ordinary differential equation for \( y(x) \), where \( a \in \mathbb{R} \) and \( y \) is a \( \pi \)-periodic function:
\[ y(x)\sin(x) - y'(x)\cos(x) = a\sin(x) \]

Answer: a


Question: In a topological space $(X, \tau)$, if $A \cup B \in \tau$, must it be true that $A, B \in \tau$? Provide a justification for your answer.

Answer: No


Question: Let $A \in \mathbb{R}^{n \times n}$ be a positive semidefinite (PSD) matrix. Suppose $x, y \in \mathbb{R}^n$ are vectors such that $x_i > 0$ and $y_i > 0$ for all $i \in \{1, 2, \ldots, n\}$. Can it be concluded that $x^T A y \geq 0$ in general?

Answer: No


Question: Is the right intersection of an oblique circular cone an ellipse?

Answer: Yes


Question: Prove or disprove the statement: If \( u \in B(H) \) is a subnormal operator, then \( u^*u \geq uu^* \).

Answer: True


Question: Calculate the second directional derivative of the function \( f(x,y) = \frac{5}{3}x^3 + 5x^2y - 5y^3 \) in the direction of the vector \( \mathbf{u} = \left\langle \frac{3}{5}, \frac{4}{5} \right\rangle \) at the point \((2,1)\). Find \( D^2_\mathbf{u}f(2,1) \).

Answer: \dfrac{54}{5}


Question: Compute the surface integral \( \int_S F \cdot n \, dS \) where \( F(x,y,z) = (x^2 + \sin y^2, y^2 + \sin z^2, z + \sin (xy)) \) and \( S \) is the surface of the bounded solid defined by the surfaces \( z = x^2 + y^2 + 1 \) and \( z = 2(x^2 + y^2) \), with the normal vector oriented outward from the solid.

Answer: \dfrac{\pi}{2}


Question: Given the function \( f \) such that \( f(1+2x) + f(1-2x) = x^{2016} \), evaluate the integral \( \int_{-1}^3 f(x) \, dx \).

Answer: \dfrac{2}{2017}


Question:
Determine the correct expression for $Y_1$ given the stochastic differential equation \( dY_t = -Y_t \, dW_t + Y_t \, dt \) with initial condition \( Y_0 = 1 \). Two potential solutions are proposed:

1. \( Y_1 = \exp \left( 1 - W_1 \right) \)
2. \( Y_1 = \exp \left( \frac{1}{2} - W_1 \right) \)

Which solution is correct?

Answer: 2


Question: Determine whether every finite ring with identity, where for all elements \( a \) in the ring \( a + a = 0 \), is a subring of \( \text{Mat}_{n \times n}(\mathbb{F}_2) \) for some integer \( n \).

Answer: Yes


Question: Find the maximum value of \( \int_{0}^{1} \{f(x)\}^3 \, dx \) subject to the following conditions: \( f(x) \) is a continuous function on \([0, 1]\), \( \int_{0}^{1} f(x) \, dx = 0 \), and \(-1 \leq f(x) \leq 1\) for all \( x \in [0, 1] \).

Answer: \dfrac{1}{4}


Question: Evaluate the infinite series \( \sum_{n = 0}^{\infty} \frac{F_n}{3^n} \), where \( F_n \) represents the Fibonacci sequence. Use the generating function for the Fibonacci sequence, \( G(t) = \frac{t}{1 - t - t^2} \), to find the sum.

Answer: \dfrac{3}{5}


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0}\frac{\ln(\sin(mx))}{\ln(\sin(nx))}. \]

Answer: 1


Question: Calculate the operator norm of the linear operator \( T: \ell^2 \rightarrow \ell^2 \) defined by \( T((x_n)_{n=1}^{\infty}) = \left( \frac{x_n}{2n} \right)_{n=1}^{\infty} \).

Answer: \dfrac{1}{2}


Question: Provide an example of an infinite field of characteristic $p$ in which the Frobenius endomorphism, defined by the $p$-th power map $x \mapsto x^p$, is not surjective.

Answer: \mathbb{F}_p(t)


Question: Let $G$ be a finite group with a normal subgroup $N$ and a subgroup $H$ such that $G = HN$ and $H \cap N = \{1\}$. Is it true that $G/N \cong H$? Justify your answer.

Answer: Yes


Question: Evaluate the set \( \bigcap_{n \in \mathbb{N}}\bigcup_{m\in\mathbb{N}}A_{m,n} \), where \( A_{m,n} \) is the interval \( \left[n-\frac{1}{m},n+m\right] \).

Answer: \emptyset


Question: Let $f(x)$ be an increasing function of $x$ and $g(x)$ be a decreasing function of $x$. If $x^\star$ maximizes $f(x)[g(x) - c]$, where $c > 0$, is it true that $x^\star$ decreases as $c$ increases?

Answer: Yes


Question:
Prove that for positive numbers \(a, b, c\) such that \(abc = 1\), the following inequality holds:
\[
\frac{1}{1+a+b^2} + \frac{1}{1+b+c^2} + \frac{1}{1+c+a^2} \leq 1.
\]

Answer: 1


Question: Determine the number of zeros of the polynomial \( z^7 - 4z^3 + z + 1 = 0 \) that lie within the interior of the circle \(|z| = 1\).

Answer: 3


Question: Suppose that $f$ is an analytic function from the Riemann sphere to the Riemann sphere. Must $f$ be a rational function? Justify your answer.

Answer: Yes


Question: Calculate \( \left( \frac{-1 + i\sqrt{3}}{1 + i} \right)^{2016} \).

Answer: 2^{1008}


Question: Find the coefficient of \(x^{12}\) in the expansion of the function \( f(x) = \frac{x^{2}+x+2}{(1-x^{2})^{3}} \).

Answer: 77


Question: Let BC be the latus rectum of the parabola \(y^2 = 4ax\) with vertex A. Determine the minimum length of the projection of BC on a tangent drawn in the portion BAC.

Answer: 2\sqrt{2}a


Question: Evaluate the limit: \[ \lim_{n \to \infty} \prod_{r=2}^{n} \cos\left(\frac{\pi}{2^{r}}\right) \]

Answer: \dfrac{2}{\pi}


Question: Determine the nature of the graph of the equation \(x^2 + y^2 - 1 = 0\) in the complex plane. Is it a circle?

Answer: Yes


Question: Evaluate the limit \( \lim_{x\to0}\frac{\Phi^{-1}(1-x)}{\Phi^{-1}(1-x/n)} \), where \( \Phi^{-1} \) is the inverse cumulative distribution function of the standard normal distribution and \( n > 0 \).

Answer: 1


Question: Let \( f : \mathbb{R} \to \mathbb{R} \) be a function such that \( f \in L^1[-a,a] \) for all \( a > 0 \), and define \( g(x) = \lim_{r \to \infty} \frac{1}{r} \int_{x-r}^{x+r} f(t) \, dt \) for every \( x \in \mathbb{R} \). Is it true that \( g \) is an affine map, i.e., \( g(x) - g(0) = x(g(1) - g(0)) \) for all \( x \in \mathbb{R} \)?

Answer: Yes


Question: Determine the values of \( p > 0 \) for which the improper integral \( \int_{0}^{1} \frac{x}{\sin(x^p)} \, dx \) converges.

Answer: (0, 2)


Question:
In the \(xy\)-plane, determine the nature of the graph of the equation \(x^{\log y} = y^{\log x}\). Choose the correct option:

(A) Empty
(B) A single point
(C) A ray in the open first quadrant
(D) A closed curve
(E) The open first quadrant

Answer: E


Question: Determine whether the following statement is true or false: If $f$ is a real-valued function such that for each $k > 0$, the set $f^{-1}((-k,k))$ is measurable, then $f$ is a measurable function.

Answer: B


Question: Let $G$ be a finite group and $H$ a subgroup of $G$ such that $|G:H|=2$. Suppose $K$ is a subgroup of $G$ with odd order. Prove that $K$ is contained in $H$.

Answer: K \subseteq H


Question:
Calculate the variance of the stochastic integral \( \int_0^4 \Phi(t) \, dW(t) \), where the stochastic process \( \Phi(t) \) is defined as follows:
\[
\Phi(t) = \begin{cases}
2, & \text{if } t \in [0,1] \\
W(1), & \text{if } t \in (1,2] \\
W(1.5), & \text{if } t \in (2,3] \\
0, & \text{if } t \in (3,4]
\end{cases}
\]
Here, \( W(t) \) represents a Brownian motion. Use Ito isometry to find the variance.

Answer: \dfrac{13}{2}


Question: Given a polynomial $P(x)$ of degree 4 with the following conditions: $P(2) = -1$, $P'(2) = 0$, $P''(2) = 2$, $P'''(2) = -12$, and $P''''(2) = 24$. Find the value of $P''(1)$.

Answer: 26


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

Answer: 1


Question: Evaluate the limit: \[ \lim_{t \to 0}\frac{1}{t^{2}\left(1+t^{2}\right)}\sin\left(\frac{1}{t}\sin\frac{1}{t}\right)e^{-\frac{1}{t^{2}}}. \]

Answer: 0


Question: Find the maximum integer value of \( n \) such that \( 4n^2 + 5n - 8 \) is a perfect square.

Answer: 9


Question: Find \( f'(0) \) for the function \( f(x) = \int_{0}^{x} \cos{\frac{1}{t}} \, dt \).

Answer: 0


Question: Determine whether there exists a linear transformation \( F: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) such that \( F \circ F = T \), where \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) is given by \( T(x,y) = (10x - 6y, -6x +10y) \). Justify your answer.

Answer: Yes


Question: Determine the integral values of \(k\) such that all roots of the polynomial \(f(x) = x^3 - (k-3)x^2 - 11x + (4k-8)\) are integers.

Answer: 5


Question: Determine the number of elements of order 3 in the symmetric group $S_7$. Provide your answer as a single integer.

Answer: 350


Question: Evaluate the limit: \[ \lim\limits_{x \uparrow 1} \exp \left(-\sum\limits_{n=0}^{\infty}x^n\right) \]

Answer: 0


Question: Does there exist a variety of groups \(\mathfrak{V}\) such that the relatively \(\mathfrak{V}\)-free group of rank 2 is finite, but the relatively \(\mathfrak{V}\)-free group of rank 3 is infinite?

Answer: Yes


Question: Let \(a_1, a_2, \ldots, a_8\) be positive integers such that the sum of any 7 numbers from this list can only yield 56, 58, 61, 63, 64, 65, or 66. What is the largest number in this list?

Answer: 15


Question: Find the values of \( n \) for which the inequality \( \frac{\log_2 n}{n} > \frac{1}{8} \) holds true.

Answer: 2 \leq n \leq 43


Question: Let $f$ be an entire function. Must there exist $R > 0$ such that either $|f(z)| \leq |f'(z)|$ for all $|z| > R$ or $|f'(z)| \leq |f(z)|$ for all $|z| > R$? Answer yes or no.

Answer: No


Question: Evaluate the limit: \[ \lim_{x \to \infty} \left(\frac{x + \log 9}{x - \log 9}\right)^x \]

Answer: 81


Question: Let \(X_1, X_2, \ldots\) be a sequence of random variables defined on the same probability space, with \(E[X_i] = 0\) and \(E[X_i^2] = 1\) for all \(i\). Show that \(P(X_n \geq n \text{ infinitely often}) = 0\).

Answer: 0


Question: Determine whether a surjective linear operator on $\mathbb{Z}^n$ is always bijective. Provide a justification for your answer.

Answer: Yes


Question: Determine if there exists a sufficiently large value of \( x \) such that the inequality \( e^{5x} > 555555e^{4.999x} + 3450000e^{3x} + 10000e^{2x} + 999999x^{500} \) holds true.

Answer: Yes


Question: Evaluate the double integral \( \int_0^1 \int_{x^2}^1 \frac{x^3}{\sqrt{1-y^6}} \, dy \, dx \).

Answer: \dfrac{\pi}{24}


Question: Suppose a function $f: \mathbb{R} \to \mathbb{R}$ has a best constant approximation at a point $x$. This means there exists a constant $c$ such that for all other constants $d$, there is a positive length interval $I$ containing $x$ where for all $y \in I$, $|f(y)-c| \leq |f(y)-d|$. Is $f$ continuous at $x$?

Answer: Yes


Question: Given an n-point metric space (A, d) and a real number $t \geq 1$, determine if there exists an integer $m$ and an embedding $f : A \rightarrow \mathbb{R}^m$ such that for all $x, y \in A$, the inequality $d(x,y) \leq d_1(f(x), f(y)) \leq t \cdot d(x,y)$ holds, where $d_1$ denotes the $l_1$ norm. Can the existence of such an embedding $f$ be verified using linear programming?

Answer: Yes


Question: Evaluate the integral \( \frac{1}{2 \pi i} \oint_C \frac{e^z}{z-2} \, dz \) where \( C \) is the circle \( |z| = 3 \).

Answer: e^2


Question: Let \( P \in \mathbb{Q}[X] \) be an irreducible polynomial of degree \( n \geq 3 \), and let \( \mathbb{L} \) be the decomposition field of \( P \). Denote the Galois group of the extension \( \mathbb{L} : \mathbb{Q} \) by \( G \). Suppose that \( G \) is solvable. Is there a function \( f(n) \), depending only on \( n \), such that the total degree \( D \) of the extension \( \mathbb{K}_r : \mathbb{Q} \) in the tower of extensions \( \mathbb{K}_0 \subseteq \mathbb{K}_1 \subseteq \ldots \subseteq \mathbb{K}_r \) satisfies \( D \leq f(n) \)?

Answer: Yes


Question: Let $E \subset \mathbb{R}^d$ be a domain and $f(x,y): E \times E \to \mathbb{R}$ be a function such that $f \in C^2(E \times E)$. Define a function $g$ on $E$ by $g(x) = f(x,x)$. Is it true that $g \in C^2(E)$?

Answer: Yes


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

Answer: \infty


Question:
Evaluate the series \( \sum_{n=1}^{15} \frac{n \cdot 2^n}{(n+2)!} \) and choose the correct value from the options below:

(a) \( 2 - \frac{16 \cdot 2^{17}}{17!} \)
(b) \( 2 - \frac{2^{17}}{17!} \)
(c) \( 1 - \frac{16 \cdot 2^{17}}{17!} \)
(d) \( 1 - \frac{2^{16}}{17!} \)

Answer: D


Question: In an equilateral triangle $ABC$, points $D$, $E$, and $F$ are located on sides $BC$, $CA$, and $AB$ respectively such that $BD/BC = 1/3$, $CE/CA = 1/3$, and $AF/AB = 1/3$. Determine the ratio of the area of triangle $DEF$ to the area of triangle $ABC$. Express your answer as a simplified fraction.

Answer: \dfrac{1}{3}


Question: Can the plane \(\mathbb{R}^2\) be expressed as a disjoint union of sequences, each converging to the point \(0\)?

Answer: Yes


Question: Let $z \in \mathbb{C}$ be an algebraic integer. Determine whether $|z| \in \mathbb{R}$ is also an algebraic integer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{t\to \infty} \int_0^{2\pi} \log \frac{2(e^{2t}+e^{-2t}-2\sin x)}{e^{2t}} \, dx = 2\pi \log 2. \]

Answer: 2\pi \log 2


Question: Let $f:(0,1)\rightarrow\mathbb{R}^n$ be a continuous curve in $\mathbb{R}^n$ with $n>1$, where $f(p)=(x_1(p),x_2(p),...,x_n(p))$. If $f$ is non-constant on every open interval of $(0,1)$, must the set of $p$ such that for some $i$, $x_i(p)$ is a rational number, be dense in $(0,1)$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_{-\infty}^\infty \frac{1}{1+x^2e^{nx^2}}\,dx. \]

Answer: 0


Question: In a dental study, 1000 children were surveyed about their brushing habits and cavities. 15% of the children never or rarely brush, 60% brush once a day, and the remaining children brush two or more times a day. The number of cavities for each child follows a Poisson distribution: those who brush infrequently average 3 cavities, those who brush once a day average 2 cavities, and those who brush two or more times average 1 cavity over a five-year period. Given that a child had exactly one cavity, calculate the probability that the child brushes once a day.

Answer: 0.587


Question: Let $C$ be a smooth projective plane curve, and let $P \in C(k)$. Let $\ell$ denote the tangent line to $C$ at $P$. Let $C^*$ denote the dual curve to $C$ in the dual plane $(\mathbb{P}^2)^*$. Let $P^*$ denote the line in $(\mathbb{P}^2)^*$ that parameterizes the lines passing through the point $P$. Is $P^*$ the tangent line to $C^*$ at its point $\ell$?

Answer: Yes


Question:
Is the following statement true for a discrete random variable $X$ with $X \ge 0$ and a new random variable $Y = X |_{X>0}$? The $n^{th}$ moment of $Y$ is given by:
\[E(Y^n) = \frac{E(X^n)}{1-P(X=0)}\]

Answer: True


Question: Is every discrete absolute value on a field non-archimedean?

Answer: Yes


Question: Let \( \gamma: [0,1] \to \{z \in \mathbb{C}: \text{Im}(z) \ge 0 \} \) be a closed and rectifiable path such that \( \gamma(0) = 1 \), \( \gamma(1) = -1 \), and \( \gamma(t) \neq 0 \) for all \( t \in [0,1] \). Suppose \( f \) is a holomorphic function on an open ball centered at 0 that contains \( \{\gamma\} \) and satisfies \( f(e^{it}) = e^{2it} + 1 \) for \( \pi \le t \le 2\pi \) and \( f'(0) = 0 \). Calculate \( \int_{\gamma} \frac{f(z)}{z^2} \, dz. \)

Answer: 0


Question: Let $f$ be a decreasing function satisfying the functional equation $f(x+y) = f(x) + f(y) - f(x)f(y)$ for all $x, y \in \mathbb{R}$, and suppose $f'(0) = -1$. Evaluate the integral $\int_0^1 f(x) \, dx$. Choose the correct answer from the following options: \(\text{A})\ 1 \quad \text{B})\ 1 - e \quad \text{C})\ 2 - e \quad \text{D})\ \text{None of these}.

Answer: C


Question: Solve the recurrence relation: \[ T(n) = T(n/3) + T(n/2) + n^3 \]

Answer: \Theta(n^3)


Question: Evaluate the quadratic variation \(\langle X \rangle_t\) of the process \(X_t = tB_t\), where \(B_t\) is a standard Brownian motion.

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


Question: If every partitioning of a set $X$ has a choice function, is $X$ necessarily well-orderable?

Answer: Yes


Question: Calculate \( [3]^{2014^{2014}} \mod 98 \) using Euler's Theorem and the Chinese Remainder Theorem.

Answer: 25


Question: Suppose $A$ is a square matrix over a finite field $F_p$, and its characteristic polynomial $C(x)$ is given by $C(x) = \prod_{i=1}^k g_i(x)$, where each $g_i(x)$ is irreducible. If one of the $g_i(x)$ divides the minimal polynomial of $A$, does this guarantee that $g_i(x)$ is an annihilating polynomial of $A$?

Answer: No


Question: Find \( \limsup_{n \to \infty} \left( \frac{2 \cdot 5 \cdot 8 \cdot \cdots \cdot (3n-4)}{3^n n!} \right)^{1/n} \).

Answer: 1


Question: Let $X$ be a topological space with the Borel $\sigma$-algebra $\mathcal{B}_X$. Suppose $Y$ is a Borel subset of $X$, meaning $Y \in \mathcal{B}_X$. Consider $Y$ as a topological space with the induced topology, and let $\mathcal{B}_Y$ be its Borel $\sigma$-algebra. If $A \in \mathcal{B}_X$, is it necessarily true that $A \cap Y \in \mathcal{B}_Y$?

Answer: Yes


Question: If the series \( \sum_{n=0}^{\infty} a_n \) converges, does it necessarily imply that the series \( \sum_{n=0}^{\infty} a_n^3 \) also converges? Provide a justification for your answer.

Answer: No


Question: Evaluate the limit: $$\lim_{n\to\infty}\sum_{r=1}^{n} ((r+1)\sin(\pi/(r+1))-r\sin(\pi/r))$$

Answer: \pi


Question: Evaluate the limit \( \lim_{t\rightarrow 0}\left(t\cot t+t\ln t\right) \) without using L'Hôpital's rule or series expansion.

Answer: 1


Question: Given that the series \( \sum a_n z^n \) has a radius of convergence of 2, determine the radius of convergence of the series \( \sum a_n z^{n^2} \).

Answer: 1


Question: If $A$ and $B$ are square matrices and the product $AB$ is invertible, is $A$ necessarily invertible? Justify your answer.

Answer: Yes


Question: A fixed line $L$ in 3-space and a square of side $r$ in a plane perpendicular to $L$ are given. One vertex of the square is on $L$. As this vertex moves a distance $h$ along $L$, the square turns through a full revolution with $L$ as the axis. Find the volume of the solid generated by this motion.

Answer: r^2 h


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

Answer: e


Question:
Determine the limits of the sequences defined by:

\[ x_n = \sum_{k=0}^{2n} 2^{-k} \cos\left(\sqrt{\frac{k}{n}}\right) \]

and

\[ y_n = \sum_{k=0}^{2n} 2^{-\frac{nk}{n+k}} \]

Answer: 2


Question: Let $\mathbb{K}$ be a field and $A, B \in M_n(\mathbb{K})$ be nilpotent matrices. Suppose that $\text{nullity}(A) \cap \text{nullity}(B) \geq 1$. Determine if there exists a regular matrix $T$ such that the first columns of the matrices $T^{-1}AT$ and $T^{-1}BT$ are zero.

Answer: Yes


Question: Evaluate the limit: $$\lim _{x\to \infty} \left|\frac{x+1}{x-2}\right|^{\sqrt{x^2-4}}$$

Answer: e^3


Question: Let \( \overline{\rm AB} \) be a diameter of a circle \( \omega \) and let \( C \) be a point on \( \omega \), different from \( A \) and \( B \). The perpendicular from \( C \) intersects \( \overline{\rm AB} \) at \( D \) and \( \omega \) at \( E (\ne C) \). The circle with center \( C \) and radius \( \rm CD \) intersects \( \omega \) at \( P \) and \( Q \). If the perimeter of \( \triangle{PEQ} \) is \( p \), find the length of \( \overline{\rm PQ} \).

Answer: \dfrac{p}{3}


Question: Determine the set of points where the function \( f(x) = \sum_{n=1}^\infty \frac{1}{x^2+n^2} \) is differentiable for \( x \in \mathbb{R} \).

Answer: \mathbb{R}


Question: Let \( f: [0,1] \to \mathbb{R} \) be defined by \( f(x) = 0 \) if \( x \) is rational, and \( f(x) = 9^n \) if \( x \) is irrational, where \( n \) is the number of zeros immediately after the decimal point in the decimal representation of \( x \). Compute the Lebesgue integral \( \int_0^1 f(x) \, dx \).

Answer: 9


Question: Suppose K is an unbounded well-ordered set that is minimal in the sense that each proper initial segment of K has strictly smaller cardinality. Let J be another unbounded well-ordered set with the same cardinality as K. Must there exist a subset M of K and an order-preserving embedding \( f: M \to J \) such that \( f(M) \) is cofinal in J? (This means for each \( j \in J \), there exists \( m \in M \) such that \( j < f(m) \).)

Answer: Yes


Question: Evaluate the 1-form \( \phi = x^2 \, dx - y^2 \, dz \) on the vector field \( V = x U_1 + y U_2 + z U_3 \).

Answer: x^3 - y^2 z


Question: Consider an infinite collection of sets, each containing two elements: \(A_1 = \{a_{1,1}, a_{1,2}\}, A_2 = \{a_{2,1}, a_{2,2}\}, \ldots, A_n = \{a_{n,1}, a_{n,2}\}, \ldots\). Determine whether the Cartesian product \(A_1 \times A_2 \times \ldots\) has the same cardinality as the set of natural numbers \(\mathbb{N}\) or the set of real numbers \(\mathbb{R}\).

Answer: \mathbb{R}


Question: Find the smallest positive integer that can be expressed as a sum of two squares in exactly three different ways.

Answer: 325


Question: Find a one-dimensional sufficient statistic for \( \lambda \) based on the observed pair \((X,Y)\), where \(X\) follows a Poisson distribution with parameter \(\lambda\) and \(Y\) follows a Bernoulli distribution with parameter \(\lambda/(1+\lambda)\).

Answer: X + Y


Question:
Determine the constant \(c\) such that the function \(f(x,y)\) defined by
\[f(x,y)=\begin{cases}cxy,&-1\leq x\leq 0\wedge 0\leq y\leq 1\wedge y-x\leq 1,\\cxy,&0\leq x\leq 1\wedge -1\leq y\leq 0\wedge x-y\leq 1,\\0,&\text{else}.\end{cases}\]
is a probability density function.

Answer: -12


Question: If $x, y, z$ are three mutually independent variables, are $x^2$ and $yz$ pairwise independent?

Answer: Yes


Question: Solve the equation: \(2^{x+3} + 3^{x-5} = 2^{3x-7} + 3^{2x-10}\) for \(x\).

Answer: 5


Question: Determine the quadratic variation $[X]_t$ of the process $X_t = \int_0^t B_s \, ds$, where $B_t$ is a standard Brownian motion.

Answer: 0


Question:
Determine the value of \( d \) for which a generalized Cantor set, constructed by retaining intervals of size \( d < 1 \) at both ends and removing an interval of size \( 1-2d \) at each stage, has Lebesgue measure zero. The process continues such that at the \( n^{th} \) stage, \( 2^{n-1} \) intervals of size \( 1-2d^n \) are removed. For which value of \( d \) does the following equation hold?
\[ \sum_{n=1}^{\infty} 2^{n-1} (1-2d^n) = 1. \]

Answer: \dfrac{1}{3}


Question:
Solve the system of linear differential equations given by \( \frac{dy}{dt} = Ay \), where \( A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} \) and \( y = \begin{pmatrix} y_1(t) \\ y_2(t) \\ y_3(t) \end{pmatrix} \). The initial condition is \( y(0) = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \). Determine which of the following expressions correctly represents \( y(t) \):

1. \( y_1(t) = 1 + t + \frac{t^2}{2} \), \( y_2(t) = 1 + t \), \( y_3(t) = 1 \)
2. \( y_2(t) = 1 + t + \frac{t^2}{2} \), \( y_1(t) = 1 + t \), \( y_3(t) = 1 \)
3. \( y_1(t) = 1 \), \( y_2(t) = 1 + t \), \( y_3(t) = 1 + t + \frac{t^2}{2} \)
4. \( y_1(t) = e^{tA} y(0) \)

Answer: 1


Question: Does the condition "every open set is a countable union of closed sets" imply that a topological space is metrizable?

Answer: No


Question: Evaluate the integral \( \int_0^{\pi/2} \frac{\sin((2n+1)t)}{\sin t} \mathrm{d}t \).

Answer: \dfrac{\pi}{2}


Question: Evaluate the limit: \[ \lim_{(x,y)\rightarrow (0,0)} xy\log(\lvert x\rvert+\lvert y\rvert) \]

Answer: 0


Question: A variable parabola with latus rectum \( l \) touches a fixed parabola \( y^2 = 4bx \), with the axes of the two parabolas being parallel. Determine the latus rectum of the parabola that is the locus of the vertex of the moving parabola.

Answer: 4b - l


Question: Determine whether the improper integral \( \int_0^\pi \frac{\cos(x)}{(1-2\sin(x))^{1/3}} \, dx \) is convergent. If it is convergent, calculate its value.

Answer: 0


Question: Given that the characteristic polynomial of a square matrix \( B \) is \( t^2(t^2+1) \), deduce all the possible values of \( \text{rank}(B^2) \).

Answer: 2


Question: Determine the number of distinct terms in the expansion of \((x^3 + \frac{1}{x^3} + 1)^{200}\).

Answer: 401


Question: Let $f(x)$ be a real-valued function defined on the interval $(-5,5)$ such that $e^{-x}f(x)=2+\int_0^x\sqrt{t^4+1}\,dt$ for all $x\in(-5,5)$. Let $f^{-1}(x)$ be the inverse function of $f(x)$. Find $\left(f^{-1}\right)^\prime(2)$. \( \text{Provide your answer in the form } \boxed{\text{answer}}. \)

Answer: \dfrac{1}{3}


Question: Determine the number of invertible 2x2 matrices with entries from the set \( \{0, 1, 2, \ldots, p-1\} \), where \( p \) is a prime number.

Answer: (p^2 - 1)(p^2 - p)


Question: Calculate the sum of the infinite series: \[ \sum_{j=0}^{\infty} (j+1) \left(\frac{1}{1.05}\right)^{j+1} \]

Answer: 420


Question: Given the ellipse \(x^2 + 4y^2 = 32\), a rectangle is inscribed such that its height is twice its width. Tangent lines to the ellipse are drawn at the vertices of the rectangle, forming a quadrilateral. What is the ratio of the area of the rectangle to the area of the quadrilateral?

Answer: \dfrac{32}{289}


Question: Evaluate \(\lim_{n\to\infty}\left(\frac {b_{n+1}}{b_n}\right)^{a_n}\) given that \(\lim_{n\to\infty}\frac{a_n}{n}=a\) and \(\lim_{n\to\infty}\frac{b_n}{n}=b\), where \(a_n\) and \(b_n\) are positive sequences.

Answer: e^a


Question:
Find a formula for the sum \(\sum\limits_{r=1}^{n} (r^2+1)(r!)\). Choose the correct option:

- \((n+1)!\)
- \((n+2)!-1\)
- \(n\cdot(n+1)!\)
- \(n\cdot(n+2)!\)

Answer: C


Question: Find a quartic polynomial with integer coefficients whose roots are the primitive 12th roots of unity.

Answer: x^4 - x^2 + 1


Question: Let $A$ and $B$ be noetherian regular domains of dimension one. If $\mathbb{A}^1_A$ is birational to $\mathbb{A}^1_B$, is $\text{Spec}(A)$ birational to $\text{Spec}(B)$? Provide a justification for your answer.

Answer: Yes


Question: Determine whether an ideal generated by linear homogeneous polynomials is a radical ideal.

Answer: Yes


Question: Evaluate the integral \( \int_{-\infty}^{\infty}\frac{\sin(5x)}{1+(x-\frac{\pi}{2})^2} \, dx \) using the residue theorem.

Answer: \pi e^{-5}


Question: Evaluate the integral: \[ I = \int_{0}^{\frac{\pi}{2}} \ln(2468^{2} \cos^2x + 990^2 \sin^2x) \, dx \]

Answer: \pi \ln 1729


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

Answer: \dfrac{\pi}{4}


Question: Determine whether the following statement is true or false: A function $f$ belongs to the space $A^n$ if and only if $f$ is in $C^{n-1}$ and its $n$-th derivative $f^{(n)}$ is integrable.

Answer: B


Question: Determine whether the number of 2x2 matrices \( A \) over \( \mathbb{Z}_p \) (where \( p > 2 \)) that satisfy \( A^TA = -I \) is equal to the number of 2x2 matrices that satisfy \( A^TA = I \).

Answer: Yes


Question: Evaluate the limit: $$L=\lim_{x \to 0}(x+2x^2+3x^3+4x^4+5x^5)^\frac{1}{\log(4x+3x^2+2x^3+x^4)}$$

Answer: e


Question:
Determine if $3 = 2 + 1$ is the only case where equality holds in the inequality:
\[ p_{n} \leq p_{1}p_{2} \cdots p_{n-1} + 1 \]
where $p_{r}$ denotes the $r^{th}$ prime number.

Answer: Yes


Question: Let $(X,d)$ be a compact metric space, and let $f: X \to X$ be an isometric embedding, meaning that for each $x, y \in X$, we have $d(f(x), f(y)) = d(x, y)$. Is $f$ automatically surjective? In other words, is it true that $\inf_{y \in X} d(x, f(y)) = 0$ for every $x \in X$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\left(\int_{0}^x e^{t^2}dt\right)^2 }{\int_{0}^{x} e^{2t^2}dt} \]

Answer: 0


Question: How many people are needed in a room to have a 50% chance that three people share the same birthday, assuming birthdays are equally distributed?

Answer: 88


Question:
Given the joint density function of the two-dimensional variable \((X, Y)\):
\[
f(x,y) = \begin{cases}
2x^2 & \text{if } -1 < x < 1 \text{ and } 0 < y < |x| \\
0 & \text{otherwise}
\end{cases}
\]
Compute the conditional expected value \(E(X-Y \mid X+Y = -0.5)\).

Answer: -\dfrac{31}{28}


Question:
Let $f \in L^1(\mathbb{R}) \cap C(\mathbb{R})$, meaning $f$ is integrable and continuous. For $z \in \mathbb{C}$ with $\text{Im}(z) \neq 0$, define
\[ g(z) = \int_{-\infty}^\infty \frac{f(t)}{t-z} \, dt. \]
Find \( \lim_{\text{Im}(z) \to 0} \left( g(z) - g(\bar{z}) \right) \).

Answer: 2\pi i f(x)


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} \frac{\sum_{r=0}^{n} \binom{2n}{2r} \cdot 3^r}{\sum_{r=0}^{n-1} \binom{2n}{2r+1} \cdot 3^r} \]

Answer: \sqrt{3}


Question: Let $X$ and $Y$ be metric spaces. Consider a convergent sequence $\{x_n\}$ in $X$ with distinct terms and limit $x$, and a convergent sequence $\{y_n\}$ in $Y$ with limit $y$. Define the function $f: \{x\} \cup \{x_n : n \in \mathbb{N}\} \to \{y\} \cup \{y_n : n \in \mathbb{N}\}$ by $f(x_n) = y_n$ for all $n \in \mathbb{N}$ and $f(x) = y$. Is the function $f$ continuous?

Answer: Yes


Question: A particle P starts at the origin and moves along the parabola \( y = x^2 \) in the first quadrant such that its distance from the y-axis increases at a rate of 3 units per second. Let Q be the point where the tangent to the parabola at P intersects the x-axis, and R be the foot of the perpendicular from P to the x-axis. Let \( \phi \) be the angle in radians subtended at Q of the triangle PQR. How fast is \( \phi \) changing when P is 4 units away from the y-axis?

Answer: \dfrac{6}{65}


Question: Evaluate the sum of the infinite series: \[ \sum_{k=1}^{\infty} \frac{2k^2}{5^k} \]

Answer: \dfrac{15}{16}


Question: A point moves around the circle given by the equation \(x^2 + y^2 + 8x + 4y - 5 = 0\) with center \(C\). The point breaks away from the circle at either point \(A\) or point \(B\) and moves along a tangent to the circle passing through the point \(D(3, -3)\). Find the area of the quadrilateral \(ABCD\).

Answer: 25


Question: Determine the values of $n$ for which the equation $x^2 - (2n+1)xy + y^2 + x = 0$ has no solution in natural numbers. Note that for $n=1$, the equation has a trivial solution.

Answer: n \geq 2


Question: Calculate the sum of the infinite series: \[ S = \sum_{n=0}^{\infty} \frac{(2n+1)!}{8^n (n!)^2} \]

Answer: 2\sqrt{2}


Question: Find the inverse of the element \((2x+3)+I\) in the ring \(\mathbb{Z}_5[x] / \langle x^2+x+2\rangle\).

Answer: 3x + 1


Question: Find the coefficient of \(x^7\) in the Taylor series expansion of \(\arcsin(x)\) around \(x=0\).

Answer: \dfrac{5}{112}


Question: Evaluate the limit: $$\lim_{n\to \infty} \frac{n!(3n)!}{((2n)!)^2}$$

Answer: \infty


Question: Determine whether every finite group can be presented by a finite set of generators and a finite set of relations. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit \( \lim_{n \to \infty} \int_{0}^\infty \cos^{2n}(\pi f(x)) \, dx \), where \( f \) is a measurable function on \([0, \infty)\) such that \( f(x) \in \mathbb{Z} \) if and only if \( x \in [0, 1] \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \to -\infty} \sum_{k=1}^{1000} \frac{x^k}{k!} \]

Answer: \infty


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

Answer: (-\infty, -1) \cup (-1, 1) \cup (1, \infty)


Question: Let $\alpha_1, \alpha_2, \alpha_3$ be the roots of the polynomial $x^3 + px^2 + qx + r$, where $p, q, r \in \mathbb{C}$. Express the sum $\sum_{i,j=1,i\neq j}^{3} \alpha_i^2 \alpha_j$ in terms of $p$, $q$, and $r$. Provide your answer in the simplest form.

Answer: 3r - pq


Question: Evaluate the limit without using L'Hopital's rule: \[ \lim_{x\to0}\frac{1}{x^3}\left(\left(\frac{2+\cos x}{3}\right)^x-1\right) \]

Answer: -\dfrac{1}{6}


Question: Evaluate the limit: \[ \lim_{h \to 0} \frac{\sin\left(\frac{h}{2}\right)-\frac{h}{2}}{h\sin\left(\frac{h}{2}\right)} \] without using l'Hospital's rule.

Answer: 0


Question: What is the minimal cardinality of a generating set for the symmetric group of permutations? Determine the smallest number of permutations needed such that all other permutations can be expressed as products of these permutations.

Answer: 2


Question: Place 5 points in a rectangle of size $17 \times 32$ such that the minimum distance between any two points is maximized. What is the highest possible minimum distance between any two points?

Answer: 17


Question: Is there an infinite torsion group $G$ such that for every subgroup $H \leq G$, there exists some $n \in \mathbb{N}$ with the property that for all $x \in H$, $x^n = 1$, and for all $x \in G \setminus H$, $x^n \neq 1$?

Answer: No


Question: Determine whether the following inequality is true for any real square matrix \( A \): \[ \det \left (\sum_{i=1}^{n} {A_{i}}^{t}A_{i} \right ) \geq 0 \]

Answer: True


Question: Evaluate the integral \( \int \int 4(x^{4} - y^{4}) e^{-xy} \, dx \, dy \) by changing variables, where \( u = x^{2} - y^{2} \) and \( v = 2xy \).

Answer: 0