Question: Calculate the determinant of the matrix \( \det \left((A^2 B^{-1})^{-1}+BC \right) \) given that \( A, B, C \in \mathbb{R}^{n\times n} \) with \( \det(A) = 2 \), \( \det(B) = -8 \), and \( \det(A^2 C+I)=1 \). Assume that \( C \) is invertible.

Answer: -2


Question: Determine the Galois group of the field extension \(\mathbb{Q}(\sqrt[3]{3},\sqrt{-3})\) over \(\mathbb{Q}\).

Answer: S_3


Question: Determine the minimum value of the function \( f(a) = \arccos(a) + \arcsin(a) + \arctan(a) \) for \( a \in [-1, 1] \).

Answer: \dfrac{\pi}{4}


Question: Find the particular solution of the differential equation \(y' = y^2 - sy - s\) given that the general solution is \(y(s) = ks + l\).

Answer: s + 1


Question: Calculate \( h'(2) \) given that \( h(x) = g(f(x, x)) \), where \( g(x) = f(x, f(x, x)) \). The function \( f(x, y) \) is differentiable, and the following values are provided: \( f(2, 2) = 4 \), \( f_x(2, 2) = 3 \), and \( f_y(2, 2) = 7 \).

Answer: 730


Question: Does a bijection on the vertices of a graph always induce a symmetry on the graph?

Answer: No


Question: Given the set \(\{1, 2, 3, \ldots, 20\}\), where each element has a probability \(p_x\) of being chosen, determine the minimum probability that two independently chosen numbers have a common factor.

Answer: 0


Question: Consider a continuously differentiable bounded function $f(t)$ defined on $[0, \infty)$ with a bounded, uniformly continuous derivative. If $\displaystyle\lim_{t \to \infty} [f(t)\dot{f}(t)] = 0$, where $\dot{f}(t)$ denotes the derivative of $f(t)$, determine whether it must be true that $\displaystyle\lim_{t \to \infty} |f(t+T) - f(t)| = 0$ for all finite $T > 0$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the matrix $X = A^*(AA^*)^{-1}$ is the Moore-Penrose inverse of the matrix $A$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{\infty}\frac{dx}{x^{4}+2x^{2}+1} \) using the residue theorem.

Answer: \dfrac{\pi}{4}


Question: Evaluate the limit \( \lim_{n \to \infty} n^2\int_0^{1/n} x^{x+1}\, dx \).

Answer: \dfrac{1}{2}


Question: Determine all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the following conditions: \( f(x-y) = f(x) - f(y) \), \( f(x+1) = f(x) + 2 \), and \( f\left(\frac{x+y}{2}\right) = \frac{f(x) + f(y)}{2} \).

Answer: 2x


Question: Determine the value of $q$ for which the integral \( \int^{1}_{0}\frac{1}{(\tan (x))^{q}}dx \) converges.

Answer: q < 1


Question: Determine whether the following statement is true or false: Every infinite algebraic extension of a complete field has countable degree.

Answer: True


Question: In triangle $\Delta ABC$, where $a \ge b \ge c$, it is given that \( \frac{a^3 + b^3 + c^3}{\sin^3 A + \sin^3 B + \sin^3 C} = 7 \). Using the Law of Sines and considering the restrictions on the sides, determine the maximum possible value of $a$.

Answer: \sqrt[3]{7}


Question: Let $f(x)$ be a function that is positive almost everywhere on a set $X$ with Lebesgue measure $m$. Suppose there is a constant $c > 0$ such that for all $n$, $m(A_n) > c$, where $A_n$ is a sequence of subsets of $X$. Can the integral of $f(x)$ over $A_n$ converge to zero?

Answer: Yes


Question: Determine whether a measurable set with positive Lebesgue measure in \(\mathbb{R}\) can always be expressed as a countable union of non-measurable sets.

Answer: Yes


Question: Determine whether the group $GL(2,\mathbb{R})$, the general linear group of 2x2 invertible matrices over the real numbers, contains a cyclic subgroup of order $n$. Provide a justification for your answer.

Answer: Yes


Question: Find the eighth derivative of the function \( f(x) = \frac{3}{1+x-2x^2} \) at \( x = 0 \) using its Maclaurin series expansion. What is \( f^{(8)}(0) \)?

Answer: 20684160


Question: Determine the degree of the expression $\rho_7+\rho_7^2+\rho_7^4$ over $\mathbb{Q}$, where $\rho_7$ is a primitive seventh root of unity.

Answer: 2


Question: Consider the Euclidean domain \( R = \{ a+b \sqrt{2} \mid a \in \mathbb{Z}, b \in \mathbb{Z} \} \) and the ideal \( I = \langle 4+\sqrt{2} \rangle \lhd R \). Given that \(|R/I| = 14\) and \(\text{char}(R) = 14\), determine the isomorphism between \(R/I\) and a well-known ring. Find the number of idempotent elements in the quotient ring \(R/I\).

Answer: 4


Question: Let $R$ be a commutative unital ring. For every maximal ideal $\mathfrak{m}$ of $R$, there is a localization map \( l_\mathfrak{m}: R \to R_\mathfrak{m} \) given by \( x\mapsto \frac{x}{1} \). Determine if it is true that an element \( x\in R \) is a unit if \( l_\mathfrak{m}(x) \) is a unit in \( R_\mathfrak{m} \) for every maximal ideal \( \mathfrak{m} \).

Answer: True


Question: Calculate the sum of the series: \[ \sum^{\infty}_{k=1}\frac{(-1)^{k-1}}{k}\sum^{\infty}_{r=0}\frac{1}{k2^{r}+1} \]

Answer: 1


Question: Calculate the surface area of the surface generated when the curve defined by \(8y^2 = x^2 - x^4\) is revolved around the x-axis, for \(-1 \leq x \leq 1\).

Answer: \dfrac{\pi}{2}


Question: Determine if there exists a natural number \( m \) such that the equation \( 8m+1 = (2p+1)^2-(2q)^2 \) holds for some natural numbers \( p \) and \( q \), with \( p \neq m \) and \( q \neq m \).

Answer: Yes


Question: Does a nonempty, quasi-affine variety $Y$ always have an open cover consisting of dense affine subsets? Justify your answer.

Answer: Yes


Question: Let $X$ be a complex Banach space and $A: X \rightarrow \mathbb{C}^n$ be a continuous linear operator. If $B_X = \{x \in X : \|x\| \leq 1\}$ denotes the closed unit ball in $X$, is the image $A(B_X)$ necessarily compact in $\mathbb{C}^n$?

Answer: Yes


Question: Given a Riemannian metric on a simply connected open domain $U \subseteq \mathbb{R}^2$ with zero curvature, determine if there always exists a smooth orthonormal frame of vector fields $E_1, E_2$ such that the associated dual coframe of one-forms $E^i$ are closed.

Answer: Yes


Question: Given the functions \( D(\rho) = g'(\rho)e^{ia\rho} - g'(-\rho)e^{-ia\rho} \) and \( S(\rho) = g'(\rho)e^{ia\rho} + g'(-\rho)e^{-ia\rho} \), determine if the ratio \( \frac{D(\rho)}{S(\rho)} \) is independent of the function \( g'(\rho) \) when \( g'(\rho) \neq g'(-\rho) \). If it is independent, find the ratio.

Answer: -i \cot(a\rho)


Question: Given a random variable $X$ with probability density function $f(x) = K|x|$ for $-1 < x < 2$, derive the probability density function of $Y = |X|$. Also, determine the value of the constant $K$.

Answer: \dfrac{2}{5}


Question: Calculate the limit using the Squeeze Theorem: \[ \lim_{x \rightarrow 0} -2\int_{0}^{x^{2}} \cos t^{2} \, dt. \]

Answer: 0


Question: Calculate the flux of the vector field \( F = (y^2 - z)i + (x + z^2)j + (x^2 + y)k \) through the closed surface formed by the lateral surface of a pipe and the two circular discs at its ends. The pipe has a volume of 12, with the radius of the larger circle being 3 and the radius of the smaller circle being 1.

Answer: 0


Question: Let \( x, y, z \) be non-zero real numbers such that \( \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 7 \) and \( \frac{y}{x} + \frac{z}{y} + \frac{x}{z} = 9 \). Find the value of \( \frac{x^3}{y^3} + \frac{y^3}{z^3} + \frac{z^3}{x^3} \).

Answer: 157


Question: Determine whether there exists a non-zero element in the quotient ring $R = (\mathbb{Z}/2)[x] / (x^{17} - 1)$ such that the square of that element is zero.

Answer: No


Question: Determine the largest natural number \( n \) such that \( n^{6033} < 2011^{2011} \).

Answer: 12


Question:
Find the value of \( p > 1 \) such that the series \( \sum_{n=1}^{\infty} f_n \) converges in \( L^p(\mathbb{R}) \), where:
\[
f_n(x) = \frac{1}{1+n^2\sqrt{x}}\chi_{[\exp(2n),2\exp(2n+2)]}(x)
\]

Answer: 2


Question: Find the maximum value of the real number \( t \) such that the inequality \[ \sum_{i=1}^n a_i \sum_{i=1}^n \frac{1}{a_i} \ge n^2 + t \cdot \frac{\displaystyle\sum_{1\le i<j \le n} (a_i - a_j)^2 }{\left( \displaystyle\sum_{i=1}^n a_i\right)^2} \] holds for all positive numbers \( a_i \) where \( n \ge 2 \).

Answer: 4


Question: Determine whether the following statement is true or false: For any matrix $A \in \mathbb{R}^{n \times n}$, if $\sigma$ is a singular value of $A$, then $\sigma^2$ is a singular value of $A^2$. Provide a justification for your answer.

Answer: False


Question: Find an integer \( m \geq 2 \) such that the congruence equation \( x^2 \equiv 1 \pmod{m} \) has more than two distinct solutions in the residue class ring \( \mathbb{Z}/m\mathbb{Z} \).

Answer: 8


Question:
Evaluate the limit as \( n \) approaches infinity:
\[ \lim_{{n}\to{\infty}} \left(\dfrac{2^x+3^x+\cdots+n^x}{n-1}\right)^{1/x} \]
where \( n > 1 \) and \( x \geq 1 \).

Answer: \infty


Question: Let $D$ be a principal ideal domain (PID), and let $E$ be a domain containing $D$ as a subring. If $d$ is a greatest common divisor (gcd) of $a$ and $b$ in $D$, determine whether $d$ is also a gcd of $a$ and $b$ in $E$. Provide a justification for your answer.

Answer: Yes


Question: Find all prime numbers \( x \) such that \( x^x + x \) is a factorial.

Answer: 2


Question: Consider a square matrix with orthogonal, non-zero rows that are not orthonormal. Determine if it is necessary for each row to have exactly one non-zero element for the columns of the matrix to be orthogonal as well. Answer with 'Yes' or 'No'.

Answer: No


Question: Given an integer lattice $\mathcal{L}(B) = \{ \sum_{i=1}^n x_i b_i : x_i \in \mathbb{Z} \}$, where $B$ is an $n \times n$ matrix with linearly independent columns, and a point $t$ in the linear span of the lattice, determine if it is possible to check in polynomial time whether $t$ lies inside the fundamental parallelepiped generated by the lattice.

Answer: Yes


Question: In an experiment where a fair coin is tossed an infinite number of times, determine the probability that the sequence of heads and tails will never repeat itself.

Answer: 1


Question: Determine the number of real solutions to the equation \(6x^2 - 77\lfloor x \rfloor + 147 = 0\), where \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \(x\).

Answer: 4


Question: Find a function $g(n)$ such that $f(n) \in \Theta(g(n))$ for the given recurrence relation: $$ f(n) = f(5n/13) + f(12n/13) + n \;\;\; \forall n \geq 0 $$.

Answer: n^2


Question: Suppose that $(x_n)$ is a sequence of complex numbers. If the sequence $(S_N(y))_{N\geq1}$ with \[S_N(y) =\sum_{n=1}^N x_ny_n \] converges for every $y = (y_n) \in \ell^1$, does it follow that $x\in \ell^1$? Answer yes or no.

Answer: No


Question: Determine the smallest integer that can replace 13 in the statement: "Every simple planar graph with minimum degree \(\delta \geq 3\) has an adjacent pair with \(\deg(u) + \deg(v) \leq\) [integer]," such that the statement remains true for all planar graphs.

Answer: 13


Question: Calculate the remainder when \(177^{20^{100500}}\) is divided by 60.

Answer: 21


Question: Evaluate the sum of the series \( \sum_{n=-\infty}^{\infty}\frac{1}{n^2+1} \) using the Poisson summation formula.

Answer: \pi \coth(\pi)


Question: Evaluate the integral \( \int_{-\pi}^{\pi} \sin(nx) \cos(mx) \, dx \).

Answer: 0


Question: Determine the number of ways to choose $k$ numbers from the first $n$ natural numbers such that the longest consecutive sequence among the chosen numbers is exactly $m$ numbers long. For example, calculate the number of combinations when $k = 7$, $n = 14$, and $m = 3$. Provide your answer as a single integer.

Answer: 1456


Question: Let $f$ be a holomorphic function on $|z|>R$ for some large $R>0$. Suppose there exist $r_2>r_1>R$ such that $|f(z)|$ achieves its maximum value on the circles $|z|=r_1$ and $|z|=r_2$. If $\lim_{z\to\infty}f(z)$ exists and is finite, is $f$ necessarily constant? Provide a justification for your answer.

Answer: Yes


Question: Derive an expression for the derivative of the step function, \( \theta'(x) \), without using the Fourier transform.

Answer: \theta'(x) = \delta(x)


Question: For a random sample $x_1, x_2, \ldots, x_9$ drawn from a Poisson distribution with mean $\theta = 0.4$, the probability density function is given by $f(x;\theta) = \frac{\theta^x e^{-\theta}}{x!}$. A hypothesis test is conducted where the null hypothesis is accepted if $\frac{1}{3} \leq \overline{x} \leq \frac{2}{3}$, and rejected otherwise. Calculate the type I error rate for this test.

Answer: 0.376


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

Answer: 4\pi


Question: Determine whether there exists a polynomial $f(x)$ with integer coefficients such that $f(p_i) \equiv 0 \pmod{p_i}$ for all distinct prime numbers $p_1, p_2, \ldots, p_n, \ldots$. Provide a justification for your answer.

Answer: Yes


Question: Determine the limit of the sequence as \( n \to \infty \): \[ \frac{3+(-1)^n+9\sqrt{n}-7n^5-2[\sqrt[3]{n}]n}{(3n-1)(n-2)(2n-3)(n-4)(4n-5)+2^{-n}} \]

Answer: -\dfrac{7}{24}


Question: Evaluate the integral \( \int_{-\infty}^\infty \frac{\sin x}{x-i} \, dx \).

Answer: \dfrac{\pi}{e}


Question: Calculate the volume of the solid in the first octant that lies below the paraboloid $z = x^2 + y^2$ and above the region bounded by the curves $y = x^2$ and $x = y^2$.

Answer: \dfrac{6}{35}


Question: Consider the family of holomorphic functions $\mathcal F = \{f \text{ holomorphic in } \Bbb D \text{ with } f(\Bbb D)\subset\Bbb D, f\left(\frac{1}{2}\right)=f'\left(\frac{1}{2}\right)=0\}$, where $\Bbb D$ is the unit disc. Compute $\sup_{f\in\mathcal F}|f(0)|$. \( \boxed{?} \)

Answer: \dfrac{1}{4}


Question: Given an integer $n \geq 2$, consider the vector space $V = F^{2n}$ with its standard basis $e_1, \ldots, e_{2n}$. In the second exterior power, $\Lambda^2(V)$, we have the element $\omega = e_1 \wedge e_2 + e_3 \wedge e_4 + \cdots + e_{2n-1} \wedge e_{2n}$. As $\Lambda^{2n}(V)$ 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 $x$. Determine this scalar.

Answer: n!


Question: Determine whether the limit \(\lim_{(u,v)\to(-\infty,\infty)}e^{u+jv}\) exists, where \(u,v\in\mathbb{R}\).

Answer: 0


Question: Determine whether the elements $s$ and $sr$ can generate all elements of the dihedral group $D_{2n}$ without explicitly using $r$. Specifically, is the argument that starting with $s(sr) = r$ allows us to generate all elements of the form $r^k$ for $1 \leq k \leq n-1$, and subsequently all elements of the form $sr^k$, valid for generating the entire group?

Answer: Yes


Question: Determine whether the sequence of functions \(f_n(t) = \frac{nt}{n+t}\) converges uniformly on the interval \([0,1]\).

Answer: Yes


Question: Determine the values of \( n \) for which the characteristic polynomial \( \chi_{A(n)}(X) \) of the matrix \( A(n) \) has at least two real roots: one equal to the golden ratio \( \varphi = \frac{1+\sqrt{5}}{2} \) and the other equal to its conjugate \( \frac{1-\sqrt{5}}{2} \).

Answer: 1


Question: In triangle $ABC$, a point $P$ is located such that $\angle PAB = 42^{\circ}$, $\angle PBA = 54^{\circ}$, $\angle PAC = 6^{\circ}$, and $\angle PBC = 12^{\circ}$. Determine the measure of $\angle PCB$. \(\boxed{?}\)

Answer: 42


Question: Determine the value of \( \beta \in \mathbb{R} \) such that the process \( 2W_t^3 + \beta t W_t \) is a martingale, where \( W_t \) is a standard Wiener process.

Answer: -6


Question:
Evaluate the limit:
\[ \lim_{k \to \infty} \frac{1}{k} \int_{-\infty}^{\infty} h\left(\frac{x}{k}\right) f(x) \, dx, \]
where \( h:\mathbb{R} \rightarrow \mathbb{R} \) is a continuous function such that \( h(x) \leq 0 \) for all \( x \in \mathbb{R} \), \( h(x) = 0 \) if and only if \( |x| \leq 2 \), and \( \int_{-\infty}^{\infty} h(t) \, dt = -4 \). The function \( f:\mathbb{R} \rightarrow \mathbb{R} \) is also continuous.

Answer: 0


Question: Evaluate the integral \[ I = \int_{0}^2 \int_{0}^2 \int_{0}^2 \int_{0}^2 \frac{x_1 + x_2 - x_3 - x_4}{x_1 + x_2 + x_3 + x_4} \, dx_1 \, dx_2 \, dx_3 \, dx_4. \]

Answer: 0


Question: Determine whether every group action on a set results in a non-trivial permutation group. Answer with 'Yes' or 'No'.

Answer: No


Question: Evaluate the limit: $$ \lim_{x\to 3}\left(\frac{x}{x-3}\int_{3}^{x}\frac{\sin t}{t} dt\right) $$ without using L'Hopital's rule.

Answer: \sin 3


Question: Solve the equation \(4 \cdot 3^x - 9 \cdot 2^x = 5 \cdot 3^{\frac{x}{2}} \cdot 2^{\frac{x}{2}}\) for \(x\).

Answer: 4


Question: Determine the radius of convergence for the Laurent series expansion of the function \( f(z) = \frac{z^2+1}{z(z-3)} \) in the region \( 3<|z|< \infty \).

Answer: 3


Question: In a linearly ordered field, determine whether the statement "if $|a-b|<\epsilon$ for all $\epsilon >0$, then $a=b$" is true or false.

Answer: True


Question: Determine whether the sequence \(a_n = \frac{\sum\limits_{k=1}^{n} \frac{1}{k}}{\ln(n)}\) converges or diverges.

Answer: 1


Question: Determine the number of integer pairs \((a, b)\) such that \(1 \le a, b \le 42\) and \(a^9 \equiv b^7 \pmod{43}\).

Answer: 42


Question: Given the functions $f(x) = \{x\} [x]$ and $g(x) = ax^2$, where $[\cdot]$ denotes the greatest integer function and $\{\cdot\}$ denotes the fractional part function, and $a$ is a positive rational number, if the sum of all real solutions to the equation $f(x) = g(x)$ is 420, find the value of $a$.

Answer: \dfrac{29}{900}


Question: Given a $3\times3$ matrix $A$ with eigenvalues $3, 2, -1$, and $B = A^2 - A$, find $\det(B)$.

Answer: 24


Question: Evaluate the limit: $$\lim_{x\to 0^-}(x+0)\left\lfloor \frac{1}{x+0}\right\rfloor$$

Answer: 1


Question: Consider the multiplicative group $S = \{z : |z| = 1\} \subset \Bbb C$. Let $G$ and $H$ be subgroups of $S$ with orders 8 and 10, respectively. Determine the order of the intersection $G \cap H$.

Answer: 2


Question: Determine whether the following statement is true or false: For a group generated by a set of elements, the minimum number of elements in the group is equal to the product of the orders of the elements in the generating set, given that the generating set is minimal (i.e., no proper subset of the set generates the same group).

Answer: False


Question: Determine whether the following statement is true or false: Let $f_{1},f_{2},f_{3},f_{4}\in \mathbf{R}[x]$ be polynomials each of degree exactly three. Then there exist a real polynomial $p\in \mathbf{R}[x]$ of degree at most two and a subset $\{i,j\} \subseteq \{1,2,3,4\}$ with $i\neq j$ such that $f_{i}\circ p=c f_{j}$ for some $c\in \mathbf{R}$. Provide your answer as either True or False.

Answer: False


Question:
Given the equations:
1. \(a^2x^3 + b^2y^3 + c^2z^3 = p^5\)
2. \(ax^2 = by^2 = cz^2 = k\)
3. \(\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{p}\)

Find \(\sqrt{a} + \sqrt{b} + \sqrt{c}\) in terms of \(p\).

Answer: \sqrt{p}


Question: Evaluate the integral \( \oint_C \frac{z^2+1}{e^{\frac {z}{10}}-1}dz \) where \( C \) is the unit circle traversed counterclockwise.

Answer: 20\pi i


Question: Find the value of $a$ such that the limit \( \lim _{ x\to 0 } \frac { a-\sqrt { { a }^{ 2 }-{ x }^{ 2 } }-{ x }^{ 2 }/4 }{ { x }^{ 4 } } \) is finite.

Answer: 2


Question: Can the Ellipsoid method be used to find an optimal solution to a linear program?

Answer: Yes


Question: Determine the cardinality of the set \( C \subseteq \mathbb{Z} \times \mathbb{Z} \), consisting of integer pairs \((a, b)\) such that the polynomial \( p(x) = x^3 - 2x^2 + ax + b \) has three complex roots \( r_1, r_2, r_3 \) satisfying \( r_1^3 + r_2^3 + r_3^3 = 0 \).

Answer: 0


Question: Calculate the coordinates of $2P$ where $P = (7, 11)$ on the elliptic curve $E: y^2 = x^3 + x + 1 \mod 23$. Provide your answer as a point $(x, y)$.

Answer: (17, 20)


Question: Evaluate the limit: \( \displaystyle \lim_{x \to 0} \frac{x-1}{x} \cdot e^{-1/|x|} \).

Answer: 0


Question: In triangle $\triangle ABC$, medians $\overline{AD}$ and $\overline{BE}$ intersect at the centroid $G$, and they are perpendicular to each other. Given that $AD = 15$ and $BE = 20$, find the area of $\triangle ABC$. Express your answer in square units.

Answer: 200


Question: Find the remainder when \(2^{1990}\) is divided by 1990.

Answer: 1024


Question: Consider a complex sequence $(\alpha_n)$. If the sequence of partial sums $\left(\sum_{n=0}^N\alpha_n\right)$ is bounded, does it necessarily follow that the sequence $(\alpha_n)$ is bounded? Provide a justification for your answer.

Answer: Yes


Question:
Compute the limit of the following expression using Taylor series expansion around the origin:

$$ \lim_{x \to 0}\ \dfrac{x^2e^{x^2}+\dfrac{1}{2}\log(1+2x^2)-2x^2}{1+\arctan(x^6)-\cos(x^3)} $$

Answer: \dfrac{11}{9}


Question: Evaluate the triple integral \( \iiint_D (x^2+y^2) \, dxdydz \) where the domain \( D \) is defined by \( 0 \leq x^2 +y^2 \leq z^2, \quad 0 \leq z\leq 1 \). Use the substitution \( x = r \cos \theta, y = r \sin \theta \) with the functional determinant \( r \) to compute the integral as \( \int_0^{1} \left(\iint_E f(x,y) dx dy\right) dz \).

Answer: \dfrac{\pi}{10}


Question: Given topological groups $G$ and $H$ with $H \subset G$, consider the commutative diagram where $f: G \rightarrow G$ is continuous, and $p: G \rightarrow G/H$ is an open and surjective map. Does this imply that the induced map $h: G/H \rightarrow G/H$ is continuous? Express your answer in terms of the quotient topology on $G/H$.

Answer: Yes


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

Answer: 2


Question: Consider the equation \(x^7 - 4x^3 + x + 1=0\). Determine how many of its roots, when plotted on the Argand plane, have a distance from the origin that is less than 1.

Answer: 3


Question:
Consider the recurrence relation defined by:

$$ \begin{pmatrix} x_n \\ y_n \\ \end{pmatrix}=\begin{pmatrix} 1&2 \\ 2&1 \\ \end{pmatrix}\begin{pmatrix} x_{n-1} \\ y_{n-1} \\ \end{pmatrix} $$

with initial conditions $x_0=1$ and $y_0=2$. Find the limit $\displaystyle\lim_{n\to\infty} \frac{y_n}{x_n}$ if it exists.

Answer: 1


Question: Determine the number of equivalence relations \(S\) on \(\mathcal P(\mathbb N)\) such that the equivalence class \([\{8\}]_S\) is given by \(\{A \in \mathcal P(\mathbb N) \mid A \neq \{1\} \wedge A \neq \{2\}\}\).

Answer: 2


Question:
Determine the set of real numbers at which the function \( G(t) \) is continuous, where \( G(t) \) is defined as follows:
\[ G(t) = \begin{cases}
1 & \text{if } t \text{ is rational} \\
e^t & \text{if } t \text{ is irrational}
\end{cases} \] Provide a justification for your answer.

Answer: \{0\}


Question: Let $n$ be a positive integer and $M_n(\mathbb{R})$ denote the space of all $n\times n$ real matrices. Consider a linear transformation $T: M_n(\mathbb{R}) \to M_n(\mathbb{R})$ such that $T(A) = 0$ for any symmetric or skew-symmetric matrix $A$. Determine the rank of the transformation $T$. \( \text{Provide your answer as a single integer.} \)

Answer: 0


Question: Find the closure of the set $A$ in the interval $(0, \infty)$, where $A = (\mathbb{Q} \cap (0, \infty)) \setminus \left\{ \frac{1}{n} \mid n \in \mathbb{N} \right\}$.

Answer: (0, \infty)


Question: Evaluate the limit: $$\lim_{x \to 0} \frac{\sin x -\arctan x}{x^2\log(1+x)}$$

Answer: \dfrac{1}{6}


Question: Find the minimum value of the expression \( \frac{18}{a+b} + \frac{12}{ab} + 8a + 5b \) when \( a \) and \( b \) are positive real numbers.

Answer: 30


Question: If $f$ is a uniformly continuous function and $\{x_n\}$ and $\{y_m\}$ are Cauchy sequences in $D$, does it follow that $f(x_n)$ and $f(y_m)$ are also Cauchy sequences?

Answer: Yes


Question: Given a sequence of vectors $(a^n)_{n \in \mathbb{N}} \subset \mathbb{R}^k$, each with non-negative components and satisfying $\sum_{i=1}^k a^n_i = 1$ for all $n \in \mathbb{N}$, the log-sum inequality states that $\sum_{i=1}^k a_i^{n+1} \log\left(\dfrac{a_i^{n+1}}{a_i^n}\right) \geq 0$, with equality if and only if $\dfrac{a_i^{n+1}}{a_i^n}$ is constant for all $i = 1, \ldots, k$. If $\lim\limits_{n\rightarrow \infty}\sum_{i=1}^k a_i^{n+1} \log\left(\dfrac{a_i^{n+1}}{a_i^n}\right) = 0$, does it imply that $\lim\limits_{n\rightarrow \infty} \dfrac{a_i^{n+1}}{a_i^n} = 1$ for all $i$? Answer yes or no.

Answer: No


Question: Let \( X_l = \{f \in C([0,l], \mathbf{R}), 0 \leq f(x) \leq 2, \forall x \in [0,l]\}\) and let \( T: X \rightarrow C([0,l], \mathbf{R})\) be defined by \((T(f))(x) = \int_0^{x} f(t)dt\). Determine the largest number \( l_0 \) such that \( T \) maps \( X_{l_0} \) into \( X_{l_0} \).

Answer: 1


Question: Evaluate the integral \( \int_{-\pi}^{\pi}\left(4\arctan\left(e^{x}\right)-\pi\right)\mathrm{d}x \) without directly showing that the integrand is an odd function.

Answer: 0


Question: Let $a$ and $b$ be positive integers such that $a^2 + b^2 = a^2b^2$. Determine the number of possible values of $a^8 + b^8$.

Answer: 0


Question: Let $X$ and $Y$ be topological spaces, and let $q: Y \rightarrow X$ be a covering map. If the induced map $H_1(q, \mathbb{Q})$ is surjective, is the cardinality of the fiber $|q^{-1}(x)|$ constant for all $x \in X$?

Answer: No


Question:
Evaluate the limit as \( n \to \infty \) of the integral:
\[
\lim_{n \to \infty} \int_{0}^{1} \int_{0}^{1} \cdots \int_{0}^{1} \left( \frac{x_1 + x_2 + \cdots + x_n}{n} \right)^2 \, dx_1 \, dx_2 \cdots \, dx_n.
\]

Answer: \dfrac{1}{4}


Question: Find the real part of the complex number \( \frac{1}{1-e^{i\pi/7}} \).

Answer: \dfrac{1}{2}


Question: Does every Borel set with a positive Lebesgue measure contain a closed interval \([a,b]\) with \(a < b\)?

Answer: No


Question: Determine whether the series \( \sum_{y=2}^{\infty} \frac{1}{y}(1-p)^{y-1} \) and \( \sum_{y=1}^{\infty} \frac{1}{y}(1-p)^{y} \) are equivalent for any value of \( p \), or if they are equivalent only when \( p = 1 \).

Answer: p = 1


Question: If $\kappa$ is an inaccessible cardinal, determine whether every function $f:\alpha\to\kappa$ is bounded for all $\alpha<\kappa$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the line integral \( \oint_{C} xy\,dx + y^2\,dy \) where \( C \) is the closed path defined by \( y \geq 0 \) and \( x^2 + y^2 = 4 \), traced clockwise.

Answer: 0


Question: Let $G$ be the group of all $3 \times 3$ matrices $\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$ where $aei+bfg+cdh-ceg-bdi-afh \neq 0$ and $a,b,c,d,e,f,g,h,i$ are integers modulo 5. Determine the order of the subgroup of $G$ consisting of matrices with determinant equal to 1.

Answer: 372000


Question: Determine whether the following statement is true or false: If \( f:\mathbb{R} \to \mathbb{C} \) is an \( L^1 \) function and continuous at \( a \in \mathbb{R} \), then \( \int_{-\infty}^\infty |f(a+x)-f(a)|dx < \infty \).

Answer: B


Question:
Solve for \( x \) in the equation:

\[ \frac{(x+2)^{3/2}-(x-3)^{3/2}}{(x+2)^{1/2}+(x-3)^{1/2}}=5 \]

Answer: 3


Question: Find the $10$-th derivative of the function $g(x) = \frac{\sin(3x^5)+1}{x^8}$ at $x = 0$ using the Maclaurin Series.

Answer: 0


Question: Calculate the area enclosed by the curve defined by the parametric equations \(x=1+t-t^3\) and \(y=1-15t^2\). Use the symmetry of the curve about the line \(x=1\) to simplify your calculation.

Answer: 8


Question: Is the class of linear reductive algebraic groups closed under quotient groups? Specifically, if $B$ is a linear reductive algebraic group and $A$ is a normal subgroup of $B$, is the quotient group $B/A$ also a linear reductive algebraic group?

Answer: Yes


Question: Find the integer value of $x$ that satisfies the equation: $$4(x+1)! = x!(2x-6)!$$

Answer: 5


Question: Find the determinant of a matrix that is similar to matrix \( A_n \), but with non-zero elements located on the left below the main diagonal instead of the bottom left corner.

Answer: 0


Question: Calculate the average value of the function \( \cos(t - \cos t) \) over the entire real line.

Answer: 0


Question: Determine the greatest integer \( n \) such that the function \( f(x) = \begin{cases} x^3\sin(\frac{1}{x}), & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases} \) is \( n \)-times continuously differentiable on the interval \([-1, 1]\).

Answer: 1


Question: Determine whether the limit \( \lim_{x \to 0+} \left(x\lfloor \frac{a}{x} \rfloor\right) \) equals \( a \) for all real numbers \( a \).

Answer: a


Question: Evaluate the integral \( \int_0^\pi \frac{\sin(2t)}{a^2\sin^2(t)+b^2\cos^2(t)}dt \) where \(0 < b \leq a\).

Answer: 0


Question: Is the ring $\mathbb{Z}G$ Noetherian when $G$ is a finite group?

Answer: Yes


Question: Let \( f \in L^1(\mathbb{R}) \). Evaluate the limit: \[ \lim_{n \rightarrow \infty} \int_{-\infty}^\infty f(x)\sin(nx) \, dx \,. \]

Answer: 0


Question: Evaluate the limit: \[ \lim _ { n \rightarrow \infty } \int _ { 0 } ^ { \pi / 2 } e ^ { - n \sin x } \, dx. \]

Answer: 0


Question:
Consider a polynomial with integer coefficients:

$$f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$

where $f(x)=0$ has all integer roots and $a_n \neq 0$. Must it always be true that $-\frac{a_0}{a_n}$ is an integer?

Answer: Yes


Question: Let $X$ be a normed linear space and $C$ be a compact convex subset of $X$. Determine whether it is true that for every open set $V$ in $X$ containing $C$, there exists a convex open set $U$ such that $C \subseteq U \subseteq V$. Provide a proof or counterexample.

Answer: True


Question: Determine the order of the image of the homomorphism \( \varphi: \mathbb{Z}^2 \to S_{15} \) given by \( \varphi(1,0)=(2,7,3)(3,11,5)(12,13) \) and \( \varphi(0,1)=(14,15)(1,4,6,8,9,10) \).

Answer: 60


Question: Define the sequence $\{a_n\}$ for $n \in \mathbb{Z^+}$, where $a_1 = \frac{1}{3}$ and $a_{n + 1} = a_n^2 + a_n$. Let $S = \sum_{i = 2}^{2008} \frac{1}{a_i}$. Find $\lfloor S \rfloor$, the greatest integer less than or equal to $S$. \( \lfloor S \rfloor \) is the answer.

Answer: 5


Question: Let $G$ be a multiplicative group with an identity element $1$. Suppose that for all elements $a, b \in G$, there exist positive integers $m$ and $n$ such that $a^m b^n = b^n a^m$. Does it necessarily follow that $G$ is an abelian group?

Answer: No


Question: Determine whether the ordinary differential equation \( \frac{d^2u}{dx^2} + \frac{u}{A} = 0 \) has a solution that satisfies the following conditions: \( \left.\frac{d^2u}{dx^2}\right|_{x=0} = 0 \), \( u(x=0) = B \), and \( \left.\frac{du}{dx}\right|_{x \to \infty} = 0 \), where \( A \), \( B \), and \( C \) are constants.

Answer: u(x) = 0


Question: Determine the values of integers $r$ and $k$ such that the polynomial $x^{p^k} - x$ divides $x^{p^r} - x$ in the ring of polynomials $\mathbb{Z}[x]$, where $p$ is a prime integer.

Answer: k \mid r


Question: Determine if there exists a matrix that is not invertible with the following properties: 1) All diagonal entries are positive. 2) All non-diagonal entries are less than or equal to 0. 3) All rows sum to 0.

Answer: Yes


Question: Evaluate the limit: $$\lim_{n \rightarrow \infty} \int_1^\infty \frac{\sqrt{x}}{1 + nx^3} \, d\lambda(x).$$

Answer: 0


Question: Find all functions \( f : \mathbb{N} \rightarrow \mathbb{N} \) such that for every positive integer \( n \), the inequality \( 2n+2001 \leq f(f(n))+f(n) \leq 2n+2002 \) holds.

Answer: f(n) = n + 667


Question: Prove that for unit vectors \( \hat{a}, \hat{b}, \hat{c} \), the inequality \( |\hat{a} - \hat{b}|^2 + |\hat{b} - \hat{c}|^2 + |\hat{c} - \hat{a}|^2 \leq 9 \) holds.

Answer: 9


Question: Is it correct to claim that a basic feasible solution (BFS) for the standard form of a linear program remains a BFS for the original non-standard form when the slack variable values are ignored?

Answer: Yes


Question:
Evaluate the integral \( \int_0^1 r(x) \, dx \), where \( r(x) \) is defined as follows for \( x \in \mathbb{R} \):
\[
 r(x)= \begin{cases} 1 &\text{if } x \text{ is rational},\\ 0 &\text{if } x \text{ is irrational}. \end{cases} \]

Answer: 0


Question: Determine whether the function \( f(x,y) = \frac{x^my^n}{x^i+y^j} \) is continuous at \((0,0)\) if and only if \( mj+ni>ij \).

Answer: mj + ni > ij


Question: Given smooth manifolds $M$ and $N$, where $M$ has a non-empty boundary, and a smooth map $F: M \to N$, let $f$ denote the restriction of $F$ to $\partial M$. Suppose there exists a homotopy $H: \partial M \times I \to N$ between $f$ and a smooth map $g: \partial M \to N$. Does there necessarily exist a smooth map $G: M \to N$ that is homotopic to $F$ and restricts to $g$ on $\partial M$?

Answer: Yes


Question: Determine whether there exists a Tychonov space \((X, \mathcal{T})\) with points \(a, b \in X\) such that \(a\) has a countable neighborhood basis while \(b\) does not have any countable neighborhood basis.

Answer: Yes


Question:
Determine the order of accuracy of the numerical approximation:

$$ \frac{(a_{i+1} + a_i)(u_{i+1} - u_i) - (a_i + a_{i-1})(u_i - u_{i-1})}{2h^2} $$

as an estimate for the derivative \(\frac{d}{dx}\Big(a(x) \frac{du}{dx} \Big)\bigg|_{x_i}\), where \(h\) is the mesh width, and \(a_i = a(x_i)\) and \(u_i = u(x_i)\). Analyze the accuracy by expanding the functions \(a(x)\) and \(u(x)\) around \(x_i\) and considering the contribution of higher-order terms.

Answer: 2


Question: Evaluate the limit \( \lim_{n \to \infty} g_n(x) \), where \( g_n(x) = \frac{n^2x}{(1+x)^n} \) for \( x > 0 \).

Answer: 0


Question: Consider polynomials $P_1(x), P_2(x) \in \mathbb{Z}[x]$, and integers $n, a$. Suppose that the polynomial $x^n-a$ is irreducible over $\Bbb Z[i][x]$. Determine the number of ordered pairs $(P_1(x), P_2(x))$ that satisfy the equation $(P_1(x))^2+(P_2(x))^2=(x^n-a)^2$, excluding trivial pairs like $(x^n-a,0)$. How many such pairs exist?

Answer: 0


Question: Let $X_n$ be a random variable with an exponential distribution with mean $n$ for each $n \geq 1$. Define $Y_n = X_n - 10 \left[\frac{X_n}{10}\right]$, where $[u]$ denotes the greatest integer less than or equal to $u$. Determine the distribution function of $Y_n$ and find the limit of this distribution function as $n \to \infty$.

Answer: \frac{y}{10}


Question: Calculate the residue of the function \( f(z) = \frac{z^2 + 1}{z^2(z + 2)} \) at the pole of order 2 located at \( z=0 \).

Answer: -\dfrac{1}{4}


Question: Let $E, F$ be finite dimensional normed spaces, $A\subset E$ be a connected open set, $f:A\times F\to F$ be a continuously differentiable mapping. Assume that the set $\Gamma=f^{-1}(0)$ is non-empty and for each $(x,y)\in\Gamma$ the derivative $\partial_y f(x,y)$ is invertible. Let $G$ be a connected component of $\Gamma$ such that $pr_1(G)=A$ ($pr_1$ being a local homeomorphism of $G$ onto $A$). Determine whether any continuous mapping $\gamma:[0,1]\to A$ can be lifted to a continuous mapping $\tilde{\gamma}:[0,1]\to G$ (i.e. $pr_1\circ\tilde{\gamma}=\gamma$).

Answer: Yes


Question: Evaluate the limit: $$ \lim_{x \to 0^+} \frac{\ln (x)+e^{-1/x}}{1/x^2 + x\sin (-1/x^2)} $$

Answer: 0


Question: Determine the number of groups of order 45 up to isomorphism.

Answer: 2


Question: Evaluate the limit \( \lim_{x \to 0} \frac{1}{x}\int_{2x}^{3x} e^{-t^2}\, dt \) using approximation methods.

Answer: 1


Question: Determine whether the following statement is true: For any $(a,b) \in \mathbb{R}^{n+m}$ and $r>0$, does $$B((a,b),r)\subset B\left(a,\frac{r}{2}\right)\times B\left(b,\frac{r}{2}\right)$$ hold?

Answer: False


Question: Let $G$ be a finite group. Determine the index of the diagonal subgroup $\mathrm{diag}(G)$ in the direct product $G \times G$. Express your answer in terms of the order of $G$. \( \boxed{} \)

Answer: |G|


Question: Determine whether there exists a smooth function \( f: \mathbb{R} \rightarrow \mathbb{R} \) that is monotonically increasing and satisfies the conditions: \( f(x) = 0 \) for all \( x \leq 0 \) and \( f(x) = 1 \) for all \( x \geq 1 \).

Answer: Yes


Question: In a row of 10 people arranged from shortest to tallest, a "mini-shuffle move" involves three individuals leaving their positions and re-entering the line in the now empty spots, without any of them returning to their original position. What is the minimum number of "mini-shuffle moves" required to completely reverse the order of the people, transforming it from shortest to tallest to tallest to shortest?

Answer: 5


Question: Determine the minimum time after which it can be concluded that the remaining bricks in a $2N \times 2N$ grid will never be hit by a ball. The ball travels 1 unit in both the x and y directions, bounces off perpendicularly upon hitting a boundary or a brick, and starts its trajectory from a given point.

Answer: 4N


Question: Evaluate the limit: $$\lim_{x\to0}\frac{1-\cos(x^2)+\ln(1+x^2)}{\tan^2(x^2)-e^{x^2}+1}$$

Answer: -1


Question: Determine whether the sequence \( a_n = n^2 \cos\left(\frac{2}{n^2} + \frac{\pi}{2}\right) \) has a limit.

Answer: -2


Question: Let $L/K$ be a Galois extension of number fields, and let $A$ and $B$ be the rings of integers of $K$ and $L$, respectively. If $a \in A$ has norm $N_{L/K}(a) = \pm 1$, does it follow that $a$ is a unit in $B$?

Answer: Yes


Question: Evaluate the improper integral \(\int_{-\infty}^\infty x^3e^{-x^4}dx\) and determine if it converges or diverges.

Answer: 0


Question: Let $R$ be a finite ring and $x \in R$ be a zero divisor. Determine if it is possible for $x^n = x^m \neq x$ to hold for some integers $1 < m < n$. Provide a justification for your answer.

Answer: Yes


Question:
Compute the limit as \( n \to \infty \) of the expression:
\[
\frac{1}{n}\left(\sin\left(\frac{2\pi}{3n}\right)+\sin\left(\frac{4\pi}{3n}\right)+\cdots+\sin\left(\frac{(n-1)2\pi}{3n}\right)\right)
\]

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


Question: Determine whether the sequence \( \left\{\sin\left(\left(1+\sqrt{2}\right)^n\pi+(1-\sqrt{2})^n\pi\right)\right\}_{n\in\mathbb{N}} \) converges, and if so, find its limit.

Answer: 0


Question: Determine whether the composition of two absolutely continuous functions on the real line is absolutely continuous. Provide a true or false answer.

Answer: False


Question: In a circle with center $A$, point $E$ is the tangent point, and $[BA] \perp [AC]$. Given that $\angle ABD = \angle DCB$ and $\angle DBC = \angle DCA$, with $D$ on the circle $\omega$, and $|EC| = x$, find the area of triangle $BCD$ in terms of the radius $r$ of the circle and $x$.

Answer: \dfrac{r x}{2}


Question: Let \( f(x) = \sin{x} + x^2 + 1 \). Find the value of \( \frac{d^4}{dy^4}(f^{-1}(y)) \bigg|_{y=1} \).

Answer: -140


Question: Solve the equation: \[ \arccos \frac{1-x}{1+x} - \arcsin \frac{2 \sqrt{x}}{1+x} = 0, \quad x \in \mathbb{R} \]

Answer: [0, 1]


Question: Determine whether there exists a bijective function $f: S \rightarrow T$, where $S = \{0, 1, 2, \ldots, 2^{2010} - 1\}$ and $T = \{0, 1, 2, \ldots, 2^{2000} - 1\}$, such that if the binary representations of $a, b \in S$ have leading zeroes up to the $2^{2009}$th place and share 10 consecutive identical binary digits in corresponding positions, then $f(a) \neq f(b)$. If such a function does not exist, find the smallest $k$ such that there exists a bijective function $g: S' \rightarrow T$, where $S' = \{0, 1, 2, \ldots, 2^k - 1\}$, satisfying the given condition.

Answer: 2000


Question: Is the unitary group compact in the strong operator topology (SOT) for a finite-dimensional Hilbert space?

Answer: Yes


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a $T$-periodic function, satisfying $f(t+T) = f(t)$ for all $t \in \mathbb{R}$. If the improper integral $\int_0^{+\infty} |f(s)| \, ds$ converges, and $f$ is continuous, must it necessarily be true that $f = 0$ everywhere? Justify your answer.

Answer: Yes


Question:
Evaluate the limit of the following integral as \( n \to \infty \):
\[ \lim_{n \rightarrow \infty} \int_{2}^{n} \frac{\sin(nx)(x-2)}{\sqrt{\log^5(x)(2+x^3)}} \, dx \]

Answer: 0


Question:
Let $h$ and $g$ be continuous, non-decreasing, and concave functions on the interval $[0, \infty)$ with $h(0) = g(0) = 0$, $h(x) > 0$, and $g(x) > 0$ for $x > 0$. Suppose both $h$ and $g$ satisfy the Osgood condition:

$$\int_0^\infty \frac{dx}{h(x)} = \int_0^\infty \frac{dx}{g(x)} = \infty.$$

Determine whether there exists a concave function $F$ such that $F(x) \ge h(x)$ and $F(x) \ge g(x)$ for all $x$, and $F$ also satisfies the Osgood condition.

Answer: Yes


Question: Determine the values of \( \alpha \in (0,1) \) for which the operator \( A: L_2[0,1]\rightarrow L_2[0,1] \), defined by \((Ax)(t) = \int_0^1 \frac{x(s)}{|t-s|^{\alpha}}ds\), is compact.

Answer: (0,1)


Question: Compute the derivative of the function \( \frac{d(x^TWW^Tx)}{dW} \), where \( x \) is an \( n \times 1 \) vector and \( W \) is an \( n \times m \) matrix, using the fact that \( \frac{d(x^TWx)}{dW} = x^Tx \).

Answer: 2xx^TW


Question: Find the area of the largest ellipse that can be enclosed by the hyperbolas $xy=1$ and $xy=-1$ using integration.

Answer: 2\pi


Question: Determine whether it is possible to construct a $C^\infty$ function $g(x)$ such that $\left|{\frac {d^{k}g}{dx^{k}}}(0)\right| = h(k)$ for any given function $h(k)$. Provide a justification for your answer.

Answer: Yes


Question: Consider a Lagrangian \(L(x,y,y')\) that is invariant under the transformation \(x \rightarrow -x\) and \(y \rightarrow -y\). If the Euler-Lagrange equations are solved with the boundary conditions \(y(a) = -y(-a)\) and \(y'(a) = y'(-a)\), does the solution necessarily satisfy \(y(x) = -y(-x)\)? Answer with 'Yes' or 'No'.

Answer: Yes


Question: Determine the points on the elliptic curve $y^2 + y = x^3$ over the field $F_{2^4}$ such that $2P = 0$.

Answer: \mathcal{O}


Question: Let $f : [0,1] \to \mathbb{R}$ be a smooth function (class $C^\infty$) that is not necessarily real-analytic. Define the function $g : (-1, \infty) \to \mathbb{R}$ by \( g(x) = \int_0^1 f(t) \, t^x \, dt \). Is $g$ necessarily a real-analytic function on $(-1, \infty)$? Provide a justification for your answer.

Answer: Yes


Question: Given two random variables, \(X\) and \(Y\), with common variance \(\sigma^2\), where \(\mathbb{E}(X|Y) = Y - 2\), find the correlation coefficient \(\rho(X,Y)\).

Answer: 1


Question: Determine whether it is possible to find an arbitrarily close approximation to any real number using elements from the ring $\mathbb{Z}[\sqrt{2}]$. Provide a justification for your answer.

Answer: Yes


Question: If $H$ is a Hilbert space and $x \in H$, does it follow that $||x|| < \infty$? Provide a justification for your answer.

Answer: Yes


Question:
Given positive numbers $x, y, z$ such that $x + y + z = 6$, find the minimum value of the function:
\[ f(x, y, z) = \left(x + \frac{1}{y}\right)^2 + \left(y + \frac{1}{z}\right)^2 + \left(z + \frac{1}{x}\right)^2. \]

Answer: \dfrac{75}{4}


Question:
How many normal subgroups does a non-abelian group $G$ of order $21$ have, other than the identity subgroup $\{e\}$ and $G$ itself? Choose the correct answer from the options below:

A) 0
B) 1
C) 3
D) 7

Answer: B


Question: Determine whether the conditions $\mathcal{L}_{X}g_{\mu\nu} = Rg_{\mu\nu}$ and $c = -\Lambda$ can hold simultaneously for the equations $R_{\mu\nu} -\frac{1}{2}\mathcal{L}_{X}g_{\mu\nu} - cg_{\mu\nu} = 0$ and $R_{\mu\nu} -\frac{1}{2}Rg_{\mu\nu} + \Lambda g_{\mu\nu} = 0$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the value of the infinite sum \( S \) given by \( S = \sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}\frac{1}{3^{i+j+k}} \) where \( i \neq j \neq k \).

Answer: \dfrac{81}{208}


Question: Let $T$ be a bounded operator on an infinite-dimensional Banach space. Suppose that $T^{2n-1} = 0$ for some positive integer $n$. Does it follow that $T$ is nilpotent? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{1} x \sqrt{1-\sqrt{x}} \, dx \) using a substitution method different from the one typically used.

Answer: \dfrac{64}{315}


Question: Let $p$ be an odd prime and $d$ be a natural number such that $d \mid (p-1)$. If $a^{\frac{p-1}{d}} \equiv 1 \pmod{p}$, does there always exist an integer $x$ such that $x^d \equiv a \pmod{p}$? Answer "Yes" or "No".

Answer: Yes


Question: Let $\mathbb{C}^{11}$ be a vector space over $\mathbb{C}$ and $T:\mathbb{C}^{11}\to \mathbb{C}^{11}$ be a linear transformation. Given that $\dim(\ker T)=4$, $\dim(\ker T^3)=9$, and $\dim(\ker T^4)=11$, determine the possible values for $\dim(\ker T^2)$. \(\boxed{\text{Your answer here}}\)

Answer: 7


Question: Evaluate the limit: $$\lim_{x \to 0^+} \frac{\int_0^x \arcsin(t^2) \, dt}{\sin(x^2)}.$$

Answer: 0


Question: Determine whether there exists a function defined on the interval $[0,1]$ that has unbounded variation and is differentiable everywhere.

Answer: Yes


Question:
Find a continuous function \( f \) that satisfies the equation:

\[ f(x) = 1 + \frac{1}{x} \int_{1}^{x} f(t) \, dt, \quad x > 0 \]

Answer: 1 + \ln x


Question: Find a function that is integrable over any finite interval but whose limit as \( x \) approaches infinity does not exist.

Answer: \sin(x)


Question: In an additive category, determine whether an object is a zero object if and only if the diagonal morphism is an isomorphism. Answer 'true' or 'false'.

Answer: true


Question: Suppose that $f \in \mathcal{C}^1([0,1])$ and that $\lim_{x \to 0^+} \frac{f(x^3)}{\sqrt{2}x^3} = 1$. Find $\lim_{x \to 0^+} \frac{f(f(x^2))}{f^{-1}(x^2)}$. \(\text{Express your answer as a single number or expression.}\)

Answer: 2\sqrt{2}


Question: Determine if a cubic polynomial function with real coefficients can have fewer real distinct roots than its derivative function. If possible, provide an example of such a polynomial.

Answer: x^3 - 3x + 3


Question: Determine the values of \(\alpha\) for which the integral \(\int_{0}^{1} \frac{1-x^{\alpha}}{1-x}dx\) converges.

Answer: \alpha > -1


Question: Determine the cardinality of the set of regressive functions from \( \omega \) to \( \omega \), where \( \omega \) is the set of natural numbers.

Answer: 2^{\aleph_0}


Question: In triangle $ABC$, it is given that $\angle B = 2\angle C$. Point $D$ is located on side $BC$ such that $AD$ bisects $\angle BAC$ and $AB = CD$. Determine the measure of $\angle BAC$. \(\boxed{\text{?}}\)

Answer: 72^\circ


Question: Evaluate the integral \( \int_{0}^{\infty}{\frac{e^{-x}\ln{x}}{4-\cos{x}\cos{2x}\cos{4x}}}dx \) and determine if the result is equal to \(-\ln{\zeta(3)}\) up to 10 decimal places.

Answer: -\ln{\zeta(3)}


Question: Determine whether a locally compact group $G$ is amenable if it has an abelian normal subgroup $A$ of finite index.

Answer: Yes


Question: Determine the initial conditions for which \(\lim_{t \rightarrow \infty}y = 4\) for the differential equation: \[y' = y(y-4)(y+4)(y-6)\]

Answer: (0, 6)


Question:
Evaluate the determinant of the matrix:
\[\begin{vmatrix} b^2c^2 & bc & b+c \\ c^2a^2 & ca & c+a \\ a^2b^2 & ab & a+b \end{vmatrix}\]

Answer: 0


Question: Suppose $f:S^n\to \Bbb R^{n+1}$ is a continuous function satisfying $f(x)=t(x) x$ for all $x\in S^n$, where $t(x)\in \Bbb R$. Is the function $t:S^n\to \Bbb R$ necessarily continuous?

Answer: Yes


Question: Let $V$ be a finite-dimensional vector space and $U$, $W$ be subspaces of $V$. If the set of linear maps from $V$ to the field that vanish on $W$ is a subset of the set of linear maps that vanish on $U$, is it true that $W \subseteq U$?

Answer: No


Question: Evaluate the limit \( \lim _{x\to \infty} \frac{1}{x} \int_x^{2x} \left|\cos \frac{1}{t} \right| dt \).

Answer: 1


Question: Find the minimum value of the expression \(|Z_1 - Z_2|^2 + |Z_2 - Z_3|^2 + |Z_3 - Z_4|^2 + |Z_4 - Z_1|^2\) given that the complex numbers \(Z_1, Z_2, Z_3, Z_4\) satisfy \(Z_1 + Z_2 + Z_3 + Z_4 = 0\) and \(|Z_1|^2 + |Z_2|^2 + |Z_3|^2 + |Z_4|^2 = 1\).

Answer: 2


Question:
Consider a triangle with side lengths $(a, b, c)$ such that $a^2, b^2, c^2 \in \mathbb{Q}$. Determine whether this triangle is necessarily rationally realizable in some $
$-dimensional space, meaning it can have vertices with rational coordinates.

Answer: Yes


Question: Compute the integral \( \int_0^{2\pi}\frac{1}{5-4\cos x}\ dx \).

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


Question: Evaluate the limit \( \lim_{n\to\infty} e^{-\sqrt{n}x}\left(1-\frac{x}{\sqrt{n}}\right)^{\sqrt{n}x-n} \).

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


Question:
Consider the boundary value problem:

$$y'' + xy = 0; \quad x \in [a, b]; \quad y(a) = y(b) = 0.$$

If $y$ is a non-trivial solution, determine whether the statement is true or false: $y$ has exactly one zero in $(a, b)$ if $x \geq 0$ on $(a, b)$. Provide your answer as "True" or "False".

Answer: False


Question: In an isosceles triangle \( \triangle ABC \) with \( AB = AC \), a point \( D \) is located on the extension of side \( BC \) such that \( C \) is between \( B \) and \( D \) and \( BD = AD \). The angle bisector of \( \angle ACD \) intersects \( AB \) at point \( E \), forming an angle \( \angle AEC = 30^\circ \). Determine the measure of \( \angle EAD \).

Answer: 30^\circ


Question: Find an element \( \beta \) such that the set \( \mathbb{F}_9 = \{a + b\beta : a, b \in \mathbb{F}_3\} \) forms a field with exactly 9 distinct elements. Justify your choice of \( \beta \).

Answer: \beta


Question: Let $\mathscr{B}$ be a Boolean algebra and $F$ be a finite subset of $\mathscr{B}$. Is the subalgebra $\left<F\right>$, generated by $F$, necessarily a finite subalgebra of $\mathscr{B}$? Provide a justification for your answer.

Answer: Yes


Question: Determine the pointwise limit \( f(x) = \lim_{n \to \infty} f_n(x) \) for the function sequence \( f_n : [0, 2] \to \mathbb{R} \) given by \[ f_n(x) = \begin{cases} n^3 x^2, & 0 < x < 1/n; \\ n^3\left(x - \frac{2}{n}\right)^2, & 1/n \le x < 2/n; \\ 0, & \text{otherwise.} \end{cases} \]

Answer: 0


Question: Find the remainder when \(2^{1990}\) is divided by 1990.

Answer: 1024


Question: Evaluate the limit: \[ \lim_{x \to +\infty} \left[ (x+a)^{1+\frac{1}{x}}-x^{1+\frac{1}{x+a}}\right] \] using only basic limit theorems and inequalities, without employing L'Hopital's rule or Taylor's formula.

Answer: a


Question: Given the function \( f(X)=\exp\left(\frac{-\gamma^2}{a^2X+b^2}\right) \) where \( X \sim \mathrm{Binomial}(n,p) \), use the central limit theorem to find an approximate expression for the value of \( \gamma^2 \) that maximizes the variance of \( f(X) \).

Answer: a^2 np + b^2


Question: Determine the radius of convergence for the power series \( \sum_{n=0}^{\infty} a_n^{1/M}z^n \), given that the original power series \( \sum_{n=0}^{\infty} a_nz^n \) has a radius of convergence \( R \), and \( M \) is a positive integer.

Answer: R^{1/M}


Question:
Find the angle between two lines whose direction cosines \((l, m, n)\) satisfy the equations:

\[ l + m + n = 0 \]
\[ \frac{mn}{q-r} + \frac{nl}{r-p} + \frac{lm}{p-q} = 0 \]

where \(p\), \(q\), and \(r\) are constants.

Answer: 60^\circ


Question: Calculate the probability of obtaining a run of five consecutive heads before a run of two consecutive tails when flipping a fair coin indefinitely.

Answer: \dfrac{3}{34}


Question: Determine whether there exists an equiareal map between two surfaces that does not preserve the first fundamental form.

Answer: Yes


Question: In a directed graph $G=(V,E)$, each vertex $v \in V$ has an equal number of incoming and outgoing edges, and no vertex has an edge pointing to itself. Given that there are $k$ mutually disjoint paths from a node $v_1$ to a node $v_2$, does it necessarily follow that there are $k$ mutually disjoint paths from $v_2$ to $v_1$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_{\mathbb{C}} \frac{e^{-\frac{(x-im)^2}{2}}}{\sqrt{2\pi}} \, dx \) when \( m \) is replaced with \( im \), where \( i \) is the imaginary unit and \( m \) is a real number.

Answer: 1


Question: Consider the power series \(\sum_{n = 0}^{\infty}a_nz^n\) with a radius of convergence \(R\). Determine the radius of convergence for the series \(\sum_{n = 0}^{\infty}a_nc^nz^n\), where \(c \in \mathbb{C}\) and \(|c| \neq 0\).

Answer: \dfrac{R}{|c|}


Question: Find the value of $a$ such that \( f(a) = \lim_{x \to \infty}\left(\frac{x-a}{x+a}\right)^x = e^4 \).

Answer: -2


Question: Let \(E\) be a locally compact second-countable metric space, and let \(E^\ast = E \cup \{\infty\}\) denote the Alexandroff one-point extension. Consider a sequence \((x_n)_{n \in \mathbb{N}} \subseteq E^\ast\) and a point \(x \in E^\ast\) such that \(x_n \xrightarrow{n \to \infty} x\). Let \(B_n \subseteq E^\ast\) be open sets with \(x_n \in B_n\) for each \(n \in \mathbb{N}\) and \(\bigcap_{n \in \mathbb{N}} B_n = \{\infty\}\). Assume that the sequence \((B_n)_{n \in \mathbb{N}}\) is nonincreasing. Determine if it follows that \(x_0 = \infty\).

Answer: \infty


Question: A box contains some identical tennis balls. The ratio of the total volume of the tennis balls to the volume of empty space surrounding them in the box is $1:k$, where $k$ is an integer greater than one. After removing a prime number of balls, the ratio of the total volume of the remaining tennis balls to the volume of empty space surrounding them in the box becomes 1:$k^2$. Determine the original number of tennis balls in the box.

Answer: 5


Question: Determine whether the following statement is true or false: If a function $g$ is discontinuous, then the sequence $(g_n)_n$, where $g_n(x) = g(x + 1/n)$, does not converge pointwise to $g$ on $\mathbb{R}$.

Answer: B


Question: Find the minimum number of subsets required to partition a set of complex numbers such that the absolute value of the sum of elements in at least one subset is at least one-sixth of the sum of the absolute values of all elements.

Answer: 3


Question: Does every group of order $p^k$, where $p$ is a prime and $k$ is a positive integer, contain an element of order $p$? Answer yes or no and justify your answer.

Answer: Yes


Question: Find all integer values of \(x\) that satisfy the equation: \[5^{2(x+1)} + 621 \times 10^x = 100 \times 4^x\]

Answer: -2


Question:
Evaluate the limit of the integral over the semicircle as \( R \to \infty \):

\[ \lim_{R\to\infty}\int_{\Gamma_R}\frac{e^{izt}}{\pi\cos(iz)}dz \]

where \( t > 0 \) and \( \Gamma_R \) is the upper semicircle of radius \( R \) in the complex plane.

Answer: 0


Question:
Determine if a ridge regression problem can be expressed as an ordinary least squares (OLS) problem with the following restructured form:
\[ \tilde{y} = \begin{pmatrix} y\\ 0 \end{pmatrix} \quad \tilde{X} = \begin{pmatrix} X \\ -\sqrt{\lambda}I \end{pmatrix} \]
such that the objective functions are equivalent:
\[ \|y-X\beta\|^2 + \lambda\|\beta\|^2 = \|\tilde{y} - \tilde{X}\beta\|^2 \]

Answer: Yes


Question:
Let $a, b \in \mathbb{R}_+$ be real positive numbers such that $\frac{1}{2} < a < 1$, and let $I = [0, b]$ be a closed real interval. Define a sequence of functions $f_n(x) : I \to \mathbb{R}$ for all $n \in \mathbb{N}$ as follows:
\[ f_n(x) = \frac{n}{(2n-1)^{a+x}} - \frac{n}{(2n)^{a+x}} \]

And define another sequence of functions $g_n(x) : I \to \mathbb{R}$ for all $n \in \mathbb{N}$ as the derivative of $f_n(x)$:
\[ g_n(x) = \frac{n \cdot \ln(2n)}{(2n)^{a+x}} - \frac{n \cdot \ln(2n-1)}{(2n-1)^{a+x}} \]

Consider the function $h : I \to \mathbb{R}$ given by:
\[ h(x) = \sum_{n=1}^\infty \left| \frac{f_n(x) - f_n(0)}{x} - g_n(0) \right|^2 \]

Determine whether the limit $\lim_{x \to 0^+} h(x)$ converges to $0$. Provide your answer as either "Yes" or "No".

Answer: Yes


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

Answer: \infty


Question: Determine the dimension of the subspace of \( F^n \) that is annihilated by the linear functionals \( g_1, g_2, \ldots, g_n \), where each \( g_k(x_1, \ldots, x_n) = \sum_{j=1}^n (j-k)x_j \).

Answer: n - 2


Question: Find the value of the expression \( \sqrt{i}^i \).

Answer: e^{-\pi/4}


Question: Find the norm of the functional \( f(x) = \sum_{j=1}^{\infty} \xi_j \frac{1}{\sqrt{j(j+1)}} \) defined by \( f: l^2 \rightarrow \mathbb{R} \) for all \( x = (\xi) \in l^2 \).

Answer: 1


Question: Are all sets obtained by forming pairs from the sets $∅, \{∅\}, \{ \{∅\} \}$, and so on, unique from each other?

Answer: Yes


Question: Find the limit of the function \( v(x,t) \) as \( t \to \infty \), where \( v(x,t) \) is a bounded solution of the heat equation \( v_t = v_{xx} \) with the initial condition \( v(x,0) = \frac{e^{-x^2}-1}{e^{-x^2}+1} \).

Answer: -1


Question: Determine whether there exists a function \( g:[0,1]\to \mathbb{R} \) such that \( \lim_{x\to a}g(x)=\infty \) for only a finite number of points \( a \) in \([0,1]\).

Answer: Yes


Question: For any positive integer $m$, determine if it is possible to pair up a class of $2m$ students each day such that no student is left unmatched and no two students are paired together more than once until every student has been paired with every other student exactly once.

Answer: Yes


Question: Let \( A \in \mathbb{R}^{3\times3} \) be a matrix with eigenvalues \(1\), \(-1\), and \(0\). Calculate \( \det \left(I + A^{50} \right) \).

Answer: 4


Question: Let $V \subset [0,\infty)^n$ be a closed set. Consider the set $W$ of minimal elements of $V$ under the partial ordering $\leq'$ defined as $(x_1,...,x_n) \leq' (y_1,...,y_n) \iff x_k \leq y_k, \forall k$. That is, $$ W = \big\{ x \in V \ \big| \ \nexists y\in V, y<'x\big\}. $$ Determine whether $W$ is Borel measurable.

Answer: Yes


Question:
Find the radius of convergence of the Taylor series for the function \( f(x) = \frac{1}{\sqrt{x}} \) centered at \( x = 1 \), given by:

$$ f(x) = 1 + \sum_{n=1}^\infty (-1)^n \cdot \frac{(2n)!}{2^{2n} (n!)^2} \cdot (x-1)^n $$

Answer: 1


Question: Evaluate the limit: $$\lim_{{x \to 0}} \frac{1 - \cos{x}\sqrt{\cos{2x}}}{x\sin{x}}$$

Answer: \dfrac{3}{2}


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

Answer: -1


Question: Evaluate the limit: $$\lim_{x\to 0}x \left[ \frac{a}{x}\right]$$ for $a>0$.

Answer: a


Question: Evaluate the improper integral \(\int_{0}^{\infty} \frac{\cos(2x) - 1}{x^2} \, dx\).

Answer: -\pi


Question: Given a sequence $\{a_n\}$ where $a_n \geq 0$ for all $n$ and the sequence is monotonically decreasing, determine if $n\log(n)a_n \rightarrow 0$ as $n\rightarrow\infty$ when $\sum_{n=1}^\infty a_n$ converges.

Answer: 0


Question: Determine the values of \( \beta \) for which the series \( \displaystyle\sum_{k = 1}^{\infty} \frac{\log(3)^k}{k!} \sum_{n = 1}^{\infty} \frac{1}{n^{\beta k}} \) converges.

Answer: \beta > 1


Question: Find the inverse of the element $z = 324 + 171i$ in the finite field $\mathbb{F}_{431}(i)$, where elements are of the form $u + vi$ with $u, v \in \mathbb{F}_{431}$. Compute $z^{-1}$. Express your answer in the form $a + bi$ with $a, b \in \mathbb{F}_{431}$.

Answer: 90 + 168i


Question: Let $X_1 \sim \operatorname{Geo}(p)$ and $X_2 \sim \operatorname{Geo}(p)$ be independent random variables, where $\operatorname{Geo}(p)$ refers to the Geometric distribution with p.m.f. $f(k) = p(1−p)^k$ for $k = 0,1,\ldots$. Find the conditional distribution of $X_1 \mid X_1+X_2=y$. Assume $p_1=p_2=p$.

Answer: \frac{1}{y + 1}


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

Answer: \infty


Question: Determine whether all fixed points of the function \( f(z) = \frac{z-i}{iz+3} \) are in the region \( \{z \in \Bbb C: \Im(z)>0\} \). State whether the statement is TRUE or FALSE.

Answer: FALSE


Question: Do projective transformations and their properties, as used in the proof of Bézout's theorem for plane projective curves, hold true for an arbitrary algebraically closed field $k$?

Answer: Yes


Question: In a vector space of dimension $m$, determine the maximum total number of dimensions that two non-intersecting affine subspaces can have.

Answer: 2m - 2


Question: Calculate the definite integral \( \int_0^\infty \frac{2t}{(t+1)^4} \, dt \) using a suitable substitution and the Beta function.

Answer: \dfrac{1}{3}


Question: Find the limit of the probability $p_N$ as $N$ approaches infinity, where $p_N$ is the probability that the sum of the squares of two randomly chosen integers between 1 and $N$ (with replacement) is less than or equal to $N^2$. Express your answer as a number.

Answer: \dfrac{\pi}{4}


Question: Find the minimum value of the expression \(\sqrt{2x^2+2x+1} +\sqrt{2x^2-10x+13}\) and express it in the form \(\sqrt{\alpha}\), where \(\alpha\) is a positive integer.

Answer: \sqrt{20}


Question: Find the residue of \((1-e^{z})^{-n}\) at \(z = 0\) for any positive integer \(n\).

Answer: -1


Question:
Given the complex numbers $z_1, z_2,$ and $z_3$ that satisfy the condition $$\frac{z_1 + z_3}{z_2 - z_3} = \frac{1 - i\sqrt{3}}{2},$$ determine the type of triangle formed by the vertices $z_1, z_2,$ and $z_3$. The possible options are:

(a) An area of 0
(b) Equilateral
(c) Right-angled and isosceles
(d) Obtuse-angled

Use the fact that $$\arg\left(\frac{z_1 + z_3}{z_2 - z_3}\right) = -\frac{\pi}{3}$$ and $$|z_1 + z_3| = |z_2 - z_3|.$$

Answer: B


Question: Did Frobenius generalize Cayley's result for the characteristic polynomial of a matrix from $3\times 3$ to $n\times n$?

Answer: Yes


Question: Find a function \( f: \mathbb{R} \to S \subset \mathbb{R} \) such that for any two real numbers \( x \) and \( y \), if \( x < y \), then \( f(x) > f(y) \), and both \( f(x) \) and \( f(y) \) are positive.

Answer: f(x) = e^{-x}


Question: What is the expected number of fair coin flips needed to obtain the sequence HTT? Use the conditional expectation technique to solve this problem.

Answer: 8


Question: Find the residue of the function \( f(z) = \frac{z}{8-z^3} \) at \( z = 2 \).

Answer: -\dfrac{1}{6}


Question: Determine the values of the constant \( p \) for which the series \( \sum_{k=3}^{\infty}\frac{1}{n\ln(n)(\ln(\ln(n)))^p} \) converges. Justify your answer.

Answer: p > 1


Question: Evaluate the limit \(\lim_{n\to\infty} \left(f\left(x_0+\frac 1{n^3}\right)+\dots+ f\left(x_0+\frac n{n^3}\right)-nf(x_0)\right)\) where \(f:\mathbb{R}\rightarrow\mathbb{R}\) is differentiable at \(x_0\).

Answer: 0


Question: Estimate the sum \(\sum_{x = -500}^{500} \sum_{y: x^2 + y^2 < 5^6 } y^2\) to the nearest power of 10.

Answer: 10^8


Question: What is the minimum value of the Kullback–Leibler divergence $D_{KL}(P || Q)$ when optimizing with respect to $P(x)$, given that $\sum_x P(x) = \sum_x Q(x) = 1$?

Answer: 0


Question:
Evaluate the limit using the Taylor series expansion of $e^x$:

$$\lim_{x \to 0}\frac{(e^{x^2} - 1)(x^2 \times \log 2) - (e^x - 1)^2}{(e^x - 1)^2(x^2 \times \log 2)}$$

Answer: -\infty


Question: Determine whether the inequality $a^{m-1}(m-a)<m-1$ holds for all integers $m \geq 2$, where $0<a<1$. Provide a justification for your answer.

Answer: Yes


Question: Given the functional equation \(f(x+y) = f(x)f(y)\) for all real numbers \(x\) and \(y\), with \(f(5) = -2\) and \(f'(0) = 3\), find the value of \(f'(5)\).

Answer: -6


Question: Evaluate the integral: $$\int^{(2,2,2)}_{(1,1,1)}5x^4y^{15}z^{15}dx+15x^5y^{14}z^{15}dy+15x^5y^{15}z^{14}dz$$

Answer: 2^{35} - 1


Question: For a positive integer \( n \), let \( f(n) \) represent the number of times each positive divisor of \( n \) is counted when summing the number of divisors of all positive integers less than or equal to \( n \). Determine an expression for \( f(n) \).

Answer: \sigma(n)


Question: Determine whether there exists a permutation matrix $P'$ such that for any $n \times n$ matrix $A$, the matrix $P'A$ is a permutation of $A$'s columns and $AP'$ is a permutation of $A$'s rows.

Answer: Yes


Question: Evaluate the limit: \[ \lim _{x\to 2}\left(\frac{\left(3x^2-11\right)^{\frac{1}{4}}-(x-1)^{\frac{1}{3}}}{x-2}\right). \]

Answer: \dfrac{8}{3}


Question: Let $A \in \mathbb{R}^{4 \times 4}$ have a characteristic polynomial $(x^2+1)^2$. Which of the following statements is true about matrix $A$?\n\n1. $A$ is diagonalizable over $\mathbb{C}$ but not over $\mathbb{R}$.\n2. $A$ is nilpotent.\n3. $A$ is invertible.\n4. There is no such matrix $A$.\n\nChoose the correct statement.

Answer: 3


Question: Let $X_i$, $1 \le i \le n$, be smooth vector fields on the open set $U \subset \mathbb{R}^n$, which are linearly independent at each point. Set $[X_i, X_j] = \sum_{k=1}^n c_{ij}^kX_k$. Suppose that the coefficients $c_{ij}^k$ are all constant. Let $\overline{X}_i$, $1 \le i \le n$, on $\overline{U} \subset \mathbb{R}^n$ be another copy (with the same coefficients). Set $Y_i = X_i + \overline{X}_i$, $1 \le i \le n$, on $U \times \overline{U}$. Do the vector fields $Y_i$ necessarily span an integrable distribution $E \subseteq T(U \times \overline{U})$?

Answer: Yes


Question: Find the equation of the ellipse obtained by rotating the ellipse \(x^2+y^2+xy= \frac{1}{2}\) using the transformation \(\sqrt{2}x = u+v\) and \(\sqrt{2}y=u-v\).

Answer: 3u^2 + v^2 = 1


Question: Determine whether the following statement is true or false: In group theory, for any group presentation, all relations between elements of the generators can be determined from the relations in the presentation.

Answer: A


Question: Given the identity \(\cos^6(x) = m\cos(6x) + n\cos(5x) + o\cos(4x) + p\cos(3x) + q\cos(2x) + r\cos(x) + a\), determine the value of \(a\).

Answer: \dfrac{5}{16}


Question: Let $K \subset L \subset M$ be a tower of fields. If $L/K$ and $M/L$ are separable extensions, is the extension $M/K$ also separable?

Answer: Yes


Question: Find the expected value of the maximum absolute value of the coordinates of a point chosen uniformly at random within a square with side length 1 centered at the origin, with edges parallel to the axes.

Answer: \dfrac{1}{3}


Question: Find the greatest common divisor (GCD) of the polynomial \(4x^4 + 4x^3 - 11x^2 - 6x + 9\) and its derivative using the Euclidean algorithm.

Answer: 2x^2 + x - 3


Question: In an additive category $\mathcal{A}$, if $A \oplus C \cong A$ for objects $A$ and $C$, must it be true that $C \cong 0$?

Answer: No


Question: Find the value of $x > 0$ such that the matrix \( \begin{pmatrix} 100\sqrt{2} & x & 0 \\ -x & 0 & -x \\ 0 & x & 100\sqrt{2} \end{pmatrix} \) has two equal eigenvalues.

Answer: 50


Question: If $X$ and $Y$ are random variables with the same distribution, does it imply that the distribution of $X-Y$ is the same as the distribution of the constant $0$? Answer yes or no and justify your answer.

Answer: No


Question: In the isosceles triangle $\triangle ABC$, where $AB = 4\sqrt{2}$ units and $\angle B$ is a right angle, the incenter $I$ is equidistant from all sides. Find the length of $BI$ in the form $a + b\sqrt{c}$, where $a$, $b$, and $c$ are integers, and $c$ is not divisible by any perfect square integers other than $1$. Express your answer in the form $a + b\sqrt{c}$.

Answer: 8 - 4\sqrt{2}


Question: Find the smallest integer k such that the number n = 90k has exactly ninety positive divisors, including 1 and n. Use elementary methods such as modular arithmetic and/or Euler's totient function to determine k.

Answer: 280


Question: Let $A$ be a real $n \times n$ matrix with rank $r$. Find the dimension of the vector space $V$ consisting of all real $n \times n$ matrices $X$ such that $AX=O$, where $O$ is the $n \times n$ zero matrix.

Answer: n(n - r)


Question: Let $g:[0,1] \to \mathbb{R}$ be an increasing function. Define $f:[0,1] \to \mathbb{R}$ by $f(x)=0$ for $x=0$, and $f(x)=\frac{1}{x} \int_0^x g(t)~dt$ for $x \in (0,1]$. Is $f$ necessarily of bounded variation?

Answer: Yes


Question: Determine whether the following statement is true or false: For any random variables $X$ and $Y$ and any non-increasing function $f$, if $E(X) \leq E(Y)$, then $E(f(X)) \leq E(f(Y))$. If the statement is false, provide a counterexample.

Answer: False


Question: Given that X and Y are independent standard Gaussian random variables, find the expected value of X conditioned on the product XY.

Answer: 0


Question: Determine whether there exists a function \( f : \mathbb{R} \rightarrow \mathbb{R} \) such that \( f(x) \cdot f(y) = f(x + y) \) for all \( x, y \in \mathbb{R} \), but \( f \) is not everywhere continuous.

Answer: Yes


Question: Consider a monic polynomial $p(x)$ with no roots within the unit circle. Given that $p(-1)=0$ and $\displaystyle{\lim_{x\to \infty}}p(x)=\infty$, determine whether $p(x)$ necessarily has a root at $x=1$. Answer with 'yes' or 'no' and justify your answer.

Answer: no


Question: Determine the number of distinct ellipses that define the domain of the function: \[ f(x,y)=\frac{\sqrt{x}\sqrt{9-x^{2}-y^{2}}}{\ln(\frac{x^{2}}{4}+\frac{y^{2}}{16}-1)} \] without using a graphical representation.

Answer: 3


Question:
Calculate the expectation of the random variable $f(w) = 1 - w$ on the probability space $(\Omega, B, \mu)$, where $\Omega = [0,1]$, $\mu$ is the Lebesgue measure, and $B$ is the Borel $\sigma$-algebra. Use the measure $\mu_f$ defined by:

$$\mu_f = \begin{cases} 0 & x < 0 \\ x & 0 \leq x \leq 1 \\ 1 & x > 1 \end{cases}$$

Express the expectation as:

$$E(f) = \int_{\mathbb R}x\mu_f(dx) = \int_{[0,\infty)}x\mu_f(dx) - \int_{(-\infty,0)}x\mu_f(dx)$$

What is the value of $E(f)$?

Answer: \dfrac{1}{2}


Question: Determine the ordinal that is order isomorphic to $n \times \omega$ with the lexicographic order, where $n$ is a finite ordinal.

Answer: \omega \cdot n


Question: Given that $E[X^n] = n^2$, determine the value of $E[e^{2X}]$. Express your answer in terms of known constants or functions.

Answer: 1 + 6e^2


Question: Find the smallest positive integer \(x\) such that \(2^x \equiv 43 \pmod{59}\).

Answer: 33


Question: Evaluate the limit: $$\lim_{n \to \infty} (2^n - n^2)^{\frac{1}{n}}$$.

Answer: 2


Question: Let $\bf{F}$ be a smooth 3D vector field and let $S_{\epsilon}$ denote the sphere of radius $\epsilon$ centered at the origin. Evaluate the limit: $$ \lim_{\epsilon \to 0+} \frac{1}{4 \pi \epsilon^2} \int \int_{S_{\epsilon}} \bf{F} \cdot d \bf{S} $$ where the limit represents the flux per unit surface area as $\epsilon$ approaches zero.

Answer: 0


Question: Let $E$ be a Hausdorff topological space and consider a decreasing sequence of compact subsets $(K_{n})_{n \in \mathbb{N}}$ in $E$. Suppose there exists an open set $U \subset E$ such that $\bigcap_{n \in \mathbb{N}} K_{n} \subset U$. Determine whether the following statement is true: For some $n_{0} \in \mathbb{N}$, all $n \geq n_{0}$ imply $K_{n} \subset U$. Provide either a proof or a counterexample.

Answer: True


Question: Consider the ring of all continuous functions $C[0,1]$. Let $f \in C[0,1]$ be such that $f$ has a countable number of zeros. Can $f$ be a zero divisor in this ring?

Answer: No


Question: Let $G$ be a group and $H \le G$. Define $X = \{aH \mid a \in G\}$ and $Y = \{Ha \mid a \in G\}$. Consider the actions $\circ: G \times X \to X$ and $\star: G \times Y \to Y$, given by $g \circ aH = gaH$ and $g \star Hb = Hbg^{-1}$, respectively. These are left transitive actions with pointwise stabilizers $\mathrm{Stab}_\circ(aH) = aHa^{-1}$ and $\mathrm{Stab}_\star(Hb) = b^{-1}Hb$. Suppose the two orbits have prime size $[G:H] = p$ and intersect in more than one point, i.e., $\tilde aH = H\tilde a$ for some $\tilde a \in G \setminus H$. Can you conclude that the two orbits coincide, i.e., $H \lhd G$?

Answer: H \lhd G


Question: Determine the largest natural number \( m \) such that the polynomial \( P(x) = x^5 - 3x^4 + 5x^3 - 7x^2 + 6x - 2 \) is divisible by \((x-1)^m\).

Answer: 3


Question:
If $g(x)$ is a differentiable real-valued function satisfying $g''(x) - 3g'(x) > 3$ for all $x \geq 0$ and $g'(0) = -1$, determine the nature of the function $h(x) = g(x) + x$ for $x > 0$:

(A) $h(x)$ is an increasing function of $x$.
(B) $h(x)$ is a decreasing function of $x$.
(C) There is insufficient data to determine.
(D) None of the above.

Answer: A


Question: Let $X = K^n$ be equipped with the Euclidean norm $\|\cdot\|_2$, and let $Y = \ell_1(\mathbb{N})$. Define the linear transformation $T: X \to Y$ by $T(x) = (x_1, \ldots, x_n, 0, 0, \ldots)$ for all $x = (x_1, \ldots, x_n) \in K^n$. Calculate the operator norm $\|T\|$.

Answer: \sqrt{n}


Question:
Evaluate the Lebesgue integral \( \int_{[0, 1]} f \, d\mu \), where the function \( f(x) \) is defined as follows:

\[ f(x)= \begin{cases} \sin \pi x,& \text{if } x \in [0,1/2] -C\\ \cos \pi x , &\text{if } x\in (1/2, 1]-C\\ x^2, & \text{if } x \in C \end{cases} \]

Here, \( C \) is a Cantor set.

Answer: 0


Question: Evaluate the limit: $$\lim_{x \to -\infty} \frac{\sqrt[3]{2+3x}}{\ln(1-x)}.$$

Answer: -\infty


Question: Suppose $X$ is a set with a measure defined on it. Let $(B_n)_n$ be a pairwise disjoint sequence of subsets of $X$ with positive measure. For sets $P, Q \subset \mathbb{N}$ where $P \neq Q$, determine the $L^\infty$-norm of the difference between the characteristic functions of $\cup_{n\in P}B_n$ and $\cup_{n\in Q}B_n$. Provide your answer as a single number.

Answer: 1


Question: Let $X$ and $Y$ be two independent random variables uniformly distributed on the interval $(0, 1)$. Determine the probability density function of the random variable $W = XY$. Provide your answer in terms of $w$.

Answer: -\ln w


Question: Given the equation \( z - c = \rho\frac{1 + it}{1 - it} \), where \( c \) is a complex number, \( \rho \) is a real number, and \( t \) is a real parameter, find the relationship between \( z \), \( c \), \( \rho \), and \( t \). Assume \( z = x + iy \) and \( c = a + ib \), where \( x \), \( y \), \( a \), and \( b \) are real numbers.

Answer: |z - c| = \rho


Question: Determine the number of real solutions to the equation \(\sin(4x) = \frac{x}{100}\) for \(x \in (0, 2\pi)\) without using graphical methods.

Answer: 7


Question: Given 10 subsets of a set X, determine the maximum number of unique subsets that can be generated using the operations of union, intersection, complement, and symmetric difference, even if some subsets might be empty.

Answer: 2^{1024}


Question:
Given the matrix $A_n \in \mathbb{R}^{n \times n}$ defined as follows:

$$
A_n =
\begin{cases}
1 & \text{if } i=j \\
-j & \text{if } i=j+1 \\
i & \text{if } i=j-1 \\
0 & \text{otherwise}
\end{cases}
$$

It has been shown that the determinant of $A_n$ satisfies the recurrence relation:

$$
\det(A_n) = \det(A_{n-1}) + (n-1)^2 \cdot \det(A_{n-2})
$$

Derive an explicit expression for $\det(A_n)$.

Answer: n!


Question: Determine if the set \( \{1, 2, 3, 5, 7, 11, 2310\} \) under the relation "divides" forms a lattice in which every element, except the least and greatest elements, has four complements.

Answer: Yes


Question:
Given a Markov chain with state space $E = \{1,2,3,4,5\}$ and the transition matrix:

$$ P = \begin{bmatrix} 0 & \frac{1}{2} & \frac{1}{2} & 0 & 0 \\ \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 \\ \frac{1}{4} & \frac{1}{4} & 0 & \frac{1}{4} & \frac{1}{4} \\ 0 & 0 & \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} \end{bmatrix} $$

Assuming $X_1 = 1$, calculate the expected number of transitions between successive visits to state 1, given that state 5 is never visited.

Answer: 4


Question: Suppose $f: \mathbb{R^n} \to \mathbb{R}$ is a twice differentiable function with a Lipschitz-continuous gradient, satisfying the condition $$\| \nabla f(x) - \nabla f(y) \|_2 \leq L \|x-y\|_2 , \quad \forall x,y \in \mathbb{R}^n.$$ Does it follow that the Hessian of $f$, denoted by $\nabla^2 f(x)$, has a bounded maximum singular value, i.e., $$\| \nabla^2 f(x) \|_2 \leq L, \quad \forall x \in \mathbb{R}^n?$$

Answer: Yes


Question: Determine whether the equation \(x=\int_{0}^{x} \cos^2(t^3) dt\) has a unique solution. If it does, find the solution.

Answer: 0


Question: Evaluate the limit \( \lim\limits_{n\to \infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!} \).

Answer: \dfrac{1}{2}


Question: Given that $f$ is analytic in a bounded domain $D$ and continuous on the boundary $\partial D$, with $|f(z)| \leq M$ for $z \in \partial D$, determine the maximum value of $|f(z)|$ for $z \in D$.

Answer: M


Question: Calculate the limit of the sum: \[ \lim_{n\to\infty}\sum\limits_{i=1}^n\frac{2i^2-1}{2^i} \]

Answer: 11


Question: Find the smallest integer value of $x$ that satisfies the equation $|x| + \left|\frac{x}{x - 1}\right| = \frac{x^2}{|x - 1|}$. \(\text{Provide your answer as an integer.}\)

Answer: 0


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

Answer: \dfrac{\pi}{4}


Question: Let $X$ be a normally distributed random variable with mean $0$ and variance $1$. Let $\Phi$ be the cumulative distribution function of the variable $X$. Find the expectation of $\Phi(X)$. \( \boxed{\text{Answer}} \)

Answer: \dfrac{1}{2}


Question: Determine if the expected value of the sum of the two largest values from a uniform distribution on the interval [0,1] for n random variables \((x_1, x_2, x_3, \ldots, x_n)\) is equal to the sum of the expected values of the maximum \((x^{(n)})\) and the second maximum \((x^{(n-1)})\).

Answer: Yes


Question: Find the maximum value of the expression \(x^4y + x^3y + x^2y + xy + xy^2 + xy^3 + xy^4\) given that \(x + y = 3\). Determine the value of \(\lfloor M \rfloor\), where \(M\) is this maximum value.

Answer: 36


Question: Determine whether a separable Hilbert space can have a finite orthonormal basis. Provide a yes or no answer.

Answer: Yes


Question: Find the equation of the osculating plane to the parametric curve defined by \( x = a\cos(\theta) \), \( y = a\sin(\theta) \), \( z = a(1+\sin(\theta)) \) at the point where \( \theta = \frac{\pi}{4} \).

Answer: z - y = a


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

Answer: 0


Question: Find all positive integer solutions \((p, q)\) to the equation: \[2^3 \cdot 3^2 = (p-1)(3q+p)\]

Answer: (3, 11)


Question: What is the maximum number of real zeros that a polyquaral with a degree of 5 and 3 nested square roots can have?

Answer: 5


Question: Evaluate the line integral \( \int_{\gamma} F \cdot d\gamma \) where \( F(x,y) = (2x + 3y, 3x + 4y) \) and \( \gamma \) is the upper part of the circle of radius \( \sqrt{5} \) centered at the point \( (2,0) \), running from the point \( (1,2) \) to \( (3,2) \).

Answer: 20


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

Answer: 1


Question: Given a surjective ring homomorphism $f: R \rightarrow S$, determine whether the preimage $f^{-1}(M)$ is a maximal left ideal of $R$ for every maximal left ideal $M$ of $S$. Provide a justification for your answer.

Answer: Yes


Question: Consider the function \( f(x) = \int_x^{2x} \frac{\sin t}{t} dt \). Determine the value of \( x \) on the interval \((0, \frac{3\pi}{2})\) at which this function attains a local maximum.

Answer: \dfrac{\pi}{3}


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

Answer: \dfrac{1}{2}


Question: Evaluate the limit: $$\lim_{x \rightarrow 0^{+}} \frac{1}{9} \left(\frac{1}{\arctan x} - \frac{1}{x}\right).$$

Answer: 0


Question: Determine the value of \( n \) for the slant asymptote of the function \( f(x) = xe^{\frac{1}{x-2}} \), given that the slant asymptote can be expressed as \( y = x + n \).

Answer: 1


Question: Find a function $\phi$ of class $C^2$ that minimizes the functional \( I(\phi) = \int_0^1 \frac{\phi''(t)}{\phi(t)} \, dt \) subject to the boundary conditions $\phi(0) = 1$ and $\phi(1) = 4$. Use the Euler-Lagrange equation to solve this problem.

Answer: 4^t


Question: Evaluate the limit of the function \( f(z) = (z-2)\log|z-2| \) as \( z \) approaches 2, or explain why the limit does not exist. Assume \( z \neq 2 \) and \( z = x + iy \), where \( x \) and \( y \) are real numbers.

Answer: 0


Question: Determine the largest integer $n$ such that $2013^n$ divides $2012!$. Provide your answer as a single integer $n$. \( \boxed{n} \)

Answer: 32


Question: Determine the value of \(a \in \mathbb{R}\) such that the function \(f : \mathbb{R}^3 \to \mathbb{R}\) defined by \[ f(x,y,z)=\begin{cases} \frac{z}{x}+y & \text{if } x \neq 0 \\ a & \text{if } x=0 \end{cases} \] is continuous at the origin.

Answer: 0


Question: Calculate the expected length of the projection of a random unit vector in \(\mathbb{R}^3\) onto a given plane.

Answer: \dfrac{\pi}{4}


Question: Determine whether the set of zeroes of the polynomial \( p(x) = x^4 + 4 \) in \( \mathbb{C} \) forms a group under multiplication. Answer true or false.

Answer: False


Question: Determine all real values of \(\alpha\) such that for every continuous function \(f: [0,1] \to [0,1]\), there exists a point \(c \in [0,1]\) satisfying \(f(c) = \alpha c\).

Answer: [1, \infty)


Question: A function $g(x)$ is twice differentiable and satisfies $|g(x)| \leq 5$ for all $x \in \mathbb{R}$. Given that $g(\beta) = 3$ and $g'(\beta) = 4$ for some $\beta$, determine the maximum possible value of $g(10)$. \( \boxed{} \)

Answer: 5


Question: Calculate the volume of the region bounded by the equation \((x^2 + y^2 + z^2)^2 = x\).

Answer: \dfrac{\pi}{3}


Question: Evaluate the limit: $$\displaystyle \lim_{\alpha \to -1}\left(2\Gamma(-\alpha-1)+\Gamma\left(\frac{\alpha + 1}{2}\right) \right)$$

Answer: -3\gamma


Question: Evaluate the limit: $$\lim_{n \to \infty }\sqrt[n]{ \frac{\left | \sin1 \right |}{1}\cdot\frac{\left | \sin2 \right |}{2}\cdot\cdot\cdot\frac{\left | \sin n \right |}{n}}$$

Answer: 0


Question: Find all functions \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) such that \( f(0) = 2 \) and \( f(x + f(x + 2y)) = f(2x) + f(2y) \) for all integers \( x \) and \( y \).

Answer: f(x) = x + 2


Question: Evaluate the limit: \[ \lim_{(x,y)\rightarrow (0,0)}\frac{e^{-\frac{1}{x^2+y^2}}}{x^4+y^4}. \]

Answer: 0


Question: Find the limit of the function \( f(x) = \frac{1}{a+\frac{1}{b+\frac{1}{c+\frac{1}{d+\ldots}}}} \) as \( x \to \infty \), where \( a, b, c, d \) are integers \( \geq 1 \) and \( f(x) = \frac{1}{\pi}\arctan(x) \).

Answer: \dfrac{1}{2}


Question: Determine the real positive value \( r \) for which the function \( y_2(x) = x^r \) is linearly independent of \( y_1(x) = \frac{1}{x} \) with respect to the differential equation \( 2x^2y'' + 3xy' - y = 0 \). Use the Wronskian determinant to verify linear independence.

Answer: \dfrac{1}{2}


Question: Determine if the function \( f(x) = \sqrt{x^5 + 4x^3 + 3x + 1} \) is bijective on the interval \([0, \infty)\). If it is bijective, find the range of its inverse.

Answer: [0, \infty)


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

Answer: 1 - \cos 1


Question: Find the value of the polynomial \( W(x) \) of degree 2015 such that \( W(n) = \frac{1}{n} \) for \( n = 1, 2, \ldots, 2016 \). What is \( W(2017) \)?

Answer: 0


Question: In a right triangle $ABC$ with $\angle CAB = 90^\circ$, let $D$ be the midpoint of side $BC$. Point $E$ is on side $AD$ such that $\angle BED = 90^\circ$. Given that $\angle BAD = 2\angle DAC = 2\alpha$ and $DE = 3$, determine the length of $AB$. Express your answer as $x$. \( \text{(Geometric solutions are preferred.)} \)

Answer: 6


Question: Let $X$ be a space and $\{U_\lambda\}_{\lambda \in \Lambda}$ be a collection of subsets of $X$. Construct the coarsest topology $\mathcal{T}$ such that each $U_\lambda$ is open. Is the topology $\mathcal{T}$ given by the following construction: \(\mathcal{S}=\left\{\bigcap_{\lambda \in \Gamma}U_\lambda \mid \Gamma\subset \Lambda, |\Gamma|<\infty \right\}\) and \(\mathcal{T} =\left\{\bigcup_{U\in \mathcal{G}}U\mid \mathcal{G}\subset \mathcal{S}\right\}\) the coarsest topology? Justify your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim _{x \rightarrow 0^{-}}\left(\cos 3 x\right)^{\cot 3 x}$$

Answer: 1


Question: Does there exist a positive integer \( N \) such that there are infinitely many positive integers \( n \) for which the Euler's totient function satisfies \( \varphi(n) = N \)?

Answer: No


Question: Determine the number of ordered pairs \((m, n)\) of positive integers that satisfy the equation \(4^m - 3^n = 1\).

Answer: 1


Question: Calculate the probability that the maximum of the independent binomial random variables $x_1, x_2, \ldots, x_n$ with parameters $p$ and $n$ is less than $n$. That is, find $P[\text{max}(x_{1}, x_{2}, \ldots, x_{n})<n]$. Express your answer in terms of $p$ and $n$.

Answer: (1 - p^n)^n


Question: Evaluate the integral \( \int \frac{e^{-x} \cos x}{\sinh x - \cosh x} \, dx \).

Answer: -\sin x + C


Question: Consider a Markov chain with state space $S = \{0, 1, 2, \dots \}$ and given transition probabilities. Determine the values of $p$ for which the chain is recurrent.

Answer: p \leq \dfrac{1}{2}


Question: Evaluate the integral \( \int_{0}^{1} \frac{r^3}{ \sqrt {1-r^2}}dr \).

Answer: \dfrac{2}{3}


Question: For a singular matrix A with dimensions greater than 2x2, does the sum of the squared elements of A correspond to an eigenvalue of ATA?

Answer: No


Question: Solve the equation \((26-x)\cdot\sqrt{5x-1} -(13x+14)\cdot\sqrt{5-2x} + 12\sqrt{(5x-1)\cdot(5-2x) }= 18x+32\) in the set of real numbers.

Answer: 2


Question: Determine an integrating factor for the differential equation \((x^2-y^2+2xy)dx+(x^2-y^2-2xy)dy=0\).

Answer: 1


Question: Find the value of $a$ that maximizes the integral \( \int_{a-1}^{a+1} \frac{1}{1+x^8} \, dx \).

Answer: 0


Question: Evaluate the limit \( \lim_{n\rightarrow\infty} 2^{-2n} \binom{2n}{n} \) using Stirling's approximation.

Answer: 0


Question: Solve the equation \( \frac{z-1}{z+1}=|z|i \) for complex numbers \( z \), where \( |z| \) denotes the modulus of \( z \).

Answer: i


Question: Evaluate the limit: $$\lim_{n\rightarrow \infty}n \int_{-\frac{1}{n}}^{\frac{1}{n}} f(x)(1-n|x|)dx$$ where $f: [-1, 1] \rightarrow \mathbb{R}$ is a continuous function.

Answer: f(0)


Question: Determine whether the differential equation \( \sin(t)y' + y\cos(t) = (\sin(t))^2 \) has a singular point at \( t = 0 \).

Answer: Yes


Question: In a right-angled triangle with a 15-degree angle and a hypotenuse of length 7, a point is randomly chosen on the hypotenuse. Two perpendicular lines are drawn from this point to the other two sides of the triangle. Calculate the probability that the area of the rectangle formed by these perpendicular lines is greater than 3.

Answer: \dfrac{1}{7}


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

Answer: 1


Question: In a right triangle, the circumcircles of the two smaller triangles formed by the altitude intersect at two points. If the midpoint of the hypotenuse is equidistant from the points where the circumcircles intersect the hypotenuse, determine the ratio of the angles formed by the hypotenuse and the lines connecting the intersection points to the midpoint of the hypotenuse.

Answer: 1


Question: Determine whether the following statement is true or false: Every infinite set has an infinite countable subset.

Answer: A


Question: Determine the supremum of the set \(\{(\omega+2)^k \mid k \in \omega\}\).

Answer: \omega^{\omega}


Question: Calculate the contour integral \( \int_C \vec{F}\cdot\vec{dr} \), where \( \vec{F}=(y,x) \) and \( C \) is the closed path \( \vec{r}(t)=(\cos t, \sin t) \) for \( 0\leq t\leq 2\pi \).

Answer: 0


Question: Given $a, b, c \in \mathbb{R}$ such that $a^2 + b^2 + c^2 = 1$, find the minimum value of $ab + bc + ac$.

Answer: -\dfrac{1}{2}


Question: Calculate the area of the region that lies inside the polar curve \( r = 3\sin\theta \) and outside the polar curve \( r = 1 + \sin\theta \).

Answer: \pi


Question: Suppose $A$ and $B$ are nonempty sets. If there exist sets $C$ and $D$ such that $(A\times B)\cup (B\times A)=C\times D,$ prove that $A=B=C=D.$

Answer: A = B = C = D


Question: Evaluate the limit: $$\lim_{n\to\infty} \frac{1}{n \log(n)} \frac{\sin\left(\frac{\pi}{n}+\frac{2 \pi \log(n)}{\sqrt n}\right)}{\sin\left(\frac{\pi}{n}\right)}$$

Answer: 0


Question: Let $R$ be a commutative ring with identity and $I$ be an ideal of $R$. If $R/I$ is a field of characteristic $p$, where $p$ is a prime number, does there exist an ideal $J$ of $R$ such that $I = (p) + J$ and $J$ does not contain any nonzero constants?

Answer: Yes


Question:
Find a function $b(x)$ such that for some $\beta > 0$, the following limit holds:
\[ \lim_{{x \to \infty}} \frac{1}{b(x)}\left(-\frac{1}{{\ln(1-\frac{1}{{xt}})}} + \frac{1}{{\ln(1-\frac{1}{x})}}\right) = \frac{t^{\beta}-1}{\beta} \]

Answer: x


Question: Given the harmonic function \(u(x,y) = x^3 + ax^2y + bxy^2 + 2y^3\) and its harmonic conjugate \(v(x,y)\), where \(v(0,0) = 1\), find the value of \(|a + b + v(1,1)|\).

Answer: 10


Question: Find the splitting field and its degree over \( \mathbb{Q} \) for the polynomial \( f(x) = x^4 - 2 \).

Answer: 8


Question: Evaluate the limit: $$\lim\limits_{n\to +\infty}\frac{\left(1+\frac{1}{n^3}\right)^{n^4}}{\left(1+\frac{1}{(n+1)^3}\right)^{(n+1)^4}}$$

Answer: \dfrac{1}{e}


Question: Calculate the probability that the sum of four independent random variables $X_1$, $X_2$, $X_3$, and $X_4$, each uniformly distributed on the interval (0, 1), is greater than or equal to 3.

Answer: \dfrac{1}{24}


Question: Find the minimum natural number \( m \) such that the limit \( \lim_{(x,y)\to(0,0)} \frac{x^{\frac{m}{3}}|x-y|}{\sqrt{x^2+y^2}} \) exists and is a real number.

Answer: 1


Question: Determine the number of integer solutions to the congruence equation \(x^5 \equiv -1 \pmod{400}\).

Answer: 5


Question: Let $f$ be a real-valued function that is differentiable on the open interval $(0,1)$. Assume that $\sup_{0<r<1} |f(r)| < \infty$ and $\liminf_{r \to 0^+} f'(r) \geq 0$. Does this imply that the one-sided limit $f(0^+)$ exists? Answer 'Yes' or 'No'.

Answer: Yes


Question: Calculate the limit: $$\lim_{x \rightarrow 0}\frac{x^{2}\ln\left(1+2x\right)}{2\sin\left(x\right)\left[ \cos\left(3x\right)-1\right]} $$

Answer: -\dfrac{2}{9}


Question: Determine the characteristic polynomial of the linear operator $T$ on the vector space $P_3(\mathbb{R})$, where $T(f(x)) = f'(x) + f''(x)$. Here, $P_3(\mathbb{R})$ denotes the space of all real polynomials of degree at most three.

Answer: \lambda^4


Question: Determine the smallest positive number $T$ such that the function $f(x) = \sin(k\cos{x})$, where $k > 0$, is periodic with period $T$. What is $T$?

Answer: 2\pi


Question: Let \(G\) be a finite group and \(\phi: G \to K\) be a surjective homomorphism. If \(K\) has an element of order \(n\), does \(G\) necessarily have an element of order \(n\)? Provide a justification for your answer.

Answer: Yes


Question: Consider a Markov chain on a ring with 16 numbers, where each step involves moving to one of the two adjacent numbers with equal probability. Calculate the expected number of steps for the chain to return to its starting position.

Answer: 16


Question: Determine the number of real solutions to the equation \(12^x + 14^x + 18^x = 20^x\).

Answer: 1


Question: Determine the value of \(a < 0\) for which the inequalities \(2\sqrt{ax} < 3a - x\) and \(x - \sqrt{\frac{x}{a}} > \frac{6}{a}\) have at least one solution in common.

Answer: -\dfrac{2}{3}


Question:
Consider the Burgers equation \(u_t + uu_x = 0\) with the initial condition given by:
\[
u(x,0) = \phi(x) = \begin{cases} 2, & x \leq \pi/2 \\ \sin x + 1, & \pi/2 \leq x \leq 3\pi/2 \\ 0, & x \geq 3\pi/2 \end{cases}.\]
Determine the breaking time \(t_B\).

Answer: 1


Question: Let \(X\) be a metric space and let \(U,V\subset X\) be nonempty open and bounded subsets. Determine whether the Hausdorff metric \( h(U,V)=\max(\sup_{x\in U}\inf_{y\in V}d(x,y),\sup_{x\in V}\inf_{y\in U}d(x,y)) \) is defined for all possible choices of \(U\) and \(V\).

Answer: Yes


Question: In the matching problem, where $n$ letters are randomly matched with $n$ envelopes, let $X$ be the total number of matches. Given that $\mathsf{E}(X) = 1$, find the expected value $\mathsf{E}(X^2)$. Express your answer as a single number.

Answer: 2


Question: Let $X$ be a normal projective variety over an algebraically closed field of arbitrary characteristic. Assume that $X$ has rational singularities, meaning there exists a resolution of singularities $f:Y\to X$ such that for all $i>0$, $R^if_*\mathcal{O}_Y=0$ and $R^if_*\omega_Y=0$. Determine if $f_*\omega_Y=\omega_X$, i.e., if the trace is an isomorphism.

Answer: f_*\omega_Y = \omega_X


Question: Determine the element of the form \((x, y)\) that generates the group \(\mathbb{Z}/ m\mathbb{Z} \times \mathbb{Z}/ n\mathbb{Z}\), given that \(m\) and \(n\) are coprime and the group is isomorphic to the cyclic group \(\mathbb{Z}/mn\mathbb{Z}\). Use the fact that \(\mathbb{Z}/mn\mathbb{Z} = \langle1\rangle\) to identify this generator.

Answer: (1, 1)


Question: Suppose $(X, \mathcal{M}, \mu)$ is a measure space such that $\dim_{\mathbb{R}}L^p(X, \mathcal{M}, \mu) > 1$. Determine whether there exist disjoint sets $A, B \in \mathcal{M}$ such that $0 < \mu(A), \mu(B) < \infty$. Provide a justification for your answer.

Answer: Yes


Question: Consider the binary operation \(x*y=\dfrac{2xy+3(x+y)+2}{3xy+2(x+y)+3}\) defined on the interval \((-1, 1)\). Given that the function \(f:(-1, 1) \rightarrow (0, \infty)\) defined by \(f(x) = \dfrac{1}{5} \cdot \dfrac{1-x}{1+x}\) satisfies \(f(x * y) = f(x)f(y)\) for all \(x,y \in (-1, 1)\), find the value of \(\underbrace{x*x*\ldots*x}_{x\text{ 5 times}}\) if \(f(x) = \frac{1}{2}\).

Answer: \dfrac{27}{37}


Question: Determine the radius of convergence for the series \( \sum_{n=0}^{\infty} \frac{n}{5^{n-1}} z^{\frac{n(n+1)}{2}} \).

Answer: 1


Question: Given a partial order \( \langle P, \preceq \rangle \) where \( P = \{ \langle k, a_k \rangle : k = 1, \ldots, n^2 + 1 \} \) and \( \langle k, a_k \rangle \preceq \langle \ell, a_\ell \rangle \) if and only if \( k \le \ell \) and \( a_k \le a_\ell \), determine what can be concluded about the sequence \( a_1, a_2, \ldots, a_{n^2+1} \) given that each antichain in \( P \) has length at most \( n \).

Answer: n + 1


Question:
Given the joint characteristic function \(\phi_{X,Y}(t_1, t_2)\) of two random variables \(X\) and \(Y\):

\[\phi_{X,Y}(t_1, t_2) = \frac{1}{2}e^{i(t_1+t_2)} + \frac{1}{4}e^{i(2t_1+t_2)} + \frac{1}{12}e^{it_2} + \frac{1}{6}e^{i(4t_1+3t_2)}\]

Calculate the probability \(P(X < Y)\).

Answer: \dfrac{1}{12}


Question: Suppose $f(x)=x^2-3x+2$. Find the sum of all the roots of the equation $f^{[5]}(x)=0$. Express your answer as a single number.

Answer: 48


Question: If the series \( \sum_{n=1}^{\infty} b_n \), formed by grouping consecutive terms of the same sign from the series \( \sum_{n=1}^{\infty} a_n \), is convergent, does it imply that the series \( \sum_{n=1}^{\infty} a_n \) also converges to the same limit?

Answer: Yes


Question: In triangle ABC, where \( \angle A = 70^\circ \), the angle bisector AE intersects side AC at point D and side BD at point H. Given the ratios \( \frac{AH}{HE} = \frac{3}{1} \) and \( \frac{BH}{HD} = \frac{5}{3} \), find the measure of \( \angle C \).

Answer: 55^\circ


Question: Determine whether it is possible for a sequence of continuous functions to converge pointwise to a discontinuous function on a compact set. Provide a justification for your answer.

Answer: Yes


Question: Let $G$ be a connected regular graph. Consider two distinct vertices $u$ and $v$ in $G$. If we construct $H_1$ by removing vertex $u$ and $H_2$ by removing vertex $v$, must $H_1$ and $H_2$ be isomorphic graphs? Answer yes or no.

Answer: No


Question: Determine the rank of the linear transformation $A$ defined by left multiplication by a matrix $P$ of rank $m$ on the space of all linear transformations on an $n$-dimensional vector space $V$. Provide your answer as a single integer.

Answer: mn


Question:
Determine the index \([C : D]\) of the subgroup \(D\) in \(C\), where \(C\) and \(D\) are subgroups of \(\mathbb{Q}[x]\) defined as follows:

\[C = \left\{p(x) \in \mathbb{Q}[x] \mid p(x) = a + bx + c\frac{x(x-1)}{2}, a,b,c \in \mathbb{Z}\right\}\]

\[D = \left\{p(x) \in \mathbb{Z}[x] \mid p(x) = a + bx + c\frac{x(x-1)}{2}, a,b,c \in 2\mathbb{Z}\right\}\]

Answer: 8


Question: Determine whether the function \(f(x,y,z) = (x+y+z)\sin\left(\frac{1}{x}\right)\sin\left(\frac{1}{y}\right)\) has a limit at the origin \((0,0,0)\). If the limit exists, find its value.

Answer: 0


Question: Find the exact real solutions of the equation \(2 \cdot 3^x + 2 = 5 \cdot 2^x\) without using Newton's method.

Answer: 2


Question: Find the minimum value of the functional \( J(y) = \int_a^b \sqrt{1+y'(x)^2} \, dx \) for functions \( y \) that are continuously differentiable on the interval \([a, b]\).

Answer: b - a


Question: Evaluate the contour integral \(\displaystyle\oint_{\gamma}\frac{e^z}{z-\log(z)}dz\), where \(\gamma\) is a positively oriented contour consisting of the straight line segments connecting the points \(z=2\), \(z=4\), \(z=4+3i\), and \(z=2+3i\).

Answer: 0


Question: Compute the limit \( \lim_{c \rightarrow 0} \iint_{R} \frac{1}{(x^2+y^2)^{3/4}}\,dA \), where \( R = \{(x,y): x^2+y^2 \leq 1\} \setminus ([-c,c] \times [-c,c]) \) for each \( 0 \leq c \leq \frac{1}{\sqrt{2}} \).

Answer: 4\pi


Question: Are all orthogonal operators on finite-dimensional subspaces of \(\mathbb{R}^n\) surjective?

Answer: Yes


Question: Suppose $X_1, X_2, ..., X_n$ and $Y_1, Y_2, ..., Y_m$ are random samples from normal distributions with means $\mu_X$ and $\mu_Y$, and standard deviations $\sigma_X$ and $\sigma_Y$, respectively. Given that $n = 3m$, find the smallest aggregate sample size ($n + m = 4m$) such that the probability that the sample mean of the $Y$'s exceeds the sample mean of the $X$'s is no greater than 0.01.

Answer: 32


Question: Evaluate the determinant of the matrix \( A = [a_{ij}] \) where \( a_{ij} = 1 + x_i x_j \) for \( i = j \) and \( a_{ij} = x_i x_j \) for \( i \neq j \), given that \( x_1, x_2, x_3, x_4, x_5 \) are the roots of the polynomial \( x^5 - 2x^4 + x^3 + 1 = 0 \).

Answer: 3


Question: Compute the limit \(\lim_{n\to \infty} \frac{\ln^2n}{n}\sum_{k=2}^{n-2}\frac{1}{\ln k\ln(n-k)}\).

Answer: 1


Question: Find the smallest natural number \( n \) such that \( n < e^{6 \sqrt{n}} \).

Answer: 1


Question: Does a function \(f: [a,b] \rightarrow \mathbb{R}\) that is continuous almost everywhere (a.e.) guarantee the existence of \(c,d \in [a,b]\) such that \(f(c) \leq f(x) \leq f(d)\) for all \(x \in [a,b]\)? Provide a justification for your answer.

Answer: No