Question: Suppose that $(X,Y)$ is a bivariate normal distribution where both $X$ and $Y$ have mean $0$ and variance $1$. Given that the correlation between $X$ and $Y$ is $\rho$, find the correlation $\operatorname{corr}(X^2,Y^2)$. Express your answer in terms of $\rho$.

Answer: \rho^2


Question: Let $x$ be a complex number such that $x^N = 1$ and $x^2 \ne 1$. Determine if the following statement is true: \( \sum_{n=0}^{N-1} x^{2n} = 0 \).

Answer: True


Question: Evaluate the limit \( \lim\limits_{h\to 0}\frac{f(2-h^2)-f(2)}{h^2} \), where \( f(x) = (x^3+2x)\left[\frac{x}{2}\right] \) and \([\cdot]\) denotes the floor function.

Answer: -\infty


Question: Given an infinite set of positive integers, determine if it is always possible to find a subset of this set that contains infinitely many numbers such that any two numbers in this subset are either relatively prime or have a greatest common divisor greater than 1.

Answer: Yes


Question: Evaluate the limit: $$ \lim_{x\to\infty}\frac{\ln(1+e^{ax})}{\ln(1+e^{bx})} $$ as \( x \) approaches infinity.

Answer: \dfrac{a}{b}


Question: In triangle $ABC$, where $AB=30$, $BC=51$, and $CA=63$, points $P$ and $Q$ are on $\overline{BC}$, $R$ is on $\overline{CA}$, and $S$ is on $\overline{AB}$ such that $PQRS$ forms a parallelogram. The center of parallelogram $PQRS$ coincides with the centroid of $\triangle ABC$. Determine the area of parallelogram $PQRS$.

Answer: 336


Question: Determine the number of real solutions for the equation \( \sqrt{3x^2 - 18x + 52} + \sqrt{2x^2 - 12x + 162} = \sqrt{-x^2 + 6x + 280} \) without using the method of squaring both sides.

Answer: 1


Question: Consider the function \( f(x) = \frac{cx}{2x+3}, \ x \neq -\frac{3}{2} \). Find all values of \( c \) for which \( f(f(x)) = x \).

Answer: -3


Question: Determine whether it is possible for a satisfiable sentence to have no models of size 4 or greater.

Answer: Yes


Question: A biased coin with a probability of 0.51 for heads is tossed repeatedly. The tossing stops when there are more heads than tails. Determine if the probability of ultimately getting more heads than tails in this series of tosses is 1.

Answer: 1


Question: Determine the degree of the splitting field of the polynomial \( f(x) = x^4 + 4 \) over \( \mathbb{Q} \).

Answer: 2


Question: Consider a sequence of numbers $s_n$ defined as the sum of the first $n$ terms, $s_n=\sum_{k=0}^{n}a_k$. If the $n$th term satisfies $n\log n \ a_n\rightarrow 0$ as $n$ approaches infinity, does it follow that $\frac{s_n}{\log n}$ converges to 0 as $n$ approaches infinity? Answer yes or no and justify your answer.

Answer: Yes


Question: Determine the degree of the covering map from the surface $R = T^{2}\#T^{2}\#T^{2}\#T^{2}\#T^{2}\#T^{2}$ to the surface $S = T^{2}\#T^{2}$, given that the genus of $R$ is $g_{R} = 6$ and the genus of $S$ is $g_{S} = 2$. Provide your answer as a single integer.

Answer: 5


Question: Evaluate the line integral \( \int_C \vec{G} \cdot d\vec{r} \) where \( \vec{G} = \langle xz, y+z, xy \rangle \) and \( C \) is the boundary curve of the surface \( S \) that is the portion of the paraboloid \( z = 4 - x^2 - y^2 \) above the plane \( z = 0 \).

Answer: 0


Question: Determine the boundary of the boundary of the set $A = [0,1]$. What is $Bd(Bd(A))$?

Answer: \{0, 1\}


Question: Determine the order of the smallest nonabelian simple group that is not isomorphic to $A_5$. Provide your answer as a single number.

Answer: 168


Question: Let $R$ be a commutative ring, and $R'$ another commutative ring. If $\{b_1, \ldots, b_n\}$ are elements of $R'$ that are integral over $R$, does it follow that the ring $R[\{b_1, \ldots, b_n\}]$ is a finitely generated $R$-module?

Answer: Yes


Question: Evaluate the limit \( L_1 = \lim_{x \to 0} \frac{\text{erf}(x^2)}{\text{erf}(x)} \) using L'Hopital's rule and Leibniz's integral rule.

Answer: 0


Question: Thirteen people wish to have dinner together at a round table for several nights, ensuring that each person has different neighbors every night. What is the maximum number of nights they can achieve this arrangement?

Answer: 6


Question: Find the fourth derivative of the inverse function \(f^{-1}(y)\) evaluated at \(y=1\), where \(f(x) = \sin{x} + x^2 + 1\).

Answer: -140


Question: Determine whether the following statement is true: For any function $f(x)$ such that $\lim_{x \to a} f(x) = 0$, the limit \( \lim_{x \to a} \frac{\sin(\ln(1+f(x)))}{f(x)} = 0 \) holds.

Answer: False


Question: Evaluate the limit \(\lim\limits_{h \rightarrow 0^{+}} \int_{-\infty}^{\infty} \frac{h}{h^{2}+x^{2}} \, dx\) using a probability density function (pdf).

Answer: \pi


Question: Let $\mathcal{C}$ be the set of all finite closed connected intervals of $\mathbb{R}$. Define the evaluation map $E : \mathcal{C} \longrightarrow \mathbb{R}$ by $[b,c] \mapsto \int_b^c 1\,dx$. Let Span $\mathcal{C}$ be the vector space of all linear combinations of intervals in $\mathcal{C}$, and extend $E$ linearly to this space. Consider the equivalence relation $X \sim \tilde{X}$ if and only if $E(X) = E(\tilde{X})$. What is the dimension of Span $\mathcal{C} /\sim$?

Answer: 1


Question: Determine the possible values of the positive integer $n$ such that $2018^n = a^4 + b^4 + (b^2 + c^2)^2$, where $a$, $b$, and $c$ are positive integers.

Answer: 1


Question: Let $M$ and $N$ be monoids. If the category $C(M)$ is isomorphic to the category $C(N)$, is $M$ necessarily isomorphic to $N$? Provide a justification for your answer.

Answer: Yes


Question: Does the converse of the Cayley-Hamilton Theorem hold? Specifically, if there exists a polynomial \( P \in K[t] \) such that \( P(F) = 0 \) for some linear operator \( F \) on a vector space \( V \), is \( P \) necessarily the characteristic polynomial of \( F \)? Answer yes or no.

Answer: No


Question: Let $A$ be a square matrix of order $2$ with $\lvert A \rvert \ne 0$. Given that $\big\lvert A-\lvert A \rvert \operatorname{adj} (A)\big\rvert = 2\det(A) [1+\det^2(A)]$, find the value of $\big\lvert A+\lvert A \rvert \operatorname{adj} (A)\big\rvert$. \( \text{Express your answer as a single number.} \)

Answer: 0


Question: Compute \( \text{Tor}_2(R/(y,z), R/(x) \oplus (R/(y,z))) \) for the formal power series ring \( R = [|x,y,z|] \).

Answer: R/(y,z)


Question: Is the measure $h\mu$ finite if $h$ is a probability density on a measure space $(\Omega,\mathcal{A},\mu)$?

Answer: Yes


Question: In the vector space $V$ of polynomials with degree at most 2 over the real numbers, a bilinear form $\psi: V \times V \rightarrow \mathbb{R}$ is defined by \( (p, q) \mapsto \int_{0}^{1}p(x)q(x)dx \). Given a linear form \( \phi: p \mapsto p(0) \), find a polynomial \( q \in V \) such that \( \phi(p) = \psi(p, q) \) for all \( p \in V \).

Answer: 30x^2 - 36x + 9


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

Answer: \dfrac{1}{e}


Question: Given an open interval $(a, b) \subseteq \mathbb{R}$ and a Lebesgue measurable function $r : (a, b) \to \mathbb{R}$ that is positive almost everywhere, determine if there must exist $k \in \mathbb{N}$ and $a < c < d < b$ such that $r(x) > \frac{1}{k}$ for almost all $x \in (c, d)$. Answer with "Yes" or "No".

Answer: No


Question: Given $n$ real matrices $A_1, A_2, \ldots, A_n$, if their $k$-th power sum is zero for all $k \in \mathbb{N}_{>0}$, are all the matrices nilpotent? Answer yes or no.

Answer: Yes


Question: Evaluate the integral \( \int_0^2 \frac{2}{\ln(x)}\,dx \) given that the function \( F(x) \) satisfies \( F(0)=1 \), \( F(1)=2 \), and \( F(2)=5 \), where \( F'(x)=\frac{1}{\ln(x)} \). Note that there is a discontinuity at \( x=1 \).

Answer: 8


Question: Determine whether a finite abelian group can have a nontrivial subgroup that is contained in every proper subgroup, yet the group itself is not cyclic. Answer yes or no.

Answer: No


Question: Find the smallest natural number \( n \) such that the group \( \mathbb{Z}_{6} \times \mathbb{Z}_{20} \times \mathbb{Z}_{45} \) is isomorphic to a direct product of \( n \) cyclic groups.

Answer: 2


Question: Determine whether a Riemann integrable function on an interval \([a, b]\) necessarily has a primitive function on the same interval. Answer with "Yes" or "No".

Answer: No


Question: Suppose that $a_{n+1} = \log(1 + a_n)$ with $a_1 > 0$. Find $\lim_{n \rightarrow \infty} n \cdot a_n$. (Note: $\log$ denotes the natural logarithm.)

Answer: 2


Question: Determine the value of $k$ such that the limit \( \lim_{x\rightarrow0} \frac{(e^{x^2}-x^2-1)(\cos(x)-1)}{x^k} \) is finite and non-zero.

Answer: 6


Question: Determine the number of unique group combinations possible when 8 people play 4 rounds of golf in groups of 4, ensuring that each player plays with every other player exactly twice.

Answer: 0


Question: Evaluate the limit: $$\lim_{n\to\infty}\frac{\int_1^{n^{1/n}} y^{n^2}(\arctan y) ny^{n-1}\,dy}{n^n}$$

Answer: \dfrac{\pi}{4}


Question: Suppose that a monic rational polynomial \(f(x)\) of degree \(p-1\) has a \(p\)-th root of unity as a root, where \(p\) is a prime number. Is \(f(x)\) necessarily the cyclotomic polynomial \(x^{p-1} + x^{p-2} + \cdots + x + 1\)? Answer "Yes" or "No".

Answer: No


Question: Consider the series \(\sum_{i=1}^\infty c_i^{(0)}\) which is a positive convergent series. Define the series \(\sum_{i=1}^\infty c_i^{(1)} = \sum_{i=1}^\infty \ln(1+c_i^{(0)})\). Generally, let \(c_i^{(N+1)} = \ln(1+c_i^{(N)})\) and define the sum \(S^{(N)} = \sum_{i=1}^\infty c_i^{(N)}\). Determine the nature of the limit \(\lim_{N\rightarrow\infty} S^{(N)}\). Is this limit zero or nonzero?

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\to0}{\frac{\ln(\sin(3x))}{\ln(\sin(2x))}} \] without using L'Hopital's rule.

Answer: 1


Question: In Laguerre's method for finding roots of polynomials, what does the assumption about the distribution of roots imply about the value of $n$?

Answer: n


Question: Given a finite dominant morphism $f:X \rightarrow Y$ of integral, locally noetherian schemes, if $f$ is flat at a point $y$, determine whether there exists an open neighborhood $V$ of $Y$ containing $y$ such that the induced morphism $f^{-1}(V) \rightarrow V$ is flat.

Answer: Yes


Question: Determine the center of the quaternion group \( Q = \langle i, j, k \mid i^2 = j^2 = k^2 = ijk = -1 \rangle \). List the elements of the center.

Answer: \{\pm 1\}


Question: Given two line segments in the hyperbolic plane with equal hyperbolic lengths, determine if there exists an isometry of the hyperbolic plane that maps one segment onto the other while preserving the upper half-plane model.

Answer: Yes


Question: Given that $E(Y|X) = aX + b$, express $E(YX)$ as a function of the first two moments of $X$.

Answer: aE(X^2) + bE(X)


Question: Solve for $x$ in the equation $3\sin\left(\frac{x^2-x}{8}\right)=3^x-3^{-x}$.

Answer: 0


Question: Find the expected value of $Z = \min(X,Y)$ for random variables $X$ and $Y$ with the joint density function: $$f(x,y) = x + y, \ 0 \leq x \leq 1, \ 0 \leq y \leq 1.$$

Answer: \dfrac{5}{12}


Question: Evaluate the contour integral \( \frac{1}{2\pi i} \oint_{C} \operatorname{Re}\{z\} \, dz \), where \( C \) denotes the counterclockwise unit circle. What is the value of this integral?

Answer: \dfrac{1}{2}


Question: Determine the cardinality of the set of all subsets of an infinite set $A$ with cardinality $a$, where each subset also has cardinality $a$. What is this cardinality?

Answer: 2^a


Question: Evaluate the limit: $$\lim_{n\rightarrow \infty}\int_1^\infty \frac{e^{\cos(\ln(x))}}{1+\sqrt{n}x^{21}}\,dx$$

Answer: 0


Question: Let $G$ be an abelian group, $S = \{ x^2 : x \in G\}$, and $T = \{ a \in G : a^2 = e\}$. Prove that $G/T$ is isomorphic to $S$.

Answer: G/T \cong S


Question: On a $2\times2$ chessboard, a king starts from the upper left corner and moves with equal probability to any adjacent square. Calculate the approximate probability that the king will be in the bottom right corner after exactly 1000 moves.

Answer: \dfrac{1}{4}


Question: Find the residue of the function \( \frac{\sin z}{z^2} \) at \( z = 0 \) using the formula for residues.

Answer: 1


Question: Determine whether there exists a positive real number $a$ and a continuous function $f: [0,1] \to \mathbb{R}$, which is not identically zero, such that $f(1) = 0$ and the integral \( \int_0^1 \cosh(ax) f(x) \, dx = 0 \) holds true. Here, \( \cosh(x) \) denotes the hyperbolic cosine function, defined by \( \cosh(x) = \frac{e^x + e^{-x}}{2} \).

Answer: Yes


Question: In a unit square, a point \((X, Y)\) is randomly chosen, with its coordinates \(X\) and \(Y\) being independent and identically distributed (i.i.d) uniform random variables on the interval \([0,1]\). A circle is drawn with a radius \(R\) around this point, where \(R\) is an independent random variable, also uniformly distributed on \([0,1]\). What is the probability that this circle intersects the boundary of the unit square?

Answer: \dfrac{5}{6}


Question: Determine if the complex version of the fundamental theorem of calculus can be used to evaluate the integral \( \int_{\Gamma} (z-1)^{-3} \, dz \), where \( \Gamma = C_2(1) \) is the circle of radius 2 centered at 1, with its usual parametrization.

Answer: 0


Question: Is it possible to have a proper open subset $O$ of the interval $[a,b]$ such that the measure of $O$ is equal to $b-a$?

Answer: Yes


Question: Determine whether a compact metrizable abelian group has enough characters to separate points. A character on a topological group is a continuous homomorphism from the group to the circle group. Separating points means that if \( g \) is not 0, then there exists a character \( f \) such that \( f(g) \) is not 0.

Answer: Yes


Question: Let \( f:X\rightarrow Y \) be a continuous function. If \( V\cap f(X)\neq \emptyset \) and \( W\cap f(X)\neq \emptyset \) and \( f(X)\subseteq V\cup W \), is it true that \( f^{-1}(V)\cup f^{-1}(W)=X \)?

Answer: Yes


Question: Find the integer value of $x$ that satisfies the equation $x^{99} + x^{98} + \dots + x + 1 \equiv 0 \pmod{101}$. Provide your answer as an integer between 0 and 100 inclusive.

Answer: 100


Question: Evaluate the integral \( \int_{|z|=3}\frac{\mathrm{e}^{1/(1-z)}}{z}\,\mathrm{d}z \) using the Residue Theorem.

Answer: 2\pi i


Question:
Determine the domain of definition for the functions \(\Gamma_1(x)\) and \(\Gamma_2(x)\), where:

\[\Gamma_1(x) = \int_0^{+\infty} e^{-t} t^x \ln t \, dt\]

\[\Gamma_2(x) = \int_0^{+\infty} e^{-t} t^x (\ln t)^2 \, dt\]

Answer: (-1, \infty)


Question: In a finite metric space, if one singleton set is open, determine whether all singleton sets are open.

Answer: Yes


Question: Given that \(\alpha, \beta, \gamma, \delta\) are the roots of the polynomial equation \(z^4 - 2z^3 + z^2 + z - 7 = 0\), find the value of \((\alpha^2 + 1)(\beta^2 + 1)(\gamma^2 + 1)(\delta^2 + 1)\).

Answer: 58


Question: Determine whether the infinite product \( \prod_{k=3}^\infty \left(1 - \frac{1}{k^\alpha}\right) \) converges or diverges for \( \alpha \in \mathbb{R} \).

Answer: \alpha > 1


Question: Let $a_1, a_2, \ldots, a_n$ be $n$ points in the $d$-dimensional Euclidean space. Suppose that $x$ and $y$ are two points outside the convex hull of $a_1, a_2, \ldots, a_n$. Determine whether there exists a vector $v$ such that $\langle v, x-a_i\rangle < 0$ and $\langle v, y-a_i\rangle < 0$ for all $i$. Answer with "yes" or "no."

Answer: no


Question: Given the joint mass function $P(k, n) = \frac{2^{-k}C}{n}$ for $k = 1, 2, \ldots$ and $n = 1, 2, \ldots, k$, where $C$ is a suitable constant, compute the conditional expectation $E(X|Y = y)$.

Answer: y + 1


Question: Find the greatest common divisor of \(a^{2^m}+1\) and \(a^{2^n}+1\) given that \(a\) is an odd positive integer and \(m\) and \(n\) are positive integers with \(m \neq n\).

Answer: 2


Question: Determine whether the integral of a holomorphic function over the boundary of a region is zero, given that the function is continuous in the region and has a singularity outside the region.

Answer: 0


Question: Compute the integral \( \int_N i^* \alpha \), where \( \alpha = \frac{xdy - ydx}{x^2 + y^2} \) is a 1-form on \( M := \mathbb{R}^2 \setminus \{(0,0)\} \), \( N \) is a smooth curve around the origin (excluding the origin), and \( i: N \to M \) is the inclusion map. The curve \( N \) is oriented counterclockwise and is parameterized by \( \gamma(t) = (\cos(t), \sin(t)) \) for \( t \in [0, 2\pi) \).

Answer: 2\pi


Question: Calculate the operator norm of the linear transformation \( T: \mathbb{R}^m \rightarrow \ell_2(\mathbb{N}) \) defined by \( T(x) = (x_1, \ldots, x_m, 0, 0, \ldots) \), where \( \mathbb{R}^m \) is equipped with the Euclidean norm \( \|\cdot\|_2 \) and \( \ell_2(\mathbb{N}) \) is equipped with the \( \ell_2 \) norm \( \|\cdot\|_2 \).

Answer: 1


Question: Let \( n > 1 \) and consider \( \mathbb{R}^n \) with the standard volume form \( \omega_0 = dx_1 \wedge \ldots \wedge dx_n \). Let \( r, R \) be two positive real numbers with \( R > r \). Is it true that for every \( a, b \in B(0; r) \), there exists a diffeomorphism \( f: \mathbb{R}^n \to \mathbb{R}^n \) such that \( f(a) = b \), \( f(x) = x \) for all \( x \in \mathbb{R}^n \) with \( ||x|| > R \), and \( f^*\omega_0 = \omega_0 \)?

Answer: Yes


Question: Given \(x = 3^{a_1}5^{a_2}7^{a_3}\) and \(y = 3^{b_1}5^{b_2}7^{b_3}\), where \(a_i, b_i\) are nonnegative integers with \(a_k + b_k \leq 5\) for all \(k\), determine the sum of the greatest common divisors (gcd) of all possible pairs \((x, y)\).

Answer: 1477773


Question: Determine whether there exists a closed relation \( \sim \) on a Hausdorff space \( X \) such that the quotient map \( f: X \to X/\sim \) is not open.

Answer: Yes


Question: In a bounded star-shaped domain $\Omega$ in $\mathbb{R}^n$, consider a function $f: \Omega \rightarrow \mathbb{R}$. If $f$ is Lipschitz continuous in $\Omega$, does it imply that there exists a constant $M > 0$ such that $\left | \frac{\partial f}{\partial x_i} \right | < M$ almost everywhere in $\Omega$?

Answer: Yes


Question: Let \( \{f_n\} \) be a sequence of integrable functions defined on an interval \([a,b]\). If \( f_n(x) \to 0 \) almost everywhere, does it imply that \( \int_{a}^{b} f_n(x) \, dx \to 0 \)? Provide a justification for your answer.

Answer: No


Question: For any matrix norm and any matrix \( A \in \mathbb{C}^{d \times d} \), does there exist a constant \( \gamma \in (0,1) \) such that the inequality \( \rho_{\epsilon}(A) \ge \gamma^{(1+\ln n)/n}\|A^{n}\|^{1/n} \) holds for all \( n \in \mathbb{N} \), where \( \rho_{\epsilon}(A) \) is the \( \epsilon \)-pseudospectral radius of \( A \)?

Answer: Yes


Question: Let $m$ be an odd prime and consider the set $A = \{0, 1, 2, \ldots, 2m-1\}$, which consists of all residues modulo $2m$. Determine the number of elements $x \in A$ that satisfy the congruence $x^2 \equiv 1 \pmod{2m}$. Provide your answer as a single integer.

Answer: 2


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

Answer: 0


Question: Let $G$ be a group with surjective group morphisms $p_1: G \twoheadrightarrow G_1$ and $p_2: G \twoheadrightarrow G_2$. Define $H_i = \text{ker}(p_i)$ for $i = 1,2$ such that $H_1 \cap H_2 = \{1\}$. Assume $p: G \rightarrow G_1 \times G_2$, given by $p(g) = (p_1(g), p_2(g))$, is injective. Let $K_1 = p_1(H_2) \unlhd G_1$ and $K_2 = p_2(H_1) \unlhd G_2$, with $H = H_1 \cdot H_2 \unlhd G$ and $p(H) = K_1 \times K_2$. There exist isomorphisms $\bar{p_i}: G/H \rightarrow G_i/K_i$ such that $\bar{p_i}(gH) = p_i(g)K_i$. Define $f_i: G_i \rightarrow G/H$ by $f_i = \bar{p_i}^{-1} \circ \pi$, where $\pi: G_i \twoheadrightarrow G_i/K_i$. Find the isomorphism induced by $p$ mapping $G$ to $L = \{(x_1, x_2) \in G_1 \times G_2 \mid f_1(x_1) = f_2(x_2)\}$. \[ \boxed{\text{Isomorphism}} \]

Answer: p


Question: Determine whether there exists a natural number $k$ such that for every even integer $n \ge k$, there exist positive integers $a, b, x, y$ satisfying $n = a^x + b^y$, where $\gcd(a, b) = 1$, $a, b > 1$, and at least one of $x, y$ is greater than 1.

Answer: Yes


Question: Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be an entire function satisfying $|f(z)| \leq C|\cos(z)|$ for all $z \in \mathbb{C}$, where $C \in \mathbb{R}$ is a constant. Determine the possible form of $f(z)$. Provide your answer in the form of a function or expression.

Answer: A \cos(z)


Question: Let $g(x)$ be a differentiable function such that $\frac{d^4g}{dx^4}=2$ for all $x\in [-1, 2].$ If $q(x)$ is a cubic polynomial that interpolates $g(x)$ at $x=-1, x=0, x=1,$ and $x=2,$ find the value of $g(0.5)-q(0.5)$. \( \boxed{} \)

Answer: \dfrac{3}{64}


Question: Let the sequence \((a_n)_{n \geq 1}\) be defined by \(a_1 >0 \), and the recurrence relation \[a_n = 2a_{n-1}*\frac{(n-1)^{(n-1)}}{(n-1)!}\] Find \(\lim_{n\to\infty} \frac {{a_{n+1}}^{1/{n+1}}}{{a_n}^{1/n}} \).

Answer: \sqrt{e}


Question: Given the function $f(x) = \sin\left(\frac{\pi x}{2}\right)$ on $\mathbb{R}$, consider the sequence $\{x_n(y)\}_{n\geqslant0}$ defined recursively by $x_0(y) = y$ and $x_{n+1}(y) = f(x_n(y))$ for all $n\geqslant1$. Let $g(y) = \lim\limits_{n\to\infty}x_{n}(y)$. Determine $\displaystyle\int_{0}^{3}g(y)\,\mathrm dy$. \(\text{Express your answer as a single number.}\)

Answer: 1


Question: Calculate the condition number of the matrix \( A = \begin{bmatrix} 0 & 0 & -10^4 & 0 \\ 0 & 0 & 0 & -10 \\ 0 & 10^{-3} & 0 & 0 \\ 10^{-2}& 0& 0& 0 \\ \end{bmatrix} \).

Answer: 10^7


Question: Evaluate the contour integral \( \oint_{|z|=3} \frac{1}{2-z} e^{\frac{1}{z}} \,dz \) by computing the residue at the singularities inside the contour, particularly focusing on the residue at the essential singularity at \( z=0 \).

Answer: -2\pi i


Question: Evaluate the limit \( \lim_{x \to 0^+} x^{1/2} \log x \).

Answer: 0


Question: Evaluate the limit: $$\lim_{n\to\infty}\int^{\infty}_0n\ln\Big(1+\frac{e^{-x}}{n}\Big)dx.$$ Find the value of this limit using appropriate techniques.

Answer: 1


Question: Calculate the definite integral: \[ \int_{0}^{2\pi}\sqrt{2-2\cos (\varphi_\alpha-\varphi_\beta)}\,d\varphi_\alpha. \]

Answer: 8


Question: Find all pairs of distinct positive integers \((m, n)\) such that the equation \(m^n - n^m = m+n\) holds.

Answer: (2, 5)


Question: Determine whether every function that satisfies the Cauchy functional equation \( f(a+b) = f(a)f(b) \) is always exponential.

Answer: No


Question: Find the Fréchet derivative of the function \( F[u] = u^2 \) at a point \( u \) in a commutative Banach algebra.

Answer: 2u \cdot h


Question: Determine the general formula for the $(2k)$th order derivative of $y=\arccos(x)$ evaluated at $x=0$.

Answer: 0


Question: Can the fractional Fourier transform be computed with the same computational complexity as the Fast Fourier Transform (FFT)?

Answer: Yes


Question: Can a finite ordinal number be compared with an infinite ordinal number using the "<" operator?

Answer: Yes


Question: Let $f(z)$ be an entire function such that $|f(z)| \le K|z|$ for all $z \in \mathbb{C}$, where $K > 0$. Given that $f(1) = i$, determine the value of $f(i)$. \( \boxed{} \)

Answer: -1


Question: Given the equation \( \int_0^\infty {f(x,r)g(x)dx} = 0 \) for all \( r \in \mathbb{R} \), where \( g(x) \) does not depend on \( r \) and \( f(x,r) \) is a function of \( x \) and \( r \), determine if it can be concluded that \( g(x) = 0 \) for all \( x \in \mathbb{R} \).

Answer: No


Question: Determine rigorously the limit \(\lim_{{x\to\infty}}\left(1 + a^x\right)^{\frac{1}{x}}\) for \(a > 1\). Prove that the limit is \(a\) using an \(\epsilon-\delta\) proof or alternative rigorous methods.

Answer: a


Question:
Minimize the function \( f(x,y) = \dfrac{e^y}{e^{x+1}} \) subject to the constraints:

\[ e^y - e^{x+1} \geq 10 \]
\[ y \geq 0 \]

Answer: 1


Question: Given the function $f(x, y, z) = x^2 + 2xz + y$, determine the set $f([1,3]^3)$ and characterize it in terms of openness, closedness, completeness, compactness, and connectedness.

Answer: [4, 30]


Question: Find the expected value of $X$ given $X^2 + Y^2$, where $X$ and $Y$ are independent random variables, both normally distributed with mean $0$ and variance $1$. Express your answer as $E(X \mid X^2 + Y^2)$.

Answer: 0


Question: Calculate the commutator \([p^2, x]\), where \(p = -i\hbar\frac{d}{dx}\).

Answer: -2i\hbar p


Question: Evaluate the integral \( \frac{1}{2 \pi i} \oint_{C} \frac{z^{2} \, d z}{z^{2}-4} \) using Cauchy's integral formula, where \( C \) is the circle with center at the origin and radius 3.

Answer: 0


Question: Determine whether there exists an infinite topological space $X$ such that there are only countably many continuous functions from $X$ to itself.

Answer: Yes


Question: Suppose there is a function \( f: \mathbb{R} \rightarrow \mathbb{C} \) such that its derivative \( \frac{d}{dx}f(x) \) exists and \( \lim_{x\rightarrow\infty}\frac{d}{dx}f(x)=0 \). Does this imply that \( f(x) \) is bounded as \( x \rightarrow \infty \)? Answer yes or no.

Answer: No


Question: Determine whether there exists an automorphism \( g: \mathcal{M}_{3}(\mathbb{C}) \to \mathcal{M}_{3}(\mathbb{C}) \) of \( \mathbb{C} \)-algebras such that \( g^3(M) = M \) for all matrices \( M \). If such an automorphism exists, describe how to construct it.

Answer: Yes


Question: Let $A$ be a Lebesgue measurable subset of $\mathbb{R}$. Consider the subspace topology on $A$ and the Borel sigma algebra under that topology. Is every Borel set under this topology a Lebesgue measurable subset of $\mathbb{R}$?

Answer: Yes


Question: Suppose $f \in L_1(-\pi,\pi)$ satisfies $\hat{f}(n) = 0$ for all $n \in \mathbb{Z}$. Is it true that $f = 0$ almost everywhere in $(-\pi,\pi)$?

Answer: Yes


Question: A pool of problems is used to create 20 tests. Each test must contain the same number of problems, and no problem can appear in more than 10 tests. Additionally, for every 5 tests, there must be at least 2 problems common to all of them. Determine the minimum number of problems required in the pool.

Answer: 124


Question: Determine the number of integer values of \( n \) such that the equation \( 2x^3 - 15x + n = 0 \) has three distinct real roots.

Answer: 31


Question: Determine the maximum value of the expression \( \frac{x^{100}}{1+x+x^2+x^3+\cdots+x^{200}} \) for \( x > 0 \), without using calculus. Simplify the expression using the formula for the sum of a geometric series.

Answer: \dfrac{1}{201}


Question: Calculate the expected value of the minimum of two random variables, $U_1$ and $U_2$, where $U_1$ is uniformly distributed over $[0,1]$. The joint distribution of $U_1$ and $U_2$ is such that $U_2 = U_1$ with probability $1/2$ and $U_2 = 1-U_1$ with probability $1/2$. Determine the expected value of $\min(U_1, U_2)$. \( \text{Express your answer as a single number.} \)

Answer: \dfrac{3}{8}


Question: Given that $x$ is a prime number and $y$ is an integer, find the maximum possible value of $y$ such that the expression \( \sum_{i=1}^{55}\frac{i}{xy} \) is an integer.

Answer: 770


Question: Determine the radius of convergence of the function \( f(x) = \int_{0}^{x} \ln{(t+\sqrt{t^2+1})} \, dt \) using its Maclaurin series.

Answer: 1


Question: Evaluate the double integral \( \iint_Q \frac{5y-x}{y^2(y-2)^2} \, dA \), where \( Q \) is the region enclosed by the quadrilateral with vertices \((2, 4)\), \((6, 3)\), \((8, 4)\), and \((4, 8)\).

Answer: \dfrac{9}{4}


Question: Evaluate the limit: $$\lim_{x\to0}\left(\frac{1}{x^{\frac{1}{x+6}}}-1\right)$$

Answer: \infty


Question: Determine the smallest dimension $d$ such that a central sphere, inscribed in a $2\times2\times...\times2$ unit cube in $n$-dimensional space and touching $2n$ spheres of radius $\frac{1}{2}$ placed at the cube's corners, extends beyond the cube.

Answer: 3


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

Answer: 2\ln 3


Question: Let \(ABCD\) be a square with side length \(x\). If \(P\) is a point on the circumcircle of \(ABCD\), find all positive integers \(n\) such that the sum \[ T^n(P) = PA^n - PB^n + PC^n - PD^n \] is constant with respect to the point \(P\).

Answer: 2


Question: In the context of functional analysis, determine whether a Banach space $X$ having a Fréchet differentiable dual norm implies that $X$ is reflexive.

Answer: Yes


Question: Evaluate the integral \(\displaystyle \int _{-2}^{2} \left[\arctan \left(\dfrac{x}{x^2+4}\right) + \arctan \left(\dfrac{x^2 + 4}{x}\right)\right]dx\) using an appropriate trigonometric identity.

Answer: 0


Question:
Given complex numbers $z_1, z_2, z_3$ such that $z_1 + z_2 + z_3 = 2$, $z_1^2 + z_2^2 + z_3^2 = 3$, and $z_1z_2z_3 = 4$, evaluate the expression:

$$\frac{1}{z_1z_2 + z_3 - 1} + \frac{1}{z_2z_3 + z_1 - 1} + \frac{1}{z_3z_1 + z_2 - 1}$$

Answer: -\dfrac{2}{9}


Question: Given a field $k$, suppose every polynomial in $k[x]$ is a factor of the characteristic polynomial of some matrix over $k$. Does it follow that $k$ is algebraically closed? Provide a justification for your answer.

Answer: No


Question:
Evaluate the limit \( \lim_{x \to 0} \frac{x - \sin x}{x^3} \) without using advanced techniques such as de l'Hôpital's Rule or Taylor series. Verify if the following solution is rigorous:

\[ \begin{align*}
\lim\limits_{x \to 0} \dfrac{x-\sin x}{x^3}&=\lim\limits_{x \to 0} \dfrac{2x-\sin 2x}{(2x)^3}\\[4pt]&=\lim\limits_{x \to 0} \dfrac{2x-2\sin x\cos x}{(2x)^3}\\[4pt]&=\frac{1}{4}\lim\limits_{x \to 0} \dfrac{x-\sin x+\sin x-\sin x\cos x}{x^3}\\[4pt] &=\frac{1}{4}\lim\limits_{x \to 0} \dfrac{x-\sin x}{x^3}+\frac{1}{4}\lim\limits_{x \to 0} \dfrac{\sin x(1-\cos x)}{x^3}\\[4pt] &=\frac{1}{4}\lim\limits_{x \to 0} \dfrac{x-\sin x}{x^3}+\frac{1}{4}\lim\limits_{x \to 0} \dfrac{\sin x}{x}\cdot\lim_{x \to 0}\frac{1-\cos x}{x^2}\\[4pt] &=\frac{1}{4}\lim\limits_{x \to 0} \dfrac{x-\sin x}{x^3}+\frac{1}{4}\cdot 1\cdot \frac{1}{2}\\[4pt] &=\frac{1}{4}\lim\limits_{x \to 0} \dfrac{x-\sin x}{x^3}+\frac{1}{8},\end{align*} \]

which sets up an equation on the limit. Solve the equation to find \( \lim_{x \to 0} \frac{x-\sin x}{x^3} = \frac{1}{6} \). Is this solution correct?

Answer: \dfrac{1}{6}


Question: Evaluate the limit: $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n\frac{\cos(i)}{i^2}$$

Answer: 0


Question: Given two norms $\Vert \cdot \Vert_1$ and $\Vert \cdot \Vert_2$ on a complex vector space $V$, each arising from an inner product and turning $V$ into a separable Hilbert space, with the property that for all $\phi \in V$, $c^{-1}\Vert \phi \Vert_1 \leq \Vert \phi \Vert_2 \leq c \Vert \phi \Vert_1$ for some constant $c > 0$, determine whether there exists a bounded linear operator $S: V \rightarrow V$ such that $\Vert S\phi \Vert_1 = \Vert \phi \Vert_2$ for all $\phi \in V$. Answer with 'Yes' or 'No'.

Answer: Yes


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{x \cos (kx)}{x^2+a^2} \, \mathrm{d}x \) using real analysis, where \( k > 0 \).

Answer: 0


Question: Let $K=\mathbb{Q}$ and $L$ be the splitting field of the polynomial $f(x)=x^4-6x^2+4$. Determine the Galois group $\text{Gal}(L/K)$. Given that the zeros of $f(x)$ are $x_{1,2}=\pm\sqrt{3+\sqrt{5}}$ and $x_{3,4}=\pm\sqrt{3-\sqrt{5}}$, express $L$ as $L=\mathbb{Q}(x_1,x_2,x_3,x_4)$. Investigate whether the extension can be generated by just one of the zeros and find the structure of the Galois group.

Answer: V_4


Question: Suppose that $f$ is a holomorphic function in the open unit disc and that the limit $\lim_{z\to 1}f(z)$ exists. Determine the radius of convergence $R$ of the Taylor series of $f$ about $z=0$. Express your answer in terms of $R$. \( \boxed{R} \)

Answer: 1


Question: Find the smallest value of the sum of squares \( P = \sum_{i=1}^{19} x_i^2 \), where \( x_1, x_2, \ldots, x_{19} \) are positive integers satisfying \( \sum_{i=1}^{19} x_i = 2020 \) and \( x_i \geq 2 \) for all \( i \).

Answer: 214762


Question: How many points at infinity are there in the complex plane?

Answer: 1


Question:
Find the determinant of the matrix $e^M$, where $M$ is a $2\times2$ matrix given by:
$$M=\begin{bmatrix}1&\frac{1-i}{\sqrt{2}}\\ \frac{1+i}{\sqrt{2}}&0 \end{bmatrix}$$

Answer: e


Question: Let $R$ be a commutative ring with identity, and let $I$ be a proper ideal of $R$ (i.e., $I \neq R$). Determine whether $I$ is contained in a maximal ideal of $R$.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x \rightarrow 0} \frac{(1-\cos x)^{\cot x}-e^{2x-1}}{\tan (2x)}$$ without using L'Hôpital's rule.

Answer: -\infty


Question:
Demonstrate that for a smooth function $u: \Omega \rightarrow \mathbb{R}$ defined on a domain $\Omega \subset \mathbb{R}^n$, the following equality holds:

$$\int_\Omega u(x)\delta(x-x_0)d\Omega = u(x_0)$$

Answer: u(x_0)


Question:
Determine the limit of the function as \(x\) approaches infinity:
\[\lim_{x \to \infty}\left(\frac{x+\ln x}{x}\right)^{\frac{x}{\ln x}}\]

Answer: e


Question: Given topological spaces $X$ and $Y$, let $F: X \to Y$ be a set-valued map from $X$ to $Y$. A function $f: X \to Y$ is called a selection of $F$ if $f(x) \in F(x)$ for all $x \in X$. Is a selection $f$ of $F$ also contained in $F$, meaning $f(x) \subset F(x)$ for all $x \in X$?

Answer: No


Question: Determine the maximum number of integer solutions \( k \) to the equation \( p(k) = k^3 \), where \( p(x) \) is a polynomial with integer coefficients and \( p(100) = 100 \).

Answer: 10


Question: Determine all analytic functions \( f \) in the open unit disk \( \{z:|z|<1\} \) such that \( f\left(\frac{1}{n}\right)=\frac{1}{n^3} \) for all positive integers \( n \).

Answer: z^3


Question: Determine the largest power of 7 that divides the binomial coefficient \( \binom{2016}{1003} \).

Answer: 343


Question: Evaluate the limit: $$\displaystyle \lim_{n\to\infty}\left(\prod_{k=1}^n\left(1+\frac{k}{n}\right)\right)^{1/n}$$

Answer: \dfrac{4}{e}


Question: Determine whether the following statement is true or false: For any positive integers \(n\) and \(m\), if \(n\) has a partition into two coprime parts greater than 1, and \(n<m<n!\), then \(S_n\) has a cyclic subgroup of order \(m\).

Answer: False


Question: Compute the value of the integral: $$ \int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\ dx\,dy $$

Answer: \dfrac{1}{2}


Question: Evaluate the limit: $$\lim_{x\rightarrow -\infty}{e^{\frac {1}{2-x}}\cdot\frac{x^2+2x-1}{x-2}}-x$$

Answer: 3


Question: Determine the values of $p$ for which the curve $y = x^p \cos \frac{\pi}{x}$, defined for $0 < x \leq 1$, has finite length.

Answer: p > 1


Question: Let $f:[a,b]\to \mathbb{R}$ be a function that is continuous on $[a,b]$ and differentiable on $(a,b)$. Suppose there exists $c\in (a,b)$ such that $f'(c)=0$, $f'(x)<0$ for all $x\in (a,c)$, and $f'(x)>0$ for all $x\in (c,b)$. Determine whether it is true that $f$ is strictly decreasing on $[a,c]$ and strictly increasing on $[c,b]$. Provide a justification for your answer.

Answer: True


Question: Find the radius of convergence of the Taylor series for the function \( f(x) = (x-1) \cdot \ln(x-1) \) using the Taylor series of \( \log(z+1) \).

Answer: 1


Question: Determine whether there exists a surface $S$ of degree $d \geq 4$ in the complex projective space $\mathbb{P}^3_{\mathbb{C}}$ that does not contain any lines.

Answer: Yes


Question: Is it always true that the derivative of a differentiable and increasing function is Lebesgue integrable?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to\infty} \underbrace{\sin(\sin(\cdots\sin(x)))}_{n}, \quad x\in\mathbb{R} \]

Answer: 0


Question: Let $X$ be a metric space. Suppose that every subsequence of a sequence $(a_n)$ in $X$ has a subsequence that converges to the same limit $l$. Is it necessary that the sequence $(a_n)$ itself converges to $l$? Provide a justification for your answer.

Answer: Yes


Question: Given the equation $2(x^3+y^3+z^3) = 3(x+y+z)^2$, where $x, y, z$ are non-negative real numbers, find the minimum value of $x+y+z$.

Answer: \dfrac{3}{2}


Question: Compute the dimension of the spline space \(\mathcal{S}_\Delta^{3,2}\) on the interval \([a,b]\) with \(n\) subintervals, where \(\Delta\) denotes the grid on \([a,b]\).

Answer: n + 3


Question: Given $n$ complete graphs, each with $m$ nodes, determine the minimum number of edges required to connect these graphs such that there exists a path of length 2 between any two nodes.

Answer: m(n - 1)


Question: Determine whether there exist two rational numbers $x$ and $y$ such that $x^4 + y^2 = 1$. Provide a justification for your answer.

Answer: Yes


Question: Consider the linear operator $L = \partial_x^2 + c\partial_x + f'(v)$ derived from the reaction-diffusion equation with a traveling wave ansatz. Does this operator $L$ generate a $C_0$-semigroup?

Answer: Yes


Question: Evaluate the integral \( \int_{-3}^2 \frac{\lfloor x\rfloor}{\lfloor x\rfloor^2+2\lfloor x \rfloor+x^3} \, dx \) using a method other than partial fractions.

Answer: \dfrac{3}{4}


Question: Find the best lower bound for \( \max_{0\le x\le 1} g(x) \) given that \( g \) is a continuous function on \([0,1]\) and \( \int_0^x g(t)dt \le [g(x)]^2 \) for all \( x \in[0,1]. \)

Answer: \dfrac{1}{2}


Question: Determine the degree and structure of the splitting field $E$ for the polynomial $x^6 - 4 = (x^3 - 2)(x^3+2)$ over $\mathbb{Q}$, and find the dimension of $E$ over $\mathbb{Q}$.

Answer: 6


Question: Let $S$ be a simply connected subset of $\mathbb{R}^2$ and let $x$ be an interior point of $S$, meaning that $B_r(x)\subseteq S$ for some $r>0$. Is it necessarily the case that $\pi_1(S\setminus\{x\})\cong\mathbb{Z}$?

Answer: \mathbb{Z}


Question: Determine whether the following statement is correct: "The limit superior of $x_{n}$ (if finite) is the smallest real number $b$ such that, for any positive real number $\varepsilon$, there exists a natural number $N$ such that $x_n < b + \varepsilon$ for all $n > N$." Provide a justification for your answer.

Answer: Correct


Question: Evaluate the limit: $$\lim_{x\to \pi}\left(\frac{1}{x-\pi}\right)\int_{\pi}^{x}\frac{\cos t}{1-\cos t}\,dt.$$

Answer: -\dfrac{1}{2}


Question: Determine whether the following statement is true or false: If $p$ is an irreducible polynomial in $n$ indeterminates, then the ideal $(p)$ generated by it is prime.

Answer: A


Question: Consider a commutative ring $R$ with 1 and an $R$-module $M$. Given an ideal $I$ of $R$, the quotient $M/IM$ can be viewed as both an $R$-module and an $R/I$-module, where the $R/I$-module operation is defined by $$(r+I)(m+IM)=rm+IM.$$ Are the $R$-module $M/IM$ and the $R/I$-module $M/IM$ essentially the same?

Answer: Yes


Question: Evaluate the integral \( I = \int_{0}^{a}\frac{1}{1+f(x)}\,dx \), given that \( f(x)f(a-x) = 1 \), \( a > 0 \), and \( f(x) \) is continuous and positive on the interval \([0, a]\).

Answer: \dfrac{a}{2}


Question: Is every function \( f: \mathbb{R}^n \to \mathbb{R}^m \) that maps bounded, convex sets (excluding straight line segments) in \( \mathbb{R}^n \) to bounded, convex sets in \( \mathbb{R}^m \) necessarily continuous for \( n \geq 2 \)? Answer yes or no.

Answer: Yes


Question: Evaluate the improper integral \(I=\int_{-\infty}^{\infty} \frac{\sin^2{x}}{x^2} \, dx\) using Parseval's theorem or other appropriate methods.

Answer: \pi


Question: For a family of topological spaces \((X_\alpha, \tau_\alpha)_{\alpha\in A}\), does the product topology on \(\prod_{\alpha\in A}X_\alpha\) induce the same subspace topology on each factor space \(X_\alpha\) as the original topology \(\tau_\alpha\)? Answer with 'Yes' or 'No'.

Answer: Yes


Question: Evaluate the double integral: $$\int_{1}^{4}\int_{0}^{2}e^{x^2}\sin[(y-1)^3]\,dy\,dx$$

Answer: 0


Question:
Suppose we have two pairs of integers \((a_1, b_1)\) and \((a_2, b_2)\) such that

\[a_1b_1 \equiv 0 \pmod{n}, \quad a_2b_2 \equiv 0 \pmod{n}, \quad a_1b_2 + a_2b_1 \equiv 0 \pmod{n}.\]

Does this imply that

\[a_1b_2 \equiv 0 \pmod{n}?\]

Answer: Yes


Question: Let $b$ be an element in a field $K$ that is algebraic of degree $n$ over a field $F$. Prove that the degree of the field extension $[F(b):F]$ is $n$.

Answer: n


Question: If $a, b, c$ are natural numbers such that $(\overline{ab})^3 = \overline{1c8ab}$, find the value of $2b - a - c$. \( \overline{ab} \) denotes the two-digit number formed by the digits $a$ and $b$, and \( \overline{1c8ab} \) denotes the five-digit number formed by the digits 1, $c$, 8, $a$, and $b$. \( a \) and \( b \) are the tens and units digits of a two-digit number, respectively.

Answer: 3


Question: Let \( G(x) \) be the antiderivative of the function \( g(x) = |\cos x| \). Calculate the limit: \[ \lim_{x \to \infty} \frac{G(x)}{x} \]

Answer: \dfrac{2}{\pi}


Question: Let $A$ and $B$ be two positive semi-definite $n \times n$ matrices such that \( \mathrm{Range}(B^{1/2}AB^{1/2}) = \mathrm{Range}(B) \) and \( \mathrm{Rank}(A) = \mathrm{Rank}(B) = n-1 \). Determine whether there exists a real symmetric matrix $X$ such that \( A = XBX \).

Answer: Yes


Question: In a circle with a radius of 3 units, two secants, $ABC$ and $ADE$, intersect outside the circle at point $A$. Point $A$ is located 5 units away from the center of the circle. The angle formed by the secants $ABC$ and $ADE$ at point $A$ is $30^\circ$. If the area of triangle $\triangle ACE$ is 10 square units, find the area of triangle $\triangle ADB$. \( \boxed{\text{Area of } \triangle ADB} \)

Answer: \dfrac{8}{5}


Question: Given that \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{a+b+c} \), determine the value of \( \frac{1}{a^5} + \frac{1}{b^5} + \frac{1}{c^5} \). Choose from the following options: (a) 1, (b) 0, (c) \( \frac{1}{a^5 + b^5 + c^5} \), (d) None of them.

Answer: C


Question: Evaluate \( \sum_{d\mid N}\Lambda(d) \) for a given positive integer \( N \), where \( \Lambda(n) = \log p \) if \( n = p^r \) for some prime \( p \) and integer \( r \), and \( \Lambda(n) = 0 \) otherwise.

Answer: \log N


Question:
Consider the expression \( A = \sqrt{x + \frac{2}{x}} - \sqrt{x - \frac{2}{x}} \) for \( x >> 1 \). Which of the following provides the most accurate approximation for \( A \)?

1) \( \frac{1}{x\sqrt{x}} \) for \( x > \sqrt{\frac{2}{u}} \)
2) \( \frac{2}{x\sqrt{x}} \) for \( x > \sqrt{\frac{2}{u}} \)
3) \( \frac{1}{x\sqrt{x}} \) for \( x > \frac{2}{u} \)
4) \( \frac{2}{x\sqrt{x}} \) for \( x > \frac{1}{u} \)

Assume \( u \) represents the maximum acceptable error in the approximation. Choose the correct option.

Answer: 2


Question: Determine the cardinality of the set of functions from \(\mathbb{Z}\) to \(\mathbb{Z}\).

Answer: 2^{\aleph_0}


Question: For two circles with equal radii that intersect, determine if the intersection area is always contained within the circle formed by connecting the two intersection points. Answer with 'yes' or 'no'.

Answer: yes


Question: Determine whether the solution to the differential equation \(x'(t) = x \ln(x)\) with the initial condition \(x(0) = x_0 > 0\) is unique.

Answer: Yes


Question: Find the parent function $f(x,y)$ such that its gradient is given by the vector $$ \nabla f = <4y\sin(xy),4x\sin(xy)> $$.

Answer: -4\cos(xy)


Question: Determine whether the image of any non-compact subset of $\mathbb{R}^n$ under the map $f:\mathbb{R}^n \rightarrow \mathcal{H}(\mathbb{R}^n)$, defined by $f(x) = \{x\}$, is non-compact. Answer true or false.

Answer: A


Question: Determine whether the following statement is true or false: If $A$ is closed in $\mathbb{R}^{n}$, then $f(A)$ is also closed in $\mathbb{R}^{n}$ for a continuous function $f: A \rightarrow \mathbb{R}$.

Answer: B


Question: Find the coefficient of \(x^{15}\) in the generating function \(g(x)=\frac{(1+x)(x^2+x^3+x^4+x^5)^2}{(1-x)^4}\).

Answer: 4940


Question: Determine whether the following property is true for two languages $L_1$ and $L_2$ over the alphabet $A$: $$ L_{1}^{*} \cap L_{2}^{*} = (L_{1}^{*} \cap L_{2}^{*})^{*} $$

Answer: True


Question: Given the linear programming problem: Minimize \(x+y+z-\alpha t\) subject to \(2x - y + z = 6\), \(-x + y + t = 3\), and \(x, y, z, t \geq 0\). If \((3, 0, 0, \beta)\) is an optimal solution, determine the maximum value of \(\beta - \alpha\).

Answer: 7


Question: Given a positive semi-definite matrix $B$, determine whether there exists a non-zero vector $z \geq 0$ with non-negative components such that all components of $Bz$ are also non-negative.

Answer: Yes


Question: Determine the smallest possible value of the correlation coefficient \( \rho \) for an \( n \times n \) correlation matrix \( M \), where every pair of random variables has a correlation of \( \rho \).

Answer: -\dfrac{1}{n-1}


Question:
Determine which of the following conditions are sufficient for the function \( f(x) = x^m \sin\left(\frac{1}{x^n}\right) \) to be differentiable at \( x=0 \):

A) \( m > 1 \)
B) No condition on \( m \) or \( n \)

Answer: A


Question: Find the value of \( \lim_{n\rightarrow \infty}e^{-n}\sum_{k=0}^{n-1} \frac{n^k}{k!} \).

Answer: \dfrac{1}{2}


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

Answer: 0


Question: Determine whether the minimum spectrum of a positive element $x$ in a von Neumann algebra $M$ is less than or equal to the minimum spectrum of $exe$ in the algebra $eMe$, where $e$ is a projector in $M$. Is this statement true or false?

Answer: True


Question: Determine whether a function $f:[0,1] \to [0,1]$ that attains every value between $0$ and $1$ within the interval $[0,1]$ must be continuous on $[0,1]$. Provide a justification for your answer.

Answer: No


Question: Determine whether the general solution of the partial differential equation \(xp^2+yq^2+(x+y)(pq)-u(p+q)+1=0\) can be expressed as \(u(x,y)=g(x+y)+\frac{1}{x+y}\), where \(g\) is an arbitrary function. State whether the statement is true or false.

Answer: False


Question: Calculate the limit: $$ \lim_{x \to \frac{\pi}{2}^-} \ln\left(\frac{2x}{\pi}\right)\cdot e^{\frac{1}{\cos x}}$$

Answer: -\infty


Question: Evaluate the integral \( \iint_D \left(\sqrt{x^2+y^2}-\sqrt{a^2-x^2-y^2}~\right)dxdy \) where \(D_{xy}\) is the disk \(x^2+y^2\le a^2\), using a change of variables different from polar coordinates.

Answer: 0


Question: In a cooperative game with symmetric players, is it possible for the core to contain allocations where the symmetric players receive unequal payoffs? Provide a yes or no answer.

Answer: Yes


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

Answer: 0


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

Answer: \dfrac{1}{e}


Question: Determine whether the series \( \sum_{k=0}^{\infty} \frac{z^k}{(k!)^2} \) has a closed-form expression.

Answer: I_0(2\sqrt{z})


Question: Determine whether the following statement is true or false: Any Riemann integrable function has at most countably many discontinuities.

Answer: B


Question: Determine whether a real skew-symmetric matrix with purely imaginary eigenvalues can be diagonalized over the complex numbers.

Answer: Yes


Question: Determine whether the following statement is true or false: \( \mathbb{E}[\max_{s \in [0,1]} B_s] = B_1 \), where \( B_s \sim \mathcal{N}(0,s) \) is a normal random variable with variance \( s \), i.e., \( \{B_s\} \) is a standard Brownian motion process.

Answer: False


Question: Find the integrating factor that makes the differential equation \( y^{2}dy = x\left ( x dy - y dx \right ) e^{\frac{x}{y}} \) exact.

Answer: \dfrac{1}{y^3}


Question: Calculate the sum of the series \(\sum_{n=1}^{\infty} \left(\cos\frac{a}{n^3}\right)^{n^3}\), where \(a\) is a constant.

Answer: \infty


Question: Solve the congruence equation $x^{45} \equiv 7 \pmod{113}$. Provide the solution for $x$.

Answer: 83


Question: Determine whether there exist matrices $P_{m\times r}$ and $Q_{r\times n}$ such that $B = PQ$, given that $B \in M_{m,n}(\mathbb{F})$ has rank $r$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit \( \lim_{x\to 0^+}\frac{\int_{x}^{1}\frac{\sin t}{t}\,\mathrm{d}t}{x} \) using L'Hopital's rule.

Answer: -1


Question: If the graph of a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is a smooth manifold, does it imply that the function itself is smooth? Answer yes or no.

Answer: Yes


Question: Find 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 relatively prime and the group is isomorphic to the cyclic group \(\mathbb{Z}/mn\mathbb{Z}\).

Answer: (1, 1)


Question: Determine the range of values for $r$ such that the series \(-\sum\limits_{n=1}^{\infty} \frac{1}{n^{r-1}}\sin(nx)\) does not converge uniformly for \(x \in \mathbb{R}\).

Answer: (-\infty, 2]


Question: Calculate the limit of \( \frac{\Gamma(x+1)}{x\ln x} \) as \( x \to \infty \), where \( \Gamma(x) \) is the gamma function.

Answer: \infty


Question: Find the limit of \( \frac{f(a)\ln a}{a} \) as \( a \to \infty \), where \( f(a) \) is the real solution of the equation \( x(1+\ln x)=a \) for any real number \( a \geq 1 \).

Answer: 1


Question: Determine whether a simple closed contour integral of the form $\oint_C f(z)dz$ can evaluate to $0$ if $f(z)$ is not holomorphic at some interior point of the contour $C$. Provide a justification for your answer.

Answer: Yes


Question: Find the set of values of $z$ for which the expression \( \frac{w - wz}{1 - z} \) is purely real, given that \( w = \alpha + i\beta \) with \( \beta \neq 0 \) and \( z \neq 1 \).

Answer: \emptyset


Question: In the polynomial ring \(\mathbb{Z}_5[x]\), determine all monic divisors of degree two for the polynomial \(2x^3 + 2x^2 + 2x + 4\).

Answer: x^2 + 2x + 3


Question: Let $k$ be an integer greater than $1$. Initially, Tita the frog is sitting at the point $k$ on the number line. In one movement, if Tita is located on the point $n$, then she jumps to the point $f(n)+g(n)$, where $f(n)$ and $g(n)$ are the greatest and the least prime divisors (both positive) of $n$, respectively. Determine all the values of $k$ for which Tita will visit infinitely many distinct points on the number line.

Answer: \emptyset


Question: Determine the smallest positive integer $n$ such that $2^n + 3^n$ is divisible by 625.

Answer: 125


Question: Consider the curve \(x^3+y^3=3axy\) and the line \(x+y+a=0\). Using Bezout's theorem, determine the number of intersections between these curves and describe the nature of these intersections.

Answer: 3


Question: Provide a formal epsilon-delta argument to prove that \( \lim\limits_{x\to 0} e^{-1/x^2} = 0 \).

Answer: 0


Question: Determine the number of real solutions to the equation \(5^{x^{10}}+16^{(1-x)^4} = 2023\) without using calculus.

Answer: 2


Question: Find the radius of convergence for the power series representation of the function \( f(z) = \frac{\mathrm{e}^z}{z-1} \) centered at \( z = 0 \).

Answer: 1


Question:
Determine the number of 4-tuples \((a,b,c,d) \in \mathbb{N}^{4}\) such that:

\[\mathrm{lcm}(a,b,c) = \mathrm{lcm}(b,c,d)=\mathrm{lcm}(c,d,a)= \mathrm{lcm}(d,a,b)=2^{5}3^{4}5^{3}7^{2}.\]

Answer: 42723153


Question: Describe the group of homomorphisms $f: C_m \rightarrow A$, where $C_m$ is the cyclic group of order $m$ and $A$ is a finite abelian group, with group composition defined by $(f * g)(x) = f(x) + g(x)$. What is this group?

Answer: A[m]


Question: Determine the minimum positive integer value of $n$ for which the expression \(\dfrac{(2i)^n}{(1+i)^{n-4}}\) is a negative integer.

Answer: 8


Question:
Evaluate the following expression:
\[ 2\sum^{\infty}_{n=1}-\frac{n^{2}(-1)^{n-1}}{n^{3}}\lim_{p\rightarrow \infty}\frac{np^{2}+n2p+2}{e^{np}} \]

Answer: 0


Question: Determine if there are cases where both $f'(a)=0$ and $f''(a)=0$, yet the point $(a, f(a))$ is a point of inflection.

Answer: Yes


Question: Determine the range of values for $a$ such that the inequality \[ a(x_1^2 + x_2^2 + x_3^2) + 2x_1x_2 + 2x_2x_3 + 2x_1x_3 \geq 0 \] holds for all real numbers $x_1, x_2, x_3$.

Answer: [1, \infty)


Question: Evaluate the limit: $$\lim_{x\rightarrow{0^{+}}}{\bigg(x\int_x^1{\frac{f(t)}{\sin^2(t)}}\: dt\bigg)}$$ where $f$ is continuous on $[-1,1]$ and differentiable on $(-1,1)$.

Answer: f(0)


Question: Find the largest value of \( y \) in the interval \([0, 0.1]\) such that \( P(P > y \mid X = 0) \leq 0.01 \), where \( n = 528174 \), \( X \sim \text{Binomial}(n, P) \), and \( P \sim U[0, 0.1] \).

Answer: 0.00000872


Question: Find a closed-form expression for the sequence $(a_{m})_{m=1}^\infty$ defined recursively by $a_{1}=1$ and $a_{m} = \lceil \frac{m}{m-1} a_{m-1} \rceil$ for $m \geq 2$. Provide your answer in terms of $m$.

Answer: m


Question: Is it possible to define a metric on the real numbers \( \mathbb{R} \) such that every non-empty open set in the resulting topology contains an interval of the form \((a,b)\)?

Answer: Yes


Question: If the Cesaro mean of a sequence of non-negative real numbers \((a_n)\) is bounded, does the series \(\sum_{n=1}^{\infty} \frac{a_n}{n^2}\) converge? Specifically, if \(\frac{1}{n}\sum_{k=1}^n a_k \leq C\) for a constant \(C\) and all \(n\), does it follow that \(\sum_{n=1}^{\infty} \frac{a_n}{n^2}\) converges?

Answer: Yes


Question: Let $G$ be an abelian group, and consider the sets $H = \{g^2 \mid g \in G\}$ and $K = \{g \in G \mid g^2 = e\}$. Define the homomorphism $f : G \to G$ by $f(g) = g^2$. Use the first isomorphism theorem to show that $G/K \cong H$. Provide a detailed explanation of your reasoning.

Answer: G/K \cong H


Question: Is the ideal $(x)$, considered as a module over the polynomial ring $k[x]$, a free module? Justify your answer.

Answer: Yes


Question:
Determine the dimension of the image of the function \(\xi: F_n \rightarrow M_n(\mathbb{R})\), where \(F_n\) is the set of real integrable functions on the interval \([1, n]\) and \(M_n(\mathbb{R})\) is the set of \(n \times n\) square matrices. The function \(\xi\) maps a function \(f \in F_n\) to a matrix \(M = (m_{i,j})\) such that:

\[ m_{i,j} = \int_{i}^{j} f(t) \, dt \]

The matrix \(M\) satisfies the following properties:

1. \(m_{i,i} = 0\)
2. \(m_{i,j} = -m_{j,i}\)
3. \(m_{i,j} + m_{j,k} = m_{i,k}\)

Find the dimension of the image of \(\xi\) within the set of antisymmetric matrices \(A_n(\mathbb{R})\).

Answer: n-1


Question: Determine the least upper bound of the function \( g(A,B) = A - \sqrt{A^2 + B} \) for \( A \in \mathbb{R} \) and \( B > 0 \), where the upper bound is independent of \( A \) and \( B \).

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to 2}\frac{\cos(\frac \pi x)}{x-2} $$

Answer: \dfrac{\pi}{4}


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

Answer: 0


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

Answer: -2i


Question: Compute \( \lim_{n\to \infty} \int_1^\infty \frac{1 + (-1)^n \exp(-nx^2)}{x^2} \, dx \).

Answer: 1


Question: Let $G$ act on a set $\Omega$. A subset $\Delta \subseteq \Omega$ is called a block if, for every $x \in G$, either $\Delta^x \cap \Delta = \emptyset$ or $\Delta^x = \Delta$, where $\Delta^x := \{ \delta^x : \delta \in \Delta \}$. The singletons of $\Omega$ and $\Omega$ itself are considered trivial blocks. The action of $G$ on $\Omega$ is primitive if it has no non-trivial blocks. Suppose $G$ acts primitively on $\Omega$ and $\Gamma \subseteq \Omega$ is not a block. Prove that there exists $g \in G$ such that either $\alpha \in \Gamma^g$ and $\beta \notin \Gamma^g$, or $\alpha \notin \Gamma^g$ and $\beta \in \Gamma^g$, for any arbitrary $\alpha, \beta \in \Omega$. \(\text{True or False?}\)

Answer: True


Question:
Evaluate the integral \( \iiint _V \frac{\sin x-\sin y}{xy+1} \, dx\,dy\,dz \) over the region \( V \) defined by the surfaces \( x^2 = y \), \( y^2 = x \), \( z = 0 \), and \( z = 1 \). The region \( V \) is bounded by:

1. \( 0 \le x \le 1 \)
2. \( x^2 \le y \le \sqrt{x} \)
3. \( 0 \le z \le 1 \)

Use symmetry to simplify the calculation.

Answer: 0


Question: Let \( f(x) = \frac{2x+3}{x-1} \) and \( h(x) = \frac{6x^2+8x+11}{(x-1)^2} \). Find the polynomial \( g(x) \) such that \( g \circ f = h \).

Answer: x^2 + 2


Question:
Given the sequence of functions $(f_n)$ defined recursively by:

$$ f_{0}(x)=\frac{1}{(1+x)^3} $$
$$ f_{n}(x)=\int^{x}_{0}f_{n-1}(y)\;dy, \quad n\ge1 $$

Determine the limit:

$$ \lim_{n\to \infty}f_{n}(1) $$

Answer: 0


Question: Find the coefficient of the term $a^8b^4c^9d^9$ in the expansion of $(abc+abd+acd+bcd)^{10}$. Provide your answer as a single integer.

Answer: 2520


Question:
Determine the range of values of \( \alpha \) for which the solution \((3,0,0,\beta)\) is optimal for the linear programming problem: Minimize \(x+y+z-\alpha t\) subject to the constraints:

\[2x-y+z=6\]
\[-x+y+t=3\]
\[x,y,z,t \ge 0\]

Answer: [-1, \infty)


Question: Find the closed form of the series \( \sum_{k=0}^{n/2} (2k)\binom{n}{2k}2^{2k} \) where \( n \) is an even integer.

Answer: n(3^{n-1} + 1)


Question: Does there exist a time-varying probability function for an event such that the probability is nonzero at all times, yet the probability of the event occurring over an infinite time period is less than 1?

Answer: Yes


Question: In a series of coin tosses, each with an equal probability of heads or tails, calculate the expected number of tosses required to observe a sequence of 80 heads within a window of 120 consecutive tosses.

Answer: 2^{81} - 2


Question: Determine the values of $p$ for which the improper integral \( \int_0^\infty \frac{\log(x)}{1+x^p} \, dx \) converges.

Answer: p > 1


Question: Determine the order of the pole of the function \( f(z)=\frac{1}{z^2\sin z} \) at \( z_0=0 \).

Answer: 3


Question: In a convex optimization problem with Slater's constraint qualification, if an optimal solution satisfies $f_j(x^*)=0$, what can be concluded about the corresponding Lagrange multiplier $\lambda_j^*$?

Answer: \lambda_j^* \geq 0


Question: Evaluate the limit \( L = \lim_{n \to \infty} \left(\sqrt[n+1]{(n+1)!} - \sqrt[n]{n!}\right) \).

Answer: \dfrac{1}{e}


Question: Determine the number of real roots of the equation \(2\sin\left(\frac{x^2-x}{4}\right) = 2^{x^2} + 2^{-x^2}.\)

Answer: 0


Question: Consider two real numbers $x$ and $y$ such that \(\left(x^2+1\right)\left(y^2+1\right)+9=6\left(x+y\right)\). Find the value of \(x^2+y^2\).

Answer: 7


Question: Evaluate the integral \(\int_{\gamma}\frac{z^2-4z}{z^2-1}dz\) where \(\gamma\) is the circle of radius 3 centered at \(-1\), traversed once clockwise.

Answer: 8\pi i


Question: Determine all values of \(\alpha\) for which the series \(\sum_{n=1}^\infty {((-1)^n + \alpha^3) (\sqrt{n+1} - \sqrt{n})}\) converges. Use Dirichlet's test to find these values.

Answer: 0


Question: What is the algorithmic complexity of the most efficient method for median smoothing a time series with T observations using a window of width n?

Answer: O(T \log n)


Question: Solve the modular equation $x^5 \equiv 4 \pmod{7}$. Determine if a solution exists and find the solution if it does.

Answer: 2


Question: Let $A = B - C$, where $B$ and $C$ are $p \times p$ symmetric positive semidefinite matrices, and the rank of $C$ is $r < p$. Determine the maximum number of negative eigenvalues that $A$ can have.

Answer: r


Question: Find the infimum of the set of integrals \( \int fg \, d\mu \) where \( f \in L_1(\mu) \) with \( \|f\|_1 = 1 \) and \( g \) satisfies \( \|g\|_\infty \leq 1 \).

Answer: -1


Question: Let $p$ be a prime number and consider a group $G$ with order $p^4$ such that the center $Z(G)$ has order $p^2$. Determine the number of conjugacy classes of $G$.

Answer: p^3 + p^2 - p


Question: Calculate the greatest common divisor (GCD) of the expressions $2022^{1729} + 3$ and $2022^{1729} - 3$. Provide your answer as a single number.

Answer: 3


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

Answer: \ln 2


Question:
Given the equations:
$$x + y = 2010$$
$$(\sqrt[3]{x} + \sqrt[3]{y})(\sqrt[3]{x} + 11)(\sqrt[3]{y} + 11) = 2942$$
Find the value of $\sqrt[3]{x} + \sqrt[3]{y}$.

Answer: 12


Question: Determine whether the rank of a positive semi-definite operator is equal to the rank of its square root.

Answer: Yes


Question: Let $B \in M_{6\times 7}(\mathbb{R})$ be a matrix with rank 5. Consider the block matrix \( E = \begin{pmatrix} I_6 & B \\ B^T & 0 \end{pmatrix} \). Determine the rank of the matrix $E$. \( \text{Provide your answer as a single integer.} \)

Answer: 11


Question: Evaluate the integral: $$\int^{1/2}_0 \int^{1-x}_x (x+y)^9(x-y)^9 \, dy \, dx $$

Answer: -\dfrac{1}{400}


Question: Verify the identity: \[ \sum_{k=0}^n \binom{n}{k} \binom{(p-1)n}{k} \binom{pn+k}{k} = \binom{pn}{n}^2, \] for all natural numbers \( p \) such that \( p \ge 2 \).

Answer: \binom{pn}{n}^2


Question: Is it always possible to find a vector $x$ in a non-zero vector space $X$ such that the norm of $x$ is 1 and a linear functional $f$ on $X$ satisfies $f(x) = \|f\|$?

Answer: Yes


Question: Determine whether a graph with 2n vertices and n^2 edges, which contains no triangles, must be bipartite.

Answer: Yes


Question: Calculate the integral \( \int_T db \wedge dt \) in \( \mathbb{R}^5 \) with coordinates \((p_1,q_1,p_2,q_2,t)\), where \( db = dp_1 \wedge dp_2 \) and \( T = \{ t \in [0,1], p_1^2 + q_1^2 = p_2^2 + q_2^2 \leq t^2 \} \).

Answer: 0


Question: Determine the order of the group $G$ given by the presentation $G = \langle a, b \mid a^3 = b^2 = (ab)^3 = e \rangle$, where $e$ is the identity element.

Answer: 12


Question: When $P(x+4)$ is divided by $P(x)$, the remainder is $3x+m$. When $P(x)$ is divided by $P(x+4)$, the remainder is $nx-6$. Compute $m-n$. \( \boxed{} \)

Answer: 9


Question: Calculate the expected value of the natural logarithm of a random variable X, where X follows a Beta distribution with parameters \(1/y\) and 1, i.e., \(X \sim \text{Beta}(1/y, 1)\). Find \(E(\ln(X))\).

Answer: -y


Question: Verify the identity \((1-y^2)\frac{d^2x}{dy^2}-y\frac{dx}{dy}+q^2x=0\) given that \(x=\cos(qt)\) and \(y=\sin(t)\).

Answer: 0


Question: Determine whether the following theorem is true or false: Given an interval $I$ and a function $y:I\rightarrow\mathbb R$ such that $y'(x)=y$ for all $x\in I$, the function $y=0$ is the unique solution satisfying $y(x)=0$ for some $x\in I$. Provide a justification for your answer.

Answer: True


Question: Two independent random variables $A$ and $B$ are uniformly distributed over $[0,1]$, representing the moments two bulbs are ignited. Two other independent random variables $S$ and $T$ are uniformly distributed over $[0,2]$, representing the durations for which the bulbs remain lit. Calculate the probability that both bulbs are lit simultaneously.

Answer: \dfrac{5}{6}


Question: Determine whether the following statement is true or false: A subspace $W$ of a vector space $V$ is a representative system of the quotient space $V/U$ if and only if $W$ is a complement of $U$ in $V$. Provide a proof or counterexample to justify your answer.

Answer: True


Question: Find the maximum value of $a+b+c$ such that $2^n = a! + b! + c!$ for $n, a, b, c \in \mathbb{N}$.

Answer: 10


Question: Find the solution to the partial differential equation \( \frac{\partial}{\partial t} f(x,y,t) = x \frac{\partial f}{\partial y} - y \frac{\partial f}{\partial x} \) with the initial condition \( f(x,y,0) = x^2 + y^2 \).

Answer: x^2 + y^2


Question: The Happy Animals Kennel has 18 cages in a row. They allocate these cages at random to 6 dogs, 6 cats, and 6 pot-bellied pigs, with one animal per cage. All arrangements are equally likely. Let \( A \) be the number of times in the row of cages that two animals of the same species are adjacent. What is the expected value of \( A \)?

Answer: 5


Question:
If $f$ is a differentiable function satisfying \(f\left(\frac{1}{n}\right)=0\; \text{ for all } n\in\mathbb{N}\cup\{0\},\) which of the following is correct?

(a) \(f(0)=f'(0)=0\)
(b) \(\left|f(x)\right|\leq 1, x\in (0,1)\)
(c) \(f(x)=0,x\in (0,1)\)
(d) \(f(0) =0\) but \(f'(0)\) not necessarily zero.

Answer: a


Question: Find the value of the infinite series \( \sum\limits_{n=0}^{\infty}\frac{n^2}{n!} \).

Answer: 2e


Question: Does there exist a function that is strictly increasing ($f'(x) > 0$) and strictly concave down ($f''(x) < 0$) for all $x$ in the first quadrant, and has a limit that approaches infinity as $x$ approaches infinity?

Answer: Yes


Question: Evaluate the limit: $$ \lim\limits_{x \to \infty} \frac{x^2 + \mathrm{e}^{4x}}{2x - \mathrm{e}^x} $$

Answer: -\infty


Question: An insurance company receives claims of two types. Let $N_i(t)$ represent the number of type $i$ claims by time $t$, where $\\{N_1(t), t \ge 0\\}$ and $\\{N_2(t), t \ge 0\\}$ are independent Poisson processes with rates $\lambda_1 = 10$ and $\lambda_2 = 1$, respectively. Type 1 claim amounts are independent exponential random variables with a mean of 1000 dollars, and type 2 claim amounts are independent exponential random variables with a mean of 5000 dollars. Given a claim of 4000 dollars, calculate the probability that it is a type 1 claim.

Answer: 0.671


Question: Given the group $G = S_7$ and a subgroup $H = \langle (1\ 2\ 3\ 4\ 5\ 6\ 7) \rangle$, determine the number of elements in the intersection of $A_7$ and the normalizer $N_G(H)$. It is given that $|N_G(H)| = 42$ and $|A_7| = 2520$. Use the internal product formula to find $|A_7 \cap N_G(H)|$.

Answer: 21


Question:
Determine whether there exists a real differentiable function $f$ such that:
1. The domain of $f$ includes all natural numbers, $D_f \supseteq \mathbb{N}$.
2. The sequence $f_n = f(n)$ is convergent as $n \to \infty$.
3. The sequence $f'_n = f'(n)$ is not convergent as $n \to \infty$.

Provide a justification for your answer.

Answer: Yes


Question: Find the limit of the function \( \frac{e^x-x^x}{x^2} \) as \( x \) approaches 0.

Answer: \infty


Question: Do all unitary matrices square to the identity matrix? Provide a justification for your answer.

Answer: No


Question: Evaluate the integral \( \int_0^\infty \frac{\sin(x)}{x^3} \, dx \) using complex analysis.

Answer: \dfrac{\pi}{4}


Question: Evaluate the integral \( \int_0^{2\pi} \frac{1}{5 + 3\cos x} \, dx. \)

Answer: \dfrac{\pi}{2}


Question: Evaluate the integral \( \int_0^{\pi/2} \frac{1}{1+\tan^{\sqrt{2}}{x}}\,\mathrm{d}x \).

Answer: \dfrac{\pi}{4}


Question: When evaluating the integral \(I=\iint(x^3y^3)(x^2+y^2)dA\) over the region bounded by the curves \(xy=1\), \(xy=3\), \(x^2-y^2=1\), and \(x^2-y^2=4\), a transformation is used: \(u=xy\), \(v=x^2-y^2\). The Jacobian determined is \(J=\frac{1}{-2y^2-2x^2}\). Should the absolute value of the Jacobian be considered in this transformation?

Answer: Yes


Question: Determine all real numbers \(a\) for which the limit \(\lim_{x \rightarrow a} f(x)\) exists, where the function \(f(x)\) is defined as follows: \[f(x)= \begin{cases} x & \text{if } x \in \mathbb{Q} \\ 0 & \text{if } x \in \mathbb{R} \setminus \mathbb{Q} \end{cases} \]

Answer: 0


Question:
In the inner product space $P([0,1])$ of all real polynomials on $[0,1]$ with the inner product \( \langle f,g\rangle=\int_0^1f(x)g(x)dx \), consider the subspace \( V=\text{span}\{t^2\} \). Determine the polynomial \( h(t) \in V \) such that \( \|(2t-1)-h(t)\| \le \|(2t-1)-x(t)\| \) for any \( x(t) \in V \). Choose the correct option:

\(a) \frac{5t^2}{6} \)

\(b) \frac{5t^2}{3} \)

Answer: a


Question: Calculate the line integral \( \oint_{C} F \cdot d\vec{r} \) for the vector field \( F(x,y)=\left(\frac{-y}{(x+1)^2+y^2}, \frac{x+1}{(x+1)^2+y^2}\right) \), where \( C \) consists of the boundary of the square with vertices at (4,4), (-4,4), (-4,-4), and (4,-4) (oriented counterclockwise) and the circle of radius 1 centered at (-1,0) (oriented clockwise). Use the definition of the line integral and Green's Theorem for the square and circle parts, respectively.

Answer: 0


Question: Find the limit of \( P\left(Y_n \geq \frac{n}{2} - n^{2/3} \right) \) as \( n \to \infty \), where \( Y_n \) is the sum of \( n \) i.i.d. random variables each uniformly distributed over \( (0, 1) \).

Answer: 1


Question: Can a lower semicontinuous function take the value $+\infty$?

Answer: Yes


Question: Determine whether the Cantor set is homeomorphic to the unit interval. Provide a yes or no answer.

Answer: No


Question: Determine whether the two Escher prints shown are tessellations of the hyperbolic plane when interpreted as half-plane models.

Answer: Yes


Question: Determine the fixed field of the automorphism \(\sigma\) in \(\text{Aut}(\mathbb{R}(t)/\mathbb{R})\) where \(\sigma(t) = -t\).

Answer: \mathbb{R}(t^2)


Question: Determine whether a countable linearly independent set in a separable Hilbert space necessarily forms a Schauder basis for the space. Answer with 'Yes' or 'No' and justify your response.

Answer: No


Question: Evaluate the limit \(\lim_{h\to 0}\frac{\arctan\left(\frac{4h}{\sqrt{16h^2+1}}\right) - \arctan\left(\frac{h}{\sqrt{h^2+1}}\right)}{h}\) using L'Hôpital's Rule or appropriate approximation techniques.

Answer: 3


Question: Determine the number of integral solutions to the equation \(4y^{3}=3x^{2}+1\).

Answer: 2


Question: Determine the last two digits of \((15!+16!)^{20}\).

Answer: 00


Question: Let $R_1 \subset R_2$ be two commutative rings with identity. Suppose $f \in R_1[x]$ is a polynomial that factors as $gh$ with $g,h \in R_2[x]$, and both $g$ and $h$ are monic. Are the coefficients of $g$ and $h$ integral over $R_1$?

Answer: Yes


Question: Given the equation: \[\frac{\cos{x}\cos{\frac{y}{2}}}{\cos{(x-\frac{y}{2})}}+\frac{\cos{y}\cos{\frac{x}{2}}}{\cos{(y-\frac{x}{2})}}=1\] find the value of \(\cos{x}+\cos{y}\).

Answer: 1


Question: Determine the value of $k$ in the equation $$\sum_{\substack{i,j=1 \\ i \neq j}}^{l} (x_iy_i + x_jy_j) = k \sum_{i=1}^l x_iy_i.$$

Answer: 2(l-1)


Question: Evaluate the integral \( \int_{0}^{\infty} \frac{{\rm e}^{-2x} - {\rm e}^{-x}}{x}\, {\rm d}x \).

Answer: -\ln 2


Question: Prove that if $a + b\sqrt[3]{2} + c\sqrt[3]{4} = 0$ for integers $a$, $b$, and $c$, then $a = b = c = 0$.

Answer: a = b = c = 0


Question: In triangle ABC, segments BD, DF, and FE divide the triangle into four equal-area triangles. Points E and D are on side CA, and point F is on side BC. Given that BF = 2DE, find the ratio of AC to BC.

Answer: \dfrac{4}{9}


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

Answer: 1


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

Answer: 1


Question: Prove that the relative entropy \( h(p||q) = \sum_{i=1}^r p_i (\ln p_i - \ln q_i) \) is non-negative for any two different probability distributions \( p = (p_1, \dotsc, p_r) \) and \( q = (q_1, \dotsc, q_r) \).

Answer: h(p||q) \geq 0


Question: Let $S$ be a semisimple linear algebraic group over a field $K$ of characteristic $0$. Let $H$ be a closed subgroup of $S$ over $K$. Is $H$ necessarily semisimple? Answer yes or no and justify your answer.

Answer: No


Question: Calculate the expected value of \( \cos(aX) \) where \( X \) is a standard normal random variable and \( a \) is a scalar. Express your answer in terms of \( a \).

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


Question: Find the arc length of the curve defined by the function \( y = \int_1^x \sqrt{t^3-1} \, dt \) for \( 1 \leq x \leq 4 \).

Answer: \dfrac{62}{5}


Question: Find the derivative of \(\tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)\) with respect to \(\tan^{-1}\left(\frac{2x\sqrt{1-x^2}}{1-2x^2}\right)\) at \(x=0\).

Answer: \dfrac{1}{4}


Question: Evaluate the limit: $$\lim_{{ x \to 1}}\frac{\sqrt[n]{x^n-1}}{\sqrt[n]{nx}-\sqrt[n]{n}-\sqrt[n]{nx-n}}$$ without using L'Hôpital's Rule.

Answer: -1


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

Answer: 1


Question: Determine the smallest positive integer $N$ such that the inequality $x^N \ge \ln(x)$ holds for all $x > 0$. Provide a proof for your answer.

Answer: 1


Question: Let $X$ be a metric space such that every closed and bounded subset of $X$ is compact. Does this imply that $X$ is a finite dimensional normed space? Answer yes or no.

Answer: No


Question: In triangle \(ABC\), medians \(BD\) and \(CE\) are equal in length. Given that \(\angle BAC = 80^\circ\), determine the measure of \(\angle BDC\).

Answer: 100^\circ


Question: Determine the values of $p$ for which the integral \( \int_{0}^{1}\frac{r^{p+1}}{(1-r^{2})^{\frac{p}{2}-\frac{1}{2}}}dr \) converges.

Answer: (-2, 3)


Question: Evaluate the integral \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dfrac{x^{2}\cos(x)}{1+\exp(x^{2}\sin(x))}\,dx \) which converges to a fraction \( \dfrac{\pi^{a}-b}{c} \), where \( a, b, c \) are integers. Find the sum \( a+b+c \).

Answer: 14


Question: Compute the triple integral \( \iiint_E (2x - 3y) \, dV \) over the solid region \( E \), which is a right circular cone with height 10 and diameter 5, sitting with its base on the \( xy \)-plane centered at \((0,0,0)\). Do not use iterated integrals.

Answer: 0


Question: Given a function $f : \mathbb{R} \rightarrow [0,1]$ with $f'(x) \geq 0$, does it necessarily follow that $f'(x) \leq 1$? Provide a justification for your answer.

Answer: No


Question: Let $f_n \in L^p(B)$ be a sequence where $B$ is a ball in $\mathbb{R}^n$. Assume that $\|f_n\|_{L^p_w(B)} \rightarrow 0$ as $n\rightarrow \infty$, where $\|f\|_{L^p_w(B)}$ is the weak $L^p$-norm of $f$ in $B$. Does $\int_B f_n \phi \rightarrow 0$ as $n\rightarrow \infty$ for any $\phi \in C^\infty_0(B)$?

Answer: Yes


Question: In a regular polygon \( A_1A_2A_3\ldots A_n \), it is given that \( \frac{1}{A_1A_2} = \frac{1}{A_1A_3} + \frac{1}{A_1A_4} \). Determine the number of vertices \( n \) in the polygon.

Answer: 7


Question: Two urns, A and B, are involved in a sequence of trials. Initially, urn A contains 2 red marbles, and urn B contains 2 white marbles. In each trial, one marble is drawn from each urn and switched. Let $X_n$ represent the number of red marbles in urn A after $n$ trials. Given that urn A contains at least one red marble after a long sequence of trials, calculate the probability that urn A contained exactly one red marble before the previous trial. Find $P(X_{n-1} = 1 \mid X_n \geq 1)$ as $n \to \infty$. Provide your answer as a probability.

Answer: \dfrac{3}{5}


Question: Evaluate the limit \(\lim_{n\to \infty} \frac{1}{n^{n+1}}\sum_{k=1}^n k^p\) as an integral using a right Riemann sum.

Answer: 0


Question: Find all positive integers $n$ such that its divisors $1=d_1<d_2<\dots<d_6=n$ satisfy the equation \( \frac{1}{d_1} + \frac{1}{d_2} + \dots + \frac{1}{d_6} = 2 \).

Answer: 28


Question:
Calculate the expected value of the sequence \(a_n = \frac{1+i}{2+n}\), where \(i\) ranges from 0 to \(n\), and the probability mass function is given by:

\[p(i) = \frac{1}{4^n} \binom{2i}{i}\binom{2(n-i)}{n-i}\]

Express the expected value \(E(a_n)\) as a sum:

\[E(a_n) = \sum\limits_{i=0}^{n}\frac{1+i}{2+n} \cdot p(i)\]

Answer: \dfrac{1}{2}


Question: Determine whether there exists a continuous function \( f:[0,1] \to \mathbb{R} \) such that \( f(x) = 0 \) uncountably often and, for every \( x \) where \( f(x) = 0 \), in any neighborhood of \( x \) there exist points \( a \) and \( b \) such that \( f(a) > 0 \) and \( f(b) < 0 \).

Answer: Yes


Question: In a compact and Hausdorff uniform space \((X, \mathcal{U})\), where \(X\) is totally disconnected, and given \(E \in \mathcal{U}\), determine if there exists a \(D \in \mathcal{U}\) such that every \(D\)-chain \(\{x_i\}_{i\in\mathbb{Z}}\) satisfies \((x_i, x_j) \in E\) for all \(i, j \in \mathbb{Z}\).

Answer: Yes


Question: Determine the radius of convergence of the series: $$1+ \frac{a.b}{1.c}z + \frac{a(a+1)b(b+1)}{1.2...c(c+1)}z^2+.... $$

Answer: 1


Question:
Given the joint probability density function of two random variables $X$ and $Y$:

$$f(x,y)=\begin{cases} \frac{1}{a^2} & 0<x,y<a \\ 0, & \text {otherwise} \end{cases}$$

Find the cumulative distribution function (CDF) of $W=\max\left(\frac {X}{Y},\frac {Y}{X}\right)$. Express your answer in terms of $a$. \( \boxed{} \)

Answer: 1 - \dfrac{1}{w}


Question: Calculate the integral of the function \( f(x,y,z) = -4 \) over the region \( D = \{(r,x,\theta): r^2 \le \min \{3|x|, 4x-x^2\}\} \), where \( y = r\cos\theta \), \( z = r\sin\theta \), \( r \ge 0 \), and \( 0 \le \theta < 2\pi \).

Answer: -42\pi


Question: Evaluate the sum \( \sum^{\infty}_{n=1} \frac{n^2}{6^n} \).

Answer: \dfrac{42}{125}


Question: Find all invertible matrices $A$ over a field $\mathbb{F}$ such that $A^2 = A$. Provide your answer in terms of matrix properties.

Answer: I


Question: Let $a, b, c$ be three non-zero integers such that $$(a+b)(a^2-ab+b^2)=c^3.$$ Assume $(a+b)$ and $(a^2-ab+b^2)$ are coprime, and set $a+b=p^3$ and $a^2-ab+b^2=q^3$. Determine if it is possible to write $c=pq$ where $p$ and $q$ are integers.

Answer: Yes


Question: Determine the first digit of the number $3^{150}$ without using logarithms or Newton's binomial theorem.

Answer: 3


Question: Determine whether there exists a strictly decreasing sequence \((x_n)_{n \in \mathbb{N}}\) such that \(x_n \rightarrow 0\) and \(\frac{x_{n+1}}{x_n}\rightarrow1\).

Answer: Yes


Question: Find the last two digits of the infinite power tower \(3 \uparrow \uparrow \infty\), where the exponentiation continues indefinitely, using modular arithmetic.

Answer: 87


Question: Evaluate the integral \( \int_0^{\frac{\pi}{2}}\frac{\tan{x}}{\tan(\frac{x}{2})}\,dx \) using an appropriate substitution.

Answer: \infty


Question: Determine whether it is possible to construct a map from a bounded region in $\mathbb{R}^2$ to the unit sphere $S^2$ such that the Jacobian determinant of the map is bounded away from zero and infinity.

Answer: Yes


Question: Evaluate the contour integral \( \int_{C} e^z \, dz \) along the upper half of the circle \( |z|=1 \), from \( z=1 \) to \( z=-1 \).

Answer: e^{-1} - e


Question: Consider the sequence $(a_n)_{n\geq1}$, where $a_n$ is the sum of the digits of $n^2$ in base 10. Determine the value of the limit \( \lim_{{n\rightarrow\infty}} \frac{a_n}{n} \).

Answer: 0


Question: Determine whether the following statement is true or false: For an injective differentiable real-valued function $f$ defined on an open interval $(a, b)$, the derivative $f'$ cannot be zero anywhere within the interval.

Answer: B


Question: Determine whether the map of sheaves \( \varphi: \mathcal{F} \to \mathcal{F} \), defined by \( f \mapsto f^2 + 1 \), commutes with the restriction maps on the complex plane \( \mathbb{C} \).

Answer: Yes


Question: Is the homeomorphism group of a compactly generated, weak Hausdorff space with the compactly generated topology stemming from the CO topology necessarily a topological group?

Answer: Yes


Question: Evaluate the double integral \( \iint_D xy\,\mathrm{d}x\,\mathrm{d}y \) over the region \( D \) bounded by the curves \( y = \sqrt{1 - x^2} \), \( y = \sqrt{1 - x^2/4} \), \( x = 0 \), and \( x = 2 \).

Answer: \dfrac{3}{8}


Question: Determine whether a continuous function can be defined from the set of irrational numbers to the set of even integers.

Answer: Yes


Question: Evaluate the improper integral \( \int_{0}^{\infty}{\frac{\cos 2x - \cos x}{x^2}dx} \) using the method of indented contour.

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


Question: Find the asymptotic upper and lower bounds for the recurrence relation \( T(n) = T(\sqrt{n}) + n \).

Answer: \Theta(n)


Question: Determine whether a skew-symmetric matrix $A$ can have a non-zero non-negative vector $x$ such that the product $Ax$ is component-wise greater than or equal to the all-ones vector $e$. Provide a justification for your answer.

Answer: No


Question: Suppose $V$ is an $m$-dimensional linear subspace of $\mathbb{R}^{2m}$, and $W$ is a complement of $V$. Does there exist an orthogonal transformation $Q \in \mathbb{R}^{2m \times 2m}$ such that $Q(V) = W$ and $Q(W) = V$?

Answer: Yes


Question: Is the function $f(z)=\frac{1}{z}$ an entire function when extended to the Riemann sphere, with $f(0)=\infty$ and $f(\infty)=0$?

Answer: Yes


Question: Let $C=\{(C_n,\partial_n)\}_{n\in\mathbb{Z}}$ and $C'=\{(C_n',\partial_n')\}_{n\in\mathbb{Z}}$ be chain complexes of vector spaces, and let $f:C\to C'$ be a chain map. Define a new chain complex $C_f=\{(D_n,\overline{\partial}_n)\}$ as follows: $D_n=C_{n-1}\oplus C_n'$ and $\overline{\partial}_n(x,y)=(-\partial_{n-1}(x),\partial_n'(y)+f_{n-1}(x))$. Prove that if $f_*:H_n(C)\to H_n(C')$ is an isomorphism, then $H_n(C_f)=0$. \[ \]

Answer: H_n(C_f) = 0


Question: Find the zeroes and their order for the function \( f(z) = \sin^4{\frac{1}{2}}z \).

Answer: 4


Question: Given the function $f(x) = 6x^3 - 11x^2 - 2$ for $x \geq 1.5$, find the derivative of its inverse function, $\frac{\text{d}f^{-1}}{\text{d}x}$, at the point where $x = 1517$ and $f(7) = 1517$. Explain why the condition $x \geq 1.5$ is relevant.

Answer: \dfrac{1}{728}


Question: Given the joint mass density function \( f_{X,Y}(x,y) = \frac{{e}^{-\left(y + \frac{x}{y}\right)}}{y} \) for \( x > 0 \) and \( y > 0 \), calculate the covariance \( \text{Cov}[X,Y] = E[XY] - E[X]E[Y] \). It is known that \( Y \) follows an exponential distribution with rate parameter \( \lambda = 1 \), hence \( E[Y] = 1 \).

Answer: 1


Question: Is it possible for three random variables $X_1$, $X_2$, and $X_3$ to be such that $X_1$ and $X_2$ are pairwise independent, $X_2$ and $X_3$ are pairwise independent, but $X_1$, $X_2$, and $X_3$ are not mutually independent?

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{2\pi} e^{\cos\theta}\cos(\sin\theta)\,d\theta \).

Answer: 2\pi


Question: Calculate the approximate probability of a type-I error for a hypothesis test with a sample size of 150, where the data follows a gamma distribution with parameters θ = 3 and k = 3. The test statistic is log(X_{0.4}), and the critical region is (-∞, 0.65].

Answer: 0


Question: Determine the number of incongruent solutions to the congruence \(x^4 - 10x^2 + 20 \equiv 0 \pmod{2023}\), where \(2023 = 7 \times 17^2\).

Answer: 0


Question: Determine the Girsanov kernel for the process $X_t = W_t + \cos t$, where $W_t$ is a standard $\mathbb{P}$-Brownian motion, such that $X_t$ becomes a standard $\mathbb{Q}$-Brownian motion.

Answer: -\sin t


Question: Let $X$ be a standard normal variate. Consider another variate $Y$ defined by the rule: $$Y = \begin{cases} -X & \text{for $-2 < X< 2$} \\ X & \text{otherwise}. \end{cases}$$ Determine whether the pair $(X, Y)$ follows a bivariate normal distribution and find the correlation between $X$ and $Y$. Provide your answer as a decision and a correlation coefficient.

Answer: -0.477


Question: Given the group $G=\mathbb{Z}_6 \oplus\mathbb{Z}_6$ and its cyclic subgroup $N$ generated by $(2,3)$, determine if the quotient group $G/N$ is cyclic. If it is cyclic, identify a generator of $G/N$.

Answer: (1,1) + N


Question: Determine the number of irreducible polynomials of degree 6 over the finite field \( \mathbb{Z}/3 \).

Answer: 116


Question: Determine whether a set with an operation that satisfies closure, has identity and inverse elements, can be non-associative. Provide a yes or no answer.

Answer: Yes


Question: Determine the set of real numbers \(x\) for which the series \(\sum_{k=0}^{\infty} \frac{a_k b_k x^k}{k!}\) converges, given that \((b_n)\) is a sequence converging to \(b \in \mathbb{R}\).

Answer: \mathbb{R}


Question:
Given the joint probability density function:

$$ f_{X,Y}(x,y) = \left\{ \begin{array}{ccc}
\frac{1}{x}, & 0 < y < x, & 0< x< 1 \\
0, & \text{otherwise} &
\end{array} \right. $$

Calculate the variance of the random variable $Y$. \( \text{Var}(Y) \)

Answer: \dfrac{7}{144}


Question: Determine the radius of convergence of the power series \( \sum_{n=1}^\infty b_n z^{n^3} \), where \( b_1=\frac{1}{2} \) and \( b_n=4^{-n}b_{n-1} \) for \( n \geq 2 \).

Answer: 1


Question: Determine the number of distinct real roots of the equation \((x-a)^3+(x-b)^3+(x-c)^3=0\), where \(a, b, c\) are distinct real numbers.

Answer: 1


Question: Evaluate the limit of the expression obtained by substituting \( n=2 \) in the equation \( r^{n-1}w'+\frac{1}{2}r^nw=a \), as \( r \) approaches infinity.

Answer: a


Question: Consider the sequence defined recursively by \(a_1 = 1\) and \(a_n = n(a_{n-1} + 1)\) for \(n = 2, 3, \dots\). Let \(P_n\) be the product given by \(P_n = \left(1 + \frac{1}{a_1}\right)\left(1 + \frac{1}{a_2}\right)\dots\left(1 + \frac{1}{a_n}\right)\). Determine \(\lim\limits_{n\to\infty} P_n\).

Answer: e


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

\[ f(x) = 1 + \frac{1}{x}\int_1^x f(t) \, dt \]

Answer: 1 + \ln x


Question: Determine the number of real roots of the function \( g(x) = \left(x^2-2\right)^2 \sinh (x) - \left(4-x^2\right)^2 \sinh (2x) - \left(\frac{x}{3}\right)^2 \).

Answer: 5


Question: Let $a_1, a_2, \cdots, a_{n}$ be real numbers such that $a_1^2 + a_2^2 + \cdots + a_{n}^2 = 1$. Determine the maximum value of the expression $$\large |a_1 - a_3| + |a_3 - a_5| + \cdots + |a_{2n-1} - a_{2n+1}| + |a_{2n+1} - a_1|$$ where $n \in \mathbb{N}$ and $n \geq 2$. Provide your answer as a single number.

Answer: 2\sqrt{2}


Question: Determine whether the set \(\{(m, -m)| m\in\mathbb{Z}\}\) with the operation \((m, -m) + (n, -n) = (m+n, -(m+n))\) is isomorphic to the set \(\mathbb{Z}\) with the operation of addition. Provide a justification for your answer.

Answer: Yes


Question:
Calculate the determinant of the matrix $B \in \mathbb{R}^{n \times n}$ given by:
\[ B = \begin{bmatrix} 1 & 1 & 1 & \cdots & 1 & 1 \\ x & x & x & \cdots & x & x \\ 0 & x & x & \cdots & x & x \\ 0 & 0 & x & \cdots & x & x \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & x & x \\ \end{bmatrix} \]

Answer: 0


Question: Given an associative composition \( \Delta \) on a set \( E \), and an element \( u \) in \( E \) such that for every \( a \) in \( E \), there exist elements \( x \) and \( y \) in \( E \) satisfying \( u \Delta x = a = y \Delta u \), determine whether there exists a neutral element for \( \Delta \).

Answer: Yes


Question: Find the minimum value of \( P = \frac{1}{\sqrt{2a+bc}}+\frac{1}{\sqrt{2b+ca}}+\frac{1}{\sqrt{2c+ab}} \) for non-negative values of \( a, b, c \) that satisfy \( ab + bc + ca = 1 \).

Answer: 1 + \sqrt{2}


Question: Given that $0< A< 90^{\circ}$, $\sin A$ and $\cos A$ are roots of the equation $4x^2+3x+k=0$, and $\sin 2A$ is a root of the equation $x^2 -44x +36=0$, find the value of $k$. \(\boxed{\text{value of } k}\)

Answer: 44 - 16\sqrt{7}


Question: Consider a service system where customers arrive according to a Poisson process with rate $\lambda$. Over the time interval $[0, t]$, let $N$ be the number of customers arriving, and their arrival times be $T_1, T_2, \ldots, T_N$. For each customer, the wait time within the interval is $t - T_i$. The average wait time is given by $\bar{W} = \frac{\sum_{i=1}^N (t - T_i)}{N}$. Determine the expected value $E[\bar{W}]$.

Answer: \dfrac{t}{2}


Question: Solve the inequality \(x\ln(2-x)\ge 2x\).

Answer: [2 - e^2, 0]


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

Answer: 0


Question: Calculate the Lebesgue measure of the Cantor set formed by iteratively removing the middle fifth of each subinterval, starting with the interval [0, 1].

Answer: 0


Question: Find an explicit formula for the function represented by the power series \( \sum_{n=0}^\infty\frac{x^{2n+1}}{(2n+1)!} \).

Answer: \sinh x


Question: What is the minimum number of sample points required to approximate the phase shift of a sinusoidal signal with a known frequency, assuming the amplitude is unknown?

Answer: 2


Question: Using the Mean Value Theorem, find the limit \( \lim_{x\to-\infty}\frac{g(x)-g(x-1)}{|x|} \) given that \(|g'(x)|\leq e^x\) for all \(x\).

Answer: 0


Question: Evaluate the limit: $$\lim_{x \to 0^+} \lim_{n \to \infty} \frac{1}{1 + n \sin^2 x}$$

Answer: 0


Question: A company manufactures light bulbs with lifetimes that are normally distributed with a mean of 3 months and a variance of 1 month. If a consumer wants to ensure that a succession of these bulbs lasts at least 40 months with a probability of at least 0.9772, what is the minimum number of bulbs they should purchase?

Answer: 16


Question: Determine whether there exist triangles with all sides and vertex coordinates being integers, such that none of the angles are 90 degrees. Answer with 'Yes' or 'No'.

Answer: Yes


Question: In a group $G$ of order $pq^3$, where $H$ is a normal Sylow $q$-subgroup, there exists a subgroup $P$ of order $pq^2$ and a Sylow $p$-subgroup $R$ of $P$ such that $|N_P(R)|=p$. Determine the order of $N_G(R)$. \( \text{(Express your answer as a single number.)} \)

Answer: pq


Question: If \( p(x) \) is a polynomial such that \( p(x^2+1) = (p(x))^2 + 1 \) and \( p(0) = 0 \), find the value of \( p'(0) \).

Answer: 1


Question: Is the Dirac distribution an example of a singular distribution?

Answer: Yes


Question: Evaluate the line integral \( \oint_C \mathbf{F} \cdot d\mathbf{r} \) where \( C \) is the boundary of the square with vertices \((0,0), (2,1), (1,-2), (3,-1)\) oriented clockwise, and \( \mathbf{F}(x,y) = \langle xy - \sin(x), y^2 + \cos(y) \rangle \). Use Green's theorem to solve this.

Answer: \dfrac{15}{2}


Question: Evaluate the limit \(\lim_{n\to\infty}\frac{1}{n^3}\int_{n}^{2n}\frac{x}{1+x^5}dx\) using a substitution method.

Answer: 0


Question: Evaluate the definite integral: \[ \int_{0}^{\frac{\pi}{2}}{\frac{dx}{\sqrt{\cos(x)}\cdot\sin(x)}} \]

Answer: \infty


Question: Calculate the value of the Jacobi symbol \(\left(\frac{-2p^2}{4a^2 + p^2}\right)^l\), where \(p \equiv 3 \pmod{4}\) is a prime, \(a\) is an odd integer, and \(l\) is an even integer.

Answer: 1


Question: Given an $n \times n$ matrix $A$ with real entries such that $A^2 = -I$, determine the value of $\det(A)$. \( \text{(Express your answer as a single number.)} \)

Answer: 1


Question: Evaluate the integral \( \int_{0}^{\infty} x^{5}e^{-2x^{3}}dx \).

Answer: \dfrac{1}{12}


Question: Determine whether the sequence \(a_n = \frac{(n+2)!}{(n+2)^{n+2}}\) converges or diverges.

Answer: 0


Question: Is there a polynomial-time algorithm to find the optimal point with the specified property in a linear program, even if the polytope is unbounded?

Answer: Yes


Question: Find the limit of the expression \( \frac{\sin(ax^2 + bx + c)}{(x-\beta)^2} \) as \( x \) approaches \( \beta \), where \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( ax^2 + bx + c = 0 \).

Answer: a


Question: Simplify the expression \( \frac{d\vec{r}}{d||\vec{r}||} \) in terms of the unit vector \( \hat{r} \).

Answer: \hat{r}


Question: Is the quotient space $S^1 \times [-1, 1] / \sim$ compact, where $\sim$ is defined by $(a, b) \sim (c, d)$ if and only if $(a, b) = (c, d)$, $b = d = 1$, or $b = d = -1$?

Answer: Yes


Question: Let $A \in M(n,n)$ be a doubly stochastic matrix such that \( A^k = \begin{pmatrix} 1/n & \cdots & 1/n \\ \vdots & \ddots & \vdots \\ 1/n & \cdots & 1/n \end{pmatrix} \) for some $k > 1$. Does it follow that \( A = \begin{pmatrix} 1/n & \cdots & 1/n \\ \vdots & \ddots & \vdots \\ 1/n & \cdots & 1/n \end{pmatrix} \)?

Answer: Yes


Question: Determine whether the series \( S_n = \sum_{k=1}^n \binom{n}{k}\frac{1}{k}\left(-\frac{1}{1-a} \right)^k\left(\frac{a}{1-a} \right)^{n-k} \) converges as \( n \to \infty \), where \( 0 < a < 1 \). If it converges, find the limit of \( S_n \) as \( n \to \infty \).

Answer: 0


Question: Calculate the volume of the set \(E=\{(x,y,z) \in \mathbb{R}^3 : 0<y<\pi, z^2 + (x - y^5)^2 < \sin y\}\).

Answer: 2\pi


Question: Evaluate the limit \(\lim \limits_{n\to \infty}\{(2+\sqrt{3})^n\}\), where \(\{\cdot\}\) denotes the fractional part.

Answer: 1


Question: Evaluate the limit \( \lim_{n\to\infty} \frac{n}{n+2}\int_0^1 g\left(u^{\frac{2}{n+2}}\right)du \) for a bounded function \( g \) on \([0,1]\).

Answer: g(1)


Question: Given a matrix $M$, find a formula for the eigenvalues of the matrix $M^n$.

Answer: \lambda^n


Question: Given positive real sequences $x_n$ and $y_n$ such that $\lim y_n = y > 0$ and $\liminf x_n = \infty$, determine whether $\lim x_n y_n = \infty$.

Answer: \infty


Question: Let $G$ be a graph with $n$ vertices where each edge is chosen with probability $\frac{1}{2}$. A graph $G$ is separated if we can divide the vertices into two equal-sized sets with no edges between them. It is almost-separated if we can divide the vertices into two equal-sized sets with fewer than $10n$ edges between them. Determine the probability that $G$ is either separated or almost-separated.

Answer: 0


Question: Evaluate the limit \( \lim_{n\to+\infty}\left(\frac{\pi}{2}-\arctan n\right)^n n! \) using Stirling's formula and the relation \(\arctan x + \arctan\frac{1}{x} = \frac\pi2\).

Answer: 0


Question: Determine whether the following statement is true or false: For any arbitrary Hausdorff space $X$, the set $X \setminus \{x_0\}$ is a dense subset of $X$.

Answer: B


Question: Find the number of integer values of $y$ such that the expression $\frac{9^n + 35n - 4}{y}$ is an integer for every positive integer $n$. Provide your answer as a single integer.

Answer: 2


Question: Calculate the area of the portion of the sphere \(x^{2} + y^{2} + z^{2} = a^{2}\) that lies within the cylinder \(x^{2} + y^{2} = ax\) and above the \(xy\)-plane. Use the parametrization of the upper half of the sphere and evaluate the integral correctly to find the surface area.

Answer: a^2(\pi - 2)


Question: Determine the kernel of the linear transformation \( \xi : F_n \rightarrow S \) defined by \( \xi(f) = M = (m_{ij}) \), where \( m_{ij} = \int_{i}^{j} f(t) \text{d}t \), and \( S \) is the subspace of matrices satisfying \( m_{ij} = -m_{ji} \) and \( m_{ij} + m_{jk} = m_{ik} \).

Answer: \{0\}


Question: Let $\{x_n\}$ be a sequence of positive real numbers such that $\lim_{n \to \infty} x_n^n = 1$. Determine $\lim_{n \to \infty} x_n$ and justify your answer.

Answer: 1


Question: Find the dimension of the quotient ring \( \mathbb{R}[x, y] / (x^3, y^3) \) by determining a spanning set and checking for linear independence.

Answer: 9


Question: Calculate the conditional variance \( \operatorname{Var}(Y | X = 2) \) given the joint density function: \[ f(x,y) = \begin{cases} \frac{1}{18}e^{-(x+y)/6} & \text{if } 0 < x < y \\ 0 & \text{otherwise} \end{cases} \] where \( X \) is the number of years until the first component fails and \( Y \) is the number of years until the machine fails.

Answer: 36


Question: Calculate the limit \(\lim_{x\to 0 } \frac {\tan^2x - \arctan^2x}{x^4}\) without applying L'Hopital's rule excessively.

Answer: \dfrac{4}{3}


Question: Find the value of the infinite power tower \(3^{3^{3^{\dots}}}\) modulo 100, where the power tower continues until the value stabilizes.

Answer: 87


Question:
For what values of \( \beta \) is the function \( g(x) \) differentiable at \( x = 0 \), where

\[ g(x) = \begin{cases} 0, & \text{if $x = 0$} \\ x^\beta \cos\left(\frac{1}{x}\right), & \text{if $x \neq 0$} \end{cases} \]

Answer: \beta > 1


Question:
In a two-consumer, two-commodity general equilibrium model, consider the following utilities and endowments:

- Consumer 1's utility: \(U_1(X_1,Y_1,X_2,Y_2) = \min\{X_1,Y_1,X_2,Y_2\}\)
- Consumer 2's utility: \(U_2(X_1,Y_1,X_2,Y_2) = \min\{X_2,Y_2\}\)

Endowments:
- Consumer 1: 10 units of good \(X\) and 0 units of good \(Y\)
- Consumer 2: 0 units of good \(X\) and 10 units of good \(Y\)

Determine the competitive equilibrium allocation of goods \(X\) and \(Y\) between the two consumers.

Answer: (X_1, Y_1, X_2, Y_2) = (5, 5, 5, 5)


Question: In a topological space $X$, consider a sequence of open sets $\{K_n\}$ such that $\bigcup K_n = \Omega$, where $\Omega$ is open in $X$. Given that $K_n \subset \text{int} K_{n+1}$ for each $n$, and $K$ is a compact subset of $\Omega$, does there always exist an index $N$ such that $K \subset K_N$?

Answer: Yes


Question: Consider two matrices $X$ and $Y$ with the same dimensions. If $X'X = Y'Y$, do $X$ and $Y$ necessarily have the same singular values? Answer yes or no and justify your answer.

Answer: Yes


Question: Determine the class to which the language \(L_3=\{\langle M\rangle\mid|\langle M\rangle|\le 2016\text{ and M is a TM that accepts }\varepsilon \}\) belongs.

Answer: R


Question: Find the value of \( n \) such that the expression \( 20^n + 16^n - 3^n - 1 \) is divisible by 323.

Answer: 2


Question: Let $X$ and $Y$ be abelian groups. If $\text{Hom}(X,Z)\cong \text{Hom}(Y,Z)$ for all abelian groups $Z$, does it follow that $X \cong Y$?

Answer: Yes


Question: Given the function \( f(x) = \int_{0}^{x} \frac{dt}{\sqrt{1+t^3}} \) and its inverse function \( g \), find the constant \( k \) such that \( g''(y) = kg^2(y) \).

Answer: \dfrac{3}{2}


Question: Find the minimum value of $x^2 + y^2 + z^2$ subject to the constraint $2x + 3y + 4z = 5$. Provide your answer as a fraction in simplest form.

Answer: \dfrac{25}{29}


Question: Evaluate the limit: \[ \lim_{(x,y)\rightarrow (0,0)} \frac{\sin(xy)}{ \sqrt{x^2+y^2}} \]

Answer: 0


Question: Determine whether the partial derivative $f_y$ exists at the point $(0,0)$. If it exists, find its value.

Answer: 0


Question: What is the minimum time complexity for checking whether a set of integers from 1 to n can be partitioned into two subsets with equal sums?

Answer: O(1)


Question: Let $g:[c,d]\to\mathbb{R}$ be a Riemann-integrable function such that $g(x) \leq 0$ for all $x\in[c,d]$. If $\int_{c}^{d}g=0$, is it true that $\{x\in[c,d]:g(x)=0\}$ is a dense set?

Answer: Yes


Question: Calculate the limit \( \lim_{r\to0}\int_{\sigma_{r}}\frac{e^{iz}}{z}\, dz \) where \(\sigma_{r}:[0,\pi]\to\mathbb{C}\) is defined as \(\sigma_{r}(t)=re^{it}\).

Answer: i\pi


Question: Evaluate the principal value of the improper integral \( P.V. \int_{-\infty}^{\infty} \frac{dx}{x(x^2+1)} \) using the residue theorem. Verify if the principal value is \( 0 \).

Answer: 0


Question: A cylindrical container is constructed with a fixed inner volume \(V\) mm\(^3\), a 2 mm thick solid wall, and is open at the top. The bottom is a solid circular disc with a thickness of 2 mm and a radius equal to the outer radius of the container. If the volume of the material used is minimized when the inner radius is 10 mm, determine the value of \(\frac{V}{250\pi}\).

Answer: 4