Question: Evaluate the limit: $$\lim_{n \to \infty} \left ( \sqrt[n]{n} + \frac 1 n \right )^{\frac n {\ln n}}$$ without using series expansion.

Answer: e


Question: Evaluate the integral \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\cos x}{\sqrt{|\sin x|}} \, dx \).

Answer: 4


Question: Consider a group $G$ and a subset $A$ of $G$. Suppose there is a group action $\cdot: G \times G \to G$. Determine if it is possible for a non-identity element $g \in G$ to exist such that the set $g \cdot A = \{g \cdot a \mid a \in A\}$ is a strict subset of $A$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit of the integral \(\lim_{R\to\infty}\int_C \frac{2z^2-5}{(z^2+1)(z^2+4)}\,dz\) where \(C\) is the upper half-circle \(|z|=R\), \(\operatorname{Im} z>0\).

Answer: 0


Question: Is a graph with 17 vertices and 129 edges guaranteed to be Hamiltonian?

Answer: Yes


Question: Find the maximum value of $xyz$ using the method of Lagrange multipliers, given the constraint $x+y+z=a$, where $x, y, z$ can be positive or negative.

Answer: \dfrac{a^3}{27}


Question: Evaluate the limit: $$\lim_{x\to 0} \frac{\int_{0}^{x^4}(e^{u^{10}}-1)du}{x^{44}}$$

Answer: \dfrac{1}{11}


Question: Find the residue of the function \( g(z) = e^{1/z^2} \) at \( z = 0 \).

Answer: 0


Question: Evaluate the limit for \(p \in \mathbb{N}\): \[\lim_{n\to\infty}\frac{1}{n^{p+1}}\sum_{i=1}^{n} \frac{(p+i)!}{i!}\]

Answer: \dfrac{1}{p+1}


Question: Calculate \((8^{2^{6^{4^{2^{5^{8^9}}}}}}) \mod 10000\).

Answer: 8256


Question: Let matrix $A$ be symmetric and positive semidefinite. Prove that if there exists a diagonal element $a_{ii} = 0$, then $\det(A) = 0$. Here, $a_{ii}$ denotes any diagonal element of $A$.

Answer: 0


Question: Determine whether the following statement is true or false: Every module is a submodule of a free module.

Answer: False


Question: Calculate the covariance between $Z = X + Y$ and $W = X^2$, where $X$ and $Y$ are jointly normal random variables with means 0, variances 1, and correlation coefficient $\rho$. Provide your answer in terms of $\rho$. \( \boxed{\text{Your answer here}} \)

Answer: 0


Question: Determine the number of distinct terms in the expansion of \(\left( x + \frac{1}{x} + x^2 + \frac{1}{x^2} \right)^{20}\).

Answer: 81


Question: X and Y are jointly normal random variables with a common mean of 0, a common variance of 1, and a covariance of 1/2. Calculate the probability that X + Y is less than or equal to \( \sqrt{3} \).

Answer: 0.8413


Question: Determine the height above the ground from which an observer can see exactly one-fourth of the Earth's surface, assuming the Earth is a sphere with radius \( r \).

Answer: r


Question: Calculate the sum of the series: \( \cos \frac{2\pi}{2013} + \cos \frac{4\pi}{2013} + \cdots + \cos \frac{2010\pi}{2013} + \cos \frac{2012\pi}{2013} \).

Answer: -\dfrac{1}{2}


Question: Let $p$ be a prime number. Determine whether there exists a sequence $(K_n)$ of number fields such that $\lim_{n\to\infty}[K_n:\mathbb{Q}]=\infty$ and $p$ splits completely in each integer ring $\mathcal{O}_{K_n}$.

Answer: Yes


Question: Let $A$ be an $n \times n$ matrix with complex entries such that $A^* = -A$. Define $B = e^A$. Prove that $B^* = e^{-A}$.

Answer: B^* = e^{-A}


Question: Determine the number of ring homomorphisms from the ring $\mathbb{Z}[X]/(X^2-X)$ to the ring $\mathbb{Z}/6\mathbb{Z}$. Provide your answer as a single integer.

Answer: 4


Question:
Compute the limit of the following expression as \( n \) approaches infinity:
\[ \lim_{n \to \infty}\left( \frac{1^a+2^a+3^a+\ldots+n^a}{n^a}-\frac{n}{a+1} \right) \]

Answer: \dfrac{1}{2}


Question: Determine the number of similarity classes of 6x6 matrices over the complex numbers \(\mathbb{C}\) with the minimal polynomial \((x-1)(x-2)^2\) and the characteristic polynomial \((x-1)^2(x-2)^4\).

Answer: 2


Question: Determine the radius of convergence of the power series \( \sum_{n=0}^{\infty}\frac{n^{1000}}{\sqrt{n!}}x^n \).

Answer: \infty


Question: Calculate the conditional mean \( E(X \mid X > Y) \) where \( X \) and \( Y \) are independent and identically distributed uniform random variables on the interval \((0,1)\). Use the law of total expectation to approach this problem.

Answer: \dfrac{2}{3}


Question: Evaluate the limit: \[\lim_{x \to 1}\frac{\left(1-\sqrt{x}\right)\left(1-\sqrt[3]{x}\right)...\left(1-\sqrt[n]{x}\right)}{\left(1-x\right)^{\left(n-1\right)}}\] without using Taylor series.

Answer: \dfrac{1}{n!}


Question: Determine whether the prime $p$ totally ramifies in the cyclotomic field $\mathbb{Q}(\zeta_{p^k})$ for any prime $p$ in $\mathbb{Z}$ and integer $k \geq 1$. Total ramification means that every prime $\wp$ of the ring of integers of $\mathbb{Q}(\zeta_{p^k})$ lying above $(p)$ has an inertial degree $f(\wp|p) = 1$. Provide a justification for your answer.

Answer: Yes


Question: Determine the norm in which the inequality \( ||g*f-g*s|| \leq C ||g|| \text{ } ||f-s|| \) holds, given that \( g \in L^p(\mathbb{R}) \) has compact support and \( 1\leq p<\infty \).

Answer: L^p


Question: Suppose $f: [0,1] \rightarrow \mathbb{R}$ is a continuous, strictly increasing function with $f(0) = 0$, $f(1) = 1$, and $f'(x) > 0$. If $\int_0^1 f(x) \, dx = \frac{1}{3}$, find the value of the integral $\int_0^1 f^{-1}(y) \, dy$.

Answer: \dfrac{2}{3}


Question: Determine whether the sum \((c+\frac{1}{3})^n+(d+\frac{1}{3})^n\) is divisible by \(3^n\) for only a finite number of integers \(n\), where \(c\) and \(d\) are positive integers.

Answer: Yes


Question: Use Stokes' theorem to evaluate the line integral \(\int_C (y-z)\,dx + (z-x)\,dy + (x-y)\,dz\) where \(C\) is the circle of radius 2 in the \(xy\)-plane centered at the origin, oriented counterclockwise when viewed from above.

Answer: -8\pi


Question: Evaluate the improper integral \( \int_0^{\frac{\pi}{2}} (\sec^2x - \sec x \tan x) \, dx \). Address the singularity at the upper limit to find the value of the integral.

Answer: 1


Question: In a Noetherian ring, if the product of two elements is 1, are the elements necessarily units?

Answer: Yes


Question: Let $f(x)=a_0+a_1x+a_2x^2+\dots +a_nx^n$ be a primitive non-zero polynomial with integer coefficients. Let $g=\gcd(\{f(k), k \in \mathbb{Z}\})$. Prove or disprove that $g$ divides $n!$, or equivalently, that $\gcd(f(k)) \mid c(f)\cdot (\deg f)!$.

Answer: g \mid n!


Question: Let $X_{1},\ldots,X_{n}$ be a random sample from the distribution with common pdf $$ f(x;\theta) = e^{-(x-\theta)} 1_{(\theta,+\infty)}(x), \; \theta \in \mathbb{R}. $$ Let $Y_{n} = \min \{X_{1},\ldots,X_{n}\}$. Determine whether $Y_{n}$ is a consistent estimator of $\theta$. Provide a justification for your answer.

Answer: Yes


Question: Is it true that any invertible $n \times n$ matrix with entries of $-1$, $0$, or $1$, having exactly one nonzero entry in each column, must be a permutation of a diagonal matrix?

Answer: Yes


Question: Evaluate the limit: $$\lim_{n\to\infty}\left(\sqrt{2} \cdot \frac{\Gamma\left(\frac{n+1}{2}\right)}{\Gamma\left(\frac{n}{2}\right)} - \sqrt{n}\right)$$

Answer: 0


Question: Consider the associative $\mathbb{C}$-algebra $\text{U}_+$ with generators $E$ and $H$, and the relation $HE - EH = 2E$. Let $M$ be a finite-dimensional $\text{U}_+$-module. Determine whether there exists a nonzero eigenvector $v \in M$ of $H$ such that $E(v) = 0$. Provide a justification for your answer.

Answer: Yes


Question: For $n > 1$, determine whether the complement of a hypersurface in $\mathbb{A}^n$ is an algebraic variety. Answer with 'Yes' or 'No'.

Answer: No


Question: Determine the cardinality of a basis for the vector space of the real numbers over the field of rational numbers.

Answer: 2^{\aleph_0}


Question: Find the coefficient of $x^{25}$ in the expansion of $\left(\frac{x^{6}-1}{x-1}\right)^5$, where the inner sum is truncated at $x^4$ to limit the maximum degree of each term.

Answer: 0


Question: Calculate the probability of selecting an odd number at random from the set \( S = \{1, 2, 3, \ldots\} \), given that a probability measure \( P \) is defined by \( P(\{n\}) = 2\left(\frac{1}{3}\right)^n \) for all \( n \) in \( S \).

Answer: \dfrac{3}{4}


Question: Evaluate the integral: $$\int_{-5}^5 \dfrac{x^3 \sin^2x}{x^4+2x^2+1}~dx$$

Answer: 0


Question: Evaluate the limit: $$\lim_{x \to -\infty}\left(\cos\left(e^{x^{1/3}+\sin x}\right)\right)$$

Answer: 1


Question: Evaluate the integral \( \int_{\Bbb D}(x^2 - y^2)\, dx\, dy \) over the region \( \Bbb D=\{|x|+|y|\le2\} \) using a suitable change of variables.

Answer: 0


Question: Determine whether every subset of the extended real numbers has an infimum.

Answer: Yes


Question: Find the point on the plane in \(\mathbb{R}^4\) that includes the points \((0,0,0,0)\), \((1,0,-1,1)\), and \((0,1,1,2)\), which is closest to the point \((-5,4,0,0)\).

Answer: (-2, 1, 3, 0)


Question: In how many ways can \((n+1)\) people be seated at two indistinguishable circular tables, ensuring that both tables have at least one person? Express your answer in terms of \(n\).

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


Question: Let $R$ be a commutative ring with unity and $f$ be a polynomial in $R[x,y]$. If $f(x,xy)$ is the zero polynomial in $R[x,y]$, does this imply that $f$ is the zero polynomial? Provide a justification for your answer.

Answer: Yes


Question: Calculate the expected value of the largest power of 2 that divides a randomly chosen non-zero positive even integer $E$, expressed as $E = A \cdot 2^n$, where $A$ is an odd number.

Answer: 2


Question: Let $(\Omega, \mathcal{A}, \mu)$ be a measure space and $\mu^{\star}$ be the outer measure induced by $\mu$. A subset $A \subset \Omega$ is said to be $\mu^{\star}$-measurable if $$\mu^{\star}(E)= \mu^{\star}(E\cap A)+\mu^{\star}(E\cap A^{\complement})$$ for all $E\subset \Omega$. The set of all $\mu^{\star}$-measurable sets is denoted by $\mathcal{M}(\mu)$. If $A,B\in \mathcal{M}(\mu)$ and $A\subset B$, does it follow that $B-A\in \mathcal{M}(\mu)$? Answer yes or no.

Answer: Yes


Question: Determine if the series \( g(x)=\sum_{n=1}^{+\infty} \frac{\cos (x n)}{n^\beta} \) defines a continuous function on \( \mathbb{R} \) for \( \beta>1 \).

Answer: Yes


Question: Determine the smallest cardinality of a base for the topological space \((X, \tau)\), where \(X = \mathbb{R}^2\) and \(\tau = \{\emptyset, \mathbb{R}^2, \{(x, y) \in \mathbb{R}^2 \mid x^2 + y^2 < r \}, r \in \mathbb{R}, r > 0\}\).

Answer: \aleph_0


Question: Determine the density of primes $p$ in $\\mathbb{Z}$ such that $p$ splits completely in $\\mathbb{Q}(\\sqrt{m})$ and remains inert in $\\mathbb{Q}(2^\\frac{1}{3})$, where $m$ is any non-zero, square-free integer.

Answer: \dfrac{1}{6}


Question: Determine whether the statement is true or false: A convex function in multiple dimensions is locally Lipschitz.

Answer: True


Question: Evaluate the limit \( \lim_{n \to \infty} \frac{(7^{1/n}+n^{1/n})^n-\frac{1}{7^n}}{7^n-n^7} \) using the squeeze theorem.

Answer: 0


Question: Calculate the probability that three independent exponential random variables X, Y, and Z satisfy the condition X < Y < Z.

Answer: \dfrac{1}{6}


Question: Find the sum of the series: \( \tan 2^\circ + \tan 6^\circ + \cdots + \tan 178^\circ \).

Answer: 0


Question: Calculate the value of \( I(1) \) to three decimal places, where \( I(x) = \int_0^x \frac{\sin(t)}{t} \, dt \).

Answer: 0.946


Question: Find the residue of the function \( f(z) = \frac{(z^6 - 1)^2}{z^5(2z^4 - 5z^2 + 2)} \) at \( z = 0 \) without using four derivatives or directly computing a Laurent series.

Answer: \dfrac{21}{8}


Question: Find the smallest ordinal \( \beta \) such that \( \omega + \beta = \beta \).

Answer: \omega^2


Question: Consider the sequence $\{y_n\}$ defined by the equation $y_n^3 - (n^2 - 5n)y_n + 5n^2 = 0$ for all $n \in \mathbb{N}$. Determine the limit of $y_n$ as $n$ approaches infinity.

Answer: 5


Question: Let $G$ be a finite group, and let $e \notin A$ be a subset not containing the identity. If $g \cdot A = A$, does this imply that $g = e$?

Answer: No


Question: Evaluate the limit: $$\lim_{n\to\infty}\int_0^1 \int_0^1 \cdots\int_0^1 \frac{n}{x_1 + x_2 + \cdots + x_n} \, dx_1 \, dx_2 \cdots dx_n$$

Answer: 2


Question: If a matrix $A$ satisfies $A^n = I$ for some positive integer $n$, is $A$ necessarily involutory? Answer yes or no and justify your answer.

Answer: No


Question: Evaluate the limit \( \lim_{x \to 1^-} \sum_{n=0}^\infty (-1)^n x^{n^2} \).

Answer: \dfrac{1}{2}


Question: Determine whether there exists a matrix that is considered invertible in practice but not invertible in the traditional mathematical sense.

Answer: Yes


Question:
Evaluate the limit:

\[
\lim_{r \to \infty} \frac{\int_0^{\pi/2}\left(\frac{x}{\pi/2}\right)^{r-1}\cos x\,dx}{\int_0^{\pi/2}\left(\frac{x}{\pi/2}\right)^r\cos{x}\,dx}
\]

Answer: 1


Question: Given the joint probability density function (pdf) of two continuous random variables, $X$ and $Y$, where $f(x,y)=\frac{1}{2} e^{-x}$ for $|y| \le x$ and $x > 0$, find the expected value of $X$ conditional on $Y = -1$, denoted as $E(X|Y=-1)$. Compute $E(X|Y=-1)$. \( \boxed{} \)

Answer: 2


Question: Let $X$ be a normed linear space and $Y$ be a Banach space. If there exists a bicontinuous linear map from $X$ to $Y$, does it imply that $X$ is also a Banach space? Provide a justification for your answer.

Answer: Yes


Question: Determine the range of possible values for $|z|$ if $z$ is a complex number satisfying the equation $|z^2 - 4| = |z + 2|$. Provide your answer as an interval or set of values.

Answer: [1, 3]


Question: Determine if there exists a real-valued, nonconstant, differentiable function $g$ such that \( \text{Re} \int_{-\infty}^0 g(x) \mathrm{d}x = \text{Re} \int_0^{\infty} g(x) \mathrm{d}x \).

Answer: Yes


Question: Calculate the limit: $$\lim\limits_{n \to \infty} n \int_{1-\epsilon}^1 u^n e^u \, du$$ where $\epsilon$ is a fixed number such that $0<\epsilon <1$. Provide your answer in terms of $\epsilon$ if necessary.

Answer: e


Question: Determine whether the Fourier series for the function $g(x) = x^2 \sin(x)$, defined on the interval $[-\pi, \pi]$, converges in $L^2$.

Answer: Yes


Question: Evaluate the double integral \( \int \int_{R} \frac{1}{(x^2+y^2)^2} \, dA \) where \( R \) is the region enclosed by the circle \( x^2+y^2 = 2 \) and the line \( x=1 \), using polar coordinates.

Answer: \dfrac{1}{4}


Question: Determine the proportion of all possible triangles that are acute.

Answer: \dfrac{1}{4}


Question: If the size of an extension's automorphism group is equal to the degree of the extension, is the extension necessarily Galois?

Answer: Yes


Question: Evaluate the line integral \(\int_\Gamma \vec F \cdot d\vec r\) where \(\vec F (x,y,z) = (6xy + 4xz)\vec i + (3x^2 + 2yz)\vec j + (2x^2 + y^2)\vec k\) and \(\Gamma\) is the path \(y=x^2, z=0\) from \((0,0,0)\) to \((2,4,0)\) followed by the line segment from \((2,4,0)\) to \((1,1,2)\).

Answer: 9


Question: Determine the Lebesgue measure of the Luzin set \( A \), which consists of numbers in the interval \([0,1]\) whose continued fraction expansion contains an infinite subsequence of integers such that each integer is divisible by the previous one.

Answer: 1


Question: Find the derivative of the function \( f(x) = \langle Cx, x \rangle \), where \( C \) is a symmetric matrix.

Answer: 2Cx


Question: Evaluate the expression $a + 2b + 3c$ given that the limit \[ \lim_{{h \to 0}} \frac{{\int_0^h f(x) \, dx - h(af(0) + bf(h/3) + cf(h))}}{{h^4}} \] is a finite non-zero number.

Answer: \dfrac{9}{4}


Question: Evaluate the infinite product \( \prod_{n=2}^{\infty}\left(1+\dfrac{1}{n^2}+\dfrac{1}{n^4}+\dfrac{1}{n^6}+\dfrac{1}{n^8}+\ldots\right) \), given that \( \left(1+\dfrac{1}{n^2}+\dfrac{1}{n^4}+\dfrac{1}{n^6}+\dfrac{1}{n^8}+\ldots\right) = \dfrac{1}{1-\dfrac{1}{n^2}} \).

Answer: 2


Question: In a math competition, a four-digit natural number 'a' is selected. Another number 'b' is chosen, squared, and added to 'a'. A third number 'c' is chosen, squared, and multiplied by 'a'. The product of these two results is 123456789. Given the equation $$(b^2 + a)(c^2 \cdot a) = 123456789,$$ find the value of 'a' without using a calculator.

Answer: 3607


Question: If $f(x)$ satisfies the conditions of the converse of L'Hôpital's rule and \( \lim\limits_{x\to +\infty}\dfrac{f(x+1) - f(x)}{e^{x+1} - e^x}=l \), does it follow that \( \lim\limits_{x\to +\infty}\dfrac{f'(x)}{e^x}=l \)? Answer 'yes' or 'no'.

Answer: yes


Question: Let $f:\mathbb{R} \to \mathbb{R}$ be a differentiable function. If $f$ is Lipschitz continuous, does it imply that $f'$ is bounded? Provide a justification for your answer.

Answer: Yes


Question: Given a ring \( R \) and an ideal \( \mathbf{a} \), let \( \pi: R \rightarrow R/\mathbf{a} \) be the natural projection map. Define \( \pi^*:\mathrm{Spec}(R / \mathbf{a}) \rightarrow \mathrm{Spec}(R) \) as the pullback of \( \pi \). Determine the image of \( \pi^* \) in \( \mathrm{Spec}(R) \).

Answer: V(\mathbf{a})


Question: Calculate the sum of the alternating series \(\sum_{n=0}^\infty (-1)^n\binom{1/2}{n}\frac{x^{4n+1}}{4n+1}\) such that the error is less than or equal to \(\frac{0.4^9}{72}\).

Answer: 0.398976


Question: Determine the last two digits of the number \(3^{3^{100}}\).

Answer: 03


Question: Determine whether a prime number $p$ of the form $4k-1$ can be expressed as a sum of integer powers, such as $p = y^3 + (-z^9) + 7$, where $y$ and $z$ are integers.

Answer: Yes


Question: Determine the minimum distance between the curves $x^2 = y + 2$ and $y^2 = x + 2$ using the method of common normals.

Answer: \dfrac{9\sqrt{2}}{4}


Question: Compute the linking number $l(K, M)$, where $K$ is the unit circle in the $xy$-plane centered at $(-1,0,0)$ and oriented clockwise, and $M$ is the unit circle in the $xy$-plane centered at $(2,0,0)$ and oriented clockwise.

Answer: 0


Question: Determine the range of values for $c$ such that the equation $x(x^2-7) = 2x+c$ has exactly two solutions. Express your answer in the form $c_1 < c < c_2$, where $c_1$ and $c_2$ are integers.

Answer: -10 < c < 10


Question: Determine the degree of the splitting field of the polynomial $x^3 - \pi$ over the field $\mathbb{R}(\pi)$. Provide your answer as a single integer.

Answer: 6


Question: Calculate \( g'(0) \) for the function \( g = f \circ f \circ f \), where \( f(x) = x \cdot (x+1)(x+2)\cdots(x+100) \).

Answer: (100!)^3


Question: Find the last two nonzero digits of $700^{2^{88}}$.

Answer: 01


Question: Evaluate the improper integral: $$\int_{-\infty}^0 t^5 e^{t^2} \, dt$$

Answer: -\infty


Question: Evaluate the limit: $$\lim_{x\rightarrow 0} \left(1+\frac{x}{(x-1)^2}\right)^{\frac{1}{\sqrt{1+x}-1}}$$.

Answer: e^2


Question: Determine the optimal way to divide a square into four equal areas using curved lines, such that the total length of the lines is minimized.

Answer: 2


Question: Compare the growth rates of the functions $f(n) = n^{\log n}$ and $g(n) = (\log n)^n$. Which function grows faster as $n$ approaches infinity?

Answer: g(n)


Question: Find the least significant digit of the product \( \prod\limits_{i=1}^{n/3} (i)(n-2i+1) \) for multiples of 3, where \( n \geq 9 \).

Answer: 0


Question: Determine the number of solutions to the congruence equation \(x^2 \equiv 605 \pmod{1010}\).

Answer: 2


Question: Evaluate the integral \( \int_{-\infty}^{0} \frac{\ln(-x)}{x^2+x+1} \, dx \) without using advanced techniques like calculus of residues.

Answer: 0


Question: Given two distinct orthogonal matrices A and B, if there exists a sequence of applications of these matrices such that each matrix appears an equal number of times and the application of this sequence is equal to the application of the sequence's dual (where each matrix is replaced by the other), do A and B necessarily commute? Answer with 'Yes' or 'No'.

Answer: No


Question: How many distinct labeled simple undirected graphs with 10 vertices can be constructed if each vertex has an even degree?

Answer: 2^{36}


Question: Determine whether the following limit exists: $$\lim_{(x,y)\to (1,1)} \frac{(x-1)(y-1)}{|x-1|+|y-1|}$$

Answer: 0


Question: Let $F=\langle (1,0,0,2),(0,1,0,0) \rangle$ and $G=\langle (-1,1,0,0),(1,0,1,0),(0,0,0,1) \rangle$. Given that the basis of $F \cap G$ is $\{(-1,1,0,-2)\}$, find the basis of the quotient space $F/(F \cap G)$. Provide your answer as a set of vectors.

Answer: \{(1,0,0,2)\}


Question: Given a number $k$, determine if it is possible to identify the term in the $n$-Simplex sequence that corresponds to $k$. Provide a 'yes' or 'no' answer.

Answer: Yes


Question: Determine the value of \(3^{3^{3^{\cdots}}}\) modulo 100, where there are 2021 threes in the power tower.

Answer: 87


Question: Let $X$ and $Y$ be two normed linear spaces. Let $T : X \longrightarrow Y$ be a bijective linear map with a closed graph. Determine which one of the following statements is necessarily true: \((\text{A})\) The graph of $T$ is equal to $X \times Y.$ \((\text{B})\) $T^{-1}$ is continuous. \((\text{C})\) The graph of $T^{-1}$ is closed. \((\text{D})\) $T$ is continuous.

Answer: C


Question: Solve the equation \( \sqrt{x+3}+\sqrt{5-x}-2\sqrt{15+2x-x^2}=-4 \) without using the substitution method.

Answer: 1


Question: For a function \( f(x,y) \), if the partial derivatives \( f_x \) and \( f_y \) exist at a point \( (a,b) \) but are not continuous at that point, can \( (a,b) \) still be a critical point?

Answer: Yes


Question: Let $C$ and $D$ be square matrices of size $m \times m$. Determine whether the determinant of the matrix $C - xD$ is a polynomial function of $x$. If it is, find the degree of this polynomial.

Answer: m


Question: Find a positive constant \( c \) such that \( \frac{S_{n}}{n^{c}} \) converges in distribution to some random variable \( A \), where \( X_{1},...,X_{n} \) are iid with characteristic function \( \phi(t)= 1-\sqrt{|t|(2-|t|)} \) for \( t\in[-1,1] \) and zero elsewhere.

Answer: 2


Question:
Given the joint moment generating function of random variables $X$ and $Y$:
\[ M(t_1,t_2) = \frac{1}{2}e^{t_1+t_2} + \frac{1}{4}e^{2t_1+t_2} + \frac{1}{12}e^{t_2} + \frac{1}{6}e^{4t_1+3t_2} \]
Find the probability $P(X < Y)$.

Answer: \dfrac{1}{12}


Question: Evaluate the expression \(\sqrt {1+\sqrt {4+\sqrt {16+\sqrt {64+\sqrt {256+\cdots }}}}}\).

Answer: 2


Question: Let $K(X)$ be the field of rational functions of $X$ over some field $K$. Define the $K$-morphism $\phi: K(X) \rightarrow K(X)$ by $\phi(X) = 1 - X$. Determine an element $Y \in K(X)$ such that $K(Y) = \{ f \in K(X) : \phi(f) = f \}$.

Answer: X(1 - X)


Question:
Determine the value of $q$ for which the following limit is a finite non-zero number:

$$\lim_{x \to \infty} \left( {x^q\left( \sqrt[4]{x + 2} + \sqrt[4]{x - 2} - 2\sqrt[4]{x} \right)} \right)$$

Answer: \dfrac{7}{4}


Question: Evaluate the limit: $$\lim_{x \to -\infty}\frac{(x-2) e^{(\ln (x^2))^2}}{((x-1)^2)^{\beta +1}}$$ where $\beta > 0$. \(\text{Express your answer in terms of } \beta.\)

Answer: -\infty


Question: Find the values of $z$ for which the series \( \sum_{n=0}^\infty \frac{1}{n^2}\left(z^n+\frac{1}{z^n}\right) \) converges.

Answer: |z| = 1


Question: Determine the crystallographic restriction for a discrete group of isometries whose translation group is generated by a single non-zero vector \( a \).

Answer: 2


Question: Suppose that $F$ is a function defined on an open interval containing $a$. If for any sequence $\{x_n\}$ that converges to $a$ but is never equal to $a$, we have $\lim_{n \rightarrow \infty} F(x_{2n}) = L$, does it follow that $\lim_{x \rightarrow a} F(x) = L$? Answer yes or no and justify your answer.

Answer: Yes


Question:
Evaluate the limit as \( n \to \infty \) of the integral:
\[ \int_0^1 f_n(x) \, dx \]
where \( f_n(x) \) is given by:
\[ f_n(x) = \begin{cases} \frac{nx}{2n-1} & , x \in \left[0, \frac{2n-1}{n}\right] \\
1 & , x \in \left[\frac{2n-1}{n}, 2\right] \end{cases} \]

Answer: \dfrac{1}{4}


Question: Find the limit of the sequence \( \lim_{n \to \infty} \frac{n!}{a^{n + 1}} \) where \( a \) is a positive real number, without using Stirling's approximation.

Answer: \infty


Question: Let $a, b, c, d$ be real numbers such that $a + b + c + d = 0$ and $abcd = 1$. Find the minimum value of the expression $a^4 + b^4 + c^4 + d^4 + a^2 + b^2 + c^2 + d^2$. \( \text{Provide your answer as a single number.} \)

Answer: 8


Question:
Evaluate the limit of the Laplace transform of the function $f_\epsilon (x)$ as $\epsilon$ approaches 0, where $f_\epsilon (x)$ is defined as follows:
$$f_\epsilon (x)= \begin{cases} \frac{1}{\epsilon}, & 0\leq x\leq \epsilon \\ 0, & x> \epsilon \end{cases}$$
The Laplace transform of $f_\epsilon (x)$ is given by:
$$L(f_\epsilon (x))=\frac{1-e^{-p\epsilon}}{p\epsilon}$$
Find:
$$\lim_{\epsilon \to 0} L(f_\epsilon(x))$$

Answer: 1


Question: Given three lines $a$, $b$, and $c$ passing through a point $O$, there exists a unique line $d$ such that the composition of reflections $\sigma_c\sigma_b\sigma_a=\sigma_d$. If $A\in a$, $B$ is the reflection of $A$ across $b$, $C$ is the reflection of $B$ across $c$, and $d$ is the perpendicular bisector of $AC$, does $d$ also pass through $O$ such that the angles between lines $a$ and $d$ are congruent to the angles between $b$ and $c$?

Answer: Yes


Question: Find the function $f(x, y)$ with continuous partial derivatives that satisfies the following conditions: \( \frac{\partial f}{\partial y} = xf \) and \( f(x, 0) = 1 \) for all \( x \).

Answer: e^{xy}


Question: Determine whether the following statement is true or false: For any sequence \( \{ a_{n} \} \) and its supremum \( A \), there exists an integer \( K \) such that for all \( m > K \), we have \( |a_{m} - A| < x \) for any given \( x > 0 \).

Answer: B


Question: Let $x:[0,\infty)\to[0,\infty)$ be an absolutely continuous function and $a:[0,\infty)\to[0,1]$ be a measurable function. If \( \lim_{t\to\infty}\int_t^{t+T}|\dot{x}(s)+a(s)x(s)|ds = 0 \) for all \( T>0 \), does it imply that \( \lim_{t\to\infty}\int_t^{t+T}a(s)x(s)ds = 0 \) for all \( T>0 \)?

Answer: Yes


Question: What is the proof-theoretic ordinal of Primitive Recursive Arithmetic (PRA)?

Answer: \omega^\omega


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

Answer: -1


Question: Let $G$ be a group with order $|G| = p^a m$, where $p$ is the smallest prime divisor of $|G|$. If $P$ is a cyclic Sylow $p$-subgroup of $G$, prove that the normalizer of $P$ in $G$ is equal to the centralizer of $P$ in $G$, i.e., $N_G(P) = C_G(P)$.

Answer: N_G(P) = C_G(P)


Question: Suppose a sequence of functions $\{f_j\}$ converges to $f$ in $L^2(\mathbb{R}^n)$, and each $f_j$ satisfies the inequality \( \int_{\mathbb{R}^n}{|f_j(x)|e^{a|x|^2} dx}\leq M \) for some $a>0$ and a constant $M>0$. Does it follow that \( \int_{\mathbb{R}^n}{|f(x)|e^{b|x|^2} dx}<\infty \) for some $0<b<a$? Provide a justification for your answer.

Answer: Yes


Question: What is the minimum number of queries required to determine the number in each cell of a $7\times7$ board with $49$ distinct numbers, given that you can query any $3$ cells at a time to find out their set of numbers?

Answer: 33


Question: Given the function \( f(x) = e^{-x} \) and the point \( x_0 = 0 \), determine the smallest integer \( n \) such that the inequality \( |P_n(x) - f(x)| < 10^{-5} \) is satisfied for all \( x \) in the interval \([0, 1]\).

Answer: 8


Question: Determine the points at which the function \( f: \mathbb{R} \to \mathbb{R} \), defined as \[ f(x) = \begin{cases} x & \text{if } x \notin \mathbb{Q} \\ \frac{m}{2n+1} & \text{if } x = \frac{m}{n}, \ m\in \mathbb{Z}, n \in \mathbb{N}, \text{ and } m \text{ and } n \text{ are coprimes} \end{cases} \] is continuous.

Answer: 0


Question: Determine whether there exists an isomorphism \( T : \mathbb{R}^2 \to W \) where \( W \) is the vector space spanned by \( \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \).

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_0^{\frac{\pi}{2}-\epsilon} \frac{(\sin(x))^{n}}{1-\sin{(x)}}\,\mathrm{d} x, \] where \(\epsilon\) is a small positive number.

Answer: 0


Question: Calculate the arc length of the curve defined by the equation \( y = \sqrt{1-x^2} \).

Answer: \pi


Question: Determine whether the following statement is true or false: If an algebra $A$ is solvable with solvable index $n$, then $A$ is also nilpotent with a nilpotency index of $n$ or a smaller integer $k < n$. Specifically, does it hold that $A^{n} \subseteq A^{(n)}$?

Answer: False


Question: Determine the degree of nilpotency of the dihedral group $G = D_{16}$ using only the upper central series.

Answer: 3


Question: Is the direct product of a free group of finite rank with a finite group Hopfian?

Answer: Yes


Question: Calculate the limit of the series: \[ L = \sum_{k=1}^{\infty} \frac{1}{k^k} \]

Answer: \int_{0}^{1} x^{-x} \, dx


Question: Given random variables $X \sim U(0,1)$ (uniform distribution from 0 to 1) and $Z \sim B(1,0.5)$ (Bernoulli distribution with success probability 0.5), find the covariance $\operatorname{Cov}(XZ,Z)$.

Answer: \dfrac{1}{8}


Question: Calculate the limit: $$\lim_{x\to\infty} x^{3/2}\left(\sqrt{x+1}+\sqrt{x-1}-2\sqrt{x}\right)$$

Answer: -\dfrac{1}{4}


Question: Find the limit of \( \frac{n}{\sqrt[n]{n!}} \) as \( n \to \infty \).

Answer: e


Question: Determine the number of ways to distribute $k$ different items among $n$ different people such that each item is shared among exactly $r$ people.

Answer: \dbinom{n}{r}^k


Question: Consider two distinct ideals $I$ and $J$ of a ring $R$. Determine if it is possible for the quotient rings $R/I$ and $R/J$ to be isomorphic.

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{\frac{\pi}{2}} \frac{x}{\sin x} d x \) and show that it equals \( 2G \), where \( G \) is the Catalan's constant.

Answer: 2G


Question: Determine the number of roots of the polynomial equation \( f(z) = z^4 + 8z^3 + 3z^2 + 8z + 3 = 0 \) that lie in the right half of the complex plane.

Answer: 2


Question: Determine whether a morphism of ringed spaces \((f,\theta) :(X, \mathcal{O}_{X})\rightarrow (Y,\mathcal{O}_{Y})\), where \(f\) is a homeomorphism and \(\theta\) is an isomorphism of sheaves of rings, is necessarily an isomorphism of ringed spaces.

Answer: Yes


Question:
Determine whether the following limit holds given that $\lim_{n \to \infty} x_n = a \in \mathbb{R}$:

$$\lim_{n \to \infty} \frac{2}{n(n+1)}\sum_{k = 1}^n kx_k = a$$

Answer: a


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

Answer: 2\sqrt{3}


Question: Determine the number of integer values of $k$ for which the polynomial $x^3 - 27x + k$ has at least two distinct integer roots.

Answer: 2


Question: Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all real numbers \( x \) and \( y \), the equation \[ f(f(y)) + f(x - y) = f(xf(y) - x) \] holds.

Answer: 0


Question: Calculate the number of ways to roll 10 distinct dice such that the sum of the numbers rolled equals 25.

Answer: 831204


Question: Determine whether a number composed of exactly $3^n$ ones is a multiple of $3^{n-1}$. Provide a proof or counterexample.

Answer: Yes


Question: Determine if there exist any subsets of the rational numbers \( \mathbb{Q} \) that are both open and closed, other than the empty set and \( \mathbb{Q} \) itself.

Answer: Yes


Question: Determine whether the inequality $3 < a < b$ implies that $a^b > b^a$. Provide a justification for your answer.

Answer: Yes


Question: Consider the initial value problem (IVP): $$ \frac{dy}{dt}=y^{2021}-y^{2019},$$ with the initial condition $y(-2021)=\frac{1}{2019}$. Determine the limit of $y(t)$ as $t$ approaches infinity.

Answer: 0


Question: Evaluate the integral \( \int_{|z|=3} \frac{dz}{\sin^2 z} \) using the residue theorem.

Answer: 0


Question: Find the function $f(x)$ that satisfies the equation \( \int_{x^{1/2}}^{x} f(t) \, dt = \int_{1}^{x^{1/2}} f(t) \, dt \) for all \( x > 0 \).

Answer: \dfrac{C}{x}


Question: Find the approximate value of $x$ in the equation \(36^x + 6^x = 42\) using Taylor expansions and series reversion.

Answer: 1


Question: Determine whether a matrix $A$ is nilpotent if it satisfies the equation $A = AB - BA$ for some matrices $A$ and $B$. Provide a justification for your answer.

Answer: A


Question: If a sequence of functions $f_k$ epi-converges to a function $f$, does it imply that $f$ is lower semicontinuous?

Answer: Yes


Question: Suppose $X$ is a normed space and $T$ is a bounded linear operator on $X$. If there exists a vector $v$ with $\|v\| = 1$ such that $\|Tv\| = \|T\|$, does it imply that $v$ is an eigenvector of $T$? Answer yes or no and justify your answer.

Answer: No


Question: Calculate the perimeter of the curve given by \( r = 1 + \cos \theta \) for \( 0 \le \theta < 2\pi \) in 2D polar coordinates using the integral formula: \[ \int_P \sqrt{(dx)^2 + (dy)^2} \] Show that \( (dx)^2 + (dy)^2 = (dr)^2 + (rd\theta)^2 \) as part of your solution.

Answer: 8


Question: Evaluate the limit \(\lim_{(x,y) \to (0,0)} \dfrac{x^2y^2}{x^4 + |y|^2}\) without using polar coordinates.

Answer: 0


Question: Find the coefficient of \( t^{q^2+q+1} \) in the expansion of the polynomial \( P(t) = \left(\frac{1-t^{q+1}}{1-t}\right)^{2q-1} \), where \( q \) is a positive integer. Assume \( q \) is a prime power and the polynomial is in the ring \( \mathbb{F}_q[t] \).

Answer: 0


Question: Given that \(x^{x^{x+1}}=\sqrt{2}\), find the value of \(W\) if \(W=x^{x^{p}}\) where \(p = 2x^{x+1}+x+1\).

Answer: 2


Question: Determine the set of all complex numbers $w$ that satisfy the inequality $|z-w|^2 \geq |1-\overline{w}z|^2$ for all complex numbers $z$ with $|z|=1$. Provide your answer as a set of complex numbers.

Answer: \mathbb{C}


Question: Find the function $g(x, y)$ that satisfies the partial differential equation \( \frac{\partial g}{\partial x} = yg \) with the initial condition \( g(0, y) = 1 \) for all \( y \).

Answer: e^{xy}


Question: In a finite-dimensional normed space $(X, \|\cdot\|_X)$ with a closed proper subspace $M$, consider the quotient space $X/M$ and the orthogonal complement $M^\perp$ in the dual space $X^*$. If $\dim X < \infty$, are the spaces $M^\perp$ and $X/M$ isomorphic?

Answer: Yes


Question: Determine if the integral \( \int_0^L \delta(x-v)H(x-d) \, dx \) has a closed-form solution, where \( \delta \) is the Dirac delta function, \( H \) is the Heaviside function, and \( v \) and \( d \) are in the interval \([0, L]\).

Answer: H(v - d)


Question: Let $f:[0,\infty) \rightarrow \mathbb{R}$ be a differentiable function such that $f(0)=1$ and $f'(x) \geq f(x)$ for all $x \geq 0$. Prove that $f'(x) \geq e^x$ for all $x \geq 0$.

Answer: f'(x) \geq e^x


Question: Consider a set of $n$ linearly independent vectors $\{v_1, \dots, v_n\}$. If we construct a set of $m < n$ orthogonal vectors $\{\phi_1, \dots, \phi_m\}$ as linear combinations of $\{v_1, \dots, v_n\}$, can $\{\phi_1, \dots, \phi_m\}$ be linearly dependent? Answer yes or no.

Answer: No


Question: In the Hilbert space $L_2[-1,1]$ with the inner product $(f,g)=\int_{-1}^1 f(t)\overline{g(t)}dt$, let $\{p_n\}_{n=0}^\infty$ be the orthonormal basis defined by $p_n(t)=\frac{\sqrt{2n+1}}{2}P_n(t)$, where $P_n(t)$ is the Legendre polynomial of degree $n$. Calculate the value of the inner product $(t^2,p_3)$, where $t^2 \in L_2[-1,1]$. \( \boxed{} \)

Answer: 0


Question: Find the solution set of the equation \( f^{-1}(x) = f(x) \) for the function \( f(x) = -(x+2)^3 - 2 \).

Answer: \{-3, -2, -1\}


Question: Evaluate the limit: \[ \lim_{n\to\infty} \int_{\{|x| > n\}} X \,dP \] where $X$ is a random variable on the probability space $(\Omega,\mathbf{F},P )$ and $E|X|<+\infty$. Determine the value of this limit.

Answer: 0


Question: Evaluate the integral \( I = \int_{0}^{\infty} \frac{\sin\left(x\right)\sin\left(\frac{x}{3}\right)}{x\left(\frac{x}{3}\right)}\,dx \) using Laplace Transforms.

Answer: \dfrac{\pi}{2}


Question: Given a symmetric and positive definite matrix $A \in M_n(\mathbb{R})$ with full rank and a matrix $B \in M_n(F)$ with rank $1$, determine if it is true that ${\rm rank} \left( A^{{-1}/{2}} B A^{{-1}/{2}} \right) = 1$. Provide a justification for your answer.

Answer: True


Question:
Given the system of equations:
\[
\begin{align*}
a^2 - bc &= 3 \\
b^2 - ca &= 4 \\
c^2 - ab &= 5
\end{align*}
\]
Determine the possible value of \( a+b+c \). Choose from the following options: \( 0 \), \( 1 \), \( -1 \), or \( \frac{1}{2} \).

Answer: \frac{1}{2}


Question: Calculate the expected value of the area under the trajectory of a Wiener process, $W_t$, above the line $y=t$. Provide your answer as a single numerical value.

Answer: \dfrac{1}{4}


Question: Find the coefficient $a_{-1}$ in the Laurent series expansion of the function \( f(z) = \frac{1}{2-e^z} \) around the singularity \( z_0 = \log(2) \).

Answer: -\dfrac{1}{2}


Question: Given that the radii of convergence of the series \(\sum a_n x^n\) and \(\sum b_n x^n\) are \(R_1\) and \(R_2\), respectively, determine the radius of convergence of the series \(\sum a_n b_n x^n\).

Answer: R_1 R_2


Question: Determine the limit of the expression \(\lim_{(x,y)\to (\alpha,0)}\left(1+\frac{x}{y}\right)^y\) along any general path, given that the limit along certain specific paths is known to be 1.

Answer: 1


Question: Estimate the probability that the sum of 32 rounding errors, each uniformly distributed on the interval \((-\frac{1}{2}, \frac{1}{2})\), differs from zero by less than $1. Assume the sum of the rounding errors can be approximated by a normal distribution.

Answer: 0.46


Question: Given a matrix A with entries 0, 1, or -1 and exactly 2012 nonzero entries, determine the sum of the entries on the main diagonal of the matrix product A · A^T.

Answer: 2012


Question: Is the function \( g: \mathbb{C}\setminus\{0\} \to \mathbb{C} \) defined by \( z=re^{i \phi} \mapsto \ln (r)-i\phi \) with \( \phi \in (-\pi, \pi] \) measurable? Consider the injectivity and continuity of the exponential function on the intervals \([-n,n]\times[-\pi+\frac{1}{n},\pi-\frac{1}{n}]\).

Answer: Yes


Question: Given two $n \times n$ nilpotent matrices $G$ and $F$ of order $k$, where $G^{k-1} \neq O$ and $F^{k-1} \neq O$, but $G^{k} = O$ and $F^{k} = O$, determine whether $G$ and $F$ are necessarily similar.

Answer: No


Question: Let $x_1, x_2, \ldots, x_k$ be distinct elements of a group $G$ with $k \geq 4$. If $x_i$ commutes with $x_{i+1}$ for $1 \leq i < k$, and $x_k$ commutes with $x_1$, determine whether all $x_i$ must commute with each other.

Answer: No


Question: Calculate the degree of the field extension \(|\mathbb{Q}(\sqrt[m]{p}, \sqrt[n]{q}, \sqrt[r]{s}):\mathbb{Q}|\) for distinct prime numbers \(p\), \(q\), and \(s\).

Answer: mnr


Question: Let \(\Omega \subset \mathbb{R}^3\) be a bounded domain, and let \(\gamma \colon [0,1] \to \Omega\) be a smooth and closed path in \(\Omega\). Determine if there exists a constant \(C > 0\) such that for every \(v \in C^2(\Omega) \cap C^0(\overline{\Omega})\) with \(\text{curl}\, v = 0\) in \(\Omega\), if \(\int_{\Omega} \lvert v(x) \rvert^2 \, dx = 0\), then \(\bigg\lvert \int_0^1 v(\gamma(t)) \cdot \dot\gamma(t) \, dt \bigg\rvert \leq C\).

Answer: Yes


Question: Determine whether the quotient group \((\mathbb{R},+)/ (\mathbb{Q},+)\) contains a non-zero element of order 10.

Answer: No


Question: Given the function \( f(x) = x^{x^x} \), find the value of \( f''(1) \).

Answer: 2


Question: Can the term $p(1-p)$ in the inequality $$Pr\left(|\frac{Y}{n}-p|< e\right) \geq 1-\frac{p(1-p)}{e^2n},$$ where $p(1-p) \leq \frac{1}{4}$, be directly replaced with $\frac{1}{4}$ without rearranging the terms?

Answer: Yes


Question: Determine whether the following statement is true or false: If $A$ is an $n \times n$ non-singular complex matrix, then the eigenvalues of the matrix $A^*A$ are real and positive. Justify your answer.

Answer: True


Question: Determine whether there exists a sequence of step functions \( \phi_n :[0,1] \rightarrow \mathbb{R} \) for \( n \in \mathbb{N} \) such that \( \int^1_0 \phi_n(t) \, dt > 0 \) and \( \frac{\int^1_0 \phi_n (t) \, dt}{||\phi_n||_{\infty}} \rightarrow 0 \) as \( n \rightarrow \infty \).

Answer: Yes


Question:
Determine the value of \(a\) such that the following limit holds:

\[ \lim_{t \to 0} \left(\frac{a}{t^2} - \frac{\sin 6t}{t^3 \cos^2 3t}\right) = -18. \]

Answer: 6


Question: Determine the number of real solutions to the equation \( e^{2x} - x^2 = 1 \).

Answer: 1


Question: Let $X_1, \ldots, X_n$ be independent and identically distributed (i.i.d.) random variables, where each $X_k$ follows a Bernoulli distribution with parameter $\theta \in (0,1)$. Define $Y := \sum_{k=1}^n X_k$. Find the conditional expectation $E(X_k \mid Y)$. \(\text{Express your answer in terms of } n, \theta, \text{ and } Y.\)

Answer: \dfrac{Y}{n}


Question: Consider the set $st_0$ of all bounded statistically convergent complex sequences. Define a linear functional $g: st_0 \to \mathbb{C}$ by $g(x) = \stlim x_n$, where $\stlim x_n$ denotes the statistical limit of the sequence $(x_n)$, and $x = (x_n) \in st_0$. Is the linear functional $g$ bounded and continuous?

Answer: Yes


Question: For a random variable $X$ with differential entropy $h(X)$, determine the change in differential entropy when $X$ is multiplied by a negative constant $a$. Express your answer in terms of $h(X)$ and $a$.

Answer: h(X) + \log |a|


Question: Let $R$ be a ring and $a \in R$. If $a$ is a root of a polynomial $F$ in $R[X]$, does there exist a polynomial $G$ in $R[X]$ such that $F = (X-a)G$? Provide a justification for your answer.

Answer: Yes


Question: Find the limit of the sequence \(a_n = \left[\frac{(n+1)(n+2)\cdots(n+n)}{n^n}\right]^{1/n}\) using Cauchy's second limit theorem.

Answer: \dfrac{4}{e}


Question: Determine whether the limit \( \lim_{x\to\infty}\left(\frac{x+n}{\ln(x+n)}-\frac{x}{\ln x}\right) \) has a closed-form expression for any integer \( n > 1 \).

Answer: 0


Question: Determine the shock curve for the weak solution \( u(x,t) = \begin{cases} \frac{x-2}{t+2} & ; x > \xi(t) \\ 0 & ; x < \xi(t) \end{cases} \) to the equation \( u_t + \left(\frac{u^2}{2}\right)_x = 0 \) in \( \mathbb{R} \times (0, \infty) \), where the shock curve \( x = \xi(t) \) starts at the origin. Draw a diagram showing the characteristics in the \( xt \)-plane.

Answer: x = 2 - \sqrt{2(t + 2)}


Question: Find the 100th derivative of the function \( g(x) = (x^3 - 4x + 7)^{30} \).

Answer: 0


Question: Find the simple closed curve $C$ in the first quadrant that maximizes the work done by the vector field $F = \left(\frac{y^2}{2} + x^2y, -x^2 + 8x\right)$. What is the maximum value of the work done by $F$ along this curve?

Answer: \dfrac{136}{5}


Question: Determine whether the set of minimal prime ideals in any closed subset of \( \operatorname{Spec} A \) is finite for a Noetherian ring \( A \).

Answer: Yes


Question: Determine whether the following statement is true or false: Let $R$ be a commutative ring and $I$ be an ideal of $R$. $I$ is a principal ideal if and only if $I$ is a free $R$-module.

Answer: B


Question: Evaluate the double integral \( \iint_\mathcal{D} \sqrt{x^2+y^2} \, dx \, dy \) where \( \mathcal{D} = \{(x,y) \mid x^2+y^2 \leq 2x\} \).

Answer: \dfrac{32}{9}


Question: Calculate the value of the sum \( \sum^{2020}_{n=1} n^{2019} \pmod{2019} \). Use Fermat's little theorem and the factorization \( 2019 = 673 \times 3 \) to find the residue of the sum when divided by 2019.

Answer: 1


Question: Calculate the probability of obtaining tails on at least three consecutive tosses in an infinite sequence of independent, fair coin tosses.

Answer: 1


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

Answer: \dfrac{1}{2}


Question: Determine whether there exists a surjective function from the real numbers \( \mathbb{R} \) to a compact metric space \( X \).

Answer: Yes


Question: What is the probability that the sequence $(1,2,3,4,5,6)$ appears infinitely many times when a fair six-sided die is rolled infinitely many times?

Answer: 1


Question: Determine whether the sum \( \sum_{k=1}^N \frac{(k+2)^3 c_k}{(N+3)^4} \) converges, given that \( \{c_k\} \) is a convergent sequence with limit \( c \).

Answer: \dfrac{c}{4}


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

Answer: 0


Question: In a sequence constructed by concatenating binary numbers with an odd number of digits followed by those with an even number of digits, find the number of '1's in the 1717th term. For example, the sequence starts with 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, ... How many '1's are in the 1717th term?

Answer: 7


Question: Determine the number of zeros of the function \( f(z) = e^{2z} - 2ze^z - 1 \) within the annulus defined by \( 1 < |z| < 2 \).

Answer: 2


Question: Let $(X,m)$ be an outer measure space with finite measure. Suppose that for any $A \subseteq X$, there exists a Caratheodory measurable subset $B \supseteq A$ such that $m(A) = m(B)$. If $F \subseteq X$ satisfies $m(F) + m(X \setminus F) = m(X)$, is $X \setminus F$ necessarily Caratheodory measurable?

Answer: Yes


Question: Evaluate the definite integral: $$\int_{-\pi}^{\pi}\frac{e^{\sin(x)+\cos(x)}\cos(\sin(x))}{e^{\sin(x)}+e^x}dx$$

Answer: \pi


Question: Evaluate the limit using Taylor's Theorem and little-o notation: $$\lim\limits_{x\to 0} \frac{\cos{(x^2)}-\cos^2{(x)}}{\cos{(x^2)}\cos^2{(x)}}$$

Answer: 0


Question: Find the minimum value of the function \( f(x,y,z) = x^2 + y^2 + z^2 \) subject to the constraints \( x^2 + y^2 + z^2 - 5 = 0 \) and \( x^2 + y^2 + z^2 - 2x - 3 = 0 \) using the Lagrange multiplier method.

Answer: 5


Question: What is the maximum number of attracting and neutral orbits for a complex polynomial of degree $d$?

Answer: 2d - 2


Question: Determine whether Besicovitch's theorem, which states that \([\mathbb{Q}(\sqrt[n]{p_1},\ldots,\sqrt[n]{p_k}):\mathbb{Q}]=n^k\) for distinct primes \(p_1,\ldots,p_k\), holds true for even values of \(n\).

Answer: Yes


Question: Determine whether a countably infinite torsion group can have a different proportion of squares compared to non-squares, given the definition of density.

Answer: Yes


Question: Determine whether the ordinary differential equation \( \dot{x}(t) = v - AA^T x(t) \) has a unique globally asymptotically stable equilibrium given that \( AA^T \) is non-singular, where \( A \) is an \( m \times n \) matrix and \( v \) is an \( m \times 1 \) vector. Answer "true" if it does, and "false" otherwise.

Answer: true


Question: Evaluate the limit: $$\lim_{a\to0}\int\limits_{-x}^x\frac{e^t\cdot(1-\sin^2(at))}{1+e^t}dt$$

Answer: x


Question: Find the limit of the function \((x^2+y^2)^{x^2y^2}\) as \((x, y)\) approaches \((0, 0)\).

Answer: 1


Question: Suppose for any vector $y$ in $\mathbb{R}^n$, there exists a vector $x$ such that the linear system $Ax = y$ is satisfied, where $A$ is an $n \times n$ real matrix. Can we conclude that the rank of $A$ is $n$?

Answer: Yes


Question: For a prime number \( p \), \( p \) people wish to sit at a round table for several nights such that each person has different neighbors every night. Determine the maximum number of nights they can achieve this arrangement.

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


Question: Fred receives emails while he sleeps. Let X be the number of legitimate emails and Y be the number of spam emails he receives. Suppose X and Y are independent, with X following a Poisson distribution with parameter 10, and Y following a Poisson distribution with parameter 40. When he wakes up, he observes that he has 30 new emails in his inbox. Given this information, what is the expected value of the number of legitimate emails he received?

Answer: 6


Question: In a simplicial triangulation, must all sub-triangles share a vertex with the main outer triangle? Answer yes or no.

Answer: No


Question: A cube has an octahedron inscribed within it by connecting the centers of the cube's faces. Another cube is then inscribed inside this octahedron by connecting the centers of the octahedron's faces. Determine the ratio of the side lengths of the inner cube to the outer cube.

Answer: \dfrac{1}{3}


Question: Let $S$ be an $n$-dimensional convex set in $\mathbb{R}^n$. Consider the set $X$ of points on the boundary of $S$ where there is more than one supporting hyperplane. Determine whether $X$ has zero $(n-1)$-dimensional Lebesgue measure.

Answer: Yes


Question: Evaluate the limit \(\lim\limits_{x \to \infty}(f(x+1) - f(x))\) using Taylor expansions, given that \(f(x)\) satisfies the equation \(f'(x) = x^{a-1}\) for \(a \in (0,1)\).

Answer: 0


Question: Given a measurable function $v_1:(0,1)\rightarrow \mathbb{R}^2$ such that $v_1(x)\neq 0$ for all $x \in (0,1)$, determine if it is possible to construct another measurable function $v_2:(0,1)\rightarrow \mathbb{R}^2$ such that $\{v_1(x),v_2(x)\}$ forms a basis of $\mathbb{R}^2$ for each $x \in (0,1)$. Answer "Yes" or "No".

Answer: Yes


Question: Determine the number of non-real roots of the polynomial \(x^7 - 10x^5 + 15x + 5\).

Answer: 2


Question: Evaluate the surface integral \(\iint \text{curl} \ \vec{F} \cdot d\vec{S}\) using Stokes' theorem, where \(\vec{F}=[-z, 5x, -y]\) and the surface is the portion of the plane \(z=x+2\) that lies inside the ellipse \(x^2+y^2=4\).

Answer: 24\pi


Question: Evaluate the limit: $$\lim_{n\to\infty}\left(\frac{2n^3-n^2+3}{3n^4-2n^3+n-5}\right)^{\frac{n^2-2n+1}{n^3-n}}$$

Answer: 1


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

Answer: 1


Question: Determine whether the following conjecture is true: \( \sum_{n=1}^k\frac{1}{\pi^{1/n}p_n}\stackrel{k\to\infty}{\longrightarrow}2 \), where \( p_n \) denotes the \( n \)th prime.

Answer: False


Question: Find a connected graph $G$ with $n$ nodes, where $n \ge 3$, such that $G$ is isomorphic to its line graph $L(G)$. Provide an example of such a graph.

Answer: C_3


Question: In a circle with a radius of 3 units, secants ABC and ADE intersect at point A, which is 5 units from the center of the circle. The angle between secants ABC and ADE is 30 degrees, and the area of triangle ACE is 10 square units. Determine the area of triangle ADB.

Answer: \dfrac{8}{5}


Question:
Find the analytical steady-state solution to Laplace's equation inside a square domain with the following conditions:

Laplace's equation:
$$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$$

Domain: $-L<x<L, -L<y<L$

Boundary conditions: $u(L,y)=u(-L,y)=u(x,L)=u(x,-L)=0$

Additional condition: $u(0,0)=1$

Answer: 0


Question: Find the number of distinct prime values that the polynomial \( n^3 - 10n^2 - 84n + 840 \) takes when \( n \) is an integer.

Answer: 2


Question: Determine the range of values for the parameter $a$ such that the inequality \( x^{-3} e^{x} - a \ln x \geq x + 1 \) holds for all \( x \in (1, +\infty) \).

Answer: (-\infty, -3]


Question: Evaluate the integral \( \int_{[0,1]\times[0,1]} \max\{x, y\} \, dx \, dy \).

Answer: \dfrac{2}{3}


Question: Let $f: \mathbb{R}^n \to \mathbb{R}$ be a convex function. Given $u \in \mathbb{R}^n$ and $v \in \mathbb{R}^n \setminus \{0\}$, assume that the directional derivative of $f$ at $u$ in the direction of $v$ is 0. Does this imply that $f$ has a global minimum at $u$? Answer yes or no.

Answer: No


Question: Determine the dimension of the null space of the linear transformation $T: P_{n} \rightarrow P_{n-1}$ given by $T(p(x)) = p'(x) - \int_{0}^{x} p(t) dt$, where $P_{n}$ is the vector space of all polynomials with real coefficients of degree at most $n$. Provide your answer as a single integer.

Answer: 0


Question: Determine the number of maximal ideals in the quotient ring \( \mathbb{R}[X,Y]/(X^2+Y^2-1) \).

Answer: \infty


Question: Determine whether the following statement is true or false: If $X$ is a topological space and $Y \subseteq X$ is homeomorphic to $X$ and $Y$ is countable, then $X = Y$. Provide a proof or counterexample.

Answer: False


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

Answer: \sqrt{e}


Question: Given two $3\times3$ rotation matrices $A$ and $B$, determine the rotation matrix $X$ such that $AX=XB$.

Answer: X


Question: Determine the number of elements in the set $D_n = \{ a^ib^j \mid \text{ order}(a)=n, \text{ order}(b)=2, a^ib = ba^{-i} \}$ that have order $n$.

Answer: \phi(n)


Question: Evaluate the contour integral \( \int_c \frac{z}{2z^2+1}dz \) where the contour \( c \) is the unit circle.

Answer: \pi i


Question: Find the fractional part of \( \frac{5^{200}}{8} \) using the Binomial Theorem.

Answer: \dfrac{1}{8}


Question: Determine the highest power of 3 that divides the sum of the factorials from $27!$ to $75!$. Express your answer as an integer.

Answer: 13


Question: Determine the range of values for |z|, where z is a complex number satisfying the inequality \(||z|-|\frac{3}{z}|| \leq 2\).

Answer: [1, 3]


Question: Calculate the limit \(\lim\limits_{n\to\infty} \frac{\log(n!)}{n\log(n)}\) when \(n\) is a positive integer.

Answer: 1


Question: Determine whether the statement is true or false: If a function $f$ is differentiable on the set of real numbers $\mathbb{R}$, then the composition $(f \circ f)$ is also differentiable on $\mathbb{R}$. Provide a proof or a counterexample to support your answer.

Answer: True


Question: Let the sequence $\{a_n\}$ satisfy \(a_1=1,a_{n+1}=a_n+[\sqrt{a_n}]\quad(n\geq1)\), where \([x]\) is the integer part of \(x\). Find the limit \(\lim\limits_{n\to\infty}\frac{a_n}{n^2}\).

Answer: \dfrac{1}{4}


Question: Find the missing digit in the number $3^{37}$ when expressed in base 10 using modular arithmetic.

Answer: 1


Question: Let $f:\mathbb{R}^{3}\to\mathbb{R}^{2}$ be defined by $f(r,s,t)=(r^{3}s+t^{2},rst)$. Let $g:\mathbb{R}^{2}\to\mathbb{R}^{11}$ and $h:\mathbb{R}^{11}\to\mathbb{R}^{4}$ be differentiable functions. Compute the Jacobian matrix $J_{h\circ g\circ f}(0,21,0)$. What is the size of the resulting Jacobian matrix?

Answer: 4 \times 3


Question: Evaluate the improper integral \(\int_0^\infty \frac{\sin x(1-\cos x)}{x^2}dx\) and determine whether it converges.

Answer: \ln 2


Question: Given a finite-dimensional vector space $V$, determine whether it is always possible to define an inner product on $V$ such that any given basis of $V$ is orthonormal. Provide a justification for your answer.

Answer: Yes


Question: Determine the number of distinct intersection points between the affine curves \( y^{3}-2 y x+1=0 \) and \( 3 x+2 y+3=0 \) over \( \mathbb{C} \).

Answer: 3


Question: Evaluate the alternating sum: $$\sum_{j=0}^{98} (-1)^j \binom{99}{j} (99-j)^{50}.$$

Answer: 0


Question: Does there exist a binary operation * on the set of non-negative integers \( \mathbb{N} \) that is associative and non-commutative?

Answer: Yes


Question: In an acute-angled triangle ABC, let D, E, and F be the feet of the altitudes from vertices A, B, and C, respectively, and let H be the orthocenter of triangle ABC. Given that the area of triangle ABC is S, determine the value of \( \frac{AH}{AD} + \frac{BH}{BE} + \frac{CH}{CF} \) in terms of S.

Answer: 2


Question: Find the remainder when \( \frac{289 \times 144^{25}}{71^{71}} \) is divided by 71.

Answer: 12


Question: Let $V,W \subset \mathbb{C}$ be open sets. Let $f\colon V \to W$ be a bijective holomorphic map. Determine if the set $Z:= \left\{v\in V \colon f'(v)=0\right\}$ is empty.

Answer: \emptyset


Question: Find the function $g:\mathbb{R}\to\mathbb{R}$ that satisfies the differential equation $x^3 g'(x) + g(x) = 2$ and the initial condition $g(0.5)=2$. Provide your answer as an explicit function $g(x)$.

Answer: 2


Question: Determine whether the spectrum of a densely defined unbounded self-adjoint operator $A$ on a Hilbert space $H$ spans an unbounded set in the real numbers. Answer true or false.

Answer: True


Question: Evaluate the line integral \( \int_{\beta} g \, ds \) where \( g(x,y) = x^3 y^3 \) and \( \beta(t) = (\sin t, \cos t) \) for \( 0 \leq t \leq 2\pi \).

Answer: 0


Question: Determine if $0$ is an element of the diagonal intersection of a family of sets $X_\alpha \subseteq \kappa$ for all $\alpha < \kappa$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right) \]

Answer: \dfrac{1}{2}


Question: Determine whether deleting a clique from a perfect graph results in another perfect graph. Answer with 'Yes' or 'No'.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to \infty}\int_{0}^{1}x^nf(x)\,dx \] where \( f(x) \) is a continuous function on the interval \([0,1]\).

Answer: 0


Question: Given that $X$ is uniformly distributed over the interval $(-1,1)$ and $Y = X^2$, find the expected value of $X$ given $Y$. Express your answer in terms of $Y$. \( \text{E}[X|Y] = ? \)

Answer: 0


Question: Determine whether it is possible to construct the fourth root of two using only a compass and straightedge. If it is possible, describe the construction process.

Answer: \sqrt[4]{2}


Question: Determine the number of solutions to the equation $e^z = 3z + 1$ within the unit disk $|z| < 1$. Provide your answer as an integer.

Answer: 1


Question: Let \(\alpha = \sqrt[16]{2}\) be a positive real number and \(K = \mathbb{Q}[\alpha]\) be the algebraic extension over \(\mathbb{Q}\) by \(\alpha\). Determine the number of intermediate fields \(F\) such that \(\mathbb{Q} \subseteq F \subseteq K\).

Answer: 5


Question: Assume we have two continuous integrable functions $f, g \in L^1(\mathbb{R})$ such that $f(x_0) \leq g(x_0)$ for some $x_0 \in \mathbb{R}$. Define the Gaussian function with variance $\sigma > 0$ as $p_\sigma(x) = \frac{1}{\sqrt{2 \pi} \sigma} e^{-\frac{x^2}{2\sigma^2}}$. Can we conclude that $f \star p_\sigma (x_0) \leq g \star p_\sigma (x_0)$?

Answer: No


Question: Evaluate the limit: $$\lim_{n\to\infty}\sqrt{n}\left(\sqrt[n]{3}-\sqrt[n]{2}\right)$$ without using L'Hôpital's rule.

Answer: 0


Question: Determine whether the sequence \( \frac{\sum_{k=1}^{n}\left(\frac{k}{k+1}\right)^k}{n} \) converges.

Answer: \dfrac{1}{e}


Question:
Let $(a_n)_{n \ge 1}$ be a sequence of positive real numbers such that the sequence $(a_{n+1}-a_n)_{n \ge 1}$ converges to a non-zero real number $L > 0$. Determine the limit:

$$ \lim_{{n \to \infty}} \left( \frac{a_{n+1}}{a_n} \right)^n $$

Answer: e


Question:
Consider a standard Borel space $X$, homeomorphic to a Borel subset of a complete separable metric space, with the Borel $
abla$-algebra $
abla(X)$ and the set of Borel probability measures $
abla(X)$ endowed with the topology of weak convergence. Let $Y$ be another standard Borel space, $f:X\to Y$ a Borel map, $\alpha\in\nabla(X)$ a probability measure, and $P:X\to\nabla(X)$ a stochastic kernel. Define the pushforward kernel $Q:X\to\nabla(Y)$ by $Q(B|x) := P(f^{-1}(B)|x)$ for all $x\in X$ and $B\in\nabla(Y)$. Determine if there exists a stochastic kernel $R:X\times Y\to\nabla(X)$ satisfying:

1. For $\alpha\otimes Q$-almost all $(x,y)\in X\times Y$, $R(f^{-1}(y)|x,y) = 1$.
2. For all $A\in\nabla(X)$ and $\alpha$-almost all $x\in X$, $P(A|x) = \int_Y R(A|x,y) \, Q(\d y|x)$.

Can this be established under the assumption of the existence of conditional expectations for general measurable spaces?

Answer: Yes


Question: Calculate \( \log_{3} (4) \mod 11 \). If 4 is not a power of 3 modulo 11, express the result as "undefined".

Answer: 4


Question:
Determine all entire functions \( f: \mathbb{C} \to \mathbb{C} \) that satisfy the following conditions for all complex numbers \( z \):
1. \( f(z+i) = f(z) \)
2. \( f(z+1) = e^{2\pi i} f(z) \)

Answer: f(z) = c


Question: Find an integer \( \lambda \neq 0 \) such that the sequence \( \lambda + \prod_{k=1}^n(\varphi(k)+1) \) contains infinitely many square-free terms.

Answer: 1


Question: Determine the fundamental group of the space \(X = \left\{\left(\sqrt{x^2+y^2}-2\right)^2 + z^2 = 1\right\} \cup \left\{(x,y,0) : x^2 + y^2 \leq 9\right\} \subset \mathbb{R}^3\).

Answer: \mathbb{Z}


Question: Determine the maximum number of elements in a subset \(X\) of \(\mathbb{R}^n\) such that the discrete metric on \(X\) is induced by the Euclidean metric in \(\mathbb{R}^n\).

Answer: n + 1


Question: Let $H$ be a nontrivial subgroup of the alternating group $\mathfrak{A}_n$ for $n \geq 5$. Determine if $H$ necessarily contains an element $\sigma$ of prime order $p$, which can be expressed as the product of disjoint $p$-cycles: $\sigma = c_1c_2...c_r$, where each $c_i$ is a $p$-cycle and the order of $\sigma$, denoted $o(\sigma)$, is the least common multiple of the cycle lengths, $l_1, l_2, ..., l_r$, satisfying $o(\sigma) = p$. Answer "yes" or "no."

Answer: yes


Question: Is every continuous image of the interval $[0,1]$ guaranteed to be locally connected?

Answer: Yes


Question: Evaluate the integral \( \int\limits_{z=-1}^{1} \int\limits_{y=-\sqrt{1-z^2}}^{y=\sqrt{1-z^2}} \int\limits_{x=-\sqrt{1-y^2}}^{x=\sqrt{1-y^2}} 1 \, dx\,dy\,dz \) to find the volume of the region in Cartesian coordinates.

Answer: \dfrac{16}{3}


Question: Find the limit of the expression \( \lim\limits_{n \to \infty} \cos^2\left(\pi \sqrt[3]{n^3+n^2+2n}\right) \) as \( n \) approaches infinity, where \( n \in \mathbb{N} \).

Answer: \dfrac{1}{4}


Question: Let \( p \) be an odd prime, and consider an abelian group \( G \) of order \( p^3 \) that is not cyclic. The \( p \)th power map \( \phi:G\to G \), defined by \( g\mapsto g^p \), is a group homomorphism. Determine the order of the image of \( \phi \), denoted as \( \operatorname{im}\phi \), given that every non-identity element of \( G \) has order \( p \) or \( p^2 \), and every non-identity element of \( \operatorname{im}\phi \) has order \( p \).

Answer: p


Question: Consider a set of \( n \) independent and identically distributed (i.i.d.) random variables \(\{X_1, X_2, \ldots, X_n\}\), each uniformly distributed on the interval \([0, k]\). Calculate the probability that the sum of these random variables is less than \( k \).

Answer: \dfrac{1}{n!}


Question: Compute the limit \( \displaystyle\lim_{x \to +\infty} f(x) \), where \( f(x) = \frac{1}{\Gamma(x)}\int_x^{+\infty}t^{x-1}e^{-t}dt \) for all \( x \in \Bbb R_+^* \).

Answer: \dfrac{1}{2}


Question: Determine the structure of all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \( f(x+y) = f(x) + f(y) + 2xy \) for all real numbers \( x \) and \( y \), given that \( f(0) = 0 \) and \( f(x+1) = f(x) + f(1) + 2x \).

Answer: f(x) = x^2 + kx


Question: Determine whether the following statement is true or false: If $A$ and $B$ are solid tori such that $S^3 = A \cup B$ and $A \cap B$ is a torus, then $S^3$ with an "unknotted circle" removed from $A$ deformation retracts to $B$. \( \text{True or False?} \)

Answer: True


Question: Evaluate the limit: $$\lim_{n\to \infty}\frac{\sum\limits_{k=2}^n k\cos\left(\frac{\pi}{k}\right)}{n^2}$$

Answer: \dfrac{1}{2}


Question: Let \( A = \{1, 2, 3, 4, 5, 6\} \). Determine the number of distinct bijective functions \( f: A \rightarrow A \) such that \( f(f(f(n))) = n \) for all \( n \in A \).

Answer: 81


Question: Find the derivative of the function \( f(x) = \int_{-\infty}^x e^{-(x-y)} \eta(y) \, dy \). What is \( \frac{d}{dx}f(x) \)?

Answer: \eta(x) - f(x)


Question: Verify the optimality of the solution obtained using the Lagrange multiplier method for minimizing the function \( f(x,y,z) = x^2 + y^2 + z^2 \) subject to the constraints \( x^2 + y^2 + z^2 = 5 \) and \( x^2 + y^2 + z^2 - 2x - 3 = 0 \). The solution provided by MATLAB is \( x = \lambda = 1 \), \( y = z = 1.4142 \), and \( \mu \approx 0 \).

Answer: 5


Question: Find the remainder when \((177 + 10^{15})^{166}\) is divided by 1003.

Answer: 234


Question: Find a primitive root of $65537$.

Answer: 3


Question: Given that $U$ and $V$ are independent standard normal variables, calculate $\text{Cov}(U^2 - V^2 \mid U > 0)$. Provide your answer as a single value.

Answer: 0


Question: Find the integer $x$ such that $x^5 \equiv 3 \pmod{221}$ using the Chinese Remainder Theorem and Euler's Theorem.

Answer: 165


Question: Determine the eigenvalues of the linear map \( \mathbf{L}:\mathbf{P_3}\rightarrow\mathbf{P_3} \) defined by \( \mathbf{L}f=f'' \), where \( \mathbf{P_3} \) is the vector space of real polynomials of degree at most three.

Answer: 0


Question: Determine whether every Lie group that is homeomorphic to an n-dimensional torus \( \mathbb{T}^n \) is necessarily abelian.

Answer: Yes


Question: Find the coefficient of degree \(-2\) in the differential form \(\frac{df}{f}\) for the formal Laurent power series \(f=\sum_{i\ge m} a^i t^i\), where \(m\in \mathbb{Z}\).

Answer: 0


Question: Evaluate the contour integral \( \int_C ze^{3/z}\,dz \), where \( C \) is the contour given by \( |z|=2 \), oriented in a counter-clockwise direction.

Answer: 9\pi i


Question: Determine the value of \( a \) such that the gradient of the function \( f(x,y) = \ln(2x^4 + ax^2y^2 + 2y^4) \) is orthogonal to the circle centered at the origin with radius \( r = \sqrt{x^2 + y^2} \) at every point \((x,y) \neq (0,0)\).

Answer: 4


Question: Find the point on the circle of intersection between the plane $2x + y + 2z = 27$ and the sphere of radius 15 centered at the origin that is closest to the point $P(33, 30, 6)$. Assume an ON-system.

Answer: (10, 11, -2)


Question: Evaluate \( x = \sqrt{3} \cos \left(\frac{1}{3} \cos^{-1}\left(\frac{1}{3 \sqrt{3}}\right)\right) \) to three decimal places without using trigonometric functions.

Answer: 1.553


Question: Determine if there exist non-zero solutions to the equation $x^2 = x$ in a ring that is not an integral domain.

Answer: Yes


Question: Calculate the expected first-passage time \( \mathbf{E}[T_{a,b}] \) for a standard Brownian motion \((B_t)_t\) with \(B_0=0\), where \(T_{a,b} = \inf\{t>0 : B_t \not\in(a,b)\}\) for \(a<0<b\).

Answer: -ab


Question: Determine the degree of the image of a hypersurface $H \subset \mathbb{P}^n$ of degree $b$ under the Veronese map $\nu_{n,d}$. Provide your answer as a single integer.

Answer: b d^{n-1}


Question: Let $X$ be a Hilbert space and $\varphi \in X' \setminus \{0\}$. Define the set \( C = \{ x \in X : \varphi(x) = 1 \} \). Find the orthogonal complement \( C^{\perp} \).

Answer: \{0\}


Question: A submarine has three navigational devices, labeled $A$, $B$, and $C$, with mean lifetimes of 1, 1.5, and 3 years, respectively. The submarine can continue operating if at least two devices are functioning. Assuming the failure times for these devices follow an exponential distribution, calculate the expected average length of time the submarine can remain at sea using the conditional expectation approach. Let $T$ represent the length of time the submarine can remain operational.

Answer: \dfrac{27}{20}


Question: Determine the values of $s$ for which the series \( \sum_{n=1}^{\infty} \frac{1}{(q_n)^s} \) converges, where \( q_n \) is the \( n \)-th prime number.

Answer: s > 1


Question: Determine whether there exist non-homeomorphic closed subsets \(A\) and \(B\) of \(\mathbb{R}^2\) such that their complements, \(\mathbb{R}^2 \setminus A\) and \(\mathbb{R}^2 \setminus B\), are homeomorphic.

Answer: Yes


Question: Determine the operator to which $T_t: \varphi(x) = \varphi(x+t)$ on $L^2(\mathbb{R})$ converges as $t \to \infty$, and specify the topology of this convergence.

Answer: 0


Question: Is there a real differentiable function whose derivative's graph is not topologically connected?

Answer: No


Question: Determine whether there exists an analytic function $f(z)$ in the annulus $A = \{z \in\mathbb{C} : 1 < |z| < 2\}$ such that $f'(z) = \frac{1}{z^2 + 1}$.

Answer: Yes


Question: Find the limit of the function \(f(r, \theta) = r^2 \arctan(1/r^2)\) as \((r, \theta)\) approaches \((0, 0)\) using polar coordinates.

Answer: 0


Question: Given a function $f(x,y)$ defined on the domain $D = [0,1] \times [0,1]$, where $f$ is continuous with respect to each variable separately, and $f$ vanishes on a dense subset of $D$, does $f$ vanish on the entire domain $D$? Answer with 'Yes' or 'No'.

Answer: Yes


Question: Determine whether the following statement is true in general for random variables $X, Y, Z$ with $X = f(Y, Z)$: \[ E[X|Y] = 0 \iff E[X|Y,Z] = 0 \]

Answer: False


Question:
Let $f(x)$ be a function such that both $f(x)$ and $f(x)g(x)$ belong to the Sobolev space $\mathcal{W}^{s+1}$ with $s \geq 1$, where the space is defined in terms of the $\mathbb{L}^2$-norm. Assume $g(x) \in \mathbb{L}^{\infty}(\mathbb{R}_+)$ with $\mathbb{R}_+ = [0, \infty)$, and for all $k \geq 0$, $\int x^k f(x) dx < \infty$ and $0 < \int f(x)g(x) dx < \infty$. Determine whether it follows that:

$$\lim_{x\rightarrow \infty} xf(x)g(x) = 0.$$

Answer: 0


Question: Calculate the limit: $$ \lim_{y\to\infty}\int_{0}^{\infty} \frac{y\cos^2(x/y)}{y+x^4} \, dx. $$

Answer: \infty


Question: Calculate the value of the infinite sum: $$\sum_{n=1}^\infty \frac{3n^2-4n+2}{n!}$$

Answer: 4e - 2


Question: Find the sum of all numbers from 1 to 25199 that are coprime to 25200, given that the sum of all numbers from 1 to 25199 is 317507400.

Answer: 72576000


Question: Determine whether it is possible for the closure of a subset $A$ of a topological space $B$ to contain no limit points of $A$, given that $B$ is not equal to $A$. Provide a justification for your answer.

Answer: Yes


Question: Suppose $f: \mathbb{Z}_p \to \mathbb{Q}_p$ is a continuous function such that for all non-negative integers $n$, $f(n) = (-1)^n$. Determine the value of the prime $p$.

Answer: 2


Question: Evaluate the integral \( \iiint_{x^2+y^2+z^2 \le z}\sqrt{x^2+y^2+z^2}\,dx\,dy\,dz \) using spherical coordinates. Provide your answer to two decimal places.

Answer: 0.31


Question: Evaluate the limit using polar coordinates: $$\lim_{(x, y)\rightarrow(0, 0)}\dfrac{3e^{-x^2 - y^2}-3}{x^2+y^2}$$

Answer: -3


Question: Calculate the line integral \( \oint_L p \, dx + q \, dy \) for the vector field \( F = [p(x,y), q(x,y)] = [y^3 + e^{x^2}, x^3 + \tan^2 y] \), where \( L \) is the positively oriented circle defined by \( x^2 + (y+2)^2 = 4 \). Use Green's theorem to evaluate the integral.

Answer: -48\pi


Question: Determine whether the limit of the sequence \( \{u_n\}_{n\ge 0} \), defined by the recurrence relation \( u_{n+1} = -\frac{1}{e} + (n+1)u_n \) with the initial condition \( u_0 = 1 - \frac{1}{e} \), can be found directly.

Answer: 0


Question: Determine whether the following statement is true or false: If a group is not cyclic, then its quotient group is also not cyclic.

Answer: False


Question: Determine the conic hull of the set \( T = \left\{(x_1,x_2) : (x_1 + 1)^2 + x_2^2 = 4 \right\} \).

Answer: \mathbb{R}^2


Question: Determine the number of functions \( h: \{1, 2, 3, \ldots, 2n+1\} \to \{-1, 1\} \) such that \( \sum_{j=1}^{2n+1} h(j) > 0 \).

Answer: 2^{2n}


Question: Consider two random variables $X_n$ and $Y_n$ defined on the same probability space. Suppose that $X_n$ converges to a constant in probability, and $Y_n$ converges to zero in probability. Determine if the total variation distance between the probability distributions of $X_n$ and $X_n + Y_n$ converges to zero.

Answer: No


Question: Let $\Omega$ be a convex open set in $\mathbb{C}^n$ and $f$ be an analytic function without zero points on $\Omega$. Can an analytic branch of $\ln f$ be defined on $\Omega$?

Answer: Yes


Question: Find the minimal polynomial of an $n \times n$ matrix $A$ with $a$ on the diagonal and $1$ on the diagonal above the actual diagonal, given that the characteristic polynomial of $A$ is $(x-a)^n$. Express the minimal polynomial in the form $(x-a)^k$, where $k \in \{0, 1, \ldots, n\}$.

Answer: (x - a)^n


Question: Evaluate the limit: $$\lim_{x\to\infty} \frac{x^2-(1-x^3)^\frac{2}{3} + x(1-x^3)^\frac{1}{3}-(1-x^3)}{x-(1-x^3)^\frac{2}{3}}$$

Answer: -\infty


Question: Given distinct integers $a$, $b$, $c$, and $d$, if the equation $(x-a)(x-b)(x-c)(x-d)=4$ has an integral root $r$, express $a+b+c+d$ in terms of $r$.

Answer: 4r


Question: Let $X$ and $Y$ be Banach spaces. Consider a sequence of invertible operators $T_n \in B(X,Y)$ such that $T_n \xrightarrow[]{\| \cdot \|} T$ as $n \to \infty$, where the convergence is with respect to the operator norm. Additionally, assume the inverses $T_n^{-1} \in B(Y,X)$ form a Cauchy sequence in $B(Y,X)$ and converge to some $S \in B(Y,X)$. Is it necessarily true that $T$ is invertible with $T^{-1} = S$?

Answer: Yes


Question: Find the minimal polynomial of the linear map $T_P: M_{2\times 2}^{\mathbb{C}} \to M_{2\times 2}^{\mathbb{C}}$ defined by $T_P(X) = P^{-1}XP$, where $P = \begin{pmatrix} i & 2 \\ -1 & -i \end{pmatrix}$ and $P^2 + 3I = 0$. Provide your answer as a polynomial.

Answer: x^2 - 1


Question: Given the limit \(\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\left(\frac{n}{n-i}\right)^{\gamma}=c\), where \(c\) is a constant, determine if for large \(n\), the sequence \(\sum_{i=1}^{n}\left(\frac{n}{n-i}\right)^{\gamma}\) is equivalent to \(n \times c\).

Answer: Yes


Question: Given that \(T_1\) is a lower \(\alpha \times 100\%\) confidence limit and \(T_2\) is an upper \(\alpha \times 100\%\) confidence limit for the parameter \(\theta\), such that \(P(T_1 < T_2) = 1\), determine the \((2\alpha - 1) \times 100\%\) confidence interval for \(\theta\). Assume \(T_1\) and \(T_2\) are not independent.

Answer: [T_1, T_2]


Question: Find the maximum number of linearly independent homogeneous quadratic forms in 4 variables over \( \mathbb{C} \) such that their intersection has a connected component that is positive dimensional and does not lie on a hyperplane.

Answer: 3


Question: Determine the number of ways to distribute 6 distinguishable objects into 4 indistinguishable boxes such that each box contains at least one object.

Answer: 65


Question: Determine the values of $p$ for which the Fourier transform $\mathcal{F}: L^p(\mathbb{T}) \rightarrow l^{p'}(\mathbb{Z})$ is a continuous and surjective mapping.

Answer: 2


Question: Let $g(x)$ be a bounded function on the interval $[0, \\infty)$. If the integral $\int_1^{\\infty} f(x)g(x) \, dx$ converges, does it imply that the integral $\int_1^{\\infty} f(x) \, dx$ also converges? Answer "yes" or "no."

Answer: no


Question: Find the limit of the sequence \( \lim_{n \to \infty}\left\{2n^2\cos\left(\frac{1}{n}\right)\right\} \cos\left(\frac{1}{n}\right) \), where \(\left\{x\right\}= x - \left\lfloor x\right\rfloor\) is the fractional part.

Answer: 0


Question: If the radius of convergence of \(\sum_{n=1}^\infty a_n z^n\) is \(R_1\) and the radius of convergence of \(\sum_{n=1}^\infty b_n z^n\) is \(R_2\), where \(R_1 < R_2\), is the radius of convergence of \(\sum_{n=1}^\infty (a_n - b_n) z^n\) equal to \(R_1\)?

Answer: R_1


Question: Calculate the value of the integral \( \int_{-\infty}^{\infty} e^{-2t} \delta'(t) \, dt \), where \( \delta(t) \) is the Dirac delta function.

Answer: 2


Question: Determine whether the statement "The derivative of $f(x)=\int_0^x[t]dt$ exists for all $x > 1$" is true or false, where $[t]$ denotes the largest integer less than or equal to $t$.

Answer: B


Question: Find the minimum value of \( \sum_{i=1}^{k} a_i^2 \) given that \( a_1 + a_2 + \cdots + a_k = A \) and \( a_i > 0 \) for all \( i \).

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


Question: Calculate the mass of a half-disc with radius $R$ centered at the origin $(0,0)$ in the upper half-plane ($y > 0$) given the density function $\rho = \frac{\rho_0y}{\sqrt{x^2+y^2}}$, where $\rho_0$ is a constant with units of $\text{kg.m}^{-3}$.

Answer: \rho_0 R^2


Question: Determine the minimum value of the expression \( \sqrt{\frac{b+c}{a}}+\sqrt{\frac{c+a}{b}}+\sqrt{\frac{a+b}{c}} \) for positive numbers \( a \), \( b \), and \( c \).

Answer: 3\sqrt{2}


Question: Determine the values of $n$ for which the binomial expansion of $(A+B)^n$ implies that $A$ and $B$ commute.

Answer: n \geq 2


Question: Evaluate the limit: $$\lim_{n\rightarrow \infty }\sum_{k=n}^{5n}\binom{k-1}{n-1}\left( \frac{1}{5} \right )^n\left( \frac{4}{5} \right )^{k-n}$$

Answer: \dfrac{1}{2}


Question: Let $D$ be a Noetherian topological space and $E$ be a closed equivalence relation on $D \times D$. If $D \times D$ has a Noetherian topology finer than the product topology with respect to closed sets, determine whether the quotient map $p: D \to D/E$ is a closed function when $D$ is not necessarily Hausdorff.

Answer: Yes


Question: Given the function $f$ where $f(1) = 2005$ and $f(1) + f(2) + \ldots + f(n) = n^2f(n)$ for all $n > 1$, determine the value of $f(2004)$. \( \boxed{} \)

Answer: \dfrac{1}{1002}


Question:
Consider an infinite sequence of smooth, compactly supported functions $\{f_n\}_{n \geq 1}$ on $\mathbb{R}$ such that each $f_n$ is orthogonal to $C$ other functions in the sequence, where $C$ is an absolute constant. Determine if there exists an absolute constant $C'$ such that the following inequality holds:
\[\left\|\sum_{n \geq 1}f_{n}\right\|_{L^{2}(\mathbb{R})} \leq C'\left(\sum_{n \geq 1}\|f_{n}\|_{L^{2}(\mathbb{R})}^{2}\right)^{1/2}\]

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to 0} \left( \lfloor n \cdot \frac{\sin x}{x}\rfloor + \lfloor n \cdot \frac{\tan x}{x} \rfloor \right) \] where \( n \in \mathbb{N} \).

Answer: 2n - 1


Question: Determine whether a basis for the vector space of $2\times2$ matrices can consist entirely of non-invertible matrices.

Answer: Yes


Question: In a topological space, given an open set $A$ and a closed set $S$, is it possible for the set $S \setminus A$ to be compact even if both $A$ and $S$ are not compact?

Answer: Yes


Question: How many onto functions exist from a set \( A \) with 6 elements to a set \( B \) with 6 elements, such that the function maps the element 1 in \( A \) to the element 2 in \( B \), and no element in \( A \) maps to itself in \( B \)?

Answer: 53


Question: Determine the value of the infinite series \( \sum_{p \in \mathbb{P}} q^p \), where \( p \) ranges over all prime numbers and \( 0 < q < 1 \). Does this series have a closed-form expression?

Answer: \sum_{p \in \mathbb{P}} q^p


Question: Determine whether the sequence \( \sqrt{3}, \sqrt{3}^{\sqrt{3}}, \sqrt{3}^{\sqrt{3}^{\sqrt{3}}}, \sqrt{3}^{\sqrt{3}^{\sqrt{3}^{\sqrt{3}}}}, \ldots \) converges. If it converges, find its limit.

Answer: \infty


Question: Determine the limit of the sequence \((a_n)\) as \(n\) approaches infinity, where \(a_n = \frac{\sqrt[3n]{4} - 6\sqrt[3n]{2} + 9}{\sqrt[2n]{9} - 4\sqrt[2n]{3} + 4}\).

Answer: 4


Question: Find the minimal polynomial of the endomorphism $\phi: V \to V$ defined by $\phi(v_1,v_2,\dots,v_n)^t = (v_n,\dots,v_2,v_1)^t,$ where $V$ is the $\mathbb R$-vector space $\mathbb R^n$. Provide your answer as a polynomial.

Answer: x^2 - 1


Question: Let $\theta = \arg{z}$, where $z$ is a complex number satisfying $|z - 3i| = 3$. Assuming $0 < \theta < \frac{\pi}{2}$, find all possible values of the expression $\cot{\theta} - \frac{6}{z}$. Determine the values of this expression.

Answer: i


Question: Given a probability measure $P$ defined as \( P(A) := \alpha \delta_0 (A) + (1-\alpha) \int_{A \cap (0, \infty)} e^{-t} dt \) for \( A \in \mathcal{B} (\mathbb{R}) \), where \( \alpha \in [0,1] \), find the expected value \( \mathbf{E} [X] \) in terms of \( \alpha \).

Answer: 1 - \alpha


Question: Determine the value of $k$ such that the limit of the integral \( I_p = \int_{1}^{p} \left(\frac{5}{2x+6} - \frac{k}{x}\right)\,dx \) exists as \( p \to \infty \).

Answer: \dfrac{5}{2}


Question: Given the functions $f(x)=2x+e^x$ and $g(x)=f^{-1}(x)$, where $(0,1)$ is on the graph of $f(x)$, find $g'(1)$.

Answer: \dfrac{1}{3}


Question:
Let $X_1, X_2, \ldots, X_n$ be a random sample of size $n \geq 2$ from an exponential distribution with parameter $\lambda$. Given $X_{(1)} = \min(X_1, X_2, \ldots, X_n)$, find the expected value of $X_{(1)}$ conditional on $T = \sum_{i=1}^n X_i$. The cumulative distribution function (CDF) of $X_{(1)}$ is given by:
$$
F_{X_{(1)}}(x) = \begin{cases}
0, & x < 0 \\
1 - e^{-n\lambda x}, & x \geq 0
\end{cases}
$$

Answer: \dfrac{T}{n^2}


Question: Determine whether a homeomorphism $f:U\rightarrow U$ of the unit disc $U$ extends to a continuous map $f:\overline{U}\rightarrow\overline{U}$ if $f$ is holomorphic.

Answer: Yes


Question: Evaluate the limit \( \lim_{n \to \infty} I_{n} \), where \( I_{n} = \int_{0}^{1} x^n e^{\sqrt{x}} \, dx \) and \( n = 0, 1, 2, \ldots \).

Answer: 0


Question: Find the limit: \[ \lim_{x \to \infty} \left(\sum_{n=1}^{\infty} \left(\frac{x}{n}\right)^{n}\right)^{\frac{1}{x}} \]

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


Question: Let $V$ be the vector space of polynomials of degree at most $99$ with real coefficients. Define a linear map $$T:V\to\mathbb{R}^{1000},\quad T(p)=(p(1),p(2),\ldots,p(1000)).$$ Find the dimension of the null space of $T$. \( \text{Provide your answer as a single integer.} \)

Answer: 0


Question: Solve the ordinary differential equation using a substitution: $$y\dfrac{d^2y}{dx^2} - \left(\dfrac{dy}{dx}\right)^2 = 0$$

Answer: y = C_1 e^{C_2 x}


Question: Determine the asymptotic behavior of the ratio of the $n$th nice prime to the $n$th prime.

Answer: 2


Question: Let $V$ be the vector space of continuous functions from $[0,1]$ to $\mathbb{R}$, with the inner product defined as $\langle f, g\rangle := \int_0^1 f(x) g(x) \, dx$. Define the subspace $U \subset V$ as $U := \{ f \in V : f(0) = 0 \}$. Determine the orthogonal complement of $U$, denoted as $U^\perp$. Provide your answer in terms of a set of functions.

Answer: \{0\}


Question: Evaluate the limit \( \lim_{x\to0}\frac{\ln(1+x^2)-x^2+\frac{x^4}{2}}{\sin(x^6)} \) without using l'Hospital's rule.

Answer: \dfrac{1}{3}


Question: Let \( n \) and \( m \) be fixed natural numbers, and let \( N, X_1, X_2, \ldots \) be independent random variables such that \( N \sim \text{Poisson}(n) \) and \( X_k \sim \text{Poisson}(m) \) for all \( k \). Define \( S_N = \sum_{k=1}^N X_k \) with the assumption that \( S_0=0 \). Find \( E(S_N) \).

Answer: mn


Question: Let $\{f_n\}$ be a sequence of bounded real-valued functions on $[0,1]$ converging at all points of this interval. If $\int^1_0 |f_n(t)-f(t)|dt\, \to 0$ as $n \to \infty$, does $\lim_{n \to \infty} \int^1_0 f_n(t) dt\,=\int^1_0 f(t)dt\,$? Answer yes or no and justify your answer.

Answer: Yes


Question: Differentiate the expression $x^TAx$ with respect to the vector $x$, where $x$ is a $3 \times 1$ vector and $A$ is a $3 \times 3$ matrix.

Answer: (A + A^T)x


Question: Evaluate the limit: $$\lim_{x\to-\infty}\frac{\ln(f^{-1}(x))}{x}$$ where $f(x)=e^x+x^3-x^2+x$. Provide your answer as a single value.

Answer: 0


Question: Calculate the covariance of $X(\omega)$ and $Y(\omega)$, where $X(\omega)=\sum_{i=1}^{n} \chi_{i}(\omega)$ and $Y(\omega)=\sum_{j=1}^{m} \psi_{j}(\omega)$. Assume that $\chi_{i}(\omega)$ and $\psi_{j}(\omega)$ are independent random variables uniformly distributed over the sets $\{1,...,n\}$ and $\{1,...,m\}$, respectively.

Answer: 0


Question: Find the positive value of \( t \) such that the derivative \( \frac{dR}{dt} = 27t^{0.5} e^{-3t} - 54t^{1.5} e^{-3t} \) equals zero.

Answer: \dfrac{1}{2}


Question: Determine the cardinality of the set \( T = \{ X \in P(\mathbb{Q}) \mid X \cup \mathbb{N} = \mathbb{Q} \} \).

Answer: 2^{\aleph_0}


Question: Evaluate the limit using Taylor expansion: $$\lim_{x\to0^+} \frac{\log(\vert x \vert (1+x^2)^{1/3}-\sin{x})}{\log(\vert x \vert)}$$

Answer: 3


Question: In square ABCD with side length 10, a semicircle is inscribed with AB as its diameter. A tangent from point C on the square's edge intersects the semicircle at point F. Point E is where side BD intersects the semicircle. Find the area of triangle BEF.

Answer: 10


Question: Consider the function $F(X)=(C-1/d X_t)^d$. Using Ito's Lemma, derive an expression for $dF$ in terms of $dX$ and find the value of $d$.

Answer: 1


Question: Given that the volume of a regular tetrahedron is \( V=\frac{1}{12}t^3\sqrt{2} \), where \( t \) is the length of each side, find the rate of increase of the volume when the surface area is \( 192\sqrt{3} \text{ cm}^2 \) and is increasing at a rate of \( 24\sqrt{3} \text{ cm}^2/\text{s} \).

Answer: 24\sqrt{6}


Question: Solve the equation \( \ln (x+1)=\frac{x}{4-x} \) for the exact value of \( x \).

Answer: 0


Question: Are all connected components of a topological space \((X, \mathcal{T})\) open if the clopen subsets form a topology?

Answer: Yes


Question: Find the sum of the infinite series \(\sum_{k=0}^{\infty}\frac{(k+1)(k+3)(-1)^k}{3^k}\) using the hint: \(\Sigma_{k=0}^\infty x^k = \frac{1}{1-x}\) for \(|x|<1\).

Answer: \dfrac{45}{32}


Question: Evaluate the limit \(\lim_{k\rightarrow\infty}k\int_{[1-\frac{1}{k}, 1]}f(x) \text{d}x\) for a continuous function \(f\) with compact support in \([0, 2]\).

Answer: f(1)


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

Answer: \dfrac{1}{e}


Question: Find the derivative of the inverse function of \( f(x) = \frac{2x^3 + 7x}{x^2 + 4} \) at \( y_0 = \frac{-9}{5} \) using the theorem \((Df^{-1})(y) = \frac{1}{f'(x)}\).

Answer: \dfrac{25}{47}


Question: Given that $\left|z+2i\right|\le3$ and $z_{2}=2-5i$, find the minimum value of $\left|z+iz_{2}\right|$. Express your answer as a single number.

Answer: 2


Question: Determine whether the following statement is true or false: If there is a bijection between an infinite set $A$ and the Cartesian product $A \times A$, then there exists an injective function from the set of natural numbers $\mathbb{N}$ to $A$. Provide a justification for your answer.

Answer: True


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

Answer: 0


Question: Determine whether the following statement is true or false: If a $k$-regular graph is $1$-factorable, then it has chromatic index $k$. Provide a justification for your answer.

Answer: True


Question: Determine the degree of the splitting field of the polynomial \( p(x) = x^5 - 3x^3 + x^2 - 3 \) over \( \mathbb{Q} \).

Answer: 4


Question: Evaluate the integral \( \oint_{|z| = 1} \frac{z+\bar{z}}{(2z-i)^2} \, dz \) using the Residue theorem.

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


Question: Given that the roots of the quadratic equation \(ax^2 + bx + c = 0\) are \(\frac{\lambda}{\lambda - 1}\) and \(\frac{\lambda + 1}{\lambda}\), find the value of \((a+b+c)^2\) in terms of \(a\), \(b\), and \(c\).

Answer: b^2 - 4ac


Question: Determine whether a function can be bounded on the interior of the closure of a bounded open set, even if the set is disconnected.

Answer: Yes


Question: Evaluate the integral \( \int_{-\infty}^\infty \left(\frac{\cos x}{x}\right)^2dx \) using Fourier transforms and Parseval's Theorem.

Answer: \pi


Question: Determine whether there exists a bijection between the set of all binary strings and the set of real numbers between 0 and 1.

Answer: Yes


Question: Determine the order of growth of the sum \(\sum_{k=1}^n\frac{1}{\sqrt{k+2}}\) as \(n\) approaches infinity.

Answer: \Theta(\sqrt{n})


Question:
Verify if the limits of integration for the volume of the solid \( W = \{ (x,y,z) \in \mathbb{R}^3 : x^2+y^2 \leq z \leq \sqrt{x^2 + y^2} \} \) in cylindrical coordinates are correct. The integral is given by:
\[ V = \int _{\theta = 0} ^{2\pi} \int _{r = 0} ^1 \int_{z=r^2 }^r r \ dzdrd\theta \]

Answer: V = \frac{\pi}{6}


Question: Solve the equation $x = 2^{-18} \mod 143$ without using Fermat's little theorem or the Euclidean algorithm to find the inverse of $2^{18} \mod 143$. What is the value of $x$?

Answer: 103


Question: If a set is not bounded, does there exist a sequence contained in the set that is also unbounded? Answer yes or no and justify your answer.

Answer: Yes


Question: Let $f(z)$ be an analytic function in the domain $D = \{z \in \mathbb{C} : |z| < 1\}$, such that $f(z) = 1$ for $z$ with $Im(z) = 0$ and $-\frac{1}{2} \leq Re(z) \leq \frac{1}{2}$. Determine the value of $f(\frac{1}{2}+i\frac{1}{2})$. \( \boxed{1} \)

Answer: 1


Question: Determine the conditions on the real numbers $a$ and $b$ such that there exist real numbers $c$ and $d$ for which the polynomial $ax^3 + bx^2 + cx + d$ has three distinct real roots.

Answer: a \neq 0


Question: Evaluate the integral \( \int_{\Omega} f\, dA \) for a holomorphic function \( f(z) \) over the unit disk \( \Omega=\{|z| < 1\} \), and determine its relation to \( f(0) \).

Answer: \pi f(0)


Question: Find a function $y=f(x)$ such that the arc length of the curve from $x=0$ to $x=a$ is equal to $\sqrt{2}a$ for any constant $a$. Describe the function $y=f(x)$ that satisfies the condition: $$\int_{0}^{a}\sqrt{1+\left(f'(x)\right)^2}dx=\sqrt{2}a.$$

Answer: y = \pm x + C


Question: In a locally convex topological vector space $X$ over the real or complex numbers, let $A$ and $B$ be two non-empty, disjoint, and convex subsets of $X$ with $A$ compact and $B$ closed. Is it possible to find a continuous linear functional $g$ on $X$ such that $$ \sup_{x\in B} \Re g (x) < \inf_{x \in A} \Re g (x)?$$

Answer: Yes


Question: Evaluate the sum \( \sum_{n=-\infty}^{\infty} \text{sinc}(\pi(2t-n)) \).

Answer: 1


Question: Evaluate the limit \( \lim_{n \rightarrow \infty} \frac{\log n}{n} \int_2^n \frac{1}{\log x} \, dx \) using L'Hôpital's rule.

Answer: 1


Question: Is the closed upper hemisphere of the 2-sphere, denoted as $M$, diffeomorphic to a closed disk in $\mathbb{R}^2$?

Answer: Yes


Question: Given that $x, y, z > 0$ and $x^2 + 7y^2 + 16z^2 = 1$, determine the maximum value of $xy + yz + zx$.

Answer: \dfrac{1}{4}


Question: Evaluate the integral \( \int_C \omega \) where \( \omega = xy^2dy - x^2ydx \) and \( C \) is the circle defined by \( x^2 + (y-1)^2 = 1 \). Use the parametrization \( \alpha(t) = (\cos(t), \sin(t) + 1) \) for \( C \). Verify the result using Green's theorem and compare it with the direct evaluation of the integral \( \int_0^{2\pi} \cos^2(t)(2\sin^2(t) + 3\sin(t) + 1)dt \). Determine the correct value of the integral.

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


Question: In a metric space $(X, d)$, let $B_d(x, r)$ be the open ball centered at $x$ with radius $r$. Given that $G_1$ is an open dense subset of $X$ and $B_d(x, r) \cap G_1$ is non-empty, with $x_1 \in B_d(x, r) \cap G_1$, determine if there exists a positive radius $r_2 > 0$ such that $B_d(x_1, r_2) \subset G_1$ and $B_d(x_1, r_2) \subset B_d(x, r)$. Provide a justification for your answer.

Answer: Yes


Question:
Consider an $n \times n$ real matrix $A = (a_{ij})$ with the following properties:
1. $a_{ii} > 0$ for all $i = 1, 2, \ldots, n$.
2. $a_{ij} < 0$ for all $i \neq j$.
3. $a_{ii} > -\sum_{i \neq j} a_{ij}$ for all $i = 1, 2, \ldots, n$.

Determine whether there exists an eigenvalue $\lambda$ of $A$ such that $\operatorname{Re}(\lambda) \geq |\operatorname{Im}(\lambda)|$. Provide a justification for your answer.

Answer: Yes


Question: Let $A$ be an $n \times n$ matrix over the field $\mathbb{C}$ and $X \in \mathbb{C}^{n}$. Let $k$ be the least positive integer such that the set of vectors $\{X, AX, A^{2} X, \dots , A^{k} X \}$ are linearly dependent. Consider the relation $\sum_{i=1}^{k} c_{i} A^{i} X = 0$ and the polynomial $g(t) = \sum_{i=1}^{k} c_{i} t^{i}$. Is it true that each root of the equation $g(t) = 0$ is an eigenvalue of $A$ corresponding to an eigenvector in the span of $\{X, AX, A^{2} X, \dots , A^{k-1} X \}$?

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to \infty} x^2\left(\arctan(x)-\frac{\pi}{2}\right)+x$$ as $x$ approaches infinity.

Answer: 0


Question: Determine the smallest positive integer $t$ such that $113^t \equiv 1 \pmod{455}$, using basic properties of modular arithmetic and divisibility.

Answer: 12


Question: Determine whether every function $f$ such that $f \geq 0$ and $\int_\mathbb{R} f(t) \, dt = 1$ is a probability density function of some distribution function.

Answer: Yes


Question: Given that \(\lim_{x\to \infty}xf(x^2+1) = 2\), find \(\lim_{x\to 0}\dfrac{2f'(1/x)}{x\sqrt{x}}\).

Answer: -2


Question: Let $F$ be a field and consider the field extensions $L = F(xz, yz, xy)$ and $K = F(x, y, z)$. Determine if the degree $[K:L]$ of the extension $K$ over $L$ is finite, and if so, compute its value.

Answer: 2


Question: Given the equations \(f(x+y)=f(x)f(y)-g(x)g(y)\) and \(g(x+y)=f(x)g(y)+f(y)g(x)\), calculate the derivative of \((f^2(x)+g^2(x))\).

Answer: 0


Question: Let $f: X \rightarrow Y$ be an open, continuous map and let $X' \subseteq X$ be a closed subspace such that the restriction $f|_{X'}: X' \rightarrow f(X')$ is a bijection. Does this imply that $f|_{X'}$ is proper?

Answer: Yes


Question: Simplify the expression: $$\sqrt{\dfrac{\sqrt[4]{x^3}-8}{\sqrt[4]{x}-2}+2\sqrt[4]{x}}\left(\dfrac{\sqrt[4]{x^3}+8}{\sqrt[4]{x}+2}-\sqrt{x}\right).$$

Answer: 8 - 2\sqrt{x}


Question: If $f: E \to \mathfrak{M}$ (where $\mathfrak{M}$ is the Lebesgue measurable sets) is continuous almost everywhere, is $f$ necessarily Lebesgue measurable? Answer true or false.

Answer: A


Question: Find the limit of the sequence \(a_n = n\left(\sqrt[n]{ea} - \sqrt[n]{a}\right)\) as \(n\) approaches infinity, given that \(a > 0\).

Answer: 1


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

Answer: -a


Question: Evaluate the integral \( \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(5x^2+6xy+5y^2)}dxdy \) using the hint that \( \int_{-\infty}^{\infty}e^{-ax^2}dx=\sqrt{\frac{\pi}{a}} \).

Answer: \dfrac{\pi}{4}


Question: Does there exist a finite topological space with a fundamental group isomorphic to the infinite cyclic group $\mathbb{Z}_2$?

Answer: Yes


Question: Is the set $C^\infty_c((0,1])$ of compactly supported smooth functions on the interval $(0,1]$ dense in the Sobolev space $H^1(0,1)$, restricted to functions that vanish at the origin, i.e., $\{u \in H^1(0,1): u(0) = 0\}$?

Answer: Yes


Question: Find the sum of the infinite series \( \sum_{k=0}^{\infty} \frac{k^2}{2^k} \).

Answer: 6


Question: Find a polynomial \( f(x) \in \mathbb{Q}[x] \) of degree less than 4 such that \( f(x) \equiv 1 \pmod{(x-1)^2} \) and \( f(x) \equiv x \pmod{x^2} \).

Answer: -x^3 + x^2 + x


Question: Determine whether it is always true that for two Sylow p-subgroups H and K of a group G, the product HK equals KH.

Answer: False


Question: Determine the number of elements of order \( p \) in the group \( \text{GL}(2, \mathbb{Z}_p) \).

Answer: p^2 - 1


Question: Determine the values of \( \alpha \in \mathbb{R} \) for which the double integral \( \iint_{\mathbb{R}^2} \frac{\sin(x^2+y^2)}{(x^2+y^2+1)^\alpha} \, dx \, dy \) converges.

Answer: \alpha > 0


Question: Find the number of ring homomorphisms from the ring \( \mathbb{Z}_{12} \) to the ring \( \mathbb{Z}_{24} \).

Answer: 2


Question: Among all non-negative real numbers $a, b, c, d, e, f, g$ such that $a + b + c + d + e + f + g = 1$, determine the minimum possible value of $M = \max \{a+b+c, b+c+d, c+d+e, d+e+f, e+f+g\}$.

Answer: \dfrac{1}{3}


Question: Compute Tor$_1(\mathbb{Q} /\mathbb{Z}, \mathbb{Q} /\mathbb{Z})$. Provide your answer as a group.

Answer: \mathbb{Q}/\mathbb{Z}


Question: If $f(x)$ is a continuous function on the interval $[a,b]$ such that for every continuous function $g(x)$ defined on $[a,b]$, \(\int_a^b f(x)g(x)dx = 0,\) does it follow that $f(x) \equiv 0$ on $[a,b]$? Answer with 'Yes' or 'No' and justify your answer.

Answer: Yes


Question: Determine whether the derivative of the function \( f(x) \), defined as \( f(x) = \tan(x) \) for all rational \( x \) and \( f(x) = x \) for all irrational \( x \), is continuous at \( x=0 \).

Answer: No


Question: In the linear model \( Y_i = \beta + U_i \) for \( i = 1, 2, 3 \), with conditions \( E(U_1) = E(U_2) = E(U_3) = 0 \), \( \text{Var}(U_1) = 1 \), \( \text{Var}(U_2) = 2 \), and \( \text{Var}(U_3) = 4 \), determine the best linear unbiased estimate (BLUE) of the variance of \( \beta \).

Answer: \dfrac{4}{7}


Question: Find a seven-digit number where each digit represents the count of that digit within the number. For example, in the number 21200, there are 2 zeros, 1 one, 2 twos, 0 threes, and 0 fours.

Answer: 3211000


Question: Determine whether a holomorphic function $f: \mathbb{C} \rightarrow \mathbb{C}$ with $\lim_{z \to \infty} f(z) = 0$ is necessarily bounded.

Answer: Yes


Question: Given a random sample \( X_1, X_2, \ldots, X_n \) from a population with density function \( f(x) = \begin{cases} e^{\theta-x}, & x \geq \theta \\ 0, & \text{otherwise} \end{cases} \), find the maximum likelihood estimator for \( \theta \).

Answer: \min\{X_1, X_2, \ldots, X_n\}


Question: Evaluate the integral \( \int_0^\infty \int_0^\infty \exp(-x^2 - y^2) \, dx \, dy \) using polar coordinates.

Answer: \dfrac{\pi}{4}


Question: Let $G$ be a group with a subgroup $H$ such that $ \frac{H} {Z(G)\cap H} \cong Z_3 $ and $ [G:H] = n$. If $h \in H - (Z(G) \cap H)$, find the minimum possible value of $|C_G(h)|$.

Answer: 3


Question: Consider two functions $f(t)$ and $g(t)$ where $f(t)$ approaches infinity and $g(t)$ approaches negative infinity as $t$ approaches some value $t_0$. If $g(t) = o(f(t))$, can we conclude that $e^{g(t)} = o(e^{f(t)})$?

Answer: Yes


Question: Determine an angle $a$ for the rotation of axes that transforms the equation $x^2 + xy + 2y^2 = 4$ into a conic form without an $XY$ term, and results in exact coefficients for $X^2$ and $Y^2$. Use the transformations $x = X\cos(a) - Y\sin(a)$ and $y = X\sin(a) + Y\cos(a)$.

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


Question: Evaluate the limit: $$\lim_{(x,y)\to (0,0)} \frac{\arctan(x+y)}{\arccos(x+2y)}$$

Answer: 0


Question:
Determine the number of functions \( f: A \rightarrow A \) where \( A = \{1, 2, \ldots, 2014\} \), satisfying the following conditions:
1. There exists a \( k \in A \) such that \( f \) is non-decreasing on \( \{1, 2, \ldots, k\} \) and non-increasing on \( \{k, k+1, \ldots, 2014\} \).
2. The difference \(|f(n+1) - f(n)|\) is at most 1 for all \( n = 1, 2, \ldots, 2013 \).
3. \( f(1) = f(2014) = 1 \).

Answer: 2^{2012}


Question: Evaluate the limit: $$\lim_{n\rightarrow \infty}\frac{\frac{n}{1}+\frac{n-1}{2}+\frac{n-3}{3}+\cdots+\frac{2}{n-1}+\frac{1}{n}}{\ln(n!)}$$

Answer: 1


Question: Evaluate the limit \( \lim_{{(x, y) \to (0,0)}} \arctan(x - y)(x + y) \) and prove that it equals zero using an epsilon-delta argument.

Answer: 0


Question: Let $f$ be a positive, even, $2\pi$ periodic, and continuous function on $\mathbb{R}$. Define $a_n = \int_0^{2\pi} f(x) \cos(2nx) \, dx$. Determine whether $a_n \geq 0$ for all $n \in \mathbb{Z}$.

Answer: No