Question: Compute the limit: $$\lim _{x\to 0}\:\frac{\left(\sqrt{1+\frac{1}{x^2(1+x)}}-1\right)\ln \frac1x}{\left(\frac1{x^x}-1\right)+\sqrt{x}\ln^2\frac1x}$$ without using L'Hopital's rule.

Answer: \infty


Question: Determine whether the following statement is true or false: If two norms on a vector space are equivalent, then the unit balls in these norms are equal.

Answer: B


Question: Evaluate the contour integral \( \int_\gamma \frac{1}{1+z^2} \,\mathrm{d}z \) where \( \gamma \) is the circle of radius 1 centered at \( i \).

Answer: \pi


Question: If \( f(x) = (2x-3 \pi)^5 + \frac{4}{3}x + \cos(x) \) and \( g \) is the inverse function of \( f \), find the value of \( g'(2 \pi) \).

Answer: \dfrac{3}{7}


Question: Determine whether a non-Hermitian matrix $X$, where $X^\dagger \ne X$, can be expressed as a linear combination of Hermitian matrices.

Answer: Yes


Question: Evaluate the limit \( \lim_{n \to \infty} \left(\cos\left(\frac{3\pi n}{4n+2}\right)\right)^n \) rigorously.

Answer: 0


Question: Determine whether every sequence of coefficients, where the $i$th coefficient is twice the $(i-1)$th coefficient plus or minus 1, necessarily contains a term divisible by the length of the sequence minus 1.

Answer: No


Question: Given that $a$, $b$, and $c$ are positive numbers such that $a + b + c = 5$, find the maximum value of the expression $a^2b + b^2c + c^2a$ without using calculus.

Answer: \dfrac{500}{27}


Question: Find the length of the curve defined by the parametric equations \( x = \sqrt{2}t,\ y = e^t,\ z = e^{-t} \) for \( 0 \leq t \leq 1 \).

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


Question: Let \( m = 2023^2 + 2^{2023} \). Determine the last digit of \( m^2 + 2^m \).

Answer: 1


Question: Given the sequence \((x_n)_{n \ge 0}\) of real numbers with \(x_0 > 0\) defined by \(x_{n+1} = x_n + \frac{1}{\sqrt{x_n}}\), determine the limit \(L = \lim_{n \rightarrow \infty} \frac{x_n^3}{n^2}\) and establish its existence.

Answer: \dfrac{9}{4}


Question: Evaluate the series \( \displaystyle \sum^{n}_{k=0}(-4)^k\binom{n+k}{2k} \).

Answer: (-1)^n (2n + 1)


Question: Let $V$ be a square real matrix with rank 1 that is positive semidefinite. Determine if $V$ can be expressed as $V = vv^T$, where $v$ is a column vector in $\mathbb{R}^n$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} g^{-1} (x) \, dx \) where the function \( g: \mathbb{R} \to \mathbb{R} \) is defined by \( g(x) = \sin x + x \).

Answer: 0


Question: Determine the type of singularity and find the residue of the function \( f(z)=\frac{e^{z}+1}{e^{z}-1} \) at \( z=0 \), using its Maclaurin series expansion.

Answer: 2


Question: Determine the radius of convergence for the series \( \sum _{{k=0}}^{\infty }{\frac {x^{k}}{p_{k+1}\#}} \), where \( p_k\# \) is the product of the first \( k \) primes.

Answer: \infty


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

Answer: \dfrac{1}{e}


Question: Determine the number of real solutions for the equation: \[ \begin{vmatrix}x^2-12&-18&-5\\10&x^2+2&1\\-2&12&x^2\end{vmatrix}=0 \]

Answer: 2


Question: Does the condition that there are no points $x \in \mathbb{Q}$ such that $f(x) \in \mathbb{R} \setminus \mathbb{Q}$ and no points $x \in \mathbb{R} \setminus \mathbb{Q}$ such that $f(x) \in \mathbb{Q}$ imply that $f$ is a linear function?

Answer: No


Question: Evaluate the integral \(\int_{-\infty}^{\infty}\frac{x^3}{(x^2+4)(x^2+1)}\;dx\) using complex analysis and the residue theorem.

Answer: 0


Question: Determine all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(x) = f(\sin(x)) \) for every real number \( x \).

Answer: f(x) = c


Question: Calculate the volume of the solid under the surface \( z = \sqrt{x^2 - 2xy + y^2} \) within the rectangular region \( R = \{(x, y) | 0 \leq x \leq 1, 0 \leq y \leq 1\} \).

Answer: \dfrac{1}{3}


Question: Let $x$ be a complex number such that $x^N = 1$ and $x^3 \neq 1$. Determine whether the following statement is true: \( \sum_{n=0}^{N-1} x^{3n} = 0 \).

Answer: True


Question: What is the value of $d(dt)$ in the context of standard calculus?

Answer: 0


Question: Determine whether the localization of a cogenerator module with respect to a multiplicative set is always a cogenerator module over the localized ring. Answer true or false.

Answer: False


Question: Determine the value of \( \limsup_{n \to \infty} \frac{\omega(n)}{\log n} \), where \( \omega(n) \) is the number of distinct prime factors of \( n \).

Answer: 0


Question: Calculate the expected value of the points in the set \( P \) as \( N \to \infty \), where \( P = \{ x_i \mid x_i > 2 \} \) and \( x_i \) are independently generated from a normal distribution with mean 0 and standard deviation \( \sigma = 2 \).

Answer: 3.05


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

Answer: \dfrac{\pi}{3}


Question: Given the sequence \(x_n\) in \(\mathbb{R}\) defined by the recurrence relation \(x_{n+1} = 1 - \sqrt{1 - x_n}\) with \(x_1 < 1\), determine if the ratio \(\frac{x_{n+1}}{x_n}\) converges as \(n\) approaches infinity, knowing that \(\lim_{n \to \infty} x_n = 0\).

Answer: \dfrac{1}{2}


Question:
Find the value of \( a \) such that the following limit holds:
\[ \lim_{n\to\infty} \frac{1^a + 2^a + 3^a + \cdots + n^a}{(n+1)^{a-1}\left((na+1) + (na+2) + \cdots + (na+n)\right)} = \frac{1}{60}. \]

Answer: 7


Question: Determine the value of $A$ such that the function $f(x, y) = Axe^{-x(y+1)^2}$ for $x > 0$ and $y > 0$, and $0$ otherwise, satisfies the condition: $$\int_{0}^{\infty}\int_{0}^{\infty}Axe^{-x\left(y+1\right)^{2}}dxdy=1.$$

Answer: 3


Question: Determine whether an $n \times n$ matrix $A$, where all entries are equal to 1 and the matrix is over a field of characteristic 0, is diagonalizable.

Answer: Yes


Question: Let \( P(x) \) be a third-degree polynomial with integer coefficients such that \( \cos(\pi/7) \) is one of its roots. Compute the value of \( \frac{P(1)}{P(-1)} \).

Answer: -\dfrac{1}{7}


Question: Find the general solution of the differential equation $y = xy' - {y'}^2$.

Answer: y = Cx - C^2


Question: Determine whether the following statement is true or false: If a ring \( R[x] \) is a Unique Factorization Domain (UFD), then the base ring \( R \) is also a UFD.

Answer: True


Question: Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that \( \lim_{x \to \infty} f(x) \) exists and for any arithmetic progression \((a_n)\), the sequence \(\bigl(f(a_n)\bigr)\) is also an arithmetic progression.

Answer: f(x) = c


Question: Calculate the conditional expectation \( E(Z|W) \) for a random vector \((W, Z)\) that is uniformly distributed over the parallelogram with vertices \((0, 0)\), \((1, 0)\), \((2, 1)\), and \((1, 1)\).

Answer: \dfrac{W}{2}


Question: Compare $17^{31}$ and $31^{17}$ using the facts that $17 = 16 + 1 = (2^4) + 1$ and $31 = (2^5) - 1$. Which is greater?

Answer: 17^{31}


Question: Let $f$ be an entire function such that $f(0) = 1$ and $|f(z) - e^z \cdot \sin(2z)| < 4$ for all $z \in \mathbb{C}$. Find the value of $f(1)$. \( \boxed{} \)

Answer: 1 + e \sin 2


Question: Given that $A$ and $P$ are $n \times n$ matrices satisfying $A^2P = AP^2$, $A^3 = P^3$, and $A^2 + P^2$ is singular, find the value of $|A^4 + P^4|$. \( \text{(Express your answer as a single number.)} \)

Answer: 0


Question: Calculate the map \( f_* : H_2(\hat{\mathbb{C}}, \mathbb{Z}) \to H_2(\hat{\mathbb{C}}, \mathbb{Z}) \) induced by the function \( f(z) = z^3 + 2z + 7 \), where \( f: \hat{\mathbb{C}} \to \hat{\mathbb{C}} \) and \( f(\infty) = \infty \).

Answer: 3


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

Answer: e


Question: Consider a fat matrix $A \in \mathbb{F}^{m\times n}$, where $m < n$. Determine if it is true that the matrix $A^tA$ has a zero eigenvalue with multiplicity at least $n-m$. Provide a justification for your answer.

Answer: True


Question: Calculate the volume of the solid enclosed by the paraboloid \(x^2 + 4z^2 + 8y = 16\) that is situated above the positive \(xz\)-plane.

Answer: 8\pi


Question: Let $T: W \rightarrow W$ be a linear operator on a finite-dimensional vector space $W$. If the minimal polynomial of $T$ on $W$ is irreducible, is it possible to decompose $W$ into a direct sum of subspaces $W = W_1 \oplus W_2 \oplus \cdots \oplus W_n$ such that each $W_i$ is a subspace of $W$ and has no non-trivial $T$-invariant subspaces? Here, non-trivial means neither $\{0\}$ nor $W_i$. Answer yes or no.

Answer: Yes


Question: Let $S$ be the ring $\mathbb{F}_2[x]/((x^2+x+1)(x^3+x+1))$, where $\mathbb{F}_2$ is the finite field of two elements. If $I$ is the ideal generated by $\overline{x}$, where $\overline{x}$ is the residue of $x$, determine the cardinality of the ring $S/I$. \( \boxed{} \)

Answer: 1


Question: Evaluate the limit: $$\lim_{n\to\infty} \frac{\int_0^{\pi/4}(\tan x)^{2n}dx}{\int_0^{\pi/4}(\tan x)^{2n+1}dx}$$

Answer: 1


Question: Evaluate the regularized improper integral \( \int_{-\infty}^{\infty} te^{-2it} e^{-\epsilon t^2} \, dt \) as \( \epsilon \to 0^+ \).

Answer: 0


Question: Find the conditional probability that a standard Gaussian random variable $x$ is greater than $0$ given that $x$ is greater than another independent standard Gaussian random variable $y$. Express your answer in terms of standard Gaussian distribution properties.

Answer: \dfrac{3}{4}


Question: Compute the 10th derivative \( f^{(10)}(0) \) of the function \( f(x) = \frac{\cos(3x^2) - 1}{x^2} \).

Answer: -3674160


Question: Evaluate the limit \( \lim_{x\to 1}\left(\frac{x^{x^{x{^{.^{.^{.}}}}}}-1}{x-1}\right)^{\displaystyle\frac{1}{\sqrt{x-1}}} \), where \(x^{x^{x{^{.^{.^{.}}}}}}\) represents an infinite tower of \(x\)'s.

Answer: 1


Question: Determine the cardinality of the set $A$, where $A$ consists of all sequences $a = (a_n)_{n \ge 1}$ of natural numbers satisfying the following conditions for all $n \ge 1$: (a) $a_n < a_{n+1}$ (strictly increasing), and (b) the limit $\lim_{n\to \infty} \frac{\#\{j \mid a_j \le n\}}{n}$ exists.

Answer: 2^{\aleph_0}


Question: Evaluate the integral \( \iint_R |x+y|\,dA \) over the region \( R \) bounded by \( x \geq 0 \), \( y \geq 0 \), and \( 2x+y \leq 4 \).

Answer: 8


Question: Let \( N \in \mathcal{B}(H) \) be a normal operator on a complex Hilbert space \( H \) such that its spectrum \( \sigma(N) \) is contained in the set \( \Gamma^+ := \{z \in \mathbb{C} \mid \text{Re}(z) > 0\} \). It is known that the closure of the numerical range \( \overline{NR(N)} \) is a subset of \( \Gamma^+ \). Determine if there exists a \( \delta > 0 \) such that for every operator \( E \) in the set \( B(N, \delta) := \{A \in \mathcal{B}(H) \mid \|N-A\| < \delta \} \), the closure of the numerical range \( \overline{NR(E)} \) is also a subset of \( \Gamma^+ \).

Answer: Yes


Question: Find the maximum value of \( \int_0^1 f^3(x) \, dx \) given that \(-1 \le f(x) \le 1\) and \(\int_0^1 f(x) \, dx = 0\).

Answer: \dfrac{1}{4}


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

Answer: 1


Question: Determine the condition under which the points \((x_1,0)\) and \((x_2,0)\) divide the segment between \((x_3,0)\) and \((x_4,0)\) harmonically. Here, \(x_1\) and \(x_2\) are the roots of the quadratic equation \(ax^2+bx+c=0\), and \(x_3\) and \(x_4\) are the roots of the quadratic equation \(Ax^2+Bx+C=0\).

Answer: 2(Ac + aC) = Bb


Question: Let \( K \) be a field and consider \( a \in K \) with \( f(x) = x^n - a \) being an irreducible polynomial. If \( m \in \mathbb{N} \) is a divisor of \( n \), and \( \alpha \) is a root of \( f(x) \) in an extension field of \( K \), determine the minimal polynomial of \( \alpha^m \).

Answer: x^{n/m} - a


Question: Calculate the area of the two regions, A1 and A2, formed by the intersection of the folium of Descartes, given by the equation \(x^3 + y^3 - 3xy = 0\), and the triangle with vertices at the origin \((0,0)\), the point \((\sqrt[3]{4}, 0)\) on the x-axis, and the point \((\sqrt[3]{4}, \sqrt[3]{2})\) where the vertical tangent intersects the curve.

Answer: \dfrac{1}{2}


Question: Let \(X\) be a Hausdorff vector space whose topology is induced by a countable family of seminorms \(\{\rho_n\}_{n\in\mathbb{N}}\). Define a new metric \(d'(x,y)\) as follows: \[d'(x,y)= \sum_{n=1}^{\infty}2^{-2n}\frac{\rho_n(x-y)}{1+\rho_n(x-y)}.\] Does this metric induce the same topology as the original metric? Provide a justification for your answer.

Answer: Yes


Question: Given that \((L^{-1}M)(L^{-1}M)^T=I\) and both \(L\) and \(M\) are invertible lower triangular matrices with positive real diagonal elements, does this imply that \(L = M\)? If so, state the relevant property.

Answer: L = M


Question: Determine the maximum value of the expression \( 72\int\limits_{0}^{y}\sqrt{x^4+(y-y^2)^2}dx \) for \( y \in [0,1] \).

Answer: 24


Question: Determine the shortest distance from the ellipsoid \(x^2 + y^2 + 4z^2 = 4\) to the plane \(x + y + z = 6\).

Answer: \sqrt{3}


Question: Find the value of $a$ such that the graphs of the functions $f(x) = a^x$ and $f^{-1}(x) = \log_a x$ touch at only one common point.

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


Question: Determine the point on the curve \(x^2 + 2y^2 = 6\) that is closest to the line \(x + y - 7 = 0\).

Answer: (2, 1)


Question: Find a non-zero element in the set \( S = \mathbb{C} \times \mathbb{C} \) that is not invertible, where the multiplication on \( S \) is defined as \((a,b) \cdot (c,d) = (ac - db, ad + bc)\).

Answer: (1, i)


Question: Find the dimension of the subspace of quadric hypersurfaces in \(\mathbb{P}^4\) that vanish at eight points in general position on a rational normal curve of degree 4.

Answer: 7


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

Answer: 1


Question: Given any $\epsilon>0$, if $$ \sum_{n=0}^\infty P [|X| > (n+1) \epsilon ]< \infty, $$ does this imply that $$ E| X| < \infty \ ?$$

Answer: Yes


Question: Determine whether there exists a function \( f \) such that \( f(f(r)\sin \theta) = f(r)\cos \theta \) for all \( \theta \).

Answer: f(r) = 0


Question: Evaluate the integral \( \int_{-\pi}^\pi \int_{-\pi}^\pi |\sin(x-y)| \, dx \, dy \) using a suitable transformation.

Answer: 8\pi


Question: In a normed linear space $X$, let $C$ be a compact convex subset. For every open set $V$ containing $C$, does there exist a convex open set $U$ such that $C \subseteq U \subseteq V$?

Answer: Yes


Question: Find all prime numbers $p$ such that $2p + 1$ is a perfect cube.

Answer: 13


Question: Solve for $b$ in the congruence equation: $44 \equiv 7^b \pmod{71}$. Provide the smallest non-negative integer solution for $b$.

Answer: 43


Question: Does there exist a non-negative infinitely differentiable function \( \phi \) with compact support in a ball \( B(x,r) \) in \( \mathbb{R}^m \) (where \( m \geq 2 \)) such that \( \phi(x) > 0 \)?

Answer: Yes


Question: If \(|x-a|\) is a positive constant and it is given that \(|x-a| < \frac{1}{n}\) for every positive integer \(n\), what can be concluded about \(x\) and \(a\)?

Answer: x = a


Question: Consider the quotient ring \(S=\mathbb{R}[x_1,x_2,\dots,x_n]/(x_1^2-x_1,x_2^2-x_2,\dots,x_n^2-x_n)\). Determine the least degree of a polynomial \(p\in S\) that satisfies the idempotent property \(p^2=tp\) modulo the ideals \((x_i)\) and \((x_i-1)\) for all \(i=1,2,\dots,n\). Assume that finding the result for \(t=1\) is sufficient, as \(p^2=tp\Rightarrow \left(\frac{p}{t}\right)^2=\frac{p}{t}\Rightarrow q^2=q\text{ for some }q\in S.\) What is the least degree that such a polynomial \(p\) can have?

Answer: 1


Question: Determine whether there exists a function \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that \( f(0) = 0 \), \( f'(0) = 0 \), and \( f(x) \) cannot be bounded by any quadratic function in any neighborhood of the origin.

Answer: Yes


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

Answer: 0


Question: Determine whether there exists an infinite sequence of positive integers \( \{x_n\} \) such that \( \lim_{{n \to \infty}}{\sin(x_n)} = t \) for any real number \( t \) within the interval \([-1, 1]\).

Answer: Yes


Question: Is it always possible to find a filter base of a filter $F$ that is not $F$ itself? Provide a justification for your answer.

Answer: No


Question: Find the value of $n$ such that $n! = 2^{25} \times 3^{13} \times 5^6 \times 7^4 \times 11^2 \times 13^2 \times 17 \times 19 \times 23$. What is $n$?

Answer: 28


Question: Find the limit of \((x!)^{1/x}\) as \(x\) approaches infinity.

Answer: \infty


Question: Evaluate the integral \( \int_{0}^{1}P_n(2\sqrt{k}-1)P_{n+2}(2\sqrt{k}-1)\,dk \) using the expansion of \( K(k) \) with respect to the base of \( L^2(0,1) \) given by the shifted Legendre polynomials.

Answer: 0


Question:
Let \( g(x) \) be a function defined as follows:

\[
g(x) = \begin{cases} x^4\cos\left(\frac{1}{x}\right), & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases}
\]

Determine the greatest value of \( n \) such that \( g \in C^n([-1,1]) \).

Answer: 1


Question: Determine the infimum of the expression \( \sqrt{(1+k^{2n})^2 + (1+k^{2n-1})^2} \) for all natural numbers \( n \) and \( k \) in the interval \([0,1)\).

Answer: \sqrt{2}


Question: Compute the integral \( \int_{-\infty}^{\infty} f(\delta(x-a)) \, dx \), where \( f(x) \) is a function. If it is possible, what is the value of the integral?

Answer: f(1)


Question:
Evaluate the value of the infinite sum:

$$\sum_{j=1}^{\infty }\left ( \sum_{n=1}^{\infty }\frac{1}{\sqrt{nj}} \left (\frac{\sin(j/n)}{n}-\frac{\sin(n/j)}{j} \right )\right )$$

Answer: 0


Question: Find the shortest distance from the point \((5,5)\) to the curve defined by the equation \(xy = 4\).

Answer: \sqrt{17}


Question: Evaluate the integral \( \int_0^1\int_0^1 dx_1 dx_2 \delta(1-x_1-x_2) \) and determine if the result is \(1\) or \(\sqrt{2}\).

Answer: 1


Question: Determine whether there exists an integer $k \geq 4$ such that the polynomial $P_k(n) = \sum_{m=1}^n m^k$ is a perfect $\ell$-th power for some integer $\ell \geq 2$. Provide a justification for your answer.

Answer: No


Question: Evaluate the limit of the sequence \( \sin^2(\pi\sqrt{n^2 + n}) \) as \( n \to \infty \). Use the identity \( \sin^2(\pi\sqrt{n^2 + n}) = \frac{1}{2}(1 - \cos(2\pi\sqrt{n^2 + n})) \) to assist in your evaluation.

Answer: 1


Question:
Determine whether the following statement is true or false for an extended real-valued function $f: \mathbb{R} \to \overline{\mathbb{R}}$:
\[ \lim_{x \to a} |f(x)| = |\lim_{x \to a}f(x)| \]

Answer: B


Question: Determine whether the limit \( \lim_{{(x,y)\to (2,-2)}} \frac{\sin(x+y)}{x+y} \) exists.

Answer: 1


Question: Is a real-valued function \( F \), defined on a closed interval \([a, b]\), with bounded variation and the intermediate value property, guaranteed to be continuous?

Answer: Yes


Question: Consider a polynomial \( a(x) \) in \( F[x] \), where \( F \) is a field. Let \( F(c_1, c_2, \ldots, c_n) \) be the root field of \( a(x) \). Determine if the degree of the extension \( [F(c_1, c_2, \ldots, c_n):F] \) can be greater than the number of distinct roots of \( a(x) \).

Answer: Yes


Question:
Given a set of $n$ unique items, each permutation can be labeled with a number from $0$ to $(n!-1)$. These labels can be normalized to be between $0$ and $1$. For example, with $n=4$, the normalized labels for permutations are:

0: 1234
0.0434: 1243
0.0869: 1324
0.1304: 1342
0.1739: 1432
0.2173: 1423
0.2608: 2134
0.3043: 2143
0.3478: 2314
0.3913: 2341
0.4347: 2431
0.4782: 2413
0.5217: 3214
0.5652: 3241
0.6086: 3124
0.6521: 3142
0.6956: 3412
0.7391: 3421
0.7826: 4231
0.8260: 4213
0.8695: 4321
0.9130: 4312
0.9565: 4132
1: 4123

Given $n$ and a normalized label $m$, determine the permutation corresponding to the $m^{th}$ label without computing $n!$. For instance, if the normalized label is $0.9$, find the closest permutation, which is 4312.

Answer: 4312


Question:
Verify Stokes' theorem for the vector field \( \vec{v} = zi + xj + yk \) over the hemispherical surface defined by \( x^2 + y^2 + z^2 = 1 \) with \( z > 0 \). Stokes' theorem states:

$$\int\limits_C \vec{F} \cdot d\vec{r} = \iint\limits_S \mathrm{curl}\ \vec{F} \cdot d\vec{S}$$

Calculate both sides of the equation and verify if they equal \( \pi \).

Answer: \pi


Question: Find all functions \( g: \mathbb{R} \rightarrow \mathbb{R}_0^+ \) that satisfy the functional equation: \[ \forall p,q,r,s \in \mathbb{R}: pq+qr+rs=0 \Rightarrow g(p-q)+g(r-s)=g(p)+g(q+r)+g(s) \]

Answer: g(x) = ax^2


Question: Find the 6th derivative of the function \( f(x) = \frac{1}{1+\sin(x^2)} \) evaluated at \( x = 0 \).

Answer: -600


Question: Evaluate the integral \( \int_{-3}^{3}\frac{\mathrm{d}x}{3+f(x)} \) given that \( f(x)\cdot f(-x)=9 \).

Answer: 1


Question: Solve the exponential equation \(2^x + 3^x = 10000\) for \(x\).

Answer: 8.354


Question: For a differentiable radial function $f(x,y)$, determine if the gradient $\nabla f(x,y)$ always has the same magnitude along any circle of fixed radius centered at the origin. Provide a justification for your answer.

Answer: Yes


Question: Determine the value of $u$ for which a supercritical Hopf bifurcation occurs in the system given by the equations: $$\frac{dr}{dt} = ur - r^3$$ and $$\frac{d\theta}{dt} = 1.$$ Justify your answer.

Answer: 0


Question: Determine all values of $b$ such that \[ \lim_{x \to \infty} \left( \frac{x-b}{x+b} \right)^x = e. \]

Answer: -\dfrac{1}{2}


Question: Determine whether the sequence \( \frac{n^n}{e^{n^2}} \) converges as \( n \to \infty \), and provide a proof for your conclusion.

Answer: 0


Question: Evaluate the integral \( \int_S (x^2 + y^2) \, dS \) where \( S \) is the unit sphere in \( \mathbb{R}^3 \).

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


Question: Determine whether there exists a locally conformal map from the unit disc onto a simply-connected domain that contains two distinct points but does not contain their corresponding critical points.

Answer: Yes


Question: Compute the surface integral \(\iint_{D}^{}(\nabla\times \vec{F})\cdot \hat{n}dS\) where \(D\) is the upper part of the paraboloid \(z=1-x^2-y^2\), oriented upwards, and \(\vec{F}=\langle xy^2,-x^2y,xyz\rangle\).

Answer: 0


Question: Simplify the expression: \( \sqrt{5 + \sqrt{5} + \sqrt{3 + \sqrt{5} + \sqrt{14 + \sqrt{180}}}} \)

Answer: 1 + \sqrt{5}


Question: Determine whether the convergence of the integral \( \int_1^\infty f(x) \,dx \) implies that \( \lim_{x\to\infty} f(x) = 0 \).

Answer: No


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

Answer: \dfrac{5}{16}


Question: Find the value of the limit: $$\lim_{n\to\infty}\left(\frac{3}{2}\cdot\frac{5}{3}\cdot\dots\cdot\frac{2n+1}{n+1}\right)^\frac{1}{n}. $$

Answer: 2


Question: A box initially contains a ball marked $1$. At 10:30, the ball marked $1$ is removed, and two balls, labeled $2$ and $3$, are added. At 10:45, the balls labeled $2$ and $3$ are removed, and four balls, marked $4$, $5$, $6$, and $7$, are added. At 10:52:30, the balls labeled $4$, $5$, and $6$ are removed, and eight balls, labeled $8$, $9$, $10$, $11$, $12$, $13$, $14$, and $15$, are added. This pattern continues, with each action occurring halfway between the previous action and 11 o'clock. Each time, one more ball is removed than was removed last time, but twice as many balls are added as were added last time. Determine the number of balls in the box when the clock strikes 11.

Answer: 0


Question: Are the valuation rings of \(\mathbb{Q}\) exactly the subrings of the form \(\{a/b \in \mathbb{Q} \mid p \nmid b\}\) for some prime \(p\)?

Answer: Yes


Question: Evaluate the line integral \( \oint_C Rdx + Sdy \) using Green's theorem, where \( R = (z-1)w^{2} \) and \( S = (w+1)z^{2} \). The curve \( C \) is the triangle with vertices at \((0,0)\), \((0,1)\), and \((\frac{1}{2}, 0)\).

Answer: \dfrac{1}{4}


Question: Calculate the value of $63^{63^{63}} \bmod 100$. Provide your answer as a single integer.

Answer: 67


Question: Determine whether the probability that a random walk constrained to a finite set in $\Bbb Z^d$ visits the starting point infinitely often is 1. Answer true or false.

Answer: True


Question: Is the determinant of an $n \times n$ matrix, where each element is either $+1$ or $-1$, divisible by $2^{n-2}$ for $n \geq 3$?

Answer: Yes


Question: Find the number of roots of the polynomial \( Q(z) = z^4 + 3z^2 + 2 \) that lie inside the annulus defined by \( \{ z : 1 < |z| < 2 \} \).

Answer: 2


Question: Evaluate the limit: $$\lim_{x \to 0} \frac{x - \sin x \cos x}{\tan x - x}$$ without using L’Hospital’s rule and without using series expansion.

Answer: 2


Question: Suppose $M$ and $N$ are manifolds without boundary, and $f: M \to N$ and $g: M \to N$ are diffeotopic embeddings. If the complement of $f(M)$ in $N$ is connected, is the complement of $g(M)$ in $N$ also connected?

Answer: Yes


Question: Determine the number of distinct closed paths on the edges of an icosahedron, where each edge is traversed exactly once and paths that are identical under rotation are considered the same.

Answer: 0


Question: Find a lower bound for the quantity \(|\langle Ax, Ay \rangle|\) given that the vectors \(x, y \in \mathbb{R}^n\) have overlapping supports, i.e., \(S_1 \cap S_2 \neq \emptyset\).

Answer: 0


Question: Determine the cardinality of the set of all functions that can be constructed as iterations of pointwise convergence starting from continuous functions.

Answer: 2^{\aleph_0}


Question: Let $X$ be an infinite set. Under ZFC, it is true that $X \notin X$. Is there another element distinct from $X$ which is not a member of $X$?

Answer: Yes


Question: Let $Q(b)$ denote the right-tail probability of a standard Gaussian random variable. Suppose that $X$ and $Y$ are jointly-distributed Gaussian random variables with $X \sim N(3, 8)$ and $Y \sim N(10, 5)$. If the correlation coefficient between $X$ and $Y$ is 0.8, find the value of $b$ such that $P[Y > 0|X = 3] = Q(b)$. Choose from the following options: -7.453, -4.431, 2.639, -4.671.

Answer: -7.453


Question: Evaluate the integral: $$\int_{0}^{2022} \left( x^{2} - \lfloor{x}\rfloor \lceil{x}\rceil \right) \, dx$$

Answer: 674


Question: Let \( q(x) \) be a differentiable function on \( \mathbb{R} \) for which \( \lim_{x \to \infty} q'(x) = 0 \). Determine \( \lim_{x \to \infty} \left[ q(x+1) - q(x) \right] \).

Answer: 0


Question: Fill in the appropriate symbol from \( \{>,<,\leq,\geq\} \) to make the following statement true: \( \arctan(x) \) __ \( \frac{x}{1+x^2} \) for all \( x > 0 \). Justify why equality cannot occur.


Question: Determine whether the series \( \sum_{n=0}^{\infty} 2^{(-1)^n - n} \) converges to exactly 3 or approximately 3.

Answer: 3


Question: Determine the smallest positive integer $m$ such that \( y^m \equiv 1 \pmod{125} \) for every integer $y$ between 3 and 50.

Answer: 100


Question: Evaluate the limit \( \lim_{x\to+\infty}\int_{x}^{x+\pi}|\sin t|dt \).

Answer: 2


Question: Determine the values of $a$ for which the infinite series \( \sum_{n=1}^{\infty} \frac{a^n \log_e n}{n^2} \) converges.

Answer: [-1, 1]


Question: Calculate the line integral \( \oint_{C} \frac{x^2 - y^2}{2} \, dx + \left(\frac{x^2}{2} + y^4\right) \, dy \) using Green's theorem, where \( C \) is the boundary of the region \( D \) defined by \( 1 \leq x^2 + y^2 \leq 4 \) and \( x \geq 0 \), \( y \geq 0 \). The boundary \( C \) consists of four curves: \( C_1 : \alpha(t) = (t, 0) \) for \( t \in [1, 2] \), \( C_2 : \alpha(t) = (2\cos{t}, 2\sin{t}) \) for \( t \in [0, \frac{\pi}{2}] \), \( C_3 : \alpha(t) = (0, 4 - 2t) \) for \( t \in [0, 1] \), and \( C_4 : \alpha(t) = (\sin{t}, \cos{t}) \) for \( t \in [0, \frac{\pi}{2}] \).

Answer: \dfrac{14}{3}


Question: Find the number of elements of order 3 in the group \(G = \mathbb{Z}/30\mathbb{Z} \times \mathbb{Z}/20\mathbb{Z} \times \mathbb{Z}/15\mathbb{Z} \times \mathbb{Z}/40\mathbb{Z}\).

Answer: 8


Question: Evaluate the limit \( \lim _{x\to -\infty }\left(\int _{x}^0e^{t^2}\,dt\right)^{\frac{1}{x^2}} \).

Answer: e


Question: Determine the relationship between the number of sides of a convex polygon and the maximum number of acute angles it can have.

Answer: 3


Question: Let $\rho_1$ and $\rho_2$ be two representations of a group $G$, with $V$ and $W$ as the corresponding $FG$-modules. The tensor product $\rho_1 \otimes \rho_2$ is defined by $\rho_1 \otimes \rho_2: G \to GL(V \otimes_F W)$, where $(\rho_1 \otimes \rho_2)(g)(v \otimes w) = (\rho_1(g)(v), \rho_2(g)(w))$. Suppose the $FG$-module $V \otimes_{FG} W$ affords the representation $\rho_3$. Determine if $\rho_3 \sim \rho_1 \otimes \rho_2$. Provide your answer as 'yes' or 'no'.

Answer: no


Question: Determine whether every irreducible polynomial of degree 5 in the finite field $F_2[x]$, where $F_2$ has characteristic 2, has distinct roots in any algebraic closure of $F_2$. Provide a proof or counterexample.

Answer: Yes


Question: Given that $E(X) = 4$, $E(Y) = 10$, $V(X) = 5$, $V(Y) = 3$, and $V(X+Y) = 6$, find $E(XY)$. Assume $X$ and $Y$ are not independent.

Answer: 39


Question: Calculate the mass of a solid body in the first octant bounded by the planes $z = 0$, $y = 0$, $z = y$, and the parabolic cylinder $y = 4 - x^2$, with density given by $\rho = xz$. What is the mass of the body?

Answer: \dfrac{16}{3}


Question: Evaluate the limit: $$\lim_{n\to\infty}\frac{n\sum_{k=1}^n \frac{1}{k}-\sum_{k=1}^n \frac{k}{k+1}}{\ln(n!)}$$ using the relationship between the harmonic series and the natural logarithm.

Answer: 1


Question: For semiprimes $M$ and $N$ with $M > N$, determine if it is possible for the Euler totient of $M$ to be greater than the Euler totient of $N$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the sum \( \sum_{n=1}^{\infty}\frac{3n^2-2n+1}{n2^n} \) using the known sums \( \sum_{n=1}^{\infty}\frac{n}{2^n}=2 \) and \( \sum_{n=1}^{\infty}\frac{1}{n2^n}=\ln(2) \).

Answer: 4 + \ln 2


Question: Given the sequence defined by $x_{n+1} = 3x_n + \sqrt{8x_n^2 + 2}$, and knowing that $x_{2017} + x_{2023} = 990$, find the value of $x_{2020}$. \( \boxed{?} \)

Answer: 5


Question: Evaluate the integral \( \int^{\pi}_0 \log{(\sin{x})} \, dx \).

Answer: -\pi \ln 2


Question: Find the supremum of the set \( T = \left\{ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e} : a+b+c+d+e = \beta \right\} \), where \( \beta \) is a positive constant.

Answer: \infty


Question: Evaluate the surface integral \( I = \int_S z \, dS \), where \( S \) is the surface of the hemisphere defined by the equation \( x^2 + y^2 + z^2 = 4 \) with \( z \geq 0 \).

Answer: 8\pi


Question: Evaluate the integral \( \int_C \sin(z) \, dz \) where \( C \) is any curve joining \( 0 \) and \( \pi \). Use a parameterization of the curve to find the result.

Answer: 2


Question: Let $X$ be a random variable with the density function: $$f(x)= \frac{1}{\sqrt{18\pi}}e^{a(c-x)^2}$$ where $-\infty<x<\infty$. Given that $\mathbb{E}[X^2] = 9$, find $\mathbb{E}[X^8]$. \(\boxed{?}\)

Answer: 688905


Question: Evaluate the limit: $$\lim_{n\to \infty}\left(\lim_{x\to 0}\left( 1+\sum_{k=1}^n\sin^4(kx)\right)^\frac{1}{n^3x^2} \right)$$

Answer: 1


Question: Suppose that $a_n$ and $b_n$ are two real sequences satisfying $a_nb_n \to 1$ and $a_n + b_n \to 2$. Determine whether it follows that $a_n$ and $b_n$ both converge to $1$. Provide a justification for your answer.

Answer: Yes


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

Answer: 0


Question: Evaluate the limit \( \lim_{\epsilon\to 0^+} \frac{3}{4\pi\epsilon^3}\oint_{S_\epsilon} G \cdot\hat{N}dS \), where \( S_\epsilon \) is the sphere centered at \((a,b,c)\) with radius \( \epsilon \), and the vector field is given by \( G(x,y,z) = (x^2y, z^2x, y^2z) \). Here, \( \hat{N} \) is the outward unit normal vector to the sphere.

Answer: 2ab + b^2


Question: Find the limit superior of the sequence \((n+2^n)^{1/n}\) as \(n\) approaches infinity.

Answer: 2


Question: In a topological space $(\mathbb{R}^N,\tau)$, if the closure of the union of any family of closed sets contains a compact set, is a finite union of closed sets necessarily closed?

Answer: Yes


Question: Simplify the distance formula \( \operatorname{dist} (\langle a, r \rangle, \langle a, R \rangle) = \operatorname{arcosh} \left( 1 + \frac{ {(R - r)}^2 }{ 2 r R } \right) \) in the Poincaré half-plane model to show that it equals \( \ln(R) - \ln(r) \).

Answer: \ln(R) - \ln(r)


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

Answer: 0


Question: Find the kernel of the homomorphism \( f: R \rightarrow R/I \times R/J \) defined by \( f(a) = (a+I, a+J) \), where \( I \) and \( J \) are ideals of a ring \( R \).

Answer: I \cap J


Question: Determine whether there exists an uncountable set of subsets of the natural numbers such that for any two subsets \( A \) and \( B \) in the set, either \( A \) is a subset of \( B \) or \( B \) is a subset of \( A \).

Answer: Yes


Question: Determine whether every discrete-time stochastic process is a Markov chain. Provide a true or false answer.

Answer: False


Question: Determine whether the function \( w(x,y) = \frac{1}{2} \ln(x^2+y^2) \) is harmonic.

Answer: Yes


Question: Given the joint distribution of $X_1$ and $X_2$ with the density function $h(x_1, x_2) = 8x_1x_2$ for $0 < X_1 < X_2 < 1$, find the joint distribution of $Y_1 = \frac{X_1}{X_2}$ and $Y_2 = X_2$. Then, determine the marginal distribution of $Y_2$. Provide your answer in terms of the probability density function.

Answer: 4y_2^3


Question: Given that $a, b, c$ are real numbers such that $a+b+c = 1$, $a^2+b^2+c^2 = 2$, and $a^3+b^3+c^3 = 3$, determine the value of $a^5+b^5+c^5$. Provide your answer if possible.

Answer: 6


Question: Find the derivative of the function \( f(x) = \begin{cases} x^2 \sin \left( \frac{1}{x}\right), & x \neq 0, \\0, & x=0\end{cases} \) at \( x=0 \).

Answer: 0


Question: Suppose that \(f: D_{18} \to GL(2,\mathbb{R})\) is a homomorphism, where \(\lvert r \rvert = 18\) and \(f(r) = R := \begin{pmatrix}1 & 1 \\ -1 & 0\end{pmatrix}\). Determine the size of the kernel of \(f\), denoted as \(\lvert \ker(f) \rvert\).

Answer: 3


Question: Find the original function \( f(x, y) \) given its gradient components: \( \frac{\partial f}{\partial x} = 2xy - y \) and \( \frac{\partial f}{\partial y} = x^2 - x \).

Answer: x^2 y - x y


Question: Find the value of $k$ such that the expression \( \binom{404}{4} - \binom{4}{1} \cdot \binom{303}{4} + \binom{4}{2} \cdot \binom{202}{4} - \binom{4}{3} \cdot \binom{101}{4} = (101)^k \).

Answer: 4


Question: Determine whether the dihedral group $D_n$ of order $2n$ is isomorphic to a subgroup of the symmetric group $S_n$ for all $n > 2$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the function \( f(x) = \begin{cases} \lfloor \sin(\frac{1}{x}) \rfloor & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \) is integrable on the interval [0, 1].

Answer: Yes


Question: Given that the area of the initial triangle is \( A \), calculate the sum of the areas of all triangles formed in a sequence where each triangle is created from the medians of the previous triangle.

Answer: 4A


Question: Let $R$ be a ring (not necessarily with identity or commutative). Suppose that $K$ is a nil ideal, and let $M$ be a minimal right ideal of $R$. Does it follow that $MK = 0$?

Answer: Yes


Question: Suppose $f(x) = g(x) + h(x)$ where $g(x)$ is a non-differentiable function. If $h(x) \ne -g(x)$, can $f(x)$ be differentiable? Assume $f, g, h$ are real-valued functions. Provide a justification for your answer.

Answer: Yes


Question: Given three coplanar unit vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, and three vectors $\vec{p}$, $\vec{q}$, and $\vec{r}$ are parallel to $\vec{a}$, $\vec{b}$, and $\vec{c}$ respectively, with integral and distinct magnitudes, determine whether $\left| \vec{p} + \vec{q} + \vec{r} \right|$ can be $\sqrt{3}$, $2$, $1$, or $0$. Provide a justification for each case.

Answer: \sqrt{3}


Question:
Given functions $f(x)$ and $g(x)$ such that $f(x+y) = f(x)g(y) + g(x)f(y)$, calculate the determinant of the matrix:

$$\begin{vmatrix} f(\alpha) & g(\alpha) & f(\alpha + \theta) \\ f(\beta) & g(\beta) & f(\beta +\theta) \\ f(\gamma) & g(\gamma) & f(\gamma + \theta) \\ \end{vmatrix}$$

Answer: 0


Question: Find the value of \( f\left(\frac{1}{4}\right) \) if \( f:[0,1] \to \mathbb{R} \) is a continuous function satisfying \( \int_{0}^{1} f(x) \, dx = \frac{1}{3} + \int_{0}^{1} \left(f\left(x^{2}\right)\right)^{2} \, dx \).

Answer: \dfrac{1}{2}


Question: Let $B = \sum_{j=1}^l \vec{x}_j \vec{y}_j^T$, where $\vec{x}_j \in \mathbb{R}^p$ and $\vec{y}_j \in \mathbb{R}^q$. Assume that $\{ \vec{x}_j \}_{j=1}^l$ and $\{ \vec{y}_j \}_{j=1}^l$ are both linearly independent sets. If $p < q < l$, determine the rank of $B$. Provide your answer as a single integer.

Answer: p


Question: Determine the value of $m > 0$ that minimizes the area of the region bounded by the lines $x = m$, $x = 2m$, and the curve $y = x + \frac{6}{x^2}$. Avoid using double integrals in your solution.

Answer: 1


Question: Consider an $n \times n$ matrix with real elements in the interval $[0,1]$. Is it always possible to achieve the maximum determinant using a matrix with entries that are exclusively $0$ or $1$?

Answer: Yes


Question: Calculate the expected distance between two points chosen randomly and independently on the circumference of a circle with radius 1.

Answer: \dfrac{4}{\pi}


Question: Determine whether the following statement is true or false: If a function \( f:(a,b)\to \mathbb{R} \) has a limit at every point in the interval \((a,b)\), then \( f \) is bounded.

Answer: B


Question: Evaluate the limit \( \lim_{n\to\infty}\dfrac{1}{n^{2+\beta}}\sum_{i=1}^{n}i^{\beta} \) using the Riemann integral of suitably chosen functions.

Answer: 0


Question: Determine whether the bounded sequence \( \frac{n}{n+\cos n} \) has a convergent subsequence.

Answer: Yes


Question: Is it possible for the expected value of the product of two integrable real random variables, where one has a non-negative expected value and the other is positive, to be negative?

Answer: Yes


Question: Evaluate the integral \( \int\limits_{-\infty}^{\infty}\frac{\sin(mx)dx}{x^2+a^2} \) using contour integration, where \( m \) and \( a \) are positive constants.

Answer: 0


Question: Find the minimum value of the function \( f(x, y, z) = x^2 + y^2 + z^2 \) subject to the constraints \( x + y + z = 1 \) and \( xyz + 1 = 0 \).

Answer: 3


Question: Evaluate the line integral \( \int_C \mathbf{G} \cdot \, \mathrm{d} \mathbf{r} \) where \( \mathbf{G} = y \mathbf{i} + x \mathbf{j} + yx \mathbf{k} \) and \( C \) is parameterized by \( \mathbf{r}(t) = \sin t \mathbf{i} + \cos t \mathbf{j} + t \mathbf{k}, \ t \in [0, \frac{\pi}{2}] \).

Answer: \dfrac{1}{2}


Question: Determine whether there exists an $L^1$ function $f(x)$ such that the limit \( \lim_{\omega \to \infty} \int_{\mathbb{R}} f(x) \cdot e^{-\imath \, x \omega} \, dx \neq 0 \).

Answer: No


Question: Determine whether a topology on a set can have a cardinality that is not a power of 2. Answer with 'Yes' or 'No'.

Answer: Yes


Question: Given a planar graph $G=(V,E)$ with a specific planar embedding that satisfies Euler's formula $v-e+f=2$, where $v$, $e$, and $f$ denote the number of vertices, edges, and faces respectively, determine if this ensures that $G$ is a connected graph.

Answer: Yes


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

Answer: (-1, 1)


Question: Determine the largest constant \( t \) such that for all positive real numbers \( x, y, z \), the inequality \((xy + xz + yz)\left(\frac{1}{x+y} + \frac{1}{x+z} + \frac{1}{y+z}\right)^2 \geq t\) holds.

Answer: \dfrac{25}{4}


Question: Can the maximum capacity cut in a network be determined in O(m) time complexity using the residual graph obtained after finding the maximum flow?

Answer: No


Question: Determine whether the function $f(x) = \frac{x}{2} + \frac{1}{x}$, defined on the set $X = \{x \in \mathbb{R} : x \ge 1\}$, is a contraction map with respect to the usual metric on $\mathbb{R}$. If it is, find the smallest contraction constant $\lambda$.

Answer: \dfrac{1}{2}


Question: Find the constant \( c \) such that the expression \( c[(X_1 - X_2)^2 + (X_3 - X_4)^2 + (X_5 - X_6)^2] \) is an unbiased estimator of the population variance \( \sigma^2 \), given a random sample \( X_1, X_2, \ldots, X_6 \) drawn from a normally distributed population with mean \( \mu \) and variance \( \sigma^2 \). The sample is independent but not necessarily identically distributed.

Answer: \dfrac{1}{6}


Question: Determine the convergence rate of the Newton iteration near the root \( r = 1 \) for the polynomial \( g(x) = 4x^{3} - 6x^{2} + 4x - 1 \).

Answer: 2


Question: Determine the maximum number of regular tetrahedrons with edge length 1 that can be packed inside a cube with edge length 2.

Answer: 16


Question:
Determine whether the following expression remains greater than zero as \( n \to \infty \):
\[ \frac{\sum_{i=0}^{n/2-1}{\frac{(an^2)^{i}}{i!}}}{e^{an^2}} \]
where \( a > 0 \).

Answer: No


Question: Evaluate the limit: \[ \lim_{n\to\infty} \int_0^1 n f(x) e^{-nx} \, dx \] where \( f: \mathbb{R} \to \mathbb{R} \) is a continuous function.

Answer: f(0)


Question: Determine the best integer value of $k$ to estimate the function \( e^{\sin x} = 1+x+\frac{x^2}{2!}-\frac{x^4}{4!}+\frac{x^6}{6!}-\quad... \) for \(|x|\le\frac{\pi}{2}\) as \(x \to 0\), using o-notation and retaining three terms.

Answer: 3


Question: Calculate the surface integral \( \iint_Y G \cdot N\, dS \) where \( Y \) is the surface defined by \((x+y)^2+(y+z)^2=4+z^2\) for \( 0 \leq z \leq 2 \), \( N \) is the outward-pointing normal vector, and \( G = (x^2y, y^2z, z^2x) \).

Answer: -128\pi


Question: If $f(t)$ is differentiable on $[0,1]$, with $f'(t) > 0$ and $f''(t) > 0$ for $t \in (0,1)$, is $f'(t)$ strictly increasing on $[0,1]$?

Answer: Yes


Question: Find a near identity change of coordinates \( y = \psi(x) = x + c_2x^2 + c_3x^3 \) for the scalar differential equation \( x' = \frac{2x – 12x^2 – 8x^3}{1 – 6x – 6x^2} \) such that close to \( y = 0 \), we have \( y' = Df(x^*)y \), where \( x^* \) is a fixed point of the equation and \( Df(x^*) \) is known.

Answer: y = x - x^3


Question: Using Rouché's theorem, determine how many roots of the polynomial \( p(z) = z^8 + 3z^7 + 6z^2 + 1 \) are contained in the annulus \( 1 < |z| < 2 \).

Answer: 5


Question: Determine the conditions on the function $F(x)$ such that the solution $u(x,t)$ to the partial differential equation $u_{t} - u_{xx} = F(x)$ on the interval $(0,\pi)$ with homogeneous Neumann boundary conditions $u_{x} = 0$ for $x=\{0,\pi\}$ has a limit as $t \to \infty$. Assume $F(x)$ is independent of $t$ and can be expressed as a Fourier cosine series $F(x) = \sum_{n=0}c_{n}\cos(nx)$. Provide the conditions on the coefficients $c_n$ that ensure the solution $u(x,t)$ approaches a limit as $t \to \infty$. \(u(x,t)\) has the Fourier cosine expansion: \(u(x,t) = \frac{1}{2}A_{0} + \sum_{n=1}A_{n}e^{-n^{2}t}\cos (nx)\).

Answer: c_0 = 0


Question: Given the equations \(\alpha^3 - 6\alpha^2 + 13\alpha = 1\) and \(\beta^3 - 6\beta^2 + 13\beta = 19\), where \(\alpha, \beta \in \mathbb{R}\), find the value of \(\alpha + \beta\).

Answer: 4


Question: Find the area enclosed by the line $x + y = a$ and the curve $\sqrt{x} + \sqrt{y} = \sqrt{a}$, given that $a > 0$. What could be the issue with the initial integration approach that resulted in a negative area?

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


Question: Determine whether there exists a prime number $p$ such that $p \equiv 3 \pmod{4}$ and $p$ is not equal to \( \text{rad}(n^{2}-1) \) nor \( \text{rad}(n^{2}+1) \) for any integer $n$. Provide a proof or counterexample.

Answer: 7


Question: Verify Stokes' theorem for the vector field \( \vec{G}(x,y,z) = (xz, -yz, xy) \) on the surface defined by \( z = 4 - x^2 - y^2 \) with \( z \geq 0 \).

Answer: 0


Question: Determine the value of \( x = a \) at which a plane flying along the curve \( f(x) = \frac{1}{x^2} \) should fire its projectile to hit a bunker located at \( (10, 0) \). The projectile will follow the tangent line to the curve at the point of firing. Find the point \( (x_0, \frac{1}{x_0^2}) \) on the curve such that the tangent from this point intersects the bunker's position.

Answer: \dfrac{20}{3}


Question: Determine whether the linear transformation $T: \mathbb{C}^2 \rightarrow \mathbb{C}^2$ defined by $T(a,b) = (2a + i b, a + 2b)$ is normal.

Answer: Yes


Question: Given positive integers $a$ and $m$ such that $\gcd(a,m)=1$, and a positive integer $n$ such that $10^n$ is divisible by the Euler totient function of $m$, find the remainder when $a^{10^n}$ is divided by $m$. \( \text{(Assume } \phi(m) \text{ is the Euler totient function of } m.\)

Answer: 1


Question: In a queuing theory scenario, two facilities, A and B, offer identical services with single servers. The service times follow negative exponential distributions with mean service times of 1 minute for facility A and 4 minutes for facility B. A total of 60 customers per hour seek service at both facilities, with arrivals following a Poisson distribution. Customers, uninformed of queue lengths, independently choose a facility aiming to minimize their average waiting time plus service time. Eventually, a steady state distribution of customers across both facilities is reached. Determine the probability \( p \) that a customer selects facility A.

Answer: \dfrac{7}{8}


Question: Find the real value \( \alpha \) such that the coordinate transformation \( \varphi(p,q) \rightarrow (P,Q) \), where \( Q = q^2 + \alpha\sqrt{q^2+p} \) and \( P = q + \sqrt{q^2+p} \), preserves the symplectic form \( dp \wedge dq \).

Answer: 2


Question: Determine the number of distinct Jordan forms for a $6 \times 6$ matrix with the characteristic polynomial $(x+2)^4 (x-1)^2$. Is the approach of calculating $P(4) \times P(2)$, where $P(n)$ represents the number of partitions of $n$, correct?

Answer: 10


Question: Determine the number of different factorizations of the polynomial \(x^{3000} - 27\) into nonconstant polynomial factors, each with a leading coefficient of 1.

Answer: 1


Question: Calculate the sum of the series \( \sum\limits_{n=1}^{\infty}\frac{n^2}{2^n} \) using the Vn method.

Answer: 6


Question: Determine the number of roots of the polynomial \( f(z) = z^4 + z^3 - 2z^2 + 2z + 4 \) that lie in the first quadrant of the complex plane.

Answer: 1


Question: Find the maximum value of the function \( f(x, y, z) = x^2 + y^2 + 2z^4 \) subject to the constraint \( x^2 + y^2 + z^2 = 1 \).

Answer: 2


Question: Identify a group $G$ that contains elements $a$ and $b$ such that $a^2 = e$, $b^2 = e$, and the order of the element $ab$ is infinite.

Answer: D_\infty


Question: Determine the number of different bases that can be formed from the set of all vectors in \(\mathbb{C}^3\) where each coordinate is either 0 or 1.

Answer: 29


Question: Determine whether an infinite sum of rational numbers can result in a rational number. Provide a justification for your answer.

Answer: Yes


Question: Find the limit of the sequence \(\left(\frac{n+1}{2n!}\right)^{\frac{1}{n}}\) as \(n\) approaches infinity.

Answer: 0


Question: Find the last five digits of the number $5^{1981}$, given that the last three digits are $125$.

Answer: 03125


Question: Evaluate the limit: $$\lim_{n \to \infty} \int_{0}^{1} f(nx+1) \, dx$$ where $f(x)$ is a continuous, strictly increasing, and positive function on the interval $[0, \infty)$, and $\lim_{x \to \infty} f(x) = M$. Determine the value of this limit.

Answer: M


Question: Let $B$ be the set of irrational numbers in the open interval $(0,7)$ and $g:B\rightarrow \mathbb{R}$ be a uniformly continuous function. Is $g$ necessarily bounded? Provide a justification for your answer.

Answer: Yes


Question: Calculate the limit: \[ \lim_{n \to \infty} \frac{\ln (\log_a (n)) - \ln (\log_n (a))}{\ln (n)} \]

Answer: 0


Question: Consider the probability simplex in $\mathbb{R}^n$, where each point $p = (p_1, p_2, p_3, \ldots, p_n)$ represents a distribution for a random variable $X$ taking one of $n$ values, with $\mathbb{P}(X = a_i) = p_i$ and $a_j < a_k$ for $j < k$. Determine whether the region where $\text{Var}(X) \geq \alpha$ for some positive $\alpha$ is a convex region.

Answer: Yes


Question: Find the values of $q$ for which the limit \( \lim_{(x,y)\to (0,0)} \frac{x^q \cdot y^q}{x^4 + y^4} \) exists.

Answer: q > 2


Question: Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed random variables with probability density function $f(x) = e^{-x}$ for $x > 0$. Define $X_n = \max(X_1, X_2, \ldots, X_n)$. Find \( \lim_{n \to \infty} P(X_n < 2 + \log_e n) \).

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


Question: Evaluate the integral: $$\int_{-1}^1 \frac{d}{dx}\ \frac{1}{1 + 2^{1/x}}\ dx$$

Answer: \dfrac{2}{3}


Question: In a class of 30 students, each student writes down the names of 15 different counties from a state with 300 counties. Assuming all students choose counties randomly and independently, calculate the probability that at least 30% of the counties are chosen by the students.

Answer: 1


Question: Let \( f: \mathbb{R} \to (0, +\infty) \) be a surjective function. Define the function \( g(x, y) = \frac{\ln(f(x)) + \ln(f(y))}{\ln(2)} - 2 \) where both \( x \) and \( y \) belong to the domain of \( f \). Is the function \( g(x, y) \) surjective?

Answer: Yes


Question: If $n$ is an odd integer, determine the number of real roots of the polynomial equation $p_n(x)=1+2x+3x^2+\ldots+(n+1)x^n$. \(\boxed{\text{Answer}}\)

Answer: 1


Question: In a semicircle with diameter \( AB \) and center \( S \), points \( C \) and \( D \) are on the arc \( AD \) such that \( \angle CSD = 120^\circ \). Let \( E \) be the intersection of \( AC \) and \( BD \), and \( F \) be the intersection of \( AD \) and \( BC \). Given that \( F \) is the orthocenter of triangle \( ABE \), find the ratio \( \frac{EF}{AB} \).

Answer: \sqrt{3}


Question: Determine which of the following cannot be the order of the quotient group \( \frac{G}{Z(G)} \): \(4\), \(6\), \(15\), \(25\).

Answer: 15


Question: Consider the functional equation \(g(x)=\frac{1}{4}\left(g(x+2) + \frac{3}{g(x+4)}\right)\). If \(g(x) > 0\) for all real values of \(x\), find \(\lim_{x\rightarrow -\infty} g(x)\).

Answer: 1


Question: Given any $n$ consecutive positive integers, where $n$ is even, determine if there exists one integer which is relatively prime to the product of the rest.

Answer: Yes


Question:
Given a matrix \( B \) and a matrix \( \Psi \) defined as:

\[\Psi=\begin{bmatrix} e_{2}^{T} & \\ e_{2}^{T}B \\ e_{2}^{T}B^2 \\ .\\ .\\ .\\ e_{2}^{T}B^{m-1}\end{bmatrix}\]

where \( e_{2} \) is the standard basis vector with the second entry 1 and others 0. If \( B \) is a block diagonal matrix with at least two \( 2\times 2 \) rotation matrices as blocks, determine if the determinant of \( \Psi \) is zero.

Answer: 0


Question: Find the inverse function of \( f(x) = \frac{\ln(x)}{x} \) for \( x > e \).

Answer: -\frac{W_{-1}(-x)}{x}


Question:
Given the system of equations:

\[ x\cos p-y \sin p+z=\cos q+1 \\
   x\sin p+y\cos p+z=1-\sin q \\
   x\cos(p+q)-y\sin(p+q)+z=2 \]

where $\sin q\ne \cos q$, derive an expression for $z$ in terms of $p$ and $q$.

Answer: 1


Question: Determine whether a real-valued nonsingular matrix of dimension $n \times n$ with strictly positive entries always has $n$ distinct eigenvalues. Answer with 'True' or 'False'.

Answer: False


Question: Let \( f \) be an entire function that is real on the lines \( \{z\in\mathbb{C}: \text{Re}(z)=1\} \) and \( \{z\in\mathbb{C}: \text{Re}(z)=2\} \). Additionally, \( \lim_{1\leq x \leq 2, y \to \infty} f(x,y) = 0 \), where \( z=x+iy \). Prove that \( f(z)=0 \) for all \( z \in \mathbb{C} \), given that \( f \) is a periodic function with a period of 1.

Answer: 0


Question: Determine the value of $k$ such that \[ \lim _{n\to \infty }\left(\frac{n^{k+1}}{n^{k+2}-(n-2)^{k+2}}\right)=\frac{1}{100}. \]

Answer: 48


Question: Evaluate the integral: \[ 2\pi\int_0^{\infty}r^3e^{-\frac{1}{2}r^2}dr \]

Answer: 4\pi


Question: Determine the number of real solutions to the equation \(12^x + 14^x + 18^x - 20^x = 0\).

Answer: 1


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

Answer: 0


Question: Determine the minimum number of terms required in the Maclaurin series expansion of \( f(x) = \log \left(\frac{1+x}{1-x}\right) \) to approximate \( \log(2) \) with six exact decimal places when \( x = \frac{1}{3} \). Assume \( \log \) represents the natural logarithm (\( \ln \)).

Answer: 6


Question:
Given the joint density function of random variables $X$ and $Y$:
$$f_{X,Y}(x,y)= \begin{cases} e^{-y}, & 0\leq x \leq y, \ y \geq 0 \\ 0, & \text{otherwise} \end{cases}$$
Evaluate the conditional expectation $E[X|Y=y]$. \(\boxed{\text{Your answer here}}\)

Answer: \dfrac{y}{2}


Question: Compute the vector-induced \( \| \cdot \|_1 \) norm of the matrix \( A = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix} \) using the definition \( \|A\|_1=\sup_{v\neq 0}\frac{\|Av\|_1}{\|v\|_1} \).

Answer: 5


Question: Determine whether the following statement is true or false: A topological space $X$ is compact if there exists a basis $B$ for $X$ such that every open cover of $X$ consisting only of elements from $B$ has a finite subcover.

Answer: A


Question: Let $f: \{z:|z|<1\} \to \{z:|z|<1\}$ be an analytic function with $f(0)=0$. Suppose there exists some $0<r<1$ such that $f(\{z:|z|<r\}) = \{z:|z|<r\}$. Determine whether it follows that $f(z)=cz$ for some $|c|=1$, or if $f$ is of some other simple form.

Answer: f(z) = cz


Question: Determine whether there exist factor-square polynomials with complex coefficients that are reducible over the complex numbers. Provide a justification for your answer.

Answer: Yes


Question: Determine the possible eigenvalues of a $3\times3$ matrix $P$ given that $P^3 = P$ and $P^2 \neq P$. Provide your answer as a set of numbers.

Answer: \{-1, 0, 1\}


Question: Determine the number of ways to distribute 12 blue, 13 white, and 14 red balls into 4 different boxes, such that each box contains at least 2 balls of each color. Assume that balls of the same color are indistinguishable.

Answer: 164640


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{\log(1^1 +2^2 + \cdots + n^n)}{\sqrt{n^4 + 2n^3\log(n)}-\sqrt{n^4-n^3}} \] After simplifying, the expression becomes: \[ \lim_{n\to\infty}\frac{n^2\log(1^1 +2^2 + \cdots + n^n)\left(\sqrt{1+\frac{ 2\log(n)}{n}}+\sqrt{1-\frac{1}{n}}\right)}{n^3 (2\log(n)-1)} \] Determine how to proceed to evaluate this limit.

Answer: 1


Question: Let \( f(x) \) be a real-valued function defined on the interval \((0,\infty)\). Determine the smallest possible value of \( \alpha \) such that \(|f'(x)| < |f(x)|\) for all \( x \in (\alpha,\infty)\).

Answer: 0


Question: For a cylindrical can, determine the ratio of the height to the radius that maximizes the expression \( \frac{V^2}{A^3} \), where the volume \( V = \pi r^2 h \) and the surface area \( A = 2\pi r(r+h) \).

Answer: 2


Question: In a strictly increasing sequence of positive integers $a_1, a_2, a_3, \ldots$, each subsequence $a_{2k-1}, a_{2k}, a_{2k+1}$ forms a geometric sequence, and each subsequence $a_{2k}, a_{2k+1}, a_{2k+2}$ forms an arithmetic sequence for every positive integer $k$. Given that $a_{13} = 539$, determine the value of $a_5$. \( \text{Express your answer as an integer.} \)

Answer: 99


Question: In an equilateral triangle ABC, a point D is located inside the triangle. Given that the sum of the squares of the distances DB, DC, and BC is 100, and the area of triangle DBC is \(5\sqrt{3}\) square units, find the value of \(AD^2\).

Answer: 20


Question: Given that the composition of linear transformations $T_1: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $T_2: \mathbb{R}^m \rightarrow \mathbb{R}^m$ is bijective, find the ranks of $T_1$ and $T_2$.

Answer: n


Question: Evaluate the integral \( \frac{1}{2\pi i}\int_{\gamma} \frac{\cos(z)}{z^7-2z^6}dz \), where \( \gamma \) is the path that traverses the circle centered at \(1/4\) with radius \(1/2\) counter-clockwise.

Answer: \dfrac{1}{192}


Question: Suppose $L_1$ and $L_2$ are two Lie algebras, and $f: L_1 \to L_2$ is a Lie algebra homomorphism. If $L_1$ is nilpotent, does it follow that $f(L_1)$ is nilpotent? Answer yes or no.

Answer: Yes


Question: Given the income distributions in two cities are represented by the Pareto-type probability density functions \( f(x) = \frac{2}{x^3} \) for \( x > 1 \) and \( g(y) = \frac{3}{y^4} \) for \( y > 1 \), calculate the probability \( P(X < Y) \) that a randomly selected person from the first city has a lower income than a randomly selected person from the second city.

Answer: \dfrac{2}{5}


Question: Let $\mathcal{R}$ be a non-type-I factor acting on a Hilbert space $H$. Consider a projection $p \not\in \mathcal{R}$, which is called $\mathcal{R}$-finite if it is the join of finitely many minimal projections in $(\mathcal{R} \cup \{p\})''$. If $p$ and $q$ are $\mathcal{R}$-finite projections, is their join $p \vee q$ also $\mathcal{R}$-finite? Assume that the ranges of $p$ and $q$ are not finite-dimensional subspaces of $H$. Provide a justification for your answer.

Answer: No


Question:
Given a linear transformation $T: \Bbb{P}_3 \to \Bbb{P}_3$ defined by:

$$T\left(-2 x^2\right) = 3 x^2 + 3 x$$

$$T(0.5 x + 4) = -2 x^2 - 2 x - 3$$

$$T\left(2 x^2 - 1\right) = -3 x + 2$$

Find the matrix representation of $T$ with respect to the basis $\beta = \{-2x^2, 0.5x+4, 2x^2-1\}$ and use it to calculate $T(1)$. Provide your answer for $T(1)$.

Answer: -3x^2 - 2


Question: Evaluate the integral \( I = \int_0^1 \int_0^1 f(x,y) \, dx \, dy \), where the function \( f(x,y) \) is defined as \( f(x,y) = 1 \) if \( x = y \) and \( f(x,y) = 0 \) if \( x \neq y \).

Answer: 0


Question: Evaluate the infinite series: $$\sum_{r=2}^{\infty} \frac{2-r}{r(r+1)(r+2)}$$ Given that the partial sum is $$\sum_{r=1}^{n} \frac{2-r}{r(r+1)(r+2)} = \frac{n}{(n+1)(n+2)}$$ determine why the infinite series equals $-\frac{1}{6}$.

Answer: -\dfrac{1}{6}


Question: Determine the maximum value of the function \( g(x) = (x_1x_2\cdots x_n)^{4/n} \) for \( x \in S_{n-1} \), where \( S_{n-1} \) is the set of all \( x \in \mathbb{R}^n \) with \( \|x\|_2 = 1 \).

Answer: \dfrac{1}{n^2}


Question: For the function \( f(x) = \frac{1}{x} \), determine the maximum possible error when approximating it using a Taylor polynomial of degree 6 centered at \( x_0 = \frac{3}{4} \) for \( x \) in the interval \( \left[\frac{1}{2}, 1\right] \).

Answer: \dfrac{1}{64}


Question:
Given the space $V = \{u \in H^1((0,1)) : u(0) = 0\}$ with the norm defined by the inner product \((u,v)_V = \int_0^1 u'v' \, dx + \int_0^1 \frac{u}{x}\frac{v}{x} \, dx\), determine the dual pairing \(\left<f,v\right>_{<V^*,V>}\) for a function \(f \in L^2((0,1))\) such that \(\frac{1}{x}f(x) \in L^2((0,1))\). Is it given by:

1. \(\left<f,v\right>_{<V^*,V>} = \int_0^1 \frac{f}{x}\frac{v}{x} \, dx\)
2. \(\left<f,v\right>_{<V^*,V>} = \int_0^1 fv \, dx\)

Which expression correctly represents the dual pairing? Justify your answer.

Answer: 2


Question: In a measure space $(X,\mathcal{A},\mu)$, consider a sequence of sets $\{A_n\}_{n\in\mathbb{N}} \subset \mathcal{A}$ with $A_{n+1} \subseteq A_{n}$ for all $n$, and such that $\mu(A_n) = 1$ for each $n\in \mathbb{N}$. Determine if it is always true that $\mu(\bigcap_{n\in\mathbb{N}} A_n) = \lim_{n \to \infty}\mu(A_n) = 1$. Provide a justification for your answer.

Answer: Yes


Question: Find a lower bound for the function \( f(x,y) = x^4 + y^4 - 3xy \) that is independent of the variables \( x \) and \( y \).

Answer: -\dfrac{9}{8}


Question: Given that $a_n(X_n-\theta)-a_n(Y_n-\theta)\xrightarrow[d]{}Z$, $a_n\xrightarrow[]{}\infty$, and $X_n, Y_n\xrightarrow[p]{}\theta$, can we conclude that $a_no(X_n-\theta)-a_no(Y_n-\theta)\xrightarrow[d]{}0$? Assume $o(X_n-\theta)= (X_n-\theta)\cdot o(1)$, and $o(1) \supset o\left((X_n-\theta)^k\right)$ for $k\geq1$. Provide a justification for your conclusion.

Answer: Yes


Question: Let $w=\sqrt{-14}$. Given the ideals $a=(5+w,2+w)$ and $b=(4+w,2-w)$ in $\mathbb{Z}[w]$, find the product of these ideals and express it in the form $(6,3w)$. Verify that this is the correct form of the product.

Answer: (6, 3w)


Question: Evaluate \( \frac{\mathrm{d}}{\mathrm{d}a} \int_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)} \).

Answer: 0


Question: Evaluate \( \lim_{x \to \infty} \int_{0}^{1} \frac{\log(x)}{\sqrt{t+x}}\, dt \).

Answer: 0


Question: Evaluate the residue of the function \( f(z) = (ze^{2-z}+(2-z)e^z)e^{(z-1)^{2}+(z-1)^{-2}} \) at \( z = 1 \) using a method other than Laurent series expansion.

Answer: 0


Question: Find the value of the limit \( L = \lim\limits_{n\to\infty} \dfrac{2^{n^{1.1}}}{n!} \).

Answer: \infty


Question: Evaluate the limit: \[ A = \lim_{x\to\infty}\left(\frac{x^2-3x+1}{x^2+x+2}\right)^{2x-5} \]

Answer: e^{-8}


Question: Determine the Galois group of the splitting field over \( \mathbb{Q} \) for the polynomial \( x^7 - 10x^5 + 15x + 5 \).

Answer: S_7


Question: Evaluate the line integral \( \int\limits_C \left(2y - \sin(x^2) \right) dx + \left(x^2 + \cos(y^2) \right) dy ,\) where \(C\) is the boundary of the region in \(\mathbb{R}^2\) enclosed by the curves \(y = x^2\) and \(y = 2x\).

Answer: 0


Question: Determine the smallest integer \( n \) such that the limit \( \lim_{(x,y) \rightarrow (0,0)}\frac{x^r}{\vert x \vert^2+\vert y \vert^2} \) exists for all \( r \geq n \).

Answer: 3


Question: Find the limit of the complex sequence \( \sqrt{-n^4+4n^2+4}-in^2 \) as \( n \to \infty \).

Answer: -2i


Question: Calculate the limit of the sequence \(a_n=\left(1-\frac{1}{\sqrt{2}}\right)\left(1-\frac{1}{\sqrt{3}}\right)\cdots\left(1-\frac{1}{\sqrt{n+1}}\right)\) as \(n\) approaches infinity.

Answer: 0


Question: Find the maximum value of the function \( f(x, y) = x^2 + xy + 2y^2 \) subject to the constraint \( g(x, y) = x^2 + 3y^2 = 16 \) using the method of Lagrange multipliers.

Answer: \dfrac{56}{3}


Question: Is it possible to construct a non-zero linear map $T: \mathbb{R}^4 \to \mathbb{R}^4$ such that $T^4 + 2T = 0$?

Answer: Yes


Question: Determine whether the following statement is true or false: A function \( f: \mathbb{Q} \to \mathbb{R} \) is continuous at a point \( x = a \) if and only if, for any \( \epsilon > 0 \), there exists some \( \delta > 0 \) such that \( |x-a|<\delta \implies |f(x)-f(a)|<\varepsilon \).

Answer: A


Question: Consider the recursive sequence defined by \(a_{n+1}=a_n+\frac{n}{a_1+\dots+a_n},\) with \(a_1>0.\) Determine the limit \(\lim_{n\to\infty} a_n.\)

Answer: \infty


Question: Let $G$ be a finite group with a subgroup $H$ and a normal subgroup $N$. Given that $|H| = n$ and $|G| = m|N|$, where $m$ and $n$ are coprime, determine the order of the element $hN$ in the quotient group $G/N$ for any $h \in H$.

Answer: 1


Question: Given an $L^1(\mathbb{R})$ function $f$ such that $f(x) = -f(-x)$, does it follow that $\lim_{\xi \to \infty} \hat{f}(\xi) = \hat{f}(0)$, given that $\hat{f}(\xi) = -\hat{f}(-\xi)$ and $\lim_{\xi \to \infty} \hat{f}(\xi) = 0$?

Answer: 0


Question: Determine whether the symmetric group $S_{10}$ has a subgroup with index 4. Answer with True or False.

Answer: False


Question: Let $X$ be a random variable with a given distribution function. Define $Y = X^2$. Calculate the probability $P(X - Y \geq 1)$.

Answer: 0


Question: Find the limit: \[ \lim_{n\to\infty}\left(\frac{\pi^2}{6}-\sum_{k=1}^n\frac{1}{k^2}\right)n \]

Answer: 1


Question: Calculate the value of the expression \(2(|AB - I_5| + |BA - I_4|) + 1\), where \(A\) is a \(5 \times 4\) matrix and \(B\) is a \(4 \times 5\) matrix. Here, \(I_n\) denotes the \(n \times n\) identity matrix.

Answer: 1


Question: Calculate the volume of the solid formed by revolving the region bounded by the curve \( y = e^{x^2} \), the lines \( y = 0 \), \( x = 0 \), and \( x = 1 \) around the \( y \)-axis.

Answer: \pi(e - 1)


Question: Find the gradient of the expression \( \nabla_X\|AXB\|_F^2 \) where \( A \) is a wide, full-rank matrix, \( B \) is a tall, full-rank matrix, and \( X \) is a square matrix. All matrices are real-valued.

Answer: 2 A^\top A X B B^\top


Question: Determine whether the infinite product \( P = \prod_{k=1}^{\infty} \left(1+\frac{A}{3^{m/2}(3^{k}s)^{b / 2}}\right) \) converges, given that \( A > 0 \), and \( m \), \( b \), and \( s \) are constants.

Answer: b > 0


Question: Determine the range of values for $b$ such that the equation $x^4 - 2bx^2 + 1 = 0$ has four distinct real roots.

Answer: (1, \infty)


Question: Evaluate the integral \( \int^{\pi}_0\int^{\pi}_x \frac{\sin(y)} {y} \, dy \, dx \).

Answer: 2


Question: Evaluate the limit of the function \(f(x,y) = (x^2+y^2)^{x^2y^2}\) as \((x,y)\) approaches \((0,0)\), using polar coordinates.

Answer: 1


Question: Given a set \((A, \cdot)\) with a closed and associative operation, where every element has an inverse, does this automatically imply that \((A, \cdot)\) is a group? Assume that the existence of an identity element can be deduced from the definition of inverses.

Answer: Yes


Question: Determine whether the polynomial \((x_1 + x_2 + \cdots + x_n)^n\) can be expressed as a sum of nonnegative terms for any positive integer \(n\) using the Bricks Throwing Method.

Answer: Yes


Question: Let $X$ be an algebraic curve over an algebraically closed field $k$. Determine whether there exists a polynomial $f \in k[x,y]$ such that $X$ is birational to the curve defined by $\{f(x,y)=0\}$.

Answer: Yes


Question: Which function grows faster as $n$ becomes larger: $f(n)=100^{100^{100^n}}$ or $g(n)=2^{2^{2^{2^n}}}$?

Answer: g(n)


Question: Determine whether the inequality \( \prod\limits_{i=1}^n\left(3 + \dfrac{1}{3i}\right) < \left(3 + \dfrac{1}{n}\right)^n \) holds for all positive integers \( n \).

Answer: No


Question: Determine whether a connected, simply connected, nilpotent \(n\)-dimensional Lie group \(G\) is homeomorphic to \(\mathbb{R}^n\).

Answer: Yes


Question: Evaluate the contour integral \( \oint_{|z|=2} \frac{\sum_{k=0}^\infty (k^2/2^k)z^k}{z^3} \, dz \).

Answer: 2\pi i


Question: Suppose $a$, $b$, $p$, and $q$ are natural numbers such that $a < 300$, $\gcd(p, q) = 1$, and $a \neq 300b^2$. Determine if it is possible for \( k = \frac{900q^2 + ap^2}{3q^2 + b^2 p^2} \) to be an integer when $a$ and $b$ are fixed.

Answer: Yes


Question: Solve the partial differential equation \( y\frac{\partial{u}}{\partial{x}} + x\frac{\partial{u}}{\partial{y}} = 0 \) with the initial condition \( u(0,y) = e^{-y^2} \).

Answer: e^{x^2 - y^2}


Question: Determine whether the fact that a singleton \( \{x\} \) is a strong deformation retract of a topological space \( X \) implies that \( X \) is locally path connected at \( x \).

Answer: Yes


Question: Let $p$ be a prime number. Determine whether $p$ always divides the binomial coefficient $\binom{p^2+1}{p}$.

Answer: Yes


Question: Given two power series \(\sum _{n=1}^{ \infty} a_nz^n\) and \(\sum _{n=1}^{ \infty} b_nz^n\) with radii of convergence \(R_1\) and \(R_2\) respectively, find the radius of convergence for the power series \(\sum _{n=1}^{ \infty} d_nz^n\), where \(d_n = \sum _{k=1}^{n-1} a_kb_{n-k}\).

Answer: \min(R_1, R_2)


Question: Calculate the integral: \[ \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \]

Answer: \dfrac{\pi}{4}


Question: Find all homomorphisms from the dihedral group of order 14 to the cyclic group of order 2.

Answer: 2


Question: In a commutative ring with identity $R$, a local ring has exactly one maximal ideal, denoted by $m$. If $I$ is a proper ideal of $R$, is $I$ necessarily contained within $m$? Answer yes or no.

Answer: Yes


Question: Evaluate the integral \( \int_C \vec{F} \cdot d\vec{r} \) using Stokes' theorem, where \( \vec{F} = [-z, 5x, -y] \) and \( C \) is the ellipse defined by \( x^2 + y^2 = 4, z = x + 2 \). The direction of integration along \( C \) is clockwise as seen by a person standing at the origin.

Answer: -24\pi


Question: Given a compact, self-adjoint operator $T$ on a Hilbert space $\mathcal{H}$, decompose it as $T = T_+ + T_-$, where $T_+$ is positive, $T_-$ is negative, and $T_+T_- = 0$. Provide the method for this decomposition without using results about $C^*$-algebras.

Answer: T = T_+ + T_-


Question: Determine the cardinality of the set \( P(P(\Bbb{N})) \). Is it \( C \) or \( 2^C \)?

Answer: 2^C


Question: Find the characteristic of a ring $R$ (not necessarily unital) that satisfies the equation $r^4 - 2r^3 + r^2 = 0$ for all $r \in R$. Provide your answer as a single integer.

Answer: 2


Question: Compute the conditional expectation \( E(W_t \,|\, |W_T|) \) where \( W_t \) is a one-dimensional Brownian motion and \( t < T \).

Answer: 0


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

Answer: e


Question: Is it true that in the determinant expansion of an $n \times n$ matrix, no term contains exactly $n-1$ diagonal components?

Answer: True


Question: Let $(X_{n})_{n\geq1}$ be a sequence of i.i.d. random variables such that $\mathbb{P}(X_{n} = -1) = \frac{1}{3}$ and $\mathbb{P}(X_{n} = 1) = \frac{2}{3}$. Let $S_{0} = 0$ and for $n\geq 1$ let $S_{n} = \sum_{k=1}^{n} X_{k}$. Determine the value of $\mathbb{P}(\liminf S_{n} = -\infty)$.

Answer: 0


Question: Let $f$ be an integrable function on $[0,1]$. Determine $\lim_{n\to\infty} I_n$, where $I_n=\int_{0}^{1} \frac{f(x)}{1+nx}dx$. Provide your answer as a single value.

Answer: 0


Question: Determine the smallest integer \( n \) such that the Lagrange interpolation error \(|f(x) - Q_n(x)|\) for the function \( f(x) = e^x \) on the interval \([-4, 0]\) using equally spaced nodes is less than 0.1. Assume \( M_{n+1} = 1 \) and use the error bound formula \(|f(x) - P_n(x)| \leq \frac{M_{n+1}}{(n+1)!}|(x - x_0)\cdot...\cdot(x - x_n)|\).

Answer: 4


Question: Let $\mathcal{O}$ be a Dedekind domain and $I=(x_1,\ldots, x_n),J=(y_1,\ldots, y_m)\subseteq \mathcal{O}$ be two ideals. Determine if it is possible that $IJ\neq K$, where $K$ is the ideal generated by the products $x_iy_j$.

Answer: No


Question: Suppose $a: \mathbb{R}_+ \to \{-1,1\}$ is a measurable function. Given $X_0 = \frac{1}{2}$, consider a particle whose position $X_t$ on the $X$-axis evolves according to the integral equation $$X_t = X_0 + \int_0^t a_s \, ds$$ where the integral is a Lebesgue integral. Let $T = \frac{1}{2}$. Thus, $X_t \in [0,1]$ for all $t \le T$. Given a set $S \subset [0,1]$ with Lebesgue measure $\ell(S) = 1$, define $$G := \{t \le T: X_t \in S\}.$$ Determine if $\ell(G) = \ell([0,T]) = \frac{1}{2}$, implying the particle spends almost no time outside $S$. Provide a justification for your answer.

Answer: \ell(G) = \frac{1}{2}


Question: Determine whether a continuous function can take both finite and infinite values within a closed interval. Provide a justification for your answer.

Answer: No


Question: Determine if the limit of the expression \(|g(x)|^{f(x)}\) can be different from 1 as both \(g(x)\) and \(f(x)\) approach zero at a point \(c\).

Answer: Yes


Question: Determine the number of non-negative integer solutions to the equation \(x_1 + x_2 + x_3 + x_4 + x_5 = 45\) given the following conditions: \(x_1 + x_2 > 0\), \(x_2 + x_3 > 0\), \(x_3 + x_4 > 0\), and \(x_4 + x_5 > 1\).

Answer: 205669


Question: Evaluate the integral \( \int_0^{\pi/2} \frac{1}{1+\tan^{2017} x} \, dx \).

Answer: \dfrac{\pi}{4}


Question: Evaluate the double integral \( \int \int_R xy \, dA \), where \( R \) is the square with vertices \((0,0)\), \((1,1)\), \((2,0)\), and \((1,-1)\). Can this integral be simplified to \( \int_0^2 \int_{-1}^1 xy \, dy \, dx \)? Justify your answer.

Answer: 0


Question: Let $\{a_n\}$ be a positive sequence such that $a_{n+1} = a_n + \frac{n^4}{a_n}$. Determine the rate of growth of $a_n$. Provide your answer in terms of asymptotic notation.

Answer: \Theta(n^{5/2})


Question: Calculate the work done by the vector field \( \overrightarrow{F} = \left(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right) \) along a straight line from the point \((0,1)\) to the point \((1,0)\). Note that the potential function \( \phi = \arctan\left(\frac{x}{y}\right) \) is undefined at \((1,0)\).

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


Question: Evaluate the limit: $$\lim_{n \to \infty}n\sin(2\pi\sqrt{1+n^2}),n\in \mathbb{N}$$

Answer: \pi


Question: Calculate the integral \( \int_{D}||\nabla f||^2 \, dA \) where \( f(x,y)=y-x^2+1 \), and \( D \) is the region defined by \( \{(x,y)|f(x,y)\geq 0, y \leq 0\} \). Express \( \nabla f \) and \( ||\nabla f||^2 \), and determine the correct limits of integration for the double integral.

Answer: \dfrac{12}{5}


Question: Evaluate the limit using Taylor series expansion: $$\lim_{x\to0} \frac{xe^{x^2}-\sin x}{x^3}$$

Answer: \dfrac{7}{6}


Question: Given that \( \left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right)=1 \), find the value of \( x+y \).

Answer: 0


Question: Find the number of units in a finite integral domain that contains 25 elements.

Answer: 24


Question: Evaluate the integral \( \int_{-\pi/2}^{\pi/2}\tan(\sin(x))dx \) without using advanced techniques such as complex analysis or series representations.

Answer: 0


Question: Evaluate the limit: $$\lim_{x \to 0}\frac{1}{x^2} \int_{0}^{x^2}f(y)dy$$ where $f(x)$ is a continuous function on $\mathbb{R}$ and $\lim_{x \to 0}f(x)=a$. Find the value of the limit.

Answer: a


Question: Determine the number of real roots of the function \( f(x,\epsilon) = x^{2} + e^{-\epsilon x} - 2 + \sin(\epsilon x) \), where \( 0 < \epsilon \ll 1 \).

Answer: 2


Question: Determine the image of the ring homomorphism from \( \mathbb{R}[t] \) to \( \mathbb{C} \) given in the example.

Answer: \mathbb{C}


Question: Consider a function defined on an interval $[a,b]$. If the function is differentiable everywhere except at a countable number of points and satisfies the condition $\int_a^b|f'(x)|\ dx<\infty$, can we conclude that the function is of bounded variation? Provide a justification for your answer.

Answer: No


Question: Given a self-adjoint operator $T$ on a separable Hilbert space $H$ with a point spectrum consisting solely of eigenvalues, where eigenvectors corresponding to distinct eigenvalues are orthogonal, can a basis for $H$ be constructed consisting of eigenvectors of $T$?

Answer: Yes


Question: If \( A, B^T, C^T \in \mathbb{R}^{D \times d} \) are full-rank matrices with \( D \geq d \), does the equation \( BA = CA \) necessarily imply \( B = C \)?

Answer: No


Question: Given a joint probability density function $f_{X,Y}(x,y)$, find the probability $P(Y > 3X)$. Provide your answer as a numerical value.

Answer: \dfrac{1}{6}


Question: Given a natural number $N$, determine if there exists a positive integer $k$ and natural numbers $N < n_1 < n_2 < \cdots < n_k$ such that \( \frac{1}{n_1} + \frac{1}{n_2} + \cdots + \frac{1}{n_k} = 1 \).

Answer: Yes


Question: Find the points of inflection for the function \( f(x) = \frac{x(x-2)(x+2)}{(16-x^2)^{\frac{3}{2}}} \).

Answer: (0, 0)


Question: Solve the exponential equation for \(x\): $$1000.00116=\frac{1000}{(1+x)^{16}}+\frac{1-(1+x)^{-16}}{x}$$

Answer: 0.001


Question: Find a number that is divisible by all integers less than or equal to half of itself, but not divisible by any integer greater than half of itself.

Answer: 6


Question: Does there exist an infinite sequence of distinct primes \((q)\), with \(p \neq q\), such that the modular product \(\prod_{q \leq Q}q \pmod p\) asymptotically stays in a small subset of \(\{1, 2, \dots, p-1\}\) as \(Q\) increases, say a subset of size \(\delta p\) for some small constant \(\delta > 0\)?

Answer: Yes


Question: Is it true that for a sequence $(a_k)_{k\in\mathbb{Z}^n}$ in $C_0(\mathbb{Z}^n)$, if the $l^1$ norm of a function $f$ is infinite, then the supremum of $|f(a)|$ over all sequences $a$ in $C_0(\mathbb{Z}^n)$ with $\infty$-norm less than or equal to 1 is also infinite?

Answer: Yes


Question: Let $f: V \to V$ be a linear map where $V$ is a vector space with possibly infinite dimension, but $\dim \operatorname{Im}(f) < \infty$. Does there exist a polynomial $P$ such that $P(f) = 0$?

Answer: Yes


Question: Let $A/B$ be a cyclic group with infinite order, where $B$ is a normal subgroup of $A$. Prove that there exists a normal subgroup $C$ of $A$ such that the index $[A : C] = i$ for any positive integer $i$. Provide a construction or argument to demonstrate this.

Answer: C


Question: What is the remainder when \(43!\) is divided by \(89\)?

Answer: 21


Question: Evaluate the surface integral \(\iint\limits_S xy \sqrt{x^2+y^2+1}\,\mathrm d\sigma,\) where \(S\) is the surface cut from the paraboloid \(2z=x^2+y^2\) by the plane \(z=1\).

Answer: 0


Question: Evaluate the limit: $$ \lim_{x\to0}\sum_{n=1}^{\infty}\frac{\sin(x)}{4+n^2x^2} $$

Answer: \dfrac{\pi}{4}


Question: Determine the upper boundary of the function \( f_n(x) = \frac{\sin nx}{\sin x} \) for \( n \in \mathbb{Z} \), where \( x \) is not a rational multiple of \( \pi \). Express your answer in terms of a function.

Answer: |n|


Question: Determine the value of $c$ such that the equation \[ v(x) = 1 + 2 \int_0^1 xy v(y) \, dy \] has a solution of the form $v(x) = 1 + cx$. \( \text{Find } c. \)

Answer: 3


Question: Let $f$ be a real-valued differentiable function on $(0,1)$. Suppose that $f$ is bounded on $(0,1)$ and $\liminf_{r \to 0+} \,f'(r) = c$, where $c$ is a positive constant. Can we conclude that $f(0+)$ exists? Provide a justification for your answer.

Answer: Yes


Question: Find the maximum dimension of a subspace of endomorphisms of a 4-dimensional vector space over the complex numbers, where all elements of the subspace commute and at least one element has three distinct eigenvalues.

Answer: 4


Question: Determine whether there exist four natural numbers $a, b, c,$ and $d$ such that the following conditions are satisfied: \[ a^2 + b^2 + 2d^2 = c^2 \] \[ \sqrt{a^2 + d^2} \in \mathbb{N} \] \[ \sqrt{b^2 + d^2} \in \mathbb{N} \]

Answer: Yes


Question: Find the range of values of $k$ for which the equations $f(x)=x^2+kx=0$ and $f(f(x))=(x^2+kx)(x^2+kx+k)=0$ have the same set of real solutions.

Answer: [0, 4)


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

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


Question: If the category $C(M)$ is isomorphic to the category $C(N)$, does it follow that the monoid $M$ is isomorphic to the monoid $N$?

Answer: Yes


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

Answer: \dfrac{1}{6}


Question:
Evaluate the asymptotic expression for large $n$:
\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos^n(x) \tan(x) \, dx \]

Answer: 0


Question: The plane \(P_1\) is defined by the equation \(4x + 7y + 4z + 81 = 0\). It rotates through a right angle about its line of intersection with another plane \(P_2\) given by \(5x + 3y + 10z = 25\). The resulting plane after rotation is denoted by \(P\). If the distance of plane \(P\) from the origin is \(\sqrt{d}\), find the value of \(d\).

Answer: 212


Question: Evaluate the convolution integral: $$\int_{-\infty}^{+\infty} \delta'(a-t) g(t) \, dt$$ where \( \delta' \) is the derivative of the Dirac delta function and \( g(t) \) is a given function.

Answer: g'(a)


Question: Determine the possible range of values for the horizontal asymptote of the function \( f(x) \) as \( x \to -\infty \), given that \( \left| f(x)-\frac{1}{x}\right| \le \frac{2x^2+x|x|+2}{x^2+1} \) for all \( x \in \mathbb{R} \).

Answer: [-1, 1]


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

Answer: -\infty


Question: Determine the values of $s$ and $k$ for which there exists a ring isomorphism between a subset of size $s^k$ of the Galois ring $\Bbb Z_2^{s^k}$ and the Galois ring $\Bbb Z_s^k$. Provide your answer as a pair $(s, k)$. \( \boxed{} \)

Answer: (2, 1)


Question: Evaluate the integral: $$ \int_{-5}^5\int_{-5}^5 -\frac{3}{2}|x+y| - \frac{3}{2}|x-y| + 15 \,\mathrm{d}x\,\mathrm{d}y $$

Answer: 500


Question: Let $f:M\to\mathbb{R}$ be a function on a manifold $M$. If the Lie derivative of $f$ along a vector field $v$ vanishes at every point $p\in M$, does this imply that the one-parameter group of diffeomorphisms $\phi_t$ generated by $v$ preserves $f$, i.e., $f(\phi_t(p))=f(p)$ for all $t\in\mathbb{R}$ and $p\in M$? Provide a justification for your answer.

Answer: Yes


Question: Let $(X,d)$ be a metric space, and let $f_n: X \to \mathbb{R}$ be a sequence of continuous functions such that $f_n \to f$ uniformly on $X$. Suppose $x_n$ is a sequence of points in $X$ with $x_n \to x$. Prove that $f_n(x_n) \to f(x)$.

Answer: f_n(x_n) \to f(x)


Question: Determine the cardinality of the quotient set of the equivalence relation defined on the set $\mathbb{Z}^{\mathbb{N}}$ of all infinite sequences of integers, where $f \equiv g$ if and only if $((f(2n) = g(2n)) \wedge (f(n) \cdot g(n) > 0 \vee f(n) = 0 = g(n)))$ for $f, g \in \mathbb{Z}^{\mathbb{N}}$. What is the power of this quotient set?

Answer: 2^{\aleph_0}


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

Answer: -2\pi i


Question: Determine whether the number of non-zero quadratic residues modulo $2p$ or $3p$ (where $p$ is an odd prime) is always equal to $p$.

Answer: Yes


Question: Are the open unit disk and the upper half-plane diffeomorphic when using polar coordinates?

Answer: Yes


Question: Determine whether the problem of choosing $x$ and $y$ to minimize $(x - 1/y)^2$ with the constraint $y > 0$ can always be solved by instead minimizing $(xy - 1)^2$ under the same constraint.

Answer: Yes


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

Answer: \max(a, 1)


Question:
Consider a weighted graph $G$ with edge weights greater than one. A new graph $G'$ is obtained by squaring the weights of the edges in $G$. Let $T$ and $T'$ be the minimum spanning trees of $G$ and $G'$, respectively, with total weights $t$ and $t'$. Which of the following statements is generally true?

A) $T' = T$ with total weight $t' = t^2$
B) $T' = T$ with total weight $t' < t^2$
C) $T' \neq T$ but total weight $t' = t^2$
D) None of the above

Answer: D


Question: Given the vector $y = \begin{bmatrix} R(t) \\ F(t) \end{bmatrix}$ representing the populations of rabbits $R(t)$ and foxes $F(t)$, which evolve according to the system of differential equations $\dot y = Ay$, where $A = \begin{bmatrix} 98 & -1170 \\ 7 & -83 \end{bmatrix}$. The initial rabbit population is $R(0) = 53,300$. Determine the initial number of foxes $F(0)$ needed for the rabbit population to grow exponentially as $R(t) = R_0 e^{8t}$, assuming the eigenvalues of $A$ are $\lambda = 8$ and $7$. Find $F(0)$.

Answer: 4100


Question: Let $F$ be a field, $V$ a vector space over $F$ with finite dimension, and $T$ a linear operator on $V$. Suppose $T$ is diagonalizable and $c_1,c_2,\ldots,c_n$ are distinct eigenvalues of $T$. If the set $\{id_V, T, T^2, \ldots, T^d\}$ is linearly independent, prove that $d < r$, where $r$ is the number of distinct eigenvalues of $T$.

Answer: d < r


Question: Given a polynomial function $P(x)$ such that $P(x^2) = x^2(x^2 + 1)P(x)$ and $P(2) = 3$, find the value of $P(3)$. Express your answer as a single number.

Answer: 18


Question: Compute the limit: \[ \lim_{n\to\infty}\frac{1}{\log n}\sum_{k=1}^n\left(1-\frac{1}{n}\right)^k\frac{1}{k} \] given that \[ \log x = \sum_{k=1}^{\infty}\left(1-\frac{1}{x}\right)^k\frac{1}{k},\quad x\ge \frac{1}{2}. \]

Answer: 1


Question: Evaluate the limit: $$\lim_{n \rightarrow \infty} \int_0^1 \frac{\sqrt{n}\sin(x)}{1+n\sqrt{x}}\,dx$$

Answer: 0


Question: In a cyclic quadrilateral $ABCD$, where $AB = AD$, $AC = 6$, and the ratio $\frac{AB}{BD} = \frac{3}{5}$, determine the maximum possible area of quadrilateral $ABCD$. Express your answer as a single number.

Answer: 5\sqrt{11}


Question: Solve the equation \( \sqrt{x}^{\sqrt{x}^{\sqrt{x}}} = 2^{512} \).

Answer: 16


Question:
Find the limit of the sequence \(s_n\) as \(n\) approaches infinity, where \(s_n\) is defined for \(n \geq 1\) as:
\[s_n = \int_0^1 \frac{nx^{n-1}}{1+x} \, dx\]

Answer: \dfrac{1}{2}


Question: Given a real matrix $A$ such that $x^T A x > 0$ for every non-zero real vector $x$, determine whether it is true that $\det(A) > 0$. (Note that $A$ is not required to be symmetric.)

Answer: True


Question: Determine the number of integer values of $k$ such that the equation $x^3 - 27x + k = 0$ has at least two distinct integer roots.

Answer: 2


Question: Let $G$ be a finite group and $p$ be a prime divisor of the order of $G$. Determine whether it is true that for all $1 \leq j \leq k$, where $p^k$ is the highest power of $p$ dividing the order of $G$, there exists a subgroup $H$ of $G$ such that $|H| = p^j$. Provide a justification for your answer.

Answer: True


Question: Determine whether the sequence \(\left(\dfrac{1}{n^3 \cos n}\right)\) converges or diverges. If it converges, find its limit.

Answer: 0


Question: Determine the minimum distance between the centers of two ellipses, $E_1$ and $E_2$, with centers $c_1$ and $c_2$, and semi-major axes $m_1$ and $m_2$, respectively, such that the ellipses do not touch or intersect.

Answer: m_1 + m_2


Question: Calculate the limit: $$\lim_{n \to \infty} \frac{\frac{\sin(1)}{1} + \frac{\sin(2)}{2} + \frac{\sin(3)}{3} + \ldots + \frac{\sin(n)}{n}}{n}$$

Answer: 0


Question: Evaluate the limit \( \lim_{x\to\infty}\frac{e}{x}\left(1-\frac{1}{2 x}+\frac{11}{24 x^2}+O\left(\frac{1}{x^3}\right) \right) \).

Answer: 0


Question: Find all sets of three distinct integers \(a, b, c\) such that \(W(a) = W(b) = W(c)\), where \(W(x) = x^4 - 3x^3 + 5x^2 - 9x\).

Answer: \emptyset


Question: Consider a sequence of real numbers $\{a_n\}$ such that the limit $\lim_{n\to \infty}\frac{\sum_{j=1}^n a_j^2}{n}=\rho$ exists, where $0\le\rho<1$. Determine whether the infinite product $\prod_{j=1}^n a_j$ converges to $0$ as $n$ approaches infinity.

Answer: 0


Question: Evaluate the line integral \(\int_C (x+y)^2\,dx - (x^2+y^2)\,dy\) where \(C\) is the positively oriented triangle with vertices at \((1,1)\), \((3,2)\), and \((2,5)\).

Answer: -\dfrac{140}{3}


Question: Given two injective functions $f : A \to B$ and $g : B \to A$, determine whether there exists a bijective function $h : A \to B$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the limit: $$\lim_{x \to 0^+}\frac{\sin^{3/2}x}{x^{3/2}}\frac{\cos^{3/2}x-1}{x^2}$$ without using L'Hôpital's Rule.

Answer: -\dfrac{3}{4}


Question: Determine the minimum value of the positive integer $N$ given by the equation $N = 3a^2 - ab^2 - 2b - 4$, where $a$ and $b$ are positive integers.

Answer: 2


Question: Determine whether the map \( \varphi: \mathbb{R} \rightarrow \mathbb{R}[x]/\langle x^2+1 \rangle \) defined by \( \varphi(a) = a+\langle x^2+1 \rangle \) is a ring homomorphism. If it is, find its kernel.

Answer: \{0\}


Question:
Find the minimum value of the integral:

$$\int_{0}^{\infty} |x^3−a−bx−cx^2|^2 e^{-x}dx$$

where $a, b, c \in \mathbb{R}$. Use concepts from Hilbert spaces and functional analysis to determine the solution.

Answer: 36


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

Answer: 0


Question: Calculate the surface area of the region of the plane \(2x + y + 2z = 16\) that is bounded by the surfaces \(x = 0\), \(y = 0\), and \(x^2 + y^2 = 64\).

Answer: 24\pi


Question: Determine the maximum order of a permutation in the symmetric group $S_7$. Additionally, provide a specific permutation in $S_7$ that achieves this maximum order.

Answer: 12


Question: Calculate the limit: $$\lim_{n \to \infty}\sum\limits_{k=1}^n \left(\dfrac {n-k}{n}\right)^{tk}$$ where $t\in \mathbb{R}$. \(t\) is a real number.

Answer: \infty


Question: Suppose $A$ is a $3 \times 3$ non-singular matrix such that $(A−3I)(A−5I)=O$, where $I$ is the identity matrix and $O$ is the zero matrix. If $\alpha A + \beta A^{-1} = 4I$, find the value of $\alpha + \beta$. \( \boxed{\ } \)

Answer: 8


Question: Calculate the residue of the function \( f(z) = ze^{\frac{1}{z-1}} \) at the point \( z = 1 \). Use this residue to evaluate the integral \( I = \int_{C} ze^{\frac{1}{z-1}}dz \) where \( C = C(1, \frac{1}{2}) \) and \( I = 2\pi i \operatorname{Res}(f(z), 1) \).

Answer: 3\pi i


Question: On the Argand plane, let $a, b, c,$ and $d$ be complex numbers representing points $A, B, C,$ and $D$, respectively, which lie on a circle centered at the origin. If the chords $AB$ and $CD$ are perpendicular, determine the value of $ab + cd$. \( \boxed{} \)

Answer: 0


Question: Evaluate the integral \( \frac{i}{4-\pi} \int_{|z|=4} \frac{dz}{z \cos z} \) using the residue theorem, given that the function \( f(z)=\frac{1}{z \cos z} \) has simple poles at \( z=0 \) and \( z=\frac{(2n-1)\pi}{2} \) for \( n\in\mathbb{Z} \).

Answer: 2


Question: Let $(X, \mathcal{U})$ be a uniform space. If the diagonal $\Delta_X$ is a closed $G_\delta$ subset of $X\times X$, is $X$ necessarily metrisable? Answer yes or no.

Answer: Yes


Question: Evaluate the series \( \frac{1}{2} \sum_{n \in B} \frac{(\log 3)^n}{n!} \), where \( B \) is the set of all positive even numbers.

Answer: \dfrac{1}{3}


Question: Evaluate the improper integral \(\int_0^\infty \left( \frac{x^{10}}{1+x^{14}} \right)^{2} \, dx\) by rewriting it as a limit: \(\lim \limits_{b \to \infty} \int_0^b \left(\frac{x^{10}}{1+x^{14}} \right)^{2} \, dx\). Find the value of this integral.

Answer: \dfrac{\pi}{28}


Question: Find the maximum value of the function \( f(x,y,z) = y \) subject to the constraints \( y^2 + x^2 + z^2 = 3 \) and \( y + x + z = 1 \).

Answer: \dfrac{5}{3}


Question: Given that \( X \sim \mathrm{Geo}(0.09) \) and \( Y|X=x \sim \mathrm{Geo}(1/(x+1)) \), calculate \( \mathbb{P}(\mathbb{E}(Y|X)=3) \).

Answer: 0.0819


Question: Evaluate the limit: $$\lim_{k \to \infty} \int_0^\infty e^{-kx^2} \, dx.$$

Answer: 0


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

Answer: 0


Question: Find the minimal polynomial of $\sqrt[5]{2}$ over the field $\mathbb{Q}(e^{2\pi i/5})$. Provide your answer as a polynomial with rational coefficients.

Answer: x^5 - 2


Question: Evaluate the improper integral \( \int_{0}^{\infty }\!{\frac {1-\cos \left( x \right) }{{x}^{2}}} \,{\rm d}x \).

Answer: \dfrac{\pi}{2}


Question: If a sequence of functions \( \{f_n\} \) converges uniformly to a function \( f \) on \( \mathbb{R} \), and there exists a constant \( M > 0 \) such that \( |f_n(x)| \leq M \) for all \( n \in \mathbb{N} \) and \( x \in \mathbb{R} \), does it follow that the sequence \( (f_n)^2 \) converges uniformly to \( f^2 \) on \( \mathbb{R} \)? Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: Let $(x_n)_{n \in \mathbb{N}}$ be a sequence such that for all $\gamma > 0$, there exists $m_\gamma \in \mathbb{N}$ with $\frac{1}{x_n^\gamma} < n$ for all $n > m_\gamma$. Then there exists $c > 1$ such that $$\sum_{n\geq 1} c^n x_n = \infty.$$

Answer: True


Question: Given two independent random variables $X$ and $Y$, both uniformly distributed between $0$ and $1$, with joint density function $f_{xy}(x, y) = 1$ for $0 \leq x \leq 1$ and $0 \leq y \leq 1$, and $f_{xy}(x, y) = 0$ otherwise. If $W = \frac{X + Y}{2}$, determine the conditional distribution of $X$ given $W = w$. Express $X$ in terms of $W$ and $Y$.

Answer: X = 2W - Y


Question: In a Hilbert space, if \( \| x - x_n \|^2 \to 0 \), does it imply that \( x_n \to x \)? Provide a justification for your answer.

Answer: Yes


Question: For a fixed positive integer $m$, determine if the inequality \( H_m = \liminf_{n \to \infty} (p_{n + m} - p_n) > p_{m + 2} \) holds for sufficiently large $n$. Answer "yes" or "no".

Answer: no


Question: Consider the polynomial \( f(x) = x^7 - 4x^3 + x + 1 \). Determine the number of roots of this polynomial that have a modulus less than 1 in the complex plane, taking into account the multiplicity of repeated roots.

Answer: 3


Question: Evaluate the limit: $$\lim _{n\to \infty }\left(n\cdot \sqrt[n]{\cos\left(\frac{\pi}{2}\right)\cdot \cos\left(\frac{\pi}{4}\right)\cdot \cdot \cdot \cos\left(\frac{\pi}{2^n}\right)}\right)$$

Answer: 0


Question: Calculate the area of the region in the first quadrant of the Cartesian plane defined by the inequalities $0 \leq x \leq \sqrt{y}$, $0 \leq y \leq \sqrt{x}$, and $x + y \leq \frac{3}{4}$.

Answer: \dfrac{13}{96}


Question: Determine whether any open subset of the interval [0, 1] can be expressed as a countable union of disjoint open intervals.

Answer: Yes


Question: Let $A$ be a Lebesgue null set in $\mathbb{R}$. Determine whether there exists a set $B$ such that: 1) $A \subset B$, 2) $B$ has measure $0$, and 3) $B$ is an $F_{\sigma}$ set (i.e., a countable union of closed sets).

Answer: No


Question: In an acute triangle $\triangle ABC$, where $\angle C = 60^{\circ}$, perpendiculars $AA_1$ and $BB_1$ are drawn from vertex $A$ to side $BC$ and from vertex $B$ to side $AC$, respectively. If $M$ is the midpoint of $AB$, find the ratio $\frac{\angle A_1MB_1}{\angle A_1CB_1}$.

Answer: 1


Question: Evaluate the limit: $$\lim\limits_{x\to 0^+}[1+[x]]^{\frac2x}$$ where $[x]$ denotes the greatest integer function less than or equal to $x$.

Answer: 1


Question: Suppose \( f(z) \) is analytic at \( z=0 \) and satisfies the equation \( f(z) = z + (f(z))^2 \). Determine the radius of convergence of the power series expansion of \( f(z) \) about \( z=0 \).

Answer: \dfrac{1}{4}


Question: Evaluate the integral \( \displaystyle \int_{0}^{\frac{\pi}{2}} \frac{1}{1+\tan(x)^{\beta}}\,dx \), where \( \beta \) is a real number.

Answer: \dfrac{\pi}{4}


Question: Given the matrix \( T = \begin{pmatrix} k-3 & 2 & 0 \\ 2 & 0 & k \\ 0 & k & -3 \end{pmatrix} \) and a matrix \( B \) with \( \det B = 5 \), let \( M = 3B + BT \). Find the value of \( k \) for which \( M \) is invertible.

Answer: k \neq 0


Question: Determine the number of non-symmetric and non-singular matrices $A$ of size $n \times n$ that satisfy the equation $A^{T} = A^2 - I$, where $I$ is the identity matrix of size $n \times n$. Provide your answer as a single integer.

Answer: 0


Question: Does the uniform continuity of a function $f$ on $\mathbb{R}$ imply the existence of a constant $C$ such that the supremum of $|f(y) - f(x) - C|x - y||$ over all $(x, y) \in \mathbb{R} \times \mathbb{R}$ is finite?

Answer: No


Question: Consider the differentiable function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ such that $f(0, -2) = 3$ and $\nabla f(0, -2) = (-6, 2)$. Let $g(t) = ||(t, t^2, f(3t, t - 2))||$. Determine $g'(0)$. \( \boxed{} \)

Answer: -16


Question: Determine whether there exists a $C^{\infty}$ function $f$ such that the radius of convergence of the Taylor series \( \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n \) is 0.

Answer: Yes


Question: Consider the extension \( \mathbb{F}_2(t)(\alpha) \) of degree 4 over \( \mathbb{F}_2(t) \), where the minimal polynomial of \( \alpha \) is given by \( \alpha^4 + a\alpha^2 + b = 0 \) with \( a, b \in \mathbb{F}_2(t) \). Determine whether \( \mathbb{F}_2(t)(\alpha) \) always contains an intermediate field \( L \) such that \([L: \mathbb{F}_2(t)] = 2\) and \( L \) is inseparable over \( \mathbb{F}_2(t) \).

Answer: No


Question: Determine whether the sequence \(x_n = (n + i\pi)^n n^{-n + \frac{1}{n}}\) converges. If it converges, find its limit.

Answer: -1


Question: Determine whether there exists a polynomial with rational coefficients \( P(x) \) such that \( \frac{P(n+1)}{P(n)} \in \mathbb{N} \) for every positive integer \( n \). If such a polynomial exists, describe how it can be constructed.

Answer: P(x) = c


Question: Determine whether the sequence of fractional parts \( \{\log_{10}(n!)\}_n \) is dense in the interval \([0,1]\).

Answer: Yes


Question: Find the limit of \( \lim_{n\to \infty}\sin(\pi(2+\sqrt{3})^n) \) where \( n \in \mathbb{N} \).

Answer: 0


Question: Suppose $p(x)$ is an irreducible polynomial in $\mathbb{Z}[x]$. If $p(x)$ has a root $z$ such that $|z| \neq 1$, determine whether $p(x)$ can be a reciprocal polynomial.

Answer: Yes


Question: Given that \(|ax^2 + bx + c| \le 1\) for all \(x\) in the interval \([0,1]\), determine the values of \(|a|, |b|, |c|\), and compute \(|a| + |b| + |c|\).

Answer: 17


Question: Is it sufficient to consider only the elements \( \beta \) with \( \beta_k \in \{-1, 1\} \) in the supremum when computing the norm of \( x \in \ell^1 \)?

Answer: Yes


Question: Find a closed-form formula for the sequence defined by \( f(n) = 1\binom{n}{1} + 2\binom{n}{2} + 3\binom{n}{3} + \dots + n\binom{n}{n} \).

Answer: n \cdot 2^{n-1}


Question: Determine whether there exists a vector $y$ such that for any matrix $B$, the inequality \( \|By\| \geq \|B\|\|y\| \) holds.

Answer: 0


Question: Determine all values of the parameter $a$ such that the integral \( \int_0^{+\infty} \frac{dx}{1+x^a \sin^2x} \) converges.

Answer: a > 2


Question: Let $x$ be a real number. Suppose that the identity $f(x)-g(x)=h(x)$ holds if and only if $x>a$. Suppose also that for $x>a$, $g(x)$ is identical to some function $j(x)$ such that $f(x)-j(x)=h(x)$ holds for all $x>b$. Does it necessarily follow that $a\leq b$? Answer yes or no.

Answer: No


Question: Find the element in \( \mathbb{Z}/2550\mathbb{Z} \) that corresponds to \((\bar{14}, \bar{32})\) under the Chinese Remainder Theorem, given the system of congruences: \(x \equiv 14 \pmod{50}\) and \(x \equiv 32 \pmod{51}\).

Answer: 1664


Question: Find all values of \( \cos(i) \), where \( i = \sqrt{-1} \).

Answer: \dfrac{e + e^{-1}}{2}


Question: Find the limit of the function \(f(x,y) = \frac{x^3y^4}{(x^4 + y^2)^2}\) as \((x,y) \to (0,0)\).

Answer: 0