Question: Given the function for accumulated value over time, \( a(t) = xt^{2} + yt + z \), where \( 100a(2) = 152 \) and \( 200a(4) = 240 \), find the accumulated value at time 8 of an investment of $1600 made at time 6. Use the conditions to determine the constants \( x, y, \) and \( z \), and then calculate the accumulated value at time 8.

Answer: -78400


Question: Calculate the volume of the torus formed when the region defined by \((x-4)^2 + y^2 \leq 4\) is revolved around the y-axis.

Answer: 32\pi^2


Question: Given that \(a^{40} + 6a = 7\) and \(a\) is a negative real number, find the sum \(1 + a + a^2 + \ldots + a^{39}\).

Answer: -6


Question: In the abelian free group $A$ with basis $x_1, x_2, x_3$, let $B$ be the subgroup generated by $x_1 + x_2 + 4x_3$ and $2x_1 - x_2 + 2x_3$. Determine the order of the coset $(x_1 + 2x_3) + B$ in the quotient group $A/B$. Provide your answer as an integer.

Answer: 3


Question: Determine the number of roots of the polynomial \( p(x) = x^{12} + x^8 + x^4 + 1 \) in the finite field \( \mathbb{F}_{11^2} \).

Answer: 4


Question: Two players each have a randomly shuffled deck of 52 cards. They compare the cards in order, one at a time, checking if there is a match in rank (suit doesn't matter). What is the probability that there is no match in rank for any of the 52 card comparisons?

Answer: e^{-4}


Question: Determine the number of integers \( n \) such that \( 4 \leq n \leq 2019 \) and both \( \lfloor \sqrt{n} \rfloor \mid n \) and \( \lfloor \sqrt{n+1} \rfloor \mid n+1 \) hold true.

Answer: 42


Question: Does the sifting property of the Dirac delta measure hold at the limits of integration? Specifically, evaluate whether \( \int_{[a,b]} f(x) \delta_b(dx) = f(b) \).

Answer: f(b)


Question: Find the limit of the series: \[ \lim_{n\to\infty} \sum^n_{k=0} \frac{k+1}{3^k} \]

Answer: \dfrac{9}{4}


Question: Determine the values of \( \alpha > 0 \) for which the sequence of functions \( f_n(x) = n^{\alpha} h(nx) \) converges to 0 \( \lambda \)-almost everywhere on \( \mathbb{R} \), where \( \lambda \) is the Lebesgue measure and \( h \in C^{\infty} (\mathbb{R}) \) with compact support.

Answer: \alpha > 0


Question: Calculate the number of possible arrangements of 5 tigers, 5 lions, and 5 cheetahs in 15 consecutive cages, given the following restrictions: tigers cannot be placed in the 5 leftmost cages, lions cannot be placed in the 5 middle cages, and cheetahs cannot be placed in the 5 rightmost cages.

Answer: 2252


Question:
Consider the set \( S = \{ T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} \mid T \text{ is a linear transformation with } T(1,0,1)=(1,2,3), T(1,2,3)=(1,0,1) \} \). Determine the nature of the set \( S \):

(a) A Singleton Set
(b) A finite set containing more than one element
(c) A countable set
(d) An uncountable set

Select the correct option.

Answer: D


Question: Find the set of all real values of \(a\) for which the equation \((a-4)\sec^4x+(a-3)\sec^2x+1=0, (a\neq4)\) has real solutions.

Answer: [3, 4)


Question: In a metric space $(M, d)$, consider two closed sets $A$ and $B$ such that $A \cap B = \emptyset$. Is it possible for the distance between $A$ and $B$, defined as $\text{dist}(A,B) = \inf\{d(x,y) : x \in A, y \in B\}$, to be zero?

Answer: Yes


Question: Evaluate the integral \( \iint_{R}(x+y)^2 \, dx \, dy \) over the region defined by \( 0 \leq r \leq 1 \) and \( \frac{\pi}{3} \leq \theta \leq \frac{2\pi}{3} \) in polar coordinates. Express your answer in terms of \( \pi \).

Answer: \dfrac{\pi}{12}


Question: Given the function \( f(x) = \cos x \cdot \cos 2x \cdot \cos 4x \cdot \cos 8x \cdot \cos 16x \), find \( f'(\pi/4) \).

Answer: \sqrt{2}


Question: Calculate \( g'(0) \) for the function \( g(x) = \int_{0}^{2^x} \sin(t^2) \, dt \).

Answer: \ln(2) \sin(1)


Question: Determine the maximum possible number of connected components formed by the complement of the union of four hyperplanes in 3-dimensional space over the set of real numbers \( \mathbb{R} \).

Answer: 15


Question: Let $H$ be a hyperplane in an $n$-dimensional complex projective space, and let $C$ be a smooth curve of positive genus. Determine if $\deg((H|_C)^{\otimes 2}) = 2\deg(H|_C)$. Provide a justification for your answer.

Answer: Yes


Question: Compute \( \lim_{n\to\infty }\int_E \sin^n(x)\,dx \) where \( E \) is a Lebesgue measurable set of finite measure.

Answer: 0


Question: Find the limit \( L = \lim\limits_{x \to \infty} \left( x \ln x + 2 x \ln \sin \frac{1}{\sqrt{x}} \right) \).

Answer: -\dfrac{1}{3}


Question: Let $a_1, a_2, \ldots, a_n$ be real numbers such that \( \sqrt{a_1} + \sqrt{a_2 - 1} + \cdots + \sqrt{a_n - (n-1)} = \frac{1}{2}(a_1 + a_2 + \cdots + a_n) - \frac{n(n-3)}{4} \). Find the sum of the first 100 terms of the sequence.

Answer: 5050


Question: Determine whether there exists a strictly increasing sequence $\{a_k\}_{k\in\mathbb N}\subset\mathbb N$ such that \( \sum_{k=1}^{\infty}\frac{1}{(\log a_k)^{1+\delta}} < \infty \), where $\delta>0$ is given, and \( \lim_{k\to \infty }\frac{a_{k+1}}{a_k}=1.\)

Answer: Yes


Question: Determine the degree of the field extension $[F(\alpha) : F(\alpha^3)]$, where $E$ is a field extension of $F$ and $\alpha \in E$. Provide your answer as an integer.

Answer: 3


Question: Find the maximum value of the function \( f(x,y,z) = x+y+z \) on the intersection of the surfaces \( x^2 + y^2 + z^2 = 2 \) and \( x^2 + y^2 - z = 0 \).

Answer: 1 + \sqrt{2}


Question: Let \( g(x) = \sin(1/x) \) for \( x \neq 0 \) and \( g(0) = 0 \). Define \( G(x) = \int_{0}^{x} g(t) \, dt \). Show that \( G'(0) = g(0) \).

Answer: 0


Question: Let \( f \) be an analytic function. Determine the value of \( \int_{0}^{2\pi} f(e^{it}) \cos t \, dt \).

Answer: \pi f'(0)


Question:
Calculate the curvature \( k(t) \) of the vector function \( \mathbf{r}(t) = \langle \cos(2t), -\sin(2t), 4t \rangle \) using the formula:
\[ k(t) = \frac{||\mathbf{r}'(t) \times \mathbf{r}''(t)||}{||\mathbf{r}'(t)||^3}. \]

Answer: \dfrac{1}{5}


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

Answer: 12


Question: Determine whether there exists a nonzero function \( f \) that is analytic on \( \mathbb{C} \setminus \mathbb{D} \) and belongs to \( L^p(\mathbb{C} \setminus \mathbb{D}) \) for \( 1 \leq p < \infty \). Here, \( \mathbb{D} \) is the closed unit disk in \( \mathbb{C} \), defined as the set of all \( z \in \mathbb{C} \) such that \( |z| \leq 1 \).

Answer: Yes


Question: Evaluate the integral: \[ \int_0^\infty \frac{1}{(1+x^{2015})(1+x^2)} \, dx \]

Answer: \dfrac{\pi}{4}


Question: Let $X$ be a compact Hausdorff space and $f: X \rightarrow X$ be a continuous function. Suppose there exists a point $x \in X$ such that $\bigcap_{n \geq 0} f^n(X) = \{x\}$. Prove that for all $y \in X$, the sequence $f^n(y)$ converges to $x$ as $n \to \infty$.

Answer: x


Question:
Consider the function defined as follows:
\[ f(x) = \begin{cases} 0 & x\;\text{is computable} \\ 1 & \text{otherwise} \end{cases} \]
Evaluate the definite integral:
\[ I = \int_0^1 f(x)\;\mathrm{d}x \]
Given that the set of computable numbers is countable, it follows that \( I = 1 \). Determine whether any numerical method will always yield \( I = 0 \) due to the computable nature of any \( x \) a computer can generate.

Answer: 0


Question: Find the coefficient of \(z^{-1}\) in the Laurent series expansion of \(f(z) = \frac{e^z}{\sin^2(z)}\) centered at the origin.

Answer: 1


Question: Are all faces of a closed convex cone in finite dimensions closed? Let $C$ be a convex closed cone in $\mathbb{R}^n$. A face of $C$ is a convex sub-cone $F$ such that whenever $\lambda x + (1-\lambda)y\in F$ for some $\lambda \in (0,1)$ and $x,y\in C$, it holds that $x,y\in F$. Determine if all faces of $C$ are closed.

Answer: Yes


Question: Given a set of vectors \(\{v_1, \dots , v_m\}\) in \(\mathbb{R}^n\) with \(m \geq n\) that generates the entire space, determine if there exist indices \(i_1, \dots , i_n\) such that the subset \(\{v_{i_1}, \dots , v_{i_n}\}\) forms a basis for \(\mathbb{R}^n\).

Answer: Yes


Question:
Consider the random vector \( Y = \begin{bmatrix} Y_1 \\ Y_2 \end{bmatrix} \) and the set of random variables \( X = \{X_1, \ldots, X_n\} \). Determine the correct expression for \( Cov(Y) \) using the law of total covariance. Is it:

1. \( Cov(Y) = E(Cov(Y|X)) + Cov(E(Y|X)) \)
2. \( Cov(Y) = E(Cov(Y|X)) + Cov(E(Y_1|X), E(Y_2|X)) \)

Choose the correct expression.

Answer: 1


Question: Is it possible for a sequence of continuous functions to converge pointwise to a non-continuous function? Provide a justification for your answer.

Answer: Yes


Question: Determine the number of group structures that can make the circle $S^1$, considered as a subspace of $\mathbb{R}^2$ with the usual topology, into a topological group.

Answer: 1


Question: Show that the function \( g(x) = -x\sin^2\left(\frac{1}{x}\right) \) for \( 0 < x \leq 1 \) has a unique fixed point on the interval \([0, 1]\).

Answer: 0


Question: Let $G$ be a semi-simple Lie group with a bi-invariant Riemannian metric. For a fixed element $g \in G$, does the left-translation map $h \mapsto g \cdot h$ define an isometry on $G$ with respect to this metric?

Answer: Yes


Question: Given a probability density function (PDF) with a minimum value of $243.483K$ and a maximum value of $308.05K$, calculate the probability that the temperature lies between $281K$ and $291K$. Assume the PDF is defined over the range from $243.483K$ to $308.05K$.

Answer: 0.155


Question: Prove that the integral \( I(f) = \int_\mathbb{R} f(x) \log \left(f(x) \sqrt{2\pi} e^{\frac{x^2}{2}}\right) dx \) is non-negative, given that \( \int_{\mathbb{R}} f(x)\, dx =1 \) and \( f \geq 0 \) almost everywhere. Use the hint that \( x\log x \geq x -1 \) for \( x\geq 0 \).

Answer: I(f) \geq 0


Question: Compute the value of the integral \( \int_0^1 \int_0^1 f(x,y) \, dx \, dy \), where \( f(x,y) = \begin{cases} xy & \text{if } x = y \\ 0 & \text{if } x \neq y \end{cases} \).

Answer: 0


Question:
Consider the following statements about non-negative definite matrices. Determine which of the following statements are true:

(a) If a real symmetric $n \times n$ matrix is non-negative definite, then all of its eigenvalues are non-negative.

(b) If a real symmetric $n \times n$ matrix has all its eigenvalues non-negative, then it is non-negative definite.

(c) If $A \in \mathbb{M}_n(\mathbb{R})$, then $AA^T$ is non-negative definite.

Select the correct answer from the options below:

1. Only (a) is true.
2. Only (b) is true.
3. Only (c) is true.
4. All of the above statements are true.

Answer: D


Question: Determine the largest number of non-isomorphic abelian groups of order 10,000.

Answer: 25


Question: If $G_0=G$ is a group generated by $X_0=X$, and for each $n>0$, $G_n=[G_{n-1},G_{n-1}]$ and $X_n=[X_{n-1},X_{n-1}]$, determine if $G_n$ is trivial whenever $X_n$ is trivial.

Answer: Yes


Question: Determine the values of $z \in \mathbb{C}$ for which the series $\sum_{n = 1}^{\infty} \frac{z^n}{n^2}$ converges.

Answer: |z| \leq 1


Question: Determine whether the statement \( 2^{\frac{1}{3}} \in \mathbb{Q}(4^{\frac{1}{3}}) \) is true or false.

Answer: True


Question: Show that $z$ can be expressed as a function of $x$ and $y$ in the equation $x + y + z = \sin(xyz)$ near $x = y = z = 0$. Then, find the partial derivatives of $z$ with respect to $x$ and $y$ at this point.

Answer: -1


Question: Find the residue of the function \( f(z) = \frac{e^z - 1}{\sin z} \) at \( z_0 = 0 \).

Answer: 0


Question: Evaluate the line integral of the vector field \( F(x,y) = \langle -y, x \rangle \) over the path defined by the triangle with vertices \( A(-2,-2) \), \( B(2,-2) \), and \( C(0,1) \).

Answer: 12


Question: Evaluate the limit: \[ \lim_{x \to 0} \left( \frac{e^x - x^x}{x^2} \right) \] without using L'Hôpital's rule.

Answer: \infty


Question: Calculate the limit: \[ \lim_{x\to\infty}\frac{x^{\ln x}}{(\ln x)^x} \]

Answer: 0


Question: Is every orientable circle bundle a principal bundle?

Answer: Yes


Question: Suppose $N$ is a Poisson random variable with parameter $\mu$. Given $N=n$, the random variables $X_1, X_2, \ldots, X_n$ are independent and uniformly distributed over $(0,1)$. What is the probability that all the $X_i$ are less than $t$, given $N=n$?

Answer: t^n


Question: Determine whether every bilinear symmetric form on a finite-dimensional vector space admits a self-adjoint map representation. Specifically, given a bilinear symmetric form \( Q(x,y) \) on a vector space \( V \), does there exist a linear self-adjoint map \( K \) such that \((K(x),y) = Q(x,y)\) for all \( x, y \in V \)?

Answer: Yes


Question: Evaluate the infinite series \( S = \sum_{k=1}^\infty \frac{k^{k-1} e^{-k}}{k!} \) and determine its value.

Answer: 1


Question:
Find the value of \(x+y+z\) for integer solutions \((x, y, z)\) to the Diophantine equation:
\[ xy^2 + yz^2 + zx^2 = x^2y + y^2z + z^2x + x + y + z \]
where \((x-y)(y-z)(z-x) = x+y+z\).

Answer: 0


Question: In triangle \(ABC\), let \(I\) be the incenter and \(I_1, I_2, I_3\) be the excenters opposite to vertices \(A, B, C\) respectively. If \((II_1)^2 + (I_2I_3)^2 = \lambda R^2\), where \(R\) is the circumradius of triangle \(ABC\), find the value of \(\lambda\).

Answer: 16


Question: Consider the linear regression model \( y = X\beta + \varepsilon \), where \( y \) and \( \varepsilon \) are \( n \times 1 \) vectors, \( X \) is an \( n \times k \) matrix, and \( \beta \) is a \( k \times 1 \) vector. The error term \( \varepsilon \) is correlated with the data such that \( E(\varepsilon|X) = \gamma \), where \( \gamma \) is a non-zero column vector. The OLS estimator is given by \( \hat{\beta} = ((X'X)^{-1})(X'y) \). Does this regression model produce biased predictions of the outcome variable?

Answer: Yes


Question: Let $a_i \in \{-1,1\}$ for all $i=1,2,3,...,2014$. Define \( M = \sum_{1 \leq i < j \leq 2014} a_i a_j \). Find the least possible positive value of \( M \).

Answer: 51


Question: Calculate the convolution \( f_n = \chi_1 \ast \chi_n \) where \( \chi_1 = \chi_{[-1,1]} \) and \( \chi_n = \chi_{[-n,n]} \) for \( n \in \mathbb{N} \). Show that \( \|f_n\|_\infty = 2 \).

Answer: 2


Question:
Determine which of the following sets are the correct generators of the syzygies for $g_1=x^2, g_2=y^2, g_3=xy+yz \in k[x,y,z]$ with a field $k$ and reverse lexicographic order $x>y>z$:

1. $(y^2,-x^2,0),(0,x+z,-y),((x+z)y,0,-x^2)$
2. $(y^2,-x^2,0),(0,x+z,-y),(y,0,-x+z)$

Answer: 1


Question: Evaluate the limit: $$\lim_{h\to0}\int_{-h}^h \delta(i+x)f(x)dx.$$

Answer: 0


Question: Evaluate the integral \( \frac{1}{\sqrt{4\pi kt}}\int\limits_{\mathbb{R}}e^{-\sigma^2/(4kt)}(\sigma^2 +2\sigma x+x^2)d\sigma \) and show that it equals \( 2kt+x^2 \).

Answer: 2kt + x^2


Question: In a three hat experiment, the probabilities of different permutations are given as follows: \(\frac{1}{5}\) for \((1,2,3)\), \((1,3,2)\), \((2,1,3)\), and \((3,2,1)\), and \(\frac{1}{10}\) for \((2,3,1)\) and \((3,1,2)\). Let \(X_1\) be the indicator variable for person 1 getting their own hat back, which occurs in the permutations \((1,2,3)\) and \((1,3,2)\). Let \(N\) be the number of people who get their own hats back. Calculate \(\operatorname{Cov}(X_1, N)\).

Answer: \dfrac{8}{25}


Question: Evaluate the integral \(\int_{\gamma} z^n \, dz\) for all integers \(n\), where \(\gamma\) is any circle not containing the origin.

Answer: 0


Question: Evaluate the integral \( \int_0^{2\pi} (u^2 (re^{i\theta}) - v^2 (re^{i\theta}))d\theta \) for the curve \( |z| = r \), where \( f(z) = u(z) + iv(z) \) is an analytic function on \( D = \{z \in \mathbb{C} \mid |z| < 1 \} \) with \( f(0) = 2+i \). Here, \( 0 < r < 1 \).

Answer: 6\pi


Question: Given the function \( x(t) = \int_0^t (t-s)^q \, ds \) where \(-1 < q < 0\), find the derivative \( x'(t) \).

Answer: t^q


Question: Consider a connected graph $G$ with a countable set of vertices and a finite number of edges meeting at each vertex, such that $G$ is of bounded geometry. Is there a decoration $\alpha$ of $G$ with a finite image such that the decorated graph $(G, \alpha)$ is aperiodic?

Answer: Yes


Question:
Compute the Lebesgue integral \( \int_{[0,\pi/2]} f \, d\mu \) where the function \( f(x) \) is defined as follows:

\[
f(x) = \begin{cases}
\sin(x) & \text{if } \cos(x) \text{ is rational} \\
\sin^2(x) & \text{if } \cos(x) \text{ is irrational}
\end{cases}
\]

Evaluate the integral and provide the result.

Answer: \dfrac{\pi}{4}


Question: Determine whether the map \( \phi : \ell^{1}(\mathbb{N}) \to L^{1}(\mathbb{R}) \) defined by \( \{ a_n \} \mapsto \sum_n a_n 1_{[n,n+1]} \) is norm-preserving.

Answer: Yes


Question: Let $N$ be a random variable chosen uniformly at random from the set \( \{1, 2, 3, 4\} \). Given that $N = n$, coin A is flipped $n$ times and coin B is flipped $(5 - n)$ times. Calculate the variance of the total number of heads, $X$, obtained from flipping both coins.

Answer: \dfrac{5}{4}


Question: Given the cubic equation \( z^3 + az^2 + bz + c = 0 \) with \( z \in \mathbb{C} \), set \( z = w + h \) where \( h = -\frac{1}{3}a \) to obtain the equation \( w^3 + Bw + C = 0 \). Here, \( B = -\frac{1}{3}a^2 + b \) and \( C = \frac{2}{27}a^3 - \frac{1}{3}ab + c \). Transform \( w = kv \) such that the equation becomes \( 4v^3 - 3v + A = 0 \). Determine the value of \( k \) that allows this transformation.

Answer: \sqrt[3]{4}


Question: Let \( \sum _{n=-\infty}^\infty a_nz^n \) be the Laurent series expansion of \( f(z)=\dfrac{1}{2z^2-13z+15} \) in the annulus \( \frac{3}{2}<|z|<5 \). Find the value of \( \frac{a_1}{a_2} \).

Answer: 5


Question: Find the integer value of \( x \) such that \( x^{701} \equiv 3 \pmod{139} \) using Fermat's Little Theorem.

Answer: 88


Question:
Let $a$ and $c$ be real numbers. If $\lim_{x\to \infty} M(x)=a$, does the following hold?
\[\limsup_{x\to\infty}\int_{1}^{x}M(s)s^c\,\mathrm ds=a\limsup_{x\to\infty}\int_{1}^{x}s^c\,\mathrm ds\]

Answer: No


Question: Let $S$ be a non-empty Lebesgue measurable subset of $\mathbb{R}$ such that every subset of $S$ is measurable. What is the measure of $S$?\n\n1. Measure of any bounded subset of $S$.\n2. Measure of any closed subset of $S$.\n3. Measure of any subset of $S$.\n4. Measure of any countable subset of $S$.\n\nChoose the correct option.

Answer: 3


Question: Consider the ideal $I=\left\langle X(X+1)^{2}\right\rangle$ in the polynomial ring $\mathbf{Q}[X]$. Determine the number of ideals $J$ in $\mathbf{Q}[X]$ such that $I \subseteq J$. Provide your answer as a single integer.

Answer: 6


Question: Evaluate the sum \( \sum_{i=0}^{k-1} 2^i \cdot 4(k-i-1) \) and express it as a formula in terms of \( k \).

Answer: 4(2^k - k - 1)


Question: Let $G = \Bbb Z \times \Bbb Z$ with group law given by addition. Let $H$ be the subgroup generated by $(2,3)$, so $H$ consists of all elements of the form $(2a,3a)$ for some $a \in \Bbb Z$. To which group is $G/H$ isomorphic?

Answer: \mathbb{Z}


Question:
Let $X_1, X_2, \ldots, X_{36}$ be a random sample of size $n=36$ from the geometric distribution with the probability density function:
\[ f(x) = \left(\frac{1}{4}\right)^{x-1} \left(\frac{3}{4}\right), \quad x = 0, 1, 2, \ldots \]
If $W = \sum_{i=1}^{36} X_i$, then $W$ follows a negative binomial distribution. Determine the parameter $r$ for this distribution.

Answer: 36


Question: Determine the topology to which the quotient space $S^1/\sim$, where $x \sim y$ if and only if $x$ and $y$ are diametrically opposite on $S^1$, is homeomorphic.

Answer: S^1


Question: Find the area of the ellipse defined by the inequality \(x^{2} + 2xy + 2y^{2} \leq 1\) using the substitution \(s = x+y\) and \(t = y\).

Answer: \pi


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

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


Question: Calculate the limit: $$\lim_{x \to 0^+} (1+\sin x)^{\frac{1}{\sqrt{x}}}$$

Answer: 1


Question: Given the sequences \(a_n=\int_0^1 (1-x^2)^n dx\) and \(b_n=\int_0^1 (1-x^3)^n dx\) for \(n \in \mathbb{N}\), find the value of \(L=\lim_{n\to \infty} \left(10 \sqrt[n]{a_n} + 5\sqrt[n]{b_n}\right)\).

Answer: 15


Question: Determine the smallest positive integer \( k \) such that there exist integers \( x_1, x_2, \ldots, x_k \) satisfying \( x_1^3 + x_2^3 + \cdots + x_k^3 = 2002^{2002} \).

Answer: 4


Question: Evaluate the contour integral \( \oint_{\gamma} \frac{1}{z} \, \mathrm{d}z \), where \( \gamma \) is the unit circle in the complex plane.

Answer: 2\pi i


Question:
Determine the exact value of the norm \( \Vert f \Vert \) for the functional \( f \) defined on the normed space \( C[-1,1] \) by:
\[ f(x) = \int_{-1}^0 x(t) \, \mathrm{d}t - \int_0^1 x(t) \, \mathrm{d}t \]
for all \( x \in C[-1,1] \), where \( C[-1,1] \) is the space of all continuous functions on \([-1,1]\) with the norm \( \Vert x \Vert_{C[-1,1]} = \max_{-1 \leq t \leq 1} |x(t)| \).

Answer: 2


Question: Determine if there exists a closed form for the alternating series: \[ \sum_{n=2}^{\infty} \frac{(-1)^n}{n!}. \]

Answer: \dfrac{1}{e}


Question: Evaluate the integral \( \int_0^\infty \frac{dx}{1+x^6} \).

Answer: \dfrac{\pi}{3}


Question: Compute \( \int_{0}^{2\pi} f(re^{i\theta}) \, d\theta \) where \( f : \Bbb C \to \Bbb C \) is an entire function and \( r > 0 \).

Answer: 2\pi f(0)


Question: Let \( f:[a,b] \rightarrow \mathbb{R} \) be continuous on \([a,b]\) with \( f > 0 \) on \((a,b)\), and let \( g:[a,b] \rightarrow \mathbb{R} \) be non-negative and integrable on \([a,b]\). If \( \int_a^b g > 0 \), is it true that \( \int_a^b fg > 0 \)? Justify your answer.

Answer: Yes


Question: Determine the values of \( \beta \) for which the series \( \sum_{n=1}^{\infty} \left( \sqrt{1+\frac{(-1)^n}{n^\beta}} - 1 \right) \) converges.

Answer: \beta > \frac{1}{2}


Question:
Determine which of the following sets of boolean functions is functionally complete:

1. \( F_1 = (M \setminus T_0) \cup (S \setminus L) \)
2. \( F_2 = (M \setminus T_0) \cup (L \setminus S) \)

where:
- \( M \) is the set of all monotonic functions.
- \( T_0 \) is the set of all falsity-preserving functions.
- \( S \) is the set of all self-dual functions.
- \( L \) is the set of all linear functions.

Provide your answer as either \( F_1 \) or \( F_2 \).

Answer: F_1


Question: Let $A$ be a finite subset of the positive integers $\mathbb{N}$ such that $\sum_{a \in A} \frac{1}{a} < 1$. Is it possible to find a finite set $A' \subseteq \mathbb{N}$ with $A \subseteq A'$ such that $\sum_{a \in A'} \frac{1}{a} = 1$?

Answer: Yes


Question:
Consider a hypergraph $H=(V,E)$ that represents a finite Fano plane, where $V$ is a finite set and $E$ satisfies the following properties:

1. For any two distinct edges $e_1$ and $e_2$ in $E$, $|e_1| = |e_2|$ and $|e_1 \cap e_2| = 1$.
2. For any two distinct vertices $v$ and $w$ in $V$, there exists a unique edge $e$ in $E$ such that $\{v, w\} \subseteq e$.

Determine whether there always exists an injective map $f: E \to V$ such that $f(e) \in e$ for all $e \in E$. Provide a justification for your answer.

Answer: Yes


Question:
Find the value of $X$ in the expression:
$$X=\sqrt{2\sqrt[3]{2\sqrt[4]{2\sqrt[5]{2\sqrt[6]{2 \ldots}}}}}$$

Answer: 2^{e-2}


Question: Determine whether the curve defined by the equation \( f(x,y) = x^4 - y^2x^2 + 9y^2 = 0 \) passes through the origin \((0,0)\).

Answer: Yes


Question: Solve the equation: \[ 2^{1+\sin(3x)} - 8^{\sin(x)+\frac{1}{3}} + \sin(x) \cos(2x) = \sin(x) \]

Answer: x = k\pi


Question: Consider a regular pentagon with vertices numbered from 1 to 5 in a clockwise direction. Initially, two ladybugs are placed at vertices 1 and 3. At each time step, each ladybug moves independently to one of its two adjacent vertices with equal probability. Calculate the expected number of time steps for the two ladybugs to meet at the same vertex.

Answer: 8


Question: Determine the number of positive integer values of \( n \) such that \( \frac{4^n + 2^n + 1}{n^2 + n + 1} \) is a positive integer.

Answer: 3


Question: Determine the number of integer pairs \((x, y)\) with \(0 \leq x, y \leq 100\) such that \(x^{20} \equiv y^{50} \pmod{101}\).

Answer: 1001


Question: Consider a quadratic complex polynomial \(f(z) = az^2 + bz + c\) where \(a, b, c \in \mathbb{C}\). If \(f(z)\) is non-zero everywhere in the unit disc \(|z| \leq 1\), prove that \(|c| > |a|\).

Answer: |c| > |a|


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

Answer: |z| < 1


Question: Given the joint probability density function \(f(x,y,z) = kxyz^2\) for \(0 < x < 1\), \(0 < y < 1\), and \(0 < z < 2\), find the probability \(P(Z > X + Y)\).

Answer: \dfrac{13}{20}


Question: Let \( U \) be an open set in \( \mathbb{R}^2 \) and \( F: U \to \mathbb{R}^3 \) be a one-to-one differentiable function such that its inverse from \( F(U) \) to \( \mathbb{R}^2 \) is continuous. Is it possible for the derivative \( dF \), as a matrix, to not have rank 2 at some point in \( U \)?

Answer: Yes


Question:
Find a closed form expression in terms of known constants for the sum of the following integrals, which converges to approximately 0.1746809:
\[-\frac{1}{2}\int _0^{1}\frac{\left(\frac{1-t}{1+t}\right)\arctan \left(t\right)}{t\left(\ln t\right)^2}dt-\frac{\pi }{4}\int _0^{1}\frac{1}{\ln \left(t\right)\left(1+t\right)^2}dt\]

Answer: \dfrac{\pi}{18}


Question: Let $G$ be a finite graph with $E$ as its set of edges. For each vertex $a$, let $E_{a}$ denote the set of edges of $G$ incident to $a$. Suppose for any vertex $a$ in $G$, the number of edges incident to $a$, denoted as $\sharp E_{a}$, is greater than 3. Consider a function $f: E \to \{1, 2\}$. Does there exist such a function $f$ such that for each vertex $a$, the number of edges in $E_{a}$ mapped to 1 by $f$, denoted as $\sharp (f^{-1}(1) \cap E_{a})$, is a nonzero even number?

Answer: Yes


Question: Find the value of \(a \in \mathbb{R}\) such that there exists a linear map \(f\) satisfying the following conditions: \(f(1,-1,1) = (2,a,-1)\), \(f(1,-1,2) = (a^2,-1,1)\), and \(f(1,-1,-2) = (5,-1,-7)\).

Answer: -1


Question: Given that $y = f(x)$ is a differentiable function and $g(x) = f(x - x^2)$, if $y = g(x)$ has a local maximum at $x = \frac{1}{2}$ but the absolute maximum occurs at a different point, determine the minimum number of solutions to $g'(x) = 0$. Provide your answer as a single number.

Answer: 3


Question: Find the smallest positive integer \( m \) such that the polynomial \( x^4 + x^3 + 1 \) divides \( x^m - 1 \) over the field \( \mathbb{F}_2 \).

Answer: 15


Question: Determine the number of associative rings with unity of order 35, up to isomorphism.

Answer: 1


Question: Let $x_1$ and $x_2$ ($x_1 < x_2$) be the two solutions of the equation \( \left|2\left(x^2+\frac{1}{x^2}\right)+|1-x^2|\right|=4\left(\frac{3}{2}-2^{x^2-3}-\frac{1}{2^{x^2+1}}\right) \). Calculate the value of the integral \( \int_{x_1+x_2}^{3x_2-x_1}\left\{\frac{x}{4}\right\}\left(1+\left[\tan\left(\frac{\{x\}}{1+\{x\}}\right)\right]\right)dx \), where \( \{*\} \) denotes the fractional part function and \( [*] \) denotes the greatest integer function.

Answer: 2


Question: Evaluate the integral of \( \log(z) \) over the unit circle in the complex plane.

Answer: 2\pi i


Question: Determine whether there exists an irreducible homogeneous polynomial $h \in \mathbb{C}[X_0,...,X_n]$ with degree greater than 2 such that the Euler characteristic of the complement of its zero set in $\mathbb{CP}^n$ is zero, i.e., $\chi(\mathbb{CP}^n \setminus Z(h)) = 0$. Note: This is not possible if $Z(h)$ is smooth and deg$h > 2$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit \( \lim\limits_{n \to \infty}\frac{\int_0^{1/2} x^{nx}\,\mathrm dx}{\int_{1/2}^1 x^{nx}\,\mathrm dx} \).

Answer: 0


Question: Let \(a\) and \(b\) be positive real numbers such that \(a\sqrt{a} + b\sqrt{b} = 183\) and \(a\sqrt{b} + b\sqrt{a} = 182\). Find the value of \(\frac{9}{5}(a+b)\).

Answer: 73


Question:
Let $f,g:\mathbb{R}^3\to\mathbb{R}$ be twice continuously differentiable scalar fields. Determine which of the following statements is false:

A: If $S$ is any simple piecewise-smooth surface with unit normal vector $\mathbf{n}$, then $\iint_S (\nabla\times\nabla f)\cdot\mathbf{n} \,dS=0$.

B: If $S$ is the unit sphere with the outward pointing normal vector $\mathbf{n}$, then $\iint_S \nabla f\cdot\mathbf{n} \,dS=4\pi$.

C: If the directional derivative of $f$ along the unit tangent vector of a simple piecewise-smooth curve $C$ is zero at every point on $C$, then $\int_C \nabla f\cdot d\mathbf{r}=0$.

D: The vector field $\nabla f \times \nabla g$ is solenoidal.

Answer: B


Question: Evaluate the definite integral: \[ \int_{0}^{\frac{\pi}{2}} \frac{1}{1+\tan^{\sqrt{2}}(x)} \, \mathrm{d}x \]

Answer: \dfrac{\pi}{4}


Question: Calculate \( \iint_S \operatorname{curl}F \cdot n \, dS \) over the semi-sphere \( x^2 + y^2 + z^2 \leq 1 \) for \( z \geq 0 \), where \( F = \langle xyz, x, e^{xy} \cos z \rangle \) and \( n \) is the outward normal vector to the surface. Use the divergence theorem to simplify the calculation. Verify that \( \operatorname{div}(\operatorname{curl}F) = 0 \) and compute the integral over the base of the semi-sphere. What is the value of the integral?

Answer: \pi


Question: Let \( B(t) \) be a standard Brownian motion. Evaluate the conditional probability \( P(B(3) \leq 3 \mid B(1) = 1) \).

Answer: \Phi(\sqrt{2})


Question: Determine the limit \( l \in \mathbb{R} \) to which the sequence \( \left( \frac{2n^2 - 1}{2n^2 + 1} \right) ^ { \left( \frac{2n^3 - n}{n + 3} \right) } \) converges as \( n \to \infty \).

Answer: e^{-2}


Question: Find \( \lim_{x \to \infty} e^x \sin (e^{-x^2}) \).

Answer: 0


Question: Find the maximum value of the expression \(3^x + 5^x - 9^x + 15^x - 25^x\) as \(x\) varies over the real numbers.

Answer: 1


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

Answer: 2 \ln 2


Question: Does there exist an even number greater than 36 that has more even divisors than 36, with each even divisor being one less than a prime number?

Answer: No


Question: Use Rouche's Theorem to determine the number of roots of the polynomial \(z^5 + 3z^2 + 1\) within the annulus \(1 < |z| < 2\).

Answer: 3


Question: Given the conditional density functions $f_x(x)=\frac{2x}{9}$ for $0 < x < 3$ and $f_{y|x}(y|x)=\frac{3y^2}{x^3}$ for $0 < y < x$, calculate the probability $P(Y<2)$. Express your answer as an integral and evaluate it.

Answer: \dfrac{20}{27}


Question: Evaluate the integral \( \int_0^{2\pi} \frac{d\theta}{1+8\cos^2\theta} \) using the residue theorem.

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


Question: Compute \( \sum_{i=1}^{2n} \frac{x^{2i}}{x^i-1} \) where \( x \) is a complex number such that \( x^{2n+1} = 1 \) and \( x \neq 1 \).

Answer: n - 1


Question:
Let \( f: [0, 1] \rightarrow \mathbb{R} \) be a continuous function. Prove that:
\[ n \cdot \int_{0}^{1} x^n \cdot f(x) \, \mathrm{d}x \underset{n\to+\infty}{\longrightarrow} f(1) \]

Answer: f(1)


Question:
Evaluate the limit:
\[
\lim_{x\to 0} \frac{e^{x^2} + 2\cos x -3}{\sin^2 x^2}.
\]

Answer: \dfrac{7}{12}


Question: Determine the number of functions \( f: A \to A \), where \( A = \{1, 2, 3, 4, 5, 6, 7\} \), such that \( f(f(f(x))) = x \) for all \( x \in A \).

Answer: 351


Question: Evaluate the limit \( \lim_{x \to +\infty} \frac{\ln(x)}{x} \) using Taylor series.

Answer: 0


Question: Let $x$ be an irrational number such that $x^2 - 2x$ and $x^3 - 5x$ are rational numbers. Determine the value of $x^3 - 5x$.

Answer: 2


Question: Given a differentiable function $f: \mathbb{R} \to \mathbb{R}$, and an open interval $(a, b)$ contained in the range of the derivative $g = f'$, does the inverse image $g^{-1}((a, b))$ necessarily contain a non-empty interior? Justify your answer.

Answer: Yes


Question: Four lines are drawn on a plane such that no two are parallel and no three are concurrent. Lines are then drawn joining the points of intersection of these four lines. How many new lines are formed?

Answer: 3


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

Answer: \ln 2


Question: Evaluate the definite integral \( \int_{0}^{2\pi} \frac{e^{|\sin x|}\cos(x)}{1+e^{\tan x}} \, dx \).

Answer: 0


Question: Given that \(\cos^{10}x + \sin^{10}x = \frac{11}{36}\), find the value of \(\cos^{12}x + \sin^{12}x\).

Answer: \dfrac{13}{54}


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

Answer: 33


Question: Let $X$ be a normal complex projective variety, and let $V$ be a closed subset of $X$ with $\dim V \leq \dim X - 2$. Suppose $A$ is an ample divisor on $X$. For $m \gg 0$, is it possible to find an irreducible divisor $A' \in |mA|$ such that $A'$ contains $V$?

Answer: Yes


Question: Let \( \sigma \) be an automorphism in \( \text{Aut}(\mathbb{R}(t)/\mathbb{R}) \) such that \( \sigma(t) = -t \). Determine the fixed field of \( \langle \sigma \rangle \).

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


Question: Compute the Kullback-Liebler divergence \( \mathrm{KL}(F;G) \) when \( F \) is the standard normal distribution and \( G \) is the normal distribution with mean \( \mu \) and variance 1. For what value of \( \mu \) is the divergence minimized?

Answer: 0


Question:
Evaluate the limit by expressing it as a definite integral using a Riemann sum:
\[
\lim\limits_{n\to\infty}\sqrt[n]{\left(\dfrac{1 + n}{n^2} \right)\left(\dfrac{4 + 2n}{n^2} \right)...\left(\dfrac{n^2 + n^2}{n^2} \right)}
\]

Answer: \dfrac{4}{e^2}


Question: For what values of \( V \) does the function \( y(t) = -(V+2)e^{-3t} + (V+3)e^{-2t} \) remain nonnegative for all \( t \geq 0 \)?

Answer: [-3, \infty)


Question: Determine the discrete logarithm of $x^2 + 1$ in the quotient ring $\Bbb Z_5[x]/\langle x^3 + x + 1 \rangle$. Express your answer in terms of the basis elements of the ring.

Answer: 30


Question: Let \( f: \mathbb{R} \to (0, \infty) \) be a function such that \( f(x+y) \le f(x)f(y) \) for all \( x, y \in \mathbb{R} \), and \( f \) is bounded on \([0,1]\). Does the limit \( \lim_{x \to \infty} (f(x))^{1/x} \) exist?

Answer: Yes


Question: Let $X$ be a connected scheme and $Y$ be an affine, Noetherian scheme of finite Krull dimension. Suppose there exists a proper morphism $f: X \to Y$ that is also a monomorphism in the category of schemes. Is $X$ necessarily affine?

Answer: Yes


Question: Determine whether there exists a Banach space $X$, along with elements $x_i \in X$ and radii $r_i > 0$ indexed by some set $I$, possibly infinite, such that for each $n \in \mathbb{N}$, the intersection \( \bigcap_{i \in I} B(x_i, (1+\tfrac{1}{n})r_i) \neq \varnothing \) holds, but \( \bigcap_{i \in I} B(x_i, r_i) = \varnothing \). Here, $B(x,r)$ denotes the closed ball centered at $x$ with radius $r$. Note that $X$ cannot be isometric to a dual space.

Answer: Yes


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

Answer: 1


Question: Find the values of \( x \) for which the distance between the vector \( A = (1, 1, 1) \) and the subspace \( L \) spanned by the vectors \( (1, 0, -1) \) and \( (3, 5, x) \) in \( \mathbb{R}^3 \) is maximal.

Answer: -8


Question: Using the identity \(1 + \sum_{ n \geq 1} \frac {u^{n}}{n!} \sum_{\pi \in S_n} \prod_{i}x_{i}^{N_i(\pi)} = \prod_{m\geq1} \exp(u^m x_m / m)\), where \(N_i(\pi)\) is the number of length \(i\) cycles of a permutation \(\pi\) on \(n\)-symbols, compute the expected number of \(j\) cycles in a random permutation \(\pi\).

Answer: \dfrac{1}{j}


Question: Consider a velocity field $v \in \mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ for $n \geq 1$, where every solution $(x_t)_{t \geq 0}$ of the differential equation \( \frac{d}{dt}x_t = v(x_t) \) is periodic. For a non-stationary point $x \in \mathbb{R}^n$ (i.e., $v(x) \neq 0$), let $T(x)$ denote the period of the solution $(x_t)_{t \geq 0}$ with initial condition $x(0) = x$. Is the function $T$ continuous over the set of non-stationary points?

Answer: Yes


Question: Is the $K$-theory space $K(A)$ of a unital C*-algebra $A$ equivalent to the group completion of the space of projections $Pr(A \otimes \mathcal K)$, where $\mathcal K$ is the C*-algebra of compact operators on an infinite dimensional Hilbert space?

Answer: Yes


Question: Let $X$ and $Y$ be compact metric spaces with Borel $\sigma$-algebras $\mathcal{B}_X$ and $\mathcal{B}_Y$, respectively. Let $\mu$ be a Borel probability measure on $X$, and let $\mathcal{B}^*_X$ be the $\mu$-completion of $\mathcal{B}_X$. If $E$ belongs to the product $\sigma$-algebra $\mathcal{B}^*_X \otimes \mathcal{B}_Y$, does there exist a $\mu$-null set $A$ in $\mathcal{B}_X$ such that $E \cup (A \times Y) \in \mathcal{B}_X \otimes \mathcal{B}_Y$?

Answer: Yes


Question: Suppose $E_1$ and $E_2$ are two elliptic curves over $\mathbb{Q}$ such that there exists a $\mathbb{Q}$-isogeny from $E_1$ to $E_2$. If the Main Conjecture for elliptic curves is true for one of these curves, is it also true for the other? Provide a justification for your answer.

Answer: Yes


Question: Is it possible to construct a triangle if only the lengths of its medians \(m_1, m_2, m_3\) are given? Provide a justification for your answer.

Answer: Yes


Question: Let $X_1, X_2, X_3, X_4$ be independent standard normal random variables and define $Y = X_1^2 + X_2^2 + X_3^2 + X_4^2$. Find the probability that $Y \leq 3$. Enter your answer as a decimal with at least 10 digits after the decimal point.

Answer: 0.4421745996


Question: Let $f: A \to A$ be a function such that $f(f(x)) + 3x = 4f(x)$ for every $x \in A$, where $A$ is a finite set of positive real numbers. Determine the function $f$.

Answer: x


Question: Given a triangle \( \triangle ABC \) with \( \angle A = 20^\circ \), and points \( N \) and \( M \) such that \( AN = MC \) and \( NB = AM \), determine the measure of \( \angle X \).

Answer: 30^\circ


Question:
Calculate the value of the limit:
\[\lim_{x\to0} \frac{2\sin\left ( e^{-\frac{x^2}{2}} -\cos x \right)}{(\arctan(\sinh(x^2)))^2}\]

Answer: \dfrac{1}{6}


Question: Evaluate the infinite product \( \prod_{n=1}^{\infty} \left[ 1 + \frac{1}{a_n} \right] \), where \( a_1 = 1 \) and \( a_{n+1} = (n+1)(1 + a_n) \) for all \( n \in \mathbb{N} \).

Answer: e


Question: Calculate the n-th Lebesgue measure of the set \(\{ (x_1, x_2, \dots, x_n) \in \mathbb{R}^n : x_i \geq 0, i = 1, 2, \dots, n, \sum_{i=1}^n x_i \leq 1 \}\) using Fubini's theorem.

Answer: \dfrac{1}{n!}


Question: Determine whether the sequence of functions \( f_n(x) = x^n \) defined over \([0, 1]\) converges to its pointwise limit under the \(L^1\) metric \(d_1\).

Answer: Yes


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

Answer: \pi


Question: Provide an example of a function that is uniformly continuous on the interval \([0, 1]\), differentiable on the interval \((0, 1)\), but has an unbounded derivative on \((0, 1)\).

Answer: \sqrt{x}


Question: Given that $a^2 + b^2 + c^2 + d^2 = 4$, determine the range of the expression $a^3 + b^3 + c^3 + d^3$. Express your answer as an interval.

Answer: [-8, 8]


Question: Find the smallest positive integer \( n \) such that \( 125 \mid 2^n + 3^n \).

Answer: 25


Question: Find the number of solutions to the complex equation \(|z|^5 - 2z|z|^3 = \bar{z}^2 |z|^3 - 1\), where \(z = x + iy\), \(x, y \in \mathbb{R}\), and \(x \neq 1\).

Answer: 1


Question: Consider a subalgebra $\cal S$ of the algebra of linear operators on a finite-dimensional vector space $V$ over a field. If \( \bigcap_{A\in\cal S}\ker A=\{0\} \) and \( \bigcup_{A\in\cal S}A(V)=V \), does it follow that there exists some $A\in\cal S$ that is non-singular?

Answer: Yes


Question: Calculate \( \lim\limits_{n \to \infty} \frac{n^k}{k^n} \) for \( k > 1 \).

Answer: 0


Question: A series of natural numbers is divided into groups as follows: \((1), (2,3,4), (5,6,7,8,9,10,11), (12,13,\ldots,26)\), and so on. The sum of numbers in the \(n^{th}\) group is given by the expression \(a \cdot 2^{2n-1} - (2n+b) \cdot 2^{2n-1} + n + 1\). Determine the value of \(a + b\).

Answer: 8


Question: Differentiate the function \( f(x) = x^T A x \) with respect to \( x \), where \( A \) is a symmetric matrix. Express the result in terms of \( x \) and \( A \).

Answer: 2Ax


Question: Evaluate the integral: $$\int_{-1}^{1} \frac{dx}{(e^x+1)(x^2+1)}$$

Answer: \dfrac{\pi}{4}


Question: Evaluate the polynomial \( 3x^{233} + 4x^6 + 2x^{37} + 3 \) at \( x = 11 \) in the ring \( \mathbb{Z}_{13} \).

Answer: 3


Question: Evaluate the limit \( \lim_{n \to \infty} n \int_{0}^{1} (f(t))^n \, dt \), where \( f: [0,1] \to \mathbb{R} \) is a non-negative function such that \( f(t) < 1 \) for all \( t \in [0,1] \).

Answer: 0


Question:
Given that \( f'(0) = 3 \), evaluate the limit:
\[
\lim_{x\rightarrow1}\left(\frac{f(x^2 -1) - f(0)}{x^3 -1}\right)
\]

Answer: 2


Question: Let $a, b, c, d$ be real numbers such that $a + b + c + d = 8$ and $ab + ac + ad + bc + bd + cd = 12$. Determine the maximum possible value of $d$. \( \text{(Express your answer as a real number.)} \)

Answer: 2 + 3\sqrt{2}


Question: Let \( K = \mathbb{Q}(\alpha) \) be a field extension of \( \mathbb{Q} \), where \( \alpha \notin \mathbb{Q} \) but \( \alpha^k \in \mathbb{Q} \). If the minimal polynomial \( m_{\alpha, \mathbb{Q}} \) has degree \( k \), determine the dimension of the field extension \( \mathbb{Q}(\alpha)/\mathbb{Q} \). Justify your answer.

Answer: k


Question: Does there exist an increasing unbounded sequence in \(\omega_1\)? Specifically, is there a sequence \((\alpha_k)_{k \in \mathbb{N}}\) such that \(\alpha_k < \alpha_{k+1}\) for all \(k \in \mathbb{N}\), and for every \(\beta \in \omega_1\), there exists a \(k \in \mathbb{N}\) such that \(\alpha_k \geq \beta\)?

Answer: No


Question: Let $X$ be a non-empty set and $F: X \to \mathcal{P}(X)$ be a function such that for any subset $A \subseteq X$, we have $|A| \leq |\bigcup F(A)|$. Does this condition imply the existence of an injective function $f: X \to X$ such that $f(x) \in F(x)$ for all $x \in X$?

Answer: Yes


Question:
Let \( T \) be a linear operator on a vector space \( V \) such that \( T^2 - T + I = 0 \). Determine which of the following statements is true:

- \( T \) is one-to-one but not onto.
- \( T \) is onto but not one-to-one.
- \( T \) is invertible.
- No such \( T \) exists.

Answer: C


Question: Does there exist an invariant non-Lebesgue probability measure for the doubling map \( T: [0,1) \rightarrow [0,1) \) defined by \( T(x) = 2x \mod 1 \)?

Answer: Yes


Question: Evaluate the definite integral \( \int_{2}^{3} \sqrt{(x^2+x+1)+\sqrt{(x^2+x+1)-\sqrt{(x^2+x+1)+\sqrt{(x^2+x+1)\cdots\cdots}}}} \, dx \).

Answer: \dfrac{7}{2}


Question: Find the sum of the infinite series \( S = \sum_{n=50}^{\infty} \frac{1}{\binom{n}{50}} \).

Answer: \dfrac{50}{49}


Question:
Given the probability density function of a continuous random variable $X$:
\[ f_{X}(x) = \begin{cases}\frac{x+1}{a+1} & -1 \leq x < a \\
1 & x=a \\
\frac{x-1}{a-1} & a < x \leq 1 \\
0 & \text{otherwise}\end{cases}\]
Calculate the expected value $E(X)$. Consider the implications of the probability being 1 at $x=a$.

Answer: \dfrac{a}{3}


Question: Let $U$ be a simply connected open set and $z_1, \dots, z_n$ be points in $U$. Define $U^* = U \setminus \{z_1, \dots, z_n\}$. Let $f$ be an analytic function on $U^*$. For each $k$, let $\gamma_k$ be a small circle centered at $z_k$, and define \[ a_k = \frac{1}{2\pi i} \int_{\gamma_k} f(\xi) \, d\xi. \] Define the function \[ h(z) = f(z) - \sum \frac{a_k}{z - z_k}. \] Prove that there exists an analytic function $H$ on $U^*$ such that $H' = h$.

Answer: H(z)


Question: Given the sequence of equations \(a_1a_2 = 1, a_2a_3 = 2, a_3a_4 = 3, \ldots\) and the condition \(\lim_{n \to \infty} \frac{a_n}{a_{n+1}} = 1\), find \(|a_1|\).

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


Question: If $f$ is a twice differentiable convex function, is the second derivative $f''$ also a convex function? Provide a justification for your answer.

Answer: No


Question: Evaluate the limit: \[ \lim _{n \rightarrow \infty} n \int _0 ^{\frac{2}{n}} f\left(x+\frac{1}{n}\right)\sin (nx) \, dx, \] where \( f \) is an odd real continuous function defined on \( \mathbb{R} \).

Answer: 0


Question: Let $X = (X, \|\cdot\|_X)$ and $Y = (Y, \|\cdot\|_Y)$ be Banach spaces such that $X$ is compactly embedded in $Y$. Suppose $A \subset X$ is a closed subspace of $X$. Is it true that $A$ is compactly embedded in $Y$?

Answer: Yes


Question: Find the largest radius \( r \) such that the function \( f(z) = \sum_{p\text{ prime}} z^p \) defines an analytic function on the open ball \( B_r(0) \).

Answer: 1


Question: Find a function \( f(x) \) that satisfies the functional equation \( f(x \cdot y) = f(x) + f(y) \) for all \( x, y > 0 \).

Answer: \ln x


Question: Find an equivalent expression for the remainder of the series \( \sum_{n=1}^{\infty} \frac{1}{n^3} \) given by \( R_n = \sum_{k=n+1}^{\infty} \frac{1}{k^3} \).

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


Question: In the Laurent series for \(f(z) = \frac{1}{z-4}\) centered at \(z=1\), find the coefficient of \((z-1)^{-2}\).

Answer: 3


Question: Determine the number of distinct terms in the expansion of \( \left(x+\frac{1}{x}+x^2+\frac{1}{x^2}\right)^{15} \).

Answer: 61


Question: Let \( \{X_n\} \) be a sequence of independent exponential random variables with means \( E(X_n) = \frac{1}{\lambda_n} \), where \( 0 < \lambda_n < \infty \). If \( \sum \frac{1}{\lambda_n} = \infty \), prove that \( P\left(\sum X_n = \infty\right) = 1 \).

Answer: 1


Question: Evaluate the integral \( \int_{0}^{+\infty} \frac{x - \sin{x}}{x^{3}}\, dx \) using the Dirichlet integral \( \int_{0}^{+\infty} \frac{\sin \alpha x}{x}\,dx = \frac{\pi}{2}\mathrm{sgn}\,\alpha \).

Answer: \dfrac{\pi}{4}


Question: Consider the set of all programs, each assigned a unique Godel number. The halting problem is known to be undecidable, meaning there exists at least one program for which we cannot determine if it halts. Define \( s \) as the number of programs with Godel numbers from 1 to \( G \) for which the halting problem can be solved, and \( ns \) as the number of programs for which it cannot be solved. Determine the value of the limit as \( G \) approaches infinity of the ratio \( \frac{s}{ns} \).

Answer: 0


Question: Determine the number of solutions to the equation \(x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 25\) where each \(x_i\) is a non-negative integer, subject to the constraints \(3 \leq x_1 \leq 10\), \(2 \leq x_2 \leq 7\), and \(x_3 \geq 5\).

Answer: 12716


Question: Let $(X,d)$ be a compact metric space and $A \subset X$ be a closed set with $A \neq X$. Does there necessarily exist an open set $B$ such that $A \subset B$ and $\overline{B} \neq X$?

Answer: Yes


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a non-constant, three times differentiable function such that \( f\left(1+\frac{1}{n}\right) = 1 \) for all integers \( n \). Find \( f''(1) \).

Answer: 0


Question: Let $\phi: \mathbb{C}^n \rightarrow \mathbb{C}^m$ be a polynomial map, and let $S \subset \mathbb{C}^n$ be a locally closed set. Assume that $\phi$ induces an isomorphism of $S$ onto a nonempty open subset. Is it true that $S$ is irreducible, meaning that if $S = (C_1 \cap S) \cup (C_2 \cap S)$ with closed $C_1$ and $C_2$, then $C_1 \cap S = S$ or $C_2 \cap S = S$?

Answer: Yes


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

Answer: \dfrac{1}{2}


Question: Determine the radius of convergence for the series \( \sum_{n=1}^\infty \frac{z^{n^2}}{n!} \), where \( z \in \mathbb{C} \).

Answer: 1


Question: Determine whether the torsion is zero for an osculating curve in Euclidean space. A curve is called an osculating curve if its position vector lies on its osculating plane. The osculating plane for the curve \( \alpha(s) \) at some point is generated by the tangent vector and normal vector of \( \alpha \) at that point, i.e., \( \alpha(s)=\lambda_1(s)t(s)+\lambda_2(s)n(s) \) for some functions \( \lambda_1(s) \) and \( \lambda_2(s) \). Is the torsion zero for such a curve?

Answer: Yes


Question: Find a positive integer \( b \) such that \( f(n) \leq bn \) for all \( n > 0 \), where \( f(n) \) is defined recursively by \( f(n) = 2f(\lfloor \frac{n}{9} \rfloor) + 3f(\lceil \frac{n}{4} \rceil) + 5n \) for \( n \geq 9 \) and initial conditions \( f(1) = 6, f(2) = 12, \ldots, f(8) = 48 \). Use induction to determine the value of \( b \).

Answer: 180


Question: Given that \(G\) is the centroid of \(\Delta ABC\), and the distances from \(G\) to the vertices are \(GA = 2\sqrt{3}\), \(GB = 2\sqrt{2}\), and \(GC = 2\), find the area \([\Delta ABC]\).

Answer: 6\sqrt{2}


Question: Let \(X \in \mathbb{R}^{m}\) and \(W \in \mathbb{R}^{p}\) be random vectors. Determine whether the statement \(E[X|W] = 0\) implies \(E[XW^{T}]=0\).

Answer: E[XW^T] = 0


Question: Find an upper bound for the recursive function \( T(n) = T(n-1) + n^2 + n^2 \log n \).

Answer: O(n^3 \log n)


Question:
Let $G$ be a soluble group with a trivial center, and let $\pi$ be the canonical homomorphism from $G$ to $G/F(G)$. Define
\[ Z=\pi^{-1}\left(Z\left(F\left(\frac{G}{F(G)}\right)\right)\right) \quad \text{and} \quad J=\pi^{-1}\left(F\left(\frac{G}{F(G)}\right)\right). \]
Is it true that if $x \notin Z$, then there exists a prime $p$ such that the $p$-Sylow subgroup of $J/F$ is not centralized by $x$?

Answer: Yes


Question:
Find all functions \( f: \mathbb{R}^2 \rightarrow \mathbb{R} \) such that for all \( \{x_1, x_2, \ldots, x_{2016}\} \subset \mathbb{R} \):
\[ f(f(f(\ldots(f(x_1, x_2), x_3), \ldots), x_{2016})) = x_1 + x_2 + \ldots + x_{2016} \]

Answer: f(x, y) = x + y


Question: Evaluate the limit as a definite integral: \[ \lim_{n\to\infty}\frac{(\sqrt[n]{e})+(\sqrt[n]{e^2})+\cdots+(\sqrt[n]{e^{2n}})}{n} \]

Answer: e^2 - 1


Question: An 8x8x8 cube is painted on the outside and then divided into 1x1x1 unit cubes. One of these unit cubes is randomly selected and rolled on a table such that an unpainted face is on the bottom. What is the probability that the entire unit cube is unpainted?

Answer: \dfrac{27}{56}


Question: Determine the maximum number of non-attacking $2 \times 2$ squares that can be placed on a $1000 \times 1000$ board. Two $2 \times 2$ squares are said to attack each other if they lie in the same pair of adjacent rows (or columns), and the cells between them are all empty. What is the largest number of such squares that can be placed on the board without any two attacking each other?

Answer: 250000


Question: Evaluate the sum: \[ \sum_{r=0}^n 2^{n-r} \binom{n+r}{r} \]

Answer: 4^n


Question: Find the residue of the function \( \frac{z^5}{(z-i)^3} \) at \( z = i \).

Answer: -10i


Question: Determine the period of the function \( f(x) = \frac{\sin x}{2 + \cos x} \).

Answer: 2\pi


Question: Determine the value of $C$ for which the improper integral \( \int_{1}^{\infty} \left(\frac{1}{\sqrt{x^2+4}}-\frac{C}{x+2}\right)dx \) is convergent.

Answer: 1


Question: Find the remainder when \(x^{203} - 1\) is divided by \(x^4 - 1\).

Answer: x^3 - 1


Question: Determine whether there exists a non-zero continuous function \( f(x) \) such that \( \int_0^1 x^k (1-x)^{n-k} f(x) \, dx = 0 \) for every integer \( k = 0, 1, 2, \ldots, n \), where \( n \) is a non-negative integer.

Answer: Yes


Question:
Let $R$ be a commutative ring with unity. Determine which of the following statements is true:

1. If $R$ has finitely many prime ideals, then $R$ is a field.
2. If $R$ has finitely many ideals, then $R$ is finite.
3. If $R$ is a PID, then every subring of $R$ with unity is a PID.
4. If $R$ is an integral domain which has finitely many ideals, then $R$ is a field.

Answer: 4


Question: Given that $x, y, z$ are positive real numbers satisfying $\frac{1}{3} \le xy + yz + zx \le 3$, determine the range of possible values for $x + y + z$. Provide your answer as an interval.

Answer: [1, \infty)


Question: Evaluate the line integral \( \int _{C} (z^3 + 2z + \text{Re}(z))\,dz \) where \( C \) is the triangle with vertices \( z=0 \), \( z=1+2i \), and \( z=1 \).

Answer: -i


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

Answer: -\infty


Question: Find the value of \(x\) that satisfies the equation: \[ 1 + x + \frac{1\cdot3}{2!}x^2 + \frac{1\cdot3\cdot5}{3!}x^3 + \ldots = \sqrt{2} \]

Answer: \dfrac{1}{4}


Question: Calculate the expected value \(E[X^3Y]\) for the continuous random variables \(X\) and \(Y\) with the joint density function \(f(x,y) = 0.25\) for \(0 \leq x \leq 2\) and \(x-2 \leq y \leq x\).

Answer: \dfrac{6}{5}


Question:
Let \( f : \mathbb{C}^∗ \rightarrow \mathbb{C} \) be the function defined by \( f(z) = z \sin\left( \frac{1}{z} \right) \). Determine \( \lim_{z \rightarrow 0} f(z) \). Choose the correct option:

(a) is equal to zero.
(b) does not exist.
(c) is infinite.
(d) is finite but not equal to zero.

Answer: B


Question: Does there exist a measurable subset $T$ of $[0, \infty)$ with finite measure and some $\epsilon > 0$ such that for every $r$ with $0 < r < \epsilon$, $nr$ is in $T$ for infinitely many positive integers $n$? Note: The integers $n$ such that $nr$ lie in $T$ can depend on $r$. Answer with 'Yes' or 'No'.

Answer: No


Question: Solve the differential equation \((1-t^2)\frac{d^2x}{dt^2}-2t\frac{dx}{dt}+6x=0\) with the initial conditions \(x(0)=1\) and \(\frac{dx}{dt}(0)=0\). Verify that the solution is \(x = 1-3t^2\).

Answer: 1 - 3t^2


Question: Evaluate the limit: \[ \lim_{x\to 3}\frac{3^x-x^3}{x-3} \] without using L'Hôpital's rule.

Answer: 27(\ln 3 - 1)


Question: Evaluate the limit: \[ \lim_{x \rightarrow 1^{-} } \frac{e^{1/(x^2 -1)}}{x-1} \]

Answer: 0


Question: Evaluate the limit: \( \lim_{n \to +\infty } \{ en! \} \), where \( \{ a \} = a - \lfloor a \rfloor \) denotes the fractional part of \( a \).

Answer: 0


Question: Let \( \phi \) be the solution of the differential equation \( y' + iy = x \) with the initial condition \( \phi(0) = 2 \). Find \( \phi(\pi) \).

Answer: -i \pi


Question: Find the value of \( \hat{x} \) that minimizes the function \( f(x) = x + \exp(-x)a \) for a given \( a \in \mathbb{R} \).

Answer: \ln(a)


Question: Evaluate the definite integral: \[ \int_{0}^{\pi /2} \cot \left(\frac{x}{2} \right) (1-\cos ^4x)\,dx \]

Answer: \dfrac{25}{12}


Question: Use a triple integral to find the volume of the solid enclosed between the elliptic cylinder \(x^2 + 9y^2 = 9\) and the planes \(z = 0\) and \(z = x + 3\).

Answer: 9\pi


Question: Find the number of valid strings of length \( n \) formed from the set \( \{A, B, C, D, E, F, G\} \) that contain at least one of the letters A, B, C, or D using the Principle of Inclusion-Exclusion.

Answer: 7^n - 3^n


Question: Find the sum function of the power series \( \sum\limits _{n=1}^{\infty}\frac{x^{2n+1}}{n} \).

Answer: -x \ln(1 - x^2)


Question: Given a finite field extension $F/K$ that is not separable, determine whether there always exists a subfield $L$ such that $K \subset L \subset F$ and $[F:L] = p$, where $p = \text{char}(K)$. Provide a justification for your answer.

Answer: Yes


Question: Find the value of \( a \) such that the plane \( x + y + z = -1 \) is tangent to the surface \( z = x^2 + ay^2 \).

Answer: \dfrac{1}{3}


Question: Find the value of \( x \) such that the matrix \( H = \begin{bmatrix} 2 & 2 & 1 \\ -4 & -1 & 4 \\ 4 & 5 & x \end{bmatrix} \) has at least one eigenvalue equal to 2.

Answer: -1


Question: Determine the number of points at which a line in \(\mathbb{CP}^n\) intersects \(\mathbb{CP}^2\). Assume the line is a copy of \(\mathbb{CP}^1\).

Answer: 1


Question: In a group $G$ of order $400 = 2^4 \cdot 5^2$, assume there are sixteen Sylow 5-subgroups. Let $P_5$ be one of them. Is the normalizer $N_G(P_5)$ equal to $P_5$?

Answer: Yes


Question: Evaluate the limit \( \lim_{x \to 0^+}\frac{4e^{x^2}-4\cos(x^3)-2\sin(x^4)-8\sqrt{1+x^2}-x^4+8}{1-\sqrt[3]{1+13x^6}} \) without using L'Hôpital's rule.

Answer: -\dfrac{1}{2}


Question: Determine the value of \(a\) such that the limit \( \lim_{x \to \infty} x^a \left ( \left(x+\frac{1}{x}\right)^\frac{2}{7} - x^\frac{2}{7} \right ) \) exists and is neither 0 nor infinity.

Answer: \dfrac{12}{7}


Question: Use the orbit-stabilizer theorem to find the order of the group of automorphisms of \( \mathbb{Z}_6 \). Consider the action of the symmetric group \( S_6 \) on \( \mathbb{Z}_6 \) defined by \( \sigma([a]) = [\sigma(a)] \) for \( \sigma \in S_6 \) and \( a \in \{1, \ldots, 6\} \). Determine the size of the stabilizer of the element \([1]\) and use it to calculate the order of the automorphism group.

Answer: 2


Question: What is the maximum length of a chain of subgroups \( \{e\} \subset H_1 \subset \ldots \subset H_k = S_n \) in the symmetric group \( S_n \), where each inclusion is proper?

Answer: \sum_{p \leq n} \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor


Question: Determine the proportion of positive integers \( n \) that satisfy the inequality \( I(n^2) < \left(1 + \frac{1}{n}\right)I(n) \), given that both \( n \) and \( n^2 \) are deficient numbers. Here, \( I(x) = \frac{\sigma(x)}{x} \) is the abundancy index, where \( \sigma(x) \) is the sum-of-divisors function.

Answer: 0


Question:
Determine whether the following determinant is non-zero:
\[
D = \begin{vmatrix}
1 & 2 & 3 & \cdots & 2008 & 2009 & 2010 & 2011 \\
2^2 & 3^2 & 4^2 & \cdots & 2009^2 & 2010^2 & 2011^2 & 2012^2 \\
3^3 & 4^3 & 5^3 & \cdots & 2010^3 & 2011^3 & 2012^3 & 2012^3 \\
\cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\
k^k & (k+1)^k & \cdots & 2011^k & 2012^k & \cdots & 2012^k & 2012^k \\
\cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\
2010^{2010} & 2011^{2010} & 2012^{2010} & \cdots & 2012^{2010} & 2012^{2010} & 2012^{2010} & 2012^{2010} \\
2011^{2011} & 2012^{2011} & 2012^{2011} & \cdots & 2012^{2011} & 2012^{2011} & 2012^{2011} & 2012^{2011}
\end{vmatrix}
\]

Answer: D \neq 0


Question: Evaluate the limit: $$\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{k=1}^{N-1}\left(\frac{k}{N}\right)^N$$

Answer: 0


Question: Two sealed bags, A and B, each initially contain either a bead or a pearl with equal probability. A pearl is then added to bag B, and a pearl is randomly drawn from one of the bags. You are offered the choice to open one of the bags. Which bag is more likely to contain a pearl?

Answer: B


Question: Determine the order of the element $x + x^2$ in the multiplicative group of the field $\mathbb{F}_2[x]/(x^4 + x^3 + 1)$. Provide your answer as a single integer.

Answer: 15


Question: Consider a matrix $A$ of size ${n\choose 3} \times {n\choose 2}$, where each row corresponds to a triplet $\{i,j,k\}$ and each column corresponds to a pair $\{s,t\}$. The entry $A_{\{i,j,k\},\{s,t\}}$ is defined to be 1 if $\{s,t\} \subseteq \{i,j,k\}$ and 0 otherwise. For $n \ge 5$, does the matrix $A$ have a nonzero maximal minor?

Answer: Yes


Question: Find all positive integer solutions \((x, y)\) to the equation \(2x^2 - y^{14} = 1\).

Answer: (1, 1)


Question: Suppose $f: \mathbb{R} \to \mathbb{R}$ is a continuous function such that for any real $x$, \(|f(x) - f(f(x))| \leq \frac{1}{2} |f(x) -x|\). Must $f$ have a fixed point? Justify your answer.

Answer: Yes


Question: Determine the complexity of the recurrence relation \( T(n) = T(n - \sqrt{n}) + n \).

Answer: O(n^{3/2})


Question: Solve for \( x \) in the equation: \[ 10^{\log_{10} {\sqrt2}} - e^{\ln4} = {\sqrt2}{x^2} - 10^{\log_{10} x} -3 \]

Answer: 1


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

Answer: \dfrac{\pi}{4}


Question: Given the equation \( z^2 + (\Im z)^2 = i(z - 18) \), where \( z = a + bi \) is a complex number, find the value of \( \Im z \).

Answer: -4


Question: Calculate the line integral \( \int_{A}\frac{y\,dx-(x+1)\,dy}{x^2+y^2+2x+1} \) where \( A \) is the line \(|x|+|y|=4\), travelling clockwise and making one rotation. Use Green's theorem to find the value of the integral.

Answer: 2\pi


Question: Evaluate the limit \( \lim_{x \to 0} \int_0^x \frac{\cos(t^3)}{t+x} \, dt \).

Answer: \ln 2


Question: Let an integer \( n \) be a 'square-in', meaning it is not a square, a prime power, or square-free. Is there always another 'square-in' number between \( n \) and \( 2n \)?

Answer: Yes


Question: Does there exist a set \( S \subset \mathbb{R}^2 \) such that its set of limit points in \( \mathbb{R}^2 \) contains the line \([0,1] = \{(x,0): 0 \le x \le 1\}\) and no three points in \( S \) are collinear?

Answer: Yes


Question: Given the functional equation \( f(x) = 3f(x+1) - 3f(x+2) \) and the condition \( f(3) = 3^{1000} \), find the value of \( f(2019) \).

Answer: \dfrac{1}{6561}


Question: Two positive numbers \(x\) and \(y\) satisfy the equation \(128x^2 - 16x^2y + 1 = 0\). Find the minimum value of \(x + y\).

Answer: \dfrac{35}{4}


Question:
Given the permutation \( \pi \in S_{10} \) defined by the table:
\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
 i & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
\pi(i) & 9 & 7 & 10 & 4 & 8 & 1 & 2 & 5 & 6 & 3 \\
\hline
\end{array}
\]
Determine the number of functions \( f: \Bbb{N_{10}} \rightarrow \Bbb{N_{10}} \) such that \( f(\pi(i)) = \pi(f(i)) \) for all \( i \in \Bbb{N_{10}} \).

Answer: 1372


Question: Let $I = A \times B$, where $A, B \subset \mathbb{R}$ are closed sets of positive Lebesgue measure, and let $E \subset \mathbb{R}^2$ be a set of zero Lebesgue measure. Is it true that $\overline{I \setminus E} = I$?

Answer: Yes


Question: Let $(a_n)$ be a decreasing sequence with $a_n \to 0$. Determine the value of $$\lim_{n\to \infty}\sum_{k=0}^n(-1)^ka_{n+k}.$$

Answer: 0


Question:
Determine the smallest integer $N$ such that the remainder $R_N$ of the series \( \sum_{n=2}^{\infty} \frac{1}{n(\ln n)^2} \) after summing the first $N$ terms is less than 0.1. That is, find $N$ such that:
\[
R_N = \sum_{n=N+1}^{\infty} \frac{1}{n(\ln n)^2} < 0.1.
\]

Answer: 22027


Question:
Given that \(a, b, c, d \in \left[\frac{1}{2}, 2\right]\) and \(abcd = 1\), find the maximum value of the expression:
\[
(a+\frac{1}{b})(b+\frac{1}{c})(c+\frac{1}{d})(d+\frac{1}{a})
\]

Answer: 25


Question:
Determine whether there exists an element of order \((p-1)/2\) in the group \(\mathbb{Z}^{\ast}_p\), where \(p\) is the large prime number given below:

\[ p = 863190718341645771003614298695024893990207192546856205218226372307466528127232538364530032707660046143282952063932885298283116284157856527266539861069097142611570511777538875474993819960760889160802939 \]

Answer: Yes


Question:
Given the vector space \( \mathbb{R}^2 \) over \( \mathbb{R} \) with the ordered basis \( \alpha = \{f_1, f_2\} \) where \( f_1 = (1,0) \) and \( f_2 = (1,1) \), consider the linear transformation \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) defined by \( T((x,y)) = (-x+2y, -x+y) \). The matrix representation of \( T \) with respect to the basis \( \alpha \) is given by:
\[ [T]^{\alpha}_{\alpha} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \]
Compute \( T^{65}(f_1) \).

Answer: (-1, -1)


Question: Given the sequence \(a_{n}\) defined as the solution to the equation \(x^{n} = \cos x\) for \(x > 0\), find the value of \(\lim_{n\rightarrow\infty}n\left ( 1- a_{n} \right )\).

Answer: -\ln(\cos 1)


Question: Evaluate the sum \( \sum_{x \in \mathbb{Q}^+} \frac{1}{f(x)^2} \), where \( f\left(\frac{a}{b}\right) = ab \) and \( \frac{a}{b} \) is an irreducible fraction in \( \mathbb{Q}^+ \).

Answer: \dfrac{5}{2}


Question: Consider the function \( f(x) = 4x^4 + 6x^3 + 2x^2 + 203x - (203)^2 \). Determine whether the local extrema of \( f'(x) \) is positive, negative, or zero. Choose the correct option: (a) positive, (b) negative, (c) zero, (d) data insufficient.

Answer: a


Question: Find the smallest distance between the point \((0,0,1)\) and the ellipsoid defined by the equation \(x^2 + 2y^2 + z^2 = 16\).

Answer: \sqrt{7}


Question: Evaluate \( f^{2011}g^{1994}\left(\frac{1}{2}\right) \) given \( f(x) = \frac{1}{1-x} \) and \( g(x) = 1 - \frac{1}{x} \).

Answer: -1


Question: Find the limit of the sequence of functions \( f_n(x) = \frac{1}{(1 + \frac{x}{n})^n x^{\frac{1}{n}}} \) as \( n \to \infty \).

Answer: e^{-x}


Question: Evaluate the limit using the Squeeze Theorem: \[ \lim_{n\to\infty} \dfrac{1^3+2^3+3^3+\ldots+n^3}{(n^2+1)^2} \]

Answer: \dfrac{1}{4}


Question: Determine the nature of singularities and calculate the residue of the function \( f(z) = \frac{e^z - \sin z - 1}{z^5 + z^3} \) at \( z = 0 \).

Answer: \dfrac{1}{2}


Question: Determine the value of $b$ ($b \neq a$) such that the causal linear shift-invariant system described by the difference equation \( y(n) - ay(n-1) = x(n) - bx(n-1) \) is an allpass system, meaning the magnitude of its frequency response is constant, independent of frequency.

Answer: \dfrac{1}{a}


Question: Let $T$ be a metric space. A subset of $T$ is regularly open if it is equal to the interior of its closure. Given a proper inclusion $A \subset B$ of two regularly open sets in $T$, must the difference $B \setminus A$ have a non-empty interior? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to\infty}\left(\binom{n}{0}\binom{n}{1}\dots\binom{n}{n}\right)^{\frac{1}{n(n+1)}} \]

Answer: \sqrt{e}


Question: Prove that \(11\) divides \(3^{2n+2} + 2^{6n+1}\) for all natural numbers \(n\).

Answer: 11


Question: Determine the number of subfields of a finite field with $2^{36}$ elements.

Answer: 9


Question: Let $a_2$, $a_1$, and $a_0$ be three digits. When the 3-digit number $a_2 a_1 a_0$ is read in base $b$ and converted to decimal, the result is 254. When the 3-digit number $a_2 a_1 a_0$ is read in base $b + 1$ and converted to decimal, the result is 330. Finally, when the 3-digit number $a_2 a_1 a_0$ is read in base $b + 2$ and converted to decimal, the result is 416. Find the 3-digit number $a_2 a_1 a_0$ in decimal form.

Answer: 512


Question: Find the asymptotic complexity in Theta notation for the recurrence relation \( T(n) = T\left(\frac{4}{5}n + 1\right) + n^2 \).

Answer: \Theta(n^2)


Question: Prove that the expression \( \|A\| = \underset{\|x\| = 1}{\sup} \frac {\|Ax\|} {\|x\|} \) defines a norm for a matrix \( A \in \mathbb{R}^{n,n} \).

Answer: \|A\|


Question: Minimize the function $x^2 + y^2$ subject to the constraint $(x+1)^3 = -y^2$. Find the values of $x$ and $y$ that achieve this minimum.

Answer: (-1, 0)


Question: Find a polynomial \( g(t) \) such that the linear map \( g(T) \) has the characteristic polynomial \( p(t) = (11-t)^r(17-t)^{n-r} \).

Answer: 17 - 6t


Question: Determine whether the limit \( \lim_{n\to\infty} d_n \) exists for the edge density of triangle-free graphs, where \( d_n \) is defined as the maximum edge density of a triangle-free graph with \( n \) vertices. If the limit exists, find its value. If not, find \( \lim \sup_{n\to\infty} d_n \).

Answer: \dfrac{1}{2}


Question: Evaluate the line integral \( \int_C G(x,y,z) \, ds \) on the curve defined by the parametric equations \( x= \frac{1}{3} t^3, \quad y=t^2, \quad z=2t \), where \( t \) ranges from 0 to 1, for the function \( G(x, y, z) = 4xyz \). Round your answer to the nearest hundredth.

Answer: 1.06


Question: Let \( A \) be a positive definite matrix. Determine whether the equation \( (I+A) A (I+A)^{-1} = A \) holds true.

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{140\pi} | \sin^2(x) \cos(x)|dx \).

Answer: \dfrac{280}{3}


Question: Determine the fundamental group of the complement of a circle in 3D space, \( \Bbb{R}^3 \setminus S^1 \).

Answer: \mathbb{Z}


Question: Determine the values of \( r > 0 \) for which the derivative \( f'(0) \) exists, where the function \( f: \mathbb{R} \to \mathbb{R} \) is defined by \( f(x) = x^r \sin(1/x) \) for \( x \neq 0 \) and \( f(0) = 0 \).

Answer: r > 1


Question: Evaluate the sum \( \sum_{n \in A} \sin^2\left( \frac{n\pi}{3003}\right) \), where \( A = \{ x \in \mathbb{N}^+ : x < 3003 \text{ and } (x, 3003) = 1 \} \).

Answer: \dfrac{1439}{2}


Question: Find the product of the roots of the polynomial with integer coefficients of least degree such that \(7^{1/3} + 49^{1/3}\) is a root.

Answer: 56


Question: Find the degree of the extension $[L:\mathbb{Q}]$, where $L$ is the splitting field of the polynomial $f(x)=x^6-9x^4+3x^2+5$ over $\mathbb{Q}$. The roots of $f(x)$ are $x_{1}=-1$, $x_{2}=1$, $x_{3}=-\sqrt{4+\sqrt{21}}$, $x_{4}=\sqrt{4+\sqrt{21}}$, $x_{5}=-i\sqrt{\sqrt{21}-4}$, $x_{6}=i\sqrt{\sqrt{21}-4}$. The splitting field is $L=\mathbb{Q}[\sqrt{4+\sqrt{21}},i\sqrt{\sqrt{21}-4}]$. Determine $[L:\mathbb{Q}]$. \(\boxed{?}\)

Answer: 8


Question: Evaluate the limit without using L'Hospital's Rule: \[ \lim_{x \to 2} \frac{\cos{\left(\frac{\pi}{x}\right)}}{x-2} \]

Answer: \dfrac{\pi}{4}


Question: A rat is trapped in a maze with three doors. If the rat takes door one, it will wander around the maze for 2 minutes and return to the starting point. If it takes door two, it will wander for 3 minutes and return to the starting point. If it takes door three, it will find the cheese after 1 minute. Upon returning to the starting point, the rat immediately picks a door uniformly at random. Calculate the average time the rat will wander before finding the cheese.

Answer: 6


Question: Use the divergence theorem to evaluate the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \), where \( \mathbf{F}(x,y,z) = x^3 \hat{\imath} + y^3 \hat{\jmath} + z^3 \hat{k} \) and \( S \) is the surface of the solid bounded by the cylinder \( x^2 + y^2 = 1 \) and the planes \( z = 0 \) and \( z = 2 \).

Answer: 11\pi


Question: Solve for \( n \) in the equation: \[ 1 = \binom{n}{3} \left(\frac{1}{2}\right)^{n-2} \]

Answer: 4


Question: Calculate \( \lim\limits_{x\to a}\frac{a^{a^{x}}-{a^{x^{a}}}}{a^x-x^a} \).

Answer: a^{a^{a}} \ln a


Question: Find the least value of $m+n$, where $1 \le m < n$, such that $1978^n - 1978^m$ is divisible by $1000$. Determine the smallest possible sum of $m$ and $n$.

Answer: 106


Question: Let $\mathscr{U}$ be a collection of half-open intervals $[a,b)$ such that $\mathscr{U}$ covers $\mathbb{R}$. Does there exist a countable subcover for any such $\mathscr{U}$?

Answer: Yes


Question: Find the smallest positive integer \( e \) such that \( 13^e \equiv 32 \pmod{37} \).

Answer: 7


Question: Determine the radius of convergence for the series \( \sum_{n=0}^{\infty}\sin(2n)x^n \).

Answer: 1


Question: Find the minimum value of \(|z-i| + |z+i| + |z-7i| + |z+3i|\) for \(z \in \mathbb{C}\).

Answer: 12


Question: Evaluate the integral \( \int_C \frac{1}{z^{2017}-1} \, dz \) using the Residue Theorem, where \( C \) is the contour \( |z|=2 \) traversed in the counterclockwise direction.

Answer: 0


Question:
Find a function \( f(x) \) whose Taylor series expansion is given by the series:
\[ \sum\limits_{n = 1}^\infty  {\dfrac{{{{( - 1)}^{n - 1}}}}{2n - 1}{x^{2n}}} \]

Answer: x \arctan(x)


Question: For any given integer $n \geq 1$, does there exist a stochastic matrix $P \in [0,1]^{n \times n}$ such that the kernel of $(P-I)^T$ contains only nonpositive vectors?

Answer: No


Question: Evaluate the limit: $$\lim_{n\to\infty} \,\sum_{k=1}^n\left(\frac{k}{n^2}\right)^{\frac{k}{n^2}+1}.$$

Answer: \dfrac{1}{2}


Question: Let $p$ and $q$ be two distinct odd primes such that $q$ divides $p-1$. Determine whether there always exists a primitive root modulo $p$ that is not divisible by $q$. Provide a proof or counterexample.

Answer: Yes


Question: Evaluate the integral: $$I = \int_1^2 \left(e^{1-\frac{1}{(x-1)^2}} +1\right) + \left(1 +\frac{1}{\sqrt{1- \log(x-1)} }\right)\mathrm{d}x.$$

Answer: 3


Question: Find the multiplicative factor \( \beta(x,y) = x^n y^m \) such that the differential equation \((2y^2-6xy) dx+(3xy-4x^2) dy=0\) becomes exact.

Answer: xy


Question: Determine if there exists a positive integer \( n > 0 \) such that \( f(n) = 0 \), where \( f(n) = n - \sum_{k=1}^{m} \lfloor r_k n \rfloor \) and \( r_1, r_2, \ldots, r_m \) are positive rational numbers with \( r_1 + r_2 + \cdots + r_m = 1 \).

Answer: Yes


Question:
Determine the radius of convergence for the power series representation of the function \( f(x) = \int_0^x \frac{\sin(t)}{t} \, dt \), given by:
\[
f(x) = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)((2n+1)!)}.
\]

Answer: \infty


Question: Let $G$ be a group and $H$ a subgroup of $G$. The commutator of $a, b \in G$ is defined as $[a, b] = aba^{-1}b^{-1}$. The commutator subgroup of $G$ is defined as $G^{'} = \langle [a, b]: a, b \in G \rangle$, which is the subgroup generated by the commutators of $G$. Determine whether it is true that if $H \leq G$, then $H^{'} \leq G^{'}$. Provide a proof or counterexample.

Answer: True


Question: Find all natural number solutions \((a, b)\) to the equation \(a^3 + b^3 = p^4\), where \(p\) is a prime number.

Answer: (2, 2)


Question: Let $Y$ be uniformly distributed on $(0,1)$, and let $X|Y=y$ be a binomial random variable with parameters $(n,y)$. Find the probability $P(X=0)$. Express your answer in terms of $n$.

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


Question: You have collected sand from the beach 100 times, with the weight of sand collected on each trip following a uniform distribution $\mathrm{U}[10-\sqrt{48}/2, 10+\sqrt{48}/2]$. Assuming each collection is independent, estimate the probability that the total weight of sand collected is between 980 and 1030 ounces, inclusive.

Answer: 0.7745


Question: Consider two touching unit balls, referred to as central balls. Determine the maximum number \( k \) of non-overlapping unit balls such that each ball touches at least one of the two central balls.

Answer: 9


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

Answer: \dfrac{1}{e}


Question: Find the volume of the solid with unit density lying under the surface \( z = xy \) and above the triangular region in the \( xy \)-plane with vertices \((0, 1)\), \((1, 1)\), and \((0, 2)\).

Answer: \dfrac{5}{24}


Question: Find a generator of the ideal generated by $3$ and $2-2\sqrt{-2}$ in $\mathbb{Z}[\sqrt{-2}]$. Given that $\mathbb{Z}[\sqrt{-2}]$ is a Euclidean domain, determine the generator with minimal norm.

Answer: 1 - \sqrt{-2}


Question: Evaluate the limit: \[ \lim _{x \rightarrow 0^{+}} \frac{\tan 3 x \sqrt{1-\cos 4 x}}{x^{2}+x^{3}} \]

Answer: 6\sqrt{2}


Question: Is there an algorithm that, given rational matrices $A_{1}, \ldots, A_{k}$ generating a finite group, outputs a matrix $T$ such that $T^{-1}A_{1}T, \ldots, T^{-1}A_{k}T$ are integer matrices?

Answer: Yes


Question:
Assume that $f:\mathbb{R} \rightarrow \mathbb{R}$ is a continuous function and $h \in \mathbb{R}$. Let $\Delta_h^n f(x)$ be the $n$-th finite difference of $f$, defined as:
\[ \Delta_h^n f(x) = \sum_{k=0}^n (-1)^{n-k} \frac{n!}{k!(n-k)!} f(x+kh). \]
Suppose for each $x \in \mathbb{R}$, the following limit holds:
\[ \lim_{h \to 0} \frac{\Delta_h^n f(x)}{h^n} = 0. \]
Is $f$ necessarily a polynomial of degree $\leq n-1$?

Answer: Yes


Question: Given the equations \(a + b + c + d + e = 12\) and \(a^4b^3c^3de = 4 \times 6^6\), where \(a, b, c, d, e\) are positive integers, find the value of \(4a + 3b + 3c + d + e\).

Answer: 36


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{(4n^3 + 1)(4n - 2)!n\sin{\frac{2}{n}}}{(4n + 1)!+3} \]

Answer: \dfrac{1}{8}


Question: Determine the number of steps, Tn, required to solve the Towers of Hanoi problem with n disks, where all disks start on pillar A and must be moved to pillar C. It is not allowed to move a disk directly from pillar A to pillar C; all disks must pass through pillar B. What is Tn?

Answer: 3^n - 1


Question: Find the smallest integer $n > 4$ such that $n$ points can be placed in the plane with no three points collinear, each point connected to exactly three other points, and for any two points $A$ and $B$ that are not directly connected, there exists a point $C$ connected to both $A$ and $B$. What is the least value of $n$?

Answer: 6


Question: Let $f(z)$ be an entire function that is real on the real axis, with $f(0) = 1$. Assume there exists a real constant $M$ such that $|f(z)| \le M$ for all $z$ with $\text{Im}(z) \ge 0$. Evaluate \( \lim\limits_{x \rightarrow + \infty}{f(x)}. \)

Answer: 1


Question: Let \( X, Y, Z \) be independent random variables, each with a standard normal distribution. Compute the probability \( P(X + Y < Z - W) \), where \( W \) is also an independent standard normal random variable.

Answer: \dfrac{1}{2}


Question:
Given the probability matrix of a Markov chain:
\[ P = \begin{bmatrix}0&1/2&1/2\\1/3&0&2/3\\1/2&1/2&0\end{bmatrix} \]
find the limit:
\[ \lim_{n \to \infty} P(X_n = 2 \mid X_0 = 1) \] analytically.

Answer: \dfrac{1}{3}


Question:
Given that \( \cos\alpha + \cos\beta + \cos\gamma = \sin\alpha + \sin\beta + \sin\gamma = 0 \) and \( \cos 3\alpha = \frac{3}{4} \), find the value of \( \cos^8\alpha + \cos^8\beta + \cos^8\gamma \). Choose the correct answer from the following options:

A) \( \frac{3}{128} \)

B) \( \frac{27}{32} \)

C) \( \frac{3}{16} \)

D) \( \frac{5}{128} \)

Answer: B


Question: Evaluate the double integral \( \int\int_D (1+xy) \, dA \) over the region \( D = \{(x,y) \mid 1 \leq x^2 + y^2 \leq 2 \text{ and } y \geq 0\} \).

Answer: \dfrac{\pi}{2}


Question: Given a yearly 3x3 transition matrix with eigenvalues \( \lambda = 1, \frac{1}{2}, 0 \), and corresponding eigenvectors \( x_1 = (60, 20, 30) \) and \( x_2 = (-60, -30, -90) \), the initial state vector is \( v_0 = 500x_1 + 200x_2 + 100x_3 \), where \( x_3 \) is the eigenvector for \( \lambda = 0 \). Determine the number of people who will be sick with the disease 2 years later.

Answer: 8500


Question:
Dan has 13 different cards, each numbered uniquely from 1 to 13. Every day, he draws one card at random from the remaining cards, without replacement. If Dan draws the card numbered 13, he stops drawing. Let \( X_i \) be the number on the card drawn on the \( i \)-th day, where \( X_i = 0 \) if the card numbered 13 is drawn before the \( i \)-th day. What is the expected value \( E[X_4] \)?

Options:

- \( \frac{9}{2} \)
- \( \frac{11}{2} \)
- \( \frac{70}{13} \)
- None of the above.

Answer: B


Question: Evaluate the limit: \[ \lim_{x \to 2} \frac{e^x - e^2 - (x^2 - 4x + 4)\sin (x - 2)}{\sin [(x - 2)^2] + \sin (x-2)} \]

Answer: e^2


Question: Let \( f(x) \) be a polynomial of degree three that leaves a remainder of 1 when divided by \((x-1)^2\) and a remainder of -1 when divided by \((x+1)^2\). If the roots of \( f(x) = 0 \) are \( \alpha, \beta, \gamma \), find the value of \( \alpha\beta + \beta\gamma + \gamma\alpha \).

Answer: -3


Question: Let $X, Y, Z$ be independent exponential random variables with parameter $\lambda$. Calculate the probability $P(X + Y > Z)$. Assume $X, Y, Z \sim \text{Exp}(\lambda)$. Provide your answer in terms of $\lambda$.

Answer: \dfrac{3}{4}


Question: If a Lie group $G$ acts smoothly and freely on a smooth manifold $M$, and $M/G$ is a topological manifold, is the dimension of $M/G$ necessarily equal to $\dim M - \dim G$?

Answer: \dim M - \dim G


Question: Evaluate the triple integral \( \iiint_T xz \, dV \) where \( T \) is the solid tetrahedron with vertices at \((0, 0, 0)\), \((1, 0, 1)\), \((0, 1, 1)\), and \((0, 0, 1)\).

Answer: \dfrac{1}{30}


Question: Solve the system of equations: \[\large \left \{ \begin{aligned} \sqrt{x + 2}(x - y + 3) &= \sqrt{y} \\ x^2 + (x + 3)(2x - y + 5) &= x + 16 \end{aligned} \right.\]

Answer: (1, 3)


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0}\frac{(1+x)^{1/x}-e+\frac{ex}{2}}{x^2} \]

Answer: \dfrac{11e}{24}


Question: Evaluate the definite integral: \[ I = \int_{-2}^{2} \frac{x^2}{1+5^x}\,\mathrm{d}x. \]

Answer: \dfrac{8}{3}


Question: Evaluate the double sum: $$\sum_{j=1}^{\infty} \sum_{i=1}^{\infty} \frac{j^2i}{3^j(j3^i+i3^j)}.$$

Answer: \dfrac{9}{32}


Question: Let $C \subseteq \mathbb{R}^n$ be a compact convex set. Determine if there exists a convex function $f : \mathbb{R}^n \to \mathbb{R}$ and real numbers $a \leq b$ such that $C = f^{-1}([a, b])$. Provide a proof or counterexample.

Answer: Yes


Question: Given a 3x3 invertible matrix $A$ with $\operatorname{tr}(A) = \operatorname{tr}(A^2) = 0$ and $\det(A) = 1$, determine the characteristic polynomial of $A$.

Answer: \lambda^3 - 1


Question: Find the coefficient of \(x^n\) in the expansion of \(\left(1 + \frac{x}{1!} + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!} \right)^2\).

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


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

Answer: 0


Question: Is it possible to have a group automorphism of the additive group of real numbers that fixes a subset of real numbers but is not the identity? The subset may be infinite.

Answer: Yes


Question: Determine the number of unique combinations of sprinkles that can be used on a cookie cake with 4 slices, given 20 types of sprinkles. You can use none, one, two, or three types of sprinkles on each slice. Note that combinations that can be obtained by rotating another combination should be counted as the same. Calculate the total number of unique combinations.

Answer: 832840129576


Question: Assume that $(f_n)$ is a sequence of continuous functions with $f_n:[0,1] \to \mathbb{R}$. Suppose $(x_n)$ is a sequence of elements from the interval $[0,1]$ such that $x_n \to 0$ and $f_n(x_n)=0$ for each $n$. Does it follow that $f_n(0) \to 0$ as $n \to \infty$?

Answer: No


Question:
Determine whether the limit \( \lim_{n \to \infty} x_n \) exists for the sequence defined by:
\[ x_n = \sqrt[3]{n+1}\cos{\sqrt{n+1}} - \sqrt[3]{n}\cos{\sqrt{n}}, \forall n\in \mathbb{N}. \]

Answer: 0


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sum_{j=0}^{n}{{j+n-1} \choose j}\frac{1}{2^{j+n}} \]

Answer: \dfrac{1}{2}


Question: Calculate the angle between the line \((6, 8, 5) + t (3, 4, 5)\), where \(t \in \mathbb{R}\), and the sphere \(x^2 + y^2 + z^2 = 25\) at their points of intersection.

Answer: 45^\circ


Question:
Consider the sequence of functions \(f_n : \mathbb{N} \to \mathbb{N}\) defined by \(f_n(k) = k \ \text{(mod n)}\). Determine which of the following statements is true:

1. \((f_n)\) is not pointwise convergent.
2. \((f_n)\) is pointwise convergent to a bounded function.
3. \((f_n)\) is pointwise convergent to an unbounded function.
4. \((f_n)\) is uniformly convergent on \(\mathbb{N}\).

Answer: 3


Question: Determine the region of uniform convergence for the series \( \sum_{n=0}^\infty \frac{\cos nx}{n^3+x^2n} \) using the Weierstrass M-test.

Answer: \mathbb{R}


Question: Determine the positive integer \(a\) such that the polynomial \(x^2 - x + a\) evenly divides \(x^8 + 5x^6 + 13x^4 + 20x^2 + 36\).

Answer: 3


Question: Determine the starting value of $x_0 \in \mathbb{R}$ for which the sequence $x_n = \cos(x_{n-1})$ converges.

Answer: \mathbb{R}


Question: Find the value of \(a\) such that \[ L = \lim_{n\to\infty}\frac{1^a+2^a+\cdots+n^a}{(n+1)^{a-1}[(na+1)+(na+2)+\cdots+(na+n)]} = \frac{1}{60} \] for \(a \in \mathbb{R} \setminus \{-1\}\).

Answer: 7


Question: Determine the number of distinct values the function \( f(z) = \sqrt{\frac{z^2-1}{\sqrt{z}}}(z-i)^{1/3} \) can take for a given value of \( z \).

Answer: 12


Question: Determine whether the sequence \( \frac {2 \cdot 4 \cdot 6 \cdot \dots \cdot (2n)} {1 \cdot 3 \cdot 5 \cdot \dots \cdot (2n-1) (n^2)} \) converges.

Answer: 0


Question: Let $Q$ be a smooth hyperquadric in $\mathbb{P}^5_{\mathbb{C}}$. Determine the dimension of the variety $F_2(Q)$, which parametrizes all 2-planes contained in $Q$. Assume that each hyperquadric contains at least one plane and that the dimension of the Grassmannian $G(k,n)$ is given by $(n-k)(k+1)$. Find $\dim(F_2(Q))$. \( \text{Provide your answer as a single integer.} \)

Answer: 3


Question: In how many ways can 12 people, each representing a unique astrological sign, be lined up in front of 12 doors labeled with the astrological signs such that no person stands in front of the door labeled with their own sign?

Answer: 176214841


Question: Determine whether the following proposition is true or false: "For a function $f$ and a fixed radius $r_0 > 0$, if $\|f\|_{L^p(B(x,r_0))} = 0$ for all $x \in \mathbb{R}^n$, then $\|f\|_{L^p(\mathbb{R}^n)} = 0$."

Answer: True


Question: Let $z_N$ be the complex $N$-th root of unity. Given $N = r \cdot t$, let $H^\bot$ be the set of multiples of $t$ in $\mathbb{Z}_N$. Prove that for any $x \notin H^\bot$, the following holds: \[ \sum_{i=0}^{t-1} z_N^{rxi} = 0. \]

Answer: 0


Question: Does there exist a complex number $z_0 \notin \{a_n\}_{n \geq 1}$ such that $f(z_0) = 0$, where $f(z) = \prod\limits_{n=1}^{\infty} E_{p_n} (z/a_n)$ is an entire function defined by the Weierstrass Factorization Theorem?

Answer: No


Question:
Find the integer \( n \) such that the following product holds:
\[
\left(1 + \frac{1}{1^2}\right) \left(1 + \frac{1}{(1 + i)^2}\right)
\left(1 + \frac{1}{(1 + 2i)^2}\right) \cdots \left(1 + \frac{1}{(1 + (n-1)i)^2}\right) = \frac{10 + 8i}{1 + 8i}.
\]

Answer: 9


Question:
Given the following congruences for a polynomial \( f(x) \):
\[ f(x) \equiv 2x + 1 \pmod{x^2 + 4} \]
\[ f(x) \equiv 6x - 1 \pmod{x^2 + 6} \]
Define \( r(x) \) such that:
\[ f(x) \equiv r(x) \pmod{(x^2 + 4)(x^2+6)} \]
Find the value of \( r(4) \).

Answer: -131


Question: Is the space of continuously differentiable functions $C^1[a,b]$ a closed subset of the space of continuous functions $C[a,b]$ under the supremum metric?

Answer: No


Question: Let \( F_n \) be the DFT (Discrete Fourier Transform) matrix defined as \( F_n = (\exp(2 \pi i /n)^{j\cdot k}) \in \mathbb{C}^{n,n} \) with \( j,k \in \{1,...,n\} \). Compute the product \( F_n^H \cdot F_n \), where \( F_n^H \) is the conjugate transpose of \( F_n \).

Answer: n I_n


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

Answer: \infty


Question: Let \( g: [0,1] \rightarrow \mathbb{R} \) be a continuous function such that \( g(0) > 0 \) and \( g(1) = 0 \). Prove that there exists a point \( a \in (0,1] \) such that \( g(a) = 0 \) and \( g(x) > 0 \) for all \( 0 \le x < a \).

Answer: a


Question: Determine whether there exists an element of order 420 in the symmetric group \( S_{19} \).

Answer: Yes


Question: Does there exist a continuous linear functional $A \in l_\infty^*$ such that for all $x \in l_\infty$, $\lVert Ax \rVert \neq \lVert A \rVert \lVert x \rVert$?

Answer: Yes


Question: Find the isomorphism between \( \mathrm{Hom}(\mathbb{Z}_n, \mathbb{Q}/\mathbb{Z}) \) and \( \mathbb{Z}_n \).

Answer: \mathbb{Z}_n


Question: Determine whether the function \( f(z) = e^{\frac{1+z}{z-1}} \) can be uniformly approximated by harmonic continuous functions in the unit disc. Specifically, is it true that \( \inf\{||f-u|| : u \text{ is harmonic in the unit disc and continuous on the closed unit disc}\} = 0 \)? The infimum is taken over the interior \( |z|<1 \).

Answer: No


Question: Consider a full-dimensional convex polytope $P \subset \mathbb{R}^n$ with $n \ge 3$, having a bipartite edge-graph $G = (V_1 \cup V_2, E)$. Suppose there are concentric spheres $S_1$ and $S_2$ in $\mathbb{R}^n$ such that all vertices of $V_i$ lie on $S_i$. If $|V_1| = |V_2|$, does it necessarily follow that $S_1 = S_2$, meaning that $P$ is inscribed?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \to \infty}\left(\frac{(3n)!}{(2n)!n^n}\right)^{1/n} \]

Answer: \dfrac{27}{4e}


Question: Find the function \( f(X) \) that minimizes the conditional expectation \( E[(Y-f(X))^2|X] \), where \( X \) and \( Y \) are random variables with \( E(Y)=\mu \) and \( E(Y^2)<\infty \). Deduce that the function \( f(X) \) that minimizes this expression is \( f(X)=E[Y|X] \).

Answer: E[Y|X]


Question: Let $X$ be a discrete random variable following a Geometric distribution with parameter $\frac{1}{2}$, and let $Y$ be a continuous random variable following an Exponential distribution with rate 1. Assume $X$ and $Y$ are independent. Define $Z = X + Y$. Calculate the probability $P(Z \le 3.5)$. Provide your answer as a numerical value.

Answer: 0.7024


Question: Calculate the operator norm \( \|T\| \) for the linear transformation \( T: K^n \to \ell_1(\mathbb{N}) \) defined by \( Tx = (x_1, \ldots, x_n, 0, 0, \ldots) \) for all \( x = (x_1, \ldots, x_n) \in K^n \), where \( K^n \) is equipped with the Euclidean norm \( \|\cdot\|_2 \).

Answer: \sqrt{n}


Question: Determine the number of zeros of the polynomial \( p(z) = z^8 - 20z^4 + 7z^3 + 1 \) that lie within the open disk \( D(0,2) \).

Answer: 4


Question: Use Stokes' theorem to evaluate the integral \( \int_C (x + 2y) \, dx + (2z + 2x) \, dy + (z + y) \, dz \), where \( C \) is the intersection of the sphere \( x^2 + y^2 + z^2 = 1 \) and the plane \( y = z \). Assume the curve \( C \) is oriented counterclockwise when viewed from above.

Answer: 0


Question: Given that $0 \leq x_i \leq 1$ for all $i \in \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and $\sum_{i=1}^{10} x_i^2 = 9$, find the minimum value of $\sum_{i=1}^{10} x_i$. Provide your answer as a single number.

Answer: 9


Question: Let $X$ be a bivariate Normal random variable with mean $μ=(1,1)$ and covariance matrix $$\Sigma=\begin{bmatrix}3&1\\1&2\end{bmatrix}.$$ Find the variance of the conditional distribution of $Y=X_1+X_2$ given $Z=X_1−X_2=0$. Provide your answer to three decimal places.

Answer: 6.667


Question: Determine the number of real solutions to the differential equation \( y'' = -4yy' \).

Answer: \infty


Question:
Given a continuous function \( f: (0, \infty) \to \mathbb{R} \) such that for all \( x > 0 \), \( y > 0 \), and any \( x' \) in \((0, \infty)\), the following holds:
\[ \int_{x}^{xy} f(u) \, du = \int_{x'}^{x'y} f(u) \, du. \]
If \( f(17) = 17 \), calculate \( A(x) = \int_{1}^{x} f(t) \, dt \) for \( x > 0 \).

Answer: 289 \ln x


Question: Evaluate the limit: \[ \lim_{x\to \frac{\pi}{2}} \frac{(1-\tan(\frac{x}{2}))(1-\sin(x))}{(1+\tan(\frac{x}{2}))(\pi-2x)^3} \]

Answer: \dfrac{1}{32}


Question: Let $T_1$ and $T_2$ be the first two inter-arrival times in a Poisson process with rate $\lambda$. Define the random variable $Y = \frac{T_1}{T_1 + T_2}$. Find the probability density function of $Y$. Express your answer in terms of $\lambda$.

Answer: 1


Question: Let \( T, U, V, W \) be vector spaces over a field \( \mathbb{F} \), and let \( \alpha: T \to U \) and \( \beta: V \to W \) be fixed linear maps with ranks \( r \) and \( s \) respectively. If these spaces are finite-dimensional, determine the rank of the linear map \( \Phi: \mathcal{L}(U,V) \to \mathcal{L}(T,W) \) defined by \( \Phi(\theta) = \beta \circ \theta \circ \alpha \). Express your answer in terms of \( r \) and \( s \).

Answer: rs


Question: Find the integrating factor of the differential equation \((y\log y)dx = (\log y-x)dy\).

Answer: \dfrac{1}{y}


Question: Estimate the limit of \( (\cos(4x))^{\frac{1}{x^2}} \) as \( x \to 0 \).

Answer: e^{-8}


Question: Compute the integral \( \int_{\mathbb{R}} \frac{i \omega - \omega^2}{1+\omega^6} \, d\omega \).

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


Question: Consider a Markov chain where $X_0 = x$ and each subsequent state $X_i$ is drawn from a discrete uniform distribution on the set $\{0, \dots, X_i-1\}$. Determine the expected absorption time for the process to reach the state 0.

Answer: \sum_{k=1}^{x} \frac{1}{k}


Question: Find the rightmost digit of \( \left \lfloor \frac{10^{20000}}{10^{100}+3} \right\rfloor \).

Answer: 3


Question: Find the Laurent series of \( f(z) = \frac{\sinh(z + 3i)}{z(z + 3i)^3} \) at \( z = -3i \) and calculate the residue at this point.

Answer: \dfrac{1}{9}


Question: Find the value of $x$ such that $x^{a^2} = a^x$ when $a = 2$.

Answer: 16


Question:
Determine whether the following statement is true for all functions \(f\) and \(g\):

If \(\lim_{x\to c} \frac{f(x)}{g(x)}=1\), then \(\lim_{x\to c}f(x)-g(x)=0\).

Answer: False


Question: Find the residue of the function \( \frac{z}{z^6+1} \) at the point \( z=i \).

Answer: \dfrac{1}{6}


Question: Evaluate the limit \( \lim _{x\to 0} \frac{e^x-\ln\left(1+x\right)-1}{x^2} \) without using L'Hospital's rule or Maclaurin's series.

Answer: 1


Question: Evaluate the integral \( \iiint (x+y+z)(x-y-z)(x+y-z) \, dV \) using a change of variables under the planes: \( x+y+z=0 \), \( x+y-z=0 \), and \( x-y-z=0 \).

Answer: 0


Question: Find a function \( f:[0,\infty) \rightarrow [0,\infty) \) that is convex and increasing, such that \( (1/z)f(z) \) is non-convex in \((0,\infty)\).

Answer: f(z) = z^{3/2}


Question: Let $X$ be a Banach space and $Y$ a closed subspace of $X$. Assume there exists a closed subset $Z$ of $X$ such that $Z \cap Y = \{0\}$ and every $x \in X$ can be uniquely expressed as $x = y + z$ with $y \in Y$ and $z \in Z$. Can we conclude that $Y$ is complemented in $X$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to\infty} \left( \sqrt{x+1} - 5\ln(x+1) \right) \]

Answer: \infty


Question: What is the order of the symmetry group of a cube?

Answer: 48


Question: Determine whether the series \( \sum_{n=1}^\infty \frac{1}{n^2\sqrt{|x-r_n|}} \) converges for almost all \( x \) in \([0,1]\), where \((r_n)_{n=1}^\infty\) is an arbitrary sequence of numbers in \([0,1]\). Answer 'true' if it converges for almost all \( x \), otherwise answer 'not true'.

Answer: true


Question: Determine the number of parameters needed to completely define a 2D n-gon without over-specifying, given that the n-gon is defined by its side lengths and internal angles.

Answer: 2n - 3


Question: Let $M$ be a finitely generated module over a commutative ring $A$. Suppose there exists a positive integer $n$ and homomorphisms $\pi: A^n \rightarrow M$ and $\phi: A^n \rightarrow M$ such that $\pi$ is surjective and $\phi$ is injective. Is it true that $M \cong A^n$?

Answer: M \cong A^n


Question: Consider a sequence of real polynomials $p_k(z)$, each having real and distinct zeros that are totally interlacing. If we form a linear combination of these polynomials, given by $\sum c_{k}p_k(z)$, where the coefficients $c_{k}$ are all real, will the zeros of this linear combination also be real?

Answer: Yes


Question: Determine the operator norm of the linear operator \( T: L^2([0,1], \mathbb{R}) \to L^2([0,1], \mathbb{R}) \) defined by \( Tf(x) = x^2f(x) \).

Answer: 1


Question: Let $M$ be an infinite compact metric space. Prove that for any $\varepsilon > 0$, there exists a positive integer $n$ such that any subset of $M$ with at least $n$ points must contain two distinct points whose distance apart is at most $\varepsilon$.

Answer: n


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} \frac{1}{n}\sqrt[n]{\frac{(2n)!}{n!}} \] using the definition of the Riemann Integral.

Answer: \dfrac{4}{e}


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

Answer: \dfrac{\pi}{e}


Question: Evaluate the integral \( \int_X +\infty\ d\mu \) given that \( \mu(X) = 0 \) in a measure space \((X, \mathcal{S}, \mu)\).

Answer: 0


Question: Find the extremals of the functional \( \mathcal{F}(u) = \int_0^1 (u(t)^2 - u(t)u'(t)) \, dt \), where \( u \in C^1([0, 1], \mathbb{R}) \).

Answer: u(t) = 0


Question: Evaluate the limit: \[ \lim_{x \to 0} \left( \frac{\ln (\cos x)}{x\sqrt {1 + x} - x} \right) \]

Answer: -1


Question: Find the circulation along the boundary of the half annulus defined by \( \{(r,\theta)|1\le r\le 3, 0\le \theta \le \pi\} \) in the force field \( F = \langle -y, x \rangle \) using Green's Theorem.

Answer: 8\pi


Question: Suppose the sequence \((x_n)_{n \ge 1}\) converges to \(x \in \mathbb{R}\). Does the sequence \(\frac{1}{n}\{x_{mn+1}+x_{mn+2}+ \cdots +x_{2mn-1}+x_{2mn}\}\), where \(m \in \mathbb{Z^+}\), converge?

Answer: mx


Question: Determine whether there exists an interval $I \subset (0.8856, +\infty)$ such that the derivative $(\Gamma^{-1})'(x) > 1$ for all $x$ in $I$. The function $\Gamma^{-1}$ is the principal inverse of the gamma function, which is increasing and concave on $(0.8856, +\infty)$. Provide a justification for your answer.

Answer: Yes


Question: Given that $A$ is an $n \times n$ orthogonal matrix with $AA^t = I$ and $\det(A) < 0$, find $\det(A + I)$. Note: $A^t$ is the transpose of $A$, $\det A$ is the determinant of $A$, and $I$ is the identity matrix.

Answer: 0


Question: Given a compact set $A \subset \mathbb{R}^n$ with $n \geq 2$, where $A$ has measure zero and $0 \not\in A$, determine if there exists a line $L \subset \mathbb{R}^n$ passing through the origin that intersects $A$ in a finite number of points.

Answer: Yes


Question: Determine the values of $a$ for which the function $f(x) = \cos(ax)$ is a contraction mapping.

Answer: (-1, 1)


Question: Identify an exceptional prime below 100 that does not satisfy the three conditions outlined by Kummer for the impossibility of solving Fermat's equation.

Answer: 37


Question: In a cyclic group of order 15, there is an element \( x \) such that the set \( \{x^3, x^5, x^9\} \) contains exactly two distinct elements. Determine the number of distinct elements in the set \( \{ x^{13n} : n \text{ is a positive integer} \}. \)

Answer: 3


Question: Find the value of \(x\) that satisfies the equation: \[ \int^{2[x+14]}_0\left\{\frac{x}{2}\right\}dx = \int^{\{x\}}_0[x+14]dx \] where \([.]\) denotes the greatest integer function and \(\{.\}\) denotes the fractional part function.

Answer: -14


Question: Determine all real numbers \( \beta \in \mathbb{R} \) such that the equation \[ x(t) + \int_0^1 (1+\alpha ts)x(s)\,ds = \beta + t^2 \] is solvable in the space \( L_2[0,1] \) for any real number \( \alpha \in \mathbb{R} \).

Answer: -\dfrac{2}{3}


Question: Determine the number of digits, \(\mathbb{L}\), in the sum of divisors of the number \(2^2 \cdot 3^3 \cdot 5^3 \cdot 7^5\) that are of the form \(4n+1\), where \(n\) is a natural number. What is \(\mathbb{L}\)?

Answer: 8


Question: Evaluate the flux of the vector field \( \vec{F} = xi + yj + zk \) across the surface \( x^2 + y^2 + z^2 = 1 \) with the unit normal \( \hat{n} \) pointing outward. Use the Gauss divergence theorem to find the value of \( \iint_S \vec{F} \cdot \hat{n} \, dS \).

Answer: 4\pi


Question: Determine whether there exists a finite graph $G$ such that the Hadwiger number $\eta(G)$ is less than the coloring number $\text{Col}(G)$. Provide a justification for your answer.

Answer: Yes


Question: Find a value of $R > 0$ such that for all complex numbers $z$ with $|z| > R$, the inequality $|z^{10} + 100z + 9| > 1$ holds.

Answer: 2