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

Answer: -\dfrac{1}{2}


Question: Let $k$ be an arbitrary field, and let $\varphi:A\to B$ be a morphism of abelian varieties over $k$. Determine whether $\varphi(A)$ has the structure of an abelian subvariety of $B$ which is defined over $k$.

Answer: Yes


Question: Consider an indexed family of topological spaces \(\{ X_\alpha \}_{\alpha \in J}\) with an arbitrary non-empty index set \(J\). Let \(\Pi_{\alpha \in J} X_\alpha\) denote their Cartesian product. If \(\Pi_{\alpha \in J} X_\alpha\) is a Hausdorff space in either the box or the product topologies, does it follow that each space \(X_\alpha\) is also Hausdorff?

Answer: Yes


Question: Can 18 consecutive integers be divided into two groups such that the product of the numbers in each group is equal? Note that each integer must be used exactly once, and no integer can be used more than once.

Answer: No


Question: A device contains two components, A and B. The lifespans of A and B are exponentially distributed with expected lifespans of 5 years and 10 years, respectively. The device functions as long as both components are operational. What is the expected lifespan of the device?

Answer: \dfrac{10}{3}


Question: Find the maximum value of \( \int_{-\frac{\pi}{2}}^{\frac{3\pi}{2}} \sin x f(x) \, dx \) given the condition \(|f(x)| \leq 5\).

Answer: 20


Question: Given a probability measure \( \mathbb{Q} \) defined by \( \mathbb{Q}(A) := \mathbb{E}[ Z_t \mathbb{1}_A ] \) where \( Z_t = \exp( \theta B_t - \frac{1}{2}\theta^2t ) \) and \( B_t \) is a Brownian motion, calculate \( \mathbb{E}^\mathbb{Q}[B_t] \). Is \( B_t \) a Brownian motion under the measure \( \mathbb{Q} \)? Justify your answer.

Answer: \theta t


Question:
Compute the expected value of the normal distribution given by the integral:
\[ \frac{1}{\sqrt{2\pi\sigma^{2}}}\int_{-\infty}^{\infty} e^{-{(x - \mu)^2/(2\sigma^{2})}} x\,dx. \]
Can this be done without using the error function (erf)?

Answer: \mu


Question: Evaluate the limit: \[ \lim_{n\to\infty} \frac{\sqrt[n]{n!}}{n} \]

Answer: \dfrac{1}{e}


Question: Using the explicit Euler's method, calculate the expression \(A = (x_n)^2 + (y_n)^2\) for the system of differential equations given by \(x' = -y\) and \(y' = -x'\) with initial conditions \(x(0) = 1\) and \(y(0) = 0\). Derive a general formula for \((x_n)^2 + (y_n)^2\) after \(n\) steps.

Answer: 1


Question: Let the real number \( x \) satisfy \( \{ x \} + \{ \frac{1}{x} \} = 1 \), where \( \{ \cdot \} \) denotes the fractional part function. If \( k \) denotes the value of \( \{ x^3 \} + \{ \frac{1}{x^3} \} \), find the value of \( k \).

Answer: 1


Question: Consider a graph with 111 nodes labeled from 1 to 111. Two nodes are connected if their product is divisible by 3 but not by 9, or if their product is divisible by 2 but not by 4 or 6. Determine if this graph can be colored using three colors such that no two adjacent nodes share the same color.

Answer: Yes


Question: Given the function \( f(x) = \int_0^x f(t) \, \mathrm{d}t \), find the value of \( f(\ln(5)) \).

Answer: 0


Question: Find the sum of the infinite series: \[ \frac{1}{6} + \frac{13}{6^2} + \frac{19}{6^3} + \frac{97}{6^4} + \cdots + \frac{3^n + (-2)^n}{6^n} + \cdots \]

Answer: \dfrac{3}{4}


Question: Find the fundamental group of the space \( \mathbb{R}^3 \setminus S \), where \( S \) is the unit circle in the \( x-y \) plane.

Answer: \mathbb{Z}


Question: What is the cardinality of the set of all intervals of real numbers \( \{(a,b) \mid a,b \in \mathbb{R}\} \)?

Answer: \mathfrak{c}


Question: Given the volume \( V = xyz \) and the constraint on surface area \( S(x,y,z) = xy + yz + zx - 5 = 0 \), confirm that \( \lim_{x \to \infty} V = 0 \).

Answer: 0


Question: In a commutative ring $R$ with identity, let $I$ be an ideal of $R$. Consider the one-to-one correspondence between the set of all ideals of $R$ containing $I$ and the set of all ideals of the quotient ring $R/I$. Does a principal ideal in $R/I$ necessarily correspond to a principal ideal in $R$? Answer yes or no.

Answer: No


Question:
The random variable X has the moment generating function:
\[ m_X(t) = \frac{6}{\sqrt{(1-t)(4-t)(9-t)}} \]
for \( t < 1 \). Compute \( E(X) \).

Answer: \dfrac{49}{72}


Question: Calculate the derivative $DF(0,0,0)$ using the chain rule, where $F(x,y,z)=f(g(x,y),h(z),f(x,h(x),g(x,x)))$. Given that $f:\mathbb{R}^{3}\rightarrow \mathbb{R}$, $g:\mathbb{R}^{2} \rightarrow \mathbb{R}$, and $h:\mathbb{R} \rightarrow \mathbb{R}$ are functions of class $C^{\infty}$ with $f(0,0,0)=g(0,0)=h(0)$, $Df(0,0,0)=(1,2,3)$, $Dg(0,0)=(4,5)$, and $h'(0)=6$. Determine $DF(0,0,0)$. Use the substitutions $u=g(x,y)$, $v=h(z)$, and $w=f(x,h(x),g(x,x))$ if necessary.

Answer: (124, 5, 12)


Question: Find the 1000th decimal digit of the number \((8+\sqrt{63})^{2012}\).

Answer: 9


Question: Let $S = \{ h: D \rightarrow D \mid h \text{ is analytic in } D \text{ and } h(z)^2 = \overline{h(z)} \text{ for all } z \in D \}$, where $D = \{ z \in \mathbb{C} : |z| < 1 \}$. Determine the cardinality of the set $S$. \( \text{Provide your answer as a number.} \)

Answer: 1


Question: Let $T: V \to V$ be a linear transformation defined by $T(A) = BAB^*$, where $V$ is the vector space of all $n \times n$ matrices over $\mathbb{C}$ and $B$ is a fixed complex matrix. Show that $\det T = |\det B|^{2n}$.

Answer: |\det B|^{2n}


Question: For a $7 \times 7$ complex matrix $A$ with $A^3=0$ and \( \text{Rank}(A^2)=2 \), determine the nullity of $A$. \( \text{Nullity}(A) = \boxed{?} \)

Answer: 3


Question: Verify the following statement using the law of iterated expectations: \[ \mathbb{E}[\mathbb{E}[\mathbb{E}[X\mid Y] \mid  Z ]] = \mathbb{E}[X] \] where \(X, Y, Z\) are random variables.

Answer: \mathbb{E}[X]


Question: Consider the set \( S = \{ a^{b^c} \; | \; a, b, c \in \{ \tan x, \cot x \} \} \) for some given \( x \in (0, \frac{\pi}{4}) \). If all the elements of \( S \) are written in ascending order, determine the number of elements of \( S \) that lie between \( \cot(x)^{\cot(x)^{\tan(x)}} \) and \( \tan(x)^{\tan(x)^{\tan(x)}} \). Choose the correct option: \( \text{A. 1} \), \( \text{B. 3} \), \( \text{C. 4} \), \( \text{D. 5} \).

Answer: B


Question: Determine the smallest integer \( n \) such that the inequality \[ \sum_{i=n}^{100} \binom{100}{i} \left(\frac{80}{100}\right)^i \left(\frac{20}{100}\right)^{100-i} \geq 0.9 \] holds.

Answer: 75


Question: Evaluate the double integral \( \iint\limits_{\Omega}|xy|\,\mathrm{d}x\,\mathrm{d}y \), where \( \Omega \) is the region inside the circle of radius \( a \) centered at the origin.

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


Question: Let $A$ and $B$ be two open sets in $\mathbf{R}^2$ with the Euclidean topology. If $A\setminus B$ is non-empty, does there exist an open ball contained entirely within $A\setminus B$?

Answer: No


Question:
Determine the dimension of the affine variety \( V \subset \mathbb{A}_{\mathbb{C}}^n \) defined by the equations:
\[ f(x) = x_1^2 + \cdots + x_{n-2}^2 = 0, \quad g(x) = x_{n-1} = 0, \quad h(x) = x_n = 0. \]

Answer: n - 3


Question: If permutation matrices are conjugate in \(\operatorname{GL}(n,\mathbb{F})\), are the corresponding permutations conjugate in the symmetric group \(S_n\)? Assume there is a standard embedding of \(S_n\) into \(\operatorname{GL}(n,\mathbb{F})\) that maps each permutation to its corresponding permutation matrix. Given that conjugate permutations map to conjugate matrices, does the converse hold? Specifically, if \(A\) and \(B\) are similar permutation matrices in \(\operatorname{GL}(n,\mathbb{F})\) corresponding to permutations \(\alpha\) and \(\beta\) in \(S_n\), does it follow that \(\alpha\) and \(\beta\) are conjugate in \(S_n\)?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to \infty }{e^{-x^2}\int_{x}^{x+\frac{1}{x}} e^{t^2}dt}. \]

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty} \int_{-\pi}^\pi x^2 \frac{\sin(2nx)}{\sin x} \, dx \]

Answer: -\pi^3


Question: Let $G$ be an abelian group of order 72. Determine the number of subgroups of order 8 that $G$ can have.

Answer: 1


Question: Find the value of the 2013th derivative of the function \( f(x) = 6x^7\sin^2(x^{1000}) e^{x^2} \) evaluated at \( x = 0 \), i.e., find \( f^{(2013)}(0) \).

Answer: 2013!


Question:
Determine the signature of the quadratic form given by:
\[ f(x,y,z) = 3x^2 + 4y^2 + z^2 - 9(5xy + 3xz + yz) \]

Answer: (2, 1)


Question: In \(\triangle ABC\), \(AB = AC\) and \(\angle B = 40^\circ\). Point \(D\) is on \(AB\) extended such that \(AD = BC\). Find \(\angle DCB\).

Answer: 10^\circ


Question: Find the degree of the field extension \( \mathbb{Q}(\sqrt{2} + \sqrt[3]{2}) \) over \( \mathbb{Q}(\sqrt{2}) \).

Answer: 3


Question: Evaluate the limit: \[ \lim_{t \to 1^{-}} (t^{p-q} -1) \log(1 -t) \] where \(0 < p < q\).

Answer: 0


Question: Calculate the area of the region \( A = \{ (x,y) \in \mathbb{R}^2 : x^2 + (e^x + y)^2 \leq 1 \} \).

Answer: \pi


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{1-\sqrt[3]{\cos x}}{x(1-\cos\sqrt{x})} \] without using L'Hôpital's Rule.

Answer: \dfrac{1}{3}


Question: Evaluate the integral \( \int_{|z|=3} \frac{z^9}{z^{10} - 1} \, dz \) using the residue theorem.

Answer: 2\pi i


Question: Let \( f(z) \) be an analytic function on the open unit disk \( D \) such that \( |f(z)| \le 1 - |z| \) for all \( z \in D \). Prove that \( f \equiv 0 \).

Answer: 0


Question: Determine the degree of the extension \(\mathbb{Q}(\sqrt{3 + 2 \sqrt{2}})\) over \(\mathbb{Q}\).

Answer: 2


Question: Determine the number of surjective functions from the set \( \{1, 2, \ldots, 10\} \) to the set \( \{1, 2, \ldots, 9\} \) such that \( f(2i+1) > f(2i-1) \) and \( f(2i) > f(2i+2) \) for \( i = 1, 2, 3, 4 \).

Answer: 630


Question: Find the expected distance from the origin of a point chosen uniformly at random from the unit disk, defined by the condition \(x^2 + y^2 \le 1\).

Answer: \dfrac{2}{3}


Question: Evaluate the integral of the function \( f(z) = z + \overline{z} \) over the path \( \gamma(t) = 2e^{2 \pi t i} \) for \( t \in [0, 1] \).

Answer: 8\pi i


Question: Given the integral equation \( f(x) = x + \int_0^1 t(x+t)f(t) \, dt \), find the value of \( \eta = \int_0^1 f(x) \, dx \).

Answer: \dfrac{42}{23}


Question: Evaluate the integral \( \int_{0}^{2} \sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}\,dx \).

Answer: \dfrac{19}{6}


Question: Solve the inequality: $x + 3^x < 4$. Determine the range of $x$ that satisfies this inequality.

Answer: (-\infty, 1)


Question: A light bulb is turned on at time $t=0$. It switches off at $t=1$, on again at $t=1+\frac{1}{2}$, off at $t=1+\frac{1}{2}+\frac{1}{3}$, and continues this pattern. As $t$ approaches infinity, what proportion of the time is the light bulb on?

Answer: \dfrac{1}{2}


Question: Determine whether the first sheaf cohomology group $H^1(X, E)$ is zero for a holomorphic vector bundle $E$ on $X = \mathbb{C}$. Justify your answer.

Answer: 0


Question: Find the number of integer solutions to the equation \(x_1 + x_2 + x_3 + x_4 = 25\) subject to the constraints \(1 \leq x_1 \leq 6\), \(2 \leq x_2 \leq 8\), \(0 \leq x_3 \leq 8\), and \(5 \leq x_4 \leq 9\).

Answer: 79


Question:
Let \( U \) be a real function from \( \mathbb{R}^+ \rightarrow \bar{\mathbb{R}} \) satisfying the following conditions:
1. \( U \) is concave, continuous, and strictly increasing.
2. \( \limsup_{x\rightarrow +\infty}\frac{xU^{'}(x)}{U(x)} <1. \)
3. \( U^{'}(0+) = +\infty, \) and \( U^{'}(+\infty) = 0. \)

Is the following statement true? For any non-negative random variable \( \xi, \) if \( E[U(\xi)] < +\infty, \) then \( E[\xi U^{'}(\xi)] < +\infty. \)

Answer: True


Question: Consider a linear transformation $T : V \longrightarrow V$ with distinct eigenvalues $r_1, r_2, ..., r_m$ and corresponding eigenvectors $v_1, v_2, ..., v_m$. Determine if any of the eigenvalues $r_1, r_2, ..., r_m$ can be equal to 0.

Answer: Yes


Question: Given a line segment of length \(n\), construct a line segment of length \(n^n\) using only a straight edge and compass.

Answer: n^n


Question: Find the smallest natural number \( r \) such that \( 6^{83} + 8^{83} = 49q + r \) for some integer \( q \).

Answer: 35


Question: Let $K$ be an ordered field with an embedding into $\mathbb{R}$, \( f:K\hookrightarrow\mathbb{R} \), where \( f \) is order preserving. Is \( f \) unique?

Answer: Yes


Question: Determine the number of roots of the polynomial \( f(x) = x^{134} + x^{127} + x^7 + 1 \) over the field \( \mathbb{F}_{2^{1463}} \).

Answer: 127


Question: Given a matrix $A$ such that $A^2 = A - I$, find the expression for $A^{15}$.

Answer: -I


Question: Calculate the probability \( P(\limsup_{n\rightarrow\infty} S_n=\infty) \) for the symmetric random walk \( S_n \) on \( \mathbb{Z} \).

Answer: 1


Question: Find an algebraic solution to the equation \(2x + \ln x = 2\).

Answer: 1


Question: Find all real values of \( m \) such that the inequality \( x^2 - (m-3)x + m > 0 \) holds for all \( x \) in the interval \([1, 2]\).

Answer: (-\infty, 10)


Question: Find a function \( F \) that satisfies the equation \( F(a \cdot b) = F(a) + F(b) \) for all real numbers \( a \) and \( b \), and is not a one-to-one function.

Answer: F(x) = 0


Question: Evaluate the integral \( \int_\gamma \frac{|z|e^z}{z^2}dz \), where \( \gamma \) is the circle of radius 2 centered at the origin in the complex plane.

Answer: 4\pi i


Question: Calculate the flux of the vector field \( F = (x, y, z) \) through the surface \( \Sigma \), where \( \Sigma \) is the union of the upper hemisphere \( K = \{(x, y, z) | z = \sqrt{1-x^2-y^2},\; z \geq 0\} \) and the disk \( C = \{(x, y, z) | x^2 + y^2 \leq 1,\; z = 0\} \). Use an outward pointing normal for the calculation.

Answer: 2\pi


Question: Let $V$ be a finitely generated vector space over the field $\mathbb{F}$, and let $T: V \to V$ be a linear transformation. Given a non-zero vector $v \in V$, prove that there exists a polynomial $m_{T,v} \in \mathbb{F}[x]$ such that $m_{T,v}$ divides any polynomial $f \in \mathbb{F}[x]$ for which $f(T)v = 0$.

Answer: m_{T,v}


Question: Let \(z\) be a complex number such that \(|z-3| = \text{Re}(z)\). Determine the value of \(|z-3|\) for the complex number \(z\) that has the maximum amplitude (argument).

Answer: 3


Question: Consider a monic polynomial \( f = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \in \mathbb{C}[x] \) with algebraic integer coefficients, where \( n > 1 \). Let \( K \) be the number field \( \mathbb{Q}(a_{n-1}, \dots, a_0) \). Suppose the discriminant of \( f \), denoted \( D(f) \), is a unit in \( K \), i.e., \( D(f) \in \mathcal{O}_K^* \), where \( \mathcal{O}_K \) is the ring of integers of \( K \). Is it possible for such a polynomial \( f \) to be irreducible over \( K \)?

Answer: Yes


Question: Determine the values of \( z_0 \) for which the function \( f(z) = (\bar{z})^2 \) is complex differentiable or holomorphic.

Answer: 0


Question: Does there exist a sequence $a_n$ such that $\lim_{n\rightarrow\infty} \left|\frac{a_{n+1}}{a_n}\right|$ does not exist, but the series $\sum_{n=1}^{\infty} a_n$ is convergent?

Answer: Yes


Question:
Given the probability density function \( f_X(x / \theta) = \begin{cases}
2\theta/x^3 & x\geq \theta \\
0 & x<\theta
\end{cases} \) with \( \theta>0 \), compute the maximum likelihood estimate of \( \theta \) based on a sample \( x_1, x_2, \ldots, x_n \).

Answer: \min\{x_1, x_2, \ldots, x_n\}


Question: Determine whether an odd number $n > 1$ can divide $2^n - 1$. Provide a justification for your answer.

Answer: No


Question: Determine the largest known natural number $n$ such that every sequence of length $n$ appears in the base 10 expansion of $\pi$. Provide the value of $n$.

Answer: 6


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

Answer: \dfrac{1}{3}


Question: In a certain city, earthquake occurrences are modeled by a Poisson process with an intensity of \( \lambda = 2 \) per year. The damage caused by each earthquake is exponentially distributed with a parameter \( \mu = 1000 \) Euros. Calculate the expected damage per decade.

Answer: 20000


Question:
Given the following information:

1. The current price of a stock is $43.00.
2. The continuously compounded risk-free interest rate is 5% per annum.
3. The stock pays a dividend of $1 in three months.
4. A 3-month European call option on the stock with a strike price of $44 costs $1.90.

Determine the amount of money you should lend to create a synthetic stock using a combination of 44-strike options expiring in 3 months and lending. Use the put-call parity and present value concepts to find the solution.

Answer: 44.44


Question: Let $n$ be a positive integer and let $V$ be the vector space of all polynomial functions over the real numbers with degree at most $n$, i.e., functions of the form $f(x) = c_0 + c_1x + \dots + c_nx^n$. Let $D$ be the differentiation operator on $V$. Find a basis for the null space of the transpose operator $D^t$. Provide your answer as a set of polynomials.

Answer: \{x^n\}


Question: Evaluate the limit \( \lim_{n \to \infty} \frac{\log \left( \sum_{k=2}^n \varphi(k) \right)}{\log(n)} \). Does this limit approach 2?

Answer: 2


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

Answer: 3


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a nonnegative function such that $f \in L^1(\mathbb{R})$. Does this imply that $f$ is bounded almost everywhere? Provide a justification for your answer.

Answer: No


Question:
Consider the polynomial \( P(x) = x^4 + ax^3 + bx^2 + cx + d \) such that \( P'(0) = 0 \) and \( x = 0 \) is the only root of \( P'(x) = 0 \). If \( P(-1) < P(1) \), determine the nature of the extrema of \( P(x) \) in the interval \([-1, 1]\). Choose the correct option:

(a) \( P(-1) \) is the minimum and \( P(1) \) is the maximum of \( P(x) \).
(b) \( P(-1) \) is not the minimum and \( P(1) \) is the maximum of \( P(x) \).
(c) \( P(-1) \) is the minimum and \( P(1) \) is not the maximum of \( P(x) \).
(d) Neither \( P(-1) \) is the minimum nor \( P(1) \) is the maximum of \( P(x) \).

Answer: B


Question: Let $F: \mathbb{R} \to \mathbb{R}$ be a non-decreasing function, and define $G: \mathbb{R} \to \mathbb{R}$ by $G(x) = F(x+)$, the right limit of $F$ at $x$. Suppose $G$ is differentiable almost everywhere. Is it true that $F$ is differentiable almost everywhere?

Answer: Yes


Question: Given a semi-simple Lie Algebra $L$ with dimension $d$ over $\mathbb{R}$, determine the dimension of its complexification $L_{\mathbb{C}} = L \otimes_{\mathbb{R}} \mathbb{C}$.

Answer: d


Question: Determine the number of terms needed in the series \(\sum_{k=0}^{\infty}{\frac{(3i)^k}{k!}}\) to ensure the error is less than \(10^{-4}\).

Answer: 14


Question: A manufacturer receives a lot of 100 parts from a vendor. The lot will be deemed unacceptable if more than five parts are defective. The manufacturer randomly selects K parts from the lot for inspection, and the lot will be accepted if no defective parts are found in the sample. Determine the smallest value of K such that the probability of accepting an unacceptable lot is less than 0.10.

Answer: 32


Question: In how many ways can 7 distinct balls be packed into 16 boxes, where 7 of the boxes are unlabeled and green, and 9 of the boxes are unlabeled and yellow?

Answer: 14214


Question:
Find the smallest natural number \( n \) such that \( A^n = I \), where \( A \) is the following \( 2 \times 2 \) matrix:
\[
A = \begin{pmatrix}
\sin\frac{\pi}{18} & -\sin\frac{4\pi}{9} \\
\sin\frac{4\pi}{9} & \sin\frac{\pi}{18}
\end{pmatrix}
\]

Answer: 9


Question: Given a Riemannian manifold $\overline{M}^{n+1}$ with metric $g$ and an embedded orientable hypersurface $M^n$ with unitary normal $\eta : M \to T\overline{M}$, consider a fixed point $p$ of $M$. There exists a local chart $\varphi : U \ni p \to V \subset \mathbb{R}^{n+1}$ such that $\varphi(p) = 0$ and $\varphi(U \cap M) = V \cap (\mathbb{R}^n \times \{0\})$. Can we further assume that $d \varphi_p (\eta(p)) = e_{n+1} = (0, \dots, 0, 1)$?

Answer: Yes


Question:
Calculate the determinant of the matrix \(X\), where \(X = A + A^2B^2 + A^3 + A^4B^4 + \cdots\) for 100 terms. The matrices \(A\) and \(B\) are given by:
\[A = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}\]
\[B = \begin{bmatrix} -4 & -3 & -3 \\ 1 & 0 & 1 \\ 4 & 4 & 3 \end{bmatrix}\]

Answer: 0


Question: Prove that \( \lceil (\sqrt{3} + 1)^{2n} \rceil \) is divisible by \( 2^{n+1} \) for all natural numbers \( n \).

Answer: 2^{n+1}


Question: A fair coin is tossed repeatedly until two consecutive tails appear. What is the expected number of tosses required to achieve this outcome?

Answer: 6


Question: Find the residue at infinity for the function \( f(z) = \frac{\sin(\pi z)}{z^2(z-1)} \).

Answer: \pi


Question: Evaluate the limit \( \lim_{x\rightarrow 0^{+}}\frac{\ln(\sin 2x)}{\ln(\sin x)} \) without using l'Hôpital's rule.

Answer: 1


Question: Determine if the sequence \(x_n=\left(\frac{n^n}{n!}\right)^{\frac{1}{n}}\) is convergent, and if it is, find its limit.

Answer: e


Question: Find the fundamental period of the function \( f: \mathbb{R} \to \mathbb{R} \) given that \( f(x+13) + f(x+630) = 0 \) for all \( x \in \mathbb{R} \).

Answer: 1234


Question:
Given the recursive probability function defined as follows:

\[ p(1) = 0 \]
\[ p(n+1) = (1-10^{-8}n)p(n)+10^{-8}n \]

Determine the maximum integer \( n \) such that \( p(n) \leq 0.1 \).

Answer: 4590


Question: Evaluate the expression \( \left\lfloor \lim_{x \to 0} \frac{e - (1+x)^{1/x}}{\tan x} \right\rfloor \), where \( \lfloor \cdot \rfloor \) denotes the floor function.

Answer: 1


Question: Consider the Hilbert Space \( l^2 \). Let \( S = \{(x_1, x_2, \ldots) \in l^2 : \sum \frac{x_n}{n} = 0 \} \). Find the number of interior points of \( S \).

Answer: 0


Question: Compute the probability that a standard Brownian motion hits 1 before hitting -1, and then hits -1 before hitting 2. Formally, find \( P(T_1 < T_{-1} < T_2) \), where \( T_a = \inf\{t \geq 0 \mid B_t = a\} \).

Answer: \dfrac{1}{6}


Question: Find the remainder when \(9^{{10}^{{11}^{12}}} - 5^{9^{10^{11}}}\) is divided by 13.

Answer: 4


Question: Evaluate the integral \( \int_0^{2\pi} e^{e^{ix}} \, dx \).

Answer: 2\pi


Question: Construct a complete graph with 7 vertices on a 2D plane. How many edges does this graph have?

Answer: 21


Question: Let \( P(x) \) be a polynomial of degree 4, having extrema at \( x=1 \) and \( x=2 \). Given that \( \lim_{x\to 0}\frac{x^2+P(x)}{x^2}=2 \), find the value of \( P(2) \).

Answer: 0


Question: Calculate the limit \( \lim _{n \rightarrow \infty} \frac{[\ln (n)]^{2}}{n^{\frac{1}{\ln (\ln (n))}}} \).

Answer: 0


Question: Evaluate the limit: \[ \lim\limits_{(r,\theta)\to(1,0)}\dfrac{r^{\frac{3}{2}}e^{3i\theta}-1}{re^{2i\theta}-1}. \]

Answer: \dfrac{3}{2}


Question: Calculate the arc length of the curve defined by \( y = \int_1^x \sqrt{t^2-1} \, dt \) for \( 4 \leq x \leq 7 \).

Answer: \dfrac{33}{2}


Question:
Consider a sequence $\{a_n\}_n$ with $a_n \in \mathbb{R}^k$ for all $n$ and $k > 1$. Are the following expressions equivalent when replacing the absolute value with the Euclidean distance?

1. $\lim_{n \rightarrow \infty} a_n = a$
2. $\lim_{n \rightarrow \infty} a_n - a = 0$
3. $\lim_{n \rightarrow \infty} \|a_n - a\| = 0$

Answer: Yes


Question: Calculate the sum of the series \( \sum_{n=0}^{\infty} \frac{n^2}{n!} \).

Answer: 2e


Question: Let $R$ be a commutative ring with unity. An ideal $I$ of $R$ is called a regular ideal if $I$ contains an element which is not a zero-divisor. If $I$ and $J$ are ideals of $R$ such that their product $IJ$ is a regular ideal, must $I$ and $J$ also be regular ideals?

Answer: Yes


Question: Calculate the volume inside the paraboloid \(x^2 + 4z^2 + 8y = 16\) that lies on the positive side of the xz-plane.

Answer: 8\pi


Question: Evaluate the limit using the squeeze theorem: \[ \lim_{x\to 0} \frac{1+\sin x-\cos x+\log(1-x)}{x^3} \]

Answer: -\dfrac{1}{2}


Question: Let \( \{X_i\}_{i=1}^6 \) be independent and identically distributed standard normal random variables. What is the expected number of indices \( i \) such that \( X_i \geq X_j \) for all \( j < i \)?

Answer: \dfrac{49}{20}


Question: Determine the asymptotic rate of \( \log\left|1+\frac{O_p(M^{-1/2})}{fu}\right| \) in terms of \( O_p \) notation, where \( f \) is a real-valued constant and \( u \) follows a standard normal distribution.

Answer: O_p\left(M^{-1/2}\right)


Question: Determine the value of $c$ for which the sequence $a_n$ converges, where $a_n = \sum_{k=0}^n \frac{2^k}{k!}$ if $n$ is even and $a_n = \left(\sum_{k=0}^n \frac{1}{k!}\right)^c$ if $n$ is odd.

Answer: 2


Question: Determine the sub-differential of the function \( f(x) = -\sqrt{x} \) at \( x = 0 \), where \( f: [0, 1] \rightarrow \mathbb{R} \).

Answer: \emptyset


Question: If \( \beta = e^{i\frac{2\pi}{7}} \) and \( \left|\sum_{r=0}^{3n-1}\beta^{2^r}\right| = 4\sqrt{2} \), find the value of \( n \).

Answer: 4


Question: Evaluate the complex integral \( \oint_{|z|=1} \sin\left(\frac{1}{z}\right) dz \).

Answer: 2\pi i


Question: Determine the Galois group of the polynomial \( f = x^5 + x^4 + x^3 + x \) over the field \( \mathbb{Z}_2 \).

Answer: \mathbb{Z}_3


Question: Consider a strictly positive smooth real function $a_n(x)$ such that as $n \to \infty$, $a_n(x) \to 0$ for $0 < x < 1$, and $a_n(x) \to \infty$ for $1 < x$. Define the number $L$ by $L = \lim\limits_{n \to \infty} a_n(1)$. Is it true that for all such $a_n(x)$, the limit $L$ is strictly positive and finite, or can there exist sequences $\{ a_n(x) \}_{n=1}^\infty$ such that $L = 0$ or $L = \infty$?

Answer: L


Question: Given $n$ nonoverlapping closed line segments and $n+2$ distinct points in the plane $\mathbb{R}^2$, where none of the points lie on the line segments, determine if there exist two points among the $n+2$ points such that the line segment connecting them does not intersect any of the given $n$ line segments.

Answer: Yes


Question: Determine the number of distinct Jordan normal forms (up to the order of the Jordan blocks on the diagonals) for matrices $B \in M_5 (\mathbb{C})$ with the characteristic polynomial $x_B(\lambda)=-(\lambda-2)^2(\lambda-3)^3$. Consider the algebraic and geometric multiplicities of the eigenvalues.

Answer: 6


Question:
Given a $3 \times 3$ matrix $N$ such that $N^3 = 0$, determine which of the following statements is true:

1. $N$ has a non-zero eigenvector.
2. $N$ is similar to a diagonal matrix.
3. $N$ has 3 linearly independent eigenvectors.
4. $N$ is not similar to a diagonal matrix.

Select the correct statement.

Answer: 1


Question: Find the volume of the cylinder defined by the equation \(x^2 + y^2 = 9\) that is cut by the planes \(z = x + y - 4\) and \(z = 1\).

Answer: 45\pi


Question: Is there a pseudofinite structure $M$ that does not have the finite model property? A pseudofinite structure is one where any sentence it satisfies is also satisfied in a finite structure, and the finite model property means that the finite structure can be taken as a substructure of $M$.

Answer: Yes


Question: How many natural numbers between 2 and 60 have no primitive roots and require at least four squares to be expressed as a sum of squares?

Answer: 5


Question: Consider a linear transformation $A: \mathbb{R}^{m} \longrightarrow \mathbb{R}^{n}$ that transforms an orthonormal set of $\mathbb{R}^{m}$ into an orthonormal set of $\mathbb{R}^{n}$. Is the adjoint $A^{\ast}: \mathbb{R}^{n} \longrightarrow \mathbb{R}^{m}$ a left inverse of $A$?

Answer: Yes


Question: Find the exact functional form of $g(y)$ for all $y > 0$ if $g(y) = \lim_{x \to \infty} \frac{f(xy)}{x^3}$ and $g(1) = 1$, where $f: \mathbb{R} \to \mathbb{R}$ is a non-zero function such that the limit exists for all $y > 0$. Assume $f(x)$ is differentiable.

Answer: y^3


Question: Consider two distinct points \(s = (s_i)_{i \in \mathbb{Z}}\) and \(u = (u_i)_{i \in \mathbb{Z}}\) in the Thue-Morse subshift \(X\). Is it true that the set of indices \(\{i \in \mathbb{Z} : s_i \neq u_i\}\) is necessarily infinite?

Answer: Yes


Question: Determine whether there exist three irrational linear forms \( f_1, f_2, f_3 \) in three variables such that the product \( f_1 f_2 f_3 \) has integral coefficients.

Answer: Yes


Question: Let $\mathcal{B}(F)$ be the algebra of all bounded linear operators on a complex Hilbert space $F$. Suppose $M \in \mathcal{B}(F)^+$, meaning $\langle Mx, x \rangle \geq 0$ for all $x \in F$. The square root of $M$ is defined as the unique operator $N$ such that $N^2 = M$, denoted as $N = M^{1/2}$. If $M$ is an invertible operator, is $M^{1/2}$ also invertible?

Answer: Yes


Question: Let $f(x)$ be a differentiable function such that $\frac{d^3f}{dx^3}=1$ for all $x\in [0, 3].$ If $p(x)$ is a quadratic polynomial that interpolates $f(x)$ at $x=0, x=2,$ and $x=3,$ find the value of $f(1)-p(1).$

Answer: \dfrac{1}{3}


Question: Two machines, A and B, have lifetimes that are exponentially distributed with the same rate parameter \( \lambda \). Given that one of the machines has stopped functioning, what is the probability that the remaining functional machine is A?

Answer: \dfrac{1}{2}


Question: Find the permutation $x$ such that for $y = (1274)(356)$, the equation $x^{-1}yx = (254)(1736)$ holds.

Answer: (2\ 3\ 7)(4\ 6)


Question: Let $V$ be a vector space over $\mathbb{Z}_2$, and let $T$ be a bijection on $V$ such that for all subspaces $W_1, W_2 < V$ with $W_1 \oplus W_2 = V$, we have $T(W_1) \oplus T(W_2) = V$. Is $T$ a linear transformation?

Answer: Yes


Question: Find the order of the inner automorphism \(\phi_{(1\,2\,4\,3)}\) in the group \(\text{Inn}(S_5)\), where \(\phi_{(1\,2\,4\,3)} = (1\,2\,4\,3)x(3\,4\,2\,1)\) for \(x \in \text{Inn}(S_5)\). Determine the smallest positive integer \(n\) such that \((\phi_{(1\,2\,4\,3)})^n = e\), where \(e\) is the identity element of \(\text{Inn}(S_5)\).

Answer: 4


Question: Determine the smallest integer $n$ such that the error in approximating $\pi$ using the series expansion $\pi = 4\arctan(1/2) + 4\arctan(1/3)$ is less than $10^{-10}$. Use the remainder integral $R_n = \int_0^x \frac{(-1)^n t^{2n}}{1+t^2} \, dt$ for the series expansion of $\arctan(x)$, and ensure that the sum of the remainders for $\arctan(1/2)$ and $\arctan(1/3)$ is less than $\frac{10^{-10}}{4}$. Approximate as necessary to find $n$. \[ \frac{(1/2)^{2n+1}}{2n+1} + \frac{(1/3)^{2n+1}}{2n+1} < \frac{10^{-10}}{4}. \]

Answer: 15


Question:
Determine whether there exists a constant \(c = c_s\) such that for any function \(u(x) = \sum_{k \in \mathbb{Z}} \hat{u}_k \, \mathrm{e}^{ikx}\) in the Sobolev space \(H^s(\mathbb{T})\) with \(s \in \left(\frac{1}{2}, \frac{3}{2}\right)\), the following inequality holds:
\[
\lvert u(x) - u(y) \rvert \le c \lvert u \rvert_{H^s} \lvert x - y \rvert^{s - \frac{1}{2}},
\]
where
\[
\lvert u \rvert_{H^s}^2 = \sum_{k \in \mathbb{Z}} \lvert k \rvert^{2s} \lvert \hat{u}_k \rvert^2.
\]

Answer: Yes


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

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to 0} \left( \frac{1}{\log(x+1)} - \frac{1}{\log(x+\sqrt{x^2+1})} \right) \] using notable limits without derivatives.

Answer: \dfrac{1}{2}


Question: Given the joint probability density function \( f_{(X,Y)}(x,y) = k \) for \((x,y) \in T\) and \(0\) otherwise, where \( T = \{(x,y) : x > 0, y > 0, x + y < 9\} \), find the covariance \( \text{cov}(X,Y) \). Recall that \( \text{cov}(X,Y) = E[XY] - E[X]E[Y] \).

Answer: -\dfrac{9}{4}


Question: Let \( n = 2^7 \cdot 3^5 \cdot 11^3 \cdot 35 \). Determine the number of ways the cyclic group \( C_n \) can be expressed as a direct product of two or more nontrivial groups.

Answer: 51


Question: Evaluate the line integral \( I = \oint_\Gamma \sin x^3\,dx + 6xy\,dy \) over the triangular path \( \Gamma \) connecting the points (0,0), (2,2), and (0,2) in the counter-clockwise direction in \( \mathbb{R}^2 \).

Answer: 16


Question: Suppose $N$ is a nontrivial normal subgroup of a group $G$, with $N \neq \{e\}$. Can the quotient group $G/N$ be isomorphic to $G$? Provide a justification for your answer.

Answer: Yes


Question: Given that $g$ is a continuous function on $[0, \infty) \rightarrow \mathbb{R}$ satisfying \( \int_{0}^{x^2(1+x)} g(t) \, dt = x \) for all \( x \in [0, \infty) \), find the value of \( g(2) \).

Answer: \dfrac{1}{5}


Question: Evaluate the contour integral \( \int_{C} \frac{z}{i} \, dz \), where \( C \) is the part of the circle \( z = e^{i\theta} \) for \( 0 \leq \theta \leq \frac{\pi}{2} \).

Answer: i


Question: Given that \(a_n + b_n\sqrt{3} = (2+\sqrt{3})^n\), where \(a_n\) and \(b_n\) are integers, compute \(\lim_{n\rightarrow\infty}\frac{a_n}{b_n}\).

Answer: \sqrt{3}


Question: Let \( T: \mathbb{C}^n \rightarrow \mathbb{C}^m \) be a linear map and \( W \) be a \( d \)-dimensional subspace of \( \mathbb{C}^n \) such that the nullity of \( T|_W \) is \( s \). Determine if the nullity of \( T \) is at most \( n-d+s \). Is this statement true or false?

Answer: True


Question:
Calculate the derivative of the function defined by the improper integral:
\[ y(t) = \int_{-\infty}^t f(x) \, dx \]
Find \( y'(t) \).

Answer: f(t)


Question: Are the braid groups $B_n$ conjugacy separable? A group $G$ is called conjugacy separable if for every two elements $x, y \in G$ that are not conjugate in $G$, there exists a finite index normal subgroup $N$ such that $xN$ and $yN$ are not conjugate in $G/N$. Determine if this property holds for braid groups.

Answer: Yes


Question: Determine all nonnegative integer pairs \((x, y)\) such that \(3^x + 7^y\) is a perfect square and \(y\) is even.

Answer: (1, 0)


Question: Evaluate the improper integral: $$\int_0^\infty \frac{x}{x^4 + 9} \, dx$$

Answer: \dfrac{\pi}{12}


Question: Evaluate the integral \( \int_{0}^{\infty} \frac{1-x^2}{x^4+3x^2+1}\ dx \) using a more efficient method.

Answer: 0


Question: Given that \( f(1)=2 \), \( f(2)=8 \), and \( f(a+b)-f(a)=kab-2b^2 \), find \( f'(x) \) for any arbitrary \( x \).

Answer: 4x


Question: Let $A$ be a real symmetric positive semi-definite matrix. If $v^TAv = 0$ for $v \in \mathbb{R}^n$, does it imply that $Av = 0$? Justify your answer.

Answer: Yes


Question:
Consider a sequence of independent and identically distributed random variables \(X_i\). Let \(N\) be a random variable representing the number of events. Compute the expected value of the sum of these random variables up to \(N\):
\[
E\left[\sum_{i=1}^N X_i\right]
\]
Can this expression be simplified to \(N E[X_i]\)?

Answer: E[N] E[X_i]


Question: Find the largest number \( k \) such that the limit \[ \lim_{(x,y)\to(0,0)} \frac{x^{15}y^{23}}{(x^2 +y^2)^p} \] exists for all \( p < k \).

Answer: 19


Question: Is there a function $f(x)$ defined on the interval $(a, b)$ such that $f'(c_i) = 0$ for an infinite ordered set of points $\{c_i\}$ where $c_i \in (a, b)$, and for each pair of consecutive points $c_i$ and $c_{i+1}$, there exists at least one point $d \in (c_i, c_{i+1})$ where $f'(d) \neq 0$?

Answer: Yes


Question: Calculate the volume of the solid $B$ in the first octant, bounded by the surfaces $y = x^2$, $x = y^2$, and the plane $x + y - z = -5$. Assume $x, y, z \geq 0$. Provide your answer as a single numerical value.

Answer: \dfrac{59}{30}


Question: Given a vector space $V$ with $\dim(V)=n$, an affine subspace $W$ with $\dim(W)=n-1$, a point $x\in V$, and a compact convex set $K\subset V$ such that $W\cap K = \{x\}$, find a linear functional $f:V\to\Bbb R$ such that $W=\{y\in V~|~f(y)= f(x) \}$ and $f(x)=\sup_{y\in K}f(y)$. Determine the linear functional $f$.

Answer: f


Question:
Consider the determinant map \( \det :\mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R} \) defined by \( \det(V,W) = V_1W_2 - V_2W_1 \) for \( V = (V_1,V_2) \in \mathbb{R}^2 \) and \( W = (W_1,W_2) \in \mathbb{R}^2 \). What is the derivative of the determinant map at \((V,W)\) evaluated on \((H, K)\) where \( (H, K) \in \mathbb{R}^2 \times \mathbb{R}^2 \)?

A) \( \det (H,W) + \det(V, K) \)

B) \( \det (H, K) \)

C) \( \det (H,V) + \det(W, K) \)

D) \( \det (V,W) + \det(K,W) \)

Answer: A


Question: Determine whether the function \( f(x) = \frac{x \cos x}{\sin x} \) is analytic at \( x = 0 \) if we define \( f(0) = 1 \).

Answer: Yes


Question: Given a function \( f: \mathbb{R}^2 \to \mathbb{R} \) such that \( \int_{\mathbb{R}} \left[ \int_{\mathbb{R}} f(x,y) \, dy \right] \, dx = 1 \) and \( \int_{\mathbb{R}} \left[ \int_{\mathbb{R}} f(x,y) \, dx \right] \, dy = -1 \), determine the possible range of values for \( \iint_{\mathbb{R}^2} |f| \, dm \), where \( m \) refers to the Lebesgue measure on \( \mathbb{R}^2 \).

Answer: \infty


Question: Determine if there is any divisor of 576, other than 24, such that the group \( \mathtt{SmallGroup}(576,8661) \) has no subgroup of that order.

Answer: 48


Question: Calculate \(9^{47^{51}} \mod 67\).

Answer: 59


Question: In the acute triangle $ABC$, let $H$ be the orthocenter and $O$ the circumcenter. The altitude from $A$ meets $BC$ at $P$. Given that the distance from $O$ to line $AC$ is $2$ and $PH = 1$, calculate the power of point $P$ with respect to the circumcircle of triangle $ABC$.

Answer: -5


Question: Evaluate the integral: \[ \int_{1/b}^{b} \frac{\arctan(x) + \arctan\left(\frac{1}{x}\right)}{x} \, dx. \]

Answer: \pi \ln b


Question: Determine whether there exists a prime $p$ such that there is a prime $\ell$ and an integer $n$ for which $PSL_n(\mathbb{F}_{\ell})$ contains nontrivial $p$-torsion, and for any element $x \in PSL_n(\mathbb{F}_{\ell})$ of order $p$, $x$ is conjugate to $x^k$ for all $0 < k < p$. Provide a justification for your answer.

Answer: Yes


Question: For all real numbers $x$, let the mapping $f(x)=\frac{1}{x-i}$, where $i=\sqrt{-1}$. If there exist real numbers $a, b, c,$ and $d$ such that $f(a), f(b), f(c),$ and $f(d)$ form a square on the complex plane, find the area of the square.

Answer: \dfrac{1}{2}


Question: A triangle has an inradius of 4 cm and a circumradius of \( \frac{65}{8} \) cm. Find the area of the triangle.

Answer: 84


Question: Find the maximum value of the function \( f(z) = |z^3 - z + 2| \) where \( |z| = 1 \) and \( z \) is a complex number.

Answer: \sqrt{13}


Question: Find the covariance \( \text{Cov}(X, Y) \) for the jointly distributed random variables \( X \) and \( Y \), where \( Y|X=x \sim \text{Bin}(n, x) \) and \( X \sim U(0,1) \). Given that \( E(X) = \frac{1}{2} \), \( E(Y) = \frac{n}{2} \), and \( \text{Var}(Y) = \frac{n}{6} + \frac{n^2}{12} \), calculate \( \text{Cov}(X, Y) = E(XY) - E(X)E(Y) \).

Answer: \dfrac{n}{12}


Question: Given that \( g(t) = \int_{-1}^{1} e^{tx} f(x) \, dx \) for every \( t \in \mathbb{R} \), find the value of \( \int_{-1}^{1} x f(x) \, dx \).

Answer: g'(0)


Question:
Find all functions \( f : \mathbb{R}_{>0} \to \mathbb{R}_{>0} \) such that for all positive real numbers \( x \) and \( y \), the following equation holds:
\[ f \big ( x y + f ( x ) \big) = f \big( f ( x ) f ( y ) \big) + x. \]

Answer: f(x) = x


Question: Find the value of \( \int_0^1 f(x) \, dx \) if \( f \) is a real, non-constant, differentiable function satisfying \( f(f(x)) = 1-x \) for all \( x \in (0,1) \).

Answer: \dfrac{1}{2}


Question:
Evaluate the following limit:
\[ \lim_{s\rightarrow\frac{1}{2}^+}\sum_{n,m\geq1}\frac{2s-1}{n^sm^s(n+m)} = \pi. \] Determine if this statement is true, and if so, provide a justification.

Answer: \pi


Question: Suppose $f: \mathbb{R}^3 \setminus B(0,1) \to \mathbb{R}$ is a smooth function such that $f(S^2) = 0$, where $S^2$ is the unit sphere. Does it necessarily follow that $f$ is a spherically symmetric function? Answer yes or no and justify your answer.

Answer: No


Question:
Consider an arithmetic progression $(x_i)$ of length $M$. The $i$th element of the Prouhet-Thue-Morse sequence $(t_i)$ is defined as follows:

- $t_0 = 0$
- $t_{2n} = t_n$ for $n \in \mathbb{N}_0$
- $t_{2n+1} = 1 - t_n$ for $n \in \mathbb{N}_0$

Determine if the value of $M$ can be arbitrarily large such that all elements of $(t_{x_i})$ are 0.

Answer: Yes


Question: Determine the number of 3-to-1 functions from the set \( A \) to the set \( B \), where \( |A| = 15 \) and \( |B| = 5 \).

Answer: \dfrac{15!}{(3!)^5}


Question: A cuboid has three of its sides in the coordinate planes and one of its corners in the first octant on the paraboloid \( z = 8 - 2x^2 - y^2 \). What is the largest volume the cuboid can have?

Answer: 4\sqrt{2}


Question: Given any collection of $n$ distinct circles in the plane, each with fixed but possibly distinct radii, is it always possible to rearrange the circles so that any two of them intersect exactly twice? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{\cos(x)-1}{x^2}\,dx \) using contour integration and the residue theorem.

Answer: -\pi


Question: Calculate the conditional expectation \( E(2W(s) + W(u) | W(u) = 2) \), where \( W(t) \) is a standard Brownian motion and \( u < s \).

Answer: 6


Question: Let \(T\) be a topological space with Borel \(\sigma\)-algebra \(B(T)\), generated by the open sets of \(T\). If \(S \in B(T)\), consider the set \(C = \{A \subset S : A \in B(T)\}\). This set forms a \(\sigma\)-algebra on \(S\). Now, generate the Borel \(\sigma\)-algebra \(B(S)\) by treating \(S\) as a topological subspace with the inherited topology from \(T\). Is it true that \(B(S) = C\)?

Answer: Yes


Question: Determine the number of orbits in the action of the Galois group of a field with nine elements over its subfield with three elements on the field with nine elements.

Answer: 6


Question: In 3-dimensional Euclidean space, does there exist an arc whose orthogonal projection onto a fixed plane forms a non-degenerate square?

Answer: Yes


Question: Prove the equality: \[ 1=(-1)^n\sum_{k=0}^n\frac{x_k^n}{\prod_{\substack{l=0 \\ l \neq k}}^n(x_l-x_k)} \]

Answer: 1


Question: Evaluate the integral \( \int_\alpha \left( e^{z^2+z+1} + e^{\text{Im}(z)} \right) \, dz \), where \( \alpha \) is the path around the square with vertices at \( 0, 1, i, i+1 \).

Answer: 1 - e


Question: Determine the real values of \( k \) for which the matrix \( B = \begin{bmatrix} 1 & k^2+k & 1 \\ -k & k^2 & k \\ k & k & -k \end{bmatrix} \) is equivalent by rows to a matrix of the form \( \begin{bmatrix} 1 & 0 & a \\ 0 & 1 & b \\ 0 & 0 & 0 \end{bmatrix} \).

Answer: -1


Question: Find the probability density function of the random variable \( V = XY \), where \( X \) and \( Y \) are independent random variables uniformly distributed on the interval \([0, 1]\).

Answer: -\ln v


Question: Consider a discrete metric space $V$ consisting of $n$ points $p_0, p_1, \ldots, p_{n-1}$ with a metric $d$. Define a function $f: V \rightarrow \mathbb{R}$ such that $f(p_0) = 0$, $f(p_1) = d(p_0, p_1)$, and for all $i, j$, $d(p_i, p_j) \geq |f(p_i) - f(p_j)|$. Does such a function $f$ exist for any metric space $V$?

Answer: Yes


Question: Determine the number of subgroups of order 3 in the symmetric group $S_6$. Provide your answer as a single integer.

Answer: 40


Question: Determine the smallest constant \(c_0 > 0\) such that for every series \(\sum a_n\) with \(a_n \geq 0\), the convergence of \(\sum a_n\) implies the convergence of \(\sum_{n=1}^\infty \frac{\sqrt{a_n}}{n^c}\) for all \(c > c_0\), and there exists a convergent series \(\sum a_n\) with \(a_n \geq 0\) such that \(\sum_{n=1}^\infty \frac{\sqrt{a_n}}{n^{c_0}}\) diverges.

Answer: \dfrac{1}{2}


Question: If $(u,v)$ is a point on the ellipse $4x^2 + a^2y^2 = 4a^2$, where $4 < a^2 < 8$, that is farthest from the point $(0, -2)$, find the value of $u + v$.

Answer: 2


Question:
Let \(X_1, X_2, \ldots, X_n\) be a random sample from a continuous distribution with the probability density function \[f(x)=\frac{1}{2\sqrt{2\pi}}\left[e^{-\frac{1}{2}(x-2\mu)^2}+e^{-\frac{1}{2}(x-4\mu)^2}\right],\quad-\infty<x<\infty\] If \(T=\sum\limits_{i=1}^n X_i\), which one of the following is an unbiased estimator of \(\mu\)?

\[(A)\quad\frac{T}{n}\qquad\quad(B)\quad\frac{T}{2n}\qquad\quad(C)\quad\frac{T}{3n}\qquad\quad(D)\quad\frac{T}{4n}\]

Answer: C


Question: Determine whether every $n \times n$ matrix can be expressed as the limit of a sequence of invertible $n \times n$ matrices.

Answer: Yes


Question: Calculate the surface integral \( \int_{T} \vec{A} \cdot \hat{n} \, dS \) where \( \vec{A} = 4 \, \hat{i} \) and \( T \) is the region of the plane \( x+y+2z=1 \) that lies in the first octant (i.e., where \( x \ge 0, y \ge 0, z \ge 0 \)).

Answer: 1


Question: Determine the number of integral values of $x$ such that the expression \( \frac{4^n + 15n - 1}{x} \) is an integer for every positive integer \( n \).

Answer: 6


Question: Let $P$ be a plane in $\mathbb{R}^3$ parallel to the $xy$-plane at a distance $h$ above it. Let $\Omega$ be a closed, bounded set in the $xy$-plane with area $B$. A point $Q$ is located in the plane $P$. A pyramid is formed by joining each point in $\Omega$ to $Q$ with a straight line segment. Find the volume of this pyramid in terms of $B$ and $h$. \( \boxed{\frac{1}{3}Bh} \)

Answer: \dfrac{1}{3}Bh


Question: Evaluate the limit \( \lim_{n\to \infty}\sqrt[n]{n!} \).

Answer: \infty


Question: For any given positive number \( \varepsilon > 0 \), does there exist a positive integer \( n \) such that \( |\sin(n)| < \varepsilon \)?

Answer: Yes


Question: Consider the affine plane $X = \mathbb{A}^2$. First, perform a blow-up at the origin $(0,0)$. Then, blow up the intersection of the line $(x=0)$ with the exceptional divisor from the first blow-up. Determine the subscheme along which the resulting space is the blow-up of $\mathbb{A}^2$. What is this subscheme?

Answer: (x, y^2)


Question:
A checker is placed on a square of an infinite checkerboard, where each square is 1 cm by 1 cm. It moves according to the following rules:
1. In the first move, the checker moves 1 square North.
2. All odd-numbered moves are North or South, and all even-numbered moves are East or West.
3. In the n-th move, the checker makes a move of n squares in the same direction.

The checker makes 12 moves so that the distance between the centers of its initial and final squares is as small as possible. What is this minimum distance?

Answer: 2


Question:
Determine the correct form of a 0.99 confidence interval for \( \mu_i \) given the model \((x_1,y_1),\ldots,(x_n,y_n)\) with \( n=23 \), where \( \mu_i^* = \frac{x_i+y_i}{2} \) is the MLE estimator for \( \mu_i \), and \( \sigma^* = \frac{1}{4n}\sum_{i=1}^{23}(x_i-y_i)^2 \). Choose between the following two forms:

1. \[ \left[ \mu_i^* \pm \frac{\sqrt{\sigma^*}}{\sqrt{23}}t_{22,.975} \right] \]
2. \[ \left[ \mu_i^* \pm \frac{\sqrt{\sigma^*}}{\sqrt{2}}t_{22,.975} \right] \]

Which form is correct?

Answer: 2


Question: Evaluate the infinite product: \[ \prod_{n=1}^\infty \frac{3}{1+2 \cos\left(\frac{\pi}{3^n}\right)} \] and find its integral part.

Answer: 1


Question: Let $(X, \mathfrak{B}, \mu)$ be a measurable space, possibly not $\sigma$-finite, and let $f_1, \cdots, f_n \colon X\to (-\infty, +\infty)$ be integrable functions on $X$. Determine whether the inequality \[ (\int f_1 \, d\mu)^2 + \cdots + (\int f_n \, d\mu)^2 \leq (\int \sqrt{f_1^2 + \cdots + f_n^2} \, d\mu)^2 \] holds. Justify your answer.

Answer: The inequality holds.


Question: Determine the degree of the map \( f: \mathbb{P}_{\mathbb{R}}^3 \rightarrow \mathbb{P}_{\mathbb{R}}^3 \) given by \( f([x_0:x_1:x_2:x_3])=[x_0^2: x_1^2: x_2^2 :x_3^2] \).

Answer: 0


Question: Find an analytic formula for the maximum value of \( \|Xa\|_2 \) over all \( k \times m \) matrices \( X \) such that \( \|X\|_F \le 1 \), where \( a \in \mathbb{R}^m \).

Answer: \|a\|_2


Question: What is the smallest positive integer \( n \) such that there are exactly four nonisomorphic Abelian groups of order \( n \)? List the four groups.

Answer: 36


Question: Find the volume of the region in \( \mathbb{R}^3 \) described by the inequalities \( x^2 + y^2 + z^2 \le 1 \) and \( |y| \le x^2 + y^2 \).

Answer: \dfrac{16}{9}


Question:
Determine whether the following limit is equal to zero, given that $np_n \rightarrow \lambda$ and $p_n \rightarrow 0$:
\[ \lim_{n \rightarrow \infty}\sum_{k=0}^{n} \frac{|(1-\frac{n p_n}{n})|^{n-k}- e^{- \lambda}|}{k!}=0. \]

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty}\frac{n!}{(n-i)!}\left(\frac{c}{n}\right)^{n-i} \] where \(c\) and \(i\) are constants.

Answer: 0


Question: Suppose $t_{n}$ is a sequence of positive real numbers such that $c_{1} \geq \lim \sup_{n\to \infty}t_{n}/n \geq \lim \inf_{n\to \infty}t_{n}/n \geq c_{2} > 0$, where $c_{1} \geq c_{2} > 0$ are positive constants. Does it follow that $\lim \inf_{N\to \infty}\dfrac{m(\cup^{N}_{n=1}[t_{n}-1,t_{n}+1])}{N} > 0$, where $m$ is the Lebesgue measure?

Answer: Yes


Question: Let $p$ be a subadditive functional on a normed space $X$ such that $p$ is continuous at $0$ and $p(0) = 0$. Prove that $p$ is continuous for all $x \in X$. Specifically, show that if $(x_n) \to x$, then $p(x_n) \to p(x)$.

Answer: p(x_n) \to p(x)


Question: Let \( F(x) = \sum_{n=0}^{\infty} a_n x^n \) be a power series that converges in some neighborhood of the origin. Compute \( \sup\left\{\delta > 0 : \text{ there exists } \epsilon > 0 \text{ such that } \int_{-\epsilon}^{\epsilon} |F(x)|^{-\delta} \, dx < \infty \right\} \). Interpret the integrals as improper Riemann integrals if \( F(0) = 0 \).

Answer: \frac{1}{k}


Question: Evaluate the limit: $$\lim_{x\to 0}\frac{1-\cos^{1/3} x}{1-\cos(x^{1/3})}$$

Answer: 0


Question: Determine the degree of the field extension \( [K:\mathbf{Q}] \) where \( K = \mathbf{Q}[\sqrt{2}, i] \) is the splitting field of the polynomial \( x^4 + 1 \) over the rationals.

Answer: 4


Question: Find the value of $x$ such that \( \sum_{k=1}^{2014} k^k \equiv x \pmod{10} \).

Answer: 3


Question: Given the polynomial equation \( x^4 - 7x^3 + (13+m)x^2 - (3+4m)x + m = 0 \), where \( x_1, x_2, x_3, x_4 \) are the roots, and it is known that \( x_1 = 2 + \sqrt{3} \) and \( x_3 = 2x_4 \), determine the value of \( m \).

Answer: 2


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sin^2(\pi\sqrt{n^2+n}) \]

Answer: 1


Question:
Consider a random sample of size $n$ from a Poisson population with mean $\lambda$. The uniformly minimum variance unbiased estimator (UMVUE) of $e^{-\lambda}$ is $\left(\frac{n-1}{n}\right)^{n \bar x}$, and the maximum likelihood estimator (MLE) of $e^{-\lambda}$ is $e^{-\bar x}$. Find the asymptotic relative efficiency (ARE) of the MLE with respect to the UMVUE, given that:

- $\sqrt{n}[e^{-\bar x} - e^{-\lambda}] \rightarrow N[0, \lambda e^{-2\lambda}]$ in distribution.
- $\sqrt{n}[\left(\frac{n-1}{n}\right)^{n \bar x} - \left(\frac{n-1}{n}\right)^{n \lambda}] \rightarrow N\left[0, \lambda \left(\left(\frac{n-1}{n}\right)^{n \lambda} \ln\left(\frac{n-1}{n}\right)^n\right)^2\right]$ in distribution.

Calculate the ARE of $e^{-\bar x}$ with respect to $\left(\frac{n-1}{n}\right)^{n \bar x}$. \[ \text{ARE}(e^{-\bar x}, \left(\frac{n-1}{n}\right)^{n \bar x}) = \frac{\lambda \left(\left(\frac{n-1}{n}\right)^{n \lambda} \ln\left(\frac{n-1}{n}\right)^n\right)^2}{\lambda e^{-2\lambda}} \]

Answer: 1


Question: Find the root of the equation $x^3 - 5x + 1 = 0$ in the interval $(0,1)$ using a sequence that satisfies the condition $|a_{n+1} - a_n| \leq c|a_n - a_{n-1}|$ with $0 < c < 1$. Approximate the root to within $10^{-4}$.

Answer: 0.2016


Question: Find the exact result of the infinite sum \( S(x) = \sum_{i=1}^\infty \frac{x^i}{i} \) where \( |x| < 1 \).

Answer: -\ln(1 - x)


Question: Cards are drawn without replacement from a standard deck of 52 cards until all four aces are chosen. Find the expected number of draws needed to achieve this using conditioning.

Answer: \dfrac{212}{5}


Question: Calculate the limit: \( \lim_{x\to 1} x^{(\frac{1}{x-1})} \)

Answer: e


Question: Evaluate the integral \( \int_{-\infty}^\infty x^2 \cdot \frac{1}{2}Ce^{-C|x|}\,dx \), where \( C \) is a constant.

Answer: \dfrac{2}{C^2}


Question: Determine whether the following statement is true or false: Let $A \in \mathbb{C}^{n \times n}$ be a matrix such that $A^2 = A$. Then $\text{rank}(A) = \text{trace}(A)$. Provide a justification for your answer.

Answer: True


Question: Show that there exists a \( \beta \in [0,1] \) such that \( \lim_{n\rightarrow \infty} \left(\sum^{n}_{k=2} \dfrac{1}{k \cdot \log(k)} - \log(\log(n))\right) = \beta \).

Answer: \beta


Question: Let \( f(x) = \int_{0}^x \frac{dt}{\sqrt{1+t^3}} \) and \( g \) be the inverse function of \( f \). Determine the value of \( k \) such that \( g''(y) = kg^2(y) \).

Answer: \dfrac{3}{2}


Question: Consider a $T_1$-topological space $(X, \tau')$ with a compatible uniformity $\mathcal{U}'$ that has a countable base. Is it always possible to find a metric $d$ on $X$ such that the uniformity generated by $d$, $\mathcal{U}(d)$, is exactly $\mathcal{U}'$?

Answer: Yes


Question: In a pyramid with a square base ABCD and apex S, the height of the pyramid is 8 units, and the angle between lines SA and SC is 48 degrees. Calculate the angle between the height SO (where O is the center of the square base) and the face SBC of the pyramid.

Answer: 17.5^\circ


Question: Given two different complex numbers $z_1$ and $z_2$ with $|z_1| = 1$, find the modulus of the expression \( \frac{|z_1 - z_2|}{|1 - z_1 \overline{z_2}|} \).

Answer: 1


Question: Let $A$ be an $n \times n$ tridiagonal symmetric matrix with nonnegative entries. Assume that $A$ is positive semidefinite. Does there exist a diagonal matrix $D = \mathrm{diag}(d_1, \ldots, d_n)$, with $d_j > 0$, such that $D^T A D$ is diagonally dominant?

Answer: Yes


Question: Calculate the limit: \[ \lim_{n \to \infty} \frac{n\log_2n}{\log_2 n!} \]

Answer: 1


Question: Determine the values of the positive real number \( b \) for which the series \( \sum_{k=1}^\infty \frac{b^{k^{2}}}{k!} \) is convergent.

Answer: (0, 1]


Question: There are seven people, each with a uniquely colored hat. After entering a room, their hats are shuffled and redistributed. In how many ways can the hats be distributed such that none of the first three people receive their original hat?

Answer: 3216


Question: Compute the value of the integral \( \frac{1}{2\pi i}\int_\mathcal{C} |1+z+z^2|^2 \, dz \), where \( \mathcal{C} \) is the unit circle in the complex plane centered at the origin and taken in the anticlockwise direction.

Answer: 2


Question: Find the covariance \( \text{Cov}\left(\int_t^T (T-s) \, dW_s, W_t\right) \).

Answer: 0


Question: Find the last three digits of \( \sqrt{4^{2016} + 2 \cdot 6^{2016} + 9^{2016}} \).

Answer: 257


Question: Find the integer \(m'\) such that \(\sum _{i=1}^{m'-1} i^k = (m')^k\) for a given integer \(k\).

Answer: 3


Question: Find the number of prime numbers \( p \) such that \( p \), \( 10 + p \), \( 10p + 1 \), and their sum are all prime numbers.

Answer: 4


Question: Let a function \(f:(a,b) \rightarrow \mathbb{R}\) be continuous and satisfy the following condition: for each \(\varepsilon >0\), there exists a \(\delta >0\) such that for \(x \in (a,b)\) and \(|h|<\delta\) with \(x+nh \in (a,b)\), the inequality \[ \left| \frac{\Delta_h^n f(x)}{h^n} \right| < \varepsilon \] holds, where \(\Delta_h^n f(x)\) is the \(n\)-th divided difference of \(f\). Is \(f\) necessarily a polynomial of degree \(\leq n-1\)?

Answer: Yes


Question: Let $(X, A, m)$ be a probability space, and let $E_n$ be measurable sets such that every $x \in X$ belongs to exactly 3 of the sets $E_n$. Compute the sum $\sum m(E_n)$. Provide your answer as a single value.

Answer: 3


Question:
Let $A$ be a $10 \times 10$ real matrix. Determine which of the following statements is correct regarding the rank of powers of $A$:

(A) $\rho(A^8) = \rho(A^9)$

(B) $\rho(A^9) = \rho(A^{10})$

(C) $\rho(A^{10}) = \rho(A^{11})$

(D) $\rho(A^8) = \rho(A^7)$

Select the correct option.

Answer: C


Question: Evaluate the integral \( \int \int \int_D y \, dz \, dy \, dx \) where \( D \) is the region bounded by the planes \( x=0 \), \( y=0 \), \( z=0 \), and \( 2x+2y+z=4 \).

Answer: \dfrac{4}{3}


Question: Let \( K = \mathbb{Q} \) and consider the polynomial \( f = X^3 - 3 \). Determine the basis and the degree of the splitting field of \( f \) over \( K \).

Answer: 6


Question: Evaluate the integral \( \int_{C}\frac{y^3dx-xy^2dy}{(x^2+y^2)^2} \) over the curve \( C = \frac{x^2}{16} + \frac{y^2}{9} = 1 \), oriented clockwise, using Green's Theorem. Note that the vector field is not defined at the origin.

Answer: \pi


Question: Evaluate the line integral \( \int_{\gamma}(x+y) \, dS \) where \( \gamma \) is the path defined by the triangle connecting the points \((0,0)\), \((0,1)\), and \((1,0)\).

Answer: 1 + \sqrt{2}


Question:
For which of the following functions does the series \( \sum_{x \in S(f)} \frac{1}{x} \) converge, where \( S(f) = \{x:x>0, f(x)=x\} \)?

A.) \( \tan x \)

B.) \( \tan x^2 \)

C.) \( \tan 2x \)

D.) \( \tan\sqrt{x} \)

E.) \( \sqrt{|\tan x|} \)

Answer: D


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{5^{\frac{n!}{(2n)!}} - 4^{\frac{1}{n!}}}{3^{\frac{n!}{(2n)!}} - 2^{\frac{1}{n!}}} \]

Answer: 2


Question: Consider a finite population from which $X_1, X_2, X_3, \ldots$ are sampled without replacement. The central limit theorem suggests that for i.i.d. random variables, the distribution of the sample mean approaches a normal distribution as the sample size $n$ increases. However, when sampling without replacement, independence does not hold, but the approximation to normality can still be valid when $n$ is small relative to the population size. Determine the value of $n$ for which the probability distribution of the sample mean is closest to the normal distribution described by the central limit theorem.

Answer: 30


Question: Determine the number of zeroes of the function \( f(z) = \sin(z) + \frac{1}{z+i} \) within the region \( D = \{ z : |\operatorname{Im}(z)| \leq 1, \pi(100+\frac{1}{2}) \leq \operatorname{Re}(z) \leq \pi(200+\frac{1}{2}) \} \).

Answer: 100


Question: Given that $x$ and $y$ are positive real numbers satisfying the equation \( \sqrt{x^2 + \sqrt[3]{x^4 y^2}} + \sqrt{y^2 + \sqrt[3]{x^2 y^4}} = 512 \), find the value of $x^{2/3} + y^{2/3}$.

Answer: 64


Question:
Consider the Laplace equation in polar coordinates:
\[ \frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial u}{\partial r})+\frac{1}{r^2}\frac{\partial^2u}{\partial \theta^2} =0 \]
with the boundary condition:
\[ u(a,\theta)=\begin{cases}2\theta & 0<\theta<\pi\\0 & \pi<\theta<2\pi\end{cases} \]
Calculate the value of \( U(0, \theta) \).

Answer: \dfrac{\pi}{2}


Question: Determine the number of possible combinations for a lock with $n$ push buttons, where each button can be pressed no more than once. The combination can include any number of distinct pressings, and the order of pressing is relevant. Buttons can be pressed simultaneously, and concurrently pressed buttons have no order. Calculate the number of combinations for $n=4$ and provide a general formula for any $n \in \mathbb{N}$. Assume that pressing no buttons is also a valid combination.

Answer: 150


Question:
Consider the discrete-time state-space system given by:
\[ x(t+1) = Ax(t) + Bu(t), \quad y(t) = Cx(t), \quad t \in \mathbb{N} \]
where
\[ A = \frac{1}{2} \begin{bmatrix} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix}, \quad C = \begin{bmatrix} -1 & 0 & 1 \end{bmatrix} \]
The eigenvalues of the matrix \( A \) are \( \lambda_1 = 0, \lambda_2 = \frac{1}{2}, \) and \( \lambda_3 = 1 \).

Which of the following statements is true about the system's stability?

A) The system is Lyapunov stable and asymptotically stable, but not BIBO stable.
B) The system is Lyapunov stable, but not asymptotically stable and not BIBO stable.
C) The system is Lyapunov stable and BIBO stable, but not asymptotically stable.
D) The system is not Lyapunov stable and not asymptotically stable, but is BIBO stable.
E) The system is not Lyapunov stable, not asymptotically stable, and not BIBO stable.

Select the correct statement.

Answer: C


Question: Calculate the volume of the larger region cut from the solid sphere defined by \(x^2 + y^2 + z^2 = 4\) by the plane \(z = 1\). Use double integrals in spherical coordinates to find the solution.

Answer: 9\pi


Question: If $19a^2 \equiv b^2 \pmod{7}$, does it follow that $19a^2 \equiv b^2 \pmod{49}$? Justify your answer.

Answer: Yes


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

Answer: \infty


Question: Evaluate the integral \( \iint_M 1 \, dx \, dy \) over the region \( M \) defined by the inequality \((x^2 + y^2)^2 \leq 2y^3\).

Answer: \dfrac{5\pi}{8}


Question: Let $f$ be a real-valued function on some open subset of $\mathbb{R}$, defined as algebraic if there exists a polynomial $P \in \mathbb{R}[x,y]$ such that $P(x,f(x))=0$ identically on the domain of $f$. Assume $f$ has an inverse function $g$ on some open subset of $\mathbb{R}$ such that $f(g(y)) = y$ and $g(f(x))=x$ whenever these equations make sense. Is the inverse function $g$ also algebraic?

Answer: Yes


Question: Does there exist a universal constant $C$ such that for any Banach space $X$, the separability measure of its unit ball satisfies \( \textrm{sep}(B_{X}) \leq C \cdot \textrm{sep}(B_{X^{*}}) \)?

Answer: Yes


Question: Given an $n \times n$ bipartite graph $G$ with a minimum degree of at least $n - d$, consider the following process: Select vertices from the left side randomly with probability $p = \frac{1}{d+1}$ to form set $S$. Then, form set $T$ by including all vertices on the right side that are connected to every vertex in $S$. Determine the lower bound on the probability that the complete bipartite subgraph $S \times T$ contains a specific edge $uw$. Express this probability in terms of $p$, $n$, and $d$.

Answer: p(1 - p)^d


Question:
Given that the radius of convergence of the power series \(\sum_{n=0}^\infty a_n z^n\) is \(R\), where \(0 < R < \infty\), determine the radius of convergence \(R_1\) of the power series \(\sum_{n=1}^\infty n^{-n} a_n z^n\). Choose the correct answer from the following options:

A) \(R_1 = 0\)
B) \(R_1 = e^R\)
C) \(R_1 = \infty\)

Answer: C


Question: Determine the injective dimension of \( \mathbb{Z}_n \) as a \( \mathbb{Z} \)-module.

Answer: 1


Question: Evaluate the limit: \[ \lim_{n\to\infty} \sum_{m=1}^n \left(\left(1+\frac{m^3}{n^4}\right)^{1/8}-1\right). \]

Answer: \dfrac{1}{32}


Question: Let $f$ be a Riemann integrable function and $g$ be a Lebesgue integrable function on $[0,1]$. If $\int_{0}^{1} |f-g| = 0$, is $g$ Riemann integrable?

Answer: No


Question: Let $A \subset [0,1]$ be a countable set. Determine whether there exists a set $B \subset [0,1]$ such that $A \subset B$, $B$ is uncountable, and $B$ has Lebesgue measure 0.

Answer: Yes


Question: Evaluate the integral \( \int_{|z-1|=1}\frac{dz}{z^2-1} \) using the Cauchy Integral Formula.

Answer: \pi i


Question: Calculate the flux across the vector field out of a hemisphere using the divergence theorem. If the flux across the vector field \( v \) out of the region \( D \) is \( 4\pi \) and the triple integral over the divergence of \( v \) is \( 8\pi \), what is the final flux value according to the divergence theorem?

Answer: 8\pi


Question: Evaluate the limit \( I = \lim_{x \to 0^{+}} \frac{\int_{0}^{x^2} t \sin{(\sqrt{x^2-t^2})} \, dt}{x^3} \).

Answer: 0


Question: If an $n \times n$ matrix $\mathbf{A}$ is diagonalizable and orthogonal eigenvectors of $\mathbf{A}$ form a basis of $\mathbb{R}^n$, is $\mathbf{A}$ necessarily symmetric? Justify your answer.

Answer: Yes


Question: Find the distance between the vector \( v_2 = \begin{pmatrix} 9 \\ 0 \\ 0 \end{pmatrix} \) and the subspace \( U = (\text{span}(v))^\perp \), where \( v = \begin{pmatrix} \frac{1}{3} \\ \frac{2}{3} \\ \frac{2}{3} \end{pmatrix} \).

Answer: 3


Question: Evaluate the integral \( \int_{0}^\infty e^{-x}\sin (x)\;dx \).

Answer: \dfrac{1}{2}


Question: Let \( y=(f(u)+3x)^2 \) and \( u=x^3-2x \). Given that \( f(4)=6 \) and \( \frac{dy}{dx}=18 \) when \( x=2 \), find \( f'(4) \).

Answer: -\dfrac{9}{40}


Question: Evaluate the limit: \( \lim_{x \to \infty} \int_0^x e^{t^2} \, dt \).

Answer: \infty


Question:
Consider two sequences of real-valued $m \times m$ matrices $W_m(n)$ and $\Omega_m(n)$ such that $\Omega_m(n) \rightarrow \Omega_m$ and $W_m(n) \rightarrow \Omega_m^{-1}$ as $n \rightarrow \infty$, where $\Omega_m^{-1}$ exists for any natural number $m$. Determine if the following limit is always equal to one:
\[ \lim_{m \rightarrow \infty} \lim_{n \rightarrow \infty} \frac{1}{m} \text{Trace}(W_m(n) \Omega_m(n)) \]

Answer: 1


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

Answer: \log 2


Question: Determine whether the function \( f(x,y) = \frac{x}{\sqrt{x^2+y^2}} \) belongs to the Sobolev space \( W^{1,p}(B) \) for some \( 1 \le p < 2 \), where \( B \) is the open unit disk in \( \mathbb{R}^2 \).

Answer: Yes


Question: Determine the number of zeros of the polynomial \( z^9 + z^5 - 8z^3 + 2z + 1 \) that lie in the annular region \( 1 < |z| < 2 \). Use Rouche's theorem to justify your answer.

Answer: 6


Question: Let \((x_{n})\) be a real sequence with \(x_{n} \neq 0\). Assume \(\lim\frac{x_{n+1}}{x_{n}}=\ell\). Prove that if \(|\ell|<1\), then \(\lim{x_{n}}=0\).

Answer: 0


Question: Minimize the function \( f(x, y, z) = x^4 - 6x^2 y^2 + y^4 \) subject to the constraint \( x^2 + y^2 + z^2 = 1 \).

Answer: -1


Question: Find the sum of the coefficients in the expansion of \((x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7)^{11}\) where each variable has a degree of at least one.

Answer: 322494480


Question: Determine the number of natural numbers \( n \) such that \( 337 \leq n \leq 2022 \) and \( \binom{n}{337} \equiv 3 \pmod{337} \). Find the cardinality of the set \( A = \{ n \in \mathbb{N} \mid 337 \leq n \leq 2022, \binom{n}{337} \equiv 3 \pmod{337} \} \).

Answer: 337


Question: Find the norm of the operator \( T: C[0, \pi] \to C[0, \pi] \) defined by \((Tx)(t) = \int_0^t \cos(t-s)x(s) \, ds\).

Answer: 2


Question: Let \( \Phi : \mathbb{R} \to \mathbb{R}^2 \) be defined by \( \Phi(t) = (t^3, t^4) \). Define a differential 1-form \( \omega : \mathbb{R}^2 \to \Lambda_1(\mathbb{R}^2) \) by \( \omega(a, b) = a \, dx + b \, dy \). Compute the integral \( \int_{\Phi|_{[0,1]}} \omega \).

Answer: 1


Question: Find the least upper bound (supremum) of the set \( S = \{ x \mid x \in \mathbb{R}, x \geq 0, x^2 < c < x(c+1) \} \).

Answer: \sqrt{c}


Question: Evaluate the integral: \[ u = \int_0^\infty \frac{dx}{x^4 + 7x^2 + 1} \]

Answer: \dfrac{\pi}{6}


Question: Evaluate the limit: \[ \lim_{x\to0_+} \frac{\sqrt{2x(1-x)} \log(1-x^2)}{(1-\cos x)\sqrt{\sin x}}. \]

Answer: -2\sqrt{2}


Question: In the category of $G$-sets with $G$-equivariant maps as morphisms, are epimorphisms necessarily surjective?

Answer: Yes


Question:
Determine whether the following statement is true or false:

Let \( f: \mathbb{R} \to \mathbb{R} \) be a function such that \( \lim_{h \to 0} \frac{f(x+h) - f(x-h)}{h} \) exists for all \( x \in \mathbb{R} \). Then \( f(x) \) is differentiable for all \( x \in \mathbb{R} \).

Answer: B


Question: Identify a topological space whose fundamental group is isomorphic to \(\mathbb{Z}_3\).

Answer: L(3,1)


Question: Calculate the probability that a random variable $X$ from a Gamma distribution with parameters $(3,3)$ is greater than a random variable $Y$ from an Exponential distribution with rate $1$. That is, find $P(X > Y)$. Assume $X \sim \text{Gamma}(3,3)$ and $Y \sim \text{Exp}(1)$. Express your answer as a probability.

Answer: \dfrac{37}{64}


Question: Determine the number of group homomorphisms from \( \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/9\mathbb{Z} \) to \( \mathbb{Z}/18\mathbb{Z} \).

Answer: 54


Question: Let \((X, M, \mu)\) be a \(\sigma\)-finite measure space and let \(1<p < \infty\). Suppose \((f_n)_{n \in \mathbb{N}}\) is a bounded sequence in \(L^{p}(\mu)\) such that \(\lim_{n \rightarrow \infty} \int f_n g \, d\mu\) exists for every \(g \in L^{q}(\mu)\). Find \(f \in L^p(\mu)\) such that \(\lim_{n\rightarrow \infty}\int f_n g \, d\mu = \int f g \, d\mu\).

Answer: f


Question: Determine if there exists a number \(a\) such that all derivatives of \(x^x\) are positive on the interval \((a, \infty)\). If such a number exists, find the smallest possible value of \(a\).

Answer: 1


Question:
Given the continued fraction with 1998 terms:
\[
1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\cdots+\dfrac{1}{1}}}} = \dfrac{m}{n}
\]
where \(m\) and \(n\) are positive integers with \(\gcd(m, n) = 1\), find the value of \(m^2 - mn - n^2\).

Answer: 1


Question:
Determine the rank of the 4x4 matrix \( A \) defined by:
\[
A = \begin{pmatrix}
x_1^2 + y_1^2 & x_1x_2 + y_1y_2 & x_1x_3 + y_1y_3 & x_1x_4 + y_1y_4 \\
x_2x_1 + y_2y_1 & x_2^2 + y_2^2 & x_2x_3 + y_2y_3 & x_2x_4 + y_2y_4 \\
x_3x_1 + y_3y_1 & x_3x_2 + y_3y_2 & x_3^2 + y_3^2 & x_3x_4 + y_3y_4 \\
x_4x_1 + y_4y_1 & x_4x_2 + y_4y_2 & x_4x_3 + y_4y_3 & x_4^2 + y_4^2
\end{pmatrix}
\]
where \(x_1, x_2, x_3, x_4, y_1, y_2, y_3, y_4\) are fixed real numbers, not all zero.

Answer: 2


Question:
Let $\mathcal{S}(\mathbb{R}^N)$ be the space of Schwartz functions. Consider a bilinear functional $T: \mathcal{S}(\mathbb{R}^N)\times \mathcal{S}(\mathbb{R}^M) \to \mathbb{C}$ such that there exist constants $c_1, c_2 > 0$ satisfying:

1. For every $g \in \mathcal{S}(\mathbb{R}^M)$, \( \forall_{f \in \mathcal{S}(\mathbb{R}^N)} |T(f,g)| \leq c_1 \|f\|_{D_1} \).
2. For every $f \in \mathcal{S}(\mathbb{R}^N)$, \( \forall_{g \in \mathcal{S}(\mathbb{R}^M)} |T(f,g)| \leq c_2 \|g\|_{D_2} \).

Here, \( \|f\|_D = \sum_{|\alpha|,|\beta|<D}\|f\|_{\alpha\beta} \) and \( \|f\|_{\alpha,\beta}=\sup_{x\in\mathbf{R}^N} \left |x^\alpha D^\beta f(x) \right | \).

According to the Schwartz kernel theorem, $T$ is a distribution on $\mathcal{S}(\mathbb{R}^{N+M})$. Determine if it is true that this distribution satisfies the condition:

\[ \forall_{f\in \mathcal{S}(\mathbb{R}^N),g\in \mathcal{S}(\mathbb{R}^M)} |T(f,g)| \leq \text{const} \cdot \|f\|_{D_1}\|g\|_{D_2} \]

for some constant independent of $f$ and $g$. Answer true or false.

Answer: True


Question:
Prove that for \( \lambda > 0 \), the following series equals 1:
\[ \sum_{j=k}^{\infty} \binom{j-1}{k-1} e^{-k \lambda}(1-e^{- \lambda })^{j-k} = 1 \]

Answer: 1


Question: Evaluate the integral \( \int\limits_{-\infty}^{\infty} \frac{\cos(x)}{x} \, dx \).

Answer: 0


Question: Determine if there exists a natural number \( n \) such that \( \sqrt[n]{22-10\sqrt7} = 1-\sqrt7 \).

Answer: 3


Question: Find the continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $\lim_{x\rightarrow\infty}f(x)^{e^x}=e$ and $f(x)+F(x)=x$, where $F$ is an antiderivative of $f$. Determine $f(x)$. \( \text{(Assume } F(x) = \int f(x) \, dx \text{)} \)

Answer: 1 + e^{-x}


Question: Find all entire functions \( f \) such that \( f\left(\frac{1}{n}\right) = \frac{1}{n^2} \) for all positive integers \( n \).

Answer: z^2


Question: Determine the values of \( a \) for which the integral \( \int\int_D r^{-a} \, dA \) converges, where \( r = \sqrt{x^2 + y^2} \) and \( D \) is the unit disk defined by \( x^2 + y^2 \leq 1 \). Recall that the improper integral \( \int_0^1 x^{-a} \, dx \) converges if \( a < 1 \).

Answer: 2


Question: Let $f$ be a holomorphic function on the open unit disc $\mathcal{D} = \{z \in \mathbb{C} \mid |z| < 1\}$. Suppose that $|f| \geq 1$ on $\mathcal{D}$ and $f(0) = i$. Determine which of the following are possible values of $f\left(\frac{1}{2}\right)$: \(-i, i, 1, -1\).

Answer: i


Question: Evaluate the sum \( S(z) = \sum_{n=1}^\infty \frac{1}{n} (z-2)^n \) for complex \( z \) where \(|z-2|<1\).

Answer: -\ln(3 - z)


Question: Evaluate the sum: \[ \sum^{7}_{k=1}\frac{1}{\sqrt[3]{k^2}+\sqrt[3]{k(k+1)}+\sqrt[3]{(k+1)^2}} \]

Answer: 1


Question: Let $(x_{n})_{n \ge 1}$ be a sequence of real numbers such that \( \lim_{n\to\infty} x_n \sum_{k=1}^{n}x^2_{k}=1 \). Compute \( \lim_{n\to\infty} (3n)^{1/3} x_n \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{N\to\infty} \frac{\ln^2 N}{N}\sum_{k=2}^{N-2} \frac{1}{\ln k \cdot \ln (N-k)} \] where \(\ln\) denotes the natural logarithm.

Answer: 1


Question: Determine all real numbers \( c \) such that the equation \( x^5 - 5x = c \) has exactly 3 real roots.

Answer: (-4, 4)


Question: Given invertible matrices $A$ and $B$, and matrices $X$ and $Y$ of size $n \times n$, if $A^k X = B^k Y$ for $k=1,2,...,2n$, does it follow that $X = Y$? Provide a justification for your answer.

Answer: X = Y


Question: Evaluate the limit: \[ \lim_{n\to\infty}\sum_{k=1}^n \frac{\pi k}{2n}\int_0^1 x^{2n}\sin\frac{\pi x}{2}dx \]

Answer: \dfrac{\pi}{8}


Question: Evaluate the binomial summation: \[ \sum_{k=0}^{n} (-4)^k \cdot \binom{n+k}{2k} \]

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


Question: Evaluate the integral \( \int_{0}^{1}\sqrt{\frac{\{1/x\}}{1-\{1/x\}}}\, \frac{\mathrm{d}x}{1-x} \) and show that it equals \( \pi \), where \( \{x\} \) denotes the fractional part of \( x \).

Answer: \pi


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

Answer: (\ln 2)^2


Question:
Let $a, b, c$ be the cube roots of a negative number $p$. For any permissible values of $x, y, z$, the following equation holds:
\[ \frac{|xa+yb+zc|}{|xb+yc+za|} + (a_1^2-2b_1^2)\omega + \omega^2([x]+[y]+[z]) = 0 \]
where $\omega$ is a cube root of unity, $a_1$ and $b_1$ are real positive numbers, and $b_1$ is a prime number. Determine the value of $[x+a_1] + [y+b_1] + [z]$, where $[\cdot]$ denotes the greatest integer function.

Answer: 6


Question: Compute the integral \( \int_D x^3+y^2 \, \mathrm{d}x \, \mathrm{d}y \) where \( D = \{(x,y) \in \mathbb{R}^2 : |x| + |y| \leq 1\} \).

Answer: \dfrac{1}{3}


Question: Compute the expected value of \(E[X_1^2 + X_2^2 + \dots + X_n^2]\) where \(n\) balls are randomly thrown into \(n\) urns. Let \(X_i\) be the number of balls in the \(i\)-th urn.

Answer: 2n - 1


Question: Let $X_1, X_2, X_3, \ldots$ be a sequence of i.i.d. random variables with mean $1$. Let $N$ be a geometric random variable with probability mass function $\mathbb{P}(N=k)=\frac{1}{2^k}$ for $k=1,2,3,\ldots$, and $N$ is independent of the $X_i$'s. Find $\mathbb{E}(X_1 + X_2 + \ldots + X_N)$.

Answer: 2


Question: Find the remainder when \( \left( \sum_{k=1}^5 { ^{20} \mathrm C (2k-1) } \right)^6 \) is divided by 11.

Answer: 3


Question: Let $A \subset \mathbb{R}^d$ be a closed, strictly convex set. Denote by $\partial A$ its topological boundary and by $A^\circ$ its interior. Suppose $\mu$ is a probability measure on $\partial A$ that is not a Dirac measure. Is it true that \( \int_{\partial A} x \mu(dx) \in A^\circ \), i.e., the mean is contained in the interior of $A$?

Answer: Yes


Question: Given that the roots of the polynomial \(x^4 + 2x^3 - 5x^2 + 7x + 10 = 0\) are \(\alpha, \beta, \gamma, \delta\), and the roots of the polynomial \(x^4 + ax^3 + bx^2 + cx + d = 0\) are \(\alpha+\beta+\gamma, \alpha+\beta+\delta, \alpha+\gamma+\delta, \beta+\gamma+\delta\), find the value of \(a + b - c - d\).

Answer: 56


Question: Determine whether the limit \( \lim_{(x,y) \to (0,0)} \frac{1}{x}\sin\frac{xy^2}{\sqrt{x^2+y^2}} \) exists. If it does not exist, explain why.

Answer: 0


Question: Let $H$ be an infinite-dimensional Hilbert space over the field $K$, and let $B$ be a basis of $H$ such that $H = \overline{\text{span}(B)}$. Consider a linear functional $T: H \rightarrow K$. If $T(B)$ is a bounded subset of $K$, is the linear functional $T$ necessarily bounded (i.e., does $T$ have a finite norm)?

Answer: Yes


Question: Evaluate the limit of the infinite product: \[ \prod_{p \in \mathbb{P}} \frac{p^4+1}{p^4-1} \] where \( \mathbb{P} \) is the set of prime numbers.

Answer: \dfrac{7}{6}


Question: Given the premises \( \lnot(P \land \lnot Q) \) and \( (\lnot P \to S) \land \lnot Q \), prove the conclusion \( T \lor S \) using natural deduction rules: Implication, And, Or, Modus Tollens, Proof by Contradiction, Copy Rule, Negation, Double Negation, Contradictions, and the Law of Excluded Middle.

Answer: T \lor S


Question: Determine the order type of the set of all limit elements in \( \omega^{\omega} \).

Answer: \omega^{\omega}


Question: Consider a countable nowhere dense set $S \subseteq \mathbb{R}$. Is it possible for this set to have uncountably many accumulation points? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int\int\int_{x^2+y^2+z^2<1}(a_1x+b_1y+c_1z)(a_2x+b_2y+c_2z)\,\mathrm dx\,\mathrm dy\,\mathrm dz \) over the unit ball, given that \( a_1a_2+b_1b_2+c_1c_2=0 \).

Answer: 0


Question: In a Noetherian ring $R$, let $\mathfrak{i}$ be an ideal generated by an infinite set $\{x_{\alpha}\}_{\alpha\in\mathcal{A}}$. Can a finite subset of $\{x_{\alpha}\}$ be extracted to generate $\mathfrak{i}$?

Answer: Yes


Question:
Determine the convergence behavior of the series \( \sum_{n=1}^{\infty}\frac{3\cdot 6\cdots(3n)}{7\cdot 10\cdots (3n+4)}x^n \). Choose the correct option:

(a) Converges for \(0<x\le 1\) and diverges for \(x>1\)
(b) Converges for all \(x>0\)
(c) Converges for \(0<x<1/2\), and diverges for \(x\ge 1/2\)
(d) Converges for \(1/2<x<1\), and diverges for \(0<x\le 1/2, x\ge 1\)

Answer: a


Question: Determine the points in the complex plane where the function \( f(z) = e^z(z-\bar{z})^2 \) is differentiable.

Answer: z \in \mathbb{R}


Question:
Given the following lists of numbers:

- List 1: 5, 100, 1
- List 2: 2
- List 3: 5, 100
- List 4: 2, 5, 100

Determine the number of ways to select one number from each list such that no number is repeated across the selections.

Answer: 2


Question: Find the length of the latera recta of the ellipse given by the equation \(3x^2+3y^2+2xy-12x+12y+4=0\).

Answer: 4


Question: Evaluate the limit: \[ \lim_{n\to+\infty}\sum_{k=1}^{n}\left(e^{\frac{k^2}{n^3}}-1\right). \]

Answer: \dfrac{1}{3}


Question: Find the smallest positive common difference of a 6-term arithmetic progression consisting entirely of positive prime numbers.

Answer: 30


Question: Given a deck of cards where each card has $n$ images, and each pair of cards shares exactly one image, but no image is present on all the cards, find an upper bound $s(n)$ on the number of cards in the deck.

Answer: n^2 - n + 1


Question: Determine the trace of an n-th root of unity, \( \xi_n \), in the cyclotomic extension \( \mathbb{Q}_p(\zeta_n)/\mathbb{Q}_p \). Specifically, is \( Tr_{\mathbb{Q}_p(\zeta_n)/\mathbb{Q}_p}(\xi_n) \in \mathbb{Q} \)?

Answer: No


Question: Find \( \lim_{(x,y)\rightarrow(0,0)} \cos\left(\frac{x^2-y^2}{\sqrt{x^2+y^2}}\right) \).

Answer: 1


Question: Calculate the probability that the absolute difference between two independent uniform random variables on the interval [0,1] is less than or equal to 0.25. That is, find \( P(|X-Y| \leq 0.25) \) where \( X, Y \sim U([0,1]) \).

Answer: \dfrac{7}{16}


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

Answer: 0


Question: Calculate the limit: \[ \lim_{n \to \infty} \left( \int_0^{\infty} (\sin(x))^n e^{-x} \, dx \right) \]

Answer: 0


Question: Determine the degree of the map \( f_k: S^1 \to S^1 \) defined by \( f_k(\cos(2\pi t), \sin(2\pi t)) = (\cos(2k\pi t), \sin(2k\pi t)) \).

Answer: k


Question: Find the real number \( \alpha \) such that there exists a value \( c \) for which \[ \int^c_0 \frac{1}{1+x^\alpha}\,dx = \int^\infty_c\frac{1}{1+x^\alpha}\,dx. \]

Answer: 2


Question: Calculate the integral \( \int_{-\pi}^{\pi}\left|\sum_{n=1}^{\infty}\frac{1}{2^{n}}e^{inx}\right|^{2}dx \) using Parseval's identity.

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


Question: Let $f$ be an entire function such that $|f(z)| \le 100\log|z|$ for all $|z| \ge 2$, and $f(i) = 2i$. Determine the value of $f(1)$. \( \text{Then } f(1) = ? \)

Answer: 2i


Question: Let $G$ be a simple graph with $n$ vertices, where $n \geq 3$. Let $\omega(G)$ denote the clique number of $G$, and $\chi(G)$ denote the chromatic number of $G$. Does the condition $3 \leq \omega(G) \leq k$ imply that $\chi(G) \leq k$?

Answer: No


Question: Calculate the area of the surface defined by the equation \(x^2 + y^2 = 2x\) that is delimited by the cone given by \(z^2 = x^2 + y^2\).

Answer: 16


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: Find the value of \(a\) such that the function \(f(x) = x^3 + 4x^2 + ax + 1\) is monotonically decreasing for \(x\) in the interval \((-2, -\frac{2}{3})\).

Answer: 4


Question: Determine the minimum number of times a rectangular object with faces labeled A, B, C, and D must be tossed so that the probability of the ratio of the number of D's to the total number of tosses being between 0.15 and 0.35 is at least 0.95.

Answer: 73


Question: Determine the pole order at infinity of the function \( \frac{4}{27}\frac{(\lambda^2-\lambda+1)^3}{\lambda^2(1-\lambda)^2} \).

Answer: 2


Question: Determine the radius of convergence for the series \( \sum_{n=0}^\infty \cosh(n) \times z^n \).

Answer: \dfrac{1}{e}


Question: Let \( K \) be a closed convex cone in a Banach space \( E \). Suppose \( x \in K \) and \( y \in E \) such that \( x + \lambda y \in K \) for all \( \lambda \geq 0 \). Prove or disprove that \( y \in K \).

Answer: y \in K


Question: Let $P_1$ and $P_2$ be the orthogonal projections onto the closed subspaces $M_1$ and $M_2$ of a Hilbert space $X$, respectively. If $M_2 \subseteq M_1$, determine whether the equality $P_2P_1 = P_2$ always holds.

Answer: P_2 P_1 = P_2


Question: Let $G$ be an abelian group. Suppose there is an integer $n \ge 1$ such that $nG = 0$. Let $m$ be the smallest integer $\ge 1$ such that $mG = 0$. Is there an element in $G$ of order $m$?

Answer: Yes


Question: A person you meet has two children, and you know that one of them is a boy born on a Tuesday. Assuming equal probability for either gender and equal probability for being born on each day of the week, what is the probability that the other child is also a boy?

Answer: \dfrac{13}{27}


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

Answer: 1


Question: Determine if there exists a function \( y = f(x) \) such that \( \frac{d^3y}{dx^3} = f(x) \). If such a function exists, provide an example.

Answer: e^x


Question: Find the minimum value of the expression \( \log_{a}(bc) + \log_{b}(ac) + \log_{c}(ab) \).

Answer: 6


Question: Find the last four digits in the binary expansion of \(1234^{5555} + 4321^{5555}\).

Answer: 0001


Question: Suppose $X \sim N(\mu,1)$ and consider $Y=\dfrac{1-\Phi(X)}{\phi(X)}$, where $\phi$ and $\Phi$ are the pdf and cdf of the standard normal distribution, respectively. Compute the expected value of $Y$ in terms of $\mu$. \( \boxed{\text{E}[Y]} \)

Answer: \dfrac{1}{\mu}


Question: Calculate the area enclosed by the curve \((x^2+y^2)^2=2a^2(x^2+y^2)\) using Green's theorem. Use the substitution \(x=a\sqrt{2}\cos t\sqrt{\cos 2t}\) and \(y=a\sqrt{2}\sin t\sqrt{\cos 2t}\) for the coordinates.

Answer: 2a^2


Question: Solve the recurrence relation \( y_{n+1} = 2y_{n} + n \) using exponential generating series, given the initial condition \( y_{0} = 1 \).

Answer: 2^{n+1} - n - 1


Question: Evaluate the limit: $$\lim_{n\rightarrow \infty}\left(\frac{\pi}{2}-\arctan n^2\right)\cdot \ln n!$$

Answer: 0


Question: Evaluate \( \lim _{n \rightarrow \infty} \int_{0}^{1} n x^{n} e^{x^{2}} \, dx \).

Answer: e


Question: Given the number \( n = 2^{14} \times 3^9 \times 5^8 \times 7^{10} \times 11^3 \times 13^5 \times 37^{10} \), determine how many positive divisors of \( n \) are perfect cubes and are also multiples of \( 2^{10} \times 3^9 \times 5^2 \times 7^5 \times 11^2 \times 13^2 \times 37^2 \).

Answer: 12


Question: Does there exist a ring with 5 elements such that the characteristic of the ring is 2? Assume the definition of characteristic is the smallest positive integer \( n \) such that \( nr = 0 \) for all \( r \in R \).

Answer: No


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

Answer: \dfrac{3}{2}


Question: Find the limit: \[ \lim_{x\to(\pi/2)}(1+\cos x)^{\tan x} \]

Answer: e


Question: Is the dimension of an eigenspace of a square matrix at most the multiplicity of the corresponding eigenvalue, where the multiplicity is defined as the multiplicity of the eigenvalue as a root of the characteristic polynomial \( p(x) = \det(xI - A) \)?

Answer: Yes


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

Answer: 0


Question: Determine the number of zeros of the function \( f(z) = z^5 + 2z^3 - z^2 + 2z + 5 \) within the region \( \{ z \in \mathbb{C} : \text{Re}(z) > 0, \text{Im}(z) > 0 \} \) using the Argument Principle.

Answer: 1


Question: Let $G$ be a finite group and $p$ a prime number. Suppose $P$ and $H$ are two Sylow $p$-subgroups of $G$ such that there exists an element $g \in G$ for which $H \subseteq gPg^{-1}$. Prove that $|H| = |P|$.

Answer: |H| = |P|


Question: Let $f(x)$ be a quadratic function such that $f(0) = 1$. Given that \( \int \frac{f(x)}{x^2(x+1)^3} \, dx \) is a rational function, determine the value of $f'(0)$. \( \boxed{} \)

Answer: 3


Question: Find the minimum value of the integral \( \int_0^{\infty} |a+bx+cx^2+x^3|^2 e^{-x} \, dx \) with respect to \(a, b, c \in \mathbb{C}\).

Answer: 36


Question:
Determine the radius of convergence for the power series:
\[ \sum_{n=1}^\infty \frac{x^{n^2}}{2^{n-1} n^n} \]

Answer: 1


Question: Determine whether the following statement is true or false: Let $K$ be the smallest subfield of $\mathbb{C}$ containing all the roots of unity. Then $\mathbb{C}$ is algebraic over $K$. Provide a justification for your answer.

Answer: B


Question: If $\pi_1(B)$ is open in $E_1$ and $\pi_2(B)$ is open in $E_2$, is $B$ open in $E_1 \times E_2$?

Answer: No


Question: Determine the degree of the algebraic number \( \frac{1+\sqrt{2}+\sqrt{3}}{5} \).

Answer: 4


Question: Let $A$ be an $m \times n$ real matrix. If for two suitable matrices $B$ and $C$, the equation $AA^TB = AA^TC$ holds, can we conclude that $A^TB = A^TC$? Provide a justification for your answer.

Answer: Yes


Question: Determine the number of one-to-one functions \( f: \{1, 2, 3, 4, 5\} \to \{0, 1, 2, 3, 4, 5\} \) such that \( f(1) \neq 0, 1 \) and \( f(i) \neq i \) for \( i = 2, 3, 4, 5 \).

Answer: 256


Question: Find the value to which the series \( \sum_{n=1}^{\infty}\frac{(-1)^{n}}{2n-1} \) converges.

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


Question:
Given the sequence defined by the initial conditions \( a_1 = 10 \) and \( a_2 = 20 \), and the recursive relation \( a_{n+1} = a_{n-1} - \frac{4}{a_n} \), find the smallest integer \( k \) such that \( a_k = 0 \). Choose the correct answer from the following options:

A. Does not exist.
B. 200
C. 50
D. 52

Answer: D


Question: Determine the values of \(a \in \mathbb{R}\) for which the function \(f(x,y) = (x-1)^2 + 2a(x-1)y + y^2 + y^4\) has a maximum or minimum at the point \((1,0)\).

Answer: [-1, 1]


Question:
Let $f: M \to \mathbb{R}$ be a positive integrable function over a relatively compact domain $M$. Define the arithmetic mean of $f$ as:
$$A(f) = \frac{1}{\mu(M)}\int_M f$$
where $\mu(M)$ is the measure of $M$. Define the geometric mean of $f$ as:
$$G(f) = \exp\left(\frac{1}{\mu(M)}\int_M \log f\right)$$
Prove or disprove that the inequality $A(f) \geq G(f)$ holds for every positive integrable function $f$.

Answer: A(f) \geq G(f)


Question: Use the ML lemma to show that \( \lim_{R \to \infty} \int_{C_R} \frac{z^2}{z^4+1} \, dz = 0 \), where \( C_R \) is the semicircle in the upper half of the complex plane oriented from \( z=R \) to \( z=-R \), centered at the origin.

Answer: 0


Question: Find the limit of \( \frac{(n!)^{1/n}}{n} \) as \( n \to \infty \).

Answer: \dfrac{1}{e}


Question: Let $X$ and $Y$ be independent $Bin(3,\frac{1}{3})$ random variables. Determine the probability that the matrix \[ P=\begin{bmatrix} \frac{X}{\sqrt2}&\frac{Y}{\sqrt2}\\\frac{-1}{\sqrt2}&\frac{1}{\sqrt2}\end{bmatrix} \] is orthogonal.

Answer: \dfrac{16}{81}


Question:
Given that \( z = x + iy \in \mathbb{C} \) satisfies \( \arg \left[\frac{z-1}{z+1}\right] = \frac{\pi}{4} \), determine which of the following equations is true:

(a) \( x^2-y^2-2x-1 = 0 \)

(b) \( x^2+y^2x-1 = 0 \)

(c) \( x^2+y^2-2y-1 = 0 \)

Answer: C


Question:
Consider a market where time is measured in milliseconds, and focus on market buy orders only. The market imbalance can be in one of three states: high (H), medium (M), or low (L). The imbalance changes due to incoming market orders or changes in standing orders. If the imbalance is low, it can remain low or increase to medium, each with probability \( \frac{1}{2} \), and no market buy orders arrive. If the imbalance is medium, it can decrease to low with probability \( \frac{1}{3} \) or increase to high. The increase from medium to high can occur due to a new market buy order with probability \( \frac{1}{2} \) or without a new order with probability \( \frac{1}{6} \). If the imbalance decreases from medium to low, no market buy orders arrive. If the imbalance is high, it can decrease to medium or stay high, each with probability \( \frac{1}{2} \), and a market order arrives with probability \( \frac{2}{3} \).

Betty keeps a unit standing sell order in the exchange, and every market buy order executes her order, earning her 1 cent. What is Betty's long-run profit per unit of time?

Hint: Construct a Markov chain with state space including more than just H, M, and L.

Answer: \dfrac{25}{54}


Question: Calculate the line integral \( \int_{\gamma} y\,dx + dy + 2dz \), where \( \gamma \) is the intersection of the paraboloid \( z = x^2 + y^2 \) with the plane \( z = 2x + 2y - 1 \). The path \( \gamma(t) \) should be chosen such that its projection onto the xy-plane is traversed in an anticlockwise direction.

Answer: -\pi


Question: Find the number of solutions to the equation \( \{f(f(x))\} = 0.5 \), where the function \( f(x) \) is defined as follows: \( f(x) = 4x - 2 \) for \( 0 \leq x \leq 1 \) and \( f(x) = -x^2 - 2x \) for \( -2 \leq x < 0 \). Here, \( \{x\} \) denotes the fractional part of \( x \).

Answer: 14


Question:
Determine which set of constants \(w_1, w_2, \alpha, \beta\) makes the Runge-Kutta method accurate to order 2 for the differential equation \(y'(x)=f(x,y)\) with initial condition \(y(x_0)=y_0\). The method is given by:

\[y(x+h)=y(x)+w_1F_1(x,y)+w_2F_2(x,y)\]
\[F_1(x,y)=hf(x,y)\]
\[F_2(x,y)=hf(x+\alpha h,y+\beta F_1)\]

Which of the following sets of values is correct?

(A) \( w_1=w_2=\frac{1}{2}, \alpha=\beta=1\).
(B) \(w_1=2 , w_2=1,\alpha =\frac{1}{2}, \beta=\frac{1}{2}\).
(C) \(w_1=\frac{1}{3} , w_2=\frac{2}{3},\alpha =\frac{3}{4}, \beta=\frac{3}{4}\).
(D) \(w_1=\frac{3}{4} , w_2=\frac{1}{4},\alpha =2, \beta=2\).

Answer: A


Question: Let $n$ be a natural number $\geq 2$ and $A$ a matrix in $M_{n}(K)$. Suppose the matrices $A$ and $I_{n}+A$ are invertible. Calculate $(I_{n}+A^{-1})^{-1}+(I_{n}+A)^{-1}$.

Answer: I_n


Question: Suppose we have two covariance matrices $A$ and $B$ that satisfy the condition $AB=BA$. Is the product $AB$ also a covariance matrix?

Answer: Yes


Question: Determine whether a countable group can have an uncountable number of distinct subgroups.

Answer: Yes


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

Answer: -\dfrac{1}{2}


Question: Determine the values of \( s \in \mathbb{R} \) for which the series \( \sum_{k \in \mathbb{Z}^n} \langle k \rangle^{-s} \) converges, where \( \langle k \rangle = (1 + |k|^2)^{1/2} \).

Answer: s > n


Question: In a k-chromatic connected graph with n vertices, where the graph is colored using exactly k colors, is it true that every pair of vertices with different colors is connected by at least one path of odd length? Answer yes or no.

Answer: Yes


Question: Determine the values of \(a \in \mathbb{R}_+\) for which the series \(\sum_{n>0} \frac{a(a+1)(a+2)\cdots(a+n-1)}{n!\cdot n^2}\) is convergent.

Answer: (0, 2)


Question: Determine whether the following statement is true or false: For a nilpotent matrix $A \in M_n(\mathbb{R})$, if $\exp(A) = \text{Id}$, then $A$ is the zero matrix.

Answer: True


Question: Let $r_1$, $r_2$, and $r_3$ be the roots of the polynomial equation $2x^3 - x^2 + 3x - 6 = 0$. Calculate the value of \( \sqrt{(r_1^2 + 3)(r_2^2 + 3)(r_3^2 + 3)} \).

Answer: 3


Question: Let $a$, $b$, $c$, and $d$ be distinct integers such that the equation $(x-a)(x-b)(x-c)(x-d) = 9$ has an integer solution $h$. Determine the value of $\frac{a+b+c+d}{h}$.

Answer: 4


Question: Evaluate the double integral \( \iint \lfloor x \rfloor \lfloor y \rfloor \, dA \) over the region bounded by \( x = -2 \), \( x = 1 \), \( y = 0 \), and \( y = 2 \).

Answer: -3


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left( 10\sqrt[n]{\int_{0}^{1}(1-x^2)^n\,dx} + 5\sqrt[n]{\int_{0}^{1}(1-x^3)^n\,dx} \right) \]

Answer: 15


Question:
Given the polynomial function \( f(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e \), and the conditions:
\[ f(1) = 2, \quad f(2) = 3, \quad f(3) = 4, \quad f(4) = 5, \quad f(5) = 6 \]
Find the value of the coefficient \( b \).

Answer: 85


Question: Determine the smallest positive integer $n$ for which $\sin\left(\frac{\pi}{n}\right)$ cannot be expressed in terms of radicals of integers.

Answer: 7


Question: Let $X$ be a compact metric space and $(\Omega, F, P)$ a nonatomic probability space. Suppose that $(\Omega, F', P')$ is the completion of $(\Omega, F, P)$. If $f: \Omega \rightarrow X$ is $F'$-measurable, does there exist an $F$-measurable function $g$ such that $g=f$ almost everywhere?

Answer: Yes


Question: Determine whether there exists a subfield $F$ of the complex numbers $\mathbb{C}$ such that the multiplicative group $F^*$ satisfies $F^{*p} = F^*$ for any prime $p \neq 5$, but $F^{*5} \neq F^*$. Provide a justification for your answer.

Answer: Yes


Question: Determine the number of distinct prime ideals in the ring \( \mathbb{Z}/p^2q\mathbb{Z} \), where \( p \) and \( q \) are distinct primes.

Answer: 2


Question: Consider a set $X$ with at least two elements. Does there exist a bijection $f: X \to X$ such that for every $x \in X$, $f(x) \neq x$?

Answer: Yes


Question: Determine if the function \( f(z) = \int_0^z \zeta^{-1} \sin \zeta \, d\zeta \) can be expressed as a power series in \( \mathbb{C} \) around \( z = 0 \).

Answer: Yes


Question: Evaluate the integral: \[ \int_{0}^{2\pi} e^{\sin t} \, dt \]

Answer: 2\pi I_0(1)


Question:
Given a non-negative sequence \((a_n)\), determine whether the following statement is true or false:
\[ \varlimsup_{n\to\infty} (a_n^2) = (\varlimsup_{n\to\infty} a_n)^2. \]

Answer: True


Question:
Suppose $f$ and $g$ are non-negative functions from a Hausdorff topological space $X$ to $[0,\infty]$. Let $x_n$ be a sequence of minimizers of the problem
\[ x^n \triangleq \operatorname{arginf}_{x \in X} f(x) + ng(x). \]
If
\[ \lim_{n \to \infty} f(x^n) + ng(x^n)<\infty, \]
does this necessarily imply that
\[ \lim_{n \to \infty}g(x^n)=0? \]

Answer: Yes


Question: Given the sine function $y = \sin x$ with an arc length $L$ from $0$ to $2\pi s$, where $s \in \mathbb{Q}$ and $s > 1$, the period is changed from $2\pi$ to $\frac{4\pi}{3}$. Determine the new amplitude $|A|$ of the sine function such that the arc length from $0$ to $2\pi s$ remains $L$. Use the general form of the sine wave equation: $f(x) = y_{max} \sin \left(\frac{2\pi x}{\lambda}\right)$, and the arc length formula: $L = \int_0^{2\pi s} \sqrt{1 + y_{max}^2\frac{4\pi^2}{\lambda^2} \cos^2 \left(\frac{2\pi x}{\lambda}\right)} \, dx$. Find the value of $y_{max}$.

Answer: \dfrac{2}{3}


Question: Given that $p = 33179$ is a prime number and $2^{2p+1} \equiv 2 \pmod{2p+1}$, deduce whether $2p+1$ is a prime number.

Answer: Yes


Question: Solve for $x$ in the equation: \[ \log_{\sqrt{x}} 2 + \log_6 x^x = 4 \] without using software.

Answer: 4


Question:
Consider the Dirichlet problem for the open unit disk \( \omega = \{(x,y) : x^2 + y^2 < 1\} \) in \( \mathbb{R}^2 \) with boundary condition \( u(x,y) = 1 - 2y^2 \) on \( \delta\omega \). If \( u(x,y) \) is the solution to the problem:
\[
\begin{cases}
  u_{xx} + u_{yy} = 0 & \text{in} \ \omega \\
  u(x,y) = 1 - 2y^2 & \text{on the boundary}.
\end{cases}
\]
Find \( u(1/2, 0) \).

A) \(-1\)

B) \(-1/4\)

C) \(1\)

D) \(1/4\)

Answer: D


Question: Find all integer values of \( n \) such that the expression \( n^4 + 6n^3 + 11n^2 + 3n + 31 \) is a perfect square.

Answer: 10


Question: Find all functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that for all positive integers \( x \) and \( y \), the equation \( yf(x) + xf(y) = (x+y)f(x^2+y^2) \) holds.

Answer: f(x) = c


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left(n^3 \int_{n} ^{2n}\frac{x \, dx} {1+x^5}\right) \]

Answer: \dfrac{7}{24}


Question: Determine the number of equivalence classes under logical equivalence for the set of all formulas that can be constructed using the atoms \( \{A_1, A_2, \ldots, A_n\} \).

Answer: 2^{2^n}


Question: Find all prime numbers of the form \(2^{2^n} + 5\) for nonnegative integers \(n\).

Answer: 7


Question: Let $T(u,v) = (u^2 - v^2, 2uv)$. Define $D^*$ as the set of $(u,v)$ such that $u^2 + v^2 \leq 1$, $u \geq 0$, and $v \geq 0$. Find the image $T(D^*) = D$. Evaluate the integral $\iint_D dx \, dy$. Compute the value of this integral.

Answer: \dfrac{\pi}{2}


Question: Determine the values of $p$ and $q$ for which the series \( \sum_{n=2}^{\infty}\frac{1}{ n^p-n^q} \) converges, given that \( 0<q<p \).

Answer: p > 1


Question: Evaluate the double integral \( \iint_D (x-y)e^{x^2 + y^2} \, dy \, dx \), where \( D \) is the region in the xy-plane bounded by the coordinate axes and the line \( x+y=1 \).

Answer: 0


Question:
Determine in which of the following cases there is no continuous function $g$ from the set $A$ onto the set $B$:

1. $A=[0,1]$, $B=\mathbb{R}$
2. $A=(0,1)$, $B=\mathbb{R}$
3. $A=(0,1)$, $B=(0,1]$
4. $A=\mathbb{R}$, $B=(0,1)$

Select the correct option.

Answer: 1


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} \sum_{k = 0}^n \frac{e^{-n} n^k}{k!} \]

Answer: \dfrac{1}{2}


Question: Find the limit as \( x \) approaches zero of the expression \( \left(x+e^{2x}\right)^{\frac{3}{x}} \).

Answer: e^9