Question: Find the maximum integer \( n \) such that there exists a real number \( x \) satisfying \( \lfloor x^i \rfloor = i \) for all integers \( i = 1, 2, 3, \ldots, n. \)

Answer: 4


Question: Let $f:[a,b] \to \mathbb{R}$ be a continuous function such that $\int_c^d f(x) \, dx = 0$ for all $c, d \in [a,b]$ with $c < d$. Is it true that $f(x) = 0$ for all $x \in [a,b]$?

Answer: 0


Question: Consider a finite abelian group $G$ and numbers $a$ and $b$ such that $ab \ge |G|$. Is it true that there exist subsets $A, B \subset G$ with $|A| \le a$ and $|B| \le b$ such that $A + B = G$? Justify your answer.

Answer: Yes


Question: Consider a bounded sequence \((u_n)_n\) in the Sobolev space \(H^1(\Omega)\), where \(\Omega\) is a bounded subset of \(\mathbb{R}^d\). Is the sequence \((u_n)_n\) tight in \(L^2(\Omega)\)? Specifically, for every \(\varepsilon > 0\), does there exist a compact set \(K_\varepsilon \subset \Omega\) such that \[ \sup_{n}\int_{\Omega\setminus K_\varepsilon }|u_n(x)|^2dx<\varepsilon? \]

Answer: Yes


Question: Find all positive integer values of \( n \) such that the equation \( \frac{1}{x} + \frac{1}{y} = \frac{1}{n} \) has exactly 5 solutions in ordered pairs of positive integers \( (x, y) \).

Answer: p^2


Question: Find integers \((n, m)\) such that \(1 \leq n, m \leq 20\) and the expression \(\left|\frac{2^n}{3^m} - 1\right|\) is minimized.

Answer: (19, 12)


Question: Find all integer solutions for \(x\) such that \(x^2 + 5145x + 2332 \equiv 0 \pmod{9797}\).

Answer: 2326


Question:
Determine the maximal cardinality of a collection \( \mathcal{C} \) of projective planes on \( \omega \) such that no two distinct members of \( \mathcal{C} \) are isomorphic. A set \( L \subseteq \mathcal{P}(X) \) is a projective plane on \( X \neq \emptyset \) if:

1. For any distinct \( x, y \in X \), there is a unique \( l \in L \) such that \( x, y \in l \).
2. For any distinct \( l, m \in L \), \( |l \cap m| = 1 \).
3. There exist four distinct elements of \( X \) such that no member of \( L \) contains more than two of these four elements.

Two projective planes \( L \) and \( M \) on \( X \) are isomorphic if there is a bijection \( \varphi: X \to X \) such that \( l \in L \) if and only if \( \varphi(l) \in M \).

Answer: 2^{\aleph_0}


Question: Let $f:X \to Y$ be a diffeomorphism between smooth complex compact manifolds, and let $\omega$ be a differential form on $Y$. Is it true that the support of $f^*\omega$ is equal to $f^{-1}$ of the support of $\omega$? Answer with 'Yes' or 'No'.

Answer: Yes


Question: Consider the direct sum of left $R$-modules $A \oplus B$. Is it possible to have a non-zero submodule $X$ of $A \oplus B$ such that $X \cap A = 0$ and $X \cap B = 0$? Justify your answer.

Answer: Yes


Question: Consider the exact sequence of modules: \(0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0\), with maps \(f: A \rightarrow B\) and \(g: B \rightarrow C\). Let \(Q\) and \(P\) be submodules of \(B\). Determine whether the following statement is true: If \(g(Q) = g(P)\) and \(f^{-1}(Q) = f^{-1}(P)\), then \(P = Q\).

Answer: False


Question: Let $R$ be a commutative ring with $1$. Suppose that $I$ is an ideal of $R$ which is maximal with respect to the property that it is proper and not prime. Prove that $I$ is contained in at most two other proper ideals of $R$.

Answer: 2


Question: Calculate \( \lim_{x\to\infty}\left(x e^{\frac{1}{x}} - \sqrt{x^2+x+1} \right) \).

Answer: \dfrac{1}{2}


Question: Let $V$ be a vector space, and let $f_1, \dots, f_{d+1}$ be linearly independent homogeneous polynomials of degree $d$ on $V$. Is there a two-dimensional subspace $U$ of $V$ such that the restrictions $f_1|_U , \dots , f_{d+1}|_U$ remain linearly independent?

Answer: Yes


Question: Find the maximum value of the expression \[ \frac{1}{a^3+2b^3+6} + \frac{1}{b^3+2c^3+6} + \frac{1}{c^3+2a^3+6} \] given that \( a, b, c > 0 \) and \( abc = 1 \).

Answer: \dfrac{1}{3}


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

Answer: 2


Question: Given two integers \( x \) and \( y \), compute the lowest 8 bits of the integer division \( \left\lfloor \frac{x}{y} \right\rfloor \). Specifically, find \( r = \left( \frac{x}{y} \right) \mod 256 \). For example, if \( x = 18423 \) and \( y = 29 \), then \( r = 123 \).

Answer: 123


Question: Find the function $u(x, y)$ such that $f(z) = u(x, y) + j(8xy + 4y)$ is analytic for $z = x + jy$ and satisfies $f(0) = 0$. Then express $f(z)$ as a function of $z$.

Answer: 4z^2 + 4z


Question: Evaluate the limit: \[ \lim_{n\to\infty} n\int_{0}^{\pi/4} \tan^n(x) \,{\rm d}x \]

Answer: \dfrac{1}{2}


Question: Find the degree of the splitting field $F$ over $\mathbb{Q}$ for the polynomial $f(x) = x^3 + x^2 + 1$. Determine $[F: \mathbb{Q}]$. Assume $f$ has no rational roots.

Answer: 6


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

Answer: \dfrac{4}{3}


Question: Find the torsion of the module \( \mathbb{Z}^2/M \) over \( \mathbb{Z} \), where \( M \) is the subgroup of \( \mathbb{Z}^2 \) generated by the vector (4,6).

Answer: \mathbb{Z}/2\mathbb{Z}


Question: Determine the radius of convergence of the series \( \sum_{n = 1}^{\infty} \frac{x^n}{n^{\sqrt{n}}} \).

Answer: 1


Question: Find the limit of the product \( \prod_{k=1}^n\left(\frac{2k-1}{2k}\right) \) as \( n \to \infty \).

Answer: 0


Question: Consider the sequence defined by \(a_1 = 1, a_2 = \frac{1}{2}\), and \(a_{n+2} = a_{n+1} - \frac{a_n a_{n+1}}{2}\) for \(n\) a positive integer. Determine \(\lim_{n\to\infty} n a_n\) if it exists.

Answer: 2


Question: Evaluate the infinite series: \[ \sum_{n=1}^{\infty} \left( e - \left(1 + \frac{1}{n}\right)^n \right) \]

Answer: \infty


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

Answer: 61


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

Answer: 1


Question: A has 2 dollars and B has 3 dollars. They toss a coin with the probability of heads being \( \frac{1}{3} \). If the coin lands on heads, A gives 1 dollar to B. If it lands on tails, B gives 1 dollar to A. What is the probability that B will win all the money? Provide your answer as a fraction.

Answer: \dfrac{7}{31}


Question: Let $p$ be a prime number, and let $G$ be a finite group whose order is a power of $p$. Let $F$ be a field of characteristic $p$, and $V$ a nonzero vector space over $F$ equipped with a linear action of $G$. Does there exist a nonzero subspace $W \subset V$ such that $G$ acts trivially on $W$?

Answer: Yes


Question: Calculate the complex line integral of the function \( g(z) = \frac{1}{z} \) along the counterclockwise-oriented square with side length 2, centered at the origin, and with sides parallel to the axes.

Answer: 2\pi i


Question: Determine whether there exists a sufficiently large natural number $n$ such that the inequality $n! > A \cdot B^{2n+1}$ holds, given constants $A, B > 0$. Provide a justification for your answer.

Answer: Yes


Question: Find the area enclosed by the ellipse given by the equation \((x+y)^2+(x+3y)^2=1\).

Answer: \dfrac{\pi}{2}


Question: Determine the signature (n+, n-, no) of the symmetric bilinear form \( s(f,g) = \int_{-1}^{1} f(x)g(x)(35x^4-30x^2+3)dx \) on the real vector space \( V = \mathbb{R}[X]_{\leq3} \) of polynomials of degree \( \leq 3 \).

Answer: (2, 1, 1)


Question: Given the equation \( e^{-x} \cdot f(x) = 2 + \int_{0}^{1}\sqrt{1+t^4}dt \) for all \( x \in (-1,1) \), find the value of \((f^{-1})'(2)\).

Answer: \dfrac{1}{2}


Question: Evaluate the improper integral \( \int_{-1}^0 \frac{e^{1/x}}{x^2} \, dx \). Determine whether the integral converges or diverges.

Answer: \dfrac{1}{e}


Question: Suppose $f(x,y)$ is a 2-ary partial recursive function, and the sequence $f_0,f_1,f_2,\ldots$ (where $f_x=\lambda y\centerdot f(x,y)$) includes only finitely many distinct partial recursive functions. Is there a recursive function $r$ such that for all $x$ and $y$, $\phi_{r(x)}(y)=f(x,y)$ and the range of $r$ is finite?

Answer: Yes


Question: Let \(K\) be a field with characteristic \(charK\) such that \(charK \nmid n\), where \(n\) is a natural number with \(n \geq 2\). Consider the polynomial \(g = X^n - 1 \in K[X]\). Do there always exist roots (other than 1) in an algebraic closure of \(K\) for \(X^n - 1\)?

Answer: Yes


Question: Let \( A = \{1, 2, 3, \ldots, 12\} \). Determine the number of functions \( g: A \to A \) such that \( g(g(x)) = x \) and \( g(x) - x \) is not divisible by 3.

Answer: 1728


Question: Calculate the limit of the function as $x$ approaches 0: $$f(x) = \left(\frac{e^x-1}{x}\right)^{1/x}$$

Answer: \sqrt{e}


Question: Evaluate the integral \( \int_{|z-1|= \frac{1}{2}} \frac{1}{(1-z)^3} \, dz \).

Answer: 0


Question:
Find a number \( a \) that satisfies the following conditions:

1. \( a \equiv 7^{3003} \pmod{3750} \)
2. \( 1 \leq a \leq 5000 \)
3. \( a \) is not divisible by 7.

Answer: 4093


Question:
Find the exact x-coordinates of the points on the graph of \( f(x) \) for which the tangent line is parallel to the line \( g(x) \) over the interval \( \frac{\pi}{2} \le x \le \pi \). Given:

\[ f'(x) = \frac{2\cos(2x)}{3\sin(2x)^{\frac{2}{3}}} \]

and the line \( g(x) \) is defined by the equation:

\[ 2x - 3(6)^{\frac{1}{3}}y = 0 \]

Solve the equation:

\[ \frac{\cos(2x)}{\sin(2x)^{\frac{2}{3}}} - \frac{1}{6^{\frac{1}{3}}} = 0 \]

for \( x \) within the specified interval.

Answer: \dfrac{5\pi}{6}


Question: Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed random variables with probability density function $f_X(x) = \frac{1}{x^2}$ for $x \geq 1$ and $f_X(x) = 0$ otherwise. Find the expected value $\mathbb{E}[Y]$, where $Y = \sum_{i=1}^{n} \log X_i$. Provide your answer in terms of $n$. \( \boxed{} \)

Answer: n


Question: Evaluate the infinite series: \[ \sum_{n=0}^\infty \log\left(1+\frac{1}{2^{3^n}}+\frac{1}{2^{2\times{3^n}}}\right) \]

Answer: \log 2


Question: In an M/G/1 queue with Poisson arrivals at rate \( \lambda = 1 \) and service time being the sum of two exponentially distributed variables with rates \( \mu_1 = 1 \) and \( \mu_2 = 2 \), calculate the probability that the waiting time \( W_q \) for an average customer is zero, i.e., find \( P(W_q = 0) \).

Answer: 0


Question: Determine whether a closed compact 2-manifold embedded in Euclidean 3-space is always orientable. Provide a true or false answer.

Answer: True


Question: Evaluate the limit without using L'Hospital's rule, derivatives, or series expansion: \[ \lim_{x \to 0}{\frac{x\sin{x}+2(\cos{x}-1)}{x^3}} \]

Answer: 0


Question: Evaluate the integral \( \int_0^{2 \pi} \ln(1+e^{i \theta}) \, d\theta \).

Answer: 0


Question: Consider the set of numbers \( \{1, 1, 2, 2, 3, 3, 4, 4\} \). How many permutations of this set are there such that no two identical numbers are immediately adjacent?

Answer: 864


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

Answer: \dfrac{\pi}{2}


Question: Find the exact sum of the series \( \sum_{n=1}^{\infty}\frac{(-1)^n n^2}{3^n} \).

Answer: -\dfrac{3}{32}


Question: Given that $X \sim U(0,1)$ and $Y|X=x \sim \text{Bin}(10,x)$, calculate the conditional variance $\text{Var}(X|Y=3)$. Provide your answer as a single value.

Answer: \dfrac{2}{117}


Question: Find the two rightmost digits of \(3^{40000005}\) using the Euler Totient function.

Answer: 43


Question: Evaluate the triple integral \( \iiint_D x^2yz \,dx\,dy\,dz \) over the region \( D = \{(x,y,z):0 \leq x \leq y+z \leq z \leq 1\} \).

Answer: -\dfrac{1}{420}


Question: Determine the dimension of the space of planes in \(\Bbb R^3\).

Answer: 3


Question: Consider a sequence \( \{x_n\} \) in \( \mathbb{N} \) with \( x_1 = 1 \) such that for any prime \( p \), the set \( \{x_1, x_2, \ldots, x_p\} \) forms a complete residue system modulo \( p \). Determine whether \( \lim_{n \to \infty} \frac{x_n}{n} \) exists, and if it does, find its value.

Answer: 1


Question: Evaluate the limit using polar coordinates: \[ \lim_{r \to 0} \frac{1}{e^{\left(\frac{1}{r^2}\right)} \sin(r)} \]

Answer: 0


Question: Given that the polynomial \(P(x) = x^{2a+b-1} + x^{a-2b+5} - 2x^{a+b-1}\) is divisible by \((x-2)\), find the product \(ab\).

Answer: 0


Question:
Given that $T \sim t_7$, find the value of $t$ such that:
\[ P(-t < T) + P(0 < T) + P(t < T) + P(2t < T) = 1.75. \]

Answer: 0.356


Question: Find the integer $k$ such that $a_k = 2^{261}$, where $a_1 = 2$ and for all integers $k \geq 2$, \[ a_k = \frac{2^{(k+1)(k+2)/2}}{\prod_{j=1}^{k-1} a_j}. \]

Answer: 260


Question: Let $k$ be a finite extension of $\mathbb{Q}$ with an absolute value $|\cdot|$ (either Archimedean or non-Archimedean). Let $L$ be the completion of $k$ with respect to this absolute value. Consider an irreducible polynomial $f(X)$ in $k[X]$. Is it true that $f(X)$ remains separable in $L[X]$, meaning every irreducible factor of $f(X)$ in $L[X]$ has multiplicity 1? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the line integral \( \int_K \frac{y}{x^2+y^2} \, dx - \frac{x}{x^2+y^2} \, dy \), where \( K \) is the positively oriented unit circle defined by \( x^2 + y^2 = 1 \).

Answer: -2\pi


Question: Evaluate the limit \( \lim_{x \to 0} \frac{f(x) - \sqrt{x+9}}{x} \) given that \(|f(x) - 3| \le x^2\).

Answer: -\dfrac{1}{6}


Question: Evaluate the integral \( \int_{-\infty}^\infty (f(x+1) - f(x)) \, dx \) given that \( \lim_{x \to \infty} f(x) = 2 \) and \( \lim_{x \to -\infty} f(x) = 1 \).

Answer: 1


Question: Let \(\Delta \subset \mathbb{C}\) be a triangle, and let \(f: \Delta \rightarrow \mathbb{C}\) be a continuous function that is holomorphic in the interior of \(\Delta\). Show that \(\int _{\partial \Delta} f = 0\).

Answer: 0


Question: Determine the number of homomorphisms from the symmetric group $S_5$ to the cyclic group $C_6$.

Answer: 2


Question: Determine whether the following statement is true: \( \bigcap_{m=1}^\infty W_m = \{0\} \), where \( W_m = V_m + U_m \), \( V_m = \overline{\operatorname{span}} \{v_n\}_{n \geq m} \), and \( U_m = \overline{\operatorname{span}} \{u_n\}_{n \geq m} \), with \( \{v_n\}_{n \in \mathbb{N}} \) and \( \{u_n\}_{n \in \mathbb{N}} \) being basic sequences in \( \ell^2 \) over \( \mathbb{C} \).

Answer: True


Question: Consider a classical knot diagram $K$ that is colored by a finite quandle $X$ such that only one element in the quandle is used to color the knot $K$. This coloring is termed trivial. Determine if the following statement is true: If the knot admits only trivial coloring by any finite quandle, then the fundamental group of the knot is isomorphic to $\mathbb{Z}$. Provide a justification for your answer.

Answer: True


Question: Find the limit: \[ \lim_{n \to \infty} \frac{n^n}{e^n n!} \]

Answer: 0


Question: In an infinite sequence of tosses of a fair coin, what is the expected number of tosses required for the pattern HTTH to appear?

Answer: 18


Question: Solve the equation \((x!)! + x! + x = x^{x!}\) for the integer value of \(x\).

Answer: 3


Question: Evaluate the limit: $$\lim_{n \rightarrow +\infty} \int_{0}^{2} \arctan \left(\frac{1}{1+x^n}\right) dx.$$

Answer: \dfrac{\pi}{4}


Question: Suppose \( A \) and \( B \) are two square matrices such that \( e^{At} = e^{Bt} \) for all \( t \in (0, T) \) where \( T > 0 \) or \( T = +\infty \). Does this imply that \( A = B \)?

Answer: A = B


Question: Evaluate the integral: $$\int _{\pi/6} ^{\pi /3} \frac {\sqrt[3]{\sin x}}{\sqrt [3]{\sin x}  + \sqrt[3]{\cos x}}\,dx$$ and find its closed form.

Answer: \dfrac{\pi}{12}


Question: A bakery sells rolls in units of a dozen. The demand for rolls, in thousands of units, follows a gamma distribution with parameters \(\alpha=3\) and \(\theta=0.5\). It costs \$2 to produce a unit of rolls, which sells for \$5 on the first day when fresh. Any leftover units are sold for \$1 on the second day. Determine the number of units that should be produced to maximize the expected profit.

Answer: 1960


Question: Let $\{N(t): t \geq 0\}$ be a Poisson process with rate $1$, and let $T_1 < T_2 < \cdots$ denote the arrival times. Determine the probability density function of the random variable $Y = \frac{T_1}{T_3}$. Provide your answer in terms of $y$.

Answer: 2(1 - y)


Question: Let $z$ and $a$ be complex numbers such that $|z|=1$ and $|a|<1$. Prove that \( \left|\frac{z-a}{1-\bar{a}z}\right|=1 \).

Answer: 1


Question: Suppose $(f_n)_{n\in\mathbb{N}}$ is a sequence of continuous real-valued functions on $[0,1]$, and the sequence of partial sums $S_m(t)=\sum_{n=1}^m f_n(t)$ converges in $\mathrm{L}^2([0,1])$ and is equicontinuous. Does this imply the uniform convergence of $(S_m)_{m\in\mathbb{N}}$ on $[0,1]$?

Answer: Yes


Question: Find the base $b$ such that the number $(374)_b$ is a perfect square.

Answer: 15


Question: Suppose $g$ is the density of the standard normal distribution $N(0,1)$, and $f: \mathbb{R} \to \mathbb{R}$ is an arbitrary function. If the convolution $[f \ast g](x) = \int_{-\infty}^{\infty} f(x-t)g(t)dt = 0$ for all $x$ on a closed interval $[a,b]$ where $-\infty < a < b < +\infty$, is it true that $f = 0$ almost everywhere?

Answer: Yes


Question: Let $f_n: [0, 1] \to \mathbb{R}$ be a sequence of functions that are equibounded in $L^\infty$, meaning $\sup_{n \in \mathbb{N}} \lVert f_n \rVert_{L^\infty} < \infty$. Suppose there exists a subset $E$ of $[0, 1]$ with measure 1 such that $f_n$ are equi-Lebesgue continuous on $E$. Does there exist a subsequence $f_{n_k}$ of $f_n$ that converges almost everywhere on $[0, 1]$?

Answer: Yes


Question: Let a group \( P \) of order 81 act on a set \( \Omega \) of order 98. Define \( \Omega_0 \) as the set of elements in \( \Omega \) that are fixed by every element of \( P \), i.e., \( \Omega_0 = \{ \alpha \in \Omega \mid \alpha \cdot g = \alpha \text{ for all } g \in P \} \). Prove that \(|\Omega_0| = 3k + 2\) for some integer \( k \) with \( 0 \leq k \leq 32 \).

Answer: |\Omega_0| = 3k + 2


Question: Use Stokes' theorem to evaluate the line integral \( \int_C \textbf{F} \cdot d\textbf{r} \) for the vector field \( \textbf{F}(x,y,z) = 2y\textbf{i} + 3z\textbf{j} + x\textbf{k} \), where \( C \) is the triangle with vertices \((2,0,0)\), \((0,2,0)\), and \((0,0,2)\).

Answer: -12


Question: Evaluate the limit \( L = \lim_{x \to 0} \frac{2 + \tan(e^x - \cos x) - e^x - \cosh x}{x(\sqrt{1+2x} - \sqrt[3]{1+3x})} \).

Answer: \dfrac{2}{3}


Question: Given the equation \( \int_0^1 (1+\cos^8 x)(ax^2+bx+c) \, dx = \int_0^2 (1+\cos^8 x)(ax^2+bx+c) \, dx \), determine the interval in which the root of the quadratic equation \( ax^2 + bx + c = 0 \) lies, where \( a, b, \) and \( c \) are non-zero numbers.

Answer: (1, 2)


Question: Find all complex numbers $z$ such that $iz + \sqrt{1-z^2} \le 0$, where the square root is defined as follows: for $z = re^{i\theta}$ with $-\pi < \theta \le \pi$, $\sqrt{z} = \sqrt{r}e^{\frac{1}{2}i\theta}$ and $\sqrt{r} \ge 0$. Determine the set of all such $z \in \mathbb{C}$.

Answer: \emptyset


Question: Consider a non-empty compact convex subset $K$ of a Banach space $E$, and a continuous function $f : K \to K$. Given $u_0 \in K$, define the sequence $(u_n)$ by $u_{n+1} = \frac{1}{n+1} \sum_{j=0}^n f(u_j)$. Is it always true that $(u_n)$ converges to a fixed point of $f$?

Answer: Yes


Question:
In an open field, you alternate flipping two fair coins and taking steps based on the following rules:

- If both coins are heads, take one step north.
- If both coins are tails, take one step south.
- If the first coin is heads and the second coin is tails, take one step west.
- If the first coin is tails and the second coin is heads, take one step east.

What is the probability that after 20 steps you are back to where you started, expressed as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are relatively prime integers? Find \( p+q \).

Answer: 70852900457


Question: Find a parametrization or explicit form of the curve resulting from the intersection of the sphere \(x^2 + y^2 + z^2 = 1\) and the plane \(x + y + z = 1\). Use this to compute the integral \(\iint_{S} dS\), where \(S\) is the surface of the circle formed by the intersection.

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


Question: Suppose two vertex-deleted subgraphs of an acyclic digraph G are isomorphic, where the deleted vertices v and w are both sinks. Determine if there is an automorphism of G that maps vertex v to vertex w.

Answer: Yes


Question: In how many ways can you collect a total of $20 from 4 different children and 3 different adults, if each child can contribute up to $6 and each adult can give up to $10, with each individual giving a non-negative whole number of dollars?

Answer: 112567


Question: Determine whether there exists a function defined on all of \(\mathbb{R}\) that is infinitely differentiable and satisfies \(f'''(x) = f(x)\) for all \(x\), but is not of the form \(f(x) = ce^x\) for a real constant \(c\).

Answer: Yes


Question: Is there an infinite set \( S \) such that there is no bijection between \( S \) and either \( \mathbb{R} \) or \( \mathbb{Z} \)?

Answer: Yes


Question:
Suppose $x:[0,\infty) \to [0,\infty)$ is a continuous function with $x(0)=0$. If for all $t \geq 0$, the inequality \((x(t))^2 \leq 2+\int_{0}^{t}x(s)\,ds\) holds, determine which of the following is true about $x(\sqrt{2})$:

(a) $x(\sqrt{2}) \in [0,2]$

(b) $x(\sqrt{2}) \in [0,\frac{3}{\sqrt{2}}]$

(c) $x(\sqrt{2}) \in [\frac{5}{\sqrt{2}},\frac{7}{\sqrt{2}}]$

(d) $x(\sqrt{2}) \in [10,\infty)$

Choose the correct option.

Answer: B


Question: Given the cumulative distribution function \( F(x,y) = \frac{(x \cdot y)^2}{4} \) for \( 0 < x < 2 \) and \( 0 < y < 1 \), find the probability \( P(X+Y < 1) \).

Answer: \dfrac{1}{24}


Question: In a truel involving three players, A, B, and C, each player takes turns shooting at one of the others. Player A hits with probability \( \frac{1}{3} \), player B hits with probability 1, and player C hits with probability \( \frac{1}{2} \). The order of shooting is A, B, C, and then repeats. The game continues until only one player remains. Each player aims to maximize their own survival probability, knowing that the others will also act optimally. If all players are alive when B takes their first shot, B will choose to shoot at C. Given this, who should player A shoot at first to maximize their probability of survival, and what is that probability?

Answer: \dfrac{11}{36}


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be defined by \( f(x) = x^3 + x - 1 \). Determine the number of solutions to the equation \( f(x) = f^{-1}(x) \).

Answer: 1


Question: Let \( A \) be an \( n \times n \) matrix such that \( A^2 - 2A + I = 0 \). Prove that 1 is an eigenvalue of \( A \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{h \to 0^+} \frac{1}{h} \left( \int_{0}^{\pi} \sin^{h} x \, \mathrm{d}x - \pi \right) \]

Answer: -\pi \ln 2


Question: Consider a continuous function $f : [0,1] \to \mathbb{R}$ that has no zeros on the interval $[0,1]$. Determine whether there exists a sequence of polynomials $(p_n)$, none of which have zeros on $[0,1]$, such that the sequence uniformly converges to $f$ on $[0,1]$. Provide a justification for your answer.

Answer: Yes


Question: Let $(x_n)$ be a sequence that converges to $x$ in a metric space $X$. Is it true that for any $n$, the inequality $d(x_n,x_{n+1}) + d(x_{n+1},x_{n+2}) + \dots \geq d(x_n,x)$ holds?

Answer: Yes


Question: Determine whether the following manipulation of the integral is valid: $$\int_2^4 \frac{1}{\sqrt{\frac{\ln(3+x)}{\ln(9-x)}} +1}dx = 1$$. Consider the transformation of the integrand when $x$ goes from $2$ to $4$, and analyze the effect of reversing the interval of $9-x$. Is the resulting integral equal to 1?

Answer: 1


Question: Given that the moment generating function of a random variable \(X\) is \(M_X(t) = e^{2e^t - 2}\) and the moment generating function of a random variable \(Y\) is \(M_Y(t) = (0.2e^t + 0.8)^7\), and assuming \(X\) and \(Y\) are independent, compute \(\Pr(X + Y = 3)\).

Answer: 0.2272


Question: Evaluate the complex integral \( \int_{0}^{2\pi} e^{e^{it}} \, dt \) along the unit circle in the complex plane.

Answer: 2\pi


Question: Determine the growth rate of the longest worm-like path in a random configuration on the lattice $[1,3n]^2$ in $\mathbb{Z}^2$, where each lattice point with coordinates $\equiv 2 \pmod{3}$ is randomly assigned one of six patterns. Express the growth rate with respect to $n$.

Answer: \Theta(n)


Question: In a category \( \mathcal{A} \) that has binary products, is it possible for there to exist an extremal epimorphism that is not an epimorphism? Assume that \( \mathcal{A} \) does not have all pullbacks.

Answer: Yes


Question:
Determine whether the following system of congruences has a solution:
\[ x \equiv 9 \pmod{11} \]
\[ x \equiv 6 \pmod{13} \]
\[ x \equiv 6 \pmod{12} \]
\[ x \equiv 9 \pmod{15} \]

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \to \infty} 2^{2n+3}\left(\sqrt[3]{8^n+3}-\sqrt[3]{8^n-3}\right) \]

Answer: 16


Question: Calculate the surface integral \( \iint_R (x,y,z) \cdot n \, \mathrm{d}s \) where the region \( R \) is bounded by the cylinder \( x^2+y^2=1 \), the plane \( z=0 \), and the plane \( x+y+z=2 \). Here, \( n \) is the normal vector to the surface.

Answer: 6\pi


Question: Given two jointly normal random variables $X$ and $Y$ with means of zero, variances $\sigma^2_{X}$ and $\sigma^2_{Y}$, and covariance $\sigma_{XY}$, calculate the expected value of $Z = X \cdot Y^2$. Find $E(Z)$. \(\text{Express your answer in terms of } \sigma^2_{X}, \sigma^2_{Y}, \text{ and } \sigma_{XY}.\)

Answer: 0


Question: Let $f, g: B \rightarrow A$ be ring homomorphisms where $A$ is a reduced ring. Assume that for every prime ideal $\mathfrak{p} \subset A$, the compositions $i_{\mathfrak{p}} \circ f$ and $i_{\mathfrak{p}} \circ g$ are equal, where $i_{\mathfrak{p}}: A \rightarrow A_{\mathfrak{p}} \rightarrow k(\mathfrak{p})$ and $k(\mathfrak{p}) = A_{\mathfrak{p}} / \mathfrak{p} A_{\mathfrak{p}}$. Is it true that $f = g$?

Answer: Yes


Question: Given that \(Z_1, Z_2, \ldots, Z_n\) are independent identically distributed standard Gaussian random variables with mean 0 and variance 1, find the distribution of \[ X = \frac{(Z_1 + Z_2)^2}{(Z_1 - Z_2)^2} \]

Answer: F(1,1)


Question: Transform the quadratic form \( F(x) = x_1^2 - 3x_3^2 - 2x_1x_2 + 2x_1x_3 - 6x_2x_3 \) into its normal form over the real number field.

Answer: y_1^2 - y_2^2


Question: Determine the largest possible order of an element in the symmetric group $S_{10}$. Provide your answer as a single integer.

Answer: 30


Question: Given the matrix \(A = \begin{bmatrix}1 & a & a^2 & a^3 & a^4 \\ 0 & a & a^2 & a^3 & a^4 \\ 0 & 0 & a^2 & a^3 & a^4 \\ 0 & 0 & 0 & a^3 & a^4 \\ 0 & 0 & 0 & 0 & a^4 \end{bmatrix}\), where \(a = e^{\frac{2\pi i}{5}}\), find the trace of the matrix \(I + A + A^2\).

Answer: 5


Question: Find the smallest natural number \( N \) such that \( \frac{(N!)^2}{(2N - 1)!} \leq \frac{1}{25000} \).

Answer: 11


Question: Let $F(x)$ be a polynomial function of degree $2018$ with leading coefficient 1, such that $F(0) = 2017$, $F(1) = 2016$, $F(2) = 2015, \ldots$, $F(2017) = 0$. The value of $F(2018)$ is of the form $n! - a$, where $n, a \in \mathbb{N}$ and $a$ is the least possible value. Find $n + a$. \( \text{Provide your answer as a single integer.} \)

Answer: 2019


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

Answer: \dfrac{e}{2}


Question: Given a probability distribution $P_n$ over the non-negative integers with a fixed mean $\bar{n} = \sum_{n=0}^{\infty} n \, P_{n}$, determine the maximum possible value of the variance $\text{Var}[n] = \left(\sum_{n=0}^{\infty} n^2 \, P_{n}\right) - \bar{n}^2$ in terms of $\bar{n}$. Express the maximum variance as a function of $\bar{n}$.

Answer: \infty


Question: Consider a random variable $X \geq 0$ and a non-negative error term $\eta \geq 0$, where $X$ and $\eta$ are independent. We measure $X$ with a one-sided error, resulting in $\widetilde{X} = X - \eta$. Is the conditional expectation $E[X|\widetilde{X}=\widetilde{x}]$ increasing in $\widetilde{x}$?

Answer: Yes


Question: Find the probability that the sum of the squares of two independent standard normal random variables is less than 1. Specifically, calculate \( P(X^2 + Y^2 < 1) \) where \( X \) and \( Y \) are independent standard normal variables.

Answer: 1 - e^{-1/2}


Question: Find the constant $c$ that best approximates the function $X = 2 \cdot \mathbb{1}_{[0,1/3]} - \mathbb{1}_{[2/3,1]}$ in the $L^1$ norm.

Answer: 0


Question: Find the holomorphic function $f(z) = u(x,y) + iv(x,y)$ such that $xu + yv = (x^2 + y^2)e^x \cos y$.

Answer: f(z) = z e^z


Question: Compute the volume of the solid bounded by the cone \( z = 3\sqrt{x^2 + y^2} \), the plane \( z = 0 \), and the cylinder \( x^2 + (y-1)^2 = 1 \).

Answer: \dfrac{32}{3}


Question:
Let $V$ be a vector space with $\dim(V)=5$. Consider a linear operator $T: V \rightarrow V$ such that:

- $T$ is not invertible.
- $\dim(\operatorname{Ker}(T+4I))=2$.
- $\dim(\operatorname{Im}(T-I))=3$.

The associated matrix of $T$ in any basis of $V$ is denoted $A$. Determine the correct statement:

A) $A$ is diagonalizable and $\operatorname{tr}(A)=-3$.
B) $A$ is diagonalizable and $\operatorname{tr}(A)=-6$.
C) $A$ is diagonalizable and $\operatorname{tr}(A)=-5$.
D) $A$ is not diagonalizable and $\operatorname{tr}(A)=0$.

Select the correct option.

Answer: B


Question: Let $R = C[x_1, \ldots, x_n]$ and $M$ be a maximal ideal of $R$ such that $R/M$ is a finite-dimensional $C$-algebra. Is $R/M^n$ for $n > 1$ also a finite-dimensional $C$-algebra? Justify your answer.

Answer: Yes


Question: Determine if the number 404 is a palindrome when expressed in base negative 31.

Answer: Yes


Question: Given the limit equation \( \lim_{x\to 0^+} \ln(x \ln(a))\ln\left(\frac{\ln(ax)}{\ln(x/a)}\right) = 6 \), find the value of \( a \).

Answer: e^3


Question:
Let \( f \) be a holomorphic function on \( \mathbb{D} = \{z \in \mathbb{C} : |z| \leq 1\} \) such that \(|f(z)| \leq 1\). Define \( g: \mathbb{D} \to \mathbb{C} \) by
\[
g(z) =
\begin{cases}
\frac{f(z)}{z} & \text{if } z \neq 0 \\
f^{'}(0) & \text{if } z = 0 \\
\end{cases}.
\]
Determine whether \(|g(z)| \le 1\) for all \(z \in \mathbb{D}\).

Answer: No


Question: Given that \( \arccos(y_1) + \arccos(y_2) + \dots + \arccos(y_k) = k\pi \) for any value of \( k > 1 \), find the value of \( A = y_1^1 + y_2^2 + \dots + y_{2k}^{2k} \).

Answer: 0


Question: Consider the first order partial differential equation \( u_t + a u_x = 0 \) for \( x \in \mathbb{R}, t > 0 \), with initial data \( u(x,0) = u_0(x) \) where \( u_0 \) is a continuously differentiable function. Let \( a \) be a fixed real constant. Determine whether the following statement is true or false: If \( u_0 \) vanishes outside a compact set, then for each fixed \( T > 0 \), there exists a compact set \( K_T \subset \mathbb{R} \) such that \( u(x,T) \) vanishes for \( x \notin K_T \).

Answer: True


Question: Determine the value of \( C \) for which the integral \( I = \int_0^\infty \left( \frac{x}{x^2+1} - \frac{C}{3x+1} \right) \, dx \) converges.

Answer: 3


Question: Let $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ be a $2 \times 2$ matrix with the characteristic polynomial $(\lambda - 1)^2$. Compute $A^{2011} - 2011A$. Assume $A$ is not diagonalizable.

Answer: \begin{pmatrix} -2010 & 0 \\ 0 & -2010 \end{pmatrix}


Question: Let $X$ be a random variable with a distribution function such that $n^t P(|X|>n) \to 0$ as $n \to \infty$, for some $t>0$. For any $\epsilon>0$, does there exist $n_0\in \mathbb{N}$ such that \(P(|X|>x) < \dfrac{\epsilon}{x^t} \) for all \(x\in \mathbb{R}\), \(x\geq n_0\)?

Answer: Yes


Question: Determine the largest integer \( n \) such that \( 2^n \) divides \( 3^{1024} - 1 \).

Answer: 12


Question: Determine the location of the maximum of the function \( f(x) = \frac{x^3}{e^x-1} \) for \( x > 0 \). Provide your answer with two-digit precision.

Answer: 2.82


Question: Let $(X,Y)$ be a random vector uniformly distributed in the unit square $[0,1]^2$. Define the random variables $U = \min(X,Y)$ and $V = \max(X,Y)$. Calculate the covariance $\text{Cov}(U,V)$. Provide your answer as a single numerical value.

Answer: \dfrac{1}{36}


Question: Given a number $N$, count how many numbers from 1 to $N$ contain at least one prime digit (2, 3, 5, or 7). For example, if $N = 100$, the answer is 64.

Answer: 64


Question:
Let \( f: \mathbb{R} \to \mathbb{R} \) be an invertible function such that \( f(x) \) and \( f^{-1}(x) \) are symmetric about the line \( y = -x \). Determine which of the following statements is correct:

(a) \( f(x) \) is odd.
(b) \( f(x) \) and \( f^{-1}(x) \) may not be symmetric about \( y = x \).
(c) \( f(x) \) may not be odd.
(d) None of these.

Answer: a


Question: Evaluate the limit: \[ \lim_{x\to a} \frac{x^x-a^a}{x-a} \] without using L'Hospital's Rule.

Answer: a^a (1 + \ln a)


Question: Find the limit of the sequence \( I_n = \int_{0}^{\pi/2} (\tan x)^{1/n} \, dx \) as \( n \to \infty \).

Answer: \dfrac{\pi}{2}


Question: Determine the sample size \( n \) such that the proportion of smokers in the sample is within 0.01 of the true proportion \( p \) with a probability of 0.95.

Answer: 9604


Question: Suppose $Y_1$ and $Y_2$ are independent real-valued random variables with distributions that have a density with respect to the Lebesgue measure, and all moments $\mathbb{E}Y_1^a$ and $\mathbb{E}Y_2^b$ exist and are finite. Let $p(y_1, y_2)$ be a polynomial with a positive degree in $y_1$. Is it true that the random variables $p(Y_1, Y_2)$ and $Y_1$ cannot be independent?

Answer: Yes


Question: Given the polynomial \(P(x) = x^3 - 3x^2 - 7x - 1\) with roots \(a, b, c\), find the value of \(\left(\frac{1}{a-b} + \frac{1}{b-c} + \frac{1}{c-a}\right)^2\).

Answer: \dfrac{9}{13}


Question: Determine whether the following statement is true or false: In a finite group $G$, let $m_r$ be the number of elements of order $r$, and let $x$ be an element of order $r$. Then $m_r = \sum_{|x|=r} |x^G|$, where $|x|$ is the order of $x$ in $G$ and $x^G$ is the conjugacy class of $x$. Provide a justification for your answer.

Answer: False


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

Answer: \dfrac{\pi}{4}


Question: Find a formula for all even moments of the form \(E(X^{2n})\) for the random variable \(X\) with the probability density function \(f_X(x) = \frac{1}{2}e^{-|x|}\), where \(-\infty < x < \infty\). Use the moment-generating function \(M_X(t) = (1-t^2)^{-1}\) to derive the formula.

Answer: (2n)!


Question: An ellipse given by the equation \(x^2 + 4y^2 = 4\) is rotated anticlockwise through a right angle in its own plane about its center. Determine the value of \(\lambda + \mu\) if the locus of the point of intersection of a tangent to the ellipse in its original position with the tangent at the same point of the ellipse in its new position is described by the curve \((x^2 + y^2)^2 = \lambda(x^2 + y^2) + \mu xy\), where \(\mu\) and \(\lambda\) are positive integers.

Answer: 11


Question: An Abelian group $G$ is generated by elements $x$ and $y$, where $|x| = 16$, $|y| = 24$, and $x^2 = y^3$. Determine the order of the group $G$.

Answer: 48


Question: Given that $x, y, z > 0$ and $x + y + z = 1$, prove that: \[ \frac{xyz}{(1-x)(1-y)(1-z)} \le \frac{1}{8} \]

Answer: \frac{1}{8}


Question: Determine if the function \( Z(s) = \prod_{p}\frac{1}{1+p^{-s}} = \sum_{k=1}^{\infty}\frac{(-1)^{\Omega(k)}}{k^s} \), where \( \Omega(k) \) is the number of distinct prime factors of \( k \), can be analytically continued. Specifically, find the value of \( Z(0) \).

Answer: 1


Question: Calculate $E[nX(1)]$, where $X(1)$ is the smallest order statistic from a sample of $n$ iid random variables drawn from an exponential distribution with parameter $1/\theta$. The probability density function is given by $f(x) = \frac{1}{\theta}e^{-x/\theta}$ for $x > 0$. Determine the value of $E[nX(1)]$.

Answer: \theta


Question: Let \(f: \left[\frac{1}{a}, a\right] \to \left[\frac{1}{a}, a\right]\) be a bijective function such that \(f^{-1}(x) = \frac{1}{f(x)}\) for all \(x \in \left[\frac{1}{a}, a\right]\). Find \(f(1)\).

Answer: 1


Question: Determine the nth derivative of the function \( f(x) = \frac{1}{x}e^x - \frac{1}{x^2}e^x + \frac{1}{x^2} \) at \( x = 0 \). Express your answer in the form \( \frac{1}{a(n)} \), where \( a(n) \) is an expression in terms of \( n \).

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


Question: In a right triangle ABC with the right angle at A, the angle bisector BH of angle B is 9 units long, and the angle bisector CD of angle C is 8\(\sqrt{2}\) units long. Determine the length of the hypotenuse BC.

Answer: 6\sqrt{5}


Question: Fifty identical six-sided dice are rolled. How many distinct outcomes are possible if each number (1 through 6) appears an even number of times?

Answer: 142506


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

Answer: e


Question: Suppose we have a collection of $T_2$ topologies on an infinite set $X$. Is the intersection of these topologies also a $T_2$ topology?

Answer: No


Question: Determine the radius of convergence of the power series \( \sum_{n=1}^{\infty} \frac{n+1}{n!} z^{n^3} \).

Answer: 1


Question: Find the expected value of \(\ln(X)\) where \(X\) is a chi-squared random variable with 2 degrees of freedom.

Answer: \ln(2) - \gamma


Question: Determine all positive values of $a$ for which the improper integral \( \int_0^{\frac{\pi}{2}} \sec^a(x) \, dx \) converges.

Answer: (0, 1)


Question: Given the series of the inverse function $A(s) = \sum_{k>0}a_ks^k$ and the equation $A(s)+A(s)^3=s$, calculate the coefficient $a_5$.

Answer: 3


Question: Find the coefficient of \(x^3y^4z^5\) in the expansion of \((xy+yz+zx)^6\).

Answer: 60


Question: Calculate the line integral \( \oint_{\sigma} (2xydx + 3xy^2dy) \) using Green's Theorem, where \( \sigma \) is the boundary curve of the quadrilateral with vertices \((-2,1)\), \((-2,-3)\), \((1,0)\), and \((1,7)\), oriented positively with respect to the quadrilateral.

Answer: \dfrac{1329}{4}


Question: Determine whether the sequence \( \log(n) - \sum_{k=1}^{n-1} \frac{1}{k} \) is bounded.

Answer: Yes


Question: If $f \in \mathcal{R}[a,b]$ and $f = g$ almost everywhere on $[a,b]$, is it true that $g \in \mathcal{R}[a,b]$?

Answer: No


Question: Determine whether there exists a curve \( \gamma : [0,1] \longrightarrow [0,1]^2 \setminus K \) such that \( \gamma(0) = \{0,0\} \), \( \gamma(1) = \{1,1\} \), and the length of \( \gamma \), \( \ell(\gamma) \leq 2 \), given that \( \{0,0\}, \{1,1\} \notin K \subset [0,1]^2 \) and the projections of \( K \) onto the \( x \)-axis and the \( y \)-axis are 1-dimensional Lebesgue null-sets.

Answer: Yes


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a $C^1$ function. A point $c \in \mathbb{R}$ is called a mean value point of $f$ if there exists an open interval $(a,b)$ containing $c$ such that $f'(c) = \frac{f(b) - f(a)}{b-a}$. Is it true that (Lebesgue) almost every point in $\mathbb{R}$ is a mean value point of $f$?

Answer: Yes


Question: Is the Erdős–Szekeres theorem sharp? Specifically, is it possible to find a sequence of $n^2$ numbers that does not contain a monotonic subsequence of $n+1$ numbers for all $n \in \mathbb{N}$?

Answer: Yes


Question: Does there exist a function that is analytic in an open disc and has an absolute value of 1 on the boundary of the disc?

Answer: Yes


Question:
Find the expected number of times a coin is thrown until the appearance of a second "tail," given that the probability of a "tail" on one toss is $p$. The random variable $X$ represents the number of tosses until the second "tail" appears, with the probability distribution given by:
$$P\{X=k\}=\binom{k-1}{1}p^2(1-p)^{k-2}$$
Calculate the expectation of $X$.

Answer: \dfrac{2}{p}


Question: Evaluate the expression \( \left\lfloor\frac{\displaystyle\int_0^{\pi/2}\cos^{\sqrt{13}-1}x{\rm d}x}{\displaystyle\int_0^{\pi/2}\cos^{\sqrt{13}+1}x{\rm d}x}\right\rfloor \).

Answer: 1


Question: Determine the cardinality of the set of all well-formed formulas (WFF) in propositional logic, given the following grammar: \( P_n \) is a WFF for all \( n \in \mathbb{N} \); if \( \phi \) and \( \psi \) are WFFs, then \( (\phi \rightarrow \psi) \) is a WFF; if \( \phi \) is a WFF, then \( \neg \phi \) is a WFF. What is the cardinality of the set \( \{\phi: \phi \text{ is a WFF} \} \)?

Answer: \aleph_0


Question: Evaluate the limit: $$\lim_{n\to\infty}{\sum_{0 \leq k \leq n}} \binom{n}{k}\frac{1}{n^{k}(k+3)}.$$

Answer: e - 2


Question: Let \( A \in M_3(\mathbb{R}) \) be a matrix such that \( A^3 = I \) and \( A \neq I \). Determine whether the statement \( A^2 + A + I = O \) is true or false.

Answer: False


Question: Determine the interval in which the constant \( k \) must lie if the curve \( f(x) = k^3 x + k^3 - 2 \) intersects the curve \( g(x) = \frac{1}{2} \ln x^2 \) at exactly one point.

Answer: (1, \infty)


Question: Determine the maximum degree of a vertex in the graph $G$, where the $2^n$ vertices correspond to all subsets of a set of size $n$ (with $n \geq 6$), and two vertices are adjacent if and only if their corresponding sets intersect in exactly two elements.

Answer: 3 \cdot 2^{n-3}


Question: Evaluate the limit: $$\lim_{h\rightarrow0^+} \frac{1}{h^2} \left[\frac{f(v+hu)+ f(v-hu)}{2} - f(v)\right]$$ where $f(x,y,z) = (x+y^2)e^z$, $v=\langle 1,0,0 \rangle$, and $u=\langle 3,2,1 \rangle$. Provide your answer in terms of derivatives if applicable.

Answer: \dfrac{15}{2}


Question: Consider two simple closed curves $C_1$ and $C_2$ in the plane, where $C_1$ lies entirely within the interior of $C_2$. Is the region between these two curves, defined as $C_1 \cup C_2 \cup (I_2 \cap E_1)$ (where $I_2$ is the interior of $C_2$ and $E_1$ is the exterior of $C_1$), homeomorphic to the region between two concentric circles?

Answer: Yes


Question: Suppose $S_t$ satisfies the equation $$dS_t = S_t(a(X_t)dt + b(X_t)dW_t),$$ where $X_t$ is a Markov process. Is the joint process $(S_t, X_t)$ a Markov process?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to\frac{\pi}{2}}\bigl(\sqrt{\cos^2x-2\sin x+2\sin^2x}\bigr)^{[\sin x]} \] where \([\, {.} \,]\) denotes the greatest integer function.

Answer: 1


Question: Let $A, B \in M_n$ and suppose $AA^* = BB^*$. Is it true that there exists a unitary matrix $U$ such that $A = BU$?

Answer: Yes


Question:
Let \( f: \{ z \in \mathbb{C} : \|z\| = 1 \} \to \mathbb{R} \) be a continuous function. For any \( \epsilon > 0 \), does there exist an analytic polynomial \( p(z) \) with complex coefficients such that
\[
|f(z) - \Re (p(z))| < \epsilon
\]
for all \( z \) on the unit circle? Provide a justification for your answer.

Answer: Yes


Question: Calculate the limit: \[ \lim_{x\to+\infty}\left(\frac{x^2 -x +1}{x^2}\right)^{\frac{-3x^3}{2x^2-1}} \] without using the de l'Hôpital rule.

Answer: e^{\frac{3}{2}}


Question: Differentiate the function \( f(x) = x + \frac{K}{x^*} \), where \( x \) is a complex number and \( x^* \) is its complex conjugate, with respect to \( x \).

Answer: 1


Question: Determine whether it is possible to transform the metric \( \frac{dx^2+dy^2}{g(r)^2} \), where \( g \) is a function and \( r=\sqrt{x^2+y^2} \), into a form \( du^2+f(u,v) dv^2 \) through a change of variables.

Answer: Yes


Question: Determine if the prime ideal \((1+i)\) in the ring of Gaussian integers \(\mathbb{Z}[i]\) lies above the prime number 2 in \(\mathbb{Z}\).

Answer: Yes


Question: Determine the number of Banach spaces that can be defined on the set of real numbers, $\mathbb{R}$. Provide your answer as a single number.

Answer: 1


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

Answer: -\dfrac{1}{3}


Question: Let $T: \mathbb{R}^n \to \mathbb{R}^n$ be an isometry. If we do not assume that $T(0) = 0$, is $T$ still surjective?

Answer: Yes


Question: Find a closed-form formula for \( \frac{d\text{vec}(X^T)}{d\text{vec}(X)} \), where \( X \) is an \( n \times n \) matrix with its entry indices increasing along every column. For example, when \( n = 2 \), \( X = \begin{pmatrix} x_1 & x_3 \\ x_2 & x_4 \end{pmatrix} \). The vectorization \( \text{vec}(X) \) is formed by stacking the columns of \( X \) into a single vector. Provide the formula for any \( n \).

Answer: K_{n,n}


Question: Given that \( a^2 + b^2 = 1 \), find the minimum value of \( \frac{b+1}{a+b-2} \). Let \( u \) be this minimum value. Determine the value of \( u^2 \).

Answer: 9


Question: Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function such that $f(x) \geq 0$ for all $x$ and \(\int_{-\infty}^\infty f(x) \, dx = 1.\) For $r\geq0,$ define \(I_n(x) = \int\dots\int_{x_1^2+x_2^2+\dots+x_n^2 \leq r^2} f(x_1) f(x_2)\dots f(x_n) \, dx_1 \, dx_2 \dots \, dx_n.\) Find \(\lim_{n\to\infty}I_n(x)\) for a fixed $r$. \(\boxed{0}\)

Answer: 0


Question: Is it true that any lower triangular square matrix over a field is similar to an upper triangular matrix?

Answer: Yes


Question: Let $E$ be a subset of the unit circle $S^1$ with positive Lebesgue measure. Define $\Gamma$ as the group of roots of unity: $\Gamma = \{z \in \mathbb{C} \mid z^n = 1 \text{ for some } n \geq 0\}$. Consider the set $\Gamma E = \{z\zeta \mid z \in \Gamma, \zeta \in E\}$. If $E$ has positive Lebesgue measure, does $\Gamma E$ have measure $2\pi$?

Answer: Yes


Question: Let $ABCD$ be an inscribed trapezium with parallel sides $AB$ and $CD$. Given that $\angle AOD = 60^{\circ}$ and the altitude of the trapezium is 10, find the area of the trapezium.

Answer: 100\sqrt{3}


Question: Given a matrix $A \in \mathbb{R}^{m \times n}$ with $m \ge n$, consider its QR decomposition where $Q \in \mathbb{R}^{m \times n}$ and $R \in \mathbb{R}^{n \times n}$. Does the matrix $Q$ always have full column rank, i.e., $n$ independent columns?

Answer: Yes


Question:
Minimize the function \(x^2 + y^2 + z^2\) subject to the constraints:
\[ x + 2y + z = 1 \]
\[ 2x - y - 3z = 4 \]
Use the method of Lagrange multipliers to find the values of \(x, y, z\) that satisfy these conditions.

Answer: \left( \dfrac{16}{15}, \dfrac{1}{3}, -\dfrac{11}{15} \right)


Question: Suppose $f$ is a continuous function on the boundary of a disc $D \subset \mathbb{C}$, and it satisfies the condition \(\int_{\partial D} f(\zeta) \zeta^k d\zeta = 0\) for all non-negative integers \(k\). Determine if it is possible to extend $f$ to be a holomorphic function within $D$. Justify your answer.

Answer: Yes


Question: Find the x-coordinate of the center of mass of a half cylinder with radius \( r \).

Answer: \dfrac{4r}{3\pi}


Question: If a field has a cyclic multiplicative group, is the field necessarily finite? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to\infty}\int_0^\infty nx^2\sin\left(\frac{1}{nx}\right)dx \]

Answer: \infty


Question: Are all homomorphisms from the fundamental group of a genus $g$ closed orientable surface to the fundamental group of a genus $h$ closed orientable surface trivial when $g < h$?

Answer: No


Question: Determine whether there are only a finite number of connected, locally compact Hausdorff topological spaces $X$, up to homeomorphism, such that $X$ has an open set $U$ homeomorphic to $\mathbb{R}$ and $X - U$ is also homeomorphic to $\mathbb{R}$. Assume connectedness to exclude the single disconnected possibility.

Answer: Yes


Question: Compute the limit of the following integral: \[ \lim_{N \rightarrow \infty} \frac{1}{\sqrt{N}} \int_{-b}^b e^{-\frac{Nx^{2}}{2}}g(x)\,dx \] where $b > 0$ and $g(x)$ is a continuous function defined on $[-b, b]$. Use a change of variables to simplify the computation.

Answer: 0


Question: Consider a metric space \((M, d)\) where every subset \(E\) of \(M\) is compact. Is \(M\) necessarily a finite set? Provide a proof or a counterexample.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to\infty}\left(\int_{0}^{1}\left(\frac{57^{x^2+1}+3}{e^{x^2}+1113337x^2+1113339x^{\frac{3}{2}}+1113341x+1}\right)^n dx\right)^{\frac{1}{n}}. \]

Answer: 30


Question: Calculate the number of ways to throw 10 distinct dice such that each of the 6 faces appears at least once. Verify if the calculation \(6^{10} - 6 \times 5^{10} = 1,872,426\) is correct.

Answer: 16435440


Question:
Find the last two digits of the number represented by the power tower of nine 9s:
\[ 9^{9^{9^{\cdots^{9}}}} \] (where there are nine 9s in total).

Answer: 89


Question: Determine whether the following statement is true: \[ \sum_{x \in \mathbb{Z}^n} e^{-2x^Tx} = \left(\sum_{y=-\infty}^{\infty} e^{-2y^2}\right)^n \]

Answer: True


Question: Let $T: l_\infty \rightarrow l_\infty$ be a linear operator defined by $T((x_1,x_2,\ldots ))=(x_1,x_2-x_1,x_3-x_2,\ldots )$. If $R=T(l_\infty)$ and $y=(1,1,1,\ldots )$, compute $d(y,R)$, the distance from $y$ to the range $R$.

Answer: 1


Question: Determine the dimension of the unitary group $U(n)$, where $U(n) = \{A \in GL_n(\mathbb{C}) \mid A^\ast A = I\}$. Provide your answer as a single number.

Answer: n^2


Question: Determine the minimum degree $n$ of the Taylor polynomial $T_n(x)$ for the function \( f(x) = \int_{\frac{\pi}{2}}^{x} \frac{\cos(t)}{t - \frac{\pi}{2}}dt \) around \( a=\frac{\pi}{2} \) such that \( \mid f(\pi) - T_n(\pi) \mid \leq \frac{1}{100} \).

Answer: 5


Question: Given a Poisson process $N(t)$ with parameter $\lambda > 0$, and independent and identically distributed random variables $X_1, X_2, \ldots$ with a common mean $\mu$ and positive variance, let \( L(t) = \sum_{i=1}^{N(t)} X_i \). Find the expected value $E[L(t) | N(t) = n]$. Express your answer in terms of $n$ and $\mu$. \( \boxed{} \)

Answer: n\mu


Question: Determine the radius of convergence of the differential equation: $$(x^2-3)y''+6xy'+4y=0$$

Answer: \sqrt{3}


Question: Determine whether there exists a subgroup of order 24 in the symmetric group $S_7$. Provide a justification for your answer.

Answer: Yes


Question: Determine the last digit of the binomial coefficient \( {369 \choose 233} \).

Answer: 4


Question: Determine whether the following claim is true: For any probability density function $f$ with domain $(0,1)$, the differential entropy satisfies $$-\int_0^1 f(x) \log f(x)\ dx \leq 0.$$

Answer: True


Question: Calculate the limit: \[ \lim_{x\to 0^{+}}x\ln\left(x\sinh\left(\frac{1}{x}\right) \right) \]

Answer: 1


Question: Calculate the value of the sum \(\left(\frac{1 \cdot 2}{73}\right) + \left(\frac{2 \cdot 3}{73}\right) + \cdots + \left(\frac{71 \cdot 72}{73}\right)\), where each term is a Legendre symbol.

Answer: -1


Question: Evaluate the infinite series: \[ \sum_{n=1}^{\infty} \frac{(-1)^n \sin(n)}{n} \]

Answer: -\dfrac{1}{2}


Question: Evaluate the integral \( \int_0^1 \left( \left\lfloor \frac{2}{x} \right\rfloor - 2 \left\lfloor \frac{1}{x} \right\rfloor \right) \, dx \).

Answer: 2 \ln 2 - 1


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{1}{n} \cdot \left((m+1)(m+2) \ldots (m+n)\right)^{\frac{1}{n}} \] where \( m \) is a fixed positive integer.

Answer: \dfrac{1}{e}


Question: Determine the radius of convergence of the power series \( \sum_{n=0}^\infty \frac{(n!)^k}{(kn)!}z^n \).

Answer: k^k


Question: Evaluate the expression \( \frac{J_{6,3}}{J_{5,2}} \), where \( J_{m, n} = \int_{0}^{\pi/2} \cos^m x \cos nx \, dx \).

Answer: \dfrac{2}{3}


Question: Let $X$ be an integral projective scheme of dimension at least 1 over a field $k$, and let $\mathscr{L}$ be an ample invertible sheaf on $X$. Prove that $H^0(X,\mathscr{L}^{-1}) = 0$.

Answer: 0


Question: Is every recursively enumerable set $A \subseteq \mathbb{N}$ also recursive? Provide a brief argument for your answer.

Answer: No


Question: Consider a parallelogram in \(\mathbb{R}^2\). Is it possible to define an alternative inner product \((\cdot, \cdot)_a\) on \(\mathbb{R}^2\) such that the parallelogram becomes a square in the space \((\mathbb{R}^2, (\cdot, \cdot)_a)\)?

Answer: Yes


Question: Given a boolean $0/1$ cube in $n$ dimensions with $2^{n-1}$ red and $2^{n-1}$ blue points, is it possible to complement the cube (where blue becomes red and vice versa) using a series of transformations that include cutting the cube with $h=\operatorname{poly}(n)$ hyperplane inequalities, permuting and negating coordinates, and rotating and translating pieces, all within $\operatorname{poly}(n)$ operations?

Answer: Yes


Question:
Given the determinants \( D = \begin{vmatrix} a & \omega b & \omega^2 c \\ \omega^2 b & c & \omega a \\ \omega c & \omega^2 a & b \end{vmatrix} \) and \( D' = \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} \), where \( \omega \) is a non-real cube root of unity, determine which of the following statements does not hold:

1) \( D = 0 \) if \( a + b + c = 0 \) and \( a, b, c \) are all distinct.
2) \( D' = 0 \) if \( a = b = c \) and \( a + b + c \neq 0 \).
3) \( D = -D' \).
4) \( D = D' \).

Select the statement that is incorrect.

Answer: 3


Question: Prove that for \( \theta = \frac{2\pi}{5} \), the equation \( 2\cos(2\theta) + 2\cos(\theta) + 1 = 0 \) holds true.

Answer: 0


Question: Calculate the limit \( \lim_{x\to 1}\frac{\sqrt[359]{x} - 1}{\sqrt[5]{x} - 1} \) without using l'Hopital's rule.

Answer: \dfrac{5}{359}


Question: In a survey using the randomized response technique, each respondent is given a card with two questions: "Was your mother born in April?" and "Have you ever evaded tax?" Respondents toss a coin and answer the first question if it lands heads, or the second question if it lands tails. The interviewer does not know which question is answered. Out of 1000 people interviewed, 200 answered "yes." Estimate the proportion of tax evaders to two decimal places.

Answer: 0.32


Question:
Let \( W \) be the Weyl group of a semisimple simply connected group over \( \mathbb{C} \). Let \( I = \{1, \ldots, r\} \) be the set of simple roots. For \( w \in W \), denote by \( \text{supp}(w) \) the subset of \( I \) corresponding to the simple reflections that appear in a reduced decomposition of \( w \). Suppose \( w \) is an element such that \( \text{supp}(w) = I \) and \( \text{length}(w) > r + 1 \). Is it true that there exists an element \( w' \) such that:

1. \( w' \leq w \)
2. \( \text{length}(w') = r + 1 \)
3. \( \text{supp}(w') = I \)?

Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: "The reversed language of a non-regular language is non-regular." Provide a justification for your answer.

Answer: True


Question: Determine the radius of convergence for the power series \( \sum _{n=2}^{\infty }\:\frac{\sin(\frac{1}{n})}{\ln(n)}x^n \).

Answer: 1


Question: For fixed values $0 < a_1 \leq a_2 \leq a_3$, find the positive value of $x$ that minimizes the function \( g(x) = \max_{1\leq i \leq 3} \left(\frac{a_i - x}{a_i + x}\right)^2 \).

Answer: \sqrt{a_1 a_3}


Question:
Find the constant value of $g$ when $\varepsilon$ is small, given the equation:
\[3\sinc\left(-1+ \frac\varepsilon T \right)-3\sinc\left(1+\frac\varepsilon T\right)-\sinc\left(-3+\frac\varepsilon T\right)+\sinc\left(3+\frac\varepsilon T\right) = \frac\varepsilon g\]

Answer: \dfrac{3T}{16}


Question: Let $(X,\|.\|)$ be a Banach space, and let $f:X\longrightarrow \mathbb{C}$ be a function such that for any $x,y\in X$, there exists a constant $M>0$ (depending on $x$ and $y$) satisfying $|f(x)-f(y)|\leq M \|x-y\|$. Does this condition imply that $f$ is continuous?

Answer: No


Question:
Consider the boundary value problem for \( \lambda \in \mathbb{R} \):
\[ x^2\frac{d^2y}{dx^2}+2x\frac{dy}{dx}+\lambda y=0, \quad x\in [1,2], \qquad y(1)=y(2)=0 \]
Which of the following statements is true?

1. There exists a \( \lambda_0 \in \mathbb{R} \) such that the problem \( P_{\lambda} \) has a non-trivial solution for \( \lambda>\lambda_0 \).
2. \( \{\lambda \in \mathbb{R}: P_{\lambda} \text{ has a non-trivial solution}\} \) is a dense subset of \( \mathbb{R} \).
3. For any continuous function \( f:[1,2] \to \mathbb{R} \) with \( f(x)\neq 0 \) for some \( x \in [1,2] \), there exists a solution \( u \) of the problem for some \( \lambda \in \mathbb{R} \) such that \( \int\limits_1 ^2 fu \neq 0 \).
4. There exists a \( \lambda \in \mathbb{R} \) such that the problem \( P_{\lambda} \) has two linearly independent solutions.

Answer: 3


Question: Given that \( f \) is a differentiable function with \( f(8) = 1 \) and \( f'(8) = 0 \), find the value of \( \frac{d}{dx}\left(\sqrt{(x^2+x+3)f(x^3)}f(x^3)^2\right) \) at \( x=2 \).

Answer: \dfrac{5}{6}


Question: Evaluate the integral \( \int_{0}^{1} (\log(x))^2 x^{3/2} \, dx \) using the function \( f(x,t) = x^t \) and the theorem of switching differentiation and integration.

Answer: \dfrac{16}{125}


Question: For a fair coin, what is the expected number of tosses needed to get 3 consecutive heads?

Answer: 14


Question: Let $X_i \sim N(0,1)$ for $0 \leq i \leq n$ be i.i.d. standard normal distributions. Define $Y_n := \frac{\sqrt{n} X_0}{\sqrt{(\sum_{i=1}^n X_i^2)}}$. Determine the limiting distribution function for $Y_n$ as $n \to \infty$. Provide your answer in terms of a known distribution.

Answer: N(0,1)


Question: What is the probability that the sum of 15 randomly chosen three-digit numbers is divisible by 10?

Answer: \dfrac{1}{10}


Question: Show that the function \( f(x,y) = \frac{x^3\sin(x+y) - y^4\ln(x^2+y^2)}{x^2+y^2} \) is continuous at the point \((0,0)\), given that \( f(0,0) = 0 \).

Answer: 0


Question: Let $f(x)$ be a trigonometric series that converges uniformly on $[-\pi, \pi]$. If $f(x)$ is multiplied by $e^{iax}$, where $a \in \mathbb{N}$, does the resulting series converge uniformly?

Answer: Yes


Question: Find an entire function whose real part is given by \( x^4 - 6x^2y^2 + y^4 \).

Answer: z^4


Question: Find the eigenvalues of the matrix $A = (p_{ij})$ where $\sum_{i=1}^3 p_{ij} = 1$ for all $j = 1, 2, 3$. Assume $A$ is a $3 \times 3$ matrix.

Answer: 1


Question: Find the radius of convergence of the power series \( \sum_{j=1}^\infty \frac{j!}{j^j}z^j \).

Answer: e


Question: Determine the conditions on the parameters \( a \) and \( b \) such that the function \( h(x) = \frac{x^2 - x +1}{x^2 + ax+(1-b^2)} \) is bounded for all real numbers \( x \).

Answer: a^2 + 4b^2 < 4


Question: Let \(W\) be a standard Brownian motion and \(x\) be a real number. Given \(0 < s < t\), find the conditional expectation \(\mathsf{E}[W_s | W_t = x]\).

Answer: \dfrac{s}{t} x


Question: Integrate the function \( f(x,y,z) = 14xz \) over the region in the first octant \((x, y, z \ge 0)\) that lies above the parabolic cylinder \( z = y^2 \) and below the paraboloid \( z = 8 - 2x^2 - y^2 \).

Answer: \dfrac{1024}{3}


Question: Compute the inverse Laplace transform of \( F(s) = \frac{1}{s+1} \) using Mellin's inverse formula. Verify that the result is \( f(t) = e^{-t} \).

Answer: e^{-t}


Question: Evaluate the integral: $$ 4n(n-1)\int_0^1 \frac{1}{8t^3}\left[\frac{(2t-t^2)^{n+1}}{(n+1)}-\frac{t^{2n+2}}{n+1}-t^4\{\frac{(2t-t^2)^{n-1}}{n-1}-\frac{t^{2n-2}}{n-1} \} \right] dt.$$ For $n \geq 3$, verify that the value of the integral is 1.

Answer: 1


Question: Consider a non-square matrix $A$ of size $m \times n$ where $m \gg n$. Can the matrix product $AA^T$ be a diagonal matrix, meaning all the diagonal entries are non-zero and the off-diagonal entries are zero?

Answer: No


Question: Find a number $n$ such that $100 < n < 2000$ and $2^n + 2$ is divisible by $n$. What is the value of $n$?

Answer: 946


Question:
Determine the set of possible ranks of the matrix \( A \) given by:
\[
A = \begin{pmatrix}
a^2 & b^2 & c^2 \\
a^5 & b^5 & c^5 \\
a^{11} & b^{11} & c^{11}
\end{pmatrix}
\]
where \(a, b, c\) are distinct integers. Choose from the following options:

- \(\{0, 1, 2, 3\}\)
- \(\{1, 2, 3\}\)
- \(\{2, 3\}\)
- \(\{3\}\)

Answer: C


Question: Compute the covariance $Cov(X_i, X_j)$ for $i \neq j$, where $X_i = \sum_{l=1}^n Y_l$ and $X_j = \sum_{l=1}^n Z_l$, given that $Y_l$ is 1 if the $l$-th marble lands in the $i$-th box and 0 otherwise, and $Z_l$ is 1 if the $l$-th marble lands in the $j$-th box and 0 otherwise. Use the formula $Cov(X_i, X_j) = E[(X_i - E[X_i])(X_j - E[X_j])]$ and show that $Cov(X_i, X_j) = -np_ip_j$. Assume $p_i$ is the probability that a marble lands in the $i$-th box, with $\sum_{i=1}^k p_i = 1$.

Answer: -np_ip_j


Question: Determine whether there exists a set $A \subseteq \mathbb{N}$ such that the upper power density $\bar{d_p}(A) = \bar{d_p}(A^C) = 1/2$. If no such set exists, find the maximal $\alpha \in [0,1]$ such that for every $A \subseteq \mathbb{N}$, either $\bar{d_p}(A) \ge \alpha$ or $\bar{d_p}(A^C) \ge \alpha$. Note that $\bar{d_p}(\text{evens}) = \bar{d_p}(\text{odds}) = 2/3$. Provide your answer in the form of a set or a value for $\alpha$. \( \boxed{} \)

Answer: \dfrac{2}{3}


Question: Find the limit of the sequence \( X_n = \cos\left(\left(\frac{3^n+\pi^n}{3^n+\left(\pi-\frac{1}{4} \right)^n} \right)^{1/n}\right) \) as \( n \to \infty \), where \( n \in \mathbb{N} \).

Answer: \dfrac{1}{2}


Question: Let $K$ be a number field. Determine whether it is true that for every narrow ideal class $\mathcal{C}$, there exist infinitely many prime ideals $\mathfrak{p}$ such that $\mathfrak{p} \in \mathcal{C}$. Provide a justification for your answer.

Answer: True


Question: Let $G = \{a^k, a^k b \mid 0 \le k < 9\}$ where $o(a) = 9$ and $o(b) = 2$, and $ba = a^{-1}b$. If $Z(G)$ denotes the center of the group $G$, find the order of the quotient group $G/Z(G)$. Express your answer as an integer.

Answer: 18


Question: Let $v \in \mathbb{C}^n$ be a complex unit norm vector, and let $G$ be an $m \times n$ matrix with i.i.d. $\mathcal{CN}(0,1)$ elements. Compute the expectation $\mathbb{E}\{G v v^H G^H\}$, where $v^H$ denotes the conjugate transpose of $v$. This expectation is a $m \times m$ matrix.

Answer: I_m


Question: Compute the flux of the vector field \( \nabla \ln \sqrt{x^2 + y^2 + z^2} \) across the surface of an icosahedron centered at the origin. Let \( S \) be the surface of the icosahedron, and \( f(x,y,z) = \ln \sqrt{x^2+y^2+z^2} \). Calculate the flux \( \iint_S (\nabla f \cdot n) \, d\sigma \), where \( n \) is the outward unit normal vector on \( S \).

Answer: 4\pi


Question: Evaluate the surface integral \( \iint_S F(x,y,z) \, dS \) where \( F(x,y,z) = x + y + z \) and \( S \) is the cylindrical surface defined by \( x^2 + y^2 = 1 \) with \( 0 \leq z \leq 2 \).

Answer: 4\pi


Question:
Given the recurrence relation \(a_n = \left(1 - \frac{1}{n}\right)a_{n-1} + \frac{1}{n}a_{n-2}\) with initial conditions \(a_1, a_2 \in \mathbb{N}\), determine \(\lim_{n \to \infty} a_n\). Choose the correct answer from the following options:

(A) \(2(a_2-a_1) + a_1 e^{-1}\)

(B) \(2(a_1-a_2)e^{-1} + a_2\)

(C) \(2(a_1-a_2)e^{-1} + a_1\)

(D) \(2(a_2-a_1)e^{-1} + a_1\)

Answer: D


Question: Given that $f(4) = \frac{1}{4}$, $f(8) = \frac{1}{2}$, and \( \int_4^8 \frac{(f'(x))^2}{(f(x))^4} \, dx = 1 \), find the value of $f(6)$. Assume $f(x)$ is a differentiable function on $\mathbb{R}$ and $f(x) \neq 0$ for all $x \in \mathbb{R}$. \( \)

Answer: \dfrac{1}{3}


Question: Consider the set $X$ of all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that for any positive rational number $\epsilon$, there exists a positive rational number $\delta$ such that for all rational numbers $x, y > \delta$, we have $|f(x) - f(y)| < \epsilon$. Define an equivalence relation $\sim$ on $X$ by $f \sim g$ if the limit of $f(x) - g(x)$ as $x \rightarrow \infty$ is $0$. Is the set of equivalence classes of elements of $X$ under $\sim$ isomorphic to the set of real numbers?

Answer: Yes


Question: Can a CW complex have a 2-dimensional cell without any 1-dimensional cells, such that the 1-skeleton is equal to the 0-skeleton?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to \infty} n(H(n) - \ln(n) - \gamma) \] where \( H(n) = 1 + \frac{1}{2} + \cdots + \frac{1}{n} \) is the \( n^{th} \) harmonic number and \( \gamma \) is the Euler-Mascheroni constant.

Answer: \dfrac{1}{2}


Question: Let $P(z)$ be a polynomial of degree $n > 0$. Prove that there exists a radius $R > 0$ such that for the circle $C$ defined by $|z| = R$ and oriented anticlockwise, the following holds: \[ \int_{C} \frac{P'(z)}{P(z)} \, dz = 2n\pi i. \]

Answer: 2n\pi i


Question: Evaluate the definite integral: \[ \int_{-\sqrt{3}}^{\sqrt{3}} \frac{e^x}{(e^x+1)(x^2+1)} \, dx \]

Answer: \dfrac{\pi}{3}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left(1+\frac{1}{4\pi}\int _{0}^{2\pi}\log(n^2-2n\cos t+1)dt\right)^{\log(n+1)} \] where \( n \geq 1 \).

Answer: \infty


Question: Determine the number of homomorphisms from the dihedral group $D_6$ to the dihedral group $D_5$. Provide your answer as a single integer.

Answer: 16


Question:
Find the KKT point for the following optimization problem:

Minimize \( f(x_1, x_2) = (x_1 - 3)^2 + (x_2 - 2)^2 \)

Subject to:

\( x_1^2 + x_2^2 \le 5 \)

\( x_1 + 2x_2 = 4 \)

\( x_1 \ge 0, x_2 \ge 0 \)

Answer: (2, 1)


Question: Find the inverse of the polynomial \(2x^2 + x + 1\) in the field \(\mathbb{Z}_3[x]/(x^3 + x^2 + x + 2)\).

Answer: 2x^2


Question: Find the splitting field of the polynomial \( f(x) = x^7 + 3 \) over \( \mathbb{Q} \), and determine the degree \([E: \mathbb{Q}]\) of the extension, where \( E \) is the splitting field.

Answer: 42


Question: Determine the number of ways to arrange the letters $A, B, C, D, E, F, G, H, I, J$ in a line such that $A$ is not adjacent to $B$, $B$ is not adjacent to $G$, and $G$ is not adjacent to $D$. Use the inclusion-exclusion principle to find the solution.

Answer: 1764000


Question: Calculate the limit: \[ \lim_{n \to \infty}\sqrt[n]{\frac{2^n+3^n}{3^n+4^n}} \]

Answer: \dfrac{3}{4}


Question:
In a triangle with angles \(A, B, C\) and sides \(a, b, c\) opposite to their respective angles, find \(\tan C\) given that:
\[ c^2 = \frac{a^3 + b^3 + c^3}{a + b + c} \]

Answer: \sqrt{3}


Question: Evaluate the integral \( \int_{|z| = 2} \frac{1}{z^{741} + 1} \, dz \) using the residue theorem.

Answer: 0


Question: Find the value of \(a \in \mathbb{R}\) such that the integral \(\int_1^{\infty}\left(\frac{1}{x+2} - \frac{ax}{x^2+1}\right) dx\) converges.

Answer: 1


Question: In the RSA cryptosystem, both the original message M and the coded message R are numbers in the range [0, n-1]. Can the coded message R be any number within this range?

Answer: Yes


Question: Suppose for each $i = 1, \ldots, n$, the sequence $\xi_{ni}$ converges in probability to 0, i.e., $\xi_{ni} \overset{p}{\to} 0$. Does it follow that $\frac{1}{n} \sum_{i=1}^{n} \xi_{ni} \overset{p}{\to} 0$? Justify your answer.

Answer: No


Question: Find all integer pairs \((x, y)\) such that \(4y^2 = x^7 + 1\).

Answer: (-1, 0)


Question: Evaluate the integral \( \int_{\gamma} d\omega \), where \( \omega = (y+z)\,dx + (z+x)\,dy + (x-y)\,dz \) and \( \gamma \) is the intersection of the spherical surface \( x^2 + y^2 + z^2 = 1 \) with the plane \( y = z \). Use the parametrization \( x = \cos t, \ y = \frac{1}{\sqrt{2}} \sin t, \ z = \frac{1}{\sqrt{2}} \sin t \) for \( t \in [0, 2\pi] \).

Answer: 0


Question:
Calculate Spearman's rho for the comonotonicity copula, defined as \( Cm(u_1, u_2) = \min(u_1, u_2) \). Spearman's rho is given by the formula:
\[ \rho = 12 \int_{0}^{1}\int_{0}^{1} \min(u_1, u_2) \, du_1 \, du_2 - 3 \]
Solve the integral to find the value of \( \rho \).

Answer: 1


Question: Consider the differential equation \( f''(t) - e^{-2t} f(t) = 0 \). Determine whether there exist constants \( A \in \mathbb{R} \) and a positive \( B \) such that \( |f(t)| \sim A + Bt \) for all sufficiently large \( t \).

Answer: Yes


Question: Solve the equation \(2\arcsin\frac{x}{2}+\arcsin(x\sqrt{2})=\frac{\pi}{2}\) for \(x\).

Answer: 2 - \sqrt{2}


Question: Is the multiplication of infinite cardinals defined in Zermelo-Fraenkel set theory without the Axiom of Choice?

Answer: Yes


Question: Determine the Galois group of the polynomial \( P(X) = (X^4 - 2)(X^2 + 2) \) over \( \mathbb{Q} \).

Answer: D_4


Question: Find all positive integers \( n \) such that \( n^4 + n^3 + 1 \) is a perfect square.

Answer: 2


Question: Calculate the variance of the random variable \( X = \int_0^2 2t \, dW(t) + \int_4^6 W(t) \, dW(t) \).

Answer: \dfrac{62}{3}


Question: Determine the Lebesgue measure of the set \( S = \left\{ \frac{1}{2} \right\} \cup \left\{ \frac{1}{4}, \frac{3}{4} \right\} \cup \left\{ \frac{1}{8}, \frac{3}{8}, \frac{5}{8}, \frac{7}{8} \right\} \cup \cdots \). The set \( S \) is constructed by iteratively removing midpoints from intervals within \((0,1)\).

Answer: 0


Question: Consider the fundamental solution to the heat equation given by \( u(x)=\int_\mathbb{R} \frac{1}{\sqrt{2\pi t}}e^{-\frac{|x-y|^2}{2t}}g(y)\, dy,\quad x\in \mathbb{R} \). Suppose \( g \) is sufficiently smooth. Does there exist a \( p > 2 \) and a constant \( C = C(p, g) \) such that \( \|D_x u\|_{L^p(\mathbb{R})} \le C \) for all \( t \in (0, T] \)?

Answer: Yes


Question: Given the function \( f(x) = (2x-3\pi)^5 + \frac{4}{3}x + \cos(x) \), let \( g \) be the inverse function of \( f \). Find the value of \( g'(2\pi) \).

Answer: \dfrac{3}{7}


Question:
Given the joint distribution PDFs of random variables $X$ and $Y$ as $f(x, y) = x + y$ for $0 < x < 1$ and $0 < y < 1$, and $Y$ and $Z$ as $g(y, z) = 3(y + 0.5)z^2$ for $0 < y < 1$ and $0 < z < 1$, determine which of the following could be the PDF of the joint distribution of $X$ and $Z$:

1. $x + 1.5z^2$, for $0 < x, z < 1$
2. $x + 0.5 + 3z^2$, for $0 < x, z < 1$
3. $3(x + 0.5z^2)$, for $0 < x, z < 1$
4. $x + z$, for $0 < x, z < 1$

Select the correct option.

Answer: 3


Question:
Determine whether the following statement is true for all positive integers \( n \in \mathbb{N} \):

For every \( \varepsilon > 0 \), there exist prime numbers \( p \) and \( q \) such that \( \left| \frac{p}{q} - n \right| < \varepsilon \).

Answer: True


Question:
Convert the boundaries of the set \( \Omega = \{(x,y) \in \mathbb{R}^2: y \leq x \leq \sqrt{y}, 0 \leq y \leq 1\} \) into polar coordinates and evaluate the integral:
\[ \iint_{\Omega}(x^2+y^2)\ dxdy \]

Answer: \dfrac{3}{35}


Question: Let $G$ and $H$ be groups, and suppose there exists a bijection $\varphi: G \to H$ such that for any proper subgroup $G' \neq G$ of $G$, the image $\varphi(G')$ is a subgroup of $H$, and the restriction $\varphi|_{G'}: G' \to \varphi(G')$ is a group isomorphism. Does it follow that $G \cong H$?

Answer: Yes


Question: Let $S$ be a closed Hilbert subspace of $\mathcal{H}$ such that $S^\perp = \{0\}$. Determine if $S = \mathcal{H}$.

Answer: S = \mathcal{H}


Question: Is there a closed set \( A \) in \( \mathbb{R}^2 \) such that the projection \( P(A) \) is not a closed set in \( \mathbb{R} \), where \( P: \mathbb{R}^2 \to \mathbb{R} \) is defined by \( P(x, y) = x \)?

Answer: Yes


Question: What is the probability that the roots of the polynomial \( P(x) = \frac{1}{4} x^2 + Ux + V^2 \) are real, given that \( U \) and \( V \) are independent normally distributed random variables with mean 0 and variance 1?

Answer: \dfrac{1}{2}


Question: If $A$ and $B$ are matrices such that $AB^2 = BA$ and $A^4 = I$, find $B^{16}$. \( \text{(Express your answer in terms of } B \text{ if necessary.)} \)

Answer: B


Question: For every square matrix $A$, does there always exist a non-diagonal matrix $B$ such that $AB = BA$? Answer with 'Yes' or 'No'.

Answer: No


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

Answer: \dfrac{\pi}{3}


Question: Given the recurrence relation \( T(n) \leq 9T\left(\left\lfloor\frac{n}{3}\right\rfloor\right) + n \) with \( T(n) \) being non-decreasing and \( T(1) = 7 \), prove that \( T(n) = O(n^2) \).

Answer: O(n^2)


Question: Determine whether the following statement is true or false: A radical ideal $I$ in a commutative ring with unity is a prime ideal if and only if it is not an intersection of two radical ideals properly containing $I$. Provide a justification for your answer.

Answer: A


Question:
Compute \( \frac{d^6 f}{dx^6}(0) \) for the function \( f(x) = \arctan\left(\frac{x^2}{5}\right) \) using the series expansion:
\[ \arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \ldots \]

Answer: -\dfrac{48}{25}


Question: Determine all group homomorphisms from \( \mathbb{Z}_6 \) to \( S_3 \).

Answer: 6


Question: Suppose $X$ is a locally compact metric space which is $\sigma$-compact, and let $K$ be a compact subset of $X$. Consider a sequence of compact sets $K_{n}$ such that $K_{n} \subset \textrm{int}(K_{n + 1})$ and $X = \bigcup_{n = 1}^{\infty}K_{n}$. Must there exist an integer $N$ such that $K \subset K_{N}$?

Answer: Yes


Question: An urn contains 5 balls numbered 0, 1, 2, 3, and 4. A ball is drawn from the urn, its number is recorded, and then it is returned to the urn. This process is repeated until the balls numbered 1, 2, and 3 have each been drawn at least once. What is the probability that this process is repeated exactly 5 times?

Answer: \dfrac{66}{625}


Question: Find the closed form of the power series \( \sum\limits_{n=0} ^\infty \frac{n^2}{n!} x^n \).

Answer: x(x + 1)e^x


Question: Given that \( \xi \) and \( \eta \) are independent random variables with a normal distribution \( \sim N(0,1) \), find the variance of \( E(2\xi + \eta | \xi + \eta) \).

Answer: \dfrac{9}{2}


Question: Given that the polynomial \( f(x) \in \mathbb{Z}[x] \) has a root \( a \), and it is known that \( f(1) - f(15) = 42 \) and \( f(3) + f(4) = 181 \), find the remainder when \( a \) is divided by 3.

Answer: 2


Question: Is it true that any \(\mathbb{P}^1\)-fibration over any variety is a smooth fibration?

Answer: Yes


Question: Consider disks $s_1, \cdots, s_n$ in the plane, and let $a_{ij}$ be the area of $s_i \cap s_j$. Is the matrix $A = (a_{ij})$ always positive semidefinite? In other words, is it true that for any real numbers $x_1, \cdots, x_n$, we have \( \sum_{i,j=1}^n x_ix_j a_{ij} \geq 0 \)?

Answer: Yes


Question: Let $A$ be a positive semidefinite matrix with all entries greater than or equal to zero, and with spectral radius $\rho(A) < 1$. Determine whether there exists a vector $x > 0$ such that $Ax \le x$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_0^\infty \ln(x)e^{-x} \, dx \).

Answer: -\gamma


Question: Verify the divergence theorem for the vector field \( \vec{F}(x,y,z) = (2xy+z)\hat{i} + y^2\hat{j} - (x+3y)\hat{k} \) and the closed surface \( S \) bounded by the planes \( 2x+2y+z=6 \) and \( x=y=z=0 \). Calculate both sides of the divergence theorem and confirm that they are equal.

Answer: 27


Question: Determine whether the product of two topological groups, $G_1$ and $G_2$, with the product topology, is also a topological group. Justify your answer.

Answer: Yes


Question: Determine the number of units in the quotient ring $R/I$, where $R = \mathbb{F}_7[x]$, $f(x) = x^3 + 4$, and $I = (f(x))$. The number of elements in $R/I$ is $343$. How many of these elements are units?

Answer: 342


Question: Calculate the limit without using L'Hospital's rule: \[ \lim_{x \to 7} \frac{\sqrt{x+2} - \sqrt[3]{x+20}}{\sqrt[4]{x+9} - 2} \]

Answer: \dfrac{112}{27}


Question: Suppose X and Y are two independent exponential random variables with parameter \( \theta \), i.e., their joint probability density function is \( f(x,y; \theta) = \frac{1}{\theta^2}e^{-\frac{x+y}{\theta}}, x\geq0, y\geq0 \), where \( \theta >0 \). Find the probability density function of \( Z = \frac{X}{X+Y} \).

Answer: 1


Question: Is the inverse of an invertible lower triangular matrix also lower triangular and invertible?

Answer: Yes


Question: Consider a triangulation $T$. Is it always possible to choose a subgraph $H$ of $T$ such that $H$ shares a common edge with every face of $T$ and can be directed so that the indegree of every vertex in $H$ is exactly one?

Answer: Yes


Question: Evaluate the integral of the function \( f_{\alpha}(u) = \sum\limits_{h\in\mathbb{Z}^s}\frac{1}{r(h)^{\alpha}}\exp^{2 \pi i \langle h,u \rangle} \) over the hypercube \([0,1)^s\), where \( r(h) = \prod\limits_{i=1}^{s}\max(1,|h_{i}|) \), \( \alpha > 1 \), and \( s > 1 \). Verify that \( \int_{[0,1)^{s}}f_{\alpha}(u) \, du = 1 \).

Answer: 1


Question: Determine whether there exists a constant \(c > 0\) such that \(f(n) > \frac{c}{n}\) for all sufficiently large \(n\), where \(f(n) = \left(1-\sqrt{\frac{\log n}{n}}\right)^{\sqrt{n\log n}}\) for all real numbers \(n > 1\).

Answer: Yes


Question: Determine the order of the smallest subgroup of the symmetric group $S_4$ that contains the set of permutations \( \{ (12)(34), (14)(32), (24), (31) \} \).

Answer: 8


Question: Find the value of the parameter \(a\) such that the plane described by the equation \(ax + 2y + z = 2\) is tangent to the sphere \((x-2)^2 + y^2 + z^2 = 4\). The plane must pass through the points \((0, 1, 0)\) and \((0, 0, 2)\). Do not use calculus in your solution.

Answer: -2


Question:
Show that the function \( g(x, y) \) defined by \( g : \mathbb{R}^2 \to \mathbb{R} \) with domain \( y^2 \leq \frac{x}{3} \) and given by
\[
g(x, y) = \begin{cases}
\frac{x^3}{y^2} \left (\sqrt{x} - \sqrt{y^2 + x} \right ) & y \neq 0\\
0 & y = 0
\end{cases}
\]
is continuous at the point \((0, 0)\).

Answer: 0


Question: Calculate \( f^{(25)}(0) \) for \( f(x) = x^2 \sin(x) \).

Answer: -600


Question: In how many ways can 8 distinguishable people, consisting of 2 Americans, 2 Canadians, 2 Mexicans, and 2 Jamaicans, be seated around a round table such that at least two pairs of countrymen are seated together? Consider rotations as identical.

Answer: 1632


Question: In an infinite binary sequence, each bit is independently '0' or '1' with a probability of \( \frac{1}{2} \). However, if three consecutive '0's appear, the next bit must be '1'. Given this rule, what is the probability that a randomly chosen bit in the sequence is '1'? Express your answer as a limit as the sequence length approaches infinity.

Answer: \dfrac{8}{15}


Question: Find \( F''(1) \) if \( F(x) = \int_1^x f(t) \,\mathrm{d}t \) and \( f(t) = \int_1^{t^2} \frac{\sqrt{5+u^4}}{u} \,\mathrm{d}u \).

Answer: 2\sqrt{6}


Question:
Evaluate the limit of the following expression as \( n \to \infty \):
\[
\lim_{n\rightarrow \infty} \sum_{x=0}^{x=n/2} \varepsilon^{2x}(1-\varepsilon)^{n-2x} \frac{n!}{(2x)!(n-(2x))!}
\]
where the sum is taken over all even values of \( x \) for an even \( n \).

Answer: \dfrac{1}{2}


Question: Compute the residue of the function \( f(z) = \exp \left( 1 + \frac{1}{z} \right) \) at \( z_0 = 0 \).

Answer: e


Question:
Given the integral equation \( y(x)= f(x) + \int_{0}^x \sin(x-t)y(t) dt \), determine the correct expression for \( y(x) \) from the following options:

1. \( y(x) = f(x) + \int_{0}^x (x-t)f(t) dt \)
2. \( y(x) = f(x) - \int_{0}^x (x-t)f(t) dt \)
3. \( y(x) = f(x) - \int_{0}^x \cos(x-t)f(t) dt \)
4. \( y(x) = f(x) - \int_{0}^x \sin(x-t)f(t) dt \)

Answer: 1


Question: Find the value of \( \beta \) such that the function \( u(x_1, x_2, \ldots, x_n) = \left(\sum_{i=1}^{n}x_{i}^2\right)^{\beta/2} \) satisfies \( \nabla^2 u = 0 \).

Answer: 2 - n


Question: Let $X_1, X_2, \ldots, X_n$ be i.i.d. random variables with the probability density function $f(x) = e^{-x}$ for $x > 0$. Define $X_n = \max(X_1, X_2, \ldots, X_n)$. Find $\lim_{n \rightarrow \infty} P(X_n < 2 + \log_e n)$. Provide your answer as a single value.

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


Question: Calculate the residue of the function \( f(z) = \frac{z-\pi/2}{1-\sin z} \) at \( z=\frac{\pi}{2} \).

Answer: 2


Question: Determine the number of homomorphisms from the group \( \mathbb{Z}_3 \times \mathbb{Z}_4 \times \mathbb{Z}_9 \) to the group \( \mathbb{Z}_{18} \).

Answer: 54


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

Answer: \dfrac{1}{4}


Question: Given that \( \phi(x) = \cos(x) - \int_0^x (x-t)\phi(t) \, dt \), find the value of \( \phi(x) + \phi''(x) \).

Answer: -\cos x


Question: Evaluate the limit: \[ \lim_{\epsilon\rightarrow 0^+} \int_0^1 \frac{1}{\epsilon x^3+1} \, dx \]

Answer: 1


Question: What is the order of the global truncation error at the end of an integration using the RK4 method with a step size of 2 seconds, from $t = 0$ seconds to $t = 258492$ seconds?

Answer: 4


Question: Evaluate the infinite series \( \sum\limits_{n=1}^{\infty}\frac{a(n)}{n(n+1)} \), where \( a(n) \) is the number of 1's in the binary expansion of \( n \).

Answer: 2 \ln 2


Question: Consider a measure-equivalence class of measurable non-null sets. Does this class necessarily have a non-empty intersection? Answer "Yes" or "No".

Answer: No


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

Let \( \mathcal M \) be a W*-algebra, and \( A \subseteq \mathcal M \) be a subalgebra closed under adjoint and containing the unit of \( \mathcal M \). Then the relative bicommutant \( A'' = \{ m \in \mathcal M \mid \forall x \in \mathcal M. (\forall a \in A. ax = xa) \implies mx=xm \} \) is a W*-algebra, and is the ultraweak closure of \( A \).

Answer: True


Question: Let $G$ be a finite solvable group such that $G = PQR$, where $P \in \Syl_{p}(G)$, $Q \in \Syl_{q}(G)$, and $R \in \Syl_{2}(G)$ with $|R| = 2$. Given that $C_P(R) = P$ and $C_Q(R) = 1$, and knowing $N_G(R) = C_G(R) = PR$, determine if $R^G \leq QR$.

Answer: Yes


Question: Evaluate the Fourier series \(2 + 2\sum_{k=1}^{\infty} (-1)^k\cos(kt)\cdot\frac{\sin(2k)}{k}\) at \(t = \pi-2\).

Answer: \dfrac{\pi}{2}


Question: Does complete regularity imply regularity in topological spaces?

Answer: Yes


Question: Find the closed form of the power series \( \sum_{k=0}^{\infty} x^k \frac{(2k)!}{(k!)^3} \).

Answer: e^{2x} I_0(2x)


Question: Let $f$ be a strictly positive (almost everywhere) measurable function that is integrable over a finite measure space. Let $E_n$ be a sequence of measurable sets such that $\int_{E_n} f \to 0$ as $n \to \infty$. Is it true that $\mu(E_n) \to 0$, where $\mu$ is a positive measure with respect to which $f$ is integrable?

Answer: Yes


Question: Determine the number of integer solutions \((x, y, z)\) to the congruence equation \(x^5 + y^5 \equiv z^5 \pmod{25}\).

Answer: 1625


Question: Given that \(f(z) = u(x, y) + iv(x, y)\) is a holomorphic function where \(u(x, y) = x^3 - 3xy^2\), find the imaginary part \(v(x, y)\) of the function.

Answer: 3x^2 y - y^3


Question: Provide an example of a Banach space $X$ for which the evaluation map $ev : X \to X''$, defined by $x \mapsto (f \mapsto fx)$, is not surjective.

Answer: c_0


Question: Given the rational function \( Q(x) \) such that \( x \cdot Q(x+2018) = (x-2018)Q(x) \) for all \( x \neq 2018 \) and \( x \neq 0 \), and \( Q(1) = 1 \), find the value of \( Q(2017) \).

Answer: 2017


Question: Is a real function that maps bounded sets to bounded sets necessarily of bounded variation?

Answer: No


Question: Evaluate the limit: \[ \lim_{x \to 1} \left(\frac{x}{x-1} - \frac{1}{\ln x}\right) \] without using L'Hôpital's rule, series expansion, or integration. Use known limits, the squeeze theorem, or basic limit theorems.

Answer: \dfrac{1}{2}


Question: If $A$ and $B$ are subsets of $\mathbb{N}$ with $\delta(A) = \delta(B) = 0$, does the natural density $\delta(A \cup B)$ exist?

Answer: 0


Question: Evaluate the limit \( \lim_{r\rightarrow 0} \frac{1}{r} \int_{C_{r}} \frac{\sin(z)}{z}dz \), where \( C_r \) is the contour \( re^{i\theta} \) for \( 0 \leq \theta \leq \pi \).

Answer: -2


Question: Let $(X,Y)$ follow a bivariate normal distribution where $X$ and $Y$ are both standard normal random variables with $\operatorname{cov}(X,Y)=c$. Determine $\operatorname{cov}(X^2,Y^2)$. Provide your answer in terms of $c$.

Answer: 2c^2


Question: Evaluate the integral \( \int_{L} z \cos(2\pi z \bar{z}) \, dz \), where \( L \) is the contour defined by \( |z| = 1 \) and \( 0 \leq \arg(z) \leq \pi \).

Answer: 0


Question: Let $f:\mathbb{C}\to \mathbb{C}$ be an entire function. Suppose that for every $z\in \mathbb{C}$, there exists $n_z\in \mathbb{N}$ such that $f^{(n_z)}(z)=0$. Is $f$ necessarily a polynomial?

Answer: Yes


Question: Let $X$ be a compact Hausdorff topological space, and $C(X)$ denote the ring of all complex-valued continuous functions on $X$. Given $A \in C(X)^{m \times n}$ and $b \in C(X)^{m \times 1}$, if for all $x \in X$, $b(x)$ belongs to the range of $A(x)$, does there exist a $y \in C(X)^{n \times 1}$ such that $Ay = b$? Provide a justification for your answer.

Answer: Yes


Question: Determine the element $a \in k$ such that for all $x \in k$, the additive character $\psi_k(x)$ satisfies $\psi_k(x) = \mathrm{Tr}_k(ax)$. Is it possible for $a$ to be equal to 1?

Answer: 1


Question: Determine whether \(-2\) is a quadratic residue modulo 307.

Answer: Yes


Question: Let $f$ be a smooth function, $f: \mathbb{R}^2 \to \mathbb{R}$. Determine the Lie bracket $\left[\frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right]$. What is the result?

Answer: 0


Question: Compute the dimension of the vector space $K \otimes_A K$ over the field $K$, where $K$ is the field of fractions of a discrete valuation ring $A$.

Answer: 1


Question: Find the value of $n$ that satisfies the equation $-4450(1.05)^{n} + 240n + 4800 = 0$. Express your answer as a whole number.

Answer: 10


Question: Determine whether the following statement is true or false: A closed and bounded subset of a complete metric space is compact. Provide a justification for your answer.

Answer: False


Question: Evaluate the double integral: $$\int_0^{\pi/2}\int_x^{\pi/2}\frac{\cos y}{y} \, dy \,  dx.$$

Answer: 1


Question: Find the closed form expression for the sum \( \sum_{i=0}^n {n \choose i} D(i) \), where \( D(i) \) represents the number of derangements of \( i \) elements.

Answer: n!


Question: Evaluate the integral: \[ \int_0^1 \frac{dx}{x+\sqrt{1-x^2}} \]

Answer: \dfrac{\pi}{4}


Question: Let \(p\) and \(q\) be distinct prime numbers. Prove that there are exactly two groups of order \(pq\), up to isomorphism.

Answer: 2


Question: Evaluate the line integral \( \int \vec{F} \cdot d\vec{l} \) for the vector field \( \vec{F} = zy\hat{x} + zx\hat{y} + xy\hat{z} \) from the point \((0,0,0)\) to \((1,2,3)\) without parameterizing the path.

Answer: 6


Question: Let $V = M_{3 \times 3}(\mathbb{C})$, the set of $3 \times 3$ matrices with complex entries. Consider the matrix $A = \begin{pmatrix} 0 & -2 & 0 \\ 1 & 3 & 0 \\ 0 & 0 & 2 \end{pmatrix}$. Define $W = \{ p(A) \mid p(t) \in \mathbb{C}[t] \}$, where $\mathbb{C}[t]$ is the set of polynomials with complex coefficients. Determine the dimension of the subspace $W$ in $V$. \(\dim W\).

Answer: 2


Question: Let $R$ be a ring with right global dimension $d = \operatorname{r.gl.dim}(R)$. Define $\operatorname{InjD} = \sup\{\operatorname{id}(R/I) \mid I \subset R\}$, where $I$ runs through all ideals of $R$. Prove or disprove that $\operatorname{InjD} = d$.

Answer: \operatorname{InjD} = d


Question: Calculate the present value at time $t = 0$ of a perpetuity that provides continuous payments. The annual rate of payment at time $t$ is $1$ if $0 \le t < 10$ and $(1.03)^{t-10}$ if $t \ge 10$. Use an annual effective interest rate of $6\%$. The correct present value is $27.03$. Verify this calculation.

Answer: 27.03


Question: Compute the integral \( \int_{C_r(0)} \sin\left(\frac{1}{z}\right)dz \) using residues, where \( r > 0 \).

Answer: 2\pi i


Question:
Let \(X_1, X_2, X_3\) be independent Poisson random variables with mean 1. Calculate \(P(\max(X_1, X_2, X_3) = 1)\). Choose the correct answer from the following options:

\((A)\ 1-e^{-3}\)
\((B)\ e^{-3}\)
\((C)\ 1-8e^{-3}\)
\((D)\ 7e^{-3}\)

Answer: D


Question: Evaluate the contour integral \( \int_C \frac{z}{2z^2+1} \, dz \) where \( C \) is the unit circle centered at the origin.

Answer: \pi i


Question: If a function $f$ is continuous on a real interval except for a single point where it is discontinuous, can $f$ be differentiable at that point?

Answer: No


Question: Is there a random variable $X$ with a probability density function that is neither even nor odd, such that $E(X) = 0$?

Answer: Yes


Question: If there are 6 matrices in \( M_{6,6}(\Bbb C) \) such that each satisfies \( A^2 = 0 \), does this imply that at least two of them are similar? Provide a true or false answer.

Answer: True


Question:
Let $x$ be a positive real number. Determine which of the following inequalities is true:

(A) $ x^{2} + \pi^{2} + x^{2\pi}> x\pi+ (\pi+x)x^{\pi} $

(B) $ x^{\pi} + \pi^{x}> x^{2\pi}+ \pi^{2x} $

(C) $ \pi x +(\pi+x)x^{\pi}>x^2 + \pi^2 + x^{2\pi} $

(D) None of the above.

Answer: A


Question: In triangle △PQR, ∠PRQ = 60° and PS = SR. If ∠PQS = x°, determine whether x is greater than, equal to, or less than 60°.

Answer: x < 60^\circ


Question: Compute the value of the infinite series: \[ \sum_{n=1}^\infty \frac{2n-1}{2 \cdot 4 \cdots (2n)}. \]

Answer: 1


Question: Find the smallest constant \( M \in \mathbb{N} \) such that for all \( n \in \mathbb{N} \), the inequality \( \pi(n) \leq \frac{Mn}{\log n} \) holds, where \( \pi(n) \) is the prime-counting function.

Answer: 2


Question: Consider a sequence of coin tosses where the probability of tails on the $n$-th toss is given by $P(H_{n}=0)=\sin\left(\frac{1}{n}\right)$ for $n = 1, 2, \ldots$. Determine the probability of observing infinitely many heads in this sequence.

Answer: 1


Question: Determine the values of the parameter $a$ for which the inequality \(|x^2 - ax + 1| < 3(x^2 + x + 1)\) holds for every real number $x$. Provide your answer in terms of $a$.

Answer: (-5, 1)


Question: In an unknown base system, two numbers are written as 402 and 302. In base 9, the product of these two numbers is 75583. Determine the unknown base.

Answer: 8


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left( \sin^4{x} + \frac{1}{4} \sin^4{2x} + \cdots + \frac{1}{{4}^n} \sin^4{2nx} \right) \]

Answer: \sin^2 x


Question: Calculate the volume under the paraboloid \(x^2+y^2=z\) and inside the cylinder \((x-1)^2+(y-1)^2=2\) above the plane \(z=0\).

Answer: 6\pi


Question: Determine the order of the group \( \text{Aut}(\Bbb Q(\sqrt[7]{3}, \zeta)/ \Bbb Q(\sqrt[7]{3})) \), where \( \zeta \) is a seventh root of unity. Given that the splitting field of the polynomial \( x^7 - 3 \) is \( E = \Bbb Q(\sqrt[7]{3}, \zeta) \), and the order of \( \text{Aut}(E/\Bbb Q) \) is 42, find the order of \( \text{Aut}(E/\Bbb Q(\sqrt[7]{3})) \).

Answer: 6


Question: Let \( A \) be a \( 3 \times 3 \) matrix with integer entries such that \( \det(A) = 1 \). What is the maximum possible number of even entries in \( A \)?

Answer: 6


Question:
Let $R$ be a commutative ring with unity and $1 \neq 0$. Let $a$ be a nilpotent element, and $x$ be a unit. Determine which of the following statements is true:

(A) $1+a$ is not a unit.
(B) $a-x$ is a nilpotent element.
(C) $a+x$ is a unit.
(D) None of the above.

Answer: C


Question: Consider compact subsets $X_i$ of a topological vector space (t.v.s) Hausdorff for each $i=1,\ldots,N$. Let $M_i$ be the set of regular probability measures on $X_i$, and $u_i: \prod X_i \rightarrow \mathbb{R}$ be a measurable bounded function. If $\lambda=(\lambda_1,\ldots,\lambda_N) \in \prod M_i$, does there exist a $\tilde{x}_i \in X_i$ such that \[ \int_{X_{-i}}u_i(\tilde{x}_i,y)d\lambda_{-i}(y) \geq \int_{X}u_i(x,y)d\lambda_i(x)d\lambda_{-i}(y) \] holds true?

Answer: Yes


Question:
Let $\Omega, \Omega^*$ be bounded domains in $\mathbb{R}^n$ and $u_0$ be a uniformly convex function defined on $\Omega$. Suppose the gradient of $u_0$ maps $\Omega$ into a subdomain of $\Omega^*$, i.e., $\omega^* = Du_0(\Omega) \subset \Omega^*$. The Legendre transform of $u_0(x)$ is given by the function
\[ v_0(y) = \sup_{x \in \Omega} \{ x \cdot y - u_0(x) \} : y \in \omega^* \]
Now, on $\Omega^\delta = \{ x \in \mathbb{R}^n \mid \text{dist}(x, \Omega) < \delta \}$, define a function $u_1$ which is also convex and extends $u_0$ to $\Omega^\delta$ (i.e., $u_1 = u_0$ in $\Omega$). Consider the Legendre transform of $u_1$ to give a function $v$,
\[ v(y) = \sup_{x \in \Omega^\delta} \{ x \cdot y - u_1(x) \}, y \in \Omega^* \]
Is $v$ the same as $v_0$ in $\omega^*$?

Answer: Yes


Question: Let $X$ and $Y$ be independent random variables, each uniformly distributed on $(0,1)$. Find the probability $P\left(\left|\frac{X}{Y} - 1\right| \leq 0.5\right)$. Express your answer as a fraction.

Answer: \dfrac{5}{12}


Question: Find the last three digits of the sum \(1^{2013} + 2^{2013} + 3^{2013} + \ldots + 1000^{2013}\).

Answer: 000


Question: Compute the limit: $$ \lim_{x\to 0^+} \left(\cfrac {\tan x}{x}\right)^{\cfrac{1}{x^3}}. $$

Answer: \infty


Question: For which values of $k$ is the blow-up $X_k = Bl_{p_1,...,p_k}\mathbb{P}^2$ of $\mathbb{P}^2$ at $k$ general points a Mori Dream Space?

Answer: k \leq 8


Question: Find all triples of non-negative integers \((a, b, c)\) such that \(a!b! = a! + b! + c!\).

Answer: (3, 3, 4)


Question:
Determine whether there exists a function \( f(x) \) that is defined and infinitely differentiable near \( x = c \) and satisfies the following properties:

For any positive real number \( \delta \), there exist real numbers \( x, x' \) such that \( c - \delta < x, x' < c \) and \( f(x) > f(c) \) and \( f(x') < f(c) \).

For any positive real number \( \delta \), there exist real numbers \( x, x' \) such that \( c < x, x' < c + \delta \) and \( f(x) > f(c) \) and \( f(x') < f(c) \).

Answer: Yes


Question: Solve the equation \( \operatorname{Log}(z) = -2+3i \) for \(z\), where \( \operatorname{Log} \) denotes the principal value of the complex logarithm.

Answer: e^{-2 + 3i}


Question: Given the equation \( \int_b^{x^2} f(t) \, dt = x \sin(\pi x) \), find the value of \( f(4) \).

Answer: \dfrac{\pi}{2}


Question: Evaluate whether the integral \( \int_C dz \ g(z) \), where \( g(z) = \frac{\partial f}{\partial z} \) and \( f(z) \) is holomorphic inside a closed contour \( C \) except for one pole, is zero or non-zero.

Answer: 0


Question: In a repeatable bet, the odds of winning a dollar are 51% and the odds of losing are 49%. If you start with $100, what is the probability that you will eventually go broke?

Answer: \left( \dfrac{49}{51} \right)^{100}


Question: What is the Lebesgue measure of the set of transcendental numbers in the interval $[0,1]$?

Answer: 1


Question: Does there exist a compact Hausdorff space that is not second countable and has finite dimension?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_0^n \frac{dx}{1+n^2\cos^2x} \]

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\tan^3 x - \sin^3 x}{x^5} \] without using l'Hôpital's rule.

Answer: \dfrac{3}{2}


Question: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f(0)=0$ and $f'(x) = f(x)^2$ for all $x \in \mathbb{R}$. Prove that $f(x) = 0$ for all $x \in \mathbb{R}$.

Answer: 0


Question: Find the greatest common divisor of \(2^{a}+1\) and \(2^{b}+1\), where \(a\) and \(b\) are prime numbers greater than 3.

Answer: 3


Question: Evaluate the limit: \[ \lim_{n\to\infty}\sum_{k=1}^n\left(\frac{2^{f(k)}+2^{-f(k)}}{2^{n}}\right) \] where \( f(n) = \left[\sqrt{n}+\frac{1}{2}\right] \) and \([\cdot]\) denotes the greatest integer function.

Answer: 0


Question: Let \( G \) be a topological group and \( H \) be a closed subgroup such that \( G/H \) is homeomorphic to \( \mathbb{R}^n \). Is it necessary that \( H \) is a normal subgroup of \( G \)?

Answer: No


Question: Given that $\frac{1}{n}\sum_{i=1}^n \hat{\theta}_i^2 = O_p(a_n)$ and $||f(X)||_{\infty}$ is bounded, determine the big $O_p$ of $\frac{1}{n}\sum_{i=1}^n (\hat{\theta}_i f(X_i))^2$.

Answer: O_p(a_n)


Question: Determine the Lipschitz constant for the canonical projection \( \pi^D_d(x) \) from \( \mathbb{R}^D \) onto \( X = \{x \in \mathbb{R}^D: x_{i}=0\, \forall d<i\leq D\} \).

Answer: 1


Question: Find the equation of the curve for which the part of the tangent cut off by the coordinate axes is bisected at the point of tangency.

Answer: xy = C


Question: Calculate the limit: $$\lim_{n\to\infty} \dfrac{\log\left( \dfrac{1}{\log(n+1)}-\dfrac{1}{\log(n+2)} \right)}{\log(n^{-2})}$$

Answer: \dfrac{1}{2}


Question: Given a weighted graph $G(V,E)$ where edges may have negative values, and a source node $s$, but no negative cycles exist, determine if there is an algorithm that finds all minimum weighted paths from $s$ to each node $v$ using at most $k$ edges in $k\cdot|E|$ time complexity.

Answer: Yes


Question: Find the degree of the field extension \( \mathbb{Q}( \sqrt[3]{2}, \zeta_3 \sqrt[3]{2}) \) over \( \mathbb{Q} \).

Answer: 6


Question: Let $p$ be an odd prime number, $G$ a finitely generated nonabelian profinite group, and $L \lhd_o G$ a pro-$p$ group with $[G : L] = 2$. Suppose there is a continuous surjection from $G$ onto a free pro-$p$ group of rank $d(G)$. Must $L$ be a free pro-$p$ group? Here, $d(G)$ is the smallest cardinality of a generating set of $G$. Provide a justification for your answer.

Answer: Yes


Question: Let \( G \) be a simply connected semisimple algebraic group over \( \mathbb{Q}_p \). Is any continuous homomorphism \( G(\mathbb{Q}_p) \to S^1 \) trivial?

Answer: Yes


Question: Evaluate the limit: \( \lim_{x\to 0}\frac{e^{-1/x^2}}{x^k}=0 \) for any positive integer \( k \).

Answer: 0


Question: Let \(X\) be a countable set of countable ordinals. Determine if there exists an ordinal \(\alpha < \omega_1\) that is an upper bound for all elements of \(X\).

Answer: Yes


Question: Let $f$ be a function such that \( \sqrt {x - \sqrt { x + f(x) } } = f(x) \) for \(x > 1\). In that domain, $f(x)$ has the form \(\frac{a+\sqrt{cx+d}}{b}\), where $a,b,c,d$ are integers and $a,b$ are relatively prime. Find $a+b+c+d.$

Answer: 2


Question: Determine whether every normal operator $T \in B(H)$ has a nontrivial invariant subspace when $\dim(H) > 1$. Provide a justification for your answer.

Answer: Yes


Question: Let \(a, b, c,\) and \(d\) be non-negative real numbers such that \(a + b + c + d = 1\). Determine the number of different ways to choose these numbers such that \(a^2 + b^2 + c^2 + d^2 = \max\{a, b, c, d\}\).

Answer: 15


Question: Determine whether there exist integers $a, b, c, d$ greater than 1, such that $a$ shares no factors with $b$, and a non-integer $n > 1$ satisfies the equations $a^n = c$ and $b^n = d$. Provide a justification for your answer.

Answer: Yes