Question: Find the 2021st derivative of the function \(f(x)=\frac{x}{(x-1)(1+x^2)}\) evaluated at \(x = 0\).

Answer: -2021!


Question:
In an acute triangle ABC, let $r_1$, $r_2$, and $r_3$ be the exradii opposite sides A, B, and C, respectively. Define $k$ as follows:
$$k = \frac{\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3}}{3}$$
Find the minimum value of the expression:
$$t = \frac{\sqrt{abc(a+b+c)}}{\delta}$$
where $a$, $b$, and $c$ are the side lengths of the triangle, and $\delta$ is the area of triangle ABC.

Answer: 4


Question: Evaluate the expression \(\sum_{k=1}^4 k \sum_{1 \le j_1 < j_2 < \ldots < j_k \le 4} (n_{j_1} + n_{j_2} + \ldots + n_{j_k})\) given that \(n_1 = 2\), \(n_2 = 3\), \(n_3 = 4\), and \(n_4 = 5\).

Answer: 280


Question: Given a matrix $A \in M(2,\mathbb{Z})$, define $\sin A$ as the infinite series $\sin A := \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}A^{2n+1}$, which converges absolutely in the Banach space $\mathcal{L}(\mathbb{R}^2)$ under the operator norm. If $\sin A$ is also a matrix in $M(2,\mathbb{Z})$, must it be true that $A^2 = O$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{n\rightarrow \infty} \frac{n}{(1^{1\over n}+2^{1\over n}+\dots+n^{1\over n})^n}.$$

Answer: 0


Question: In the RSA encryption system with modulus $m = 536813567 = 8191 \cdot 65537$ and public exponent $e = 3602561$, determine the maximum value of a plaintext block (encoded as a number in the "26-system") that can be encrypted without exceeding the modulus $m$. Assume blocks of length 5 are used for encoding the plaintext.

Answer: 11881375


Question: Calculate the determinant of an $n \times n$ matrix $B$ where each element $b_{ij}$ is given by $b_{ij} = c^{|i-j|}$, with $0 < c < 1$. Provide your answer in terms of $n$ and $c$.

Answer: (1 - c^2)^{n-1}


Question: Evaluate the limit \( \lim_{x\to 0} \frac{\arctan x+\arctan \frac{1}{x}}{\frac{1}{x}} \).

Answer: 0


Question: Determine if the property (*) is preserved for metrics that are Lipschitz equivalent to $d_1$.

Answer: Yes


Question: Determine if the element $x^3$ is invertible in the ring $R = \mathbb{Z}_3[x]/(x^4 - x^3 + x - 1)$. If it is invertible, find its inverse.

Answer: x^3


Question: Let $X_{(1)}, X_{(2)}, X_{(3)}$ be the order statistics of three independent random variables $X_1, X_2, X_3$ each uniformly distributed on $(0,1)$. Find the probability that $X_{(1)} + X_{(2)} \leq 1$. \( \text{Express your answer as a fraction.} \)

Answer: \dfrac{3}{4}


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

Answer: 1


Question: Compute the integral of the function \( f(x) \) from 0 to 2, where \( f(x) \) is defined as 0 at points \( \frac{n}{n+1} \) for natural numbers \( n \) and 1 everywhere else.

Answer: 2


Question: Find the number of integer pairs \((x, y)\) such that \(x^2 + y^2 = 5^k\), where \(k\) is a fixed positive integer.

Answer: 4(k + 1)


Question: Is the bounded derivative $f'$ of a continuous and differentiable function $f:[a,b] \to \mathbb{R}$, with one-sided derivatives at $a$ and $b$, necessarily Riemann integrable on $[a,b]$? Answer yes or no.

Answer: No


Question: Evaluate the limit \( \lim_{x\to 1}f(x) \), where \( f(x) = \int_{x}^{x^2}\frac{1}{\ln t}dt \).

Answer: \ln 2


Question: A 2D Brownian motion starts at the point \((2,2)\). Calculate the probability that it hits the line \(x=0\) before it hits the line \(y=0\).

Answer: \dfrac{1}{2}


Question: For a function $f$ defined on all real numbers, if $f''(p) = 0$ and $f'''(p) \neq 0$, does the graph of $f$ have an inflection point at $x = p$? Answer with 'Yes' or 'No' and justify your answer.

Answer: Yes


Question: Calculate the volume of the solid bounded by the paraboloids \(z = 2x^2 + y^2\) and \(z = 12 - x^2 - 2y^2\) in the first octant.

Answer: 6\pi


Question: Suppose $R[t, t^{-1}]$ is Noetherian. Is $R[t]$ necessarily Noetherian? (Here, $R$ is a commutative ring with an identity.)

Answer: Yes


Question: Solve the equation: $$5 \log_{4}a + 48\log_{a}4 = \frac{a}{8}$$

Answer: 256


Question: Calculate the limit: $$\lim_{x\to \infty} \left(\frac{x+2}{x-1}\right)^{x+1}$$ without using L'Hôpital's rule.

Answer: e^3


Question: Determine whether the following statement is true or false: Any complex matrix in \(GL(m,\Bbb{C})\) can be embedded into \(GL_+(2m,\Bbb{R})\) by treating it as a block matrix.

Answer: True


Question: Consider three finitely generated modules $A$, $B$, and $C$ over a principal ideal domain (PID) $F$. If $A \oplus B$ is isomorphic to $A \oplus C$, is $B$ necessarily isomorphic to $C$? Determine whether this statement is true or false.

Answer: True


Question: Determine whether the following statement is true or false: If $p(x)$ is a polynomial with a negative leading coefficient and odd degree such that $p(x) \le p''(x)$ for all real $x$, then $p(x) \le 0$ for all real $x$. Provide a justification for your answer.

Answer: True


Question: Find the number of integer solutions to the congruence \(x^3 \equiv 3y^2 \pmod{11}\) using the theory of primitive roots and indices.

Answer: 11


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

Answer: 0


Question: Determine the possible cardinalities of the equivalence classes formed by the relation \( x \sim y \iff d(x,y) \in \mathbb{Q} \) on the set of real numbers \( \mathbb{R} \).

Answer: \aleph_0


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

Answer: \ln 2


Question: Evaluate the integral \( \int_{0}^{2\pi} \ln \left(|e^{ix}-1| \right)\,dx \) and express the result in terms of \( \pi \).

Answer: 0


Question: Evaluate the integral \( \int_{0}^{\infty}\frac{\sin(3x)\sin(x)}{x^2}\,\mathrm{d}x \) using integration by parts.

Answer: \dfrac{\pi}{2}


Question: Determine whether there exist fields $F$ and $K$ such that every element in $F$ has zero trace relative to $K$. Provide a justification for your answer.

Answer: Yes


Question: Given a constant $C$ where $0 < C < \frac{\pi}{2}$, determine the values of $A_1, A_2, \ldots, A_n > 0$ that minimize the sum $\sum_{i=1}^n \tan(A_i)$ subject to the constraint $\sum_{i=1}^n A_i = C$. Provide the values of $A_1, A_2, \ldots, A_n$ that achieve this minimum.

Answer: \frac{C}{n}


Question: Calculate the sum of the infinite series: \[ \sum_{n=0}^\infty \frac{(n+1)^2}{2^n} \]

Answer: 12


Question:
Given two independent random variables, $X$ and $Y$, where $X \sim N(1,4)$ and $Y \sim N(1,9)$, find the value of $a$ such that:

$$P(2X + Y \le a) = P(4X - 2Y \ge 4a)$$

Answer: \dfrac{4}{3}


Question: Given a set \( S = \{1, 2, \ldots, 2005\} \), find the maximum size of a subset \( A = \{a_1, a_2, \ldots, a_k\} \subset S \) such that for any distinct elements \( a_i \) and \( a_j \) in \( A \), the sum \( a_i + a_j \) is not a multiple of 125. What is the maximum possible value of \( k \)?

Answer: 998


Question: If the integral of a vector field over every closed curve is zero, does this imply that the vector field is conservative?

Answer: Yes


Question: Given triangle KMN with KM = 6, MN - KN = 2 \cos(\angle KMN), and \cos(\angle KMN) = \frac{3}{5}, determine the possible areas of triangle KMN.

Answer: \dfrac{432}{25}


Question: Calculate the rate of mass flow through the sphere defined by \( \rho = 1 \) in the outward direction, given the fluid density \( f(x, y, z) = x^{2} + 2y^{2} + z^{2} \) and velocity \( v(x, y, z) = x^{2}i + xy^{2}j + zk \).

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


Question: Evaluate the definite integral \(\int_{0}^{1} \dfrac{1}{1 + \left(1 - \dfrac{1}{x}\right)^{2015}} \, dx\). Consider the behavior at the lower limit \(x = 0\).

Answer: \dfrac{1}{2}


Question: Determine the limit of the function \( f(a) = (\sqrt{a})^{f(\sqrt{a})} \) as \( a \) approaches 2. If the limit exists, what is its value?

Answer: 2


Question: Determine if it is valid to find the radius of convergence $R$ for the power series $$\sum_{n=1}^{\infty}(-1)^{n}n^{-\frac{2}{3}}x^{n}$$ by using the terms $a_{n} := (-1)^{n}n^{-\frac{2}{3}}x^{n}$ and calculating the limit $$\lim\limits_{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right| = \lim\limits_{n \rightarrow \infty}\left|x\left(\frac{n}{n+1}\right)^{\frac{2}{3}}\right| = |x|.$$ Conclude whether the radius of convergence is $|x|$.

Answer: 1


Question: Determine whether the following statement is true or false: A sequence $(x_n)$ converges if every subsequence of $(x_n)$ has a further subsequence that converges, without assuming that all convergent subsequences have the same limit.

Answer: B


Question: Given a positive integer $m > 0$ where $m \neq 2$, let $c_1, c_2, \ldots, c_{\phi(m)}$ be a reduced residue system modulo $m$. Determine the value of $x \equiv c_1 + c_2 + \cdots + c_{\phi(m)} \pmod{m}$.

Answer: 0


Question: Does the result of diagonalizing a matrix change if the eigenvectors used as columns of the matrix U are multiplied by different non-zero constants?

Answer: No


Question: Consider the sequence of real numbers \((a_n)_{n \ge 1}\), where the expression \(1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} - a_n \ln n\) is bounded for all \(n \ge 1\). Determine the limit of the sequence \((a_n)_{n \ge 1}\).

Answer: 1


Question: Evaluate the integral \( \int_\gamma (z+1)e^{z+1}dz \), where \( \gamma \) is the semicircle of radius 1 centered at the origin, traversed from \(-1\) to \(1\).

Answer: e^2 + 1


Question: Given an orthonormal positively oriented basis $B=(\mathbf{i}, \mathbf{j}, \mathbf{k})$ and a real number $h$, let \( \mathbf{v} = \mathbf{i}-h\mathbf{k} \), \( \mathbf{w} = 2\mathbf{i}-\mathbf{j}+\mathbf{k} \), and \( \mathbf{z}= h\mathbf{i}+\mathbf{j}-\mathbf{k} \). Determine the values of $h$ for which $(\mathbf{v},\mathbf{w},\mathbf{z})$ forms a positively oriented basis.

Answer: (-2, 0)


Question: Is $co^{-1}(V)$ an open set in $\mathcal{F}$, where $\mathcal{F}$ is the set of all finite subsets of $\mathcal{R}^n$ equipped with the Hausdorff metric, $\mathcal{C}=\{co(F):F\in\mathcal{F}\}$ is the set of convex hulls of elements in $\mathcal{F}$ also endowed with the Hausdorff metric, and $V$ is an open set in $\mathcal{C}$?

Answer: Yes


Question: Determine the dual space of the space of all bounded functions on an arbitrary set $X$, where the sigma-algebra is the power set $2^X$.

Answer: ba(X, 2^X)


Question: Determine the number of ordered pairs \((x, y)\) of positive integers that satisfy the equation \(x^2 - 2y^2 = 1\), where \(y\) is a prime number.

Answer: 1


Question: Compute the contour integral \( \int_{\Gamma} z^{-2} \, dz \) where \( \Gamma = C_1(0) \), a circle of radius 1 centered at the origin, traced counterclockwise. Verify the result using the anti-derivative \( F(z) = -z^{-1} \).

Answer: 0


Question: Evaluate the limit of the ratio obtained in the ratio test for the series \(\sum\limits_{n=1}^\infty \frac{n^n(n!)}{(2n)!}\) as \(n\) approaches infinity.

Answer: \dfrac{e}{4}


Question: Determine whether every quotient of a direct sum of cyclic $p$-groups is also a direct sum of cyclic $p$-groups. Answer with true or false.

Answer: True


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left(\int_0^2 |2x-1|^n \, \mathrm{d}x\right)^{1/n}. \]

Answer: 3


Question: Determine whether the following statement is true or false: Let $k$ be a field of characteristic zero and size $\le\mathfrak{c}$. If $X$ is a $k$-variety, then to any pair of embeddings $\sigma,\tau:k\hookrightarrow\mathbb{C}$ we can associate a homotopy class of homotopy equivalences $\varphi_{\sigma,\tau}:X\times_\sigma\operatorname{Spec}\mathbb{C}\to X\times_\tau\operatorname{Spec}\mathbb{C}$ (analytic topologies) satisfying the cocycle conditions $\varphi_{\sigma,\upsilon}=\varphi_{\tau,\upsilon}\circ\varphi_{\sigma,\tau}$, $\varphi_{\sigma,\sigma}=\mathrm{Id}$.

Answer: False


Question: For a finite group $G$ with a normal 2-Sylow subgroup $H$, determine the probability that a randomly chosen element from the group algebra $F_2[G]$ is a unit, where $F_2$ is the field of characteristic 2.

Answer: \dfrac{1}{2}


Question:
Find the coordinates of point $p$ given the following conditions: Point $a = (3,2,5,1)$, point $b = (3,4,7,1)$, and point $c = (5,8,9,3)$. Point $p$ lies on the same line as $a$ and $b$, and shares two coordinates with them, so $p = (3, x, y, 1)$ where $x, y \in \mathbb{R}$ are unknown. Additionally, triangles $\triangle acp$ and $\triangle bcp$ are right-angled. Use these conditions to determine the coordinates of $p$. The equations for the right-angled triangles are:

1. For $\triangle acp$: $x^2 + y^2 - 10x - 14y + 61 = 0$.
2. For $\triangle bcp$: $x^2 + y^2 - 12x - 16y + 95 = 0$.

Solve these equations to find the coordinates of point $p$. \(x + y - 17 = 0\) is derived from the equations. Use this to find the exact values of $x$ and $y$.

Answer: (3, 7, 10, 1)


Question: Evaluate the integral \( \int_{0}^{1} x^{1/2} (1 - x)^{-1/2} \, dx \) using the properties of the Beta function.

Answer: \dfrac{\pi}{2}


Question: Does there exist a countable separating family for the space of probability measures on a Polish space \((E, \mathcal{A})\)?

Answer: Yes


Question: Given any solution \(u(x, y)\) to the partial differential equation \(au_{x} + bu_{y} = 0\), where \(u(1, 2) = u(3, 6)\), determine the value of \(\frac{b}{a}\) based on the fact that \(u(x, y)\) is constant along lines defined by \(bx - ay = c\), for all \(c \in \mathbb{R}\).

Answer: 2


Question: Find the inverse Laplace transform of the function \( H(s) = \frac{1}{s^4 - s^2} \).

Answer: \sinh(t) - t


Question: Given that \( s = \sqrt{a} + \sqrt{b} + \sqrt{c} \), where \( a \), \( b \), and \( c \) are the roots of the cubic equation \( x^3 - 9x^2 + 11x - 1 = 0 \), find the value of the expression \( s^4 - 18s^2 - 8s \). Use Vieta's formulas and the relationship \( \rho = \sqrt{ab} + \sqrt{bc} + \sqrt{ac} \) to assist in your calculations.

Answer: -37


Question: Evaluate the limit of the series: $$ \lim_{n \rightarrow \infty} \sum_{k=1}^n{\frac{k \sin k}{n^3}} $$

Answer: 0


Question: Evaluate the limit of the function \( g(x,y)=\frac{x^4-x^2y^2}{x^2+y^2} \) as \((x,y)\) approaches \((0,0)\) using a method other than polar coordinates.

Answer: 0


Question: Differentiate the integral \( \int_{2x}^{0} f(u) \, du + \int_{0}^{3x} f(u) \, du \) with respect to \( x \).

Answer: 3f(3x) - 2f(2x)


Question: Evaluate the limit: $$\lim_{x\to0} [\ln(1+\sin^2(x))\cdot\cot(\ln^2(1+x))].$$

Answer: 1


Question: Find the angle at which the parametric curve \( \gamma(t) = \begin{bmatrix} t^2-1 \\ t(t^2-1) \end{bmatrix} \) intersects itself at the origin \((0,0)\).

Answer: \dfrac{\pi}{2}


Question: Determine the number of real roots of the equation \((x^2-4x+1)\exp(x) = 1\).

Answer: 3


Question: Let $F/K$ be a finite field extension and $p(x) \in K[x]$ be an irreducible polynomial with a root in $F$. Does this imply that the degree of the extension $[F:K]$ is a multiple of the degree of $p(x)$?

Answer: Yes


Question: Evaluate the sum \( \displaystyle \sum^{10}_{r=0}\frac{1}{1+w^r+w^{2r}} \) given that \( w^{11}=1 \) and \( w\neq 1 \).

Answer: \dfrac{22}{3}


Question: Find the smallest element in the set \( S = \left\{ s \mid s = \frac{\text{lcm}(a, a+d, a+2d, \ldots, a+10d)}{a+10d} \right\} \), where \( a \) and \( d \) are positive integers and \( \text{lcm} \) denotes the least common multiple.

Answer: 2310


Question: In triangle $\triangle ABC$, points $D$ and $E$ are located on sides $CA$ and $AB$ such that $BE = 6$ and $CD = 10$. Let $M$ and $N$ be the midpoints of segments $BD$ and $CE$, respectively, with $MN = 7$. Find the measure of $\angle BAC$. \(\text{(Express your answer in degrees.)}\)

Answer: 120


Question: Is $K$, the kernel of $G_{\alpha}$ acting on $\Omega - \alpha$, also the kernel of $G$ acting on $\Omega$?

Answer: Yes


Question: Find the value of $k$ such that for a standardized normal variable $Z$, the probability $P(0 < Z < k)$ is equal to twice the probability $P(k < Z < 2k)$. Express your answer in terms of $k$.

Answer: 0.86


Question: If every proper closed subset of a metric space is complete, does it imply that the entire metric space is complete?

Answer: Yes


Question: Calculate the limit \(\lim_{r\rightarrow 0^+}\int_{D_r} \frac{x+1}{y} \, dx \, dy\) where \(D_r=\{(x,y)\in \mathbb{R}^2 : x^2+y^2\ge r^2, 0\le x\le y^2, r\le y\le 1\}\).

Answer: \dfrac{5}{8}


Question: Given 4 equilateral triangles on a plane, each with distinct vertices, forming a set S of 12 vertices, and no three points in S are collinear, determine the maximum number of squares that can be formed using the vertices from S.

Answer: 3


Question: Determine the range of \( \alpha \) for which the function \( c(u_1, u_2) = 1 + \alpha(1- 2u_1)(1- 2u_2) \), where \( u_1, u_2 \in (0,1) \), is a copula density for the random variables \((U_1, U_2)\).

Answer: [-1, 1]


Question: Determine which of the following numbers does not divide $2^{1650} - 1$: $3$, $7$, $31$, $127$, $2047$.

Answer: 127


Question: Determine whether there exists a bijective function \( g:(0,1) \to (0,1) \) such that its graph is a dense subset of \((0,1) \times (0,1)\).

Answer: Yes


Question: Find the length of the arc in the first quadrant of the semi-cubical parabola $y^2 = x^3$ from the point where $x=0$ to the point where $x = \frac{5}{9}$. Provide your answer to three decimal places.

Answer: 0.704


Question: Find the radius of convergence for the Taylor series of the function \( g(z) = \frac{12z}{z^2 - z - 2} \) about the point \( z = -i \).

Answer: \sqrt{2}


Question: Evaluate the expression $i^{2i}$.

Answer: e^{-\pi}


Question: Find the maximum value of the function \( f(t) = 16 \cos t \cdot \cos 2t \cdot \cos 3t \cdot \cos 6t \).

Answer: 16


Question: Evaluate the integral \( \int_{0}^{3} (x^2+1) \, d[x] \), where \([x]\) denotes the greatest integer less than or equal to \(x\).

Answer: 17


Question: Let $f:[0,1] \to \mathbb{R}$ be a continuous function such that $f(x) = f(x^3)$ for all $x \in [0,1]$. Find the value of $f(1) - f(0)$. \( \boxed{0} \)

Answer: 0


Question: In a hemigroup with the following properties: (i) Associativity: $(x\odot y)\odot z = x\odot (y\odot z)$, (ii) Commutativity: $x\odot y = y\odot x$, (iii) Cancellation property: $(x\odot y) = (x\odot z) \implies y = z$, and (iv) Existence of an idempotent element: $\exists_e e\odot e = e$, consider the set $\mathcal{S} = \{\xi | \xi = x\odot e \land e, x\in \mathcal{H}\}$, where $\mathcal{H}$ is the set of the hemigroup and $\odot$ is preserved in $\mathcal{S}$. Does $e$ become an identity element for $\mathcal{S}$?

Answer: Yes


Question: Solve the equation involving a floor function: \[ a\left(\big \lfloor\frac {a} {2\pi}\big \rfloor + 1\right) = 100 \]

Answer: 25


Question: Determine whether the following statement is true or false: For any two finite sets $X$ and $Y$, if $\mu$ is a probability distribution over $Y^X$, and $\sigma$ is a permutation on $X$, then $\mu \circ f_\sigma^{-1} = \mu$ for all $\sigma$, if and only if $\mu$ is the uniform distribution on $Y^X$. Justify your answer.

Answer: False


Question: Evaluate the integral \( \int_C{\frac{\sin z}{z^2(z-2)}}dz \) where \( C \) is the contour defined by the equation \( x^2+xy+y^2=1 \).

Answer: -\pi i


Question: Find the length of the curve given by the polar equation \( r = \cos^2\left(\frac{\theta}{2}\right) \).

Answer: 4


Question: Is the base field $F$ uniquely determined in the definition of a CM-field?

Answer: Yes


Question: Does there exist a set within the interval [0,1] that has zero Lebesgue measure but a Hausdorff dimension of 1?

Answer: Yes


Question: Calculate the limit: $$\lim \limits_{x \to +\infty}\cos(x)\left(\ln\left(\sqrt{x-1}+2\right)-\ln\left(\sqrt{x+2}\right)\right)$$

Answer: 0


Question: Let \(X_1, X_2, \ldots, X_n \sim \text{Unif}[0, \beta]\). Define the estimators \(\hat{\beta_1} = (n+1)X_{\min}\) and \(\hat{\beta_2} = \frac{n+1}{n} X_{\max}\). Compute the ratio of the mean squared errors of \(\hat{\beta_1}\) and \(\hat{\beta_2}\).

Answer: n^2


Question: Consider a morphism of schemes $(f,f^\sharp): (X,\mathcal O_X)\to (Y,\mathcal O_Y)$. We have $f^\sharp: \mathcal O_Y\to f_*\mathcal O_X$. This induces a morphism $f_* f^{-1} \mathcal O_Y\to f_*f^{-1}f_*\mathcal O_X$. Given the canonical morphisms of sheaves $\mathcal O_Y\to f_*f^{-1} \mathcal O_Y$ and $f^{-1}f_*\mathcal O_X\to \mathcal O_X$, we obtain a morphism $f_*f^{-1}f_*\mathcal O_X\to f_*\mathcal O_X$. By composing these three morphisms, we derive a morphism $\mathcal O_Y\to f_*\mathcal O_X$. Determine if this morphism is the same as $f^\sharp$. \(\text{Yes or No}\)

Answer: Yes


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

Answer: 4


Question: Given the functions \( f(x) = x^3 - 5x^2 + x - 7 \) and \( f(g(x)) = 27x^3 + 90x^2 + 78x - 2 \), find the function \( g(x) \) without calculating the inverse of \( f(x) \).

Answer: 3x + 5


Question: Find the last two digits of the term $T_4$ in the sequence defined by $T_n = 3^{T_{n-1}}$ with $T_1 = 3$. Assume that for $n > 10$, the last 10 digits of the sequence remain constant.

Answer: 87


Question: Let $A$ be a $3 \times 3$ matrix with real entries that commutes with all other $3 \times 3$ matrices with real entries. What is the maximum number of distinct roots that the characteristic polynomial of $A$ can have? Provide your answer as a single integer.

Answer: 1


Question: Given three complex numbers $z$, $w$, and $u$ such that $|z|=1$, $|w|=2$, $|u|=3$, and $|z+w-u|=|u+z-w|$, determine the greatest possible value of $|z-u|$. Provide your answer as a single number.

Answer: \sqrt{14}


Question: Determine the ring to which the quotient ring \( R[X,Y,Z,T]/(X-Y^2,T-Y^4,T^3-Z) \) is isomorphic.

Answer: R[Y]


Question: Determine whether the polynomial \(1+xy+(xy)^2+\dots+(xy)^n\) can be expressed as a sum of products of polynomials in \(x\) and \(y\) for all \(x, y \in \mathbb{R}\).

Answer: Yes


Question: Determine the number of elements in the quotient group \((\mathbb{Z}\times\mathbb{Z})/M\), where \(M\) is the subgroup generated by the elements \((2,4)\) and \((4,2)\).

Answer: 12


Question: Evaluate the polynomial congruence \(\sum_{j=(p-1)/2}^{p-1}\left(\frac{j}{p}\right)j(j-1)...(j-(p-1)/2+1) \ (\textrm{mod} \ p)\), where \(p\) is an odd prime.

Answer: -1


Question: Given the sequence of real numbers $(x_n)_{n \geq 1}$, where $\lim_{n \rightarrow \infty} (x_{2n} + x_{2n+1} )= 315$ and $\lim_{n \rightarrow \infty} (x_{2n} + x_{2n-1}) = 2003$, evaluate $\lim_{n \rightarrow \infty} \frac{x_{2n}}{x_{2n+1}}$. \( \text{Express your answer as a single number.} \)

Answer: -1


Question:
Find the minimum value of the function:

$$f(x, y) = \sqrt{4 + y^2} + \sqrt{(x - 2)^2 + (2 - y)^2} + \sqrt{(4 - x)^2 + 1}$$

where \(x\) and \(y\) are real numbers.

Answer: 5


Question: If \( f(x)=\frac{1}{x^2+x+1} \), find \( f^{(36)} (0) \).

Answer: 36!


Question: Let $X$ be a scheme, and $\mathcal A$ be a quasi-coherent $\mathcal O_X$-algebra. Suppose there is a morphism $\mu: W \rightarrow X$ and a morphism of sheaves from $\mu_{*}(\mathcal O_W)$ to $\mathcal A$ that is a morphism of $\mathcal O_X$-algebras. If $\mu$ is affine, does this imply there exists an $X$-morphism from $\operatorname{Spec}(\mathcal A)$ to $W$?

Answer: Yes


Question: Given the matrix \( A= \begin{pmatrix} 0 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{pmatrix} \), determine the number of values \( n \) for which there exists a matrix \( X \) such that \( X^n = A \).

Answer: 1


Question: Determine whether there exist any limit ordinals that are finite cardinals. Provide a yes or no answer.

Answer: No


Question: Evaluate the integral \( \iiint_{H}(x+y+z)^{2} \, dV \) where \( H \) is the region defined by the inequalities: \( 0 \leq x+y+z \leq 1 \), \( 0 \leq x+2z \leq 3 \), and \( 0 \leq x+2y+3z \leq 2 \). Use the change of variables \( u = x+y+z \), \( v = x+2z \), and \( w = x+2y+3z \) to evaluate the integral.

Answer: \dfrac{2}{3}


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

Answer: \dfrac{1}{4}


Question: Let $V$ be a finite dimensional vector space over the field with $p$ elements, where $p$ is a prime number, and let $d$ be the dimension of $V$. Two subsets $A$ and $A'$ of $V$ are equivalent if there exists an automorphism $f$ such that $f(A) = A'$. This defines an equivalence relation on the set of subsets of $V$. Determine the number of equivalence classes containing generating subsets of $V$ with $d+1$ elements.

Answer: 2


Question: Suppose $f \in L^2(\mathbb{R}^n)$ and consider a function $h \in L^2(\mathbb{R}^n)$ with its Fourier transform denoted by $\hat{h}$. Given that the function \(g(\xi) = \sum_{j=1}^n \xi_j \hat{f}(\xi)\) belongs to $L^2(\mathbb{R}^n)$, determine if it is necessarily true that each component $\xi_j \hat{f}(\xi)$ is in $L^2(\mathbb{R}^n)$. Justify your answer.

Answer: Yes


Question: For the set \( A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \), determine the number of functions \( f: A \to A \) such that \( f^{30}(x) = x \) for all \( x \in A \), where 30 is the smallest positive integer for which this condition holds.

Answer: 120960


Question: Determine the largest possible value of the modulus of the entire function $f(z)$ at $z = \ln(2)$, given that $f(z)$ is analytic for all $z \in \mathbb{C}$ and satisfies $|f(z)| \leq |e^z|$ along the contour $|z| = 1$. Can the Maximum Modulus Principle provide a better upper bound than $e$?

Answer: 2


Question: Calculate the sum of the infinite series \( \sum_{n=0}^{\infty} \frac{(n+1)^3}{3^n} \).

Answer: \dfrac{99}{8}


Question: Evaluate the integral \( \int_0^{\infty} \frac{x^4}{e^x} \, dx \).

Answer: 24


Question: Evaluate the line integral \( \oint_C \mathbf{F} \cdot \, \mathrm{d} \mathbf{r} \) where \( \mathbf{F} = (e^x + xy) \mathbf{i} + (e^y - xy^2) \mathbf{j} \) and \( C \) is the circle \( x^2 + y^2 = 4 \), oriented clockwise. Use Green's Theorem to compute the integral as a double integral over the interior of the circle.

Answer: 4\pi


Question: Evaluate the integral \( \int_{-\infty}^\infty \frac{x \cdot \cos(x)}{x^2+4} \, dx \).

Answer: 0


Question: Evaluate the integral \( \int_0^{100}\frac{e^{-\sqrt{x}}}{\sqrt{x}+100}dx \) and determine if the result is greater than 0.005.

Answer: Yes


Question: Given the joint density function \( f_{X,Y}(x,y) = \frac{1}{4}xy \) for \( 0 < x < 2 \) and \( 0 < y < 2 \), find the probability \( P(X-Y<1) \).

Answer: \dfrac{89}{96}


Question: Evaluate the integral: $$\int_{0}^{\pi/2}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\, \mathrm{d}x.$$

Answer: \dfrac{\pi}{4}


Question:
Evaluate the limit of the complex function as \( z \to 0 \):

\[ \lim_{z\to 0} \frac{z \cdot \text{Re}(z)}{\overline{z}} \]

where \( z = x + yi \) and \((x, y) \to (0, 0)\).

Answer: 0


Question: Calculate the limit of the expression \(\lim_{n \to \infty}(\alpha^{1/(n \cdot \ln n)}-1)^{1/n}\) as \(n\) approaches infinity, where \(2<\alpha<3\) is a real number.

Answer: 1


Question: Determine whether the following statement is true or false: If two polynomials of degree $n > 2$ have a common root in $\mathbb{Z}/p\mathbb{Z}$, where $p$ is prime and $n|(p-1)$, then they have a common root in $\mathbb{C}$. Justify your answer.

Answer: False


Question: Consider a fibre bundle \( F \to E \xrightarrow{\pi} B \) over a connected and compact base \( B \), where both the fibre \( F \) and the base \( B \) have the homotopy type of a CW complex. Does the total space \( E \) necessarily have the homotopy type of a CW complex?

Answer: Yes


Question: Suppose a sequence of functions $f_n(x)$ converges pointwise to $+\infty$ as $n \rightarrow +\infty$ on a set $D \subset \mathbb{R}$. Given that $\arctan(f_n(x))$ converges uniformly to $\frac{\pi}{2}$ on $D$, does it follow that $f_n(x)$ converges to $+\infty$ uniformly on $D$?

Answer: Yes


Question: Evaluate the limit: $$\lim\limits_{x \to +\infty} \frac{\cos(2x^{-\frac{1}{2}})-1}{\frac{\pi}{2}-\arctan(3x)}$$ without using integrals, derivatives, or L'Hopital's rule.

Answer: -6


Question: Is there a metric $d$ on the open interval $(0,1)$ such that the topology induced by $d$ is the same as the topology induced by the usual Euclidean metric on $(0,1)$?

Answer: Yes


Question: Can a real-valued square matrix \(\textbf{A} \in \mathbb{R}^{n \times n}\) have no real eigenvalues when \(n\) is odd?

Answer: No


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

Answer: x^{n/m} - a


Question: Solve for $x$ in the equation \( \log_a(3x-4a)+\log_a(3x)=\frac{2}{\log_2a}+\log_a(1-2a) \) given that \( 0<a<\frac{1}{2} \).

Answer: \dfrac{2}{3}


Question: Given the matrix \(A=\begin{pmatrix} 1 & -1\\ 1 &1 \end{pmatrix}\), where \(A^{n}=\begin{pmatrix} x_{n} & -y_{n}\\ y_{n} &x_{n} \end{pmatrix}\), and the properties \(2A-A^{2}=2I_{2}\) and \(A^{48}=2^{24}I_{2}\), find the value of \(\frac{x_{12}^{2}+y_{12}^2}{x_{10}^{2}+y_{10}^2}\).

Answer: 4


Question: Let $A$ be a $5 \times 5$ matrix with real entries such that $A^2 - 4A - I = 0$, where $I$ is the identity matrix. If $a_1, a_2, a_3, a_4, a_5$ are the eigenvalues of $A$, find the value of the expression: \[ \left(a_1-\frac{1}{a_1}\right)+\left(a_2-\frac{1}{a_2}\right)+\left(a_3-\frac{1}{a_3}\right)+\left(a_4-\frac{1}{a_4}\right)+\left(a_5-\frac{1}{a_5}\right) \]

Answer: 20


Question: Evaluate the integral \( \iint_{x^2+y^2<\sqrt{x}}\sqrt{x^2+y^2}\,dxdy \) over the region where \( x^2 + y^2 < \sqrt{x} \).

Answer: \dfrac{2}{3}


Question: Find the integer value of \( a \equiv 19^{10562} \pmod{40320} \).

Answer: 361


Question: Find the number of solutions to the equation $x^d = e$ in a cyclic group of order $n$, where $d = \gcd(m,n)$ and $m$ does not divide $n$. Provide your answer as a single integer.

Answer: d


Question: Determine whether the sequence \( \frac{e^n \cdot n!}{n^n} \) converges or diverges.

Answer: \infty


Question: Evaluate the triple integral \( \iiint_K e^z \, dV \) where \( K \) is the region inside the sphere \( x^2+y^2+z^2 \leq 1 \), considering only the portion of the sphere above the \( xy \) plane.

Answer: \pi


Question: Find an \( \mathbb{R} \)-basis for the quotient ring \( \mathbb{R}[x,y]/(x^2-x,y^2-y) \).

Answer: \{1, x, y, xy\}


Question: Let $X_1, X_2, \ldots, X_{10}$ be independent and identically distributed random variables, each uniformly distributed on $(-5,5)$. Determine the distribution of the random variable \( Y = -2\sum\limits_{i=1}^{10}\log\left(\frac{|X_i|}{5}\right) \).

Answer: \chi^2(20)


Question: Evaluate \((25^{1202} + 3)^2 \mod 61\).

Answer: 19


Question: Determine the length of the longest snake-like path that can be packed into a 4x4x5 cuboid.

Answer: 80


Question: Find the number of non-trivial solutions to the congruence equation \(x(x+1) \equiv 0 \pmod{2^k3^m}\) for any positive integers \(k\) and \(m\).

Answer: 2


Question: Find the minimum value of the integral \( \int_{0}^{1}|t^3-a-bt-ct^2|^2dt \) by using the projection of \( t^3 \) onto the subspace of polynomials of degree at most 2.

Answer: \dfrac{1}{2800}


Question:
Determine whether the following equality holds for convex compact subsets $X$ and $Y$ in $\mathbb{R}^n$ and the standard inner product $\langle ., . \rangle$:
\[ \max_{y \in Y} [\langle y, z \rangle - \max_{x \in X} \langle y, x \rangle] = \max_{y \in Y} \min_{x \in X} \langle y, z-x \rangle. \]

Answer: Yes


Question: Find the maximum value of the determinant of a $3 \times 3$ real matrix $A$, where each element of $A$ is either $0$, $1$, $2$, or $3$. \( \boxed{} \)

Answer: 54


Question: Calculate the area under the curve of the function \( f(x) = \begin{cases} 0, & x \in \mathbb{Q} \\ 1, & x \notin \mathbb{Q} \end{cases} \) from \( x = 0 \) to \( x = 1 \).

Answer: 1


Question: Determine whether the condition \( \int_0^\infty f(x)^p e^{-x}dx < \infty \) implies \( \int_0^\infty F(x)^p e^{-x}dx < \infty \) for functions \( f \) with \( f(x) > 0 \) and \( F(x) = \int_0^x f(t) dt \), where \( p > 1 \).

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\rightarrow 0}\frac{1}{x}\int_{0}^{x}\sin(t^3){\rm d}t$$ using the fundamental theorem of calculus and L'Hopital's rule.

Answer: 0


Question: Given that for all positive integers $x$, $f(f(x)) = 4x+3$, and there exists a unique positive integer $k$ such that $f(5^k) = 5^k \times 2^{k-2} + 2^{k-3}$, determine the value of $f(2015)$. Provide hints without giving the full solution.

Answer: 4031


Question: Let $f:M\to N$ be a smooth bijection between manifolds with the same dimension. Does the condition $df_p(T_pM)=T_{f(p)}N$ necessarily hold?

Answer: No


Question: Find the greatest common factor between the polynomials \((x-w)^{4n+3} + x^{2n}\) and \(x^3 - 1\), where \(w\) is a primitive cube root of unity.

Answer: 1


Question: Evaluate \( \lim_{n \rightarrow \infty} \frac{x_{2n+1}}{x_{2n}} \) for the sequence \((x_n)_{n \geq 1}\) that satisfies \(\lim_{n \rightarrow \infty} (x_{2n} + x_{2n+1} )= 315\) and \(\lim_{n \rightarrow \infty} (x_{2n} + x_{2n-1}) = 2003\).

Answer: -1


Question: Determine the largest integer $n$ such that a $C^n$ approximate embedding of the hyperbolic plane $\Bbb H^2$ into $\Bbb R^3$ exists for all $\epsilon > 0$, where the geodesic distance on the embedded surface is within an arbitrarily small $\epsilon$ of the true hyperbolic distance.

Answer: 1


Question: Determine the values of \( n \in \mathbb{N} \) for which the proposition \( \exists x\;\exists y:(xy \notin n\mathbb{N}) \land (x \notin n\mathbb{N}) \land (y \notin n\mathbb{N}) \) is false.

Answer: 1


Question: Evaluate the double integral of the function \( f(x,y) = \min(x,y) \) over the region \( A = [0,a]\times[0,a] \) using the Lebesgue measure.

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


Question: Determine whether the following statement is true for a polynomial $f(x)$ over the finite field $GF[q]$, where $q = p^m$ and $p$ is a prime number: $$f^{p^m}(X) = f(X^{p^m}).$$

Answer: True


Question: Evaluate the integral \( \oint \dfrac{\sin(z)}{z(z^2-4)} \, dz \) using Cauchy's integral formula, given that \((z^2-4) = (z+2)(z-2)\).

Answer: 0


Question: Let $G$ be a group with order $n = \prod_{i=0}^{k} a_i$, where all $a_i$ are mutually coprime. Suppose that for $j > 0$, $a_j \equiv 1 \pmod{\prod_{i=0}^{j-1} a_i}$. Determine whether there always exist subgroups of order $a_0, a_1, \dots, a_k$ in $G$.

Answer: Yes


Question: Given the second-order differential equation \(y'' - 2xy' + (\sin x^2)y = 0\), two solutions are \(y_1 = \phi(x)\) and \(y_2 = \psi(x)\) with initial conditions \(\phi(0) = 1\), \(\phi'(0) = 1\), \(\psi(0) = 1\), and \(\psi'(0) = 2\). The Wronskian \(W(\phi, \psi)\) is given by \(W[\phi, \psi](x) = Ce^{f(x)}\). Find the value of \(W(\phi, \psi)(1)\).

Answer: e


Question: Given a vector space $V$ of real $n \times n$ symmetric matrices with $\text{dim } V = \frac{n(n+1)}{2}$, and a subspace $W$ of matrices with trace $0$ where $\text{dim } W = \frac{n(n+1)}{2} - 1$, determine the dimension of the orthogonal complement $W^{\perp}$.

Answer: 1


Question: Determine the total number of ways to achieve four in a row in a 4x4x4 tic-tac-toe game.

Answer: 76


Question: Can a finite game be designed using an honest coin (with equal probabilities of heads and tails) such that the probability of one player winning is an irrational number \( \alpha \), where \( 0 < \alpha < 1 \)? A finite game is one that inevitably ends in a finite number of moves with probability 1.

Answer: Yes


Question: Evaluate the triple integral \( \int\limits_0^5 \int\limits_{\sqrt{25-x^2}}^{-\sqrt{25-x^2}} \int\limits_{\sqrt{25-x^2-z^2}}^{-\sqrt{25-x^2-z^2}} \frac{1}{x^2+y^2+z^2} \,\mathrm dy\,\mathrm dz\,\mathrm dx \) by converting to spherical coordinates.

Answer: 10\pi


Question: Let $\mathcal{H}$ be a Hilbert space. If $T \in \mathcal{L}(\mathcal{H})$ is compact, is the set $\{ x \in S_{\mathcal{H}} \; |\; \| Tx\| =\| T\| \}$ nonempty, where $S_{\mathcal{H}}$ denotes the unit sphere?

Answer: Yes


Question: Compute the ratio of variances between the estimators \(\hat{\beta_1} = (n+1)X_{\min}\) and \(\hat{\beta_2} = \frac{n+1}{n}X_{\max}\), where \(X_1,X_2,...,X_n\) are i.i.d. random variables following a uniform distribution on the interval \([0,\beta]\). Determine how this ratio depends on the sample size \(n\).

Answer: n^2


Question: Calculate \( \mathbb{E}[N(X)] \), where \( N(\cdot) \) is the cumulative distribution function (CDF) of the standard normal distribution, and \( X \) is a standard normal random variable.

Answer: \dfrac{1}{2}


Question: Evaluate the integral: $$\int_{-\infty}^\infty x^3e^{-ax^2}dx$$

Answer: 0


Question: Find the kernel of the evaluation map \( f: \mathbb{Z}[x] \longrightarrow \mathbb{Z}[2^{1/3}] \) given by \( x \mapsto 2^{1/3} \).

Answer: (x^3 - 2)


Question: Evaluate the surface integral \(\iint_{S}(z-x^2-y^2)\,dS\) over the surface \(S\) of the cylinder \(x^2+y^2=4\) for \(-2\le z\le2\) using symmetry and geometry.

Answer: -64\pi


Question: In an exchange economy with two types of goods and n consumers, half of the consumers have a utility function given by \( U(x) = 5\ln{x} + m \) and the other half have \( U(x) = 3\ln{x} + m \). Each consumer has an initial endowment of 20 units of good \( x \) and 10 units of good \( m \). Determine the maximum amount of good \( x \) that the first type of consumers can obtain at a Pareto Optimal allocation, given the constraint \( m > 0 \) for all consumers.

Answer: 25


Question: Let $F$ be a closed subset of the complex plane whose complement is connected and locally connected at infinity. Determine whether there exists an entire function that can approximate any function holomorphic on an open subset containing $F$ arbitrarily well.

Answer: Yes


Question: Is the bounded component of the complement of a Jordan curve in the plane always simply connected?

Answer: Yes


Question: Determine the dimension of the subspace $V \subset \mathbb{R}^{100}$ defined by the conditions $x_1 = 2x_2 = 3x_3$ and $x_{51} - x_{52} - \dots - x_{100} = 0$ for tuples $(x_1, x_2, \dots, x_{100})$. What is the dimension of $V$?

Answer: 97


Question: Evaluate the limit \( \lim_{x\to +\infty}\frac {\zeta(1+\frac 1x)}{\Gamma(x)} \), where \(\zeta\) is the Riemann zeta function and \(\Gamma\) is the gamma function.

Answer: 0


Question: Find the value of \( \int_{-2}^{2} q(x) \, dx \), where \( q(x) \) is a quartic polynomial satisfying \( q(-2)=1, q(-1)=2, q(0)=3, q(1)=4, \) and \( q(2)=5 \).

Answer: 12


Question: Evaluate the definite integral \( \int_{0}^{\pi/4}\frac{\tan^{2}{x}}{\cos^{4}{x}}dx \).

Answer: \dfrac{8}{15}


Question: Evaluate the infinite series \( \sum_{x=0}^\infty P(X>x) \), where \( X \) is a discrete random variable defined on the non-negative integers (0, 1, 2, 3, ...).

Answer: E[X]


Question: Find the limit of \( z_n=\sum_{k=1}^n \frac{n^3}{n^4+n^2+k} \) as \( n \to \infty \) using harmonic numbers.

Answer: 1


Question: Calculate the work done by the vector field \( F = (x^2 + y^2)\mathbf{i} + (x^2 - y^2)\mathbf{j} \) along the closed curve formed by the line segments from \((0, 0)\) to \((2, 0)\), from \((2, 0)\) to \((2, 2)\), and from \((2, 2)\) to \((0, 0)\).

Answer: \dfrac{8}{3}


Question: Evaluate the integral \[ \iint_R \sqrt{\frac{x+y}{x-2y}} \, dA \] where the region $R$ is bounded by the lines $y = \frac{x}{2}$, $y = 0$, and $x + y = 1$. Use a coordinate transformation and the Jacobian to find the value of the integral.

Answer: \dfrac{1}{3}


Question: Determine the number \( x \) such that \( 579^{65} \equiv x \pmod{679} \) and \( x < 676 \). Given the pattern: \( 579^2 \equiv 494 \pmod{679} \), \( 494^2 \equiv 275 \pmod{679} \), \( 275^2 \equiv 256 \pmod{679} \), \( 256^2 \equiv 352 \pmod{679} \), \( 352^2 \equiv 326 \pmod{679} \), and \( 326^2 \equiv 352 \pmod{679} \), use this pattern to find \( x \).

Answer: 108


Question: Evaluate the limit: $$\lim_{n \to \infty} n \int_{-10}^{10}f(x)h(nx)dx$$ where $h: \mathbb{R} \to \mathbb{R}$ is a continuous function with $h(z)=0$ for all $z \notin [-1,2]$ and $\int_{-1}^{2}h(z)dz=2$, and $f: \mathbb{R} \to \mathbb{R}$ is a twice differentiable function.

Answer: 2f(0)


Question: Find the quadratic Taylor approximation of the function \( f(x,y) = \int_0^{x+y^2} e^{-t^2} \, dt \) around the point \((0,0)\).

Answer: x + y^2


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

Answer: (-1, 1)


Question: Solve the initial value problem given by the partial differential equation $u_x^2u_t - 1 = 0$ with the initial condition $u(x,0) = x$. Provide the solution function $u(x,t)$.

Answer: u(x, t) = x + t


Question: Determine whether an unbounded solution is guaranteed when a tableau in the Simplex method has more columns than rows, requiring pivoting on each column. Answer with 'True' or 'False'.

Answer: False


Question: Determine whether the complement of any arbitrary solid genus-2-handlebody in $S^3$ is also a solid genus-2-handlebody.

Answer: No


Question: Determine the real number \( m \) such that the quadratic equation \( 2z^2 - (3+8i)z - (m+4i) = 0 \) has a real root.

Answer: 2


Question: Determine whether the set $X=\{A \in P : 1 \in A\}$, where $P=\{A \subseteq \mathbb{N} : A \text{ is non-empty, finite, and has an even number of elements}\}$, has a lower bound in the partially ordered set $(P,\subseteq)$. Provide your answer as "yes" or "no".

Answer: no


Question: Suppose $g(z)$ is analytic in the unit disc $D$ and continuous in $\overline{D}$. If $g(z)$ can be extended to be analytic at the origin, and $|g(z)|\leq 12$ in $\overline{D}$, determine the maximum possible value of $|g(0.3+0.6i)|$. \(\boxed{\text{Your answer here}}\)

Answer: 12


Question: Determine the value of $p_i$ that maximizes the probability of obtaining a sum of $k$ when summing $n$ independent Bernoulli random variables $X_1, X_2, \ldots, X_n$ with success probabilities $p_1, p_2, \ldots, p_n$. Provide your answer for $p_i$.

Answer: \dfrac{k}{n}


Question: Evaluate the integral \( \int_0^{2\pi} \sin(\cos(z))\sinh(\sin(z)) \, dz \) using Cauchy's Integral Formula.

Answer: 0


Question: Determine the action of the Lie algebra of vector fields $\mathfrak{X}(M)$ on the space of smooth functions $C^\infty(M)$, corresponding to the representation $\rho: \mathrm{Diff}(M) \to \mathrm{GL}(C^\infty(M))$ where $\rho(F)(f) = f \circ F^{-1}$.

Answer: -X(f)


Question: Let $G$ be a finite group, and let $H$ be a normal subgroup of $G$ with order $p$, where $p$ is a prime number. Let $K$ be a subgroup of $G$ such that $\gcd(|K|, p-1) = 1$. Determine the structure of the subgroup $HK$.

Answer: H \times K


Question: Given a complex Borel measure $\mu$ on $\mathbb{R}$, if \( \int_a ^{b} e^{itx} \, d\mu (x) \to 0 \) as \( |t| \to \infty \) for all \( a<b \), does it imply that \( \mu \) is absolutely continuous with respect to the Lebesgue measure? Answer yes or no.

Answer: No


Question: Determine the maximum value of the expression \(72\int_0^y \sqrt{x^4+(y-y^2)^2}dx\) for \(y\in [0,1]\) using single-variable calculus.

Answer: 24


Question: Determine whether the function defined by the equation \( xy = \cot{xy} \) is differentiable. If it is differentiable, find the correct expression for \( \frac{dy}{dx} \).

Answer: -\dfrac{y}{x}


Question: Find the inverse Z-transform of \( \frac{z}{z+1} \).

Answer: (-1)^n u[n]


Question: Let $X$ be a locally compact, Hausdorff space and $Y = X \cup \{\infty\}$ its one-point compactification. If $K$ is a compact subset of $X$ and $V$ is an open neighborhood of $\infty$ in $Y$, does $V$ intersect $X$?

Answer: Yes


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

Answer: \dfrac{1}{e}


Question: Given a sequence $(a_n)$ such that $a_n > 0$ and $\frac{a_{n+1}}{a_n} > 0$ for all $n$, does the condition $\lim_{n\to \infty} \frac{a_{n+1}}{a_n} < 1$ imply that $\lim_{n\to \infty} a_n = 0$? Answer 'yes' or 'no' and justify your answer.

Answer: yes


Question: Calculate the sum of the series: $$ \zeta(2) + \zeta(3) + \zeta(4) + \dots + \zeta(2013) + \zeta(2014) $$ where \( \zeta(s) \) is the Riemann zeta function.

Answer: 2014


Question: In an acute triangle $\triangle ABC$, $AD$ is the altitude from $A$ to $BC$, and $CE$ is the angle bisector of $\angle ACB$. Given that $\angle CEA = 45^\circ$, find the measure of $\angle EDB$. \(\text{Express your answer in degrees.}\)

Answer: 45


Question: Determine the number of solutions to the equation \((x-2)+2\log_{2}(2^x+3x)=2^x\).

Answer: 2


Question: Evaluate the integral \( \int_0^{\infty}\frac{e^{-x}-e^{-2x}}{x}dx \) using a double integral approach.

Answer: \ln 2


Question: Find the measure of the subset $A$ of $[0,1]$ consisting of all numbers that do not have the digit $4$ in their decimal expansion.

Answer: 0


Question: Does there exist a continuous real-valued function \( g(x) \) such that \( g(g(x)) = x \) for all real numbers \( x \)?

Answer: Yes


Question: Suppose $C$ is a nonempty finite intersection of open halfspaces in $\mathbb{R}^n$. Determine whether the closure of $C$ is equal to the intersection of the associated closed halfspaces. Answer true or false.

Answer: True


Question: Find the modular inverse of $x+1$ with respect to $x^5 + x^4 + 1$ in the ring $\mathbb{Z}_2[x]$ using the Euclidean algorithm.

Answer: x^4


Question: Determine whether there exist finite sets $G$ with a composition law such that $G$ is non-associative, possesses a two-sided identity element $e$, each element has a unique two-sided inverse, and the composition table resembles a Sudoku puzzle.

Answer: Yes


Question: Determine whether the following statement is true or false: If $F \subseteq (0,1)$ is a set of full measure in $(0,1)$, then $g(F)$ is a set of full measure in ${\rm Im}(g)$ for absolutely continuous functions $g$. Provide a justification for your answer.

Answer: True


Question: Evaluate the integral \(\int_C \frac{(z-1)^2}{z^2 -1} dz\) where \(C\) is the circle centered at the origin with radius 2, traversed in the clockwise direction.

Answer: 4\pi i


Question: Solve the differential equation: $$y\cdot y'' + (y')^2 = 0$$.

Answer: y^2 = C_1 x + C_2


Question: Determine the set of values of $x$ for which the series \( \sum^{\infty}_{n=1} \frac{e^n+1}{e^{2n}+n} x^n \) converges.

Answer: (-e, e)


Question: Identify the algebraic structure defined by the following properties: Let $R$ be a ring and $G$ be a group of ring automorphisms of $R$. Consider the additive group $P=\bigoplus_{\sigma\in G}R$ and define multiplication for $x,y\in P$ as $(xy)_\omega=\sum_{\sigma\tau=\omega}x_\sigma \sigma^{-1}(y_\tau)$ for every $\omega\in G$, where the sum runs over $\sigma,\tau\in G$ such that $\sigma\tau=\omega$. What is the name of this algebraic structure?

Answer: skew group ring


Question: Find the constant $a$ such that the function \( f(x) = \begin{cases} x\left(1+2x\sin\left(\frac{1}{x}\right)\right) & \text{if } x \neq 0 \\ a & \text{if } x = 0 \end{cases} \) is differentiable everywhere.

Answer: 0


Question: Given $Z_1$ as a standard normal variable, find the moment generating function $M_{Z^2_1}(t)$ of $Z_1^2$ and evaluate $M_{Z^2_1}(0.14)$. Provide your answer.

Answer: \dfrac{5\sqrt{2}}{6}


Question: Evaluate the limit: $$ \lim_{x\rightarrow 0} \frac{\left( \cosh x \right) ^{\sin x}-1}{\sinh x(\cos \left( \sin \left( x \right) \right)-1)} $$

Answer: -1


Question: Evaluate the limit \(\lim_{n\to\infty}\frac{x_n-n}{\ln n}\) where \(x_1, x_2 > 0\) are real numbers, and for \(n > 2\), \(x_{n+1}\) is defined as \(\displaystyle x_{n+1}=\sum_{k=1}^{n}\sqrt[n]{x_k}\).

Answer: 1


Question: Solve for \(x\) in the equation: \[ \frac{60_{10}!}{12_2!} = 5^x \]

Answer: 14


Question: Consider a sequence of $n$ independent discrete random variables $X(k)$, where $k$ is an integer from $1$ to $n$. Each $X(k)$ can take the values $-\sqrt{k}$ with probability $1/k$, $0$ with probability $1 - 2/k$, and $\sqrt{k}$ with probability $1/k$. Determine whether the sequence $X(k)$ satisfies the weak law of large numbers.

Answer: Yes


Question: Determine the maximum value of the function \( g(u,v) = \frac{u\sin(\log u) - v\sin(\log v)}{v-u} \) for \( 0 < u < v \).

Answer: \sqrt{2}


Question: Let $f$ be a complex function such that $f^{(n)} = 0$ for all $n \geq 0$. Does this imply that $f$ is a polynomial? Justify your answer.

Answer: Yes


Question: Evaluate the definite integral: $$\int_0^1((1-x)^8x^{11}-(1-x)^{11}x^8)dx$$

Answer: 0


Question: Evaluate the limit: $$\lim_{n \to \infty} \frac{1}{n} \int_{0}^{n} \frac{x \ln(1 + \frac{x}{n})}{1 + x} \, dx.$$

Answer: 2 \ln 2 - 1


Question: Suppose $\epsilon$ and $X$ are random variables, and $\epsilon_i$ and $x_i$ are the $i$-th realizations of these random variables. Determine if the condition $E[\epsilon_i|x_i] = 0$ for all $i$ implies $E[\epsilon | X] = 0$. Answer 'yes' or 'no'.

Answer: no


Question: Find the smallest positive integer \( n \) such that \( 1,000,000^{{1,000,000}^{1,000,000}} < n^{n^{n^n}} \). Simplify the inequality to \( 6 + \log_{10}(6) < n\log_{10}(n) \) and determine the value of \( n \).

Answer: 8


Question: Determine whether the function \( f(x) = e^{\frac{x+1}{x}} \) has a unique fixed point in the set of positive real numbers.

Answer: Yes


Question: What is the minimum number of non-adjacent swaps required to sort a list of distinct integers in ascending order?

Answer: n - c


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

Answer: 1


Question: Given the circle defined by the equation \(x^2 + y^2 - 6x + 8y + 24 = 0\), determine the maximum value of the expression \(\frac{16\cos^2(\sqrt{x^2+y^2})}{5} - \frac{24\sin(\sqrt{x^2+y^2})}{5}\) for all points \((x, y)\) on the circle.

Answer: 5


Question: Let $f: (a, b) \to \mathbb{R}$ be a differentiable function such that there exists an $M > 0$ with $|f'(x)| \leq M$ for all $x \in (a, b)$. Prove that $f$ is uniformly continuous on $(a, b)$. \(\text{True or False?}\)

Answer: A


Question: Let \(k\) be a real number, and let \(A\) denote the ring \(\mathbb{R}[x]/(x^2+k)\). Determine an \(\mathbb{R}\)-linear basis for \(A\) and describe the multiplication law in terms of this basis.

Answer: \{1, x\}


Question: Find a harmonic conjugate of the harmonic function \( u(x, y) = x^3 - 3xy^2 \) and express the resulting analytic function in terms of the complex variable \( z = x + iy \).

Answer: z^3


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

Answer: 2e


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

Answer: \infty


Question: Evaluate the limit: $$ \lim_{x \to\infty} \frac{\log\log x }{ \sqrt{\log x}}. $$

Answer: 0


Question: Provide an example of an invertible linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that there is no linear transformation $S: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with $e^S = T$. What is the transformation $T$?

Answer: \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}


Question: Determine whether the statement $\pi_n(D^{n+1}) = 0$ holds true for all $n \in \mathbb{N}$, where $D^n$ is the $n$-dimensional disk and $\pi_n$ is the $n$-th homotopy group.

Answer: Yes


Question: Consider an open set $\Omega \subset \mathbb{R}^N$ with the Lebesgue measure and the set of natural numbers $\mathbb{N}$ with the counting measure. Let $f: \Omega \times \mathbb{N} \to \mathbb{R}$ be a function such that for each fixed $i \in \mathbb{N}$, the slice function $f(\cdot, i)$ is measurable in $\Omega$. Is the function $f$ itself measurable?

Answer: Yes


Question:
Let $f(x)=x^{A_0}(x+1)^{A_1}(x+2)^{A_2}\cdots(x+n)^{A_n}$. If

$$\frac{f'(x)}{f(x)}=\frac{n!}{x(x+1)(x+2)(x+3)\cdots(x+n)},$$

find the value of $A_8$. \(\boxed{\text{Your answer here}}\)

Answer: \dbinom{n}{8}


Question: Evaluate the integral \(\int_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^d)}\) for an arbitrary value of \(d > 0\).

Answer: \dfrac{\pi}{4}


Question: Consider a countably infinite semilattice \((S,<)\) with the following properties: \(S\) is nowhere dense, meaning no subset \(T\) exists such that for all \(a, b \in T\) with \(a < b\), there is a \(c \in T\) such that \(a < c < b\). Additionally, \(S\) has finite width \(n\), meaning the largest size of an anti-chain in \(S\) is \(n\). Given elements \(a, b \in S\) with \(a < b\), is the set \(\{x \in S : a < x < b \}\) necessarily finite?

Answer: No


Question: Find the function that represents the infinite series \( \sum\limits_{k=0} ^\infty \frac{n^2}{n!} x^n \).

Answer: x(x + 1)e^x


Question: Determine whether every infinite group has a maximal subgroup.

Answer: No


Question: Determine the number of group homomorphisms from $\Bbb Z/2\Bbb Z \times \Bbb Z/2\Bbb Z$ to $S_3$. Provide your answer as a single integer.

Answer: 10


Question: Find the trace of a $227 \times 227$ matrix $A$ with entries in $\mathbb{Z}_{227}$, given that all its eigenvalues are distinct.

Answer: 0


Question: Let $A$ be a unital $C^*$-algebra with a trace $\tau$, and let $\lambda: A \to B(L^2(A, \tau))$ be the left representation, where $L^2(A, \tau)$ is the GNS space. Let $\lambda(A)^{''}$ be the double commutant of $\lambda(A)$. Suppose there is a central projection $p$ in the center of $\lambda(A)^{''}$. Is the map $A \to p\lambda(A)^{''}$ surjective?

Answer: No


Question: Given that $P$ is an idempotent operator satisfying $P^2 = P$ and $P \neq \mathbb{1}$, and that $P.L |\psi\rangle = L|\psi\rangle$ holds for all $|\psi\rangle$, determine the relationship between the operators $P$ and $L$.

Answer: PL = L


Question: Given an integral domain $A$ and its field of fractions $K$, consider a subfield $k$ of $K$ such that $K$ is a simple finite extension of $k$. Determine whether there exists an element $a \in A$ such that $K = k[a]$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int^{1}_{0}(f(x))^3\,dx \) given that \( f(x) \) is defined on \([0,1]\) and satisfies the following conditions: \( \int^{1}_{0}(f(x))^2\,dx=4 \), \( \int^{1}_{0}f(x)\,dx=1 \), and \( \int^{1}_{0}x\cdot f(x)\,dx=1 \). Assume there is a unique solution for \( f(x) \).

Answer: 10


Question: Let $A$ be a $7 \times 7$ matrix satisfying $2A^2 - A^4 = I$. If $A$ has two distinct eigenvalues and each eigenvalue has geometric multiplicity 3, determine the number of non-zero entries in the Jordan Canonical Form of $A$. \( \text{(Express your answer as a single integer.)} \)

Answer: 8


Question: Consider an nth-degree polynomial $p(z)$ with real coefficients. For $z$ within the unit disc, expressed as $z = re^{it}$ with $0 \leq r \leq 1$ and $0 \leq t \leq 2\pi$, determine if $|p(re^{it})|$ is a continuous function with respect to $r$ and $t$.

Answer: Yes


Question: Given $P$ and $A$ are $3 \times 3$ real matrices satisfying $PAP^t = -A^t$, determine if it is possible to find the determinant of $A$.

Answer: 0


Question: Given a conditionally convergent series $\sum a_n$, determine if there exists an increasing sequence of integers $k_1, k_2, k_3, \ldots$ such that the sequence of partial sums $$\left(\sum_{n=1}^{k_1} a_n, \sum_{n=k_1+1}^{k_2} a_n, \sum_{n=k_2+1}^{k_3} a_n, \ldots\right)$$ is alternating and decreasing in absolute value.

Answer: Yes


Question: Find the power of the prime $p$ that divides the binomial coefficient $\binom{np^\beta}{p^\beta}$, given that $p^s$ divides $n$ and $p^{s+1}$ does not divide $n$. Express your answer as an integer.

Answer: s


Question: Determine the number of automorphisms of the group \( \Bbb Z \oplus \Bbb Z_3 \).

Answer: 12


Question: Evaluate the double integral \( \int_0^\pi \int_0^\pi |\cos(x+y)| \,dx\,dy \).

Answer: 2\pi


Question: Determine the minimum value of \( \det(I + R^T R) \) where \( R \) is an arbitrary real \( n \times n \) matrix.

Answer: 1


Question: Find all complex numbers $z$ such that $e^{iz} = 1$.

Answer: 2\pi k


Question: Determine the range of values of \(\alpha\) for which the function \(f(x) = |x|^{\alpha}\) belongs to the Sobolev space \(H^1(B(0,1))\) in \(\mathbb{R}^2\).

Answer: \alpha \geq 0


Question: Consider a group where every subgroup of prime power order is normal. Does this group satisfy Cauchy's theorem?

Answer: Yes


Question: Line l is tangent to circle O at point A. Let D be the foot of the perpendicular from A to chord BC. The perpendiculars from points B and C to line l are drawn with feet at E and F, respectively. Given that CF = 12 and AD = 9, find the length of BE.

Answer: \dfrac{27}{4}


Question: Find all natural number solutions for the equation: $$2^{\pi-x}\cos{\pi x}=(-1)^{x+10}\cdot 4^{x+\frac{\pi}{2}-3}$$

Answer: 2


Question: Evaluate the limit: $$\lim_{n \to \infty} \sqrt[n^3]{1!^2 + 2!^2 + 3!^2 + \dots + n!^2}$$

Answer: 1


Question: Calculate the volume enclosed by the surfaces $x^2+z^2=4$, $y=z$, and $y+z=4$ using a triple integral in cylindrical coordinates.

Answer: 16\pi


Question: Given $n$ entire functions $f_1, f_2, \ldots, f_n$ without any common zeros, determine whether there exist $n$ entire functions $g_1, g_2, \ldots, g_n$ such that the function $F = f_1g_1 + f_2g_2 + \cdots + f_ng_n$ is zero-free.

Answer: Yes


Question: Determine if there exists a continuous function \( g: \mathbb{R} \to \mathbb{R} \) such that \( \lim_{|x| \to \infty} g(x) = -\infty \) and there exists some \( x_0 \in \mathbb{R} \) such that \( g(x_0) \geq g(x) \) for every \( x \in \mathbb{R} \).

Answer: Yes


Question: Determine whether the following statement is true or false: For a system of $n$ linear equations with $n$ indeterminates, $Ax = b$ has no solution if and only if $A^{\top}x = 0$ has infinitely many solutions.

Answer: B


Question: Find the maximum value of $a+b+c$ given that $a^2+b^2+c^2=48$.

Answer: 12


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

Answer: \dfrac{1}{2}


Question: Evaluate the contour integral \( \frac{1}{2\pi i}\int_{\gamma}\frac{p'(z)}{p(z)} dz \) where \( p(z) \) is a polynomial with a simple zero \( z_0 \) of multiplicity 1, and \( \gamma \) is a simple, \( C^1 \) curve that surrounds \( z_0 \) only.

Answer: 1


Question: Given that the polynomial \( p(x) = x^4 + ax^3 + bx^2 + ax + 1 \) has \( x = 1 \) as a root, find the discriminant of the cubic factor of \( p(x) \) in terms of \( a \).

Answer: a(a + 4)^3


Question: Given the curve \( f(t) = (t, 0) \) on an interval \( J \subset \mathbb{R} \), determine if the derivative \( f'(t) \) can be represented as \( (1, 0) \) in the context of standard velocity for curves in Euclidean spaces.

Answer: (1, 0)


Question: Evaluate the limit \( \lim_{n\to\infty}\int_0^n\frac{\left(1-\frac{x}{n}\right)^n}{e^{x/2}}\:dx \) using the dominated convergence theorem.

Answer: \dfrac{2}{3}


Question: Determine whether there exists a strictly increasing sequence in a non-empty subset of \( \Bbb R \) that is bounded above and converges to a limit outside the subset. Provide a justification for your answer.

Answer: Yes


Question: Does there exist a shortest possible derivation in natural deduction, where the number of steps needed to derive a conclusion is minimized?

Answer: Yes


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

Answer: \dfrac{4}{e}


Question: In a tetrahedron ABCD, points M, N, and K are located on edges AD, AB, and BC, respectively, such that AM:MD = 2:3, BN:AN = 1:2, and BK:KC = 1:1. Determine the ratio in which the plane MNK divides edge CD.

Answer: 1:3


Question: A particle's position is uniformly distributed on a sphere centered at the origin with a radius of 7. What is the expected value of the particle's distance from the origin? \( \boxed{7} \)

Answer: 7


Question: Determine the values of $p$ for which the closure of $L^1(\mathbb{R}) \cap L^p(\mathbb{R})$ is equal to $L^p(\mathbb{R})$. Provide your answer as a set of values for $p$.

Answer: [1, \infty)


Question: Evaluate the limit \( \lim\limits_{x \to +\infty} x \cdot \sin\left(\sqrt{x^2+3}-\sqrt{x^2+2}\right) \).

Answer: \dfrac{1}{2}


Question: Find the whole number value of $x$ that maximizes the expression \[ \frac{100\left\lfloor\frac{10000}{\left\lfloor \frac{100000}{2x}\right\rfloor+1}\right\rfloor}{x} \] given that $100000$ is divisible by $x$. \(x\) must be a divisor of $100000$. What is the maximum value of $x$?

Answer: 500


Question: Consider a continuously differentiable bounded function $f(t)$ with a bounded, uniformly continuous derivative defined on $[0,\infty)$. Given that $$\lim_{t\rightarrow\infty}\left[f(t)\dot{f}(t)\right]=0,$$ determine whether the following claim is true for all finite $T>0$: $$\lim_{t\rightarrow\infty}|f(t+T)-f(t)|=0.$$

Answer: True


Question:
Given two dependent Bernoulli random variables $X$ and $Y$ with the following probabilities:
\[ \begin{align} P(X=0)&=P(X=1)=1/2 \\
P(Y=1)&=11/24 \\
P(Y=0|X=0)&=1/3 \\
P(Y=1|X=1)&=1/4 \\
\end{align} \]
Determine if it is possible to calculate the expectation $\mathbb{E}[XY]$. Provide a justification for your answer.

Answer: \dfrac{1}{8}


Question: Find the minimum value of the expression \((a+b+c+d+e)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}\right)\), given that \(a, b, c, d, e > 0\).

Answer: 25


Question: Calculate the flux of the vector field \( \vec{G}=\operatorname{curl} \vec{F} \), where \( \vec{F}(x, y, z)=y^{3} \vec{i}+x^{3} \vec{j}+z^{3} \vec{k} \), through the upward-oriented portion of the paraboloid \( z=1-x^{2}-y^{2} \) that lies above the plane \( z=0 \).

Answer: 0


Question: Determine whether a function \( f:\mathbb{R}^2\rightarrow\mathbb{R} \) is a polynomial if, for every linear function \( \varphi:\mathbb{R}\rightarrow\mathbb{R}^2 \), the composition \( f\circ\varphi \) is a polynomial.

Answer: No


Question: Consider two continuous and convex functions, \(f(x)\) and \(g(x)\), defined on the domain \([0, +\infty)\) with the condition that \(f(x)\) and \(g(x)\) take non-negative values. If \(f(x)\) approaches infinity faster than \(g(x)\) as \(x\) goes to infinity, i.e., \(\lim_{x\rightarrow +\infty}\frac{f(x)}{g(x)}=+\infty\), determine the maximum number of points at which their graphs can intersect.

Answer: 2


Question:
In a one-kilometer race, the completion times of three individuals are modeled by random variables. Let $X_1$, $X_2$, and $X_3$ represent the completion times in minutes for each person, where:
- $X_1$ is uniformly distributed over [2.9, 3.1]
- $X_2$ is uniformly distributed over [2.7, 3.1]
- $X_3$ is uniformly distributed over [2.9, 3.3]

Assuming the completion times are independent, calculate the expected value of the latest completion time, $Y = \max\{X_1, X_2, X_3\}$, to the nearest 0.1 minute.

Answer: 3.1


Question: Determine whether there exist finite groups where not every element of the derived subgroup can be expressed as a commutator.

Answer: Yes


Question: In a village, the chief collects a grain tax from all male villagers based on their age, with each man paying an amount of grain equal to his age in pounds. The chief uses a platform with 7 different weights and a large balance scale. By tradition, the chief can only use 3 of these weights at a time to measure the grain. Determine the weights of these 7 weights and find the maximum age of a man that can be measured using any combination of 3 weights.

Answer: 1053


Question: Determine the minimum number of distinct intersections of $n$ independent events, each with a probability between zero and one, that a sample space must contain.

Answer: 2^n


Question: Let \( P \) be a non-zero \( n \times n \) matrix. Define the vector space \( L \) as the set of all \( n \times n \) matrices \( X \) such that \( \text{tr}(PX) = 0 \). Determine the dimension of \( L \).

Answer: n^2 - 1


Question: Calculate the number of elements in the quotient group \(\mathbb{Z}^2/\!\ker\varphi\), where \(\varphi: \mathbb{Z}^2 \to S_{15}\) is the homomorphism defined by \(\varphi(1,0)=(2,7,3)(3,11,5)(12,13)\) and \(\varphi(0,1)=(14,15)(1,4,6,8,9,10)\).

Answer: 60


Question: Determine the degree of the field extension \(\mathbb{Q}(\sqrt{5},i)\) over \(\mathbb{Q}\). Given that \(\mathbb{Q} \subseteq \mathbb{Q}(\sqrt{5}) \subseteq \mathbb{Q}(\sqrt{5},i)\), and the minimal polynomial of \(\sqrt{5}\) over \(\mathbb{Q}\) is \(x^2 - 5\), while the minimal polynomial of \(i\) over \(\mathbb{Q}(\sqrt{5})\) is \(x^2 + 1\), find \([\mathbb{Q}(\sqrt{5},i) : \mathbb{Q}]\).

Answer: 4


Question: Determine whether the function $-f(z)$, where $f(z) = \frac{1}{z}$, is a primitive function of $g(z) = \frac{1}{z^2}$ on the domain $\mathbb{C}\setminus\{0\}$.

Answer: Yes


Question: Find the minimum value of \( n \) such that the Chebyshev interpolation of \( f(x) = e^x \) on the interval \([-1, 1]\) has an error bound less than \(10^{-10}\).

Answer: 10


Question: If $R$ and $S$ are graded rings, and $h_n:R_n\longrightarrow S_n$ is an isomorphism of abelian groups for each $n$, is $h$ a ring isomorphism in the natural way? Answer 'yes' or 'no'.

Answer: no


Question:
Consider the function \( f(x) = x^2\sin\left(\frac{1}{x}\right) \), defined with \( f(0) = 0 \). Determine the behavior of \( f'(x) \) at \( x = 0 \). Choose the correct option:

A) \( f \) is differentiable at every \( x \) but \( f' \) is discontinuous at \( 0 \).
B) \( f \) is not differentiable at \( 0 \).

Answer: A


Question: Does the maximum modulus principle hold for a connected analytic subset $X$ of a domain $U \subset \mathbb{C}^n$, even when $X$ has singularities? Specifically, if $f : U \to \mathbb{C}$ is a holomorphic function whose modulus attains its maximum on $X$, must $f$ be constant on $X$?

Answer: Yes


Question: Evaluate the definite integral: $$ \int_{-\sqrt{2}}^{\sqrt{2}} x\sqrt{4 - x^{4}}\,dx $$

Answer: 0


Question: Calculate the sum of the infinite series: \[ \sum_{n=1}^\infty (-1)^{n+1}\frac{(n+1)^2}{2^n}. \]

Answer: \dfrac{23}{27}


Question:
Given $N\in\mathbb{N}$ and $0<\alpha<1$, we define for each $0\leq n \leq N-1$ the sequences:
\[a_{n+1,N}=a_{n,N}+\dfrac{1}{N}a_{n,N}^2\left(\dfrac{2}{b_{n,N}}\right)^\alpha\]
\[b_{n+1,N}=b_{n,N}-\dfrac{1}{N}a_{n,N}^2\left(\dfrac{2}{b_{n,N}}\right)^\alpha.\]
Do there exist any initial values $a_0=a_{0,N}$ and $b_0=b_{0,N}$ such that the sequences $\{a_{N,N}\}$ and $\{b_{N,N}\}$ converge to positive numbers?

Answer: Yes


Question: Calculate the limit: $$\lim_{n \to \infty} \left(\cos \left(\frac{a}{n}\right)\right)^{n^2}$$ for $n \in \mathbb{N}$ and $a \in \mathbb{R} \setminus \{0\}$.

Answer: e^{-\frac{a^2}{2}}


Question: Calculate the limit \( \displaystyle \lim_{n\rightarrow\infty}\frac{(3n)!}{(n!)^3} \) without using Stirling's Approximation.

Answer: \infty


Question: Given the functions $y=at^2+2bt+c$ and $t=ax^2+bx+c$, find the third derivative of $y$ with respect to $x$, denoted as $\frac{d^3y}{dx^3}$.

Answer: 24a^3x + 12a^2b


Question:
Determine if the list of elements \( 1, a, b, c, ab, bc, ca, a^2 b, a^2c, a^2 bc, abc, ba, bac \) is complete for the group \( G \) defined by the presentation:
\[ G = \big\langle \, a, b, c \colon a^3 = b^2 = c^2 = (ab)^2 = (bc)^2 = 1, ac = ca \big\rangle. \]
What is the order of the group \( G \)?

Answer: 12


Question: Evaluate the limit: \[ L = \lim_{n \to \infty} \sqrt[n]{\int_0^1 (\arctan(1 + x^n))^n \, dx} \]

Answer: \arctan 2


Question: Evaluate the limit: \[ \lim_{n\to \infty}\left[31^{1/(2n+1)}+(-93)^{1/(2n+3)}-(-75)^{1/(2n+5)}\right]^{n} \]

Answer: 5


Question:
Given three types of computer components with lifetimes following exponential distributions with parameters \(\lambda_1 = 1\), \(\lambda_2 = 2\), and \(\lambda_3 = 3\), respectively, calculate the probability that the lifetime of the first component is greater than five times the lifetime of the second component, which in turn is greater than ten times the lifetime of the third component. Specifically, find the probability:

\[ P(X_1 > 5X_2 > 10X_3) \]

Answer: \dfrac{6}{119}


Question: Given a vector space $V$ over a field $F$, determine whether there always exists a set $X$ such that $V$ is isomorphic to a subspace of $F^X$. Provide a justification for your answer.

Answer: Yes


Question: Suppose $S$ and $T$ are simplicial sets such that $S_0 = T_0$, $S_1 = T_1$, and $S_2 = T_2$. The face maps $d_0, d_1 : S_1 \to S_0$ agree with the maps $d_0, d_1 : T_1 \to T_0$, and the degeneracy map $s_0 : S_0 \to S_1$ agrees with $s_0 : T_0 \to T_1$. Additionally, the face maps $d_0, d_1, d_2 : S_2 \to S_1$ agree with $d_0, d_1, d_2 : T_2 \to T_1$, and the degeneracy maps $s_0, s_1: S_1 \to S_2$ agree with $s_0, s_1: T_1 \to T_2$. Given that $S$ and $T$ are simplicial sets of dimension \( \leq 2 \), does it follow that $S$ and $T$ are isomorphic?

Answer: Yes


Question: Find the entire function \( f: \mathbb{C} \to \mathbb{C} \) such that \( f'(z) = 3f(z) \) for all \( z \in \mathbb{C} \) and \( f(0) - f'(0) = 2 \).

Answer: -e^{3z}


Question:
Let $X=\{(x,y)\in \mathbb{R^2}:0\le|x|=|y|\le1\}$, $Y=\{(x,y)\in \mathbb{R}^{2}:|x|+|y|=1\}$, and $Z=\{(x,y)\in \mathbb{R}^{2}:x^{2}+y^{2}=1\}$ be subspaces of the Euclidean space in $\mathbb{R}^{2}$. Determine which of the following statements is true:

(a) $X$ is homeomorphic to $Y$ and $Z$
(b) $X$ is homeomorphic to $Y$, but not $Z$
(c) $Y$ is homeomorphic to $Z$, but not $X$
(d) $Z$ is homeomorphic to $X$, but not $Y$

Answer: C


Question: Let \( l^2= \{ (x_1,x_2,x_3,\dots):x_n \in \mathbb{R} \text{ for all } n \in \mathbb{N} \text{ and } \sum_{n=1}^{\infty} x_n^2 < \infty \}. \) For a sequence \((x_1,x_2,x_3,\dots) \in l^2,\) define \[\lvert \lvert (x_1,x_2,x_3,\dots)\rvert\rvert_2= \bigg( \sum_{n=1}^{\infty} x_n^2\bigg)^{\frac{1}{2}}.\] Consider the subspace \(M=\{ (x_1,x_2,x_3,\dots) \in l^2 :\sum_{n=1}^{\infty} \frac{x_n}{4^n}=0 \}.\) Let \(M^{\perp}\) denote the orthogonal complement of \(M\) in the Hilbert space \((l^2, \lvert \lvert \cdot \rvert \rvert_2).\) Consider \(\big(1,\frac{1}{2},\frac{1}{3},\frac{1}{4}, \dots \big) \in l^2.\) If the orthogonal projection of \(\big(1,\frac{1}{2},\frac{1}{3},\frac{1}{4}, \dots \big)\) onto \(M^{\perp}\) is given by \(\alpha\big( \sum_{n=1}^{\infty} \frac{1}{n4^n}\big)\big(\frac{1}{4},\frac{1}{4^2},\frac{1}{4^3}, \dots \big)\) for some \(\alpha \in \mathbb{R},\) find the value of \(\alpha\).

Answer: 15


Question: Evaluate the integral \( \int_{\mathbb{R}^2} e^{-(x^2+2xy+2y^2)} \, dA \), where \( dA \) is the area element in the plane.

Answer: \pi


Question: Determine whether the perimeter of a polygon can be an odd number given that the polygon is not a triangle, has no pair of parallel sides, and all its vertices have integer coordinates with sides of positive integer lengths.

Answer: No


Question: Is every subgroup of \((\mathbb{R},+)\) that has a finite intersection with \([-1,1]\) and contains a positive element necessarily cyclic?

Answer: Yes


Question:
Evaluate the limit:
\[\lim_{n \to \infty} \sum_{k=0}^{n} \frac{a_{n-k}}{3^k}\]
where \(\lim_{n \to \infty} a_n = a\).

Answer: \dfrac{3a}{2}


Question: Determine whether the following equality holds true: \[ \sqrt[3]{\frac{1}{2} \left(56-\sqrt{\frac{84640}{27}}\right)} + \sqrt[3]{\frac{1}{2} \left(\sqrt{\frac{84640}{27}}+56\right)} = 4. \] Provide a proof suitable for an oral examination if the equality is true.

Answer: 4


Question: Given the function \( f(x) = \cos{2x} \) defined on the interval \((0, \pi)\), find the $2\pi$-periodic even extension \( \tilde{f} \) of \( f \) and compute the cosine Fourier series representation of \( \tilde{f} \).

Answer: \cos(2x)


Question: Determine the number of idempotent elements in the ring $\mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{Z}[i]$. An element $e$ is idempotent if $e^2 = e$. Provide your answer as a single integer.

Answer: 2


Question:
For positive real numbers \(x, y, z\), if \(xy + yz + xz = \sqrt{\frac{p}{q}}\) and given the system of equations:
\[x^2 + xy + y^2 = 2\]
\[y^2 + yz + z^2 = 1\]
\[z^2 + zx + x^2 = 3\]
find \(p - q\), where \(p\) and \(q\) are relatively prime integers.

Answer: 5


Question: Find the polynomial $R(x)$ of smallest degree with rational coefficients and leading coefficient $1$ such that $R\left(\sqrt[3]{7}+\sqrt[3]{49}\right)=2$. What is the polynomial $R(x)$?

Answer: x^3 - 21x - 54


Question:
Given the relations for parameters $a$ and $b$ in the function $y = ax^p + b$ that best fits a set of data $\{x_i\}$ and $\{y_i\}$, determine if it is possible to extract the exponent $p$ from the following equations:

1. \[ a = \frac{N\sum_i(y_ix_i^p)-\sum_iy_i\cdot\sum_ix_i^p}{(\sum_ix_i^p)^2+N\sum_i(x_i^p)^2} \]
2. \[ b = \frac{1}{N}[\sum_iy_i-a\sum_ix_i^p] \]
3. \[ \frac{1}{N}[N\sum_i(y_ix_i^p\ln x_i)-\sum_iy_i\cdot\sum_ix_i^p\ln x_i]=\frac{a}{N}[N\sum_i(x_i^p)^2\ln x_i-\sum_ix_i^p\cdot\sum_ix_i^p\ln x_i] \]

Answer: Yes


Question: Given independent Gamma random variables $G_1, G_2 \sim \Gamma(n,1)$, does the independence of $\frac{G_1}{G_1+G_2}$ from $G_1+G_2$ imply the independence of $G_1+G_2$ from $G_1-G_2$? Answer yes or no.

Answer: No


Question: Simplify the series: $ \frac{1}{2^1}+\frac{3}{2^2}+\frac{5}{2^3}+\frac{7}{2^4}+\cdots $.

Answer: 3


Question: In triangle $ABC$, the circumcenter is $O$ and the orthocenter is $H$. Given that $AH = 10$ and $BC = 16$, determine the radius of the circumcircle of triangle $ABC$. Express your answer as a single number.

Answer: \sqrt{89}


Question:
Given the binomial coefficients defined as:
\begin{align*}
a &= \binom{2015}{0} + \binom{2015}{3} + \binom{2015}{6} + \dots \\
b &= \binom{2015}{1} + \binom{2015}{4} + \binom{2015}{7} + \dots \\
c &= \binom{2015}{2} + \binom{2015}{5} + \binom{2015}{8} + \dots
\end{align*}
where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ for all $n, k \in \mathbb{N}$ with $k \leq n$. It is known that $a + b + c = 2^{2015}$. Determine the value of $(b - c)^2 + (c - a)^2$. \( \text{Express your answer as a single integer.} \)

Answer: 1


Question: Solve the modular congruence equation: \(29x^{33} \equiv 27 \pmod{11}\). Given that the primitive root modulo 11 is 2, and the order of 2 modulo 11, denoted as \(\text{ord}_{11}(2)\), is 10.

Answer: 6


Question:
Determine the value of $K$ in the integral equation below, given that $e^2 < 1$:
\[ \int_{0}^{\pi}\frac{x \, dx}{1+ e \sin x}=K\frac{\arccos e}{\sqrt{1-e^{2}}} \]

Answer: \pi


Question: Calculate the correlation coefficient \( \rho_{XY}(\alpha) \) for the joint distribution \( f_\alpha (x,y)=k(\alpha)e^{-|x+y|/\alpha} \) where \( 0\leq x\leq 1, 0\leq y \leq 1 \) and \( f(x,y)=0 \) elsewhere.

Answer: 0


Question: Find the value of $d$ such that the range of the function \( f(x) = \frac{2x^2 + x + d}{x^2 - x + d} \) is \([1, 3]\).

Answer: 2


Question: A point P inside a square ABCD is located at distances of 3, 4, and 5 meters from vertices A, B, and C, respectively. If A, B, and C are vertices of the square with sides parallel to the x- and y-axes, determine the area of square ABCD.

Answer: 17 + 4\sqrt{14}


Question: Calculate the volume integral \( \iiint_D (1-z^2) \, dx \, dy \, dz \) over the pyramid \( D \) with vertices at \((0,0,0)\), \((1,0,0)\), \((0,1,0)\), \((1,1,0)\), and \((0,0,1)\). Determine the appropriate limits for \(x\), \(y\), and \(z\) for the integrals and find the value of the integral.

Answer: \dfrac{3}{10}


Question: Evaluate the limit: $$\lim_{n \rightarrow \infty} \left[n \; \arccos \left( \left(\frac{n^2-1}{n^2+1}\right)^{\cos \frac{1}{n}} \right)\right]$$

Answer: 2


Question: Evaluate the integral \( \int_{7}^{10}\sqrt{-40+14x-x^2}dx \).

Answer: \dfrac{9\pi}{4}


Question: Determine whether the orthogonal projection of an ellipsoid onto a subspace is always a lower-dimensional ellipsoid.

Answer: Yes


Question: Compute the residue of the function \( \frac{z}{e^z - 1} \) at its singularity.

Answer: 0


Question: Find the maximum value of the expression \(x\sqrt{y}+y\sqrt{z}+z\sqrt{w}+w\sqrt{x}\) given that \(x,y,z,w>0\) and \(x+y+z+w=1\).

Answer: \dfrac{1}{2}


Question: Determine whether a differential system with functions $u$ and $v$ holomorphic in the punctured disk $D^{*}(0,a)$ can have a non-trivial holomorphic solution, where "non-trivial" means that the solution is not identically zero.

Answer: Yes


Question:
Suppose $A_{1}$, $A_{2}$, $
ots$, $A_{30}$ are thirty sets each having $5$ elements and $B_{1}$, $B_{2}$, $
ots$, $B_{n}$ are $n$ sets each having $3$ elements. Let $igcup\limits_{i=1}^{30} A_{i} = \bigcup\limits_{i=1}^{n} B_{i} = S$ and each element of $S$ belongs to exactly $10$ of the $A_{i}$'s and exactly $9$ of the $B_{j}$'s. Find the value of $n$. \( \boxed{} \)

Answer: 45


Question: What is the norm of the difference between the projection operators $P_n$ and $P_m$ on $l^2(\mathbb{N})$ when $n = m$?

Answer: 0


Question: Evaluate the limit of \( G(x) = \int_x^{x^2} \frac{1}{\ln t} \, dt \) as \( x \to 0^+ \).

Answer: 0


Question: Compute the surface area of the boundary of the solid defined by the inequalities $x^2 + z^2 \leq 1$ and $y^2 + z^2 \leq 1$. What is the surface area of this solid?

Answer: 16


Question:
Determine whether there exists a Riemann integrable function \( f: [a,b] \to \mathbb{R} \) that satisfies the following conditions:

1. \( f \) is a positive function, meaning \( f(x) \geqslant 0 \) for all \( x \in [a,b] \).
2. There exists an infinite subset \( E \) of \( [a, b] \) such that \( f(x) > 0 \) for each \( x \in E \).
3. \( \int_a^b f(x) \, dx = 0 \).

Answer: Yes


Question: Calculate the probability that a sequence \((x_n)_{n \ge 1}\), defined by \(x_1 = x_2 = 1\) and \(x_{n+2} = ax_{n+1} - bx_n\), where \(a\) and \(b\) are distinct numbers chosen randomly from the set \(\{1, 2, \ldots, 10\}\), converges.

Answer: \dfrac{1}{10}


Question:
Evaluate the limit as \( n \to \infty \) of the expression:
\[
\lim_{n\to\infty}\dfrac{1}{n} \left( \left\{ \frac{n}{1} \right\} - \left\{ \frac{n}{2} \right\} + \ldots + (-1)^{n+1} \left\{ \frac{n}{n} \right\} \right),
\]
where \(\left\{ x \right\} \) denotes the fractional part of \(x\).

Answer: 0


Question: Find the complex function $f(z)$ satisfying the condition $\Im(f'(z)) = 6x(2y - 1)$, where $z = x + iy$, with initial conditions $f(0) = 3 - 2i$ and $f(1) = 6 - 5i$. Determine $f(1 + i)$. \(\boxed{\text{Your answer here}}\)

Answer: 6 + 3i


Question: Calculate the first homology group \( H_1(K, \mathbb{Z}) \) for the simplicial complex \( K = \{\{v_0\}, \{v_1\}, \{v_2\}, \{v_0, v_1\}, \{v_0, v_2\}, \{v_1, v_2\}\} \) with vertex set \( V_K = \{v_0, v_1, v_2\} \).

Answer: \mathbb{Z}


Question: Does Exercise 4 of Chapter 6 in Rudin's Functional Analysis, which states that every "positive" distribution \(\Lambda\in D'(\Omega)\) (such that \(\Lambda\psi\geq 0\) for all \(\psi\in D(\Omega)\)) is a positive measure in \(\Omega\), where \(\Omega\subseteq\mathbb{R}^d\) is open and \(D'(\Omega)\) denotes the space of test functions, directly follow from the Riesz Representation theorem asserting that every positive linear functional on the space of compactly supported smooth functions on a locally compact Hausdorff space corresponds to a positive Radon measure?

Answer: Yes


Question: Tank A initially contains 100 gallons of brine with 100 pounds of salt, and Tank B contains 100 gallons of water. Water flows into Tank A at a rate of 2 gallons per minute, and the uniform mixture flows from Tank A to Tank B at the same rate. The mixture in Tank B, also kept uniform, flows out at 2 gallons per minute. Calculate the amount of salt in Tank B at the end of 1 hour.

Answer: 36.14


Question: The number $110179$ can be expressed as the product of two distinct prime numbers $p$ and $q$. Given that the number of integers less than $110179$ and relatively prime to it (Euler's totient function $\phi(110179)$) is $109480$, find the sum $p+q$. What is the sum $p+q$?

Answer: 700


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

Answer: 0


Question: Determine the number of points of continuity of the function \( g(x) \), defined as \( g(x) = \sin x \) for rational \( x \) and \( g(x) = x^3 - 1 \) for irrational \( x \), within the interval \([-\pi, 2\pi]\).

Answer: 1


Question: Find the angle between the vector \( v = (2,2,1,1) \) and the linear subspace \( S \) spanned by the vectors \( u_1 = (3,4,-4,-1) \) and \( u_2 = (0,1,-1,2) \).

Answer: \dfrac{\pi}{3}


Question: Determine whether there exists a noncommutative ring of finite order that has a proper subring which is a division ring.

Answer: Yes


Question: Evaluate the improper integral \( \int^{+\infty}_0\frac{e^{-t}} {\sqrt t} \, dt\) and determine if it converges or diverges.

Answer: \sqrt{\pi}


Question: Using only a ruler and compass, construct the point on line MN that maximizes the difference in distances from two given points A and B, which are located on opposite sides of MN.

Answer: P


Question: Determine whether the following statement is true or false: If $\lim_{x\to\infty}\frac{f(x)}{g(x)}=\infty$, then there exists a function $h(x)$ such that $\lim_{x\to\infty}\frac{h(x)}{g(x)}=\infty$ and $\lim_{x\to\infty}\frac{f(x)}{h(x)}=\infty$. Provide a justification for your answer.

Answer: True


Question: How many distinct ways can the numbers $1, 2, 3, 4, 5, 6, 7$ be inserted into an empty binary search tree such that the resulting tree has a height of $6$? Note that a tree with a single node has a height of $0$. Provide your answer as a single integer.

Answer: 64


Question: Given that $y_1 = x$ is a solution to the differential equation $$(2x^2 + 1)y'' - 4xy' + 4y = 0,$$ find the second linearly independent solution $y_2$ using the reduction of order method.

Answer: 2x^2 - 1


Question: Find the limit of the sequence \( \frac{x_{n+1}}{x_1x_2\cdots x_n} \), where \( x_1=5 \) and \( x_{n+1}=x_n^2-2 \).

Answer: \sqrt{21}


Question: Evaluate the limit: $$\lim_{n\to\infty}\sum_{r=0}^n(-1)^r\binom{n}{2r}\left(\frac{x}{n}\right)^{2r}$$

Answer: \cos x


Question: Determine the cardinality of the set of all power series with real coefficients.

Answer: \mathfrak{c}


Question: Find the sum of the areas of all triangles with vertices at lattice points (0, 0), (a, 0), and (0, b), where a and b are positive integers, such that each triangle contains exactly 100 lattice points in its interior. Exclude lattice points on the sides of the triangles.

Answer: 3118


Question: Determine the order of a $15 \times 15$ matrix that satisfies the same properties as matrix A described in the original context.

Answer: 15


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

Answer: \dfrac{1}{12}


Question: Does there exist a function that is continuous everywhere except on a countable dense subset and is bounded, meaning the supremum of the function is a finite number?

Answer: Yes


Question: Calculate the volume enclosed by the hyperboloid \(-x^2-y^2+z^2=1\) and the plane \(z=2\) using polar coordinates.

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


Question: In a ring $R$ with a cyclic additive group $(R, +, 0)$, where $I$ is an ideal, must $I$ be a principal ideal? Assume that $R$ is not necessarily unital or commutative. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left( 1 + \sqrt{2} + \sqrt[3]{3} + \dots + \sqrt[n]{n} \right) \ln{\frac{2n+1}{n}} \]

Answer: \infty


Question: Calculate the value of \(\lim_{n\to\infty}\frac{\Gamma\left(n+\frac{7}{4}\right)n^{-3/2}}{\Gamma\left(n+\frac{5}{4}\right)}.\)

Answer: 0


Question: Determine the number of real solutions for the equation \( g(x) = x^4 - 5\sin 3x + \arctan (x-2) \) for \( x > 2 \).

Answer: 0


Question: Given that $x$, $y$, and $z$ are positive real numbers satisfying $x^3 + y^3 + z^3 = 81$, determine the maximum value of $x^2 + y^2 + z^2$.

Answer: 27


Question: In base 10, a number has the property that its square and the digit reversal of its square produce the same reverse order number. Are there any positive integers with this property that have more than two digits?

Answer: Yes


Question: Identify a nonabelian group in which every subgroup of order 2 or 4 is normal.

Answer: Q_8


Question: Suppose we have a collection of non-empty bounded open sets, $A_{0},A_{1},\dots$ that are subsets of $\mathbb{R}^n$, such that $A_{i}\supseteq A_{i+1}$, and the intersection $A=\bigcap_{i=0}^{\infty}A_{i}$ is a non-empty closed set. If there is a convergent sequence $a_{0},a_{1},\dots$ with limit $a$ such that $a_{i}\in A_{i}$, determine whether $a\in A$. Justify your answer.

Answer: a \in A


Question: Is the normed space \((C([a,b]), \|\cdot\|_1)\) complete if a sequence of functions converges uniformly on a compact subset of \([a,b]\)?

Answer: No


Question: Find an example of a field (not algebraically closed) such that its algebraic closure is a finite extension, or prove that no such fields exist.

Answer: \mathbb{R}


Question: Evaluate the sum \( S_n = \sum_{k \geq 1} \left\lfloor \frac{n}{2^k} + \frac{1}{2} \right\rfloor \).

Answer: n


Question: Let $A$ be a nonsingular matrix in $M_n(\mathbb{F})$. Determine whether there exists a polynomial $f \in \mathbb{F}[x]$ such that $f(A) = I$, where $I$ is the identity matrix.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to1} (\tan(\pi x/4))^{\tan(\pi x/2)}$$

Answer: \dfrac{1}{e}


Question: Evaluate the triple integral \( \iiint_B \sqrt{x^2+y^2+z^2} \, dV \) over the region \( B \) in \( \mathbb{R}^3 \) defined by the inequalities \( 0 \leq x \), \( 0 \leq y \), \( 0 \leq z \), and \( x^2 + y^2 + z^2 \leq 4 \). Express the integral in spherical coordinates and find its value.

Answer: 2\pi


Question: Find the Fourier transform of the product $f(x)\delta(x)$, where $f(x)$ is a continuous function and $\delta(x)$ is the Dirac delta function.

Answer: \dfrac{f(0)}{\sqrt{2\pi}}


Question: Find the smallest positive integer \( k \) such that for any \( k \) vectors \( v_1, v_2, \ldots, v_k \in \mathbb{R}^n \), there exist real numbers \( a_1, a_2, \ldots, a_k \), not all zero, satisfying \( a_1v_1 + a_2v_2 + \cdots + a_kv_k = 0 \) and \( a_1 + a_2 + \cdots + a_k = 0 \).

Answer: n + 2


Question: Determine the isomorphism class of the quotient group \( \frac{\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}}{\langle (2,0,1),(0,1,1) \rangle} \).

Answer: \mathbb{Z}


Question: Given a 2 by 2 matrix $B$ with eigenvalues $\frac{4}{5}$ and $\frac{7}{3}$, determine if there exists a non-zero vector $w$ such that $\|Bw\|^2 = 2\|w\|^2$. Provide a justification for your answer.

Answer: Yes


Question:
Let $f:\Bbb{R} \rightarrow \Bbb{R}$ be a differentiable function whose derivative is continuous. Evaluate the limit as $n$ approaches infinity of the expression:
$$(n+1)\int_0^1x^{n}f(x)dx$$

Answer: f(1)


Question: What is the result of the sum of three Cantor sets, denoted as $C+C+C$, where $C$ is the usual Cantor set obtained from the interval $[0,1]$?

Answer: [0, 3]


Question: Evaluate the line integral \( \oint_C \mathbf{B} \cdot \,\d\mathbf{r} \), where \( \mathbf{B} = \mathbf{e}_x(y^2-b(x+z))+\mathbf{e}_y(x^2-bz)+\mathbf{e}_z(z^2-b(x+y)) \), and \( C \) is the curve formed by the intersection of the cone \( x^2+y^2=z^2, \ z\geq0 \) and the sphere \( x^2+y^2+z^2=S^2 \) with \( S^2>2b^2 \).

Answer: 0


Question: Simplify the expression \(\frac{1}{4(2+m\pi i)} \left( e^{-(2+m\pi i )}- e^{2+m\pi i }\right)\) and find the values of \(m\) for which the expression is real.

Answer: 0


Question: Let $A$ be an $n \times n$ matrix which is both Hermitian and unitary. Which of the following statements is true?\n\n(a) $A^2=I$\n(b) $A$ is real\n(c) The eigenvalues of $A$ are $0$, $1$, $-1$\n(d) The minimal and characteristic polynomials are the same.

Answer: a


Question: In the binomial expansion of \(\left(\sqrt[3]{\frac{a}{\sqrt{b}}} + \sqrt{\frac{b}{\sqrt[3]{a}}}\right)^{21}\), find the value of \(r\) for which the \((r+1)^{th}\) term has the same power of \(a\) and \(b\).

Answer: 9


Question: If $x$ and $y$ are real numbers such that $(x+y)(x+1)(y+1) = 3$ and $x^3 + y^3 = \frac{45}{8}$, find the value of $xy$. \( \boxed{} \)

Answer: -\dfrac{1}{2}


Question: Compute the induced group homomorphism $(f_n)_* : \pi_1(S^1, 1) \to \pi_1(S^1, 1)$ for the map $f_n : S^1 \to S^1$ defined by $f_n(z) = z^n$, where $S^1$ is the unit circle $\{z \in \mathbb{C} \mid |z| = 1\}$ and $\pi_1(S^1, 1)$ is identified with $\mathbb{Z}$.

Answer: n


Question: Calculate the limit \( L = \lim_{a \to 0} \int_{-x}^x \frac{e^t \cdot \cos(at)}{1+e^t} \, dt \) for \( a \in \mathbb{R} \) and \( x \in \mathbb{R} \), without using the dominated convergence theorem or the monotone convergence theorem.

Answer: x


Question: Determine the number of times the number 3 appears in all compositions of a natural number \( n \geq 4 \).

Answer: n \cdot 2^{n - 5}


Question: Find all functions \( g : \mathbb{Q} \to \mathbb{Q} \) that satisfy the equation \( g \left( \frac{x+y}{2} \right) = \frac{g(x) + g(y)}{2} \) for all \( x, y \in \mathbb{Q} \).

Answer: g(x) = ax + b


Question:
Let \( f: \mathbb{R} \to [0, \infty) \) be a non-negative measurable function such that for all \( t > 0 \),
\[ \mu\left(\{x: f(x) > t\}\right) = \frac{1}{t^2 + 1} \]
Compute the integral \( \int_{\mathbb{R}} f\,d\mu \).

Answer: \dfrac{\pi}{2}


Question: Determine the number of natural numbers \( n \) such that \( 5^n \equiv 1 \pmod{p} \) does not hold for any prime number \( p > 3 \).

Answer: 2


Question: Determine whether the following expression has a limit as \( k \) approaches infinity, given that \( 2^mf\notin \mathbb{Z} \) for every \( m\in\mathbb{Z} \): \[ \lim_{k\rightarrow \infty}\frac{\sin^2(2^{k}\pi f)}{\sin^2 (2^{k-1}\pi f)}\prod_{j=1}^{k} \cos^2(2^{k-j}\pi f) \]

Answer: 0


Question:
Determine whether the function \( f(x) \) defined by

\[ f(x)=\begin{cases}
0, & x=0 \\
1, & \lfloor\frac{1}{x}\rfloor \text{ is even} \\
-1, & \lfloor\frac{1}{x}\rfloor \text{ is odd}
\end{cases} \]

(where \( \lfloor x\rfloor \) denotes the floor function) is Riemann-integrable on the interval \([0,2]\).

Answer: Yes


Question: Let $V$ be a vector space over a field $F$, and let $W$ be a non-zero vector space over the same field. Consider a subset $S_v \subset V$. If every mapping $f: S_v \to W$ can be uniquely extended to a homomorphism $g: V \to W$, is $S_v$ necessarily a basis of $V$? Answer "yes" or "no" and justify your answer.

Answer: yes


Question: Consider a continuous function \( f: [0, \infty) \to (0, 1) \). Define a sequence \( a_n \) recursively by \( a_1 = 1 \) and \( a_{n+1} = \int_0^{a_n} f(x) \, \mathrm{d}x \) for \( n \geq 1 \). Determine the limit of \( a_n \) as \( n \to \infty \). Provide a detailed and rigorous proof of your result.

Answer: 0


Question: Find a single polynomial that generates the ideal $J = (h,k)$ in $\mathbb{R}[x]$, where $h = (x^2 - 9)(x - 3)$ and $k = (x^4 - 81)(x + 2)$. Provide your answer as a polynomial expression.

Answer: x^2 - 9


Question: Evaluate the limit \( \lim\limits_{n\to\infty}\frac{\sum_{i=1}^n i^r}{n^{r+2}} \) for \( r>0 \).

Answer: 0


Question: Find the number of distinct ways to color a regular hexagon using 5 colors, taking into account all rotational and reflectional symmetries of the hexagon.

Answer: 1505


Question: Evaluate the integral \( \int_0^\frac{\pi}{3}\ln^n\left(2\sin\frac{\theta}{2}\right)d\theta \) at \( a=0 \).

Answer: \dfrac{\pi}{3}


Question:
Evaluate the limit for a continuous function $f 	ext{ on } [-1,1]$:
$$\lim_{h \to 0} \frac{ \int^{h}_{-h} f(t) \, dt }{h} $$

Answer: 2f(0)


Question: Find the non-constant polynomial $f(x)$ such that $f(g(x)) = g(f(x))$, where $g(x) = 3x^4 - 1$. Express $f(x)$ in its simplest form.

Answer: 3x^4 - 1


Question: Does the existence of the integral \( I = \int_2^{\infty} f(x)g(x) \, \mathrm{d}x \), where \( f(x) \) and \( g(x) \) are nonconstant functions, imply that both \( f(x) \) and \( g(x) \) are necessarily differentiable functions of \( x \)? Answer yes or no.

Answer: No


Question: Determine the number of real roots of the equation \(x^2 + 10x + (\sin(x))^2 = \cos(x)\).

Answer: 2


Question: Determine the radius of convergence of the power series: $$\sum_{k = 0}^{\infty}\frac{ k^{3 k - 2} \log^{8} k}{\left((k+1)!\right)^2} \,x^k$$

Answer: 0


Question:
Find the smallest positive integer value of $x$ that satisfies the following system of equations, where $x, y, z$, and $u$ are positive integers:
\[\begin{cases} x+y=3(z+u)\\ x+z=4(y+u)\\ x+u=5(y+z)\\ \end{cases}\]

Answer: 83


Question: Find the limit of the expression: $$\lim_{n\to\infty}{\frac{1}{n} {\sum_{k=3}^{n}{\frac{3}{k^2-k-2}}}}$$

Answer: 0


Question: Determine if the function \( y(x) = e^{x^3} \) can be a solution to the differential equation \( y'' + p(x)y' + q(x)y = 0 \) in an interval containing 0, given that \( p(x) \) and \( q(x) \) are continuous functions.

Answer: Yes


Question: Evaluate the second derivative of \( f(x)=\int_x^{2x}\frac{\sin t}{t}dt \) at the critical points \( x=\pi \) and \( x=\pi/3 \). Determine which point corresponds to a local maximum.

Answer: \frac{\pi}{3}


Question: Determine the minimum value of the expression \( \left|7x-1\right|+\left|7y-5\right|+\left|7z-1\right| \) for non-negative real numbers \(x\), \(y\), and \(z\) such that \(x + y + z = 1\) and \(y^2 \leq 3xz\).

Answer: 52 - 28\sqrt{3}


Question:
Evaluate the limit as \( N \to \infty \):
\[
\lim_{{N \to \infty}} \left( \left(\frac{\sin \frac{1}{N}}{\frac{1}{N}}\right)^{N} -1 \right) \cdot \frac{1}{N}.
\]

Answer: 0


Question: Given that $X \sim N(0,1)$ and conditional on $X = x$, $Y \sim N(x,1)$, find the covariance $\text{Cov}(X,Y)$ using the relationship $E[XY] = E[E[XY|X]]$. Provide your answer as a single numerical value.

Answer: 1


Question: Determine whether there exists a Boolean complete lattice that is not atomic.

Answer: Yes


Question: Find the remainder when $2^{1990}$ is divided by $1990$. Express your answer as a single integer.

Answer: 1024


Question: Compute the double integral \( \iint_{M} y \, dx \, dy \) where \( M \) is the region defined by \( y \geq 0 \), \( x^2 + y^2 \leq 1 \), and \( (x - 1)^2 + y^2 \geq 1 \).

Answer: \dfrac{11}{24}


Question: Find the last four digits of the number \(4 \times 3^{50} + 5 \times 4^{30}\).

Answer: 5876


Question: Evaluate the limit: $$\lim_{x \to 0} \frac{1}{x^3} \int_{0}^{x} (t^2f(t)-t^2f(0)) \space dt$$ using L'Hôpital's Rule or an alternative method.

Answer: 0


Question: Determine if the function \(g_n(x)=\begin{cases}n(1-|nx|)&x \in \left(-\dfrac{1}{n},\dfrac{1}{n}\right)\\[6pt] 0 & \text{otherwise} \end{cases}\) is a nascent delta function at 0.

Answer: Yes


Question: Evaluate the double integral \( \int_{1}^{\infty}\int_{0}^{a} e^{-xy}dxdy \) by reversing the order of integration.

Answer: \infty


Question: Is the derivative of a real differentiable function $h$, defined on the open interval $(0,1)$ and having a one-to-one derivative, necessarily continuous on the interval?

Answer: Yes


Question: Determine the number of different complex structures that can be defined on the real plane.

Answer: 1


Question:
Determine the maximum possible value of \(a \in \mathbb{R}\) for which there exist twice differentiable nondecreasing functions \(f(x)\) and \(g(x)\) satisfying the following conditions:
1. \(f^2(x) + g^2(x) \geq a\)
2. \(f''(x) = g(x)\) and \(g''(x) = f(x)\)
3. \(f(x)g(x)\) is a linear function of \(x\).

Answer: 0


Question: Let \( g: \ell_2 \rightarrow \mathbb{R} \) be defined by \( g(x) = x_2 + 3x_4 - x_6 + 2x_8 \). Compute \( \|g\| \) using the Cauchy-Schwarz Inequality.

Answer: \sqrt{15}


Question: In triangle $XYZ$, where $3x^2 = y^2 + z^2$ and the measure of angle $X$ is $45^{\circ}$, calculate the value of $\sin 2Y + \sin 2Z$. \( \text{Express your answer as a single number.} \)

Answer: 1


Question: Let $f$ be an analytic function on $|z|<2$ such that $f(0)=f'(0)=0$, $f''(0) \neq 0$, $f(1/3)=i/12$, and $|f(z)|\le3$ for $|z|<2$. Determine the value of $f(2i/3)$. \( \boxed{} \)

Answer: -\dfrac{i}{3}


Question: Find the values of \(\alpha\) for which the series \(\sum_{k=0}^{\infty}(-1)^{k}\frac{x^{2k}}{(2k)!^{\alpha}}\) converges for all real numbers \(x\).

Answer: \alpha > 0


Question: Let $X$ and $Y$ be random variables defined on the probability space $(\Omega,\mathcal{F},\mathbb{P})$ with values in $(\mathbb{R},\mathcal{B})$. If $\mathbb{P}(X \in B) = \mathbb{P}(Y \in B)$ for all Borel sets $B \in \mathcal{B}$, does it follow that $X = Y$ almost surely?

Answer: No


Question:
Given the equations:
1. $w + x + y + z = 25$
2. $wx + wy + wz + xy + xz + yz = 2y + 2x + 193$

Find the largest possible value of $w$ expressed in lowest terms as $\frac{w_1}{w_2}$, where $w_1, w_2 > 0$. What is the value of $w_1 + w_2$?

Answer: 27


Question:
Determine the value of the following expression:

$$\inf \left\{\sup \left\{ \frac{2n}{2+n(-1)^n}:n>k\right\}:k\in\mathbb N\right\}$$

Answer: 2


Question: Determine whether the Fourier inversion theorem holds for a function $f$ such that $\int_{-\infty}^{\infty}|f(x)|dx<\infty$. Specifically, is it true that for almost every $x\in\mathbb{R}$, the Fourier transform of $f$, denoted by $\hat{f}$, satisfies $$f(x)=\lim_{R\rightarrow+\infty}\int_{-R}^{R}\hat{f}(w)e^{2i\pi x w}dw?$$

Answer: No


Question: Calculate the value of $g(e) + g\left(\frac{1}{e}\right)$ where $g(x) = \int_{1}^{x}\frac{\ln t}{t+1}dt$. Provide your answer in simplest form.

Answer: \dfrac{1}{2}


Question: Calculate the limit of the sum as follows: $$\lim_{n\to \infty} \sum _{i=1}^{n} \left(\sqrt{n}+\frac{i}{\sqrt{n}}\right)^{-2}$$ Interpret this expression as a definite integral and solve it using Riemann sum techniques.

Answer: \dfrac{1}{2}


Question: Consider a hat containing slips numbered from \(kr+1\) to \((k+1)r\), where \(k\) is a positive integer. What is the expected number of slips that need to be drawn from the hat until all numbers drawn are unique?

Answer: r H_r


Question: Let \(X\) be a compact star-shaped subset of \(\mathbb{R}^n\) centered at \(x_0 \in X\), such that for any \(x \in X\), the line segment \([x_0,x]\) is contained in \(X\). Suppose that \(X\) has a nonempty interior and its boundary \(\partial X\) is homeomorphic to \(\mathbb{S}^{n-1}\). Determine whether \(X\) admits a center point in its interior.

Answer: Yes