Question: Suppose that \( h: \mathbb{R} \to \mathbb{R} \) is a function with a fixed point at \( x^* \). If \( h'(x^*) = 0 \) and \( |h''(x)| < 2 \), determine the smallest value of \( n \) such that the sequence \( x_0, x_1, x_2, \ldots \) defined by \( x_n = h(x_{n-1}) \) satisfies \( |x_n - x^*| < 10^{-50} \).

Answer: 6


Question: In a free group $G$ generated by two elements $a$ and $b$, consider a random walk starting at the identity element $1$. At each step, multiply the current word by one of the generators $a$, $b$, $a^{-1}$, or $b^{-1}$, each with probability $\frac{1}{4}$. What is the probability $\rho$ that the reduced word $1$ is ever reached during this process?

Answer: \dfrac{1}{3}


Question: Let $X, Y,$ and $Z$ be topological spaces. Consider a continuous function $f: X \times Y \to Z$ with the product topology on $X \times Y$. Given a point $y \in Y$, define a function $f_y: X \to Z$ by $f_y(x) := f(x, y)$. Is $f_y$ continuous for every $y \in Y$?

Answer: Yes


Question: Determine whether any closed cone in \(\mathbb{R}^n\) can be described as the closure of a cone fully represented by a countable number of vectors, where each vector is a linear combination of non-negative scalar multiples of a finite set of basis vectors.

Answer: Yes


Question: Determine the order of the quotient group $G/Z(G)$, where $G = \{a^k, a^k b \mid 0 \le k < 9\}$ with $o(a) = 9$, $o(b) = 2$, and $ba = a^{-1}b$. The center $Z(G)$ consists of elements that commute with all elements in $G$. What is the order of $G/Z(G)$?

Answer: 18


Question: Evaluate the sum: $$\sum\limits_{k = 0}^{250} (-1)^k \binom{2000 - 4k}{1500} \binom{1501}{k}$$ using a binomial expansion.

Answer: 0


Question:
Evaluate the limit as \( x \) approaches infinity:

\[
\lim_{x \to \infty} \frac{1}{e^{x^2}} \int_{x}^{x + \frac{\ln x}{x}} e^{t^2} dt
\]

Answer: \infty


Question: Determine the dimensions $n$ for which the existence of an $n \times n$ complex matrix $A$ with $\det(I-A^*A)<0$ implies that the operator norm of $A$ is greater than 1.

Answer: n \geq 1


Question: Given a set $A$ with $n$ elements and a natural number $k$, determine the number of $k$-tuples $(S_1, S_2, \ldots, S_k)$ such that each $S_i$ is a subset of $A$, $S_i \subseteq S_{i+1}$ for $i = 1, 2, \ldots, k-1$, and $S_1 \cup S_2 \cup \ldots \cup S_k = A$. Provide your answer as a single number.

Answer: k^n


Question: Calculate the integral \( \int_{\mathbb{R}} e^{-x^2}\, d\lambda (x) \) using Fubini's theorem. First, evaluate the double integral \( \int_{\mathbb{R}^2} e^{-(x^2+y^2)}\,d(\lambda \times \lambda) \) by converting to polar coordinates and applying Tonelli's theorem.

Answer: \sqrt{\pi}


Question: Determine whether the circle \(S^1 = \{X^2 + Y^2 = 1\} \subseteq k^2\), where \(k\) is a field, is a rational curve.

Answer: Yes


Question:
Determine the correct equation for the slope of a tangent line to an ellipse given by the equation \(Ax^2 + By^2 + Cx + Dy + Exy + F = 0\). The two candidate equations are:

1) \(\frac{dy}{dx} = -\frac{2Ax+C+Ey}{2By+D+Ex}\)

2) \(M = \frac{2By + D + Ex}{2Ax + C + Ey}\)

Which equation correctly represents the slope of the tangent line?

Answer: 1


Question: Consider a set $B$ with four elements and operations defined by tables. Can $B$ form a field if $y = 1$ but $xy \neq x$ for some $x$?

Answer: No


Question: If $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function with $x^2f'(x)\to 0$ as $x\to\infty,$ determine whether it is true that $f(x)\to c\in\mathbb{R}$ as $x\to\infty$. Provide a justification for your answer.

Answer: Yes


Question: Does the derivative of a differentiable function within a closed domain $J$ guarantee that the derivative is continuous throughout the domain $J$? Answer yes or no.

Answer: No


Question: Let \( H = \{(x,y,z) \in \mathbb{R}^{3}: x^2+y^2 - z^2 + 4 = 0\} \). Compute the shortest distance between \( H \) and the point \( P \equiv (2,4,0) \).

Answer: \sqrt{14}


Question: Find an explicit formula for the operator $TT^*$, where $T: H \rightarrow \ell^{2}$ is the analysis operator $T x=\left\{\left\langle x, e_{n}\right\rangle\right\}_{n \in \mathbb{N}}$ and $T^{*}: \ell^{2} \rightarrow H$ is the synthesis operator $T^{*} c=\sum_{n=1}^{\infty} c_{n} e_{n}$, given an orthonormal sequence $\{e_{n}\}_{n \in \mathbb{N}}$ in a Hilbert space $H$.}

Answer: TT^* = I


Question: Given that \((B_2 - B_1) \sim N(0, 3)\), find \(\mathbb{E}[B_1 | B_2]\).

Answer: \dfrac{B_2}{2}


Question: Evaluate the limit: $$\lim_{x\to\infty}\frac{1-\cos{x}\sqrt{\cos{2x}}}{x\sin{x}}$$

Answer: 0


Question: Find the remainder when the polynomial $P(x,y) - P(y,x)$ is divided by $(x - y)$ in the polynomial ring $K[x,y]$, where $K$ is a field.

Answer: 0


Question: Find the solution to the boundary value problem: $$\Delta u = 0 \text{ in } \Omega,\quad u|_{\partial \Omega} = y,$$ where $\Omega=\{(x,y)|x^2+y^2<4\}$.

Answer: y


Question: Given that $A$ is a $5 \times 7$ matrix of rank $5$ and $B$ is a $7 \times 5$ matrix of rank $5$, and $AB = I_5$, determine the rank and nullity of the matrix $BA$. Provide your answer as a pair $(\text{rank}, \text{nullity})$. \( \boxed{(5, 2)} \)

Answer: (5, 2)


Question: Evaluate the limit: $$\lim_{x\to0}\frac{x^2\sin\left(\frac{1}{x}\right)}{\sin x}$$

Answer: 0


Question: Determine the set of all real numbers $x$ that satisfy the equation $x^2 + 3x - 9 + 2^{x^2 + 3x - 2} + \log_2{(x^2 + 3x - 2)} = 0$. Find the set $A = \{x \in \mathbb{R} \mid x^2 + 3x - 9 + 2^{x^2 + 3x - 2} + \log_2{(x^2 + 3x - 2)} = 0\}$. \( \text{Express your answer in set notation.} \)

Answer: \{-4, 1\}


Question: Determine the cardinality of the set $S$ of all convergent sequences of real numbers.

Answer: \mathfrak{c}


Question: Evaluate the integral \( \int_{1}^{2} x^2f(x^3) \, dx \) given that \( \int_{1}^{4} f(x) \, dx = 6 \).

Answer: 2


Question: Evaluate the integral \( \int_0^{\pi/2} \frac{1}{1+\tan^{\beta}{x}}\,\mathrm{d}x \) for any positive real number \( \beta \).

Answer: \dfrac{\pi}{4}


Question: Suppose $f$ is analytic in a punctured disk and $f'$ is bounded in the punctured disk. Is $f$ necessarily bounded in that disk? Provide a justification for your answer.

Answer: Yes


Question: If $f(n)$ is an injective function and the sequence $p_n$ converges to $L$, does the sequence $p_{g(n)}$ also converge to $L$ for any injective function $g(n)$?

Answer: Yes


Question: Can any angle be trisected using only a compass and straightedge? Provide a justification for your answer.

Answer: No


Question: Suppose $f(x,y)$ is differentiable at $(1,1)$, and $f(1,1)=\frac{\partial f}{\partial x}(1,1)=\frac{\partial f}{\partial y}(1,1)$. Find the value of $\frac{d}{dx}f(x,f(x,x))$ at $x=1$. \( \boxed{?} \)

Answer: 3


Question: Calculate the value of $2S + 1$ for the infinite series $S = \sum_{n=1}^{\infty}\frac{\sin (n)}{n}$ using the Euler Method and the logarithmic series.

Answer: \pi


Question: Find the minimum value of \( c \) such that the inequality \( \frac{e^{2x} - e^{-2x}}{2e^{cx^2}} \geq 1 \) holds for all real numbers \( x \).

Answer: 2


Question:
Demonstrate that the following equality holds:

$$\displaystyle\sum_{k=1}^{\infty} (-1)^{k-1}\frac{1}{k!}=1-\frac{1}{e}$$

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


Question: Consider a sequence of bounded real-valued random functions \((f_n)\) defined on the interval \([0,1]\). For each \(x \in [0,1]\), assume that \(f_n(x)\) converges in probability (or almost surely) to a limit function \(f(x)\). Determine whether the integral \(\int_{0}^{1} |f_n(x) - f(x)| dx\) converges in probability (or almost surely) to \(0\).

Answer: Yes


Question: Determine whether the inequality \( \|f*g\|_{L^\infty} \leq C\|f\|_{H^1}\|g\|_{BMO} \) holds for all functions \( f \in H^1 \) and \( g \in BMO \), where \( C \) is a constant.

Answer: Yes


Question: Simplify the expression: \[ \tan^{-1}\left(\sqrt{\frac{x(x+y+z)}{yz}}\right) + \tan^{-1}\left(\sqrt{\frac{y(x+y+z)}{zx}}\right) + \tan^{-1}\left(\sqrt{\frac{z(x+y+z)}{xy}}\right) \] using the properties of inverse trigonometric functions.

Answer: \pi


Question: Let $X$ be a Polish space, and consider a sequence of Borel probability measures $(\mu_n)_{n \geq 1}$ on $X$. If, for every continuous and bounded function $f : X \to \mathbb{R}$, the limit $L(f) := \lim_{n \to \infty} \int_X f \, d \mu_n$ exists, does there exist a Borel probability measure $\mu$ such that $L(f) = \int_X f \, d \mu$ for all such functions?

Answer: Yes


Question: Evaluate the limit: $$\lim_{q\to 1^-}\frac{\pi K'} {K} \cdot\frac{K} {2\pi^2}\cdot(E-k'^2K)$$ where $K, K', E$ are elliptic integrals and $k, k'$ are moduli corresponding to the nome $q$. Provide your answer as a single value.

Answer: \dfrac{1}{4}


Question: Given a number field $L$ and a non-zero ideal $I$ of the ring of integers $O$ of $L$, determine if there exists a prime ideal $\mathcal{P} \subseteq O$ in the ideal class of $I$ such that $p = \mathcal{P} \cap \mathbb{Z}$ splits completely in $O$. Provide a justification for your answer.

Answer: Yes


Question: Find the greatest common divisor of \(^{72}C_{36}\) and \(36!\).

Answer: 159068


Question: In a deck of cards numbered from 1 to 200, calculate the expected number of cycles formed when performing a card-flipping process. What is the expected number of cycles?

Answer: H_{200}


Question: Evaluate the polynomial \(x^{16} + 8x^8 + 1\) at all 32nd roots of unity. How many distinct values are obtained?

Answer: 4


Question: Does there exist a function \( f : A \rightarrow A \) (where \( A \) is a finite set) such that there is an element \( a \) in \( A \) for which \( f^{n+1}(a) \neq f(a) \) for every positive integer \( n \)?

Answer: Yes


Question: Find the maximum value of the linear expression \(2x - y + 5z\) for a unit vector \((x, y, z)\).

Answer: \sqrt{30}


Question: Evaluate the limit: $$ \lim_{x\to\infty} \left( (x+2017)^{1+\frac{1}{x}} \: -\: x^{1+\frac{1}{x+2017}} \right) $$

Answer: 2017


Question: Evaluate the limit: \[ \lim_{x\rightarrow\infty} \frac{f(x)}{x} \] where $f(x)$ is the number of prime powers less than $x$. A prime power is defined as any number of the form $p^n$ where $p$ is a prime and $n \ge 1$. Determine the value of this limit.

Answer: 0


Question: Determine the covariance function \( \text{Cov}(Y(t), Y(s)) \) for the process \( Y(t) = X^2(t) \), where \( X(t) \) is a Brownian motion with variance parameter \( \sigma^2 \) for \( t \geq 0 \) and \( E(Y(t)) = t \).

Answer: 2 (\min(t, s))^2


Question: Determine the number of solutions in natural numbers \((x, y)\) to the equation \(\frac{1}{x} + \frac{1}{y} = \frac{1}{2023}\) such that \(x + y\) is a multiple of 101.

Answer: 0


Question: Evaluate the contour integral \( \int_{|z-1|=1} \frac{2z-1}{z^2 - 1} \, dz \) using the residue theorem, where the contour is a circle of radius 1 centered at 1, traversed in the anticlockwise direction.

Answer: \pi i


Question: Determine whether there exists an exact sequence of the form $0 \to K \to A^m \to P \to 0$, where $P$ is a finitely generated projective $A$-module and $K$ is also finitely generated.

Answer: Yes


Question: Determine whether the integral \( \int_t^\infty \exp(-x+o(x))\,dx \) approaches \( \exp(-t+o(t)) \) as \( t \to \infty \).

Answer: \exp(-t + o(t))


Question: Find the limit of the sequence \(a_n = \frac{1}{n^2} \sum _{k=1}^n \left( \sqrt{\left(n+k\right)^2+n+k}\, \right)\) as \(n\) approaches infinity.

Answer: \dfrac{3}{2}


Question: Is it necessary for a non-constant matrix function $A(t)$, such that $\exp(A)$ solves $\dot{\Phi} = \dot{A} \Phi$, to satisfy the condition $A\dot{A} = \dot{A}A$? Answer with 'Yes' or 'No'.

Answer: Yes


Question: Consider an open subset $G$ of the real numbers that is not bounded above. Determine whether there exists a positive real number $x$ such that the set of all integer multiples of $x$ has infinitely many points in common with $G$. In other words, is it true that there exists $x > 0$ for which the set $\{mx \mid m \in \mathbb{Z}\} \cap G$ is infinite?

Answer: Yes


Question: Evaluate the integral \( \int_{(-1, 1)} |x| \, d\mu \), where \( \mu((a, b]) = \int_a^b \frac{1}{1+x^2} \, dx \) for all intervals \((a, b]\) on the measurable space \((\mathbb{R}, \mathcal{B})\).

Answer: \ln 2


Question: If a $C^*$-algebra $A$ has the property that all its irreducible representations are faithful, is $A$ necessarily simple?

Answer: Yes


Question:
Evaluate the limit of the double integral as the radius of the circular region approaches 0, given that $f$ is a differentiable function such that $f(0) = 0$:
$$\lim_{r\to 0^{+}} \iint_{x^2 + y^2 \leq r^2} \frac{3}{2\pi r^3} f(\sqrt{x^2+y^2}) \, dx \, dy$$

Answer: f'(0)


Question: Find the argument \( \arg z \) in its simplest form for \( z=(\sin\theta+i(1-\cos\theta))^2 \), where \( 0<\theta<\frac{\pi}{2} \).

Answer: \theta


Question: If a function $f$ is differentiable at a point $x$ according to the condition $\overline Df(x) = \underline Df(x)$, does this imply that the standard definition of differentiability holds at $x$? Specifically, does it imply that \( \lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h} \) exists?

Answer: Yes


Question: Suppose $f$ and $g$ are two analytic power series centered at $z=0$, both with a radius of convergence equal to $R > 0$. Assume that $g$ is nowhere vanishing in $B(0,R)$, so that the quotient $f/g$ is defined and analytic there. Can $f/g$ be analytic in a domain strictly larger than $B(0,R)$?

Answer: Yes


Question: Can a full set of orthonormal eigenvectors be constructed for a Hermitian matrix $A \in \mathbb{C}^{n \times n}$ if it has repeated eigenvalues?

Answer: Yes


Question: Determine the range of values for \( a \) such that the equation \( 2x^3 - 3x^2 - 12x + a = 0 \) has three real and distinct roots.

Answer: (-7, 20)


Question: Express the function \( f \) as a Taylor series around 0, given that \( f \) is infinitely differentiable on an open interval containing 0, \( f'(x) = 2xf(x) \), and \( f(0) = 1 \). What is the radius of convergence of this Taylor series?

Answer: \infty


Question: Given that \(x, y, z\) are the sides of a triangle where \(x \neq y \neq z\), find the range of \(\mu\) such that the roots of the equation \(x^{2} + 4(x+y+z)x + 6\mu(xy+yz+zx) = 0\) are real.

Answer: (-\infty, 2]


Question: For a twice differentiable function \(f(x)\), it is given that \(|f(x)| \leq 3\) for all \(x \in \mathbb{R}\). Additionally, for some \(\alpha\), we have \(f(\alpha) + \bigl(f'(\alpha)\bigr)^2 = 80\). Determine the number of integral values that \((f'(x))^2\) can take in the interval \((0, 77)\).

Answer: 76


Question: Given the polynomial \(f(x) = x^3 + x^2 - 5x - 1\), where \(\alpha, \beta, \gamma\) are its roots, find the value of \(\lfloor\alpha\rfloor + \lfloor\beta\rfloor + \lfloor\gamma\rfloor\), where \(\lfloor.\rfloor\) represents the floor function.

Answer: -3


Question: Let $\{a_n\}$ be an unbounded sequence of positive real numbers such that $\lim\limits_{n\to\infty}(a_{n+1}-a_n)=0$. Determine whether $a_n$ can be expressed as a partial sum $\sum_{k=1}^n b_k$ of some sequence $\{b_k\}$ such that $\lim\limits_{k\to\infty}b_k=0$, but $\sum_{k=1}^\infty b_k=\infty$. Provide a justification for your answer.

Answer: Yes


Question: If $A \in M_n$ is a normal matrix with $k$ nonzero eigenvalues, determine the dimension of the nullspace of $A$. Express your answer in terms of $n$ and $k$. \( \boxed{} \)

Answer: n - k


Question: Find the limit of the function \( f(x) = x\left(\left(1 + \frac{1}{x}\right)^x - e\right) \) as \( x \to \infty \).

Answer: -\dfrac{e}{2}


Question: In a separable compact Hausdorff space $X$, determine whether there exist two open sets $V, U$ and a continuous function $f: X \to \mathbb{R}$ such that $x \in V$, $y \in U$, $f(t) = 1$ for all $t \in V$, $f(t) < 0$ for all $t \in U$, and $f^{-1}(\{0\})$ has empty interior.

Answer: Yes


Question: In a Poisson process with rate parameter $\lambda > 0$, where $N_0 = 0$, $N_{t_2} - N_{t_1}$ and $N_{t_4} - N_{t_3}$ are independent for all $0 \leq t_1 < t_2 \leq t_3 < t_4$, and $P(N_{t+h} - N_t = 1) = \lambda h + o(h)$ and $P(N_{t+h} - N_t \geq 2) = o(h)$, does it logically follow that $N_t \geq 0$ for all $t$? Provide a justification for your answer.

Answer: Yes


Question: Calculate the rank of the symmetric bilinear form \( f(A,B) = \operatorname{tr}A\operatorname{tr}B + n\operatorname{tr}AB \).

Answer: n^2


Question: Evaluate the limit: $$\lim\limits_{x \to 0} \dfrac{8 [1-\cos\left(\frac{x^2}{2}\right)-\cos\left(\frac{x^2}{4}\right)+\cos\left(\frac{x^2}{2}\right)\cos\left(\frac{x^2}{4}\right)]}{x^8} $$

Answer: \dfrac{1}{32}


Question: If two regular unit speed curves have the same non-vanishing curvature and torsion, are the curves necessarily congruent?

Answer: Yes


Question: Evaluate the integral: $$\int_0^{\infty } {\frac{(x^7)(1-x^{12})}{(1+x)^{28}}}dx$$

Answer: 0


Question:
Solve the system of equations for $m=1$:
\[x^{x+y} = y^m\]
\[y^{x+y} = x^{2m} \cdot y^m\]

Answer: (1, 1)


Question: Can any elliptic curve over an algebraically closed field be embedded into the projective plane $\mathbb{P}^2$ as a cubic curve?

Answer: Yes


Question: Evaluate the integral \(\oint_C \sec z \, \mathrm{d}z\) where \(C\) is the unit circle centered at the origin, traversed counterclockwise.

Answer: 0


Question:
Given the cubic equation \(8x^3 - ax^2 + bx - 1 = 0\) with three real roots in geometric progression, determine the ordered pair \((\lambda_1, \lambda_2)\) such that \(\lambda_1 \leq a \leq \lambda_2\). Assume the roots are \(a', a'r, a'r^2\) where \(a', r \in \mathbb{R}\) and \(r \neq 1\). The following conditions hold:
1. \(a' + a'r + a'r^2 = \frac{a}{8}\)
2. \(a'^2(r + r^2 + r^3) = \frac{b}{8}\)
3. \(a'^3r^3 = \frac{1}{8}\)
Find the relationship between \(a, a', r\) and determine the ordered pair \((\lambda_1, \lambda_2)\).

Answer: (-4, 12)


Question: Evaluate the expression: \( \sqrt{2+\sqrt{3}} \cdot \sqrt{2+\sqrt{2+\sqrt{3}}} \cdot \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}} \cdot \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}} \)

Answer: 1


Question: Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra, and consider its realification, $\mathfrak{g}_{\mathbb{R}}$. The radical of $\mathfrak{g}$, denoted $\mathrm{rad}(\mathfrak{g})$, is the unique maximal solvable ideal. Determine if it is possible that $\mathrm{rad}(\mathfrak{g}) \subsetneq \mathrm{rad}(\mathfrak{g}_{\mathbb{R}})$. Answer with 'Yes' or 'No'.

Answer: No


Question: Evaluate the integral of the function $⌊x^2⌋ + \{x^2\}$ from $0$ to $2$.

Answer: \dfrac{8}{3}


Question: Find the stationary point of the functional \( J[y]=\int \left( x^2y'^2+2y^2 \right) dx \) subject to the condition \( y(1)=2.\)

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


Question: Given a set of sentences \(\Sigma = \{\gamma_{\geq n} : n\) is even\}, where \(\gamma_{\geq n}\) is a sentence such that its models have the cardinality of at least \(n\), determine the smallest model of \(\Sigma\).

Answer: \aleph_0


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

Answer: 1


Question: Determine whether the 1-dimensional circle is a one-dimensional manifold, despite its ability to be embedded into a two-dimensional object.

Answer: Yes


Question: In a von Neumann algebra $M$, determine if there exists a central projection $z$ for a projection $p$ such that $pz$ is infinite and $p(1-z)$ is finite.

Answer: Yes


Question: Determine the value(s) of $c \in \mathbb{R}$ for which the function $f: \mathbb{R} \rightarrow \mathbb{R}$, defined by \[ f(x) = \begin{cases} x & x \in \mathbb{Q} \\ 1-x & x \notin \mathbb{Q} \end{cases} \] is continuous.

Answer: \dfrac{1}{2}


Question: Calculate the limit: $$\lim_{x \to 0} \frac{1}{x^3}\int_0^x \frac{t^2}{1 + t^4}dt$$

Answer: \dfrac{1}{3}


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

Answer: \dfrac{1}{2}


Question: Evaluate the infinite series \( S = \sum_{n=0}^{\infty} (-1)^n\frac{\pi^{4n}}{n!} \).

Answer: e^{-\pi^4}


Question: Evaluate the limit of the sequence as \( n \to \infty \): \[ \lim_{n \to \infty} \left( \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)} \right) \]

Answer: 0


Question: Evaluate the integral: $$\int_0^{\frac{\pi}{2}} \frac{\sin^3(t)}{\sin^3(t)+\cos^3(t)}\,dt$$

Answer: \dfrac{\pi}{4}


Question: Determine the number of subrings of cardinality $p^2$ in the field $\mathbb{F}_{p^2}$, where $p$ is a prime number.

Answer: 1


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

Answer: e^{-2}


Question: Evaluate the limit: $$\lim_{x\to 0}\left(\frac {e^x}{x}-\frac {1}{\arctan{x}}\right)$$

Answer: 1


Question: Evaluate the limit: $$\lim_{{(x,y)\rightarrow (0,0)}}{(x^2+y^2)}^{x^2y^2}.$$

Answer: 1


Question: Determine whether there exists a compact complex manifold in $n$ complex dimensions that is also parallelizable. Specifically, does there exist $n$ holomorphic sections that form a basis for the holomorphic tangent space at every point?

Answer: Yes


Question: Let $X$ be an inner product space and $M\subset X$. Determine whether the following statement is true or false: $$(M^{\perp})^{\perp}=\overline{\operatorname{span} M}.$$

Answer: False


Question: Determine whether a continuous function on the interval $[0,1]$ can be constructed that is not differentiable exactly at two specific points within the interval.

Answer: Yes


Question: Solve the differential equation \( \ddot{x} = (\tan x) \dot{x}^2 \) using integration.

Answer: \sin x = C_1 t + C_2


Question: Let $\{Y_n\}$ be a sequence of random variables such that $\sup_n\text{Var}(Y_n)<\infty$. If $T_n=\sum_{k=1}^n Y_k$ and $\frac{T_n}{n^2}$ converges in probability as $n\to\infty$, determine the limit of $\frac{T_n}{n^2}$ in probability.

Answer: 0


Question: Given a linear mapping $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ such that $f \circ f = 2f$, determine if $f(v) = 2v$ for any $v \in \text{Im}(f)$. Provide a justification for your answer.

Answer: Yes


Question: Determine the largest prime number \( p \) such that when \( 2012! \) is expressed in base \( p \), there are at least \( p \) consecutive trailing zeroes.

Answer: 43


Question: Calculate the volume of the solid in \(\mathbb{R}^{3}\) that is bounded by the cylinders \(y^{2}+z^{2}=1\), \(y^{2}+z^{2}=4\) and the planes \(x=-1\), \(x+z=3\).

Answer: 12\pi


Question: Let $A$ and $B$ be $n \times n$ matrices such that $A^2 = A$, $B^2 = B$, and $(A + B)^2 = A + B$. Additionally, it is given that $AB + BA = 0$. Prove that $AB = 0$.

Answer: AB = 0


Question: Determine whether the following statement is true or false: If \( G \) is a group and \( \text{Aut}(G) \) is cyclic, then \( G \) is cyclic.

Answer: True


Question: Evaluate the limit: $$\lim_{n\to\infty} \frac {e^\frac{1}{n} + e^\frac{2}{n} +\ldots+e^\frac{n}{n}}{n}$$

Answer: e - 1


Question: Find the limit of the function \( f(x,y) = \frac{y^3(x^2+2xy-y^2)}{(x^2+y^2)^2} \) as \((x, y)\) approaches the origin along the path \(y = x^{2/3}\).

Answer: 0


Question: Let \(f(x)=\int_{0}^{x}|\log_{2}(\log_{3}(\log_{4}(\cos t+a)))|dt\). Determine the smallest integer value of \(a\) such that \(f(x)\) is increasing for all \(x \in \mathbb{R}\).

Answer: 6


Question: Given the recurrence relation \(b_{n+1}=b_n+\frac{1}{b_n}\) with initial condition \(b_0=1\), find a function \(f(n)\) such that \(0<\lim_{n \to \infty}\frac{b_n}{f(n)}<\infty\).

Answer: \sqrt{2n}


Question: Determine whether the following statement is true: If $X$ and $Y$ are topological spaces, $A$ is a subspace of $X$, and $f: A \rightarrow Y$ is a function such that there exists a continuous function $g: X \rightarrow Y$ satisfying $g(x) = f(x)$ for all $x \in A$, then $f$ is continuous. If true, provide a formal proof or reference.

Answer: True


Question: Given a quadratic function \(f(x) = ax^2 + bx + c\) such that \(f(0) = 1\) and the integral \(\int\frac{f(x)}{x^2(x+1)^3}dx\) is a rational function, determine the value of \(f'(0) = b\).

Answer: 3


Question: Determine all real numbers \( a \) such that the equation \[ x^2a - 2x + 1 = 3|x| \] has exactly three distinct real solutions for \( x \).

Answer: \dfrac{1}{4}


Question: Let $X$ and $Y$ be two independent random variables, each uniformly distributed on the interval $[0,1]$. Define $U = \min(X,Y)$ and $V = \max(X,Y)$. Find the joint density function of $U$ and $V$.

Answer: 2


Question: Determine whether the sequence \( b_n = \int_0^n \int_0^{\frac{1}{n}} \int_0^1 \int_0^{\frac{1}{n}} \log^2\left[(x-t)^2+(y-s)^2\right] \, dx \, dy \, dt \, ds \) converges or diverges as \( n \to \infty \).

Answer: 0


Question: A necklace is composed of 3 identical beads of one color and 6n beads of another color. Determine the number of distinct arrangements for the necklace, considering that clockwise and counterclockwise arrangements are equivalent, but reflections are not. [Note: The necklace is a circular arrangement where rotations are considered equivalent, but reflections are not.]

Answer: 6n^2 + 3n + 1


Question: Evaluate the triple integral \( \iiint_E dV \), where the region \( E \) is defined as \( E = \{(x,y,z) \in \mathbb{R}^3 : x^2z^2 + y^2z^2 \leq 1, \frac{1}{2} \leq z \leq 1 \} \).

Answer: \pi


Question: Calculate the surface integral of the curl of the vector field \( \vec{F} = y^2 \vec{i} + xy^2 \vec{j} + xz \vec{k} \) over the surface \( S \) defined by \( x^2+y^2 \leq 1 \) and \( z = y^2 \). The edge of \( S \) is oriented counter-clockwise when projected onto the \( xy \)-plane.

Answer: \dfrac{\pi}{4}


Question: Determine whether it is possible to find a vector field on a surface in \(\mathbb{R}^3\) that has a zero with index 0.

Answer: Yes


Question: If a function $f$ has a pole of order $k+1$ at $z_0$, find the residue of the function $\frac{f'}{f}$ at $z_0$ in terms of $k$. Provide your answer in terms of $k$. \( \boxed{} \)

Answer: - (k + 1)


Question: Evaluate the limit: $$\lim_{x \to \infty}\left(\left(x+\frac{1}{x}\right)\arctan(x)-\frac{\pi}{2}x\right)$$

Answer: -1


Question: In the context of Lebesgue measure \(\mu\) on a phase space, if \(\sum_{i=0}^\infty \mu(A_i) \leq 1\), does it imply that \(\mu(A_i) = 0\) for all \(i\)? Answer yes or no and justify your answer.

Answer: No


Question:
Find the closed-form expression for the continued fraction:
\[ \pi = \cfrac{4} {2+\cfrac{2^2} {4+\cfrac{4^2} {6+\cfrac{6^2} {8+\ddots}}}} \]

Answer: \pi


Question: Calculate the value of the infinite series \( \sum_{k=1}^{\infty} \frac{(2k-1)(-3)^{k-2}}{2^{k-1}} \).

Answer: \dfrac{2}{75}


Question: Determine the maximum possible value of the expression $ac + bd$, given that $a^2 + b^2 \leq 2$ and $c^2 + d^2 \leq 4$ for real numbers $a, b, c,$ and $d$. Apply the Cauchy-Schwarz inequality to solve this problem.

Answer: 2\sqrt{2}


Question: If a graph $G$ has an edge $n$-coloring, does it necessarily follow that every subgraph $H$ of $G$ also has an edge $n$-coloring? Answer yes or no.

Answer: Yes


Question: Evaluate the integral: $$\int_{-\pi/2}^{\pi/2}\ln(\sin x + \sqrt{\sin^2x+e^{\cos x}})dx.$$

Answer: 1


Question: Determine the number of codewords of weight 4 in a binary $[31,22,5]$ linear code, given that each weight 2 coset can contain at most one weight 4 codeword.

Answer: 0


Question: Determine whether there exists a unitary matrix \( U \) such that \( U^*AU \) is upper triangular for the given matrix \( A = \begin{pmatrix} 0 & 0 & 0 & 0 & a\\ 0 & 0 & 0 & b & c\\ 0 & 0 & d & e & f \\ 0 & g & h & i & j \\ m & n & o & p & q \\ \end{pmatrix} \).

Answer: Yes


Question: Determine the conditions on the parameters $c$ and $d$ such that the following limit is infinite: $$\lim\limits_{y\to0^+}\frac{\int_{\frac{1}{2}}^{y}\frac{g(x)}{x^c(1-x)^d}dx}{\int_{\frac{1}{2}}^{y}z^{c-1}(1-z)^{d-1}dz}=\infty$$ where $g(x)=\displaystyle{\int_{\frac{1}{2}}^{x}}z^{c-1}(1-z)^{d-1}dz$. Provide the conditions on $c$ and $d$.

Answer: c \geq 1


Question: Evaluate the integral \( \int_{-\infty}^{\infty}\frac{x^2}{(1+x^2)^2}dx \) using contour integration.

Answer: \dfrac{\pi}{2}


Question: Determine the powers $m$ for which the polynomial $(n+1)^m - 1$ is divisible by $n^2$ for any positive integer $n$.

Answer: 0


Question: Is the intersection of a countable collection of dense, open subsets of a complete metric space also dense in the space?

Answer: Yes


Question: Evaluate the line integral \(\int_C (1+\cosh(y),x\sinh(y))d\vec{s}\) where \(C\) is the curve from \((0,0)\) to \((1,1)\).

Answer: 1 + \cosh(1)


Question: In a metric space, is the distance from a point to a closed subset always equal to the distance from the point to some element in the subset? Answer yes or no.

Answer: No


Question: Given positive integers \(a, b, c,\) and \(d\) such that \(a^5 = b^6\), \(c^3 = d^4\), and \(d - a = 61\), determine the smallest possible value of \(c - b\).

Answer: 593


Question: Find the conditions on \( \alpha \) such that the integral \( \int_0^1 \frac{x^{\alpha-1}}{e^x-1} \, dx \) converges.

Answer: \alpha > 1


Question: Determine which viewpoint is correct regarding the homeomorphism between the closed sets $T$ and $L$ in the Euclidean space $\mathbb{R}^2$. View A claims that $T$ and $L$ are homeomorphic, while View B argues they are not. Consider the implications of removing a point from each set and the resulting connectivity. Which viewpoint is correct?

Answer: B


Question:
Consider the system of differential equations:
\[\frac{\mathrm{d}x}{\mathrm{d}t}=ax+by\]
\[\frac{\mathrm{d}y}{\mathrm{d}t}=cx+dy\]
where $a, b, c, d$ are constants. Suppose that $x(t), y(t)$ is a solution to this system and that the solution lies on a straight line. Given $y(0)=5$, $x(0)=6$, and $x(10)=2$, find the value of $y(10)$. \(\boxed{\text{Your answer here}}\)

Answer: \dfrac{5}{3}


Question: Evaluate the limit: \[ \lim_{x \to +\infty} \int_{0}^{\frac{\pi}{2}} x e^{-x^2 \cos(t)} \, dt. \]

Answer: 0


Question:
In a metric space $X$, consider two disjoint subsets $A$ and $B$, where $A$ is compact and $B$ is closed. Prove or disprove that there exists a $
abla > 0$ such that $|\alpha - \beta| \geq \nabla$ for all $\alpha \in A$ and $\beta \in B$. Provide a justification for your answer.

Answer: \nabla > 0


Question: Determine the expression for the fourth mixed partial derivative of the probability generating function \( Q(x,y) \) in terms of the expected values of \( X \) and \( Y \).

Answer: E[X(X-1)Y(Y-1)]


Question: Is the space $C^2[0,1]$ with the norm $\|f\|=\|f\|_\infty + \|f^\prime\|_\infty + \|f^{\prime\prime}\|_\infty$ a Banach space?

Answer: Yes


Question: Let $p(x)$ and $q(x) \neq 0$ be polynomials with coefficients in $\mathbb{Q}$. If \( \frac{p(\pi)}{q(\pi)} = \frac{p(\pi+n)}{q(\pi+n)} \) for all integers \( n \), determine whether \( \frac{p(\pi)}{q(\pi)} \) must be a rational number.

Answer: Yes


Question: Let $K$ be an algebraic number field, i.e., a finite extension of $\mathbb{Q}$. Let $G$ be a finite abelian group. Determine whether there exists a Galois extension of $K$ whose Galois group is isomorphic to $G$.

Answer: Yes


Question: Given a parameterized curve \( \gamma : I \rightarrow \mathbb{R}^{3} \) with arc length parameterization, curvature \( \| k \| = 2 \), and torsion \( \| t \| = 3 \), find the magnitude of the derivative of the normal vector, \( \| N' \| \).

Answer: \sqrt{13}


Question: Determine whether the following statement is true or false: If a set $A$ is larger than the set of natural numbers $\mathbb{N}$, then it must be uncountably infinite.

Answer: A


Question: Determine whether the series \( \sum_{n=1}^\infty \left(\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}\right) x^n \) converges using the Cauchy Root Test.

Answer: |x| < 1


Question: Calculate the spectral radius of the Volterra operator \( A \) defined on \( L^2[0,1] \) by \( A(f)(x) = \int_0^x f(t) \, dt \). Use the formula \( \rho(A) = \lim_{n\rightarrow \infty} \|A^n\|^{1/n} \) and the fact that \( \|(x-s)^n\|_\infty = 1 \) for \( s \in [0,x] \), \( x \leq 1 \).

Answer: 0


Question: If $x = a-\frac{1}{a}$, $y = b-\frac{1}{b}$, and $z = c-\frac{1}{c}$ satisfy the equation $x + y + z = 0$, find the value of $\frac{1}{xy} + \frac{1}{yz} + \frac{1}{xz}$.

Answer: 0


Question: Find the fundamental period of the exponential function given by \( x(t) = \sum_{k=2}^{14} e^{j k \frac{2\pi}{7} t} \).

Answer: 7


Question: In a poset $(X, \le)$, define the set $A_n$ of $n$-cells of $X$ recursively: an element is a $0$-cell if it is the bottom element, and an element is an $n+1$-cell if it covers some $n$-cell. If the lattice is modular, does it follow that $A_n$ is an antichain for each $n$?

Answer: Yes


Question: Evaluate the line integral \( \int_{K}\left(x^{2}+y z\right) d x+\left(y^{2}+x z\right) d y+\left(z^{2}-x y\right) d z \) where \( K \) is a closed curve, oriented positively, consisting of an arc defined by the parametric equation \( x=a\cos t, y=a\sin t, z=\frac{1}{2\pi}t \) and the segment \( BA, A=(a,0,0), B=(a,0,1) \).

Answer: 0


Question: Determine a bound on the difference between $\sin x$ and its degree 5 Taylor polynomial for $x \in [0,1]$. Provide your answer as a numerical value or expression.

Answer: \dfrac{1}{5040}


Question:
Find the value of $a^2+b^2+c^2$ given that $a, b,$ and $c$ are positive integers satisfying the following system of equations:

$$a^2+b^2+\sqrt{c^2}=148$$
$$a^2+\sqrt{b^2}+c^2=82$$
$$\sqrt{a^2}+b^2+c^2=98$$

Answer: 154


Question: Determine the number of ways to choose $k$ numbers from the first $n$ natural numbers such that the longest string of consecutive numbers is exactly $m$ numbers long. For instance, if $k = 7$, $n = 14$, and $m = 3$, find the number of combinations of numbers that satisfy these conditions.

Answer: 1456


Question: Let $f(x): \mathbb{R}^+ \to \mathbb{R}^+$ be an invertible function such that $f'(x)>0$ and $f''(x)>0$ for all $x \in [1,5].$ Given that $f(1)=1$, $f(5)=5$, and the area under the curve $y=f(x)$ from $x=1$ to $x=5$ is 8 square units, find the area bounded by $y=f^{-1}(x)$ from $x=1$ to $x=5$.

Answer: 16


Question: Solve the equation \(xe^x = 2\) using the Lambert-W function.

Answer: W(2)


Question: Consider the recursive sequence defined by \( a_{n+1} = -ta_n^2 + (t+1)a_n \) with initial condition \( a_0 \in \left(0, \frac{1}{t} + 1\right) \) and parameter \( t \in (0,1] \). Determine whether the limit \( \lim\limits_{n\to\infty}a_n \) exists for all such \( t \) and \( a_0 \), and if it does, find the limit.

Answer: 1


Question: Find the length of segment AB, where A and B are the feet of the perpendiculars from the two foci of the ellipse \(\frac{x^2}{6} + \frac{y^2}{3} = 1\) onto a tangent drawn at the point \((2,1)\).

Answer: \sqrt{6}


Question: Find the limit of \((x^2 + y^2)^{x^2 y^2}\) as \((x, y)\) approaches \((0, 0)\).

Answer: 1


Question: Determine whether the limit \( \lim_{n\to\infty}\dfrac{p_{n}}{n} \) converges to a defined value, where \( p_{n} \) denotes the nth prime number.

Answer: \infty


Question:
Determine whether there exists a continuous, strictly increasing function \( f: [0,1] \rightarrow [0,1] \) that satisfies the following conditions:

1. \( f(0) = 0 \) and \( f(1) = 1 \).
2. There is no interval \( A \subset [0,1] \) where the derivative \( f'(x) \) exists, is continuous, and is finite.
3. There exists a set \( B \subset [0,1] \) dense in \([0,1]\) where the derivative \( f'(x) \) exists, is positive, and is finite.

Answer: Yes


Question: Calculate the value of the series \(\sum_{n=0}^{\infty} \frac{a_{n}b_{n}}{7^{n}}\), where \(z = (2 + i)\) and \(a_{n} + ib_{n} = z^{n}\). Use de Moivre's formula and the properties of geometric series to find the result.

Answer: \dfrac{7}{16}


Question: Calculate the expected value of \(\frac{\max\{|X|,|Y|\}}{\min\{|X|,|Y|\}}\) where \(X\) and \(Y\) are independent random variables each following a standard normal distribution.

Answer: \infty


Question: Find the radius of convergence for the power series: $$\sum _{n=0}^{\infty }\frac{\left(-1\right)^{n+1}\left(n-1\right)}{n!}\left(x-1\right)^n$$

Answer: \infty


Question: Evaluate the integral \( \int_0^\pi \frac{x}{1+\sin x} \, dx \) using the substitution \( t = \tan(x/2) \) and the identity \( \sin x = \frac{2\tan(x/2)}{1+\tan^2(x/2)} \).

Answer: \pi


Question: Suppose $f:[a,b]\to\mathbb{R}$ is a strictly increasing and left-continuous function. Determine whether $f$ maps Borel subsets of $[a,b]$ to Lebesgue measurable subsets of $\mathbb{R}$. Provide a justification for your answer.

Answer: Yes


Question: Determine the number of integer values of $k$ for which the cubic equation $2x^3 + 3x^2 + 6x + k = 0$ has exactly three distinct real roots.

Answer: 0


Question: Let $x_1, x_2, x_3, \dotsc, x_{2560}$ be the roots of the polynomial \(x^{2560} + 2x^{2559} + 3x^{2558} + \dotsb + 2561.\) Find the value of \((x_1^2 + 1)(x_2^2 + 1)(x_3^2 + 1) \dotsb (x_{2560}^2 + 1).\)

Answer: 3279361


Question: Using the parametric equations for the folium of Descartes, \(x = \frac{3t}{1+t^3}\) and \(y= \frac{3t^2}{1+t^3}\), calculate the condition for a vertical tangent at the point \((0,0)\).

Answer: t \to \infty


Question: Evaluate the limit: $$\lim_{x \to 0}\frac{4^{\operatorname{arcos}^2 \left(\frac{1}{1 + x^2}\right)} - 1}{\log_4(1 + x)}$$

Answer: 0


Question: Given an open set \(\Omega\) in \(\mathbb{R}^n\) and a compact subset \(K \subset \Omega\), consider a sequence of harmonic functions \(\{f_i\}\) satisfying \(\Delta f_i = 0\) in \(\Omega\), with a uniform \(L^2(\Omega)\) norm bound \(\|f_i\|_{L^2(\Omega)} \leq C\). Does there exist a constant \(\lambda\) such that the inequality \( \|f_i\|_{L^{\infty}(K)} \leq \lambda \) holds for all \(i\)?

Answer: Yes


Question: Find the minimum value of the expression: \[ E(x)= |x| - |x+1| + |x+2| - |x+3| + \dots + |x+2016| \] for any real number \(x\).

Answer: 1008


Question: Evaluate the integral: $$\int_{-\infty}^{\infty} x^2 e^{-\frac{1}{2}x^2} \, dx$$

Answer: \sqrt{2\pi}


Question: Calculate the probability $P(B_1 > 0, B_2 > 0)$, where $B_t$ is a Brownian Motion at time $t$. Given that $P(B_1 > 0, B_2 > 0) = \frac{3}{8}$, explain how symmetry is applied to arrive at this result.

Answer: \dfrac{3}{8}


Question:
Let \(X_{1}, X_{2}, \ldots\) be a sequence of independent random variables, where \(X_{2k-1}\) follows a Rademacher distribution with \(P(X_{2k-1}=1) = P(X_{2k-1}=-1) = \frac{1}{2}\), and \(X_{2k}\) follows a Cauchy distribution with probability density function \(f(x) = \frac{1}{\pi(1+x^2)}\). Determine the limit:

\[ \lim_{n \to \infty} P\left(\frac{X_{1} + \cdots + X_{2n}}{\sqrt{n}} \ge 2 \right) \]

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_{-\infty}^{+\infty} e^{-x^2/2} \, dx \) without using a table.

Answer: \sqrt{2\pi}


Question: Find a constant bound on the variance of the number of fixed points in a random permutation of n elements, given that the mean is 1.

Answer: 1


Question: Determine the minimum number of detectors required to uniquely identify the location of a radiation point in an n-dimensional space using a given metric $d$. Provide your answer as a function of $n$.

Answer: n + 1


Question: Consider three subspaces $A$, $B$, and $C$ of $\mathbb{R}^4$ such that each pair of subspaces sums to a direct sum. If $A$ and $B$ are 1-dimensional subspaces and $C$ is a 2-dimensional subspace, determine the maximum possible dimension of $A + B + C$. \( \text{Provide your answer as a single number.} \)

Answer: 4


Question: Evaluate the integral \( \int_{\partial D}\ln |x-y| \, dy \) where \( D \) is the unit disc in \( \mathbb{R}^2 \) and \( x \in \partial D \). Determine if this integral can be solved analytically.

Answer: 0


Question: Evaluate the limit \(\lim_{n \to \infty}\left(\frac{(3n)!}{(2n)!n^n}\right)^{1/n}\) using Stirling's approximation.

Answer: \dfrac{27}{4e}


Question: Determine the maximum peak of the function \( \phi(r) = a_1 J_0(\alpha r) + a_2 J_0(\beta r) \) within the interval \([0, l]\), given that \( a_1 + a_2 = 1 \), \( \alpha \) and \( \beta \) are positive parameters, and \( \phi(l) = 0 \) with \( \phi(r) \) being non-negative for \( r \in [0, l] \). Here, \( J_0 \) denotes the Bessel function of the first kind.

Answer: 1


Question: Given four points $\{x_1, x_2, x_3, x_4\}$ on a sphere, determine if there exists a set of points $\{y_j\}$ (with at most four elements) such that the product of Euclidean distances from each $y_j$ to the given points $x_i$ is the same for all $j$. Provide a justification for your answer.

Answer: Yes


Question: Determine the angle between the left and right tangents to the graph of the function \(g(x)=\frac{1}{\sqrt{3}}\arcsin\left(\frac{2x}{1+x^{2}}\right)\) at the point \(P\left(1, \frac{\sqrt{3}\pi}{6}\right)\). Given the first derivative \(g'(x) = \frac{-2(x-1)(x+1)}{\sqrt3 (x^2+1)\cdot |x-1| \cdot |x+1|}\), evaluate the limits \(\lim_{x\rightarrow 1^{+}} g'(x)\) and \(\lim_{x\rightarrow 1^{-}} g'(x)\). Use these limits to calculate the angle between the tangents.

Answer: \dfrac{\pi}{3}


Question: Find the value of the infinite series \( \sum_{n=0}^{\infty}{\frac{n(n+2)}{3^n}} \) by identifying a function \( f(n) \) such that \( \dfrac{n(n+2)}{3^n}=f(n+1)-f(n) \).

Answer: 3


Question: Let \( l = \lim_{x \rightarrow a}\frac{x^x-a^x}{x-a} \) and \( m = \lim_{x \rightarrow a}\frac{a^x-x^a}{x-a} \) where \( a > 0 \). If \( l = m \), find the value of \( a \).

Answer: e^2


Question: Evaluate the integral \( \int_{-\pi/2}^{\pi/2} \frac{\sin(kn)}{1+k^2} \, dk \).

Answer: 0


Question: Calculate the area enclosed by the curves \(y=\frac{4}{x}\), \(y=\frac{6}{x}\), \(y=4x\), and \(y=8x\) using a suitable coordinate transformation.

Answer: \ln 2


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

Answer: e


Question: Let \( g(v(x)) = \int_{-1}^{1} (x^2-1) v(x) \, dx \) be a linear functional defined for all \( v \in (C[-1,1], ||.||_\infty) \). Find the norm \( ||g|| \).

Answer: \dfrac{4}{3}


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

Answer: 0


Question: In triangle ABC with vertices A(1,0,0), B(0,1,0), and C(0,0,1), a point P(X,Y,Z) is uniformly distributed within the triangle. Determine the probability density function (PDF) for the random variable X, representing the x-coordinate of point P.

Answer: 2(1 - x)


Question: Find the remainder when $70!$ is divided by $125$ using the Chinese Remainder Theorem.

Answer: 0


Question: If $X_n \xrightarrow{\text{a.s.}}X$ and $\mathbf{E}[X_n] \rightarrow \mathbf{E}[X] $, does this imply $\mathbf{E}[|X_n - X|] \rightarrow 0$? Answer yes or no and justify your answer.

Answer: No


Question: Find the maximum possible value of the sum \( \sin(a_1)\cos(a_2) + \sin(a_2)\cos(a_3) + \cdots + \sin(a_{2014})\cos(a_1) \) where \( a_1, a_2, \ldots, a_{2014} \) are real numbers.

Answer: 1007


Question: Determine whether the exceptional Lie group $E_8$ can be faithfully represented as a subgroup of the orthogonal group $O(248)$. Provide a justification for your answer.

Answer: Yes


Question: Let $f:\mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f'(x) > x^2$ for every $x < 0$. Determine $\lim_{x \to -\infty} f(x)$. Provide a justification for your answer.

Answer: -\infty


Question: Given the series \( \frac{1}{\sin 8^\circ} + \frac{1}{\sin 16^\circ} + \frac{1}{\sin 32^\circ} + \ldots + \frac{1}{\sin 4096^\circ} + \frac{1}{\sin 8192^\circ} = \frac{1}{\sin \alpha} \), where \( \alpha \in (0, 90^\circ) \), find the value of \( \alpha \) in degrees.

Answer: 4


Question: Consider topological spaces $X, Y, A, B$ and continuous functions $f, g: X \to A$ and $f', g': Y \to B$. If $f \times f' \simeq g \times g'$, does it necessarily follow that $f \simeq g$ and $f' \simeq g'$? Provide a justification for your answer.

Answer: Yes


Question: Determine if the number of ends of the right-angled Coxeter group $C(\Gamma)$ can be deduced solely from the graph $\Gamma$ when $C(\Gamma)$ is one-ended.

Answer: Yes


Question: Determine the radius of convergence of the power series \(\sum _{n=0}^{\infty }\frac{\left(-1\right)^{n+1}\left(n-1\right)}{n!}\left(x-1\right)^n\).

Answer: \infty


Question: Determine the number of connected components of the space $X = \mathbb{R}^2 \setminus (F \times F)$, where $F$ is the set of all irrational numbers in $\mathbb{R}$, with the usual subspace topology of $\mathbb{R}^2$.

Answer: 1


Question: Determine the isomorphism class of the quotient ring \( \mathcal{O}_K / \langle \sqrt{6} - 1 \rangle \) for the number field \( K = \mathbb{Q}(\sqrt{2}, \sqrt{3}) \).

Answer: \mathbb{F}_{25}


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

Answer: \mathbb{R}


Question: Evaluate the limit: $$\lim_{y\to 0} \Bigg(\frac{1 + 20y}{1+4y}\Bigg)^{\ln(\sin 2y)}$$

Answer: 1


Question: Assume that $A\in\mathbb{C}^{n\times n}$ is a Hermitian positive semi-definite matrix and $z\in\mathbb{C}^{n}$ such that $z\neq 0$. If $z^*Az=0$, is $z$ an eigenvector of $A$ associated with eigenvalue $0$?

Answer: Yes


Question:
Consider every 9-digit number $\overline{abcdefghi}$. Associate with each number a $3 \times 3$ matrix as follows:

$$ \left[ \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \\ \end{array} \right] $$

Determine the sum of the determinants for all such matrices. Which of the following options represents the sum of determinants?

(a) a number of the form $9k + 1$, where $k$ is an integer
(b) $999$
(c) $-1$
(d) $2022$
(e) other answer

Answer: E


Question: Find the value of $n$ such that the limit \( \lim_{x\to 0} \frac{(\sin x-x)(\tan x-x)}{x^n} \) is a finite non-zero number.

Answer: 6


Question: A die is tossed $n$ times. Calculate the covariance between the number of times a one appears and the sum of all the results from the tosses.

Answer: -\dfrac{5n}{12}


Question: Find the value of \( c \) such that the sequence \((S^3_n - cn)_{n \in \mathbb{N}_0}\) is a martingale, where \( S_n \) is a random walk.

Answer: 3


Question: Determine whether every absolutely continuous function defined on a compact set has a left-hand and right-hand derivative at every point in its interior.

Answer: No


Question:
Let $L^2([0,1])$ be the Hilbert space of all real-valued square integrable functions on the interval $[0,1]$, with the standard inner product. Consider the linear functional $\phi$ defined by:
\[\phi(f) = \int_{1/4}^{3/4} 3\sqrt{2} f \, \mathrm{d}\mu\]
where $\mu$ is the Lebesgue measure on $[0,1]$. Find the norm of $\phi$, denoted $\|\phi\|$. \(\boxed{\text{Your answer here}}\)

Answer: 3


Question: Using the result $\nabla_{A} (XAY) = Y^{T}X^{T}$, derive the equation $\nabla_A \mathrm{tr}(XAY)=(YX)^T$. Provide a detailed derivation.

Answer: (YX)^T


Question: Find the integral basis for the ring of algebraic integers in the quadratic field $\mathbb{Q}[\sqrt{D}]$ when $D \equiv 2 \pmod{4}$.

Answer: \left\{1, \sqrt{D}\right\}


Question: Determine the number of unique linear transformations $T$ that satisfy the following mappings: $$\begin{align}T(1,2,3)&=(1,1,1), \\ T(1,1,1)&=(1,2,3), \\ T(-2,-2,-2)&=(1,0,-1).\end{align}$$

Answer: 0


Question: Determine the number of solutions to the equation $z^{100} = 1 + i$ that lie in the third quadrant of the complex plane.

Answer: 25


Question: Determine the smallest integer $n$ such that $e^{\sqrt{n}} > n^4$.

Answer: 681


Question: Determine whether the set \( \mathbb{N} \times \mathbb{N} \) can be partitioned into four pairwise disjoint countably infinite subsets, where each subset is defined by the parity of the elements in each ordered pair.

Answer: Yes


Question: Compute the limit: $$\lim_{n\to\infty}\frac{\Gamma(n+1)\Gamma(\alpha-n+1)}{n^{\alpha+1} }$$ where $\Gamma$ is the standard gamma function and $\alpha\in\mathbb{R}$.

Answer: 0


Question: Is it always possible to color the vertices of a triangulation of a domain with a Lipschitz boundary using the numbers 0, 1, and 2 such that opposing vertices of two triangles receive the same number?

Answer: Yes


Question: A random variable $X_1$ has a uniform distribution on the interval $(0, 1)$. Another random variable $X_2$ has a uniform distribution on the interval $(0, X_1)$. A third random variable $X_3$ has a uniform distribution on the interval $(0, X_2)$. Calculate the variance of $12(X_1 + X_2 + X_3)$. \( \text{Provide your answer as a single numerical value.} \)

Answer: \dfrac{565}{12}


Question: Determine if there exists a continuous function that maps the annulus \( \Omega_1 = \{(x,y) \in \mathbb{R}^2 : 1 < x^2 + y^2 <2 \} \) onto the elliptic annulus \( \Omega_2 = \{(x,y) \in \mathbb{R}^2 : 4 < x^2 + 2y^2 < 8\} \).

Answer: Yes


Question: Find all complex triples \((x, y, z)\) such that the matrix \(A = \begin{bmatrix}1&x&y \\ 0&2 & z \\0&0&1\end{bmatrix}\) is diagonalizable. Determine the relationship between \(x, y,\) and \(z\) that ensures diagonalizability.

Answer: y = xz


Question: Given a sequence of positive real numbers $(a_i)$ such that $\sum_{i=1}^{\infty} a_i^2 = \infty$, determine whether it follows that $[0,1]^2$ can be covered by translates of the squares $[0,a_i]^2$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit of the expression \( a_{n+1}\left[1+\left(\frac{a_{n+1}}{a_n}-1\right)\right]^n \) as \( n \to \infty \), given that \( a_n \to b < \infty \) and \( L = \lim_{n \to \infty} (a_{n+1}-a_n)n \) exists.

Answer: b e^{L/b}


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

Answer: 2\pi i


Question: Determine the value of \( k \) for which the differential form \( (y^3 + k xy^4 - 2x) \, dx + (3xy^2 + 20 x^2y^3) \, dy \) is exact.

Answer: 10


Question:
Let \( \alpha \) be a root of the equation \( x^4 + x^3 + x^2 + x + 1 = 0 \). Determine the value of \( \alpha + \frac{1}{\alpha} \). Choose the correct answer from the following options:

A) \( \frac{1 \pm \sqrt{3}}{2} \)
B) \( \frac{1 \pm \sqrt{5}}{2} \)
C) \( \frac{-1 \pm \sqrt{3}}{2} \)
D) \( \frac{-1 \pm \sqrt{5}}{2} \)

Answer: D


Question: A man travels between three towns forming a triangle by walking, riding, and driving. He can walk, ride, and drive a mile in \(a\), \(b\), and \(c\) minutes respectively. If he starts from town B, he takes \(a + c - b\) hours; if he starts from town C, he takes \(b + a - c\) hours; and if he starts from town A, he takes \(c + b - a\) hours. Determine the total length of the circuit in miles.

Answer: 60


Question:
Given a 10x10 matrix A with complex entries, all of whose eigenvalues are non-negative real numbers and at least one is positive, determine which of the following statements is always false:

A. There exists a matrix B such that AB - BA = B
B. There exists a matrix B such that AB - BA = A
C. There exists a matrix B such that AB + BA = A
D. There exists a matrix B such that AB + BA = B

Identify the statement that cannot be true based on the properties of matrix A.

Answer: B


Question: Can the Tietze extension theorem be applied to extend a continuous function defined on a closed subset of a non-normal space?

Answer: No


Question:
Let \( Q: V \to V \) be a projection such that \( Q \circ Q = Q \). Find a basis \( E = \{e_1, \ldots, e_n\} \) and an integer \( k \in \{1, \ldots, n\} \) such that the matrix representation of \( Q \) with respect to \( E \) is:
\[
Q_E^E = \begin{bmatrix}
I_k & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}
\]
where \( I_k \in M_{k \times k} \) is the identity matrix, and the other blocks are zero matrices of sizes \( k \times (n-k), (n-k) \times k, \) and \( (n-k) \times (n-k) \).

Answer: k


Question: Determine the value of \(a\) for which the system of equations \(a + e^a \ln(x) = x + e^a \ln(a)\) and \(x + e^a \ln(a) = a + e^x \ln(a)\) has a second real solution at \(x \approx 5.841992948\), given that \(x = a\) is already a solution. The value of \(a\) is approximately 1.0142286126.

Answer: 1.0142286126


Question: Let $X$ be a Banach space, and let $E$ be a dense subspace of $X$. Suppose $A: X \to X$ is a bounded operator that maps $E$ into itself, and the spectral radius of $A$ restricted to $E$ is a finite $r > 0$. Is the spectral radius of $A$ on $X$ also equal to $r$?

Answer: r


Question:
Given the system matrix \( A \):
\[ A = \begin{bmatrix} 0&0&0&1\\0&-1&1&3\\0&1&-1&-1\\0&-1&1&2 \end{bmatrix} \]
with eigenvalues \( \lambda_i = 0 \) for \( i = 1,2,3,4 \), determine the correct Jordan form of \( A \) from the options below:
\[ A) \quad J=\begin{bmatrix} 0&1&0&0\\0&0&1&0\\0&0&0&1\\0&0&0&0 \end{bmatrix} \]
\[ B) \quad J=\begin{bmatrix} 0&1&0&0\\0&0&1&0\\0&0&0&0\\0&0&0&0 \end{bmatrix} \]
\[ C) \quad J=\begin{bmatrix} 0&1&0&0\\0&0&0&0\\0&0&0&1\\0&0&0&0 \end{bmatrix} \]
\[ D) \quad J=\begin{bmatrix} 0&1&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0 \end{bmatrix} \]
The eigenvectors corresponding to \( \lambda = 0 \) are \( \begin{bmatrix} 0\\1\\1\\0 \end{bmatrix} \) and \( \begin{bmatrix} 1\\0\\0\\0 \end{bmatrix} \). The dimension of the nullspace of \( A - \lambda I \) is 2, and the ranks of \( \ker(A-\lambda I)^2 \) and \( \ker(A-\lambda I) \) are 3 and 2, respectively, indicating one Jordan block of size 2 or larger. Which option represents the correct Jordan form?

Answer: B


Question: Let $\langle x_n\rangle$ be a sequence defined by the recursive relation \(x_{n+1}=x_n+a+\sqrt {b^2+4ax_n}\) for \(n\geq0\), with initial condition \(x_0 =c\), where \(a\), \(b\), and \(c\) are fixed positive integers. Determine \(\lim_{n\to\infty}\frac {x_n}{n^2}.\)

Answer: a


Question: Given that \(x = \sqrt{2} + \sqrt{5}\) is a root of the polynomial equation \(kx^4 + mx^3 + nx^2 + l = 0\), where \(k, m, n, l\) are integers, find the value of \(|k| + |m| + |n| + |l|\).

Answer: 24


Question: Compute the limit \( \lim_{n\to\infty} \int_0^{\infty}\frac{e^{-x/n}}{1+(x-n)^2}dx \).

Answer: \dfrac{\pi}{e}


Question: Find the greatest common divisor of \(3^{100!} - 1\) and 116.

Answer: 116


Question: Let \( g(t) = t \) for all \( t \in [-1,1] \) and \( f(t) = t^{1/n} \), where \( n \) is a large odd integer. Consider the operator \( T: C[-1,1] \rightarrow \mathbb{R} \), defined by \( T_g(f) = \int_{-1}^1 g(t)f(t)dt \). Given the norm \( \|f\| = \max_{t \in [-1,1]} |f(t)| \), find \( \|T\| \).

Answer: 1


Question: Given integers $f(i)$ for all integers $i=1,2,\ldots,2^{n-1}$ such that $1\le f(i)\le i$, determine if there exist integers $a_{1},a_{2},\ldots,a_{n}$ such that $1=a_{1}<a_{2}<\cdots<a_{n}\le 2^{n-1}$, and $f(a_{1})\le f(a_{2})\le \cdots\le f(a_{n})$. Answer "yes" or "no."

Answer: yes


Question: Determine whether there exists a group of order greater than 8 that cannot be decomposed into a direct product of smaller groups, and whose Sylow 2-subgroup is isomorphic to the quaternion group $Q_8$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_0^1 g(x) \, dx \) given that \( g(x) = x + \int_0^1 t(x+t)g(t) \, dt \) and that the second derivative of \( g(x) \) is zero.

Answer: \dfrac{42}{23}


Question: Determine whether the following statement is true or false: If \( \alpha < \frac{1}{3} \), then \[ \prod_{n=1}^{\infty}(1-\alpha^n) = \frac{1-3\alpha}{1-2\alpha}. \]

Answer: False


Question: Given that $a^{100} \equiv 2 \pmod{73}$ and $a^{101} \equiv 69 \pmod{73}$, find the integer $x$ such that $a \equiv x \pmod{73}$. Provide your answer as a single integer $x$.

Answer: 71


Question: Determine the number of fixed points of the iterated cosine function, \(\underbrace{\cos(\cos(\ldots(\cos x)\ldots))}_{n\text{ times }\cos}\), on the interval \([0,1]\) for a positive integer \(n\).

Answer: 1


Question: Find the area of the region bounded by the curves $y = x^2$ and $y = 4\cos(x)$, considering both the regions where $y \geq 0$ and $y < 0$. Express your answer as a single numerical value.

Answer: 6.304


Question: Determine whether there exists a function that is infinitely differentiable but cannot be completely defined by its derivatives at a single point.

Answer: Yes


Question: Verify if the equation \( x = 1 + \frac{1}{x + \frac{1}{x + \frac{1}{x + \frac{1}{x + \frac{1}{x + \ldots}}}}} \) simplifies to \( x = 1 + \frac{1}{x} \).

Answer: \dfrac{3}{2}


Question: Given the hazard function \( \phi(t) = 16t^3 \), determine the survival function \( \mathcal{R}(t) \).

Answer: e^{-4t^4}


Question: Does the Aleksandrov Theorem, which holds for convex functions, also apply to Lipschitz functions?

Answer: No


Question: Calculate the limit \(\lim_{x\to 0} \frac{2(\cosh(x)-1)^{\frac{1}{x^2}}}{x^2}.\)

Answer: 0


Question: Determine whether, for every positive integer $n$, there exists a number field of degree $n$ whose ring of integers is a unique factorization domain (UFD).

Answer: Yes


Question: Given \(\lim \limits_{x \to 3} \frac{g(x-1)-2g(x^3)}{x^3+1}=4\), find \(\lim \limits_{x \to 27} g(x)\).

Answer: -112


Question: An insurance company's monthly claims are modeled by a continuous, positive random variable $X$ with a probability density function proportional to $(x+1)^{-4}$ for $0 < x < \infty$. Calculate the expected value of $X$. \( \text{(Express your answer as a single number.)} \)

Answer: \dfrac{1}{2}


Question: Determine the cofinality of the ordinal expression \( \omega_3^{\omega_2} \).

Answer: \omega_2


Question: For any prime number \(p\), determine if every element in the set \(\mathcal{U}_p\) generates the group \((\mathbb{Z}_p,+_p)\).

Answer: Yes


Question: Which of the following conditions ensures that $f(X)$, the image of a continuous map $f: X \rightarrow Y$ between metric spaces, is a complete subset of $Y$?\n\nA. $X$ is compact.\nB. $Y$ is compact.\nC. $X$ is complete.\nD. $Y$ is complete.

Answer: A


Question: Evaluate the limit: \[ \lim_{n \to \infty} n \cos x \cos(\cos x) \cdots \underbrace{\cos(\cos(\cdots(\cos x)))}_{n \text{ times of } \cos}. \]

Answer: 0


Question: In the context of Galois extensions of number fields, determine whether the statement is true or false: The ramification index of a prime ideal in the composite field is equal to the ramification index of its restriction to one of the subfields.

Answer: False


Question: Evaluate the line integral \( \int_{\gamma} x^3 \, dx + (x^3 + y^3) \, dy \) where \( \gamma \) is the line segment from \((0, 2)\) to \((2, 0)\) along the line \( y = 2 - x \).

Answer: -4


Question: Determine the dual space of $L^\infty$ with the weak-* topology.

Answer: L^1


Question: Determine the order of the zero at $z=0$ for the function $f(z)=\frac{d^{50}}{{dz}^{50}}[z^{100}\sin(z^{10})]$. Provide your answer as an integer.

Answer: 60


Question: Evaluate the contour integral \( \int_C \frac{z^2+1}{z^3-z} \, dz \) using Cauchy's residue theorem, where \( C \) is the circle \( |z|=2 \).

Answer: 2\pi i


Question: Evaluate the triple integral \( \int \int \int_{P} (x^2 + y^2) \, dV \), where \( P \) is the pyramid with top vertex at \((0,0,1)\) and base vertices at \((0,0,0)\), \((1,0,0)\), \((0,1,0)\), and \((1,1,0)\).

Answer: \dfrac{2}{15}


Question: Determine the number of Sylow 2-subgroups in the symmetric group $S_5$.

Answer: 15


Question: Given a parameterized curve \(\gamma :I\rightarrow \mathbb{R}^{3}\) by arc length with curvature \(\| k \| = 2\) and torsion \(\| t \| = 3\), find the norm of the derivative of the normal vector \(\| {N}' \|\). Use the Frenet-Serret formulas to express \(\| {N}' \|\) in terms of \(k\) and \(t\), and calculate its precise value.

Answer: \sqrt{13}


Question: Determine the value of \( n \) such that the limit \( \lim_{x \to 0} \frac{x^n}{\cos(\sin x) -\cos x + x^2} \) is a non-zero finite real number.

Answer: 2


Question: Evaluate the line integral \( \oint_C \frac{-y}{x^2 + y^2} \,dx + \frac{x}{x^2 + y^2} \,dy \) where \( C \) is the unit circle \( x^2 + y^2 = 1 \).

Answer: 2\pi


Question: Determine whether the following statement is true or false: For any linear transformation $T:V→W$ with $V$ having finite dimension, the image of the dual map of $T$ is a subset of the annihilator of the kernel of $T$. Provide a justification for your answer.

Answer: A


Question: If $A$ is a positive definite matrix and $B$ is similar to $A$, is $B$ also positive definite? Answer with 'Yes' or 'No' and justify your answer.

Answer: No


Question: In an isosceles right triangle ABC with AB = BC, points P and Q lie on side AC such that \( AP^2 + CQ^2 = PQ^2 \). Find the measure of angle PBQ.

Answer: 45^\circ


Question: Calculate the expectation of $Y^2$ where $Y = \exp(X^2)$ and $X$ is a standard normal random variable, $X \sim N(0,1)$. Provide your answer as a numerical value.

Answer: \infty


Question: Calculate the limit using Taylor expansion: $$\lim_{x \to 0} \ (2x+1)(\log(x+1)-\log(x-1))$$

Answer: 0


Question: Let \( g(x) \) be a continuous and bijective function on \( \mathbb{R} \). Suppose that for every \( t \in \mathbb{R} \), the area bounded by \( y=g(x) \), \( x=b-t \), \( x=b \), and the \( x \)-axis is equal to the area bounded by \( y=g(x) \), \( x=b+t \), \( x=b \), and the \( x \)-axis. If \( g(b)=0 \) and \( \int_{\alpha}^{\beta}g^{-1}(x)~\mathrm{d}x = (\beta-\alpha) mb \), find the value of \( m \).

Answer: 1


Question: If $A$ is a skew-symmetric matrix with rank 4, determine the possible values of $\operatorname{rank}(A + A^T)$. \(A^T\) denotes the transpose of matrix \(A\).

Answer: 0


Question: Given that $a, b, c$ are the roots of the cubic equation $f(x) = x^3 + 3x^2 - 7x + 1$, calculate the value of $f'(a)f'(b)f'(c)$. Provide your answer in a simplified form.

Answer: -1300


Question:
Compute the limit:

\[ \lim_{n \to \infty} \frac{\sum\limits_{i = 0}^{n + 1} \binom{n + 1}{i}^3}{\sum\limits_{i = 0}^n \binom{n}{i}^3} \]

where \( \sum\limits_{i = 0}^n \binom{n}{i}^3 = \text{hypergeom}\left(\left[-n, -n, -n\right], [1, 1], -1\right) \).

Answer: 8


Question: Let \(x\) and \(y\) be vectors in an inner product space, with \(\|x\| = 4\) and \(\|y\| \le 2\). Find the minimum value of \(\langle 3x-y, x+3y\rangle\).

Answer: -28


Question: In 3-dimensional topology, determine whether a closed surface embedded within a 3-sphere (S³) necessarily bounds a handlebody.

Answer: Yes


Question: Determine the greatest common divisor (GCD) of the numbers $2023^{2024}$ and $2024^{2023}$ without using prime factorization.

Answer: 1


Question: Determine the value of \( g'(0) \) if \( g \) is differentiable and satisfies the equation \( \int^{2x}_1 (3g(u) - u^2) \, du = \ln(x^2 + 1) \).

Answer: \dfrac{1}{6}


Question: Let $F$ be a field and $E$ be a finite extension of $F$. If $E$ is complete with respect to some absolute value, is $F$ also complete with respect to the restriction of that absolute value to $F$? Answer yes or no.

Answer: Yes


Question: Is a linear function \(f:\{\lambda a+(1-\lambda)b:0\leq\lambda \leq 1\} \to \mathbb{R}\), where \(V\) is a real vector space and \(a,b\in V\) with \(a\neq b\), necessarily continuous?

Answer: Yes


Question: Solve for $z$ in the equation: $$\left(v^{\log_{b}\left(a-m\right)}+m\right)^{z}+m-m^{z}=a$$ where $v$, $b$, $a$, and $m$ are positive real numbers, and $a > m$. Provide the value of $z$ that satisfies the equation.

Answer: 1


Question: Let \( f_1:[0,4]\to [0,4] \) be defined by \( f_1(x)=3-\frac{x}{2} \). Define \( f_n(x) =f_1\left( f_{n-1}(x)\right) \) for \( n\ge 2 \). Find the set of all \( x \) such that \( \displaystyle\lim_{n\to \infty} f_n(x) \) exists and determine the corresponding limits.

Answer: 2


Question: Find the range of the function \( f(x) = \int_x^{2x} \frac{e^{-xt^2}}{t}dt \) for \( x > 0 \).

Answer: (0, \ln 2)


Question: Calculate the probability that the sum of five independent random variables, each uniformly distributed over the interval (0,1), is less than 1.

Answer: \dfrac{1}{120}


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

Answer: 1


Question: Suppose $X(k)$ is a $k \times m$ matrix with elements $x_{ij}(k)$. If \(\lim_{k \to \infty} \frac{X'(k)X(k)}{k}=R,\) where $R$ is positive semi-definite, determine if the following holds: \(\max_{i=1, \ldots, k} x_i'(X'(k)X(k))^{-1}x_i =0, \text{ as } k \to \infty.\)

Answer: 0


Question: Simplify the expression \(\sum_{a= \max(0, c-n)}^{\min(n,c)}\binom{n}{a}\binom{n}{a+n-c}\) where \(c\) and \(n\) are non-negative integers.

Answer: \dbinom{2n}{c}


Question:
Determine whether the following statement is true for any two Hermitian matrices $A$ and $B$:

If $A \geq B \geq -A$, then $|A| \geq |B|$.

Note: $A \geq B$ means $A - B$ is positive semidefinite, and $|A| = \sqrt{A^\dagger A}$ is the absolute value of matrix $A$, which is positive semidefinite.

Answer: True


Question: In the context of metric spaces, if two metrics $d$ and $d'$ on a set $X$ are topologically equivalent, do they induce the same topology on $X$? Provide a justification for your answer.

Answer: Yes


Question: Consider a field \( \mathbb{K} \) with a discrete non-archimedean absolute value, where its ring of integers \( A \) is Noetherian with ideals generated by a power of a uniformizer \( \pi \in A \), and has a unique maximal ideal \( \mathfrak{M} = \pi A \). Determine if there exists a field \( \mathbb{K} \) such that the residue field \( A/\mathfrak{M} \) is not finite.

Answer: Yes


Question: Let \((X,d)\) be a compact metric space and fix some \(\epsilon > 0\). Prove that there exists a constant \(N\) such that any subset \(S\) of \(X\) with the property that for all \(x, y \in S\), \(d(x, y) > \epsilon\), has at most \(N\) elements.

Answer: N


Question: A group of $y$ individuals is divided into five teams. Each individual is assigned to exactly two teams, and any two teams have exactly two members in common. What are the possible values of $y$?

Answer: 20


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

Answer: \sqrt{e}


Question: Let $A \in M_n$ be a normal matrix with its polar decomposition $A = PU$, where $U$ is a unitary matrix and $P$ is a positive semidefinite matrix. Determine whether it holds that $PU = UP$. Justify your answer.

Answer: PU = UP


Question: Determine whether the sequence \( a_n = \left[\prod_{k=1}^{n}\left(1+\frac{k}{n}\right)\right]^{\frac{1}{n}} \) converges or diverges.

Answer: \dfrac{4}{e}


Question: Determine if the operator $S: \ell_2 \to \ell_2$ defined by $(S(x))_n = \frac{x_{n+1}}{n+1}$ has any eigenvalues.

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to 0^+ }{\frac{x^2}{\ln\Bigl[\int_0^{x^{\frac{2}{3}}}{e^\frac{t^2}{2}dt}+1-x^{\frac{2}{3}}\Bigr]}}$$

Answer: 6


Question: Determine if the sum of the coefficients of the polynomial interpolation of the points \((1,p_1),(2,p_2),\ldots,(n,p_n)\), where \(p_n\) is the \(n\)th prime number, is always equal to two for any positive integer \(n\).

Answer: 2


Question: For $x \in [0,n]$ where $n \in \mathbb{Z}$, determine the maximum number of distinct values that the function $f(x) = \lfloor x^2 \rfloor - \lfloor x \rfloor^2$ can take.

Answer: 2n - 1


Question: Consider three linearly independent vectors $J_1, J_2, J_3 \in \mathbb{C}^3$. Let $A$ be a random matrix in $\mathbb{C}^{4\times 3}$. Are the vectors $AJ_1, AJ_2, AJ_3 \in \mathbb{C}^4$ linearly independent with probability 1?

Answer: Yes


Question: Given a continuous function $f:[-2,8]\rightarrow\mathbb{R}$ such that $f(6-x)=f(x)$ for all $x\in[-2,8]$, and knowing that \(\int_{-2}^8f(x)dx=10\), find the value of \(\int_{-2}^8xf(x)dx\).

Answer: 30


Question: Find the limit of the sequence \(a_n = \frac{1 + \sqrt{2} + \sqrt{3} + \sqrt{4} + \ldots + \sqrt{n}}{n^{2/3}}\) as \(n\) approaches infinity, using only high school level mathematics.

Answer: \infty


Question: Evaluate the integral \( \int_{\lambda}(n_1(x,y)x+n_2(x,y)y)ds \), where \( n=(n_1(x,y)+n_2(x,y)) \) is the outward unit normal at the point \( P=(x,y) \) on the curve \( \lambda \) given by \( x^2+4y^2=4 \).

Answer: 4\pi


Question: Find the minimum value of the expression: \[x^8-8x^6+19x^4-12x^3+14x^2-8x+9\]

Answer: 1


Question: Determine the order of convergence of the fixed point iteration for the function \( f(x) = \sin x^{100} \) as it converges to the fixed point 0. Justify your answer.

Answer: 100


Question: Find the maximum value of the function \( f(x,y) = 2x^3 - 3x^2 \) subject to the constraint \( g(x,y) = (3-x)^3 - y^2 = 0 \), where \( f(x,y) \) and \( g(x,y) \) are defined on \( \mathbb{R}^2 \).

Answer: 27


Question: Determine the rank of the first homology group of the double branched cover of a knot $K$ in $S^3$.

Answer: 0


Question: Evaluate the limit: $$\lim_{x \to 0+} \frac{x-\sqrt{x^2+1}\cdot \ln \big(\sqrt{x^2+1} + x \big)}{2x^2\sqrt{x^2+1}\cdot \ln\big(\sqrt{x^2+1} + x \big)}$$

Answer: -\dfrac{1}{6}


Question: Let $R$ be an integral domain with a total ordering. Determine if this ordering induces a total ordering on its field of fractions $F$. Provide a justification for your answer.

Answer: Yes


Question: Determine the behavior of the sequence \( S_n = \frac{p_n-1}{\phi(p_n-1)} \) as \( n \) approaches infinity, where \( p_n \) is the \( n \)th prime number.

Answer: \infty


Question: Determine whether the following statement is true or false: Any open set in a metric space contains uncountably many points.

Answer: B


Question: Evaluate the limit: \[ \lim_{n\to\infty}\sum_{k=1}^{n}\frac{6}{k(k+1)(k+3)} \]

Answer: \dfrac{7}{6}


Question: Let $X$ be a connected topological space and let $\mathcal{O}$ be an open cover of $X$, excluding the empty set. Define a graph where the vertices are elements of $\mathcal{O}$, and two vertices are connected by an edge if their intersection in $X$ is non-empty. Is the graph formed by $\mathcal{O}$ connected?

Answer: Yes


Question: Let $X_1, X_2, \ldots, X_n$ be i.i.d. random variables with a uniform distribution on $[0,1]$. Define $Z = \max\{X_1, X_2, \ldots, X_n\}$. Calculate the probability $P(X_2 < a \mid Z = z, X_2 < z)$.

Answer: \dfrac{\min(a, z)}{z}


Question: For varieties $X$ and $Y$ over an algebraically closed field and a morphism $f: X \rightarrow Y$, determine the minimum dimension of an irreducible component of the fiber $f^{-1}(y)$ for a closed point $y \in Y$. Provide your answer as a single integer.

Answer: \dim X - \dim Y


Question: Calculate the total number of button presses required to try all possible 4-digit codes on a 10-key keyboard, assuming you do not need to press enter after each digit and consecutive digits can form different codes.

Answer: 10003


Question: Determine whether the following statement is true or false: For any continuous function $g$ on an interval $(p,q)$ and $d\in(p,q)$, there exist points $y_1,y_2\in(p,q)$ such that $\frac{g(y_2)-g(y_1)}{y_2-y_1}=g'(d)$.

Answer: B


Question: Evaluate the line integral \(\int^{(2,1,0)}_{(0,0,0)}(x^2y-xz)dx+(xy^2-yz)dy+(xz^2-xy)dz\) along the straight line path from \((0,0,0)\) to \((2,1,0)\).

Answer: \dfrac{5}{2}


Question: Evaluate the limit: \[ \lim_{n\to \infty} \left( \lim_{x\to0} \left(1+\tan^2(x)+\tan^2(2x)+ \cdots + \tan^2(nx)\right)^{\frac{1}{n^3x^2}} \right) \]

Answer: e^{\frac{1}{3}}


Question: Calculate the conditional expectation \( \mathbb{E}[Y|X] \) where \( Z \) is a uniformly distributed random variable on the interval \([0, 2]\), \( X = \cos(\pi Z) \), and \( Y = 2\sin(\pi Z) \).

Answer: 0


Question: Consider a vector $x \in \mathbb{R}^n$ and a real $n \times n$ matrix $A$ such that $n > 2$, $x \neq 0$, and $A$ is not a scalar multiple of the identity matrix $I_n$. Additionally, assume that the column space of $aI_n - A$ is not spanned by $x$ for any real scalar $a$. Determine the Lebesgue measure of the set of $y \in \mathbb{R}^n$ for which $x, y,$ and $Ay$ are linearly dependent.

Answer: 0


Question: Given a discrete-time Markov Chain described by the equation $X(n+1) = X(n) - 1 + A(n)$, where $A(n)$ is a random variable taking values $0, 1, 2$ with probabilities $p_0, p_1, p_2$, respectively, and $X(n) = n$ with $X(n) \geq 1$, determine the condition on $p_0, p_1, p_2$ for the stability of the system.

Answer: p_1 + 2p_2 < 1


Question: Determine whether it is true that for a finite solvable group $G$, the degree of any irreducible representation over a field of positive characteristic (which divides the order of $G$) divides the order of the group itself.

Answer: True


Question: In an Eulerian graph, consider a vertex with degree $2k$. Determine the number of times an Eulerian circuit can pass through this vertex, given that the circuit does not start or end at this vertex.

Answer: k


Question: Suppose we have a collection of thirty sets \(A_1, A_2, \dots, A_{30}\), each containing 5 elements, and another collection of \(n\) sets \(B_1, B_2, \dots, B_n\), each with 3 elements. The union of the \(A_i\) sets is equal to the union of the \(B_j\) sets, denoted as \(S\). Every element in \(S\) belongs to exactly 10 of the \(A_i\) sets and exactly 9 of the \(B_j\) sets. Determine the value of \(n\).

Answer: 45


Question: Given the first-order partial differential equation (PDE) \(xu_x + (x^2 + y)u_y = u\) for all \(x, y \in \mathbb{R}\), where \(u(2, y) = y - 4\), find the value of \(u(1, 2)\).

Answer: 1


Question: Evaluate the integral \( \int_0^{\pi/2} \frac{(\log(\tan x))^2(\log(\cot x))^3}{\tan^2x\cot^2x}dx \) using the Beta function.

Answer: 0


Question: Find a linear operator on \( \mathbb{C} \) (considered as a vector space over \( \mathbb{R} \)) such that the range of the operator is a proper subset of its kernel.

Answer: 0


Question: Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed random variables with a uniform distribution on $(0, 1)$. Determine the limit as $n$ approaches infinity of the probability that the sum $S_n = X_1 + X_2 + \cdots + X_n$ satisfies $S_n \leq \frac{n}{2} + n^{3/4}$.

Answer: 1


Question: Determine whether the following limit exists: \[\lim_{(x,y)\to(0,1)} \frac{(x^2+y^2-1)^2}{\ln(1-(x^2+y^2))}.\]

Answer: 0


Question: Find the distance between the vertices of the circle \(x^2+y^2=2\) and the rectangular hyperbola \(xy=9\), given that their transverse axes coincide.

Answer: 2\sqrt{2}


Question: Suppose $x$ is a non-unit in a non-commutative ring $R$ with unity $1$. Determine whether $x$ is contained in some maximal left ideal of $R$. Provide a justification for your answer.

Answer: No


Question: Calculate the expected value of $a^b$, where $a$ and $b$ are independent random variables uniformly distributed over the interval [0, 1].

Answer: \ln 2


Question: Determine if the fraction \( \frac{\pi b}{a - e b} \), where \( e \) is Euler's constant and \( a, b \) are integers, can be equal to an integer.

Answer: Yes


Question: Find the coefficient of \(x^{10}\) in the expansion of \((x+2x^3+3x^5+\cdots)^3\) using generating functions.

Answer: 0


Question: Evaluate the integral \( \int^{\frac{\pi}{2}}_0 \frac{\cos^{2019}x}{\sin^{2019}x+\cos^{2019}x}\,\mathrm dx \).

Answer: \dfrac{\pi}{4}


Question: Find the function $f(y, e)$ that satisfies the equation $y^2 = \hat{y}^2 + f(y,e)(y - \hat{y})$, where $e = y - \hat{y}$.

Answer: 2y - e


Question: Provide an example of a functor \( F: \mathcal{A} \to \mathcal{B} \) such that \( F \) is faithful, but there exist distinct maps \( f_1 \) and \( f_2 \) with \( F(f_1) = F(f_2) \). Consider the preordered set \( (A = \{a_1, a_2, a_1', a_2'\}, \leq) \) where \( a_1 \leq a_2 \) and \( a_1' \leq a_2' \), and the preordered set \( (B = \{b_1, b_2\}, \leq) \) where \( b_1 \leq b_2 \). Regard \( (A, \leq) \) as category \( \mathcal{A} \) and \( (B, \leq) \) as category \( \mathcal{B} \). Define \( F \) by \( F(a_1) = F(a_1') = b_1 \), \( F(a_2) = F(a_2') = b_2 \), and \( F(f_1) = F(f_2) = b_1 \leq b_2 \).

Answer: F


Question:
Consider the iterative process defined by:

$$ x_0 = a $$
$$ x_1 = \frac {n*x_0*b + n*c + 2*d} {n*x_0*b + n*c + 2*x_0*e} x_0 $$
$$ ... $$
$$ x_n = \frac {n*x_{n-1}*b + n*c + 2*d} {n*x_{n-1}*b + n*c + 2*x_{n-1}*e} x_{n-1} $$

where $a, b, c, d, e$ are all rational numbers greater than 0. Determine if $x_n$ converges to a certain value, and if so, find that value.

Answer: \dfrac{d}{e}


Question: Calculate the number of permutations of the set \(\{1, 2, \.\.\., 10\}\) where no odd number is in its original position using the Generalized Inclusion-Exclusion Principle.

Answer: 2170680


Question: Given the equations: \( \prod_{i=1}^n p_i^2 = \prod_{i=1}^n (1-p_i)^2 = \frac{1}{4^n} \) where \( p_i \in (0,1) \) for \( i = 1, 2, \ldots, n \), determine if \( p_1 = p_2 = \ldots = p_n = \frac{1}{2} \) is a solution for all \( n \in \mathbb{N} \).

Answer: Yes


Question: Determine the highest dimension for which the classification of nilpotent Lie algebras is currently unknown.

Answer: 8


Question: Evaluate the limit: $$\lim_{x\to0}\frac{\ln\left(\frac{e^x}{x+1}\right)}{x^2}$$ without using l'Hospital's rule.

Answer: \dfrac{1}{2}


Question: For every singular matrix $S$, does there exist an invertible matrix $M$ such that the product $MS$ or $SM$ results in a diagonal matrix with only ones and zeros on the diagonal?

Answer: No


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

Answer: \dfrac{1}{e}


Question: Determine whether every affine $k$-variety can be expressed as $\text{Spec} \ A$ for some $k$-algebra $A$ that is finitely generated over $k$. Answer with true or false.

Answer: True


Question: Given a domain of holomorphy $U\subset\mathbb{C}^n$, determine whether the following statement is true: For every boundary point $a\in\partial U$, there exists a holomorphic function $f\in H(U)$ such that the supremum of $|f(z_j)|$ over all sequences $(z_j)_{j\ge 1}\subset U$ converging to $a$ is unbounded.

Answer: True


Question: Find the infimum of \( \frac{x^2+y^2+z^2}{w^2} \) where \( x, y, z, w \) are positive real numbers satisfying \( x + y + z > w \).

Answer: \dfrac{1}{3}


Question: Determine the condition on integers $m$ and $n$ such that the map $\varphi: G \rightarrow G$, defined by $\varphi(a) = a^m$, is surjective for a finite abelian group $G$ of order $n$. \( \text{Express your answer in terms of } m \text{ and } n. \)

Answer: \gcd(m, n) = 1


Question: What is the minimum number of clues required to create a Sudoku puzzle with a unique solution, given a completed Sudoku grid?

Answer: 17


Question: Given the joint Gaussian distribution with the density function \( f_{X,Y}(x,y) = Ce^{-x^2 -y^2 +xy} \), where \( C \) is a normalization constant, and the covariance matrix \( \Sigma = \frac{1}{3} \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} \), find the largest variance among the following expressions: \( \textsf{Var}(X+Y) \), \( \textsf{Var}(X-Y) \), \( \textsf{Var}(\frac{1}{2} Y^3) \), and \( \textsf{Var}(X^2) \).

Answer: 2


Question: In triangle $\Delta ABC$, the circumradius $R = 1$ and the perimeter satisfies $AB + BC + AC \leq 2 + \sqrt{3}$. Determine the maximum value of the product $P = AB \cdot BC \cdot AC$. Express the side lengths in terms of the angles: $a = 2\sin A$, $b = 2\sin B$, and $c = 2\sin C$. Maximize $P = 8\sin A \sin B \sin C$ subject to $A + B + C = \pi$ and $2\sin A + 2\sin B + 2\sin C \leq 2 + \sqrt{3}$. Find the maximum value of $P$. \(\boxed{\text{Maximum value of } P}\)

Answer: \sqrt{3}


Question: Find the coefficient of correlation between two random variables, $X$ and $Y$, given their joint probability density function (PDF) is $f(x,y)=e^{-(x+y)}$.

Answer: 0


Question: Calculate the flux of the vector field \( F = (3x, 4y, -z) \) through the surface \( S \) defined by the equations \( z^2 = x^2 + y^2 \) and \( z = x^2 + y^2 \), where \( 0 \leq z \leq 1 \).

Answer: \pi


Question: Estimate the smallest integer $n$ such that $\frac{10^n}{n!} < 0.001$ using Stirling's approximation.

Answer: 32


Question: Compute the volume of the solid in the first octant bounded by the surfaces $y^2 = 4 - x$ and $y = 2z$ using a double integral. Determine the limits of integration without relying on a graphical representation of the region.

Answer: 2


Question: Evaluate the integral: $$\int_0^\infty \frac{e^{2x}}{\left(e^{2x}\cdot\frac{1-x}{1+x}\right)^2+1}\frac{x^2}{(1+x)^2}dx$$

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


Question: Evaluate the limit: $$\lim_{x\to0^-} (x^{-x^x+1})$$

Answer: 1


Question: Assume that $z$ and $w$ are differentiable functions of $x$ and $y$, satisfying the equations $xw^3+yz^2+z^3=-1$ and $zw^3-xz^3+y^2w=1.$ Find the value of $\frac{\partial z}{\partial x}$ at the point $(x,y,z,w)=(1,-1,-1,1)$. \( \boxed{?} \)

Answer: -\dfrac{5}{4}


Question: Determine the necessary and sufficient condition for the convergence of the integral \( \int_0^\infty \frac{x^2+1-\cos x}{(x^2+1)x^p} \, dx \), where \( p \) is a real number. Is the condition \( 1 < p \le 2 \) correct?

Answer: 1 < p < 3


Question: Calculate \( \frac{\partial^2 g}{\partial u \, \partial v}(1,1) \) given that \( f(1,2)=4 \), \( \nabla f(1,2)=(3,-1) \), \( \frac{\partial^2 f}{\partial x^2}(1,2)= \frac{\partial^2 f}{\partial y^2}(1,2)=1 \), and \( \frac{\partial^2 f}{\partial x \, \partial y}(1,2)=-1 \), where \( z=f(x,y) \) and \( g(u,v)=uf(u^2, 2uv) \).

Answer: -4


Question: Let $C$ be the curve defined as the locus of points of intersection of the lines $x=2+m$ and $my=4-m$. A circle $S$ is given by the equation $(x-2)^2+(y+1)^2=25$. The circle $S$ intersects the curve $C$ at four points $P, Q, R, S$. If $O$ is the center of the curve $C$, calculate $OP^2 + OQ^2 + OR^2 + OS^2$. Provide your answer as a single number.

Answer: 100


Question: If $x, y, z$ are the roots of the polynomial equation $x^3 - ax^2 + bx - c = 0$, determine the value of the expression $(x - y - z)(y - z - x)(z - x - y)$. Provide your answer as a single number.

Answer: a^3 - 4ab + 8c


Question: Determine whether the second cohomology group $H^2(U, \mathbb{Z})$ is zero for $U$ being a disk, where $\mathbb{Z}$ is the locally constant integer sheaf.

Answer: 0


Question: Determine \( \lim_{n\rightarrow\infty} \frac{f(n)}{n} \), where \( f(n) \) is the number of positive divisors of \( n \). Provide a proof for your answer.

Answer: 0


Question: Determine whether every topological space can be induced by a metric. Provide a yes or no answer.

Answer: No


Question: Given that \( \lim_{x\to{0}}{}\frac{\sin(x)}{x}=\alpha \), find the value of \( \lim_{x\to1}\frac{\sin(x^2-1)}{x-1} \) in terms of \( \alpha \).

Answer: 2\alpha


Question: Let $V$ be an $n$-dimensional vector space over the field $K$ and let $T: V \to V$ be a linear operator. Assume that there exists a vector $v_0 \in V$ that is contained in no proper $T$-invariant subspace of $V$. Determine if the minimal polynomial of $T$ over $K$ has degree equal to $n$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x \to \infty}x\int_0^{1/x}e^t \cos(t) \space \text{d}t$$.

Answer: 1


Question: Determine the measure of the set \( F = \left\{x\in [-1, 1]: \text{there exist finitely many }p,q \in\mathbb{N} \text{ such that }\left|x -\frac{p}{q}\right|> \frac{1}{q^{2}} \right\} \). Denote this measure as \( m(F) \).

Answer: 0


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

Answer: 0


Question: Let $X$ be a metric space and let $E \subset X$. Consider the property: \( \forall x \in X, \forall \delta > 0 \quad E \cap B(x,\delta) \ne \emptyset \). Can $E$ have zero measure? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the Riemann-Stieltjes integral \( \int_{x_l}^{1} F(x) \, dF(x) \), given that \( F(x) \) is a continuous function with support on \([x_l, 1]\), \( F(x_l) = 0 \), and \( F(1) = 1 \).

Answer: \dfrac{1}{2}


Question: Determine whether the following statement is true or false: A plane graph with more than 3 vertices is maximally planar if and only if it is a triangulation.

Answer: True


Question: Evaluate the limit: $$\lim_{x \to +\infty} \frac{e^{\cos \frac{1}{x}}-1-\frac{1}{x}}{\ln \left(1+\frac{x^3}{(2+x)^4} \right )-\frac{x^3}{(2+x)^4}}$$

Answer: -\infty


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

Answer: \infty


Question: Determine the number of distinct roots of the equation \( f^{1992}(x) = x \), where the function \( f : [0, 1] \to [0, 1] \) is defined by \( f(x) = 4x(1-x) \) for all \( x \in [0, 1] \).

Answer: 2^{1992}


Question: Given two homeomorphic metric spaces $X$ and $Y$, where $d(X)$ represents the minimum cardinality of a subset $A \subseteq X$ such that $A$ is dense in $X$ (i.e., $\bar A = X$), prove that $d(X) = d(Y)$. Provide a proof for this case.

Answer: d(X) = d(Y)


Question: Evaluate the double integral \( \iint_A \sin(x+y) \, dx \, dy \) where the region \( A \) is defined as \( A=\{(x,y)\in\mathbb{R}^2: 0 \le x \le \pi, 0 \le y \le \pi\} \).

Answer: 0


Question:
Determine the number of real solutions for the system of equations given by:

\[ x+y+z=3 \]
\[ x^2+y^2+z^2=3 \]
\[ x^3+y^3+z^3=3 \]

Answer: 1


Question: Determine the number of real roots of the equation \(2^x = 100x\) without using graphical methods.

Answer: 2


Question: Evaluate the limit: $$\lim_{x\to0^+}\frac{\frac{1}{1+x^2}-1}{x^4}$$

Answer: -\infty


Question: Find the limit of the complex function \( \arccos(z) \) as \( z \) approaches \(-1\) from a direction where the imaginary part of \( \arccos(z) \) is zero.

Answer: \pi


Question: Evaluate the double integral \( \iint_D (x^2+y)\,dx\,dy \) where the region \( D \) is defined by \(-2 \le x \le 4\) and \(5x - 1 \le y \le 5x + 3\).

Answer: 240


Question: Evaluate the limit: $$\lim_{x\to\infty}\left(\frac{x^3+2x^2-x}{x^3-x^2+3}\right)^{3x+2}$$

Answer: e^9


Question: Compute the value of \( \cos^4 20^{\circ} + \cos^4 40^{\circ} + \cos^4 60^{\circ} + \cos^4 80^{\circ} \) without using a calculator.

Answer: \dfrac{19}{16}


Question: Calculate the integral \( \int_{\frac{5\pi}{2}}^{\frac{9\pi}{2}} \cos^4{\theta}\;d\theta \) using the identity \( \cos^4(x)=\frac{1}{8}(3+4\cos(2x)+\cos(4x)) \). Use Parseval's Theorem to determine the result.

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


Question: Find the correct solution to the integral equation $$f(x) = a - \int^x_b (x-t)f(t)dt$$ where $a$ and $b$ are constants.

Answer: a \cos(x - b)


Question: Evaluate the integral \( \int_0^\infty \frac{\sin\left(\frac{1}{bx}\right) - \sin\left(\frac{1}{ax}\right)}{x} \, dx \) using Frullani's theorem.

Answer: 0


Question: Let \( p > 2 \) be a prime that divides \( q(n) = 2^{n^3+1} - 3^{n^2+1} + 5^{n+1} \). Does there exist a positive integer \( k \) such that \( p \) also divides \( q(n+k) \)?

Answer: Yes


Question: Determine whether the sequence \(b_{n} = [\sqrt{3}n] + [\sqrt{7}n]\) contains infinitely many even numbers.

Answer: Yes


Question: Does the dominated convergence theorem apply to random variables that are not in $L^1$?

Answer: No


Question: Determine the number of different solutions to the equation \(x^2_{1}+2x^2_{2}+3x^3_{3}=4x^2_{4}\) over the field with 7 elements, \(\mathbb{F}_{7}\).

Answer: 343


Question: Is it possible to construct a sequence of functions that converges pointwise to a discontinuous function, but does not converge uniformly on any interval? Provide a justification for your answer.

Answer: Yes


Question: Let $X$ be a continuous local martingale such that $|X_t|^p$ is uniformly integrable for each $t \ge 0$ and some $p \ge 1$. Does it follow that $X$ is uniformly integrable? Answer yes or no.

Answer: No


Question: Given a permutation group \( G \) acting on a finite set \( D \), and a finite set \( R \) of colors, a permutation group \( \hat{G} \) is induced to act on \( R^D \) (the set of functions from \( D \) to \( R \)). For any \( \sigma \in G \), the function \( \hat{\sigma} \) is defined by \( \hat{\sigma}(F)(d) = F(\sigma(d)) \) for all \( F \in R^D \). Determine the relationship between \( |G| \) and \( |\hat{G}| \).

Answer: |\hat{G}| = |G|


Question: Does the sequence \( \{b_n\}_{n \geq 1} \) defined by \( b_n = \left\lfloor \frac{3^n}{\sqrt{3}} \right\rfloor \) contain an infinite number of terms that are integer powers of 3?

Answer: No


Question:
Find the value of $B$ when $B = \frac{m}{\pi}$, where $m$ is the minimum value of the function \( f(x) = \frac{4\pi^2}{3\arccos{(x^4-2x^2)}} + \frac{5\pi}{3} \). Choose the correct answer from the following options:

\[ \begin{array}{ll} 1.&4\\ 2.&3\\ 3.&5\\ 4.&2\\ \end{array} \]

Answer: 2


Question: Evaluate the integral \( \int_{-\infty}^{\infty} x^{2n+1} e^{-bx^2} \, dx \), where \( n \) is a non-negative integer and \( b \) is a positive constant.

Answer: 0


Question: Given the limit \( \lim_{h\to 0}(f(x+h)-f(x-h))=0 \), determine if the substitution \( x-h=a \) leads to \( \lim_{h\to 0}(f(a+2h)-f(a))=0 \). Justify your reasoning.

Answer: Yes


Question: Find the image of the set $G = \{ z = x+iy \mid x^2 + y^2 + 2x + 2y + 1 = 0\}$ under the function $f(z) = \frac{2z+3}{z+i}$.

Answer: 6x - 4y + 1 = 0


Question: Find the minimum value of the expression \(|1+z_1|+|1+z_2|+\cdots+|1+z_n|+|1+z_1z_2\cdots z_n|\) where \(n\) is an even integer and each \(z_i\) is a complex number.

Answer: 2


Question: Calculate the limit \( \lim_{r\rightarrow \infty} \frac{\Gamma(r\alpha)}{\Gamma((r+1)\alpha)} \) where \(0<\alpha <1\) and \(\Gamma(.)\) denotes the Gamma function.

Answer: 0


Question: Consider the functional equation \(f(x-y)=f(x)/f(y)\). If \(f'(0)= p\) and \(f'(5)=q\), find the value of \(f'(-5)\).

Answer: \dfrac{p^2}{q}


Question: Evaluate the limit \(\lim _{(x,y)\to (0,0)}\left(\frac{x^2y}{x^2+y^2}\right)\).

Answer: 0


Question: Determine whether there exists a polynomial time algorithm to check if a given point $z \in \mathbb{R^n}$ is a basic feasible solution of a polyhedron $P \subset \mathbb{R^n}$ defined by $m$ inequalities $Ax \leq b$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether a matrix with no eigenvalues can have a non-zero determinant.

Answer: Yes


Question: Suppose we have a countable set of intervals that covers the rational numbers. If each interval is extended by a factor of $1/(2^n)$, where $n$ is the index of the interval, does this new set of intervals cover all real numbers? Answer with 'Yes' or 'No'.

Answer: No


Question: Determine the value of a subgradient for the absolute value function \(|x|\) at \(x = 0\).

Answer: [-1, 1]


Question: Evaluate the limit \( \lim_{n\to\infty} \frac{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots+\frac{1}{n}}{1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\ldots+\frac{1}{2n+1}} \).

Answer: 2


Question: Let $f:[a,b] \to \mathbb{R}$ be a function such that $f(x)$ is bounded and $f^3(x)$ is integrable. Does it follow that $f$ is integrable?

Answer: Yes


Question: Evaluate the limit \( \lim_{n\to\infty} X_n \), where \( X_n = \frac{1000^n}{\sqrt{n!}} + 1 \).

Answer: 1


Question: Calculate the value of the expression \( S = 3\sqrt{\sqrt[3]{5} - \sqrt[3]{4}} - \sqrt[3]{2} - \sqrt[3]{20} + \sqrt[3]{25} \) without using a calculator.

Answer: 0


Question: Find the minimum value of the expression \(\sqrt{a^2+b^2}+\sqrt{(a-1)^2+b^2}+\sqrt{a^2+(b-1)^2}+\sqrt{(a-1)^2+(b-1)^2}\).

Answer: 2\sqrt{2}


Question: Assume $U$ is a bounded set and $G:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function with a bounded derivative. Given that a sequence of smooth functions $f_n$ converges to $f$ in $L^p(U)$, does it follow that $G(f_n)$ converges to $G(f)$ in $L^p(U)$?

Answer: Yes


Question: Calculate the Hilbert polynomial of the twisted cubic $C$ in $\mathbb{P}^3$, defined by the ideal $V(XZ-Y^2, YW-Z^2, XW-YZ)$. Denote this polynomial as $P_C(n)$. What is $P_C(n)$?

Answer: 3n + 1


Question: Does the existence of a bijective homomorphism between two groups guarantee that they are isomorphic? Provide a justification for your answer.

Answer: Yes


Question: Find the limit: $$\lim_{n\to\infty}\left(\frac{2}{1}\cdot\frac{4}{3}\cdot\dots\frac{2n}{2n-1}\right)^\frac{1}{n}.$$

Answer: 1


Question: Determine the cardinality of the set of all finite subsets of the natural numbers.

Answer: \aleph_0


Question: Is every infinite-dimensional normed space metrizable?

Answer: Yes


Question: Determine the number of nonisomorphic abelian groups of order 100.

Answer: 4


Question: Given a $3\times 3$ matrix $A$ with $\det A = 9$ and the equation $2A - A^2 = 3I_3$, where $I_3$ is the identity matrix, find the determinant of $4I_3 - 2A$. Use the observation that $4I_3 - 2A = (I_3 - A)(I_3 + A)$ to help calculate $\det(4I_3 - 2A)$. Provide your answer as a single number.

Answer: -96


Question:
Determine whether the following two statements about a topological space $X$ and its subset $E$ are equivalent:

(1) Any continuous function $f:E\longrightarrow \mathbb{R}$ can be extended to a continuous function $g: X\longrightarrow \mathbb{R}$ such that $g|_E=f$.

(2) $E$ is a deformation retract of $X$.

Are these statements equivalent?

Answer: No


Question: Determine whether the maximum value of the function \( f_k = (x-1)\,e^{(x-1)^k-x^k} \) decreases as \( k \) increases.

Answer: Yes


Question: Consider the function \( f(z) = z + 2z^2 + 3z^3 + \cdots = \sum_{n \geq 0} nz^n \) defined on the open disk \( \{z \mid |z| < 1\} \). Choose the correct option: (a) \( f \) is not injective but attains every complex value at least once. (b) \( f \) is injective but does not attain every complex value. (c) \( f \) is injective and attains every complex value. (d) None of the above.

Answer: B


Question: For an arbitrary matrix $A$, if $A^4 = 0$, does it necessarily follow that $A^2 = 0$? Answer 'yes' or 'no' and justify your response.

Answer: no


Question: Determine the winding number of the function \( f: S^1 \to \mathbb{C} \setminus \{ 0 \} \) given by \( z \mapsto 8z^4+4z^3+2z^2+z^{-1} \) about the origin.

Answer: 4


Question: Determine whether the equation \( \sum_{k=1}^{n} k^{m} = \left ( \sum_{k=1}^{n} k \right )^{m-1} \) is true for all positive integers \( n \) and \( m \).

Answer: False


Question: Prove that if \( \int_0^{\pi} f(t)\cos(nt)~dt=0 \) for all \( n \in \{0\} \cup \mathbb{N} \), then \( f \equiv 0 \).

Answer: f \equiv 0


Question: Find the covering radius of the code $C = \{0011, 1100, 2211\}$ over the field $F_3$. Express your answer as a single integer.

Answer: 3


Question: Find the value of $x$ that minimizes the function $f(x) = \int_{0}^{1} |t-x|^3 \, dt$.

Answer: \dfrac{1}{2}


Question: Determine whether a polynomial with an even degree and negative coefficients for the even powers can be positive over some interval.

Answer: Yes


Question: Find the interval of convergence for the power series \(\sum_{n=0}^{\infty} \frac{x^{2n}}{1-x^2}.\)

Answer: (-1, 1)


Question:
Evaluate the limit:

$$\lim_{n \to \infty}\frac{\left(\sqrt[3]{n^3+\sqrt{n}}-\sqrt[3]{n^3-1}\right)\sqrt{3n^3+1}}{\sqrt[3]{\left(n^3+\sqrt{n}\right)^2}+\sqrt[3]{\left(n^3+\sqrt{n}\right)\left(n^3-1\right)}+\sqrt[3]{\left(n^3-1\right)^2}}$$

Answer: 0


Question: Evaluate the limit: \( \lim_{h\rightarrow0^-}\frac{e^{-1/|h|}}{h^2} \)

Answer: 0


Question: Determine the spectrum of the operator $S$ on the Banach space $\ell^2$, where $\|x_n\|=\sqrt{\sum_{n=1}^\infty |x_n|^2}$ and $S\{x_n\}=\{x_3,x_2, x_5, x_4, x_7, x_6, x_9,x_8, \ldots \}$.

Answer: \{\lambda \in \mathbb{C} : |\lambda| \leq 1\}


Question: Evaluate the integral \( \int_0^\pi \frac{\sin x}{\sqrt{|\cos x|}} \, dx \).

Answer: 4


Question: Determine the number of zeros of the polynomial \( z^p + z - 1 \) that lie in the region \( |z| < 1 \), where \( p \) is a prime number.

Answer: 1


Question: Find an integer \(0 < x < 81\) such that \(x^{19} \equiv 8 \pmod{81}\).

Answer: 8


Question: In an infinite series $S = a_1 + a_2 + a_3 + 	ext{...}$, does the absolute value of any term $a_n$ always exceed the sum of the absolute values of all subsequent terms $a_{n+1}, a_{n+2}, 	ext{...}$? Answer yes or no.

Answer: No


Question: If the minimal polynomial of an endomorphism $u$ of a vector space $E$ does not split in the field $\mathbb{K}$, can $\mathbb{K}[u]$ be isomorphic to $\mathbb{K}^d$ for some positive integer $d$? Provide a justification for your answer.

Answer: No


Question: Determine the number of distinct prime ideals in the quotient ring \(\mathbb{Q}[x]/(x^5 - 1)\), given that \(x^5 - 1\) factors as \((x-1)(x^4 + x^3 + x^2 + x + 1)\) in \(\mathbb{Q}[x]\), with \(x^4 + x^3 + x^2 + x + 1\) being irreducible over \(\mathbb{Q}\).

Answer: 2


Question: Solve the equation: \( \sqrt{2 x^4-3 x^2+1}+\sqrt{2x^4-x^2}=4 x-3 \).

Answer: 1


Question: Evaluate the limit: $$\lim_{n\to\infty}\max_{1\leq k\leq n}\left|\frac{n(n-1)\cdots(n-k+1)}{(n-x)^k}-1\right|$$ where $x$ is a fixed real number greater than 0.

Answer: 1


Question: Determine whether the number expressed as $2 \rightarrow 17 \rightarrow 417$ in Conway chained arrow notation is larger than Graham's Number.

Answer: No


Question: Find the equation of the locus of points where the lengths of the tangents from the circles with equations \( x^2 + y^2 - 6x - 27 = 0 \) and \( x^2 + y^2 + 2x - 8 = 0 \) are equal.

Answer: 8x + 19 = 0


Question: Consider a quantum system with a Hilbert space \(\mathbb{C}^3\). The Hamiltonian is given by \(H=\begin{pmatrix} 1 & 0 & 1 \\ 0 & -1 & 0 \\ 1 & 0 & 1 \end{pmatrix}\). If the initial state is \(\psi(0)=\frac{1}{\sqrt{3}}\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\), calculate the probability of measuring the energy \(E=1\) at any time \(t\).

Answer: 0


Question: Find the maximum value of \( T > 0 \) such that the function \( g(x) = \cos{(k\sin{x})} \) satisfies \( g(x+T) = g(x) \) for all \( x \in \mathbb{R} \).

Answer: \pi


Question: Suppose $f(x)$ and $g(x)$ are continuous, differentiable, positive, and monotone decreasing functions such that $\lim_{x \to \infty} f(x) = \lim_{x \to \infty} g(x) = 0$. If $\int_0^\infty |g(x)-f(x)| \, dx=0$, does it imply that $f(x) = g(x)$ for all $x$? Answer 'yes' or 'no' and justify your answer.

Answer: yes


Question: Let \( V \) and \( W \) be finite-dimensional vector spaces over an infinite field \( k \). Fix a positive integer \( n \leq \min(\dim(W), \dim(V)) \). Given \( n \) injective linear maps \( f_i:V\rightarrow W \), such that the \( f_i \) are linearly independent as elements of \( \operatorname{Hom}(V,W) \), determine whether there necessarily exists a vector \( v\in V \) such that the vectors \( f_i(v) \) are linearly independent in \( W \).

Answer: Yes


Question: Given that $g$ is the inverse function of a differentiable function $f$, and $G(x) = \frac{-4}{g^2(x)}$, where $f(5) = 3$ and $f'(5) = \frac{1}{125}$, find $G'(3)$. \( \boxed{} \)

Answer: 8


Question: Find the tenth digit (from the right) of the sum \( \sum_{k=1}^{30} (k!) \) using modular arithmetic.

Answer: 3


Question: Find the supremum of the function \( f(x,y,z)=\frac{(x-xyz)(y-xyz)(z-xyz)}{(1-xyz)^3} \) where \( f: [0,1]^3\backslash\{(1,1,1)\} \to \mathbb{R} \).

Answer: \dfrac{8}{27}