Question: Calculate the divergence of the vector field \(r^n \vec{r}\), where \(r\) is the magnitude of the position vector \(\vec{r}\) and \(n\) is a constant, using tensor notation.

Answer: (n + 3) r^n


Question: If $A$ and $B$ are closed subspaces of a Hilbert space $H$ such that $A^\perp \cap B^\perp = \{0\}$, can we conclude that $A = B$? Provide a justification for your answer.

Answer: No


Question:
Let \( A \in \mathbb{C} \) be a \( 2 \times 2 \) matrix, and let \( f(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n \) be an arbitrary polynomial over \( \mathbb{C} \). Which of the following statements is true regarding \( f(A) \)?

(a) \( f(A) \) can be expressed as \( c_0I + c_1A \), for some \( c_0, c_1 \in \mathbb{C} \)
(b) \( f(A) \) is the zero matrix
(c) The rank of \( f(A) \) is 2
(d) \( f(A) \) can never be the zero matrix

Answer: a


Question: Is it possible for an idempotent matrix to have a transpose that is different from itself but is also idempotent?

Answer: Yes


Question: Determine the norm of the operator \(A\) in the \(\ell^1\) space, where \(A(x_n)_n = \left(\sum_{k=0}^\infty \frac{a^k}{k!} x_{n+k} \right)_n\), \((x_n)_{n\ge 1}\) is a real absolutely summable sequence, \(a > 0\) is a constant, and the norm \(\| \cdot \|\) is defined as \(\| (x_n)_n \| = \sum_{n=1}^\infty |x_n|\).

Answer: e^a


Question: Let $P(x)$ be a polynomial with real coefficients. If $M$ is a Hermitian matrix such that $P(M) = 0$, does it imply that all eigenvalues of $M$ are roots of $P(x)$? Answer yes or no and justify your answer.

Answer: Yes


Question: Suppose $X$ and $Y$ are compact subsets of $\mathbb{R}^m$ such that $X\setminus Y$ has measure zero. If $g:X\rightarrow \mathbb{R}$ is integrable on $Y$, does it follow that $\int_X g$ exists? Answer yes or no and justify your answer.

Answer: Yes


Question:
Given the system of equations:
\[\begin{align*}
a + b + c &= 0, \\
a^2 + b^2 + c^2 &= 2, \\
a^3 + b^3 + c^3 &= 19,
\end{align*}\]
compute the value of $abc$. \(\boxed{\text{Your answer here}}\)

Answer: \dfrac{19}{3}


Question: Given that \(a^2 + b^2 + c^2 = 1\) where \(a, b, c\) are real numbers, find the maximum value of the expression \((4a-3b)^2 + (5b-4c)^2 + (3c-5a)^2\).

Answer: 50


Question: Solve the integral equation: $$ \int_{0}^{x}(x-t)y(t)dt = 2x + \int_{0}^{x}y(t)dt $$

Answer: -2e^x


Question: Simplify the expression: \[ b^2(\sin^2 C-\sin^2 A)+c^2(\sin^2 A-\sin^2 B)+a^2(\sin^2 B-\sin^2 C) \]

Answer: 0


Question: Determine whether the following statement is true or false: If $E[(X-Y)^2]=0$, then $X=Y$ almost surely.

Answer: True


Question: Determine the number of elements of order 7 in a given group.

Answer: 6


Question: Compute the limit \( L = \lim_{n\to\infty}{\prod_{k=1}^{n}{\left(\frac{k}{n}\right)^{1/n}}} \). Use Stirling's approximation to simplify the expression and find the value of \( L \).

Answer: \dfrac{1}{e}


Question: In \(\mathbb{R}^n\), determine if the union of a closed ball and a finite number of isolated points is a closed set.

Answer: Yes


Question: Find the limit of the function \( g(x,y) = \frac{1+xy^2+o(xy^2)-1}{x^2+y^2} \) as \((x,y) \to (0,0)\).

Answer: 0


Question: Determine whether the following statement is true or false: The number of irreducible components of the algebraic subset $V(f_1, \\ldots , f_k)$ of $\mathbb{R}^n$ is always less than or equal to the number of irreducible components of the algebraic subset $Z(f_1, \\ldots , f_k)$ of $\mathbb{C}^n$. Justify your answer.

Answer: True


Question: Let $X$ and $Y$ be topological spaces and let $f: X \to Y$ be a homeomorphism. Suppose that there exists a metric $d_X$ on $X$ such that $(X, d_X)$ is complete. Does it follow that there exists a metric $d_Y$ on $Y$ such that $(Y, d_Y)$ is complete? Provide a justification for your answer.

Answer: Yes


Question: Find the greatest common divisor (GCD) of the numbers represented by 120 consecutive twos and 70 consecutive twos in base 3.

Answer: 59048


Question: Evaluate the limit: $$ \lim_{x\to\infty} (\sqrt[100]{(x + 3)(x + 6)...(x + 300)} - x) $$

Answer: \dfrac{303}{2}


Question: If $f,g: \mathbb{N}\rightarrow\mathbb{N}$ and $f\circ g=\operatorname{id}_\mathbb{N}$, is $f$ necessarily invertible? Provide a justification for your answer.

Answer: No


Question: Consider a collection of open sets in $\mathbb{R}^n$. Is it possible to find a subset of this collection whose boundary has a non-zero volume?

Answer: Yes


Question: A set $M$ contains $1000$ integers, with no prime factor exceeding $10$. What is the minimum number of pairs of elements in $M$ such that their product is a perfect square?

Answer: 30752


Question: Evaluate the integral \( \int_1^3 x^2[x] \, d(x[x]) \), where \([\cdot ]\) denotes the floor function.

Answer: \dfrac{374}{3}


Question: Find the limit in probability of \((\bar{X}_n)^2\), where \(\bar{X}_n\) is the sample mean of a sample of size \(n\) drawn from a continuous random variable \(X\) with cumulative distribution function \(F(x) = \begin{cases} 1 - x^{-4} & x > 1 \\ 0 & \text{elsewhere} \end{cases}\). Justify your answer.

Answer: \dfrac{16}{9}


Question: Can a function be differentiable at a point of discontinuity if the slopes of tangents at points immediately before and after that point are equal? Provide a justification for your answer.

Answer: No


Question: Determine the range of values for \( p \) such that the interval \((1,2)\) lies between the roots of the equation \(2^x + 2^{2-x} + p = 0\).

Answer: (-\infty, -5)


Question: Find a non-recursive formula for the sequence \( s_n \) defined by the recursive relation \( s_{n+1} = (n+2)s_n + n! \) with the initial condition \( s_1 = 1 \).

Answer: n \cdot n!


Question: Evaluate the limit \( \lim_{x\to0}\frac{(\sqrt{1+x}-\cos x)^2}{(e^{x^2}-1)x^4} \) using L'Hopital's rule or a mixed method.

Answer: \infty


Question: Determine the values of \( a \) for which the integral \( \int _0^{\infty }\:\frac{\sin\left(x\right)}{\left(1+x\right)^a} dx \) converges.

Answer: a > 0


Question: In the quotient ring $\mathbb{Z}_5[x]/J$, where $J=(x^4+2x^2+3x+4)$, find a polynomial $h(x)$ such that $(x^3+x+2+J)(h(x)+J)=J$. Provide your answer as a polynomial $h(x)$ in $\mathbb{Z}_5[x]$. \( \boxed{h(x)} \)

Answer: 3x^3 + 4x^2 + 2


Question: Consider separable Banach spaces $(E, \Vert \cdot \Vert_E)$ and $(F, \Vert \cdot \Vert_F)$. The space $E$ is equipped with a sigma algebra $\mathcal{F}$ generated by its open sets, and $F$ has a sigma algebra $\mathcal{G}$. Let $A: D(A) \to F$ be a closed operator where $D(A)$ is a subspace of $E$ and $D(A) \in \mathcal{F}$. Is the map $A: (D(A), \mathcal{F}|_{D(A)}) \to (F, \mathcal{G})$ necessarily measurable?

Answer: Yes


Question: A sequence $\{a_n\}$ is defined by the recurrence relation $a_n = a_{n-1} + 2a_{n-2} - a_{n-3}$ with initial conditions $a_1 = a_2 = \frac{a_3}{3} = 1$. Find the value of the infinite series $$a_1 + \frac{a_2}{2} + \frac{a_3}{2^2} + \cdots$$

Answer: 8


Question: Find the maximum area of a rectangle that can be inscribed in the ellipse \(\frac{x^2}{4}+\frac{y^2}{9}=1\).

Answer: 12


Question: Find the minimum possible value of the expression \( \frac{ab}{c} + \frac{bc}{a} + \frac{ca}{b} \) for positive real numbers \(a, b, c\) such that \(a^2 + b^2 + c^2 = 1\).

Answer: \sqrt{3}


Question: Find the coefficient of \(x^3\) in the expansion of \((1-2x+3x^2-4x^3)^{1/2}\).

Answer: -1


Question: Given the joint density function \( f(x,y) = \frac{e^{-x/y}e^{-y}}{y} \), calculate the conditional probability \( P(X > 1 \mid Y = y) \).

Answer: e^{-1/y}


Question: Given the quadratic function \( f(x) = ax^2 + bx + c \), where it is known that \( |f(x)| \le 100 \) for all \( |x| \le 1 \), determine an upper bound for \( |a| + |b| + |c| \).

Answer: 300


Question: How many normal subgroups of the symmetric group $S_4$ are isomorphic to the Klein four-group?

Answer: 1


Question: Determine the degree of the field extension \(\mathbb{Q}(5^{1/3}, 5^{1/9})\) over \(\mathbb{Q}\).

Answer: 9


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

Answer: \sqrt{e}


Question: Evaluate the limit: $$\lim_{n \to \infty}\frac{1}{\sqrt[n]{n!}}, n \in \mathbb{N}$$

Answer: 0


Question:
Determine for which of the following initial conditions the solution to the differential equation \( y' = 2\sqrt{|y-1|} \) exists and is unique:

1. \( y(0) = 1 \)
2. \( y(1) = 2 \)
3. \( y(2) = 1 \)

Answer: 2


Question: Calculate the gradient of the function \( g(X) = \text{Tr} (X^T B^T B X) \) with respect to the matrix \( X \), where \( B \) is a given matrix.

Answer: 2 B^T B X


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

Answer: 2


Question: Calculate the limit of the function \( f(x,y)=\frac{\ln(1+x^2y^2)}{x^2} \) as \((x,y)\) approaches \((0,y_0)\). Find an upper bound function that converges to \(y_0^2\) to confirm the limit.

Answer: y_0^2


Question: Given \(a_{n}\) as the positive solution to the equation \(x^{n}= \cos x\), find the limit \[ \lim_{n\rightarrow\infty}\frac{1- a_{n}}{\ln n} \]

Answer: 0


Question: Find the value of $k$ such that the curves $y=k\sqrt{x^2-1}$ and $y=\ln x$ are tangential.

Answer: 0.402


Question: Find the residue of the function \( \frac{e^{z + \frac{1}{z}}}{1-z^2} \) at \( z=1 \).

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


Question: Determine whether there exists an integer $n$ such that $2^{2^n} + 1$ is divisible by $3$ but not by $9$. If such an integer exists, provide an example.

Answer: 0


Question: Evaluate the integral \( \int_{|z| = 106} \cos\left(\frac{1}{6z+1}\left(\frac{1}{(z-1)(z-2)\cdots(z-105)}\right)\right)\,dz \).

Answer: 0


Question: Find the area of the region enclosed by the curve parameterized by \( g(t) = (t^4 - t^2, t^6 - t^2) \), where \( 0 \leq t \leq 1 \), using Green's Theorem.

Answer: \dfrac{1}{60}


Question: Let $a, b, c$ be the lengths of the sides of a triangle such that $a + b + c = 3$. Determine the minimum value of the expression \( a^4 + b^4 + c^4 + 2(a-1)(b-1)(c-1) \).

Answer: 3


Question: Determine whether there exists an uncountable commutative ring with identity \( R \) such that \( R/I \) is countable for some two-sided proper ideal \( I \).

Answer: Yes


Question: Evaluate the infinite series \( \sum_{n=1}^{\infty} \frac{2n+1}{(n^{2}+n)^{2}}. \)

Answer: 1


Question: Let $H$ be the Hilbert space $L^2[-1,1]$, and let $S$ be the subspace of functions $f \in H$ satisfying $\int_{-1}^{1} x f(x) \, dx = 0$. Find the element of $S$ that is closest to the function $g(x) = x^2$. Provide your answer as a function.

Answer: x^2


Question: Let $p_1 < p_2 < \cdots < p_{40}$ be prime numbers such that $30$ divides $p_1^4 + p_2^4 + \cdots + p_{40}^4$. Find the value of $p_4$. \( \boxed{\text{Your answer here}} \)

Answer: 13


Question: Given the joint probability density function of $X$ and $Y$ as \( f(x, y) = \frac{4}{y} e^{-4y} \) for \( y>0 \) and \( 0 \leq x< y \), find the covariance of $X$ and $Y$. Express your answer as a single numerical value.

Answer: \dfrac{1}{32}


Question: Evaluate the limit: $$ \lim_{n \to \infty} \int_0^1 \frac{x^n e^x}{1 + e^x} \, dx $$

Answer: 0


Question: Calculate \( \operatorname{Cov}(X, Y^2) \) given that \( X \sim N(0,1) \), \( Y \sim N(0,1) \), and \( \operatorname{Cov}(X, Y) = p \) where \( -1 < p < 1 \).

Answer: 0


Question: Does there exist a divergent sequence \((z_n)_{n\in\mathbb{N}}\) of complex numbers such that the sequence of absolute values \((|z_n|)_{n\in\mathbb{N}}\) converges?

Answer: Yes


Question: Find the value of \(15^{100!} \mod 5000\) using basic number theory techniques.

Answer: 625


Question: Evaluate the outward flux of the vector field \( V = x(x^2 + y^2 + z^2)^{-3/2}\mathbf{i} + y(x^2 + y^2 + z^2)^{-3/2}\mathbf{j} + z(x^2 + y^2 + z^2)^{-3/2}\mathbf{k} \) across the boundary of the rectangular box defined by the points \((-1, -1, -1)\) to \((1, 1, 1)\). Note that \( V \) is not defined at the origin.

Answer: 4\pi


Question: Calculate the volume enclosed by the surfaces defined by the equations: \(x^2 + y^2 = z\), \(y = x^2\), \(z=0\), and \(y = 1\).

Answer: \dfrac{88}{105}


Question: Find the chromatic number of a graph $H$ where the vertices represent non-empty subsets of $[n]$, and two vertices are connected if their corresponding subsets have a non-empty intersection.

Answer: 2^{n-1}


Question: Given an irreducible polynomial $p(t) \in \mathbb{Q}[t]$, represented as $p(t) = (t - \alpha_1)(t - \alpha_2) \dotsb (t - \alpha_n)$, where $\alpha_1, \alpha_2, \dots, \alpha_n$ are its roots, determine if the polynomial $q(t) = (t - r(\alpha_1))(t - r(\alpha_2)) \dotsb (t - r(\alpha_n))$ belongs to $\mathbb{Q}[t]$, for some $r(t) \in \mathbb{Q}[t]$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following equality is true: \[ N([s_1, s_2] | [0,0], [[\sigma, 0],[0, \sigma]]]) = N(s_1 | 0, \sigma) \times N(s_2 | 0, \sigma) \]

Answer: True


Question: Find the remainder when $55^{142}$ is divided by 143.

Answer: 55


Question: Determine the value of $q$ for which the integral \( \int_{0}^{1} \frac{1}{(\tan(x))^{q}} \, dx \) converges.

Answer: q < 1


Question: Evaluate the improper integral \( \int_0^4 \frac{dx}{\sqrt{|x-3|}} \).

Answer: 2 + 2\sqrt{3}


Question: Determine whether the following statement is true or false: Let \( f:\mathbb{R}^2\to\mathbb{R}^2 \) be a smooth function. If the determinant of the derivative matrix is non-zero at every point, then the function is globally one-to-one.

Answer: B


Question: Find the smallest natural number \( n \) that satisfies the following system of congruences: \( n \equiv 1 \pmod{2} \), \( n \equiv 0 \pmod{3} \), \( n \equiv 1 \pmod{4} \), \( n \equiv 1 \pmod{5} \), \( n \equiv 3 \pmod{6} \), \( n \equiv 0 \pmod{7} \), \( n \equiv 1 \pmod{8} \), and \( n \equiv 0 \pmod{9} \).

Answer: 441


Question: Given a symmetric matrix $M \in \mathbf{R}^{n \times n}$ that is positive semidefinite, and assuming the leading principal $i \times i$ minor $M_i$ has a non-zero determinant ($\det (M_i) \neq 0$), determine if it follows that if the determinant of the leading principal $(i+1) \times (i+1)$ minor, $\det (M_{i+1})$, is zero, then the rank of $M$ is $i$. Provide a justification for your answer.

Answer: No


Question: Evaluate the infinite series \( \sum_{n=0}^\infty \frac{n^3}{n!} \).

Answer: 5e


Question: Determine if the integral \(\int_{h}^{R}\int_{0}^{2\pi}\int_{0}^{\sqrt{R^2-z^2}}A(2-\sin^2\theta)r \, dr \, d\theta \, dz\) can be rewritten as the product of two integrals: \(\int_{h}^{R}\int_{0}^{\sqrt{R^2-z^2}}r \, dr \, dz \cdot \int_{0}^{2\pi}A(2-\sin^2\theta) \, d\theta\).

Answer: Yes


Question: Let \( f = \sum_{i \ge m} a^i t^t \) (for \( m \in \mathbb{Z} \)) be a formal Laurent power series. Consider its formal differential \( df \). What is the coefficient of degree \(-1\) of the differential form \( \frac{df}{f} \)? Provide an explicit expression for this coefficient.

Answer: m


Question: Determine whether there exists a map \( \phi: \mathbb{R}^2 \rightarrow S^2 \), where \( S^2 \) is the unit sphere in \( \mathbb{R}^3 \), such that for any Borel set \( A \) in \( \mathcal{B}([0,10]\times[0,10]) \), the spherical measure \( \sigma(\phi(A)) \) can be estimated as a constant multiple of the Lebesgue measure \( |A| \) of \( A \), with the constant independent of \( A \)?

Answer: Yes


Question: Let $X$ be a metric space such that every continuous function $f: X \to \mathbb{R}$ is uniformly continuous. Determine whether it is true that for any metric space $Y$, every continuous function $f: X \to Y$ is also uniformly continuous.

Answer: True


Question: Determine whether there exists a well-ordered uncountable set of real numbers.

Answer: No


Question: Evaluate the infinite product \(\prod_{n=2}^\infty\dfrac{n^3 - 1}{n^3 + 1}\) by simplifying the terms and demonstrating that the product telescopes.

Answer: \dfrac{2}{3}


Question: Evaluate the expression \( \frac {(x + \frac{1}{x})^6 - (x^6 + \frac{1}{x^6}) - 2} {(x + \frac{1}{x})^3 - (x^3 + \frac{1}{x^3})} \) and determine its minimum value.

Answer: 10


Question: What is the smallest value of $a$ such that a Möbius strip can be constructed using a rectangular piece of paper with dimensions $[0,1]\times[0,a]$, where $a > 1$?

Answer: \sqrt{3}


Question:
Given a symmetric positive semi-definite matrix $A \in \mathbb{R}^{n \times n}$ with $\operatorname{Tr}(A) \leq n$, determine a lower bound for the expression:

$$\operatorname{Tr} \left( \left( (A+I )^{-1} \right)^2 \right)$$

where $I$ is the identity matrix.

Answer: \dfrac{n}{4}


Question:
In an electromagnetic plane wave, the electric field is given in phasor form as \(\vec{E}=E_{x 0} e^{-\gamma z} \hat{x}\), where \(\gamma\) is the propagation constant. The wave propagates in the positive \(z\) direction through a conducting medium with conductivity \(\sigma=0.2 \mathrm{~S} / \mathrm{m}\), permittivity \(\varepsilon=\varepsilon_{0}=8.8534 \times 10^{-12} \mathrm{~F} / \mathrm{m}\), and permeability \(\mu=\mu_{0}=4 \pi \times 10^{-7} \mathrm{H} / \mathrm{m}\). The angular frequency is \(\omega=4.518 \times 10^{10} \mathrm{rad} / \mathrm{s}\). The propagation constant \(\gamma\) is expressed as:

\[ \gamma=j \omega \sqrt{\mu_{0} \varepsilon_{0}} \sqrt{1-\frac{j \sigma}{\omega \varepsilon_{0}}} \]

Calculate the distance \(z\) (in meters) from the source where the magnitude of the electric field reduces to \(0.01 E_{x 0}\).

Answer: 0.126


Question: Solve the partial differential equation \( u_t + u u_x = 0 \) with the initial condition \( u(x,0) = h(x) = \begin{cases} 1 & \text{for} \; x < 0 \\ 0 & \text{for} \; x \geqslant 0 \end{cases} \). Determine the breaking time of the solution.

Answer: 0


Question: Solve the initial value problem (IVP) for the partial differential equation: \[ 2xyu_x + u_y - u = 0 \] with the initial condition \( u(x,0) = x \).

Answer: u(x, y) = x e^{y - y^2}


Question: Find all functions \( f: \mathbb{R}^+ \to \mathbb{R}^+ \) such that \[ f(x^2 + xf(y)) = f(f(x))(x+y) \] for all positive real numbers \( x \) and \( y \).

Answer: f(x) = x


Question: Calculate the surface area of the intersection of a circle with radius $R=1$ and the set of points above the parabola $f(x) = 2x^2$ in the positive $x,y$-plane. The circle is parameterized in polar coordinates as $x=R\cos(\phi)$ and $y=R\sin(\phi)$. The points of intersection are at $x=0.68$ and $x=-0.68$, corresponding to $\phi = 0.896$ radians and $\phi = 2.356$ radians. Determine the surface area of this region.

Answer: 0.85


Question: Determine the dual group of the multiplicative group of positive real numbers, denoted by $\mathbb{R}_{>0}^*$.

Answer: \mathbb{R}


Question: Determine whether it is true or false that free modules over non-commutative rings can have more than one generator.

Answer: A


Question: Determine whether the statement "All Eulerian graphs are also Hamiltonian" is true or false.

Answer: False


Question: Calculate the volume of the half-cone defined by the inequalities \( \sqrt{x^{2}+y^{2}}<z<1 \) and \( x>0 \).

Answer: \dfrac{\pi}{6}


Question: Determine the number of polynomial functions \( f \) of degree \( \geq 1 \) that satisfy the equation \( f(x^2) = (f(x))^2 = f(f(x)) \) for all real \( x \).

Answer: 1


Question: Evaluate the contour integral \( \int_{\gamma} \frac{\log(z)}{z} \, dz \) where \( \gamma(t) = e^{it} \) for \( 0 \le t \le 2\pi \).

Answer: -2\pi^2


Question: Evaluate the integral \( \int_{0}^{2\pi} f(z_0 + re^{i\theta}) e^{ki\theta} \, d\theta \) for an entire function \( f \) and \( k \in \mathbb{N} \) with \( k \geq 1 \).

Answer: 0


Question: For a Poisson distribution with a double mode at $x=0$ and $x=1$, determine the probability that $X=1$. Express your answer as a decimal or a fraction.

Answer: \dfrac{1}{e}


Question: Determine the radius of a circle centered at \((15, -3)\) that is tangent to the curve \(y = \frac{x^2}{3}\) at a point in the first quadrant by minimizing the distance between the center of the circle and the curve.

Answer: 6\sqrt{5}


Question: Determine whether there exists at least one integer among any 10 consecutive positive integers that is coprime to the product of the remaining integers.

Answer: Yes


Question: Let $B$ be a compact linear operator on a Banach space $X$. Given that $\dim\mathcal N(1-B)=0$ and $\mathcal R(1-B)$ is closed, consider $B:=\frac{A}{\lambda}$ for $\lambda \neq 0$. If for all $y \in X$, there is at most one $x \in X$ such that $(\lambda - A)x = y$, can we guarantee that there always exists such an $x$?

Answer: Yes


Question: Determine the radius of convergence of the complex series \(\sum_{n=0}^{\infty} a_n(z-z_0)^n\) if it converges for all positive real values of \(z\).

Answer: \infty


Question: Find all complex numbers $z = a + ib$ that satisfy the equation $|3 - \sqrt{2 - |z|}| = 2$.

Answer: a^2 + b^2 = 1


Question: Let $M$ and $N$ be two closed subspaces of a Hilbert space $H$ such that $N \subset M$. Let $P_M$ and $P_N$ be the orthogonal projections onto $M$ and $N$, respectively. Determine the range of the orthogonal projection $P_M - P_N$. \( \text{Provide your answer as a subspace of } H. \)

Answer: M \ominus N


Question: Evaluate the integral \( \int_{x=-1}^1\int_{y=-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\frac{x^2+y^2}{\sqrt{1-x^2-y^2}}\,dy\,dx \) using polar coordinates.

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


Question: Identify all ideals in the polynomial ring \(\Bbb C[x]\) that are both prime and maximal.

Answer: (x - a)


Question: Determine the number of isomorphism classes of countable models of the empty theory over one binary relation. In other words, how many countable unlabeled graphs are there?

Answer: 2^{\aleph_0}


Question: Let $A$ be a unital C*-algebra and $\{b_{n}\}$ be a positive invertible sequence in $A$. If $||1_{A}-b_{n}||\rightarrow 0$, determine whether $||1_{A}-b_{n}^{-\frac{1}{2}}||\rightarrow 0$.

Answer: Yes


Question: Given a curve \(C\subset\mathbb{R}^2\), determine the number of angles \(\theta\) such that \(C\) can be parameterized as a function from the rotated \(X\) axis with angle \(\theta\).

Answer: 2


Question: Find the minimum distance between the ellipse given by \(2x^2 + y^2 = 2\) and the line \(2x + y = 2\).

Answer: 0


Question: Determine whether the sequence \( \frac{n}{\sum_{k=1}^{n}\left(\frac{k}{k+1}\right)^k} \) converges.

Answer: e


Question: Find the coefficient of \( x^{24} \) in the expansion of \( (1 - x)^{-1}(1 - x^2)^{-1}(1 - x^3)^{-1} \). Use the binomial theorem for negative exponents, \((1 - x)^{-n} = \sum_{r=0}^{\infty} \binom{n+r-1}{r} x^r\), to assist in your calculation.

Answer: 61


Question: Evaluate the surface integral \( \int_S{\bf F}\cdot{\bf\hat n}\,dS \) for the vector field \( {\bf F}=2y{\bf i}-z{\bf j}+x^2{\bf k} \), where \( S \) is the surface of the parabolic cylinder \( y^2=8x \) in the first octant, bounded by the planes \( y=4 \) and \( z=6 \). The unit normal vector \( \bf\hat n \) points in the direction of increasing \( x \). Project the integral onto the plane \( x=0 \) to find the result.

Answer: 132


Question: Evaluate the double integral \( \int \int_{x^2 + y^2 \leq 1} \frac{(x-y)^2}{x^2 + y^2} \, dx \, dy \) by converting to polar coordinates. Verify if the result is \( \pi - 2 \).

Answer: \pi


Question: Calculate the arc length of the astroid curve defined by the equation \(\sqrt[3]{x^2} + \sqrt[3]{y^2} = \sqrt[3]{9}\).

Answer: 18


Question: Evaluate the supremum of the set \( B = \left\{-\dfrac{x}{y} - q \dfrac{y}{x} : x, y \in \mathbb{R}^+ \right\} \), where \( q \) is a positive constant.

Answer: -2\sqrt{q}


Question:
Use the 4th degree Maclaurin polynomial of $\cos(x^2)$ to approximate the definite integral $\int_0^1 \cos(x^2)\,dx$. The given Maclaurin polynomial is:

$$\cos(x^2) \approx 1 - \frac{x^4}{2}$$

Calculate the approximate value of the integral using this polynomial.

Answer: \dfrac{9}{10}


Question: If $\phi:M_1\rightarrow M_2$ is a local diffeomorphism and $M_1$ is orientable, is $M_2$ necessarily orientable? Answer yes or no.

Answer: No


Question: Fifty identical six-sided dice are rolled. Determine the number of distinct outcomes where each face (1 through 6) appears an even number of times.

Answer: 142506


Question: Evaluate the contour integral \( \oint_{|z|=2} \tan{z}\,dz \). Here, \( |z|=2 \) represents a circle of radius 2 centered at the origin in the complex plane.

Answer: -4\pi i


Question: Determine the range of values for \( r \) such that the curves \( C_1 = \arg(z) = \frac{\pi}{6} \) and \( C_2 = |z - i2\sqrt{3}| = r \) intersect at exactly two points.

Answer: (3, 2\sqrt{3})


Question: For a connected topological manifold without boundary, determine if the path-connected component of the identity in the group of homeomorphisms always acts transitively on the manifold.

Answer: Yes


Question: Evaluate the limit \(\lim_{(x,y) \to (0,0)}\frac{(x^2y^3+x^3y^2)\ln(1+x^4+y^2)}{x^6+2x^4y^2+y^6}\) using polar coordinates and the inequality \(\ln(1+u) \leq u\) for \(u>-1\).

Answer: 0


Question: Evaluate the surface integral \( \iint z^2 \, dS \) over the outside of the unit sphere centered at the origin.

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


Question: Let $X$ be a Banach space and $\{ f_n \}_{n \ge 1} \in X'$. If there exists $C \ge 0$ such that for each $F \in X''$, \( \sum_{n = 1}^\infty | F(f_n) | \le C \left\Vert F \right\Vert \), does it follow that \( \sum_{n = 1}^\infty | f_n(x) | < \infty \) for all $x \in X$? Answer yes or no and justify your answer.

Answer: Yes


Question: Let $J$ be an $n \times n$ matrix with all entries equal to 1. Define $P$ as an $n \times n$ matrix given by $P = \begin{bmatrix} v_1 | v_2 | \ldots | v_{n-1} | v_n \end{bmatrix}$, where the columns $v_i = e_i - e_n$ for $1 \le i \le n-1$ and $v_n = \sum e_i$, with $e_i$ being the $i$-th column of the identity matrix. The vectors $v_i$ for $1 \le i \le n-1$ are eigenvectors of $J$ corresponding to the eigenvalue 0, and $v_n$ is an eigenvector corresponding to the eigenvalue $n$. Find $\det P$. \( \boxed{} \)

Answer: n


Question: In a metric space, if a sequence \( \{p_n\} \) has the property that for every point in a sequence \( \{q_n\} \) converging to \( q \), \( \{p_n\} \) is arbitrarily close to \( q_n \), does \( \{p_n\} \) also become arbitrarily close to \( q \)?

Answer: Yes


Question: Let $T: \mathbb{R}^2 \to \mathbb{R}^2$ be the rotation transformation of 120 degrees counterclockwise around the origin. Define $P(t) = t^7 - t^4 + t^3$. Determine if $P(T)(x,y) = (x,y)$ for every $(x,y) \in \mathbb{R}^2$. Provide a justification for your answer.

Answer: Yes


Question: Find the minimum distance between a point on the curve \( y = e^x \) and a point on the curve \( y = \ln(x) \).

Answer: \sqrt{2}


Question: Calculate the limit: $$\lim_{x\to0}\frac{\tan(\tan(\tan x))-\sin(\sin(\sin x))}{\tan(\tan x)-\sin(\sin x)}.$$

Answer: \dfrac{3}{2}


Question: In a finite-dimensional vector space $V$, let $L \subset \mathfrak{gl}(V)$ be an abelian Lie algebra consisting only of nilpotent maps. Determine whether there exists a nonzero vector $v \in V$ such that $x(v) = 0$ for all $x \in L$. Provide a proof or counterexample.

Answer: Yes


Question: Evaluate the integral \( \int_{-\infty}^{+\infty}\int_{0}^{+\infty}xe^{-x^{2}e^{y}} \Gamma \left(0,x^2\right)\, dydx. \)

Answer: 0


Question: Find the closure of the set \( M = \left\{ \frac{2}{n} \mid n \in \mathbb{N} \right\} \) in the upper limit topology generated by \( \beta = \{(-\infty, a)\} \).

Answer: [0, \infty)


Question: Calculate the limit: $$\lim_{x \to 0}\frac {\cos x- \sqrt {\cos 2x} \times \sqrt[3] {\cos 3x}}{x^2}$$ without using L'Hôpital's rule.

Answer: 2


Question: Determine the radius of convergence for the alternating series \( \sum_{n=1}^\infty (-1)^n \frac{1\cdot3\cdot5\cdot\cdot\cdot(2n-1)}{3\cdot6\cdot9\cdot\cdot\cdot(3n)}x^n \) using convergence tests.

Answer: \dfrac{3}{2}


Question: Calculate the average value of the function \( f(x, y, z) = xyz \) over the region \( W \), where \( W \) is defined by the inequalities \( 0 \le z \le y \le x \le 1 \).

Answer: \dfrac{1}{8}


Question: Determine the set of all accumulation points of the sequence of sets in \(\mathbb{R}^2\) defined by \(K_n = \left\{(x,y) \in \mathbb{R}^2: \left\|(x,y) - \left(\frac{1}{n},\frac{1}{n}\right)\right\|_\infty = \frac{1}{n}, xy \neq 0\right\}\), where \(\| \cdot \|_\infty\) denotes the infinity norm, and \(K = \bigcup\limits_{n\in\mathbb{N}} K_n\).

Answer: \{(0, 0)\}


Question: Let \( \mu \) be a complex measure on \( \mathbb{R} \) such that \( \int_{\mathbb{R}} f(x) \, d\mu(x) = 0 \) for all \( f \in C_c(\mathbb{R}) \). Prove or disprove that \( \mu = 0 \).

Answer: \mu = 0


Question: Let $f$ be a continuous function on the closure of $U$, where $U=\{ (x,y) \in \mathbb{R}^2 : x^2+y^2<1 \}$, and harmonic on $U$. Given that \(f(x,y)=x^2y^2\) on \(\partial U\), find the value of \(f(0,0)\).

Answer: \dfrac{1}{8}


Question: Calculate the number of distinct colorings of a cube using Burnside's orbit counting theorem, given that each face can be colored with one of 4 colors and rotations are considered equivalent.

Answer: 240


Question: Find the asymptote of the function \( g(x) = \frac{x^2 + f(x)}{x + 1 + 2f(x)} \), given that \( \lim_{x \to \infty} x^n f(x) = 0 \) for any \( n \in \mathbb{N} \).

Answer: y = x - 1


Question: Calculate \( \mathbb{E}\left[\left(\int_{0}^{\infty}g(t)\,dW_t\right)^2\right] \) where \( g(t) = (W_3 - W_2)\mathbf{1}_{[3,4)}(t) + (W_4 - W_2)\mathbf{1}_{[4,6)}(t) \) and \((W_t)_{t \ge 0}\) is a Brownian motion.

Answer: 5


Question: Evaluate the line integral \( \int_{L} (x+2xy)dy + y^{2}dx \) along the circle \( L=\{(x,y)|x^2+y^2=y\} \) using Green's theorem.

Answer: \dfrac{\pi}{4}


Question:
Find the length of the circumference of the unit circle \((x^2+y^2=1)\) using the limit formula for arc length:

\[L = \lim\limits_{h\rightarrow 0} \sum\limits_{r=1}^{\frac{b-a}{h}} \sqrt{\left\{ f\Big(a+rh\Big)-f\Big(a+(r-1)h\Big)\right\}^2 +h^2}\]

Answer: 2\pi


Question: Find the limit as \( n \) approaches infinity for the expression \( \lim_{n \to \infty}\frac{(n^2)!}{(2n)!} \).

Answer: \infty


Question:
For every natural number $n$, let $b_n$ be the smallest natural non-zero number such that:

$$\binom {b_n}{n-1} < \binom {b_n+1}{n}.$$

Find the value of

$$\lim_{n \to \infty} \frac {b_n}{n}. $$

Answer: 1


Question: Find the order of the element $\mathbb{Z} + \frac{s}{n}$ in the quotient group $\frac{\Bbb{Q}}{\Bbb{Z}}$, where $n$ is a natural number, $1 \leq s \leq n$, and $\gcd(s,n) = 1$. Express your answer as a single integer.

Answer: n


Question: Find the equation of the plane tangent to the surface defined by \(7 = x^2 - 3y^2 + z^2\) at the point \((1,1,3)\).

Answer: x - 3y + 3z = 7


Question: In a finite field $\mathbb{F}_p$ of prime order $p$, calculate the probability that a randomly chosen polynomial $P(x)$ of degree $d$ satisfies $P(\alpha) = \beta$, where $\alpha$ and $\beta$ are fixed non-zero elements of $\mathbb{F}_p$. Express your answer as a fraction or a decimal.

Answer: \dfrac{1}{p}


Question: Evaluate the limit \(\lim_{n \to \infty} \frac{m(m - 1)\cdots(m - n + 1)}{(n - 1)!}x^n,\) where \(x \in (-1, 1)\).

Answer: 0


Question: Consider a bounded domain $\Omega\subset\mathbb{R}^N$ and a sequence of functions $(u_n)_n\subset H^1_0(\Omega)$ such that $u_n$ weakly converges to $u\in H^1_0(\Omega)$, denoted by $u_n\rightharpoonup u$. Does this imply that the partial derivatives $\partial_i u_n$ weakly converge to $\partial_i u$ in $L^2(\Omega)$ for all $i=1,2,\dots,N$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the surface integral \( \iint\limits_{S}\mathbf{f\cdot \mathbf{n}}dS \) for the triangle \( S \) with vertices at the origin and at vectors \( \mathbf{a},\mathbf{b} \in \mathbf{R}^3 \) such that \( \mathbf{a} \times \mathbf{b} = (0, 3, 4) \). The unit normal vector \( \mathbf{n} \) points in the direction of \( \mathbf{a} \times \mathbf{b} \), and the vector field \( \mathbf{f} = (1, 2, 1) \). Find the flux of \( \mathbf{f} \) through \( S \) with respect to the orientation \( \mathbf{n} \).

Answer: 5


Question: Evaluate the limit \( \lim_{x \to 0}\left(\frac{e^{ax}-e^{bx}}{\sin(ax)-\sin(bx)}\right) \) for any constants \( a \) and \( b \) without using L'Hopital's rule.

Answer: 1


Question: Calculate the integral of $z^2$ over the circle of radius $3$ centered at the origin using a parametrization of the circle.

Answer: 0


Question: Is it possible for a matrix $A$ to have singular values that are not equal to the singular values of $A^T$?

Answer: No


Question: Evaluate the integral \( \displaystyle \int_{-1}^{0}\frac{x^2+2x}{\ln(x+1)}dx \) using substitution and logarithmic differentiation.

Answer: \ln 3


Question: Determine whether the sequence $a_n = n^k x^n$ converges for complex $x$ with $|x| < 1$, where $k$ and $n$ are positive integers.

Answer: 0


Question: Determine the dimension of the tangent space at a point where multiple circles intersect in \(\mathbb{R}^2\).

Answer: 2


Question: Determine whether $spin(3)$, the group of unit quaternions, is a homogeneous space of $SL(3, \mathbb{R})$. Provide a yes or no answer.

Answer: No


Question: Determine whether the complete graph $K_8$ can be embedded on a torus without any edges crossing.

Answer: No


Question: Use continued fractions to find a rational number that approximates \(\sqrt{11}\) to within \(10^{-4}\). Determine where to terminate the continued fraction expansion to achieve this level of accuracy.

Answer: \dfrac{199}{60}


Question: Evaluate the integral: $$\int_{-\pi}^\pi\left(\sum_{n=1}^\infty\frac{\cos(nx)}{2^n}\right)^2dx$$

Answer: \dfrac{\pi}{3}


Question: Let $A$ be a real $3 \times 3$ matrix such that $A^3 = I$ and $A \neq I$. Determine the trace of $A$. \( \text{(Express your answer as a single integer.)} \)

Answer: 0


Question: Determine the maximum value of the function \( g(x) = |e^{3ix} + 3e^{ix} + 2i|^2 \) for real numbers \( x \).

Answer: 27


Question: Find the natural number \( n \) such that \( n^5 = 27^5 + 84^5 + 110^5 + 133^5 \).

Answer: 144


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

Answer: \dfrac{\pi}{4}


Question: Evaluate the limit: $$\lim_{n\rightarrow\infty}\int_{-\infty}^{\infty}e^{-x^{2n}}\,dx$$

Answer: 2


Question: Is it possible to construct a square with three times the area of a given square using only an unmarked straightedge? Provide a yes or no answer.

Answer: No


Question: Find the positive real number $x$ that satisfies the equation $2^x + 4^x + 2^{\lfloor x \rfloor} + 4^{\lfloor x \rfloor} + 2^{x - \lfloor x \rfloor} - 4^{x - \lfloor x \rfloor} = 50 + \sqrt{50}$. \( \lfloor x \rfloor \) denotes the floor function of \( x \).

Answer: \dfrac{5}{2}


Question: Determine if the limit \( \lim_{(x,y) \rightarrow (0,0)} \frac{x^2 + y^2 - x^{3}y^{3}}{ x^2 + y^2} \) exists.

Answer: 1


Question: Evaluate the limit: $$ \lim_{x\to 0}\frac{\sin(x)\arcsin(x)-x^2}{x^6} $$

Answer: \dfrac{1}{18}


Question: Determine if there exists a function \( g \colon \mathbb{R}^n \to \mathbb{R}^m \) such that \( g(a+b) = g(a)g(b) \) for all \( a, b \in \mathbb{R}^n \), but \( g \) is not an exponential function.

Answer: Yes


Question: Determine whether there exists a complete metric space without isolated points that has a cardinality greater than the continuum.

Answer: Yes


Question: Determine the value of $x$ for which the series \( \sum_{n=1}^\infty \left(1+\frac{1}{2}+ \frac{1}{3}+\cdots+\frac{1}{n}\right) x^n \) converges.

Answer: (-1, 1)


Question: Determine whether the derivative $g'(x)$ is integrable on the interval $[0,1]$ for the function $g(x) = \sin(x^2)\chi_{(0,1]}$ where $g(x) = 0$ if $x = 0$. Provide a justification for your answer.

Answer: Yes


Question: Find the real part of $z$ when $z = \left( \cot \frac{(2k + 1)\pi}{2n} + i \right)^n$, where $n$ is a positive integer and $k$ is an integer such that $0 \leq k < n$. Use De Moivre's theorem to express $z$ in the form $\cos \phi + i \sin \phi$ before computing its $n$-th power.

Answer: 0


Question: Does a real sequence \((u_{k})_{k \geq 1}\) with infinitely many zeros necessarily change its sign infinitely many times?

Answer: No


Question: Evaluate the limit \( \lim_{t \to \infty} t^z e^{-t} \), where \( z = x + iy \) with \( x, y \) being real numbers.

Answer: 0


Question: Given the joint probability density function $f(x,y) = 2$ for $0 \le y \le x \le 1$, find the covariance between the random variables $X$ and $Y$. \( \text{Cov}(X, Y) = ? \)

Answer: \dfrac{1}{36}


Question: Determine if the generators of the ideal \( I = \langle y_1 - \psi_1, \ldots, y_m - \psi_m \rangle \) form a Gröbner basis with respect to the lexicographic order \( y_1 > y_2 > \cdots > y_m > x_1 > \cdots > x_n \), where \( \psi_1, \ldots, \psi_m \in k[x_1, \ldots, x_n] \).

Answer: Yes


Question: Determine whether for every positive integer $n$, there exists a polynomial $P$ with integer coefficients such that the set $\{P(1), P(2), \dots, P(n)\}$ consists of distinct prime numbers.

Answer: Yes


Question: Determine the number of terms required in the series representation of $e = \sum^∞_{k=0}{1\over k!}$ to ensure that the error is at most $0.06$ units in the $20$th decimal place.

Answer: 23


Question: Given that $f(x)$ is a differentiable function of $x$ satisfying the equation $f(x) \cdot f(y) = f(x) + f(y) + f(xy) - 2$ and that $f(2) = 5$, find the value of $f(3)$. \( \boxed{} \)

Answer: 10


Question: Let $X$ be a Hausdorff first countable topological space. Consider a countable decreasing local base of open neighborhoods $(U_n:n\ge 1)$ of a point $x \in X$. Does the existence of an integer $n$ such that $\overline{U_n}\subseteq U_1$ imply that $X$ is regular?

Answer: No


Question: Find the smallest possible value of the expression \(4x^2y^2 + x^2 + y^2 - 2xy + x + y + 1\) for real numbers \(x\) and \(y\).

Answer: \dfrac{1}{4}


Question:
Given the real numbers $x_1, x_2, \dots, x_{10}$ that satisfy the equation:

$$\sum_{i=1}^{10} i\sqrt{x_i - i^2} = \frac{1}{2} \sum_{i=1}^{10} x_i$$

Find the minimum value of the sum $x_1 + x_2 + x_3 + \dots + x_{10}$. \(\boxed{\text{Minimum value}}\)

Answer: 770


Question: Calculate the integral \( \int_{-\pi}^\pi \sin^4(x) \, dx \) using the Fourier series of \( \sin^2(x) \) and Parseval's identity.

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


Question: Find the minimum value of $z + d$ where $z$ and $d$ are 3-digit positive integers such that $24\text{gcd}(z, d) = \text{lcm}(z, d)$.

Answer: 374


Question: Determine the maximum number of critical points a solution of the differential equation \(y''(t) = a(t)y(t)\) can have, where \(a(t)\) is continuous and positive for all real numbers \(t\).

Answer: 1


Question: Given a prime number $p$ and an integer $c$ in the field $\mathbb{Z}_p$, determine whether there always exists a solution to the equation $a^2 + b^2 \equiv c \pmod{p}$ for some integers $a, b \in \mathbb{Z}_p$. Provide a justification for your answer.

Answer: Yes


Question: Determine the number of primitive quartic polynomials over the field \(\mathbb{Z}_5\).

Answer: 48


Question: Find the closure of the set \( A \) in \((0, \infty)\), where \( A = (\mathbb{Q} \cap (0, \infty)) \setminus \{n^{-1} \mid n \in \mathbb{N} \}\).

Answer: (0, \infty)


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

Answer: 0


Question: Let $R$ be a commutative ring with identity such that $R^{n} \simeq R^{n} \oplus M$ (as $R$-modules), where $n$ is a fixed natural number. Determine if it is true that $M = \{0\}$.

Answer: M = \{0\}


Question:
In triangle $\triangle ABC$, let $G$, $S$, and $I$ be the centroid, circumcenter, and incenter, respectively. If $R$ and $r$ are the circumradius and inradius, determine which of the following statements is incorrect:

(A) $SI^2 = R^2 (1 - \cos A \cos B \cos C)$, where $A, B, C$ are the angles of the triangle.

(B) $SI^2 = R^2 - 2Rr$.

(C) $SG^2 = R^2 - \frac{a^2 + b^2 + c^2}{9}$, where $a, b, c$ are the sides of the triangle.

(D) $SG \leq SI$.

Choose the incorrect statement.

Answer: A


Question: Determine the real number $R$ such that the flux of the vector field $F(x, y, z) = (xz - x\cos(z), -yz + y\cos(z), -4 - (x^2 + y^2))$ across the upper hemisphere $S_R$ defined by $x^2 + y^2 + z^2 = R^2$ and $z \geq 0$ can be evaluated using the divergence theorem and the lower disk on the $xy$-plane.

Answer: R


Question: Solve the differential equation: $$ \frac{dy}{dx}=\frac{x^{2}+y^{2}}{2xy} $$

Answer: x^2 - y^2 = Cx


Question:
Calculate Kendall's Tau for the random variable vector $(X_1, X_2)$, where $X_1$ follows an Exponential distribution with parameter 1 (Exp$(1)$), $X_2$ follows a standard Normal distribution (N$(0,1)$), and the dependence structure is defined by the copula:

$$ C(u_1, u_2) = \frac{1}{3}W(u_1,u_2) + \frac{2}{3}\Pi(u_1,u_2), \text{ } u \in [0,1]^2 $$

with the contra-montone copula $W(u_1,u_2) = (u_1+u_2+1)_+$ and the independence copula $\Pi(u_1,u_2) = u_1u_2$. Use the formula for Kendall's Tau:

$$ \rho_{\tau}(X_1,X_2) = 4 \int_0^1 \int_0^1 C(u_1,u_2)dC(u_1,u_2)-1 $$

Evaluate this integral to find Kendall's Tau.

Answer: -\dfrac{7}{27}


Question: Consider a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(f(x)) = x$ for all $x \in \mathbb{R}$. Does this function necessarily have a fixed point?

Answer: Yes


Question: For a random walk in $\mathbb{Z}^2$ with increments having finite support and an expectation vector on the horizontal axis, determine whether the probability of first intersecting a vertical line at a given height scales with $t$ or $\sqrt{t}$ as $t$ approaches infinity.

Answer: \sqrt{t}


Question: Find the integer part of the sum \(\sum_{i=2}^{9999} \frac{1}{\sqrt{i}}\).

Answer: 197


Question: Find the limit of the function \( f(x) = C + \sum_{n=1}^{+\infty} \frac{x^n}{(n+1)^n} \) as \( x \) approaches positive infinity.

Answer: \infty


Question: Compute the limit: \[ \lim_{n\to\infty}(n^8+2016^n)\left(1-\sqrt{1-\frac{1}{n!}}\right). \]

Answer: 0


Question: Determine the real number \( c \) such that \[ \lim_{x\to\infty} \left(\frac{x-c}{x+c}\right)^x = \frac{1}{4}. \]

Answer: \ln 2


Question: Determine whether the function \( f: G \rightarrow H \) defined by \( f(A) = (A^{-1})^T \) is a homomorphism, where \( G = GL(n, \mathbb{R}) \) and \( H = GL(n, \mathbb{R}) \). If it is a homomorphism, find the kernel of \( f \).

Answer: \{I\}


Question: Prove that the identity \( \binom{2n}{n} = \sum_{k=0}^n\binom{n}{k}^2 \) holds, using either an algebraic or combinatorial argument.

Answer: \binom{2n}{n} = \sum_{k=0}^n \binom{n}{k}^2


Question: Determine whether the following statement is true or false: If a k-regular graph is 1-factorable, then its chromatic index is k.

Answer: True


Question: In the Zariski topology on the affine line $\mathbb{A}_F^1$ over a field $F$ of characteristic zero, prove that if a closed set $A$ contains the set of integers $\mathbb{Z}$, then $A$ must be the entire space $\mathbb{A}_F^1$. Provide a justification for your conclusion.

Answer: \mathbb{A}_F^1


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

Answer: 2


Question: Let $X$ be a metric space and $f: X \to \mathbb{R}$ be a function. If $F_r = \{x \in X \mid f(x) > r \}$ is open for all $r \in \mathbb{Q}$, does it follow that $F_r$ is open for all $r \in \mathbb{R}$? Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: "If \(\sum_{n=1}^\infty \frac{a_n}{n} = a > 0\), then \(\lim_{M\to\infty} \frac{a_1+a_2+\ldots+a_M}{M} = 0\)."

Answer: True


Question: Evaluate the limit: $$\lim_{x\to 0^+} \frac{1}{x} \int_0^{x} \sin\frac{1}{t} \,\d t$$ using basic calculus techniques.

Answer: 0


Question: Evaluate the integral \( \int_0^{\infty} e^{-x} f(x) \, dx \), where the function \( f(x) \) is defined as \( f(x) = \sum_{k=0}^{\infty} \frac{(-1)^k (2015x)^{2k}}{(2k)!^2} \).

Answer: \cos(2015)


Question: Suppose $q(x)$ is a polynomial in $\mathbb{R}[x]$ such that $q(1) = 3$, $q(x) \leq q(2)$ for all $x$, and $\lim_{x \rightarrow \infty} q''(x) = -2$. Find the value of $q(3)$. \( \boxed{} \)

Answer: 3


Question: Evaluate the limit: \[ \lim_{x\to0}\left[1 + \left(\frac{\log \cos x}{\log \cos(x/2)}\right)^2 \right]^2 \]

Answer: 289


Question: Find the coefficient of $x^3$ in the expansion of $(1-2x+3x^2-4x^3)^{\frac{1}{2}}$. Use appropriate methods to handle the fractional exponent.

Answer: -1


Question: Calculate the expected value of $\exp(Y)$, where $Y$ is a binomial random variable with parameters $x$ (number of trials) and $p$ (probability of success).

Answer: (1 - p + pe)^x


Question: Find the relationship between the initial conditions \(x(0) = x_0\) and \(\dot{x}(0) = y_0\) that ensures the solution \(x(t) = Ae^{k_1t} + Be^{-k_2t}\) of the differential equation approaches zero as \(t \to \infty\).

Answer: y_0 = -k_2 x_0


Question: Demonstrate that the improper integral \( \int_0^\infty \sin^2\left[\pi\left(x + \frac{1}{x}\right)\right]dx \) diverges.

Answer: \infty


Question:
Given an open, bounded, and connected set $\Omega \subset \mathbb{R}^n$ with a $C^2$ boundary, and a function $\rho \in C^2(\mathbb{R}^n)$ such that:
$$
\Omega = \{ x \in \mathbb{R}^n : \rho(x) < 0 \}, \quad \partial \Omega = \{ x \in \mathbb{R}^n : \rho(x) = 0 \}, \quad |\nabla \rho(x)| = 1 \quad \text{on} \quad \partial \Omega,
$$
determine if there exists an $\epsilon > 0$ small enough such that the sets:
$$
\{ x \in \mathbb{R}^n : \rho(x) < \epsilon \}, \quad \{ x \in \mathbb{R}^n : \rho(x) < -\epsilon \}
$$
remain connected.

Answer: Yes


Question: Let $F$ be an infinite field such that $F^*$, the multiplicative group of $F$, is a torsion group. Determine whether $F^*$ has a proper subgroup with finite index.

Answer: No


Question: Determine the limit of the sequence \(a_n=\sqrt{1+\sqrt{5+\sqrt{11+\sqrt{19+\sqrt{29+\cdots+n^2-n-1}}}}}\) as \(n\) approaches infinity.

Answer: 2


Question: Consider a rational function $w = f(x, y, z)$ that is symmetric in $x, y, z$ and satisfies $f(x, y, z) = f(y, z, x) = f(z, x, y)$ for all positive reals $x, y, z$ such that $x + y + z = 3$. Determine if it is possible for $f(1, 1, 1)$ to be neither the maximum nor the minimum possible value of $w$. Provide a justification for your answer.

Answer: Yes


Question: Find the residue of the function \( \frac{1}{z^2 \sin(z)} \) at \( z=0 \).

Answer: \dfrac{1}{6}


Question: Evaluate \( \lim_{n\to \infty} \int_{n^2+1}^{n^2+n+1}\frac{\sin^{-1}x}{\sqrt{x}}\,dx \) without performing the integration.

Answer: 0


Question:
Evaluate the limit using a Taylor series centered at 0:

$$\lim_{x\to \infty}4x^3\left(e^{-\frac{2}{x^3}}-1\right)$$

Answer: -8


Question: Evaluate the line integral \(\int_{C} (x^2+y)dx+(x-y^2)dy\) along the parabola \(x=y^2\) from \((4,2)\) to \((1,-1)\).

Answer: -27


Question: Does $L^1(\mathbb{X})$ have the Schur property, where weak convergence implies strong convergence in a Banach space?

Answer: No


Question: Given a commutative ring with identity $R$, if there exists a natural number $n$ such that the $R$-modules $R^n$ and $R^n \oplus M$ are isomorphic, must $M$ be the trivial module $\{ 0 \}$? Provide a justification for your answer.

Answer: Yes


Question: In rectangle ABCD, an interior point P is chosen such that ∠APD + ∠BPC = 180°. Determine the value of ∠DAP + ∠BCP.

Answer: 90^\circ


Question: Determine the number of ways to distribute 40 jelly beans among ten children such that exactly three of them receive an even number of jelly beans. Use generating functions to solve the problem.

Answer: 0


Question: Determine whether an entire function with two linearly independent periods can be unbounded. Provide a yes or no answer.

Answer: No


Question: Find the range of values of $r$ for which the function $f(x) = e^{r\ln(x)}\sin\left(\frac{1}{x}\right)$ is differentiable on the interval $[0, \infty)$.

Answer: (1, \infty)


Question: Find the range of values of $m$ such that the inequality $\sqrt{x^2-1} \geq m-x$ holds for all real numbers $x$.

Answer: (-\infty, -1]


Question: In a $\mathrm{T}_1$ space, consider an open set $U$. If a finite number of points $u_0, u_1, \ldots, u_n$ are removed from $U$, is the resulting set still open?

Answer: Yes


Question: Evaluate the integral \( \int_0^5 \dfrac{v^3}{\sqrt{25-v^2}} \, dv \).

Answer: \dfrac{250}{3}


Question: Calculate the sum of the series: \[ \sum^{\infty}_{n=1} (-1)^{n+1}\frac{n-1}{(n-1)!} \]

Answer: -\dfrac{1}{e}


Question: Evaluate the definite integral \( \int_0^1 \frac{x^7-1}{\log(x)}\, \mathrm{d}x \).

Answer: \ln 8


Question: Given the joint density function \( f_{X,Y}(x,y)=6e^{-x}e^{-2y} \) for \( 0<x<y< \infty \), where \( X \) and \( Y \) are the fail times for two components of a machine, find the expected time of failure of the machine. The component \( Y \) starts working if and only if component \( X \) fails, and the machine fails when component \( Y \) fails.

Answer: \dfrac{5}{6}


Question: Given the parabola \(y^2 = 3x\), two perpendicular lines are drawn from the origin, intersecting the parabola at points \(P\) and \(Q\). Find the equation, in Cartesian form, of the locus of the centroids of all triangles \(OPQ\), where \(O\) is the origin.

Answer: y^2 = x - 2


Question: Determine whether the sequence $X_n = \frac{\omega}{n}$, where $\omega$ is a random variable uniformly distributed over $[0,1]$, converges in probability.

Answer: 0


Question: Determine the dimension of the vector space of multilinear forms \( \Delta: V^n \rightarrow K \), where \( V \) is a finite-dimensional vector space over the field \( K \).

Answer: (\dim V)^n


Question:
Consider the function $g : [0,\infty) \rightarrow \mathbb{R}$ defined as follows:

$$ g(x) = \begin{cases} 1 & \text{if } x \in \mathbb{Q} \\ \frac{1}{4^{\lfloor x \rfloor}} & \text{if } x \not\in \mathbb{Q} \end{cases} $$

where $\lfloor x \rfloor$ denotes the floor function of $x$. Determine if $g$ is Lebesgue measurable and compute the integral $\int_{[0,\infty)} g(x) \, dx$. Provide your answer as a single value or expression.

Answer: \dfrac{4}{3}


Question: Determine the number of complex numbers \( z \in \mathbb{C} \) that satisfy the equation \((2z+i\overline{z})^3=27i\) and the inequality \( \text{Re}(z) \geq \text{Im}(z) \).

Answer: 2


Question: Given the functional equation \(f(x)f(y) = f(x+y)\) and the condition \(f(0) = 1\), find the derivative \(f'(0)\) if \(f\) is differentiable and continuous at \(0\), and there exists a function \(g(x)\) such that \(f(x) = 1 + \sin(3x)g(x)\).

Answer: 0


Question: Find the residue of the function \( z^2 \sin\left(\frac{1}{z}\right) \) at \( z = 0 \).

Answer: -\dfrac{1}{6}


Question:
Let $X$ be a discrete random variable with the following probability mass function (PMF):

$$P\{X=z\} = \left\{ \begin{array}{ll}
\frac{\lambda^{|z|}e^{-\lambda}}{|z|!\cdot 2} & \text{if } z \neq 0 \text{ and } z \text{ is an integer} \\
e^{-\lambda} & \text{if } z = 0
\end{array}\right.$$

where $\lambda > 0$. Compute the expectation of $X$, denoted as $E[X]$. \(\lambda\) is a positive parameter.

Answer: 0


Question: Determine whether the statement is true or false: The tensor product of any finite non-abelian group with a rational number is undefined.

Answer: A


Question: Solve the equation \( \frac{x^2}{2^x} = 2 \) algebraically.

Answer: -1


Question: Evaluate the limit \( L = \lim_{x \rightarrow \infty} \left(\sqrt[11]{\prod_{n=1}^{11}\left(2x+\frac{n(2n+1)}{2}\right)}-2x\right) \).

Answer: 49


Question: Determine the norm \( ||A|| \) of an operator \( A \) that satisfies \( AA = I \) in a Hilbert space \( H \), where the norm is defined as \( ||A|| = \sup_{x \in H} \frac{|Ax|}{|x|} \).

Answer: 1


Question: In some well-orderings of the real numbers, if $a$ is the $i$-th smallest number and $b$ is the $j$-th smallest number, is it true that $a+b$ is the $(i+j)$-th smallest number? Answer yes or no.

Answer: No


Question: Is it possible to reduce the joint entropy \( H(X,Y) \) while keeping the individual entropies \( H(X) \) and \( H(Y) \) constant?

Answer: Yes


Question: If $a, b, c$ are the roots of the polynomial $x^3 - px + q = 0$, find the trace of the matrix $$ \begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{pmatrix}. $$

Answer: 0


Question: Determine whether the polynomial $x^3 + x + c$ has exactly one real root for any real number $c$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the contour integral \( \int_C(z^2+4iz)\sin\left(\frac{3}{z+2i}\right)\,dz \) where \(C\) is the circle \(|z| = 3\) oriented in the counterclockwise direction.

Answer: 15\pi i


Question: Calculate the limit: \[ \lim_{{x \to 0}} \frac{\sqrt{x^3+2x+1}-\sqrt{x^2-3x+1}}{\sqrt{4x^2-3x+1} - \sqrt{2x^3+6x^2+5x+1}} \]

Answer: -\dfrac{5}{8}


Question: Determine whether a continuous, bijective function can have a discontinuous inverse function.

Answer: Yes


Question: Calculate the remainder when $1234^{567}$ is divided by $45$. Use the Chinese Remainder Theorem and Fermat's Little Theorem to find the solution.

Answer: 19


Question: Calculate the curvature tensor of the standard Euclidean metric in 2-dimensions using the Christoffel symbols.

Answer: 0


Question: What is the value of the ceiling function applied to an infinitesimal number?

Answer: 1


Question: Evaluate the limit: $$\lim_{n \rightarrow \infty} \int_0 ^{\infty} \left(1+ \frac{x}{n}\right)^n e^{-nx} dx.$$

Answer: 0


Question: Evaluate the line integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \), where \( C \) is the circle \( x^2+y^2=4 \) in the \( xy \)-plane centered at the origin, and \( \mathbf{F} = (x-z)\mathbf{i} + (x^3+yz)\mathbf{j} - (3xy^2)\mathbf{k} \).

Answer: 12\pi


Question: Determine the maximum length of a repeating loop that can be obtained by repeatedly applying the function \( f(x) \) to any natural number \( x \).

Answer: 8


Question: Determine whether the infinite multiplication of rank-one matrices always results in a rank-one matrix. Answer with 'yes' or 'no' and justify your reasoning.

Answer: no


Question: Determine if the number $87180$ can be expressed as the sum of two triangular numbers without explicitly finding the values of the triangular numbers.

Answer: Yes


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

Answer: \sqrt{e}


Question: If $M$ is a martingale with respect to the filtration $(F_t)$, is the stopped process $M^T$ also a martingale with respect to the filtration $(F_{t \wedge T})$?

Answer: Yes


Question: Determine the number of ways to distribute 20 identical blue balls and 15 identical green balls into 5 distinct boxes such that each box contains more blue balls than green balls.

Answer: 3876


Question: Determine whether the following statement is true or false: If $G$ has a composition series and $N$ is a normal subgroup of $G$, then the factor group $(G_iN)/G_{i+1}N$ is isomorphic to $G_i/G_{i+1}$.

Answer: False


Question: Determine the radius of convergence of the infinite series \( \sum_{n=1}^\infty \frac{(2n)!}{(n!)^2}z^n \).

Answer: \dfrac{1}{4}


Question: Evaluate the integral \( \oint_C \frac{\sinh(z)\cos(z)}{\cos(z)}dz \) using the Residue Theorem, where \( C \) is the contour defined by \( |z|=(n+1/2)\pi \) with \( n \in \mathbb{N}_0 \).

Answer: 0


Question:
Calculate the sum of the series:

$$\sum_{n=2}^{\infty}\frac{3n-5}{n(n^2-1)}.$$

Answer: 1


Question: Find the correlation coefficient between the random variables $X$ and $Y$ given the joint distribution: $$f(x, y) = \begin{cases} 2,& 0 < x \leq y < 1,\\ 0,&\text{otherwise} \end{cases}.$$

Answer: \dfrac{1}{2}


Question: Consider a sequence consisting of $2n$ consecutive natural numbers, all of which are less than $n^2 + 2n$. Determine whether there exists a number in this sequence that is not divisible by any prime less than or equal to $n$.

Answer: Yes


Question: In a probability space \((\Omega, \mathcal{A}, P)\), consider a sequence of identically independently distributed (i.i.d.) random variables \((X_n)_{n \in \mathbb{N}}\), which are not integrable. Define \(C_n = \{|X_n| \geq nK\}\) for each \(n \in \mathbb{N}\), where \(K > 0\). Determine whether it is true that \(P\left(\limsup_{n \to \infty} C_n \right) = 1\).

Answer: 1


Question: Calculate the norm of the operator $A: C[-\pi/2,\pi/2] \to C[-\pi/2,\pi/2]$ defined by $Ax(t) = \cos(t)x(t)$. Provide your answer as a single value.

Answer: 1


Question: Determine whether every open set in the standard topology on \( \mathbb{R} \) is also an open set in the \( K \)-topology on \( \mathbb{R} \).

Answer: Yes


Question: Find the quadratic function $f(x)$ such that $f(0) = 1$ and the integral \( \int \frac{f(x)}{x^2(x + 1)^3} \, dx \) is a rational function.

Answer: 2x^2 + 3x + 1


Question: Evaluate the limit \(\lim_{\substack{n\to\infty\\x\to\infty}}\prod_{k=0}^{n-1}\left(1+\frac{k}{x}\right)\) and express your answer in its exact form.

Answer: \infty


Question:
Determine the values of $p \in [1, \infty]$ for which the function \( u(x) \) belongs to the Sobolev space \( W^{1,p}(\Omega) \), where \( \Omega = \{(x_1, x_2) \in \mathbb{R}^2 \mid |x_1| < 1, |x_2| < 1\} \) and \( u(x) \) is defined as:

\[
 u(x) =
 \begin{cases}
 1 - x_1, & \text{if } |x_2| < x_1, \\
 1 + x_1, & \text{if } |x_2| < -x_1, \\
 1 - x_2, & \text{if } |x_1| < x_2, \\
 1 + x_2, & \text{if } |x_1| < -x_2.
 \end{cases}
\]

Is the observation that the \( W^{1,p} \) norm is finite for all \( p \) correct?

Answer: [1, \infty]


Question: Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an entire function. If $f(\mathbb{C}) \cap [a, b] = \emptyset$ for some interval $[a, b] \subseteq \mathbb{R}$, is $f$ necessarily constant?

Answer: Yes


Question:
Find the best asymptotic expansion for \(x \to 0\) of the expression:

\[ \sqrt{1-2x+x^2+o(x^3)} \]

Answer: 1 - x + o(x^3)


Question: For any matrix $A$ that can be written in block diagonal form, is the induced 2-norm of $A$ equal to the maximum induced 2-norm of its diagonal blocks? Answer with 'Yes' or 'No'.

Answer: Yes


Question:
Evaluate the expression \(4\sum^{30}_{n=1} n \cdot T(n)\), where \(T(n)\) is defined as:

\[T(n) = \cos^2(30^\circ - n) - \cos(30^\circ - n)\cos(30^\circ + n) + \cos^2(30^\circ + n)\]

Compute the sum using trigonometric identities.

Answer: 1395


Question:
Determine whether there exists a differentiable function \( g : \Bbb{R} \to \Bbb{R} \) that satisfies the following conditions:

1. \( g'(x) = g(g(x)) \) for all \( x \in \Bbb{R} \).
2. \( g \) is not strictly increasing.
3. \( g \) is not surjective.

Answer: Yes


Question: Determine the values of $p$ for which the integral \( \int_{0}^{1} \frac{1}{\sqrt{y^{-p}-1}} \, dy \) converges.

Answer: p > 0


Question: Evaluate the definite integral: $$\int_{1}^{3} \frac{2x^3}{1+e^{-x}}\mathrm dx$$

Answer: 20


Question: For a finite-dimensional vector space $U$ over a field $\mathbb{F}$, there is a bijection between subspaces of dimension $m$ and subspaces of dimension $n-m$. Determine if this property can be generalized to any field $\mathbb{F}$, assuming $\dim U$ is finite.

Answer: Yes


Question: Find the maximum value of the function \( f(x,y) = 4x - 9y \) subject to the constraint \( x^2 + 9y^2 - 4x + 6y + 4 = 0 \).

Answer: 16


Question: Evaluate the triple integral \( \int_T xyz \, dV \) over the tetrahedron \( T \) with vertices \( (0,0,0), (1,0,0), (1,1,0), (1,0,1) \).

Answer: \dfrac{1}{144}


Question: If $S$ is a 5-dimensional subspace of $\mathbb{R}^6$, can every basis of $\mathbb{R}^6$ be reduced to a basis of $S$ by removing a single vector? Answer yes or no.

Answer: No


Question: Let $P$ be an invertible $n \times n$ matrix. Consider the vector space $L$ consisting of $n \times n$ matrices $X$ such that $\operatorname{tr}(PX) = 0$. Determine the dimension of the vector space $L$. Express your answer as a single integer.

Answer: n^2 - 1


Question: Evaluate the limit: \[ \lim_{x\to0} \frac{\cos(x^{20})-\cos(x^{10})}{\sin^{10}(2x)}. \]

Answer: 0


Question: In a Hilbert space with the norm defined as \( ||\cdot||=\sqrt{(\cdot,\cdot)} \), suppose the inequality \( ||x+\alpha y|| \ge ||x|| \) holds for all \( \alpha \in \mathbb{C} \). Prove that this implies \( (x,y) = 0 \).

Answer: (x, y) = 0


Question: Determine whether every positive integer can be expressed as a sum of distinct unit fractions.

Answer: Yes


Question: Determine the maximum value of $n$ such that there exists a sequence $a_1, a_2, \ldots, a_n$ of positive integers satisfying the condition that for every $2 \leq k \leq n$, the expression \( \sqrt[k]{a_1^k + a_2^k + \cdots + a_k^k} \) is an integer. If no such maximum exists, prove the existence of an infinite sequence with this property.

Answer: 3


Question: Find all pairs of natural numbers \(a\) and \(b\) such that \(\gcd(a, b) = 1\) and \(\frac{a}{b} = b.a\), where \(b.a\) represents a decimal number. For example, \(\frac{5}{2} = 2.5\).

Answer: (5, 2)


Question: Given the solutions \( t^3 \) and \( t e^t \sin(t) \) to an nth-order homogeneous differential equation with constant coefficients, determine a restriction on \( n \).

Answer: n \geq 8


Question: In a non-atomic measure space, determine if it is possible to split a measurable set into two disjoint parts of equal measure.

Answer: Yes


Question: In a right-angled triangle $ABC$ with $\angle A = 90^{\circ}$, it is given that $AB = AC = 2AD$. The line $AE$ is perpendicular to $BD$, and the side lengths are denoted by $a$. Let point $O$ be the intersection of $BD$ and $AE$. Given that $AO = \frac{a}{\sqrt 5}$, determine the ratio $BE : CE$. Express your answer as a simplified ratio.

Answer: 2:1


Question: Find all positive integers $n$ such that the sum of the squares of the divisors of $n$, denoted as $\sigma_2(n)$, equals $n^2 + 2n + 37$, and $n$ is not of the form $p(p+6)$ where $p$ and $p+6$ are prime numbers.

Answer: 27


Question: Find the degree of the field extension \( \mathbb{Q}(\sqrt{c-\sqrt{d}}) \) over \( \mathbb{Q} \), where \( c \) and \( d \) are distinct positive integers.

Answer: 4


Question: Given that $f(x)$ is a function with its first derivative $f'(x) = f^2(x) + x$, determine how many times $f(x)$ is differentiable, assuming it is differentiable at least once.

Answer: \infty


Question: Find the general solution of the differential equation \(xy^{''}+y^{'}+xy=0\) using the power series method.

Answer: y = C_1 J_0(x) + C_2 Y_0(x)


Question: Evaluate the limit: $$\lim_{x\to\infty} (2x+1)^{x-\sqrt{x^2-2}}$$ Is the limit equal to $1$ or $e^2$?

Answer: 1


Question:
Given that $a,b,c,d$ are solutions to the equation $x^4-bx^3-3=0$, determine the form of the equation whose solutions are $\frac{a+b+c}{d^2},\frac{a+b+d}{c^2},\frac{a+c+d}{b^2},\frac{b+c+d}{a^2}$. Choose from the following options:

A) $x^4-3bx^3+1=0$

B) $3x^4-bx^3-1=0$

C) $3x^4+bx^3-1=0$

D) $3x^4-bx+1=0$

E) $3x^4+bx+1=0$

F) There's no right answer

Answer: F


Question: Let $\mathcal{O}$ be a Dedekind domain, and consider two ideals $I=(x_1,\ldots, x_n)$ and $J=(y_1,\ldots, y_m)$ within $\mathcal{O}$. Determine whether the product ideal $IJ$ is always equal to the ideal $K$ generated by the products $x_iy_j$. Provide a proof or counterexample.

Answer: IJ = K


Question: Given that for any prime factors $p_i$ and $q_i$, if $p_i$ divides $q_i$ ($p_i | q_i$), then $p_i$ does not divide any other $q_j$ where $j \neq i$. Also, if $p_i$ does not divide $q_i$, then it is false that $p_i$ divides some $q_j$ where $j \neq i$. Does this imply that $p_i$ never divides any $q_j$ when $j \neq i$, regardless of the scenario? Answer yes or no.

Answer: Yes


Question:
Let $p(x) = a_0 + a_1x + \cdots + a_nx^n$ be the characteristic polynomial of an $n \times n$ matrix $A$ with entries in $\mathbb{R}$. Determine which of the following statements is true:

(a) $p(x)$ has no repeated roots.

(b) $p(x)$ can be expressed as a product of linear polynomials with real coefficients.

(c) If $p(x)$ can be expressed as a product of linear polynomials with real coefficients, then there is a basis of $\mathbb{R}^n$ consisting of eigenvectors of $A$.

Choose the correct statement.

Answer: C


Question: Determine the angle of rotation for the rotation matrix $$ A=\left[\begin{array}{ccc} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right] $$ given its diagonal form in a new basis $(x', y', z')$: $$ D=\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & -\left(\frac{1+\sqrt{3} i}{2}\right) & 0 \\ 0 & 0 & \left(\frac{-1+\sqrt{3} i}{2}\right) \end{array}\right] $$ and the rotation axis $\hat{n}=\frac{\hat{x}+\hat{y}+\hat{z}}{\sqrt{3}}$. Use the transformation rules: \begin{aligned} x &=\frac{1}{\sqrt{3}}\left(x^{\prime}+\left(\frac{-1+\sqrt{3} i}{2}\right) y^{\prime}+\left(\frac{-1-\sqrt{3} i}{2}\right) z^{\prime}\right) \\ y &=\frac{1}{\sqrt{3}}\left(x^{\prime}+\left(\frac{-1-\sqrt{3} i}{2}\right) y^{\prime}+\left(\frac{-1+\sqrt{3} i}{2}\right) z^{\prime}\right) \\ z &=\frac{1}{\sqrt{3}}\left(x^{\prime}+y^{\prime}+z^{\prime}\right) \end{aligned}

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


Question: Let $B$ be a $(n-1) \times (n-1)$ matrix such that all elements on the diagonal equal $0$ and all other elements are either $1$ or $-1$. Let $A = \begin{bmatrix}B&(1,...,1)^T\\ (1,...,1)&1\end{bmatrix}$, so $A$ is an $n \times n$ matrix derived from $B$ by adding a row and column with $1$. Determine the rank of the matrix $A$.

Answer: n


Question: Determine whether every non-zero Lie algebra over an algebraically closed field has an ad-nilpotent element.

Answer: Yes


Question: Evaluate the limit: $$\lim_{n \to \infty} n\left(\frac{{\pi}^2}{6} - \sum_{k=1}^{n} \frac{1}{k^2} \right).$$

Answer: 1


Question: Let $(b_n)_{n \ge 1}$ be a sequence of real numbers such that the sequence $(b_{n+1}+b_n)_{n \ge 1}$ converges to a non-zero real number. Determine the value of the limit: $$ \lim_{n \to \infty} \left( \frac{b_{n+1}}{b_n} \right)^{n^2}. $$

Answer: 1


Question: Calculate the integral \( \oint_{c(0,2)} \frac{z^{3}-2}{z(z^{2}-1)}\,\mathrm{d}z \) over the circle \( c(0,2) \) with the parametrization \( t \rightarrow 0 + 2\mathrm{e}^{\mathrm{i}t} \). Determine the poles of the integrand that lie within the circle.

Answer: 0


Question: Calculate the length of the module $A = \mathbb{R}[X] / (X^3 - X^2 + X -1)$ as an $\mathbb{R}[X]$-module.

Answer: 2


Question: Find the orthogonal trajectories of the family of curves given by the equation \(x^2 + 2y^2 = k^2\).

Answer: y = Cx^2


Question: Determine whether angles that are not multiples of 15° can be constructed using only a compass and a straightedge. Provide a justification for your answer.

Answer: Yes


Question: Simplify the algebraic fraction: $$\frac{(2^4+2^2+1)(4^4+4^2+1)(6^4+6^2+1)(8^4+8^2+1)(10^4+10^2+1)}{(3^4+3^2+1)(5^4+5^2+1)(7^4+7^2+1)(9^4+9^2+1)(11^4+11^2+1)}$$

Answer: \dfrac{3}{133}


Question: Find the limit of the iterated sine function as the number of iterations approaches infinity: $$\lim_{n \to \infty} \underbrace{\sin(\sin(\cdots\sin(x)))}_{n \text{ times}}, \quad x \in \mathbb{R}$$

Answer: 0


Question: Evaluate the integral: \[ \int_{0}^{\pi/4} \frac{\ln(1+\tan(x))}{\cos^2(x)} \, \mathrm{d}x \]

Answer: 2 \ln 2 - 1


Question: Determine the values of $p$ for which the improper integral \( \int_1^\infty \frac{1}{x^p(1+\ln x)}\, dx \) converges.

Answer: p > 1


Question: If $X$ and $Y$ are subspaces of $Z$, $X \cong Y$, and $X$ is a retract of $Z$, is $Y$ also a retract of $Z$? Provide a justification for your answer.

Answer: No


Question: Determine the rank of the $n \times n$ matrix $A_n$, where the $(i, j)$ entry is given by $i - j$ for $n \geq 2$. Express the rank as a function of $n$.

Answer: 2


Question:
Given a Markov jump process for a gas station with a single gas pump, cars arrive at a rate of 20 vehicles per hour. Each car takes 6 minutes (1/10 hour) to fill up. The system states are 0, 1, or 2 cars at the pump. The transition rate matrix is:
$$
Q = \begin{bmatrix}
-20 & 20 & 0 \\
10 & -30 & 20 \\
0 & 10 & -10
\end{bmatrix}
$$
The stationary distribution is $(u_1, u_2, u_3) = \left(\frac{1}{7}, \frac{2}{7}, \frac{4}{7}\right)$. Calculate the average number of cars served by the gas station per hour.

Answer: \dfrac{60}{7}


Question: Compute the line integral \( \int_{C}{x\,\mathrm{d}z - z\,\mathrm{d}y + y\,\mathrm{d}z} \) where \( C \) is the path in \( \mathbb{R}^3 \) given by the parametric equation \( \gamma(t) = (\cos t, \sin t, 2t) \) for \( t \in [0, 2\pi] \).

Answer: 0


Question: What is the term for a lattice that is isomorphic to its dual lattice?

Answer: self-dual lattice


Question: Let $V$ be a finite-dimensional inner product space with orthonormal basis $\{v_1,\cdots,v_n\}$. If $\{u_1,\cdots,u_n\} \subset V$ satisfy $\sum_{j=1}^n||u_j||^2>1$, can $\{v_1+u_1,\cdots,v_n+u_n\}$ still form a basis for $V$? Provide a justification for your answer.

Answer: Yes


Question: Find an upper bound for $l_n$, the length of the curve $y=x^n$ in the interval $[0,1] \times [0,1]$, using the function $f_n(x) = \sqrt{1+n^2(1-x)^{2n-2}}$ and the approximation provided in the original answer.

Answer: 2


Question: Determine the number of continuous functions $g(x)$ defined on the interval $[-1, 1]$ that satisfy the following integral equations: \[ \int_{-1}^1 g(x) \, dx = 0, \] \[ \int_{-1}^1 x^2 g(x) \, dx = b, \] \[ \int_{-1}^1 x^4 g(x) \, dx = b^2. \]

Answer: \infty


Question: Find the determinant of a 4x4 matrix \( M \) such that \( M^2 = -I_4 \), where \( I_4 \) is the 4x4 identity matrix.

Answer: 1


Question: Determine the values of $a$ for which the function \( f(x) = \begin{cases} x^a & \text{if } x \geq 0 \\ 0 & \text{if } x < 0 \end{cases} \) is three times differentiable at $x = 0$. Provide your answer as a set of values for $a$.

Answer: (3, \infty)


Question: Evaluate the limit: \[\lim_{n \to \infty}\left( \frac{1^{a+1}+2^{a+1}+3^{a+1}+...+n^{a+1}}{n^{a+1}}-\frac{n}{a+2} \right)\] where \(a \in \mathbb{R}\) and \(a > 0\).

Answer: \dfrac{1}{2}


Question: Find the remainder when \((205 - 15^{13})^{144}\) is divided by 962.

Answer: 222


Question:
Given the system of equations:
\[ ax + by = 7 \]
\[ ax^2 + by^2 = 49 \]
\[ ax^3 + by^3 = 133 \]
\[ ax^4 + by^4 = 406 \]
find the value of \(2023(x+y+xy) + 50(a+b)\).

Answer: 3073


Question: Find the minimum value of the norm of the matrix $P = a \cdot b^*$, where $a$ and $b$ are column vectors in $\mathbb{C}^n$ and their inner product $b^* \cdot a = 1$. Express your answer in terms of $a$ and $b$.

Answer: 1


Question: Evaluate the improper integral \( I = \int^{\infty}_0 (y^{\frac{1}{2}}e^{-\frac{1}{2}y})dy \) using the fact that \( \int^{\infty}_0 (e^{-x^2})dx = \frac{\sqrt{\pi}}{2} \).

Answer: \sqrt{2\pi}


Question: Determine the number of terms required in the series \( e = \sum_{k=0}^{\infty} \frac{1}{k!} \) to approximate \( e \) with an error of at most \( \frac{6}{10} \) unit in the 20th decimal place.

Answer: 22


Question: Find the imaginary part of the expression \( \frac{z'(t)}{z(t)} \) using the polar coordinate representation of \( z(t) \).

Answer: \theta'(t)


Question: Let \( p(x) \) be a polynomial of degree strictly less than 100 such that it does not have \( x^3-x \) as a factor. If \( \frac{d^{100}}{d x^{100}} \frac{p(x)}{x^3-x} =\frac{f(x)}{g(x)} \) for some polynomials \( f(x) \) and \( g(x) \), find the smallest possible degree of \( f(x) \).

Answer: 200


Question: Determine the number of partial orders on a set of 4 elements, given that the number of partial orders on a set of 3 elements is 19.

Answer: 219


Question: Evaluate the limit \( \lim\limits_{x\rightarrow-\infty}e^{2x}\int f(x)e^{-x}dx \) given that \( \lim\limits_{x\rightarrow-\infty}f(x)=F \).

Answer: 0


Question: Determine the convergence and find the limit of the sequence \( b_n = \sqrt[n]{n^2 \cdot 3^{4n}+4^{3n}} \).

Answer: 81


Question: Find the other point of intersection of the line normal to the curve \( x^2 + xy - 2y^2 = 0 \) at the point \((4,4)\) with the curve.

Answer: (16, -8)


Question: If a sequence of functions \( \{f_n\} \) converges to \( f \) uniformly on \( \mathbb{R} \), does \( (f_n)^2 \) converge to \( f^2 \) uniformly on \( \mathbb{R} \)?

Answer: No


Question: Let $X$ be a topological space where each connected component is open. Consider the sheaf $\mathcal{O}_X$ associated with the constant presheaf $\mathbb{Z}$. For an open subset $U$ of $X$, determine if $\mathcal{O}_X(U) \cong \mathbb{Z}^i$, where $i$ is the number of connected components of $U$. Is this statement true or false?

Answer: True


Question: Does Proposition 5.47 in John Lee's "Introduction to Smooth Manifolds" (p. 121) remain valid if the manifold $M$ is a smooth manifold with boundary?

Answer: Yes


Question: Determine if the formula \( \mathcal{L}[f](s) = e^{-A\sqrt{s}} \) is valid for complex values \( s \in \mathbb{C} \) with \( \operatorname{Re}(s) \geq 0 \), using the principal branch of the square root function (\( \sqrt{s} \) with positive real part).

Answer: Yes


Question: Given that \(\sqrt{2}+\sqrt{3}+\sqrt{6}\) is a root of the polynomial \(x^4+ax^3+bx^2+cx+d=0\), where \(a, b, c, d\in\mathbb{Z}\), determine the value of \(|a+b+c+d|\).

Answer: 93


Question: Determine the number of 25th power residues modulo 343, given that \(\gcd(25, 342) \neq 1\).

Answer: 294


Question: Evaluate the integral \( \frac{1}{2\pi} \int_{0}^{2\pi} e^{\cos\theta} \cos(n\theta) \,d\theta \) using the residue theorem.

Answer: I_n(1)


Question: In the context of weak convergence of probability measures on $\mathbb{R}$, consider a sequence $(\mu_n)_n$ of probability measures. For each $k \in \mathbb{N}$, there exists a subsequence $(\nu^{(k)}_n)_n$ that converges weakly to a probability measure $\nu^{(k)}$, where $\nu^{(k)}_n = \mu_n \ast \mathcal{N}(0, k^{-1})$. Does this imply that a subsequence of the original sequence $(\mu_n)_n$ converges weakly to some probability measure $\mu$?

Answer: Yes


Question: Let \(f\) be a holomorphic function on the unit disk \(\mathbb{D}\). Given that the diameter of \(f(\mathbb{D})\) is \(M\), find an upper bound for \(|f''(0)|\).

Answer: M


Question: Evaluate the infinite series \(\sum_{n=0}^\infty \frac{n!}{2^n\prod_{i=1}^n(1+\frac{i}{2})}\) by rewriting it as an alternate sum of the form \(\sum_{n=1}^\infty a_{n+1}-a_n\). Find a formula for \(a_n\) that allows this transformation.

Answer: 2


Question: The numbers \(\frac{1}{3}\), \(\frac{\ln(y)}{3\ln(x)}\), \(\frac{\ln(z)}{3\ln(y)}\), \(\frac{\ln(x)}{7\ln(z)}\) are in harmonic progression. If \(y = x^r\) and \(z = x^s\), find the value of \(4(r+s)\).

Answer: 6


Question: Suppose that $f:\Bbb R\to\Bbb R$ is a differentiable function such that $\lim_{x\to-\infty}f'(x)=-2$. Is the statement "$f$ is unbounded on the interval $(-\infty, \infty)$" true or false?

Answer: A


Question: Let $\{f_n\}_{n=0}^{\infty}$ be a sequence of holomorphic functions defined on a domain containing the unit disk. Suppose that $\lim_{n\rightarrow \infty}f_n(z)=0$ uniformly on the boundary of the unit disk. Determine whether $\lim_{n\rightarrow \infty}f_n(z)=0$ uniformly on the whole unit disk.

Answer: Yes


Question: Determine the minimum distance between the curves \( y = x^3 - 2 \) and \( y = 3x^3 \).

Answer: 0


Question: In a ring $R$, if $p \cdot c = p$ for all nonzero elements $p, c \in R$, must it be true that $c = 1$? Assume that $R$ is not necessarily an integral domain.

Answer: Yes


Question: Evaluate the integral \( \int^a_0 \frac{dx}{x + \sqrt{a^2 - x^2}} \).

Answer: \dfrac{\pi}{4}


Question:
Consider $\mathbb{R}^2$ with the $\|\;.\,\|_1$-norm and let $M=\{(x,0) \mid x \in \mathbb{R}\}$. Define $g:M \to \mathbb{R}$ by $g(x,0)=x$. Determine which of the following functions is not a Hahn-Banach extension of $g$:

(a) $f(x,y)=2x$
(b) $f(x,y)=x+y$
(c) $f(x,y)=x-2y$
(d) $f(x,y)=x+2y$

Answer: A


Question: Find all pairs of positive integers \((x, y)\) such that \(1 + 4^x = 5^y\).

Answer: (1, 1)


Question: Find the inverse function of the function defined by the equation \( 2y(x) + y(1/x) = 4x + \frac{2}{x} \).

Answer: \dfrac{x}{2}


Question:
Evaluate the integral \( \int_{[0,1]^3} \max (x,y,z)\,dx\,dy\,dz \) using the hint:

$$\max(x,y,z)=x\cdot (\boldsymbol 1_{y\leq x\leq z }+\boldsymbol 1_{z\leq x\leq y})+y\cdot (\boldsymbol 1_{x\leq y\leq z }+\boldsymbol 1_{z\leq y\leq x})+z\cdot (\boldsymbol 1_{x\leq z\leq y }+\boldsymbol 1_{y\leq z\leq x}).$$

Answer: \dfrac{3}{4}


Question: Let $f$ be a function defined on the real numbers. Suppose that for any sequence of rational numbers converging to 2, the sequence of function values converges to 5. Similarly, for any sequence of irrational numbers converging to 2, the sequence of function values converges to 5. Can we conclude that for any sequence of real numbers converging to 2, the sequence of function values converges to 5?

Answer: Yes


Question: Evaluate the limit: $$\lim_{n \to \infty} \frac{\frac{\log (n+2)}{(n+2)} + \frac{\log (n+3)}{(n+3)} + \cdots + \frac{\log (2n)}{2n}}{\log^2(2n)}$$

Answer: 0


Question: Determine whether every orthogonal transformation over \( \mathbb{R} \) is diagonalizable. Answer with true or false.

Answer: False


Question: Consider the function \(f(z)=\frac{z^{2}+1}{z^{10}-2}\). Show that for its Taylor series around \(z_0 = 0\), the remainder term satisfies \(|R_{n}(z)|<\frac{1}{2^{n}}\) for all \(|z|<\frac{1}{2}\).

Answer: |R_{n}(z)| < \frac{1}{2^{n}}


Question: Evaluate the limit: $$\lim_{n \to \infty} \left(\frac{ \sqrt{n^4+1} - \sqrt{n^4-1}}{ \frac1{(2n+1)^2}} \right)$$

Answer: 4


Question: Evaluate the sum \(\sum_{n=2}^\infty \frac{1}{n(n-1)(n+1)}\) using the integral representation of the zeta function.

Answer: \dfrac{1}{4}


Question: Determine whether it is possible for a metric space to have a subset that is totally bounded while the entire space itself is not.

Answer: Yes


Question: Calculate the limit of the expression \( f(x) = \lim \limits_{x \to 0} \left(\frac{\sin x}{x}\right)^{\frac{1}{x}} \) without using Taylor's theorem.

Answer: 1


Question: Consider a random vector $(x_1, \ldots, x_k)$ that follows a multinomial distribution with parameters $n$ and $(p_1, \ldots, p_k)$. Given that $p_i > p_j$ for some $i, j$, determine if the probability of the event $x_i > x_j$ approaches 1 as the sample size $n$ tends to infinity.

Answer: Yes


Question: Given a complex matrix $A$, if $\operatorname{tr}(A^*A - AA^*) = 0$, can we conclude that $A$ is a normal matrix?

Answer: No


Question: Find the z-coordinate of the center of mass of a cone with constant mass density, where the cone is defined as \(C:= \{ (x,y,z) \in \mathbb{R^3}: \sqrt{x^2+y^2}\leq z \leq 1\}\).

Answer: \dfrac{3}{4}


Question: Given that $P(A_n^c) \rightarrow 1$ and $\sum_{n=1}^{\infty}{P(A_n\cap A_{n+1}^c)}<\infty$, determine if $P(A_n \text{ i.o.})=0$.

Answer: 0


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

Answer: \dfrac{4}{e}


Question: Find the $p$-adic valuation of the product $(1-	heta)(1-	heta^2)\cdots(1-	heta^{p-1})$, where $\theta$ is a primitive $p$th root of unity.

Answer: 1


Question: Determine whether the Borel $\sigma$-algebra for the interval $(0,1]$ contains the set of all irrational numbers within that interval.

Answer: Yes


Question: Determine the order of convergence of the sequence \(\{y_n\}_{n=0}^{\infty }\) defined by \(y_{n+1}=3y_n-2 {y_n}^3\) with \(y_0=1/2\), given that the limit of the sequence is \(1\).

Answer: 1


Question: Evaluate the sum \( \sum_{k=0}^{n} k \binom{n}{k} (-1)^{k-1} 2^{n-k} \).

Answer: n


Question: Find the order of the group generated by elements $x$ and $y$ satisfying the relations $x^2y = 1$ and $xy^3 = 1$. Express your answer as a single integer.

Answer: 5


Question: Find the limit of \( \displaystyle{ \lim \limits_{n \to \infty} \sqrt[2^n]{\sqrt[n]{b}+1}} \) given \(b>0\).

Answer: 1


Question: Find the greatest integer not exceeding the sum of the cube roots of the reciprocals of the first 1000 positive integers.

Answer: 149


Question: Evaluate the Cauchy principal value of the integral \( \int_{1}^{3} \frac{1}{x-2} \, dx \).

Answer: 0


Question: Solve the equation $x^3 = 2^{-x}$ for $x$.

Answer: 0.826


Question: Find the limit of the sequence \( \left(\frac{3n^2-n+1}{2n^2+n+1}\right)^{\frac{n^2}{1-n}} \) as \( n \to \infty \).

Answer: 0


Question: Find a formula to calculate the number of functions \( f: P_k \rightarrow \{1, \ldots, n\} \) that satisfy the condition \( f(A \cup B) = \max(f(A), f(B)) \) for any \( A, B \in P_k \). Express the formula in a simplified form for specific values of \( n \) and \( k \), such as when \( k = 3 \) and \( n = 4 \).

Answer: 100


Question: Determine whether there exists a continuous function \( f: \mathbb{R} \to \mathbb{R} \) such that for every algebraic number \( q \), \( f(q) \) is a rational number.

Answer: Yes


Question: Find the number of solutions to the congruence $x^k \equiv 45 \pmod{97}$ for $k = 7, 8, 9$, given that $5$ is a primitive root of $97$ and $\text{ind}_5 (45) = 45$. Provide the total number of solutions.

Answer: 4


Question: Evaluate the integral \( \int_{C(0;2)} \frac{z^3}{z^2+i} \, dz \), where \( C(0;2) \) is the circle centered at the origin with radius 2.

Answer: 2\pi


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

Answer: \sin 1


Question: Determine if metrics $d_1$ and $d_2$ on a space $X$ are equivalent given that for any sequence $\{x_n\}_{n=1}^\infty \subset X$ and point $x_0 \in X$, the condition \( \lim_{n \to \infty}d_1(x_n,x_0)=0 \iff \lim_{n \to \infty}d_2(x_n,x_0)=0 \) holds. Equivalence implies that they induce the same metric topology.

Answer: Yes


Question: Evaluate the double integral \( \int\int_D \sqrt{x^2+y^2} \, dA \) over the region \( D \) bounded by the circles \( x^2+y^2=36 \) and \( (x-3)^2+y^2=9 \). Determine the appropriate limits of integration in polar coordinates and compute the integral.

Answer: 144\pi - 96


Question: Given a Vandermonde matrix, replace its last column with the vector \((1, 1, \ldots, 1)^T\). Determine if the determinant of the resulting matrix is an integer.

Answer: Yes


Question: Simplify the expression \( \dfrac{\Gamma(\beta)\Gamma(u)}{\Gamma(\beta+u)} \), given that \( \beta \) is a variable and assuming \( \text{Re}(\beta), \text{Re}(u) > 0 \).

Answer: B(\beta, u)


Question: Determine the number of integer solutions to the equation \(x+y+z=120\) where \(x, y, z \geq 5\) and \(x \leq y \leq z\). Use the action of a group to find the solution.

Answer: 972


Question:
Evaluate the limit using the telescoping series method:
\[ \lim_{k\to \infty} \sum_{n=1}^k \left(\frac{4}{3} \cdot \frac{(2n+1)}{(n^2+n)^2}\right) \]

Answer: \dfrac{4}{3}


Question: Find the limit of the function \( \frac{\sin(x^2y^2)}{(x^2+y^2)^{3/2}} \) as \((x,y)\) approaches \((0,0)\).

Answer: 0


Question: Solve the differential equation $\sin(x)\cos(2y)\;dx+\cos(x)\sin(2y)\;dy=0$ with the initial condition $y(0)=\frac{\pi}2$.

Answer: y = \frac{\pi}{2}


Question: Find a matrix $T$ that transforms vectors in a 2D vector space with a weighted inner product defined by a positive-definite matrix $Q$ such that the distance between vectors in the transformed space matches the distance in the Euclidean space.

Answer: Q^{1/2}


Question: Let $A$ be a discrete valuation ring with maximal ideal $\mathfrak{m}$ and fraction field $K$. Suppose $L$ is a finite separable extension of $K$ with degree $n$, which is unramified over $A$. Let $B$ denote the integral closure of $A$ in $L$. Determine whether there exists a monic polynomial $f \in A[x]$ such that $B$ is isomorphic to $A[x]/(f)$. Provide a justification for your answer.

Answer: Yes


Question: Determine the interior of the set of even numbers $E$ in the topological space $(\mathbb{N}, \text{Cofin})$, where $\text{Cofin}$ is the topology on the set of natural numbers $\mathbb{N}$ consisting of all cofinite subsets of $\mathbb{N}$. Express the interior using the interior operator $\text{Int}$ as follows: $\text{Int} E = \bigcup \{ M \subseteq \mathbb{N} \mid M \text{ is open in } (\mathbb{N}, \text{Cofin}) \wedge M \subseteq E \}$.

Answer: \emptyset


Question: Evaluate the integral \( \int_0^1 \int_x^1 6e^{\left(\frac{x}{y}\right)} \, dydx \). Rearrange the order of integration to simplify the computation.

Answer: 3(e - 1)


Question: In a measurable space $(\Omega, \Sigma, \mu)$, if a real-valued function $f: \Omega \rightarrow \mathbb{R}$ has its absolute value $\vert f \vert$ measurable with respect to $\mu$, is $f$ also $\mu$-measurable? Provide a justification for your answer.

Answer: No


Question:
For real numbers \( z \), consider the following functions:

\[ f_1(z) = \frac{1}{\sqrt{2\pi}}\exp\left(-\frac{\max(1,z^2)}{2}\right) \]
\[ f_2(z) = \frac{1}{\sqrt{2\pi}}\exp\left(-\frac{\min(1,z^2)}{2}\right) \]

Can either function serve as a valid probability density function (PDF) or be modified into one by multiplying by a non-negative constant \( c \)?

Answer: f_1


Question: Determine the cardinality of the set \(\{f: \mathbb{N} \to \mathbb{N} \mid f(f(a)) = a \text{ for all } a \in \mathbb{N}\}\). Is this set finite, countably infinite (\(\aleph_0\)), or uncountably infinite?

Answer: 2^{\aleph_0}


Question:
Given the parametric curve defined by the equations:
\[ x(t) = \cos(t) \]
\[ y(t) = \cos(2^k t) \]
where \( k \) is a natural number, determine the order of the polynomial obtained by eliminating the parameter \( t \).

Answer: 2^k


Question: Let $A$ be an $n \times n$ matrix with entries in $\mathbb{C}$ such that its minimal polynomial is $(t - \lambda)^{k}$ for some $k \leq n$. Determine the maximum possible dimension of the eigenspace corresponding to the eigenvalue $\lambda$. \( \text{Provide your answer as a single integer.} \)

Answer: n - k + 1


Question:
Calculate the expected value \(\mathbb{E}(Y_2)\) for the joint density function \(f(y_1, y_2)\) given by:

\[
f(y_1, y_2) = \begin{cases}
3y_1, & \text{for $0 \leq y_2 \leq y_1 \leq 1$}, \\
0, & \text{elsewhere}.
\end{cases}
\]

Verify the calculation of \(\mathbb{E}(Y_2)\) using the correct method.

Answer: \dfrac{3}{8}


Question: Let \((x_n)\) be a sequence such that \(\lim (x_n)=+\infty\) and \(\lim (x_{n+2}-x_n)=0\). Define \(v_n=x_n -[x_n]\), where \([t]\) denotes the integer part of \(t\). Find the set of adherence values of \((v_n)\).

Answer: [0,1]


Question: How many right-angled triangles with integer sides have an inradius of 2009? The inradius of a right triangle with sides $a$, $b$, and $c$ is given by $r = \frac{ab}{a+b+c}$. Find the number of such triangles.

Answer: 15


Question: Find all pairs \((m, n)\) of positive integers such that \(3^m = 8n^2 + 1\).

Answer: (2, 1)


Question: Evaluate the improper integral \( \int_0^\infty \frac{e^{-x}}{\sqrt{x}}dx \). Consider it as the sum of two integrals from \(0\) to \(1\) and from \(1\) to \(\infty\), and use appropriate comparison tests and techniques to find the value.

Answer: \sqrt{\pi}


Question: Given a partially separable 3-qubit state \( \phi = \left(a_0\left|0\right\rangle + a_1\left|1\right \rangle\right) \otimes \left(b_{00}\left|00\right \rangle + b_{01}\left|01\right \rangle + b_{10}\left|10\right \rangle + b_{11}\left|11\right \rangle\right) \), where the second and third qubits are entangled, and the first qubit is separable, the unseparated state is given by \( \phi = c_{000}\left|000\right\rangle + c_{001}\left|001\right\rangle + c_{010}\left|010\right\rangle + c_{100}\left|100\right\rangle+ c_{011}\left|011\right\rangle + c_{101}\left|101\rangle + c_{110}\left|110\right\rangle + c_{111}\left|111\right\rangle \) with \( c_{ijk} = a_i b_{jk} \). If a unitary transformation is applied to the first two qubits using a 4x4 unitary matrix \( U = (u_{nm}) \), determine the overall 8x8 matrix that represents the effect on all three qubits.

Answer: U \otimes I


Question: Is it always possible to find a square with vertices labeled with the same number when labeling points with integer coordinates in a plane using numbers from 1 to n?

Answer: Yes


Question: Given the curve defined by the function \(\mathbf{r}(t) = \langle \cos(2t), -\sin(2t), 4t \rangle\), calculate its curvature \(k(t)\) using the formula \(k(t) = \frac{||\mathbf{r}'(t) \times \mathbf{r}''(t)||}{||\mathbf{r}'(t)||^3}\).

Answer: \dfrac{1}{5}


Question: Find a natural number $x$ such that $x^{125} \equiv 7 \pmod{39}$ using Ender Wiggin's method.

Answer: 37


Question: Determine the maximum possible price difference between two American-style call options on a stock with different strike prices, assuming no dividends are paid.

Answer: K_2 - K_1


Question: Evaluate the limit: \[ \lim_{n\to\infty}\sum_{k=0}^{n} \binom{n}{k}\frac{1}{n^k(k+5)}. \]

Answer: 9e - 24


Question: Evaluate the sum \( \sum_{n=0}^{\infty}\binom{2n}{n}\left(\dfrac{3}{16}\right)^{n} \) using the Taylor expansion of \((1-x)^{-\frac{1}{2}}\).

Answer: 2


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{a^n}{n^{n/2}} \]

Answer: 0


Question:
Find the limit of the sequence of functions as \( n \to \infty \):

\[ \lim_{n\to \infty} f_n(x) = \frac{x^2}{x^2+(1-nx)^2} \]

Answer: 0


Question: Evaluate the limit: $$\lim_{x\rightarrow 0} \frac{\sin(\arctan(x))-\sin(\arcsin(x))}{x^5}$$

Answer: -\infty


Question: Determine the length of the Cantor set within the interval $[0, 1]$. Express your answer as a single number.

Answer: 0


Question: Compute the value of the complex integral \( \int_0^{2\pi}\dfrac{4ie^{it}}{4e^{it}-3}\,\mathrm{d}t \).

Answer: 2\pi i


Question: Find the supremum of the set \( \{2^n - 3^m : n, m \in \mathbb{N}\} \).

Answer: \infty


Question: Consider a sphere in \(\mathbb{R}^3\) with diameter \(d\). Secants are drawn randomly through the sphere, with tangents considered as secants of length zero. Each secant is equally likely. Let \(L\) be the length of such a randomly drawn secant inside the sphere. What is the most likely value of \(L\)?

Answer: d


Question: Is Tutte's graph a counterexample to Tait's conjecture that every cubic, polyhedral graph has a Hamiltonian cycle?

Answer: Yes


Question: Given an increasing function $f$ over the real numbers, determine if it is always possible to find some $x \ge M$ such that $f$ is continuous at $x$ for any given number $M$. Provide a justification for your answer.

Answer: Yes


Question: Compute the value of the probability generating function \( f(x) = \left(\frac{x(x^7 - 1)}{7(x - 1)}\right)^7 \) at \( x = 21 \).

Answer: 270163281^7


Question: Find the general form of entire functions \( f(z) \) that satisfy the inequality \( |f(z)| \leq e^{2\Re(z)} \) for all \( z \in \mathbb{C} \).

Answer: Ce^{2z}


Question: Consider a square $[0,1]\times[0,1]$. Construct a set $C_n$ by iteratively removing the middle third strip of width $1/3$ in both the $x$ and $y$ dimensions from each interval of $C_{n-1}$, starting with $C_0 = [0,1]\times[0,1]$. Does the Lebesgue measure of $C_n$ approach 0 as $n$ increases?

Answer: Yes


Question: Suppose $A$ is an $n \times n$ matrix and $B$ is an $m \times m$ matrix. Determine whether there always exists an $m \times n$ matrix $C$ such that $CA = BC$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_{\gamma} \frac{|dz|}{z} \) where \( \gamma \) is the unit circle.

Answer: 0


Question: Consider an abelian group $G$ of order 34. Define the subset $S = \{ g \in G \mid g = g^{-1} \}$. Determine the number of elements in the subset $S$. Provide your answer as a single integer.

Answer: 2


Question: Determine the values of $p$ and $n$ for which the function $\dfrac{1}{|x|^p}$ is integrable over the region $\mathbb{R}^n\setminus B_r(0)$, where $B_r(0)$ is the ball of radius $r$ centered at the origin.

Answer: p > n


Question: Let $K = \mathbb{Q}(\sqrt[3]{3}, \eta)$ where $\eta = (e^{\frac{\pi}{3}i})^2$. Determine the degree of the field extension $[K: \mathbb{Q}(\sqrt[3]{3})]$. \(\eta\) is a primitive sixth root of unity.

Answer: 2


Question: Consider a Toeplitz matrix $T \in \mathbb{R}^{n \times p}$ with $n < p$, where the entries are generated by a continuous random variable. If $n$ columns $T_1, \dots , T_n$ are selected from $T$, are these columns linearly independent with probability 1?

Answer: Yes


Question: In \(\mathbb{R}^n\) with the Euclidean norm, let \(\mathbb{S}^{n-1}\) be the unit sphere. Consider a function \(f: \mathbb{S}^{n-1} \rightarrow \mathbb{R}_+\) such that for each \(u \in \mathbb{S}^{n-1}\), \(f(u) < +\infty\). Is \(f\) necessarily bounded on \(\mathbb{S}^{n-1}\)?

Answer: No


Question: Let $g$ be a continuous function on $(p,q)$ and let $r \in (p,q)$. Suppose that $g$ and $g^{'}$ are non-zero on a deleted neighborhood of $r$, but \(\lim_{y \rightarrow r}g(y) = \infty.\) Calculate \(\lim_{y \rightarrow r} |{g(y)}^{-g(y)}|.\)

Answer: 0


Question: Find the values of $a_i$ such that the polynomial $\sum_{i=0}^n c_i x^i$ can be expressed as $\prod_{i=1}^n (x-a_i)$. What are the roots $a_i$?

Answer: a_1, a_2, \ldots, a_n


Question: Determine whether the following statement is true or false: If $f \in \mathbb{Q}[x]$ has a greatest common divisor of 1 with its derivative $f'$, then there are no nonzero nilpotent elements in the ring $\mathbb{C}\otimes_{\mathbb{Q}} \mathbb{Q}[x]/(f)$.

Answer: A


Question: Evaluate the integral of the conjugate of the complex variable $z$ over a circular path of radius 2 centered at the origin.

Answer: 8\pi i


Question: Let $G_1 = ( \mathbb { Z } / 7 \mathbb { Z } ) ^ { \times }$ and $G_2 = ( \mathbb { Z } / 19 \mathbb { Z } ) ^ { \times }$. Determine the number of elements in $G_1 \times G_2$ that have an order of 18.

Answer: 54


Question: Given a convergent series $\sum a_n$ with positive terms, where $\lim \inf (na_n) = 0$, and the sequence $\{a_n\}$ is decreasing, determine if $\lim (na_n) = 0$.

Answer: 0


Question:
Solve the differential equation for $y(x)$ given the boundary conditions:

$$ y''-i(\sin(x)y)'-i\omega y-\lvert y \rvert^2y'=0 , \quad y\rvert_{x=\pi}=0 \quad y'\rvert_{x=\pi}=0 $$

Answer: 0


Question: Given positive numbers $a$ and $b$ such that $\frac{2}{a} + \frac{1}{b} = 1$, determine the minimum value of the expression $a + b + \sqrt{a^2 + b^2}$.

Answer: 10


Question: Let $P(x)$ be a polynomial of degree $101$. Determine if the function $P: \mathbb{R} \to \mathbb{R}$ can be bijective.

Answer: Yes


Question: Given that \( \int_{-\infty}^{\infty} f(x) \, dx = 1 \), evaluate \( \int_{-\infty}^{\infty} f\left(x - \frac{1}{x}\right) \, dx \).

Answer: 1


Question: Evaluate the definite integral \( \int_{0}^{\frac{\pi}{2}}{\frac{\tan{x}}{1+\tan^4{x}}}\frac{dx}{\cos^2x} \) using properties of definite integrals.

Answer: \dfrac{\pi}{4}


Question: An insect is moving on the hyperbola $x^{2} - y^{2} = 1$ in the counterclockwise direction at a constant speed of 5 centimeters per second. The temperature function is given by $T(x, y) = x^{2} + y^{2}$. Calculate the rate of change of the temperature when the insect is at the point (2, 1).

Answer: 8\sqrt{5}


Question: Let $G$ be an abelian group of finite order $n$. Determine the modulus of the determinant of the character table of $G$.

Answer: n^{n/2}


Question: Find the residues of the function \( f(z) = \frac{1}{(e^{z}-1)^{2}} \) at its poles.

Answer: -1


Question: Find the coefficient of \(x^{25}\) in the expansion of \((2x^3 + 3x^2 - 5)^{15}\).

Answer: -7999230330000


Question: Let \( A = \begin{bmatrix} 1 & 3 & 4\\ 3 & 6 & 9\\ 1 & 6 & 4 \end{bmatrix} \) and \( B \) be a \( 3\times 3 \) matrix such that \( A \cdot A^{T} \cdot A +3B^{-1} =0 \). Find the value of \( \det( \operatorname{adj} (A^{-1}(B^{-1}){2B^{T}})) \).

Answer: \dfrac{64}{81}


Question: Find the projective closure of the set $\mathbb{P}^2 \setminus V(x_1^2 + x_2^2)$.

Answer: \mathbb{P}^2


Question: Let \( f(x)= (x^3+2x)\left[\frac{x}{2}\right] \) where \([x]\) denotes the floor function. Find \[ \lim\limits_{h\to 0}\frac{f(2-h^2)-f(2)}{h^2}. \]

Answer: -\infty


Question: Consider the limit \( \lim_{t \to 0}\int_t^x \frac{1}{s} \, ds \). Analyze the behavior of this limit as \( t \to 0 \).

Answer: \infty


Question: Calculate the expected value of the estimator \( \hat{\theta} = -\frac{1}{n}\sum{\ln(X_i)} \), where \( X_i \) are iid random variables with the density function \( f(x, \theta) = \frac{1}{\theta} x^{\frac{1-\theta}{\theta}} \) for \( 0 < x < 1 \) and \( \theta > 0 \).

Answer: \theta


Question: Find a positive constant \( c \) such that the normalized sum \( \frac{S_n}{n^c} \) converges in distribution to a random variable \( A \), where \( S_n = X_1 + X_2 + \ldots + X_n \) and the characteristic function of the iid random variables \( X_i \) is given by \( \phi(t) = 1 - \sqrt{|t|(2-|t|)} \) for \( t \in [-1,1] \) and \( \phi(t) = 0 \) elsewhere.

Answer: 2


Question: Calculate the sum of the infinite series \( \sum_{n=1}^\infty \frac{n(n+2)}{3^n} \).

Answer: 3


Question: Determine whether the limit \( \lim_{n\to \infty}\int_{0}^{1}nx^ng(x)dx \) exists for any continuous function \( g \) on the interval \([0, 1]\).

Answer: g(1)


Question: Find the remainder when \((2023^{45}+17)^{33}\) is divided by 101.

Answer: 30


Question: Express the sum \( \sum_{i,j\in\{0,1,2\}\atop i\neq j} x_ix_j^2 \) in terms of the elementary symmetric functions \( e_1 = x_0 + x_1 + x_2 \), \( e_2 = x_0x_1 + x_1x_2 + x_0x_2 \), \( e_3 = x_0x_1x_2 \) and the power sum functions \( p_2 = x_0^2 + x_1^2 + x_2^2 \), \( p_3 = x_0^3 + x_1^3 + x_2^3 \).

Answer: e_1 p_2 - p_3


Question: Given a symmetric linear transformation $T$ on a vector space over $\mathbb{R}^n$, represented by a matrix $M$, and a subspace $W$ of $\mathbb{R}^k$ ($k<n$) such that $T(u) \in W$ for any vector $u \in W$. Define $T^*$ as the restriction of $T$ to $W$. Can a basis for $W$ be found such that the $k \times k$ matrix of $T^*$ with respect to this basis is symmetric?

Answer: Yes


Question: Suppose $T \in \mathscr{L}(W)$ satisfies $T^4 = I$. If $T^2 = I$, does it follow that $T$ has no eigenvalues? Answer yes or no and justify your answer.

Answer: No


Question: Consider a cyclic group and elements $a$ and $b$ within it. Suppose $a^r$ generates $\langle a \rangle$ if and only if $\gcd(r, n) = 1$, where $n = \operatorname{ord}(a)$, and $a = b^k$. Determine whether the claim that $\langle a \rangle = \langle b \rangle$ is true or false.

Answer: False