Question: If $f$ is uniformly continuous over $\mathbb{R}$, is it true that for all $\alpha > 1$, $f = o_{\infty}(x^{\alpha})$?

Answer: Yes


Question: Let \( P \) and \( Q \) be \( 3 \times 3 \) matrices such that \( P \neq Q \), \( P^3 = Q^3 \), and \( P^2Q = Q^2P \). Find the value of \( \det(P^2 + Q^2) \).

Answer: 0


Question:
Evaluate the following limit for $r, q, B > 0$ and $x \in \mathbb{R}$:
\[ \lim_{d \to 0^+} \exp\left[\left(\frac{d}{1-q}\right)\log\left(\log B+\frac{1}{d}\log\left(1+\frac{x}{rq}\right)\right)\right] = 1. \]

Answer: 1


Question: Calculate the flux of the vector field \( F = (3x, 4y, -z) \) through the surface \( S \), where \( S \) is bounded by the surfaces \( z^2 = x^2 + y^2 \) and \( z = x^2 + y^2 \), with \( 0 \leq z \leq 1 \). Use Gauss's theorem to evaluate the flux.

Answer: \pi


Question: Determine the Assouad dimension of a complete unit ball in \( \mathbb{R}^D \).

Answer: D


Question: Determine the cardinality of the set \( S = \{ f: \mathbb{R} \to \mathbb{R} \mid f(x) \neq 0, \text{ only for finitely many } x \in \mathbb{R} \} \).

Answer: \mathfrak{c}


Question: Find the greatest common divisor \( \gcd(a,b) \) given that \( \gcd(a^3+6b^2,6480)=216 \) and \( 11a+8b \mid 540 \).

Answer: 6


Question: Evaluate the double integral \( I = \iint_D \cos{y^2} \, dx \, dy \), where \( D \) is the region bounded by the \( y \)-axis and the lines \( y = x \) and \( y = \sqrt{\pi/2} \).

Answer: \dfrac{1}{2}


Question:
Given positive integers \(a, b, c\) such that:
1. \(a < 2b\)
2. The remainder when \(a\) is divided by \(b\) is \(2r\)
3. The remainder when either \(a\) or \(b\) is divided by \(c\) is \(r\)

Find the smallest positive integer divisible by \(c\) among the following choices:
\(i)\ a+b\ \ \ ii)\ \frac{a+b}{2}\ \ \ iii)\ \frac{a+b}{3}\ \ \ iv)\ 2(a+b)\ \ \ v)\ 3(a+b)\)

Answer: ii


Question: Determine if 28 is the only integer that can be expressed both as the sum of a positive cube plus one and as \(2^n - 4\).

Answer: 28


Question: Evaluate the limit \( \lim_{x \to 0} \frac{1-\cos(1-\cos x)}{x^4} \) using only the fact that \( \lim_{x \to 0}\frac{\sin x}{x}=1 \).

Answer: \dfrac{1}{8}


Question: Consider a 10x10 symmetric matrix \(A\) where all diagonal entries are \(a+1\) and all off-diagonal entries are 1. The sum of the largest and smallest eigenvalues of \(A\) is 24. Find the value of \(a\).

Answer: 7


Question: Verify the identity \( i = \frac{1}{\sqrt{2i}+1} \) given that \( \sqrt{-i} = \frac{1-i}{\sqrt{2}} \) and \( \sqrt{-i} = i\sqrt{i} \).

Answer: i


Question: Evaluate the triple integral \( \iiint_D (3 + 2xy) \, dV \) over the solid hemispherical dome \( D \) defined by \( x^2 + y^2 + z^2 \le 4 \) and \( z \ge 0 \) without using spherical coordinates.

Answer: 16\pi


Question:
Let \( f(x) = ax^3 + bx^2 + cx + d \) be a polynomial function. Find the relation between the coefficients \( a, b, c, d \) such that its roots are in an arithmetic progression and a geometric progression. Choose the correct relation for the geometric progression from the following options:

(a) \( a^2b = c^2d \)
(b) \( a^2b^2 = c^2d \)
(c) \( ab^3 = c^3d \)
(d) \( ac^3 = b^3d \)
(e) \( ac = bd \)
(f) \( a^3c = b^3d \)

Answer: D


Question:
Given the linear regression model with 100 observations:
\[ y_i = b_0 + b_1 x_i + \epsilon_i \]
The estimated coefficients are:
\[ \hat{\beta} = \begin{pmatrix} 9 \\ 1 \end{pmatrix} \]
And the variance-covariance matrix of the estimated coefficients is:
\[ V(\hat{\beta}) = \hat{\sigma}^2 (X'X)^{-1} = \begin{pmatrix} 3 & 0.12 \\ 0.12 & 0.064 \end{pmatrix} \]
Calculate the sum of squared residuals, \( e'e \).

Answer: 27195


Question: Determine the number of solutions for the equation \((F(x))^2 = \frac{9x^4}{G(x)}\). Let \(f_1(x)\) and \(f_2(x)\) be twice differentiable functions, where \(F(x) = f_1(x) + f_2(x)\) and \(G(x) = f_1(x) - f_2(x)\), for all \(x \in \mathbb{R}\). Given that \(f_1(0) = 2\), \(f_2(0) = 1\), \(f_1'(x) = f_2(x)\), and \(f_2'(x) = f_1(x)\) for all \(x \in \mathbb{R}\), find the number of solutions of the equation \((F(x))^2 = \frac{9x^4}{G(x)}\).

Answer: 3


Question: Construct a polynomial with integer coefficients such that \( f(i + \sqrt{2}) = 0 \). The polynomial should be of even degree greater than 2.

Answer: x^4 - 2x^2 + 9


Question: Consider a primorial $p_k\#$. It is known that the number of elements in the reduced residue system for $p_k\#$ is divisible by $p_k - 1$. Does it follow that if you divide the elements of a reduced residue class into distinct classes modulo $p_k$, each class modulo $p_k$ has the same number of elements, excluding the class $x \equiv 0 \pmod{p_k}$? Prove or disprove this statement for all primes.

Answer: True


Question:
Let \( f : [0, 1] \rightarrow \mathbb{R} \) be a twice differentiable function such that \( f(0) = f(1) = 0 \) and \( f''(x) - 2f'(x) + f(x) \ge e^x \) for all \( x \in [0, 1] \). Which of the following is true for \( 0 < x < 1 \)?

(A) \( 0 < f(x) < \infty \)

(B) \( -\frac{1}{2} < f(x) < \frac{1}{2} \)

(C) \( -\frac{1}{4} < f(x) < 1 \)

(D) \( -\infty < f(x) < 0 \)

Select the correct option.

Answer: D


Question: Calculate the surface area of the portion of the cone defined by \( z^2 = x^2 + y^2 \), where \( z \geq 0 \), that is contained within the cylinder \( y^2 + z^2 \leq 81 \).

Answer: 81\pi


Question: Consider the Young diagram of a partition \(\lambda = (\lambda_1,\ldots,\lambda_k)\). For a square \((i,j) \in \lambda\), define the hook numbers \(h_{(i,j)} = \lambda_i + \lambda_j' -i - j +1\) where \(\lambda'\) is the conjugate of \(\lambda\). Recall the Fibonacci numbers \(F(0)=0, \, F(1)=1\) with \(F(n)=F(n-1)+F(n-2)\). Define \([0]!_F=1\) and \([n]!_F=F(1)\cdot F(2)\cdots F(n)\) for \(n\geq1\). Is it true that \([n]!_F\prod_{\square\,\in\,\lambda}\frac1{F(h_{\square})}\) is an integer?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{\int_{0}^{x}\sin{t}\ln{(1+t)}dt-\frac{x^3}{3}+\frac{x^4}{8}}{(x-\sin{x})(e^{x^2}-1)} \]

Answer: \dfrac{1}{5}


Question: Let $f(x)$ be an irreducible polynomial of degree $p$, where $p$ is a prime number, over the field of rational numbers $\mathbb{Q}$. Suppose that $f(x)$ has exactly two nonreal roots in the complex numbers $\mathbb{C}$. Prove that the Galois group of $f(x)$ over $\mathbb{Q}$ is the symmetric group $S_p$. Provide a rigorous proof for this statement.

Answer: S_p


Question: In triangle ABC, where AB = 12, BC = 20, and CA = 16, points X and Y are located on segments AB and AC, respectively. A point K is on segment XY such that the ratio XK/KY = 7/5. As X and Y vary along segments AB and AC, the point K traces out a region. Express the area of this region as a fraction \( \frac{m}{n} \) in its simplest form. Compute the sum \( m+n \).

Answer: 143


Question:
Consider a Hidden Markov Model (HMM) with 3 states and 2 output symbols. The state transition matrix is given by:
\[ A = \begin{pmatrix}0.5 & 0 & 0 \\ 0.3 & 0.6 & 0 \\ 0.2 & 0.4 & 1 \end{pmatrix} \]
where \( A_{i,j} = P(h_{t+1} = i | h_t = j) \). The emission matrix is:
\[ B = \begin{pmatrix} 0.7 & 0.4 & 0.8 \\ 0.3 & 0.6 & 0.2 \end{pmatrix} \]
where \( B_{i,j} = P(v_t = i | h_t = j) \). The initial state probability vector is \( a = (0.9, 0.1, 0)^T \). Given the observed symbol sequence \( v_{1:3} = (1, 2, 1) \), compute \( P(v_{1:3}) \).

Answer: 0.153823


Question: Is the real number line \( \mathbb{R} \) with the usual Euclidean topology homeomorphic to any topological field of positive characteristic? Provide a justification for your answer.

Answer: No


Question: Calculate the value of the infinite series \( \sum_{n=0}^{\infty}\frac{1}{(n!)^{2}} \).

Answer: I_0(2)


Question: Let $A$ be a finite-dimensional $k$-algebra that does not necessarily have a multiplicative identity. If the map \( \mu: A \otimes A \rightarrow A, \ x \otimes y \mapsto xy \) is surjective, does $A$ have a multiplicative identity?

Answer: No


Question: Determine the degree of the extension \( [ \mathbb{Q}(\zeta) : \mathbb{Q}(\zeta) \cap \mathbb{R}] \), where \( \zeta = e^{2\pi i / 7} \).

Answer: 2


Question: Find the largest radius of a disk centered at the origin in which the mapping $f(z) = z^2 + z$ is one-to-one.

Answer: \dfrac{1}{2}


Question: Determine the values of \(z \in \mathbb{C}\) for which the series \(\sum_{n=1}^\infty \frac{1}{\frac{1}{z^n}-1}\) converges absolutely.

Answer: |z| < 1


Question: A list contains 20 numbers. For each positive integer \( n \), from 1 to 20, the \( n \)-th number in the list is either \( \frac{n}{2} \) or \( 2n \). If the mean of the numbers in the list is exactly 12 and the product of the numbers is \( P \), what is the greatest possible value of \( \frac{P}{20!} \)?

Answer: 16


Question: Let $G$ be a soluble group and $K/L$ be a chief factor of $G$. Suppose $M$ is the smallest normal subgroup of $K$ such that $K/M$ is nilpotent. Show that $M$ is a characteristic subgroup of $K$, i.e., for any automorphism $\phi$ of $K$, $\phi(M) = M$. Why is this the case?

Answer: M


Question: Evaluate the integral \( J = \int^{2\pi}_{0} e^{\cos x} \cos(2x - \sin x)\, dx \).

Answer: \pi


Question: What is the last digit of \(\operatorname{lcm}(3^{2003}-1,3^{2003}+1)\)?

Answer: 4


Question: Compute the contour integral \( \oint_{C} \frac{1}{z^{3} + 9z} \, dz \), where \( C \) is the counterclockwise-oriented circle defined by \( |z| = 4 \).

Answer: 0


Question: Estimate \( \int^{1}_{-1} \left(\frac{\sin{x}}{x}\right)^{300} dx \) to one significant figure.

Answer: 0.3


Question: Find the integer $m$ such that $16 \equiv m^{19} \pmod{143}$. Use any relevant theorems or properties of modular arithmetic to solve the problem.

Answer: 42


Question: If a manifold $M$ is $\sigma$-compact, does $M$ possess a countable dense subset?

Answer: Yes


Question: A clock has 12 positions, numbered 1 through 12. Each step, the hour hand moves either clockwise or counterclockwise with equal probability (1/2), independently of previous steps. Starting at position 1, what is the expected number of steps required for the hour hand to reach position 12?

Answer: 11


Question: Find the closed form of the summation \( A = \sum_{j=0}^\infty \frac{1}{j!}\times \frac{1}{j!}x^j \).

Answer: I_0(2\sqrt{x})


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

Answer: \dfrac{1}{2}


Question:
Determine the correct value of \( S \) in the recursive integral equation:
\[ S = \frac{1}{\int_1^2 \frac{1}{\int_1^2 \frac{1}{\int_1^2 \ddots}}} \]
Assume that the variable of integration is either \( x \) or \( S \). Calculate \( S \) based on these assumptions and find the correct value.

Answer: 1


Question: Determine the points in the complex plane where the function \( f(x + yi) = x^2 - y^2 \) is complex differentiable.

Answer: 0


Question: Let \( \zeta_{2^{n+2}} \) be a \( 2^{n+2} \)th root of unity, and let \( \overline{\zeta}_{2^{n+2}} \) be its complex conjugate. Show that the degree of the field extension \([\mathbb{Q}(\zeta_{2^{n+2}}): \mathbb{Q}(\zeta_{2^{n+2}} + \overline{\zeta}_{2^{n+2}})] = 2\).

Answer: 2


Question: Let \( S = \{1, 2, 3, \ldots, 1992\} \). Find the number of subsets \( \{a, b, c\} \) such that \( 3 \mid (a + b + c) \).

Answer: 438471736


Question: Evaluate the triple integral \( J = \iiint_{Q} \frac{1}{x^2+y^2+z^2} \,dx \,dy \,dz \) where \( Q \) is the region between the spheres \( x^2+y^2+z^2 = 1 \) and \( x^2+y^2+z^2 = 9 \).

Answer: 32\pi


Question: Calculate the sum \( \sum_{w\in S_n}(-1)^{l(w)}n^{l(w)} \), where \( l(w) \) is the number of cycles in the permutation \( w \) from the symmetric group \( S_n \).

Answer: (-1)^n n!


Question: Find the coefficient of \(x^2\) in the expansion of \((x^3 + 2x^2 + x + 4)^{15}\).

Answer: 15099494400


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

Answer: 0


Question:
For what value of \(a \in \mathbb{R}\) is the function \(f(x)\) differentiable at least at one point, where:
\[ f(x) = \begin{cases} a & \text{if } x \in \mathbb{Q} \\ x^2 + ax + a & \text{if } x \in \mathbb{R} \setminus \mathbb{Q} \end{cases} \]

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{1 - \frac{x^2}{2} - \cos \left(\frac{x}{1-x^2}\right)}{x^4} \] using the fact that \( \lim_{x \to 0} \frac{1-\cos x}{x^2}=\frac{1}{2} \).

Answer: \dfrac{23}{24}


Question: Determine the values of the parameter \( p \) for which the series \( \sum_{n=2}^{\infty}\frac{1}{(\sqrt{n}+\sqrt{n+1})^p}\ln\left(\frac{n-1}{n+1}\right) \) is convergent.

Answer: p > 0


Question: Determine the cardinality of the set \( \{ (x,y) \in \Bbb{R}^2 \; | \; y > x > 0 , x^x = y^y \} \).

Answer: \mathfrak{c}


Question: Evaluate the integral \( \int_{\partial \mathbb{B}(-i,3)}\frac{\sin(z)}{(z-3)^3}\, \mathrm{d}z \) using Cauchy's Integral Formula.

Answer: 0


Question: Find the differentiation of the Lagrangian function \( L(X, \lambda) = \mathop{Trace}(XH) + \lambda \mathop{Trace}(X^{\dagger}X) \) with respect to \( X^* \), where all matrices are complex, \( \dagger \) denotes the conjugate transpose, \( * \) denotes the conjugate, and \( \mathop{Trace} \) denotes the trace of a matrix.

Answer: \lambda X


Question: Find the remainder when \((x-1)^{2017}\) is divided by \(x^2 - x + 1\).

Answer: x - 1


Question: A white Rubik's cube has 6 sides, each with 9 spaces that can be filled with numbers from 1 to 9. A 3x3 magic square is one in which the sum of the numbers in each row, column, and both main diagonals is the same, and there are 8 possible combinations of such a square. How many ways can you fill each side of the Rubik's cube with a 3x3 magic square?

Answer: 262144


Question:
Let \(ABC\) be an acute triangle. Determine the maximum value of the expression:
\[
\frac{\tan^2A+\tan^2B}{\tan^4A+\tan^4B}+\frac{\tan^2B+\tan^2C}{\tan^4B+\tan^4C}+\frac{\tan^2C+\tan^2A}{\tan^4C+\tan^4A}.
\]

Answer: 1


Question: Determine the parameters $u$ and $v$ for the surface $z = xe^y$ over the region $0 \leq x \leq 1$ and $0 \leq y \leq 1$ to evaluate the flux of the vector field $\mathbf{f}(x,y,z)=\langle xy, 4x^2, yz \rangle$ across the surface $\Sigma$ with upward orientation. Use the surface integral formula: $$\iint_{\Sigma} \mathbf{f} \cdot d\sigma = \iint_R \mathbf{f}(x(u,v),y(u,v),z(u,v)) \left|\left| \frac{\partial r}{\partial u} \times \frac{\partial r}{\partial v} \right|\right| du dv$$

Answer: 1 - e


Question: Solve the differential equation using a power series method: \[ \begin{cases} w''=(z^2-1)w \\ w(0)=1 \\ w'(0)=0 \end{cases} \] Find a closed form for the solution $w(z)$.

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


Question: Given that \( \lim_{x \to a} \frac{f(x) - f(a)}{x-a} = 2 \), find \( \lim_{h \to 0} \frac{f(a+h) - f(a-h)}{h} \) without using derivatives and integrals.

Answer: 4


Question: Find the volume of the region bounded by the planes $6x+5y+6z = 6$, $y=x$, $x=0$, and $z=0$.

Answer: \dfrac{6}{55}


Question: Consider the sequence \( \{X_n\} \) defined by the recurrence relation \( X_{n+1} = \alpha X_n - (1-\alpha)X_{n-1} \), where \( \alpha \in (0,2) \). Determine the limit of this sequence as \( n \to \infty \).

Answer: 0


Question: Calculate the heat flow through the sphere defined by the temperature function \( T(x,y,z) = x \) and the surface \( S = \{(x,y,z): x^2+y^2+z^2=1\} \). Use the formula for heat flow \( \int \int_S (-k \nabla T) \cdot dS \) for some parameter \( k \).

Answer: 0


Question: How many distinct bracelets can be made using 5 beads of one color and 9 beads of another color, considering that bracelets are identical if they can be rotated or flipped into each other? Use Burnside's Lemma to determine the number of distinct bracelets.

Answer: 79


Question: Find the distance between the skew lines $p$ and $q$. Line $p$ is determined by the point $[6, 3, -3]$ and direction vector $(-3, 2, 4)$, while line $q$ is determined by the point $[-4, -4, 12]$ and direction vector $(-3, 3, 8)$. Use orthogonal projection to find the distance.

Answer: 13


Question: If the sum \(\sum_{n=0}^{2011} \frac{n+2}{n!+(n+1)!+(n+2)!}\) can be written as \(\frac{1}{2} - \frac{1}{a!}\), find the last three digits of \(a\).

Answer: 013


Question: Let $X$ and $Y$ be topological spaces, and let $U$ be an open subset of $X \times Y$. For a given $x \in X$, define the section $U_x = \{ y \in Y : (x, y) \in U \}$. Is $U_x$ an open subset of $Y$?

Answer: Yes


Question: Let $f:[0,1]\to [0,\infty)$ be a function such that \(\int_0^1 \left|\frac{d^r}{d x^r} f(x)\right|^2 d x<\infty\) for some \(1\leq r<\infty, r\in\mathbb{N}\). Does it follow that \(\int_0^1 \left|\frac{d^{r-1}}{d x^{r-1}} f(x)\right|^2 d x<\infty\)? Provide a justification for your answer.

Answer: Yes


Question: If $K_1$ and $K_2$ are finite extensions of $F$ with degrees $m$ and $n$ respectively, and $[K_1K_2:F]=[K_1:F][K_2:F]$, must $m$ and $n$ be relatively prime?

Answer: No


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} \frac{1+\sqrt[n] {e}+\sqrt[n] {e^2}+ \dots + \sqrt[n] {e^{n-1}}  } {n} \]

Answer: e - 1


Question:
Given the formula for experience as a function of level:
\[ experience = \left\lfloor \frac{1}{4} \times \sum_{x=1}^{L-1} \left\lfloor x + 300 \cdot 2^{\frac{x}{7}} \right\rfloor \right\rfloor \]
Determine the level \( L \) for a given experience value. For example, if the experience \( e = 83 \), find the level \( L \).

Answer: 2


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

If $A$ and $B$ are two semisimple square matrices over a finite field with equal characteristic polynomials, then $A$ and $B$ are similar. That is, there exists an invertible matrix $T$ such that $T^{-1}AT = B$. Provide a justification for your answer.

Answer: True


Question: If \( f \) is a strictly increasing function over the positive reals with \( f(f(x)) = x^2 + 2 \), find the value of \( f(3) \).

Answer: 6


Question: Let $Y$ and $Z$ be two $n \times n$ matrices. Suppose there exists a unitary matrix $X$ such that $X^* Y X$ and $X^* Z X$ are upper triangular matrices. Determine the eigenvalues of the matrix $YZ - ZY$. Provide a justification for your answer.

Answer: 0


Question: Evaluate the integral \( I = \iint_R |xy|\,dx\,dy \) using polar coordinates, where \( R \) is the region bounded by \( x^2+4y^2 \ge 1 \) and \( x^2+y^2 \le 1 \).

Answer: \dfrac{3}{8}


Question: Evaluate the integral \( \frac{1}{2\pi i} \int_{|z|=r}{f(z)g(z)dz} \) for \( r > 0 \), where \( f : \mathbb{C}\setminus \{0\} \to \mathbb{C} \) is an analytic function with a simple pole of order 1 at 0 with residue \( a_1 \), and \( g : \mathbb{C} \to \mathbb{C} \) is analytic with \( g(0) \neq 0 \).

Answer: a_1 g(0)


Question: Calculate the flux of the vector field \( v = (\cos{2 \pi x}, y^3, z^3) \) through the unit sphere centered at the origin. The flux is given by the integral \( \iint_S(v \cdot n) \, ds \), where \( n \) is the outward unit normal vector to the surface \( S \), and \( ds \) is the area element on \( S \).

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


Question:
Consider the sequence of functions defined for \( n \ge 1 \) by \( g_n(x) = \sin^2(x + \frac{1}{n}) \) for \( x \in [0, \infty) \) and \( f_n(x) = \int_{0}^{x} g_n(t) \, dt \). Determine which of the following statements is true:

1) \( \{f_n\} \) converges pointwise to a function \( f \) on \([0, \infty)\) but does not converge uniformly on \([0, \infty)\).
2) \( \{f_n\} \) does not converge pointwise to any function on \([0, \infty)\).
3) \( \{f_n\} \) converges uniformly on \([0, 1]\).
4) \( \{f_n\} \) converges uniformly on \([0, \infty)\).

Answer: 4


Question: Let $n > 1$ be an integer, and let $K$ be a field such that $n$ does not divide the characteristic of $K$ and $K$ contains the $n$-th roots of unity. Consider $\Phi \in \text{Hom}(K^{\times}/K^{\times n}, \mu_n)$, where $\mu_n \subseteq K$ is the set of $n$-th roots of unity. For $x \in K^{\times}$, let $\Phi(xK^{\times n}) = \zeta_x$ for some $\zeta_x \in \mu_n$. If $x^{1/n}$ is any $n$-th root of $x$, do $x^{1/n}\zeta_x$ and $x^{1/n}$ have the same minimal polynomial over $K$?

Answer: Yes


Question: Evaluate the integral \( \int_0^{\frac{\pi}{2}} \frac{\sin x \cos x}{\sin^4 x + \cos^4 x} \, dx \).

Answer: \dfrac{\pi}{4}


Question:
Determine whether the following equality holds for all integers \( j \):
\[
\int_{-\pi}^{\pi} \left(\sum_{k = -\infty}^{\infty} c_k e^{i (k - j) x}\right) d x = \sum_{k = -\infty}^{\infty} \left(\int_{-\pi}^{\pi} c_k e^{i (k - j) x} d x\right)
\]
where \( f : \mathbb{R} \to \mathbb{C} \) is a \( 2\pi \)-periodic function such that \( \int_{-\pi}^{\pi} |f| d x < \infty \), and \( f(x) = \sum_{k = -\infty}^{\infty} c_k e^{i k x} \) for all \( x \in \mathbb{R} \).

Answer: Yes


Question: Evaluate the integral \( \int_\gamma \frac{dz}{z^2} \) where \( \gamma(t) = \cos(t) + 2i\sin(t) \) for \( 0 \le t \le 2\pi \).

Answer: 0


Question: Find all natural numbers $k$ such that $\sqrt{2^k + k^2}$ is an integer.

Answer: 6


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

Answer: 0


Question: Does there exist an $L^1$ function whose Fourier series converges almost everywhere but does not converge to the function itself?

Answer: No


Question: Determine whether the following statement is true or false: If a finite group has only 1-dimensional irreducible representations, then it is abelian.

Answer: True


Question: Determine the radius of convergence of the series \(\sum\limits_{n=0}^\infty 3^nz^{n!}\).

Answer: 1


Question: Determine the number of roots of the polynomial \( p(z) = z^{10} + 100z + 1 \) inside the unit disk \( \{z:|z|<1\} \).

Answer: 1


Question: Calculate the limit: \[ \lim_{x\to\infty} \left(x^{1/n}-\ln(x)\right) \]

Answer: \infty


Question: Determine the limit of the sequence \( a_n = \sqrt[n]{\sum_{k=1}^{n}{(2 - \frac{1}{k})^k}} \).

Answer: 2


Question: Evaluate the limit: \[ \lim_{p \to 0} \left(\int_0^1 (1+x)^p \, dx\right)^{1/p}. \]

Answer: \dfrac{4}{e}


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{\sin^2x-x\tan x}{x^4} \]

Answer: -\dfrac{2}{3}


Question: If the Laplace transform \((\mathcal{L}f)(s)\) of a piecewise continuous function \(f(t)\) is non-negative for all \(s \in \mathbb{R^+}\), does it imply that \(f(t)\) is non-negative for all \(t \geq 0\)? Provide a justification for your answer.

Answer: No


Question: If \(\lim_{x\to 0} \frac{1+a\cos 2x + b\cos 4x}{x^4}\) exists for all \(x\in\mathbb{R}\) and is equal to \(c\), find \(\lfloor a^{-1} +b^{-1} + c^{-1}\rfloor\).

Answer: 2


Question: Calculate the probability of observing two consecutive rolls of 3 before observing two consecutive rolls that sum to 8 in an infinite sequence of dice rolls. Assume each roll is independent and uniformly distributed over the numbers 1 to 6.

Answer: \dfrac{7}{47}


Question: Evaluate the contour integral \( \int_{|z|=3} \frac{z^{17}}{(z^2+2)^3(z^3+3)^4} \, dz \).

Answer: 2\pi i


Question: Given the integral \( w = \int_{xy}^{2x-3y} \frac{du}{\ln(u)} \, du \), find \( \frac{\partial y}{\partial x} \) at \( x = 3, y = 1 \).

Answer: \dfrac{1}{6}


Question: Find the value of \(x\) such that \(\sqrt{(2+\sqrt{3})^x} + \sqrt{(2-\sqrt{3})^x} = 2^x\).

Answer: 2


Question: Let $f$ and $g$ be monic polynomials in $\mathbb{Z}[x]$. Suppose there exists a polynomial $h$ in $\mathbb{R}[x]$ such that $f = gh$. Is $h$ necessarily in $\mathbb{Z}[x]$?

Answer: Yes


Question: Is it true that for any odd cycle graph $G$ of length greater than 5, $\Theta(G)^m \neq \alpha(G^{\boxtimes m})$ for any finite positive integer $m$? In other words, does $\limsup_{m\rightarrow\infty}\alpha(G^{\boxtimes m})^{\frac{1}{m}}$ not attain its value at any finite positive integer $m$?

Answer: Yes


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

Answer: 0


Question: Consider a closed contour \( C \) in the complex plane, parametrized by \( z(t) = u(t) + i v(t) \). Can there be any point on this contour where the derivative \( z'(t) = 0 \)?

Answer: Yes


Question: Evaluate the flux integral \( \int_{\delta V} F \cdot \hat{n} \, dS \) where \( F = xy^2 \hat{i} + xe^z \hat{j} + z^3 \hat{k} \), and \( \delta V \) is the surface of the cylinder \( V \) bounded by \( y^2 + z^2 = 1 \) and the planes \( x = -1 \), \( x = 2 \). The vector \( \hat{n} \) is the outward-pointing unit normal to \( V \).

Answer: 3\pi


Question: Find the expectation of the expression \( \min\{U_{1}, U_{2}\} \times \min\{U_{1}, U_{3}\} \), where \( U_{1}, U_{2}, U_{3} \) are independent random variables uniformly distributed over the interval \([0, 1]\).

Answer: \dfrac{2}{15}


Question: Suppose $f_n : D \rightarrow \mathbb{R}$ are continuous functions that converge uniformly to $f : D \rightarrow \mathbb{R}$. Is $f$ necessarily continuous?

Answer: Yes


Question: Determine the cardinality of the equivalence class of \( \frac{1}{\sqrt{2}} \) under the relation defined on \( I = [0, 1] \) by \( x - y \in \mathbb{Q} \).

Answer: \aleph_0


Question: Given the differential equation \( x \cdot \ln(x) \frac{dy}{dx} + y = 2x \cdot \ln(x) \) for \( x \ge 1 \), find the value of \( y(e) \).

Answer: 2


Question: Determine the number of distinct subgroups of the group of all automorphisms of the finite field $F_{3^{100}}$. Provide your answer as a single integer.

Answer: 9


Question: Losses relating to a type of insurance policy follow a gamma distribution with mean $30,000$ and shape parameter $\alpha = 2$. For a sample of 100 policy claims, calculate the approximate probability that the mean loss size is less than $27,000$. Provide your answer to four decimal places.

Answer: 0.0786


Question: Find the surface area of a torus obtained by rotating a circle of radius $b$ centered at $(a,0)$ about the $y$-axis, where $a > b$. Use the parametrization: $$x = (a + b \cos \psi)\cos \theta, \quad y = b \sin \psi, \quad z = (a + b \cos \psi)\sin \theta$$ where $0\leq\psi,\theta\leq 2\pi$. Compute the surface area of this torus.

Answer: 4\pi^2 ab


Question: Determine the value of \( x \) such that the matrix \( M_x = \begin{pmatrix} 1 & x+1 & 0 & 0 \\ 0 & x & x-1 & 0 \\ 0 & x-1 & x & 0 \\ 1 & 0 & x+1 & x \end{pmatrix} \) is diagonalizable.

Answer: -1


Question: Evaluate the complex line integral \( \int_\gamma \frac{z^5}{z^7+3z-10}\,dz \), where \( \gamma \) is the boundary of \( D(0,2) \) oriented counterclockwise.

Answer: 0


Question: Prove that if \( \int_0^x f(t)\, dt = f(x) \) for all \( x \), then \( f(x) \) is identically zero.

Answer: 0


Question: Find the value of \( w \ge 0 \) such that the differential equation \( y'' + y = \cos(wt) \) does not have a bounded solution.

Answer: 1


Question: In triangle \(ABC\), it is given that \(AB = 4\), \(AC = 2\), and \(BC = 3\). The bisector of angle \(A\) intersects side \(BC\) at point \(K\). A line passing through point \(B\) and parallel to \(AC\) intersects the extension of the angle bisector \(AK\) at point \(M\). Find \((AM)^2\).

Answer: 54


Question: Determine the smallest integer \( n \) such that the proportion \( p(n) = \frac{\text{number of asymmetric graphs with } n \text{ nodes}}{\text{number of graphs with } n \text{ nodes}} \) exceeds 0.99.

Answer: 15


Question: Compute the sum \( S = 1 \times 2 \times 3 \times 4 + 3 \times 4 \times 5 \times 6 + \ldots + 97 \times 98 \times 99 \times 100 \).

Answer: 974510040


Question: Consider the Cauchy problem given by the ordinary differential equation \( x' = f(t,x) \) with the initial condition \( x(t_0) = x_0 \). Suppose there exists a constant \( K \) such that \( ||f(t,x)|| \leq K \) for all \((t,x) \in \mathbb{R} \times \mathbb{R}^n\). Is it true that the solution is defined for all time?

Answer: Yes


Question: Calculate the sum of the infinite series: \[ \sum_{k=0}^\infty \frac{2^k(k+1)k}{3e^2k!} \]

Answer: \dfrac{8}{3}


Question: Calculate the surface area of the portion of the surface defined by \( z^2 = 2xy \) within the region \( 0 \le x \le 2 \) and \( 0 \le y \le 1 \).

Answer: 8


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

Answer: \dfrac{1}{2}


Question: Show that \( \lim_{n \to \infty} \left| \left( 1 - (1 - s) \frac{\lambda_n}{n}\right)^n - \left( 1 - (1 - s) \frac{\lambda}{n}\right)^n \right| \le \lim_{n \to \infty} | 1 - s | | \lambda_n - \lambda | = 0 \). This is an application of the convergence theorem in probability, demonstrating that the generating function of a sequence of binomially distributed random variables converges to the generating function of the Poisson distribution. Assume \( n p_n \to \lambda > 0 \) as \( n \to \infty \) and let \( \lambda_n = n p_n \).

Answer: 0


Question: Is it possible to find a countable subcollection of vectors \( \{s_{\iota_k}, k \in \mathbb{N}\} \) from a given collection \( \{s_\iota, \iota \in I\} \) in \( \mathbb{R}^2 \) such that the union of the corresponding open half-planes \( \bigcup_{k=1}^\infty U_{\iota_k} \) is equal to the union of all open half-planes \( \bigcup_{\iota \in I} U_\iota \)?

Answer: Yes


Question: Evaluate the definite integral \( \int_{0}^{1} \cos(x^2) \, dx \) using a power series expansion, and compute the result with an accuracy of \(10^{-4}\).

Answer: 0.9045


Question: Let \( X \) be a Poisson-distributed random variable with parameter \( n \). Prove that \( P(0 < X < 2(n+1)) \ge \frac{n}{n+1} \).

Answer: \frac{n}{n+1}


Question: Determine the characteristic of the ring $R = \Bbb{Z}_3[x] / (x^3 - x - 2)$, where $R$ consists of elements of the form $a + bx + cx^2$ with $a, b, c \in \Bbb{Z}_3$. What is the characteristic of this ring?

Answer: 3


Question: Let $A, B, C,$ and $D$ be abelian varieties over $\mathbb{C}$ such that $A \times B \cong C \times D$ and $A \cong C$. Given these conditions, are the abelian varieties $B$ and $D$ isomorphic?

Answer: Yes


Question: Let $X$ and $Y$ be two continuous random variables with the joint density function $f(x, y) = 2e^{-x-2y}$ for $x, y \geq 0$, and $f(x, y) = 0$ elsewhere. Compute the probability $P(X > Y)$. Provide your answer as a single probability value.

Answer: \dfrac{2}{3}


Question:
Given a symmetric function \( R(x, y) \) of two variables, where \( R(x, y) = R(y, x) \), and the conditions:

- \( R(x, y) \le R(x-1, y) + R(x, y-1) \)
- \( R(0, i) = n \) for a fixed natural number \( n \),

find the value of the constant \( c \) in terms of \( m \) such that \( R(m, m) \le cn \).

Answer: \dbinom{2m}{m}


Question: Let \( f : (0,1) \to \mathbb{R} \) be a continuously differentiable function such that \( f' \) has finitely many zeros in \((0,1)\) and \( f' \) changes sign at exactly two of these points. Determine the maximum number of solutions to the equation \( f(x) = y \) in the interval \((0,1)\) for any \( y \in \mathbb{R} \).

Answer: 3


Question: Find the norm of the linear functional \( f(x) = \int_{-1}^{0} x(t) \, dt - \int_{0}^{1} x(t) \, dt \) on the space \( C[-1,1] \).

Answer: 2


Question: Determine the order of convergence for the fixed point iteration function \( g(x) = e^{-x} \).

Answer: 1


Question: Given the implicit function defined by \( f(x)^3 - f(x)^2 - x^2 f(x) + x^2 = 0 \), where the maximum value of \( f(x) \) is 1 and the minimum value is 0, find the value of \( f\left( -\frac{4}{3} \right) + f(0) + f\left( \frac{1}{2} \right) \).

Answer: \dfrac{3}{2}


Question: Find the limit of the sequence \(a_n\) as \(n\) approaches infinity, where \(a_n = \left(1-\frac{1}{3}\right)^2 \cdot \left(1-\frac{1}{6}\right)^2 \cdot \ldots \cdot \left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2\) for all \(n \geq 2\).

Answer: \dfrac{1}{9}


Question:
Consider a function $f(x)$ with known values at equidistant points $x_0, x_1, x_2$, and $x_3$, separated by a distance $h$. The values at these points are $f_0, f_1, f_2$, and $f_3$, respectively. Using the two-point formula, the first derivatives at $x_0$, $x_1$, and $x_2$ are approximated as:
\[
f'_2 \approx \frac{f_3 - f_2}{h}, \\
f'_1 \approx \frac{f_2 - f_1}{h}, \\
f'_0 \approx \frac{f_1 - f_0}{h}.
\]
The second derivative at $x_1$ is approximated using the central difference scheme:
\[
(f'_1)'' \approx \frac{f'_2 - 2 f'_1 + f'_0}{h^2}.
\]
Determine if the third derivative of $f(x)$ at $x_1$ can be approximated by the formula:
\[
f'''_1 \approx \frac{1}{h^3} \left( (f_3 - f_2) - 2(f_2 - f_1) + (f_1 - f_0) \right),
\]
and find the order of convergence of this approximation.

Answer: 1


Question: Find the area of the set \( A = \{ (x, y) \in \mathbb{R}^2 \mid (x+y)^4 > 0 \} \).

Answer: \infty


Question:
Consider a semigroup \(X = (X, \cdot)\) where the operation satisfies associativity: \((a \cdot b) \cdot c = a \cdot (b \cdot c)\) for all \(a, b, c \in X\). In the context of monoids, the axiom of identity is typically stated as follows: \(\exists e \in X\), \(\forall a \in X\), \(ae = ea = a\). Suppose we replace this axiom with: \(\forall a \in X\), \(\exists e \in X\), \(ae = ea = a\).

Consider the set \(X = G \sqcup G'\), where \(G = \{e, a\}\) and \(G' = \{e', a'\}\) are copies of a group of order 2, with \(e\) and \(e'\) as identities. Define the operation between an element of \(G\) and an element of \(G'\) to be \(e\). Does \(X\) satisfy the modified axiom? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit of the improper integral: \[ \lim_{x\to \infty} \int_1^x x\,e^{t^2-x^2} \,dt. \]

Answer: \dfrac{1}{2}


Question: Consider an inverse system \( \{A_i\} \) of commutative, unital Noetherian rings with a finite index set. Determine whether the inverse limit \( \varprojlim A_i \) is also a Noetherian ring.

Answer: Yes


Question: Determine if there exists a tangent plane to the surface defined by \( z = x^2 + y^2 \) that is parallel to the vector field \( F(x, y, z) = (-x^3, -x - y, z) \). If such a tangent plane exists, specify the point(s) \((x_0, y_0)\) where this occurs.

Answer: (0, 0)


Question: Given two sets of basis vectors \( V = \{v_1, \ldots, v_n\} \) and \( U = \{u_1, \ldots, u_n\} \), each spanning \( \mathbb{R}^n \), is it possible to select a subset \( T \) such that \( T = \{v_1, \ldots, v_m\} \cup \{u_1, \ldots, u_o\} \) with \( m + o = n \) and \( T \) spans \( \mathbb{R}^n \)?

Answer: Yes


Question: Given an i.i.d. sequence \( \{X_i\} \) where each \( X_i \sim U[0,1] \), define the number \( N = \min\{n \geq 1: X_1 + \ldots + X_n > x\} \) for \( x \in (0,1) \). What is the probability \( \Pr\{N > n\} \)?

Answer: \dfrac{x^n}{n!}


Question:
Use the Newton-Raphson method to find the value of $t$ near $0.04$, correct to four decimal places, for which the amplitude $y$ is $0.880$ in the equation:
\[ y = \left(t + \frac{\pi}{4}\right) + \sin(t) + \frac{1}{8}\sin(3t) \]

Answer: 0.0399


Question: Let $M$ be a smooth manifold with or without boundary, and let $A$ be a compact subset of $M$. Does there exist a compact smooth embedding submanifold $N \subset M$ with or without boundary such that $N \supset A$?

Answer: Yes


Question: Calculate the limit: $$\lim_{(x,y)\rightarrow(0,0)} \frac{x^2y^2}{x^4+y^2+\sin^2(x^3)}.$$

Answer: 0


Question: Does the multivariate t-distribution converge to a multivariate normal distribution as the degrees of freedom increase, similar to the univariate case?

Answer: Yes


Question: What is the minimum number of orthogonal matrices required to express the matrix $M_{i,j}$, which is a standard basis matrix of size $n > 1$, as a linear combination of those orthogonal matrices?

Answer: 2


Question: Given the equations \( \sin(y-x)\cos(x+y) = \frac{1}{2} \) and \( \sin(x+y)\cos(x-y) = \frac{1}{3} \), determine the value of \( \sin(2x) \).

Answer: -\dfrac{1}{6}


Question: Given the curve \( C \) with the equation \( y = x^{1/2} - \frac{1}{3}x^{3/2} + \lambda \), where \( \lambda > 0 \) and \( 0 \le x \le 3 \), the length of \( C \) is \( s = 2\sqrt{3} \). The area of the surface generated when \( C \) is rotated about the x-axis is \( S = 3\pi + 4\pi\sqrt{3} \lambda \). The y-coordinate of the centroid of the region bounded by \( C \), the axes, and the line \( x = 3 \) is \( h \). Given that \( \int_0^3 y^2 \, dx = \frac{3}{4} + 8 \sqrt{3} \cdot \frac{\lambda}{5} + 3\lambda^2 \), show that \( \lim_{\lambda \to \infty} \frac{S}{hs} = 4\pi \).

Answer: 4\pi


Question: Evaluate the limit: $$\lim_{x \rightarrow 0} e^{-\frac{1}{x^2}}\cdot \frac{2}{x^3}$$

Answer: 0


Question: Determine whether the first cohomology group of the structure sheaf of a Grassmannian vanishes.

Answer: 0


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

Answer: 1


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

Answer: 1


Question: Evaluate the expression: \[ \csc^2\left(\frac{\pi}{9}\right) + \csc^2\left(\frac{2\pi}{9}\right) + \csc^2\left(\frac{4\pi}{9}\right) \]

Answer: 12


Question: Determine the order of the conjugacy class of the permutation \( \sigma = (12345) \) in the symmetric group \( S_5 \).

Answer: 24


Question: Evaluate the integral of the function \( f(z) = \frac{2z+3i}{z^{2}+1/4} \) over the unit circle \( C \) centered at zero.

Answer: 4\pi i


Question: Determine the set of cluster points of the set \( \left\{\frac{m}{10^n} : m \in \mathbb{Z}, n \in \mathbb{N}\right\} \).

Answer: \mathbb{R}


Question: Determine whether $10^{8}!$ is greater than $10^{10^9}$. Answer with 'yes' or 'no'.

Answer: no


Question: Find \(\lim_{x\to0}\frac{1-\cos x\cos2x\cos3x}{x^2}.\)

Answer: 7


Question: Evaluate the limit: \[ \lim_{n\to\infty} \frac{2^{\sqrt{\log n}}}{n\log^3 n} \]

Answer: 0


Question: Determine the group generated by the permutations \( \sigma = (3\ 4), \tau = (1\ 3\ 2), \nu = (1\ 2\ 3\ 4) \).

Answer: S_4


Question: Given a matrix $A \in [0,1]^{n \times m}$ that is broad ($n < m$), full-rank, and row-stochastic, i.e., $A \mathbf{1}_m = \mathbf{1}_n$, determine if there exists a square full-rank matrix $B$ such that the matrix $BA^\mathsf{T}(ABA^\mathsf{T})^{-1}$ has rows that sum to 1, i.e., $BA^\mathsf{T}(ABA^\mathsf{T})^{-1} \mathbf{1}_n = \mathbf{1}_m$. Note that the entries of the resulting matrix are not required to be in $[0,1]$, only that the row entries sum to one.

Answer: Yes


Question: Consider a non-homogeneous linear system with $n+1$ equations and $n$ unknowns. If the row echelon form of the augmented matrix of this system has exactly one row of zeros, does the system have exactly one solution? Justify your answer.

Answer: Yes


Question: Let $c \ne 0$ and $a \in \mathbb{R}$. Consider the sequence defined by $x_0 = c$ and $x_{k+1} = \frac{2}{3}(x_k + \frac{a}{x_k^2})$. Compute the limit of this sequence and prove its convergence.

Answer: \sqrt[3]{2a}


Question: Prove that \( \sum_{k=0}^{n} (-1)^k \binom{n}{k} a_{k+1} = 0 \) for \( n \geq 2 \), where \( \binom{n}{k} \) is the binomial coefficient and \( (a_k) \) is an arithmetic sequence of the form \( a_k = a_1 + (k-1)d \) with \( a_1, d \in \mathbb{R} \).

Answer: 0


Question: Let $f$ be an entire function satisfying $|f''(z) - 3| \geq 0.001$ for all $z \in \mathbb{C}$, with $f(0) = 0$, $f(1) = 2$, and $f(-1) = 4$. Find $f(i)$. \( \text{(Note: Consider if the condition on } f''(z) \text{ implies } f \text{ is a polynomial of degree 2.)} \)

Answer: -3 - i


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sum_{k=1}^{\infty}\frac{1}{k^{2}\sqrt[k]{n}}\sin^{2}\left(\frac{n \pi}{k}\right) \]

Answer: 0


Question: Suppose $S$ is a $C^k$ compact hypersurface. If $F$ is a $C^l$ diffeomorphism, is $F(S)$ also a compact hypersurface?

Answer: Yes


Question: Find the value of \( p \) such that the inequality \( \cosh(x) \leq e^{px^2} \) holds for all real numbers \( x \).

Answer: \dfrac{1}{2}


Question: Consider a field extension $K/k$ that is Galois. If $k \subset F \subset K$, is the extension $K/F$ also Galois?

Answer: Yes


Question: Calculate the Vassiliev invariant for the knot K_1.

Answer: 1


Question: Find the coefficient of \( \frac{1}{z} \) in the Laurent series expansion of \( \log\left(\frac{z}{z-1}\right) \) for \(|z| > 1\).

Answer: 1


Question: Determine the minimum possible number of distinct prime divisors of \(19^{4n} + 4\) for a positive integer \(n\).

Answer: 3


Question: Find the sum of all real values of $x$ that satisfy the equation \((x^2+4x+5)^{(x^2+4x+5)^{(x^2+4x+5)}} = 2018\).

Answer: -4


Question: Solve the Volterra type integral equation: \[ \int_0^x (1-x^2+t^2)\phi(t)\,dt = \frac{x^2}{2}. \]

Answer: x e^{x^2}


Question: Determine the number of conjugates in the symmetric group $S_7$ of the permutation \( \sigma = (2,3,1,5,6,4,7) \).

Answer: 280


Question: In the context of Fourier series, can the coefficients of the full Fourier series always be obtained from the complex form coefficients \( C_n \)? Answer "yes" or "no."

Answer: yes


Question: Determine the minimum sample size required to estimate the proportion of smokers in a population with a margin of error of 0.01 and a confidence level of 95%, using the Central Limit Theorem. Assume the proportion of smokers is unknown.

Answer: 9604


Question: Find the minimum value of the function \( f(x) = \sqrt{(1-x^2)^2+(2-x)^2} + \sqrt{x^4-3x^2+4} \).

Answer: \sqrt{5}


Question: Calculate the area that lies inside the polar curve \( r = 2a\cos(2\theta) \) and outside the circle \( r = \sqrt{2}a \).

Answer: a^2


Question: If a linear functional \( \phi(x): H\rightarrow\mathbb{C} \) on a Hilbert space \( H \) can be represented as \( \phi(x) = (x_0, x) \) for some \( x_0 \in H \), is the linear functional necessarily continuous? Justify your answer.

Answer: Yes


Question: Determine the range of \(x\) for which the series \(\sum_{n=1}^\infty \frac{x^{n^2}}{n(n+1)}\) converges.

Answer: [-1, 1]


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

Answer: \infty


Question: Twelve politicians are seated at a round table. A committee of five is to be chosen such that no two committee members are immediate neighbors. In how many ways can this committee be selected?

Answer: 36


Question: Determine whether the flow with Lagrange path trajectories given by \(x(\alpha,t)=(\alpha_1\cos(t)+\alpha_2\sin(t),\alpha_2\cos(t)-\alpha_1\sin(t),\alpha_3)\) is incompressible. Calculate the divergence of the velocity field and state whether the flow is incompressible.

Answer: 0


Question: Determine the number of terms required in the Taylor series for \( \ln(1+x) \) centered at \( x = 0 \) to estimate \( \ln(1.4) \) to within an error of 0.0005.

Answer: 6


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{e^{-x^{-2}}}{x^{100}} \]

Answer: 0


Question: Determine the values of $p$ for which the series \( \sum_{n=2}^\infty \frac{1}{n(\ln n)^p} \) converges.

Answer: p > 1


Question: Evaluate the integral \( \int\int_A(3x^2y-y^3)\, dx\, dy \), where \( A = \{(x, y) \mid x^2 + (y-1)^2 \leq 1\} \).

Answer: -\pi


Question: Evaluate the integral \( \oint_{C} \text{Re}(z) \, dz \) where \( C \) is the square path defined by \( \{x+iy: \vert x \vert \leq 1, \vert y \vert \leq 1\} \) traversed in a counterclockwise direction.

Answer: 4i


Question: Does there exist a transformation \( T: \mathbb{R}^2 \to \mathbb{R} \) such that for any vector \( v \) in the \( xy \) plane and any real constant \( r \), the equality \( T(rv) = rT(v) \) holds, but \( T \) is not a linear transformation?

Answer: Yes


Question: Consider a Markov chain model for rolling a fair six-sided die. What is the probability that the sequence "1 followed by 2" occurs before the sequence "two consecutive sixes"?

Answer: \dfrac{7}{13}


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

Answer: 0


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

Answer: \ln 4


Question: Determine the positive integer \( n \leq 5 \) such that \( \int_{0}^{1} e^x (x-1)^n \, dx = 16 - 6e \).

Answer: 3


Question: Determine the distance of the function \( \mathrm{id}_{[0,1]}(x) = x \) from the closed subspace \( \left\{ f \in C[0,1] \: : \: \int_0^{\frac{1}{2}}f(t) \, dt = f(0) \right\} \) in the space \( C[0,1] \) with the supremum norm.

Answer: \dfrac{1}{12}


Question: If every function from a topological space X to itself is continuous, must the space X be equipped with the discrete topology?

Answer: No


Question: Given a bacterium that behaves in one of two ways at the end of the day: it may die and produce 2 descendants with probability $p$, or die without producing any descendants with probability $1-p$. Let $X_k$ be the number of bacteria after $k$ days, with $X_0 = 1$. Using the Law of Total Variance, find $\text{Var}(X_{k+1})$. You are given that $E[X_{k+1} | X_k] = 2pX_k$ and $E[X_k] = (2p)^k$. Evaluate the expression $E(V(X_{k+1}|X_k))$.

Answer: 4p(1 - p)(2p)^k


Question: Evaluate the integral \( \iint_D x\,dx\,dy \) where the region \( D \) is defined by \( 4x^2+y^2 \leq 1 \) and \( 2y+4x^2-1 \leq 0 \).

Answer: 0


Question: Determine the smallest integer $n$ such that the Simpson's Rule approximation of the integral \( \int_0^1 19e^{x^2} \, dx \) is accurate to within 0.0001.

Answer: 22


Question: Determine the number of arrangements of the 10 letters ABCDEFGHIJ such that none of the patterns ABE, BED, or HID appear in the arrangement.

Answer: 3513600


Question:
Determine which of the following functions is uniformly continuous on the interval \((0,1)\):

1. \( f(x) = \cos x \cos \frac{\pi}{x} \)
2. \( f(x) = \sin x \cos \frac{\pi}{x} \)

Answer: 2


Question:
Consider two natural numbers $a \leq b$. Let $P(n)$ be a statement involving $n$. Determine if the following formula implies that $P(n)$ holds for all $n$ in the interval $[a, b]$:

$$P(b) \land (\forall n)(a \leq n < b \land (\forall m)(n < m \implies P(m)) \implies P(n))$$

Does this imply $$(\forall n)(n \in [a,b] \implies P(n))?$$

Answer: Yes


Question: Compute \( E[|X^2 - 16|] \) where \( X \sim U(-4, 7) \), with \( U \) being a continuous uniform distribution.

Answer: \dfrac{391}{33}


Question: Let \( \mathbb{F}_{16} = \mathbb{F}_2[x]/(x^4 + x^3 + 1) \) and let \( \alpha \) be a root of \( x^4 + x^3 +1 \). Compute the minimal polynomial of \( \alpha^2 \) over \( \mathbb{F}_2 \) in \( \mathbb{F}_{16} \).

Answer: x^4 + x^3 + 1


Question: Determine the value of \(c\) such that \(F(x) = c\int_{-\infty}^x e^{-|x|} \, dx\) is a distribution function.

Answer: \dfrac{1}{2}


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

Answer: 2


Question: Calculate the limit: \[ \lim_{x \to 0^+} x^{x^{x-1}} \]

Answer: 0


Question: Find the coefficient of \( x^{21} \) in the expansion of the polynomial \((x^3 + x^4 + \cdots + x^{10})^4\).

Answer: 204


Question:
Let $f: \mathbb{C} \setminus \{0\} \to \mathbb{C}$ be an analytic function such that for any closed contour $\gamma$, the line integral of $f$ over $\gamma$ is zero. Determine which of the following statements are true:

1. $f$ has a removable singularity at $0$.
2. $f$ cannot have an essential singularity at $0$.
3. There exists a holomorphic function $F: \mathbb{C} \setminus \{0\} \to \mathbb{C}$ such that $F'(z) = f(z)$ for all $z \in \mathbb{C} \setminus \{0\}$.
4. There exists a holomorphic function $F: \mathbb{C} \to \mathbb{C}$ such that $F'(z) = f(z)$ for all $z \in \mathbb{C}$.

Answer: 3


Question: Calculate the limit: \[ \lim_{x\to\infty}\dfrac{\sqrt{x+1}-2\sqrt{x+2}+\sqrt{x}}{\sqrt{x+2}-2\sqrt{x}+\sqrt{x-4}} \]

Answer: \dfrac{3}{2}


Question: Evaluate the limit: $$\lim_{x\to\infty} \frac{x^{2011} + 2010^x}{-x^{2010} + 2011^x}$$

Answer: 0


Question: Evaluate the line integral \( \int_C {F \cdot \ dr} \) for the vector field \( F(x,y,z) = \langle 2xyz {e}^{x^2y},z^2+x^2z {e}^{x^2y}, {e}^{x^2y} + 2yz-3z^2 \rangle \) over all possible smooth curves \( C \) from the point \((0,-1,1)\) to the point \((\sqrt{\ln(2)},1,1)\).

Answer: 3


Question: Let \( f:[0,1] \to \mathbb{R} \) be a twice continuously differentiable function such that \( f(0) = f(1) = 0 \) and \( f''(x) \geq -1 \) for all \( x \in [0,1] \). Determine the maximum possible value of \( f(x) \) on the interval \([0,1]\).

Answer: \dfrac{1}{8}


Question:
Solve the equation over the real numbers without using calculus:
\[ \left(\frac{1}{2}\right)^{1+x} + \left(\frac{1}{6}\right)^{x} - \sqrt{2} \cdot \left(\frac{\sqrt{2}}{6}\right)^x = 1 \]

Answer: -1


Question: Let $f(x) = \begin{cases} x^n \cos(x^{-n}) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases}$ for $x \in [0, 0.5]$. Find the value of $\lim_{n \to \infty} f'(x)$.

Answer: 0


Question: Let $N_1$ and $N_2$ be normal subgroups of a group $G$. Suppose $N_i$ has a complement $L_i$ for $i=1,2$, and $N_2 \leq L_1$. Determine whether $N_1N_2$ necessarily has a complement in $G$. Is the statement true?

Answer: True


Question: Let $I$ be an ideal of a Noetherian ring $R$. Is it true that there exists $n \in \mathbb{N}$ such that $(\sqrt{I})^n \subseteq I$? Justify your answer.

Answer: Yes


Question: Let S be a topological space that is locally compact, Hausdorff, and second countable, but not compact. Consider S*, the one point compactification of S. Is S* necessarily second countable?

Answer: Yes


Question: Find the normalization constant \( a \) for the probability density function \( f(x) = a x \sin(x) e^{-x} \) defined for \( 0 < x < \infty \) and 0 otherwise.

Answer: 2


Question: If $(X_n)_n$ converges to $T$ in probability and also $(X_n)_n$ converges almost surely to $Y$, is it true that $P(T = Y) = 1$?

Answer: Yes


Question: Calculate \( \lim_{x\to 1^{-}} \frac{e^{1 / \ln{x}}}{\ln^2{x}} \).

Answer: 0


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

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


Question: Determine whether there exists a subset \( M \subseteq S_\omega \) such that \( \langle M \rangle = S_\omega \) and for every element \( m \in M \), the group generated by \( M \setminus \{m\} \) is not equal to \( S_\omega \).

Answer: Yes


Question: Solve the equation \(x^2(\log_{10}(x))^5=100\) for \(x\).

Answer: 10


Question:
Determine whether the following equality holds for functions $f \in C_0^\infty(\mathbb{R}^n)$ and $g \in L^1(\mathbb{R}^n)$:
\[ \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} f(x-y)g(y) \, dy \, dx = \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} f(x-y)g(y) \, dx \, dy. \] If it holds, provide a justification.

Answer: Yes


Question:
Consider a field $F$ contained in a commutative ring $R$. Let $x_1, \dots, x_n, y$ be elements of $R$ such that:

- $R = F[x_1, \dots, x_n, y] = \{ \sum_{i_1 \dots i_n, j} a_{i_1 \dots i_n, j} x_1^{i_1} \dots x_n^{i_n} y^j \mid a_{i_1 \dots i_n, j} \in F \}$
- $x_1, \dots, x_n$ are algebraically independent, while $x_1, \dots, x_n, y$ are algebraically dependent.

Is it true that for any $r \in R$, the elements $x_1, \dots, x_n, r$ are algebraically dependent?

Answer: Yes


Question: Evaluate the integral \( \int_{\mathbb{R}} e^{-itx} \, dx \). Discuss the convergence of this integral and explain how it relates to finding the density of a random variable given the characteristic function \( \phi(t) = \frac{1}{4} + \frac{1}{4}e^{-it} + \frac{1}{2}\frac{3}{4-e^{2it}} \).

Answer: 2\pi \delta(t)


Question: Let $P = (0, q)$ be the hyperbolic midpoint of $A = (0, a)$ and $B = (0, a + b)$ in the Hyperbolic/Half-Plane Model. Compute the limit of $q$ as $a \rightarrow \infty$, as a function of the fixed number $b$.

Answer: \dfrac{b}{2}


Question: Evaluate the integral \( \int_{-\infty}^{+\infty} \frac{e^{\mathrm{i} x}}{x} \, \mathrm{d}x \).

Answer: i\pi


Question:
Evaluate the integral \( \int f^{-1}(x) \, dx \) given that \( \int f(x) \, dx = g(x) \). Choose the correct answer from the following options:

1. \( g^{-1}(x) \)
2. \( x f^{-1}(x) - g(f^{-1}(x)) \)
3. \( x f^{-1}(x) - g^{-1}(x) \)
4. \( f^{-1}(x) \)

Answer: 2


Question: Can any set that forms a basis for a subspace of a vector space V be extended to form a basis for V? Provide a justification for your answer.

Answer: Yes


Question: Let \( P(x) \) be a polynomial of degree 3 such that \( P(-1) = 10 \), \( P(1) = -6 \), and \( P(x) \) has a local maximum at \( x = -1 \). Additionally, \( P'(x) \) has a local minimum at \( x = 1 \). Find the distance between the local maximum and local minimum of the curve \( P(x) \).

Answer: 4\sqrt{65}


Question: Find the smallest constant \( C \) such that for all real numbers \( x \) and \( y \), the inequality \( 1+(x+y)^2 \leq C(1+x^2)(1+y^2) \) holds.

Answer: \dfrac{4}{3}


Question: Given the system of differential equations \(x' = \begin{bmatrix} 0 & 1\\ -4 & 4 \end{bmatrix}x\), one solution is \(x^1 = \begin{bmatrix} 1\\ 2 \end{bmatrix}e^{2t}\). Suppose the second solution is of the form \(x^2 = \begin{bmatrix} a\\ b \end{bmatrix}te^{2t}\). Determine the values of \(a\) and \(b\) by substituting \(x^2\) back into the original equation and showing that \(a = b = 0\).

Answer: 0


Question:
Determine the number of positive integers \( n \) less than or equal to 1988 for which the function \( f(n) = n \), where the function \( f \) is defined on the positive integers by:
\[ f(1) = 1, \quad f(3) = 3, \]
\[ f(2n) = f(n), \]
\[ f(4n + 1) = 2f(2n + 1) - f(n), \]
\[ f(4n + 3) = 3f(2n + 1) - 2f(n) \]
for all positive integers \( n \).

Answer: 92


Question: Let $(R,m)$ be a $1$-dimensional noetherian local domain and $S$ its integral closure. Prove or disprove that $mS = J(S)$, where $J(S)$ is the Jacobson radical of $S$.

Answer: mS \neq J(S)


Question: Determine the asymptotic distribution of the likelihood ratio test statistic \( \text{LR}(\mu_0,\mu_1) = -2\ln\frac{L(\mathbf{x}\mid\mu_0,\sigma^2)}{\max\{ L(\mathbf{x}\mid\mu_0,\sigma^2), L(\mathbf{x}\mid\mu_1,\sigma^2) \}} \) under the null hypothesis \( H_0: \mu = \mu_0 \).

Answer: \chi^2_1


Question: Are all Steiner Triple Systems regular hypergraphs, meaning that every vertex has the same degree?

Answer: Yes


Question: Calculate the volume of the solid obtained by rotating the region bounded by the curve \( x = -y^2 + 6y \) and the y-axis about the line \( y = -7 \).

Answer: 720\pi


Question: Evaluate the integral \( \int_{-\infty}^{\infty}\frac{1}{(1+y^2)^2}dy \) using the residue method.

Answer: \dfrac{\pi}{2}


Question: Use a triple integral to find the volume of the solid enclosed by the cylinder \(x^2+y^2=9\) and the planes \(y+z=5\) and \(z=1\).

Answer: 36\pi


Question: Compute the conditional expectation \( E\left(W_6 | W_2 , W_4\right) \) where \( W \) is a standard Brownian motion.

Answer: W_4


Question:
Given an ideal $I$ of $\mathbb{R}[X_1,X_2,X_3,X_4,X_5]$ generated by two unknown polynomials, suppose you know two homogeneous polynomials $p_1 \in I$ and $p_2 \in I$ such that:

- $p_1$ is of degree 2 and, up to a multiplicative constant, is the polynomial of smallest degree in $I$.
- $p_2$ is of degree 3 and, up to a linear combination with $p_1$, is the only polynomial of degree 3 in $I$.

Can you conclude that $p_1$ and $p_2$ generate $I$?

Answer: Yes


Question:
Provide an example of a linear operator $T \in L(V)$ on the vector space $V$ of infinite sequences of finite length, such that:

1. $T$ admits an adjoint, i.e., there exists $T^*$ such that $\langle Ta, b \rangle = \langle a, T^*b \rangle$ for all $a, b \in V$.
2. $T^*T$ is the identity operator on $V$.
3. $TT^*$ is a projection operator but not the identity operator, meaning $(TT^*)^2 = TT^*$ and $TT^* \neq I$.

Consider whether the left-shift operator $T: (a_1, a_2, a_3, \ldots) \to (a_2, a_3, \ldots)$ satisfies these conditions.

Answer: T


Question: Calculate the value of \( E[X|XY] \) where \( X \) and \( Y \) are two independent standard normal variables.

Answer: 0


Question: An insurance policy covers a loss X with a density function given by \( f(x) = \frac{3}{8}x^2 \) for \( 0 \le x \le 2 \). The time (in hours) to process a claim of size x, where \( 0 \le x \le 2 \), is uniformly distributed on the interval from x to 2x. Calculate the probability that a randomly chosen claim on this policy is processed in three hours or more.

Answer: \dfrac{11}{64}


Question: Given four points \(A, B, C, D\) in space such that \(|\vec{AB}| = 3\), \(|\vec{BC}| = 7\), \(|\vec{CD}| = 11\), and \(|\vec{DA}| = 9\), find the value of \(\vec{AC} \cdot \vec{BD}\).

Answer: 0


Question: Let \( (\mathbb{R}^2, \mathfrak{M}, m) \) be a Lebesgue measurable space, and let \( f \colon \mathbb{R} \rightarrow \mathbb{R} \) be a continuous function. Define the set \( E = \{(x, f(x)): x \in \mathbb{R}\} \). Calculate the Lebesgue measure of \( E \), denoted as \( m(E) \).

Answer: 0


Question: Evaluate the integral \( \int_{-1}^1 \frac{x}{2}\ln\frac{1+x}{1-x}\,dx \).

Answer: 1


Question: Consider a probability space $(\Omega, \Gamma, \mathbb{P})$ and a sequence of events $A_1, A_2, \dots \in \Gamma$. If $\mathbb{P}(A_i) \geq \frac{1}{2}$ for each $i$, is it true that there exists an $\omega \in \Omega$ that belongs to infinitely many of the sets $A_i$?

Answer: Yes


Question: Find the angle between the curves given by the equations \(x^2 + y^2 = 8\) and \(xy = 4\) at their intersection points.

Answer: 0


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

Answer: 0


Question: Determine the number of elements of order 5 in a group of order 100.

Answer: 24


Question: Are finitely generated submodules of a $p$-adic module direct summands?

Answer: No


Question: If $T$ is a nonlinear surjective isometry from the Lipschitz-free space $\mathcal{F}(M)$ to $\mathcal{F}(N)$, where $M$ and $N$ are metric spaces, is $M$ homeomorphic to $N$?

Answer: Yes


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a non-constant continuous function satisfying the functional equation \((e^x-1)f(2x)= (e^{2x}-1)f(x)\) and the condition \(f'(0) = 1\). Determine \( \lim_{x \to 0} \left(\frac{f(x)}{x}\right)^{\frac{1}{x}} \).

Answer: \sqrt{e}


Question: Let \( F = \mathbb{Q}(\xi_p) \) be the \( p^{th} \) cyclotomic field. Determine the norm \( N(1 + \xi_p) \).

Answer: 1


Question: Find the sum of the infinite series: \[ \sum_{n=1}^{\infty} \frac{n2^n}{(n+2)!} \]

Answer: 1


Question: Evaluate the limit: \[ \lim_{q\to\infty} \int_{0}^{\infty} \left(e^{i(q+i\delta)r} - e^{-i(q-i\delta)r}\right) dr \]

Answer: 0


Question: Let $a$ and $b$ be any two positive real numbers such that $a \geq lb$ where $l \geq 1$. Suppose $\gamma$ is any real number such that $0 \leq \gamma \leq 2\pi$. Is it true that \[|l+e^{i\gamma}|(a+mb) \leq (l+m)|a+e^{i\gamma}b|\] for any $m \geq 1?$

Answer: Yes


Question: Given a finite-dimensional vector space $V$ and its dual space $V^*$, suppose we have a basis $\{f_1, \ldots, f_n\}$ for $V^*$. Is it possible to find vectors $v_1, \ldots, v_n \in V$ such that the dual basis $\{v_1^*, \ldots, v_n^*\}$ satisfies $v_i^* = f_i$ for each $i$?

Answer: Yes


Question:
Let \(f(z)\) be an analytic function on the domain \(D = \{(x,y) : |x| \leq a, |y| \leq b, a \geq b\}\). If \(|f(z)| \leq M\) on the boundary of \(D\), which of the following can be taken as the upper bound for \(|f'(0)|\)?

(A) \(\frac{2M(a+b)}{\pi ab}\)

(B) \(\frac{2\pi ab}{M^2}\)

(C) \(\frac{2M^2}{2\pi ab}\)

(D) \(\frac{2M(a+b)}{\pi b^2}\)

Answer: A


Question: Find the area enclosed by the curve given by the parametric equations \( x = \cos^3(t) \) and \( y = \sin^3(t) \) for \( t \in [0, 2\pi] \). Use the formula \(-\int_C y \, dx\) to compute the area.

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


Question:
In an N*N grid puzzle, each cell contains a distinct numbered tile from 1 to N*N-1, except for one empty cell represented as 0. Bob can use two types of moves:

1. Move Type 1: Pull out any tile and place it in any cell of choice.
2. Move Type 2: Move any tile to an adjacent cell in any of the four directions, regardless of whether the cell is empty.

The goal is to arrange the tiles such that the first row contains tiles 0, 1, 2, ..., N-1, the second row contains N, N+1, N+2, ..., 2N-1, and the last row contains N*(N-1), ..., N*N-1.

Calculate the extra work Bob has to do if he uses Move Type 2 instead of Move Type 1. If the initial configuration can be solved optimally using Move Type 1 in X moves and using Move Type 2 in Y moves, find the extra work as (Y-X).

Example: For M=N*N=4 and initial configuration:

0 2
3 1

The answer is 1, as X=3 and Y=4, so Y-X=1.

Answer: 1


Question: Determine whether the sequence \( \left( \frac{\sum_{j=1}^n (-1)^j j^k}{\sum_{j=1}^n j^k} \right)_{n \in \mathbb{N}} \) converges or diverges for some \( k \in \mathbb{N} \). Justify your answer.

Answer: 0


Question: Compute the infimum of the expression \( \frac{\alpha - \mu}{1 - \alpha \cdot \mu} \) for distinct \( \alpha, \mu \in \beta \), where \( \beta = \{1 - e^{-n}\}_{n > 0} \).

Answer: -1


Question: Find the minimum value of the function \( (r-1)^2 + \left(\frac{s}{r} -1\right)^2 + \left(\frac{t}{s}-1\right)^2 + \left(\frac{4}{t}-1\right)^2 \) subject to the constraints \( 1 \le r \le s \le t \le 4 \), where \( r, s, t \in \mathbb{R} \).

Answer: 12 - 8\sqrt{2}


Question: Find the value to which the series \( \frac{\pi}{2}-\frac{\pi^{3}}{8 \cdot 3!} + \frac{\pi^{5}}{32 \cdot 5!} - \cdots \) converges. The series can be expressed as \( \sum_{n=1}^{\infty} (-1)^{n+1} \, \frac{\pi^{2n-1}}{2^{2n-1} \cdot (2n-1)!} \).

Answer: 1


Question: Find the volume of the solid B bounded by the parabolic cylinder \(x = (y − 4)^2 + 3\) and the planes \(z = x + 2y − 4\), \(z = x + 4y − 7\), and \(x + 2y = 11\).

Answer: 4


Question:
Determine the number of positive integer solutions \((a, b, c)\) that satisfy the equation:
\[ \frac{a^2+b^2-c^2}{ab}+\frac{c^2+b^2-a^2}{cb}+\frac{a^2+c^2-b^2}{ac}=2+\frac{15}{abc} \]

Answer: 9


Question: Let \( A \subseteq S^{d-1} = \{x \in \mathbb{R}^{d}: ||x||=1 \} \) be a subset of the unit sphere in \( \mathbb{R}^d \). Define the set \( E = \{x \in \mathbb{R}^d: ||x||<1 \} \cup A \). Is the set \( E \) convex?

Answer: Yes


Question: Given the polynomial relationship between two variables, \(b = p_1a^3 + p_2a^2 + p_3a\), find the second derivative of \(b\) with respect to \(a\), \(\frac{d^2b}{da^2}\). Verify that the expression for the second derivative is \(15p_1a^2 + 6p_2a + p_3\).

Answer: 6p_1a + 2p_2


Question:
Given the function \( g = \sqrt{1+\dot{x}^2} \), where \( \dot{x} \) is a function of time, simplify the Euler-Lagrange equation:
\[ \frac{\partial g}{\partial x} - \frac{d}{dt} \left[ \frac{\partial g}{\partial \dot{x}} \right] = 0 \]
to show that \( \ddot{x} = 0 \).

Answer: \ddot{x} = 0


Question: Given the transition matrix \( M = \begin{pmatrix} 0.25 & 0.5 & 0.25 \\ 0.5 & 0.25 & 0.25 \\ 0.5 & 0.25 & 0.25 \end{pmatrix} \) of a Markov chain with states \( S = \{0, 1, 2\} \), calculate the expected number of steps to reach state 1 starting from state 2.

Answer: \dfrac{20}{7}


Question: Given the expression \( y = w^T X + b \), where \( w \) and \( X \) are both \([13, 1]\) matrices, find the derivative \( \frac{dy}{dw} \).

Answer: X


Question: Find the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfies the equation \( f(xf(y) + f(x)) = 2f(x) + xy \) for all real numbers \( x \) and \( y \).

Answer: f(x) = x + 1


Question: Find the order of the group generated by the matrices \(\begin{pmatrix}0&1\\-1&0\end{pmatrix}\) and \(\begin{pmatrix}0&i\\-i&0\end{pmatrix}\) under multiplication.

Answer: 8


Question: For the scalar ordinary differential equation \(\frac{du}{dt} = \sin(t)u\), find a Floquet transformation \(u = p(t)z\), where \(p(t)\) is periodic, that reduces the ODE to a constant coefficient form \(\frac{dz}{dt} = rz\). Determine the value of \(r\). Use the condition \(p' = (\sin(t) - r)p\) and \(p(0) = p(2\pi)\) to find \(r\).

Answer: 0


Question:
Given an increasing and continuous function \( f: \mathbb{R} \to \mathbb{R} \), and small positive numbers \( a > 0 \) and \( \epsilon > 0 \), determine if there exist values of \( x \) and \( y \) such that for some \( \epsilon \), for all \( \delta > 0 \), there exists \( a < \delta \) satisfying the inequality:
\[ a(f(y+\epsilon)-f(y)) < (f(x+a)-f(x))(1-a). \]

Answer: Yes


Question:
Determine whether the following statement is equivalent to the definition of equicontinuity of a family of functions $F$ at a point $x_0$ in metric spaces $X$ and $Y$:

"The family $F$ is equicontinuous at a point $x_0 \in X$ if and only if \(\lim_{\delta \to 0} \sup_{f \in F, x \in \{x \in X: d(x_0, x) < \delta\}} d(f(x_0), f(x)) = 0\)."

Answer: Yes


Question: For $t > 0$, how many times does the equation $S(t) = C(t)$ hold true, where $S(t) = -1200\cos\left(\frac{\pi}{6}t\right) + 1500$ and $C(t) = 1200(1.025)^{t}$? Here, $t$ represents the number of months since January 1, 2004.

Answer: 6


Question: Evaluate the limit: $$\lim_{n \to \infty}\left(\frac{n^4-3n^3-n^2+2n-1}{n^5+n^4-n^3-3n^2-3n+1}\right)^{\left(\frac{6n^5-2n^4-2n^3+n^2-2n}{9n^4-2n^3+n^2+3n}\right)}$$

Answer: 0


Question: Let \( G \) be a finite group with order \( p^a t \), where \( p \) is a prime number and \( p \nmid t \). Suppose \( P \) is a \( p \)-subgroup of \( G \), and let \( N_G(P) \) be the normalizer of \( P \) in \( G \). Assume \( P \) is a Sylow \( p \)-subgroup in \( N_G(P) \). Is it always true that \( p \nmid [G : N_G(P)] \)?

Answer: Yes


Question:
Identify which of the following metrics is not complete on \(\mathbb{R}\):

1. \(|x-y|\)
2. \(|\arctan(x) - \arctan(y)|\)
3. \(|x^3 - y^3|\)
4. \(|x^{1/3} - y^{1/3}|\)

Answer: 2


Question: Four distinct points on the curve \( y = 2x^4 + 7x^3 + 3x - 5 \) are collinear. The arithmetic mean of the x-coordinates of these four points is expressed as \( \frac{a}{b} \), where \( a \) and \( b \) are coprime integers. Find \( a + b \).

Answer: 1


Question: Consider a sequence of independent two-dimensional vectors of random variables \((A_n, B_n)_{n=1}^{\infty}\), where each vector is uniformly distributed on the square \([-2,2] \times [-2,2]\). Define \(V_n=(S_n, T_n) = (\sum_{i=1}^n A_i, \sum_{i=1}^n B_i)\) and \(|V_n| = \sqrt{(S_n)^2+(T_n)^2}\). Determine the constant \(c\) such that \(\lim_{n \to \infty} P(|V_n|<c\sqrt{n})=0.95\).

Answer: 2.83


Question: Determine the number of subgroups of order \(p^2\) in the group \(\mathbb{Z}_{p^3} \oplus \mathbb{Z}_{p^2}\).

Answer: p^2 + p + 1


Question: Determine the number of linear Boolean functions from \({0, 1}^n\) to \({0, 1}\). A function \(F\) is linear if for all \(x, y \in {0, 1}^n\), \(F(x \oplus y) = F(x) \oplus F(y)\).

Answer: 2^n


Question: Evaluate the sum \[ 1 - \frac{1}{2} {n \choose 1} + \frac{1}{3} {n \choose 2} + \ldots + (-1)^n \frac{1}{n+1} {n \choose n}. \]

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


Question: Determine the last digit of the number \(7^{7^{7^{7^{7}}}}\).

Answer: 3


Question:
Let $B\subset \mathbb{R}^2$ be an open ball within a domain $\Omega$. Consider $y_1, y_2 \in H_0^1(\Omega) \cap H^2(\Omega)$ satisfying:
\[ -\Delta y_i = 1 \text{ for } i=1,2 \text{ on } B \]
\[ 0 \leq y_1(x) \leq y_2(x) \text{ on } \bar{B} \]
Is there a point $x \in \bar{B}$ such that $|\nabla y_1(x)| \leq |\nabla y_2(x)|$?

Answer: Yes


Question: Is it possible for a space to have two norms such that the topology induced by one norm is strictly finer than the other, while the space remains complete under both norms?

Answer: No


Question:
Given a triangle with sides and angles as described, find the angle \(x\) using the following relationships:

1. Assume \(\overline{\rm BC}=1\).
2. \(\overline{\rm BD}= 2\sin{\frac{x}{2}}\).
3. \(\overline{\rm BH}= 4\sin^2{\frac{x}{2}}= 2(1-\cos{x})\).
4. \(\overline{\rm CH} = 2\cos{x}-1\).
5. \(\overline{\rm CE}=\frac{2\cos{x}-1}{\sqrt{2-2\cos{x}}}\).
6. Let \(\overline{\rm DE}=y\), and since \(\bigtriangleup DCE = \bigtriangleup HCE\),
   \[\frac{1}{2}y\sin{50^{\circ}}=\frac{1}{2}\sin{x}\frac{(2\cos{x}-1)^2}{2-2\cos{x}}\]
7. Apply the law of cosines to \(\bigtriangleup DEC\):
   \[y^2+1-2y\cos{50^{\circ}}=\frac{(2\cos{x}-1)^2}{2-2\cos{x}}\]

Solve the system of equations to find the angle \(x\).

Answer: 40^\circ


Question: Find the residue of the function \( f(z) = e^z \csc^2 z \) at the poles \( z = k\pi \), where \( k \) is an integer.

Answer: e^{k\pi}


Question: Find the entire length of the cardioid given by the polar equation \( r = 1 - \cos\theta \).

Answer: 8


Question: Lucas and Monk play chess, and the first to win a match receives a book as a reward. Each match can result in a win, loss, or tie. The probability of there being $k$ matches until someone wins is $\frac{1}{2^{k+2}}$, where $k=1,2,...$. Lucas' chance of winning the book is twice that of Monk. What is the probability that Monk will win the book?

Answer: \dfrac{1}{12}


Question: Find the value of the limit: \[ \lim_{n\to \infty }\left(\sqrt[\leftroot{-2}\uproot{2}n+1]{(n+1)!}-\sqrt[\leftroot{-2}\uproot{2}n]{n!}\right). \]

Answer: \dfrac{1}{e}


Question: Evaluate the integral \( \int\limits_{-\infty}^{\infty} x e^{-|(x-u)|} \, dx \).

Answer: 2u


Question: Given the function \(f(x) = \frac{ax+b}{cx+d}\) with \(abcd \neq 0\) and the condition \(f(f(x)) = x\) for all \(x\) in the domain of \(f\), find the value of \(a + d\).

Answer: 0


Question: Verify that the line integral \( \int_{C(0,r)} \frac {\log(1+z)}{z} dz = 0 \) for a circle \( C(0,r) \) of radius \( r \) centered at 0. Then, conclude that for every \( r \in (0,1) \), the integral \( \int_0^{\pi}\log(1+r^2+2r\cos t)dt = 0 \).

Answer: 0


Question:
Compute \( \lim_{n\rightarrow\infty}\frac{1}{n}\left\{(2n+1)(2n+2)\cdots(2n+n)\right\}^{1/n} \). If \( f(n)=\frac{1}{n}\Big\{(2n+1)(2n+2)\cdots(2n+n)\Big\}^{1/n} \), then \( \lim\limits_{n\to\infty}f(n) \) equals:
\[
\begin{array}{}
(\mathrm{A})\ \frac{4}{e}\qquad&(\mathrm{B})\ \frac{27}{4e}\qquad&(\mathrm{C})\ \frac{27e}{4}\qquad&(\mathrm{D})\ 4e
\end{array}
\]

Answer: B


Question: Is a finite Hausdorff space necessarily discrete? Provide a justification for your answer.

Answer: Yes


Question: In a finite group of order $g$, is it necessarily true that at least one element must have an order equal to $g$? Provide a justification for your answer.

Answer: No


Question: Find the limit of the sequence \( a_n = \frac{n}{\pi}\sin(2\pi e n!) \) as \( n \to \infty \).

Answer: 2


Question:
Determine the values of $a > 0$ for which the following limit holds:
\[ \lim_{(x,y)\to(0,0)}\frac{|x-y|^{a-1}}{(x+y)\sqrt{x^2+y^2}}=0. \]

Answer: a > 3


Question: Find the range of the function \( f(x) = \sin\log_e\left( \frac{\sqrt{4-x^2}}{1-x} \right) \).

Answer: [-1, 1]


Question: Determine the Galois group of the polynomial \(x^5 - x - 1\) over \(\mathbb{Q}\).

Answer: S_5


Question: Determine whether the following statement is true or false: Let $G$ be a finite group with subgroups $H$ and $P$. If $H$ is normal in $G$, $P$ is normal in $H$, and $P$ is a Sylow subgroup of $G$, then $P$ is normal in $G$. Provide a proof or counterexample.

Answer: True


Question: Let \(A\) and \(B\) be \(n \times n\) square real matrices. Prove that if \((AB)^k = I_n\) for some integer \(k\), then \((BA)^k = I_n\).

Answer: (BA)^k = I_n


Question: Calculate the greatest common divisor: \[ \gcd\left(\frac{2^{40}+1}{2^8+1}, 2^8+1 \right) = k \]

Answer: 1


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} \int _0 ^ \pi \frac {\sin((n+1/2)t) }{t} \, dt \]

Answer: \dfrac{\pi}{2}


Question: Determine the radius of convergence for the power series \( \sum_{k=1}^{\infty}( \sqrt[k]{e}-1) {x^k} \).

Answer: 1


Question: Evaluate the integral \(\int_{|z-\pi|=\pi} \frac{\sin(\pi z)}{z^{2}-\pi^{2}} \, dz\) using Cauchy's integral formula.

Answer: i \sin(\pi^2)


Question: Let \( f(x) = \frac{-x^2}{x^2+1} \). If \( g(x) \) is the inverse function of \( f(x) \) and \( f(1) = -\frac{1}{2} \), find \( g'\left(-\frac{1}{2}\right) \).

Answer: -2


Question:
Determine the value of the expression:
\[ \inf \left\{\sup \left\{ \frac{2n}{2+n(-1)^n}:n>k\right\}:k\in\mathbb{N}\right\} \]

Answer: 2


Question: Determine whether the sequence \( \{(n+1)^{\alpha}-n^{\alpha}\}_{n=1}^{+\infty} \), where \( 0 < \alpha < 1 \), converges or diverges. If it converges, find its limit.

Answer: 0


Question: Let $K$ be a finite extension of $\mathbf{Q}_p$. Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion of $F$ at $v$, we have a topological isomorphism of topological field extensions of $\mathbf{Q}_p$: $$F_v\simeq K?$$ In other words, does any $p$-adic field come as a completion of a number field?

Answer: Yes


Question: Does there exist a continuous function \( f: [0,1] \rightarrow [0, \infty) \) such that \( \int_{0}^{1} x^{n} f(x) \, dx = 1 \) for all \( n > 0 \)?

Answer: No


Question: Calculate the limit: \[ \lim_{j \to \infty} \frac{j^{j/2}}{j!} \]

Answer: 0


Question: Given a basis of the vector space $A = \mathbb{R}^{3\times 3}$, where $A$ is the space of all $3\times 3$ matrices, and a matrix $B \in \mathbb{R}^{3\times 3}$ that defines a linear mapping $S_B: A \rightarrow A$ by $\kappa \mapsto B\kappa B^*$. Choose a basis for the vector space $A$, and find the matrix representation of $S_B$ with respect to the chosen basis.

Answer: B \otimes B


Question: Determine whether the following statement is true or false: All solutions \( \varphi(t) \) of the differential equation \( y'' + (y')^3 + y = 0 \) satisfy \( \varphi(t) \to 0 \) as \( t \to \infty \).

Answer: True


Question: Find the values of \( x \) that satisfy the equation \( \Gamma (x-\sqrt{2x})=(\sqrt{x+1})! \).

Answer: 8


Question:
Find the sum of the shifted sinc function given by:
\[ g(y) = \sum_{n=-\infty}^\infty \frac{\sin(\pi(n - y))}{\pi(n-y)} \]

Answer: 1


Question: A random point \((a,b)\) is uniformly distributed in a unit square \(K=[(u,v):0<u<1,0<v<1]\). Let \(n\) be the number of real roots of the polynomial \(f(x)=\frac{1}{3}x^3 - a^2 x +b\). Find the probability \(P(n=3)\).

Answer: \dfrac{1}{6}


Question:
Given that \(a > b > c > 0\) and \(7a + 8b = 15c + 24\sqrt[3]{abc}\), find the maximum value of the expression:
\[ \frac{a-b}{c} + \frac{b-c}{a} + \frac{c-a}{b} \]

Answer: \dfrac{32}{9}


Question: In a dice game, a standard six-sided die is rolled repeatedly until the sum of the numbers on the top face equals or exceeds 100. What is the most likely sum that will be reached or exceeded?

Answer: 100


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} (2^2 \cdot 3^3 \cdot \ldots \cdot n^n)^{\frac{2}{n(n+1)}} \]

Answer: \infty


Question: Consider a uniformly continuous function $f: X \rightarrow Y$ between metric spaces $X$ and $Y$. For a given $\varepsilon > 0$, define $\delta(\varepsilon)$ as the supremum of all $\delta > 0$ such that $d_X(x,y) < \delta$ implies $d_Y(f(x),f(y)) < \varepsilon$. Let $g: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ be the function given by $g(\varepsilon) = \sup \{ \delta > 0 \text{ that work} \}$. Excluding cases where the supremum is infinite, is $g(\varepsilon)$ a continuous function?

Answer: Yes


Question: Is it possible to cover the circle \( S = \{ (x,y) \in \mathbb{R}^{2} \mid x^{2} + y^{2} = 1 \} \) using only two coordinate patches?

Answer: Yes


Question: Determine the solution to the Cauchy problem for the partial differential equation \( y z_x - x z_y = 0 \) with the initial conditions \( x_0(s) = \cos(s), y_0(s) = \sin(s), z_0(s) = 1, s > 0 \).

Answer: 1


Question: Let \( f(x) \) be an invertible increasing function such that \( f''(x) + 2f'(x) \geq 2 \) and \( f'(0) = 2 \). Determine the maximum value of \( (f^{-1})'(f(x)) \).

Answer: 1


Question: Evaluate the limit: \[ L = \lim_{x \to \infty} \frac{\int_{1}^{x} t^{t-1} ( t + t\ln(t) +1 ) \, dt }{x^x} \]

Answer: 1


Question: Is the theory $Th(C)$ of the class of finite sets with at least $n$ elements axiomatized by the sentence that states there are at least $n$ elements?

Answer: Yes


Question: Evaluate the double integral over the first quadrant: \[ \int_0^\infty \int_0^\infty \frac{\exp(-x-y)}{(x+y)}\,dx\,dy. \]

Answer: 1


Question: Evaluate the double integral \( \int_{0}^{2}\int_{0}^{2}\lfloor x+y \rfloor \, dx \,dy \).

Answer: 6


Question: Let \( A: V \to V \) be a linear operator in an inner product space \( V \) such that for all \( x, y \in V \), if \( x \cdot y = 0 \), then \( Ax \cdot Ay = 0 \). Prove that there exists a scalar \( \lambda \in \mathbb{F} \) such that \( \lambda A \) is a unitary operator.

Answer: \lambda


Question: Find a nontrivial polynomial function \( p(x) \) such that \( p(2x) = p'(x)p''(x) \neq 0 \).

Answer: \dfrac{4}{9}x^3


Question: Let $f(x): \mathbb{R}^n \to \mathbb{R}$ be a real-valued twice continuously differentiable function with $n > 1$. Define the function $g(x) = f(x) + x^{\top} A x$, where $A$ is a random matrix with entries i.i.d. from a uniform distribution on [-1,1]. Is the Hessian of $g$ invertible for all $x$ with probability one?

Answer: Yes


Question: Let \((x_n)\) be a monotonically decreasing sequence of positive real numbers that is summable. Let \((y_n)\) be a sequence of positive real numbers such that \(\sum_n x_n y_n\) converges. Let \((z_n)\) be a monotonically increasing sequence of positive real numbers such that \(\sum_n x_n z_n =\infty.\) Assume that the sequences \(y_n\) and \(z_n\) are such that \(2^{-\varepsilon y_n}\) and \(2^{-\varepsilon z_n}\) are summable for every \(\varepsilon>0.\) Does it follow that there exists some \(\delta>0\) such that \[ \sum_n \Big(2^{-\varepsilon y_n}-2^{-\varepsilon z_n}\Big) \ge 0 \text{ for all } \varepsilon \in (0,\delta)?\]

Answer: Yes


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

Answer: 1


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

Answer: 1


Question: Find the spectrum of the operator \( T \in \mathcal{B}(\ell^2) \) given by \( T(x_1, x_2, x_3, x_4,...) = (-x_2, x_1, -x_4, x_3,...) \).

Answer: \{i, -i\}


Question: Evaluate the integral \( I = \int_{0}^{1} f(x) \, dx \) using Lebesgue integration, where \( f(x) = 1 \) if \( x \) is irrational and \( f(x) = 0 \) if \( x \) is rational.

Answer: 1


Question: Evaluate the contour integral \( \int_C \frac{[g(z)]^4}{(z-i)^3} \,\mathrm{d} z \), where \( C \) is the circle centered at the origin with radius 2. The function \( g \) is entire, and you are given the following values: \( g(i) = 2, g(4i) = 5, g'(i) = 3, g'(4i) = 6, g''(i) = 4, g''(4i) = 7 \). Use Cauchy's integral formula if applicable.

Answer: 560\pi i


Question: Calculate the arclength of the curve given by the vector function \( r(t) = i + 3t^2j + t^3k \) over the interval \( 0 \leq t \leq \sqrt{12} \).

Answer: 56


Question: Given a matrix $A \in M_4(\mathbb{R})$ with characteristic polynomial $p(\lambda) = \lambda^3(\lambda-1)$, determine whether the mapping $A: \mathbb{R}^4 \rightarrow \mathbb{R}^4$ defined by $A(x) = Ax$ is surjective. Answer true or false.

Answer: False


Question: Find the limit as \( n \) approaches infinity of the probability \( P \) that \((x-n)^2 + (y-n)^2 \leq n^2\), where \( x \) and \( y \) are chosen randomly from the set \( S = \{0, 1, \ldots, 2n\} \).

Answer: \dfrac{\pi}{4}


Question: Evaluate the limit: \[ \lim_{\lambda \to 0^+} \frac{1}{\ln \lambda}\int_{\lambda}^{a}\frac{\cos{x}}{x}dx \]

Answer: -1


Question: Calculate the volume of the solid bounded by the surfaces \( z^2 = x^2 + y^2 \) and \( 2 - z = x^2 + y^2 \) using a double integral.

Answer: 4\pi


Question: Let $G$ be a finite group and $T$ be a normal subgroup of $G$ such that $PSL(3,4) \unlhd T \leq \text{Aut}(PSL(3,4))$ and $|T|=2|PSL(3,4)|$. If $G= T\rtimes C_{11}$, determine whether $G$ has an element of order 22.

Answer: Yes


Question: Does there exist a linear operator $T$ on a Hilbert space $\mathcal{H}$ such that $0 < \dim(\ker T) < \infty$ and the range of $T$ is not closed?

Answer: Yes


Question: Let $u,v \in \mathcal{E}'(\mathbb{R}^n)$ be two compactly supported distributions. Define the convolution $u*v$ by $u*v(\phi) = u(Rv*\phi)$, where $v*\phi = v(\tau_xR\phi)$ for $\phi \in C^\infty$, $\tau_x\phi(t) = \phi(t-x)$, and $R\phi(t) = \phi(-t)$. Prove that the convolution is commutative, i.e., $u*v = v*u$. Show that $u*v(\phi) = v*u(\phi)$ for all test functions $\phi$.

Answer: u * v = v * u


Question: Given two sequences encoded as follows: a=1, b=2, c=4, d=8, ..., y=2^24, z=2^25, and a '|' symbol indicating the end of a message, determine the time at which the sequences synchronize. For example, the sequence "*ea|babab" has an offset of 17 and a length of 25, while "*d|abaca" has an offset of 8 and a length of 17. Calculate the synchronization time for these sequences.

Answer: 42


Question: Find a single continuous function \( f(x) \) such that \( f(0) = 1 \), \( \int_{-\infty}^{\infty} f(x) \, dx = 1 \), and the function decreases on both sides of \( x = 0 \) at a rate faster than or equal to \( e^{-2|x|} \).

Answer: e^{-2|x|}


Question: Consider two graphs $G$ and $G^{\prime}$ that are graph isomeric, meaning they have the same number of vertices and edges. If there exists a graph homomorphism $h \colon G \to G^{\prime}$ that preserves vertex degree, does it necessarily follow that $G$ and $G^{\prime}$ are graph isomorphic? Answer yes or no.

Answer: No


Question: Determine whether there exists an infinite connected simple undirected graph \(G=(V, E)\) such that the identity map \(\text{id}_V: V\to V\) is the only graph self-homomorphism from \(G\) to itself. A graph self-homomorphism is a map \(f: V\to V\) such that for all \(e\in E\) with \(e = \{v, w\}\), we have \(\{f(v), f(w)\} \in E\).

Answer: Yes


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

Answer: 1


Question: Given that \((1+\sqrt{2})^n = a_n + b_n\sqrt{2}\) for all \(n \in \mathbb{N}\), where \(a_n, b_n \in \mathbb{Q}\), find the value of \(\lim_{n\rightarrow \infty}\frac{a_{n}}{b_{n}}\).

Answer: \sqrt{2}


Question: Evaluate the following limit of an integral using the Dominated Convergence Theorem (DCT): $$\lim_{n \rightarrow \infty} \int_6^n \left(1+\frac{x}{n}\right)^n e^{-2x} \, dx.$$

Answer: e^{-6}


Question: Use Green's Theorem to evaluate the line integral \( \oint_C y^2 \, dx + x \, dy \), where \( C \) is the curve described by the parametric equations \( x = 2\cos^3 t \) and \( y = 2\sin^3 t \) for \( t \) ranging from \( 0 \) to \( 2\pi \).

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


Question: Let X and Y be continuous random variables with the following properties: the probability density function of Y is given by \( f_Y(y) = 2y \) for \( 0 \leq y \leq 1 \), and given \( Y = y \), X is uniformly distributed over \([0, y]\). Find the conditional variance \( \text{Var}(Y | X = 0.5) \).

Answer: \dfrac{1}{48}


Question: Evaluate the integral \( \int_{\gamma} \frac{1}{1-e^{z}}dz \) using the residue theorem, where \( \gamma(t) = e^{it} \) for \( t \in [0, 2\pi] \) is the boundary curve of the unit circle.

Answer: -2\pi i


Question: Determine the domain of the function $g(z, \bar{z}) = f(x, y)$, where $z = x + iy$ and $\bar{z} = x - iy$. Is the domain $g: \mathbb{C} \to \mathbb{C}$ or $g: \mathbb{C}^2 \to \mathbb{C}$?

Answer: \mathbb{C}


Question: Consider the initial value problem \(u'(t) = Au(t), t > 0\) with \(u(0) = u_0\), where \(u_0\) is a given vector in \(\mathbb{R}^2\) and \(A = \begin{bmatrix} 1 & -2 \\ 1 & a \end{bmatrix}\). Find the range of values of \(a\) such that \(|u(t)| \to 0\) as \(t \to \infty\).

Answer: (-2, -1)


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

Answer: 10


Question:
Given \( \int _0^{x^2}t^2f(t)\,dt = (x + 1)^2 \sin(x^2) \), find the value of \( f\left(\frac{\pi}{2}\right) \). Choose the correct option:

(a) \( \frac{\left(1+\sqrt{\frac{2}{\pi}}\right)4}{\pi^2} \)

(b) \( \frac{\left(1-\sqrt{\frac{2}{\pi}}\right)4}{\pi^2} \)

(c) \( \frac{\left(1+\sqrt{\frac{2}{\pi}}\right)1}{\pi^2} \)

(d) \( \frac{\left(1+\sqrt{\frac{2}{\pi}}\right)4}{\pi} \)

Answer: A


Question: Let $f: S^n \to X$ be a continuous map. Determine if the following statement is true for any $n > 1$: $[f] = 1 \in \pi_n(X)$ if and only if $f$ extends to $D^{n+1}$. Provide a justification for your answer.

Answer: True


Question: Let $R$ be a commutative ring and $I$ be an ideal of $R$. Determine whether the statement "$I$ is a principal ideal if and only if $I$ is a free $R$-module" is true or false.

Answer: False


Question: Evaluate the integrals \(I_1 = \int_{-1}^{1} \left( \tan^{-1}(x) + \tan^{-1} \left(\frac{1}{x}\right) \right) \, dx\) and \(I_2 = \int_{-1}^{1} \left( \cot^{-1}(x) + \cot^{-1} \left(\frac{1}{x}\right) \right) \, dx\). Determine if \(I_1 = I_2\).

Answer: I_1 \neq I_2


Question: Evaluate the triple integral \( \iiint f(x,y,z) \, dV = \int^1_0\int^{\pi z}_0\int^z_\frac{y}{\pi} \sin\left(\frac{y}{x}\right) \,dx\,dy\,dz \) by switching the order of integration to simplify the computation.

Answer: \dfrac{1}{3}


Question: Let \( f : [0,1]^k \to [0,1]^k \) be a one-to-one and onto \( C^1 \)-map. Given that the Lebesgue measure of \( \partial([0,1]^k) \) is zero, determine if the Lebesgue measure of \( f(\partial([0,1]^k)) \) is also zero.

Answer: Yes


Question: Is it possible to construct a sequence of continuous functions \( f_{n}: [0,1] \rightarrow \mathbb{R} \) such that \( |f_{n}(x)| \) converges to 0 for each \( x \in [0,1] \), but the sequence \( \{f_{n}\}_{n} \) does not converge to 0 in the normed space \( C[0,1] \) with the infinity norm?

Answer: Yes


Question:
Let $A$ be a commutative ring and let $s,t \in A$ which generate the unit ideal. Suppose $M$ and $N$ are modules over $A$. Given maps between the localizations:
\[ f_s: M_s \rightarrow N_s \]
\[ f_t: M_t \rightarrow N_t \]
\[ f_{st}: M_{st} \rightarrow N_{st} \]
such that the following diagram commutes:
\[ \begin{CD}
 M_s @>f_s  >> N_s\\
 @V  VV @V  VV\\
 M_{st} @>f_{st} >> N_{st}
\end{CD} \]
and the analogous one for $t$ also commutes, does there exist a map $f: M \rightarrow N$ making all the relevant diagrams commute?

Answer: Yes


Question: Determine the values of \( p \) for which the series \( \sum_{n=1}^\infty \frac{1}{n^p} \) converges using Raabe's test.

Answer: p > 1


Question: Let $K$ and $L$ be compact Hausdorff spaces. Suppose there exists a bijection $p: C(K) \to C(L)$ that preserves multiplication. Is $p$ a ring isomorphism?

Answer: Yes


Question: Determine the values of \( n \) for which the improper integral converges.

Answer: n > 1


Question:
Consider the linear transformation $T: M_5(\mathbb{R}) \to M_5(\mathbb{R})$ defined by $T(A) = A^T$, where $A^T$ is the transpose of the matrix $A$. Let $M$ be the matrix representation of $T$ with respect to the standard basis $B$. Determine which of the following statements is correct:

(a) $\lambda = \{-1, 1\}$ are the only eigenvalues of $M$.
(b) There exist non-zero matrices $X, Y \in M_5(\mathbb{R})$ such that $M(X + Y) = X - Y$.
(c) $\det(M) = 1$.
(d) The dimension of the eigenspace corresponding to $\lambda = -1$ is 10.

Answer: a,b,c,d


Question:
Determine whether there exist two uncountable subsets $A$ and $B$ of real numbers such that:

1. $(A-A) \cap (B-B) = \{ 0 \}$,
2. $(A-A) + B = \mathbb{R}$ or $(B-B) + A = \mathbb{R}$.

Answer: Yes


Question: Consider a bounded Lipschitz domain \(\Omega \subset \mathbb{R}^N\) and let \(p \in [1, \infty)\). Suppose \(u \in W_0^{1,p}(\Omega) \cap L^\infty(\Omega)\). Is it possible to approximate \(u\) by a sequence of functions \(u_k \in C_0^\infty(\Omega)\) such that \(\|u_k\|_\infty \leq M\) for some positive constant \(M\)?

Answer: Yes


Question: Find the smallest positive integer \( n \) such that \((-\sqrt{2} + i\sqrt{6})^n\) is an integer, where \( i \) is the imaginary unit.

Answer: 6


Question: Evaluate the limit \( \lim_{x \rightarrow 0^+}(e^{\frac{1}{x}}x^2) \) without using L'Hopital's rule.

Answer: \infty


Question: Consider the space $C^\infty(X)$ of infinitely smooth functions on a compact manifold $X$, and let $\mathcal{D}(X)$ be the space of Schwartz distributions equipped with the weak* topology. Let $F: \mathcal{D}(X) \to \mathbb{C}$ be a linear functional that is sequentially continuous in this topology. Is $F$ continuous in the weak* topology?

Answer: Yes


Question: Determine the value of \( m \) such that the differential equation \( 2x^4yy' + y^4 = 4x^6 \) becomes homogeneous when the substitution \( y = z^m \) is applied.

Answer: \dfrac{3}{2}


Question: Let $\lambda$ be the Lebesgue measure on the measurable space $(\mathbb{R}^2, \mathcal{A})$. Assume $A \in \mathcal{A}$ is such that $A = X \times Y$, where $X$ is a non-measurable subset of $\mathbb{R}$, and $Y = \emptyset$ or $Y = \{y\}$ for some $y \in \mathbb{R}$. Determine whether the Lebesgue measure of $A$ is identically zero or non-measurable.

Answer: 0


Question: Find the degree of the field extension \( \mathbb{C}(e_{1}, e_{2}, e_{3})(x_{1}+2x_{2}+3x_{3})/\mathbb{C}(e_{1}, e_{2}, e_{3}) \), where \( e_{1}=x_{1}+x_{2}+x_{3} \), \( e_{2}=x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3} \), and \( e_{3}=x_{1}x_{2}x_{3} \).

Answer: 6


Question:
Let \( T: \ell_2 \to \ell_2 \) be defined by \( T((x_1, x_2, \ldots, x_n, \ldots)) = (x_2 - x_1, x_3 - x_2, \ldots, x_{n+1} - x_n, \ldots) \). Determine the value of \( ||T|| \).

(A) \( ||T|| = 1 \)
(B) \( ||T|| > 2 \) but bounded
(C) \( 1 < ||T|| < 2 \)
(D) \( ||T|| \) is unbounded

Answer: C


Question: Evaluate the surface integral \( \iint_S \mathbf{F} \cdot \mathbf{n} \, dS \) for the vector field \( \mathbf{F}(x,y,z) = xi + yj + zk \), where \( S \) is the plane surface bounded by the triangle with vertices at \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\). The unit normal \( \mathbf{n} \) to \( S \) has a nonnegative z-component. Use the vector representation \( \mathbf{r}(u,v) = (u+v)i + (u-v)j + (1-2u)k \) to perform the evaluation.

Answer: \dfrac{1}{2}


Question: Compute the integral \( \int_T xyz(1-x-y-z)\,\mathrm{d}x\mathrm{d}y\mathrm{d}z \), where \( T \) is the simplex \( \{(x, y, z) \in \mathbb{R}^3 \mid x, y, z \ge 0, x + y + z \le 1\} \).

Answer: \dfrac{1}{5040}


Question: Assume that $H$ is a subgroup of a finite group $G$, and that $G$ contains elements $a_1, a_2, \ldots, a_n$ such that $a_i a_j^{-1} \notin H$ for $1 \leq i < n, 1 \leq j < n$, and $i \neq j$. What can you conclude about the index $[G:H]$?

Answer: [G:H] \geq n


Question: Are finite rank operators on a Hilbert space $H$ dense in $B(H)$ in the weak operator topology?

Answer: Yes


Question:
Determine the minimum radius of convergence \( R \) of the power series solution about \( x=3 \) for the differential equation:
\[
(x^2+16)y''+xy'+y=0
\]

Answer: 5


Question: Compute the fundamental group of the space \( \mathbb{R}^{3} \smallsetminus A \), where \( A = \{(x,y,z): y=0,x^{2}+z^{2}=1\} \cup \{(x,y,z): y=z=0, x \ge 1\} \).

Answer: \mathbb{Z} * \mathbb{Z}


Question: Evaluate the integral \( \oint_{|z|=2\pi} \tan(z) \, dz \) using the Cauchy Residue Theorem, where the contour is positively oriented. Determine the value of the integral by calculating the residues at the singularities within the contour.

Answer: -8\pi i


Question: Let $E/F$ be a finitely generated field extension of transcendence degree one. Suppose that $F$ is algebraically closed in $E$ and $t \in E$ is a transcendental element over $F$. Let $B$ be the integral closure of $A := F[t]$ in $E$. If $m$ is a maximal ideal of $B$ lying over $(t)$, determine if $B/m = F$.

Answer: B/m = F


Question: If a matrix $A$ satisfies $A^3 = I_n$, where $A \in \mathbb{K}^{n,n}$ and $\frac{1}{2} \in \mathbb{K}$, does it necessarily follow that $\det(A) = 1$? Justify your answer.

Answer: No


Question: Let \( f \) be a differentiable function on \((a,b)\) with \( c \in (a,b) \). Suppose \( f \) and \( f' \) are nonzero on a deleted neighborhood of \( c \), and \( \lim_{x\rightarrow c}f(x)=0 \). Find \( \lim_{x\rightarrow c}|f(x)|^{f(x)} \).

Answer: 1


Question:
Is it possible to construct a Taylor expansion for a function \( f: \mathbb{X} \to \mathbb{R} \), where \( \mathbb{X} \) is a normed space and \( f \) is twice Gâteaux differentiable, in the form:
\[
f(u) = f(\bar{u}) + f'(\bar{u})(u-\bar{u}) + \frac{1}{2}f''(\bar{u}+\theta(u-\bar{u}))(u-\bar{u})^2,
\]
with \( \theta \in (0,1) \)?

Answer: Yes


Question: In a queuing system with two facilities, A and B, each facility has a single server with service times that are negative exponential random variables. The mean service times are 1 minute for facility A and 4 minutes for facility B. A total of 60 customers per hour arrive, with arrivals following a Poisson process. Each customer independently chooses a facility without knowing the queue status, with probability \( p \) for facility A and \( 1-p \) for facility B. Determine the value of \( p \) that minimizes the expected time a customer spends in the system, assuming the system reaches a steady state.

Answer: \dfrac{5}{6}


Question: Evaluate the limit of the sequence: \[ \lim_{n \to \infty} \frac{2 \cdot 3^{2n - 1} - \left( -2 \right)^n}{2 \cdot 3^n - 3 \cdot 2^{2n + 1}}. \]

Answer: -\infty


Question: Given the position vector \( \mathbf{r}(t) = (5\cos 3t) \mathbf{i} + (6t) \mathbf{j} + (5\sin 3t) \mathbf{k} \), find the component of the acceleration in the direction tangent to the curve.

Answer: 0


Question: Calculate the volume of the region defined by the inequality \( \{(x,y,z) \in \mathbb{R}^3 \mid 2\cdot \max(\lvert x\rvert,\lvert y\rvert)^2+z^2\leq 4\} \).

Answer: \dfrac{64}{3}


Question: Let $G = \langle X_1, X_2 \rangle$ be a group. Determine whether, if $X_1$ or $X_2$ is a normal subgroup of $G$, it follows that $G = X_1X_2$.

Answer: G = X_1 X_2


Question: Calculate the volume bounded by the surface \( \sqrt{x} + \sqrt{2y} + \sqrt{3z} = 1 \) in the region where \( x, y, z \geq 0 \).

Answer: \dfrac{1}{540}


Question: Calculate the limit: $$\lim_{n\rightarrow\infty} \frac{7^{\sqrt{n}}\cdot(n/2)!\cdot(n/2)!}{n!}$$

Answer: 0


Question: Determine the value(s) of $a$ for which the functions $\cos(2x)$, $\sin(2ax)$, and $1$ are linearly independent over the real numbers. For which value(s) of $a$ are they linearly dependent?

Answer: 0


Question: Let $X \sim U(0, 1)$ and $Y \sim U(0, 2)$ be independent random variables. Calculate the expected value of $|X - Y|$. Express your answer as a single number.

Answer: \dfrac{2}{3}


Question: If $(X, d)$ is a Lindelöf metric space, is the product space $X \times X$ also Lindelöf?

Answer: Yes


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

Answer: \dfrac{1}{e}


Question: Given that \(a, b, c, d\) are positive real numbers satisfying \(a^2 + b^2 + c^2 + d^2 = 4\), determine the range of the expression \(a^3 + b^3 + c^3 + d^3\).

Answer: [4, 8)


Question: Use Taylor's formula to find the second-order approximation of the function \( F(x,y) = \sin(x)\sin(y) \) around the point \((a, b) = (0, 0)\).

Answer: xy


Question: For any group $G$ of order $36$ and any subgroup $H$ of $G$ of order $4$, is $H$ contained in the center $Z(G)$ of the group?

Answer: No


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

Answer: 3


Question: Evaluate the integral \( \int_0^{\infty} x^{4n+3} e^{-x} \sin x \, dx \).

Answer: 0


Question: Evaluate the integral: \[ \int_0^{\pi} \frac{e^{\sin x}\cos(x)}{1+e^{\tan x}} \, dx \]

Answer: 0


Question: Consider the linear subspace $c$ of $l^\infty$ consisting of sequences $\{a_k\}_k$ that converge to a finite limit as $k \to +\infty$. Define a linear functional $T: c \to \mathbb{R}$ by $T(\{a_k\}_k) = \lim\limits_{k \to +\infty} a_k$. Prove that the norm of $T$ is equal to 1. Additionally, verify that for each element $e^h$ of the canonical basis, $T(e^h) = 0$. Explain why this is the case.

Answer: 1


Question: Determine the condition under which the metric \( \frac{1}{P^2}(\mathrm{d}\theta^2+\sin^2 \theta\; \mathrm{d}\phi^2) \), where \( P=\sin^2 \theta+K\cos^2 \theta \) and \( K \) is the Gaussian curvature (constant), can define a sphere.

Answer: K = 1


Question: Evaluate the limit: \[ \lim_{x\to\infty}\left(x-\sqrt{x-1}\log\left(x^2+e^{\sqrt x}\right)\right) = \frac{1}{2} \] without using Taylor series.

Answer: \dfrac{1}{2}


Question: Calculate the limit: $$ \lim_{x \to \infty} \frac{1}{x^2}\log\left(\left(1+(\exp(2x)-1)^2\right)\right)^2. $$

Answer: 16


Question: Find the residue of the function \( f(z) = \frac{\cot(z)\coth(z)}{z^3} \) at \( z = 0 \).

Answer: -\dfrac{7}{45}


Question: Find the formula for the Tower of Hanoi numbers given by the recurrence relation \( h_n = 2h_{n-1} + 1 \) with \( h_1 = 1 \) using generating functions.

Answer: 2^n - 1


Question: Given the differential equation \( x \frac{dy}{dx} = y( \ln y - \ln x + 1) \) with the initial condition \( y(1) = 3 \), find the value of \( y(3) \).

Answer: 81


Question: Evaluate the summation: $$\sum_{1\leq l< m <n} \frac{1}{5^l3^m2^n}$$ where $1 \leq l < m < n$.

Answer: \dfrac{1}{145}


Question: Calculate $14^{2017^{2017}} \mod 3$. What is the result?

Answer: 2


Question: Evaluate the limit: \[ \lim_{n \to \infty}{\frac{1^1+2^2+3^3+\cdots+n^n}{n^n}}. \]

Answer: 1


Question: Calculate the Fourier coefficients $f_k$ for the function $f(x) = \frac{3}{5-4\cos(x)}$ on the interval $[0, 2\pi]$. Verify that the coefficients are given by $2^{-|k|}$.

Answer: 2^{-|k|}


Question: Determine whether the matrix \( \begin{pmatrix} 1+a & -a & a \\ 2+a & -a & a-1 \\ 2 & -1 & 0 \end{pmatrix} \) is diagonalizable depending on the parameter \( a \).

Answer: 0


Question: Determine whether the following statement is true: For all $\alpha,\beta\in \mathbb{R}$ and for all $\varepsilon \in \mathbb{R}_{>0}$, there exist $a,b,c\in \mathbb{Z}$ such that $|a-c\alpha|<\varepsilon$ and $|b-c\beta|<\varepsilon$. Provide a justification for your answer.

Answer: True


Question: Let $X$, $Y$, and $Z$ be independent random variables, each following an exponential distribution with parameter $\lambda$. Calculate the probability $P(X + Z > Y)$. Express your answer in terms of $\lambda$.

Answer: \dfrac{3}{4}


Question: Let X be a topological space, and let W(X) be the specialization preorder of X. Consider the Alexandrov topology T(W(X)) corresponding to W(X). If T(W(X)) = X, is X necessarily an Alexandrov topology?

Answer: Yes


Question: Consider the interval $[0,1]$. At each stage, remove the closed middle third interval from each remaining segment. For example, from $[0,1]$, remove $[rac{1}{3}, rac{2}{3}]$ to get $[0, rac{1}{3}) \cup (\frac{2}{3}, 1]$. Continue this process indefinitely. How many elements are left in the intersection of all these sets?

Answer: \mathfrak{c}


Question: Evaluate the limit: $$\lim\limits_{x \to 0} \frac{\ln(1+x^{144})-\ln^{144}(1+x)}{x^{145}}$$

Answer: 72


Question: Compute \( \lim_{n \to \infty}\int_{0}^{\infty} \frac{n \sin(\frac{x}{n})}{x(1+x^2)}dx \) using the Dominated Convergence Theorem.

Answer: \dfrac{\pi}{2}


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

Answer: 0