Question: Find the closed-form expression for the number of neighboring integer points of a given integer point in an n-dimensional space, given the recursive formula: \( n_{i-1} = 3 * n_{i-2} + 2 \).

Answer: 3^n - 1


Question: Calculate the number of ring homomorphisms from $\mathbb{Z}_{12}$ to $\mathbb{Z}_{12}$ using the formula $2^{\omega(n)-\omega(n/(m,n))}$, where $m$ and $n$ are the orders of the groups.

Answer: 4


Question: Evaluate the integral \( \int_0^1 \frac{x^{10}-1}{\ln x} \, dx \) using Feynman's method.

Answer: \ln 11


Question: How many distinct bracelets can be formed using 8 red beads, 6 blue beads, and 4 yellow beads, given that all beads must be used and two bracelets are considered identical if one can be obtained from the other through rotation or reflection? Beads of the same color are indistinguishable.

Answer: 255920


Question: Calculate the integral \( \int_{-2}^2 \left(x^3\cos\frac{x}{2}+\frac{1}{2}\right)\sqrt{4-x^2}\ dx \) and provide the first 10 digits of the result.

Answer: 3.141592653


Question: In the polynomial identity \(x^4+1=(x^2+1)(x^2+px+1)(x^2+qx+1)\), determine the value of \(p+q\).

Answer: 0


Question: Given the equation \(\alpha=\sum_{i=1}^{n}\exp(-y\beta_i)\) where \(\alpha\), \(y\), and \(\beta_i\) are positive real numbers, with \(\beta_i\) being independent random numbers following a uniform distribution within the interval \([\sqrt{2\omega_a}, \sqrt{2\omega_b}]\), where \(0\leq\omega_a<\omega_b\) and \(n\) is a large positive integer, determine if \(y\) can be solved for.

Answer: Yes


Question: Determine the rank of the subgroup $H$ of $\mathbb{Z}^3$ generated by the vectors $v_1=(2,-2,0)$, $v_2=(0,4,-4)$, and $v_3=(5,0,-5)$. Given that $10v_1+5v_2-4v_3=0$, find the exact rank of $H$. Provide your answer as a single integer.

Answer: 2


Question: Calculate the probability that the set of independent random variables $E_1, E_2, \cdots$, where $E_n \sim \text{Unif}\{1, 2, \cdots, 2n\}$, equals the set of all even numbers $\mathbb{E}$.

Answer: 0


Question: Evaluate the expression \( \delta_{ij} \epsilon_{ijk} \) using Einstein summation notation.

Answer: 0


Question: Given a strictly increasing sequence of positive integers \(a_1, a_2, a_3,...\) where for every positive integer \(k\), the subsequences \(a_{2k-1}, a_{2k}, a_{2k+1}\) are geometric and the subsequences \(a_{2k}, a_{2k+1}, a_{2k+2}\) are arithmetic, if \(a_{13} = 539\), find \(a_5\).

Answer: 99


Question: Evaluate the limit: \[ \lim_{x\to\pi/2}\frac{\sin x - (\sin x)^{\sin x}}{1 - \sin x + \ln(\sin x)}. \]

Answer: 2


Question:
Given two sequences $y_t$ and $z_t$ that satisfy the following equations:

$$y_t = 6y_{t-1} - 20z_{t-1}$$
$$z_t = -17y_{t-1} - 12z_{t-1}$$

with $a = 6$, $b = -20$, $c = -17$, and $d = -12$, a second-order difference equation can be derived in the form:

$$y_{t+1} + ny_t + ry_{t-1} = 0$$

where $n$ and $r$ are constants. The associated auxiliary equation is:

$$m^2 + nm + r = 0$$

This equation has two solutions, $m = h$ and $m = k$, with $h > k$. Determine $h$ to two decimal places.

Answer: 17.52


Question: Given a random variable $Y$ uniformly distributed over $[0,1]$, determine if it is possible to transform $Y$ into a random variable with an exponential distribution with parameter 1.

Answer: Yes


Question: Determine the infimum of the sequence \(\left\{\frac{(n+1)^2}{2^n}\right\}_{n=1}^{\infty}\). Justify why 0 is a candidate for the infimum and find a method to determine an \(N(\varepsilon)\) such that for all \(n > N(\varepsilon)\), \(\frac{(n+1)^2}{2^n} < \varepsilon\).

Answer: 0


Question: Calculate the surface area of the cone defined by the equation \(3z^2 = x^2 + y^2\) for \(0 \leq z \leq 2\).

Answer: 8\sqrt{3}\pi


Question: Find the constant $c$ such that for any point of intersection of the spheres $(x-c)^{2} + y^{2} + z^{2} = 3$ and $x^{2} + (y-1)^{2} + z^{2} = 1$, the tangent planes at the intersection are perpendicular.

Answer: \pm \sqrt{3}


Question: Provide an example of a topological space that is Fréchet-Urysohn but not countable.

Answer: \mathbb{R}


Question: Given an open set $O$ in a topological space and a point $x \in O$, determine whether there always exists a non-empty connected open set $U \subset O$ with $x \in U$. Provide a justification for your answer.

Answer: No


Question: Let \( \{f_{n}\}_{n \in \mathbb{N}} \) be a sequence of measurable functions on a measure space \( M \) such that \( f_{n} \rightarrow f \) almost surely, where \( f \) is also a measurable function. Suppose there exists a nonnegative measurable function \( g \) satisfying \( |f_{n}| \leq g \) almost surely for all \( n \in \mathbb{N} \) and \( \int_{M} g^{p} \, d\mu < \infty \) for some \( p > 0 \). Prove that \( \int_{M} |f_{n} - f|^{p} \, d\mu \rightarrow 0 \) as \( n \rightarrow \infty \).

Answer: 0


Question: Determine whether the following limit exists: $$\lim_{x\rightarrow 0^+}\frac{\sin(x\log(x))}{x\log(x)}$$ You may not use L'Hospital's rule or differentiation.

Answer: 1


Question: In which dimension does a Riemannian manifold with constant scalar curvature also have constant sectional curvature?

Answer: 2


Question: Given a differentiable scalar field \( f \), the directional derivative at the point \((1,2)\) is 2 in the direction toward \((2,2)\) and -2 in the direction toward \((1,1)\). Determine the gradient vector of \( f \) at the point \((1,2)\).

Answer: (2, 2)


Question: Calculate the limit: $$\lim_{x \to 0} \frac{\sin(x^2)-\sin^2(x)}{x^2\ln(\cos x)}$$

Answer: -\dfrac{2}{3}


Question: Determine whether an open and measurable set $G$ in a measurable space is necessarily bounded if its measure $\mu(G)$ is finite. Answer with 'True' or 'False'.

Answer: False


Question: How many 4-digit numbers exist such that deleting any one of its digits results in a 3-digit number that is a divisor of the original 4-digit number?

Answer: 14


Question: Let $J$ be a maximal right ideal of a ring $S$. If $z \in \text{rad}(S)$ and $s \in S$ such that $sz \notin J$, determine whether $S = sS + J$.

Answer: S = sS + J


Question: Determine whether the following statement is true or false: For any sequence of Lebesgue measurable functions $f_n : \mathbb{R} \rightarrow \mathbb{R}$, there exists a subsequence $\lim_{k\to \infty}f_{n_k}(x)$ that converges almost everywhere on $\mathbb{R}$. Provide a justification for your answer.

Answer: False


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

Answer: 18


Question: Determine the maximum value of \(|f'''(x)|\) for a function \(f\) that is infinitely differentiable on the real numbers, given that \(f(0) = 1\), \(f'(0) = 1\), \(f''(0) = 2\), and \(f(1) < 5\).

Answer: 12


Question: Evaluate the limit: $$\lim_{n\to\infty}\frac{2^n}{n^{\sqrt{n}}}$$

Answer: \infty


Question: Evaluate the sum: $$\frac{1}{2^{99}}\sum_{k=1}^{50} \sum_{r=0}^{k-1}\binom{50}{k}\binom{49}{r}$$

Answer: \dfrac{1}{2}


Question: Determine whether there are infinitely many positive integers $n$ such that $$\sigma(n^2)-\varphi(n)\sigma(n)-n$$ is a prime number, where $\sigma(n)$ is the sum of divisors function and $\varphi(n)$ is the Euler's totient function.

Answer: Yes


Question: Calculate the correlation between a standard normally distributed random variable $X$ and its square $X^2$, where $X \sim \mathcal{N}(0, 1)$. Use the formula for correlation: \[ \varrho_{X, X^{2}} = \frac{\mathbb{E} \left( X^{3} - X \right)}{2}. \]

Answer: 0


Question: In the category of Stone spaces, determine whether epimorphisms are always surjective.

Answer: Yes


Question: Determine the number of real roots of the equation \[ 2\cos\left(\frac{x^2+x}{6}\right) = 2^x + 2^{-x} \]

Answer: 1


Question: Determine whether the sequence defined by \(x_{n+1} = \left(1 - \frac{4x_n^2}{n+1}\right)x_n\), with initial condition \(0 < x_0 < \frac{1}{2}\), converges.

Answer: 0


Question: Determine the cardinality of the set $G$ of all entire functions $g$ such that $g(x) = x^k$ for all $x \in (-1, 1)$, where $k$ is an odd integer $\geq 1$. \( K \) is given as an odd integer $\geq 1$. \( G \) is the set of all entire functions $g$ such that $g(x) = x^k$ for all $x \in (-1, 1)$. What is the cardinality of $G$?

Answer: 1


Question: Consider a set of 100 independent random variables, $X_1, X_2, \ldots, X_{100}$, each following a standard normal distribution. Determine the approximate probability that the sum of their squares falls between 80 and 120, inclusive. In other words, find $P(80 < \sum_{j=1}^{100} X_j^2 < 120)$. \(\boxed{?}\)

Answer: 0.843


Question: Determine the locus of the midpoints of all segments $PQ$, where $P$ is a point on the line segment from $(0,1,1)$ to $(1,0,1)$, and $Q$ is a point on the line segment from $(1,1,0)$ to $(1,0,1)$. Provide your answer in the form of an equation or geometric description.

Answer: x + y + z = 2


Question: Determine whether the generating function of the language $L = \{w \in \Sigma^* \mid w \neq uvu', \text{ with } u, u' \in \Sigma^*, v \in W\}$, where $W$ is the set of all words over the alphabet $\Sigma$, is rational. Provide a justification for your answer.

Answer: Yes


Question: Let \( \phi \) be a continuous function on the interval \([0,b]\) such that \( \phi(x) > 0 \) for all \( x \in [0,b] \). Assume \( \phi(x) \sim A x^r \) for some constant \( A > 0 \) and \( r \geq 0 \). Define the function \( F(\varepsilon) = \int_0^b \frac{dx}{\varepsilon + \phi(x)} \). Find \( \lim_{\varepsilon \to 0} F(\varepsilon) \) for \( r = 1 \).

Answer: \infty


Question: Derive a differential equation for the family of curves given by the parametric equation: \( x = a \cdot \sin(t+b) \), where \( a, b \in \mathbb{R} \). Eliminate the parameter \( b \) to express the relationship solely in terms of \( x \), \( x' \), and \( x'' \).

Answer: x'' + x = 0


Question: Evaluate the integral: $$\int_0^{\pi/2}\frac{\sqrt{\sin x}}{\sqrt{\sin x}\,+\sqrt{\cos x}} \, dx$$

Answer: \dfrac{\pi}{4}


Question: Determine whether the functional differential equation \( D'(t) = 2 D(2t) - D(t) \) with the initial condition \( D(0) = 0 \) can be solved in a closed form.

Answer: D(t) = 0


Question: Evaluate the limit \( \lim_{x\to0}\frac{\int_0^x \sin(t^2) dt}{x^3}. \)

Answer: \dfrac{1}{3}


Question: Determine the limit superior of the sequence \(\left\{\frac{1}{n}\sum\limits_{k=1}^{n}(-1)^{k-1}(k+1)\right\}\).

Answer: \dfrac{1}{2}


Question: Does the property \(\mathbb{E} [f_1(X_1) \ldots f_M(X_M)] = \mathbb{E}[f_1(X_1)]\ldots\mathbb{E}[f_M(X_M)]\) for all real-valued functions \(f_m: \mathbb{R} \rightarrow \mathbb{R}\) imply that it also holds for all complex-valued functions \(g: \mathbb{R} \rightarrow \mathbb{C}\)?

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to 0}\frac{(\tan x)^{2008}-(\arctan x)^{2008}}{x^{2009}}$$ without using Taylor series.

Answer: 0


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

Answer: e^2


Question: Determine if the function \( u(x, y) = e^x \sin y \) is harmonic given the condition \( u(0, y) = 0 \). If it is harmonic, find the corresponding holomorphic function \( f = u + iv \).

Answer: -i e^z


Question: Determine the number of subspaces of a vector space that contains 128 elements.

Answer: 29212


Question: Evaluate the line integral of the vector field \( F(x, y) = \left(-\frac{y}{4x^2 + 9y^2}, \frac{x}{4x^2 + 9y^2}\right) \) along the boundary \( C \) of the region \( R \), which is exterior to the ellipse \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \) and is bounded by the curves \( \frac{x^2}{25} + \frac{y^2}{16} = 1 \) (for \( y \leq 0 \)), \( x + y = 5 \), and \( y - x = 5 \). Use Green's theorem to determine if the integral evaluates to zero.

Answer: 0


Question: Find the rank of the image of the homomorphism $H_2(Q) \to H_2(\tilde{X})$ in the blow-up of $\mathbb{CP}^{3}$ at a planar nodal cubic curve, where $Q$ is the smooth quadric surface replacing the node in the blow-up.

Answer: 2


Question: Evaluate the limit using rationalization: $$\lim_{t\to0}\frac{(1-(1-t)^{1/2})(1-(1-t)^{1/3})\cdots(1-(1-t)^{1/n})} {t^{n-1}}$$

Answer: \dfrac{1}{n!}


Question: Evaluate the limit: $$\lim_{n\to \infty}\frac{2^n n^2}{n!}$$

Answer: 0


Question: Calculate the expected value: \( \mathbb{E}(2X_1+3X_2 \mid X_1+3X_2-X_3=4) \), where \( X_i, i=1,2,3 \) are i.i.d standard normal random variables.

Answer: 4


Question: In the context of pro-$p$ groups, which are defined as projective limits of $p$-groups, if a subgroup $H$ of a pro-$p$ group $G$ has a finite quotient $G/H$, must the order $|G/H|$ be a power of $p$? Answer yes or no.

Answer: Yes


Question: Determine the radius of convergence of the series \( 1+2x+\frac{3^2x^2}{2!}+\frac{4^3x^3}{3!}+\ldots \) using the root test.

Answer: \dfrac{1}{e}


Question: Given the system of equations \( \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 7 \) and \( \frac{y}{x} + \frac{z}{y} + \frac{x}{z} = 9 \), find the value of \( \frac{x^3}{y^3} + \frac{y^3}{z^3} + \frac{z^3}{x^3} \).

Answer: 157


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

Answer: a > 1


Question: Determine whether the function \( r \mapsto M_r \), where \( M_r = \max\{|f(z)|: |z| = r\} \) for a non-constant entire function \( f \), is increasing.

Answer: Yes


Question: From a point, perpendicular tangents are drawn to the ellipse $x^2 + 2y^2 = 2$. The chord of contact touches a circle that is concentric with the ellipse. Determine the ratio of the minimum and maximum areas of this circle.

Answer: \dfrac{1}{4}


Question: Find the limit of \( \sin(x)^{\arcsin(x)} \) as \( x \) approaches 0.

Answer: 1


Question: Evaluate the sum \(\sum\limits_{(m,n) \in E, m > n} \frac{1}{n^2 m^2}\), where \(E = \{ (m,n) \in (\mathbb{N}^*)^2 \mid \gcd(m,n) = 2\}\).

Answer: \dfrac{3}{64}


Question: Using generating functions, determine the number of ways to make change for a $100 bill using only dollar coins and $1, $5, and $10 bills. The generating function is given by: \[ \frac{1}{(1-x)^2}\cdot(1-x^5)^2\cdot(1+x^5). \] Find the coefficient of $x^{100}$ in this expression.

Answer: 0


Question: Find the optimal semi-orthogonal matrix \( X \) that minimizes the norm of \( XQX^T \), where \( Q \) is a diagonal matrix with non-negative entries less than 1, and \( X \) has fewer rows than columns.

Answer: X


Question: Determine whether the following statement is true or false: If $A$ is an orthogonal matrix and $\mathcal{V}$ is a vector subspace, then the dimension of the subspace spanned by the columns of $A\mathcal{V}$ is equal to the dimension of $\mathcal{V}$. Provide your answer as either "True" or "False."

Answer: True


Question: Two independent random variables $X$ and $Y$ are uniformly distributed on $[0,1]$. Find the expected value of $X$ given that $X > Y$, i.e., $\mathbb{E}(X|X>Y)$.

Answer: \dfrac{2}{3}


Question: Given a geometric surface representing a real-world topography, such as a mountain range, and three points A, B, and C on this surface, where the geodesic distances $AB$ and $AC$ are known, determine if the geodesic distance $BC$ necessarily satisfies the triangle inequality, i.e., is $AC \leq AB + BC$?

Answer: AC \leq AB + BC


Question: Compute the derivative of the integral \( \int_0^x f(x-s)\,ds \) with respect to \( x \).

Answer: f(x)


Question: Calculate the entropy of a discrete random variable with the probability distribution given by \( p(x) = \frac{1}{2^x} \) for \( x = 1, 2, 3, \ldots \).

Answer: 2


Question: Determine whether the following statement is true or false: If $A, B \subseteq X$ are equinumerous, meaning there exists a bijection $f: A \to B$, then their complements, $X \setminus A$ and $X \setminus B$, are also equinumerous. Provide a proof if the statement is true, or a counterexample if it is false.

Answer: False


Question: Determine if there exist constants \(c \in (0,1)\) and \(n_0 \in \mathbb{N}\) such that for all \(n \geq n_0\), the inequality \(g\left(\frac{n}{2}\right) \leq c \cdot g(n)\) holds, where \(g(n) = n^5 - \left(\frac{n}{2^n}\right)^3\).

Answer: Yes


Question:
Evaluate the sum

$$\sum_{k=3}^{\infty}{\dfrac{\lfloor q^k \rceil}{3^k}}$$

where $q = \sqrt{2+\sqrt{2+\sqrt{2 + \cdots}}}$ and $\lfloor \cdot \rceil$ denotes the nearest integer function.

Answer: \dfrac{8}{9}


Question: A traveler token starts on the center tile of a hexagonal grid with 19 tiles. Each turn, a die is rolled to determine the neighboring tile the traveler moves to, with all six directions being equally likely. The game ends on the turn the traveler leaves the board. What is the expected number of turns the game will last?

Answer: \dfrac{55}{9}


Question: Determine the range of values of $x$ that satisfy the inequality \(x^2-2x\le \frac{\sqrt{1-\lfloor x\rfloor^2}}{\lfloor x \rfloor + \lfloor -x \rfloor}\) using only algebraic methods.

Answer: (1, 2)


Question: Given that $X$ follows a uniform distribution $Uniform(1, 9.9)$ and, conditionally on $X = x$, $Y$ is normally distributed $Normal(1.4, x^2)$, find the covariance between $X$ and $Y$, denoted as $Cov(X, Y)$. Calculate $E[XY]$ as part of your solution.

Answer: 0


Question:
Given the system of equations:
\[ x + y + z = 4 \]
\[ xy + yz + zx = 2 \]
\[ xyz = 1 \]
find the value of \( x^3 + y^3 + z^3 \).

Answer: 43


Question: Evaluate the limit using L'Hopital's Rule: $$\lim_{x\to0}\frac{\int_{x}^{2x}\frac{\sin u}{u}du}{\ln(1+x)}$$

Answer: 1


Question: Find all natural number solutions \((c, d)\) to the equation \[2^{c-2} + 1 = d^2 + d.\]

Answer: (2, 1)


Question: In a right triangle ABC with integer side lengths, the area is twice the perimeter. Find the sum of all possible circumradii of triangle ABC.

Answer: \dfrac{87}{2}


Question: Can the sum of a square matrix and a rectangular matrix with the same number of rows and columns be invertible?

Answer: No


Question: Evaluate the sum \( \sum_{n=0}^{\infty} n(n-1)(n-2)(n-3) \) using the method of differences and falling factorials.

Answer: \infty


Question: Evaluate the improper integral \( \int_{0}^{+\infty} \frac{\cos x \sin \sqrt{1+x^{2}}}{\sqrt{1+x^{2}}} \mathrm{~d} x \).

Answer: \dfrac{\pi}{4}


Question: What is the conditional expectation $E[Y|X,Z]$ if $Y$ is measurable with respect to the sigma-algebra generated by $X$ and $Z$?

Answer: Y


Question: Evaluate the line integral \(\int_\gamma \sin(y-x)\ dx +\left(2xy + \sin(x-y)\right) \ dy\) along the curve \(\gamma: y = \sqrt{x}\), where \(0 \leq x \leq 1\).

Answer: \dfrac{1}{2}


Question:
Suppose $a_n = n(1 + a_{n-1})$ with $a_1 = 1$. Determine if the limit exists:
\[ \lim_{n \to \infty} \left(1 + \frac{1}{a_n}\right) \left(1 + \frac{1}{a_{n-1}}\right) \cdots \left(1 + \frac{1}{a_1}\right) \]

Answer: e


Question: Is it possible to construct a metric space where every open ball is disconnected, but the space itself is connected? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_C \frac{\log z}{(z-i)^2} \, dz \), where \( C \) is the circle defined by \( |z-i| = \alpha \) and \( 0 < \alpha < 1 \).

Answer: 2\pi


Question: Calculate the expected value of $X_k 1_{\{|X_k| > n\}}$ for $x \geq e$, where $X_1, X_2, \ldots$ are independent and identically distributed random variables with $P(X_i > x) = \frac{e}{x \log(x)}$. Provide your answer in terms of $n$.

Answer: \infty


Question: Calculate the line integral of the vector field \( F = (y^2, xy) \) along the path of the parabola \( y = x^2 \) from \((-1, 1)\) to \((2, 4)\).

Answer: \dfrac{99}{5}


Question: Find the covariance of the Gaussian random variables $x[n]$ and $y[n]$ in the AR(1) model, given that $w[n]$ and $z[n]$ are independent random variables.

Answer: 0


Question: Find a sequence \( v_n \) such that \( \lim_{n\to\infty}\frac{u_n}{v_n}=1 \), where \( u_n=\int_0^1\frac{e^{-t}}{t+\frac{1}{n}}dt \).

Answer: \ln n


Question: Determine the minimum value of the expression \( e^{x+y-2} \) for points \((x, y)\) in the first quadrant that lie on the circle \(x^2 + y^2 = r^2\), where \(r\) is a positive constant.

Answer: e^{r - 2}


Question: Determine the range of values for $m$ such that the equation $|x^2-4|+x^2+mx=0$ has exactly 2 distinct roots in the interval $(1,3)$. Provide your answer as an interval for $m$.

Answer: (-4, -2)


Question: Evaluate the limit: $$S=\lim_{n\to\infty}\sum_{k=1}^n \frac{\log(k)}{nk}$$

Answer: 0


Question: For which real numbers \(a\) does the series \( \sum_{n=0}^\infty \frac{1}{u_n} \) converge, where the sequence \( \{u_n\} \) is defined recursively by \( u_0 = 2 \) and \( \int_{u_n}^{u_{n+1}} (\ln u)^a \, du = 1 \)?

Answer: a < -1


Question: Determine whether every compact topological manifold can be embedded in some Euclidean space \(\mathbb{R}^N\) for a sufficiently large \(N\).

Answer: Yes


Question: Evaluate the limit: $$\lim_{n\to\infty}{n!\cdot (2n)!\over (3n)!}$$

Answer: 0


Question: Determine whether the following statement is true or false: If the derivative of a function $f$ at $0$ does not exist, then the right-hand derivative $f'_+(0)$, defined as $\lim\limits_{n\to \infty}n\left[f\left(\dfrac{1}{n}\right)-f(0)\right]$ for $n \in \mathbb{N}$, may not exist either.

Answer: A


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

Answer: 0


Question: Evaluate the limit \( \lim_{n\rightarrow\infty} \int_{-1}^{1} \frac{x^2}{1+n^4x^4} \, dx \) using the Dominated Convergence Theorem.

Answer: 0


Question: Let $X_r$ be independent random variables such that $X_r$ follows an exponential distribution with rate $\frac{1}{r}$. Let $Z_n = \sum_{r=1}^n X_r$. Use the Monotone Convergence Theorem to find the expected value of $Z=\lim_{n\to\infty} Z_n$. What is the expected value of $Z$?

Answer: \infty


Question: Calculate the probability that the equation \( \frac{x^3}{3} - a^2x + b = 0 \) has exactly one real root, given that \( a \) and \( b \) are randomly chosen from the interval (0,1) with a uniform distribution.

Answer: \dfrac{5}{6}


Question: Evaluate the surface integral \( \iint_S \bar{N} \cdot \bar{F} \, ds \) where \( \bar{F} = 4x\mathbf{i} + 3y\mathbf{j} - 2z\mathbf{k} \) and \( S \) is the surface bounded by the planes \( x=0 \), \( y=0 \), \( z=0 \), and \( 2x+2y+z=4 \).

Answer: \dfrac{40}{3}


Question: Is it necessary for a finite field $E$ with $p^n$ elements, where $p$ is a prime number, to be a field extension of a subfield isomorphic to $\mathbb{Z}_p$, and consequently have characteristic $p$?

Answer: Yes


Question: Determine whether a ring with positive characteristic can have an infinite number of elements.

Answer: Yes


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

Answer: 1


Question: In triangle ABC, where AB = 10 cm and AC = 12 cm, the incenter (I) and barycenter (B') are on a line parallel to side BC. If B'G = GB and B'H = HC, determine the length of side BC.

Answer: 11


Question: Let $f: I \rightarrow \mathbb{R}$ be an injective and continuous function defined on a closed interval $I$, where $c \in I$. Is the inverse function $f^{-1}: f(I) \rightarrow I$ necessarily continuous at $f(c)$?

Answer: Yes


Question: Given a Poisson random variable \( Z \) with parameter \( \mathbb{E}Z \) and another random variable \( G \) that converges in distribution to \( Z \) with an error bound \( d_{TV}(G,Z) \leq c \), where \( c \) is a constant and \( d_{TV} \) is the total variation distance, find an upper bound for the probability \( P( |G-Z| > t) \) for a fixed real \( t > 0 \).

Answer: c


Question: Find the maximum value of the function \( f(x) = 12x^2\int_0^1 yf(y) \, dy + 20x\int_0^1 y^2 f(y) \, dy + 4x \).

Answer: \dfrac{1}{8}


Question: Given real numbers \(x, y, z\), solve the equation \(4^{\sqrt{5x+9y+4z}} - 68 \cdot 2^{\sqrt{5x+9y+4z}} + 256 = 0\). Find the product of the minimum and maximum values of \(x+y+z\).

Answer: 4


Question: Calculate the surface integral of the vector field \( \vec{v} = 2yi - zj + 2xk \) over the portion of the sphere \( x^2 + y^2 + z^2 = a^2 \) that lies above the plane \( x + 2y + z = 0 \).

Answer: 0


Question: For $r > 0$, $\theta \in \mathbb{R}$, and $z_0 \in \mathbb{C}$, determine if the argument $\arg{(r\text{e}^{i\theta}+z_{0})}$ approaches $\theta$ as $r$ approaches infinity.

Answer: \theta


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{1}{(e^{-x}+1)(\pi^2+x^2)}dx \) using a complex-analytic approach.

Answer: \dfrac{1}{2}


Question: Determine the range of the function \( f(x) = \log_{1-x}(x) \cdot \lg(1-x) \).

Answer: (-\infty, 0)


Question: Given that $N=q^k n^2$ is an odd perfect number with Euler prime $q$ and $q \equiv k \equiv 1 \pmod{4}$, where $\gcd(q,n)=1$, determine if the condition $I(n^2) \geq \frac{5}{3}$ implies that $k=1$. Answer with 'yes' or 'no'.

Answer: no


Question: Let $X_i$, $i = 1,\ldots,n$ be i.i.d. Poisson random variables with parameter $\lambda = 2$. Determine the minimum value of $n$ such that $P(1.9 \leq \overline{X} \leq 2.1) \geq 0.90$, where $\overline{X}$ is the sample mean.

Answer: 542


Question: Let $f(x)=8x^3+4x^2-4x-1$. If $f(x)$ has a root of the form $\cos\left(\frac{2\pi}{n}\right)$, where $n$ is a positive integer, find the value of $n$. \(\boxed{\text{Your answer here}}\)

Answer: 7


Question: Evaluate the limit: $$ \lim _{ n \to \infty }{ \int _{ 0 }^{ 1 }{ \left( \frac{1+nx^{ 2 }}{(1+x^{ 2 })^{ n }} \right) \, dx } } $$

Answer: 0


Question: Suppose $X$ is a compact Hausdorff space and $A$ and $B$ are open sets that cover $X$. Determine whether $X$ can be covered by compact sets $C$ and $D$ such that $C \subseteq A$ and $D \subseteq B$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: If $U, V \subset \mathbb{R}^n$ are bounded open subsets and $h: U \to V$ is a homeomorphism, then there exists a homeomorphism $H : \overline{U} \to \overline{V}$ extending $h$. Provide a proof or counterexample.

Answer: False


Question: In an equilateral triangle $ABC$, point $D$ is located on side $BC$. Given that $\angle BAD = 20^{\circ}$, let $I_1$ and $I_2$ be the incenters of triangles $ABD$ and $ACD$, respectively. Construct point $E$ such that triangle $I_1I_2B$ is equilateral with $D$ and $E$ on opposite sides of line $I_1I_2$. Determine the measure of $\angle ADE$. \( \text{Provide your answer in degrees.} \)

Answer: 30


Question:
Find the covariance of $X$ and $Y^2$ given that the random vector $(X,Y)$ follows a bivariate normal distribution with parameters:

$$\begin{pmatrix} X\\ Y \end{pmatrix}\sim N_{2}\left(\begin{pmatrix} 0\\ 0 \end{pmatrix},\begin{pmatrix} 1 & \rho\\ \rho & 1 \end{pmatrix}\right)$$

where $-1<\rho<1$. Provide your answer as a single value.

Answer: 0


Question: In an $n$-person ($n\ge 3$) cooperative game, if players $1$ and $2$ are symmetric, does it necessarily follow that for every element $(x_1,x_2,...,x_n)$ in the core, we have $x_1=x_2$? Answer 'yes' or 'no' and justify your answer.

Answer: No


Question: Find the coefficient of \(x^{27}\) in the expansion of \( f(x) = \frac{(1-x^{10})^6}{(1-x)^6} \).

Answer: 55252


Question: Find the sum of the infinite series: \[ \frac{4}{1!}+ \frac{8}{2!}+ \frac{14}{3!} + \frac{22}{4!}+\cdots \]

Answer: 5e - 2


Question: Calculate \(\lim_{x \to 1^{-}} \frac{\arccos{x}}{\sqrt{1-x}}\) without using L'Hôpital's rule.

Answer: \sqrt{2}


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

Answer: \dfrac{1}{e}


Question: Determine the radius of convergence of the power series \( \sum \frac{(n+1)^n}{n^n} b_n z^n \) given that the radius of convergence of the series \( \sum b_n z^n \) is \( R \).

Answer: R


Question: Given the characteristic equation of a linear transformation $T$ as $t^4 + 3t^2 + 5$, determine the trace of $T$. \( \boxed{0} \)

Answer: 0


Question:
Find the quadratic variation of the stochastic process $Y(t)$ over the interval $[-1, 2]$, where $Y(t)$ is defined as follows:

$$Y(t) = \begin{cases} 0 & -1 \le t \le 0 \\\ Z(t) - tZ(1) & 0 < t < 1 \\\ 1 - t & 1 \le t \le 2 \end{cases}$$

and $Z(t)$ is a standard Brownian motion.

Answer: 1


Question: Let $f:\mathbb{N} \to \mathbb{N}$ be a strictly increasing function such that $f(f(f(n))) = 8n$ for all $n \in \mathbb{N}$. Determine the value of $f(2024)$. \( \boxed{} \)

Answer: 4048


Question: Evaluate the surface integral \( \int \int yz \, dx \, dy + xz \, dy \, dz + xy \, dx \, dz \) over the surface of the cylinder \( x^2 + y^2 = 4 \) bounded by the planes \( x = 0 \), \( y = 0 \), \( z = 0 \), and \( z = 2 \).

Answer: 2\pi - \dfrac{16}{3}


Question: Let $M$ be a W*-algebra with a non-empty set of minimal projections. If $z(M)$ is the supremum of all minimal projections in $M$, and $M_0 = z(M)M$, does every projection in $M_0$ dominate a minimal projection?

Answer: Yes


Question: Solve the congruence equation: \(5^{{9}{^{13}}^{17}} \equiv x \pmod{11}\).

Answer: 9


Question: Determine the x-coordinate of the second intersection point where the tangent line to the curve \( y = ax^3 + bx \) at \( x = x_0 \) intersects the curve again.

Answer: -2x_0


Question: Determine the number of roots of the equation \( f(f(x)) = \int_3^4{f(x)\mathrm{d}x} \) within the interval \([1, 4]\), where \( f(x) = \frac{4x+3}{x^2+1} \).

Answer: 1


Question: Can the method of undetermined coefficients be used to solve the differential equation \(y''+2y'-y=t^{-1}e^t\)?

Answer: No


Question: Find the limit of the function \( g(x,y) = \frac{x^3y^2}{x^4 + 2y^4} \) as \((x,y)\) approaches \((0,0)\).

Answer: 0


Question: Given a random sample $X_1$ and $X_2$ of size $n=2$ drawn from an exponential population with parameter $\theta$ (where the PDF is $f(x)=\lambda e^{-\lambda x}$ and $\lambda=\frac{1}{\theta}$), determine the efficiency of the estimator $2Y_1$ compared to $\bar{X}$, where $Y_1$ is the first order statistic and both $2Y_1$ and $\bar{X}$ are unbiased estimators of $\theta$. Calculate the efficiency of $2Y_1$ relative to $\bar{X}$. \(\text{Efficiency} = \frac{\text{Var}(\bar{X})}{\text{Var}(2Y_1)}\)

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left\{ \prod_{k=1}^{n} \frac{k}{H_k + 1} \right\}^{\frac{1}{n}} \] where \( H_k = \sum_{i=1}^{k} \frac{1}{i} \).

Answer: \infty


Question: Determine whether the inequality \( f(n)^{f(n)} \leq f(n^n) \) holds for all increasing functions \( f: \mathbb{R} \rightarrow \mathbb{R} \).

Answer: No


Question: Determine the symbol and the principal symbol of the differential operator $$Lu=\frac{\partial^4}{\partial x^4}u(x,y)-2\frac{\partial^4}{\partial x^2 \partial y^2}u(x,y)+\frac{\partial^4}{\partial y^4}u(x,y)$$ for $x,y \in \mathbb{R}$. Provide your answer in terms of the variables $x$ and $y$.

Answer: (x^2 - y^2)^2


Question: Let $S^{1} \subset \mathbb{R}^2$ be the unit circle, and let $\pi_1 : S^1 \to \mathbb{R}$ be the natural projection onto the first coordinate. Determine whether the preimage $\pi_{1}^{-1} ( [0,\frac{1}{2}])$ is compact.

Answer: Yes


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

Answer: 0


Question: Let $\{a_n\}$ be a sequence such that $0\le a_n\lt 1$. If the series $\sum_{n=0}^{\infty} \frac{a_n}{1+a_n}$ converges, does it follow that the series $\sum_{n=0}^{\infty} a_n$ converges? Answer 'yes' or 'no' and justify your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to 0}\frac{\arcsin(x)-\arctan(x)}{x^3}$$

Answer: \dfrac{1}{2}


Question: Let $x, y, z$ be positive real numbers satisfying $x^3 + y^3 + z^3 = 3$. Find the minimum value of the expression \[ P = \frac{1}{2-x} + \frac{1}{2-y} + \frac{1}{2-z}. \]

Answer: \dfrac{9 + 2\sqrt[3]{3} + \sqrt[3]{9}}{5}


Question: Find the value of \(\alpha\) such that the series \(\sum_{n=1}^\infty \frac{1}{n(1+\ln(n))^\alpha}\) converges.

Answer: \alpha > 1


Question: Calculate the limit \( \lim_{z \rightarrow 0} \frac{\cos(z)-1}{\sin(2z)} \) in the complex plane.

Answer: 0


Question: Calculate the sum \( S = \sum_{k=1}^{10} t_k \), where \( t_k = k^4 - 2k^3 + 2k^2 - 1 \).

Answer: 20043


Question: Determine whether the following statement is true or false: For an infinitely differentiable function $f$, the higher the degree $n$ of the Taylor polynomial $T_{n,f,x_0}$ of $f$ around $x_0$, the better the approximation of $f$ by $T_{n,f,x_0}$.

Answer: B


Question: Evaluate the partial derivative \( \frac{\partial}{\partial g}\left( g' - g \right) \) where \( g: \mathbb{R} \to \mathbb{R} \) and \( g' = \frac{dg}{du} \). Assume that there is only one variable in the function \( g \).

Answer: -1


Question: Evaluate the limit \( \lim_{x \to 0^+} x^{x^x} - x^x \). Determine if the textbook's conclusion that the limit is \(-1\) is valid.

Answer: -1


Question: Calculate the current price of a 2-year derivative contract with a maturity payoff of £90 × I(S2 > £90), where the current share price is £100, the share price can increase or decrease by 20% annually, and the continuously compounded constant annual risk-free interest rate is such that e^r = 1.1.

Answer: 69.73


Question: Consider a power series \(P(z) = 1 + a_1 z^2 + a_2 z^4 + a_3 z^6 + \ldots\). It is given that \(P\) converges in the open unit disc centered at the origin. Also, for any \(\omega\) such that \(|\omega|=1\), we have that \(\lim_{z \to \omega} |P(z)| = 1\). Can we conclude that \(a_i=0\) for all \(i \geq 1\)? Provide a justification for your answer.

Answer: Yes


Question: Find the smallest value of \( X = xP_1 + yP_2 + zP_3 \) for consecutive primes \( P_1, P_2, P_3 \) (where \( x, y, z \) are positive integers) such that \( X \) is divisible by a prime greater than \( P_3 \).

Answer: 13


Question: As the number of sides $n$ on a dice becomes very large, determine the expected number of rolls needed for the cumulative sum of the top faces to equal or exceed $n$. How does this expectation relate to the mathematical constant $e$?

Answer: e


Question: Determine the value of $n$ such that $2^{2017} + 2^{2014} + 2^n$ is a perfect square.

Answer: 2018


Question: Determine whether the monic greatest common divisor (GCD) of two monic polynomials $f$ and $g$ in $\mathbb{Z}[x]$, when considered as a monic polynomial in $\mathbb{Q}[x]$, necessarily has coefficients in $\mathbb{Z}$. Provide a justification for your answer.

Answer: Yes


Question: Given a linear operator $A: V \rightarrow V$, where $V$ is a vector space, if $R(A) = R(A^2)$, does this imply that $N(A)$ is a trivial subspace (i.e., $N(A) = \{0\}$)? Provide a justification for your answer.

Answer: No


Question: Evaluate the limit \(\lim_{x\to0}\left(e^x+x\right)^{\frac{x}{\sin x}}\).

Answer: 1


Question: Suppose $V$ is a real vector space and $J: V \rightarrow V$ is a linear operator such that $J^2 = -1$. If the complexification of $V$, denoted $\widetilde{V}$, has dimension $n$, does $V$ admit a natural orientation using the basis $\{e_1, J \circ e_1, \ldots, e_n, J \circ e_n\}$ and the dual basis $\{\alpha_1, \alpha_1 \circ J, \ldots, \alpha_n, \alpha_n \circ J\}$?

Answer: Yes


Question: Compute the differential of $\phi(t) = e^{A+tB}$ at $0$, where $A$ and $B$ are commuting square matrices of size $n\times n$. Explain why $e^{A}o(hB)$ is $o(h)$ in the expression $\phi(h) = e^{A}+he^{A}B+e^{A}o(hB)$. Provide a justification for this result.

Answer: e^A B


Question: Evaluate the integral \( \int_{-\infty}^{+\infty} \frac{\cos(mx)}{x^2+1} \, dx \) using the residue theorem.

Answer: \pi e^{-|m|}


Question: Find the residue of the function \( f(z) = (z^2 + 4iz)\sin\left(\frac{3}{z + 2i}\right) \) at \( z_0 = -2i \) using the Laurent series expansion.

Answer: \dfrac{15}{2}


Question: Is the following claim true? If $h(t)=o(t^2)$, then for $g(n) = h\left(\frac{t}{\sigma\sqrt{n}}\right)$, we have $g(n)=o\left(\frac{1}{n}\right)$. Show that $$\lim_{n \rightarrow \infty} h\left(\frac{t}{\sigma\sqrt{n}}\right) n = 0.$$

Answer: 0


Question: Given a differentiable function \( f(x) \) on the real line such that \( \lim_{x \to \infty}f(x)=1 \) and \( \lim_{x \to \infty}f'(x)=\alpha \), determine the value of \( \alpha \).

Answer: 0


Question: Let $\{a_n\}$ be an unbounded sequence of positive real numbers such that $\lim\limits_{n\to\infty}(a_{n+1}-a_n)=0$. Determine whether there exists a sequence $\{b_k\}$ with $\lim\limits_{k\to\infty}b_k=0$ and $\sum_{k=1}^\infty b_k=\infty$ such that $a_n = \sum_{k=1}^n b_k$ for all $n$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the sum of the tangents of the angles -63°, -27°, 9°, 45°, and 81° using Vieta's formula.

Answer: 5


Question:
Given the following system of equations, find the value of \( \pi_1 \):

\[
\begin{align*}
\pi_1 &= \frac {1}{2} \pi_{23} + \frac {1}{2} \pi_2 \\
\pi_2 &= \frac {1}{2} \pi_1 + \frac {1}{2} \pi_3\\
\pi_3 &= \frac {1}{2} \pi_2 + \frac {1}{2} \pi_4\\
\pi_4 &= \frac {1}{2} \pi_3 + \frac {1}{2} \pi_5\\
&\ \ \vdots\qquad \ \ \vdots\\
\pi_{21} &= \frac {1}{2} \pi_{20} + \frac {1}{2} \pi_{22}\\
\pi_{22} &= \frac {1}{2} \pi_{21} + \frac {1}{2} \pi_{23}\\
\pi_{23} &= \frac {1}{2} \pi_{22} + \frac {1}{2} \pi_1.\\
\end{align*}
\]

It is given that \( \pi_1 = \frac{1}{23} \).

Answer: \dfrac{1}{23}


Question: Calculate the 100th derivative of \( f(x) = \frac{1}{1+x^2} \) at \( x=0 \) without using complex numbers.

Answer: 100!


Question: Determine the infimum of the functional \( \phi(f) = \int_{-1}^{1} ((f(x) - x^2)^2 - x^4) \, dx \) over the subset \( \mathcal{M} \) of \( L_{\mathbb{R}}^2[-1,1] \), where \( \mathcal{M} \) contains functions that are not necessarily odd.

Answer: -\dfrac{2}{5}


Question: Let $f : \mathbb{R} \to \mathbb{R}$ be a function with a continuous derivative such that $f(\sqrt{2}) = 2$ and \[f(x) = \lim_{t \to 0}\dfrac{1}{2t}\int_{x-t}^{x+t}sf'(s)\,ds \ \text{for all} \ x \in \mathbb{R}.\] Find the value of $f(3)$. \( \boxed{} \)

Answer: 3\sqrt{2}


Question: Determine the cosine of the angle of intersection between the curves represented by the functions \( f(x) = 3^{x-1}\log(x) \) and \( g(x) = x^x - 1 \).

Answer: 1


Question: Given two uniformities \( \mathcal{U}_1 \) and \( \mathcal{U}_2 \) on a set \( X \) that induce the same topology on \( X \), determine if the identity map \( id: (X, \mathcal{U}_1) \to (X, \mathcal{U}_2) \) is a homeomorphism.

Answer: Yes


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

Answer: 1


Question: Given that the function $f(x+y) = f(x)f(y)$ for all $x, y \in \mathbb{R}$, $f(10) = 4$, and $f'(0) = 3$, find the value of $f'(10)$. \( \boxed{} \)

Answer: 12


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: Calculate the limit using Taylor expansions: $$\lim_{x\to 0}\frac{\ln(1+x^2)\cos 2x}{e^{2x^2}\sin 3x - 3x + \frac{3}{2}x^2}. $$

Answer: \dfrac{2}{3}


Question: Find the derivative of the function \( \psi(t) = \int_0^1 e^{\sqrt{x^2 + t^2 + 1}} \, \mathrm{d}x \) at \( t = 0 \) using the dominated convergence theorem.

Answer: 0


Question: Find a particular solution to the recurrence relation \( y_{n+2} - 6y_{n+1} + 9y_n = 2 \cdot 3^{n+2} \) using the method of undetermined coefficients.

Answer: n^2 \cdot 3^n


Question: Determine whether the following statement is true or false: For a function $f \in L^1(\mathbb{R}^d)$, where $d > 1$, that is uniformly continuous, does the limit \( \lim_{R\to\infty} \int_{|x|=R} |f(x)| \ dS(x) = 0 \) hold, where $dS$ is the surface measure on the sphere of radius $R$?

Answer: True


Question: Calculate \( \lim_{n \to \infty} \frac{\ln\left( 2^{p_{n + 1}} - 1 \right)}{\ln\left( 2^{p_{n}} - 1 \right)} \), where \( \{ p_n \} \) is the sequence of consecutive prime numbers.

Answer: 1


Question: Given the generating function \( A(x) = \frac{1}{2-e^x} \), compute the number of ways to rank 3 ice-cream flavors allowing ties, denoted as \( a_3 \).

Answer: 13


Question: Evaluate the limit \( \lim_{x\rightarrow0}\frac{(e^x \cos x- (x+1)\sec x)}{(x\ln\sin(x))} \) using L'Hopital's rule.

Answer: 0


Question: Find the order of a permutation $a$ in the symmetric group $S_n$ if $a^k$ is a cycle of length $n$, where $n$ is not necessarily a prime number.

Answer: n


Question: Let $G$ be an abelian and periodic group. Consider the set of prime numbers $\mathbb{P}$, and let $\pi$ and $\pi'$ denote subsets of $\mathbb{P}$ such that $\pi$ consists of primes not in $\pi'$. Let $O_{\pi}(G)$ denote the normal subgroup generated by all $\pi$-subgroups of $G$. If $G/O_{2'}(G)$ is a $2$-group, is $G$ necessarily a $2$-group?

Answer: No


Question: Determine whether there exist two representations $U$ and $V$ of the symmetric group $S_n$ such that $Res_{S_{n-1}}^{S_n} U \cong Res_{S_{n-1}}^{S_n} V$, but $U$ and $V$ are not isomorphic as representations of $S_n$. Provide a justification for your answer.

Answer: Yes


Question: Suppose $ f $ is a continuous function from $ [0, 1] $ to $ [-1, 1] $ with $ |f(x)| \leq x $ for $ x \in [0, 1] $. Determine the maximal value of the expression \( \left| \int_{0}^{1} (f(x))^2 - f(x) \, dx \right|. \)

Answer: \dfrac{5}{6}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\int_0^n \frac{e^{-t} \sin{\frac{t}{n}} dt}{(1+\frac{t}{n}^2)\arctan{\frac{t}{n}}}. \]

Answer: 1


Question: Find \( \lim_{x \to 1^+} \frac{x-\sqrt{\arctan(x)-\frac{\pi}{4}}-1}{x-1} \).

Answer: -\infty


Question: Determine the number of elements in the Lie algebra $sl(5,F)$, where $F$ is a finite field with 5 elements.

Answer: 5^{24}


Question: Evaluate the limit for $b > 1$: $$\lim\limits_{n\to\infty}\left[\frac{1}{n}+\sum\limits_{p=1}^{\infty}\left(e^{-b^{pn}}+e^{-b^{-pn}}-1\right)\right]$$

Answer: 0


Question: Calculate the value of the series: \[ \sum_{r=0}^{50} (-1)^r \frac{\binom{50}{r}}{r+2} \]

Answer: \dfrac{1}{2652}


Question: Calculate \( \int_0^1 f(x) \, dx \) where \( f(x) = 0 \) when \( x = 0 \), and \( f(x) = n \) when \( x \in \left(\frac{1}{n+1}, \frac{1}{n}\right] \).

Answer: \infty


Question: Evaluate the limit: $$\lim _{ x \to 0} \frac{\sqrt[3]{1+6x+3x^2+3x^3+3x^4}-\sqrt[4]{1+8x+4x^2+4x^3-2x^4}}{6x^2}$$

Answer: \dfrac{1}{3}


Question:
Determine how many of the following degree sequences correspond to a valid directed graph without loops or multiple edges. The vertices are numbered from 1 to n, and the degree sequences are sorted by vertex numbers.

(a) \(d_{in}=(0,1,2,3), d_{out}=(2,2,1,1)\)

(b) \(d_{in}=(2,2,1), d_{out}=(2,2,1)\)

(c) \(d_{in}=(1,1,2,3,3), d_{out}=(2,2,3,1,2)\)

Answer: 2


Question: Find the minimum value of $x$ for which the infinite power tower $x^{x^{x^{x\cdots}}}$ converges, using the Lambert W function.

Answer: e^{-e}


Question: Find the probability that the roots of the quadratic equation \( h(t) = t^2 + 2Xt + Y = 0 \) are real, given that \( X \) and \( Y \) are uniformly distributed over the rectangle \([-1, 1] \times [0, 1]\).

Answer: \dfrac{1}{3}


Question: Suppose $g$ is a real-valued function defined on $\mathbb{R}$ such that $|g(x) - g(y)| \leq \frac{3}{4}|x-y|$ for all $x,y\in\mathbb{R}$. What is the maximum possible number of intersections between the graph of $g(x)$ and the line $y = x$?

Answer: 1


Question: Find the quadratic function $f(x) = x^2 + bx + c$, where $b, c \in \mathbb{R}$, such that $f(x)$ is a factor of both $x^4 + 6x^2 + 25$ and $3x^4 + 4x^2 + 28x + 5$. Determine the expression for $f(x)$. \( \text{Express your answer in the form } f(x) = x^2 + bx + c. \)

Answer: x^2 - 2x + 5


Question: Determine the asymptotic behavior of the function \( h(n) = \sum_{p_k < n} \frac{\log (1 + p_k)}{p_k} \), where \( p_k \) is the \( k \)th prime number.

Answer: \log n


Question: Determine the dimension of the kernel of $X^3$, where $X$ is the nilpotent matrix $J_n$. Assume $J_n$ is the standard nilpotent Jordan block of size $n$. Provide your answer as a single integer.

Answer: 3


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

Answer: 0


Question: Given the function \( f(x)= \sqrt{(x^2-2x+2)(-x^2+3x+10)}-\sqrt{(-x^{2}+x-1)(x^{2}+x-6)} \), where the domain is \( D=[a,b+c] \), find the value of \( \frac{a^3+b^3+c^3}{abc} \).

Answer: 3


Question: Determine the minimum number of bits Alice must transmit to Bob so that he can compute the approximate Hamming distances between a binary string $A$ of length $n$ and every substring of length $n$ of a binary string $B$ of length $2n$, within a factor of 2.

Answer: O(\log n)


Question: Find the value of the infinite nested radical involving the imaginary unit $i$, given by \( i\sqrt{i\sqrt{i\sqrt{i\sqrt{i\sqrt{i\sqrt{\cdots}}}}}} \).

Answer: -1


Question: Consider a convex but non-smooth function \( f : \mathbb{R}^2 \to \mathbb{R} \). At a particular point, the function increases in the directions \((1,0)\), \((-1,0)\), \((0,1)\), and \((0,-1)\). Can we conclude that this point is a local optimum, and due to the convexity of the function, also a global optimum?

Answer: Yes


Question: Consider a bijective map \( f: X \rightarrow X \) between topological spaces, where \( X \) is the same space, and \( f \circ f = \text{id} \). Must \( f \) necessarily be continuous and hence a homeomorphism?

Answer: No


Question: Determine the number of permutations in the symmetric group $S_4$ that cube to a 3-cycle.

Answer: 0


Question: Calculate the probability that a symmetric random walk returns to the origin for the first time after $2n+1$ steps.

Answer: 0


Question: Given an affine subspace $H \subset \mathbb{R^3}$ defined by the equation $x_1 + 2x_2 - x_3 = 2$, and a line $L$ in $\mathbb{R^3}$ with the parametric equation $(0,1,1) + t(2,0,1)$, find the equation of the plane $f(H_1)$, where $H_1$ is another plane defined by the equation $2x_1 - x_2 + x_3 = 1$. The symmetry $f$ reflects $L$ about $H$. What is the equation of the plane $f(H_1)$?

Answer: 7x_1 - x_2 + 2x_3 = 5


Question: Evaluate the double integral \( \int_{\mathbb{R}} \int_{\mathbb{R}} x^2 e^{-y^2} \sin(y) \, \mathrm{d}y \, \mathrm{d}x \) over the entire real plane.

Answer: 0


Question:
Evaluate the limit of the sequence as \( n \to \infty \):
\[
\lim_{n \to \infty}\left(\frac{1}{\sqrt[3]{(8n^{3}+4)}^{2}}+\frac{1}{\sqrt[3]{(8n^{3}+8)}^{2}}+\frac{1}{\sqrt[3]{(8n^{3}+12)}^{2}}+\cdots+\frac{1}{\sqrt[3]{(8n^{3}+8n^{2}-4)}^{2}}+\frac{1}{\sqrt[3]{(8n^{3}+8n^{2})}^{2}}\right)
\]

Answer: \dfrac{1}{2}


Question: Calculate the surface integral of the vector field \( F = (-y, x, z) \) over the surface \( S \) defined by the equation \( x^2 + y^2 = 9 \), \( z = 0 \), for \( x \ge 0, y \ge 0, z \ge 0 \). Use the parameterization \( \phi(r, \theta) = (r\cos\theta, r\sin\theta, 0) \).

Answer: 0


Question:
Let $k$ be a field and consider the formal power series ring $R = k[[t]]$. Suppose $A$ is a ring and there exist two ring homomorphisms $\phi, \psi: R \to Z(A)$, where $Z(A)$ denotes the center of $A$, endowing $A$ with two distinct structures of an $R$-algebra. These structures induce $R$-module structures on $A$ defined as follows:
\[ r \cdot_{\phi} a = \phi(r)a \]
\[ r \cdot_{\psi} a = \psi(r)a \]
If $A$ is finitely generated and free as an $R$-module under the first structure, is it necessarily finitely generated and free as an $R$-module under the second structure?

Answer: No


Question: Determine whether the following statement is true for distinct odd primes $p$ and $q$: $$(pq, (p-1)(q-1)) = 1 \quad \Longleftrightarrow \quad (pq, \operatorname{lcm}(p-1, q-1)) = 1$$

Answer: True


Question: Evaluate the sum \( f(x) = \sum_{n=1}^{\infty} \frac{(n-1)!}{\prod_{r=1}^{n}(x+r)} \) where \( x \in \mathbb{R}^{+} \).

Answer: \dfrac{1}{x}


Question: Determine whether the function sequence \(f_n(x)= \begin{cases} n^2x, & 0 \leq x \leq \frac{1}{n} \\ 2n - n^2x, & \frac{1}{n} < x \leq \frac{2}{n} \\ 0, & \text{elsewhere} \end{cases}\) converges pointwise.

Answer: 0


Question: In a group, let $a$ and $b$ be elements with orders 4 and 2, respectively, such that $a^{3}b=ba$. Determine the order of the element $ba$.

Answer: 2


Question: Let $V$ be a vector space of dimension $n$ over a field $F$, and let $A$ be an endomorphism of $V$. The endomorphism $A$ induces a map $\Phi_A: \text{End}(V) \rightarrow \text{End}(V)$, defined by $\Phi_A(B) = AB$. Determine all possible dimensions for the kernel of $\Phi_A$, denoted as $\ker\Phi_A$. Prove that for all $0 \leq l \leq n$, there exists an endomorphism $A$ such that $\dim(\ker\Phi_A) = ln$, and these are the only possible dimensions.

Answer: ln


Question: Evaluate the limit \( \lim_{n \to \infty} f_n(x) \) where the function sequence \( f_n:[0,\infty) \to \mathbb{R} \) is defined as \( f_n(x) = e^{-nx}\sin(nx) \) for \( n \geq 0 \).

Answer: 0


Question: Compute the limit: $$\lim_{t \rightarrow \infty} \int_{-1}^1 \frac{\cos^2(t^3x^{10})}{tx^2 + 1} \, dx.$$

Answer: 0


Question: Given that $f$ is a positive function, and for any function $g(x)$ which satisfies $g(x) > 1$ for all $x \geq x_0$ (where $x_0$ is some constant), we have \[ \lim\limits_{x \to \infty} \frac{f(x)}{x\cdot g(x)} = 0. \] Does this property imply that $f(x) = O(x)$?

Answer: Yes


Question: Evaluate the limit: $$\lim_{x \to 0} \frac{\ln(1 - \sin x) + x} {x^2}$$ without using l'Hôpital's rule or Taylor expansions.

Answer: -\dfrac{1}{2}


Question: Determine the number of elements in the poset $(F(X, Y), \subseteq)$, where $F(X, Y)$ is the set of functions from $X$ to $Y$, and $f \subseteq g$ if and only if $f(x) = g(x)$ for all $x \in X$. What is the order of this poset?

Answer: |Y|^{|X|}


Question: In triangle \(ABC\), where \(AB = \sqrt{17}\), \(BC = 5\), and \(CA = 4\sqrt{2}\), points \(D\), \(E\), and \(F\) lie on lines \(AB\), \(BC\), and \(CA\) respectively, forming triangle \(DEF\) inside \(ABC\). The minimum perimeter of \(DEF\) is given by \(\frac{q}{p}\sqrt{34}\). Find the value of \(p + q\).

Answer: 37


Question:
Evaluate the limit as \( n \) approaches infinity of the series:

\[
\lim_{n\to\infty}\left(\sum_{r=0}^{n}\left(\binom{n}{r}\frac{1}{n^r(r+3)}\right)\right)
\]

Answer: e - 2


Question: Determine the value of $k$ such that the function \( f(x)=\begin{cases} \frac{kx^3}{(1+2x)^6} & \text{for } x > 0 \\ 0 & \text{for } x \leq 0 \end{cases} \) is a valid probability density function, satisfying \( \int_{0}^{\infty} f(x) \, dx = 1 \).

Answer: 320


Question: Suppose $f$ is a non-negative measurable function and let $E = \{ x : f(x) > 0 \}$. If the Lebesgue measure $m(E) = 0$, does it follow that $f = 0$ almost everywhere? Answer yes or no.

Answer: Yes


Question: Find the probability assigned by the measure $\mu$ to the union of the top and left edges of the unit square $[0,1]^2$. Express your answer as a single value.

Answer: 0


Question: Solve for \(x\) in the equation: \[3\sqrt{3x-2}+x\sqrt{6-x}=10\]

Answer: 2


Question: Given positive real numbers $x_1,x_2,\ldots,x_n$ satisfying $\displaystyle\sum_{i=1}^{n}{x_i}=n$, determine whether the inequality \(\sum_{\text{sym}}{\prod_{i=1}^{n}{x_i^i}} \leqslant n!\) holds. Here, $\sum_{\text{sym}}$ denotes the sum over all permutations of $x_1,x_2,\ldots,x_n$.

Answer: Yes


Question: Evaluate the definite integral \( \int_0^{2\pi} \frac{\sin{nx}\cos{nx}}{\sin{x}}\,\mathrm{d}x \) for any positive integer \( n \).

Answer: 0


Question: Given that $x + y + z = xyz$ for positive integers $x, y, z$, find the value of $xy + yz + xz$.

Answer: 11


Question: Evaluate the integral \( \int_{0}^{2\pi} \frac{\sin{3\theta}}{5-4\cos\theta}d\theta \) using the method of residues.

Answer: 0


Question: Find the range of values of $k$ such that $3^k - 1$ is divisible by $2^n + 1$ for some positive integer $n$.

Answer: k \equiv 0 \pmod{4}


Question: Determine whether the statement $f + g \sim 2g$ as $x \to 0$ is true, given that $f \sim g$ as $x \to 0$. Provide a justification for your answer.

Answer: True


Question: Let $f(x)$ be a continuous function defined on $[0,1]$ such that the derivative $f'(x)$ exists and is continuous. Consider the sequence of intervals $G_n = [ a(n), b(n) ]$, where $a(n)<b(n)<x_0$ and $a(n) \to x_0$, $b(n) \to x_0$ as $n\to\infty$. Determine if $\lim_{n\to\infty} \frac{f(b(n)) - f(a(n))}{ b(n) - a(n) } = f'(x_0)$. Justify your answer.

Answer: f'(x_0)


Question: Evaluate the limit of the function \( \exp\left(\frac{|x-2y|}{(x-2y)^2}\right) \) as \((x,y)\) approaches \((2y, y)\). Consider the behavior of the function when \((x-2y)\) is less than 0 and greater than 0.

Answer: \infty


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \left(\cos^{2} x - 3 \sec x\right) \delta^{\prime\prime}(x) \, dx \) using the definition of the distributional derivative.

Answer: -5


Question:
Given a nonempty manifold $M$ with dimension $n$ where $n > 0$ and $0 < k < n$, determine the dimension of the space of $k$-forms, denoted $
abla^kM$. Choose the correct option:
1. $
abla$
2. $\binom{n}{k}$
3. $\frac{k(k-1)}{2}$

Answer: 2


Question: Let $T$ be a linear operator on $\mathbb{R}^5$ such that $T^4 \neq 0$ and $T^5 = 0$. Determine the rank of $T^2$. \( \text{Provide your answer as a single integer.} \)

Answer: 3


Question: Given that the three-digit numbers A88, 6B8, and 86C are all divisible by 72, determine the value of the determinant \(\begin{vmatrix} A & 6 & 8\\ 8 & B & 6 \\ 8 & 8 & C \end{vmatrix} \pmod{72}\).

Answer: 0


Question: Let $x$ and $y$ be real numbers satisfying $x^3 - 3xy^2 = 10$ and $y^3 - 3x^2 y = 15$. Calculate the value of $x^2 + y^2$. \( \boxed{} \)

Answer: \sqrt[3]{325}


Question: Solve the equation \( \sqrt{2x^2 + 3x +5} + \sqrt{2x^2-3x+5}=3x \) without using the square of both sides method. Consider using substitution or other creative techniques.

Answer: 4


Question: Evaluate the infinite series: \[ \sum_{n=0}^{\infty} \frac{3n-2}{n!} \]

Answer: e


Question: Determine the maximum number of perfect squares that can appear in an arithmetic progression of 10 positive integer terms.

Answer: 5


Question: Determine if there are any other perfect numbers of the form \(2^x \cdot 3^y\) besides 6, where \(x\) and \(y\) are positive integers.

Answer: 6


Question: Determine whether there exists a non-compact topology \( E \) such that for every continuous function \( f: E \to \mathbb{R} \), the maximum value \( \max(f(E)) \) exists.

Answer: Yes


Question: Determine whether an irreducible polynomial in \( \mathbb{Q}[x] \) can have more than two real roots. If it is possible, provide an example of such a polynomial.

Answer: x^3 - 3x + 1


Question: If the partial sums of a complex sequence $(\beta_n)$ are unbounded, does it imply that the sequence $(\beta_n)$ itself is unbounded? Provide a justification for your answer.

Answer: No


Question:
Given positive numbers $a$, $b$, and $c$ such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} \le 3$, find the maximum value of the expression:
\[ \frac{1}{\sqrt{a^2 - ab + 3b^2 + 1}} + \frac{1}{\sqrt{b^2 - bc + 3c^2 + 1}} + \frac{1}{\sqrt{c^2 - ca + 3a^2 + 1}} \]

Answer: \dfrac{3}{2}


Question: Evaluate the sum \(\sum_{n=2}^\infty \frac{n^2+n-1}{(n+1)!}\) using a telescoping method.

Answer: \dfrac{3}{2}


Question: Transform the infinite series \(\sum_{k=1}^\infty\frac{1}{k(k+1)(k+2)(k+3)}\) into a telescopic series and find its sum.

Answer: \dfrac{1}{18}


Question: Let $f(x)$ be an irreducible polynomial of degree $p$ (where $p$ is a prime) over the field of rational numbers $\mathbb{Q}$. Assume that $f(x)$ has exactly two nonreal complex roots in $\mathbb{C}$. Determine the Galois group of $f(x)$ over $\mathbb{Q}$.

Answer: S_p


Question: Find the residue of the function \( f(z) = \frac{\sin^2(z)}{\sinh^5(z)} \) at \( z = 0 \).

Answer: -\dfrac{7}{6}


Question: Consider the function \(f(x,y,z) = \alpha^2(x-y) + \alpha(z-y) + (x-z)\), where \(\alpha\) is a constant in \(\mathbb{R}\). If \(f(a,b,c) = f(c,a,b) = f(b,c,a) = 0\) for all \(a, b, c \in \mathbb{R}\), does this imply that \(a = b = c\)? Answer 'yes' or 'no'.

Answer: yes


Question: How many 10-digit numbers, using each digit from 0 to 9 exactly once, can be divided by any number between 2 and 15 in sequence to result in a final value between 1 and 14, without resulting in fractions at any step?

Answer: 0


Question: Suppose we have two sequences of random variables, $X_n$ and $Z_n$, such that $\limsup_{n \to \infty} X_n \leq X$ and $Z_n - X_n \to 0$ in probability. Prove or disprove that for any $\varepsilon > 0$, the following holds: $$\lim_{n \to \infty} \Pr(Z_n \leq X + \varepsilon) = 1.$$

Answer: 1


Question: Evaluate the infinite series: $$\sum_{n=1}^{\infty}\frac{f(n)}{n(n+1)}$$ where $f(n)$ is the number of ones in the binary representation of the integer $n$. What is the value of this series?

Answer: 2 \ln 2


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

Answer: 1


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

Answer: y = Cx^2


Question: Given the equation \(z^3-pz+q=0\) with roots \(\alpha\), \(\beta\), and \(\gamma\), and the conditions \(\alpha^2+\beta^2+\gamma^2=2p\) and \(\alpha^4+\beta^4+\gamma^4=2p^2\), find the value of \((\alpha+\beta+\gamma)^2(\alpha\beta+\beta\gamma+\gamma\alpha)\).

Answer: 0


Question: Given a symmetric $N \times N$ matrix $A = \{a_{ij}\}$, where $a_{ii} = 1$ for all $i$, and $a_{ij} = 1$ or $0$ for $i \neq j$, with at least one non-diagonal element being non-zero, define $E_A(\vec{\theta}) = \sum_{i,j}a_{ij}(1-\cos(\theta_i-\theta_j))$, where $\theta_i \in (0, 2\pi]$ for all $i \in \{1, \cdots, N\}$. Is it possible to find another symmetric matrix $B = \{b_{ij}\}$, which satisfies the conditions $b_{ii} = 1$ for all $i$, $b_{ij} = 0$ or $1$ for $i \neq j$, and $b_{ij} = 0$ whenever $a_{ij} = 1$ for $i \neq j$, such that $E_B(\vec{\theta}) \leq E_A(\vec{\theta})$? Provide a justification for your answer.

Answer: Yes


Question: Determine the order of the subgroup $H$ of the group $G$, consisting of $2 \times 2$ integer matrices modulo a prime number $p$, with non-zero determinants under matrix multiplication, where the elements $h \in H$ satisfy $\det(h) = 1$. For a matrix $h = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the condition $\det(h) = ad - bc = 1$ holds. Enumerate the distinct equivalence class combinations for $a, b, c,$ and $d$ such that their determinant equals $1$.

Answer: p(p^2 - 1)


Question:
Given a field $K$ and the vector space $V=K^n$, the determinant function $D:V^n\rightarrow K$ has the following properties:
1. $D$ is linear in each row.
2. Switching two rows changes the sign, i.e., $D(P) = -D(P')$ for any permutation matrix $P$ that switches two rows.
3. $D(e_1, e_2, \dots, e_n) = 1$, where $e_i$ are the standard basis vectors.

From property 2, we have $D(a_1, \dots, b, \dots, b, \dots, a_n) = -D(a_1, \dots, b, \dots, b, \dots, a_n)$. Hence, $2D(a_1, \dots, b, \dots, b, \dots, a_n) = 0$. Determine if $D(a_1, \dots, b, \dots, b, \dots, a_n) = 0$ regardless of the characteristic of $K$. Provide a justification for your answer.

Answer: 0


Question: In a graph $G = (V,E)$, each vertex $v \in V$ is randomly assigned a unique integer identifier. An edge $(u,v)$ is directed from $u$ to $v$ if $u < v$. What is the expected number of vertices that have an in-degree of 0 or an out-degree of 0?

Answer: 2


Question: Let \( \mathcal{H} = l^2(\mathbb{N}) \). If \( N \geq 1 \) and \( L: \mathcal{H} \rightarrow \mathbb{F} \) is defined by \( L(\{\alpha_n\}) = \alpha_N \), find a vector \( h_0 \in \mathcal{H} \) such that the linear functional \( L(h) = \langle h, h_0 \rangle \) holds for all \( h \in \mathcal{H} \), where \( \langle \cdot, \cdot \rangle \) denotes the inner product on \( l^2(\mathbb{N}) \) with \( \langle x, y \rangle = \sum_{n\geq1} x(n)\overline{y(n)} \).

Answer: e_N


Question: Let $G = \langle a, b \rangle \times H$, where $a$ and $b$ commute. Consider the subgroups $G_1 = \langle a \rangle \times H_1$ and $G_2 = \langle a \rangle \times H_2$, where $H_1$ and $H_2$ are subgroups of $H$. If $G_1$ and $G_2$ are conjugate in $G$, are $H_1$ and $H_2$ conjugate in $H$?

Answer: Yes


Question: Let \( n > 1 \), and let \( f: \mathbb{R}^n \to \mathbb{R} \) be a real-analytic function that is not identically zero. Is it true that \( \dim_{\mathcal{H}}(f^{-1}(0)) \le n-1 \), where \( \dim_{\mathcal{H}} \) denotes the Hausdorff dimension?

Answer: Yes


Question: Determine the value of the Ramsey number \( R(K_3, C_5) \), which is the smallest number of vertices \( n \) such that any graph on \( n \) vertices contains either a triangle \( K_3 \) or a cycle of length 5 \( C_5 \).

Answer: 9


Question: In the context of the real numbers \(\mathbb{R}\) with the Lebesgue measure, if two sets \(A\) and \(B\) have positive measure (\(m(A), m(B) > 0\)), does their sum \(A+B = \{a+b \mid a \in A, b \in B\}\) necessarily contain an open interval?

Answer: Yes


Question: Evaluate the limit \(\lim_{{n\to\infty}}\left(1+\frac{1}{a_1}\right)\left(1+\frac{1}{a_2}\right)\cdots\left(1+\frac{1}{a_n}\right)\), given that \(a_1 = 1\) and \(a_n = n(1+a_{n-1})\) for \(n > 1\).

Answer: e


Question: Determine the non-negative integer values of $a$ for which the system of equations \(x^2 - y^2 = 0\) and \((x-a)^2 + y^2 = 1\) has exactly three real solutions.

Answer: 1


Question: Find an odd integer $k$ such that the expression $\frac{(k^2 - 1)}{8}$ is both a triangular number and a prime number.

Answer: 5


Question: Determine the maximum possible order of a group $G$ in which every element $g$ satisfies $g^3 = e$, where $e$ is the identity element.

Answer: 27


Question: Evaluate the integral \( \int_{\mathbb{R}^{2}} |x| e^{-\sqrt{x^{2}+y^{2}}} \, d\lambda^{2}(x,y) \).

Answer: 8


Question: Find the limit of the sequence \(a_n=\int_n^{n+\sqrt{n}}\frac{\sin x}{x}\ dx\) as \(n\to\infty\).

Answer: 0


Question: Calculate the moment of inertia of the solid defined by the inequalities $0 \leq x$, $0 \leq y$, $0 \leq z$, and $x + y + z \leq 1$ about the axis given by the line $\{(t,t,t)|t\in \mathbb{R}\}$.

Answer: \dfrac{1}{60}


Question: Verify if the coefficient of the highest order derivative in the linear, homogeneous ordinary differential equation obtained from the determinant equation vanishes at \( x = 0 \).

Answer: Yes


Question: Evaluate the limit: $$\lim\limits_{x\to0}\frac{\left(1+x\cdot e^x\right)^{\frac{1}{x^2}}}{\left(1+x\cdot5^x\right)^{\frac{1}{x^2}}}$$

Answer: \dfrac{e}{5}


Question: In isosceles triangle \( \triangle ABC \), where \( \overline{AD} \) is a median and \( \overline{EC} \) is a cevian, it is given that \( \overline{AD} = 2\overline{EC} \). Find the value of \( \cos{B} \).

Answer: \dfrac{1}{4}


Question: Evaluate the limit \( \lim_{n\to\infty}\int_{0}^{\pi/2} \frac{\cos^n(x)}{1+x^2} \, dx \) using the Laplace method.

Answer: 0


Question: Evaluate the sum \( S_n = \sum_{k=n}^{2n-1} \binom{2n-1}{k} x^k (1-x)^{2n-1-k} \) and express it in a closed form involving special functions, where \(|x| \leq 1\).

Answer: I_x(n, n)


Question: Determine whether the inequality \((\nabla f(x) - \nabla f(y))^T (x-y) \geq 0\) is a sufficient condition for the convexity of a twice differentiable function \(f\) with a convex domain.

Answer: Yes


Question: Find all analytic functions \( g: \mathbb{C}-\{1\} \to \mathbb{C} \) such that \( g(z) = g\left(\frac{z}{2}\right) \) for all \( z \in \mathbb{C}-\{1\} \).

Answer: g(z) = c


Question: Evaluate the limit \( \lim_{x\to 0}\frac{\int_{x}^{x^2}e^{\sin t}dt}{x^3} \) using L'Hopital's Rule and the Fundamental Theorem of Calculus.

Answer: -\infty


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

Answer: 0


Question: Determine whether the limit \( \lim_{x\rightarrow 0^+}\frac{1}{\sqrt{x}}\exp\left[-\frac{a^2}{x}\right] \) exists for real \( a \).

Answer: 0


Question: Determine the fixed subfield of the field of rational functions \( \mathbb{K}(x) \) over a field \( \mathbb{K} \) of characteristic zero, under the action of the subgroup \( G \) generated by the automorphism \( \sigma: x \mapsto x+1 \).

Answer: \mathbb{K}


Question: Evaluate the limit: $$\lim_{x\to 0} \frac{\int_{x^2}^{x^6}(e^{u^{5}}-1)du}{x^{20}}$$

Answer: -\infty


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

Answer: 1


Question: Find the minimum value of $n$ such that the set of all positive integers can be partitioned into $n$ disjoint subsets, with none of the subsets containing an arithmetic progression of length 3 or more.

Answer: 3


Question: Find the limit: $$\lim_{x\rightarrow 0}|x|^{[\cos x]}$$ where $[\cdot]$ denotes the greatest integer function.

Answer: 1


Question: Evaluate the triple integral \( \iiint_\Omega{\sqrt{x^2+y^2+z^2}dxdydz} \) where \( \Omega=\{x^2+y^2+z^2\le x\} \).

Answer: \dfrac{\pi}{10}


Question: Find the interval of convergence for the power series with coefficients \(a_n= \frac{(\ln n)^2}{n}.\)

Answer: [-1, 1)


Question: Calculate the sum of the infinite series: \[ 1 + \sum_{n=1}^\infty \frac{(4n-2)}{3^n} \]

Answer: 3


Question: Identify the property of a measure space $(X, \Sigma, \mu)$ that ensures any function $f$ in $L_\infty$ also belongs to $L_p$ for all $p \geq 1$. Provide the property as your answer.

Answer: \mu(X) < \infty


Question: Given a point $P$ on the curve $b^4x + 2a^2y^2=0$ and a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} =1$, two tangents $PQ$ and $PR$ are drawn from $P$ to the hyperbola. If $QR$ always touches a fixed parabola, find the equation of this parabola.

Answer: y^2 = 8x


Question: If $\alpha , \beta, \gamma$ are the roots of the polynomial $x^3+x^2-2x+1=0$, find the value of the determinant $$\left|\begin{array}{ccc} \alpha^2 & \beta^2 & \gamma^2 \\ \beta^2 & \gamma^2 & \alpha^2 \\ \gamma^2 & \alpha^2 & \beta^2 \end{array}\right|.$$

Answer: -95


Question: Determine whether there exists an infinite sequence of nonzero rational numbers \((a_n)_{n\in \Bbb{N}}\) such that for all \(n\in \Bbb{N}\), the polynomial \(a_0+a_1X+\cdots+a_nX^n\) has a root at \(x=n\).

Answer: Yes


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

Answer: 1


Question: Determine the number of positive integers \( x \) less than 301 such that for every integer \( i \) from 1 to 2016, the greatest common divisor of \( 7x+i \) and \( i+1 \) is at most 1.

Answer: 2


Question: Evaluate the integral \( \int_{|z|=2}\frac{\cos(zt)}{(z-i)(z+i)}dz \) using the Cauchy integral formula, where \(|z|=2\) is traversed in the clockwise direction.

Answer: 0


Question:
Consider the differential equation:

$$x^2y'' + xy' - y = x^2e^x$$

What is the particular solution of this equation? Choose the correct option from the following:

(a) $y_p = e^x + \frac{1}{x}$
(b) $y_p = e^x - \frac{e^x}{x}$
(c) $y_p = e^x + \frac{e^x}{x}$
(d) $y_p = e^x - \frac{1}{x}$

Answer: B


Question: Consider the polynomial \( x(x+1)(x+2)\cdots(x+239) = \sum_{n=1}^{240}a_nx^n \). Determine the total number of coefficients \( a_n \) that are multiples of 3.

Answer: 159


Question: Let $G = D \ltimes C$ be a finite non-nilpotent group, where $D = N_G(B) = (A \ltimes B)$ is a maximal subgroup of $G$. Here, $A \cong \mathbb{Z}_r$, $B = B_1 \times B_2 \cong \mathbb{Z}_s \times \mathbb{Z}_s$, and $C \cong \mathbb{Z}_t$, with $r, s, t$ being distinct prime numbers. Assume the following conditions hold: (1) $[B, C] \neq 1$, (2) $B_2C \ntrianglelefteq G$, (3) $A \ntrianglelefteq D$, (4) $CA \ntrianglelefteq G$, and $CA$ is cyclic, (5) $B_1 = C_B(C)$, (6) $C$ is a minimal normal subgroup of $G$. Prove that $[B_1, A] = 1$.

Answer: [B_1, A] = 1


Question: Evaluate the limit: $$\lim_{x\to 0} \frac{\sin^2x-x^2}{x^4}$$ without using L'Hopital's rule or series expansion.

Answer: -\dfrac{1}{3}


Question: Determine whether the following statement is true or false: A real square matrix $A$ is diagonalizable if and only if it satisfies the equation $A^2 + A + I = 0$. Provide a justification for your answer.

Answer: False


Question: Does every locally convex topological vector space have a convex balanced local base? Provide a justification for your answer.

Answer: Yes


Question: In a field of characteristic $p > 0$, determine if there exists a $q$th root of unity different from 1, where $q$ is not a multiple of $p$. Provide a justification for your answer.

Answer: Yes


Question: Solve the system of equations: \[ \large \left \{ \begin{aligned} x^2 + y^2 &= 8 \\ \sqrt[2018]{x} - \sqrt[2018]{y} &= (\sqrt[2019]{y} - \sqrt[2019]{x})(xy + x + y + 2020) \end{aligned} \right. \]

Answer: (2, 2)


Question: In an equilateral triangle ABC, point D is inside the triangle such that \(DB^2 + DC^2 + BC^2 = 120\). If the area of triangle DBC is \(6\sqrt{3}\), find the length of side BC.

Answer: 6\sqrt{2}


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

Answer: 2(1 - y)


Question: Determine whether 0 can be a limit point of the set \(\{\sum_{j=0}^{n-1}a_{j}e^{\frac{2\pi i j}{n}}: a_{0}, ..., a_{n-1} \in \mathbb{Z}\}\) in the complex plane for \(n \in \mathbb{N}\) and \(n \geq 2\).

Answer: Yes


Question: Evaluate the integral \( \int_0^1 \frac{\sqrt{1-x}}{\sqrt{x}}dx \).

Answer: \dfrac{\pi}{2}


Question: Find the limit of the sequence \( b_n = \frac{4^n n!^2}{(2n)! n^2} \) as \( n \to \infty \).

Answer: 0


Question: Evaluate the limit \(\lim_{n\to\infty}\frac{\prod_{k=1}^n(2k-1)}{(2n)^n}\).

Answer: 0


Question: Determine the extension degree of the constructible subfield $\Omega^{\text{constr}}$ of the splitting field $\Omega^f_{\mathbb{Q}}$ for the polynomial $f=x^9-2x^7+3x^2-6$, given that it is a subfield of the largest real subfield of $\Omega^f_{\mathbb{Q}}$. Provide your answer as an integer.

Answer: 2


Question: Find the normal vector to the surface defined by the equation $xyz=1$ at the point $(a, b, c)$, where $abc \neq 0$.

Answer: (bc, ac, ab)


Question: Determine whether the function \( f(x) = \int_0^x g(t) \, dt \) is differentiable for \( x < 0 \) given that \( g(-x) \) is continuous on \( \mathbb{R} \).

Answer: Yes


Question: Let $f^{(n)}(a)$ denote the n-th derivative of $f$ at $a$. If $f(x) = (1 + x^2)^{15}$, find $f^{(27)}(1)$. \( \boxed{?} \)

Answer: 160 \cdot 28!


Question: Determine whether it is valid to interchange the limit operators in the expression involving a branching process $Z_n, n \ge 0$: \[ \lim_{n \to \infty} \lim_{k \to \infty} P(Z_n > \frac{1}{k}) = \lim_{k \to \infty} \lim_{n \to \infty} P(Z_n > \frac{1}{k}) \] given that $P(Z_n > \frac{1}{k})$ is non-decreasing in $k$, $\lim_{k \to \infty} P(Z_n > \frac{1}{k}) = P(Z_n > 0)$ converges for all $n$, and $\lim_{n \to \infty} P(Z_n > 0) = 0$. Provide a justification for your answer.

Answer: Yes


Question: Find the chromatic number of the hypercube graph $Q_n$, where $n$ is a positive integer.

Answer: 2


Question: In triangle $\triangle ABC$, the sides are $AB=3$, $BC=5$, and $AC=7$. A point $O$ is located inside the triangle such that $\vec{OA}+2\vec{OB}+3\vec{OC}=0$. Determine the ratio of the area of $\triangle ABC$ to the area of $\triangle AOC$. Express your answer as a simplified fraction.

Answer: 3


Question: Is the consequence set \(Cn(X) =\{q\mid p\vdash q,\; p\in X\}\) of a consistent set \(X\) of formulas consistent?

Answer: Yes


Question: Determine whether there exists a holomorphic function \( f \) defined on the unit disk \( \{ z : |z| < 1 \} \) such that \( f\left(\frac{1}{n}\right) \leq \frac{1}{2^n} \) for all \( n \in \mathbb{N} \).

Answer: Yes


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

Answer: -\dfrac{1}{2}


Question: If $f$ is an analytic function on a domain $D$ and the real part of $f$, denoted as $\mathrm{Re} f$, is constantly $-45$, prove or disprove that $f$ must be of the form $f = -45 + Ci$ on $D$ for some constant $C \in \Bbb{R}$.

Answer: f = -45 + Ci


Question: Determine if there exists a constant $K$ depending on $p > 1$ such that for all $a > 0$ and $b > 0$, the inequality $|a-b|^p \leq K |a^p - b^p|$ holds. If such a constant exists, find its explicit form.

Answer: 1


Question: Evaluate the expression: \[ \int\limits_2^{10} {g(y)\,dy} + \int\limits_1^2 {f(x)\,dx} \] where $g(x)$ is the inverse function of $f(x) = x^3 + x$. Provide your answer as a single numerical value.

Answer: 18


Question: Find a function $h(t)$ such that the expression \( M_t = [B_t + h(t)] \cdot e^{-B_t - \frac{t}{2}} \) is a martingale, where \( B_t \) is a Brownian motion.

Answer: t


Question: Find the distance from the point $A(1, 3, 2)$ to the line of intersection of the planes $P_1: x + y + 2z = 1$ and $P_2: x - z = 2$. Express your answer in simplest form.

Answer: \sqrt{10}


Question: Determine whether the statement is true or false: If $p$ is a prime number, then $(p)$ is a prime ideal in $\mathbb{Z}[x,y]$. Provide a justification for your answer.

Answer: A


Question: Find the value of \( k \) such that \[ \int_{0}^{\infty} \left(\dfrac{\ln x}{1-x}\right)^{2} \mathrm{d}x + k \times \int_{0}^{1} \dfrac{\ln (1-x)}{x} \mathrm{d}x = 0. \]

Answer: 4


Question: Evaluate the finite sum \( \frac{1}{n} \sum_{k=1}^{n-1} \sin\left(\frac{2\pi k}{n}j\right) \) for integers \( j \) and \( n \), where \( j < n \).

Answer: 0


Question: Determine whether the sequence \(a_n = n\left(\prod\limits_{k=1}^{n} \sin\frac{1}{k}\right)^{1/n}\) converges. If it converges, find its limit.

Answer: e


Question: Evaluate the surface integral \(\int \int_ S(x^2 +y^2 +z^2)^2dS \) where \(S\) is the sphere defined by \(x^2 +y^2 +z^2=a^2\) and \(a >0\).

Answer: 4\pi a^6


Question: Find the covariance between the random variables \( Y = \int_0^{2} B(s) \, ds \) and \( X = B(2) \), where \( (B(t))_{t \geq 0} \) is a Brownian motion process.

Answer: 2


Question: Given the differential equation \( y' = y^2 - 13y + 77 \) with the initial condition \( y(0) = \frac{13}{2} \), determine if there exists a \( t^* \in \mathbb{R} \) such that \( y(t^*) = -1 \). Provide a proof without solving the ODE.

Answer: Yes


Question: Find the value of $y$ such that $y^{13} \equiv 15 \pmod{923}$, given that $923$ is the product of two primes.

Answer: 444


Question: Let $K$ be a Galois and CM field. If an element $u \in K$ has absolute value 1 in $\mathbb{Q}$, do all conjugates of $u$ also have absolute value 1?

Answer: Yes


Question: Find the limit of the function \( f(h) = \frac{\ln^2(1+h)\sin\left( \frac{1}{h}\right)}{h} \) as \( h \) approaches 0.

Answer: 0


Question: Suppose $A$ is a real symmetric positive definite matrix. Determine whether the equation $A^{-1/2}AA^{-1/2}=I$ holds true.

Answer: True


Question: Find the radius of convergence for the power series \( \sum_{n=1}^{\infty}\left(\frac{1}{\sqrt[n]{e}}-1\right) x^n \). Avoid using methods that result in the indeterminate form \( \frac{0}{0} \).

Answer: 1


Question: Evaluate \( \bigtriangleup ^{6} p(x) \mid _{x=0} \) for the polynomial \( p(x) = \sum_{k=0}^{10} k^{2} x^{ \underline{k} } \), where \( x^{\underline{k}} \) denotes the falling factorial. Use the identity \( p(x) = \sum_{k=0}^{n} \bigtriangleup ^{k}p(0) \frac{{x}^{\underline{k}}}{k!} \) and the properties of the forward difference operator \( \Delta \) to find the solution.

Answer: 25920


Question: In how many ways can 10 people be arranged in a circle such that no person is seated next to their partner, given that they came in pairs?

Answer: 112512


Question: Evaluate the limit: $$\lim_{{n\to \infty}} \left(10\sqrt[n]{a_n}+5\sqrt[n]{b_n}\right),$$ where $$a_n=\int_0^1 (1-x^2)^n \,dx$$ and $$b_n=\int_0^1 (1-x^3)^n \,dx$$ for $n \in \mathbb{N}$."

Answer: 15


Question: What is the value of the Hausdorff measure \(\mathcal{H}^2\) for the unit cube \(A = \{(x,y,z) \in \mathbb{R}^3: x \in [0,1], y \in [0,1], z \in [0,1]\}\) in \(\mathbb{R}^3\)? Justify your answer.

Answer: \infty


Question: Let $\mathbb{k}$ be a field, and consider the vector space $\mathbb{k}_{10}[x]$ of polynomials in $x$ with degree at most 10. Define the shift operator $T: \mathbb{k}_{10}[x] \to \mathbb{k}_{10}[x]$ such that $T(f(x)) = f(x+1)$. Determine the minimal polynomial $\mu(t)$ of $T$. \( \boxed{\mu(t)} \)

Answer: (t - 1)^{11}


Question: Given that \((g'(x))^2 = g(x)\) for all real \(x\), with \(g(0) = 0\) and \(g(4) = 4\), find the value of \(g(1)\).

Answer: \dfrac{1}{4}


Question: Determine the number of permutations \( s \) in the symmetric group \( S_{10} \) such that \( bs = sa \), where \( a = (135)(2468) \) and \( b = (13)(24568) \).

Answer: 0


Question: Consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$, defined by $f(0) = 1$ and satisfying the property $f(xy + 1) = f(x)f(y) - f(y) - x + 2$ for all $x, y \in \mathbb{R}$. Determine the function $f(x)$. \( \boxed{?} \)

Answer: x + 1


Question: Let \( f_n : \mathbb{R} \rightarrow \mathbb{R} \) be defined by \( f_n(x)=2|x|(1-\sin^{2n}(\frac{\pi}{2}x)) \) for \( n \in \mathbb{N} \). Calculate \( \beta = \lim_{n\rightarrow \infty}\int_{-9}^{44}f_n(x)\,dx \) using the Dominated Convergence Theorem.

Answer: 2017


Question: Given the recursive relation $\langle x_n \rangle$ defined by \[x_{n+1} = x_n + a + \sqrt{b^2 + 4ax_n}, \quad n \geq 0, \quad x_0 = 0\] where $a$ and $b$ are fixed positive integers, determine the limit \[\lim_{n\to\infty}\frac{x_n}{n^2}.\]

Answer: a


Question: Verify Stokes's theorem for the vector field \( \vec{v} = y\vec{i} + z\vec{j} + x\vec{k} \) over the surface of the paraboloid \( z = x^2 + y^2 \) bounded by the plane \( z = 4 \).

Answer: -4\pi


Question: Determine the remainder when the integer $a$, which satisfies the equation \( \frac{1}{1}+\frac{1}{2}+\ldots+\frac{1}{23}=\frac{a}{23!} \), is divided by 13.

Answer: 7


Question: Evaluate the limit: $$\lim_{x\to0}\frac{\log_{\sin^2x}\cos x}{\log_{\sin^2(x/2)}\cos(x/2)}$$

Answer: 4


Question: Determine \( \limsup_{n\to\infty} \sqrt[n]{\frac{(n!)^3}{(3n)!}} \) using Stirling's formula \( n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n \) as \( n \to \infty \).

Answer: \dfrac{1}{27}


Question: Determine whether the statement "Any two cycles of length four are conjugate in $A_n$ for $n \geq 4$" is true or false. Justify your answer.

Answer: False


Question: Calculate Spearman's rho for the comonotonicity copula, defined as \( C_m(u_1, u_2) = \min(u_1, u_2) \). Spearman's rho is given by the formula \( \rho = 12 \int_{0}^{1}\int_{0}^{1} C(u_1,u_2) \, du_1 \, du_2 - 3 \). Evaluate this integral.

Answer: 1


Question: Find the value of $|a|+|b|$ such that the identity $|ax+by|+|bx+ay|=|x|+|y|$ holds for all real numbers $x$ and $y$. If a unique value exists, provide it.

Answer: 1


Question: Given a context-free grammar with $n$ non-terminals and a maximum production rule length of $k$, determine the maximum length of the shortest word that can be generated by a grammar with $n+1$ non-terminals and a maximum production rule length of $2k$.

Answer: (2k)^n


Question: Suppose we have a sequence of equicontinuous functions $p_n:E \rightarrow \mathbb{R}$ that converge uniformly to $p$ on a complete metric space $E$. Prove or disprove: for any sequence $\{x_n\} \subset E$ with $x_n \rightarrow x$, we have $p_n(x_n) \rightarrow p(x)$. \(\text{True or False?}\)

Answer: A


Question: Evaluate the sum: \[ \sum_{n=1}^{47} \binom{n+3}{3} \]

Answer: 249899


Question: Suppose $f$ and $g$ are two analytic functions on an open set containing the closed disc $\overline{D}$, where $D$ is the open disc in $\mathbb{C}$ with radius 1. Given that $|f(z)| = |g(z)|$ for all $z$ on the boundary of $D$, and that $f$ has a zero of order $n$ at the origin, determine the order of the zero of $g$ at the origin.

Answer: n


Question: Determine whether the sequence $X_n = \sum_{j=1}^{n} Z_j$, where $Z_i$ are independent Gaussian random variables with $Z_i \sim N(0, \frac{1}{2^i})$, converges in mean square. Given that $X_n$ converges in distribution to $N(0,1)$, does this convergence also occur in the mean square sense?

Answer: Yes


Question: Given the joint probability density function for the waiting times $W_1$ and $W_2$ as $f(w_1,w_2)=\lambda^2 e^{-\lambda w_2}$ for $0<w_1<w_2$, determine the conditional probability density function for $W_1$ given that $W_2=w_2$. Provide your answer in terms of $w_1$, $w_2$, and $\lambda$.

Answer: \dfrac{1}{w_2}


Question: Let $R$ be a ring and $M$ be a finitely generated $R$-module. If ${\rm Hom}_R(M , R)=R$, can we conclude that $M=R$? Provide a justification for your answer.

Answer: No


Question:
Consider the one-sided limits $\lim_{x\to c^-} \{ \ln x \}$ and $\lim_{x\to c^+} \{ \ln x \}$, where $\{ \cdot \}$ denotes the fractional part function. If both limits exist finitely but are not equal, determine the possible nature of the value $c$. Choose the correct option:

(a) $c$ can only be a rational number.
(b) $c$ can only be an irrational number.
(c) $c$ can take infinitely many values, with only one being irrational.
(d) $c$ can take infinitely many values, with only one being rational.

Answer: D


Question: In a finite field $F$ with $q=p^n$ elements, where $p$ is an odd prime and $a \neq 0$, determine the number of solutions to the equation $x^2 - y^2 = a$. Express your answer in terms of $q$ and $a$.

Answer: q - 1


Question: Consider a decreasing sequence $(a_n)$ such that the series $\sum_{n=1}^{\infty} a_n$ converges. Determine whether it follows that $\lim_{m\to\infty} ma_{m+1} = 0$. Provide a proof or counterexample.

Answer: 0


Question: Find the value of the limit \( \lim_{x \to \infty} \frac{N(x)}{x} \), where \( N(x) \) is the number of positive integers less than or equal to \( x \) that cannot be expressed as the sum of three squares.

Answer: \dfrac{1}{6}


Question: Determine if there exists a constant \( c > 0 \) such that the inequality \( \frac{(2n)^{cn}}{(2n)!} < 0.05 \) holds for sufficiently large values of \( n \).

Answer: Yes


Question: Determine the number of real solutions to the equation \(11^x + 13^x + 17^x = 19^x\).

Answer: 1


Question: Calculate the volume of the solid enclosed by the surface \((x^2 + y^2 + z^2)^3 = 3xyz\) using a triple integral in spherical coordinates.

Answer: \dfrac{1}{2}


Question:
Given the function \( f(n) = \sum_{r=0}^{n}\sum_{k=r}^{n}\binom{k}{r} \), where \( \binom{k}{r} \) represents the binomial coefficient, and the reformulation:

\[ f(n) = \binom{n}{0} + \binom{n+1}{1} + \binom{n+2}{2} + \ldots + \binom{2n}{n} \]

find the value of \( n \) such that \( f(n) = 2047 \).

Answer: 10


Question: Maximize the integral \( \int_{0}^{1} x^2 \, f(x) \, \mathrm{d}x \) subject to the following constraints: \( \int_{0}^{1} f(x) \, \mathrm{d}x = 1 \), \( \int_{0}^{1} x f(x) \, \mathrm{d}x = 1 \), and \( f(x) \geq 0 \) for all \( x \in [0,1] \).

Answer: 1


Question: Can Euler's equation be used to find extremals of the functional \( \int_0^1 [(y')^2 + 12 xy] \, dx \) for functions in the space \( \{f \in C^1([0,1]) , f''\in  L^1([0,1]) : f(0)=0, f(1)=1\} \)?

Answer: y = x^3


Question: Evaluate the integral \( \frac{1}{2 \pi} \int_0^{2 \pi} \frac{1 - r^2}{1 - 2r \cos(\theta) +r^2}d\theta \) for \(0 < r < 1\).

Answer: 1


Question: Determine whether there exists a continuous function \( f: \mathbb{R} \to \mathbb{R} \) that is differentiable at only one point.

Answer: Yes


Question: Evaluate the limit: $$\lim_{N\to \infty} \frac{2}{N} \sum_{j=0}^{N-1} \left(\frac{3}{2}\left[\frac{2j}{N} + 1\right]\right)$$

Answer: 6


Question: Determine if the function \( f(x) = x^2 \) ever exceeds the function \( g(x) = e^{x \ln x} \) for positive values of \( x \).

Answer: Yes


Question: Evaluate the limit: $$\lim_{x \to \infty} \left( \int_0^{\pi/6} (\sin t)^x dt \right)^{1/x}$$

Answer: \dfrac{1}{2}


Question:
Verify that the following equality holds for any three distinct values of \( x \):
\[
\frac{x^2}{(x-a)(x-b)} + \frac{a^2}{(a-x)(a-b)} + \frac{b^2}{(b-x)(b-a)} = x+a+b
\]

Answer: x + a + b


Question: Let $V$ be a two-dimensional vector space and $T, S$ be two linear transformations on $V$. Determine whether $(TS - ST)^2 = (TS - ST) \circ (TS - ST)$ commutes with every linear transformation on $V$.

Answer: Yes


Question: Given an arrangement of \( n \) circles in the plane such that each circle intersects every other circle without any three circles intersecting at a single point, derive and solve a recurrence relation for the number of regions formed by these intersections.

Answer: n^2 - n + 2


Question: Determine whether the following statement is true or false: If a normed vector space is complete with respect to one norm, then it is complete with respect to any equivalent norm.

Answer: A


Question: Find the minimum value of the trace of the matrix product $AA^T$ for an $n \times n$ non-singular matrix $A$ with integer entries.

Answer: n


Question: Use Stokes' Theorem to evaluate the surface integral \( \iint_S \operatorname{curl} F \cdot \hat{n}\, dS \) where the vector field \( F =\langle xyz, x, e^{xy} \cos(z)\rangle \) and \( S \) is the upper hemisphere \( x^2+y^2+z^2=25 \) for \( z \ge 0 \) oriented upward.

Answer: 25\pi


Question: Given two independent standard normal random variables, $X$ and $Z$, calculate the probability $P(X+Z<0, Z>0)$. Express your answer as a single probability value.

Answer: \dfrac{1}{8}


Question: Let continuous variables $X$ and $Y$ follow a uniform distribution on the domain $D = \{(x, y) \mid 0 < x < y < 1\}$. Determine the joint probability density function (pdf) for $Y$ and $Z$, where $Z = X/Y$. Given that the pdf of $(X, Y)$ is $2$ on $D$, find the joint pdf $f(Y, Z)$ by computing the correct form of the Jacobian matrix and its determinant.

Answer: 2y


Question: Determine whether the level surface \(N_k = \lbrace (x_1,...,x_n): z(x_1,...,x_n)= k, k\in \mathbb{R}\rbrace\) of the tangent plane \(z(x_1,...,x_n)\) of a function \(f(x_1,...,x_n)\) at a point \((a_1,..,a_n) \in N_k\) is equal to the tangent plane \(p(x_1,..,x_n)\) at \((a_1,..,a_n)\) of the level surface \(N_k = \lbrace (x_1,...,x_n): f(x_1,...,x_n)=k, k\in\mathbb{R}\rbrace\) of the function.

Answer: Yes


Question:
Given two linear combinations:
\[ y^{(1)} = \sum_{i=1}^{N-1} a_{i}x_{i} + a_{N}x_{N} \]
\[ y^{(2)} = \sum_{i=1}^{N-1} a_{i}x_{i} + a_{N}x_{N} \]
where $a_{i} \in \mathbb{R}_{\geq 0}$, $x_{1}, \dots, x_{N-1}$ have independent continuous density functions $f_{i}(x)$, and $x_{N}$ has density $h^{(1)}(x_{N})$ for $y^{(1)}$ and $h^{(2)}(x_{N})$ for $y^{(2)}$. Assume $h^{(1)}(x_{N})$ first-order stochastically dominates $h^{(2)}(x_{N})$, i.e., $H^{(1)}(x_{N}) \leq H^{(2)}(x_{N})$ for all $x_{N}$. Does this imply that the density function governing $y^{(1)}$ stochastically dominates the density function of $y^{(2)}$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$ \lim_{x\to 0}\sum_{i=1}^{10}x\left\lfloor\frac{i}{x}\right\rfloor $$ given that $$\lim_{x\to 0} x\left\{\frac{1}{x}\right\} = 1$$. Determine if the limit equals $55$. Provide your answer as a single value.

Answer: 55


Question: Determine the number of positive integers less than or equal to $6 \cdot 7 \cdot 8 \cdot 9$ that satisfy the following system of congruences: \[ \begin{align*} m &\equiv 5 \pmod{6}, \\ m &\equiv 4 \pmod{7}, \\ m &\equiv 3 \pmod{8}, \\ m &\equiv 3 \pmod{9}. \end{align*} \]

Answer: 0


Question: Determine the eigenvalues of the operator \( T: \ell^2 \to \ell^2 \) defined by \( T(x_1, x_2, x_3, \ldots) = \left(0, x_1, \frac{x_2}{2}, \frac{x_3}{3}, \ldots\right) \), where \( \ell^2 \) is the space of square-summable sequences.

Answer: \emptyset


Question: Determine the number of group homomorphisms from $\mathbb{Z}/3\mathbb{Z}$ to $S_3$ without using the fact that the normal subgroups of $S_3$ are $\{e\}, A_3$, and $S_3$. Provide your answer as a single number.

Answer: 3


Question: Evaluate the contour integral \( \int _0^{\infty} \frac{\ln x}{x^2+1} \, dx \) using the residue theorem.

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\rightarrow \infty } x^{2}\left(\sqrt[7]{\frac{x^{3} +x}{1+x^{3}}} -\cos\frac{1}{x}\right) \]

Answer: \dfrac{9}{14}


Question: Evaluate the limit: $$L = \lim_{x\to\infty} x(\ln(1+\sqrt{1+x^2})-\ln(x))$$

Answer: 1


Question: Determine the number of group homomorphisms from the symmetric group $S_3$ to the cyclic group $\mathbb{Z}/6\mathbb{Z}$. \( \boxed{?} \)

Answer: 2


Question: If the center of the automorphism group $Z(\operatorname{Aut}(G))$ is trivial, does it imply that the center of the group $G$ is also trivial, given that $G$ has order at least 3?

Answer: No


Question: Find the value of $a$ such that the plane $ax + 2y + z = 2$ is tangent to the sphere $(x-2)^2 + y^2 + z^2 = 4$. \( \text{Express your answer as a single value of } a. \)

Answer: -2


Question: Simplify the limit \( \lim\limits_{t\to0}\frac{(1-(1-t)^{1/2})(1-(1-t)^{1/3})\cdots(1-(1-t)^{1/n})} {t^{n-1}} \) using rationalization and the standard formula for derivatives.

Answer: \dfrac{1}{n!}


Question: Evaluate the limit: $$\lim_{x\to 0, n\to \infty}\frac{2\sin^2\left(\frac{\tan(x)}{1+\tan(x)}\right)(x+1)^n + \frac{1}{2} \sin\left(\frac{8nx}{n+\sqrt{n^2-8n}}\right)}{x^2(x+1)^n+x}$$

Answer: 2


Question: Determine the closure of the subspace $X = \mathrm{lin}\{e_n : n \in \mathbb{Z}^+\}$ in $l^\infty$, where $e_j$ is the sequence with a 1 in the $j$-th position and 0 elsewhere.

Answer: c_0


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

Answer: 1


Question:
Determine whether the following inequality holds for all continuous functions $f$ and $g$ on $[0,1]$:
\[ \sup_{x \in [0,1]} \left\{ \int_{0}^{x} |f(t) - g(t)| \, dt \right\} \leq \sup_{x \in [0,1]} \left\{ |f(x) - g(x)| \right\}. \]

Answer: Yes


Question: Find the value(s) of the real constant \( C \) such that the integral \( \int_0^\infty \left(\frac{x}{{x^2+1}} - \frac{C}{{3x+1}}\right) \, dx \) converges.

Answer: 3


Question: Consider two coprime monic polynomials \(f\) and \(g\) with rational coefficients. If there exists a Bezout relation \(f(x)u(x)+g(x)v(x)=1\), where \(u\) and \(v\) are polynomials with integer coefficients, does it imply that \(f\) and \(g\) have integer coefficients? Answer yes or no.

Answer: No


Question: Evaluate the integral: \[ \int_{0}^{\infty}\frac{\sin\left(x^{2}\right) + \cos\left(x^{2}\right) - 1} {x^{2}}\,\mathrm{d}x. \]

Answer: 0


Question: Solve the inequality \(8\cdot 3^{\sqrt{x}+\sqrt[4]{x}} + 9^{\sqrt[4]{x}+1} \geq 9^{\sqrt{x}}\) and express the solution set for \(x\).

Answer: [0, 16]


Question: Suppose \( X \), \( Y \), and \( Z \) are independent and identically distributed (i.i.d.) exponential random variables with parameter \( \lambda \). Calculate the probability \( P(X \le 2Y \le 3Z) \).

Answer: \dfrac{18}{55}


Question: Given a compact subset $C$ of a set $A$, and a collection of functions $\Phi$, suppose there exist open sets $V_x$ containing each $x \in C$ such that only finitely many functions $\phi \in \Phi$ are non-zero on each $V_x$. If a finite number of these open sets $V_x$ cover $C$, does it follow that only finitely many functions $\phi \in \Phi$ are non-zero on $C$? Answer 'yes' or 'no' and justify your answer.

Answer: yes


Question: If for any vector \( \mathbf{x} \), the equation \( \mathbf{x}^T A \mathbf{x} = 0 \) holds, does it imply that the matrix \( A \) must have zero diagonal elements? Answer yes or no and justify your answer.

Answer: Yes


Question:
Let $A$ be a positive semidefinite matrix with distinct eigenvalues. Consider its eigendecomposition with eigenpairs $(\lambda_i, v_i)$:

$$ A = \sum_{i=1}^{n} \lambda_i v_i v_i^T $$

where $v_i^T v_i = 1$ and $v_i^T v_j = 0$ for $i \neq j$. If $A = I$, does this decomposition hold for any set of orthonormal vectors $\{u_i\}_{i=1}^n$? Justify your answer.

Answer: Yes


Question: Does there exist a sequence such that its $k^{th}$ difference sequence is anti-Cauchy for all $k \ge 1$?

Answer: Yes


Question: Consider an infinite sequence of squares and circles. The first square, $S_1$, has a unit area. A circle $C_1$ is inscribed in $S_1$, and a square $S_2$ is inscribed in $C_1$. This process continues such that a circle $C_n$ is inscribed in square $S_n$, followed by a square $S_{n+1}$ inscribed in $C_n$. Let $a_n$ represent the sum of the areas of the first $n$ circles, $C_1$ to $C_n$. Find the limit of $a_n$ as $n$ approaches infinity.

Answer: \dfrac{\pi}{2}


Question: For any matrix $A \in M_n(\mathbb{C})$, if $\lambda$ is an eigenvalue of $A$, is its complex conjugate $\overline{\lambda}$ also an eigenvalue of $A$? Answer yes or no.

Answer: No


Question: Evaluate the limit: \( \lim_{x\to \infty }x\left(\sqrt[x]{a}-1\right) \) without using L'Hôpital's rule.

Answer: \ln a


Question: Evaluate the integral \( \int_{-1}^{1} \lvert f'(x) \rvert \, dx \) where \( f(x) = e^{x^2} \).

Answer: 2(e - 1)


Question: Given the equations $x^4 + y^4 = 5$ and $x^2 + xy + y^2 = 10$, find the value of $(x+y)^4$. Express your answer as a single number.

Answer: 195


Question: Calculate the flux of the vector field \( F(x, y, z) = (0, 0, z) \) through the portion of the plane \( x + y + z = 1 \) that lies within the first octant, using the divergence theorem.

Answer: \dfrac{1}{6}


Question: Evaluate the limit: $$\lim\limits_{y \to 0} \frac{{\left(3 \, y - 2\right)} \log\left(3 \, y + 1\right) - 2 \, {\left(3 \, y - 2\right)} \log\left(2 \, y + 1\right) + 3 \, y \log\left(-y + 1\right) - 2 \, \log\left(-y + 1\right)}{2 \, y}$$

Answer: 2


Question: Given a square matrix $A \in \mathbb{K}^{n \times n}$, where $a$ columns and $b$ rows are chosen such that $a + b > n$, and the entry at the intersection of any chosen row and column is zero, is the determinant of such a matrix always zero?

Answer: 0


Question: Determine whether the following statement is true or false: If a function $f$ is analytic within the open unit disk $\left\lvert z \right\rvert < 1$, then for all $n \in \mathbb{N}$, it holds that $$f\left(1-\frac{1}{n}\right) = \left(1-\frac{1}{n}\right)^2.$$

Answer: B


Question: Let $A \in \mathbb{R}^{n \times n}$ where $k_i \in \mathbb{R}^n, i = 1,2, \cdots , n$ are column vectors of $A$ and satisfy the condition $k_i = (i+2)k_{i+2}, i= 1,2, \cdots, n-2$ for $n>3$. Choose one eigenvalue of $A$ and determine the smallest possible dimension for the eigenspace of the chosen eigenvalue.

Answer: 1


Question: Let \( C[0,1] \) denote the set of all continuous real-valued functions on the interval \([0,1]\). Define the operator \( T: (C[0,1], ||.||_{\infty}) \to \mathbb{R} \) by \( T(f) = \int_0^1 t f(t) \, dt \) for all \( f \in C[0,1] \). Find \( ||T|| \).

Answer: \dfrac{1}{2}


Question: Given that \( f^3(x) - 3x^2f(x) + x^3 = 0 \), evaluate the integral \( \int \frac{f(x)}{x} \, dx \).

Answer: f(x) + C


Question: Determine whether the limit \( \lim_{{n\to\infty}}\frac{{p_{n}}}{{n}} \) exists, where \( p_{n} \) denotes the nth prime number.

Answer: \infty


Question: Evaluate the limit: $$\lim_{{(x,y)} \to {(7,3)}} (x^2 - 49)\cos\left(\frac{349}{(x-7)^2 + (y-3)^2}\right)$$

Answer: 0


Question: Given three matrices $A, B, C \in M_n(\mathbb{R})$ such that $A + B + C = 0$, determine the possible values of the determinant of the matrix $\begin{bmatrix} A & B \\ C & I_n \end{bmatrix}$, where $I_n$ is the $n \times n$ identity matrix.

Answer: \mathbb{R}


Question: Determine whether the following statement is true or false: Given $x \in \ell^{p_{0}}$ for some $1 \leq p_{0}<\infty$, it holds that $\|x\|_{p} \rightarrow\|x\|_{\infty}$ as $p \rightarrow \infty$. Here, $\ell^{p}=\left\{x=\left(x_{1}, x_{2}, \ldots\right): \sum_{j=1}^{\infty}\left|x_{j}\right|^{p}<\infty\right\}$ for $1 \leq p<\infty$, $\|x\|_{p}=\left(\sum_{j=1}^{\infty}\left|x_{j}\right|^{p}\right)^{\frac{1}{p}}$, and $\|x\|_{\infty}=\sup_{j \in \mathbb{N}}\left|x_{j}\right|$. Provide a justification for your answer.

Answer: True


Question: Find the expected value of the second largest number when three numbers are chosen randomly from the interval (0, 1).

Answer: \dfrac{1}{2}


Question: Consider a random sample $X$ with probability density function $f(x)=\frac{k(p)}{x^{p}}$ for $x>1$, where $p>0$ and $k(p)$ is a positive constant. Determine the set of values of $p$ for which the variance of $X$ is finite, but the fourth moment does not exist.

Answer: (3, 5]


Question: Determine the value of $p$ for which the series \( \sum\limits_{n=1}^\infty\left(\dfrac1{n^p}\sum\limits_{k=1}^nk^{3/2}\right) \) converges.

Answer: \dfrac{7}{2}


Question: Determine whether the following statement is true or false: If a function $f: \mathbb{R}^d \to \mathbb{R}$ is twice continuously differentiable and its gradient $\nabla f$ is $L$-Lipschitz continuous, then the Hessian of $f$, denoted by $\nabla^2 f(x)$, has a bounded norm, i.e., $||\nabla^2 f(x)|| \leq L$ for all $x$, without assuming $f$ to be convex.

Answer: True


Question: Determine whether the following statement is true or false: If $f$ is a continuous complex-valued function on the entire complex plane, is the function $G_R(w) = \int_0^w f(z)dz$ holomorphic on the closed disk of radius $R$ centered at 0?

Answer: B


Question: Given that \( \tan 3x = 4 \), find the value of \( S = \tan^2 x + \tan^2(120^\circ + x) + \tan^2(60^\circ + x) \).

Answer: 150


Question: A line from the origin intersects the lines \( \frac{x-2}{1} = \frac{y-1}{-2} = \frac{z+1}{1} \) and \( \frac{x-\frac{8}{3}}{2} = \frac{y+3}{-1}=\frac{z-1}{1} \) at points \( P \) and \( Q \) respectively. Calculate \( (PQ)^2 \).

Answer: 6


Question: Calculate the variance of the process \(X_t = W_t^2\), where \(W_t\) is a standard Wiener process.

Answer: 2t^2


Question: Determine the degree of the algebraic number \( \frac{1+\sqrt{2}+\sqrt{3}}{5} \).

Answer: 4


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

Answer: \dfrac{1}{e}


Question: Find $G'(0)$ if \[ G(x) = \int_{\cos x}^{x^2}\left( \int_0^{t^2}\sqrt{1+s^4} \ ds \right) dt. \]

Answer: 0


Question: Determine whether a ring with finite non-zero characteristic must be finite. Provide a justification for your answer.

Answer: No


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

Answer: \dfrac{\pi}{4}


Question: Determine whether the improper integral \( \int_0^{+\infty} \frac{t\ln t}{(t^2+1)^\alpha} \, dt \) converges or diverges, focusing on the behavior as \( t \to +\infty \).

Answer: \alpha > 1


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{\cosh 4u}{\sinh 5u} \, du \) using a substitution.

Answer: 0


Question: Find all continuous functions \( f : \mathbb{R} \rightarrow [1,\infty) \) such that there exist constants \( a \in \mathbb{R} \) and \( k \) a positive integer, satisfying the inequality \( f(x)f(2x)\dots f(nx) \leq an^k \) for every real number \( x \) and positive integer \( n \).

Answer: 1