Question: Let $n$ be a positive integer and let $M_n(\mathbb{R})$ be the space of all $n \times n$ real matrices. Consider the linear transformation $T: M_n(\mathbb{R}) \to M_n(\mathbb{R})$ defined by $T(A) = 0$ whenever $A \in M_n(\mathbb{R})$ is symmetric or skew-symmetric. Determine the rank of the transformation $T$. Express your answer in terms of $n$. \( \text{Rank}(T) = \boxed{\,} \)

Answer: 0


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

Answer: 2


Question: An object, initially a plane, is stretched along its Y-axis until it doubles in size along this axis, resulting in a scaling vector of (1, 2). If the object is rotated by 45° with respect to the plane's reference frame, what is the new scaling vector applied to the object?

Answer: (1, 2)


Question:
Evaluate the double integral \(\iint_{[0,1]^2} f(x,y) \, \mathrm{d}A\) where the function \(f(x,y)\) is defined as follows:
\[ f(x,y) = \begin{cases} e^{x^2}, & \text{if } x > y \\ e^{y^2}, & \text{if } y > x \end{cases} \]

Answer: e - 1


Question: Determine whether the following statement is true or false: If a function $f$ is uniformly continuous on the interval $(a,b)$, then the limits $\lim_{x\to a^+} f(x)$ and $\lim_{x\to b^-} f(x)$ exist. Assume $f$ is continuous on $(a,b)$. Provide a justification for your answer.

Answer: A


Question: Find the last three digits of \( 7^{7^7} \).

Answer: 343


Question: The roots of the quadratic equation \(x^2 + 3x - 1 = 0\) are also the roots of the quartic equation \(x^4 + ax^2 + bx + c = 0\). Find the value of \(a + b + 4c\).

Answer: -7


Question: Determine the degree of the splitting field of the polynomial \((x^3+x+1)(x^2+1)\) over \(\mathbb{Q}\).

Answer: 12


Question: Determine whether 41 is a quadratic residue modulo $10^6$. In other words, is there an integer $x$ such that $x^2 \equiv 41 \pmod{10^6}$?

Answer: Yes


Question: Evaluate the limit of the series: $$\lim_{n \to \infty} \sum_{k=0}^{\infty} \frac{(2\pi)^{2k}(-1)^k (n+1)!}{(n+2k+1)!}$$ where $n$ is an integer greater than or equal to 4.

Answer: 1


Question: Around a circle, write 1000 numbers such that each three consecutive numbers A, B, C (where B is between A and C) satisfy the inequality \(A^2 + C^2 \leq B - \frac{1}{8}\). Find the maximum and minimum value for the sum of the numbers around the circle.

Answer: 250


Question: Suppose $(X,d)$ is a complete metric space and $\mu$ is a $\sigma$-finite regular Radon measure on $X$ that does not assign measure 0 to non-empty open subsets of $X$. If $Y \subseteq X$ is a dense $G_\delta$ set, must it have positive $\mu$-measure? Answer 'yes' or 'no' and justify your answer.

Answer: No


Question: Find the Fourier Transform of the derivative of a function $f(t)$, given that the Fourier Transform of $f(t)$ is $F(s) = \int_{-\infty}^{\infty} e^{-2 \pi i s t} f(t) \, dt$. Express the Fourier Transform of $f'(t)$, denoted as $G(s)$, in terms of $F(s)$. What is $G(s)$?

Answer: 2\pi i s F(s)


Question: Does there always exist a subgroup $H$ of a finite group $G$ such that the index of $H$ in $G$ is the smallest prime dividing the order of $G$?

Answer: No


Question: Is there an algorithm that can find all paths from a source node \(s\) to a target node \(t\) in a directed, weighted graph, where each path has a total weight of exactly \(K\)? The graph has positive weights, and the runtime of the algorithm should be at most \(O(K(|V|+|E|))\). All vertices can be visited more than once.

Answer: Yes


Question: Let $1 \leq p < \infty$. Consider a von Neumann algebra $\mathcal{M}$ with a normal semifinite faithful trace $\tau$. Let $L_p(\mathcal{M}, \tau)$ be the associated noncommutative $L_p$-space. Suppose $(x_n)$ is a sequence in $L_p(\mathcal{M}, \tau)$ such that $\|x_n - x\|_p \to 0$ as $n \to \infty$ for some $x \in L_p(\mathcal{M}, \tau)$. Determine whether it is true that $\||x_n| - |x|\|_p \to 0$ as $n \to \infty$. Provide a justification for your answer.

Answer: Yes


Question: Given that $x^3[f(x+1)-f(x)]=1$ and $f$ is continuous, determine $\lim_{x\rightarrow \infty}f(x)$. Express your answer as a single value.

Answer: 0


Question: Determine the parameter interval for which the parametrized curve \( \left(\frac{3t}{1+t^3}, \frac{3t^2}{1 + t^3}\right) \) of the folium of Descartes is closed. Show that this interval is \( t \in [0, \infty) \).

Answer: [0, \infty)


Question: Determine the largest real number \( R > 0 \) such that the Laurent series of \( f(z) = \frac{1}{z-1} + \frac{2}{z-i} \) about \( z = 1 \) converges for \( 0 < |z-1| < R \).

Answer: \sqrt{2}


Question: Let $M$ be a von Neumann algebra in $B(H)$. Let $p$ and $q$ be projections in $M$. Assume that they are equivalent in $B(H)$, i.e., there exists a partial isometry $u$ in $B(H)$ such that $p = uu^*$ and $q = u^*u$. Are $p$ and $q$ equivalent in $M$?

Answer: No


Question: Given the equation \(x^2 + xy + y^3 = 1\), find the value of \(y'''\) at the point where \(x = 1\).

Answer: 42


Question: Let N be a set of n arbitrary points in the plane \(\mathbb{R}^2\). For a given \(k \in \{1, 2, \ldots, n\}\), determine whether there always exists a disk \(D\) that contains exactly \(k\) points of \(N\).

Answer: Yes


Question: In a category \( \mathbf{C} \), if for an object \( B \) it holds that \( B \times A \cong A \) for all objects \( A \), does this imply that \( B \) is a terminal object?

Answer: Yes


Question: Let \( N \) be a non-zero nilpotent matrix in \( M_n(D) \) where the characteristic of \( D \) is 2. Suppose there exists a natural number \( k \) such that \( N^k = 0 \) and \( N^{k-1} \neq 0 \). Let \( s \) be the smallest integer such that \( 2^s \geq k \). Prove that \( (N^{2^{s-1}} + I)^2 = I \).

Answer: I


Question: Let $C$ be the curve represented by the equation $5x^2 + 5y^2 - 8xy - 9 = 0$. A tangent is drawn at a point $P$ (other than the vertex) on the ellipse. If a line $AP$ intersects the line passing through $B$ and perpendicular to the tangent at $Q$, find the length of $AQ$. Here, $A = (-2, -2)$ and $B = (2, 2)$. Determine the value of $AQ$.

Answer: 6


Question: Find the minimum value of the expression \( f(a,b,c) = \left\lfloor\frac{a+b}{c}\right\rfloor + \left\lfloor\frac{b+c}{a}\right\rfloor + \left\lfloor\frac{c+a}{b}\right\rfloor \) where \( a, b, c > 0 \). For what values of \( a, b, \) and \( c \) is this minimum achieved?

Answer: 4


Question:
Given the following data:
\[\sum\limits_{i=1}^{50} x_i = 1250, \quad \sum\limits_{i=1}^{50} x_i^2 = 45000, \quad \sum\limits_{i=1}^{50} y_i = 3100, \quad \sum\limits_{i=1}^{50} y_i^2 = 250000\]
and knowing that there is a strong positive correlation between $x$ and $y$, determine if it must necessarily be true that \(\sum\limits_{i=1}^{50} x_i y_i > 45000\).

Answer: Yes


Question: Let $f:[a,b] \rightarrow \mathbb{R}$ be a function of class $C^n$. Given distinct numbers $x_0, ..., x_m$ from $[a,b]$, determine if for each $\varepsilon > 0$, there exists a polynomial $P$ such that $P^{(k)}(x_i) = f^{(k)}(x_i)$ for $i = 0, ..., m$ and $k = 0, ..., n$, and $\sup_{x \in [a,b]} |f(x) - P(x)| < \varepsilon$. Provide a justification for your answer.

Answer: Yes


Question: Find all positive integer solutions \((x, y, z)\) to the equation \(3^x - 5^y = z^2\).

Answer: (2, 1, 2)


Question: Determine the number of elements in the field $F_2(a^2+a)$, where $F_{16} = F_2(a)$ and the minimal polynomial of $a$ over $F_2$ is $a^4 + a + 1$. Provide your answer as a single number.

Answer: 4


Question:
Evaluate the convergence and find the sum of the series:
\[ \sum_{k=0}^{\infty} \frac{\left(\frac{1}{2}\right)^{2 k+4} 4\left(\begin{array}{c}
2 k+3 \\
k
\end{array}\right)}{k+4} \]

Answer: 1


Question: Calculate \( \lim_{x\to 0^+} \frac{x^x- (\sin x)^x}{x^3} \).

Answer: \dfrac{1}{6}


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\ln(1+\|x\|^2)}{\|x\|^2+\sin(x_1x_2x_3)} \] where \( x=(x_1,x_2,x_3) \) and \( \|x\|=(x_1^{2}+x_2^{2}+x_3^{2})^{\frac{1}{2}} \).

Answer: 1


Question: Evaluate the integral \( \int_{0}^{\infty} \frac{x^2}{(x^2 + 4)^2} \, dx \).

Answer: \dfrac{\pi}{8}


Question: Calculate the line integral \( \int_r (xy^2z^2-1, x^2yz^2, x^2y^2z)\cdot ds \) where \( r = r_1 \cup r_2 \). The parametric representation of \( r_1 \) is \( \sigma_1 (t) = (\sin t, \cos t, t-\frac{\pi}{2}), t \in [0,\frac{\pi}{2}] \) and the parametric representation of \( r_2 \) is \( \sigma_2 (t) = (\cos^3 \phi, \sin^3 \phi, 0), \phi \in [0,\pi] \).

Answer: 1


Question:
Given the differential equation \( f(x) y'' - 4 f'(x) y' + g(x) y = 0 \), if the transformation \( y = uv \) transforms it into the equation \( v'' + h(x) v = 0 \), determine the expression for \( u \). Choose the correct option:

- \( \frac{1}{f^2} \)
- \( xf \)
- \( \frac{1}{2f} \)
- \( f^2 \)

Answer: D


Question: If $AB$ and $BA$ are invertible matrices, where $A$ is an $n \times m$ matrix and $B$ is an $m \times n$ matrix, is it necessarily true that $m = n$?

Answer: Yes


Question: Suppose \( \mu \) is a finite measure on the Borel sets of \( \mathbb{R} \) such that for any real-valued, bounded, and integrable function \( f \), the equation \( f(x) = \int_{\mathbb{R}} f(x + t) \mu(dt) \) holds almost everywhere. Show that \( \mu(\{0\}) = 1 \).

Answer: 1


Question: Let \( \vec{f}(\vec{r}) = (y, -x, zxy) \) and let \( S \) be the surface defined by \( x^2 + y^2 + 3z^2 = 1 \) for \( z \leq 0 \), with the unit normal vector \( \vec{n} \) pointing in the positive \( z \)-direction. Calculate the value of the surface integral \[ \iint_S (\nabla \times \vec{f}) \cdot \vec{n} \, \mathrm{d}S \] which is given as \( n\pi \), where \( n \) is an integer. Determine the value of \( n \).

Answer: -2


Question: Consider a measurable subset $E$ of $\mathbb{R}$, a sequence of measurable functions $(f_n)$ from $E$ to $[-\infty, \infty]$, and a function $f$ from $E$ to $[-\infty, \infty]$. If $\lim\limits_{n \to \infty} f_n = f$ almost everywhere on $E$, does it follow that $\lim\limits_{n \to \infty} f_n = f$ in measure on $E$? Determine whether this statement is true or false.

Answer: False


Question:
Consider a system where two repairmen serve three machines, meaning at most two machines can be under repair at the same time. The time until a machine needs repair is exponentially distributed with a mean of \(1/3\), and each repair time is exponentially distributed with a mean of \(1/2\). These times are independent of each other. Let \(X_t\) represent the number of machines under repair or needing repair at time \(t\). Given the transition rate matrix:
\[
Q = \begin{bmatrix}
 -9 &  9  &  0  &  0  \\
  2 & -8  &  6  &  0  \\
  0 &  4  & -7  &  3  \\
  0 &  0  &  2  & -2
\end{bmatrix}
\]
Determine \(\lim_{t \to \infty} \mathbb{P}_i(X_t = 0)\) for \(i = 0, 1, 2, 3\).

Answer: \dfrac{8}{179}


Question: Given that the joint probability density function (PDF) of random variables \( X \) and \( Y \), denoted as \( f_{X,Y}(x,y) \), has a bounded range such that \( \max_{(x,y) \in \mathbb{R}^2} f_{X,Y}(x,y) \leq \kappa \) for some constant \( \kappa \in \mathbb{R} \), determine whether the PDF of the random variable \( Z = |X - Y| \), denoted as \( f_Z(z) \), also has a bounded range.

Answer: Yes


Question: Determine whether the following statement is correct or incorrect: If $X$ and $Y$ are topological spaces and $f: X \rightarrow Y$ is a homeomorphism, then $f$ is a quotient map and $Y$ has the quotient topology.

Answer: correct


Question:
If \( \alpha, \beta \) are the roots of the equation \( x^2 - 2x - 7 = 0 \), evaluate the following limit:
\[ \lim_{n \to \infty} \sum_{r=1}^n \left( \frac{r}{\alpha^r} + \frac{r}{\beta^r} \right) \]

Answer: \dfrac{1}{4}


Question:
Determine whether both solutions for the maximum likelihood estimator (MLE) of \(\theta\) in the beta distribution \(B(\theta, 2)\) are valid, or if one should be discarded. The solutions are:
\[\theta_1 = \frac{-a_n - 2 - \sqrt{a_n^2 + 4}}{2a_n}\]
\[\theta_2 = \frac{-a_n - 2 + \sqrt{a_n^2 + 4}}{2a_n}\]
where \(a_n = \frac{\ln(x_1 \cdot x_2 \cdot ... \cdot x_n)}{n}\) and \(X_1, X_2, ..., X_n \in [0, 1]\).

Answer: \theta_1


Question: What is the upper bound of the cumulative distribution function (CDF) for a standard normal random variable $W \sim N(0,1)$? Specifically, what is the maximum value of $P(W \le t)$ for any real number $t$?

Answer: 1


Question: Does there exist a compact metric space $(X,d)$ that is $\omega$-homogenous but not $\omega_1$-homogenous? Specifically, is there a compact metric space $(X,d)$ and countably infinite sets $X_1, X_2 \subseteq X$ that are isometric via $h$, but $h$ does not extend into an automorphism of $(X,d)$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to +\infty}\frac{n \ln\left(\frac{((n+1)!)^{2n}}{n^{2n}(n!)^{2n}}\right)+\frac{\arctan(n)}{n}+\frac{\sin(n)}{n}}{(-1)^n\sqrt{n}-4n} \]

Answer: -\dfrac{1}{2}


Question: Given the quadratic equation \(x^2 + px + q + 1 = 0\) with two distinct roots \(x_1\) and \(x_2\), where \(p\) and \(p^2 + q^2\) are prime numbers, find the largest possible value of \(x_1^{2014} + x_2^{2014}\). Assume \(p\) and \(q\) are integers, and the roots are whole numbers.

Answer: 2^{2014}


Question: Find the area of the region in the plane bounded by the curve \(x^2 - 2x + 4y^2 - 8y + 1 = 0\).

Answer: 2\pi


Question:
Given \( \theta = \frac{\pi}{6} \), evaluate the expression:
\[
\frac{\sin(9\theta)}{\left(1 + \sum_{k=1}^{8}\frac{\cos(k\theta)}{\cos^k(\theta)}\right) \sin\left(\frac{\pi}{6}\right) \left(\cos\left(\frac{\pi}{6}\right)\right)^8}
\]

Answer: 1


Question: Determine the value of \( m \) such that the polynomial \( 16t^{4} - mt^{3} + (2m+17)t^{2} - mt + 16 = 0 \) has four real roots that form a geometric sequence.

Answer: 170


Question: Find a lower bound for the conditional probability \( \mathbb{P}(Y=1|X=1) \) where \( X=\left(\sum_{i=1}^n a_iW_i\right)\mod 2 \) and \( Y=\left(\sum_{i=1}^n b_iW_i\right) \mod 2 \). Here, \( a_i, b_i \) are constants in \( \{0,1\} \), and \( W_i \) are independent and identically distributed Bernoulli random variables with \( W_i \in \{0,1\} \) each having probability \( 1/2 \). Assume \( n \) is a fixed positive integer and the case \( b_i= 0 \) for all \( 1\leq i\leq n \) is excluded. Provide a proof or reasoning for your answer.

Answer: \dfrac{1}{2}


Question: Evaluate the contour integral \( \int_{|z|=1} |z-1||dz| \) in the counterclockwise direction.

Answer: 8


Question: Evaluate the limit \( \lim_{x\rightarrow 0} \frac{(a+x)^x-a^x}{x^2} \) where \( a > 0 \).

Answer: \dfrac{1}{a}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{\sqrt[n]{n!}}{n} \]

Answer: \dfrac{1}{e}


Question: Determine the number of zeros of the polynomial \( f(z) = z^{87} + 36z^{57} + 71z^4 + z^3 - z + 1 \) in the annular region \( 1 \leq |z| \leq 2 \).

Answer: 83


Question: Let ${f_n}$ be a sequence of continuous functions on $\Bbb R$ such that $f_n \rightarrow f$ uniformly on every finite interval $[a,b]$. If each $f_n$ is bounded, must $f$ also be bounded? Justify your answer.

Answer: No


Question: If A and B are \(n \times n\) matrices such that \(AB = BA\), and A has \(n\) distinct eigenvalues, are the eigenvectors of A also eigenvectors of B?

Answer: Yes


Question: Determine whether the following statement is true: Is $H_n(X) \cong \text{Hom}(H_n(X), \mathbb{Z})$ for the nth homology group $H_n(X)$ of a topological space $X$?

Answer: False


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

Answer: G


Question: Suppose $P$ is a finite abelian $2$-group and $U$, $V$ are characteristic subgroups of $P$ such that $|V:U|=2$. Does it follow that $P$ has a characteristic subgroup of order $2$?

Answer: Yes


Question:
Evaluate the triple integral \( \iiint_W z\frac{e^{2x^2+2y^2}}{2}\,dx\,dy\,dz \) where \( W \) is the solid bounded by the right circular cylinder \( x^2+y^2=1 \) and the planes \( z=0 \) and \( z=4 \). Use cylindrical coordinates to express the integral as:
\[ \int_0^{2\pi}\int_0^1\int_0^4 z\frac{e^{2r^{2}}}{2}r\,dz\,dr\,d\theta \]

Answer: 2\pi(e^2 - 1)


Question: Use Stokes' Theorem to evaluate the line integral \( \int_{C}(y\,dx+z^2\,dy+x\,dz) \), where \( C \) is the curve of intersection of the plane \( 2x + z = 0 \) and the ellipsoid \( x^2 + 5y^2 + z^2 = 1 \). The curve \( C \) is oriented counterclockwise as seen from above. Find the exact value of the line integral.

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


Question: Determine the degree of the field extension \( [\mathbb{Q}(\alpha):\mathbb{Q}] \) where \( \alpha \) is a primitive 5th root of unity.

Answer: 4


Question: Determine the domain of the function \( f(x) = \sqrt{\frac{(e^x-2)^x + \arccos(e^x -2)}{x- \sqrt{x^2 -1}}} \).

Answer: [1, \ln 3]


Question: Calculate \(1^1 + 2^2 + \cdots + 99^{99} \pmod{3}\). Justify your answer.

Answer: 1


Question: Let $K$ be a finite extension of $\mathbb{Q}_p$, and $\mathcal{O}$ the ring of integers of $K$. Let $\mathfrak{p}$ be the maximal ideal of $\mathcal{O}$. Is the quotient ring $\mathcal{O}/\mathfrak{p}^n$ finite for all $n > 0$?

Answer: Yes


Question: Find the least nonnegative residue of \( \frac{31!}{22!} \) modulo 11.

Answer: 1


Question:
Find the Jacobian determinant of the transformation given by the equations:
\[ u = x - y \]
\[ v = xy \]

Answer: x + y


Question: Solve the equation: $xe^x + e^x = e^4$.

Answer: W(e^5) - 1


Question: Evaluate the infinite product: \[ \prod_{x=1}^{\infty} (2x)^{\frac{1}{4x}} \]

Answer: \infty


Question: Find the number of terms of the form \(2^a \times 3^b\) within the range \((L, R)\), where \(a > 0\) and \(b \ge 0\). For example, in the range \((2, 8)\), the number of such terms is 4: \(2, 4, 6, 8\).

Answer: 4


Question: Let \( f \in \mathcal{C}[0, \infty) \) be such that \( \lim\limits_{x \to \infty} f(x) \in \mathbb{R} \) and let \( a > 0 \). Compute \[ \lim_{n \to \infty} \int_0^a f(nx)\,dx. \]

Answer: a \cdot \lim_{x \to \infty} f(x)


Question: Find the extreme values of the expression \(bcx + cay + abz\) subject to the constraint \(xyz = abc\), where \(a > 0\), \(b > 0\), and \(c > 0\). Use the method of Lagrange multipliers and the inequality AM > GM to determine the solution.

Answer: 3abc


Question: Find a 5-cycle $x$ such that $x^3 = (abcde)$. Show your work and verify that $x^3$ equals $(abcde)$. What is the 5-cycle $x$?

Answer: (a\,c\,e\,b\,d)


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0} \left[ \csc^2(2x) - \frac{1}{4x^2} \right] \]

Answer: \dfrac{1}{3}


Question: Determine the range of values that the quadratic form \( q(\begin{pmatrix}x\\y\end{pmatrix}) = 11x^2 - 16xy - y^2 \) takes on the unit circle \( x^2 + y^2 = 1 \).

Answer: [-5, 15]


Question: Determine the maximum value of the expression \( \sqrt{x_1-1} + \sqrt{x_2-1} + \cdots + \sqrt{x_{2017}-1} \) given that \( x_1, x_2, \ldots, x_{2017} \geq 1 \) and \( x_1 + x_2 + \cdots + x_{2017} = 4034 \).

Answer: 2017


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

Answer: 1008


Question: Determine the rank of the linear operator on the vector space of real 3x3 matrices that maps a matrix X to AX - XA, where \( A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix} \).

Answer: 4


Question: In the figure, AD = BE = CF and triangle DEF is equilateral. Determine whether triangle ABC must be equilateral.

Answer: Yes


Question: Let \( f(x) = 6x^3 - 11x^2 - 2 \) for \( x \geq 1.5 \). Find \( \frac{\text{d}f^{-1}}{\text{d}x} \) at the point \( x = 1517 \), where \( 1517 = f(7) \).

Answer: \dfrac{1}{728}


Question: Determine if there are positive integer solutions to the equation \(a=\frac{2b^3+2c^3+d^3}{2b^2+2c^2+d^2}\) with \(d > 1\).

Answer: Yes


Question: Given that \(x_1, x_2, x_3\) are the roots of the polynomial \(x^3-2x^2+3x+5=0\), find \((x_2-x_1)^2(x_3-x_1)^2(x_3-x_2)^2\).

Answer: -1127


Question: How many ways are there to divide 100 balls into two cells such that the first cell contains at least 2 balls and the second cell contains an even number of balls? Use generating functions to solve this problem.

Answer: 50


Question: How many permutations of the set \(\{1, 2, \.\.\., 8\}\) do not leave any even number in its original position?

Answer: 24024


Question: Given a 6th degree polynomial \( f(x) \) such that \( f(0) = 0 \), \( f(1) = 1 \), \( f(2) = 2 \), \( f(3) = 3 \), \( f(4) = 4 \), \( f(5) = 5 \), and \( f(6) = 7 \), find the value of \( f(8) \).

Answer: 36


Question: A car insurance company has 2,500 policyholders. The expected claim paid to a policyholder during a year is $1,000$ with a standard deviation of $900$. What premium should the company charge each policyholder to ensure that, with a probability of 0.999, the total premium income will cover the total cost of the claims?

Answer: 1055.62


Question: Determine the subgroup of the symmetric group $S_5$ generated by the permutations $(123)$ and $(12345)$. What is this subgroup?

Answer: A_5


Question: Suppose \( f \) is an entire function such that \( f(\sqrt{n}) = 0 \) for \( n = 1, 2, 3, \ldots \), and there exists a positive constant \( \alpha \) such that \( |f(z)| < \exp(|z|^\alpha) \) for all sufficiently large \( |z| \). Determine the values of \( \alpha \) for which it follows that \( f(z) = 0 \) for all \( z \). Consider the function \( \sin(\pi z^2) \) in your analysis.

Answer: \alpha < 2


Question: Find the smallest number of terms needed in the partial sum of the series \( \sum_{n=0}^{+\infty}{(-1)^n\frac{1}{(2n+1)!(4n+3)}} \) to approximate the value of the integral \( \int_{0}^{1}\sin(x^2)\,dx \) with an accuracy of 3 decimal places.

Answer: 3


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a strictly decreasing function satisfying the equation \[ f(x+y) + f(f(x) + f(y)) = f(f(x+f(y)) + f(y+f(x))) \] for all \( x, y \in \mathbb{R} \). Find a relation between \( f(f(x)) \) and \( x \).

Answer: f(f(x)) = x


Question: Let $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ be a function that is Gâteaux differentiable, and let $J_f\in\mathbb{R}^{M\times N}$ be its Jacobian matrix. Determine if the Gâteaux derivative of $f$ along a direction $v\in\mathbb{R}^N$ is equal to the matrix-vector product $J_f \cdot v$. Is this statement true or false?

Answer: A


Question: Let \( \mathbb{F} \) be a field and \( \mathbb{F}(a) \) a simple algebraic extension of degree \( n \). Can the field \( \mathbb{F}(a) \) be embedded in the matrix ring \( M_n(\mathbb{F}) \)?

Answer: Yes


Question: Evaluate the limit \( \lim_{x \to \infty} \left( \frac{P(x)}{5(x-1)} \right)^x \) where \( P(x) = ax^2 + bx + c \) with \( a, b, c \in \mathbb{R} \), \( P(2) = 9 \), and \( P'(3) = 5 \). Given that one root \( \alpha \to \infty \), find the value of the limit.

Answer: e^{\frac{4}{5}}


Question: Prove that if \( \lim x_n = x \), then \( \lim \inf(x_n) = x \).

Answer: x


Question: Evaluate the double integral \( F(\Lambda) = \int_0^{\infty}\int_0^{\infty}\left(\frac{e^{-\Lambda x}-e^{-\Lambda y}}{x-y}\right)^2 \,dx\,dy \), where \( \Lambda \) is a positive real number.

Answer: 2 \ln 2


Question: Determine the number of distinct six-sided dice that can be manufactured if each of three different labels must be placed on exactly two of the faces, using Polya's enumeration theorem.

Answer: 6


Question: Let $M$ be a complete Riemannian manifold and $a$, $b$ two different points on it. Define the set $A = \{x \in M \mid d(x,a) = d(x,b)\}$, where $d$ is the distance induced by the metric of $M$. Is the set $A$ necessarily a measure zero set with respect to the volume form $dV$?

Answer: Yes


Question: Does there exist a smooth stochastic process with nonzero quadratic variation?

Answer: No


Question: Find the residue of the function \( f(z) = (z-2) \cos\left(\frac{1}{z-1}\right) \) at \( z_0 = 1 \).

Answer: -\dfrac{1}{2}


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

Answer: \dfrac{1}{e}


Question: Let the quadratic equation $ax^2 + bx + c = 0$ have both roots in the interval $[0,1]$. Determine the range of the expression $\frac{(a-b)(2a-b)}{a(a-b+c)}$. Provide your answer as an interval.

Answer: [2, 3]


Question: Evaluate the contour integral \( \oint_C \frac{z}{z-1} \, dz \), where \( C \) is a circle of radius 4 centered at the origin.

Answer: 2\pi i


Question: Determine the number of zeros of the function $3e^z - z$ within the unit disk $|z| < 1$.

Answer: 0


Question: Compute \( F_{1000} \mod 1001 \), where \( F_n \) denotes the Fibonacci numbers.

Answer: 616


Question: Find the smallest integer \( n_0 \) such that for all \( n > n_0 \), the inequality \( \sqrt{n^3+1} + \sqrt{n^3} > 10^3 \) holds.

Answer: 62


Question: Determine the number of solutions to the equation \( \log_3(|x-1|)\cdot\log_4(|x-1|)\cdot\log_5(|x-1|) = \log_5(|x-1|) + \log_3(|x-1|)\cdot\log_4(|x-1|) \).

Answer: 6


Question: Given a set of 6 distinct digits, how many valid sequences of six digit pairs can be generated? Each digit in the set can be used exactly twice, and each pair must consist of two different digits. Determine the total number of such sequences.

Answer: 67950


Question: Evaluate \( \lim_{x \to \infty} (x + \log_e(x) - x\log_e(x)) \). Determine whether the limit is \(-\infty\) or \(\infty\).

Answer: -\infty


Question: Find the maximum value of the harmonic function \( u \) in the closed disk \( D = \{ z \in \mathbb{C} : |z| < 2 \} \), given that on the boundary circle, \( u(2e^{it}) = 3 \sin{2t} + 1 \).

Answer: 4


Question: Find the limit: \[ \lim_{n\rightarrow \infty }\left(1+\frac{1}{n}\right)^{n^n} \]

Answer: \infty


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

Answer: \infty


Question: Evaluate the limit: $$\lim_{n \to \infty} \int_{-\infty}^{\infty} f(x-n)\left(\frac{1}{1+|x|}\right)\,dx$$ where $f$ is integrable with respect to a Lebesgue measure.

Answer: 0


Question: Find the smallest integer value of \( N \) such that \( \sum\limits_{k=N+1}^{1000}\binom{1000}{k}\left(\frac{1}{2}\right)^{1000} < \frac{1}{100} \).

Answer: 537


Question: Let $f: \Bbb R \rightarrow \Bbb R$ be a differentiable function with $f'$ uniformly continuous. Suppose $\lim\limits_{x\rightarrow \infty}f(x)=L$ for some $L$. Does $\lim\limits_{x\rightarrow \infty}f'(x)$ exist?

Answer: 0


Question: Given the sequences \( S_n = \sum_{r=0}^{n} \frac{1}{\binom{n}{r}} \) and \( T_n = \sum_{r=0}^{n} \frac{r}{\binom{n}{r}} \), find the expression for \( \frac{T_n}{S_n} \) in terms of \( n \).

Answer: \dfrac{n}{2}


Question: Find the limit inferior of the sequence \((-1-\frac{2}{n})^n\) as \(n\) approaches infinity.

Answer: -e^2


Question: A rectangular box has one corner at the origin and the opposite corner on the plane \(x + 2y + 3z = 6\). The edges of the box are parallel to the coordinate axes. What is the maximum possible volume of the box?

Answer: \dfrac{4}{3}


Question: Evaluate the limit: \[ \lim_{n\to +\infty}\frac{1}{n^2}\sum_{i=1}^{n}\log\binom{n}{i}. \]

Answer: \dfrac{1}{2}


Question: Consider the rule of existential generalization in first-order logic: \( \forall \phi,a:\phi(a)\implies \exists x:\phi(x) \). Determine whether this rule can be extended to cases where \( a \) contains bound variables. Specifically, evaluate the statement: \( y\in\left\{2x: x\in H \right\} \implies \exists a: y\in\left\{a: x\in H \right\} \). Is this statement true or false?

Answer: True


Question: Let $X$ be a compact space covered by two open sets $A$ and $B$. Determine if there exist compact sets $C \subseteq A$ and $D \subseteq B$ such that $C \cup D = X$. Provide a justification for your answer.

Answer: No


Question: Determine whether the set of simple coroots $B^{\vee} = \{\alpha_1^{\vee}, \ldots, \alpha_l^{\vee}\}$ is always a base of the dual root system $(V^{\vee}, R^{\vee})$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{1} \frac{1-x^2}{x^2} \ln\left(\frac{(1+x^2)^2}{1-x^2}\right) \, dx \) and show that it equals 2.

Answer: 2


Question: Find the highest common factor of all integer values of \( n \) such that both \( 2n + 1 \) and \( 3n + 1 \) are perfect squares.

Answer: 40


Question: Determine whether the following statement is true or false: Let \( \sum a_n\) with \(a_n > 0\) be a convergent series. Then there exists a sequence \( b_n = \frac{1}{n^{1+\alpha}}\), where \(\alpha > 0\), such that \( \lim_{n\to\infty} \frac{a_n}{b_n} = L > 0 \).

Answer: B


Question: Compute the surface area of the surface generated by revolving the astroid with parametrization \(c(t) = (\cos^3 t, \sin^3 t)\) about the \(x\)-axis for \(t \in [0, \pi/2]\).

Answer: \dfrac{6\pi}{5}


Question: Suppose circle $I$ is the incircle of $\triangle ABC$, and $D, E,$ and $F$ are the points where the incircle is tangent to $BC, CA,$ and $AB$, respectively. Given that $DI$ intersects $EF$ at $K$, and the side lengths are $BC = 10$, $AC = 8$, and $AB = 7$, find the ratio $\frac{S_{\triangle BKC}}{S_{\triangle ABC}}$. Provide your answer as a simplified fraction.

Answer: \dfrac{2}{3}


Question: Consider a planar graph $G$ with $|G| \geq 4$. Transform $G$ into a triangulation $G'$. Prove that there are at least 4 vertices in $G'$ with degree at most 5.

Answer: 4


Question: Does there exist a metric on $\mathbb{Q}$ such that $(\mathbb{Q}, d)$ is compact, without considering $\mathbb{Q}$ as a subspace of $\mathbb{R}$?

Answer: Yes


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

Answer: e


Question: Let $X$ be a random variable with probability density function $f(x) = c(3+x)$ for $-3 < x < 2$ and zero elsewhere. Find the almost sure limit of the sequence \( \frac{1}{n} \sum_{i=1}^{n} \mathbf{1}_{\{0 \leq X_i \leq 1\}} X_i^3 \), where \( \{X_1, \ldots, X_n\} \) is a random sample from $X$, and \( \mathbf{1} \) is the indicator function.

Answer: \dfrac{19}{250}


Question: Let $X$ be a noetherian projective scheme, and let $\mathcal{F}$ be a torsion-free $\mathcal{O}_X$-module on $X$. Consider a submodule $\mathcal{G} \subset \mathcal{F}$. Is it possible for $\mathcal{G}$ to be torsion-free?

Answer: Yes


Question: Consider a non-trivial cycle in the Collatz Conjecture, where each term in the cycle is an odd integer greater than 1. Let \( x_1, x_2, \ldots, x_n \) be the terms of this cycle, and let \( m = \sum_{i=1}^n \nu_2(3x_i+1) \), where \( \nu_2(x) \) is the 2-adic valuation of \( x \). Prove or disprove that \( m < 2n \).

Answer: m < 2n


Question: Given two fields $K \subset L$ and a matrix $A \in GL(n, L)$, consider the $K$-linear map $C_A: L^{n \times n} \to L^{n \times n}$ defined by $C_A(B) = ABA^{-1}$. Determine the determinant of the linear map $C_A$. Provide your answer in terms of $n$ and any relevant properties of $A$. \( \text{Hint: Consider the case } n=2, K=\mathbb{R}, L=\mathbb{C}. \)

Answer: 1


Question: Determine the cardinality of the set \( Q \), where \( Q = \{ \text{countable unions of elements of } P \} \) and \( P = \{(a,b) : a,b \in \mathbb{R} \} \). Is the cardinality of \( Q \) equal to the cardinality of the continuum \( C \)?

Answer: C


Question: Determine the symmetry group of the following configuration: Consider six points $A_1$, $A_2$, $\cdots$, $A_6$ on a circle $(O_A)$ and a point $B_1$ on another circle $(O_B)$. Let the circle $(A_iA_{i+1}B_i)$ meet $(O_B)$ again at $B_{i+1}$ for $i=1,\cdots, 5$. Then the four points $A_6$, $A_1$, $B_1$, $B_6$ lie on a circle. Denote by $(O_i)$ the circle $(A_iA_{i+1}B_{i+1}B_i)$ for $i=1,\cdots,6$, taking subscripts modulo 6. If $(O_1)$ meets $(O_4)$ at two points $C_3, C_6$, $(O_2)$ meets $(O_5)$ at two points $C_4, C_1$, and $(O_3)$ meets $(O_6)$ at two points $C_2, C_5$, then the six points $C_1$, $C_2$, $C_3$, $C_4$, $C_5$, $C_6$ lie on a circle. What is the symmetry group of this configuration?

Answer: D_6


Question: Find the residue of the function \( f(z) = \frac{\sinh(z)}{z^4(1-z^2)} \) at \( z_0 = 0 \).

Answer: \dfrac{7}{6}


Question: Determine the range of values of \( \alpha \) for which the integral \( \int_{0}^{\infty}\frac{\sin x}{x^{\alpha}} \, dx \) is convergent.

Answer: 0 < \alpha < 2


Question: Let \(A\) and \(B\) be \(C^{\ast}\)-algebras, and let \(A \otimes B\) denote the algebraic tensor product of \(A\) and \(B\). Given representations \(\pi_1\) and \(\pi_2\) of \(A\) and \(B\) respectively, satisfying \(\pi_1(a)\pi_2(b)= \pi_2(b)\pi_1(a)\), we define \(\pi_1.\pi_2: A\otimes B \to B(H)\) as \[\pi_1.\pi_2\big(\sum_i a_i \otimes b_i\big)= \sum_i \pi_1(a_i)\pi_2(b_i)\]. For \(x\in A\otimes B\), does there exist representations \(\pi_1\) and \(\pi_2\) of \(A\) and \(B\) respectively, such that \(\pi_1(a)\pi_2(b)= \pi_2(b)\pi_1(a)\) and \(\|x\|_\text{max}=\|\pi_1.\pi_2(x)\|\)?

Answer: Yes


Question: Evaluate the double integral \(\iint_{\mathbb{R}^2} (x^2+y^2+1)^{-\frac{3}{2}} \, dx \, dy\).

Answer: 2\pi


Question: Calculate the flux of the vector field \( F(x,y,z) = (zy^2-2x, \frac{1}{4}yz+z^2, xy+2x^2+2z) \) exiting the region \( \Omega = \{(x,y,z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 \leq 16, z \leq \sqrt{x^2+y^2}\} \).

Answer: -8\pi


Question: Calculate the limit: \[ \lim_{x \to 0} \frac{x - \arcsin(x)}{\sin^3(x)} \]

Answer: -\dfrac{1}{6}


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

Answer: \ln 2


Question: Find the value of the series: \[ \sum_{n=1}^{\infty} \frac{2}{4n^2+8n+3} \]

Answer: \dfrac{1}{3}


Question: Evaluate the limit: \[ \lim_{x\to 1} \left( \frac{x}{\ln x} - \frac{1}{x \ln x} \right) \] without using L'Hospital's Rule.

Answer: 2


Question: Calculate the value of the series: $$\sum^{\infty}_{n=0} (-1)^n\frac{n+1}{n!}$$

Answer: 0


Question: Evaluate the limit of the sum \( \arctan\left(\frac{1}{3}\right) + \arctan\left(\frac{1}{7}\right) + \arctan\left(\frac{1}{13}\right) + \arctan\left(\frac{1}{21}\right) + \cdots \) as \( n \to \infty \). What is the value of this sum?

Answer: \dfrac{\pi}{4}


Question: Find the order of the element \( \bar{p} \) in the group \( (\mathbb{Z}/(p^{n}-1)\mathbb{Z})^{\times} \), where \( p \) is a prime and \( n \) is a positive integer.

Answer: n


Question: Determine the minimum value of \( n \) such that it is possible to create at least 100 ordered lists of length 5 using numbers from 1 to \( n \), with the condition that any two lists share at most one common element at the same index.

Answer: 11


Question: Find the value of \( \frac{S}{t} \) if \( S = \sum_{k=1}^{\infty} \frac{1}{k^2} \) and \( t = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^2} \).

Answer: 2


Question: Evaluate the limit: $$\lim_{n\rightarrow +\infty}\left [ i^i\left ( 2i \right )^{2i} \cdots\left ( ni \right )^{ni}\right ]$$ where the branch of the logarithm function is such that the argument lies within the interval $(-\pi, \pi]$. Provide your answer with justification.

Answer: 0


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

Answer: e


Question: Evaluate the limit: \[ \lim_{x \rightarrow 0^+} (\sin x)^{\ln x} \]

Answer: \infty


Question: Evaluate the integral \( \int_{\gamma} \frac{2z}{(z+1)(z+3)} \, dz \), where \( \gamma(t) = 2e^{it} \) for \( 0 \leq t \leq 2\pi \).

Answer: -2\pi i


Question: Find the Galois group of the splitting field of the polynomial \(x^4 + 3\) over \(\mathbb{Q}\).

Answer: D_4


Question: Determine the interval \([a, b]\) that maximizes the integral \(\int_a^b (2 + x - x^2) \, dx\).

Answer: [-1, 2]


Question: Evaluate the integral \( \int_0^{2\pi} \frac{d\theta}{8\cos^2 (\theta) + 1} \) using the method of residues.

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


Question: Given the group \( G = C_{25} \times C_{45} \times C_{48} \times C_{150} \), where \( C_n \) denotes a cyclic group of order \( n \), determine the number of elements in \( G \) that have an order of 5.

Answer: 124


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a continuous function and let \( E \subseteq \mathbb{R} \) be a set with finite Lebesgue outer measure. Does the image \( f(E) \) necessarily have finite Lebesgue outer measure?

Answer: No


Question: Evaluate the double integral \( \iint_R xy \, dA \) in polar coordinates, where \( R \) is the region defined by \((x-2)^2 + y^2 = 4\) in the first quadrant.

Answer: \dfrac{32}{3}


Question: Determine the degree of the splitting field of the polynomial \(x^5 - 7\) over \(\mathbb{Q}\).

Answer: 20


Question: Let $A$ be an $n \times n$ matrix with rank $n-1$. Furthermore, let $Q$ be an orthogonal matrix. Determine an eigenvalue of $Q^T(A-I)Q$.

Answer: -1


Question: Let \(\lambda_i \in \mathbb{C}\) for \(i=1,2,\ldots,n\) be \(n\) different complex numbers, and let \(p_i \in \mathbb{C}[t]\) for \(i=1,2,\ldots,n\) be polynomials. Given the relation \[ e^{\lambda_1 t} p_1(t) + e^{\lambda_2 t} p_2(t) + \cdots + e^{\lambda_n t} p_n(t) = 0 \] for all \(t \geq 0\), determine if it follows that all \(p_i(t) = 0\).

Answer: Yes


Question: If \(z_1\) and \(z_2\) are complex numbers such that \(|z_1+z_2|=|z_1-z_2|\), prove that the arguments of \(z_1\) and \(z_2\) differ by \(\frac{\pi}{2}\).

Answer: \frac{\pi}{2}


Question: Consider two matrices $P$ (dimension $m \times 1$) and $Q$ (dimension $n \times 1$). A matrix $M$ is constructed by $M = PQ'$, where $'$ denotes the transpose, making $M$ of size $m \times n$. Determine whether $M$ has only one non-zero singular value, and explain why this is the case.

Answer: Yes


Question: Evaluate the line integral \( \int_C (z \, dx + x \, dy + xy \, dz) \) where \( C \) is the path of the helix given by \( \mathbf{r}(t) = 4\cos t \, \mathbf{i} + 4\sin t \, \mathbf{j} + 3t \, \mathbf{k} \) for \( 0 \le t \le 2\pi \).

Answer: 40\pi


Question: Determine the number of distinct roots of the equation \(A\sin^3 x + B\cos^3 x + C = 0\) within the interval \([0, 2\pi)\), where no two roots differ by \(2\pi\). Here, \(A, B, C \in \mathbb{R}\). Choose the correct option: \(\textbf{(a)}\; 3\; \textbf{(b)}\; 4\; \textbf{(c)}\; 6\; \textbf{(d)}\; \text{infinite}\).

Answer: C


Question: Find the last five digits of the number \(5^{1981}\).

Answer: 03125


Question: Find the value of \(c\) such that the areas of the regions bounded by the parabolas \(y = cx^2\) and \(y = 1 - x^2\) in the first quadrant are equal.

Answer: 2


Question: For any arbitrarily small number \( \epsilon > 0 \), is there a natural number \( n \) such that \( \frac{\phi(n)}{n} < \epsilon \)? Here, \( \phi(x) \) is the Euler's totient function, which represents the number of positive integers up to a given integer \( x \) that are relatively prime to \( x \).

Answer: Yes


Question: Is there a Lebesgue measurable function \( F: \mathbb{R}^2 \to \mathbb{R} \) such that for every Borel function \( f: \mathbb{R} \to \mathbb{R} \), there exists some \( y_f \) such that \( f(x) = F(x, y_f) \) for all \( x \)?

Answer: No


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

Answer: 1


Question: Find the limit: $$\lim_{n \to \infty} \underbrace{\sin \sin \ldots \sin }_{\text{$n$ times}}x$$

Answer: 0


Question: Determine the order of the group of 3x3 matrices over \( \mathbb{Z}/2\mathbb{Z} \) with determinant 1.

Answer: 168


Question: Let \( A \) and \( B \) be \( n \times n \) real matrices such that \( A + B = I \) and \( \text{rank}A + \text{rank}B = n \). Prove that \( \text{range}A \cap \text{range}B = \{0\} \).

Answer: \{0\}


Question: Find \(\mathbb{E}(Y\,\mid\,X)\) when \((X,Y)\) is uniformly distributed on the triangle with vertices \((0,0), (1,0), (1,1)\).

Answer: \dfrac{X}{2}


Question: A baker sells bear claws in boxes of 6, 9, and 20. Assuming you have an unlimited supply of these boxes, what is the largest number of bear claws that cannot be purchased from the baker?

Answer: 43


Question: Let \( S_n = \frac{1}{n}\sum_{i=1}^n X_{i,n} \), where \( X_{i,n} \) are random variables such that for all \( i = 1, 2, \ldots \), we have \( X_{i,n} \to 0 \) almost surely as \( n \to \infty \). Does \( S_n \to 0 \) almost surely as \( n \to \infty \)?

Answer: No


Question: Consider a projective normal and $\mathbb{Q}$-factorial variety $X$ that is log Fano if there exists an effective divisor $D$ on $X$ such that $-K_X-D$ is ample and the pair $(X,D)$ is klt. Let $f:X\dashrightarrow Y$ be a birational map that is an isomorphism in codimension one between two projective normal and $\mathbb{Q}$-factorial varieties. Assume that the strict transform of $-K_X$ on $Y$ is nef and big. Can we conclude that $X$ is log Fano?

Answer: Yes


Question:
Determine the interval of convergence for the power series:
\[ \sum_{n=1}^{\infty} \sin\left(\frac{1}{n}\right)\tan\left(\frac{1}{n}\right)x^n \]

Answer: [-1, 1]


Question: Maximize the function \( f(x,y) = (ax)^2 + abxy + (by)^2 - a^2x - b^2y \) over the interior and boundary of the convex hull of the points \((0,0), (1,0), (0,1)\), where \(a\) and \(b\) are positive real constants.

Answer: 0


Question: Let \( f(x) \) be a continuous function such that \( f(x) > 0 \) for all non-negative \( x \), and \( (f(x))^{101} = 1 + \int_{0}^x f(t) \, dt \). Given that \( (f(101))^{100} = 100A + 10B + C \), where \( A, B, C \) are integers from \([0,9]\), find the value of \( A + B + C \).

Answer: 2


Question: Consider a compact connected Riemann surface $X$ of genus $g > 0$ and a finite subset $B \subset X$. Is the fundamental group of the Riemann surface $X - B$ finitely generated?

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to+\infty}\left(\left(x^3+\frac{x}{2}-\tan \frac{1}{x}\right)\text{e}^{\frac{1}{x}}-\sqrt{1+x^6}\right)$$

Answer: +\infty


Question:
Find the tensor product \( M \otimes_A N \) of the left \( A \)-module \( M = \mathbf{k} \oplus \mathbf{k}^2 \) and the right \( A \)-module \( N = \mathbf{k} \oplus \mathbf{k} \), where the \( A \)-action is given by:
\[
\begin{bmatrix}\lambda&0\\(u_1,u_2)&\mu\end{bmatrix}\begin{pmatrix}m_1\\(m_2,m_3)\end{pmatrix}=\begin{pmatrix}\lambda m_1\\(u_1m_1+\mu m_2, u_2m_2+\mu m_3) \end{pmatrix}
\]
and
\[
\begin{pmatrix} n_1& n_2\end{pmatrix}\begin{bmatrix}\lambda&0\\(u_1,u_2)&\mu\end{bmatrix}=\begin{pmatrix}\lambda n_1+u_1n_2&\mu n_2\end{pmatrix}
\]

Answer: \mathbf{k}


Question: Consider the system of ordinary differential equations given by: \[ \begin{array}{c} x'=h(x,y)f(x,y) \\ y'=h(x,y)g(x,y) \end{array} \] where \( x, y \in \mathbb{R} \) and \( h(x,y) > 0 \) for all values of \( x, y \). Show that dividing the right-hand side by \( h(x,y) \) results in an equivalent system with unchanged orbits: \[ \begin{array}{c} x'=f(x,y) \\ y'=g(x,y) \end{array} \] Is this statement true?

Answer: True


Question: Compute the work done by the vector field \( F(x,y)=\left(\frac{y}{x^2+y^2},\frac{-x}{x^2+y^2}\right) \) along the line segment from \((0,1)\) to \((1,0)\).

Answer: \dfrac{\pi}{2}


Question: Determine whether the element $xy$ is contained in the finitely generated $R$-module $R[x,y]$, where $x, y \in S$ and $R \subseteq S$ is a subring.

Answer: Yes


Question: Find the infimum of the expression \( |\lambda-2|^2+|2\lambda-1|^2+|\lambda|^2 \) over all \( \lambda \in \mathbb{C} \).

Answer: \dfrac{7}{3}


Question: Evaluate the infinite series: \[ \sin\left(\frac{\pi}{3}\right) + \frac{1}{2}\sin\left(\frac{2\pi}{3}\right) + \frac{1}{3}\sin\left(\frac{3\pi}{3}\right) + \cdots \]

Answer: \dfrac{\pi}{3}


Question: Does the group of all permutations of natural numbers contain a subgroup isomorphic to $SO(2)$, the group of all rotations of a plane? Justify your answer.

Answer: No


Question: Calculate the line integral \( \int_{|z|=R}\frac{1}{P(z)}dz \), where \( P(z) \) is a polynomial with all roots inside a circle of radius \( R \) centered at the origin.

Answer: 0


Question: Find all integer solutions to the equation \(a^{2017} + a - 2 = (a-1)(b^{11})\).

Answer: (1, b)


Question: Evaluate the limit: \[ \lim_{a\to\infty} \int_0^1 \frac {x^2e^x}{(2+ax)} \, dx. \]

Answer: 0


Question: Determine the cluster set of the sequence \(\{\cos(2^n)\}\).

Answer: [-1, 1]


Question: Let $M$ be a topological manifold that is homeomorphic to the interior of a compact manifold with boundary. If $f \colon M' \to M$ is a finitely-sheeted covering, is $M'$ also homeomorphic to the interior of a compact manifold with boundary?

Answer: Yes


Question: Find the limit: \[ \lim_{n \to \infty} \sum_{k=n}^{\infty} e^{-k^2} \]

Answer: 0


Question: Find the last two digits of the number \( A = (2016^{2015^{2014}} + 2014^{2015^{2016}} + 2017)^{2017} \).

Answer: 77


Question: Determine whether there exists a totally disconnected set of real numbers that is Lebesgue measurable and has positive measure.

Answer: Yes


Question: Let $f \in \mathcal{C}^1[-1,1]$ be a function such that $|f(t)| \leq 1$ and $|f'(t)| \leq \frac{1}{2}$ for all $t \in [-1,1]$. Define the set $A = \{t \in [-1,1] : f(t) = t\}$. Determine whether $A$ is nonempty, and if it is, find its cardinality.

Answer: 1


Question: Compute the line integral \( \int_C (y−z)\, dx + (z−x)\, dy + (x−y)\, dz \) where \( C \) is the intersection of the cylinder \( x^2+y^2 = 1 \) and the plane \( x−z = 1 \).

Answer: 0


Question: Determine the minimum number of steps required to find the lowest floor in a 100-floor building from which a marble will break when dropped, using only two marbles.

Answer: 14


Question: Evaluate the integral: \[ \int_0^\infty \frac{1}{(x+1)(\pi^2+\ln(x)^2)}\,dx \]

Answer: \dfrac{1}{2}


Question: Find the maximum value of the line integral \( \int_{C} (4x^2y+y^3)dx+(x^3+4x-xy^2)dy \) among all smooth simple closed curves \( C \) oriented counterclockwise.

Answer: 4\pi


Question: Solve the initial value problem for the differential equation \( y' = \frac{8\cos 8x}{3+2y} \) with the initial condition \( y(0) = -1 \). Then, determine the value of \( x \) at which the solution \( y(x) \) attains a maximum.

Answer: \dfrac{\pi}{16}


Question: Given that $a, b, c$ are distinct negative real numbers, determine which of the following conditions the determinant of the matrix \[ \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} \] satisfies: \[(a) \le 0 \quad (b) > 0 \quad (c) < 0 \quad (d) \ge 0 \]

Answer: B


Question: Let \( P(x) \) be a polynomial function such that \( P(x^2) = x^2(x^2+1) P(x) \) and \( P(2) = 3 \). Find \( P(3) \).

Answer: 18


Question: Determine whether there exists a continuous injective map from the set of rational numbers, \( \mathbb{Q} \), to the set of irrational numbers, \( \mathbb{R} - \mathbb{Q} \).

Answer: Yes


Question: Determine whether the limit \( \lim_{n \to \infty} \frac{p_n}{n} \) exists, where \( p_n \) is the largest prime less than or equal to \( n \).

Answer: 1


Question: Calculate the minimum distance between the curves given by the equations \(x^2 + y^2 = 9\) and \(2x^2 + 10y^2 + 6xy = 1\).

Answer: 2


Question: Alice is an astronomer performing measurements to estimate the true distance $d$ of a star. Each measurement is independent with mean $d$, and the $n$th measurement has a standard deviation of $n^{1/3}$. As $n$ goes to infinity, does the average of her measurements converge to the actual distance $d$?

Answer: Yes


Question: Given a natural number $n$, determine whether there exists a natural number $x$ such that there are $n$ distinct Pythagorean triples $(x, y_i, z_i)$ with distinct $y_i$ values, satisfying $x^2 + y_i^2 = z_i^2$ for each $i \leq n$. Answer "Yes" if such an $x$ exists, otherwise answer "No".

Answer: Yes


Question: Prove that if $f(x)$ is a surjective function with $\frac{\text{d}}{\text{d}x}f(x) = g(x)$, and $h(x) = f(ax + b)$ where $a$ and $b$ are constants with $a \neq 0$, then $\frac{\text{d}}{\text{d}x}h(x) = ag(ax + b)$ using only the limit definition of the derivative.

Answer: a g(ax + b)


Question: Find the value of \( \lambda \) that minimizes the integral \( \int_{0}^{\pi}\sin^2(x-\lambda x) \, dx \).

Answer: 1


Question: Given the function \( f : \mathbb{Z} \rightarrow \mathbb{Z} \) satisfying the equation \( f(x) \cdot f(y) = f(x-y) + f(x+y) \) and the condition \( f(1) = 1 \), find the value of \( f(100) \).

Answer: -1


Question: Determine whether the following statement is true or false: For a commutative ring $R$ and its ideals $I$ and $J$, if the quotient $R$-modules $R/I$ and $R/J$ are isomorphic, then $I = J$. Provide a proof or counterexample.

Answer: True


Question: Let $z = 5 + 12i$ be a complex number and $w$ be a complex number such that $|w| = 13$. What is the probability that $|z - w| > 13\sqrt{3}$?

Answer: \dfrac{1}{3}


Question: Given polynomials $f_1, f_2, \ldots, f_m \in \mathbb{Z}[x_1, \ldots, x_n]$ such that $f_1(X) = f_2(X) = \ldots = f_m(X) = 0$ have no solutions over $\mathbb{C}^n$, can there always be found polynomials $g_1, g_2, \ldots, g_m \in \mathbb{Q}[x_1, \ldots, x_n]$ such that $1 = f_1 g_1 + \ldots + f_m g_m$?

Answer: Yes


Question: Find the constant \(a\) such that the primitive function \(F(x)\) of \(f(x) = x^3 - 4a^2x\) has a minimum value of 0 and satisfies \(F(2) = 4\).

Answer: \sqrt{2}


Question: Evaluate the triple integral \( \int \int \int_R (y+z) \, dx \, dy \, dz \) where \( R \) is the region of the cube defined by \( 0 \le x, y, z \le 1 \) that lies under the plane \( x+y+z=2 \) and above the plane \( y+z=1 \).

Answer: \dfrac{5}{12}


Question: Find all functions \( f: \mathbb{N} \to \mathbb{N} \) such that \( f(m^2 + f(n)) = f(m^2) + n \) for all natural numbers \( m \) and \( n \).

Answer: f(n) = n


Question:
Given that $f(x)$ is a polynomial of degree 3, and its remainders are $2x - 5$ and $-3x + 4$ when divided by $x^2 - 1$ and $x^2 - 4$ respectively, find $f(-3)$.

Note: $f(x)$ can be expressed as $f(x) = (x^2 - 1)q(x) + (2x - 5)$ and $f(x) = (x^2 - 4)r(x) + (-3x + 4)$ for some polynomials $q(x)$ and $r(x)$. Use these expressions to determine $f(-3)$.

Provide your answer in the form $\boxed{\text{answer}}$.

Answer: 53


Question:
Given that $a, b \in (1, \infty)$ and $m, n$ are natural numbers with $m, n \geq 2$, $a \leq b$, and $m \leq n$, determine which is larger:
\[ A =(a^{\frac{1}{n}}+b^{\frac{1}{n}})^{\frac{1}{m}} \] or
\[ B=(a^{\frac{1}{m}}+b^{\frac{1}{m}})^{\frac{1}{n}}. \]

Answer: A


Question:
Consider a $C^1$ function $f$ defined in a convex neighborhood of the origin. In the one-dimensional case, let $f: \Omega \to \mathbb{R}$, where $0 \in \Omega \subset \mathbb{R}$. The function can be expressed as:
\[ f(x) = f(0) + x \int_0^1 \frac{\partial f}{\partial x}(tx) \, dt. \]
In the two-dimensional case, let $f: \Omega \to \mathbb{R}$, where $(0,0) \in \Omega \subset \mathbb{R}^2$. The function can be expressed as:
\[ f(x,y) = f(0,0) + x \int_0^1 \frac{\partial f}{\partial x}(tx,ty) \, dt + y \int_0^1 \frac{\partial f}{\partial y}(tx,ty) \, dt. \]
Are the remainder terms $R_x(x,y) = x \int_0^1 \frac{\partial f}{\partial x}(tx,ty) \, dt$ and $R_y(x,y) = y \int_0^1 \frac{\partial f}{\partial y}(tx,ty) \, dt$ $C^1$?

Answer: Yes


Question: Find the determinant of the matrix \( A = \begin{bmatrix} -2 & -1 & \cdots & -1 \\ -1 & -2 & \cdots & -1 \\ \vdots & \vdots & \ddots & \vdots \\ -1 & -1 & \cdots & -2 \end{bmatrix} \) where \( A \) is a \( 53 \times 53 \) matrix.

Answer: -54


Question: What is the probability that the number \(3^a + 7^b\) has a digit equal to 8 at the units place, where \(a\) and \(b\) are randomly selected from the set \(\{0, 1, 2, \ldots, 98, 99\}\)?

Answer: \dfrac{3}{16}


Question: Is the Sobolev space $H^{s+1}(\mathbb{R}^n)$ dense in $H^s(\mathbb{R}^n)$ for $s = 0, 1, 2, \ldots$?

Answer: Yes


Question: Evaluate the value of the infinitely nested radical expression: \( \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\ldots}}}} \).

Answer: 3


Question: Find the maximum integer $k \ge 7$ such that there exists an integer $n$ for which the digit sums of the squares of $n+1, n+2, \ldots, n+k$ are all perfect squares. What is the largest such $k$?

Answer: 7


Question: Consider the integrals $A = \int_0^1 x^x \, dx$ and $B = \int_0^1 \int_0^1 (xy)^{xy} \, dx \, dy$. Determine which of the following is true: $A > B$, $A = B$, or $A < B$. Provide your answer as one of these three options.

Answer: A = B


Question: Evaluate the infinite sum: \[ \sum_{n=3}^{\infty} \binom{n}{3} \cdot 5^{-n} \]

Answer: \dfrac{5}{256}


Question: Determine the number of distinct ways to 4-color the vertices of a square such that adjacent vertices have different colors, considering colorings equivalent under rotations and reflections.

Answer: 21


Question: Determine the fundamental group of \( \mathbb{R}^3 \) with the \( z \)-axis removed.

Answer: \mathbb{Z}


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

Answer: 0


Question: Find the harmonic conjugate of the function \( u(z) = \ln(|z|) \).

Answer: \arg(z)


Question: Calculate the mass of the surface of a semisphere defined by the equation \(x^2 + y^2 + z^2 = a^2\) with \(z \geq 0\), given that the density of the surface is proportional to the distance from the plane \(XY\). Use spherical coordinates to evaluate the integral \(\int_{S} z \, dS\).

Answer: \pi a^3


Question: Determine whether the function \( f(x,y,z) = \begin{cases} (x^2+y^2+z^2) \sin \frac{1}{\sqrt{x^2+y^2+z^2}} & \text{if } (x,y,z) \neq (0,0,0) \\ 0 & \text{if } (x,y,z) = (0,0,0) \end{cases} \) is differentiable at the point \((0,0,0)\).

Answer: Yes


Question: Let \( \Omega \subset \mathbb{R}^N \) be a smooth domain with a bounded complement, and let \( a \in C(\Omega) \) be a continuous function such that \( a > 0 \) in \( \Omega \) and \( \lim_{\vert x \vert \rightarrow \infty} a(x) = \alpha > 0 \). Let \( u \in W_0^{1,2}(\Omega) \). Is the integral \( I = \int_{\Omega} a u^2 \, dx \) finite?

Answer: Yes


Question: Given the functional equation \( f(x-1) + f(x+1) = \sqrt{3} f(x) \), find the period of the function \( f(x) \).

Answer: 12


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty} \left(\int_0^n (1+\arctan^2x )\,dx \right)^ {\frac{1}{n}} \]

Answer: 1


Question: Let $J=\langle x^2+y^2-1,y-1\rangle$. Find a polynomial $f \in I(V(J))$ such that $f \notin J$. Provide your answer as a polynomial expression.

Answer: x


Question:
Prove that for a given sequence \( \sigma = (\sigma_1, \sigma_2, \ldots, \sigma_m) \) where \( \sigma_i \in \{1, -1\} \), the determinant of the matrix \( A_m \) defined by the following conditions is \( \frac{1}{2^m} \):

- \( a_{i,i+1} = -\frac{1}{2} \) for \( 1 \leq i \leq m-1 \).
- \( a_{i,i} = \frac{1}{2} \) if \( j_i = i \), and \( a_{i,i} = 1 \) if \( j_i < i \) for \( 1 \leq i \leq m \).
- \( a_{i,j_i} = -\frac{1}{2} \) if \( j_i < i \).
- \( a_{i,j} = 0 \) in all other cases.

Here, \( j_k \) is uniquely determined by the sequence \( \sigma \).

Answer: \dfrac{1}{2^m}


Question: Evaluate the integral \( \int_{0}^{\pi /2} \frac{\sin^{m}(x)}{\sin^{m}(x)+\cos^{m}(x)}\, dx \) for arbitrary positive integer \( m \).

Answer: \dfrac{\pi}{4}


Question: Provide an example of a finitely generated group that contains a subgroup which is not finitely generated.

Answer: F_2


Question: Find the real part of the expression \( \left(\frac{a+bi}{a-bi}\right)^2 - \left(\frac{a-bi}{a+bi}\right)^2 \), where \(a\) and \(b\) are real numbers.

Answer: 0


Question: Given discrete, finite random variables $X$ and $Y$, is the joint distribution $P_{XY}(x,y)$ uniquely determined by the marginal distribution $P_X(x)$ and the conditional distribution $P_{X|Y}(x|y)$? Assume $P_{XY}(x,y) = \mathbb{P}(X=x \wedge Y=y)$, $P_X(x)=\sum_y P_{XY}(x,y)$, and $P_{X|Y}(x|y)=P_{XY}(x,y)/P_Y(y)$. Determine if $P_X(x)$ and $P_{X|Y}(x|y)$ are sufficient to recover $P_{XY}(x,y)$. Provide a justification for your answer.

Answer: No


Question: Find a closed-form expression for the function \( f(x) = \int_0^{2\pi} e^{x \sin \theta} \, \text{d}\theta \).

Answer: 2\pi I_0(x)


Question: Evaluate the integral \( \int_{|z|=1} \frac{\sin(z)}{z} \, dz \).

Answer: 0


Question:
Evaluate the limit given that the function \( f(x) \) is continuous and \( \lim_{x \to 0} \frac{f(x)}{x} = 1 \):
\[ \lim_{x \to 0} \frac{x\int_0^x{f(x-t)dt}}{\int_0^x tf(x-t)dt} \]

Answer: 3


Question: Calculate the value of the complex number expression \((-1)^{3i}\).

Answer: e^{-3\pi}


Question: Let $1 < p < \infty$. Suppose $f$ is a continuous function with compact support on $\mathbb{R}$. Determine whether there exists a function $g \in L^p(\mathbb{T})$ such that the Fourier transform of $f$ restricted to $\mathbb{Z}$ equals the Fourier transform of $g$, i.e., \( \widehat{f}|_{\mathbb{Z}} = \widehat{g} \).

Answer: Yes


Question: A rod of length \( l > 1 \) has a mass density given by \( p(x) = 3x^{-1/2} \cdot |x-1| \) for \( 0 < x < l \). Given that the total mass of the rod is \( M = 8 \), determine the length \( l \).

Answer: 3


Question: Evaluate the integral \( \int_{0}^{1} \int_{0}^{2 \pi} \int_{0}^{2-r\cos{\theta} - r\sin{\theta}} yr\,dy\,dr\,d\theta \). This integral represents the function \( f(x,y,z)=y \) over the region bounded by the plane \( x+y+z=2 \), the cylinder \( x^2+z^2=1 \), and the plane \( y=0 \).

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


Question: Find the number of non-negative integer solutions to the equation \(a+b+c+d+e=79\) given the constraints \(a \geq 7\), \(b \leq 34\), and \(3 \leq c \leq 41\).

Answer: 968239


Question: Evaluate the infinite product: \[ \prod_{n=1}^{\infty} \left(1 + \frac{2}{3^n + 1}\right) \]

Answer: 2


Question: Find the most efficient sequence of digits to enter on a keycode doorlock to cover all possible 4-digit passcodes, given that the lock accepts any 4 consecutive digits as a passcode.

Answer: 10003


Question: Evaluate the limit: \[ \lim_{x\to 0^{+}} \frac{\arcsin(1-\{x\})\arccos(1-\{x\})}{\sqrt{2\{x\}}(1-\{x\})} \] where \( \{\cdot\} \) denotes the fractional part function. Provide your answer without using L'Hopital's Rule.

Answer: \dfrac{\pi}{2}


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

Answer: (-1, 1)


Question: Let $N$ and $K$ be groups, and let $x \in N \cap K$ and $k \in K$. If $kx = x'k$ for some $x' \in N$, does $kx \in N \cap K$ imply that $x' \in K$?

Answer: Yes


Question:
Let $k > 1$. Determine whether the limit of the sequence $(x_n)$ exists, where
\[ x_n := \left(k \sin \left(\frac{1}{n^2}\right) + \frac{1}{k}\cos n \right)^n. \] If the limit exists, calculate its value.

Answer: 0


Question: Approximate \( \log(7) + \cos(1) \) such that the error is less than \( 10^{-4} \).

Answer: 2.4862


Question: Determine the number of real solutions to the equation \(2^x = x^2\) without using graphing methods.

Answer: 3


Question: Evaluate the supremum of the expression \( \frac{\text{area}(M)}{\text{diam}^{2}(M)} \) for a connected set \( M \).

Answer: \dfrac{\pi}{4}


Question:
Given that \( f \in \mathcal{C}[0,1] \) and the following conditions:

- \( f(1) = 2 \).
- \( \int_{0}^{1} f(x) \, dx = \frac{1}{3} \).
- \( \int_{0}^{1} f'(x) \, dx = \frac{25}{3} \).

Find the value of \( \int_{0}^{1} x f(x) \, dx \).

Answer: \dfrac{31}{36}


Question:
Let $a$, $b$, $c$, and $d$ be distinct real numbers such that\[\begin{align*}
a &= \sqrt{4 + \sqrt{5 + a}}, \\
b &= \sqrt{4 - \sqrt{5 + b}}, \\
c &= \sqrt{4 + \sqrt{5 - c}}, \\
d &= \sqrt{4 - \sqrt{5 - d}}.\end{align*}\]Compute the product $abcd$. Provide your answer as a single number.

Answer: 11


Question: Let $x \in \mathbb{R}^+$ and define the operation $x \diamond 1 = x$ and $x \diamond (n+1) = x^{x \diamond n}$ for $n \in \mathbb{N}$. For example, $2 \diamond 3 = 2^{2^2} = 16$. Given a unique positive real number $t$ such that $t^t = 10^{10^{10}}$, it is known that $10^{10} > t$. Let $k$ be the unique positive real number such that $k \diamond 50 = 10 \diamond 51$, and $s$ be the unique positive real number such that $s \diamond 51 = 10 \diamond 52$. Determine if $k > s$.

Answer: k > s


Question: Evaluate the integral \( \int_{\frac{1}{7}}^{\frac{6}{7}} f(f(x)) \, dx \) where \( f(x) = \frac{x^3}{3} - \frac{x^2}{2} + x + \frac{1}{12} \).

Answer: \dfrac{5}{14}


Question: Evaluate the limit: \( \lim\limits_{x \to 0^+} x^{(x^x-1)} \).

Answer: 1


Question: Let $X_1, X_2, \ldots, X_n, \ldots$ be independent and identically distributed random variables with $P(X_n=0)=P(X_n=1)=P(X_n=2)=P(X_n=3)=\frac{1}{4}$. Define the sequence $\{Y_n\}$ by $Y_0=0$ and for all $n \in \mathbb{N}$, \[ Y_n=\begin{cases} 3 & \text{if } X_n=3, \\ \min\{Y_{n-1},X_n\} & \text{if } X_n<3. \end{cases} \] Compute $\lim_{n \to +\infty} E[Y_n Y_{n-1}]$. \(\text{Express your answer as a single number.}\)

Answer: \dfrac{19}{12}


Question: Determine the interval of convergence for the series \( \sum_{k=0}^{\infty} \frac{k!(x-2)^k}{k^k} \).

Answer: (2 - e, 2 + e)


Question: How many distinct cards can be constructed by punching holes in two of the small triangles of an equilateral triangle divided into 9 congruent sub-triangles?

Answer: 9


Question: Let $X$ be a closed subspace of $\ell^2$ over $\mathbb{C}$ with a basis $\{x_m\}_{m \in \mathbb{N}}$ equivalent to the canonical basis $\{e_m\}_{m \in \mathbb{N}}$ for $\ell^2$. Prove or disprove that: \[ \bigcap_{p=1}^\infty \overline{ \operatorname{span} } \{x_m\}_{m \geq p} = \{0\} \]

Answer: \{0\}


Question: Evaluate the expression: \[ \frac{2012!}{2^{2010}} - \sum_{k=1}^{2010} \frac{k^2 k!}{2^k} - \sum_{k=1}^{2010} \frac{k \cdot k!}{2^k} \]

Answer: 2


Question:
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $| f(x)-f(y)| \leq 4321|x-y|$ for all $x,y \in \mathbb{R}$. Determine which of the following statements is true:

1. $f$ is always differentiable.
2. There exists at least one such $f$ that is continuous and satisfies $\displaystyle{\lim_{x \to \pm\infty}\frac{f(x)}{|x|}} = \infty$.
3. There exists at least one such $f$ that is continuous but not differentiable at exactly $2018$ points and satisfies $\displaystyle{\lim_{x \to \pm\infty}\frac{f(x)}{|x|}} = 2018$.
4. It is not possible to find a sequence $\{x_n\}$ of reals such that $\displaystyle{\lim_{n \to \infty} x_n = \infty}$ and further satisfying $\displaystyle{\lim_{n \to \infty}|\frac{f(x_n)}{x_n}|} \leq 10000$.

Select the correct statement.

Answer: 3


Question: Let $X_1, X_2, \ldots$ be a sequence of independent and identically distributed continuous random variables. Define $N \geq 2$ such that $X_1 \geq X_2 \geq \ldots \geq X_{N-1} < X_N$. Show that $E[N] = e$.

Answer: e


Question: A binary message is emitted by a source, where the probability of a '1' appearing is 0.6 and the probability of a '0' appearing is 0.4. The interpreter of the message incorrectly interprets 10% of the symbols. If the interpreter receives the message "101", what is the probability that this is the same message that was sent by the source? Provide your answer to three decimal places.

Answer: 0.743


Question: Determine the number of solutions to the equation \(z - 2 - e^{-z} = 0\) in the right half plane \(P = \{z \in \mathbb{C} : \Re(z) > 0\}\).

Answer: 1


Question: Let $G$ be a 1-vertex-connected $d$-regular graph with $d \geq 2$. Determine the chromatic index $\chi'(G)$. Use Brooks' theorem, which states $\Delta(G) \leq \chi'(G) \leq \Delta(G) + 1$, to find the chromatic index.

Answer: d


Question: Find the value of \( k \) such that the equation \( \frac{x^3}{2} - kx^2 + 4kx - 32 = 0 \) has exactly two distinct solutions.

Answer: -2


Question: Let \( X \sim N(0,1) \). Determine the distribution of \( \Phi(X) = \int_{-\infty}^X {\exp\left(-{z^2}{/2}\right) \over \sqrt{2\pi}} \, \mathrm{d}z \).

Answer: U(0,1)


Question: Let \( f(x, y) \) be a differentiable function for all \( (x,y) \). Define \( g(u, v) = f(u^2 - v^2, u^2v) \). Given that \( \nabla f(-3, 2) = (2, 1) \), find \( \nabla g(1, 2) \).

Answer: (8, -7)


Question: Determine whether the sequence \( z_n = \sqrt[n]{ni} \) converges, and if so, find its limit.

Answer: 1


Question:
Find the determinant of the matrix \( A \) given by:
\[
A = \begin{pmatrix}
x & 0 & \cdots & 0 & -1\\
0 & x & \ddots & \vdots & 0\\
\vdots & \ddots & \ddots & 0 & \vdots\\
0 & \cdots & 0 & x & 0\\
-1 & 0 & \cdots & 0 & x
\end{pmatrix}
\]
where \( x \) is on the diagonal, \(-1\) is at the end of the first row, and \(-1\) is at the beginning of the last row, with all other elements being zero.

Answer: x^n - x^{n-2}


Question: Determine the number of solutions to the congruence equation \(x^d \equiv 1 \pmod{p^2}\) where \(p\) is an odd prime and \(d\mid p-1\).

Answer: d


Question: Calculate the integral \( P = \int_C \frac{e^{-x^2y}}{\sqrt{(1+x^2y^4)^3}} \left( xy[2(1+x^2y^4)-y^3] \, dx + x^2[e^{-x^2y}\sqrt{(1+x^2y^4)^3} + 1 + x^2y^4 - 2y^3] \, dy \right) \), where \( C \) is the half-circle defined by \( x^2 + y^2 = 1, \, y \ge 0 \), oriented from \( A(1,0) \) to \( B(-1,0) \).

Answer: 0


Question: For a $p$-group $G$ of order $p^4$, where the center of $G$ has order $p^2$, determine the number of conjugacy classes of $G$. Provide your answer as a single integer.

Answer: p^3 + p^2 - p


Question: Determine how the prime number 2 splits in the splitting field of the polynomial \(X^3 - 3X + 1 = 0\).

Answer: (2)


Question: Find the inverse of the element \(1010\) in the field \(GF(2^4)\) with multiplication modulo \(x^4 + x^3 + 1\).

Answer: 1011


Question: Determine the minimum number of times a balanced coin must be flipped so that the probability that the difference between the proportion of tails and 0.50 is at least 0.04 is at most 0.01.

Answer: 1037


Question: Determine whether the sequence of complex numbers \(z_n = \frac{\tan(in)}{n}\) is convergent or divergent.

Answer: 0


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

Answer: 159


Question: For every prime number \( p \), does there exist a natural number \( N \) such that \( p \mid (P_N + 2) \), where \( P_N \) is the \( N \)-th prime number?

Answer: Yes


Question: Determine the radius of convergence of the function \( \frac{\sinh(z)}{z} \) about the point \( z = \pi \).

Answer: \infty


Question: Evaluate the integral \( I = \int_{-b/2}^{\frac{\pi-10b}{20}}\frac{\mathrm{d}x}{x^2+bx+c} \) given that \( 4c-b^2 = \frac{\pi^2}{100} \).

Answer: 5


Question: Let $G$ be a group and $c$ be a fixed element of $G$. Define a new operation $*$ on $G$ by \( a * b = ac^{-1} b \). Given that the identity element for this operation is $c$, find the inverse of an element $a \in G$ under this operation.

Answer: c a^{-1} c


Question: Determine whether the following statement is true: \[ \int_{\mathbb{R}^N}(-\Delta)^su(x) \, dx = 0, \] where \((-\Delta)^s\) is the fractional Laplacian.

Answer: True


Question: If \( K \) is an \( n \)-dimensional extension field of \( \mathbb{Z}_p \), what is the maximum possible number of elements in \( K \)?

Answer: p^n


Question: Evaluate the integral \( \int_{C}(z-i) \,dz \) where \( C \) is the parabolic segment defined by \( z(t) = t + it^2 \) for \( -1 \le t \le 1 \). Use the method of integrating along the straight line from \(-1+i\) to \(1+i\) and apply the Closed Curve Theorem to find the value of the integral.

Answer: 0


Question: Calculate the third central moment, \( E[(X_i - \mu)^3] \), for a random variable \( X_i \) drawn from a normal distribution with mean \( \mu \) and variance \( \sigma^2 \).

Answer: 0


Question: In a ring $(R; +, -, 0, \times, 1)$, given that $a \times a = a$, prove that $a + a = 0$. Use the following lemmas if necessary: (i) $0(a) = a(0) = 0$, (ii) $(−a)b = −(ab)$, (iii) $(−a)(−b) = ab$. Provide a detailed proof.

Answer: a + a = 0


Question: Consider the product \( \frac{p(i)^2}{p(i)^2-1}\prod_{j=1}^5\frac{p(i+j)^2-1}{p(i+j)^2} \), where \( p(i) \) denotes the \( i \)-th prime number. Determine if this product is always less than or equal to 1 for \( i \geq 3 \) (i.e., when \( p(i) \geq 5 \)).

Answer: Yes


Question:
Given a Poisson distribution with parameter \( \lambda \), where \( \lambda \) is a random variable with probability density function \( f(\lambda) = e^{-\lambda} \), find the expectation of the random variable. The Poisson distribution is given by:
\[ p(x, \lambda) = e^{-\lambda} \frac{\lambda^x}{x!} \]

Answer: 1


Question: In a triangle, the perimeter and the area have the same numerical value when measured in a certain unit. If the lengths of the three altitudes of the triangle are \( p, q, \) and \( r \), find the numerical value of \( \frac{1}{q} + \frac{1}{p} + \frac{1}{r} \).

Answer: \dfrac{1}{2}


Question: Evaluate the line integral \( \oint\limits_C \frac{(x+y)\;\mathrm{d}x-(x-y)\;\mathrm{d}y}{x^2+y^2} \), where \( C \) is the circle defined by \( x^2+y^2=a^2 \).

Answer: -2\pi


Question: Find the almost sure limit of the sequence \( \frac{X_t}{t} \) as \( t \to +\infty \), where \( X_t = \sup(n: \xi_1 + \ldots + \xi_n \leq t) \) and \( \{\xi_i\} \) are i.i.d. random variables with \( \xi_i \sim \mathrm{Exp}(\lambda) \).

Answer: \lambda


Question: Evaluate the limit: \[ \lim_{x \to 0^+} \left(\frac{\tan x}{x}\right)^{1/x^2} \] using L'Hôpital's Rule.

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


Question:
Let \(X_1, X_2, \ldots, X_n\) be independent random variables where \(X_i \sim N(0, \sigma_i^2)\) for \(i = 1, 2, \ldots, n\). Define \(Z\) as follows:
\[ Z = \frac{\sum_{i=1}^{n} \frac{X_i}{\sigma_i^2}}{\sum_{i=1}^{n} \frac{1}{\sigma_i^2}}. \]

Now, consider the random variable \(\xi\) defined by:
\[ \xi = \sum_{i=1}^{n} \frac{(X_i - Z)^2}{\sigma_i^2}. \]

Find the distribution of \(\xi\).

Answer: \chi^2(n-1)


Question: Let $X$ be a symmetric random variable around $0$, meaning that for all $x \in \mathbb{R}$, $\mathbb{P}(X \leq -x) = \mathbb{P}(X \geq x)$. Given any $\varepsilon > 0$, determine whether $\mathbb{E}[X I_{\{|X| < \varepsilon\}}] = 0$, where $I$ is the indicator function.

Answer: 0


Question: Given a triple \((V, E, F)\) that satisfies the Euler characteristic equation \(V - E + F = 2\), is it always possible to construct a convex polyhedron with \(V\) vertices, \(E\) edges, and \(F\) faces? Provide a justification for your answer.

Answer: No


Question: Determine the number of homomorphisms from the symmetric group $S_n$ to the cyclic group $\mathbb{Z}_2$ for $n \geq 5$. Provide your answer as a single integer.

Answer: 2


Question: Given that \( f(x) \) is a positive function for \( x > 0 \) and satisfies the equation \( f(x)f^{-1}(x)=1 \), find the value of \( f(1) \).

Answer: 1


Question:
Find the exact value of the limit:
\[\lim_{n\to\infty}\int_0^\infty\left(\sum_{k=0}^n\frac{(-1)^kx^{2k}}{(2k)!}\right)e^{-2x}\;dx.\]
Use the dominated convergence theorem to justify your solution.

Answer: \dfrac{2}{5}


Question:
Evaluate the limit of the definite integral as \( n \to \infty \):
\[ \lim_{n\rightarrow \infty}\int^{\infty}_{0}\left(1+\frac{t}{n}\right)^{-n}\cdot \cos\left(\frac{t}{n}\right)dt \]

Answer: 1


Question: Find the residue of the function \( f(z) = \frac{(z-1)^3}{z(z+2)^3} \) at \( z = \infty \).

Answer: -1


Question: Can Basil always determine the original 8 integers if he is given all their pairwise sums? Provide a mathematical explanation for your answer.

Answer: No


Question: Evaluate the integral \( \int_{\pi/6}^{\pi/3} \frac{\sqrt[3]{\sin x}}{\sqrt[3]{\sin x} + \sqrt[3]{\cos x}} \, dx \).

Answer: \dfrac{\pi}{12}


Question: Let \( m_1, \ldots, m_{n^2} \) be \( n \times n \) matrices in \( M(n, \mathbb{C}) \). Consider these matrices as vectors in \( \mathbb{C}^{n^2} \) and form an \( n^2 \times n^2 \) matrix \( M \) with columns \( m_1, \ldots, m_{n^2} \). Prove that \( m_1, \ldots, m_{n^2} \) span \( M(n, \mathbb{C}) \) if and only if \( M \) has a non-zero determinant.

Answer: \det(M) \neq 0


Question: Can the measure \( \mu \) be chosen to be a finite positive measure such that \( C[0,1]^{*} \) is order isometric to \( L_{1}(\mu) \)?

Answer: No


Question: Evaluate the improper integral \( \int_{-\infty}^{\infty} \dfrac {e^{x}}{1+e^{2x}} \, dx \).

Answer: \dfrac{\pi}{2}


Question: Consider a Markov chain with states labeled from 1 to n, where the transition probability is 0.5 between consecutive states, except for the first and last states which have a transition probability of 1 to the next state. What is the expected number of steps to reach the rightmost state starting from the leftmost state?

Answer: (n-1)^2


Question: Determine the asymptotic density of positive integers \( n \) such that \( \gcd(n, \sigma(n^2)) = \gcd(n^2, \sigma(n^2)) \), where \( \sigma(x) \) denotes the sum of divisors of \( x \).

Answer: 1


Question: Evaluate the complex line integral \( \oint_{|z|=1}\frac{e^z}{z}\,\mathrm dz \).

Answer: 2\pi i


Question: Find the closure of the set \( \{ \frac{1}{2} \} \) in the topology generated by the basis \( B = \{ (n, n+1) : n \in \mathbb{Z} \} \cup \{ \{ k \} : k \in \mathbb{Z} \} \).

Answer: (0, 1)


Question: Evaluate the path integral \( \int_C \frac{1}{z^2} \, dz \) where \( C \) is a circle of positive radius centered at the origin.

Answer: 0


Question: Compute the limit: $$\lim_{n\to\infty} \frac{\sum_{r=0}^{n}\binom{2n}{2r}3^r}{\sum_{r=0}^{n-1}\binom{2n}{2r+1}3^r}$$

Answer: \sqrt{3}


Question: Find the limit \( \lim_{n\to+\infty}\left(n\left(1+\frac{1}{n}\right)^n-ne\right) \) using the squeeze theorem, without using differentiation, integrals, or series.

Answer: -\dfrac{e}{2}


Question: Compute the surface integral \( \iint_S{\mathbf{F} \cdot \mathbf{n}} \, dS \) for the hemisphere \( S \) defined by \( x^2 + y^2 + z^2 = 1 \) with \( z \ge 0 \), where \( \mathbf{F}(x, y, z) = x\mathbf{i} + y\mathbf{j} \) and \( \mathbf{n} \) is the unit outward normal to \( S \). Use the parameterization \( \mathbf{r}(u, v) = \sin u \cos v \mathbf{i} + \sin u \sin v \mathbf{j} + \cos u \mathbf{k} \) to evaluate the integral.

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


Question: Determine the number of roots of the polynomial \(z^4 + z^3 - 4z + 1 = 0\) within the annular region \(1 < |z| < 2\) using Rouché's theorem.

Answer: 3


Question: Let $B_1$ and $B_2$ be two Banach spaces, and let $L(B_i,B_j)$ and $K(B_i,B_j)$ denote the spaces of bounded and compact linear operators between them, respectively, for $i,j=1,2$. Given $T \in L(B_1,B_1)$, suppose there exists $S \in K(B_1,B_2)$ and a constant $c>0$ such that for any $v \in B_1$, \[ \|Tv\|_{B_1} \le c\|v\|_{B_1} + \|Sv\|_{B_2}. \] Can we find an operator $A \in K(B_1,B_1)$ such that \[ \|T - A\|_{L(B_1,B_1)} \le c? \]

Answer: Yes


Question: Determine all values of \( \alpha \) for which the improper integral \( \int_0^{+ \infty } \frac{e^x \arctan(xe^{-x})}{(x+ \sqrt[3]{x})^ \alpha } \, dx \) converges.

Answer: (2, 6)


Question: Determine whether the random variable \( \frac{S_n-nE(X_1)}{\sqrt{n \text{Var}(X_1)}} \) is standardized, given that the random variables \( X_1, X_2, X_3, \ldots \) satisfy the conditions of the central limit theorem, where \( S_n = \sum_{i=1}^n X_i \).

Answer: Yes


Question: Let \( \mathbf{F} \) be a vector field defined on \( \mathbb{R}^2 \setminus\{(0,0)\} \) by \( \mathbf{F} (x,y)=\frac{y}{x^2+y^2}i-\frac{x}{x^2+y^2}j \). Consider the curves \( \gamma(t) = (8\cos 2\pi t, 17\sin 2\pi t) \) and \( \alpha(t) = (26\cos 2\pi t, -10\sin 2\pi t) \) for \( t \in [0,1] \). If \( 3\int_{\alpha} \mathbf{F \cdot dr} - 4 \int_{\gamma} \mathbf{F \cdot dr} = 2m\pi \), find the value of \( m \).

Answer: 7


Question:
Evaluate the limit:
\[ \lim_{k \to \infty, p \to \infty} \frac{a_{k,p}}{b_{k,p}} \]
where
\[
a_{k,p} = \sum_{n=1}^{k} (2^{n}-1) + \sum_{n=1}^{k}{(2^{2n}-2^n-1)} + \sum_{n=1}^{k}{(2^{3n}-2^{2n}-2^n-1)} + \sum_{n=1}^{k}{(2^{4n}-2^{3n}-2^{2n}-2^n-1)} + \ldots + \sum_{n=1}^{k}{(2^{pn}-2^{n(p-1)}-2^{n(p-2)}-\ldots-2^{3n}-2^{2n}-2^n-1)}
\]
and
\[
b_{k,p} = \frac{1}{4} \times \left[2^{pk} - 2^{k(p-1)} - 2^{k(p-2)} - \ldots - 2^{3k} - 2^{2k} - 2^k\right]^2
\]

Answer: 0


Question: Determine the value of \( c \in \mathbb{R} \) such that the function \( \kappa(t) = c\sin{(2\pi t)} \) can represent the curvature of a regular, closed, and simple curve \( \alpha(t):[0,1]\to\mathbb{R}^2 \).

Answer: 4\pi


Question: Evaluate the limit: $$\lim_{x \to 0+} \frac{1}{x} \int_0^{2x} (\sin{t})^t\,\mathrm{d}t.$$

Answer: 2


Question: Determine the real rank of the $C^*$-algebra $C_0(\mathbb{R})$. Provide your answer as a single value.

Answer: 1


Question: There are 3 cities, and roads must be constructed so that every pair of cities is connected. Assume that all the roads connecting any two cities have independent costs that are exponentially distributed with parameter one. Find the expected value of the minimal cost to connect all cities. That is, find \( E(c_3) \).

Answer: \dfrac{7}{6}


Question: Find the equation of the tangent line to the curve \((x^2 + y^2 - 3)^3 = (xy)^3 - x^2 - y^2\) at the point \((1, 1)\).

Answer: y = -x + 2


Question: Consider a covering map \( p: \tilde{X} \to X \) where \( \tilde{X} \) is path-connected. If the fiber over some point in \( X \) is finite with cardinality \( k \), then the fiber over any point in \( X \) also has cardinality \( k \). Does this statement hold if the fiber is infinite? Specifically, if \( \tilde{X} \) is path-connected, is the cardinality of the fiber at any point in \( X \) the same, ensuring that the number of sheets is unique and equal to the cardinality of the fiber at any point in \( X \)?

Answer: Yes


Question: Evaluate the line integral \( \int_{C}{(3x+2y) \, dx + (2x-y) \, dy} \) along the curve \( y = \sin\left(\frac{\pi x}{2}\right) \) from \((0,0)\) to \((1,1)\).

Answer: 3


Question: Given that the aggregate claims \(\sum_{X=i}^N X_{i}\) follow a compound Poisson distribution with individual claim distribution probabilities \(P(X=1) = \frac{1}{3}\) and \(P(X=3) = \frac{2}{3}\), and the condition \(P(S=4)=P(S=3)+6P(S=1)\), find the expectation of the claim count \(N\), denoted as \(E[N]\).

Answer: 12


Question: Let $X$ be a continuous random variable with distribution function $F(x) = P(X \le x)$. Is it true that $F(X)$ follows a uniform distribution on the interval [0, 1]? Justify your answer.

Answer: Yes


Question: Determine the number of solutions for the equation \(x^2 - x \sin x - \cos x = 0\) over the interval \((-\infty, \infty)\).

Answer: 2


Question: Evaluate the limit: \[ \lim_{n\to\infty}\int_0^{\pi/2} \frac{\sin^n(x)}{1+x^2} \, dx \] Determine if the following is the correct answer: \[ \begin{cases} 0, & x \ne \pi/2\\[8pt] \dfrac{2\pi}{4 + \pi^2}, & x = \pi/2 \end{cases} \]

Answer: 0


Question: Let \( A = [a_{ij}] \) be a \( 4 \times 4 \) nonsingular matrix with \( \det(A) = 5 \). Find \( \det([C_{ij}]) \), where \( C_{ij} \) is the cofactor of \( a_{ij} \).

Answer: 125


Question: Consider a group $G$ and a subgroup $H$. Let $A$ be the set of all left cosets of $H$ in $G$. Define an action of $G$ on $A$ by $g \cdot aH = gaH$ for all $g \in G$ and $aH \in A$, where $gaH$ is the left coset with representative $ga$. Does this action permute the left cosets of $H$ in $G$?

Answer: Yes


Question: What is the maximum number of lines along which a quintic surface in \(\mathbb{P}^3\) can be singular?

Answer: 10


Question: Determine the degree of the field extension \([\mathbb{Q}(\sqrt{2}, e^{\frac{2\pi i}{3}}):\mathbb{Q}]\).

Answer: 4


Question: Is it true that the dual of a faithful representation of a finite group is also faithful?

Answer: Yes


Question: Let \(X_1\) and \(X_2\) be independent random variables, each uniformly distributed on \((0,1)\). Calculate \(\operatorname{Cov}(X_1+X_2,\max(X_1,X_2))\).

Answer: \dfrac{1}{12}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_{0}^{1} \frac{2nx^{n-1}}{1+x} \, dx. \]

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \to 0} x^2 \ln\left(\frac{1}{x}\right) \]

Answer: 0


Question:
Calculate the condition number of the matrix \( A \):
\[
A = \begin{bmatrix}
0 & 0 & -10^4 & 0  \\
0 & 0 & 0 & -10  \\
0 & 10^{-3} & 0 & 0 \\
10^{-2}& 0& 0& 0 \\
\end{bmatrix}
\]

Answer: 10^7


Question: Determine the values of \(a \in \mathbb{R}\) for which the equation \(e^x - a = \ln(x+a)\) has no solution.

Answer: (-\infty, 1)


Question: Evaluate the integral \( \int_{0}^{2\pi} \sin\left(\frac{\pi}{6} - 2\exp(i\theta)\right) d\theta \).

Answer: \pi


Question: Determine which is greater: \(\pi^3\) or \(3^\pi\).

Answer: 3^\pi


Question: Use the Plancherel formula to compute the integral \( \int_{-\infty}^\infty \frac{1}{c^2+y^2} \, dy \) where \( c > 0 \).

Answer: \dfrac{\pi}{c}


Question: Determine whether there exist matrices that are both skew-symmetric and orthogonal. Specifically, find if there exists a matrix $A$ such that $A^T = A^{-1} = -A$. Justify your answer.

Answer: Yes


Question: Determine the minimum number of actions required to make a given triangle equilateral. The action $P(X)$ involves moving vertex $X$ to the intersection of the perpendicular bisector of the opposite side and a line parallel to that side passing through $X$. How many such actions are needed to transform the triangle into an equilateral triangle?

Answer: 3


Question: Consider a bounded linear functional defined as \( l(v) := (f,v)_\Omega \). Prove or disprove that \( \|l\| \le \|f\| \). Additionally, determine if it is possible to have \( \|l\| = \|f\| \).

Answer: \|l\| = \|f\|


Question: Determine whether the following statement is true: \( \overline{ \sqrt{1 + i}} = \sqrt{1-i} \).

Answer: True


Question: Determine the radius of convergence of the power series \( \sum_{n=-\infty}^{\infty} 3^{-|n|} z^{2n} \), where \( z \in \mathbb{C} \).

Answer: \sqrt{3}


Question: Given two measurable disjoint subsets $A, B \subset \mathbb{R}^n$, which are surfaces of some dimension $k < n$ (e.g., two curves in $\mathbb{R}^2$ or in $\mathbb{R}^3$) with finite $k$-dimensional Lebesgue measure, and the condition that the Hausdorff distance $d_H(A, B) < \epsilon$, can we conclude that $|\mu_k(A) - \mu_k(B)| < \epsilon$?

Answer: No


Question: Find the maximum value of $ab$ given that $a, b \in \mathbb{R}$ and satisfy the equations $a^2 + b^2 = 1 + \frac{2ab}{a-b}$ and $\sqrt{a-b} = a^2 + 5b$. What is the maximum value of $ab$?

Answer: 42


Question: Suppose $U$ and $V$ are nonempty disjoint open subsets of $[0,1]^2$. Is there necessarily a continuum $K \subseteq [0,1]^2$ such that $[0,1]^2 \setminus K$ is the union of two disjoint open sets $T$ and $W$ with $U \subseteq T$ and $V \subseteq W$?

Answer: Yes


Question: Find \( \lim_{n\to\infty}\tan{\left(\tan^{-1}{\frac{2}{1^2}}+\tan^{-1}{\frac{2}{2^2}}+\ldots+\tan^{-1}{\frac{2}{n^2}}\right)} \).

Answer: -1


Question: Given that \(1, \alpha_1, \alpha_2, \ldots, \alpha_{2008}\) are the 2009 roots of unity, calculate the value of \(\sum_{r=1}^{2008} r(\alpha_r + \alpha_{2009-r})\).

Answer: -2009


Question: Calculate the line integral \( \int_\gamma G \, dl \), where \( \gamma \) is a positively-oriented circle of radius \( R \) centered at \((0,0)\). The vector field \( G: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) is given by \( G(x,y) = (2x\cos(y), -x^2\sin(y) + ye^y) \). Use the parametrization \((R\cos(\theta), R\sin(\theta))\) for the circle and evaluate the integral over the interval \( \theta = 0 \) to \( \theta = 2\pi \).

Answer: 0


Question: Determine whether the group isomorphism \((\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z})/(\mathbb{Z} \oplus \mathbb{Z}) \cong \mathbb{Z}\) holds true.

Answer: True


Question: Calculate the limit: \[ \lim_{x\to0}\frac{e^{\cos x}-e}{x^2} \] without using L'Hôpital's rule.

Answer: -\dfrac{e}{2}


Question: Given a real number $p \in (0, 1)$ and a fixed integer $k$, is there a constant $C > 0$ independent of $k$ such that for every $n \geq k$, the inequality \[ n \cdot \binom{n}{k} p^k (1-p)^{n-k} \leq C \] holds true?

Answer: No


Question: Let $n$ be a fixed positive integer. For a distribution $F$ on positive real numbers, let $a_n(F)$ be the expected value of the maximum of $n$ i.i.d. random variables drawn from $F$, and let $a_{n+1}(F)$ be the expected value of the maximum of $n+1$ i.i.d. random variables drawn from $F$. Determine the maximum value of \( \frac{a_{n+1}(F)}{a_n(F)} \) over all distributions $F$. For example, if $F$ is the uniform distribution on $(0,1)$, then $a_n(F)=\frac{n}{n+1}$ and $a_{n+1}(F)=\frac{n+1}{n+2}$. Is the maximum value \( \frac{n+1}{n} \)?

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


Question: Is the proof that the quotient space \( P = (\mathbb{R} \times \{1,2\})/\sim \) is second-countable accurate?

Answer: Yes


Question: The polynomial \( P(x) = x^4 + ax^3 + bx^2 + cx + d \) satisfies \( P(k) = 11k \) for \( k = 1, 2, 3, 4 \). Find the value of \( c \).

Answer: -39


Question: Determine the minimum number of distinct non-zero vectors required to span the vector space \(\mathbb{F}_2^n\).

Answer: n


Question: Determine the minimum value of \( n \) such that for every odd integer \( m \), \( 2^{2012} \) divides \( m^n - 1 \).

Answer: 2^{2010}


Question: Find the value of \( n \) such that a regular \( n \)-gon satisfies the property \( \frac{1}{A_1A_2} = \frac{1}{A_1A_3} + \frac{1}{A_1A_4} \), where \( A_1, A_2, A_3, A_4 \) are consecutive vertices of the polygon.

Answer: 7


Question: Evaluate the sum \( \sum_{k=1}^{1233} f\left(\frac{k}{1234}\right) \) where \( f(x) = \frac{e^{2x-1}}{1+e^{2x-1}} \).

Answer: \dfrac{1233}{2}


Question: In an ordered field, if all rational Cauchy sequences converge, must all Cauchy sequences converge? Determine whether there exists an ordered field in which all rational Cauchy sequences converge, but some Cauchy sequences do not converge.

Answer: Yes


Question: Let \(a, b, c\) be real numbers such that \(\frac{a+b}{1-ab}, b, \frac{b+c}{1-bc}\) are in arithmetic progression. If \(\alpha\) and \(\beta\) are the roots of the equation \(2acx^2 + 2abcx + (a+c) = 0\), find the value of \((1+\alpha)(1+\beta)\).

Answer: 1


Question: Find the last three digits of \(7^{12341}\).

Answer: 007


Question: Consider two sets $A$ and $B$, each containing $n$ vertices. For each pair $(a,b) \in A \times B$, an edge between $a$ and $b$ appears with probability $0.01$, independently of other edges. As $n \rightarrow \infty$, does the probability that there exists a matching between $A$ and $B$ approach 1?

Answer: Yes


Question: Determine whether the following statement is true or false: If $f$ is an infinitely differentiable function such that $f^{(k)}(0) = 0$ for all non-negative integers $k$, then $f$ is identically zero.

Answer: B


Question: Aaron samples a number from a uniform \((0,1)\) distribution. Brooke then repeatedly samples from the same distribution until she obtains a number higher than Aaron's. What is the expected number of samples Brooke needs to make?

Answer: \infty


Question: In a game where the winner is the first player to have 7 or more points and is 2 points ahead of the other player, you currently lead with a score of 6-5. Assuming both players have equal skill and each point is equally likely to be won by either player, what is the probability that you will eventually win the game?

Answer: \dfrac{3}{4}


Question: Determine the order of the pole of the function \( f(z) = \frac{z}{(1-e^z) \sin z} \) at \( z = 2\pi ik \) where \( k \in \mathbb{Z} \) and \( k \neq 0 \).

Answer: 1


Question: Find the 43rd derivative of \( \sin(x^{13}+x^{15}) \) at \( x=0 \).

Answer: -\dfrac{43!}{2}


Question: Find the matrix $A$ with integer entries such that the set $\{x_1,x_2, \dots, x_d \in \mathbb{R} : |x_1| + |x_2| + \dots + |x_d| \leq 1 \}$ can be rewritten as $\{ u \in \mathbb{R}^d : Au \leq \textbf{1} \}$, where $\textbf{1}$ is a vector with each coordinate being 1.

Answer: A


Question: Determine the largest integer \( k \) such that the function \( f(x) = |x|^3 \) is \( C^k \) but not \( C^{k+1} \).

Answer: 2


Question: Let \( f(x) \) be a polynomial function satisfying the equation \( f(x) \cdot f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right) \). Given that \( f(4) = 65 \), find the value of \( f(6) \).

Answer: 217


Question: Evaluate the integral \( \int_\gamma \frac{1}{(z-1)^2(z^2+1)} \, dz \), where \( \gamma \) is a circle of radius 2 centered at 0, traversed counterclockwise. Use the Residue Theorem to find the value of the integral.

Answer: 0


Question: Find the limit \( \lim_{t \rightarrow \infty} tx(t) \) where \( x = x(t) \) is the least positive root of the equation \( x^2 + \frac{1}{2} = \cos(tx) \).

Answer: \dfrac{\pi}{3}


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

Answer: 0


Question: Evaluate the limit \( \lim_{T \to 0} U(T) \) where \( U(T) = \frac{1 \cdot e^{-\frac{\epsilon}{k_BT}} + 2 \cdot e^{-\frac{2\epsilon}{k_BT}} + 3 \cdot e^{-\frac{3\epsilon}{k_BT}}}{1 \cdot e^{-\frac{\epsilon}{k_BT}} + 1 \cdot e^{-\frac{2\epsilon}{k_BT}} + 1 \cdot e^{-\frac{3\epsilon}{k_BT}}} \), and \( \epsilon \) and \( k_B \) are constants. Prove that \( U(T=0K) = 1 \).

Answer: 1


Question: Let $u(x,y)$ be the real part of an entire function $f(z)=u(x,y)+iv(x,y)$ for $z=x+iy\in \mathbb{C}$. If $C$ is the positively oriented boundary of a rectangular region $R$ in $\mathbb{R}^{2}$, find the value of \(\int_{C} \left[\frac{\partial u}{\partial y}dx-\frac{\partial u}{\partial x}dy\right]\).\)\n\n(a) 1\n(b) 0\n(c) $\pi$\n(d) $2\pi$

Answer: B


Question: Let \( f(z) = \sum C_k z^k \) be an entire function such that for all \( |z| = R \), \( |f(z)| \leq M \). Prove that for each coefficient \( C_k \) in the power series expansion of \( f \), the inequality \( |C_k| \leq \frac{M}{R^k} \) holds.

Answer: |C_k| \leq \dfrac{M}{R^k}


Question: Let the function $F(x,y,z)$ satisfy the equations \( \frac{\partial{F}}{\partial{z}} = \frac{\partial^2{F}}{\partial{x}^2} + \frac{\partial^2{F}}{\partial{y}^2} \) and \( \frac{\partial^2{F}}{\partial{x}\partial{y}} = \frac{\partial^2{F}}{\partial{y}\partial{x}} \). Define \( G(\gamma,s,t) = F(\gamma + s, \gamma - s, At) \). Find the value of \( A \) such that \( \frac{\partial{G}}{\partial{t}} = \frac{\partial^2{G}}{\partial{\gamma}^2} + \frac{\partial^2{G}}{\partial{s}^2} \).

Answer: 2


Question: Let $G$ be a group and $H$ be a subgroup of the direct product $G^{|G|}$, where $|H| \geq 2^{|G|}$. Can $H$ be a cyclic group?

Answer: No


Question: Find the probability that for four independent random variables \(x_1, x_2, x_3, x_4\) uniformly distributed over the interval \([0, a]\), the inequality \(x_1 > x_2 + x_3 + x_4\) holds.

Answer: \dfrac{1}{24}


Question: Using Green's theorem, evaluate the line integral of the vector field \( F(x,y) = \left(e^y+\frac{1}{y+3}, xe^y-\frac{x+1}{(y+3)^2}\right) \) over the closed curve formed by the points \((-1,0)\), \((-1,2)\), \((0,1)\), and \((1,2)\). Determine if the integral over the triangular region enclosed by these points is zero.

Answer: 0


Question:
Does the convergence of a Taylor series centered at \( x_0 \) to a function \( f(x) \) over an interval \( I \) imply that \( f(x) \) is analytic throughout \( I \)?

A function \( f: (a,b) \subset \mathbb{R} \to \mathbb{R} \) is said to be analytic at \( x_0 \in (a,b) \) if there exists \( \delta > 0 \) such that the Taylor series of \( f(x) \) centered at \( x_0 \) converges in \( (x_0-\delta, x_0+\delta) \) and its sum is \( f(x) \).

Suppose the Taylor series of a function \( f \in C^{\infty} \) centered at \( x_0 \)
\[ \sum_{n \geq 0} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n \]
converges for all \( x \in (x_0-\xi, x_0+\xi) \) for some \( \xi > 0 \) and that
\[ f(x) = \sum_{n \geq 0} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n \quad \forall x \in (x_0-\xi, x_0+\xi) \]
Can you conclude that \( f(x) \) is analytic in the entire interval \( (x_0-\xi, x_0+\xi) \)?

Answer: Yes


Question: Find a primitive root modulo $5^{10}$. Provide a method to determine such a primitive root efficiently.

Answer: 2


Question: Find the least residue of \(3^{1.1 \cdot 10^{43}}\) modulo 95.

Answer: 66


Question: Suppose $Z(t)$ is a standard Brownian motion process with $Z(0)=0$. Calculate the probability $P(Z(3) > Z(2) > 0)$. Provide your answer as a single probability value.

Answer: \dfrac{1}{4}


Question: Suppose $X \sim N(0,1)$ and $Y \sim N(0,1)$, where $X$ and $Y$ are independent. Given that $\rho(x,y) = -0.72$, compute $\mathbb{E}(3X+Y|X-Y=1)$. Provide your answer as a single value.

Answer: 1


Question:
Evaluate the integral \( \int_{\Lambda}\left(\sin\frac{1}{z}+\sin^2\frac{1}{z}\right)dz \), where \( \Lambda \) is a circle centered at zero in the complex plane excluding zero. Choose the correct value from the following options:

(a) \( 0 \)
(b) \( \pi i \)
(c) \( 2\pi i \)
(d) \( 4\pi i \)

Answer: C


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{1}{z^2+25} \, dz \).

Answer: \dfrac{\pi}{5}


Question: Given the differential equation \(2x^2 y'' - 6xy' + 10y = 0\) and a known solution \(y_1 = x^2 \cos(\ln x)\), use the method of reduction of order to find a second linearly independent solution.

Answer: x^2 \sin(\ln x)


Question: Assume we have three sequences of positive real numbers, \((a_n)_{n\in \mathbb{N}}\), \((b_n)_{n\in \mathbb{N}}\), and \((c_n)_{n\in \mathbb{N}}\), such that \(\limsup_{n\to \infty}\frac{a_n}{b_n}<\infty\) and \(\lim_{n\to \infty}\frac{b_n}{c_n}=0\). Prove or disprove that \(\limsup_{n\to \infty}\frac{a_n}{c_n}=0\).

Answer: 0


Question: Evaluate the integral \( \int_0^\infty \operatorname{Prob} (\Phi(X) \geq u) \; du \), where \( X \) is a random variable with a standard normal distribution and \( \Phi \) is its distribution function.

Answer: \dfrac{1}{2}


Question: Determine the number of distinct values that the integral \( \oint_\gamma \frac{dz}{(z-a_1)(z-a_2)...(z-a_n)} \) can take, where \( \gamma \) is a simple closed curve not passing through any of the points \( a_1, a_2, \ldots, a_n \).

Answer: 2^n - 1


Question: Evaluate the double integral \( \int \int_D \frac{1}{\sqrt{x^2+y^2}} \, dx \, dy \), where \( D \) is the domain defined by \( 1 \le x^2+y^2 \le 4 \).

Answer: 2\pi


Question: Determine whether the limit \( \lim_{(x,y)\to (0,0)} \frac {x^3y^2}{x^4+y^6} \) exists.

Answer: 0


Question: Compute the limit \( \lim\limits_{(x,y) \to(0,0)} \frac{e^{x^2-y^2} -1}{ x - y} \) without using Taylor series.

Answer: 0


Question: Determine the values of \(m\) and \(k\) for which the series \(\sum_{n=1}^\infty \frac{\sqrt[m]{n!}}{\sqrt[k]{(2n)!}}\) converges.

Answer: k \leq 2m


Question: Evaluate the limit \( \lim_{n \to \infty} \frac{m(n)}{n} \), where \( m(n) \) is the number of terms in the sequence \( \{2^1, 2^2, \ldots, 2^n\} \) that contain the digit 1 when written in base 10.

Answer: 1


Question: The curve \( y = \sin(4x) \cdot 10e^{-x/2} \) for \( x \ge 0 \) has stationary points labeled as \( T_1, T_2, T_3, \ldots \). The x-coordinates of \( T_1 \) and \( T_2 \) are \( 0.362 \) and \( 1.147 \) respectively. Given that the x-coordinate of \( T_n \) is greater than 25, find the least possible value of \( n \).

Answer: 33


Question: Determine the limit of the sequence \( \left\{ \frac{(n!)^3}{(3n)!} \right\} \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to 0^+} \left(1-a^x\right)^{\frac{1}{x}} \quad (0 < a < 1) \]

Answer: 0


Question: Consider a function \( f: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) such that for all vectors \( a, b \in \mathbb{R}^3 \), the equation \( f(a) \times f(b) = a \times b \) holds, where "\(\times\)" denotes the cross product. Must \( f \) be a linear mapping?

Answer: Yes


Question: Find the residue of \(\frac{e^z-1}{\sin^2(z)}\) at \(z = k\pi\), where \(k \in \mathbb{Z}\).

Answer: e^{k\pi}


Question: Find the directional derivative of the function \( f(x,y) = \frac{x^3}{x^2 + y^2} \) for \((x,y) \neq (0,0)\) and \( f(0,0) = 0 \) in the direction of \((1,1)\) at the point \((0,0)\).

Answer: \dfrac{\sqrt{2}}{4}


Question: Determine whether the integral \( \int_{0}^{+\infty} \frac{x \log(x)}{(x^{2}+1)^{2}} \, dx \) converges. If it does, find its value.

Answer: 0


Question:
Determine whether the following statement is true:
\[ (-1.2)^{1.6} = \sqrt[10]{(-1.2)^{16}}. \]

Answer: True


Question: Let $k$ be an integer. If the equation \((x-1)|x+1|=x + \frac{k}{2020}\) has 3 distinct real roots, how many different possible values of $k$ are there?

Answer: 4544


Question:
Determine the value of $A$ such that the function \( f(x) \) is differentiable at \( x = 0 \), where
\[
f(x)=
\begin{cases}
\frac{e^{3x}-3e^x+2}{x^2} & :x \neq 0\\
A & :x=0
\end{cases}
\]

Answer: 3


Question: Suppose \( \mu \) is a finite measure and for some \( \gamma > 0 \), we have \( \sup_n \int |f_n|^{1 + \gamma}d\mu < \infty \). Does it follow that the sequence \( \{f_n\} \) is uniformly integrable?

Answer: Yes


Question: Given the joint probability density function \(f_{X,Y}(x,y)=\begin{cases}12(y-x^2),&0<x<y<1\\0,&\text{elsewhere.}\end{cases}\), compute the probability \(P(Y > 1/2)\).

Answer: \dfrac{41}{48}


Question: Evaluate the integral \( \int_0^{2\pi}\frac{\cos^2 3\theta\, d\theta}{5-4\cos 2\theta} \).

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


Question:
Find parametric equations of the straight line tangent to the space curve \(r(t)=(8t^2+63t+46)i+(2t^3-98t-9)j+\frac{70\sqrt3}{\pi}(1+2\cos(\frac{4\pi t}{21}))k\) at the point \(P(-3,-9,0)\) on the curve. Express the tangent line in the form:
\[x(s) = -3 - 49s\]
\[y(s) = -9 + 196s\]
\[z(s) = -40s\]

Answer:
\begin{aligned}
x(s) &= -3 - 49s, \\
y(s) &= -9 + 196s, \\
z(s) &= -40s.
\end{aligned}


Question: Evaluate the limit: \[ \lim_{y\to\infty} \int^1_0 yx^y f(x) \,dx \] where $f$ is a differentiable function on $[0, 1]$ with $f(1) = 1$ and $f'$ is continuous on $[0, 1]$.

Answer: 1


Question: Consider the group \( G = \mathbb{Z}/60\mathbb{Z} \times \mathbb{Z}/45\mathbb{Z} \times \mathbb{Z}/12\mathbb{Z} \times \mathbb{Z}/36\mathbb{Z} \). Determine the number of elements in \( G \) that have order 2.

Answer: 7


Question: Given the functional equation \( f(f(x))[1+f(x)] = -f(x) \), find the value of \( f(3) \).

Answer: -\dfrac{3}{4}


Question: Calculate the product: \[ \prod_{k=0}^{2001} \left( 1 + \left( \frac{1+i}{2} \right)^{2^k} \right). \]

Answer: (1 + i)\left(1 - 2^{-2^{2001}}\right)


Question: Determine the number of tangents that can be drawn from the point \(\left(\frac{-1}{2},0\right)\) to the curve \(y = e^{\{x\}}\), where \(\{x\}\) denotes the fractional part of \(x\).

Answer: 1


Question:
Let \(f\) be a function on \([0, \infty)\) such that for each point on its graph, \((x, y) = (y^2, y)\). At how many points must such a function \(f\) have a limit?

(A) 1
(B) 2
(C) 3
(D) 4
(E) Infinitely many

Answer: A


Question: Find the angle between the one-sided tangents to the curve \( f(x) = \sqrt{1-e^{-x^2}} \) at the point \( x = 0 \).

Answer: \dfrac{\pi}{2}


Question: Evaluate the limit: $$\lim_{n\to\infty}\left(\dfrac{(n+1)^{n+1}\cdot n!}{2^{n!\cdot n}}\right).$$

Answer: 0