Let $\{\epsilon_n\}^\infty_{n=1}$ be a sequence of positive reals with $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$. Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right) \]

Let $D$ and $E$ be points on respectively $[AB]$ and $[AC]$ of $\triangle ABC$ where $|AB|=|AC|$, $m(\widehat{BAC})=40^\circ$. Let $F$ be a point on $BC$ such that $C$ is between $B$ and $F$. If $|BE|=|CF|$, $|AD|=|AE|$, and $m(\widehat{BEC})=60^\circ$, then what is $m(\widehat{DFB})$ ? $ \textbf{(A)}\ 45^\circ \qquad\textbf{(B)}\ 40^\circ \qquad\textbf{(C)}\ 35^\circ \qquad\textbf{(D)}\ 30^\circ \qquad\textbf{(E)}\ 25^\circ $

In some computer languages (such as APL), when there are no parentheses in an algebraic expression, the operations are grouped from left to right. Thus, $ a \times b \minus{} c$ in such languages means the same as $ a(b\minus{}c)$ in ordinary algebraic notation. If $ a \div b \minus{} c \plus{} d$ is evaluated in such a language, the result in ordinary algebraic notation would be $ \textbf{(A)}\ \frac{a}{b} \minus{} c \plus{} d \qquad \textbf{(B)}\ \frac{a}{b} \minus{} c \minus{} d \qquad \textbf{(C)}\ \frac{d \plus{} c \minus{} b}{a} \qquad \textbf{(D)}\ \frac{a}{b \minus{} c \plus{} d} \qquad \textbf{(E)}\ \frac{a}{b\minus{}c\minus{}d}$

Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value? $ \textbf{(A) } \frac{1}{36} \qquad\textbf{(B) } \frac{7}{72} \qquad\textbf{(C) } \frac{1}{9} \qquad\textbf{(D) }\frac{5}{36}\qquad\textbf{(E) }\frac{1}{6} \qquad $

Prove that in a finite group G the number of subgroups with index n is at most $| G |^{2 \log_2 n}$.

Find all positive integers $n$ such that $2^n-n^2+1$ is a perfect square.

Let \( I \) and \( M \) be the incenter and the centroid of a scalene triangle \( ABC \), respectively. A line passing through point \( I \) parallel to \( BC \) intersects \( AC \) and \( AB \) at points \( E \) and \( F \), respectively. Reconstruct triangle \( ABC \) given only the marked points \( E, F, I, \) and \( M \). [i]Proposed by Hryhorii Filippovskyi[/i]

Rectangle $EFGH$ with side lengths $8$, $9$ lies inside rectangle $ABCD$ with side lengths $13$, $14$, with their corresponding sides parallel. Let $\ell_A, \ell_B, \ell_C, \ell_D$ be the lines connecting $A,B,C,D$, respectively, with the vertex of $EFGH$ closest to them. Let $P = \ell_A \cap \ell_B$, $Q = \ell_B \cap \ell_C$, $R = \ell_C \cap \ell_D$, and $S = \ell_D \cap \ell_A$. Suppose that the greatest possible area of quadrilateral $PQRS$ is $\frac{m}{n}$, for relatively prime positive integers $m$ and $n$. Find $100m+n$. [i]Proposed by Yannick Yao[/i]

Find all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ such that $$f(x+f(y+1))+f(xy)=f(x+1)(f(y)+1)$$ for any $x,y$ integers.

Let $f(x)$ be a polynomial with rational coefficients, and let $\alpha$ be a real number. If \[\alpha^3-2019\alpha=(f(\alpha))^3-2019f(\alpha)=2021,\] prove that $(f^n(\alpha))^3-2019f^n(\alpha)=2021$ for any positive integer $n$. (Here, we define $f^n(x)=\underbrace{f(f(f\cdots f}_{n\text{ times}}(x)\cdots ))$.)

Call a positive integer $x$ a leader if there exists a positive integer $n$ such that the decimal representation of $x^n$ starts ([u]not ends[/u]) with $2012$. For example, $586$ is a leader since $586^3 =201230056$. How many leaders are there in the set $\{1, 2, 3, ..., 2012\}$?

Let $n \geq 2$ be an integer. Let $S$ be a subset of $\{1,2,\dots,n\}$ such that $S$ neither contains two elements one of which divides the other, nor contains two elements which are coprime. What is the maximal possible number of elements of such a set $S$?

Show that no non-zero integers $a$, $b$, $x$, $y$ satisfy $$ \begin{cases} a x - b y = 16,\\ a y + b x = 1. \end{cases} $$

Prove that for all integers $k\geq 2,$ the number $k^{k-1}-1$ is divisible by $(k-1)^2.$

On the blackboard, there are numbers $1, 2, \dots , 100$. At each move, Bob erases arbitrary two numbers $a$ and $b$, where $a \ge b > 0$, and writes the single number $\lfloor{a/b}\rfloor$. After $99$ such moves the blackboard will contain a single number. What is its maximum possible value? (Reminder that $\lfloor{x}\rfloor$ is the maximum integer not exceeding $x$.)

Prove that there is a unique $1000$-digit number $N$ in base $2022$ with the following properties: [list=1] [*] All of the digits of $N$ (in base $2022$) are $1$’s or $2$’s, and [/*] [*] $N$ is a multiple of the base-$10$ number $2^{1000}$. [/*] [/list] (Note that you must prove both that such a number exists and that there is not more than one such number. You do not have to write down the number! In fact, please don’t!)

This graph depicts the torque output of a hypothetical gasoline engine as a function of rotation frequency. The engine is incapable of running outside of the graphed range. [asy] size(200); draw((0,0)--(10,0)--(10,7)--(0,7)--cycle); for (int i=1;i<10;++i) { draw((i,0)--(i,7),dashed+linewidth(0.5)); } for (int j=1;j<7;++j) { draw((0,j)--(10,j),dashed+linewidth(0.5)); } draw((0,0)--(0,-0.3)); draw((4,0)--(4,-0.3)); draw((8,0)--(8,-0.3)); draw((0,0)--(-0.3,0)); draw((0,2)--(-0.3,2)); draw((0,4)--(-0.3,4)); draw((0,6)--(-0.3,6)); label("0",(0,-0.5),S); label("1000",(4,-0.5),S); label("2000",(8,-0.5),S); label("0",(-0.5,0),W); label("10",(-0.5,2),W); label("20",(-0.5,4),W); label("30",(-0.5,6),W); label("I",(1,-1.5),S); label("II",(6,-1.5),S); label("III",(9,-1.5),S); label(scale(0.95)*"Engine Revolutions per Minute",(5,-3.5),N); label(scale(0.95)*rotate(90)*"Output Torque (Nm)",(-1.5,3),W); path A=(0.9,2.7)--(1.213, 2.713)-- (1.650, 2.853)-- (2.087, 3)-- (2.525, 3.183)-- (2.963, 3.471)-- (3.403, 3.888)-- (3.823, 4.346)-- (4.204, 4.808)-- (4.565, 5.277)-- (4.945, 5.719)-- (5.365, 6.101)-- (5.802, 6.298)-- (6.237, 6.275)-- (6.670, 6.007)-- (7.101, 5.600)-- (7.473, 5.229)-- (7.766, 4.808)-- (8.019, 4.374)-- (8.271, 3.894)-- (8.476, 3.445)-- (8.568, 2.874)-- (8.668, 2.325)-- (8.765, 1.897)-- (8.794, 1.479)--(8.9,1.2); draw(shift(0.1*right)*shift(0.2*down)*A,linewidth(3)); [/asy] At what engine RPM (revolutions per minute) does the engine produce maximum power? (A) $\text{I}$ (B) At some point between $\text{I}$ and $\text{II}$ (C) $\text{II}$ (D) At some point between $\text{II}$ and $\text{III}$ (E) $\text{III}$

Let $a$, $b$, $c$ be the three sides of a triangle, and let $\alpha$, $\beta$, $\gamma$, be the angles opposite them. If $a^2+b^2=1989c^2$, find \[ \frac{\cot \gamma}{\cot \alpha+\cot \beta}. \]

Amy and Bob choose numbers from $0,1,2,\cdots,81$ in turn and Amy choose the number first. Every time the one who choose number chooses one number from the remaining numbers. When all $82$ numbers are chosen, let $A$ be the sum of all the numbers Amy chooses, and let $B$ be the sum of all the numbers Bob chooses. During the process, Amy tries to make $\gcd(A,B)$ as great as possible, and Bob tries to make $\gcd(A,B)$ as little as possible. Suppose Amy and Bob take the best strategy of each one, respectively, determine $\gcd(A,B)$ when all $82$ numbers are chosen.

Let $f(x)$ be a rational function with complex coefficients whose denominator does not have multiple roots. Let $u_0, u_1,... , u_n$ be the complex roots of $f$ and $w_1, w_2,..., w_m$ be the roots of $f'$. Suppose that $u_0$ is a simple root of $f$. Prove that \[ \sum_{k=1}^m \frac{1}{w_k - u_0} = 2\sum_{k = 1}^n\frac{1}{u_k - u_0}.\]

$N$ cossacks split into $3$ groups to discuss various issues with their friends. Cossack Taras moved from the first group to the second, cossack Andriy moved from the second to the third, and cossack Ostap - from the third group to the first. It turned out that the average height of the cossacks in the first group decreased by $8$ cm, while in the second and third groups it increased by $5$ cm and $8$ cm, respectively. What is $N$, if it is known that there were $9$ cossacks in the first group?

Let $ABCD$ be a parallelogram. Let $M$ be the midpoint of the arc $AC$ containing $B$ of the circumcircle of $ABC$ . Let $E$ be a point on segment $AD$ and $F$ a point on segment $CD$ such that $ME=MD=MF$. Show that $BMEF$ is cyclic. [i]Proposed by A. Tereshin[/i]

Two rational numbers \(\tfrac{m}{n}\) and \(\tfrac{n}{m}\) are written on a blackboard, where \(m\) and \(n\) are relatively prime positive integers. At any point, Evan may pick two of the numbers \(x\) and \(y\) written on the board and write either their arithmetic mean \(\tfrac{x+y}{2}\) or their harmonic mean \(\tfrac{2xy}{x+y}\) on the board as well. Find all pairs \((m,n)\) such that Evan can write 1 on the board in finitely many steps. [i]Proposed by Yannick Yao[/i]

The polynomial $ (x\plus{}y)^9$ is expanded in decreasing powers of $ x$. The second and third terms have equal values when evaluated at $ x\equal{}p$ and $ y\equal{}q$, where $ p$ and $ q$ are positive numbers whose sum is one. What is the value of $ p$? $ \textbf{(A)}\ 1/5 \qquad \textbf{(B)}\ 4/5 \qquad \textbf{(C)}\ 1/4 \qquad \textbf{(D)}\ 3/4 \qquad \textbf{(E)}\ 8/9$

Let $C$ be the curve expressed by $x=\sin t,\ y=\sin 2t\ \left(0\leq t\leq \frac{\pi}{2}\right).$ (1) Express $y$ in terms of $x$. (2) Find the area of the figure $D$ enclosed by the $x$-axis and $C$. (3) Find the volume of the solid generated by a rotation of $D$ about the $y$-axis.