Call a positive integer number $n$ [i]poor[/i] if equation \[x_1x_2 \dots x_{101}=(n-x_1)(n-x_2)\dots (n-x_{101}) \] has no solutions in positive integers $1<x_i<n$. Does there exist poor number, which has more than $100 \ 000$ distinct prime divisors?

Let $ABCD$ be a quadrilateral. The circumcircle of the triangle $ABC$ intersects the sides $CD$ and $DA$ in the points $P$ and $Q$ respectively, while the circumcircle of $CDA$ intersects the sides $AB$ and $BC$ in the points $R$ and $S$. The lines $BP$ and $BQ$ intersect the line $RS$ in the points $M$ and $N$ respectively. Prove that the points $M, N, P$ and $Q$ lie on the same circle.

Draw all the diagonals of a convex polygon of $10$ sides. They divide their angles into $80$ parts. It is known that at least $59$ of those parts are equal. Determine the largest number of distinct values ​​among the $ 80$ angles of division and how many times each of those values ​​occurs.

Let $a_1,\ldots ,a_{10}$ be distinct positive integers, all at least $3$ and with sum $678$. Does there exist a positive integer $n$ such that the sum of the $20$ remainders of $n$ after division by $a_1,a_2,\ldots ,a_{10},2a_1,2a_2,\ldots ,2a_{10}$ is $2012$?

Show that there is a digit unequal to 2 in the decimal represesentation of $ \sqrt [3]{3}$ between the $ 1000000$-th und $ 3141592$-th position after decimal point.

Prove that if the sides $ a $, $ b $, $ c $ of a triangle satisfy the inequality $$a < b < c$$then the angle bisectors $ d_a $, $ d_b $, $ d_c $ of opposite angles satisfy the inequality $$ d_a > d_b > d_c.$$

Let $0 < a < \frac{\pi}{2}$ and $x_1,x_2,...,x_n$ be real numbers such that $\sin x_1 + \sin x_2 +... + \sin x_n \ge n \cdot sin a $. Prove that $\sin (x_1 - a) + \sin (x_2 - a) + ... + \sin (x_n - a) \ge 0$ .

Consider a directed graph $G$ with $n$ vertices, where $1$-cycles and $2$-cycles are permitted. For any set $S$ of vertices, let $N^{+}(S)$ denote the out-neighborhood of $S$ (i.e. set of successors of $S$), and define $(N^{+})^k(S)=N^{+}((N^{+})^{k-1}(S))$ for $k\ge2$. For fixed $n$, let $f(n)$ denote the maximum possible number of distinct sets of vertices in $\{(N^{+})^k(X)\}_{k=1}^{\infty}$, where $X$ is some subset of $V(G)$. Show that there exists $n>2012$ such that $f(n)<1.0001^n$. [i]Linus Hamilton.[/i]

Let $ABC$ be a triangle and let $D, E$ are points on its circumscribed circle, such that $D$ lies on arc $AB, E$ lies on arc $AC$ (smaller arcs) and $BD \parallel CE$ . Let the point F be the intersection of the lines $DA$ and $CE$, and the intersection of the lines $EA$ and $BD$ is $G$. Let $P$ be the second intersection of the circumscribed circles of $\vartriangle ABG$ and $\vartriangle ACF$. Prove that the line$ AP$ passes through the mid point of the side $BC$.

Six thousand points are marked on a circle, and they are colored using 10 colors in such a way that within every group of 100 consecutive points all the colors are used. Determine the least positive integer $ k$ with the following property: In every coloring satisfying the condition above, it is possible to find a group of $ k$ consecutive points in which all the colors are used.

Let $ABC$ be an acute triangle. Let $M$ be the midpoint of side $BC$, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively. Suppose that the common external tangents to the circumcircles of triangles $BME$ and $CMF$ intersect at a point $K$, and that $K$ lies on the circumcircle of $ABC$. Prove that line $AK$ is perpendicular to line $BC$. [i]Kevin Cong[/i]

Determine the number of ways of colouring a $10\times 10$ square board using two colours black and white such that each $2\times 2$ subsquare contains 2 black squares and 2 white squares.

Let $a$ and $b$ be positive integers such that $2a - 9b + 18ab = 2018$. Find $b - a$.

[b]p1.[/b] There is a string of numbers $1234567891023...910134 ...91012...$ that concatenates the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, then $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $1$, then $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $1$, $2$, and so on. After $10$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, the string will be concatenated with $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$ again. What is the $2021$st digit? [b]p2.[/b] Bob really likes eating rice. Bob starts eating at the rate of $1$ bowl of rice per minute. Every minute, the number of bowls of rice Bob eats per minute increases by $1$. Given there are $78$ bowls of rice, find number of minutes Bob needs to finish all the rice. [b]p3.[/b] Suppose John has $4$ fair coins, one red, one blue, one yellow, one green. If John flips all $4$ coins at once, the probability he will land exactly $3$ heads and land heads on both the blue and red coins can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, Find $a + b$. [b]p4.[/b] Three of the sides of an isosceles trapezoid have lengths $1$, $10$, $20$ Find the sum of all possible values of the fourth side. [b]p5.[/b] An number two-three-delightful if and only if it can be expressed as the product of $2$ consecutive integers larger than $1$ and as the product of $3$ consecutive integers larger than $1$. What is the smallest two-three-delightful number? [b]p6.[/b] There are $3$ students total in Justin's online chemistry class. On a $100$ point test, Justin's two classmates scored $4$ and $7$ points. The teacher notices that the class median score is equal to $gcd(x, 42)$, where the positive integer $x$ is Justin's score. Find the sum of all possible values of Justin's score. [b]p7.[/b] Eddie's gym class of $10$ students decides to play ping pong. However, there are only $4$ tables and only $2$ people can play at a table. If $8$ students are randomly selected to play and randomly assigned a partner to play against at a table, the probability that Eddie plays against Allen is $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, Find $a + b$. [b]p8.[/b] Let $S$ be the set of integers $k$ consisting of nonzero digits, such that $300 < k < 400$ and $k - 300$ is not divisible by $11$. For each $k$ in $S$, let $A(k)$ denote the set of integers in $S$ not equal to $k$ that can be formed by permuting the digits of $k$. Find the number of integers $k$ in $S$ such that $k$ is relatively prime to all elements of $A(k)$. [b]p9.[/b] In $\vartriangle ABC$, $AB = 6$ and $BC = 5$. Point $D$ is on side $AC$ such that $BD$ bisects angle $\angle ABC$. Let $E$ be the foot of the altitude from $D$ to $AB$. Given $BE = 4$, find $AC^2$. [b]p10.[/b] For each integer $1 \le n \le 10$, Abe writes the number $2^n + 1$ on a blackboard. Each minute, he takes two numbers $a$ and $b$, erases them, and writes $\frac{ab-1}{a+b-2}$ instead. After $9$ minutes, there is one number $C$ left on the board. The minimum possible value of $C$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p, q$. Find $p + q$. [b]p11.[/b] Estimation (Tiebreaker) Let $A$ and $B$ be the proportions of contestants that correctly answered Questions $9$ and $10$ of this round, respectively. Estimate $\left \lfloor \dfrac{1}{(AB)^2} \right \rfloor$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f_n(x,\ y)=\frac{n}{r\cos \pi r+n^2r^3}\ (r=\sqrt{x^2+y^2})$, $I_n=\int\int_{r\leq 1} f_n(x,\ y)\ dxdy\ (n\geq 2).$ Find $\lim_{n\to\infty} I_n.$ [i]2009 Tokyo Institute of Technology, Master Course in Mathematics[/i]

Let $x,y,z$ be real numbers with $x+y+z=0$. Prove that \[ |\cos x |+ |\cos y| +| \cos z | \geq 1 . \] [i]Viorel Vajaitu, Bogdan Enescu[/i]

Call a convex quadrilateral [i]angle-Pythagorean[/i] if the degree measures of its angles are integers $w\leq x \leq y \leq z$ satisfying $$w^2+x^2+y^2=z^2.$$ Determine the maximum possible value of $x+y$ for an angle-Pythagorean quadrilateral.

Let $ (a_{n})_{n\ge 1}$ be a sequence of positive integers satisfying $ (a_{m},a_{n}) = a_{(m,n)}$ (for all $ m,n\in N^ +$). Prove that for any $ n\in N^ + ,\prod_{d|n}{a_{d}^{\mu (\frac {n}{d})}}$ is an integer. where $ d|n$ denotes $ d$ take all positive divisors of $ n.$ Function $ \mu (n)$ is defined as follows: if $ n$ can be divided by square of certain prime number, then $ \mu (1) = 1;\mu (n) = 0$; if $ n$ can be expressed as product of $ k$ different prime numbers, then $ \mu (n) = ( - 1)^k.$

Consider a fixed circle $\Gamma$ with three fixed points $A, B,$ and $C$ on it. Also, let us fix a real number $\lambda \in(0,1)$. For a variable point $P \not\in\{A, B, C\}$ on $\Gamma$, let $M$ be the point on the segment $CP$ such that $CM =\lambda\cdot CP$ . Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle. [i]Proposed by Jack Edward Smith, UK[/i]

Let $\angle XOY = \frac{\pi}{2}$; $P$ is a point inside $\angle XOY$ and we have $OP = 1; \angle XOP = \frac{\pi}{6}.$ A line passes $P$ intersects the Rays $OX$ and $OY$ at $M$ and $N$. Find the maximum value of $OM + ON - MN.$

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function that satisfies the conditions: $ \text{(i)}\quad \lim_{x\to\infty} (f\circ f) (x) =\infty =-\lim_{x\to -\infty} (f\circ f) (x) $ $ \text{(ii)}\quad f $ has Darboux’s property [b]a)[/b] Prove that the limits of $ f $ at $ \pm\infty $ exist. [b]b)[/b] Is possible for the limits from [b]a)[/b] to be finite?

Find all natural integers $n$ such that $(n^3 + 39n - 2)n! + 17\cdot 21^n + 5$ is a square.

There are exactly $77,000$ ordered quadruples $(a,b,c,d)$ such that $\gcd(a,b,c,d)=77$ and $\operatorname{lcm}(a,b,c,d)=n$. What is the smallest possible value of $n$? $\textbf{(A)}\ 13,860 \qquad \textbf{(B)}\ 20,790 \qquad \textbf{(C)}\ 21,560 \qquad \textbf{(D)}\ 27,720 \qquad \textbf{(E)}\ 41,580$

Circles $\Gamma_1$ and $\Gamma_2$ have centers $O_1$ and $O_2$ and intersect at $P$ and $Q$. A line through $P$ intersects $\Gamma_1$ and $\Gamma_2$ at $A$ and $B$, respectively, such that $AB$ is not perpendicular to $PQ$. Let $X$ be the point on $PQ$ such that $XA=XB$ and let $Y$ be the point within $AO_1 O_2 B$ such that $AYO_1$ and $BYO_2$ are similar. Prove that $2\angle{O_1 AY}=\angle{AXB}$. [i]Author: Matthew Brennan[/i]

What is the degree measure of the acute angle formed by lines with slopes $2$ and $\tfrac{1}{3}$? $\textbf{(A)}~30\qquad\textbf{(B)}~37.5\qquad\textbf{(C)}~45\qquad\textbf{(D)}~52.5\qquad\textbf{(E)}~60$