Circles $S_1$ and $S_2$ intersect at points $M$ and $N$. Prove that if vertices $A$ and $ C$ of some rectangle $ABCD$ lie on the circle $S_1$, and the vertices $B$ and $D$ lie on the circle $S_2$, then the point of intersection of its diagonals lies on the line $MN$.

Find all sets X of points in a plane, not all collinear, such that: For any two distinct circumferences, each contains three points of X, its intersection points are points of X.

How many integers $0\leq x < 121$ are there such that $x^5+5x^2 + x + 1 \equiv 0 \pmod{121}$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

Solve in the real numbers the equation $ (n+1)^x+(n+3)^x+\left( n^2+2n\right)^x=n^x+(n+2)^x+\left( n^2+4n+3\right)^x, $ wher $ n\ge 2 $ is a fixed natural number.

A straight road consists of green and red segments in alternating colours, the first and last segment being green. Suppose that the lengths of all segments are more than a centimeter and less than a meter, and that the length of each subsequent segment is larger than the previous one. A grasshopper wants to jump forward along the road along these segments, stepping on each green segment at least once an without stepping on any red segment (or the border between neighboring segments). Prove that the grasshopper can do this in such a way that among the lengths of his jumps no more than $8$ different values occur. [i]Proposed by T. Korotchenko[/i]

In the given figure hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$? $\textbf{(A) }6\sqrt{2}\qquad\textbf{(B) }9\qquad\textbf{(C) }12\qquad\textbf{(D) }9\sqrt{2}\qquad\textbf{(E) }32$ [asy] draw((-4,6*sqrt(2))--(4,6*sqrt(2))); draw((-4,-6*sqrt(2))--(4,-6*sqrt(2))); draw((-8,0)--(-4,6*sqrt(2))); draw((-8,0)--(-4,-6*sqrt(2))); draw((4,6*sqrt(2))--(8,0)); draw((8,0)--(4,-6*sqrt(2))); draw((-4,6*sqrt(2))--(4,6*sqrt(2))--(4,8+6*sqrt(2))--(-4,8+6*sqrt(2))--cycle); draw((-8,0)--(-4,-6*sqrt(2))--(-4-6*sqrt(2),-4-6*sqrt(2))--(-8-6*sqrt(2),-4)--cycle); label("$I$",(-4,8+6*sqrt(2)),dir(100)); label("$J$",(4,8+6*sqrt(2)),dir(80)); label("$A$",(-4,6*sqrt(2)),dir(280)); label("$B$",(4,6*sqrt(2)),dir(250)); label("$C$",(8,0),W); label("$D$",(4,-6*sqrt(2)),NW); label("$E$",(-4,-6*sqrt(2)),NE); label("$F$",(-8,0),E); draw((4,8+6*sqrt(2))--(4,6*sqrt(2))--(4+4*sqrt(3),4+6*sqrt(2))--cycle); label("$K$",(4+4*sqrt(3),4+6*sqrt(2)),E); draw((4+4*sqrt(3),4+6*sqrt(2))--(8,0),dashed); label("$H$",(-4-6*sqrt(2),-4-6*sqrt(2)),S); label("$G$",(-8-6*sqrt(2),-4),W); label("$32$",(-10,-8),N); label("$18$",(0,6*sqrt(2)+2),N); [/asy]

Betty Lou and Peggy Sue take turns flipping switches on a $100 \times 100$ grid. Initially, all switches are "off". Betty Lou always flips a horizontal row of switches on her turn; Peggy Sue always flips a vertical column of switches. When they finish, there is an odd number of switches turned "on'' in each row and column. Find the maximum number of switches that can be on, in total, when they finish.

Let $n, m$ be positive integers. A set $S$ of positive integers is called $(n, m)$-good, if: (1) $m \in S$; (2) for all $a\in S$, all divisors of $a$ are also in $S$; (3) for all distinct $a, b \in S$, $a^n+b^n \in S$. For which $(n, m)$, the only $(n, m)$-good set is $\mathbb{N}$?

Let $(a_n)$ be defined by: $$ a_1 = 2, \qquad a_{n+1} = a_n^3 - a_n + 1 $$ Consider positive integers $n,p$, where $p$ is an odd prime. Prove that if $p | a_n$, then $p > n$.

The number $2.5081081081081\ldots$ can be written as $\frac{m}{n}$ where $m$ and $n$ are natural numbers with no common factors. Find $m + n$.

Let $n$ be a positive integer and $(x_1, x_2, ..., x_{2n})$, $x_i = 0$ or $1, i = 1, 2, ... , 2n$ be a sequence of $2n$ integers. Let $S_n$ be the sum $S_n = x_1x_2 + x_3x_4 + ... + x_{2n-1}x_{2n}$. If $O_n$ is the number of sequences such that $S_n$ is odd and $E_n$ is the number of sequences such that $S_n$ is even, prove that $$\frac{O_n}{E_n}=\frac{2^n - 1}{2^n + 1}$$

Alice and Bob play the following game. They write some fractions of the form $1/n$, where $n{}$ is positive integer, onto the blackboard. The first move is made by Alice. Alice writes only one fraction in each her turn and Bob writes one fraction in his first turn, two fractions in his second turn, three fractions in his third turn and so on. Bob wants to make the sum of all the fractions on the board to be an integer number after some turn. Can Alice prevent this? [i]Andrey Arzhantsev[/i]

Mark writes the expression $\sqrt{d}$ for each positive divisor $d$ of $8!$ on the board. Seeing that these expressions might not be worth points on HMMT, Rishabh simplifies each expression to the form $a \sqrt{b} $ where $a$ and $b$ are integers such that $b$ is not divisible by the square of a prime number. (For example, $\sqrt{20}$, $\sqrt{16}$, and $\sqrt{6}$ simplify to $2\sqrt5$, $4\sqrt1$, and $1\sqrt6$, respectively.) Compute the sum of $a+b$ across all expressions that Rishabh writes.

For every algebraic integer $\alpha$ define its positive degree $\text{deg}^+(\alpha)$ to be the minimal $k\in\mathbb{N}$ for which there exists a $k\times k$ matrix with non-negative integer entries with eigenvalue $\alpha$. Prove that for any $n\in\mathbb{N}$, every algebraic integer $\alpha$ with degree $n$ satisfies $\text{deg}^+(\alpha)\le 2n$.

Determine all positive integers $n$ such that the following statement holds: If a convex polygon with with $2n$ sides $A_1 A_2 \ldots A_{2n}$ is inscribed in a circle and $n-1$ of its $n$ pairs of opposite sides are parallel, which means if the pairs of opposite sides \[(A_1 A_2, A_{n+1} A_{n+2}), (A_2 A_3, A_{n+2} A_{n+3}), \ldots , (A_{n-1} A_n, A_{2n-1} A_{2n})\] are parallel, then the sides \[ A_n A_{n+1}, A_{2n} A_1\] are parallel as well.

In rectangle $ABCD$, point $M$ is the midpoint of $AB$ and $P$ is a point on side $BC$. The perpendicular bisector of $MP$ intersects side $DA$ at point $X$. Given that $AB = 33$ and $BC = 56$, find the least possible value of $MX$. [i]Proposed by Michael Tang[/i]

Expanding $(1+0.2)^{1000}$ by the binomial theorem and doing no further manipulation gives \begin{eqnarray*} &\ & \binom{1000}{0}(0.2)^0+\binom{1000}{1}(0.2)^1+\binom{1000}{2}(0.2)^2+\cdots+\binom{1000}{1000}(0.2)^{1000}\\ &\ & = A_0 + A_1 + A_2 + \cdots + A_{1000}, \end{eqnarray*} where $A_k = \binom{1000}{k}(0.2)^k$ for $k = 0,1,2,\ldots,1000$. For which $k$ is $A_k$ the largest?

[url=https://artofproblemsolving.com/community/c677808][b]Cyprus IMO TST 2018[/b][/url] [url=https://artofproblemsolving.com/community/c6h1666662p10591751][b]Problem 1.[/b][/url] Determine all integers $n \geq 2$ for which the number $11111$ in base $n$ is a perfect square. [url=https://artofproblemsolving.com/community/c6h1666663p10591753][b]Problem 2.[/b][/url] Consider a trapezium $AB \Gamma \Delta$, where $A\Delta \parallel B\Gamma$ and $\measuredangle A = 120^{\circ}$. Let $E$ be the midpoint of $AB$ and let $O_1$ and $O_2$ be the circumcenters of triangles $AE \Delta$ and $BE\Gamma$, respectively. Prove that the area of the trapezium is equal to six time the area of the triangle $O_1 E O_2$. [url=https://artofproblemsolving.com/community/c6h1666660p10591747][b]Problem 3.[/b][/url] Find all triples $(\alpha, \beta, \gamma)$ of positive real numbers for which the expression $$K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma}$$obtains its minimum value. [url=https://artofproblemsolving.com/community/c6h1666661p10591749][b]Problem 4.[/b][/url] Let $\Lambda= \{1, 2, \ldots, 2v-1,2v\}$ and $P=\{\alpha_1, \alpha_2, \ldots, \alpha_{2v-1}, \alpha_{2v}\}$ be a permutation of the elements of $\Lambda$. (a) Prove that $$\sum_{i=1}^v \alpha_{2i-1}\alpha_{2i} \leq \sum_{i=1}^v (2i-1)2i.$$(b) Determine the largest positive integer $m$ such that we can partition the $m\times m$ square into $7$ rectangles for which every pair of them has no common interior points and their lengths and widths form the following sequence: $$1,2,3,4,5,6,7,8,9,10,11,12,13,14.$$

The lengths of two line segments are $ a$ units and $ b$ units respectively. Then the correct relation between them is: $ \textbf{(A)}\ \frac{a\plus{}b}{2} > \sqrt{ab} \qquad\textbf{(B)}\ \frac{a\plus{}b}{2} < \sqrt{ab} \qquad\textbf{(C)}\ \frac{a\plus{}b}{2} \equal{} \sqrt{ab}\\ \textbf{(D)}\ \frac{a\plus{}b}{2} \leq \sqrt{ab} \qquad\textbf{(E)}\ \frac{a\plus{}b}{2} \geq \sqrt{ab}$

For every $n \in \mathbb{N}_{0}$, prove that $\sum_{k=0}^{\left[\frac{n}{2} \right]}{2}^{n-2k} \binom{n}{2k}=\frac{3^{n}+1}{2}$

Let $d$ be a positive integer. Define the sequence $a_1, a_2, a_3, \dots$ such that \[\begin{cases} a_1 = 1 \\ a_{n+1} = n\left\lfloor\frac{a_n}{n}\right\rfloor + d, \quad n \ge 1.\end{cases}\] Prove that there exists a positive integer $M$ such that $a_M, a_{M+1}, a_{M+2}, \dots$ is an arithmetic sequence.

Let $ a$, $ b$, $ c$, $ d$ be positive real numbers such that $ abcd \equal{} 1$ and $ a \plus{} b \plus{} c \plus{} d > \dfrac{a}{b} \plus{} \dfrac{b}{c} \plus{} \dfrac{c}{d} \plus{} \dfrac{d}{a}$. Prove that \[ a \plus{} b \plus{} c \plus{} d < \dfrac{b}{a} \plus{} \dfrac{c}{b} \plus{} \dfrac{d}{c} \plus{} \dfrac{a}{d}\] [i]Proposed by Pavel Novotný, Slovakia[/i]

In a triangle $ABC$, $AB = AC$, and $D$ is on $BC$. A point $E$ is chosen on $AC$, and a point $F$ is chosen on $AB$, such that $DE = DC$ and $DF = DB$. It is given that $\frac{DC}{BD} = 2$ and $\frac{AF}{AE} = 5$. Determine that value of $\frac{AB}{BC}$.

Find, with proof, the maximum positive integer $k$ for which it is possible to color $6k$ cells of $6 \times 6$ grid such that, for any choice of three distinct rows $R_1$, $R_2$, $R_3$ and three distinct columns $C_1$, $C_2$, $C_3$, there exists an uncolored cell $c$ and integers $1 \le i, j \le 3$ so that $c$ lies in $R_i$ and $C_j$

Let be four distinct complex numbers $ a,b,c,d $ chosen such that $$ |a|=|b|=|c|=|d|=|b-c|=\frac{|c-d|}{2}=1, $$ and $$ \min_{\lambda\in\mathbb{C}} |a-\lambda d -(1-\lambda )c| =\min_{\lambda\in\mathbb{C}} |b-\lambda d -(1-\lambda )c| . $$ Calculate $ |a-c| $ and $ |a-d|. $ [i]Carmen Botea[/i]