Two circles $K_1$ and $K_2$ of different radii intersect at two points $A$ and $B$, let $C$ and $D$ be two points on $K_1$ and $K_2$, respectively, such that $A$ is the midpoint of the segment $CD$. The extension of $DB$ meets $K_1$ at another point $E$, the extension of $CB$ meets $K_2$ at another point $F$. Let $l_1$ and $l_2$ be the perpendicular bisectors of $CD$ and $EF$, respectively. i) Show that $l_1$ and $l_2$ have a unique common point (denoted by $P$). ii) Prove that the lengths of $CA$, $AP$ and $PE$ are the side lengths of a right triangle.

Let $a_n>0$, $n\ge1$. Consider the right triangles $\triangle A_0A_1A_2$, $\triangle A_0A_2A_3,\ldots$, $\triangle A_0A_{n-1}A_n,\ldots,$ as in the figure. (More precisely, for every $n\ge2$ the hypotenuse $A_0A_n$ of $\triangle A_0A_{n-1}A_n$ is a leg of $\triangle A_0A_nA_{n+1}$ with right angle $\angle A_0A_nA_{n+1}$, and the vertices $A_{n-1}$ and $A_{n+1}$ lie on the opposite sides of the straight line $A_0A_n$; also, $|A_{n-1}A_n|=a_n$ for every $n\ge1$.) [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi8yL2M1ZjAxM2I1ZWU0N2E4MzQyYWIzNmQ5OGM3NjJlZjljODdmMTliLnBuZw==&rn=U0VFTU9VUyAyMDEyLnBuZw==[/img] Is it possible for the set of points $\{A_n\mid n\ge0\}$ to be unbounded but the series $\sum_{n=2}^\infty m\angle A_{n-1}A_0A_n$ to be convergent? [i]Note.[/i] A subset $B$ of the plane is bounded if and only if there is a disk $D$ such that $B\subseteq D$.

A function $f:\mathbb{N} \to \mathbb{N}$ is given. If $a,b$ are coprime, then $f(ab)=f(a)f(b)$. Also, if $m,k$ are primes (not necessarily different), then $$f(m+k-3)=f(m)+f(k)-f(3).$$ Find all possible values of $f(11)$.

Given reals $x_1 \ge x_2 \ge ... \ge x_{2003} \ge 0$, show that $$x_1^n - x_2^n + x_2^n - ... - x_{2002}^n + x_{2003}^n \ge (x_1 - x_2 + x_3 - x_4 + ... - x_{2002} + x_{2003})^n$$ for any positive integer $n$.

Prove that for any triangle $ABC$ there exists a point P in the plane of the triangle and three points $A' , B'$ , and $C'$ on the lines $BC, AC$, and $AB$ respectively such that \[AB \cdot PC'= AC \cdot PB'= BC \cdot PA'= 0.3M^2,\] where $M = max\{AB,AC,BC\}$.

Points $P$ and $Q$ are both in the line segment $AB$ and on the same side of its midpoint. $P$ divides $AB$ in the ratio $2:3$, and $Q$ divides $AB$ in the ratio $3:4$. If $PQ=2$, then the length of $AB$ is: ${{ \textbf{(A)}\ 60\qquad\textbf{(B)}\ 70\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 80}\qquad\textbf{(E)}\ 85} $

Let $x_1,x_2,\dots,x_{2023}$ be pairwise different positive real numbers such that \[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\] is an integer for every $n=1,2,\dots,2023.$ Prove that $a_{2023} \geqslant 3034.$

Let $ABC$ be a triangle with incenter $I$. The line $BI$ meets $AC$ at $D$. Let $P$ be a point on $CI$ such that $DI=DP$ ($P\ne I$), $E$ the second intersection point of segment $BC$ with the circumcircle of $ABD$ and $Q$ the second intersection point of line $EP$ with the circumcircle of $AEC$. Prove that $\angle PDQ=90^\circ$. [i]Proposed by Ariel García[/i]

Which of the following are true? $\textbf{(A)}~\exists f:\mathbb N\to\mathbb Z\text{ onto and increasing}$ $\textbf{(B)}~\exists f:\mathbb Z\to\mathbb Q\text{ onto and increasing}$ $\textbf{(C)}~\exists f:\mathbb Q\to\mathbb Z\text{ onto and increasing and bounded}$ $\textbf{(D)}~\text{None of the above}$

A grid measuring $10 \times 11$ is given. How many "crosses" covering five unit squares can be placed on the grid? (pictured right) so that no two of them cover the same square? [img]https://cdn.artofproblemsolving.com/attachments/a/7/8a5944233785d960f6670e34ca7c90080f0bd6.png[/img]

Consider the function $f(n) = n^2 + n + 1$. For each $n$, let $d_n$ be the smallest positive integer with $gcd(n, dn) = 1$ and $f(n) | f(d_n)$. Find $d_6 + d_7 + d_8 + d_9 + d_{10}$.

A whole number is written on each square of a $3\times 2021$ board. If the number written in each square is greater than or equal to at least two of the numbers written in the neighboring squares, how many different numbers written on the board can there be at most? Note: Two squares are neighbors when they have a common side.

Let $a,b,c\in\mathbb{R^+}$. Prove that $$\frac{a^{11}}{b^5c^5}+\frac{b^{11}}{ c^5a^5}+\frac{c^{11}}{a^5b^5}\ge a+b+c$$ [i](Real Matrik)[/i]

Find the least positive integer $n$ for which $2^n + 5^n - n$ is a multiple of $1000$.

Let $ABCDEFG$ be a regular heptagon. We'll call the sides $AB$, $BC$, $CD$, $DE$, $EF$, $FG$ and $GA$ opposite to the vertices $E$, $F$, $G$, $A$, $B$, $C$ and $D$ respectively. If $M$ is a point inside the heptagon, we'll say that the line through $M$ and a vertex of the heptagon intersects a side of it (without the vertices) at a $\textit{perfect}$ point, if this side is opposite the vertex. Prove that for every choice of $M$, the number of $\textit{perfect}$ points is always odd.

In a party of eight people, each pair of people either know each other or do not know each other. Each person knows exactly three of the others. Determine whether the following two conditions can be satisfied simultaneously: [list] – for any three people, at least two do not know each other; – for any four people there are at least two who know each other. [/list]

Prove that there is a unique positive integer formed only by the digits $2$ and $5$, which has $ 2007$ digits and is divisible by $2^{2007}$.

Let $m,n,p$ be rational numbers such that $\sqrt{m} + \sqrt{n} + \sqrt{p}$ is a rational number. Prove that $\sqrt{m}, \sqrt{n}, \sqrt{p}$ are also rational numbers

Two candles of the same length are made of different materials so that one burns out completely at a uniform rate in $3$ hours and the other in $4$ hours. At what time P.M. should the candles be lighted so that, at 4 P.M., one stub is twice the length of the other? $\textbf{(A)}\ 1:24\qquad \textbf{(B)}\ 1:28\qquad \textbf{(C)}\ 1:36\qquad \textbf{(D)}\ 1:40\qquad \textbf{(E)}\ 1:48$

Determine all roots, real or complex, of the system of simultaneous equations \begin{align*} x+y+z &= 3, \\ x^2+y^2+z^2 &= 3, \\ x^3+y^3+z^3 &= 3.\end{align*}

The number $n{}$ cointains $k{}$ units in binary system. Prove that $2^{n-k}{}$ divides $n!$.

A chess king was placed on a square of an \(8 \times 8\) board and made $64$ moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called $\emph{pleasant}$ if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)

Let $ABC$ be an equilateral triangle. Let $P$ be a point inside of it such that the square root of the distance of $P$ to one of the sides is equal to the sum of the square roots of the distances of $P$ to the other two sides. Find the geometric place of $P$.

[b]6.[/b] Let $T$ be a one-to-one mapping of the unit square $E$ of the plane into itself. Suppose that $T$ and $T^{-1}$ are measure-preserving (i.e. if $M \subseteq E$ is a measurable set, then $TM$ and $T^{-1}M$ are also measurable and $\mu (M)= \mu (TM)= \mu (T^{-1}M)$, where $\mu$ denotes the Lebesgue measure) and, furthermore, that if $Tx \in N$ for almost all points $x$ of a measurable set $N \subseteq E$, then either $n$ or $ E \setminus N$ is of measure 0. Prove that, for any measurable set $A \subseteq E$, with $\mu (A)>0$, the function $n(x)$ defined by $$n(x)=\begin{cases} 0, \mbox{if} \quad T^k x \notin A \quad (k=1, 2, \dots),\\ \min (k: T^k x \in A; k=1,2, \dots ) &\mbox{otherwise} \end{cases} $$ is measurable and $\int_{A}n(x) d\mu(x) =1$ [b](R. 18)[/b]

Let $a_1=24$ and form the sequence $a_n$, $n\geq 2$ by $a_n=100a_{n-1}+134$. The first few terms are $$24,2534,253534,25353534,\ldots$$ What is the least value of $n$ for which $a_n$ is divisible by $99$?