Each of three lines cuts chords of equal lengths in two given circles. The points of intersection of these lines form a triangle. Prove that its circumcircle passes through the midpoint of the segment joining the centres of the circles.

Given a $24 \times 24$ square grid, initially all its unit squares are coloured white. A move consists of choosing a row, or a column, and changing the colours of all its unit squares, from white to black, and from black to white. Is it possible that after finitely many moves, the square grid contains exactly $574$ black unit squares?

Let $a=x_1\leq x_2\leq ...\leq x_n=b$. Prove the following inequality: $(x_1+x_2+...+x_n )(\frac{1}{x_1} +\frac{1}{x_2} +...+\frac{1}{x_n} )\leq \frac{(a+b)}{4ab} n^2$.

Let $a_1, a_2,\cdots , a_n$ and $b_1, b_2,\cdots , b_n$ be (not necessarily distinct) positive integers. We continue the sequences as follows: For every $i>n$, $a_i$ is the smallest positive integer which is not among $b_1, b_2,\cdots , b_{i-1}$, and $b_i$ is the smallest positive integer which is not among $a_1, a_2,\cdots , a_{i-1}$. Prove that there exists $N$ such that for every $i>N$ we have $a_i=b_i$ or for every $i>N$ we have $a_{i+1}=a_i$.

In a triangle $ABC$, right in $A$ and isosceles, let $D$ be a point on the side $AC$ ($A \ne D \ne C$) and $E$ be the point on the extension of $BA$ such that the triangle $ADE$ is isosceles. Let $P$ be the midpoint of segment $BD$, $R$ be the midpoint of the segment $CE$ and $Q$ the intersection point of $ED$ and $BC$. Prove that the quadrilateral $ARQP$ is a square.

The sequence $a_1, a_2, a_3, ...$ is defined by $a_1 = a_2 = a_3 = 1$, $a_{n+3} = a_{n+2}a_{n+1} + a_n$. Show that for any positive integer $r$ we can find $s$ such that $a_s$ is a multiple of $r$.

A privateer discovers a merchantman $10$ miles to leeward at 11:45 a.m. and with a good breeze bears down upon her at $11$ mph, while the merchantman can only make $8$ mph in her attempt to escape. After a two hour chase, the top sail of the privateer is carried away; she can now make only $17$ miles while the merchantman makes $15$. The privateer will overtake the merchantman at: $\textbf{(A)}\ 3\text{:}45\text{ p.m.} \qquad \textbf{(B)}\ 3\text{:}30\text{ p.m.} \qquad \textbf{(C)}\ 5\text{:}00\text{ p.m.} \qquad \textbf{(D)}\ 2\text{:}45\text{ p.m.} \qquad \textbf{(E)}\ 5\text{:}30\text{ p.m.}$

Let the inscribed circle of the triangle $\vartriangle ABC$ touch side $BC$ at $M$ , side $CA$ at $N$ and side $AB$ at $P$ . Let $D$ be a point from $\left[ NP \right]$ such that $\frac{DP}{DN}=\frac{BD}{CD}$ . Show that $DM \perp PN$ .

Five edges of a tetrahedron are tangent to a sphere. Prove that there are another five edges from this tetrahedron that are also tangent to a $($not necessarily the same$)$ sphere.

Let $P$ be a point in the plane. Prove that it is possible to draw three rays with origin in $P$ with the following property: for every circle with radius $r$ containing $P$ in its interior, if $P_1$, $P_2$ and $P_3$ are the intersection points of the three rays with the circle, then \[|PP_1|+|PP_2|+|PP_3|\leq 3r.\]

Determine the maximum value of $\frac{8(x+y)(x^3+y^3)}{(x^2+y^2)^2}$ for all $(x,y)\neq (0,0)$

Given acute triangle $\vartriangle ABC$ with orthocenter $H$ and circumcenter $O$ ($O \ne H$) . Let $\Gamma$ be the circumcircle of $\vartriangle BOC$ . Segment $OH$ untersects $\Gamma$ at point $P$. Extension of $AO$ intersects $\Gamma$ at point $K$. If $AP \perp OH$, prove that $PK$ bisects $BC$. [img]https://cdn.artofproblemsolving.com/attachments/a/b/267053569c41692f47d8f4faf2a31ebb4f4efd.png[/img]

Two right angled triangles are given, such that the incircle of the first one is equal to the circumcircle of the second one. Let $S$ (respectively $S'$) be the area of the first triangle (respectively of the second triangle). Prove that $\frac{S}{S'}\geq 3+2\sqrt{2}$.

Let $A_1 = (0, 0)$, $B_1 = (1, 0)$, $C_1 = (1, 1)$, $D_1 = (0, 1)$. For all $i > 1$, we recursively define $$A_i =\frac{1}{2020} (A_{i-1} + 2019B_{i-1}),B_i =\frac{1}{2020} (B_{i-1} + 2019C_{i-1})$$ $$C_i =\frac{1}{2020} (C_{i-1} + 2019D_{i-1}), D_i =\frac{1}{2020} (D_{i-1} + 2019A_{i-1})$$ where all operations are done coordinate-wise. [img]https://cdn.artofproblemsolving.com/attachments/8/7/9b6161656ed2bc70510286dc8cb75cc5bde6c8.png[/img] If $[A_iB_iC_iD_i]$ denotes the area of $A_iB_iC_iD_i$, there are positive integers $a, b$, and $c$ such that $\sum_{i=1}^{\infty}[A_iB_iC_iD_i] = \frac{a^2b}{c}$, where $b$ is square-free and $c$ is as small as possible. Compute the value of $a + b + c$

Let $\Delta$ denote the set of all triangles in a plane. Consider the function $f: \Delta\to(0,\infty)$ defined by $f(ABC) = \min \left( \dfrac ba, \dfrac cb \right)$, for any triangle $ABC$ with $BC=a\leq CA=b\leq AB = c$. Find the set of values of $f$.

Minivan and Megavan play a game. For a positive integer $n$, Minivan selects a sequence of integers $a_1,a_2,\ldots,a_n$. An operation on $a_1,a_2,\ldots,a_n$ means selecting an $a_i$ and increasing it by $1$. Minivan and Megavan take turns, with Minivan going first. On Minivan's turn, he performs at most $2025$ operations, and he may choose the same integer repeatedly. On Megavan's turn, he performs exactly $1$ operation instead. Megavan wins if at any point in the game, including in the middle of Minivan's operations, two numbers in the sequence are equal. [i](Proposed by Ho Janson)[/i]