In a quadrilateral $ABCD$ angles $A$ and $C$ are right. Two circles with diameters $AB$ and $CD$ meet at points $X$ and $Y$ . Prove that line $XY$ passes through the midpoint of $AC$. (F. Nilov )

Let \( ABC \) be an equilateral triangle with circumcenter \( O \). Let \( X \) and \( Y \) be two points on segments \( AB \) and \( AC \), respectively, such that \( \angle XOY = 60^\circ \). If \( T \) is the reflection of \( O \) with respect to line \( XY \), prove that lines \( BT \) and \( OY \) are parallel.

Consider any rectangular table having finitely many rows and columns, with a real number $a(r, c)$ in the cell in row $r$ and column $c$. A pair $(R, C)$, where $R$ is a set of rows and $C$ a set of columns, is called a [i]saddle pair[/i] if the following two conditions are satisfied: [list] [*] $(i)$ For each row $r^{\prime}$, there is $r \in R$ such that $a(r, c) \geqslant a\left(r^{\prime}, c\right)$ for all $c \in C$; [*] $(ii)$ For each column $c^{\prime}$, there is $c \in C$ such that $a(r, c) \leqslant a\left(r, c^{\prime}\right)$ for all $r \in R$. [/list] A saddle pair $(R, C)$ is called a [i]minimal pair[/i] if for each saddle pair $\left(R^{\prime}, C^{\prime}\right)$ with $R^{\prime} \subseteq R$ and $C^{\prime} \subseteq C$, we have $R^{\prime}=R$ and $C^{\prime}=C$. Prove that any two minimal pairs contain the same number of rows.

Prove that the sum of the squares of five consecutive integers cannot be a perfect square.

Let $z$ be a complex number with $|z| = 2014$. Let $P$ be the polygon in the complex plane whose vertices are $z$ and every $w$ such that $\tfrac{1}{z+w} = \tfrac{1}{z} + \tfrac{1}{w}$. Then the area enclosed by $P$ can be written in the form $n\sqrt{3},$ where $n$ is an integer. Find the remainder when $n$ is divided by $1000$.

Let be $n$ a positive integer. Denote all its (positive) divisors as $1=d_1<d_2<\cdots<d_{k-1}<d_k=n$. Find all values of $n$ satisfying $d_5-d_3=50$ and $11d_5+8d_7=3n$. (Day 1, 1st problem author: Matúš Harminc)

A circle of radius $ 2$ is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle's area is shaded? [asy]unitsize(6mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; filldraw(Arc((0,0),4,0,180)--cycle,gray,black); filldraw(Circle((0,2),2),white,black); dot((0,2)); draw((0,2)--((0,2)+2*dir(60))); label("$2$",midpoint((0,2)--((0,2)+2*dir(60))),SE);[/asy]$ \textbf{(A)}\ \frac{1}{2}\qquad \textbf{(B)}\ \frac{\pi}{6}\qquad \textbf{(C)}\ \frac{2}{\pi}\qquad \textbf{(D)}\ \frac{2}{3}\qquad \textbf{(E)}\ \frac{3}{\pi}$

Let $x_1$, ... , $x_{n+1} \in [0,1] $ and $x_1=x_{n+1} $. Prove that \[ \prod_{i=1}^{n} (1-x_ix_{i+1}+x_i^2)\ge 1. \] A. Khrabrov, F. Petrov

Let $T$ be a trapezoid with two right angles and side lengths $4, 4, 5,$ and $\sqrt{17}$. Two line segments are drawn, connecting the midpoints of opposite sides of $T$ and dividing $T$ into $4$ regions. If the difference between the areas of the largest and smallest of these regions is $d$, compute $240d$.

The numbers $x_1,...x_{100}$ are written on a board so that $ x_1=\frac{1}{2}$ and for every $n$ from $1$ to $99$, $x_{n+1}=1-x_1x_2x_3*...*x_{100}$. Prove that $x_{100}>0.99$.

How many integer values of $x$ satisfy \[\dfrac32 < \dfrac9x < \dfrac 73?\] $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

Find: a) Any quadruple of positive integers $(a, k, l, m)$ such that $a^k = a^l + a^m,$ b) Any quintuple of positive integers $(a, k, l, m, n)$ for which $a^k = a^l + a^m+a^n$

Let rectangle $ABCD$ have side lengths $AB = 8$, $BC = 6$. Let $ABCD$ be inscribed in a circle with center $O$, as shown in the diagram. Let $M$ be the midpoint of side $\overline{AB}$, and let $X$ be the intersection of ray $\overrightarrow{MO}$ with the circle. Compute the length $AX$. [img]https://cdn.artofproblemsolving.com/attachments/6/0/a13e7ec6798f57d896265f61fa42df4c6cab15.png[/img]

How many ordered triples $(x,y,z)$ of real numbers satisfy the system of equations $$x^2+y^2+z^2=9,$$ $$x^4+y^4+z^4=33,$$ $$xyz=-4?$$

Let $f,g:[0,1]\to(0,+\infty)$ be two continuous functions such that $f$ and $\frac gf$ are increasing. Prove that $$\int^1_0\frac{\int^x_0f(t)\text dt}{\int^x_0g(t)\text dt}\text dx\le2\int^1_0\frac{f(t)}{g(t)}\text dt.$$

Points $ A$ and $ B$ are on the parabola $ y \equal{} 4x^2 \plus{} 7x \minus{} 1$, and the origin is the midpoint of $ \overline{AB}$. What is the length of $ \overline{AB}$? $ \textbf{(A)}\ 2\sqrt5 \qquad \textbf{(B)}\ 5\plus{}\frac{\sqrt2}{2} \qquad \textbf{(C)}\ 5\plus{}\sqrt2 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 5\sqrt2$

Rectangle $ABCD$ is inscribed in circle $O$. Let the projections of a point $P$ on minor arc $CD$ onto $AB,AC,BD$ be $K,L,M$, respectively. Prove that $\angle LKM=45$if and only if $ABCD$ is a square.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that$$f(x+f(y^2)) + f(xy) = f(x) + yf(x+y)$$ for all $x, y \in \mathbb{R}$. [i]Proposed by Md. Fuad Al Alam[/i]

A king decides to reward one of his knights by making a game. He sits the knights at a round table and has them call out $1,2,3,1,2,3,\dots$ around the circle (that is, clockwise, and each person says a number). The people who say $2$ or $3$ immediately lose, and this continues until the last knight is left, the winner. Numbering the knights initially as $1,2,\dots,n$, find all values of $n$ such that knight $2008$ is the winner.

Note that $8^3 - 7^3 = 169 = 13^2$ and $13 = 2^2 + 3^2.$ Prove that if the difference between two consecutive cubes is a square, then it is the square of the sum of two consecutive squares.

Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

The number $ \left (2+2^{96} \right )!$ has $2^{93}$ trailing zeroes when expressed in base $B$. [b] a)[/b] Find the minimum possible $B$. [b]b)[/b] Find the maximum possible $B$. [b]c)[/b] Find the total number of possible $B$. [i]Proposed by Lewis Chen[/i]

Let $l,m$ be two parallel lines in the plane. Let $P$ be a fixed point between them. Let $E,F$ be variable points on $l,m$ such that the angle $EPF$ is fixed to a number like $\alpha$ where $0<\alpha<\frac{\pi}2$. (By angle $EPF$ we mean the directed angle) Show that there is another point (not $P$) such that it sees the segment $EF$ with a fixed angle too.

Let $P$ be a point inside a triangle $ABC$ such that $\angle PAC= \angle PCB$. Let the projections of $P$ onto $BC$, $CA$, and $AB$ be $X,Y,Z$ respectively. Let $O$ be the circumcenter of $\triangle XYZ$, $H$ be the foot of the altitude from $B$ to $AC$, $N$ be the midpoint of $AC$, and $T$ be the point such that $TYPO$ is a parallelogram. Show that $\triangle THN$ is similar to $\triangle PBC$. [i]Proposed by Sammy Luo[/i]

Let $n\geq3$ be a positive integer. Let $C_1,C_2,C_3,\ldots,C_n$ be unit circles in the plane, with centres $O_1,O_2,O_3,\ldots,O_n$ respectively. If no line meets more than two of the circles, prove that \[ \sum\limits^{}_{1\leq i<j\leq n}{1\over O_iO_j}\leq{(n-1)\pi\over 4}. \]