Find all integer solutions $(x,y)$ to the equation $\displaystyle \frac{x+y}{x^2-xy+y^2}=\frac{3}{7}$.

Prove that the only function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following is $f(x)=x$. (i) $\forall x \not= 0$, $f(x) = x^2f(\frac{1}{x})$. (ii) $\forall x, y$, $f(x+y) = f(x)+f(y)$. (iii) $f(1)=1$.

Let $CM$ be the median of an acute-angled triangle $ABC$, and $P$ be the projection of the orthocenter $H$ to the bisector of $\angle C$. Prove that $MP$ bisects the segment $CH$.

It is a fact that every set of 2016 consecutive integers can be partitioned in two sets with the following four properties: (i) The sets have the same number of elements. (ii) The sums of the elements of the sets are equal. (iii) The sums of the squares of the elements of the sets are equal. (iv) The sums of the cubes of the elements of the sets are equal. Let $S =\{n + 1; n + 2;$ [b]. . .[/b] $; n + k\}$ be a set of $k$ consecutive integers. (a) Determine the smallest value of $k$ such that property (i) holds for $S$. (b) Determine the smallest value of $k$ such that properties (i) and (ii) hold for $S$. (c) Show that properties (i), (ii) and (iii) hold for $S$ when $k = 8$. (d) Show that properties (i), (ii), (iii) and (iv) hold for $S$ when $k = 16$.

Consider $37$ distinct points in space, all with integer coordinates. Prove that we may find among them three distinct points such that their barycentre has integers coordinates.

Richard has an infinite row of empty boxes labeled $1, 2, 3, \ldots$ and an infinite supply of balls. Each minute, Richard finds the smallest positive integer $k$ such that box $k$ is empty. Then, Richard puts a ball into box $k$, and if $k \geq 3$, he removes one ball from each of boxes $1,2,\ldots,k-2$. Find the smallest positive integer $n$ such that after $n$ minutes, both boxes $9$ and $10$ have at least one ball in them. [i]Proposed by [b]vvluo[/b] & [b]richy[/b][/i]

Let $ABCD$ be a convex quadrilateral and draw regular triangles $ABM, CDP, BCN, ADQ$, the first two outward and the other two inward. Prove that $MN = AC$. What can be said about the quadrilateral $MNPQ$?

Suppose $a, b, c$ are real numbers such that $abc \ne 0$. Determine $x, y, z$ in terms of $a, b, c$ such that $bz + cy = a, cx + az = b, ay + bx = c$. Prove also that $\frac{1 - x^2}{a^2} = \frac{1 - y^2}{b^2} = \frac{1 - z^2}{c^2}$.

Consider a line segment $[AB]$ with midpoint $M$ and perpendicular bisector $m$. For each point$ X \ne M$ on m consider we are the intersection point $Y$ of the line $BX$ with the bisector from the angle $\angle BAX$. As $X$ approaches $M$, then approaches $Y$ to a point of $[AB]$. Which? [img]https://cdn.artofproblemsolving.com/attachments/a/3/17d72a23011a9ec22deb20184717cc9c020a2b.png[/img] [hide=original wording]Beschouw een lijnstuk [AB] met midden M en middelloodlijn m. Voor elk punt X 6= M op m beschouwenwe het snijpunt Y van de rechte BX met de bissectrice van de hoek < BAX . Als X tot M nadert, dan nadert Y tot een punt van [AB]. Welk? [/hide]

Let $n$ be a positive integer. - Find, in terms of $n$, the number of pairs $(x,y)$ of positive integers that are solutions of the equation : $$x^2-y^2=10^2.30^{2n}$$ - Prove further that this number is never a square

Let $n \ge 2018$ be an integer, and let $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ be pairwise distinct positive integers not exceeding $5n$. Suppose that the sequence \[ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_n}{b_n} \] forms an arithmetic progression. Prove that the terms of the sequence are equal.

Let $a,b,c$ be real numbers with sum equal to zero. Prove that \[ab^3+bc^3+ca^3\leqslant 0.\]

For a given integer $n\geq2$, a pyramid of height $n$ if defined as a collection of $1^2+2^2+\dots+n^2$ stone cubes of equal size stacked in $n$ layers such that the cubes in the $k$-th layer form a square with sidelength $n+1-k$ and every cube (except for the ones in the bottom layer) rests on four cubes in the layer below. Some of the cubes are made of sandstone, some are made of granite. The top cube is made of granite, and to ensure the stability of the piramid, for each granite cube (except for the ones in the bottom layer), at least three out of four of the cubes supporting it have to be granite. What is the minimum possible number of granite cubes in such an arrangement?

Compute $\sqrt{202 \times 3 - 20 \times 23 + 2 \times 23 - 23}$. [i]Proposed by Andy Xu[/i]

How many $2$-digit integers have an equal number of odd and even positive divisors? $\text{(A) }11\qquad\text{(B) }12\qquad\text{(C) }22\qquad\text{(D) }23\qquad\text{(E) }45$

Find all function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(x+y)-2f(x-y)+f(x)-2f(y)=y-2,\forall x,y\in \mathbb{R}$.

The largest side of the triangle $ABC$ is equal to $1$ unit. Prove that , the circles centred at $A,B$ and $C$ wit radiuses $\frac{1}{\sqrt{3}}$ can compeletely cover the triangle $ABC$.

Let $n\geq 2$ be an integer, and let $0 < a_1 < a_2 < \cdots < a_{2n+1}$ be real numbers. Prove the inequality \[ \sqrt[n]{a_1} - \sqrt[n]{a_2} + \sqrt[n]{a_3} - \cdots + \sqrt[n]{a_{2n+1}} < \sqrt[n]{a_1 - a_2 + a_3 - \cdots + a_{2n+1}}. \] [i]Bogdan Enescu, Romania[/i]

A rectangle, with sides parallel to the $x-$axis and $y-$axis, has opposite vertices located at $(15, 3)$ and$(16, 5).$ A line is drawn through points $A(0, 0)$ and $B(3, 1).$ Another line is drawn through points $C(0, 10)$ and $D(2, 9).$ How many points on the rectangle lie on at least one of the two lines? [asy] size(9cm); draw((0,-.5)--(0,11),EndArrow(size=.15cm)); draw((1,0)--(1,11),mediumgray); draw((2,0)--(2,11),mediumgray); draw((3,0)--(3,11),mediumgray); draw((4,0)--(4,11),mediumgray); draw((5,0)--(5,11),mediumgray); draw((6,0)--(6,11),mediumgray); draw((7,0)--(7,11),mediumgray); draw((8,0)--(8,11),mediumgray); draw((9,0)--(9,11),mediumgray); draw((10,0)--(10,11),mediumgray); draw((11,0)--(11,11),mediumgray); draw((12,0)--(12,11),mediumgray); draw((13,0)--(13,11),mediumgray); draw((14,0)--(14,11),mediumgray); draw((15,0)--(15,11),mediumgray); draw((16,0)--(16,11),mediumgray); draw((-.5,0)--(17,0),EndArrow(size=.15cm)); draw((0,1)--(17,1),mediumgray); draw((0,2)--(17,2),mediumgray); draw((0,3)--(17,3),mediumgray); draw((0,4)--(17,4),mediumgray); draw((0,5)--(17,5),mediumgray); draw((0,6)--(17,6),mediumgray); draw((0,7)--(17,7),mediumgray); draw((0,8)--(17,8),mediumgray); draw((0,9)--(17,9),mediumgray); draw((0,10)--(17,10),mediumgray); draw((-.13,1)--(.13,1)); draw((-.13,2)--(.13,2)); draw((-.13,3)--(.13,3)); draw((-.13,4)--(.13,4)); draw((-.13,5)--(.13,5)); draw((-.13,6)--(.13,6)); draw((-.13,7)--(.13,7)); draw((-.13,8)--(.13,8)); draw((-.13,9)--(.13,9)); draw((-.13,10)--(.13,10)); draw((1,-.13)--(1,.13)); draw((2,-.13)--(2,.13)); draw((3,-.13)--(3,.13)); draw((4,-.13)--(4,.13)); draw((5,-.13)--(5,.13)); draw((6,-.13)--(6,.13)); draw((7,-.13)--(7,.13)); draw((8,-.13)--(8,.13)); draw((9,-.13)--(9,.13)); draw((10,-.13)--(10,.13)); draw((11,-.13)--(11,.13)); draw((12,-.13)--(12,.13)); draw((13,-.13)--(13,.13)); draw((14,-.13)--(14,.13)); draw((15,-.13)--(15,.13)); draw((16,-.13)--(16,.13)); label(scale(.7)*"$1$", (1,-.13), S); label(scale(.7)*"$2$", (2,-.13), S); label(scale(.7)*"$3$", (3,-.13), S); label(scale(.7)*"$4$", (4,-.13), S); label(scale(.7)*"$5$", (5,-.13), S); label(scale(.7)*"$6$", (6,-.13), S); label(scale(.7)*"$7$", (7,-.13), S); label(scale(.7)*"$8$", (8,-.13), S); label(scale(.7)*"$9$", (9,-.13), S); label(scale(.7)*"$10$", (10,-.13), S); label(scale(.7)*"$11$", (11,-.13), S); label(scale(.7)*"$12$", (12,-.13), S); label(scale(.7)*"$13$", (13,-.13), S); label(scale(.7)*"$14$", (14,-.13), S); label(scale(.7)*"$15$", (15,-.13), S); label(scale(.7)*"$16$", (16,-.13), S); label(scale(.7)*"$1$", (-.13,1), W); label(scale(.7)*"$2$", (-.13,2), W); label(scale(.7)*"$3$", (-.13,3), W); label(scale(.7)*"$4$", (-.13,4), W); label(scale(.7)*"$5$", (-.13,5), W); label(scale(.7)*"$6$", (-.13,6), W); label(scale(.7)*"$7$", (-.13,7), W); label(scale(.7)*"$8$", (-.13,8), W); label(scale(.7)*"$9$", (-.13,9), W); label(scale(.7)*"$10$", (-.13,10), W); dot((0,0)); label(scale(.65)*"$A$", (0,0), NE); dot((3,1)); label(scale(.65)*"$B$", (3,1), NE); dot((0,10)); label(scale(.65)*"$C$", (0,10), NE); dot((2,9)); label(scale(.65)*"$D$", (2,9), NE); draw((15,3)--(16,3)--(16,5)--(15,5)--cycle,linewidth(1.125)); dot((15,3)); dot((16,3)); dot((16,5)); dot((15,5)); [/asy] $\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) } 4$

Let $ABC$ be an equilateral triangle. Denote by $D$ the midpoint of $\overline{BC}$, and denote the circle with diameter $\overline{AD}$ by $\Omega$. If the region inside $\Omega$ and outside $\triangle ABC$ has area $800\pi-600\sqrt3$, find the length of $AB$. [i]Proposed by Eugene Chen[/i]

Given is a fixed regular pentagon $ABCDE$ with side $1$. Let $M$ be an arbitrary point inside or on it. Let the distance from $M$ to the closest vertex be $r_1$, to the next closest be $r_2$ and so on, so that the distances from $M$ to the five vertices satisfy $r_1\le r_2\le r_3\le r_4\le r_5$. Find (a) the locus of $M$ which gives $r_3$ the minimum possible value, and (b) the locus of $M$ which gives $r_3$ the maximum possible value.

Suppose that $ a_1,\cdots , a_{25}$ are non-negative integers, and $ k$ is the smallest of them. Prove that $$\big[\sqrt{a_1}\big]+\big[\sqrt{a_2}\big]+\cdots+\big[\sqrt{a_{25}}\big ]\geq\big[\sqrt{a_1+a_2+\cdots+a_{25}+200k}\big].$$ (As usual, $[x]$ denotes the integer part of the number $x$ , that is, the largest integer not exceeding $x$.)

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

A rectangle with sides of lengths $a$ and $b$ ($a > b$) is cut into rightangled triangles so that any two of these triangles either have a common side, a common vertex or no common points. Moreover, any common side of two triangles is a leg of one of them and the hypotenuse of the other. Prove that $a > 2b$. (A Shapovalov)

Bradley is driving at a constant speed. When he passes his school, he notices that in $20$ minutes he will be exactly $\frac14$ of the way to his destination, and in $45$ minutes he will be exactly $\frac13$ of the way to his destination. Find the number of minutes it takes Bradley to reach his destination from the point where he passes his school.