For a finite non empty set of primes $P$, let $m(P)$ denote the largest possible number of consecutive positive integers, each of which is divisible by at least one member of $P$. (i) Show that $|P|\le m(P)$, with equality if and only if $\min(P)>|P|$. (ii) Show that $m(P)<(|P|+1)(2^{|P|}-1)$. (The number $|P|$ is the size of set $P$) [i]Dan Schwarz, Romania[/i]

Solve $(x-4)(x-5)(x-6)(x-7)=1680$

(a) For non negetive $a,b,c, r$ prove that \[a^r(a-b)(a-c) + b^r(b-a)(b-c) + c^r (c-a)(c-b) \geq 0 \] (b) Find an inequality for non negative $a,b,c$ with $a^4+b^4+c^4 + abc(a+b+c)$ on the greater side. (c) Prove that if $abc = 1$ for non negative $a,b,c$, $a^4+b^4+c^4+a^3+b^3+c^3+a+b+c \geq \frac{a^2+b^2}{c}+\frac{b^2+c^2}{a}+\frac{c^2+a^2}{b}+3$

Let $p$ be a prime number, $p> 5$. Determine the non-zero natural numbers $x$ with the property that $5p + x$ divides $5p ^ n + x ^ n$, whatever $n \in N ^ {*} $.

Using only once each of the digits $1, 2, 3, 4, 5, 6, 7$ and $ 8$, write the square and the cube of a positive integer. Determine what that number can be.

All vertices of a regular 2016-gon are initially white. What is the least number of them that can be painted black so that:\\ (a) There is no right triangle\\ (b) There is no acute triangle\\ having all vertices in the vertices of the 2016-gon that are still white?

Let $ABC$ be a triangle such that $AB\neq AC$. We denote its orthocentre by $H$, its circumcentre by $O$ and the midpoint of $BC$ by $D$. The extensions of $HD$ and $AO$ meet in $P$. Prove that triangles $AHP$ and $ABC$ have the same centroid.

One side of an equilateral triangle of height $24$ lies on line $\ell.$ A circle of radius $12$ is tangent to $\ell$ and is externally tangent to the triangle. The area of the region exterior to the triangle and the circle and bounded by the triangle, the circle, and line $\ell$ can be written as $a\sqrt{b} - c\pi,$ where $a,$ $b,$ and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is $a+b+c\,?$ $\phantom{boo}$ $\displaystyle \textbf{(A)}\; 72 \quad \textbf{(B)}\; 73 \quad \textbf{(C)}\; 74 \quad \textbf{(D)}\; 75 \quad \textbf{(E)}\; 76 $

Let $n$ be a given positive integer. (a) Do there exist $2n+1$ consecutive positive integers $a_0,a_1,\ldots,a_{2n}$ in the ascending order such that $a_0+a_1+\ldots+a_n=a_{n+1}+\ldots+a_{2n}$? (b) Do there exist consecutive positive integers $a_0,a+1,\ldots,a_{2n}$ in ascending order such that $a_0^2+a_1^2+\ldots+a_n^2=a_{n+1}^2+\ldots+a_{2n}^2$? (c) Do there exist consecutive positive integers $a_0,a_1,\ldots,a_{2n}$ in ascending order such that $a_0^3+a_1^3+\ldots+a_n^3=a_{n+1}^3+\ldots+a_{2n}^3$? [hide=Official Hint]You may study the function $f(x)=(x-n)^3+\ldots+x^3-(x+1)^3-\ldots-(x+n)^3$ and prove that the equation $f(x)=0$ has a unique solution $x_n$ with $3n(n+1)<x_n<3n(n+1)+1$. You may use the identity $1^3+2^3+\ldots+n^3=\frac{n^2(n+1)^2}2$.[/hide]

If in triangle $ABC$ , $AC$=$15$, $BC$=$13$ and $IG||AB$ where $I$ is the incentre and $G$ is the centroid , what is the area of triangle $ABC$ ?

Given is a rectangle with perimeter $x$ cm and side lengths in a $1:2$ ratio. Suppose that the area of the rectangle is also $x$ $\text{cm}^2$. Determine all possible values of $x$.

Each face of a cube is of the same size as each square of a chessboard. The cube is coloured black and white, placed on one of the squares of the chessboard and rolled so that each square of the chessboard is visited exactly once. Can this be done in such a way that the colour of the visited square and the colour of the bottom face of the cube are always the same? (A Shapovalov)

The vertices $A,B,C$ and $D$ of a square lie outside a circle centred at $M$. Let $AA',BB',CC',DD'$ be tangents to the circle. Assume that the segments $AA',BB',CC',DD'$ are the consecutive sides of a quadrilateral $p$ in which a circle is inscribed. Prove that $p$ has an axis of symmetry.

Find the last three digits of the product of the positive roots of \[ \sqrt{1995}x^{\log_{1995}x}=x^2. \]

Given $n \geq 2$ distinct integers, which are bigger than $-10$. It turned out that the amount of odd numbers among them is equal to the biggest even number, and the amount of even to the biggest of odd. a) Find the smallest $n$ possible b) Find the biggest $n$ possible

In the coordinate plane, Yifan the Yak starts at $(0,0)$ and makes $11$ moves. In a move, Yifan can either do nothing or move from an arbitrary point $(i,j)$ to $(i+1,j)$, $(i,j+1)$ or $(i+1,j+1)$. How many points $(x,y)$ with integer coordinates exist such that the number of ways Yifan can end on $(x,y)$ is odd? [i]Proposed by Yifan Kang[/i]

Two players play a game involving an $n \times n$ grid of chocolate. Each turn, a player may either eat a piece of chocolate (of any size), or split an existing piece of chocolate into two rectangles along a grid-line. The player who moves last loses. For how many positive integers $n$ less than $1000$ does the second player win? (Splitting a piece of chocolate refers to taking an $a \times b$ piece, and breaking it into an $(a-c) \times b$ and a $c \times b$ piece, or an $a \times (b-d)$ and an $a \times d$ piece.) [i]Proposed by Lewis Chen[/i]

Prove that for an arbitrary triangle $ABC$ : $sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2} < \frac{1}{4}$.

Let $P_0,P_1,P_2,\ldots$ be a sequence of convex polygons such that, for each $k\ge0$, the vertices of $P_{k+1}$ are the midpoints of all sides of $P_k$. Prove that there exists a unique point lying inside all these polygons.

Find all solutions of the system consisting of $3$ equations: $x \left(1 - \frac{1}{2^n}\right) +y \left(1 - \frac{1}{2^{n+1}}\right) +z \left(1 - \frac{1}{2^{n+2}}\right) = 0$ for $n = 1, 2, 3$.

Find, with proof, all solutions of the equation $\frac 1x +\frac 2y- \frac 3z = 1$ in positive integers $x, y, z.$

The given integers are $ a_1, a_2, \ldots , a_{11} $ . Prove that there exists a non-zero sequence $ x_1, x_2, \ldots, x_{11} $ with terms from the set $ \{-1,0,1\} $ such that the number $ x_1a_1 + \ldots x_{11}a_{ 11}$ is divisible by 1989.

The sides of a triangle are divided by the angle bisectors into two segments each. Is it always possible to form two triangles from the obtained six segments? [i]Lev Emelyanov[/i]

Let $n$ be a positive integer. The numbers $1,2,3,\dots, 4n$ are written in a board. Olive wants to make some "couples" of numbers, such that the product of the numbers in each couple is a perfect square. Each number is in, at most, one couple and the two numbers in each couple are distincts. Determine, for each positive integer $n$, the maximum number of couples that Olive can write.

On the extensions of sides $CA$ and $AB$ of triangle $ABC$ beyond points $A$ and $B$, respectively, the segments $AE = BC$ and $BF = AC$ are drawn. A circle is tangent to segment $BF$ at point $N$, side $BC$ and the extension of side $AC$ beyond point $C$. Point $M$ is the midpoint of segment $EF$. Prove that the line $MN$ is parallel to the bisector of angle $A$.