Found problems: 85335
There is secret society with $2011$ members. Every member has bank account with integer balance ( can be negative). Sometimes some member give one dollar to every his friend. It is known, that after some such moves members can redistribute their money arbitrarily. Prove, that there are exactly $2010$ pairs of friends.
A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly 1/1985.
[asy]
size(200);
pair A=(0,1), B=(1,1), C=(1,0), D=origin;
draw(A--B--C--D--A--(1,1/6));
draw(C--(0,5/6)^^B--(1/6,0)^^D--(5/6,1));
pair point=( 0.5 , 0.5 );
//label("$A$", A, dir(point--A));
//label("$B$", B, dir(point--B));
//label("$C$", C, dir(point--C));
//label("$D$", D, dir(point--D));
label("$1/n$", (11/12,1), N, fontsize(9));[/asy]
Each vertex of a finite graph can be coloured either black or white. Initially all vertices are black. We are allowed to pick a vertex $P$ and change the colour of $P$ and all of its neighbours. Is it possible to change the colour of every vertex from black to white by a
sequence of operations of this type?
Note: A finite graph consists of a finite set of vertices and a finite set of edges between vertices. If there is an edge between vertex $A$ and vertex $B,$ then $A$ and $B$ are neighbours of each other.
Integers $a$, $b$, $c$ are selected independently and at random from the set $ \{ 1, 2, \cdots, 10 \} $, with replacement. If $p$ is the probability that $a^{b-1}b^{c-1}c^{a-1}$ is a power of two, compute $1000p$.
[i]Proposed by Evan Chen[/i]
There are $172$ two-way direct airways between $20$ cities, at most one between any two cities. Prove that one can reach any city from any other city with at most one transfer.
Two bicyclists traveled the distance from $A$ to $B$, which is $100$ km, with speed $30$ km/h and it is known that the first started $30$ minutes before the second. $20$ minutes after the start of the first bicyclist from $A$, there is a control car started whose speed is $90$ km/h and it is known that the car is reached the first bicyclist and is driving together with him for $10$ minutes, went back to the second and was driving for $10$ minutes with him and after that the car is started again to the first bicyclist with speed $90$ km/h and etc. to the end of the distance. How many times will the car drive together with the first bicyclist?
[i]K. Dochev[/i]
Find all non-negative integers $m$ and $n$, such that $(2^n-1) \cdot (3^n-1)=m^2$.
Several points on the plane are given, no three of them lie on the same line. Some of these points are connected by line segments. Assume that any line that does not pass through any of these points intersects an even number of these segments. Prove that from each point exits an even number of the segments.
Two noncongruent circles $k_1$ and $k_2$ are exterior to each other. Their common tangents intersect the line through their centers at points $A$ and $B$. Let $P$ be any point of $k_1$. Prove that there is a diameter of $k_2$ with one endpoint on line $PA$ and the other on $PB$.
The $A$-excircle of a triangle $ABC$ touches the side $BC$ at the point $K$ and the extended side $AB$ at the point $L$. The $B$-excircle touches the lines $BA$ and $BC$ at the points $M$ and $N$, respectively. The lines $KL$ and $MN$ meet at the point $X$.
Show that the line $CX$ bisects the angle $ACN$.
Let $n>1$ be an integer. There are $n$ orangutoads, conveniently numbered $1,2,\dots{},n$, each sitting at an integer position on the number line. They take turns moving in the order $1,2,\dots{},n$, and then going back to $1$ to start the process over; they stop if any orangutoad is ever unable to move. To move, an orangutoad chooses another orangutoad who is at least $2$ units away from her towards them by a a distance of $1$ unit. (Multiple orangutoads can be at the same position.) Show that eventually some orangutoad will be unable to move.
Let $\mathcal M$ be the set of $n\times n$ matrices with integer entries. Find all $A\in\mathcal M$ such that $\det(A+B)+\det(B)$ is even for all $B\in\mathcal M$.
[i]Proposed by Ethan Tan[/i]
Let $AA_1, BB_1, CC_1$ be the heights of triangle $ABC$ and $H$ be its orthocenter. Liune $\ell$ parallel to $AC$, intersects straight lines $AA_1$ and $CC_1$ at points $A_2$ and $C_2$, respectively. Suppose that point $B_1$ lies outside the circumscribed circle of triangle $A_2 HC_2$. Let $B_1P$ and $B_1T$ be tangent to of this circle. Prove that points $A_1, C_1, P$, and $T$ are cyclic.
Let $a,b$ be two integers. Prove that
a) $13 \mid 2a+3b$ if and only if $13 \mid 2b-3a$;
b) If $13 \mid a^2+b^2$ then $13 \mid (2a+3b)(2b+3a)$.
[i]Mircea Fianu[/i]
Determine all natural numbers $n$ for which there are $2n$ distinct positive integers $x_1,…,x_n,y_1,…,y_n$
such that the product $$(11x^2_1+12y^2_1)(11x^2_2+12y^2_2)…(11x^2_n+12y^2_n)$$ is a perfect square.
Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]
A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square?
[i]Proposed by Evan Chen[/i]
Find all couples of natural numbers $(a,b)$ not relatively prime ($\gcd(a,b)\neq\ 1$) such that
\[\gcd(a,b)+9\operatorname{lcm}[a,b]+9(a+b)=7ab.\]
Let $ABC$ be an acute triangle. Denote by $H$ and $M$ the orthocenter of $ABC$ and the midpoint
of side $BC,$ respectively. Let $Y$ be a point on $AC$ such that $YH$ is perpendicular to $MH$ and let $Q$ be a point
on $BH$ such that $QA$ is perpendicular to $AM.$ Let $J$ be the second point of intersection of $MQ$ and the circle
with diameter $MY.$ Prove that $HJ$ is perpendicular to $AM.$
(Steve Dinh)
Let $f:\mathbb{N}\mapsto\mathbb{R}$ be the function \[f(n)=\sum_{k=1}^\infty\dfrac{1}{\operatorname{lcm}(k,n)^2}.\] It is well-known that $f(1)=\tfrac{\pi^2}6$. What is the smallest positive integer $m$ such that $m\cdot f(10)$ is the square of a rational multiple of $\pi$?
Let $p, q$ be positive integers and let $x_{0}=0$. Suppose $x_{n+1}=x_{n} + p + \sqrt{q^{2} + 4px_{n}}$. Find an explicit formula for $x_{n}$.
Find the integers $x, y, z$ for which $$\dfrac{1}{x+\dfrac{1}{y+\dfrac{1}{z}}}=\dfrac{7}{17}$$
There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold:
[i]i.)[/i] Each pair of students are in exactly one club.
[i]ii.)[/i] For each student and each society, the student is in exactly one club of the society.
[i]iii.)[/i] Each club has an odd number of students. In addition, a club with ${2m+1}$ students ($m$ is a positive integer) is
in exactly $m$ societies.
Find all possible values of $k$.
[i]Proposed by Guihua Gong, Puerto Rico[/i]
In how many ways, can we draw $n-3$ diagonals of a $n$-gon with equal sides and equal angles such that:
$i)$ none of them intersect each other in the polygonal.
$ii)$ each of the produced triangles has at least one common side with the polygonal.
$n$ points are given on a circle $\omega$. There is a circle with radius smaller than $\omega$ such that all these points lie inside or on the boundary of this circle. Prove that we can draw a diameter of $\omega$ with endpoints not belonging to the given points such that all the $n$ given points remain in one side of the diameter.