Pratyya and Payel have a number each, $n$ and $m$ respectively, where $n>m.$ Everyday, Pratyya multiplies his number by $2$ and then subtracts $2$ from it, and Payel multiplies his number by $2$ and then add $2$ to it. In other words, on the first day their numbers will be $(2n-2)$ and $(2m+2)$ respectively. Find minimum integer $x$ with proof such that if $n-m\geq x,$ then Pratyya's number will be larger than Payel's number everyday.

Let $A$ be the set of positive real numbers less than $1$ that have a periodic decimal expansion with a period of ten different digits. Find a positive integer $n$ greater than $1$ and less than $10^{10}$ such that $na-a$ is a positive integer for all $a$. of set $A$.

A graph has $n$ points and $\frac{n(n-1)}{2}$ edges. Each edge is colored with one of $k$ colors so that there are no closed monochrome paths. What is the largest possible value of $n$ (given $k$)?

In triangle $ABC$, points $D$, $E$, and $F$ are on sides $BC$, $CA$, and $AB$ respectively, such that $BF = BD = CD = CE = 5$ and $AE - AF = 3$. Let $I$ be the incenter of $ABC$. The circumcircles of $BFI$ and $CEI$ intersect at $X \neq I$. Find the length of $DX$.

In a school with $101$ students, each student has at least one friend among the other students. Show that for every integer $1<n<101$, a group of $n$ students can be selected from this school in such a way that each selected student has at least one friend among the other selected students.

All the students in an algebra class took a $ 100$-point test. Five students scored $ 100$, each student scored at least $ 60$, and the mean score was $ 76$. What is the smallest possible number of students in the class? $ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ 11 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 13 \qquad \textbf{(E)}\ 14$

Find all finite and non-empty sets $A$ of functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that for all $f_1, f_2 \in A$, there exists $g \in A$ such that for all $x, y \in \mathbb{R}$ $$f_1 \left(f_2 (y)-x\right)+2x=g(x+y)$$

Define $m(n)$ to be the greatest proper natural divisor of $n\in \mathbb{N}$. Find all $n \in \mathbb{N} $ such that $n+m(n) $ is a power of $10$. [i]N. Agakhanov[/i]

Let $f(x,y)$ be a function from ordered pairs of positive integers to real numbers such that \[ f(1,x) = f(x,1) = \frac{1}{x} \quad\text{and}\quad f(x+1,y+1)f(x,y)-f(x,y+1)f(x+1,y) = 1 \] for all ordered pairs of positive integers $(x,y)$. If $f(100,100) = \frac{m}{n}$ for two relatively prime positive integers $m,n$, compute $m+n$. [i]David Yang[/i]

Given real positive a,b , find the larget real c such that $c\leq max(ax+\frac{1}{ax},bx+\frac{1}{bx})$ for all positive ral x. There is a solution here,,,, http://www.kalva.demon.co.uk/irish/soln/sol039.html but im wondering if there is a better one . Thank you.

Two equal hard triangles are given. One of their angles is equal to $ \alpha$ (these angles are marked). Dispose these triangles on the plane in such a way that the angle formed by some three vertices would be equal to $ \alpha / 2$. [i](No instruments are allowed, even a pencil.)[/i] (E. Bakayev, A. Zaslavsky)

Let $ \alpha$ be the positive root of the equation $ x^{2} \equal{} 1991x \plus{} 1$. For natural numbers $ m$ and $ n$ define \[ m*n \equal{} mn \plus{} \lfloor\alpha m \rfloor \lfloor \alpha n\rfloor. \] Prove that for all natural numbers $ p$, $ q$, and $ r$, \[ (p*q)*r \equal{} p*(q*r). \]

Octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle where $A_1A_2=A_2A_3=A_3A_4=A_6A_7=13$ and $A_4A_5=A_5A_6=A_7A_8=A_8A_1=5. $ The sum of all possible areas of $A_1A_2A_3A_4A_5A_6A_7A_8$ can be expressed as $a+b\sqrt{c}$ where $\gcd{a,b}=1$ and $c$ is squarefree. Find $abc.$ [asy] label("$A_1$",(5,0),E); label("$A_2$",(2.92, -4.05),SE); label("$A_3$",(-2.92,-4.05),SW); label("$A_4$",(-5,0),W); label("$A_5$",(-4.5,2.179),NW); label("$A_6$",(-3,4), NW); label("$A_7$",(3,4), NE); label("$A_8$",(4.5,2.179),NE); draw((5,0)--(2.9289,-4.05235)); draw((2.92898,-4.05325)--(-2.92,-4.05)); draw((-2.92,-4.05)--(-5,0)); draw((-5,0)--(-4.5, 2.179)); draw((-4.5, 2.179)--(-3,4)); draw((-3,4)--(3,4)); draw((3,4)--(4.5,2.179)); draw((4.5,2.179)--(5,0)); dot((0,0)); draw(circle((0,0),5)); [/asy]

Suppose $ABCD$ is a cyclic quadrilateral with $\angle ABC = \angle ADC = 90^{\circ}$. Let $E$ and $F$ be the feet of perpendiculars from $A$ and $C$ to $BD$ respectively. Prove that $BE = DF$.

[b]2.[/b] Let $S$ be a given finite set of hyperplanes in $\mathbb{R}^n$, and let $O$ be a point. Show that there exists a compact set $K \subseteq \mathbb{R}^n$ containing $O$ such that the orthogonal projection of any point of $K$ onto any hyperplane in $S$ is also in $K$. ([b]G.37[/b]) [Gy. Pap]

Figure $ABCD$ is a trapezoid with $AB || DC, AB = 5, BC = 3 \sqrt 2, \measuredangle BCD = 45^\circ$, and $\measuredangle CDA = 60^\circ$. The length of $DC$ is $\textbf{(A) }7 + \frac{2}{3} \sqrt{3}\qquad \textbf{(B) }8\qquad \textbf{(C) }9 \frac{1}{2}\qquad \textbf{(D) }8 + \sqrt 3\qquad \textbf{(E) }8 + 3 \sqrt 3$

Suppose a function $ f: \mathbb R\to \mathbb R$ satisfies $ f(f(x)) \equal{} \minus{} x$ for all $ x\in \mathbb R$. Prove that $ f$ has infinitely many points of discontinuity.

How many ordered triples $(x, y, z)$ of positive integers satisfy $\text{lcm}(x, y) = 72$, $\text{lcm}(x, z)= 600$, and $\text{lcm}(y, z) = 900$? $\textbf{(A) } 15 \qquad\textbf{(B) } 16 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 27 \qquad\textbf{(E) } 64$

It is given $4$ circles in a plane and every one of them touches the other three as shown: [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC82L2FkYWQ5NThhMWRiMjAwZjYxOWFhYmE1M2YzZDU5YWI2N2IyYzk2LnBuZw==&rn=a3J1Z292aS5wbmc=[/img] Biggest circle has radius $2$, and every one of the medium has $1$. Find out the radius of fourth circle.

There is a ruler and a "rusty" compass, with which you can construct a circle of radius $R$. The point $K$ is from the line $\ell$ at a distance greater than $R$. How to use this ruler and this compass to draw a line passing through the point $K$ and perpendicular to line $\ell$? (Misha Sidorenko, Katya Sidorenko, Rodion Osokin)

Let $ABC$ be a triangle . A circle through $B$ and $C$ crosses sides $AB$ and $AC$ at $P$ and $Q$, respectively. Points $X$ and $Y$ on segments $BQ$ and $CP$, respectively, satisfy $\angle ABY=\angle AXP$ and $ACX=\angle AYQ$. Prove that $XY$ and $BC$ are parallel.

The following number is the product of the divisors of $n$. $$46, 656, 000, 000$$ What is $n$?

If the inequality \[ ((x+y)^2+4)((x+y)^2-2)\geq A\cdot (x-y)^2 \] is hold for every real numbers $x,y$ such that $xy=1$, what is the largest value of $A$? $ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 14 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 20 $

In a concert, 20 singers will perform. For each singer, there is a (possibly empty) set of other singers such that he wishes to perform later than all the singers from that set. Can it happen that there are exactly 2010 orders of the singers such that all their wishes are satisfied? [i]Proposed by Gerhard Wöginger, Austria[/i]

Amy has a six-sided die which always rolls values greater than or equal to the previous roll. She rolls the die repeatedly until she rolls a $6$. Find the expected value of the sum of all distinct values she has rolled when she finishes.