For positive integer $x>1$, define set $S_x$ as $$S_x=\{p^\alpha|p \textup{ is one of the prime divisor of }x,\alpha \in \mathbb{N},p^\alpha|x,\alpha \equiv v_p(x)(\textup{mod} 2)\},$$ where $v_p(n)$ is the power of prime divisor $p$ in positive integer $n.$ Let $f(x)$ be the sum of all the elements of $S_x$ when $x>1,$ and $f(1)=1.$ Let $m$ be a given positive integer, and the sequence $a_1,a_2,\cdots,a_n,\cdots$ satisfy that for any positive integer $n>m,$ $a_{n+1}=\max\{ f(a_n),f(a_{n-1}+1),\cdots,f(a_{n-m}+m)\}.$ Prove that (1)there exists constant $A,B(0<A<1),$ such that when positive integer $x$ has at least two different prime divisors, $f(x)<Ax+B$ holds; (2)there exists positive integer $Q$, such that for any positive integer $n,a_n<Q.$

Each of the 2001 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let $m$ be the smallest number of students who could study both languages, and let $M$ be the largest number of students who could study both languages. Find $M-m$.

Integers $ a$, $ b$, $ c$, and $ d$, not necessarily distinct, are chosen independently and at random from $ 0$ to $ 2007$, inclusive. What is the probability that $ ad \minus{} bc$ is even? $ \textbf{(A)}\ \frac {3}{8}\qquad \textbf{(B)}\ \frac {7}{16}\qquad \textbf{(C)}\ \frac {1}{2}\qquad \textbf{(D)}\ \frac {9}{16}\qquad \textbf{(E)}\ \frac {5}{8}$

Jerry buys a bottle of 150 pills. Using a standard 12 hour clock, he sees that the clock reads exactly 12 when he takes the first pill. If he takes one pill every five hours, what hour will the clock read when he takes the last pill in the bottle?

Let $E$ denote the set of $25$ points $(m,n)$ in the $\text{xy}$-plane, where $m,n$ are natural numbers, $1\leq m\leq5,1\leq n\leq5$. Suppose the points of $E$ are arbitrarily coloured using two colours, red and blue. SHow that there always exist four points in the set $E$ of the form $(a,b),(a+k,b),(a+k,b+k),(a,b+k)$ for some positive integer $k$ such that at least three of these four points have the same colour. (That is, there always exist four points in the set $E$ which form the vertices of a square with sides parallel to the axes and having at least three points of the same colour.)

A right triangle $ABC$ with hypotenuse $AB$ is inscribed in a circle. Let $K$ be the midpoint of the arc $BC$ not containing $A, N$ the midpoint of side $AC$, and $M$ a point of intersection of ray $KN$ with the circle. Let $E$ be a point of intersection of tangents to the circle at points $A$ and $C$. Prove that $\angle EMK = 90^\circ$.

Let $ n$ be a composite. Prove that there exists positive integer $ m$ satisfying $ m|n, m\le\sqrt {n},$ and $ d(n)\le d^3(m).$ Where $ d(k)$ denotes the number of positive divisors of positive integer $ k.$

Two circles $\gamma_1, \gamma_2$ are given, with centers at points $O_1, O_2$ respectively. Select a point $K$ on circle $\gamma_2$ and construct two circles, one $\gamma_3$ that touches circle $\gamma_2$ at point $K$ and circle $\gamma_1$ at a point $A$, and the other $\gamma_4$ that touches circle $\gamma_2$ at point $K$ and circle $\gamma_1$ at a point $B$. Prove that, regardless of the choice of point K on circle $\gamma_2$, all lines $AB$ pass through a fixed point of the plane.

Given is a prime number $p$ and natural $n$ such that $p \geq n \geq 3$. Set $A$ is made of sequences of lenght $n$ with elements from the set $\{0,1,2,...,p-1\}$ and have the following property: For arbitrary two sequence $(x_1,...,x_n)$ and $(y_1,...,y_n)$ from the set $A$ there exist three different numbers $k,l,m$ such that: $x_k \not = y_k$, $x_l \not = y_l$, $x_m \not = y_m$. Find the largest possible cardinality of $A$.

It is given the function $f:\mathbb{R}\rightarrow \mathbb{R}$ fow which $f(1)=1$ and for all $x\in\mathbb{R}$ satisfied $f(x+5)\geq f(x)+5$ and $f(x+1)\leq f(x)+1$ If $g(x)=f(x)-x+1$ then find $g(2016)$ .

For a real number $x,$ let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x,$ and let $\{x\} = x -\lfloor x\rfloor$ denote the fractional part of $x.$ The sum of all real numbers $\alpha$ that satisfy the equation $$\alpha^2+\{\alpha\}=21$$ can be expressed in the form $$\frac{\sqrt{a}-\sqrt{b}}{c}-d$$ where $a, b, c,$ and $d$ are positive integers, and $a$ and $b$ are not divisible by the square of any prime. Compute $a + b + c + d.$

Let $x, y, z$ be real numbers each of whose absolute value is different from $\frac{1}{\sqrt 3}$ such that $x + y + z = xyz$. Prove that \[\frac{3x - x^3}{1-3x^2} + \frac{3y - y^3}{1-3y^2} + \frac{3z -z^3}{1-3z^2} = \frac{3x - x^3}{1-3x^2} \cdot \frac{3y - y^3}{1-3y^2} \cdot \frac{3z - z^3}{1-3z^2}\]

Prove that in the set $ \{1,2, \ldots, 1989\}$ can be expressed as the disjoint union of subsets $ A_i, \{i \equal{} 1,2, \ldots, 117\}$ such that [b]i.)[/b] each $ A_i$ contains 17 elements [b]ii.)[/b] the sum of all the elements in each $ A_i$ is the same.

Find all triples of positive integers $(a,b,p)$ with $a,b$ positive integers and $p$ a prime number such that $2^a+p^b=19^a$

Define a positive integer $n$ to be a fake square if either $n = 1$ or $n$ can be written as a product of an even number of not necessarily distinct primes. Prove that for any even integer $k \geqslant 2$, there exist distinct positive integers $a_1$, $a_2, \cdots, a_k$ such that the polynomial $(x+a_1)(x+a_2) \cdots (x+a_k)$ takes ‘fake square’ values for all $x = 1,2,\cdots,2023$. [i]Proposed by Prof. Aditya Karnataki[/i]

A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars would increase sales. If the diameter of the jars is increased by $ 25\%$ without altering the volume, by what percent must the height be decreased? $ \textbf{(A)}\ 10\% \qquad \textbf{(B)}\ 25\% \qquad \textbf{(C)}\ 36\% \qquad \textbf{(D)}\ 50\% \qquad \textbf{(E)}\ 60\%$

Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$. [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

An $11\times 11\times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point? $\textbf{(A) }328\qquad \textbf{(B) }329\qquad \textbf{(C) }330\qquad \textbf{(D) }331\qquad \textbf{(E) }332\qquad$

Let $a,b,c\in R^+$ with $a+b+c=3$. Prove that $$2(ab+bc+ca)\le 5+ abc$$ [i](Real Matrik)[/i]

Let \[f(x)=\dfrac{1}{1-\dfrac{1}{1-x}}\,.\] Compute $f^{2016}(2016)$, where $f$ is composed upon itself $2016$ times.

Determine the number of positive integer solutions $(x,y, z)$ to the equation $xyz = 2(x + y + z)$.

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(n+1) > f(f(n)).\]

About $5$ years ago, Joydip was researching on the number $2017$. He understood that $2017$ is a prime number. Then he took two integers $a,b$ such that $0<a,b <2017$ and $a+b\neq 2017.$ He created two sequences $A_1,A_2,\dots ,A_{2016}$ and $B_1,B_2,\dots, B_{2016}$ where $A_k$ is the remainder upon dividing $ak$ by $2017$, and $B_k$ is the remainder upon dividing $bk$ by $2017.$ Among the numbers $A_1+B_1,A_2+B_2,\dots A_{2016}+B_{2016}$ count of those that are greater than $2017$ is $N$. Prove that $N=1008.$

A square $ABCD,AB=s=1$ is given in the plane with its center $S$. Furthermore, points $E,F$ are given on the rays opposite to $CB,DA$, respectively, $CE=a,DF=b$. Determine all triangles $XYZ$ such that $X,Y,Z$ lie in this order on segments $CD,AD,BC$ and $E,S,F$ lie on lines $XY,YZ,ZX$ respectively. Discuss conditions of solvability in terms of $a,b,s$ and unknown $x=CX$.

Let $AL$ be the bisector of triangle $ABC$, $D$ be its midpoint, and $E$ be the projection of $D$ to $AB$. It is known that $AC = 3AE$. Prove that $CEL$ is an isosceles triangle.