Suppose integers $a < b < c$ satisfy \[ a + b + c = 95\qquad\text{and}\qquad a^2 + b^2 + c^2 = 3083.\] Find $c$.

Let $a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3,\ldots$ be positive integers such that $a_{n+2} = a_n + a_{n+1}$ and $b_{n+2} = b_n + b_{n+1}$ for all $n \ge 1$. Assume that $a_n$ divides $b_n$ for infinitely many values of $n$. Prove that there exists an integer $c$ such that $b_n = c a_n$ for all $n \ge 1$.

Let $t(n)$ be the sum of the digits in the binary representation of a positive integer $n,$ and let $k \geq 2$ be an integer. [b]a.[/b] Show that there exists a sequence $(a_i)_{i=1}^{\infty}$ of integers such that $a_m \geq 3$ is an odd integer and $t(a_1a_2 \cdots a_m)=k$ for all $m \geq 1.$ [b]b.[/b] Show that there is an integer $N$ such that $t(3 \cdot 5 \cdots (2m+1))>k$ for all integers $m \geq N.$

Show that for all non-constant functions $f:\mathbb R\to\mathbb R$, there are two real numbers $x,y$ such that $$f(x+f(y))>xf(y)+x.$$

Bisectors $AA_1$ and $BB_1$ of a right triangle $ABC \ (\angle C=90^\circ )$ meet at a point $I.$ Let $O$ be the circumcenter of triangle $CA_1B_1.$ Prove that $OI \perp AB.$

Let $x,y,z\in\mathbb{R}$ and $xyz=-1$. Prove that: \[ x^4+y^4+z^4+3(x+y+z)\geq\frac{x^2}{y}+\frac{x^2}{z}+\frac{y^2}{x}+\frac{y^2}{z}+\frac{z^2}{x}+\frac{z^2}{y}. \]

Let us call a triangle “almost right angle triangle” if one of its angles differs from $90^\circ$ by no more than $15^\circ$. Let us call a triangle “almost isosceles triangle” if two of its angles differs from each other by no more than $15^\circ$. Is it true that that any acute triangle is either “almost right angle triangle” or “almost isosceles triangle”? [i](2 points)[/i]

Determine all pairs $(a,b)$ of real numbers with $a\leqslant b$ that maximise the integral $$\int_a^b e^{\cos (x)}(380-x-x^2) \mathrm{d} x.$$

In $\triangle ABC$, $\angle A=\alpha$, $CD,BE$ are height on sides $AB,AC$. Then$\frac{|DE|}{|BC|}=$________.

Let $a$ and $n$ be positive integers and $s = a + a^2 + \cdots + a^n$. Prove that the last digit of $s$ is $1$ if and only if the last digits of $a$ and $n$ are both equal to $1$.

Let $a,b,c,d$ be real numbers. Prove that the set $M=\left\{ax^3+bx^2+cx+d|x\in\mathbb R\right\}$ contains no irrational numbers if and only if $a=b=c=0$ and $d$ is rational.

The rectangular parallelepiped (box) $P$ has some special properties. If one dimension of $P$ were doubled and another dimension were halved, then the surface area of $P$ would stay the same. If instead one dimension of $P$ were tripled and another dimension were divided by $3$, then the surface area of $P$ would still stay the same. If the middle (by length) dimension of $P$ is $1$, compute the least possible volume of $P$.

Graphistan has $2011$ cities and Graph Air (GA) is running one-way flights between all pairs of these cities. Determine the maximum possible value of the integer $k$ such that no matter how these flights are arranged it is possible to travel between any two cities in Graphistan riding only GA flights as long as the absolute values of the difference between the number of flights originating and terminating at any city is not more than $k.$

The triangle $ABC$ has circumcentre $O$ and circumcircle $\Gamma$. Let $AI$ be a diameter of $\Gamma$. The ray $AI$ extends to intersect the circumcircle $\omega$ of $\triangle BOC$ for the second time at a point $P$. Let $AD$ and $IQ$ be perpendicular to $BC$, with $D$ and $Q$ on $BC$. Let $M$ be the midpoint of $BC$. (a) Prove that $|AD| \cdot |QI| = |CD| \cdot |CQ| = |BD| \cdot |BQ|$. (b) Prove that $IM$ is parallel to $PD$.

There are $9$ point in the Cartezian plane with coordinates $(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2).$ Some points are coloured in red and the others in blue. Prove that for any colouring of the points we can always find a right isosceles triangle whose vertexes have the same colour.

We have $1998$ polygon such that sum of the areas is $1997,5$ $cm^2$. These polygons placing inside a square with side lenght $1$ $cm$. (Polygons no overflow). Prove that we can find a point such that, all polygons have this point.

Consider $9$ points in the interior of a square of side $1$. Prove that there are three of them that form a triangle with an area less than or equal to $\frac18$ .

In a sequence of positive integers an inversion is a pair of positions such that the element in the position to the left is greater than the element in the position to the right. For instance the sequence $2,5,3,1,3$ has five inversions - between the first and fourth positions, the second and all later positions, and between the third and fourth positions. What is the largest possible number of inversions in a sequence of positive integers whose sum is $2014$?

Sunny wants to send some secret message to usjl. The secret message is a three digit number, where each digit is one digit from $0$ to $9$ (so $000$ is also possibly the secret message). However, when Sunny sends the message to usjl, at most one digit might be altered. Therefore, Sunny decides to send usjl a longer message so that usjl can decipher the message to get the original secret message Sunny wants to send. Sunny and usjl can communicate the strategy beforehand. Show that sending a $4$-digit message does not suffice. Also show that sending a $6$-digit message suffices. If it is deduced that sending a $c$-digit message suffices for some $c>6$, then partial credits may be awarded.

Alice and Bob live $10$ miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is $30^\circ$ from Alice's position and $60^\circ$ from Bob's position. Which of the following is closest to the airplane's altitude, in miles? $\textbf{(A)}\ 3.5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 4.5 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 5.5$

For a natural number $n{},$ determine the number of ordered pairs $(S,T)$ of subsets of $\{1,2,\ldots,n\}$ for which $s>|T|$ for any element $s\in S$ and $t>|S|$ for any element $t\in T.$

Describe all the ways to color each natural number as one of three colors so that the following condition is satisfied: if the numbers $a$, $b$ and $c$ (not necessarily different) satisfy the condition $2000(a + b) = c$, then they either all the same color or three different colors

Find the number of integers $n$ with $1\le n\le 100$ for which $n-\phi(n)$ is prime. Here $\phi(n)$ denotes the number of positive integers less than $n$ which are relatively prime to $n$. [i]Proposed by Mehtaab Sawhney[/i]

Positive integers $a,b,c$ satisfying the equation $$a^3+4b+c = abc,$$ where $a \geq c$ and the number $p = a^2+2a+2$ is a prime. Prove that $p$ divides $a+2b+2$.

Let $f(x)$ be an odd function on $\mathbb{R}$, such that $f(x)=x^2$ when $x\ge 0$. Knowing that for all $x\in [a,a+2]$, the inequality $f(x+a)\ge 2f(x)$ holds, find the range of real number $a$.