Let $P$ be an arbitrary interior point of the equilateral triangle $ABC$. From $P$ draw parallel to the sides: $A'_1A_1 \parallel AB$, $B' _1B_1 \parallel BC$ and $C'_1C_1 \parallel CA$. Prove that the sum of legths $| AC_1 | + | BA_1 | + | CB_1 |$ is independent of the choice of point $P$. [img]https://cdn.artofproblemsolving.com/attachments/5/a/15b06706c09e2458fb5938807b9f3833ffb62e.png[/img]

Let $a,b,c$ denote the sides of a triangle and $[ABC]$ the area of the triangle as usual. $(a)$ If $6[ABC] = 2a^2+bc$, determine $A,B,C$. $(b)$ For all triangles, prove that $3a^2+3b^2 - c^2 \ge 4 \sqrt{3} [ABC]$.

Let $ABCD$ be a parallelogram such that $AB = 35$ and $BC = 28$. Suppose that $BD \perp BC$. Let $\ell_1$ be the reflection of $AC$ across the angle bisector of $\angle BAD$, and let $\ell_2$ be the line through $B$ perpendicular to $CD$. $\ell_1$ and $\ell_2$ intersect at a point $P$. If $PD$ can be expressed in simplest form as $\frac{m}{n}$, find $m + n$.

Let a chain denote a row of positive integers which continue infinitely in both directions, such that for each number $n$, the $n$ numbers directly to the left of $n$ yield $n$ distinct remainders upon division by $n$. (a) If a chain has a maximum integer, what are the possible values of that integer? (b) Does there exist a chain which does not have a maximum integer?

Let $a$ and $b$ be positive integers. Prove that the numbers $an^2+b$ and $a(n+1)^2+b$ are both perfect squares only for finitely many integers $n$.

Determine all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ with $ x,y \in \mathbb{R}$ such that \[ f(x \minus{} f(y)) \equal{} f(x\plus{}y) \plus{} f(y)\]

Let $n$ be a positive integers. Equilateral triangle with sides of length $n$ is split into equilateral triangles with side lengths $1$, forming a triangular lattice. Call an equilateral triangle with vertices in the lattice "important". Let $p_k$ be the number of unordered pairs of vertices in the lattice which participate in exactly $k$ important triangles. Find (as a function of $n$) (a) $p_0+p_1+p_2$ (b) $p_1+2p_2$

a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime? b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?

Given a convex $ n $-gon ($ n \geq 5 $). Prove that the number of triangles of area $1$ with vertices at the vertices of the $ n $-gon does not exceed $ \frac{1}{3} n (2n-5) $.

Given $m,n \in N$ such that $M>n^{n-1}$ and the numbers $m+1, m+2, ..., m+n$ are composite. Prove that exist distinct primes $p_1,p_2,...,p_n$ such that $M+k$ is divisible by $p_k$ for any $k=1,2,...,n$. Tuymaada Olympiad 2004, C.A.Grimm. USA

Let $p$ be a prime number and $a_1, \ldots, a_{p-1}$ be not necessarily distinct nonzero elements of the $p$-element $\mathbb Z_p \pmod{p}$ group. Prove that each element of $\mathbb Z_p$ equals a sum of some of the $a_i$'s (the empty sum is $0$). (translated by Miklós Maróti)

Let $n$ be a positive integer. On a number line, Azer is at point $0$ in his car which have fuel capacity of $2^n$ units and is initially full. At each positive integer $m$, there is a gas station. Azer only moves to the right with constant speed and doesn't stop anywhere except the gas stations. Each time his car moves to the right by some amount, its fuel decreases by the same amount. Azer may choose to stop at a gas station or pass it. There are thieves at some gas stations. (A station may have multiple thieves) If Azer stops at a station which have $k\ge 0$ thieves while its car have fuel capacity $d$, his cars new fuel capacity becomes $\frac{d}{2^k}$. After that, Azer fulls his cars tank and leaves the station. Find the minimum number of thieves needed to guarantee that Azer will eventually run out of fuel. Proposed by[i] Mehmet Can Baştemir[/i] and [i]Deniz Can Karaçelebi[/i]

Prove that a)$\sum_{k=1}^{1008}kC_{2017}^{k}\equiv 0$ (mod $2017^2$ ) b)$\sum_{k=1}^{504}\left ( -1 \right )^kC_{2017}^{k}\equiv 3\left ( 2^{2016}-1 \right )$ (mod $2017^2$ )

A triangle has area 114 and sides of integer length. What is the perimeter of the triangle?

Let $n \ge 2$ be a positive integer. Prove that the following assertions are equivalent: a) for all integer $x$ coprime with n the congruence $x^6 \equiv 1$ (mod $n$) hold, b) $n$ divides $504$.

$(a_{n})$ is sequence with positive integer. $a_{1}>10$ $ a_{n}=a_{n-1}+GCD(n,a_{n-1})$, n>1 For some i $a_{i}=2i$. Prove that these numbers are infinite in this sequence.