Let $AD$ be the inner angle bisector of the triangle $ABC$. The perpendicular on the side $BC$ at the point $D$ intersects the outer bisector of $\angle CAB$ at point $I$. The circle with center $I$ and radius $ID$ intersects the sides $AB$ and $AC$ at points $F$ and $E$ respectively. $A$-symmedian of $\Delta AFE$ intersects the circumcircle of $\Delta AFE$ again at point $X$. Prove that the circumcircles of $\Delta AFE$ and $\Delta BXC$ are tangent.

Each of the $36$ squares of a $6 \times 6$ table is to be coloured either Red, Yellow or Blue. (a) No row or column is contain more than two squares of the same colour. (b) In any four squares obtained by intersecting two rows with two columns, no colour is to occur exactly three times. In how many dierent ways can the table be coloured if both of these rules are to be respected?

For each integer $n\geq 1$, let $S_n$ be the set of integers $k > n$ such that $k$ divides $30n-1$. How many elements of the set \[\mathcal{S} = \bigcup_{i\geq 1}S_i = S_1\cup S_2\cup S_3\cup\ldots\] are less than $2016$?

Find all positive numbers $x$ such that $20\{x\}+0.5\lfloor x\rfloor = 2005$.

Find x, y, z $\in Z$\\$x^2+y^2+z^2=2^n(x+y+z)$\\$n\in N$

Right triangle $\vartriangle ABC$ with its right angle at $B$ has angle bisector $\overline{AD}$ with $D$ on $\overline{BC}$, as well as altitude $\overline{BE}$ with $E$ on $\overline{AC}$. If $\overline{DE} \perp \overline{BC}$ and $AB = 1$, compute $AC$.

Find all triples $(p,M, z)$ of integers, where $p$ is prime, $m$ is positive and $z$ is negative, that satisfy the equation $$p^3 + pm + 2zm = m^2 + pz + z^2$$

Let $a_1,a_2,\cdots,a_n$ be a sequence of positive numbers, so that $a_1 + a_2 +\cdots + a_n \leq \frac 12$. Prove that \[(1 + a_1)(1 + a_2) \cdots (1 + a_n) < 2.\] [hide="Remark"]Remark. I think this problem was posted before, but I can't find the link now.[/hide]

Construct a geometric gure in a sequence of steps. In step $1$, begin with a $4\times 4$ square. In step $2$, attach a $1\times 1$ square onto the each side of the original square so that the new squares are on the outside of the original square, have a side along the side of the original square, and the midpoints of the sides of the original square and the attached square coincide. In step $3$, attach a $\frac14\times  \frac14$ square onto the centers of each of the $3$ exposed sides of each of the $4$ squares attached in step $2$. For each positive integer $n$, in step $n + 1$, attach squares whose sides are $\frac14$ as long as the sides of the squares attached in step n placing them at the centers of the $3$ exposed sides of the squares attached in step $n$. The diagram shows the gure after step $4$. If this is continued for all positive integers $n$, the area covered by all the squares attached in all the steps is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$. [img]https://cdn.artofproblemsolving.com/attachments/2/1/d963460373b56906e93c4be73bc6a15e15d0d6.png[/img]

Given the list \[A=[9,12,1,20,17,4,10,7,15,8,13,14],\] we would like to sort it in increasing order. To accomplish this, we will perform the following operation repeatedly: remove an element, then insert it at any position in the list, shifting elements if necessary. What is the minimum number of applications of this operation necessary to sort $A$?

[b]p1.[/b] A right triangle has a hypotenuse of length $25$ and a leg of length $16$. Compute the length of the other leg of this triangle. [b]p2.[/b] Tanya has a circular necklace with $5$ evenly-spaced beads, each colored red or blue. Find the number of distinct necklaces in which no two red beads are adjacent. If a necklace can be transformed into another necklace through a series of rotations and reflections, then the two necklaces are considered to be the same. [b]p3.[/b] Find the sum of the digits in the decimal representation of $10^{2016} - 2016$. [b]p4.[/b] Let $x$ be a real number satisfying $$x^1 \cdot x^2 \cdot x^3 \cdot x^4 \cdot x^5 \cdot x^6 = 8^7.$$ Compute the value of $x^7$. [b]p5.[/b] What is the smallest possible perimeter of an acute, scalene triangle with integer side lengths? [b]p6.[/b] Call a sequence $a_1, a_2, a_3,..., a_n$ mountainous if there exists an index $t$ between $1$ and $n$ inclusive such that $$a_1 \le a_2\le ... \le a_t \,\,\,\, and \,\,\,\, a_t \ge a_{t+1} \ge ... \ge a_n$$ In how many ways can Bishal arrange the ten numbers $1$, $1$, $2$, $2$, $3$, $3$, $4$, $4$, $5$, and $5$ into a mountainous sequence? (Two possible mountainous sequences are $1$, $1$, $2$, $3$, $4$, $4$, $5$, $5$, $3$, $2$ and $5$, $5$, $4$, $4$, $3$, $3$, $2$, $2$, $1$, $1$.) [b]p7.[/b] Find the sum of the areas of all (non self-intersecting) quadrilaterals whose vertices are the four points $(-3,-6)$, $(7,-1)$, $(-2, 9)$, and $(0, 0)$. [b]p8.[/b] Mohammed Zhang's favorite function is $f(x) =\sqrt{x^2 - 4x + 5} +\sqrt{x^2 + 4x + 8}$. Find the minumum possible value of $f(x)$ over all real numbers $x$. [b]p9.[/b] A segment $AB$ with length $1$ lies on a plane. Find the area of the set of points $P$ in the plane for which $\angle APB$ is the second smallest angle in triangle $ABP$. [b]p10.[/b] A binary string is a dipalindrome if it can be produced by writing two non-empty palindromic strings one after the other. For example, $10100100$ is a dipalindrome because both $101$ and $00100$ are palindromes. How many binary strings of length $18$ are both palindromes and dipalindromes? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $\forall x\in \mathbb{R} \ \ f(x) = max(2xy-f(y))$ where $y\in \mathbb{R}$.

The plane has been partitioned into $N$ regions by three bunches of parallel lines. What is the least number of lines needed in order that $N > 1981$?

There are given $20$ points in the plane, no three of which lie on a single line. Prove that there exist at least $969$ quadrilaterals with vertices from the given points.

Mattis is hosting a badminton tournament for $40$ players on $20$ courts numbered from $1$ to $20$. The players are distributed with $2$ players on each court. In each round a winner is determined on each court. Afterwards, the player who lost on court $1$, and the player who won on court $20$ stay in place. For the remaining $38$ players, the winner on court $i$ moves to court $i + 1$ and the loser moves to court $i - 1$. The tournament continues until every player has played every other player at least once. What is the minimal number of rounds the tournament can last?

Find the number of different positive integer triples $(x, y,z)$ satisfying the equations $x^2 + y -z = 100$ and $x + y^2 - z = 124$:

Prove that there exist infinitely many odd positive integers $n$ for which the number $2^n + n$ is composite. (V Senderov)

For any positive numbers $a,b,c$ and natural numbers $n,k$ prove the inequality $$\frac{a^{n+k}}{b^n}+\frac{b^{n+k}}{c^n}+\frac{c^{n+k}}{a^n}\ge a^k+b^k+c^k.$$

What is the largest number of acute angles that a convex polygon can have?

Find the smallest positive real $\alpha$, such that $$\frac{x+y} {2}\geq \alpha\sqrt{xy}+(1 - \alpha)\sqrt{\frac{x^2+y^2}{2}}$$ for all positive reals $x, y$.

Let $n$ be natural number and $1=d_1<d_2<\ldots <d_k=n$ be the positive divisors of $n$. Find all $n$ such that $2n = d_5^2+ d_6^2 -1$.

Find all the integers $x$ in $[20, 50]$ such that $6x + 5 \equiv -19 \mod 10,$ that is, $10$ divides $(6x + 15) + 19.$

Let $a_i, b_i$ be coprime positive integers for $i = 1, 2, \ldots , k$, and $m$ the least common multiple of $b_1, \ldots , b_k$. Prove that the greatest common divisor of $a_1 \frac{m}{b_1} , \ldots, a_k \frac{m}{b_k}$ equals the greatest common divisor of $a_1, \ldots , a_k.$

Find all polynomials $P(x),Q(x)$ which have integer coefficients and satify the following condtion: For the sequence $(x_n )$ defined by \[x_0=2014,x_{2n+1}=P(x_{2n}),x_{2n}=Q(x_{2n-1}) \quad n\geq 1\] for every positive integer $m$ is a divisor of some non-zero element of $(x_n )$

In triangle $ABC$, the angle at $C$ is $30^o$, side $BC$ has length $4$, and side $AC$ has length $5$. Let $ P$ be the point such that triangle $ABP$ is equilateral and non-overlapping with triangle $ABC$. Find the distance from $C$ to $ P$.