An equilateral triangle $\bigtriangleup ABC$ is split into $4$ triangles with equal area; three congruent triangles $\bigtriangleup ABX,\bigtriangleup BCY, \bigtriangleup CAZ$, and a smaller equilateral triangle $\bigtriangleup XYZ$, as shown. Prove that the points $X, Y, Z$ lie on the incircle of triangle $\bigtriangleup ABC$. [i]Proposed by Josef Tkadlec - Czech Republic[/i]

Let $P$ be a point inside triangle $ABC$ such that $\angle CPA = 90^o$ and $\angle CBP = \angle CAP$. Prove that $\angle P XY = 90^o$, where $X$ and $Y$ are the midpoints of $AB$ and $AC$ respectively.

Find all functions $f:\mathbb{R^{+}} \rightarrow \mathbb{R^{+}}$, such that $$xf(1+xf(y))=f(f(x)+f(y))$$ for all positive reals $x, y$.

Two lines $m$ and $n$ intersect in a unique point $P$. A point $M$ moves along $m$ with constant speed, while another point $N$ moves along $n$ with the same speed. They both pass through the point $P$, but not at the same time. Show that there is a fixed point $Q \ne P$ such that the points $P,Q,M$ and $N$ lie on a common circle all the time.

Prove that for tetrahedron $ABCD$; vertex $D$, center of insphere and centroid of $ABCD$ are collinear iff areas of triangles $ABD,BCD,CAD$ are equal.

Find all prime numbers $p$ and $q$ such that $3p^2q+2pq^2=483$

Given that $\{a_n\}$ is a sequence of integers satisfying the following condition for all positive integral values of $n$: $a_n+a_{n+1}=2a_{n+2}a_{n+3}+2016$. Find all possible values of $a_1$ and $a_2$

How many ways are there to insert plus signs $+$ between the digits of number $111111 ...111$ which includes thirty of digits $1$ so that the result will be a multiple of $30$?

How many real numbers $x$ are there such that $\sqrt{ x + 1 - 4\sqrt{x-3}} + \sqrt{ x + 6 - 6\sqrt{x-3}} = 1$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ \text{None of the preceding} $

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?

$50$ silver coins ordered by weight and $51$ gold coins also ordered by weight are given. All coins have different weights. You are given a balance to compare weights of any two coins. How can you find the “middle” coin (that occupying the $51$st place in weight among all $101$ coins) using $7$ weighings? (A. Andjans)

[asy] for (int i=0; i<9; ++i) { draw(dir(10+40*i)--dir(50+40*i)); } draw(dir(50) -- dir(90)); label("$a$", dir(50) -- dir(90), N); draw(dir(10) -- dir(90)); label("$b$", dir(10) -- dir(90), SW); draw(dir(-70) -- dir(90)); label("$d$", dir(-70) -- dir(90), E); //Credit to MSTang for the diagram[/asy] If $a,b,$ and $d$ are the lengths of a side, a shortest diagonal and a longest diagonal, respectively, of a regular nonagon (see adjoining figure), then $\textbf{(A) }d=a+b\qquad\textbf{(B) }d^2=a^2+b^2\qquad\textbf{(C) }d^2=a^2+ab+b^2\qquad$ $\textbf{(D) }b=\frac{a+d}{2}\qquad \textbf{(E) }b^2=ad$

let $x,y$ be non-zero reals such that : $x^3+y^3+3x^2y^2=x^3y^3$ find all values of $\frac{1}{x}+\frac{1}{y}$

Let $1=d_1<d_2<\ldots<d_{2m}=n$ be the divisors of a positive integer $n$, where $n$ is not a perfect square. Consider the determinant $$D=\begin{vmatrix}n+d_1&n&\ldots&n\\n&n+d_2&\ldots&n\\\ldots&\ldots&&\ldots\\n&n&\ldots&n+d_{2m}\end{vmatrix}.$$ (a) Prove that $n^m$ divides $D$. (b) Prove that $1+d_1+d_2+\ldots+d_{2m}$ divides $D$.

There are $100$ points on the plane, all pairwise distances between which are different. Is there always a polyline with vertices at these points, passing through each point once, in which the link lengths increase monotonously?

For her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First she plants six tree saplings at the end of Day $0$. On each day afterwards, Emilia attempts to use her magic to turn each sapling into a perfectly-shaped apple tree, and for each sapling she succeeds in turning it into a perfectly-shaped apple tree that day with a probability of $\frac{1}{2}$. (Once a sapling is turned into a perfectly-shaped apple tree, it will stay a perfectly-shaped apple tree.) The expected number of days it will take Emilia to obtain six perfectly-shaped apple trees is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$. [i]Proposed by Yannick Yao[/i]

[b]p1.[/b] Two players play the following game. On the lowest left square of an $8\times 8$ chessboard there is a rook. The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second player is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? [b]p2.[/b] In Crocodile Country there are banknotes of $1$ dollar, $10$ dollars, $100$ dollars, and $1,000$ dollars. Is it possible to get 1,000,000 dollars by using $250,000$ banknotes? [b]p3.[/b] Fifteen positive numbers (not necessarily whole numbers) are placed around the circle. It is known that the sum of every four consecutive numbers is $30$. Prove that each number is less than $15$. [b]p4.[/b] Donald Duck has $100$ sticks, each of which has length $1$ cm or $3$ cm. Prove that he can break into $2$ pieces no more than one stick, after which he can compose a rectangle using all sticks. [b]p5.[/b] Three consecutive $2$ digit numbers are written next to each other. It turns out that the resulting $6$ digit number is divisible by $17$. Find all such numbers. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a$ denote the positive real root of the polynomial $x^2 -3x-2$. Compute the remainder when $\lfloor a^{1000}\rfloor $ is divided by the prime number $997$. Here, $\lfloor r\rfloor$ denotes the greatest integer less than $r$.

Prove that if $n$ can be written as $n=a^2+ab+b^2$, then also $7n$ can be written that way.

Let $n>1$ be a given integer. The Mint issues coins of $n$ different values $a_1, a_2, ..., a_n$, where each $a_i$ is a positive integer (the number of coins of each value is unlimited). A set of values $\{a_1, a_2,..., a_n\}$ is called [i]lucky[/i], if the sum $a_1+ a_2+...+ a_n$ can be collected in a unique way (namely, by taking one coin of each value). (a) Prove that there exists a lucky set of values $\{a_1, a_2, ..., a_n\}$ with $$a_1+ a_2+...+ a_n < n \cdot 2^n.$$ (b) Prove that every lucky set of values $\{a_1, a_2,..., a_n\}$ satisfies $$a_1+ a_2+...+ a_n >n \cdot 2^{n-1}.$$ Proposed by Ilya Bogdanov

Show that $$\log_2\pi+\log_4\pi<\frac52.$$

Determine the largest positive integer $n$ such that the following statement is true: There exists $n$ real polynomials, $P_1(x),\ldots,P_n(x)$ such that the sum of any two of them have no real roots but the sum of any three does.

In how many distinguishable ways can $10$ distinct pool balls be formed into a pyramid ($6$ on the bottom, $3$ in the middle, one on top), assuming that all rotations of the pyramid are indistinguishable?

Let $A$ be the largest subset of $\{1,\dots,n\}$ such that for each $x\in A$, $x$ divides at most one other element in $A$. Prove that \[\frac{2n}3\leq |A|\leq \left\lceil \frac{3n}4\right\rceil. \]

[b]Q.[/b] Determine all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x) \geqslant x+1, \forall\ x \in \mathbb{R}\quad \text{and}\quad f(x+y) \geqslant f(x) f(y), \forall\ x, y \in \mathbb{R}$$ [i]Proposed by TuZo[/i]