Prove that for all $n\in \mathbb{N}$ there exist natural numbers $a_1,a_2,...,a_n$ such that: $(i)a_1>a_2>...>a_n$ $(ii)a_i|a^2_{i+1},\forall i\in\{1,2,...,n-1\}$ $(iii)a_i\nmid a_j,\forall i,j\in \{1,2,...,n\},i\neq j$

Does there exist an operation $*$ on the set of all integers such that the following conditions hold simultaneously: $(1)$ for all integers $x,y,z$, $(x*y)*z=x*(y*z)$; $(2)$ for all integers $x$ and $y$, $x*x*y=y*x*x=y$?

In a trapezoid $ABCD$ with $AB<CD$ and $AB \parallel CD$, the diagonals intersect each other at $E$. Let $F$ be the midpoint of the arc $BC$ (not containing the point $E$) of the circumcircle of the triangle $EBC$. The lines $EF$ and $BC$ intersect at $G$. The circumcircle of the triangle $BFD$ intersects the ray $[DA$ at $H$ such that $A \in [HD]$. The circumcircle of the triangle $AHB$ intersects the lines $AC$ and $BD$ at $M$ and $N$, respectively. $BM$ intersects $GH$ at $P$, $GN$ intersects $AC$ at $Q$. Prove that the points $P, Q, D$ are collinear.

Q. A light source at the point $(0, 16)$ in the co-ordinate plane casts light in all directions. A disc(circle along ith it's interior) of radius $2$ with center at $(6, 10)$ casts a shadow on the X-axis. The length of the shadow can be written in the form $m\sqrt{n}$ where $m, n$ are positive integers and $n$ is squarefree. Find $m + n$.

Let $a,b,c,d$ be four non-negative numbers satisfying \[ a+b+c+d=1. \] Prove the inequality \[ a \cdot b \cdot c + b \cdot c \cdot d + c \cdot d \cdot a + d \cdot a \cdot b \leq \frac{1}{27} + \frac{176}{27} \cdot a \cdot b \cdot c \cdot d. \]

Square $ABCD$ has side length $4$. Points $P$ and $Q$ are located on sides $BC$ and $CD$, respectively, such that $BP = DQ = 1$. Let $AQ$ intersect $DP$ at point $X$. Compute the area of triangle $P QX$.

Let $ n $ be a positive integer and $ n $ real numbers $ a_1,a_2,\ldots ,a_n $ such that $ a_1^2+a_2^2+\cdots +a_n^2=1. $ Show that $$ \sum_{1\le ij\le n} a_ia_j<2\sqrt n. $$ [i]Russian math competition[/i]

Subsets $S$ of the first 3$5$ positive integers $\{1, 2, 3, ..., 35\}$ are called [i]contrived [/i] if $S$ has size $4$ and the sum of the squares of the elements of $S$ is divisible by $7$. Find the number of contrived sets.

Let $ x, y, z $ be positive real numbers. Prove that \[ \frac{2x^2 + xy}{(y+ \sqrt{zx} + z )^2} + \frac{2y^2 + yz}{(z+ \sqrt{xy} + x )^2} + \frac{2z^2 + zx}{(x+ \sqrt{yz} +y )^2} \ge 1 \]

The number of solutions of $2^{2x}-3^{2y}=55$, in which $x$ and $y$ are integers, is: ${{ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3}\qquad\textbf{(E)}\ \text{More than three, but finite} } $

$DEB$ is a chord of a circle such that $DE=3$ and $EB=5$. Let $O$ be the centre of the circle. Join $OE$ and extend $OE$ to cut the circle at $C$. (See diagram). Given $EC=1$, find the radius of the circle. [asy] size(6cm); pair O = (0,0), B = dir(110), D = dir(30), E = 0.4 * B + 0.6 * D, C = intersectionpoint(O--2*E, unitcircle); draw(unitcircle); draw(O--C); draw(B--D); dot(O); dot(B); dot(C); dot(D); dot(E); label("$B$", B, B); label("$C$", C, C); label("$D$", D, D); label("$E$", E, dir(280)); label("$O$", O, dir(270)); [/asy]

[color=darkred]A row of a matrix belonging to $\mathcal{M}_n(\mathbb{C})$ is said to be [i]permutable[/i] if no matter how we would permute the entries of that row, the value of the determinant doesn't change. Prove that if a matrix has two [i]permutable[/i] rows, then its determinant is equal to $0$ .[/color]

For how many nonempty subsets $S \subseteq \{1, 2, \ldots , 10\}$ is the sum of all elements in $S$ even? [i]Proposed by Andrew Wen[/i]

Let $n$ be a positive integer. We are given a $2n \times 2n$ table. Each cell is coloured with one of $2n^2$ colours such that each colour is used exactly twice. Jana stands in one of the cells. There is a chocolate bar lying in one of the other cells. Jana wishes to reach the cell with the chocolate bar. At each step, she can only move in one of the following two ways. Either she walks to an adjacent cell or she teleports to the other cell with the same colour as her current cell. (Jana can move to an adjacent cell of the same colour by either walking or teleporting.) Determine whether Jana can fulfill her wish, regardless of the initial configuration, if she has to alternate between the two ways of moving and has to start with a teleportation.

Let $\mathbb{R}^+ = (0, \infty)$ be the set of all positive real numbers. Find all functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ and polynomials $P(x)$ with non-negative real coefficients such that $P(0) = 0$ which satisfy the equality $f(f(x) + P(y)) = f(x - y) + 2y$ for all real numbers $x > y > 0$. [i]Proposed by Sardor Gafforov, Uzbekistan[/i]

In an acute triangle $ABC$ with $\overline{AB} > \overline{AC}$, let $D, E, F$ be the feet of the altitudes from $A, B, C$, respectively. Let $P$ be an intersection of lines $EF$ and $BC$, and let $Q$ be a point on the segment $BD$ such that $\angle QFD = \angle EPC$. Let $O, H$ denote the circumcenter and the orthocenter of triangle $ABC$, respectively. Suppose that $OH$ is perpendicular to $AQ$. Prove that $P, O, H$ are collinear.

Are there positive integers $m>4$ and $n$, such that a) ${m \choose 3}=n^2;$ b) ${m \choose 4}=n^2+9?$

Let $a$ and $b$ be two real numbers, with $0<a,b<1$. Prove that \[\sqrt{ab^2+a^2b}+\sqrt{(1-a)(1-b)^2+(1-a)^2(1-b)}<\sqrt{2}\]

Show that if the points of an isosceles right triangle of side length $1$ are each colored with one of four colors, then there must be two points of the same color which are at least a distance $2-\sqrt 2$ apart.

In the figure above, the large triangle and all four shaded triangles are equilateral. If the areas of triangles $A, B,$ and $C$ are $1, 2,$ and $3,$ respectively, compute the smallest possible integer ratio between the area of the entire triangle to the area of triangle $D.$ [Insert Diagram] [i]Proposed by Alex Li[/i]

An $8 \times 8$ chessboard is covered completely and without overlaps by $32$ dominoes of size $1 \times 2$. Show that there are two dominoes forming a $2 \times 2$ square.

Let $x_1, x_2, …, x_n$ and $y_1, y_2, …,y_n$ be positive reals such that $$x_1 + x_2 +.. + x_n \ge y_i \ge x^2_i$$ for all $i = 1, 2, …, n$. Prove that $$\frac{x_1}{x_1y_1 + x_2}+ + \frac{x_2}{x_2y_2 + x_3} + ...+ \frac{x_n}{x_ny_n + x_1}> \frac{1}{2n}.$$

Let $n$ be positive integer, set $M = \{ 1, 2, \ldots, 2n \}$. Find the minimum positive integer $k$ such that for any subset $A$ (with $k$ elements) of set $M$, there exist four pairwise distinct elements in $A$ whose sum is $4n + 1$.

A square $ABCD$ is divided into $n^2$ equal small (fundamental) squares by drawing lines parallel to its sides.The vertices of the fundamental squares are called vertices of the grid.A rhombus is called [i]nice[/i] when: $\bullet$ It is not a square $\bullet$ Its vertices are points of the grid $\bullet$ Its diagonals are parallel to the sides of the square $ABCD$ Find (as a function of $n$) the number of the [i]nice[/i] rhombuses ($n$ is a positive integer greater than $2$).

In a convex $n$-gon all angles are equal from a certain point, located inside the $n$-gon, all its sides are seen under equal angles. Can we conclude that this $n$-gon is regular?