Determine all triples $(x,y,z)$ of positive integers such that \[\frac{13}{x^2}+\frac{1996}{y^2}=\frac{z}{1997} \]

In the expansion of $(2x +3y)^{20}$, find the number of coefficients divisible by $144$. [i]Proposed by Hannah Shen[/i]

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

Find the greatest real $k$ such that, for every tetrahedron $ABCD$ of volume $V$, the product of areas of faces $ABC,ABD$ and $ACD$ is at least $kV^2$.

For an ordered pair $(m,n)$ of distinct positive integers, suppose, for some nonempty subset $S$ of $\mathbb R$, that a function $f:S \rightarrow S$ satisfies the property that $f^m(x) + f^n(y) = x+y$ for all $x,y\in S$. (Here $f^k(z)$ means the result when $f$ is applied $k$ times to $z$; for example, $f^1(z)=f(z)$ and $f^3(z)=f(f(f(z)))$.) Then $f$ is called \emph{$(m,n)$-splendid}. Furthermore, $f$ is called \emph{$(m,n)$-primitive} if $f$ is $(m,n)$-splendid and there do not exist positive integers $a\le m$ and $b\le n$ with $(a,b)\neq (m,n)$ and $a \neq b$ such that $f$ is also $(a,b)$-splendid. Compute the number of ordered pairs $(m,n)$ of distinct positive integers less than $10000$ such that there exists a nonempty subset $S$ of $\mathbb R$ such that there exists an $(m,n)$-primitive function $f: S \rightarrow S$. [i]Proposed by Vincent Huang[/i]

Suppose $log_x (2x^2+x-1)>log_x 2-1$. Then the range of $x$ is ${ \textbf{(A)}\ \frac{1}{2}<x<1\qquad\textbf{(B)}\ x>\frac{1}{2} \text{and} x \not= 1\qquad\textbf{(C)}\ x>1\qquad\textbf{(D)}}\ 0<x<1\qquad $

Let $a$ be a real number in the open interval $(0,1)$. Let $n\geq 2$ be a positive integer and let $f_n\colon\mathbb{R}\to\mathbb{R}$ be defined by $f_n(x) = x+\frac{x^2}{n}$. Show that \[\frac{a(1-a)n^2+2a^2n+a^3}{(1-a)^2n^2+a(2-a)n+a^2}<(f_n \circ\ \cdots\ \circ f_n)(a)<\frac{an+a^2}{(1-a)n+a}\] where there are $n$ functions in the composition.

The midpoints of the opposite sides of the hexagon are connected by segments. It turned out that the points of pairwise intersection of these segments form an equilateral triangle. Prove that the drawn segments are equal. (M. Volchkevich)

Triangle $ABC$ has $AB = 13$, $BC = 14$, and $CA = 15$. The internal angle bisector of $\angle ABC$ intersects side $CA$ at $X$. The circumcircles of triangles $AXB$ and $BXC$ intersect sides $BC$ and $AB$ at $M$ and $N$, respectively. The value of $MN^2$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m+n$ is divided by $1000$. [i]Proposed by [b] Th3Numb3rThr33[/b][/i]

On a circle there are written 2005 non-negative integers with sum 7022. Prove that there exist two pairs formed with two consecutive numbers on the circle such that the sum of the elements in each pair is greater or equal with 8. [i]After an idea of Marin Chirciu[/i]

Let $f,g:\mathbb R\to\mathbb R$ be continuous functions such that $g$ is differentiable. Assume that $(f(0)-g'(0))(g'(1)-f(1))>0$. Show that there exists a point $c\in (0,1)$ such that $f(c)=g'(c)$. [i]Proposed by Fereshteh Malek, K. N. Toosi University of Technology[/i]

What is the greatest number of positive integers lesser than or equal to 2016 we can choose such that it doesn't have two of them differing by 1,2, or 6?

There is exactly one astronomer on every planet of a certain system. He watches the closest planet. The number of the planets is odd and all of the distances are different. Prove that there one planet being not watched.

Find all natural numbers $n$ that are smaller than $128^{97}$ and have exactly $2019$ divisors.

Let $S \subset [0, 1]$ be a set of 5 points with $\{0, 1\} \subset S$. The graph of a real function $f : [0, 1] \to [0, 1]$ is continuous and increasing, and it is linear on every subinterval $I$ in $[0, 1]$ such that the endpoints but no interior points of $I$ are in $S$. We want to compute, using a computer, the extreme values of $g(x, t) = \frac{f(x+t)-f(x)}{ f(x)-f(x-t)}$ for $x - t, x + t \in [0, 1]$. At how many points $(x, t)$ is it necessary to compute $g(x, t)$ with the computer?

Find all polynomials $P(X)$ of fourth degree with real coefficients that verify the properties: [b]a)[/b] $P(-x)=P(x), \forall x\in\mathbb{R};$ [b]b)[/b] $P(x)\geq0, \forall x\in\mathbb{R};$ [b]c)[/b] $P(0)=1;$ [b]d)[/b] $P(X)$ has exactly two local minimums $x_1$ and $x_2$ such that $|x_1-x_2|=2.$