Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.

Find all real numbers $x$ such that $4^x-2^{x+2}+3=0$.

The altitudes through the vertices $ A,B,C$ of an acute-angled triangle $ ABC$ meet the opposite sides at $ D,E, F,$ respectively. The line through $ D$ parallel to $ EF$ meets the lines $ AC$ and $ AB$ at $ Q$ and $ R,$ respectively. The line $ EF$ meets $ BC$ at $ P.$ Prove that the circumcircle of the triangle $ PQR$ passes through the midpoint of $ BC.$

Exactly $n$ chords (i.e. diagonals and edges) of a regular $2n$-gon are coloured red, satisfying the following two conditions: (1) Each of the $2n$ vertices occurs exactly once as the endpoint of a red chord. (2) No two red chords have the same length. For which positive integers $n \ge 2$ is this possible?

Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers in which $\lim_{n\to\infty} a_n = 0$ such that there is a constant $c >0$ so that for all $n \geq 1$, $|a_{n+1}-a_n| \leq c\cdot a_n^2$. Show that exists $d>0$ with $na_n \geq d, \forall n \geq 1$.

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

There are 2010 boxes labeled $B_1,B_2,\dots,B_{2010},$ and $2010n$ balls have been distributed among them, for some positive integer $n.$ You may redistribute the balls by a sequence of moves, each of which consists of choosing an $i$ and moving [i]exactly[/i] $i$ balls from box $B_i$ into any one other box. For which values of $n$ is it possible to reach the distribution with exactly $n$ balls in each box, regardless of the initial distribution of balls?

Let $ABCD$ be a tangential and cyclic quadrilateral. Let $S$ be the intersection point of diagonals $AC$ and $BD$ of the quadrilateral. Let $I$, $I_1$, and $I_2$ be the incenters of quadrilateral $ABCD$ and triangles $ACD$ and $BCS$, respectively. Let the ray $II_2$ intersect the circumcircle of quadrilateral $ABCD$ at point $E$. Prove that the points $D$, $E$, $I_1$, and $I_2$ are collinear or concyclic. [i]Proposed by Teodor von Burg[/i]

We say an integer number $n \ge 1$ is conservative, if the smallest prime divisor of $(n!)^n+1$ is at most $n+2015$. Decide if the number of conservative numbers is infinite or not.

Let $n \ge 2$ be a positive integer, with divisors \[1=d_{1}< d_{2}< \cdots < d_{k}=n \;.\] Prove that \[d_{1}d_{2}+d_{2}d_{3}+\cdots+d_{k-1}d_{k}\] is always less than $n^{2}$, and determine when it divides $n^{2}$.

Let $n$ be a positive integer and $\mathcal{P}_n$ be the set of integer polynomials of the form $a_0+a_1x+\ldots +a_nx^n$ where $|a_i|\le 2$ for $i=0,1,\ldots ,n$. Find, for each positive integer $k$, the number of elements of the set $A_n(k)=\{f(k)|f\in \mathcal{P}_n \}$. [i]Marian Andronache[/i]

The problem is about real polynomial functions, denoted by $f$, of degree $\deg f$. a) Prove that a polynomial function $f$ can`t be wrriten as sum of at most $\deg f$ periodic functions. b) Show that if a polynomial function of degree $1$ is written as sum of two periodic functions, then they are unbounded on every interval (thus, they are "wild"). c) Show that every polynomial function of degree $1$ can be written as sum of two periodic functions. d) Show that every polynomial function $f$ can be written as sum of $\deg f+1$ periodic functions. e) Give an example of a function that can`t be written as a finite sum of periodic functions. [i]Dan Schwarz[/i]

Let $p(x)$ be an $n$-degree $(n \ge 2)$ polynomial with integer coefficients. If there are infinitely many positive integers $m$, such that $p(m)$ at most $n -1$ different prime factors $f$, prove that $p(x)$ has at most $n-1$ different rational roots . [color=#f00]a help in translation is welcome[/color]

What is $\frac{2^3+2^3}{2^{-3}+2^{-3}}?$ ${ \textbf{(A)}\ \ 16\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 32\qquad\textbf{(D)}}\ 48\qquad\textbf{(E)}\ 64 $

Let $P =\{(a, b) / a, b \in \{1, 2, ..., n\}, n \in N\}$ be a set of point of the Cartesian plane and draw horizontal, vertical, or diagonal segments, of length $1$ or $\sqrt 2$, so that both ends of the segment are in $P$ and do not intersect each other. Furthermore, for each point $(a, b)$ it is true that i) if $a + b$ is a multiple of $3$, then it is an endpoint of exactly $3$ segments. ii) if $a + b$ is an even not multiple of $3$, then it is an endpoint of exactly $2$ segments. iii) if $a + b$ is an odd not multiple of $3$, then it is endpoint of exactly $1$ segment. a) Check that with $n = 6$ it is possible to satisfy all the conditions. b) Show that with $n = 2015$ it is not possible to satisfy all the conditions.

each cells in a $9\times 9 $ grid is painted either blue or red.two cells are called [i]diagonal neighbors[/i] if their intersection is exactly a point.show that some cell has exactly two red neighbors,or exactly two blue neighbors, or both.

Is that true that there exist a polynomial $f(x)$ with rational coefficients, not all integers, with degree $n>0$, a polynomial $g(x)$, with integer coefficients, and a set $S$ with $n+1$ integers such that $f(t)=g(t)$ for all $t \in S$?

Ada is taking a math test from 12:00 to 1:30, but her brother, Samuel, will be disruptive for two ten-minute periods during the test. If the probability that her brother is not disruptive while she is solving the challenge problem from 12:45 to 1:00 can be expressed as $\frac{m}{n}$, find $m+n$. [i]Proposed by Ada Tsui[/i]

In the cartesian plane consider rectangles with sides parallel to the coordinate axes. We say that one rectangle is [i]below[/i] another rectangle if there is a line $g$ parallel to the $x$-axis such that the first rectangle is below $g$, the second one above $g$ and both rectangles do not touch $g$. Similarly, we say that one rectangle is [i]to the right of[/i] another rectangle if there is a line $h$ parallel to the $y$-axis such that the first rectangle is to the right of $h$, the second one to the left of $h$ and both rectangles do not touch $h$. Show that any finite set of $n$ pairwise disjoint rectangles with sides parallel to the coordinate axes can be enumerated as a sequence $(R_1,\dots,R_n)$ so that for all indices $i,j$ with $1 \le i<j \le n$ the rectangle $R_i$ is to the right of or below the rectangle $R_j$

Let $D$ and $E$ be points on $[BC]$ and $[AC]$ of acute $\triangle ABC$, respectively. $AD$ and $BE$ meet at $F$. If $|AF|=|CD|=2|BF|=2|CE|$, and $Area(\triangle ABF) = Area(\triangle DEC)$, then $Area(\triangle AFC)/Area(\triangle BFC) = ?$ $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 2\sqrt2 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \sqrt2 \qquad \textbf{(E)}\ 1$

On a line, there are $n$ closed intervals (none of which is a single point) whose union we denote by $S$. It's known that for every real number $d$, $0<d\le 1$, there are two points in $S$ that are a distance $d$ from each other. [list](a) Show that the sum of the lengths of the $n$ closed intervals is larger than $\frac{1}{n}$. (b) Prove that, for each positive integer $n$, the $\frac{1}{n}$ in the statement of part (a) cannot be replaced with a larger number.[/list]

Find all values ​​of the real parameter $a$, for which the equation $(x -6\sqrt{x} + 8)\cdot \sqrt{x- a} = 0$ has exactly two distinct real solutions.

Suppose $P(x)$ is a polynomial with real coefficients, satisfying the condition $P(\cos \theta+\sin \theta)=P(\cos \theta-\sin \theta)$, for every real $\theta$. Prove that $P(x)$ can be expressed in the form$$P(x)=a_0+a_1(1-x^2)^2+a_2(1-x^2)^4+\dots+a_n(1-x^2)^{2n}$$for some real numbers $a_0, a_1, \dots, a_n$ and non-negative integer $n$. [i]Proposed by C.R. Pranesacher[/i]

Let a regular polygon with $p$ vertices be given, where $p$ is an odd prime number. At every vertex there is one monkey. An owner of monkeys takes $p$ peanuts, goes along the perimeter of polygon clockwise and delivers to the monkeys by the following rule: Gives the first peanut for the leader, skips the two next vertices and gives the second peanut to the monkey at the next vertex; skip four next vertices gives the second peanut for the monkey at the next vertex ... after giving the $k$-th peanut, he skips the $2 \cdot k$ next vertices and gives $k+1$-th for the monkey at the next vertex. He does so until all $p$ peanuts are delivered. [b]I.[/b] How many monkeys are there which does not receive peanuts? [b]II.[/b] How many edges of polygon are there which satisfying condition: both two monkey at its vertex received peanut(s)?

A ticket to a school play costs $ x$ dollars, where $ x$ is a whole number. A group of 9th graders buys tickets costing a total of $ \$48$, and a group of 10th graders buys tickets costing a total of $ \$64$. How many values of $ x$ are possible? $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$