Four numbers are written in a row. The average of the first two is $21$, the average of the middle two is $26$, and the average of the last two is $30$. What is the average of the first and last of the numbers? $\textbf{(A)} ~24\qquad\textbf{(B)} ~25\qquad\textbf{(C)} ~26\qquad\textbf{(D)} ~27\qquad\textbf{(E)} ~28\qquad$

Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ where $n$ is an integer.

Given a finite number of boys and girls, a [i]sociable set of boys[/i] is a set of boys such that every girl knows at least one boy in that set; and a [i]sociable set of girls[/i] is a set of girls such that every boy knows at least one girl in that set. Prove that the number of sociable sets of boys and the number of sociable sets of girls have the same parity. (Acquaintance is assumed to be mutual.) [i](Poland) Marek Cygan[/i]

Natural numbers are placed in an infinite grid. Such that the number in each cell is equal to the number of its adjacent cells having the same number. Find the most distinct numbers this infinite grid can have. (Two cells of the grid are adjacent if they have a common vertex)

In a convex quadrilateral it is known $ABCD$ that $\angle ADB + \angle ACB = \angle CAB + \angle DBA = 30^{\circ}$ and $AD = BC$. Prove that from the lengths $DB$, $CA$ and $DC$, you can make a right triangle.

On a spherical surface there is a set $M{}$ with $n{}$ points with the property: for every point $A{}$ from $M{}$ there exist points $B$ and $C$ from $M{}$ such that the triangle $ABC$ is equilateral. For every equilateral triangle with vertexes in $M{}$ the perpendicular on its plane that goes through the geometric center of the other points from $M{}$. Prove that all these perpendiculars are concurrent.

Consider the set $ S_n$ of all the $ 2^n$ numbers of the type $ 2\pm \sqrt{2 \pm \sqrt {2 \pm ...}},$ where number $ 2$ appears $ n\plus{}1$ times. $ (a)$ Show that all members of $ S_n$ are real. $ (b)$ Find the product $ P_n$ of the elements of $ S_n$.

Positive integers $a$, $b$, and $c$ satisfy $a^2 +b^2 = c^3 -1$ where $c \le 40$. Find the sum of all distinct possible values of $c$.

Let a real function $f$ defined on the real numbers satisfy the following conditions: (i) $f(10+x) = f(10- x)$ (ii) $f(20+x) = - f(20- x)$ for all $x$. Prove that f is odd and periodic.

Let $x_{1}$ be a positive real number and for every integer $n \geq 1$ let $x_{n+1} = 1 + x_{1}x_{2}\ldots x_{n-1}x_{n}$. If $x_{5} = 43$, what is the sum of digits of the largest prime factors of $x_{6}$?

Find the sum of the non-repeated roots of the polynomial $P(x) = x^6-5x^5-4x^4-5x^3+8x^2+7x+7$.

In a square of sidelength $7$, $51$ points are given. Show that there's a disk of radius $1$ covering at least $3$ of these points.

Find all positive integer $n(\ge 2)$ and rational $\beta \in (0,1)$ satisfying the following: There exist positive integers $a_1,a_2,...,a_n$, such that for any set $I \subseteq \{1,2,...,n\}$ which contains at least two elements, $$ S(\sum_{i\in I}a_i)=\beta \sum_{i\in I}S(a_i). $$ where $S(n)$ denotes sum of digits of decimal representation of $n$.

Naomi has three colors of paint which she uses to paint the pattern below. She paints each region a solid color, and each of the three colors is used at least once. If Naomi is willing to paint two adjacent regions with the same color, how many color patterns could Naomi paint? [asy] size(150); defaultpen(linewidth(2)); draw(origin--(37,0)--(37,26)--(0,26)--cycle^^(12,0)--(12,26)^^(0,17)--(37,17)^^(20,0)--(20,17)^^(20,11)--(37,11)); [/asy]

Find the integer numbers $a, b, c$ such that the function $f: R \to R$, $f(x) = ax^2 +bx + c$ satisfies the equalities : $$f(f(1) ))= f (f(2 ) )= f(f (3 ))$$

In triangle $ABC$, point $M$ is the midpoint of $BC$, $P$ the point of intersection of the tangents at points $B$ and $C$ of the circumscribed circle of $ABC$, $N$ is the midpoint of the segment $MP$. The segment $AN$ meets the circumcircle $ABC$ at the point $Q$. Prove that $\angle PMQ = \angle MAQ$.

Given five real numbers $u_0, u_1, u_2, u_3, u_4$, prove that it is always possible to find five real numbers $v0, v_1, v_2, v_3, v_4$ that satisfy the following conditions: $(i)$ $u_i-v_i \in \mathbb N, \quad 0 \leq i \leq 4$ $(ii)$ $\sum_{0 \leq i<j \leq 4} (v_i - v_j)^2 < 4.$ [i]Proposed by Netherlands.[/i]

In the diagram, $OB_i$ is parallel and equal in length to $A_i A_{i + 1}$ for $i = 1$, 2, 3, and 4 ($A_5 = A_1$). Show that the area of $B_1 B_2 B_3 B_4$ is twice that of $A_1 A_2 A_3 A_4$. [asy] unitsize(1 cm); pair O; pair[] A, B; O = (0,0); A[1] = (0.5,-3); A[2] = (2,0); A[3] = (-0.2,0.5); A[4] = (-1,0); B[1] = A[2] - A[1]; B[2] = A[3] - A[2]; B[3] = A[4] - A[3]; B[4] = A[1] - A[4]; draw(A[1]--A[2]--A[3]--A[4]--cycle); draw(B[1]--B[2]--B[3]--B[4]--cycle); draw(O--B[1]); draw(O--B[2]); draw(O--B[3]); draw(O--B[4]); label("$A_1$", A[1], S); label("$A_2$", A[2], E); label("$A_3$", A[3], N); label("$A_4$", A[4], W); label("$B_1$", B[1], NE); label("$B_2$", B[2], W); label("$B_3$", B[3], SW); label("$B_4$", B[4], S); label("$O$", O, E); [/asy]

Three fair six-sided dice are rolled. The expected value of the median of the numbers rolled can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$. [i]Proposed by [b]AOPS12142015[/b][/i]

Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of $10$ hours per week helping around the house for $6$ weeks. For the first $5$ weeks, she helps around the house for $8$, $11$, $7$, $12$ and $10$ hours. How many hours must she work during the final week to earn the tickets? $\textbf{(A)}\ 9 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 13$

The edge lengths of a tetrahedron are a, b, c, d, e, f, the areas of its faces are S1, S2, S3, S4, and its volume is V . Prove that 2 [S1 S2 S3 S4](1/2) > 3V [abcdef](1/6) this problem comes from: http://www.imomath.com/othercomp/jkasfvgkusa/MonMO99.pdf I was just wondering if someone could write it in LATEX form. [color=red]_____________________________________ EDIT by moderator: If you type[/color] [code]The edge lengths of a tetrahedron are $a, b, c, d, e, f,$ the areas of its faces are $S_1, S_2, S_3, S_4,$ and its volume is $V.$ Prove that $2 \sqrt{S_1 S_2 S_3 S_4} > 3V \sqrt[6]{abcdef}$[/code] [color=red]it shows up as:[/color] The edge lengths of a tetrahedron are $ a, b, c, d, e, f,$ the areas of its faces are $ S_1, S_2, S_3, S_4,$ and its volume is $ V.$ Prove that $ 2 \sqrt{S_1 S_2 S_3 S_4} > 3V \sqrt[6]{abcdef}$

Let $ABC$ be an acute angled triangle and let $D$ be an interior point of the segment $BC$. Let the circumcircle of $ACD$ intersect $AB$ at $E$ ($E$ between $A$ and $B$) and let circumcircle of $ABD$ intersect $AC$ at $F$ ($F$ between $A$ and $C$). Let $O$ be the circumcenter of $AEF$. Prove that $OD$ bisects $\angle EDF$.

Let $a,b,c,$ and $d$ be real numbers and $k\geq l\geq m$ and $p\geq q\geq r$. Prove that $f(x)=a(x+1)^k (x+2)^p+b(x+1)^l (x+2)^q+c(x+1)^m (x+2)^r-d=0$ has no more than 14 positive roots.

Bengt wants to put out crosses and rings in the squares of an $n \times n$-square, so that it is exactly one ring and exactly one cross in each row and in each column, and no more than one symbol in each box. Mona wants to stop him by setting a number in advance ban on crosses and a number of bans on rings, maximum one ban in each square. She want to use as few bans as possible of each variety. To succeed in preventing Bengt, how many prohibitions she needs to use the least of the kind of prohibitions she uses the most of?

Find with proof all positive $3$ digit integers $\overline{abc}$ satisfying \[ b\cdot \overline{ac}=c \cdot \overline{ab} +10 \]