For natural numbers $a, b c$ it holds that $(a + b + c)^2 | ab (a + b) + bc (b + c) + ca(c + a) + 3abc$. Prove that $(a + b + c) |(a - b)^2 + (b - c)^2 + (c - a)^2$

Let $a_1,a_2,a_3,\dots,a_6$ be an arithmetic sequence with common difference $3$. Suppose that $a_1$, $a_3$, and $a_6$ also form a geometric sequence. Compute $a_1$.

Laura won the local math olympiad and was awarded a "magical" ruler. With it, she can draw (as usual) lines in the plane, and she can also measure segments and replicate them anywhere in the plane; but she can also divide a segment into as many equal parts as she wishes; for instance, she can divide any segment into $17$ equal parts. Laura drew a parallelogram $ABCD$ and decided to try out her magical ruler; with it, she found the midpoint $M$ of side $CD$, and she extended $CB$ beyond $B$ to point $N$ so that segments $CB$ and $BN$ were equal in length. Unfortunately, her mischievous little brother came along and erased everything on Laura's picture except for points $A, M$, and $N$. Using Laura's magical ruler, help her reconstruct the original parallelogram $ABCD$: write down the steps that she needs to follow and prove why this will lead to reconstructing the original parallelogram $ABCD$.

Each of $3600$ subscribers of a telephone exchange calls it once an hour on average. What is the probability that in a given second $5$ or more calls are received? Estimate the mean interval of time between such seconds $(i, i + 1)$.

For which of the following value of $n$, there exists integers $a,b$ such that $a^2 + ab-6b^2 = n$? $ \textbf{(A)}\ 17 \qquad\textbf{(B)}\ 19 \qquad\textbf{(C)}\ 29 \qquad\textbf{(D)}\ 31 \qquad\textbf{(E)}\ 37 $

A number is written in the each vertex of a cube. It is allowed to add one to two numbers written in the ends of one edge. Is it possible to obtain the cube with all equal numbers if the numbers were initially as on the pictures:

One hundred pins are arranged to form a square grid as shown. Jimmy wants to mark these pins using four letters $a,b,c,d,$ so that: (I) every horizontal line and every vertical line contains all four letters; (ii) each small square (such as the one shown) has its vertices marked by four different letters. [asy] unitsize(.5cm); for(int a=1; a<11; ++a) { for(int b=1; b<11; ++b) { draw(Circle((a,b),.1)); } } draw((5.1, 5)--(5.9,5)); draw((5.1, 6)--(5.9,6)); draw((5, 5.1)--(5, 5.9)); draw((6, 5.1)--(6, 5.9)); [/asy] Can he do this?

Call a triple $(a, b, c)$ of positive integers a [b]nice[/b] triple if $a, b, c$ forms a non-decreasing arithmetic progression, $gcd(b, a) = gcd(b, c) = 1$ and the product $abc$ is a perfect square. Prove that given a nice triple, there exists some other nice triple having at least one element common with the given triple.

Let $ABCD$ be a parallelogram, such that the point $M$ is the midpoint of the side $CD$ and lies on the bisector of the angle $\angle BAD$. Prove that $\angle AMB = 90^o$.

Find all the functions $f : \mathbb{R} \to \mathbb{R}$ satisfying for all $x,y \in \mathbb{R}$ $f(f(x)-y^2) = f(x)^2 - 2f(x)y^2 + f(f(y))$.

We will say that two pyramids touch each other by faces if they have no common interior points and if the intersection of a face of one of them with a face of the other is either a triangle or a polygon. Is it possible to place $8$ tetrahedra in such a way that every two of them touch each other by faces? (A Andjans, Riga)

Let $ \Psi\equal{}\langle A;...\rangle$ be an arbitrary, countable algebraic structure (that is, $ \Psi$ can have an arbitrary number of finitary operations and relations). Prove that $ \Psi$ has as many as continuum automorphisms if and only if for any finite subset $ A'$ of $ A$ there is an automorphism $ \pi_{A'}$ of $ \Psi$ different from the identity automorphism and such that \[ (x) \pi_{A'}\equal{}x\] for every $ x \in A'$. [i]M. Makkai[/i]

The quadrilateral $ABCD$ with perpendicular diagonals is inscribed in the circle with center $O$, the points $M,N$ are the midpoints of $[BC]$ and $[CD]$ respectively. Find the ratio of areas of the figures $OMCN$ and $ABCD$

Prove that if the two altitudes of a tetrahedron intersect, then the other two atltitudes intersect also.

$12$ schoolchildren are engaged in a circle of patriotic songs, each of them knows a few songs (maybe none). We will say that a group of schoolchildren can sing a song if at least one member of the group knows it. Supervisor the circle noticed that any group of $10$ circle members can sing exactly $20$ songs, and any group of $8$ circle members - exactly $16$ songs. Prove that the group of all $12$ circle members can sing exactly $24$ songs.

Find the smallest positive integer $k$ with the following property: if each cell of a $100\times 100$ grid is dyed with one color and the number of cells of each color is not more than $104$, then there is a $k\times1$ or $1\times k$ rectangle that contains cells of at least three different colors.

$a,b,c,d$ are fixed positive real numbers. Find the maximum value of the function $f: \mathbb{R^{+}}_{0} \rightarrow \mathbb{R}$ $f(x)=\frac{a+bx}{b+cx}+\frac{b+cx}{c+dx}+\frac{c+dx}{d+ax}+\frac{d+ax}{a+bx}, x \geq 0$

Points $A, B, C$ are vertices of an equilateral triangle inscribed in a circle. Point $D$ lies on the shorter arc $\overarc {AB}$ . Prove that $AD + BD = DC$.

Prove that there exists a function $f : \mathbb{N} \rightarrow \mathbb{N}$ that satisfies the following (1) $\{f(n) : n\in\mathbb{N}\}$ is a finite set; and (2) For nonzero integers $x_1, x_2, \ldots, x_{1000}$ that satisfy $f(\left|x_1\right|)=f(\left|x_2\right|)=\cdots=f(\left|x_{1000}\right|)$, then $x_1+2x_2+2^2x_3+2^3x_4+2^4x_5+\cdots+2^{999}x_{1000}\ne 0$.

An [i]animal[/i] with $n$ [i]cells[/i] is a connected figure consisting of $n$ equal-sized cells[1]. A [i]dinosaur[/i] is an animal with at least $2007$ cells. It is said to be [i]primitive[/i] it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur. (1) Animals are also called [i]polyominoes[/i]. They can be defined inductively. Two cells are [i]adjacent[/i] if they share a complete edge. A single cell is an animal, and given an animal with $n$ cells, one with $n+1$ cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.

If $f : [a, b] \to (0,\infty)$; $0 < a \leq b$; $f$ derivable; $f'$ continuous then:$$\int \limits_{a}^{b}\frac{f'(x)\sqrt{f(x)}}{f^3(x) + 1}\leq \tan^{-1}\left(\frac{f(b)-f(a)}{1 + f(a)f(b)}\right)$$

Ryan Murphy is playing poker. He is dealt a hand of $5$ cards. Given that the probability that he has a straight hand (the ranks are all consecutive; e.g. $3,4,5,6,7$ or $9,10,J,Q,K$) or $3$ of a kind (at least $3$ cards of the same rank; e.g. $5, 5, 5, 7, 7$ or $5, 5, 5, 7,K$) is $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m +n$. [i]Proposed by Aditya Rao[/i]

If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola, then the possible number of points of intersection with the hyperbola is: $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 2\text{ or }3 \qquad\textbf{(C)}\ 2\text{ or }4 \qquad\textbf{(D)}\ 3\text{ or }4 \qquad\textbf{(E)}\ 2,3,\text{ or }4$

Let $n$ be a positive integer and $(x_1, x_2, ..., x_{2n})$, $x_i = 0$ or $1, i = 1, 2, ... , 2n$ be a sequence of $2n$ integers. Let $S_n$ be the sum $S_n = x_1x_2 + x_3x_4 + ... + x_{2n-1}x_{2n}$. If $O_n$ is the number of sequences such that $S_n$ is odd and $E_n$ is the number of sequences such that $S_n$ is even, prove that $$\frac{O_n}{E_n}=\frac{2^n - 1}{2^n + 1}$$

The quadrilateral $ABCD$ has perpendicular diagonals that meet at $O$. The incenters of triangles $ABC, BCD, CDA, DAB$ form a quadrilateral with perimeter $P$. Show that the sum of the inradii of the triangles $AOB, BOC, COD, DOA$ is less than or equal to $\frac{P} {2}$.