Show that $abc \le (ab + bc + ca)(a^2 + b^2 + c^2)^2$ for all positive real numbers $a, b$ and $c$ such that $a + b + c = 1$.

If $n$ is a positive integer and $n+1$ is divisible with $24$, prove that sum of all positive divisors of $n$ is divisible with $24$

There exist right triangles with integer side lengths such that the legs differ by $ 1$. For example, $3-4-5$ and $20-21-29$ are two such right triangles. What is the perimeter of the next smallest Pythagorean right triangle with legs differing by $ 1$?

Using squares of side 1, a stair-like figure is formed in stages following the pattern of the drawing. For example, the first stage uses 1 square, the second uses 5, etc. Determine the last stage for which the corresponding figure uses less than 2014 squares. [img]http://www.artofproblemsolving.com/Forum/download/file.php?id=49934[/img]

Find all functions $f : \mathbb{R} \mapsto \mathbb{R}$ such that $f(xy+f(x)) = xf(y) +f(x)$ for all $x,y \in \mathbb{R}$.

Find the largest real number $c$ such that $$\sum_{i=1}^{n}\sum_{j=1}^{n}(n-|i-j|)x_ix_j \geq c\sum_{j=1}^{n}x^2_i$$ for any positive integer $n $ and any real numbers $x_1,x_2,\dots,x_n.$

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

Let $f_0, f_1,...$ be the Fibonacci sequence: $f_0 = f_1 = 1, f_n = f_{n-1} + f_{n-2}$ if $n \geq 2$. Determine all possible positive integers $n$ so that there is a positive integer $a$ such that $f_n \leq a \leq f_{n+1}$ and that $a( \frac{1}{f_1}+\frac{1}{f_1f_2}+\cdots+\frac{1}{f_1f_2...f_n} )$ is an integer.

Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}}$, $y=\frac{1}{n}\sum\limits_{i=1}^n{{y_i}}$ and $z_i=x{y_i}+y{x_i}-{x_i}{y_i}$. Prove that $\sum\limits_{i=1}^n{\left| {{z_i}}\right|}\leqslant n$.

An acute triangle has unit area. Show that there is a point inside the triangle whose distance from each of the vertices is at least $\frac{2}{\sqrt[4]{27}}$.

Each face of a regular tetrahedron with edge length $2011$ is divided into $2011^2$ equilateral triangles of side length $1$, created by drawing lines parallel to each edge. Bruno and Mariano take turns marking one of the unit triangles. Except for the first move, every triangle marked must share at least one point with the triangle marked in the previous move. Bruno plays first. The game ends when a player cannot make a move, and that player loses. Determine which of the two players has a winning strategy and describe the strategy.

[b]6.1 / 7.1[/b] There are $8$ rooks on the chessboard such that no two of them they don't hit each other. Prove that the black squares contain an even number of rooks. [b]6.2 [/b] The natural numbers are arranged in a $3 \times 3$ table. Kolya and Petya crossed out 4 numbers each. It turned out that the sum of the numbers crossed out by Petya is three times the sum numbers crossed out by Kolya. What number is left uncrossed? $$\begin{tabular}{|c|c|c|}\hline 4 & 12 & 8 \\ \hline 13 & 24 & 14 \\ \hline 7 & 5 & 23 \\ \hline \end{tabular} $$ [b]6.3 [/b] Misha and Sasha left at noon on bicycles from city A to city B. At the same time, I left from B to A Vanya. All three travel at constant but different speeds. At one o'clock Sasha was exactly in the middle between Misha and Vanya, and at half past one Vanya was in the middle between Misha and Sasha. When Misha will be exactly in the middle between Sasha and Vanya? [b]6.4[/b] There are $35$ piles of nuts on the table. Allowed to add one nut at a time to any $23$ piles. Prove that by repeating this operation, you can equalize all the heaps. [b]6.5[/b] There are $64$ vertical stripes on the round drum, and each stripe you need to write down a six-digit number from digits $1$ and $2$ so that all the numbers were different and any two adjacent ones differed in exactly one discharge. How to do this? [b]6.6 / 7.6[/b] Two brilliant mathematicians were told in natural terms number and were told that these numbers differ by one. After that they take turns asking each other the same question: “Do you know my number?" Prove that sooner or later one of them will answer positively. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988085_1969_leningrad_math_olympiad]here[/url].

Construct for the triangle $ABC$ a circle $S$ passing through the point $B$ and touching the line $CA$ at the point $A$, a circle $T$ passing through the point $C$ and touches the line $BA$ at the point $A$. The second intersection point of the circles $S$ and $T$ is denoted by $D$. The intersection point of the line $AD$ and the circumscribed circle $\Delta ABC$ is denoted by $E$. Prove that $D$ is the midpoint of the segment $AE$.

Compute $$\lim \limits_{n \to \infty}\frac{1}{n}\sum \limits_{k=1}^n\frac{\sqrt[n+k+1]{n+1}-\sqrt[n+k]{n}}{\sqrt[n+k]{n+1}-\sqrt[n+k]{n}}$$

Let $E \subset [0,1]$ be a Lebesgue measurable set having Lebesgue measure $| E |<\frac{1}{2}$. Let $$h (s) = \int _ {\overline {E}} \frac{dt}{{(s-t)}^2}$$ where $\overline {E} = [0,1] \backslash E$. Prove that there is one $t \in \overline {E}$ for which $$\int_E \frac {ds} {h (s) {(s-t)} ^ 2} \leq c {| E |} ^ 2$$ with some absolute constant c .

Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow? [i]Proposed by Gerhard Woeginger, Netherlands[/i]

Friends Alice, Betty, and Cathy are playing a game. Betty and Cathy are each given a square number, such that Betty knows Cathy's number and Cathy knows Betty's, but neither of them know their own. Alice then says: "The sum of the numbers is less than 100." Betty says: "If Cathy knew the number of possibilities for my number, she would know her own." Cathy then says: "Now I know my number." What is Cathy's number? $\textbf{(A) }16\qquad\textbf{(B) }25\qquad\textbf{(C) }36\qquad\textbf{(D) }49\qquad\textbf{(E) }64$

Set \[A_n=\sum_{k=1}^n \frac{k^6}{2^k}.\] Find $\lim_{n\to\infty} A_n.$

Let $S$ denote the set of fractions $\dfrac mn$ for relatively prime positive integers $m$ and $n$ with $m+n\le 10000$. The least fraction in $S$ that is strictly greater than \[\prod_{i=0}^\infty \left(1-\dfrac{1}{10^{2i+1}}\right)\] can be expressed in the form $\dfrac pq$, where $p$ and $q$ are relatively prime positive integers. Find $1000p+q$. [i]Proposed by James Lin[/i]

Baron Munchausen has a set of $50$ coins. The mass of each is a distinct positive integer not exceeding $100$, and the total mass is even. The Baron claims that it is not possible to divide the coins into two piles with equal total mass. Can the Baron be right?

Let $\Omega$ be a circle, and let $A$, $B$, $C$, $D$ and $K$ be distinct points on it, in that order, and such that lines $BC$ and $AD$ are parallel. Let $A'\neq A$ be a point on line $AK$ such that $BA=BA'$. Similarly, let $C'\neq C$ be a point on line $CK$ such that $DC=DC'$. Prove that segments $AC$ and $A'C'$ have the same length.

Find all integers $ b$ and $ c$ such that the equation $ x^2 - bx + c = 0$ has two real roots $ x_1, x_2$ satisfying $ x_1^2 + x_2^2 = 5$.

The two squares shown share the same center $O$ and have sides of length 1. The length of $\overline{AB}$ is $43/99$ and the area of octagon $ABCDEFGH$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ [asy] real alpha = 25; pair W=dir(225), X=dir(315), Y=dir(45), Z=dir(135), O=origin; pair w=dir(alpha)*W, x=dir(alpha)*X, y=dir(alpha)*Y, z=dir(alpha)*Z; draw(W--X--Y--Z--cycle^^w--x--y--z--cycle); pair A=intersectionpoint(Y--Z, y--z), C=intersectionpoint(Y--X, y--x), E=intersectionpoint(W--X, w--x), G=intersectionpoint(W--Z, w--z), B=intersectionpoint(Y--Z, y--x), D=intersectionpoint(Y--X, w--x), F=intersectionpoint(W--X, w--z), H=intersectionpoint(W--Z, y--z); dot(O); label("$O$", O, SE); label("$A$", A, dir(O--A)); label("$B$", B, dir(O--B)); label("$C$", C, dir(O--C)); label("$D$", D, dir(O--D)); label("$E$", E, dir(O--E)); label("$F$", F, dir(O--F)); label("$G$", G, dir(O--G)); label("$H$", H, dir(O--H));[/asy]

Let $S = \{1, 2, \ldots, n\}$ for some positive integer $n$, and let $A$ be an $n$-by-$n$ matrix having as entries only ones and zeroes. Define an infinite sequence $\{x_i\}_{i \ge 0}$ to be [i]strange[/i] if: [list] [*] $x_i \in S$ for all $i$, [*] $a_{x_kx_{k+1}} = 1$ for all $k$, where $a_{ij}$ denotes the element in the $i^{\text{th}}$ row and $j^{\text{th}}$ column of $A$. [/list] Prove that the set of strange sequences is empty if and only if $A$ is nilpotent, i.e. $A^m = 0$ for some integer $m$.

Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP>CP$. Let $O_1$ and $O_2$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB=12$ and $\angle O_1 P O_2 = 120^\circ$, then $AP=\sqrt{a}+\sqrt{b}$ where $a$ and $b$ are positive integers. Find $a+b$.