Let $a, b,c$ real numbers that are greater than $ 0$ and less than $1$. Show that there is at least one of these three values $ab(1-c)^2$, $bc(1-a)^2$ , $ca(1- b)^2$ which is less than or equal to $\frac{1}{16}$ .

Supppose that there are roads $AB$ and $CD$ but there are no roads $BC$ and $AD$ between four cities $A$, $B$, $C$, and $D$. Define [i]restructing[/i] to be the changing a pair of roads $AB$ and $CD$ to the pair of roads $BC$ and $AD$. Initially there were some cities in a country, some of which were connected by roads and for every city there were exactly $100$ roads starting in it. The minister drew a new scheme of roads, where for every city there were also exactly $100$ roads starting in it. It's known also that in both schemes there were no cities connected by more than one road. Prove that it's possible to obtain the new scheme from the initial after making a finite number of restructings. [i] (Т. Зубов)[/i] [hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]

If the sum of digits in decimal representaion of positive integer $n$ is $111$ and the sum of digits in decimal representation of $7002n$ is $990$, what is the sum of digits in decimal representation of $2003n$? $ \textbf{(A)}\ 309 \qquad\textbf{(B)}\ 330 \qquad\textbf{(C)}\ 550 \qquad\textbf{(D)}\ 555 \qquad\textbf{(E)}\ \text{None of the preceding} $

Albert has a very large bag of candies and he wants to share all of it with his friends. At first, he splits the candies evenly amongst his $20$ friends and himself and he finds that there are five left over. Ante arrives, and they redistribute the candies evenly again. This time, there are three left over. If the bag contains over $500$ candies, what is the fewest number of candies the bag can contain?

Let a sequences: $ x_0\in [0;1],x_{n\plus{}1}\equal{}\frac56\minus{}\frac43 \Big|x_n\minus{}\frac12\Big|$. Find the "best" $ |a;b|$ so that for all $ x_0$ we have $ x_{2009}\in [a;b]$

Find the number of all unordered pairs $\{A,B \}$ of subsets of an $8$-element set, such that $A\cap B \neq \emptyset$ and $\left |A \right | \neq \left |B \right |$.

Let $S$ be the set of all polygonal areas in a plane. Prove that there is a function $f : S \to (0,1)$ which satisfies $f(S_1 \cup S_2) = f(S_1)+ f(S_2)$ for any $S_1,S_2 \in S$ which have common points only on their borders

Let $r,s,t$ positive integers which are relatively prime and $a,b \in G$, $G$ a commutative multiplicative group with unit element $e$, and $a^r=b^s=(ab)^t=e$. (a) Prove that $a=b=e$. (b) Does the same hold for a non-commutative group $G$?

In the figure, the triangles $ABC$ and $CDE$ are equilateral, with side lengths $1$ and $4$, respectively. Moreover, $B$, $C$ and $D$ are collinear and $F$ and $G$ are midpoints of $BC$ and $CD$, respectively. Let $P$ be the intersection point of $AF$ and $BE$. Determine the area of the shaded triangle $BPG$. [img]https://fv5-4.failiem.lv/thumb_show.php?i=qmpfykxcek&view&v=1&PHPSESSID=1f433228a75b4117c35f707722c547c423d3d671[/img]

Two bags contain some numbers, and the total number of numbers is prime. When we tranfer the number 170 from 1 bag to bag 2, the average in both bags increases by one. If the total sum of all numbers is 2004, find the number of numbers.

Find all triplets $(\lambda_1,\lambda_2,\lambda_3) \in \mathbb{R}^3$ such that there exists a matrix $A_{3 \times 3}$ with all entries being non-negative reals whose eigenvalues are $\lambda_1,\lambda_2,\lambda_3$.

$ABCD$ is a cyclic quadrilateral such that $AB = BC = CA$. Diagonals $AC$ and $BD$ intersect at $E$. Given that $BE = 19$ and $ED = 6$, find the possible values of $AD$.

Consider any three consecutive natural numbers. Prove that the cube of the largest cannot be the sum of the cubes of the other two.

Given a positive integer $ n\geq 2$, let $ B_{1}$, $ B_{2}$, ..., $ B_{n}$ denote $ n$ subsets of a set $ X$ such that each $ B_{i}$ contains exactly two elements. Find the minimum value of $ \left|X\right|$ such that for any such choice of subsets $ B_{1}$, $ B_{2}$, ..., $ B_{n}$, there exists a subset $ Y$ of $ X$ such that: (1) $ \left|Y\right| \equal{} n$; (2) $ \left|Y \cap B_{i}\right|\leq 1$ for every $ i\in\left\{1,2,...,n\right\}$.

For a natural number $ n$, let $ a_n\equal{}\int_0^{\frac{\pi}{4}} (\tan x)^{2n}dx$. Answer the following questions. (1) Find $ a_1$. (2) Express $ a_{n\plus{}1}$ in terms of $ a_n$. (3) Find $ \lim_{n\to\infty} a_n$. (4) Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{(\minus{}1)^{k\plus{}1}}{2k\minus{}1}$.

A real function $f$ is defined for $0\le x\le 1$, with its first derivative $f'$ defined for $0\le x\le 1$ and its second derivative $f''$ defined for $0<x<1$. Prove that if $f(0)=f'(0)=f'(1)=f(1)-1 =0$, then there exists a number $0<y<1$ such that $|f''(y)|\ge 4$.

Let $ a, b, c$ be positive reals. Find the smallest value of \[ \frac {a \plus{} 3c}{a \plus{} 2b \plus{} c} \plus{} \frac {4b}{a \plus{} b \plus{} 2c} \minus{} \frac {8c}{a \plus{} b \plus{} 3c}. \]

Let $m$ and $n$ are relatively prime integers and $m>1,n>1$. Show that:There are positive integers $a,b,c$ such that $m^a=1+n^bc$ , and $n$ and $c$ are relatively prime.

A round robin tournament is held with $2016$ participants. Each player plays each other player once and exactly one game results in a tie. Let $W$ be the sum of the squares of each team's win total and let $L$ be the sum of the squares of each team's loss total. Find the maximum possible value of $W-L$. [i]Proposed by Matthew Weiss

A square is inscribed into a triangle (one side of the triangle contains two vertices and each of two remaining sides contains one vertex. Prove that the incenter of the triangle lies inside the square.

A parallelogram with an angle of $60^o$ has $a$ as the longest side and a shortest side $b$. Let's take the perpendiculars down from the vertices of the obtuse angles to the longest diagonal, then it is divided into three equal parts. Determine the ratio $\frac{a}{b}$.

A triangle is constructed on each side of a convex polygon in a manner that the third vertex of each triangle is the meet point of bisectors of the angles adjacent to this side. Prove that these triangles cover all the polygon. Egor Bakaev

Where is the number $35 351$ in the sequence $1, 8, 22, 43,...$?

Given a natural number $n>4$ and $2n+4$ cards numbered with $1, 2, \dots, 2n+4$. On the card with number $m$ a real number $a_m$ is written such that $\lfloor a_{m}\rfloor=m$. Prove that it's possible to choose $4$ cards in such a way that the sum of the numbers on the first two cards differs from the sum of the numbers on the two remaining cards by less than $$\frac{1}{n-\sqrt{\frac{n}{2}}}$$.

You are given the existence of an unsorted sequence $a_1,\ldots, a_5$ of five distinct real numbers. The Erdos-Szekeres theorem states that there exists a subsequence of length $3$ which is either strictly increasing or strictly decreasing. You do not have access to the $a_i$, but you do have an oracle which, when given two indexes $1\leq i < j\leq 5$, will tell you whether $a_i < a_j$ or $a_i > a_j$. What is the minimum number of calls to the oracle needed in order to identify an ordered triple of integers $(r,s,t)$ such that $a_r,a_s,a_t$ is one such sequence?