Let $ n$ be a positive integer such that $ n\geq 3$. Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2n$ positive real numbers satisfying the equations \[ a_1 \plus{} a_2 \plus{} ... \plus{} a_n \equal{} 1, \quad \text{and} \quad b_1^2 \plus{} b_2^2 \plus{} ... \plus{} b_n^2 \equal{} 1.\] Prove the inequality \[a_1\left(b_1 \plus{} a_2\right) \plus{} a_2\left(b_2 \plus{} a_3\right) \plus{} ... \plus{} a_{n \minus{} 1}\left(b_{n \minus{} 1} \plus{} a_n\right) \plus{} a_n\left(b_n \plus{} a_1\right) < 1.\]

In the following, the point of intersection of two lines $ g$ and $ h$ will be abbreviated as $ g\cap h$. Suppose $ ABC$ is a triangle in which $ \angle A \equal{} 90^{\circ}$ and $ \angle B > \angle C$. Let $ O$ be the circumcircle of the triangle $ ABC$. Let $ l_{A}$ and $ l_{B}$ be the tangents to the circle $ O$ at $ A$ and $ B$, respectively. Let $ BC \cap l_{A} \equal{} S$ and $ AC \cap l_{B} \equal{} D$. Furthermore, let $ AB \cap DS \equal{} E$, and let $ CE \cap l_{A} \equal{} T$. Denote by $ P$ the foot of the perpendicular from $ E$ on $ l_{A}$. Denote by $ Q$ the point of intersection of the line $ CP$ with the circle $ O$ (different from $ C$). Denote by $ R$ be the point of intersection of the line $ QT$ with the circle $ O$ (different from $ Q$). Finally, define $ U \equal{} BR \cap l_{A}$. Prove that \[ \frac {SU \cdot SP}{TU \cdot TP} \equal{} \frac {SA^{2}}{TA^{2}}. \]

Prove that in a group of $50$ people there are always two who have an even number (possibly zero) of common acquaintances within the group. (SI Tokarev)

Find the number of subsets of $\{1,2,\ldots,7\}$ that do not contain two consecutive integers.

We string $2n$ white balls and $2n$ black balls, forming a continuous chain. Demonstrate that, in whatever order the balls are placed, it is always possible to cut a segment of the chain to contain exactly $n$ white balls and $n$ black balls.

An acute triangle $\triangle ABC$ has incenter $I$, and the incircle hits $BC, CA, AB$ at $D, E, F$. Lines $BI, CI, BC, DI$ hits $EF$ at $K, L, M, Q$ and the line connecting the midpoint of segment $CL$ and $M$ hits the line segment $CK$ at $P$. Prove that $$PQ=\frac{AB \cdot KQ}{BI}$$

For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.

The natural numbers are written in sequence, in increasing order, and by this we get an infinite sequence of digits. Find the least natural $k$, for which among the first $k$ digits of this sequence, any two nonzero digits have been written a different number of times. [i]Aleksandar Ivanov, Emil Kolev [/i]

Let $n$ be an integer greater than two, and let $A_1,A_2, \cdots , A_{2n}$ be pairwise distinct subsets of $\{1, 2, ,n\}$. Determine the maximum value of \[\sum_{i=1}^{2n} \dfrac{|A_i \cap A_{i+1}|}{|A_i| \cdot |A_{i+1}|}\] Where $A_{2n+1}=A_1$ and $|X|$ denote the number of elements in $X.$

If $a$, $b$ and $c$ are real numbers such that $ab = 31$, $ac = 13$, and $bc = 5$, compute the product of all possible values of $abc$.

Let $ a,b,c $ be positive real numbers such that $ a^4+b^4+c^4=3 $. Prove that \[ \frac{9}{a^2+b^4+c^6}+\frac{9}{a^4+b^6+c^2}+\frac{9}{a^6+b^2+c^4}\leq\ a^6+b^6+c^6+6 \]

Vanya has chosen two positive numbers, x and y. He wrote the numbers x+y, x-y, x/y, and xy, and has shown these numbers to Petya. However, he didn't say which of the numbers was obtained from which operation. Show that Petya can uniquely recover x and y.

Prove that if the function $f$ is defined on the set of positive real numbers, its values are real, and $f$ satisfies the equation $$f\left( \frac{x+y}{2}\right) + f\left(\frac{2xy}{x+y} \right) =f(x)+f(y)$$ for all positive $x,y$, then $$2f(\sqrt{xy})=f(x)+f(y)$$ for every pair $x,y$ of positive numbers.

Given a prime number $p$ such that $2p$ is equal to the sum of the squares of some four consecutive positive integers. Prove that $p-7$ is divisible by 36.

Let $p > 3$ be a given prime number. For a set $S \subseteq \mathbb{Z}$ and $a \in \mathbb{N}$ , define $S_a = \{ x \in \{ 0,1, 2,...,p-1 \}$ | $(\exists_s \in S) x \equiv_p a \cdot s \}$ . $(a)$ How many sets $S \subseteq \{ 1, 2,...,p-1 \} $ are there for which the sequence $S_1 , S_2 , ..., S_{p-1}$ contains exactly two distinct terms? $(b)$ Determine all numbers $k \in \mathbb{N}$ for which there is a set $ S \subseteq \{ 1, 2,...,p-1 \} $ such that the sequence $S_1 , S_2 , ..., S_{p-1} $ contains exactly $k$ distinct terms. [i]Proposed by Milan Basic and Milos Milosavljevic[/i]

A knight is placed on each square of the first column of a $2017 \times 2017$ board. A [i]move[/i] consists in choosing two different knights and moving each of them to a square which is one knight-step away. Find all integers $k$ with $1 \leq k \leq 2017$ such that it is possible for each square in the $k$-th column to contain one knight after a finite number of moves. Note: Two squares are a knight-step away if they are opposite corners of a $2 \times 3$ or $3 \times 2$ board.

Prove that there exist at least two non-congruent quadrilaterals, both having a circumcircle, such that they have equal perimeters and areas.

Let \(\mathcal{S}=\left\{S_1,S_2,\ldots,S_n\right\}\) be a set of \(n\geq 2020\) distinct points on the Euclidean plane, no three of which are collinear. Andy the ant starts at some point \(S_{i_1}\) in \(\mathcal{S}\) and wishes to visit a series of 2020 points \(\left\{S_{i_1},S_{i_2},\ldots,S_{i_{2020}}\right\}\subseteq\mathcal{S}\) in order, such that \(i_j>i_k\) whenever \(j>k\). It is known that ants can only travel between points in \(\mathcal{S}\) in straight lines, and that an ant's path can never self-intersect. Find a positive integer \(n\) such that Andy can always fulfill his wish. (Lower n will be awarded more marks. Bounds for this problem may be used as a tie-breaker, should the need to do so arise.) [i]Proposed by the ICMC Problem Committee[/i]

In all rectangles with same diagonal $d$ find that one with bigger area .

Let $n$ be a positive integer. Jadzia has to write all integers from $1$ to $2n-1$ on a board, and she writes each integer in blue or red color. We say that pair of numbers $i,j\in \{1,2,3,...,2n-1\}$, where $i\leqslant j$, is $\textit{good}$ if and only if number of blue numbers among $i,i+1,...,j$ is odd. Determine, in terms of $n$, maximal number of good pairs.

In triangle $ABC$, points $P$ and $Q$ are projections of point $A$ onto the bisectors of angles $ABC$ and $ACB$, respectively. Prove that $PQ\parallel BC$.

Portia’s high school has $3$ times as many students as Lara’s high school. The two high schools have a total of $2600$ students. How many students does Portia’s high school have? $\textbf{(A) }600 \qquad \textbf{(B) }650 \qquad \textbf{(C) }1950 \qquad \textbf{(D) }2000 \qquad \textbf{(E) }2050$

Find all real numbers $a$ such that exists function $\mathbb {R} \rightarrow \mathbb {R} $ satisfying the following conditions: 1) $f(f(x)) =xf(x)-ax$ for all real $x$ 2) $f$ is not constant 3) $f$ takes the value $a$

There are 100 points on a circle that are about to be colored in two colors: red or blue. Find the largest number $k$ such that no matter how I select and color $k$ points, you can always color the remaining $100-k$ points such that you can connect 50 pairs of points of the same color with lines in a way such that no two lines intersect.

A postman has to deliver five letters to five different houses. Mischievously, he posts one letter through each door without looking to see if it is the correct address. In how many different ways could he do this so that exactly two of the five houses receive the correct letters?