Let $ABC$ be a triangle such that $AB = 26, AC = 30,$ and $BC = 28$. Let $C'$ and $B'$ be the reflections of the circumcenter $O$ over $AB$ and $AC$, respectively. The length of the portion of line segment $B'C'$ inside triangle $ABC$ can be written as $\frac{p}{q}$, where $p,q$ are relatively prime positive integers. Compute $p+q$.

Euler is selling Mathematician cards to Gauss. Three Fermat cards plus $5$ Newton cards costs $95$ Euros, while $5$ Fermat cards plus $2$ Newton cards also costs $95$ Euros. How many Euroes does one Fermat card cost? $$ \mathrm a. ~ 10\qquad \mathrm b.~15\qquad \mathrm c. ~20 \qquad \mathrm d. ~30 \qquad \mathrm e. ~35 $$

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 $a$