A right that passes through the incircle $ I$ of the triangle $ \Delta ABC$ intersects the side $ AB$ and $ CA$ in $ P$, respective $ Q$. We denote $ BC\equal{}a\ , \ AC\equal{}b\ ,\ AB\equal{}c$ and $ \frac{PB}{PA}\equal{}p\ ,\ \frac{QC}{QA}\equal{}q$. i) Prove that: \[ a(1\plus{}p)\cdot \overrightarrow{IP}\equal{}(a\minus{}pb)\overrightarrow{IB}\minus{}pc\overrightarrow{IC}\] ii) Show that $ a\equal{}bp\plus{}cq$. iii) If $ a^2\equal{}4bcpq$, then the rights $ AI\ ,\ BQ$ and $ CP$ are concurrents.

The positive real numbers $a, b, c$ with $abc = 1$ Show that: $\sqrt{a + \frac{1}{a}} + \sqrt{b + \frac{1}{b}} + \sqrt{c + \frac{1}{c}}\geq 2(\sqrt{a} + \sqrt{b} + \sqrt{c})$

In a tournament with $n$ players, $n$ < 10, each player plays once against each other player scoring 1 point for a win and 0 points for a loss. Draws do not occur. In a particular tournament only one player ended with an odd number of points and was ranked fourth. Determine whether or not this is possible. If so, how many wins did the player have?

In the lower left corner of the $8 \times 8$ chessboard there is a king. He can move one cell either to the right, or up, or diagonally - to the right and up. How many ways can the king go to the upper right corner of the board if his route does not contain cells located on opposite sides of the diagonal going from the lower left to the upper right corner of the board?

In a circle $C_1$ with centre $O$, let $AB$ be a chord that is not a diameter. Let $M$ be the midpoint of this chord $AB$. Take a point $T$ on the circle $C_2$ with $OM$ as diameter. Let the tangent to $C_2$ at $T$ meet $C_1$ at $P$. Show that $PA^2 + PB^2 = 4 \cdot PT^2$.

$a_1,a_2,\dots,a_{14}$ are different positive integers. All 196 numbers of the form $a_k+a_l$ with $1\leq k,l\leq 14$ are written on a board. Is it possible that for any two-digit combination, there exists a number among all 196 that ends with that combination (i.e., there exist numbers ending with $00, 01, \dots, 99$)? (Author: P. Kozhevnikov)

Let $n\ge 2$ be an integer. Given numbers $a_1, a_2, \ldots, a_n \in \{1,2,3,\ldots,2n\}$ such that $\operatorname{lcm}(a_i,a_j)>2n$ for all $1\le i<j\le n$, prove that $$a_1a_2\ldots a_n \mid (n+1)(n+2)\ldots (2n-1)(2n).$$

Find all positive integers $a$ for which $a^{2000}-1$ is divisible by $10$.

Rectangles $BCC_1B_2,$ $CAA_1C_2,$ and $ABB_1A_2$ are erected outside an acute triangle $ABC.$ Suppose that \[\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.\] Prove that lines $B_1C_2,$ $C_1A_2,$ and $A_1B_2$ are concurrent.

Let circles $\Omega$ and $\omega$ touch internally at point $A$. A chord $BC$ of $\Omega$ touches $\omega$ at point $K$. Let $O$ be the center of $\omega$. Prove that the circle $BOC$ bisects segment $AK$.

A rectangle inscribed in a triangle has its base coinciding with the base $b$ of the triangle. If the altitude of the triangle is $h$, and the altitude $x$ of the rectangle is half the base of the rectangle, then: $\textbf{(A)}\ x=\dfrac{1}{2}h \qquad \textbf{(B)}\ x=\dfrac{bh}{b+h} \qquad \textbf{(C)}\ x=\dfrac{bh}{2h+b} \qquad \textbf{(D)}\ x=\sqrt{\dfrac{hb}{2}} \qquad \textbf{(E)}\ x=\dfrac{1}{2}b$

The numbers $\frac{1}{1}, \frac{1}{2}, ... , \frac{1}{2010}$ are written on a blackboard. A student chooses any two of the numbers, say $x$, $y$, erases them and then writes down $x + y + xy$. He continues to do this until only one number is left on the blackboard. What is this number?

Consider all the ways of writing exactly ten times each of the numbers $0, 1, 2, \ldots , 9$ in the squares of a $10 \times 10$ board. Find the greatest integer $n$ with the property that there is always a row or a column with $n$ different numbers.

Kate multiplied all the integers from $1$ to her age and got $1,307,674,368,000$. How old is Kate? $\text{(A) }14\qquad\text{(B) }15\qquad\text{(C) }16\qquad\text{(D) }17\qquad\text{(E) }18$

Consider $D$ the midpoint of the base $[BC]$ of the isosceles triangle ABC in which $\angle BAC < 90^o$. On the perpendicular from $B$ on the line $BC$ consider the point $E$ such that $\angle EAB= \angle BAC$, and on the line passing though $C$ parallel to the line $AB$ we consider the point $F$ such that $F$ and $D$ are on different side of the line $AC$ and $\angle FAC = \angle CAD$. Prove that $AE = CF$ and $BF = EF$

Given integer $n \geq 3$. There are $n$ dots marked $1$ to $n$ clockwise on a big circle. And between every two neighboring dots, there is a light. At first, every light were dark. A and B are playing a game, A pick up $n$ pairs from $\{ (i,j)|1 \leq i < j \leq n \}$ and for every pairs $(i,j)$. B starts from the point marked $i$ and choose to walk clockwise or counterclockwise to the point marked $j$. And B invert the status of all passing lights (bright $\leftrightarrow$ dark) A hopes the number of dark light can be as much as possible while B hopes the number of bright light can be as much as possible. Suppose A, B are both smart, how many lights are bright in the end? [i]Proposed by BlessingOfHeaven[/i] [img]https://pbs.twimg.com/profile_images/1014932415201120256/u9KAaMZ4_400x400.jpg[/img]

25. Chris and Paul each rent a different room of a hotel from rooms $1 - 60$. However, the hotel manager mistakes them for one person and gives ”Chris Paul” a room with Chris’s and Paul’s room concatenated. For example, if Chris had $15$ and Paul had $9$, ”Chris Paul” has $159$. If there are $360$ rooms in the hotel, what is the probability that ”Chris Paul” has a valid room? 26. Find the number of ways to choose two nonempty subsets $X$ and $Y$ of $\{1, 2, \ldots , 2001\}$, such that $|Y| = 1001$ and the smallest element of $Y$ is equal to the largest element of $X$. 27. Let $r_1, r_2, r_3, r_4$ be the four roots of the polynomial $x^4 - 4x^3 + 8x^2 - 7x + 3$. Find the value of $$\frac{r_1^2}{r_2^2+r_3^2+r_4^2}+\frac{r_2^2}{r_1^2+r_3^2+r_4^2}+\frac{r_3^2}{r_1^2+r_2^2+r_4^2}+\frac{r_4^2}{r_1^2+r_2^2+r_3^2}$$