Let $ABC$ be a triangle inscribed in a circle of radius $O$. The angle bisectors $AD,BE,CF$ are concurrent at $I$. The points $M,N, P$ are respectively on $EF, FD$, and $DE$ such that $IM, IN, IP$ are perpendicular to $BC,CA,AB$ respectively. Prove that the three lines $AM,BN, CP$ are concurrent at a point on $OI$. Nguyễn Minh Hà

Let $p$ be a given prime. For each prime $r$, we defind the function as following $F(r) =\frac{(p^{rp} - 1) (p - 1)}{(p^r - 1) (p^p - 1)}$. 1. Show that $F(r)$ is a positive integer for any prime $r \ne p$. 2. Show that $F(r)$ and $F(s)$ are coprime for any primes $r$ and $s$ such that $r \ne p, s \ne p$ and $r \ne s$. 3. Fix a prime $r \ne p$. Show that there is a prime divisor $q$ of $F(r)$ such that $p| q - 1$ but $p^2 \nmid q - 1$.

The figure below is used to fold into a pyramid, and consists of four equilateral triangles erected around a square with area nine. What is the length of the dashed path shown? [asy] real r = 1/2 * 3^(1/2); size(45); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((0,0)--(-r,0.5)--(0,1)--(0.5,1+r)--(1,1)--(1+r,0.5)--(1,0)--(0.5,-r)--cycle,dashed); [/asy] $\textbf{(A) }18\qquad\textbf{(B) }20\qquad\textbf{(C) }21\qquad\textbf{(D) }24\qquad\textbf{(E) }27$

Suppose $ABC$ is an acute triangle. Points $A_1, B_1$ and $C_1$ are chosen on sides $BC, AC$ and $AB$ respectively so that the rays $A_1A, B_1B$ and $C_1C$ are bisectors of triangle $A_1B_1C_1$. Prove that $AA_1, BB_1$ and $CC_1$ are altitudes of triangle $ABC$. (6)

Find all pairs of integers $(a,b)$ satisfying the equation $a^7(a-1)=19b(19b+2)$.

Find all values of $b$ such that the difference between the maximum and minimum values of $f(x) = x^2-2bx-1$ on the interval $[0, 1]$ is $1$.

Alex Lishkov is trying to guess sequence of $2009$ random ternary digits ($0, 1$, or $2$). After he guesses each digit, he finds out whether he was right or not. If he guesses incorrectly, and $k$ was the correct answer, then an oracle tells him what the next $k$ digits will be. Being Bulgarian, Lishkov plays to maximize the expected number of digits guessed correctly. Let $P_n$ be the probability that Lishkov guesses the nth digit correctly. Find $P_{2009}$. Write your answer in the form $x + yRe(\rho^k)$, where $x$ and $y$ are rational, $\rho$ is complex, and $k$ is a positive integer

Three men, Alpha, Beta, and Gamma, working together, do a job in $6$ hours less time than Alpha alone, in $1$ hour less time than Beta alone, and in one-half the time needed by Gamma when working alone. Let $h$ be the number of hours needed by Alpha and Beta, working together to do the job. Then $h$ equals: $\text{(A)}\ \dfrac{5}{2}\qquad \text{(B)}\ \frac{3}{2}\qquad \text{(C)}\ \dfrac{4}{3}\qquad \text{(D)}\ \dfrac{5}{4}\qquad \text{(E)}\ \dfrac{3}{4}$

Consider a square of side of length 50. A polygonal chain $L$ is given in the square such that for every point $P$ of the square there is a point $Q$ of the chain with the property $PQ\le 1.$ Show that the length of $L$ is greater than 1248.

Let a; b; c be positive real numbers such that $\frac{1}{bc}+\frac{1}{ca}+\frac{1}{ab} \geq 1$. Prove that $\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab} \geq 1$.

In Eldorado a year has $20$ months, and each month has $20$ days. One day Brigi asked Adél who lives in Eldorado what day her birthday is. Adél answered that she is only going to tell her the product of the month and the day in her birthday. (For example, if she was born on the $19$th day of the $4$th month, she would say $4 \cdot 19 = 76$.) From this, Brigi was able to tell Adél’s birthday. Based on this information, how many days of the year can be Adél’s birthday?

Let $a, b, c$ are natural numbers such that $a^{2}+b^{2}=c^{2}$ and $c-b=1$ Prove that $(i)$ $a$ is odd. $(ii)$ $b$ is divisible by $4$ $(iii)$ $a^{b}+b^{a}$ is divisible by $c$

Let $I$ be the incenter in the acute triangle $ABC.$ Rays $BI$ and $CI$ intersect the circumcircle of triangle $ABC$ at points $S$ and $T,$ respectively. The segment $ST$ intersects the sides $AB$ and $AC$ at points $K$ and $L,$ respectively. Prove that $AKIL$ is a rhombus.

Find a number that increases by a factor of $1996$ if the digits in the first and fifth places after the decimal place are swapped in its decimal notation.

Does there exist a set $\mathcal{R}$ of positive rational numbers such that every positive rational number is the sum of the elements of a unique finite subset of $\mathcal{R}$? [i]Proposed by Tony Wang[/i]

In the Flatland Congress there are senators who are on committees. Each senator is on at least one committee, and each committee has at least one senator. The rules for forming committees are as follows: $\bullet$ For any pair of senators, there is exactly one committee which contains both senators. $\bullet$ For any two committees, there is exactly one senator who is on both committees. $\bullet$ There exist a set of four senators, no three of whom are all on the same committee. $\bullet$ There exists a committee with exactly $6$ senators. If there are at least $25$ senators in this Congress, compute the minimum possible number of senators $s$ and minimum number of committees $c$ in this Congress. Express your answer in the form $(s, c)$.

Given two integers $x$ and $y$, let $(x \| y)$ denote the [i]concatenation[/i] of $x$ by $y$, which is obtained by appending the digits of $y$ onto the end of $x$. For example, if $x=218$ and $y=392$, then $(x \| y) = 218392$. (a) Find 3-digit integers $x$ and $y$ such that $6(x \| y) = (y \| x)$. (b) Find 9-digit integers $x$ and $y$ such that $6(x \| y) = (y \| x)$.

In triangle $ABC$, the altitude through $B$ intersects $AC$ at $E$ and the altitude through $C$ intersects $AB$ at $F$. Point $T$ is such that $AETF$ is a parallelogram and points $ A$ ,$T$ lie on different half-planes wrt the line $EF$. Point $D$ is such that $ABDC$ is a parallelogram and points $ A$ ,$D$ lie in different half-planes wrt line $BC$. Prove that $T, D$ and the orthocenter of $ABC$ are collinear.

A square $ABCD$ with sude length 1 is given and a circle with diameter $AD$. Find the radius of the circumcircle of this figure.

Does existes a function $f:N->N$ and for all positeve integer n $f(f(n)+2011)=f(n)+f(f(n))$

The function $ f$ satisfies \[f(x) \plus{} f(2x \plus{} y) \plus{} 5xy \equal{} f(3x \minus{} y) \plus{} 2x^2 \plus{} 1\] for all real numbers $ x$, $ y$. Determine the value of $ f(10)$.

The complex numbers $z_1$ and $z_2$ are given such that $z_1=-1+i$ and $z_2=2+4i$. Find the complex number $z_3$ such that $z_1,z_2,z_3$ are the points of an equilateral triangle. How many solutions do we have ?

Let $A$ and $B$ be $2\times 2$ matrices with integer entries, such that $AB=BA$ and $\det B=1$. Prove tht if $\det(A^3+B^3)=1$, then $A^2=O$.

Let $a,b,c$ be positive real numbers with $a^2+b^2+c^2 \geq 3$. Prove that: $$\frac{(a+1)(b+2)}{(b+1)(b+5)}+\frac{(b+1)(c+2)}{(c+1)(c+5)}+\frac{(c+1)(a+2)}{(a+1)(a+5)} \geq \frac{3}{2}.$$

Let $a, b$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc-ab-bc-ca$ is the largest integer which cannot be expressed in the form $xbc+yca+zab$, where $x, y, z \in \mathbb{N}_{0}$