Let $AH_A$ and $BH_B$ be the altitudes of a triangle $ABC$. The line $H_AH_B$ meets the circumcircle of $ABC$ at points $P$ and $Q$. Let $A'$ be the reflection of $A$ about $BC$, and $B'$ be the reflection of $B$ about $CA$. Prove that $A',B', P,Q$ are concyclic.

Let $ABC$ be a acute triangle in which $O$ and $H$ are the center of the circumscribed circle, respectively the orthocenter. Let $M$ be a point on the small arc $BC$ of the circumscribed circle (different from $B$ and $C$) and be $D, E, F$ be the symmetrical of the point $M$ to the lines $OA, OB, OC$. We note with $K$ the intersection of $BF$ and $CE$ and $I$ is the center of the circle inscribed in the triangle $DEF$. a) Show that the segment bisectors of the segments $EF$ and $IK$ intersect on the circle circumscribed to triangle $ABC$. a) Prove that points $H, K, I$ are collinear.

Let $p$ be a prime number. Determine all triples $(a,b,c)$ of positive integers such that $a + b + c < 2p\sqrt{p}$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{p}$

[b]Q6.[/b] At $3:00$ AM, the temperature was $13^o$ below zero. By none it has risen to $32^o$. What is the average hourly increase in temperature ?

In triangle $\vartriangle ABC$, $M$ is the midpoint of $\overline{AB}$ and $N$ is the midpoint of $\overline{AC}$. Let $P$ be the midpoint of $\overline{BN}$ and let $Q$ be the midpoint of $\overline{CM}$. If $AM = 6$, $BC = 8$ and $BN = 7$, compute the perimeter of triangle $\vartriangle NP Q$.

At the 4 PM show, all the seats in the theater were taken, and 65 percent of the audience was children. At the 6 PM show, again, all the seats were taken, but this time only 50 percent of the audience was children. Of all the people who attended either of the shows, 57 percent were children although there were 12 adults and 28 children who attended both shows. How many people does the theater seat?

What is the sum of distinct remainders when $(2n-1)^{502}+(2n+1)^{502}+(2n+3)^{502}$ is divided by $2012$ where $n$ is positive integer? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 1510 \qquad \textbf{(C)}\ 1511 \qquad \textbf{(D)}\ 1514 \qquad \textbf{(E)}\ \text{None}$

In a cyclic quadrilateral $ABCD$ we have $|AD| > |BC|$ and the vertices $C$ and $D$ lie on the shorter arc $AB$ of the circumcircle. Rays $AD$ and $BC$ intersect at point $K$, diagonals $AC$ and $BD$ intersect at point $P$. Line $KP$ intersects the side $AB$ at point $L$. Prove that $\angle ALK$ is acute.

Find the smallest number $n$ such that there exist polynomials $f_1, f_2, \ldots , f_n$ with rational coefficients satisfying \[x^2+7 = f_1\left(x\right)^2 + f_2\left(x\right)^2 + \ldots + f_n\left(x\right)^2.\] [i]Proposed by Mariusz Skałba, Poland[/i]

Let $u, v$ be real numbers. The minimum value of $\sqrt{u^2+v^2} +\sqrt{(u-1)^2+v^2}+\sqrt {u^2+ (v-1)^2}+ \sqrt{(u-1)^2+(v-1)^2}$ can be written as $\sqrt{n}$. Find the value of $10n$.

Let $P$ be an arbitrary variable point in the plane of a triangle $ABC. A_1$ is the projection of $P$ onto $BC, A_2$ is the midpoint of line segment $PA_1, A_2P$ meets $BC$ at $A_3, A_4$ is the reflection of $P$ about $A_3$. Prove that $PA_4$ has a fixed point. Trần Quang Hùng

Andrew writes down all of the prime numbers less than $50$. How many times does he write the digit $2$?

Show that there is precisely one sequence $ a_1,a_2,...$ of integers which satisfies $ a_1\equal{}1, a_2>1,$ and $ a_{n\plus{}1}^3\plus{}1\equal{}a_n a_{n\plus{}2}$ for $ n \ge 1$.

Incircle of $\triangle ABC$ touch $AC$ at $D$. $BD$ intersect incircle at $E$. Points $F,G$ on incircle are such points, that $FE \parallel BC,GE \parallel AB$. $I_1,I_2$ are incenters of $DEF,DEG$. Prove that angle bisector of $\angle GDF$ passes though the midpoint of $I_1I_2 $.

Suppose that the function $f:\mathbb{R}\to\mathbb{R}$ satisfies \[f(x^3 + y^3)=(x+y)(f(x)^2-f(x)f(y)+f(y)^2)\] for all $x,y\in\mathbb{R}$. Prove that $f(1996x)=1996f(x)$ for all $x\in\mathbb{R}$.

To connect to the OFM site, Alice must choose a password. The latter must be consisting of $n$ characters among the following $27$ characters: $$A, B, C, . . ., Y , Z, \#$$ We say that a password $m$ is [i]redundant [/i] if we can color in red and blue a block of consecutive letters of $m$ in such a way that the word formed from the red letters is identical to the word formed from blue letters. For example, the password $H\#ZBZJBJZ$ is redundant, because it contains the [color=#00f]ZB[/color][color=#f00]Z[/color][color=#00f]J[/color][color=#f00]BJ[/color] block, where the word $ZBJ$ appears in both blue and red. At otherwise, the $ABCACB$ password is not redundant. Show that, for any integer $n \ge 1$, there exist at least $18^n$ passwords of length $n$, that is to say formed of $n$ characters each, which are not redundant.

Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positive integers. What is $a+b?$ $\textbf{(A) }31 \qquad \textbf{(B) }47 \qquad \textbf{(C) }62\qquad \textbf{(D) }72 \qquad \textbf{(E) }85$

Let $ABC$ be a triangle with circumcircle $\Gamma$, circumcenter $O$, and orthocenter $H$. Assume that $AB\neq AC$ and that $\angle A \neq 90^{\circ}$. Let $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$ in $\triangle ABC$, respectively. Let $P$ be the intersection of line $MN$ with the tangent line to $\Gamma$ at $A$. Let $Q$ be the intersection point, other than $A$, of $\Gamma$ with the circumcircle of $\triangle AEF$. Let $R$ be the intersection of lines $AQ$ and $EF$. Prove that $PR\perp OH$. [i]Proposed by Ray Li[/i]

A positive integer is called [i]saturated [/i]i f any prime factor occurs at a power greater than or equal to $2$ in its factorisation. For example, numbers $8 = 2^3$ and $9 = 3^2$ are saturated, moreover, they are consecutive. Prove that there exist infinitely many saturated consecutive numbers.

Two non-zero real numbers, $ a$ and $ b,$ satisfy $ ab \equal{} a \minus{} b$. Which of the following is a possible value of $ \frac {a}{b} \plus{} \frac {b}{a} \minus{} ab$? $ \textbf{(A)}\minus{}\!2 \qquad \textbf{(B)}\minus{}\!\frac {1}{2} \qquad \textbf{(C)}\ \frac {1}{3} \qquad \textbf{(D)}\ \frac {1}{2} \qquad \textbf{(E)}\ 2$

How many ordered pairs of integers $(x,y)$ are there such that $2011y^2=2010x+3$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{Infinitely many} $

Let $t\in R$. Prove that $\exists A:R \times R \to R$ such that A is a symmetric, biadditive, nonzero function and $A(tx,x)=0 \,\forall x\in R$ iff t is transcendental or (t is algebraic and t,-t are conjugates over $\mathbb{Q}$).

Let $P\left(X\right)=X^5+3X^4-4X^3-X^2-3X+4$. Determine the number of monic polynomials $Q\left(x\right)$ with integer coefficients such that $\frac{P\left(X\right)}{Q\left(X\right)}$ is a polynomial with integer coefficients. Note: a monic polynomial is one with leading coefficient $1$ (so $x^3-4x+5$ is one but not $5x^3-4x^2+1$ or $x^2+3x^3$). [i]2016 CCA Math Bonanza Individual #9[/i]

$100$ distinct natural numbers $a_1, a_2, a_3, \ldots, a_{100}$ are written on the board. Then, under each number $a_i$, someone wrote a number $b_i$, such that $b_i$ is the sum of $a_i$ and the greatest common factor of the other $99$ numbers. What is the least possible number of distinct natural numbers that can be among $b_1, b_2, b_3, \ldots, b_{100}$?

For positive integers $n$, let $S_n$ be the set of integers $x$ such that $n$ distinct lines, no three concurrent, can divide a plane into $x$ regions (for example, $S_2=\{3,4\}$, because the plane is divided into 3 regions if the two lines are parallel, and 4 regions otherwise). What is the minimum $i$ such that $S_i$ contains at least 4 elements?