Let $O$ be the circumcenter of triangle $ABC$ , a line passes through $O$ intersects sides $AB,AC$ at points $M,N$ , $E$ is the midpoint of $MC$ , $F$ is the midpoint of $NB$ , prove that : $\angle FOE= \angle BAC$

Show that there cannot be four squares in arithmetical progression.

$n$ people are in the plane, so that the closest person is unique and each one shoot this closest person with a squirt gun. If $n$ is odd, prove that there exists at least one person that nobody shot. If $n$ is even, will there always be a person who escape? Justify that.

Hernan wants to paint a $8\times 8$ board such that every square is painted with blue or red. Also wants to every $3\times 3$ subsquare have exactly $a$ blue squares and every $2\times 4$ or $4\times 2$ rectangle have exactly $b$ blue squares. Find all couples $(a,b)$ such that Hernan can do the required.

Mark and William are playing a game with a stored value. On his turn, a player may either multiply the stored value by $2$ and add $1$ or he may multiply the stored value by $4$ and add $3$. The first player to make the stored value exceed $2^{100}$ wins. The stored value starts at $1$ and Mark goes first. Assuming both players play optimally, what is the maximum number of times that William can make a move? (By optimal play, we mean that on any turn the player selects the move which leads to the best possible outcome given that the opponent is also playing optimally. If both moves lead to the same outcome, the player selects one of them arbitrarily.)

The centre of the circumcircle of quadrilateral $ABCD$ is $O$ and $O$ is not on any of the sides of $ABCD$. $P=AC \cap BD$. The circumecentres of $\triangle{OAB}$, $\triangle{OBC}$, $\triangle{OCD}$ and $\triangle{ODA}$ are $O_1$, $O_2$, $O_3$ and $O_4$ respectively. Prove that $O_1O_3$, $O_2O_4$ and $OP$ are concurrent.

For all positive real numbers $a, b,c.$ Prove the folllowing inequality$$\frac{a}{c+5b}+\frac{b}{a+5c}+\frac{c}{b+5a}\geq\frac{1}{2}.$$

If $n$ is an integer greater than $7$, prove that ${n \choose 7} - \left[ \frac{n}{7} \right]$ is divisible by $7$.

Let $ n$ be a positive integer and let $ a_1,a_2,\ldots,a_n$ be positive real numbers such that: \[ \sum^n_{i \equal{} 1} a_i \equal{} \sum^n_{i \equal{} 1} \frac {1}{a_i^2}. \] Prove that for every $ i \equal{} 1,2,\ldots,n$ we can find $ i$ numbers with sum at least $ i$.

The eighth grade class at Lincoln Middle School has $93$ students. Each student takes a math class or a foreign language class or both. There are $70$ eighth graders taking a math class, and there are $54$ eighth graders taking a foreign language class. How many eighth graders take [i]only[/i] a math class and [i]not[/i] a foreign language class? $\textbf{(A) }16\qquad \textbf{(B) }23\qquad \textbf{(C) }31\qquad \textbf{(D) }39\qquad \textbf{(E) }70\qquad$

Triangle $ABC$ has right angle at $B$, and contains a point $P$ for which $PA = 10$, $PB = 6$, and $\angle APB = \angle BPC = \angle CPA$. Find $PC$. [asy] pair A=(0,5), B=origin, C=(12,0), D=rotate(-60)*C, F=rotate(60)*A, P=intersectionpoint(A--D, C--F); draw(A--P--B--A--C--B^^C--P); dot(A^^B^^C^^P); pair point=P; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$P$", P, NE);[/asy]

Find all triples $(a,b,c)$ of integers satisfying $a^2 + b^2 + c^2 = 20122012$.

For a positive integer $n$, $(a_0, a_1, \cdots , a_n)$ is a $n+1$-tuple with integer entries. For all $k=0, 1, \cdots , n$, we denote $b_k$ as the number of $k$s in $(a_0, a_1, \cdots ,a_n)$. For all $k = 0,1, \cdots , n$, we denote $c_k$ as the number of $k$s in $(b_0, b_1, \cdots ,b_n)$. Find all $(a_0, a_1, \cdots ,a_n)$ which satisfies $a_0 = c_0$, $a_1=c_1$, $\cdots$, $a_n=c_n$.

[b]p1.[/b] A tennis net is made of strings tied up together which make a grid consisting of small squares as shown below. [img]https://cdn.artofproblemsolving.com/attachments/9/4/72077777d57408d9fff0ea5e79be5ecb6fe8c3.png[/img] The size of the net is $100\times 10$ small squares. What is the maximal number of sides of small squares which can be cut without breaking the net into two separate pieces? (The side is cut only in the middle, not at the ends). [b]p2.[/b] What number is bigger $2^{300}$ or $3^{200}$ ? [b]p3.[/b] All noble knights participating in a medieval tournament in Camelot used nicknames. In the tournament each knight had combats with all other knights. In each combat one knight won and the second one lost. At the end of tournament the losers reported their real names to the winners and to the winners of their winners. Was there a person who knew the real names of all knights? [b]p4.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $10$ rocks in the first pile and $12$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p5.[/b] There is an interesting $5$-digit integer. With a $1$ after it, it is three times as large as with a $1$ before it. What is the number? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are six lines in the plane. No two of them are parallel and no point lies on more than three lines. What is the minimum possible number of points that lie on at least two lines?

Given two real numbers $a, b$ with $a \neq 0$, find all polynomials $P(x)$ which satisfy \[xP(x - a) = (x - b)P(x).\]