The average of the numbers $ 1,2,3,...,98,99$, and $ x$ is $ 100x$. What is $ x$? $ \textbf{(A)}\ \frac{49}{101} \qquad\textbf{(B)}\ \frac{50}{101} \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ \frac{51}{101} \qquad\textbf{(E)}\ \frac{50}{99}$

On the table lay a pencil, sharpened at one end. The student can rotate the pencil around one of its ends at $45^{\circ}$ clockwise or counterclockwise. Can the student, after a few turns of the pencil, go back to the starting position so that the sharpened end and the not sharpened are reversed?

We call a set of points [i]free[/i] if there is no equilateral triangle with the vertices among the points of the set. Prove that every set of $n$ points in the plane contains a [i]free[/i] subset with at least $\sqrt{n}$ elements.

Let $ABCD$ be the cyclic quadrilateral. Suppose that there exists some line $l$ parallel to $BD$ which is tangent to the inscribed circles of triangles $ABC, CDA$. Show that $l$ passes through the incenter of $BCD$ or through the incenter of $DAB$. [i](Proposed by Fedir Yudin)[/i]

[b]a)[/b] Among the $21$ pairwise distances between the $7$ points of the plane, prove that one and the same number occurs not more than $12$ times. [b]b)[/b] Find a maximum number of times may meet the same number among the $15$ pairwise distances between $6$ points of the plane.

Find all $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, $f(x)+f(y) = f(x+y)$ and $f(x^{2013}) = f(x)^{2013}$. [i]Proposed by Calvin Deng[/i]

Given that $x$, $y$ and $z$ are positive integers such that $x^3y^5z^6$ is a perfect 7th power of a positive integer, show that also $x^5y^6z^3$ is a perfect 7th power.

What is the minimum number of times you have to take your pencil off the paper to draw the following figure (the dots are for decoration)? You must lift your pencil off the paper after you're done, and this is included in the number of times you take your pencil off the paper. You're not allowed to draw over an edge twice. [center][img]http://i.imgur.com/CBGmPmv.png[/img][/center]

Let $p>$ be a prime number and $S$ be a set of $p+1$ integers. Prove that there exist pairwise distinct numbers $a_1,a_2,...,a_{p-1}\in S$ that $$ a_1+2a_2+3a_3+...+(p-1)a_{p-1}$$ is divisible by $p$.

For a given positive integer $n$, there are a total of $5n$ balls labeled with numbers $1$, $2$, $3$, $\cdots$, $n$, with 5 balls for each number. The balls are put into $n$ boxes, with $5$ balls in each box. Show that you can color two balls red and one ball blue in each box so that the sum of the numbers on the red balls is twice the sum of the numbers on the blue balls.

As shown in the figure, let point $E$ be the intersection of the diagonals $AC$ and $BD$ of the cyclic quadrilateral $ABCD$. The circumcenter of triangle $ABE$ is denoted as $K$. Point $X$ is the reflection of point $B$ with respect to the line $CD$, and point $Y$ is the point on the plane such that quadrilateral $DKEY$ is a parallelogram. Prove that the points $D,E,X,Y$ are concyclic. [img]https://cdn.artofproblemsolving.com/attachments/3/4/df852f90028df6f09b4ec1342f5330fc531d12.jpg[/img]

Determine the period of orbit for the star of mass $3M$. (A) $\pi \sqrt{\frac{d^3}{GM}}$ (B) $\frac{3\pi}{4}\sqrt{\frac{d^3}{GM}}$ (C) $\pi \sqrt{\frac{d^3}{3GM}}$ (D) $2\pi \sqrt{\frac{d^3}{GM}}$ (E) $\frac{\pi}{4} \sqrt{\frac{d^3}{GM}}$

$a,b,c$ are real numbers. The sufficient and necessary condition of $\forall x\in\mathbb{R},a\sin x+b\cos x+c>0$ is $\text{(A)}$ $a=b=0,c>0$ $\text{(B)}$ $\sqrt{a^2+b^2}=c$ $\text{(C)}$ $\sqrt{a^2+b^2}<c$ $\text{(D)}$ $\sqrt{a^2+b^2}>c$

Consider any permutation $\sigma$ of $\{0,1,2,\dots,9\}$ and for each of the 8 triples of consecutive numbers in this permutation, consider the sum of these three numbers. Let $M(\sigma)$ be the largest of these 8 sums. (For example, for the permutation $\sigma = (4, 6, 2, 9, 0, 1, 8, 5, 7, 3)$ we get the 8 sums 12, 17, 11, 10, 9, 14, 20, 15, and $M(\sigma) = 20$.) (a) Find a permutation $\sigma_1$ such that $M(\sigma_1) = 13$. (b) Does there exist a permutation $\sigma_2$ such that $M(\sigma_2) = 12$?

Three friends Archie, Billie, and Charlie play a game. At the beginning of the game, each of them has a pile of $2024$ pebbles. Archie makes the first move, Billie makes the second, Charlie makes the third and they continue to make moves in the same order. In each move, the player making the move must choose a positive integer $n$ greater than any previously chosen number by any player, take $2n$ pebbles from his pile and distribute them equally to the other two players. If a player cannot make a move, the game ends and that player loses the game. $\hspace{5px}$ Determine all the players who have a strategy such that, regardless of how the other two players play, they will not lose the game. [i]Proposed by Ilija Jovčeski, Macedonia[/i]

Let $M$ be the midpoint of side $BC$ of acute triangle $ABC$. The circle centered at $M$ passing through $A$ intersects the lines $AB$ and $AC$ again at $P$ and $Q$, respectively. The tangents to this circle at $P$ and $Q$ meet at $D$. Prove that the perpendicular bisector of $BC$ bisects segment $AD$.

From the point $P$ outside a circle $\omega$ with center $O$ draw the tangents $PA$ and $PB$ where $A$ and $B$ belong to $\omega$.In a random point $M$ in the chord $AB$ we draw the perpendicular to $OM$, which intersects $PA$ and $PB$ in $C$ and $D$. Prove that $M$ is the midpoint $CD$.

The greatest common divisor $d$ and the least common multiple $u$ of positive integers $m$ and $n$ satisfy the equality $3m + n = 3u + d$. Prove that $m$ is divisible by $n$.

Let $ABC$ be an acute triangle such that $AH = HD$, where $H$ is the orthocenter of $ABC$ and $D \in BC$ is the foot of the altitude from the vertex $A$. Let $\ell$ denote the line through $H$ which is tangent to the circumcircle of the triangle $BHC$. Let $S$ and $T$ be the intersection points of $\ell$ with $AB$ and $AC$, respectively. Denote the midpoints of $BH$ and $CH$ by $M$ and $N$, respectively. Prove that the lines $SM$ and $TN$ are parallel.

The point $ M$ is chosen inside parallelogram $ ABCD$. Show that $ \angle MAB$ is congruent to $ \angle MCB$, if and only if $ \angle MBA$ and $ \angle MDA$ are congruent.

If $p$ is a prime number and $a>1$ is a natural number , then show that the greatest common divisor of the two numbers $a-1$ and $\frac{a^p-1}{a-1}$ is either $1$ or $p$ .

A regular polygon of 10 sides (a regular decagon) may be inscribed in a circle in the following two distinct ways: Divide the circumference into $10$ equal arcs and (1) join each division point to the next by straight line segments, (2) join each division point to the next but two by straight line segments. (See figures). Prove that the difference in the side lengths of these two decagons is equal to the radius of their circumscribed circle. [img]https://cdn.artofproblemsolving.com/attachments/7/9/41c38d08f4f89e07852942a493df17eaaf7498.png[/img]

Let $m$ and $n$ be fixed positive integers. Tsvety and Freyja play a game on an infinite grid of unit square cells. Tsvety has secretly written a real number inside of each cell so that the sum of the numbers within every rectangle of size either $m$ by $n$ or $n$ by $m$ is zero. Freyja wants to learn all of these numbers. One by one, Freyja asks Tsvety about some cell in the grid, and Tsvety truthfully reveals what number is written in it. Freyja wins if, at any point, Freyja can simultaneously deduce the number written in every cell of the entire infinite grid (If this never occurs, Freyja has lost the game and Tsvety wins). In terms of $m$ and $n$, find the smallest number of questions that Freyja must ask to win, or show that no finite number of questions suffice. [i]Nikolai Beluhov[/i]

(a) A set of lines is drawn in the plane in such a way that they create more than 2010 intersections at a particular angle $\alpha$. Determine the smallest number of lines for which this is possible. (b) Determine the smallest number of lines for which it is possible to obtain exactly 2010 such intersections.

It is known that the quadratic equations $x^2 + 6x + 4a = 0$ and $x^2 + 2bx - 12 = 0$ have a common solution. Prove that then there is a common solution to the quadratic equations $x^2 + 9x + 9a = 0$ and $x^2 + 3bx - 27 = 0$.