(a) For which positive numbers $n$ does there exist a sequence $x_1, x_2, ..., x_n$, which contains each of the numbers $1, 2, ..., n$ exactly once and for which $x_1 + x_2 +... + x_k$ is divisible by $k$ for each $k = 1, 2,...., n$? (b) Does there exist an infinite sequence $x_1, x_2, x_3, ..., $ which contains every positive integer exactly once and such that $x_1 + x_2 +... + x_k$ is divisible by $k$ for every positive integer $k$?

Let $a$ be a real number. Evaluate \[\int _{-\pi+a}^{3\pi+a} |x-a-\pi|\sin \left(\frac{x}{2}\right)dx\]

The incircle of a triangle $ABC$ touches $AC, BC$ , and $AB$ at $M , N$, and $R$, respectively. Let $S$ be a point on the smaller arc $MN$ and $t$ be the tangent to this arc at $S$ . The line $t$ meets $NC$ at $P$ and $MC$ at $Q$. Prove that the lines $AP, BQ, SR, MN$ have a common point.

Five friends named Ella, Jacob, Muztaba, Peter, and William are suspicious of their friends for having secret group chats. Call a group of three people a "secret chat" if there is a chat with just the three of them (there cannot be multiple chats with the same three people). They have the following perfectly logical conversation in this order: [list] [*] Ella: I am part of $5$ secret chats. [*] Jacob: I know all of the secret chats that Ella is in. [*] Muztaba: Peter is in all but one of my secret chats. [*] Peter: I am in a secret chat that William cannot know exists. [*] William: I share exactly two secret chats with Jacob and two secret chats with Peter. [/list] Let $E$ be the number of chats Ella is in, $J$ the number of chats Jacob is in, $M$ the number of chats Muztaba is in, $P$ the number of chats Peter is in, and $W$ the number of chats William is in. Find $10000E$ $+$ $1000J$ $+$ $100M$ $+$ $10P+W$.

A mathematics competition had $9$ easy and $6$ difficult problems. Each of the participants in the competition solved $14$ of the $15$ problems. For each pair, consisting of an easy and a difficult problem, the number of participants who solved both those problems was recorded. The sum of these recorded numbers was $459$. How many participants were there?

Prove that $$\sum_{n=1}^\infty\frac{n^2}{(7n)!}=\frac1{7^3}\sum_{k=1}^2\sum_{j=0}^6e^{\cos(2\pi j/7)}\cdot\cos\left(\frac{2k\pi j}7+\sin\frac{2\pi j}7\right).$$

$2013$ people run a marathon on a straight road $4m$ wide broad. At any given moment, no two runners are closer $2$ m from each other. Prove that there are two runners that at that moment are more than $1052$ m from each other. Note: Consider runners as points.

Let $\alpha$ be a complex number such that both $\alpha$ and $\alpha+1$ have modulus $1$. If for a positive integer $n$, $1+\alpha$ is an $n$-th root of unity, then show that $\alpha$ is also an $n$-th root of unity and $n$ is a multiple of $6$.

We consider all partitions of a positive integer n into a sum of (nonnegative integer) exponents of $2$ (i.e. $1$, $2$, $4$, $8$ , $ . . .$ ). A number in the sum is allowed to repeat an arbitrary number of times (e.g. $7 = 2 + 2 + 1 + 1 + 1$) and two partitions differing only in the order of summands are considered to be equal (e.g. $8 = 4 + 2 + 1 + 1$ and $8 = 1 + 2 + 1 + 4$ are regarded to be the same partition). Let $E(n)$ be the number of partitions in which an even number of exponents appear an odd number of times and $O(n)$ the number of partitions in which an odd number of exponents appear an odd number of times. For example, for $n = 5$ partitions counted in $E(n)$ are $5 = 4 + 1$ and $5 = 2 + 1 + 1 + 1$, whereas partitions counted in O(n) are $5 = 2 + 2 + 1$ and $5 = 1 + 1 + 1 + 1 + 1$, hence $E(5) = O(5) = 2$. Find $E(n) - O(n)$ as a function of $n$.

We are given a convex four-sided pyramid with apex $S$ and base face $ABCD$ such that the pyramid has an inscribed sphere (i.e., it contains a sphere which is tangent to each race). By making cuts along the edges $SA,SB,SC,SD$ and rotating the faces $SAB,SBC,SCD,SDA$ outwards into the plane $ABCD$, we unfold the pyramid into the polygon $AKBLCMDN$ as shown in the figure. Prove that $K,L,M,N$ are concyclic. [i] Tibor Bakos and Géza Kós [/i]

Let $D$, $E$, $F$ be the midpoints of the arcs $BC$, $CA$, $AB$ on the circumcircle of a triangle $ABC$ not containing the points $A$, $B$, $C$, respectively. Let the line $DE$ meets $BC$ and $CA$ at $G$ and $H$, and let $M$ be the midpoint of the segment $GH$. Let the line $FD$ meet $BC$ and $AB$ at $K$ and $J$, and let $N$ be the midpoint of the segment $KJ$. a) Find the angles of triangle $DMN$; b) Prove that if $P$ is the point of intersection of the lines $AD$ and $EF$, then the circumcenter of triangle $DMN$ lies on the circumcircle of triangle $PMN$.

A tourist office was investigating the numbers of sunny and rainy days in a year in each of six regions. The results are partly shown in the following table: Region , sunny or rainy , unclassified $A \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 336 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,29$ $B \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 321 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,44$ $C \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 335 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,30$ $D \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 343 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,22$ $E \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 329 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,36$ $F \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 330 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,35$ Looking at the detailed data, an officer observed that if one region is excluded, then the total number of rainy days in the other regions equals one third of the total number of sunny days in these regions. Determine which region is excluded.

A game is played with tokens according to the following rules. In each round, the player with the most tokens gives one token to each of the other players and also places one token into a discard pile. The game ends when some player runs out of tokens. Players $ A$, $ B$, and $ C$ start with $ 15$, $ 14$, and $ 13$ tokens, respectively. How many rounds will there be in the game? $ \textbf{(A)}\ 36 \qquad \textbf{(B)}\ 37 \qquad \textbf{(C)}\ 38\qquad \textbf{(D)}\ 39\qquad \textbf{(E)}\ 40$

The circles $k_1$ and $k_2$ intersect at points $D$ and $P$. The common tangent of the two circles on the side of $D$ touches $k_1$ at $A$ and $k_2$ at $B$. The straight line $AD$ intersects $k_2$ for a second time at $C$. Let $M$ be the center of the segment $BC$. Show that $ \angle DPM = \angle BDC$ .

Suppose there exist constants $A$, $B$, $C$, and $D$ such that \[n^4=A\binom n4+B\binom n3+C\binom n2 + D\binom n1\] holds true for all positive integers $n\geq 4$. What is $A+B+C+D$? [i]Proposed by David Altizio[/i]

The numbers $1,2,\ldots,100$ are written in the cells of a $10\times 10$ table, each number is written once. In one move, Nazar may interchange numbers in any two cells. Prove that he may get a table where the sum of the numbers in every two adjacent (by side) cells is composite after at most $35$ such moves. [i]N. Agakhanov[/i]

Are there two such quadrangles that the sides of the first are less than the corresponding sides of the second, and the corresponding diagonals are larger? (Arseniy Akopyan)

Pinocchio claims that he can take some non-right-angled triangles , all of which are similar to one another and some of which may be congruent to one another, and put them together to form a rectangle. Is Pinocchio lying? (A Fedotov)

(A.Zaslavsky, 9--10) Given four points $ A$, $ B$, $ C$, $ D$. Any two circles such that one of them contains $ A$ and $ B$, and the other one contains $ C$ and $ D$, meet. Prove that common chords of all these pairs of circles pass through a fixed point.

Pentagon $ABCDE$ has $AB = BC = CD = DE$, $\angle ABC = \angle BCD = 108^o$, and $\angle CDE = 168^o$. Find the measure of angle $\angle BEA$ in degrees.

Find the smallest positive integer $n$ such that it is possible to paint each of the $64$ squares of an $8 \times 8$ board of one of $n$ colors so that any four squares that form an $L$ as in the following figure (or congruent figures obtained through rotations and/or reflections) have different colors. [img]https://cdn.artofproblemsolving.com/attachments/a/2/c8049b1be8f37657c058949e11faf041856da4.png[/img]

Suppose we have a convex quadrilateral $ABCD$ such that $\angle B = 100^\circ$ and the circumcircle of $\triangle ABC$ has a center at $D$. Find the measure, in degrees, of $\angle D$. [i]Note:[/i] The circumcircle of a $\triangle ABC$ is the unique circle containing $A$, $B$, and $C$.

Let $\{a_n\}$ be a bounded real sequence. (a) Prove that if X is a positive-measure subset of $\mathbb R$, then for almost all $x\in X$, there exist a subsequence $\{y_n\}$ of X such that $$\sum_{n=1}^\infty (n(y_n-x)-a_n)=1$$ (b) construct an unbounded sequence $\{a_n\}$ for which the above equation is also true.

A light has been placed on every lattice point (point with integer coordinates) on the (infinite) 2$D$ plane. Dene the Chebyshev distance between points $(x_1,y_1)$ and $(x_2, y_2)$ to be $\ max (|x_1 - x_2|, |y_1 -y_2|)$. Each light is turned on with probability $\frac{1}{2^{d/2}}$ , where $d$ is the Chebyshev distance from that point to the origin. What is expected number of lights that have all their directly adjacent lights turned on? (Adjacent points being points such that $|x_1-x_2|+|y_1- y_2| =1$.)

Find all functions $f : \mathbb{N} \rightarrow \mathbb{R}$ such that for all triples $a,b,c$ of positive integers the following holds : $$f(ac)+f(bc)-f(c)f(ab) \ge 1$$ Proposed by [i]Mojtaba Zare[/i]