Eighteen students participate in a team selection test with three problems, each worth up to seven points. All scores are nonnegative integers. After the competition, the results are posted by Evan in a table with 3 columns: the student's name, score, and rank (allowing ties), respectively. Here, a student's rank is one greater than the number of students with strictly higher scores (for example, if seven students score $0, 0, 7, 8, 8, 14, 21$ then their ranks would be $6, 6, 5, 3, 3, 2, 1$ respectively). When Richard comes by to read the results, he accidentally reads the rank column as the score column and vice versa. Coincidentally, the results still made sense! If the scores of the students were $x_1 \le x_2 \le \dots \le x_{18}$, determine the number of possible values of the $18$-tuple $(x_1, x_2, \dots, x_{18})$. In other words, determine the number of possible multisets (sets with repetition) of scores. [i]Proposed by Yang Liu[/i]

Let $I$ be the center of the circle inscribed in triangle $ABC$ which has $\angle A = 60^o$ and the inscribed circle is tangent to the sideBC at point $D$. Choose points X andYon segments $BI$ and $CI$ respectively, such than $DX \perp AB$ and $DY \perp AC$. Choose a point $Z$ such that the triangle $XYZ$ is equilateral and $Z$ and $I$ belong to the same half plane relative to the line $XY$. Prove that $AZ \perp BC$. (Matthew Kurskyi)

A factory packs its products in cubic boxes. In one store, they put $512$ of these cubic boxes together to make a large $8\times 8 \times 8$ cube. When the temperature goes higher than a limit in the store, it is necessary to separate the $512$ set of boxes using horizontal and vertical plates so that each box has at least one face which is not touching other boxes. What is the least number of plates needed for this purpose?

If $p$ and $q$ are natural numbers so that \[ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, \] prove that $p$ is divisible with $1979$.

Let $x_1, x_2, \dots, x_n$ be different real numbers. Prove that \[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll} 0, & \text { if } n \text { is even; } \\ 1, & \text { if } n \text { is odd. } \end{array}\right.\]

In how many ways we can choose 3 non empty and non intersecting subsets from $ (1,2,\ldots,2008)$.

$(FRA 3)$ A polygon (not necessarily convex) with vertices in the lattice points of a rectangular grid is given. The area of the polygon is $S.$ If $I$ is the number of lattice points that are strictly in the interior of the polygon and B the number of lattice points on the border of the polygon, find the number $T = 2S- B -2I + 2.$

Let $ABC$ be a triangle . Let $A'$ be the center of the circle through the midpoint of the side $BC$ and the orthogonal projections of $B$ and $C$ on the lines of support of the internal bisectrices of the angles $ACB$ and $ABC$ , respectively ; the points $B'$ and $C'$ are defined similarly . Prove that the nine-point circle of the triangle $ABC$ and the circumcircle of $A'B'C'$ are concentric.

A three-digit number $N$ is equal to $36$ times the sum of its digits. Find the sum of all possible values of $N$. $\textbf{(A) }576\qquad\textbf{(B) }648\qquad\textbf{(C) }972\qquad\textbf{(D) }1152\qquad\textbf{(E) }1620$

Compute the number of ordered quadruples of positive integers $(a,b,c,d)$ such that $$a!\cdot b!\cdot c!\cdot d!=24!$$ [list=1] [*] 4 [*] 4! [*] $4^4$ [*] None of these [/list]

Give an example of a second degree polynomial $P\in \mathbb{R}[x]$ such that \[\forall x\in \mathbb{R}\text{ with } |x|\leqslant 1: \; \left|P(x)+\frac{1}{x-4}\right| \leqslant 0.01.\] Are there linear polynomials with this property? [i]Octavian Stănășilă[/i]

Jeff rolls a standard $6$ sided die repeatedly until he rolls either all of the prime numbers possible at least once, or all the of even numbers possible at least once. Find the probability that his last roll is a $2$.

In the parallelogram $ABCD$, $\angle BAD =60 ^ \circ$. Let $E $ be the intersection point of the diagonals. The circle circumscribed to the triangle $ACD$ intersects the line $AB$ at the point $K$ (different from $A$), the line $BD$ at the point $P$ (different from $D$), and to the line $BC$ in $L$ (different from $C$). The line $EP$ intersects the circumscribed circle of the triangle $CEL$ at the points $E$ and $M$. Show that the triangles $KLM$ and $CAP$ are congruent.

Let $ABCD$ be a parallelogram and $M,N$ be points in the plane such that $C \in (AM)$ and $D \in (BN)$. Lines $NA,NC$ meet lines $MB,MD$ at points $E,F,G,H$. Show that points $E,F,G,H$ lie on a circle if and only if $ABCD$ is a rhombus.

To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y<0$, then the following operation is allowed: $x,y,z$ are replaced by $x+y,-y,z+y$ respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.

a) Prove that when you divide any prime number by $30$, the remainder is either $1$ or is a prime number! b) Does this also apply when dividing a prime number by $60$? Justify your answer!

1. The numbers $1,2, \ldots$ are placed in a triangle as following: \[ \begin{matrix} 1 & & & \\ 2 & 3 & & \\ 4 & 5 & 6 & \\ 7 & 8 & 9 & 10 \\ \ldots \end{matrix} \] What is the sum of the numbers on the $n$-th row?

The diagonals of an cyclic quadrilateral $ABCD$ intersect at $O$. The circumcircles of triangle $AOB$ and $COD$ intersect lines $BC$ and $AD$, for the second time, at points $M, N, P$and $Q$. Prove that the $MNPQ$ quadrilateral is inscribed in a circle of center $O$.

Let $\alpha\leq 22$ be non-negative integer. For which $\alpha$ does the equation $$8x^{23}-5^{\alpha}y^{23}=1$$ have the most integer solutions (x,y)? What can we say about $\alpha\geq 23$? [hide=Note]I believe the eqn has solutions only when $\alpha=0$. taking modulo 47, $\alpha\equiv 9,17\pmod{23}$ or ($23|\alpha$ and $47|x$). taking modulo 139 and 277 eliminates the $\alpha\equiv 9,17\pmod{23}$ cases. 139=23*6+1 , 277=23*12+1[/hide]

Let $ABCD$ be a parallelogram. Let $\omega_1$ be the circle passing through $D$ tangent to $AB$ at $A$. Let $\omega_2$ be the circle passing through $A$ tangent to $CD$ at $D$. The tangents from $B$ to $\omega_1$ touch it at $A$ and $P$. The tangents from $C$ to $\omega_2$ touch it at $D$ and $Q$. Lines $AP$ and $DQ$ intersect at $X$. The perpendicular bisector of $BC$ intersects $AD$ at $R$. Show that the circumcircles of triangles $\triangle PQX$, $\triangle BCR$ are concentric.

An equilateral triangle of side $n$ has been divided into little equilateral triangles of side $1$ in the usual way. We draw a path over the segments of this triangulation, in such a way that it visits exactly once each one of the $\frac{(n+1)(n+2)}{2}$ vertices. What is the minimum number of times the path can change its direction? The figure below shows a valid path on a triangular board of side $4$, with exactly $9$ changes of direction. [asy] unitsize(30); pair h = (1, 0); pair v = dir(60); pair d = dir(120); for(int i = 0; i < 4; ++i) { draw(i*v -- i*v + (4 - i)*h); draw(i*h -- i*h + (4 - i)*v); draw((i + 1)*h -- (i + 1)*h + (i + 1)*d); } draw(h + v -- v -- (0, 0) -- 2*h -- 2*h + v -- h + 2*v -- 2*v -- 4*v -- 3*h + v -- 3*h -- 4*h, linewidth(2)); draw(3*h -- 4*h, EndArrow); fill(circle(h + v, 0.1)); [/asy] [i]Proposed by Oriol Solé[/i]

Given $100$ integers, is it always possible to choose $15$ of them such that the difference of any two of the chosen numbers is divisible by $7$? What is the answer if $15$ is replaced by $16$?

In the diagram below, $ABCD$ is a square. Point $E$ is the midpoint of $\overline{AD}$. Points $F$ and $G$ lie on $\overline{CE}$, and $H$ and $J$ lie on $\overline{AB}$ and $\overline{BC}$, respectively, so that $FGHJ$ is a square. Points $K$ and $L$ lie on $\overline{GH}$, and $M$ and $N$ lie on $\overline{AD}$ and $\overline{AB}$, respectively, so that $KLMN$ is a square. The area of $KLMN$ is 99. Find the area of $FGHJ$. [asy] pair A,B,C,D,E,F,G,H,J,K,L,M,N; B=(0,0); real m=7*sqrt(55)/5; J=(m,0); C=(7*m/2,0); A=(0,7*m/2); D=(7*m/2,7*m/2); E=(A+D)/2; H=(0,2m); N=(0,2m+3*sqrt(55)/2); G=foot(H,E,C); F=foot(J,E,C); draw(A--B--C--D--cycle); draw(C--E); draw(G--H--J--F); pair X=foot(N,E,C); M=extension(N,X,A,D); K=foot(N,H,G); L=foot(M,H,G); draw(K--N--M--L); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,NE); label("$E$",E,dir(90)); label("$F$",F,NE); label("$G$",G,NE); label("$H$",H,W); label("$J$",J,S); label("$K$",K,SE); label("$L$",L,SE); label("$M$",M,dir(90)); label("$N$",N,dir(180)); [/asy]

Let $m$ and $n$ be positive integers and $p$ be a prime number. Find the greatest positive integer $s$ (as a function of $m,n$ and $p$) such that from a random set of $mnp$ positive integers we can choose $snp$ numbers, such that they can be partitioned into $s$ sets of $np$ numbers, such that the sum of the numbers in every group gives the same remainder when divided by $p$.

Let the real numbers $a, b, c$ be such that $a \ge b \ge c > 0$. Show that $$\frac{a^2-b^2}{c}+ \frac{c^2-b^2}{a}+ \frac{a^2-c^2}{b}\ge 3a - 4b + c.$$ When does equality hold?