Let $m> 3$ be an integer. At a camp there are more than $m$ participants. The camp manager discovers that every time he picks out the camp participants, they say they have exactly one mutual friend among the participants. Which is the largest possible number of participants at the camp? (If $A$ is a friend of $B, B$ is also a friend of $A$. A person is not considered a friend of himself.)

Let $ABC$ be an acute triangle with orthocenter $H$. A point $L \ne A$ lies on the plane of $ABC$ such that $\overline{HL} \perp \overline{AL}$ and $LB : LC = AB : AC$. Suppose $M_1 \ne B$ lies on $\overline{BL}$ such that $\overline{HM_1} \perp \overline{BM_1}$ and $M_2 \ne C$ lies on $\overline{CL}$ such that $\overline{HM_2} \perp \overline{CM_2}$. Prove that $\overline{M_1M_2}$ bisects $\overline{AL}$.

Find all positive integer $n$ and nonnegative integer $a_1,a_2,\dots,a_n$ satisfying: $i$ divides exactly $a_i$ numbers among $a_1,a_2,\dots,a_n$, for each $i=1,2,\dots,n$. ($0$ is divisible by all integers.)

The least positive angle $\alpha$ for which $$\left(\frac34-\sin^2(\alpha)\right)\left(\frac34-\sin^2(3\alpha)\right)\left(\frac34-\sin^2(3^2\alpha)\right)\left(\frac34-\sin^2(3^3\alpha)\right)=\frac1{256}$$ has a degree measure of $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Prove that the equation $x^2+y^2+z^2+t^2=2^{2004}$, where $0 \leq x \leq y \leq z \leq t$, has exactly $2$ solutions in $\mathbb Z$. [i]Mihai Baluna[/i]

One of the factors of $ x^4\plus{}4$ is: $ \textbf{(A)}\ x^2\plus{}2 \qquad\textbf{(B)}\ x\plus{}1 \qquad\textbf{(C)}\ x^2\minus{}2x\plus{}2 \qquad\textbf{(D)}\ x^2\minus{}4\\ \textbf{(E)}\ \text{none of these}$

Basil needs to solve an exercise on summing two fractions $\dfrac{a}{b}$ and $\dfrac{c}{d}$, where $a$, $b$, $c$, $d$ are some non-zero real numbers. But instead of summing he performed multiplication (correctly). It appears that Basil's answer coincides with the correct answer to given exercise. Find the value of $\dfrac{b}{a} + \dfrac{d}{c}$.

The $52$ cards of a standard deck are placed in a $13\times 4$ array. If every two adjacent cards, vertically or horizontally, have the same suit or have the same value, prove that all $13$ cards of the same suit are in the same row.

In the diagram below $ABCDE$ is a regular pentagon, $\overline{AG}$ is perpendicular to $\overline{CD}$, and $\overline{BD}$ intersects $\overline{AG}$ at $F$. Find the degree measure of $\angle AFB$. [asy] import math; size(4cm); pen dps = fontsize(10); defaultpen(dps); pair A,B,C,D,E,F,G; A=dir(90); B=dir(162); C=dir(234); D=dir(306); E=dir(18); F=extension(A,G,B,D); G=(C+D)/2; draw(A--B--C--D--E--cycle^^A--G^^B--D); label("$A$",A,dir(90)*0.5); label("$B$",B,dir(162)*0.5); label("$C$",C,dir(234)*0.5); label("$D$",D,dir(306)*0.5); label("$E$",E,dir(18)*0.5); label("$F$",F,NE*0.5); label("$G$",G,S*0.5); [/asy]

A Yule log is shaped like a right cylinder with height $10$ and diameter $5$. Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\pi$. Compute $a$.

Let $ABC$ be a triangle with $\angle BAC = 120^\circ$ and let $O$ be its circumcenter. Let $P$ and $D$ be the feet of the altitudes from $B$ to the lines $CO$ and $AO$, respectively. Let $M$ be the midpoint of $AO$. Prove that the circumcircle of $MPD$ is tangent to the line $AC$.

In the triangle $ABC$, the inscribed circle touches the sides $CA{}$ and $AB{}$ at the points $B_1{}$ and $C_1{}$, respectively. An arbitrary point $D{}$ is selected on the side $AB{}$. The point $L{}$ is the center of the inscribed circle of the triangle $BCD$. The bisector of the angle $ACD$ intersects the line $B_1C_1$ at the point $M{}$. Prove that $\angle CML=90^\circ$. [i]Proposed by Chan Quang Heung (Vietnam)[/i]

There are 16 boxers in a tournament. Each boxer can fight no more often than once per day. It is known that the boxers are of different strength, and the stronger man always wins. Prove that a 1$0$ day tournament can be organised so as to determine their classification (put them in the order of strength). The schedule of fights for each day is fixed on the evening before and cannot be changed during the day. (A. Andjans, Riga)

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 $2$.