Define $\phi^{!}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the number of integers $2 \le n \le 50$ such that $n$ divides $\phi^{!}(n)+1$.

For $-1 < r < 1$, let $S(r)$ denote the sum of the geometric series \[12 + 12r + 12r^2 + 12r^3 + \ldots.\] Let $a$ between $-1$ and $1$ satisfy $S(a)S(-a)=2016$. Find $S(a) + S(-a)$.

Prove that for every nonnegative integer $n$, the number $7^{7^{n}}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.

The license plate of each automobile in Iran consists of a two-digit and a three-digit number as well as a letter of the Persian alphabet. The digit $0$ is not used in the two numbers. To each license plate, we assign the product of the two numbers on it. For example, if the two numbers are $12$ and $365$ on a license plate, the assigned number would be $12 \times 365 = 4380$. What is the average of all the assigned numbers to all possible license plates?

Two triangles $ABC, A’B’C’$ have the same orthocenter $H$ and the same circumcircle with center $O$. Letting $PQR$ be the triangle formed by $AA’, BB’, CC’$, prove that the circumcenter of $PQR$ lies on $OH$.

For which maximal $N$ there exists an $N$-digit number with the following property: among any sequence of its consecutive decimal digits some digit is present once only? Alexey Glebov

Alex and Brian take turns shooting free throws until they each shoot twice. Alex and Brian have 80% and 60% chances of making their free throws, respectively. What is the probability that after each free throw they take, Alex has made at least as many free throws as Brian if Brian shoots first?

You are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2\times 1$ and $3\times 1$. The playing field should be seamless with such dominoes and without overlapping, with no domino protruding beyond the playing field may. Furthermore, all dominoes must be aligned in the same way, i. e. their long sides must be parallel to each other. How many such coverings are possible? (Walther Janous)

A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have $ 3$ rows of small congruent equilateral triangles, with $ 5$ small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of $ 2003$ small equilateral triangles? [asy]unitsize(15mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair Ap=(0,0), Bp=(1,0), Cp=(2,0), Dp=(3,0), Gp=dir(60); pair Fp=shift(Gp)*Bp, Ep=shift(Gp)*Cp; pair Hp=shift(Gp)*Gp, Ip=shift(Gp)*Fp; pair Jp=shift(Gp)*Hp; pair[] points={Ap,Bp,Cp,Dp,Ep,Fp,Gp,Hp,Ip,Jp}; draw(Ap--Dp--Jp--cycle); draw(Gp--Bp--Ip--Hp--Cp--Ep--cycle); for(pair p : points) { fill(circle(p, 0.07),white); } pair[] Cn=new pair[5]; Cn[0]=centroid(Ap,Bp,Gp); Cn[1]=centroid(Gp,Bp,Fp); Cn[2]=centroid(Bp,Fp,Cp); Cn[3]=centroid(Cp,Fp,Ep); Cn[4]=centroid(Cp,Ep,Dp); label("$1$",Cn[0]); label("$2$",Cn[1]); label("$3$",Cn[2]); label("$4$",Cn[3]); label("$5$",Cn[4]); for (pair p : Cn) { draw(circle(p,0.1)); }[/asy] $ \textbf{(A)}\ 1,\!004,\!004 \qquad \textbf{(B)}\ 1,\!005,\!006 \qquad \textbf{(C)}\ 1,\!507,\!509 \qquad \textbf{(D)}\ 3,\!015,\!018 \qquad \textbf{(E)}\ 6,\!021,\!018$

Anumber of schools took part in a tennis tournament. No two players from the same school played against each other. Every two players from different schools played exactly one match against each other. A match between two boys or between two girls was called a [i]single[/i] and that between a boy and a girl was called a [i]mixed single[/i]. The total number of boys differed from the total number of girls by at most 1. The total number of singles differed from the total number of mixed singles by at most 1. At most how many schools were represented by an odd number of players?

Find integer solutions $(x,y)$ of the equation ($1964$ times "$\sqrt{}$"): $$\sqrt{x+\sqrt{x+\sqrt{....\sqrt{x+\sqrt{x}}}}}=y$$

Consider the sequence of integers $ \left( a_n\right)_{n\ge 0} $ defined as $$ a_n=\left\{\begin{matrix}n^6-2017, & 7|n\\ \frac{1}{7}\left( n^6-2017\right) , & 7\not | n\end{matrix}\right. . $$ Determine the largest length a string of consecutive terms from this sequence sharing a common divisor greater than $ 1 $ may have.

Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. [i]Proposed by Finland[/i]

We are given a chessboard with 100 rows and 100 columns. Two squares of the board are said to be adjacent if they have a common side. Initially all squares are white. a) Is it possible to colour an odd number of squares in such a way that each coloured square has an odd number of adjacent coloured squares? b) Is it possible to colour some squares in such a way that an odd number of them have exactly $4$ adjacent coloured squares and all the remaining coloured squares have exactly $2$ adjacent coloured squares? c) Is it possible to colour some squares in such a way that an odd number of them have exactly $2$ adjacent coloured squares and all the remaining coloured squares have exactly $4$ adjacent coloured squares?

Sherry and Val are playing a game. Sherry has a deck containing $2011$ red cards and $2012$ black cards, shuffled randomly. Sherry flips these cards over one at a time, and before she flips each card over, Val guesses whether it is red or black. If Val guesses correctly, she wins $1$ dollar; otherwise, she loses $1$ dollar. In addition, Val must guess red exactly $2011$ times. If Val plays optimally, what is her expected profit from this game?

Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$ with $1$ and $2$ appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designed.

Two segments slide along two skew lines. On each straight line there is a segment. Consider the tetrahedron with vertices at the endpoints of the segments. Prove that the volume of the tetrahedron does not depend on the position of the segments

Jose is $4$ years younger than Zack. Zack is $3$ years older than Inez. Inez is $15$ years old. How old is Jose? $\text{(A)}\ 8 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 14 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 22$

Let $ABCD$ be a convex quadrilateral. Is it possible to find a point $P$ such that the segments drawn between $P$ and the midpoints of the sides of $ABCD$ divide the quadrilateral into four sections of equal area? If $P$ exists, is it unique?

In a table with $n$ rows and $2n$ columns where $n$ is a fixed positive integer, we write either zero or one into each cell so that each row has $n$ zeros and $n$ ones. For $1 \le k \le n$ and $1 \le i \le n$, we define $a_{k,i}$ so that the $i^{\text{th}}$ zero in the $k^{\text{th}}$ row is the $a_{k,i}^{\text{th}}$ column. Let $\mathcal F$ be the set of such tables with $a_{1,i} \ge a_{2,i} \ge \dots \ge a_{n,i}$ for every $i$ with $1 \le i \le n$. We associate another $n \times 2n$ table $f(C)$ from $C \in \mathcal F$ as follows: for the $k^{\text{th}}$ row of $f(C)$, we write $n$ ones in the columns $a_{n,k}-k+1, a_{n-1,k}-k+2, \dots, a_{1,k}-k+n$ (and we write zeros in the other cells in the row). (a) Show that $f(C) \in \mathcal F$. (b) Show that $f(f(f(f(f(f(C)))))) = C$ for any $C \in \mathcal F$.

Is it possible to place 100 consecutive numbers around a circle in some order such that the product of each pair of adjacent numbers is always a perfect square? (Recall that a number is a perfect square if it is the square of an integer.)

Petya and Vasya are given equal sets of $ N$ weights, in which the masses of any two weights are in ratio at most $ 1.25$. Petya succeeded to divide his set into $ 10$ groups of equal masses, while Vasya succeeded to divide his set into $ 11$ groups of equal masses. Find the smallest possible $ N$.

Let $ABCD$ be a cyclic quadrilateral with $|AB|=26$, $|BC|=10$, $m(\widehat{ABD})=45^\circ$,$m(\widehat{ACB})=90^\circ$. What is the area of $\triangle DAC$ ? $ \textbf{(A)}\ 120 \qquad\textbf{(B)}\ 108 \qquad\textbf{(C)}\ 90 \qquad\textbf{(D)}\ 84 \qquad\textbf{(E)}\ 80 $

On the extension of the hypotenuse $AB$ of the right-angled triangle $ABC$, the point $D$ after the point B is marked so that $DC = 2BC$. Let the point $H$ be the foot of the altitude dropped from the vertex $C$. If the distance from the point $H$ to the side $BC$ is equal to the length of the segment $HA$, prove that $\angle BDC = 18$.

How many angles $\theta$ with $0\le\theta\le2\pi$ satisfy $\log(\sin(3\theta))+\log(\cos(2\theta))=0$? $ \textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4 \qquad $