Eddy and Moor play a game with the following rules: [list=a] [*] The game begins with a pile of $N$ stones, where $N$ is a positive integer. [/*] [*] The $2$ players alternate taking turns (e.g. Eddy moves, then Moor moves, then Eddy moves, and so on). [/*] [*] During a player's turn, given $a$ stones remaining in the pile, the player may remove $b$ stones from the pile, where $\gcd(a,b)=1$ and $b\leq a$. [/*] [*] If a player cannot make a move, they lose. [/*] [/list] For example, if Eddy goes first and $N=4$, then Eddy can remove $3$ stones from the pile (since $3\leq4$ and $\gcd(3,4)=1$), leaving $1$ stone in the pile. Moor can then remove $1$ stone from the pile (since $1\leq1$ and $\gcd(1,1)=1$), leaving $0$ stones in the pile. Since Eddy cannot remove stones from an empty pile, he cannot make a move, and therefore loses. Both Eddy and Moor want to win, so they will both always make the best possible move. If Eddy moves first, for how many values of $N<2016$ can Eddy win no matter what moves Moor chooses?

Let $n\geqslant3$ be an integer. Let $A_1A_2\ldots A_n$ be a convex $n$-gon. Consider a line $g$ through $A_1$ that does not contain a further vertice of the $n$-gon. Let $h$ be the perpendicular to $g$ through $A_1$. Project the $n$-gon orthogonally on $h$. For $j=1,\ldots,n$, let $B_j$ be the image of $A_j$ under this projection. The line $g$ is called admissible if the points $B_j$ are pairwise distinct. Consider all convex $n$-gons and all admissible lines $g$. How many different orders of the points $B_1,\ldots,B_n$ are possible?

In an acute triangle $ABC$ with $AC > AB$, let $D$ be the projection of $A$ on $BC$, and let $E$ and $F$ be the projections of $D$ on $AB$ and $AC$, respectively. Let $G$ be the intersection point of the lines $AD$ and $EF$. Let $H$ be the second intersection point of the line $AD$ and the circumcircle of triangle $ABC$. Prove that \[AG \cdot AH=AD^2\]

Let $ABC$ be a triangle with $AC> AB$. Let $P$ be the intersection of $BC$ and the tangent through $A$ around the triangle $ABC$. Let $Q$ be the point on the straight line $AC$, so that $AQ = AB$ and $A$ is between $C$ and $Q$. Let $X$ and $Y$ be the center of $BQ$ and $AP$. Let $R$ be the point on $AP$ so that $AR = BP$ and $R$ is between $A$ and $P$. Show that $BR = 2XY$.

Three chords of a sphere, each having length $5,6,7$, intersect at a single point inside the sphere and are pairwise perpendicular. For $R$ the maximum possible radius of this sphere, find $R^2$.

Numbers $1,2,\ldots,1997^2$ are written in the cells of a $1997\times1997$ table. It is allowed to apply the following transformations: exchange places of any two rows or any two columns, or reverse a row or column. (When a row or column is reversed, the first and last entry exchange their positions, so do the second and second last, etc.) Is it possible that, after finitely many such transformations, arbitrary two numbers exchange their positions and no other number changes its position?

Let $1=d_1<d_2<d_3<\dots<d_k=n$ be the divisors of $n$. Find all values of $n$ such that $n=d_2^2+d_3^3$.

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$. Let the internal angle bisectors at $A$ and $B$ meet at $X$, the internal angle bisectors at $B$ and $C$ meet at $Y$, the internal angle bisectors at $C$ and $D$ meet at $Z$, and the internal angle bisectors at $D$ and $A$ meet at $W$. Further, let $AC$ and $BD$ meet at $P$. Suppose that the points $X$, $Y$, $Z$, $W$, $O$, and $P$ are distinct. Prove that $O$, $X$, $Y$, $Z$, $W$ lie on the same circle if and only if $P$, $X$, $Y$, $Z$, and $W$ lie on the same circle.

Let $N$ be a positive integer. Alberto and Barbara write numbers on a blackboard taking turns, according to the following rules. Alberto starts writing $1$, and thereafter if a player has written $n$ on a certain move, his adversary is allowed to write $n+1$ or $2n$ as long as he/she does not obtain a number greater than $N$. The player who writes $N$ wins. $(a)$ Determine which player has a winning strategy for $N=2005$. $(b)$ Determine which player has a winning strategy for $N=2004$. $(c)$ Find for how many integers $N\le 2005$ Barbara has a winning strategy.

Let be a real $ 4\times 4 $ real matrix with $ \text{det} \left( A^2-I\right) <0. $ Prove that there is a number $ \alpha\in (-1,1) $ so that $ A+\alpha I $ is singular. [i]Mihai Haivas[/i]

Let $ABC$ be an acute-angled triangle inscribed in a circle $\mathcal C (O, R)$ and $F$ a point on the side $AB$ such that $AF < AB/2$. The circle $c_1(F, FA)$ intersects the line $OA$ at the point $A'$ and the circle $\mathcal C$ at $K$. Prove that the quadrilateral $BKFA'$ is cyclic and its circumcircle contains point $O$.

The chemical potential energy stored in a battery is converted into kinetic energy in a toy car that increases its speed first from $0 \text{ mph}$ to $2 \text{ mph}$ and then from $2 \text{ mph}$ up to $4 \text{ mph}$. Ignore the energy transferred to thermal energy due to friction and air resistance. Compared to the energy required to go from $0$ to $2 \text{ mph}$, the energy required to go from $2$ to $4 \text{ mph}$ is $ \textbf{(A)}\ \text{half the amount.}$ $ \textbf{(B)}\ \text{the same amount.}$ $ \textbf{(C)}\ \text{twice the amount.}$ $ \textbf{(D)}\ \text{three times the amount.}$ $ \textbf{(E)}\ \text{four times the amount.} $

For every $m$ and $k$ integers with $k$ odd, denote by $[\frac{m}{k}]$ the integer closest to $\frac{m}{k}$. For every odd integer $k$, let $P\left(k\right)$ be the probability that \[ [\frac{n}{k}] + [\frac{100-n}{k}] = [\frac{100}{k}] \] for an integer $n$ randomly chosen from the interval $1 \le n \le 99!$. What is the minimum possible value of $P\left(k\right)$ over the odd integers $k$ in the interval $1 \le k \le 99$? $ \textbf{(A)}\ \frac{1}{2} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{44}{87} \qquad \textbf{(D)}\ \frac{34}{67} \qquad \textbf{(E)}\ \frac{7}{13} $

In a parlor game, the magician asks one of the participants to think of a three digit number (abc) where a, b, and c represent digits in base 10 in the order indicated. The magician then asks this person to form the numbers (acb), (bca), (bac), (cab), and (cba), to add these five numbers, and to reveal their sum, $N$. If told the value of $N$, the magician can identify the original number, (abc). Play the role of the magician and determine the (abc) if $N= 3194$.

In the convex pentagon $ABCDE$ we know that $AB=DE, BC=EA$ but $AB \neq EA$. $B,C,D,E$ are concyclic . Prove that $A,B,C,D$ are concyclic if and only if $AC=AD.$

Consider the polynomial $P(X)=X^{p-1}+X^{p-2}+\ldots+X+1$, where $p>2$ is a prime number. Show that if $n$ is an even number, then the polynomial \[-1+\prod_{k=0}^{n-1} P\left(X^{p^k}\right)\] is divisible by $X^2+1$. [i]Mircea Becheanu[/i]

Is there a positive integer $m$ such that the equation \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{abc}= \frac{m}{a+b+c}\] has infinitely many solutions in positive integers $a, b, c \;$?

Let $AB$ be a chord of circle $\Gamma$. Let $O$ be the centre of a circle which is tangent to $AB$ at $C$ and internally tangent to $\Gamma$ at $P$. Point $C$ lies between $A$ and $B$. Let the circumcircle of triangle $POC$ intersect $\Gamma$ at distinct points $P$ and $Q$. Prove that $\angle{AQP}=\angle{CQB}$.

Memories all must have at least one out of five different possible colors, two of which are red and green. Furthermore, they each can have at most two distinct colors. If all possible colorings are equally likely, what is the probability that a memory is at least partly green given that it has no red? [i]Proposed by Matthew Weiss

Consider quadrilateral $ABCD$. It is given that $\angle DAC=70^\circ$, $\angle BAC=40^\circ$, $\angle BDC=20^\circ$, $\angle CBD=35^\circ$. Let $P$ be the intersection of $AC$ and $BD$. Find $\angle BPC$.

Consider a triangle $ABC$. Let $A_1$ be the symmetric point of $A$ with respect to the line $BC$, $B_1$ the symmetric point of $B$ with respect to the line $CA$, and $C_1$ the symmetric point of $C$ with respect to the line $AB$. Determine the possible set of angles of triangle $ABC$ for which $A_1B_1C_1$ is equilateral.

Find all the real roots of the system of equations: $$ \begin{cases} x^3+y^3=19 \\ x^2+y^2+5x+5y+xy=12 \end{cases} $$

Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.

In the triangle $ABC$, we have that $M$ and $N$ are points on $AB$ and $AC$, respectively, such that $BC$ is parallel to $MN$. A point $D$ is chosen inside the triangle $AMN$. Let $E$ and $F$ be the points of intersection of $MN$ with $BD$ and $CD$, respectively. Show that the line joining the centers of the circles circumscribed to the triangles $DEN$ and $DFM$ is perpendicular to $AD$.

Let $p$ be a prime and $k$ a positive integer such that $k \le p$. We know that $f(x)$ is a polynomial in $\mathbb Z[x]$ such that for all $x \in \mathbb{Z}$ we have $p^k | f(x)$. [b](a)[/b] Prove that there exist polynomials $A_0(x),\ldots,A_k(x)$ all in $\mathbb Z[x]$ such that \[ f(x)=\sum_{i=0}^{k} (x^p-x)^ip^{k-i}A_i(x),\] [b](b)[/b] Find a counter example for each prime $p$ and each $k > p$.