Suppose we have a $n$-gon. Some $n-3$ diagonals are coloured black and some other $n-3$ diagonals are coloured red (a side is not a diagonal), so that no two diagonals of the same colour can intersect strictly inside the polygon, although they can share a vertex. Find the maximum number of intersection points between diagonals coloured differently strictly inside the polygon, in terms of $n$. [i]Proposed by Alexander Ivanov, Bulgaria[/i]

There are 100 visually identical coins of three types: golden, silver and copper. There is at least one coin of each type. Each golden coin weighs 3 grams, each silver coins weighs 2 grams and each copper coin weighs 1 gram. How to find the type of each coin performing no more than 101 measurements on a balance scale with no weights.

[list] [*] A power grid with the shape of a $3\times 3$ lattice with $16$ nodes (vertices of the lattice) joined by wires (along the sides of squares. It may have happened that some of the wires have burned out. In one test technician can choose any two nodes and check if electrical current circulates between them (i.e there is a chain of intact wires joining the chosen nodes) . Technicial knows that current will circulate from any node to another node. What is the least number of tests required to demonstrate this? [*] Previous problem for the grid of $7\times 7$ lattice.[/list]

In the 75th Annual Putnam Games, participants compete at mathematical games. Patniss and Keeta play a game in which they take turns choosing an element from the group of invertible $n\times n$ matrices with entries in the field $\mathbb{Z}/p\mathbb{Z}$ of integers modulo $p,$ where $n$ is a fixed positive integer and $p$ is a fixed prime number. The rules of the game are: (1) A player cannot choose an element that has been chosen by either player on any previous turn. (2) A player can only choose an element that commutes with all previously chosen elements. (3) A player who cannot choose an element on his/her turn loses the game. Patniss takes the first turn. Which player has a winning strategy?

Given a triangle $ABC$, a point $P$ is chosen on side $BC$. Points $M$ and $N$ lie on sides $AB$ and $AC$, respectively, such that $MP \parallel AC$ and $NP \parallel AB$. Point $P$ is reflected across $MN$ to point $Q$. Show that triangle $QMB$ is similar to triangle $CNQ$. [i]Brian Hamrick.[/i]

Several children were playing in the ugly tree when suddenly they all fell.  $\bullet$ Roger hit branches $A$, $B$, and $C$ in that order on the way down.  $\bullet$ Sue hit branches $D$, $E$, and $F$ in that order on the way down.  $\bullet$ Gillian hit branches $G$, $A$, and $C$ in that order on the way down.  $\bullet$ Marcellus hit branches $B$, $D$, and $H$ in that order on the way down.  $\bullet$ Juan-Phillipe hit branches $I$, $C$, and $E$ in that order on the way down. Poor Mikey hit every branch A through $I$ on the way down. Given only this information, in how many different orders could he have hit these 9 branches on the way down?

For $p_0,\dots,p_d\in\mathbb{R}^d$, let \[ S(p_0,\dots,p_d)=\left\{ \alpha_0p_0+\dots+\alpha_dp_d : \alpha_i\le 1, \sum_{i=0}^d \alpha_i =1 \right\}. \] Let $\pi$ be an arbitrary probability distribution on $\mathbb{R}^d$, and choose $p_0,\dots,p_d$ independently with distribution $\pi$. Prove that the expectation of $\pi(S(p_0,\dots,p_d))$ is at least $1/(d+2)$.

Let $A = \{a_1, a_2, a_3, a_4, a_5\}$ be a set of $5$ positive integers. Show that for any rearrangement of $A$, $a_{i1}$, $a_{i2}$, $a_{i3}$, $a_{i4}$, $a_{i5}$, the product $$(a_{i1} -a_1) (a_{i2} -a_2) (a_{i3} -a_3) (a_{i4} -a_4) (a_{i5} -a_5)$$ is always even.

$4$ equilateral triangles of side length $1$ are drawn on the interior of a unit square, each one of which shares a side with one of the $4$ sides of the unit square. What is the common area enclosed by all $4$ equilateral triangles?

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]