Find all pairs $(m, n)$ of positive integers such that the $m \times n$ grid contains exactly $225$ rectangles whose side lengths are odd and whose edges lie on the lines of the grid.

Show that a polynomial of odd degree $2m+1$ over $\mathbb{Z}$, \[f(x)=c_{2m+1}x^{2m+1}+\cdots+c_{1}x+c_{0},\] is irreducible if there exists a prime $p$ such that \[p \not\vert c_{2m+1}, p \vert c_{m+1}, c_{m+2}, \cdots, c_{2m}, p^{2}\vert c_{0}, c_{1}, \cdots, c_{m}, \; \text{and}\; p^{3}\not\vert c_{0}.\]

A square is cut into 25 smaller squares, exactly 24 of which are unit squares. Find the area of the original square. (V Proizvolov)

Let $n$ be an integer greater than or equal to $3$, and let $P_n$ be the collection of all planar (simple) $n$-gons no two distinct sides of which are parallel or lie along some line. For each member $P$ of $P_n$, let $f_n(P)$ be the least cardinal a cover of $P$ by triangles formed by lines of support of sides of $P$ may have. Determine the largest value $f_n(P)$ may achieve, as $P$ runs through $P_n$.

Let $A$ be a set consisting of 6 points in the plane. denoted $n(A)$ as the number of the unit circles which meet at least three points of $A$. Find the maximum of $n(A)$

Given any six points in the plane, no three collinear, show that we can always find three which form an obtuse-angled triangle with one angle at least $120^o$.

Let $k$ be a fixed positive integer. For each $n = 1, 2,...,$ we will call [i]configuration [/i] of order $n$ any set of $kn$ points of the plane, which does not contain $3$ collinear, colored with $k$ given colors, so that there are $n$ points of each color. Determine all positive integers $n$ with the following property: in each configuration of order $n$, it is possible to select three points of each color, such that the $k$ triangles with vertices of the same color that are determined are disjoint in pairs.

In the interior of the convex 2011-gon are $2011$ points, such that no three among the given $4022$ points (the interior points and the vertices) are collinear. The points are coloured one of two different colours and a colouring is called "good" if some of the points can be joined in such a way that the following conditions are satisfied: 1) Each segment joins two points of the same colour. 2) None of the line segments intersect. 3) For any two points of the same colour there exists a path of segments connecting them. Find the number of "good" colourings.

In space $4$ points are given. How many planes equidistant from these points are there? Consider separately (a) the generic case (the points given do not lie on a single plane) and (b) the degenerate cases.

Let $A{}$ be a point in the Cartesian plane. At each step, Ann tells Bob a number $0\leqslant a\leqslant 1$ and he then moves $A{}$ in one of the four cardinal directions, at his choice, by a distance of $a{}.$ This process cotinues as long as Ann wishes. Amongst every 100 consecutive moves, each of the four possible moves should have been made at least once. Ann's goal is to force Bob to eventually choose a point at a distance greater than 100 from the initial position of $A.{}$ Can Ann achieve her goal? [i]Selected from an Argentine Olympiad[/i]

There are pieces in the shape of an equilateral triangle with sides $1, 2, 3, 4, 5$ and $6$ ($50$ pieces of each size). You want to build an equilateral triangle of side $7$ using some of these pieces, without gaps or overlaps. What is the least number of pieces needed?

All internal angles of a convex octagon $ABCDEFGH$ are equal to each other and the edges are alternatively equal: $$AB = CD = EF = GH,BC = DE = FG = HA$$ (we call such an octagon semiregular). The diagonals $AD$, $BE$, $CF$, $DG$, $EH$, $FA$, $GB$ and $HC$ divide the inside of the octagon into certain parts. Consider the part containing the centre of the octagon. If that part is an octagon, then this central octagon is semiregular (this is obvious). In this case we construct similar diagonals in the central octagon and so on. If, after several steps, the central figure is not an octagon, then the process stops. Prove that if the process never stops, then the initial octagon was regular. (A. Tolpygo, Kiev)

A lake is in the shape of a regular hexagon of side length \(1\). Initially there is a single lotus leaf somewhere in the lake, sufficiently far from the shore. Each day, from every existing leaf a new leaf may grow at distance \(\sqrt{3}\) (measured between centers), provided it does not overlap any other leaf. If the lake is large enough that edge effects never interfere, what is the least number of days required to have \(2025\) leaves in the lake?

Find the largest possible number of intersection points of the diagonals of a convex $n$-gon.

(a) A point $A$ is marked inside a circle. Two perpendicular lines drawn through $A$ intersect the circle at four points. Prove that the centre of mass of these four points does not depend on the choice of the lines. (b) A regular $2n$-gon ($n \ge 2$) with centre $A$ is drawn inside a circle (A does not necessarily coincide with the centre of the circle). The rays going from $A$ to the vertices of the $2n$-gon mark $2n$ points on the circle. Then the $2n$-gon is rotated about $A$. The rays going from $A$ to the new locations of vertices mark new $2n$ points on the circle. Let $O$ and $N$ be the centres of gravity of old and new points respectively. Prove that $O = N$.

$ n > 2 $ points are chosen on a circle and each of them is connected to every other by a segment. Is it possible to draw all of these segments in one sequence, i.e. so that the end of the first segment is the beginning of the second, the end of the second - the beginning of the third, etc., and so that the end of the last segment is the beginning of the first?

Prove that you can't split a square into finitely many hexagons, whose inner angles are all less than $180^o$.

In the plane we are given two circles intersecting at $ X$ and $ Y$. Prove that there exist four points with the following property: (P) For every circle touching the two given circles at $ A$ and $ B$, and meeting the line $ XY$ at $ C$ and $ D$, each of the lines $ AC$, $ AD$, $ BC$, $ BD$ passes through one of these points.

Towns $A_1, A_2, . . . , A_{30}$ lie on line $MN$. The distances between the consecutive towns are equal. Each of the towns is the point of origin of a straight highway. The highways are on the same side of $MN$ and form the following angles with it: [img]https://cdn.artofproblemsolving.com/attachments/a/f/6cfcac497bdd729b966705f1060bd4b1caba25.png[/img] Thirty cars start simultaneously from these towns along the highway at the same constant speed. Each intersection has a gate. As soon as the first (in time, not in number) car passes the intersection the gate closes and blocks the way for all other cars approaching this intersection. Which cars will pass all intersections and which will be stopped? Note: This refers to angles measured counterclockwise from straight MN to the corresponding road.

There are $n$ pieces on the squares of a $5 \times 9$ board, at most one on each square at any time during the game. A move in the game consists of simultaneously moving each piece to a neighboring square by side, under the restriction that a piece having been moved horizontally in the previous move must be moved vertically and vice versa. Find the greatest value of $n$ for which there exists an initial position starting at which the game can be continued until the end of the world.

Julián chooses $2007$ points of the plane between which there are no $3$ aligned, and draw with red all the segments that join two of those points. Next, Roberto draws several lines. Its objective is for each red segment to be cut inside by (at least) one of the lines. Determine the minor $\ell$ lines such that, no matter how Julián chooses the $2007$ points, with the properly chosen $\ell$ lines, Roberto will achieve his objective with certainty.

Let $M$ be the set of five points in space, none of which four do not lie in a plane. Let $R$ be a set of seven planes with properties: a) Each plane from the set $R$ contains at least one point of the set$ M$. b) None of the points of the set M lie in the five planes of the set $R$. Prove that there are also two distinct points $P$, $Q$, $ P \in M$, $Q \in M$, that the line $PQ$ is not the intersection of any two planes from the set $R$.

There are $10000$ equal tiles in the shape of an equilateral triangle. With these little triangles, regular hexagons are formed, without overlaps or gaps. If the regular hexagon that wastes the fewest triangles is formed, how many triangles are left over?

Let $A$ and $B$ be two points inside a given circle $k$. Prove that there exist (infinitely many) circles through $A$ and $B$ which lie entirely in $k$.

You are given some equilateral triangles and squares, all with side length 1, and asked to form convex $n$ sided polygons using these pieces. If both types must be used, what are the possible values of $n$, assuming that there is sufficient supply of the pieces?