Let $a$, $b$, $c$, and $d$ be positive real numbers such that \[a^2 + b^2 - c^2 - d^2 = 0 \quad \text{and} \quad a^2 - b^2 - c^2 + d^2 = \frac{56}{53}(bc + ad).\] Let $M$ be the maximum possible value of $\tfrac{ab+cd}{bc+ad}$. If $M$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $100m + n$. [i]Proposed by Robin Park[/i]

The map of a town shows a plane divided into equal equilateral triangles. The sides of these triangles are streets and their vertices are intersections; $6$ streets meet at each junction. Two cars start simultaneously in the same direction and at the same speed from points $A$ and $B$ situated on the same street (the same side of a triangle). After any intersection an admissible route for each car is either to proceed in its initial direction or turn through $120^o$ to the right or to the left. Can these cars meet? (Either prove that these cars won’t meet or describe a route by which they will meet.) [img]https://cdn.artofproblemsolving.com/attachments/2/d/2c934bcd0c7fc3d9dca9cee0b6f015076abbdb.png[/img]

A round table has room for n diners ( $n\ge 2$). There are napkins in three different colours. In how many ways can the napkins be placed, one for each seat, so that no two neighbours get napkins of the same colour?

Consider all quadratic functions $f(x) = ax^2 +bx+c$ with $a < b$ and $f(x) \ge 0$ for all $x$. What is the smallest possible value of the expression $\frac{a+b+c}{b-a}$?

Find all continuous functions $f$ such that $f(x)+ f(x^2) = 0$ for all real numbers $x$.

Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$. The lines $AB$ and $CD$ meet at $P$, the lines $AD$ and $BC$ meet at $Q$, and the diagonals $AC$ and $BD$ meet at $R$. Let $M$ be the midpoint of the segment $PQ$, and let $K$ be the common point of the segment $MR$ and the circle $\omega$. Prove that the circumcircle of the triangle $KPQ$ and $\omega$ are tangent to one another.

Triangle $ABC$ is inscribed in a circle. Chords $AM$ and $AN$ intersect side $BC$ at points $K$ and $L$ respectively. Prove that if a circle passes through all of the points $K, L, M$ and $N$, then $ABC$ is an isosceles triangle. (V Zhgun)

Given an odd integer $n \geq 3$. Let $V$ be the set of vertices of a regular $n$-gon, and $P$ be the set of all regular polygons formed by points in $V$. For instance, when $n=15$, $P$ consists of $1$ regular $15$-gon, $3$ regular pentagons, and $5$ regular triangles. Initially, all points in $V$ are uncolored. Two players, $A$ and $B$, play a game where they take turns coloring an uncolored point, with player $A$ starting and coloring points red, and player $B$ coloring points blue. The game ends when all points are colored. A regular polygon in $P$ is called $\textit{good}$ if it has more red points than blue points. Find the largest positive integer $k$ such that no matter how player $B$ plays, player $A$ can ensure that there are at least $k$ $\textit{good}$ polygons.

Let $\triangle{ABC}$ be an acute angled triangle. The circle with diameter AB intersects the sides AC and BC at points E and F respectively. The tangents drawn to the circle through E and F intersect at P. Show that P lies on the altitude through the vertex C.

Triangle $ABC$ is inscribed in circle $C.$ The bisectors of the angles $A,B$ and $C$ meet the circle $C$ again at the points $A', B', C'$. Let $I$ be the incenter of $ABC,$ prove that \[\frac{IA'}{IA} + \frac{IB'}{IB}+\frac{IC'}{IC} \geq 3\]\[, IA'+IB'+IC' \geq IA+IB+IC\]

A circle of radius $2$ is centered at $A$. An equilateral triangle with side $4$ has a vertex at $A$. What is the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle? $ \textbf {(A) } 8-\pi \qquad \textbf {(B) } \pi + 2 \qquad \textbf {(C) } 2\pi - \frac {\sqrt{2}}{2} \qquad \textbf {(D) } 4(\pi - \sqrt{3}) \qquad \textbf {(E) } 2\pi + \frac {\sqrt{3}}{2} $

For any positive reals $a,b,c,d$ that satisfy $a^2 + b^2 + c^2 + d^2 = 4,$ show that $$\frac{a^3}{a+b} + \frac{b^3}{b+c} + \frac{c^3}{c+d} + \frac{d^3}{d+a} + 4abcd \leq 6.$$

Let $(a_n), (b_n)$, $n = 1,2,...$ be two sequences of integers defined by $a_1 = 1, b_1 = 0$ and for $n \geq 1$ $a_{n+1} = 7a_n + 12b_n + 6$ $b_{n+1} = 4a_n + 7b_n + 3$ Prove that $a_n^2$ is the difference of two consecutive cubes.

Given a convex polygon with 20 vertexes, there are many ways of traingulation it (as 18 triangles). We call the diagram of triangulation, meaning the 20 vertexes, with 37 edges(17 triangluation edges and the original 20 edges), a T-diagram. And the subset of this T-diagram with 10 edges which covers all 20 vertexes(meaning any two edges in the subset doesn't cover the same vertex) calls a "perfect matching" of this T-diagram. Among all the T-diagrams, find the maximum number of "perfect matching" of a T-diagram.

Is there a power of $2$ such that it is possible to rearrange the digits giving another power of $2$?

Partition $\{1,2,3, ... ,2024\}$ into $506$ sets $\{a_i, b_i, c_i, d_i\}$ such that $a_i<b_i<c_i<d_i$. Find the maximum of \[\sum_{i=1}^{506} (a_i-b_i-c_i+d_i)\] over all partitions. [i]Tiebreaker #2[/i]

Suppose that $\omega$ is a primitive $2007^{\text{th}}$ root of unity. Find $\left(2^{2007}-1\right)\displaystyle\sum_{j=1}^{2006}\dfrac{1}{2-\omega^j}$.

We call a table of size $n \times n$ self-describing if each cell of the table contains the total number of even numbers in its row and column other than itself. How many self-describing tables of size a) $3 \times 3$ exist? b) $4 \times 4$ exist? c) $5 \times 5$ exist? Two tables are different if they differ in at least one cell.

Let $0 < x_i < \pi$ for $i=1,2,\ldots, n$ and set $$x= \frac{ x_1 +x_2 + \ldots+ x_n }{n}.$$ Prove that $$ \prod_{i=1}^{n} \frac{ \sin x_i }{x_i } \leq \left( \frac{ \sin x}{x}\right)^{n}.$$

Larry always orders pizza with exactly two of his three favorite toppings: pepperoni, bacon, and sausage. If he has ordered a total of $600$ pizzas and has had each topping equally often, how many pizzas has he ordered with pepperoni? $\text{(A) }200\qquad\text{(B) }300\qquad\text{(C) }400\qquad\text{(D) }500\qquad\text{(E) }600$

Let $ABCD$ be any convex quadrilateral and let $P, Q, R, S$ be points on the segments $AB, BC, CD$, and $DA$, respectively. It is given that the segments $PR$ and $QS$ dissect $ABCD$ into four quadrilaterals, each of which has perpendicular diagonals. Show that the points $P, Q, R, S$ are concyclic.

Suppose that $x$, $y$, and $z$ are real numbers such that $x + y + z = 3$ and $x^2 + y^2 + z^2 = 6$. What is the largest possible value of $z$?

What is the largest $n$ such that $n! + 1$ is a square?

Let $n$, $k$ be fixed integers. On a $n \times n$ board, label each square $0$ or $1$ such that in each $2k \times 2k$ sub-square of the board, the number of $0$'s and $1$'s written are the same. What is the largest possible sum of numbers written on the $n\times n$ board?

In triangle $ABC$ we have $\angle C = 90^o$ and $AC = BC$. Furthermore $M$ is an interior pont in the triangle so that $MC = 1 , MA = 2$ and $MB =\sqrt2$. Determine $AB$