A function from the positive integers to the positive integers satisfies these properties 1. $f(ab)=f(a)f(b)$ for any two coprime positive integers $a,b$. 2. $f(p+q)=f(p)+f(q)$ for any two primes $p,q$. Prove that $f(2)=2, f(3)=3, f(1999)=1999$.

Let there be a unit square initially tiled with four congruent shaded equilateral triangles, as seen below. The total area of all of the shaded regions can be expressed in the form $\frac{a-b\sqrt{c}}{d}$ , where $a, b, c$, and $d$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c + d$. [img]https://cdn.artofproblemsolving.com/attachments/b/b/34883cf73da568ca237a13fbc2e0fb9322c2e5.png[/img]

Let $ZHAO$ be a square with area $2024$. Let $X$ be the center of this square and let $C$, $D$, $E$, $K$ be the centroids of $XZH$, $XHA$, $XAO$, and $XOZ$, respectively. Find $[ZHAO]$ $+$ $[CZHAO]$ $+$ $[DZHAO]$ $+$ $[EZHAO]$ $+$ $[KZHAO]$. (Here $[\mathcal P]$ denotes the area of the polygon $\mathcal P$.)

Circle $C$ center $O$ touches externally circle $C'$ center $O'$. A line touches $C$ at $A$ and $C'$ at $B$. $P$ is the midpoint of $AB$. Show that $\angle OPO' = 90^o$.

A school has three senior classes of $M$ students each. Every student knows at least $\frac{3}{4}M$ people in each of the other two classes. Prove that the school can send $M$ non-intersecting teams to the olympiad so that each team consists of $3$ students from different classes who know each other. [i]Proposed by C. Magyar, R. Martin[/i]

Solve for real $x,y$: $x+y=2$ $x^5+y^5=82$

Point $ M$ is taken on side $ BC$ of a triangle $ ABC$ such that the centroid $ T_c$ of triangle $ ABM$ lies on the circumcircle of $ \triangle ACM$ and the centroid $ T_b$ of $ \triangle ACM$ lies on the circumcircle of $ \triangle ABM$. Prove that the medians of the triangles $ ABM$ and $ ACM$ from $ M$ are of the same length.

A right hexagon with side length $n$ is divided into tiles of three types, which are shown in the image, which are rhombuses with side length $1$ each and the acute angle $60$. In one move you can choose three tiles, arranged as shown in the image on the left, and rearrange them, as shown in the image on the right [img]https://iili.io/dxEvyqN.jpg[/img] Moves are made until it is impossible to make a move. a) Prove that for the fixed initial arrangement of tiles the same amount of moves would be made independent of the moves. b) For each positive integer $n$ find the maximum number of moves among all possible initial arrangements [i]M. Zorka[/i]

With $21$ pieces, some white and some black, a rectangle is formed of $3 \times 7$. Prove that there are always four pieces of the same color located at the vertices of a rectangle.

Let $p,q, r, s$ be real numbers with $q \ne -1$ and $s \ne -1$. Prove that the quadratic equations $x^2 + px+q = 0$ and $x^2 +rx+s = 0$ have a common root, while their other roots are inverse of each other, if and only if $pr = (q+1)(s+1)$ and $p(q+1)s = r(s+1)q$. (A double root is counted twice.)

Let $c_i$ denote the $i$th composite integer so that $\{c_i\}=4,6,8,9,...$ Compute \[\prod_{i=1}^{\infty} \dfrac{c^{2}_{i}}{c_{i}^{2}-1}\] (Hint: $\textstyle\sum^\infty_{n=1} \tfrac{1}{n^2}=\tfrac{\pi^2}{6}$)

Let $ n$ and $ k$ be positive integers such that $ \frac{1}{2} n < k \leq \frac{2}{3} n.$ Find the least number $ m$ for which it is possible to place $ m$ pawns on $ m$ squares of an $ n \times n$ chessboard so that no column or row contains a block of $ k$ adjacent unoccupied squares.

It is known that, for all positive integers $k,$ \[1^{2}+2^{2}+3^{2}+\cdots+k^{2}=\frac{k(k+1)(2k+1)}{6}. \]Find the smallest positive integer $k$ such that $1^{2}+2^{2}+3^{2}+\cdots+k^{2}$ is a multiple of $200.$

A positive integer is called [i]cool[/i] if it can be expressed in the form $a!\cdot b!+315$ where $a,b$ are positive integers. For example, $1!\cdot 1!+315=316$ is a cool number. Find the sum of all cool numbers that are also prime numbers. [i]Proposed by Evan Fang

In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $O$. Denote by $P, Q, R, S$ the orthogonal projections of $O$ onto $AB$ , $BC$ ,$CD$ , $DA$, respectively. Prove that $$PA \cdot AB + RC \cdot CD =\frac12 (AD^2 + BC^2)$$ if and only if $$QB \cdot BC + SD \cdot DA = \frac12(AB ^2 + CD^2)$$

Ten million fireflies are glowing in $\mathbb{R}^3$ at midnight. Some of the fireflies are friends, and friendship is always mutual. Every second, one firefly moves to a new position so that its distance from each one of its friends is the same as it was before moving. This is the only way that the fireflies ever change their positions. No two fireflies may ever occupy the same point. Initially, no two fireflies, friends or not, are more than a meter away. Following some finite number of seconds, all fireflies find themselves at least ten million meters away from their original positions. Given this information, find the greatest possible number of friendships between the fireflies. [i]Nikolai Beluhov[/i]

The ray of light starts from the center of a square and reflects from its sides with the principle that the angle of reflection is equal to the angle of incidence. After some time the ray returns to the center of the square. The ray never reached the vertex and has never returned to the center of the square before. Prove that the ray reflected from the sides of the square an odd number of times.

A circle of radius $5$ is inscribed in an isosceles right triangle, $ABC$. The length of the hypotenuse of $ABC$ can be expressed as $a+a\sqrt{2}$ for some $a$. What is $a$?

Four consecutive natural numbers are divided into two groups of $2$ numbers. It is known that the product of numbers in one group is $1995$ greater than the product of numbers in another group. Find these numbers.

Let $\displaystyle ABCD$ be a a convex quadrilateral and $\displaystyle O$ be a point in its interior. Let $\displaystyle a,b,c,d,e,f$ be the areas of the triangles $\displaystyle OAB,OBC,OCD,ODA,OAC,OBD$. Prove that \[ \displaystyle \left| ac - bd \right| = ef . \]

Find all non-zero real numbers $x$ such that \[\min \left\{ 4, x+ \frac 4x \right\} \geq 8 \min \left\{ x,\frac 1x\right\} .\]

Consider the closed plane curves $C_i$ and $C_o,$ their respective lengths $|C_i|$ and $|C_o|,$ the closed surfaces $S_i$ and $S_o,$ and their respective areas $|S_i|$ and $|S_o|.$ Assume that $C_i$ lies inside $C_o$ and $S_i$ inside $S_o.$ (Subscript $i$ stands for "inner," $o$ for "outer.") Prove the correct assertions among the following four, and disprove the others. (i) If $C_i$ is convex, $|C_i| \le |C_o|.$ (ii) If $S_i$ is convex, $|S_i| \le |S_o|.$ (iii) If $C_o$ is the smallest convex curve containing $C_i,$ then $|C_o| \le |C_i|.$ (iv) If $S_o$ is the smallest convex surface containing $S_i,$ then $|S_o| \le |S_i|.$ You may assume that $C_i$ and $C_o$ are polygons and $S_i$ and $S_o$ polyhedra.

Compute the $100^{\text{th}}$ smallest positive integer $n$ that satisfies the three congruences \[\begin{aligned} \left\lfloor \dfrac{n}{8} \right\rfloor &\equiv 3 \pmod{4}, \\ \left\lfloor \dfrac{n}{32} \right\rfloor &\equiv 2 \pmod{4}, \\ \left\lfloor \dfrac{n}{256} \right\rfloor &\equiv 1 \pmod{4}. \end{aligned}\] Here $\lfloor \cdot \rfloor$ denotes the greatest integer function. [i]Proposed by Michael Tang[/i]

It is given that the roots of the polynomial $P(z) = z^{2019} - 1$ can be written in the form $z_k = x_k + iy_k$ for $1\leq k\leq 2019$. Let $Q$ denote the monic polynomial with roots equal to $2x_k + iy_k$ for $1\leq k\leq 2019$. Compute $Q(-2)$.

Let $ABC$ be a triangle with incenter $I$. Let $D$ be the point of intersection between the incircle and the side $BC$, the points $P$ and $Q$ are in the rays $IB$ and $IC$, respectively, such that $\angle IAP=\angle CAD$ and $\angle IAQ=\angle BAD$. Prove that $AP=AQ$.