Is it possible to divide $\mathbb{N}$ into six disjoint sets $A_1, A_2, A_3, A_4, A_5, A_6$, such that $x,y,z$ are not in the same set if $x+2y=5z$?

Let $P(x)$ and $Q(x)$ be integer polynomials of degree $p$ and $q$ respectively. Assume that $P(x)$ divides $Q(x)$ and all their coefficients are either $1$ or $2002$. Show that $p+1$ is a divisor of $q+1$. [i]Mihai Cipu[/i]

Prove that for any finite bipartite planar graph, a circle can be assigned to each vertex so that all circles are coplanar, the circles assigned to any two adjacent vertices are tangent to one another, while the circles assigned to any two distinct, non-adjacent vertices intersect in two points.

Let $ABCD$ be a cyclic quadrilateral. Points $E$ and $F$ lie on the sides $AD$ and $CD$ in such a way that $AE = BC$ and $AB = CF$. Let $M$ be the midpoint of $EF$. Prove that $\angle AMC = 90^{\circ}$.

Let $S$ be a finite set of positive integers. Assume that there are precisely 2023 ordered pairs $(x,y)$ in $S\times S$ so that the product $xy$ is a perfect square. Prove that one can find at least four distinct elements in $S$ so that none of their pairwise products is a perfect square. [i]Note:[/i] As an example, if $S=\{1,2,4\}$, there are exactly five such ordered pairs: $(1,1)$, $(1,4)$, $(2,2)$, $(4,1)$, and $(4,4)$. [i]Proposed by Sutanay Bhattacharya[/i]

Determine the largest positive integer $n$ for which there exists a set $S$ with exactly $n$ numbers such that [list][*] each member in $S$ is a positive integer not exceeding $2002$, [*] if $a,b\in S$ (not necessarily different), then $ab\not\in S$. [/list]

Given $19$ red boxes and $200$ blue boxes filled with balls. None of which is empty. Suppose that every red boxes have a maximum of $200$ balls and every blue boxes have a maximum of $19$ balls. Suppose that the sum of all balls in the red boxes is less than the sum of all the balls in the blue boxes. Prove that there exists a subset of the red boxes and a subset of the blue boxes such that their sum is the same.

Let $\omega_1$ and $\omega_2$ be intersecting circles in the plane with radii $12$ and $15$, respectively. Suppose $\Gamma$ is a circle such that $\omega_1$ and $\omega_2$ are internally tangent to $\Gamma$ at $X_1$ and $X_2$, respectively. Similarly, $\ell$ is a line that is tangent to $\omega_1$ and $\omega_2$ at $Y_1$ and $Y_2$, respectively. If $X_1X_2=18$ and $Y_1Y_2=9$, what is the radius of $\Gamma$?

A plane contains $40$ lines, no $2$ of which are parallel. Suppose there are $3$ points where exactly $3$ lines intersect, $4$ points where exactly $4$ lines intersect, $5$ points where exactly $5$ lines intersect, $6$ points where exactly $6$ lines intersect, and no points where more than $6$ lines intersect. Find the number of points where exactly $2$ lines intersect.

Prove that in every polygon there is a diagonal that cuts off a triangle and lies within the polygon.

We are given three non-negative numbers $A , B$ and $C$ about which it is known that $$A^4 + B^4 + C^4 \le 2(A^2B^2 + B^2C^2 + C^2A^2)$$ (a) Prove that each of $A, B$ and $C$ is not greater than the sum of the others. (b) Prove that $A^2 + B^2 + C^2 \le 2(AB + BC + CA)$ . (c) Does the original inequality follow from the one in (b)? (V.A. Senderov , Moscow)

Find the least positive integer $m$ such that $105| 9^{(p^2)} -29^p +m$ for all prime numbers $p > 3$.

Let $ABC$ be a triangle with $\angle BAC > 90^o$ and with $AB < AC$. Let $r$ be the internal bisector of $\angle ACB$ and let $s$ be the perpendicular, through $A$, on $r$. Denote by $F$ the intersection of $r$ and $ s$, and denote by $E$ the intersection of $s$ with the segment $BC$. Let also $D$ be the symmetric of $A$ with respect to the line $BF$. Assuming that the circumcircle of triangle $EAC$ is tangent to line $AB$ and $ D$ lies on $r$, determine the value of $\angle CDB$.

The altitudes $AN$ and $BM$ are drawn in triangle $ABC$. Prove that the perpendicular bisector to the segment $NM$ divides the segment $AB$ in half.

Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that the following two conditions hold: [b]1. [/b] For all real numbers $a$ and $b$ satisfying $a^2 + b^2 = 1$, We have $f(x) + f(y) \geq f(ax + by)$ for all real numbers $x, y$. [b]2.[/b] For all real numbers $x$ and $y$, there exist real numbers $a$ and $b$, such that $a^2 + b^2 = 1$ and $f(x) + f(y) = f(ax + by)$. [i]Proposed by Zaza Melikidze, Georgia[/i]

A point $P$ and a segment $AB$ with length $20$ are randomly drawn on a plane. Suppose that the probability that a randomly selected line passing through $P$ intersects segment $AB$ is $\tfrac{1}{2}$. Next, randomly choose point $Q$ on segment $AB$. What is the probability with respect to choosing $Q$ that a circle centered at $Q$ passing through $P$ contains both $A$ and $B$ in its interior?

Find all perfect powers whose last $ 4$ digits are $ 2,0,0,8$, in that order.

Find the number of ordered triplets $(a,b,c)$ of positive integers such that $a<b<c$ and $abc=2008$.

Given a rectangular grid with some cells containing one letter, we say a row or column is [i]edible [/i] if it has more than one cell with a letter and all such cells contain the same letter. Given such a grid, the hungry, hungry letter monster repeats the following procedure: he nds all edible rows and all edible columns and simultaneously eats all the letters in those rows and columns, removing those letters from the grid and leaving those cells empty. He continues this until no more edible rows and columns remain. Call a grid a [i]meal [/i] if the letter monster can eat all of its letters using this procedure. In the $7$ by $7$ grid to the right, ll each empty space with one letter so that the grid is a meal and there are a total of eight Us, nine Ss, ten As, eleven Ms, and eleven Ts. Some letters have been given to you. You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justication acceptable.) [img]https://cdn.artofproblemsolving.com/attachments/9/a/d1886720796e4befd9d3ce0cbd2868d1b649d1.png[/img]

Let $a_1$, $a_2$, $\cdots$ be a sequence such that $a_1=a_2=\frac 15$, and for $n \ge 3$, $$a_n=\frac{a_{n-1}+a_{n-2}}{1+a_{n-1}a_{n-2}}.$$ Find the smallest integer $n$ such that $a_n>1-5^{-2022}$.

Consider the shape shown below, formed by gluing together the sides of seven congruent regular hexagons. The area of this shape is partitioned into $21$ quadrilaterals, all of whose side lengths are equal to the side length of the hexagon and each of which contains a $60^{\circ}$ angle. In how many ways can this partitioning be done? (The quadrilaterals may contain an internal boundary of the seven hexagons.) [asy] draw(origin--origin+dir(0)--origin+dir(0)+dir(60)--origin+dir(0)+dir(60)+dir(0)--origin+dir(0)+dir(60)+dir(0)+dir(60)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)+dir(240)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)+dir(240)+dir(300)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)+dir(240)+dir(300)+dir(240)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)+dir(240)+dir(300)+dir(240)+dir(300)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)+dir(240)+dir(300)+dir(240)+dir(300)+dir(0)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)+dir(240)+dir(300)+dir(240)+dir(300)+dir(0)+dir(300)--cycle); draw(2*dir(60)+dir(120)+dir(0)--2*dir(60)+dir(120)+2*dir(0),dashed); draw(2*dir(60)+dir(120)+dir(60)--2*dir(60)+dir(120)+2*dir(60),dashed); draw(2*dir(60)+dir(120)+dir(120)--2*dir(60)+dir(120)+2*dir(120),dashed); draw(2*dir(60)+dir(120)+dir(180)--2*dir(60)+dir(120)+2*dir(180),dashed); draw(2*dir(60)+dir(120)+dir(240)--2*dir(60)+dir(120)+2*dir(240),dashed); draw(2*dir(60)+dir(120)+dir(300)--2*dir(60)+dir(120)+2*dir(300),dashed); draw(dir(60)+dir(120)--dir(60)+dir(120)+dir(0)--dir(60)+dir(120)+dir(0)+dir(60)--dir(60)+dir(120)+dir(0)+dir(60)+dir(120)--dir(60)+dir(120)+dir(0)+dir(60)+dir(120)+dir(180)--dir(60)+dir(120)+dir(0)+dir(60)+dir(120)+dir(180)+dir(240)--dir(60)+dir(120)+dir(0)+dir(60)+dir(120)+dir(180)+dir(240)+dir(300),dashed); [/asy]

Let $a,b,c>0$ and $abc\ge1.$ Prove that $a^4+b^3+c^2\ge a^3+b^2+c.$

a) Fold a $10 \times 10$ square from a $1 \times 118$ rectangular strip. b) Fold a $10 \times 10$ square from a $1 \times (100+9\sqrt3)$ rectangular strip (approximately $1\times 115.58$). The strip can be bent, but not torn.

$a_1,a_2,...,a_{81}$ are nonzero, $a_i+a_{i+1}>0$ for $i=1,...,80$ and $a_1+a_2+...+a_{81}<0$. What is sign of $a_1*a_2*...*a_{81}$?

Define a function $f$ as follows. For any positive integer $i$, let $f(i)$ be the smallest positive integer $j$ such that there exist pairwise distinct positive integers $a,b,c,$ and $d$ such that $\gcd(a,b)$, $\gcd(a,c)$, $\gcd(a,d)$, $\gcd(b,c)$, $\gcd(b,d)$, and $\gcd(c,d)$ are pairwise distinct and equal to $i, i+1, i+2, i+3, i+4,$ and $j$ in some order, if any such $j$ exists; let $f(i)=0$ if no such $j$ exists. Compute $f(1)+f(2)+\dots +f(2019)$. [i]Proposed by Edward Wan[/i]