Let $d$ be a real number such that $d^2=r^2+s^2$, where $r$ and $s$ are rational numbers. Prove that we can color all points of the plane with rational coordinates with two different colors such that the points with distance $d$ have different colors.

Let $ ABCD$ be a convex quadrilateral such that $ AB$ and $ CD$ are not parallel and $ AB\equal{}CD$. The midpoints of the diagonals $ AC$ and $ BD$ are $ E$ and $ F$, respectively. The line $ EF$ meets segments $ AB$ and $ CD$ at $ G$ and $ H$, respectively. Show that $ \angle AGH \equal{} \angle DHG$.

All naturals from $1$ to $2002$ are placed in a row. Can the signs: $+$ and $-$ be placed between each consecutive pair of them so that the corresponding algebraic sum is $0$?

Prove that in every $16$-digit number there is a chain of one or more consecutive digits such that the product of those digits is a perfect square. For example, if the original number is $7862328578632785$ we can take the digits $6$, $2$ and $3$ whose product is $6^2$ (note that these appear consecutively in the number).

Let $ p,q $ be the two most distant points (in the Euclidean sense) of a closed surface $ M $ embedded in the Euclidean space. [b]a)[/b] Show that the tangent planes of $ M $ at $ p $ and $ q $ are parallel. [b]b)[/b] What happened if $ M $ would be a closed curve of $ \mathcal{C}^{\infty } \left(\mathbb{R}^3\right) $ class, instead?

Let $d(n)$ denote the number of positive integers that divide $n$, including $1$ and $n$. For example, $d(1)=1,d(2)=2,$ and $d(12)=6$. (This function is known as the [i]divisor function[/i].) Let \[f(n)=\frac{d(n)}{\sqrt[3]{n}}.\] There is a unique positive integer $N$ such that $f(N)>f(n)$ for all positive integers $n\ne N$. What is the sum of the digits of $N?$ $\textbf{(A) }5 \qquad \textbf{(B) }6 \qquad \textbf{(C) }7 \qquad \textbf{(D) }8\qquad \textbf{(E) }9$

Given finitely many points in a plane, it is known that the area of the triangle formed by any three points of the set is less than 1. Show that all points of the set lie inside or on boundary of a triangle with area less than 4.

A $5 \times 5$ Latin Square is a $5 \times 5$ grid of squares in which each square contains one of the numbers $1$ through $5$ such that every number appears exactly once in each row and column. A partially completed grid (with numbers in some of the squares) is puzzle-ready if there is a unique way to fill in the remaining squares to complete a Latin Square. Below is a partially completed grid with seven squares filled in and an additional three squares shaded. Determine what numbers must be filled into the shaded squares to make the grid (now with ten squares filled in) puzzle-ready, and then complete the Latin Square. There is a unique solution, but you do not need to prove that your answer is the only one possible. You merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] unitsize(1.5cm); defaultpen(font("OT1","cmss","m","n")); defaultpen(fontsize(48pt)); for (int i=0; i<6; ++i) { draw((i,0)--(i,5)); draw((0,i)--(5,i)); } label(scale(2)*"1",(0.5,4.5)); label(scale(2)*"1",(1.5,3.5)); label(scale(2)*"3",(2.5,3.5)); label(scale(2)*"2",(0.5,2.5)); label(scale(2)*"3",(1.5,2.5)); label(scale(2)*"5",(4.5,2.5)); label(scale(2)*"5",(3.5,1.5)); path p = (0,0)--(1,0)--(1,1)--(0,1)--cycle; filldraw(shift(0,1)*p,gray,black); filldraw(shift(4,1)*p,gray,black); filldraw(shift(2,2)*p,gray,black); [/asy]

Let $ n$ and $ k$ be given natural numbers, and let $ A$ be a set such that \[ |A| \leq \frac{n(n+1)}{k+1}.\] For $ i=1,2,...,n+1$, let $ A_i$ be sets of size $ n$ such that \[ |A_i \cap A_j| \leq k \;(i \not=j)\ ,\] \[ A= \bigcup_{i=1}^{n+1} A_i.\] Determine the cardinality of $ A$. [i]K. Corradi[/i]

A list consists of all positive integers from $1$ to $2020$, inclusive, with each integer appearing exactly once. Define a move as the process of choosing four numbers from the current list and replacing them with the numbers $1,2,3,4$. If the expected number of moves before the list contains exactly two $4$'s can be expressed as $\frac{a}{b}$ for relatively prime positive integers, evaluate $a+b$. [i]Proposed by Richard Chen and Taiki Aiba[/i]

Let $a$ and $b$ be given positive real numbers, with $a<b.$ If two points are selected at random from a straight line segment of length $b,$ what is the probability that the distance between them is at least $a?$

Let $f$ be a quadratic function which satisfies the following condition. Find the value of $\frac{f(8)-f(2)}{f(2)-f(1)}$. For two distinct real numbers $a,b$, if $f(a)=f(b)$, then $f(a^2-6b-1)=f(b^2+8)$.

Suppose $\theta_{i}\in(-\frac{\pi}{2},\frac{\pi}{2}), i = 1,2,3,4$. Prove that, there exist $x\in \mathbb{R}$, satisfying two inequalities \begin{eqnarray*} \cos^2\theta_1\cos^2\theta_2-(\sin\theta\sin\theta_2-x)^2 &\geq& 0, \\ \cos^2\theta_3\cos^2\theta_4-(\sin\theta_3\sin\theta_4-x)^2 & \geq & 0 \end{eqnarray*} if and only if \[ \sum^4_{i=1}\sin^2\theta_i\leq2(1+\prod^4_{i=1}\sin\theta_i + \prod^4_{i=1}\cos\theta_i). \]

Consider the function $f : N \to N$ given by (i) $f(1) = 1$, (ii) $f(2n) = f(n)$ for any $n \in N$, (iii) $f(2n+1) = f(2n)+1$ for any $n \in N$. (a) Find the maximum value of $f(n)$ for $1 \le n \le 1995$; (b) Find all values of $f$ on this interval.

Let $n\ge 2$ be positive integer. Find the least possible number of elements of tile set $A =\{1,2,...,2n-1,2n\}$ that should be deleted in order to the sum of any two different elements remained be a composite number.

Given an isosceles triangle $ABC$ with a right angle at $C,$ construct the center $M$ and radius $r$ of a circle cutting on segments $AB, BC, CA$ the segments $DE, FG,$ and $HK,$ respectively, such that $\angle DME + \angle FMG + \angle HMK = 180^\circ$ and $DE : FG : HK = AB : BC : CA.$

The sides of triangles $ABC$ and $ACD$ satisfy the following conditions: $AB = AD = 3$ cm, $BC = 7$ cm, $DC = 11$ cm. What values can the side length $AC$ take if it is an integer number of centimeters, is the average in $\Delta ACD$ and the largest in $\Delta ABC$?

Let $A_1 B_1 C_1$ and $A_2 B_2 C_2$ be two oppositely oriented concentric equilateral triangles. Prove that the lines $A_1 A_2$ , $B_1 B_2$ , and $C_1 C_2$ intersect in one point.

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

A car and a motorcyclist left point $A$ in the direction of point $B$ at $10$ o'clock, and half an hour later a cyclist left point $B$ (in the direction of point A) and a pedestrian left (in the direction of point $A$) The car met the cyclist at $11$ o'clock hour and half an hour later overtook the pedestrian, and the motorcyclist overtook the pedestrian at $12:30$ p.m. At what time did the motorcyclist and the cyclist meet? (Speeds and directions of movement of ALL participants)

Triangle $ABC$ has circumcenter $O$. $D$ is the foot from $A$ to $BC$, and $P$ is apoint on $AD$. The feet from $P$ to $CA, AB$ are $E, F$, respectively, and the foot from $D$ to $EF$ is $T$. $AO$ meets $(ABC)$ again at $A'$. $A'D$ meets $(ABC)$ again at $R$. If $Q$ is a point on $AO$ satisfying $\angle ABP = \angle QBC$, prove that $D, P, T, R$ lie on acircle and $DQ$ is tangent to it.

Let $m,n \in Z, m\ne n, m \ne 0, n \ne 0$ . Find all $f: Z \to R$ such that $f(mx+ny)=mf(x)+nf(y)$ for all $x,y \in Z$ .

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

With three different digits, six-digit numbers are written, multiples of $3$. One of the the digits are in the unit's place, another in the hundred's place, and the third in the remaining places. If we take out two units from the hundred's digit and add these to the unit's digit, the number is left with all the same digits. Find the numbers.

find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $f(-f(x)-f(y)) = 1-x-y$ $\quad \forall x,y \in \mathbb{Z}$