Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that $$f(x + f(y)) = f(x) + f(y)$$ for every $x,y\in\mathbb R$ Find all rational numbers $q$ such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying $f(z)=qz$.

Determine all the possible positive integer $n,$ such that $3^n+n^2+2019$ is a perfect square.

Split a face of a regular tetrahedron into four congruent equilateral triangles. How many different ways can the seven triangles of the tetrahedron be colored using only the colors orange and black? (Two tetrahedra are considered to be colored the same way if you can rotate one so it looks like the other.)

You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

How many whole numbers between 1 and 1000 do [b]not[/b] contain the digit 1? $ \textbf{(A)}\ 512 \qquad \textbf{(B)}\ 648 \qquad \textbf{(C)}\ 720 \qquad \textbf{(D)}\ 728 \qquad \textbf{(E)}\ 800$

For any integer $n\geq 3$, let $A\left(n\right)$ denote the maximal number of self-intersections a closed broken line $P_1P_2...P_nP_1$ can have; hereby, we assume that no three vertices of the broken line $P_1P_2...P_nP_1$ are collinear. Prove that [b](a)[/b] if n is odd, then $A\left(n\right)=\frac{n\left(n-3\right)}{2}$; [b](b)[/b] if n is even, then $A\left(n\right)=\frac{n\left(n-4\right)}{2}+1$. [i]Note.[/i] A [i]self-intersection[/i] of a broken line is a (non-ordered) pair of two distinct non-adjacent segments of the broken line which have a common point.

Let $ ABC$ be a triangle with $ \angle BAC\equal{}60^{\circ}$. The incircle of $ ABC$ is tangent to $ AB$ at $ D$. Construct a circle with radius $ DA$ and cut the incircle of $ ABC$ at $ E$. If $ AF$ is an altitude, prove that $ AE\ge AF$.

Let $ A$ and $ B$ be the intersections of two circumferences $\Gamma_1$, and $\Gamma_2$. Let $C$ and $D$ points in $\Gamma_1$ and $\Gamma_2$ respectively such that $AC = AD$. Let $E$ and $F$ be points in $\Gamma_1$ and $\Gamma_2$, such that $\angle ABE = \angle ABF = 90^o$. Let $K_1$ and $K_2$ be circumferences with centers $E$ and $F$ and radii $EC$ and $FD$ respectively. Let $T$ be a point in the line $AB$, but outside the segment, with $T\ne A$ and $T \ne A'$, where $A'$ is the point symmetric to $A$ with respect to $ B$. Let $X$ be the point of tangency of a tangent to $K_1$ passing through $T$, such that there arc two points of intersection of the line $TX$ to $K_2$. Let $Y$ and $Z$ be such points. Prove that $$\frac{1}{XT}=\frac{1}{XY} + \frac{1}{XZ}.$$

When the decimal point of a certain positive decimal number is moved four places to the right, the new number is four times the reciprocal of the original number. What is the original number? $ \textbf{(A) }0.0002\qquad\textbf{(B) }0.002\qquad\textbf{(C) }0.02\qquad\textbf{(D) }0.2\qquad\textbf{(E) }2$

J has $53$ cheetahs in his hair, which he will put in $10$ cages. Let $A$ be the number of cheetahs in the cage with the largest number of cheetahs (there could be a tie, but in this case take the number of cheetahs in one of the cages involved in the tie). Find the least possible value of $A.$

Vijay picks two random distinct primes $1\le p, q\le 10^4$. Let $r$ be the probability that $3^{2205403200}\equiv 1\bmod pq$. Estimate $r$ in the form $0.abcdef$, where $a, b, c, d, e, f$ are decimal digits.

Suppose hops, skips and jumps are specific units of length. If $b$ hops equals $c$ skips, $d$ jumps equals $e$ hops, and $f$jumps equals $g$ meters, then one meter equals how many skips? $ \textbf{(A)}\ \frac{bdg}{cef}\qquad\textbf{(B)}\ \frac{cdf}{beg}\qquad\textbf{(C)}\ \frac{cdg}{bef}\qquad\textbf{(D)}\ \frac{cef}{bdg}\qquad\textbf{(E)}\ \frac{ceg}{bdf} $

King Radford of Peiza is hosting a banquet in his palace. The King has an enormous circular table with $2021$ chairs around it. At The King's birthday celebration, he is sitting in his throne (one of the $2021$ chairs) and the other $2020$ chairs are filled with guests, with the shortest guest sitting to the King's left and the remaining guests seated in increasing order of height from there around the table. The King announces that everybody else must get up from their chairs, run around the table, and sit back down in some chair. After doing this, The King notices that the person seated to his left is different from the person who was previously seated to his left. Each other person at the table also notices that the person sitting to their left is different. Find a closed form expression for the number of ways the people could be sitting around the table at the end. You may use the notation $D_{n},$ the number of derangements of a set of size $n$, as part of your expression.

Of the vertices of a cube, $7$ of them have assigned the value $0$, and the eighth the value $1$. A [i]move[/i] is selecting an edge and increasing the numbers at its ends by an integer value $k > 0$. Prove that after any finite number of [i]moves[/i], the g.c.d. of the $8$ numbers at vertices is equal to $1$. Russian M.O.

Let $a$ and $b$ be real numbers such that$$\frac{a}{a^2-5} = \frac{b}{5-b^2} = \frac{ab}{a^2b^2-5}$$where $a+b \neq 0$. $a^4 + b^4 =$ ?

On the $ AB $ side of the triangle $ ABC $, points $ X $ and $ Y $ are chosen, on the side of $ AC $ is a point of $ Z $, and on the side of $ BC $ is a point of $ T $. Wherein $ XZ \parallel BC $, $ YT \parallel AC $. Line $ TZ $ intersects the circumscribed circle of triangle $ ABC $ at points $ D $ and $ E $. Prove that points $ X $, $ Y $, $ D $ and $ E $ lie on the same circle.

Consider a very simple model for an open self-replicative system such as a cell, or an economy. A system $\mathbf{S}$ is comprised of two kinds of mass: one kind is $\mathbf{S}_R$ that is capable of of taking raw material that comes from outside the system, and converting it into components of the system, and the other is $\mathbf{S}_O$ that takes care of other things. Think of $\mathbf{S}_R$ like factories and $\mathbf{S}_O$ like street sweepers. One part is creating new things and the other is doing maintenance on what already exists. The catch is, the material in $\mathbf{S}_R$ must not only make the rest of the system, but also itself! Suppose that the materials in $\mathbf{S}_R$ and the materials in $\mathbf{S}_O$ cost the same amount of energy for $\mathbf{S}_R$ to make per unit amount. Suppose the material in $\mathbf{S}_R$ can convert raw material from the environment into system mass at the rate $\gamma_R = 3 \mbox{ kg } \mathbf{S}/\mbox{hr}/\mbox{kg }\mathbf{S}_R$. If the system doubles in size once every 2 $\mbox{hrs}$, what fraction of the material in $\mathbf{S}$ is devoted to $\mathbf{S}_O$? $\textbf{Assumptions:}$ The fact that the system continuously doubles in size in a fixed time means that this system is in exponential growth, i.e. $\dot{\mathbf{S}} = \lambda \mathbf{S}$. [i]Problem proposed by Josh Silverman[/i]

A computer is programmed to randomly generate a string of six symbols using only the letters $A,B,C$. What is the probability that the string will not contain three consecutive $A$'s?

The quadrilateral $ABCD$ is a rhombus with acute angle at $A.$ Points $M$ and $N$ are on segments $\overline{AC}$ and $\overline{BC}$ such that $|DM| = |MN|.$ Let $P$ be the intersection of $AC$ and $DN$ and let $R$ be the intersection of $AB$ and $DM.$ Prove that $|RP| = |PD|.$

Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $af(a)^3+2abf(a)+bf(b)$ is a perfect square for all positive integers $a,b$.

The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is $ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$

The continuous function $f(x)$ satisfies $c^2f(x + y) = f(x)f(y)$ for all real numbers $x$ and $y,$ where $c > 0$ is a constant. If $f(1) = c$, find $f(x)$ (with proof).

Find all real numbers $x, y, z$ that satisfy the following system $$\sqrt{x^3 - y} = z - 1$$ $$\sqrt{y^3 - z} = x - 1$$ $$\sqrt{z^3 - x} = y - 1$$

Let $\mathbb{N}$ be the set of all natural numbers. Let $f:\mathbb{N} \to \mathbb{N}$ be a bijective function. Show that there exists three numbers $a$, $b$, $c$ in arithmatic progression such that $f(a)<f(b)<f(c)$

Cube $ABCDEFGH$, labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\overline{AB}$ and $\overline{CG}$ respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. [asy] draw((0,0)--(10,0)--(10,10)--(0,10)--cycle); draw((0,10)--(4,13)--(14,13)--(10,10)); draw((10,0)--(14,3)--(14,13)); draw((0,0)--(4,3)--(4,13), dashed); draw((4,3)--(14,3), dashed); dot((0,0)); dot((0,10)); dot((10,10)); dot((10,0)); dot((4,3)); dot((14,3)); dot((14,13)); dot((4,13)); dot((14,8)); dot((5,0)); label("A", (0,0), SW); label("B", (10,0), S); label("C", (14,3), E); label("D", (4,3), NW); label("E", (0,10), W); label("F", (10,10), SE); label("G", (14,13), E); label("H", (4,13), NW); label("M", (5,0), S); label("N", (14,8), E); [/asy]