Every positive integer is marked with a number from the set $\{ 0,1,2\}$, according to the following rule: $$\text{if a positive integer }k\text{ is marked with }j,\text{ then the integer }k+j\text{ is marked with }0.$$ Let $S$ denote the sum of marks of the first $2019$ positive integers. Determine the maximum possible value of $S$. [i]Proposed by Ivan Novak[/i]

\[\int_0^\infty x e^{-\sqrt[3]{x}}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Given points $O, A_1, A_2, ..., A_n$ on the plane. For any two of these points the square of distance between them is natural number. Prove that there exist two vectors $\vec{x}$ and $\vec{y}$, such that for any point $A_i$, $\vec{OA_i }= k\vec{x}+l \vec{y}$, where $k$ and $l$ are some integer numbers. (A.Glazyrin)

In triangle $\vartriangle ABC$, we have marked points $A_1$ on side $BC, B_1$ on side $AC$, and $C_1$ on side $AB$ so that $AA_1$ is an altitude, $BB_1$ is a median, and $CC_1$ is an angle bisector. It is known that $\vartriangle A_1B_1C_1$ is equilateral. Prove that $\vartriangle ABC$ is equilateral too. (Note: A median connects a vertex of a triangle with the midpoint of the opposite side. Thus, for median $BB_1$ we know that $B_1$ is the midpoint of side $AC$ in $\vartriangle ABC$.)

Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq m \leq n$ either the sum of the first $m$ terms of the sum or the sum of the last $m$ terms is integral. Determine the minimum number of integers in a complete sequence of $n$ numbers.

Let $ABF$ be a right-angled triangle with $\angle AFB = 90$, a square $ABCD$ is externally to the triangle. If $FA = 6$, $FB = 8$ and $E$ is the circumcenter of the square $ABCD$, determine the value of $EF$

Let $D_1$, $D_2$, ..., $D_n$ be closed discs in the plane. (A closed disc is the region limited by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most $2003$ discs $D_i$. Prove that there exists a disc $D_k$ which intersects at most $7\cdot 2003 - 1 = 14020$ other discs $D_i$.

For any integer $n\geq1$, we consider a set $P_{2n}$ of $2n$ points placed equidistantly on a circle. A [i]perfect matching[/i] on this point set is comprised of $n$ (straight-line) segments whose endpoints constitute $P_{2n}$. Let $\mathcal{M}_{n}$ denote the set of all non-crossing perfect matchings on $P_{2n}$. A perfect matching $M\in \mathcal{M}_{n}$ is said to be [i]centrally symmetric[/i], if it is invariant under point reflection at the circle center. Determine, as a function of $n$, the number of centrally symmetric perfect matchings within $\mathcal{M}_{n}$. [i]Proposed by Mirko Petrusevski[/i]

Farmer John has $ 5$ cows, $ 4$ pigs, and $ 7$ horses. How many ways can he pair up the animals so that every pair consists of animals of different species? (Assume that all animals are distinguishable from each other.)

Among any $79$ consecutive natural numbers there exists one whose sum of digits is divisible by $13$. Find a sequence of $78$ consecutive natural numbers for which the above statement fails.

There are two boxes containing balls. One of them contains $m$ balls, and the other contains $n$ balls, where $m, n > 0$. Two actions are permitted: (i) Remove an equal number of balls from both boxes. (ii) Increase the number of balls in one of the boxes by a factor $k$. Is it possible to remove all of the balls from both boxes with just these two actions, 1. if $k = 2$? 2. if $k = 3$?

Let $S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$. Then $S$ equals: ${{ \textbf{(A)}\ (x-2)^4 \qquad\textbf{(B)}\ (x-1)^4 \qquad\textbf{(C)}\ x^4 \qquad\textbf{(D)}\ (x+1)^4 }\qquad\textbf{(E)}\ x^4+1} $

Given: \[ S = 1 + \frac{1}{1 + \frac{1}{3}} + \frac{1}{1 + \frac{1}{3} + \frac{1} {6}} + \cdots + \frac{1}{1 + \frac{1}{3} + \frac{1}{6} + \cdots + \frac{1} {1993006}} \] where the denominators contain partial sums of the sequence of reciprocals of triangular numbers (i.e. $k=\frac{n(n+1)}{2}$ for $n = 1$, $2$, $\ldots$,$1996$). Prove that $S>1001$.

Maisy is at the origin of the coordinate plane. On her first step, she moves $1$ unit up. On her second step, she moves $ 1$ unit to the right. On her third step, she moves $2$ units up. On her fourth step, she moves $2$ units to the right. She repeats this pattern with each odd-numbered step being $ 1$ unit more than the previous step. Given that the point that Maisy lands on after her $21$st step can be written in the form $(x, y)$, find the value of $x + y$. Proposed by Audrey Chun

Consider in the plane a circle $\Gamma$ with centre O and a line l not intersecting the circle. Prove that there is a unique point Q on the perpendicular drawn from O to line l, such that for any point P on the line l, PQ represents the length of the tangent from P to the given circle.

Find all integer solutions $(a,b)$ of the equation \[ (a+b+3)^2 + 2ab = 3ab(a+2)(b+2)\]

Determine all natural $ n$ for which there exists natural $ m$ divisible by all natural numbers from 1 to $ n$ but not divisible by any of the numbers $ n \plus{} 1$, $ n \plus{} 2$, $ n \plus{} 3$.

Prove the inequality: $$\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{1974}-\frac{1}{1975}<\frac{2}{5}$$

There are two distinct pairs of positive integers $a_1 < b_1$ and $a_2 < b_2$ such that both $(a_1 + ib_1)(b_1 - ia_1) $ and $(a_2 + ib_2)(b_2 - ia_2)$ equal $2020$, where $i =\sqrt{-1}$. Find $a_1 + b_1 + a_2 + b_2$.

The grid to the right consists of 74 unit squares, marked by gridlines. Partition the grid into five regions along the gridlines so that the areas of the regions are 1, 13, 19, 20, and 21. A square with a number should be contained in the region with that area. [asy]unitsize(20); path p = (5,0)--(3,0)--(3,1)--(1,1)--(1,2)--(0,2)--(0,7)--(1,7)--(1,8)--(3,8)--(3,9)--(5,9); draw(p^^reflect((5,0),(5,3.14))*p); int[] v = {0,1,1,0,0,0,0,0,1,1}; int[] h = {0,1,0,0,0,0,0,1,1}; for(int i=1; i<10; ++i) { draw((i,v[i])--(i,9-v[i]),dotted); } for(int i=1; i<9; ++i) { draw((h[i],i)--(10-h[i],i),dotted); } int[][] dord = {{1,4,21},{2,4,19},{3,4,1},{4,4,13},{5,4,20},{6,4,19},{7,4,20},{8,4,19},{2,2,21},{2,6,19},{4,7,19},{5,1,21},{7,6,13},{7,2,13}}; for(int i=0; i<14; ++i){ label(string(dord[i][2]),(dord[i][0]+.5,.5+dord[i][1])); } [/asy] 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.)

A moving point $P(x,y)$ rotate anticlockwise around unit circle, who seangular speed is $\omega$. Then how does $Q(-2xy,y^2-x^2)$ moves? $\text{(A)}$ Rotate clockwise around unit circle, who seangular speed is $\omega$. $\text{(B)}$ Rotate anticlockwise around unit circle, who seangular speed is $\omega$. $\text{(C)}$ Rotate clockwise around unit circle, who seangular speed is $2\omega$. $\text{(D)}$ Rotate anticlockwise around unit circle, who seangular speed is $2\omega$.

Let $A_1,A_2,...$ be a sequence of sets such that for any positive integer $i$, there are only finitely many values of $j$ such that $A_j\subseteq A_i$. Prove that there is a sequence of positive integers $a_1,a_2,...$ such that for any pair $(i,j)$ to have $a_i\mid a_j\iff A_i\subseteq A_j$.

Consider $\triangle XPQ$ and $\triangle YPQ$ such that $X$ and $Y$ are on the opposite sides of $PQ$ and the circumradius of $\triangle XPQ$ and the circumradius of $\triangle YPQ$ are the same. $I$ and $J$ are the incenters of $\triangle XPQ$ and $\triangle YPQ$ respectively. Let $M$ be the midpoint of $PQ$. Suppose $I, M, J$ are collinear. Prove that $XPYQ$ is a parallelogram.

In CMI, each person has atmost $3$ friends. A disease has infected exactly $2023$ peoplein CMI . Each day, a person gets infected if and only if atleast two of their friends were infected on the previous day. Once someone is infected, they can neither die nor be cured. Given that everyone in CMI eventually got infected, what is the maximum possible number of people in CMI?

Indicate whether it is possible to write the integers $1, 2, 3, 4, 5, 6, 7, 8$ at the vertices of an regular octagon such that the sum of the numbers of any $3$ consecutive vertices is greater than: a) $11$. b) $13$.