Consider the graph on $1000$ vertices $v_1, v_2, ...v_{1000}$ such that for all $1 \le i < j \le 1000$, $v_i$ is connected to $v_j$ if and only if $i$ divides $j$. Determine the minimum number of colors that must be used to color the vertices of this graph such that no two vertices sharing an edge are the same color.

Show that there are infinitely many positive integers n such that $2\underbrace{555...55}_{n}3$ is divisible by $2553$.

Let $\{a_{n}\}_{n \ge 1}$ be a sequence of positive numbers such that \[a_{n+1}^{2}= a_{n}+1, \;\; n \in \mathbb{N}.\] Show that the sequence contains an irrational number.

A spider built a web on the unit circle. The web is a planar graph with straight edges inside the circle, bounded by the circumference of the circle. Each vertex of the graph lying on the circle belongs to a unique edge, which goes perpendicularly inward to the circle. For each vertex of the graph inside the circle, the sum of the unit outgoing vectors along the edges of the graph is zero. Prove that the total length of the web is equal to the number of its vertices on the circle.

Find the real roots of the equation $x^2 + 2ax + \frac{1}{16} = -a +\sqrt{ a^2 + x - \frac{1}{16} }$ , $\left(0 < a < \frac14 \right)$ .

An ellipse with focus $F$ is given. Two perpendicular lines passing through $F$ meet the ellipse at four points. The tangents to the ellipse at these points form a quadrilateral circumscribed around the ellipse. Prove that this quadrilateral is inscribed into a conic with focus $F$

You have $8$ friends, each of whom lives at a different vertex of a cube. You want to chart a path along the cube’s edges that will visit each of your friends exactly once. You can start at any vertex, but you must end at the vertex you started at, and you cannot travel on any edge more than once. How many different paths can you take?

A number $n$ is [i]interesting[/i] if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.

$2023$ balls are divided into several buckets such that no bucket contains more than $99$ balls. We can remove balls from any bucket or remove an entire bucket, as many times as we want. Prove that we can remove them in such a way that each of the remaining buckets will have an equal number of balls and the total number of remaining balls will be at least $100$.

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

Suppose that $a$ and $b$ are natural numbers such that \[p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}\] is a prime number. What is the maximum possible value of $p$?

What is the maximum number of subsets, having property that none of them is a subset of another, can a set with 10 elements have? $\textbf{(A)}\ 126 \qquad\textbf{(B)}\ 210 \qquad\textbf{(C)}\ 252 \qquad\textbf{(D)}\ 420 \qquad\textbf{(E)}\ 1024$

Let N be a positive integer greater than 2. We number the vertices of a regular 2n-gon clockwise with the numbers 1, 2, . . . ,N,−N,−N + 1, . . . ,−2,−1. Then we proceed to mark the vertices in the following way. In the first step we mark the vertex 1. If ni is the vertex marked in the i-th step, in the i+1-th step we mark the vertex that is |ni| vertices away from vertex ni, counting clockwise if ni is positive and counter-clockwise if ni is negative. This procedure is repeated till we reach a vertex that has already been marked. Let $f(N)$ be the number of non-marked vertices. (a) If $f(N) = 0$, prove that 2N + 1 is a prime number. (b) Compute $f(1997)$.

We call a square table of a binary, if at each cell is written a single number 0 or 1. The binary table is called regular if each row and each column exactly two units. Determine the number of regular size tables $n\times n$ ($n> 1$ - given a fixed positive integer). (We can assume that the rows and columns of the tables are numbered: the cases of coincidence in turn, reflect, and so considered different).

The function $ƒ$ is defined at $[0,1]$, and $f\{f(x)\} = ƒ(x)$. $\exists _{c\in [0,1]} \left[f(c) =\frac12 \right]$ Determine $f\left(\frac12 \right).$ $\forall _{t\in [0,1]}\exists _{s\in [0,1]}[f(s) = t]$. Determine $f$. Prove that the function $g$, with $g(x) = x$,$0 \le x \le k$, $g(x) = k$, $k \le x \le 1$ satisfies the relation $g\{g(x)\} = g(x)$.

Let $O$ be the center of a regular triangle $ABC$. From an arbitrary point $P$ of the plane, the perpendiculars were drawn on the sides of the triangle. Let $M$ denote the intersection point of the medians of the triangle , having vertices the feet of the perpendiculars. Prove that $M$ is the midpoint of the segment $PO$.

A line initially $ 1$ inch long grows according to the following law, where the first term is the initial length. \[ 1 \plus{} \frac {1}{4}\sqrt {2} \plus{} \frac {1}{4} \plus{} \frac {1}{16}\sqrt {2} \plus{} \frac {1}{16} \plus{} \frac {1}{64}\sqrt {2} \plus{} \frac {1}{64} \plus{} \cdots. \]If the growth process continues forever, the limit of the length of the line is: $ \textbf{(A)}\ \infty \qquad\textbf{(B)}\ \frac {4}{3} \qquad\textbf{(C)}\ \frac {8}{3} \qquad\textbf{(D)}\ \frac {1}{3}(4 \plus{} \sqrt {2}) \qquad\textbf{(E)}\ \frac {2}{3}(4 \plus{} \sqrt {2})$

The word "'''HELP'''" in block letters is painted in black with strokes $1$ unit wide on a $5$ by $15$ rectangular white sign with dimensions as shown. The area of the white portion of the sign, in square units, is [asy] unitsize(12); fill((0,0)--(0,5)--(1,5)--(1,3)--(2,3)--(2,5)--(3,5)--(3,0)--(2,0)--(2,2)--(1,2)--(1,0)--cycle,black); fill((4,0)--(4,5)--(7,5)--(7,4)--(5,4)--(5,3)--(7,3)--(7,2)--(5,2)--(5,1)--(7,1)--(7,0)--cycle,black); fill((8,0)--(8,5)--(9,5)--(9,1)--(11,1)--(11,0)--cycle,black); fill((12,0)--(12,5)--(15,5)--(15,2)--(13,2)--(13,0)--cycle,black); fill((13,3)--(14,3)--(14,4)--(13,4)--cycle,white); draw((0,0)--(15,0)--(15,5)--(0,5)--cycle); label("$5\left\{ \begin{tabular}{c} \\ \\ \\ \\ \end{tabular}\right.$",(1,2.5),W); label(rotate(90)*"$\{$",(0.5,0.1),S); label("$1$",(0.5,-0.6),S); label(rotate(90)*"$\{$",(3.5,0.1),S); label("$1$",(3.5,-0.6),S); label(rotate(90)*"$\{$",(7.5,0.1),S); label("$1$",(7.5,-0.6),S); label(rotate(90)*"$\{$",(11.5,0.1),S); label("$1$",(11.5,-0.6),S); label(rotate(270)*"$\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.$",(1.5,4),N); label("$3$",(1.5,5.8),N); label(rotate(270)*"$\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.$",(5.5,4),N); label("$3$",(5.5,5.8),N); label(rotate(270)*"$\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.$",(9.5,4),N); label("$3$",(9.5,5.8),N); label(rotate(270)*"$\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.$",(13.5,4),N); label("$3$",(13.5,5.8),N); label("$\left. \begin{tabular}{c} \\ \end{tabular}\right\} 2$",(14,1),E); [/asy] $\text{(A)}\ 30 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 34 \qquad \text{(D)}\ 36 \qquad \text{(E)}\ 38$

Given a positive real number $t$, find the number of real solutions $a, b, c, d$ of the system \[a(1 - b^2) = b(1 -c^2) = c(1 -d^2) = d(1 - a^2) = t.\]

Given an initial integer $ n_0 > 1$, two players, $ {\mathcal A}$ and $ {\mathcal B}$, choose integers $ n_1$, $ n_2$, $ n_3$, $ \ldots$ alternately according to the following rules : [b]I.)[/b] Knowing $ n_{2k}$, $ {\mathcal A}$ chooses any integer $ n_{2k \plus{} 1}$ such that \[ n_{2k} \leq n_{2k \plus{} 1} \leq n_{2k}^2. \] [b]II.)[/b] Knowing $ n_{2k \plus{} 1}$, $ {\mathcal B}$ chooses any integer $ n_{2k \plus{} 2}$ such that \[ \frac {n_{2k \plus{} 1}}{n_{2k \plus{} 2}} \] is a prime raised to a positive integer power. Player $ {\mathcal A}$ wins the game by choosing the number 1990; player $ {\mathcal B}$ wins by choosing the number 1. For which $ n_0$ does : [b]a.)[/b] $ {\mathcal A}$ have a winning strategy? [b]b.)[/b] $ {\mathcal B}$ have a winning strategy? [b]c.)[/b] Neither player have a winning strategy?

For triangle $\vartriangle ABC$, define its $A$-excircle to be the circle that is externally tangent to line segment $BC$ and extensions of $\overleftrightarrow{AB}$ and $\overleftrightarrow{AC}$, and define the $B$-excircle and $C$-excircle likewise. Then, define the $A$-[i]veryexcircle [/i] to be the unique circle externally tangent to both the $A$-excircle as well as the extensions of $\overleftrightarrow{AB}$ and $\overleftrightarrow{AC}$, but that shares no points with line $\overleftrightarrow{BC}$, and define the $B$-veryexcircle and $C$-veryexcircle likewise. Compute the smallest integer $N \ge 337$ such that for all $N_1 \ge N$, the area of a triangle with lengths $3N^2_1$ , $3N^2_1 + 1$, and $2022N_1$ is at most $\frac{1}{22022}$ times the area of the triangle formed by connecting the centers of its three veryexcircles. If your submitted estimate is a positive number $E$ and the true value is $A$, then your score is given by $\max \left(0, \left\lfloor 25 \min \left( \frac{E}{A}, \frac{A}{E}\right)^3\right\rfloor \right)$.

Let $ABC$ be a triangle, and $P$ be a variable point inside $ABC$ such that $AP$ and $CP$ intersect sides $BC$ and $AB$ at $D$ and $E$ respectively, and the area of the triangle $APC$ is equal to the area of quadrilateral $BDPE$. Prove that the circumscribed circumference of triangle $BDE$ passes through a fixed point different from $B$.

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]

In triangle $ABC$ the midpoints of sides $AC, BC$, vertex $C$ and the centroid lie on the same circle. Prove that this circle touches the circle passing through $A, B$ and the orthocenter of triangle $ABC$.

George has $150$ cups of flour and $200$ eggs. He can make a cupcake with $3$ cups of flour and $2$ eggs, or he can make an omelet with $4$ eggs. What is the maximum number of treats (both omelets and cupcakes) he canmake?