On a blackboard the numbers $1,2,3,\dots,170$ are written. You want to color each of these numbers with $k$ colors $C_1,C_2, \dots, C_k$, such that the following condition is satisfied: for each $i$ with $1 \leq i < k$, the sum of all numbers with color $C_i$ divide the sum of all numbers with color $C_{i+1}$. Determine the largest possible value of $k$ for which it is possible to do that coloring.

Let $B(n)$ be the number of ones in the base two expression for the positive integer $n.$ Determine whether $$\exp \left( \sum_{n=1}^{\infty} \frac{ B(n)}{n(n+1)} \right)$$ is a rational number.

For all positive integers $ n$ less than $ 2002$, let \[ a_n \equal{} \begin{cases} 11 & \text{if }n\text{ is divisible by }13\text{ and }14 \\ 13 & \text{if }n\text{ is divisible by }11\text{ and }14 \\ 14 & \text{if }n\text{ is divisible by }11\text{ and }13 \\ 0 & \text{otherwise} \end{cases} \]Calculate $ \sum_{n \equal{} 1}^{2001} a_n$. $ \textbf{(A)}\ 448 \qquad \textbf{(B)}\ 486 \qquad \textbf{(C)}\ 1560 \qquad \textbf{(D)}\ 2001 \qquad \textbf{(E)}\ 2002$

For any two different real numbers $x$ and $y$, we define $D(x,y)$ to be the unique integer $d$ satisfying $2^d\le |x-y| < 2^{d+1}$. Given a set of reals $\mathcal F$, and an element $x\in \mathcal F$, we say that the [i]scales[/i] of $x$ in $\mathcal F$ are the values of $D(x,y)$ for $y\in\mathcal F$ with $x\neq y$. Let $k$ be a given positive integer. Suppose that each member $x$ of $\mathcal F$ has at most $k$ different scales in $\mathcal F$ (note that these scales may depend on $x$). What is the maximum possible size of $\mathcal F$?

The lines $\ell_1$ and \ell_2 are parallel. The points $A_1,A_2, ...,A_7$ are on $\ell_1$ and the points $B_1,B_2,...,B_8$ are on $\ell_2$. The points are arranged in such a way that the number of internal intersections among the line segments is maximized (example Figure). The [b]greatest number[/b] of intersection points is [img]https://cdn.artofproblemsolving.com/attachments/4/9/92153dce5a48fcba0f5175d67e0750b7980e84.png[/img] A. $580$ B. $585$ C. $588$ D. $590$ E. $593$

$3.$ Let $S$ be the set of all positive integers from $1$ to $100$ included. Two players play a game. The first player removes any $k$ numbers he wants, from $S$. The second player's goal is to pick $k$ different numbers, such that their sum is $100$. Which player has the winning strategy if : $a)$ $k=9$? $b)$ $k=8$?

$\triangle ABC$ satisfies $\angle A=60^{\circ}$. Call its circumcenter and orthocenter $O, H$, respectively. Let $M$ be a point on the segment $BH$, then choose a point $N$ on the line $CH$ such that $H$ lies between $C, N$, and $\overline{BM}=\overline{CN}$. Find all possible value of \[\frac{\overline{MH}+\overline{NH}}{\overline{OH}}\]

Determine all possible values of positive integer $n$, such that there are $n$ different 3-element subsets $A_1,A_2,...,A_n$ of the set $\{1,2,...,n\}$, with $|A_i \cap A_j| \not= 1$ for all $i \not= j$.

Five points are given on the plane. Prove that among all the triangles with vertices at these points there are no more than seven acute-angled ones.

If I roll three fair $4$-sided dice, what is the probability that the sum of the resulting numbers is relatively prime to the product of the resulting numbers?

Let $\tau(n)$ denote the number of positive integer divisors of $n$. For example, $\tau(4) = 3$. Find the sum of all positive integers $n$ such that $2 \tau(n) = n$.

Let $p$ be a prime number of the form $4k+1$. Suppose that $r$ is a quadratic residue of $p$ and that $s$ is a quadratic nonresidue of $p$. Show that $p=a^{2}+b^{2}$, where \[a=\frac{1}{2}\sum^{p-1}_{i=1}\left( \frac{i(i^{2}-r)}{p}\right), b=\frac{1}{2}\sum^{p-1}_{i=1}\left( \frac{i(i^{2}-s)}{p}\right).\] Here, $\left( \frac{k}{p}\right)$ denotes the Legendre Symbol.

Given a regular tetrahedron with edge $a$, its edges are divided into $n$ equal segments, thus obtaining $n + 1$ points: $2$ at the ends and $n - 1$ inside. The following set of planes is considered: $\bullet$ those that contain the faces of the tetrahedron, and $\bullet$ each of the planes parallel to a face of the tetrahedron and containing at least one of the points determined above. Now all those points $P$ that belong (simultaneously) to four planes of that set are considered. Determine the smallest positive natural $n$ so that among those points $P$ the eight vertices of a square-based rectangular parallelepiped can be chosen.

Given that circle $ \omega$ is tangent internally to circle $ \Gamma$ at $ S.$ $ \omega$ touches the chord $ AB$ of $ \Gamma$ at $ T$. Let $ O$ be the center of $ \omega.$ Point $ P$ lies on the line $ AO.$ Show that $ PB\perp AB$ if and only if $ PS\perp TS.$

Rhombus $ABCD$ has side length $2$ and $\angle B = 120 ^\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$? $ \textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad \textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad \textbf{(D)}\ 1+\frac{\sqrt{3}}{3} \qquad \textbf{(E)}\ 2 $

Let $k$ be a positive integer and $r_n$ be the remainder when ${2 n} \choose {n}$ is divided by $k$. Find all $k$ for which the sequence $(r_n)_{n=1}^{\infty}$ is eventually periodic.

The function $F$ is a one-to-one transformation of the plane into itself that maps rectangles into rectangles (rectangles are closed; continuity is not assumed). Prove that $F$ maps squares into squares.

Let $ABC$ an acute triangle with orthocenter $H$. Let $M$ be a point on segment $BC$. The line through $M$ and perpendicular to $BC$ intersects lines $BH$ and $CH$ in points $P$ and $Q$ respectively. Prove that the orthocenter of triangle $HPQ$ lies on the line $AM$.

If $46$ squares are colored red in a $9\times 9$ board, show that there is a $2\times 2$ block on the board in which at least $3$ of the squares are colored red.

What is the maximal number of checkers that can be placed on an $8\times 8$ checkerboard so that each checker stands on the middle one of three squares in a row diagonally, with exactly one of the other two squares occupied by another checker?

A quadruple $(a,b,c,d)$ of distinct integers is said to be $balanced$ if $a+c=b+d$. Let $\mathcal{S}$ be any set of quadruples $(a,b,c,d)$ where $1 \leqslant a<b<d<c \leqslant 20$ and where the cardinality of $\mathcal{S}$ is $4411$. Find the least number of balanced quadruples in $\mathcal{S}.$

The quadrilateral $ABCD$ is such that $AB=AD=1$ and $\angle A=90^\circ$. If $CB=c$, $CA=b$, and $CD=a$, then prove that $(2-a^2-c^2 )^2+(2b^2-a^2-c^2 )^2=4a^2 c^2$ and $(a-c)^2\leq 2b^2\leq (a+c)^2$.

On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the lateral sides of all such trapezoids share a common point.

Some lottery is played as follows. A lottery participant buys a card with $10$ numbered cells. He has the right to cross out any $4$ of these $10$ cells. Then a drawing occurs, during which some $7$ out of $10$ cells become winning. The player wins when all $4$ squares he crosses out are winning. The question arises, what is the smallest number of cards that can be used so that, if filled out correctly, at least one of these cards will win in any case? We do not suggest that you answer this question (we ourselves do not know the answer), although, of course, we will be very glad if you do and will evaluate this achievement accordingly. The task is; to indicate a certain number $n$ and a method of filling n cards that guarantees at least one win. The smaller $n$, the higher the rating of the work.

Several straight lines such that no two are parallel, cut the plane into several regions. A point $A$ is marked inside of one region. Prove that a point, separated from $A$ by each of these lines, exists if and only if $A$ belongs to an unbounded region.