Consider two overlapping regular tetrahedrons of side length $2$ in space. They are centered at the same point, and the second one is oriented so that the lines from its center to its vertices are perpendicular to the faces of the first tetrahedron. What is the volume encompassed by the combined solid?

What is the greatest number of balls of radius $1/2$ that can be placed within a rectangular box of size $10 \times 10 \times 1 \ ?$

Let $V=[(x,y,z)|0\le x,y,z\le 2008]$ be a set of points in a 3-D space. If the distance between two points is either $1, \sqrt{2}, 2$, we color the two points differently. How many colors are needed to color all points in $V$?

The director has found out that six conspiracies have been set up in his department, each of them involving exactly $3$ persons. Prove that the director can split the department in two laboratories so that none of the conspirative groups is entirely in the same laboratory.

We call a set of squares with sides parallel to the coordinate axes and vertices with integer coordinates friendly if any two of them have exactly two points in common. We consider friendly sets in which each of the squares has sides of length $n$. Determine the largest possible number of squares in such a friendly set.

In next year's finals in Schools Mathematics competition, $20$ finalists will participate. The final exam contains six problems. Emil claims that regardless of results, there must be five contestants and two problems such that either all the five contestants solve both problems, or neither of them solve any of the two problems. Is he right?

Given $120$ unit squares arbitrarily situated in the $20\times 25$ rectangle. Prove that you can place a circle with the unit diameter without intersecting any of the squares.

There are 12 people in a line in a bank. When the desk closes, the people form a new line at a newly opened desk. In how many ways can they do this in such a way that none of the 12 people changes his/her position in the line by more than one?

Let $X$ be a subset of $R_{+}^{*}$ with $m$ elements. Find $X$ such that the number of subsets with the same sum is maximum.

A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary. [i]Author: Kei Irie, Japan[/i]

Triangular grid divides an equilateral triangle with sides of length $n$ into $n^2$ triangular cells as shown in figure for $n=12$. Some cells are infected. A cell that is not yet infected, ia infected when it shares adjacent sides with at least two already infected cells. Specify for $n=12$, the least number of infected cells at the start in which it is possible that over time they will infected all the cells of the original triangle. [asy] unitsize(0.25cm); path p=polygon(3); for(int m=0; m<=11;++m){ for(int n=0 ; n<= 11-m; ++n){ draw(shift((n+0.5*m)*sqrt(3),1.5*m)*p); } } [/asy]

$n$ players took part in badminton tournament, where $n$ is positive and odd integer. Each two players played two matches with each other. There were no draws. Each player has won as many matches as he has lost. Prove that you can cancel half of the matches s.t. each player still has won as many matches as he has lost.

For the team, power, and tournament rounds, BMT divided up the teams into $14$ rooms. You sign up to proctor all $3$ rounds, but you cannot proctor in the same room more than once. How many ways can you be assigned for rooms for the $3$ rounds?

It is known that $4$ people $A, B, C$, and $D$ each have a $1/3$ probability of telling the truth. Suppose that $\bullet$ $A$ makes a statement. $\bullet$ $B$ makes a statement about the truthfulness of $A$’s statement. $\bullet$ $C$ makes a statement about the truthfulness of $B$’s statement. $\bullet$ $D$ says that $C$ says that $B$ says that $A$ was telling the truth. What is the probability that $A$ was actually telling the truth?

Bob plays a game on the whiteboard. Initially, the numbers $\{1, 2, ...,n\}$ are shown. On each turn, Bob takes two numbers from the board $x$, $y$, erases them both, and writes down $2x + y$ onto the board. In terms of n, what is the maximum possible value that Bob can end up with?

Is it possible to color each square on a $9\times 9$ board so that each $2\times 3$ or $3\times 2$ block contains exactly $2$ black squares? If so, what is/are the possible total number(s) of black squares?

A set of $n$ points in Euclidean 3-dimensional space, no four of which are coplanar, is partitioned into two subsets $\mathcal{A}$ and $\mathcal{B}$. An $\mathcal{AB}$-tree is a configuration of $n-1$ segments, each of which has an endpoint in $\mathcal{A}$ and an endpoint in $\mathcal{B}$, and such that no segments form a closed polyline. An $\mathcal{AB}$-tree is transformed into another as follows: choose three distinct segments $A_1B_1$, $B_1A_2$, and $A_2B_2$ in the $\mathcal{AB}$-tree such that $A_1$ is in $\mathcal{A}$ and $|A_1B_1|+|A_2B_2|>|A_1B_2|+|A_2B_1|$, and remove the segment $A_1B_1$ to replace it by the segment $A_1B_2$. Given any $\mathcal{AB}$-tree, prove that every sequence of successive transformations comes to an end (no further transformation is possible) after finitely many steps.

[b]p1.[/b] Suppose Mitchell has a fair die. He is about to roll it six times. The probability that he rolls $1$, $2$, $3$, $4$, $5$, and then $6$ in that order is $p$. The probability that he rolls $2$, $2$, $4$, $4$, $6$, and then $6$ in that order is $q$. What is $p - q$? [b]p2.[/b] What is the smallest positive integer $x$ such that $x \equiv 2017$ (mod $2016$) and $x \equiv 2016$ (mod $2017$) ? [b]p3.[/b] The vertices of triangle $ABC$ lie on a circle with center $O$. Suppose the measure of angle $ACB$ is $45^o$. If $|AB| = 10$, then what is the distance between $O$ and the line $AB$? [b]p4.[/b] A “word“ is a sequence of letters such as $YALE$ and $AELY$. How many distinct $3$-letter words can be made from the letters in $BOOLABOOLA$ where each letter is used no more times than the number of times it appears in $BOOLABOOLA$? [b]p5.[/b] How many distinct complex roots does the polynomial $p(x) = x^{12} - x^8 - x^4 + 1$ have? [b]p6.[/b] Notice that $1 = \frac12 + \frac13 + \frac16$ , that is, $1$ can be expressed as the sum of the three fractions $\frac12 $, $\frac13$ , and $\frac16$ , where each fraction is in the form $\frac{1}{n}$, with each $n$ different. Give a $6$-tuple of distinct positive integers $(a, b, c, d, e, f)$ where $a < b < c < d < e < f$ such that $\frac{1}{a} +\frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{1}{e} + \frac{1}{f} = 1$ and explain how you arrived at your $6$-tuple. Multiple answers will be accepted. [b]p7.[/b] You have a Monopoly board, an $11 \times 11$ square grid with the $9 \times 9$ internal square grid removed, where every square is blank except for Go, which is the square in the bottom right corner. During your turn, you determine how many steps forward (which is in the counterclockwise direction) to move by rolling two standard $6$-sided dice. Let $S$ be the set of squares on the board such that if you are initially on a square in $S$, no matter what you roll with the dice, you will always either land on Go (move forward enough squares such that you end up on Go) or you pass Go (you move forward enough squares such that you step on Go during your move and then you advance past Go). You randomly and uniformly select one square in $S$ as your starting position. What is the probability that you land on Go? [b]p8.[/b] Using $L$-shaped triominos, and dominos, where each square of a triomino and a domino covers one unit, what is the minimum number of tiles needed to cover a $3$-by-$2017$ rectangle without any gaps? [b]p9.[/b] Does there exist a pair of positive integers $(x, y)$, where $x < y$, such that $x^2 + y^2 = 1009^3$? If so, give a pair $(x, y)$ and explain how you found that pair. If not, explain why. [b]p10.[/b] Triangle $ABC$ has inradius $8$ and circumradius $20$. Let $M$ be the midpoint of side $BC$, and let $N$ be the midpoint of arc $BC$ on the circumcircle not containing $A$. Let $s_A$ denote the length of segment $MN$, and define $s_B$ and $s_C$ similarly with respect to sides $CA$ and $AB$. Evaluate the product $s_As_Bs_C$. [b]p11.[/b] Julia and Dan want to divide up $256$ dollars in the following way: in the first round, Julia will offer Dan some amount of money, and Dan can choose to accept or reject the offer. If Dan accepts, the game is over. Otherwise, if Dan rejects, half of the money disappears. In the second round, Dan can offer Julia part of the remaining money. Julia can then choose to accept or reject the offer. This process goes on until an offer is accepted or until $4$ rejections have been made; once $4$ rejections are made, all of the money will disappear, and the bargaining process ends. If Julia or Dan is indifferent between accepting and rejecting an offer, they will accept the offer. Given that Julia and Dan are both rational and both have the goal of maximizing the amount of money they get, how much will Julia offer Dan in the first round? [b]p12.[/b] A perfect partition of a positive integer $N$ is an unordered set of numbers (where numbers can be repeated) that sum to $N$ with the property that there is a unique way to express each positive integer less than $N$ as a sum of elements of the set. Repetitions of elements of the set are considered identical for the purpose of uniqueness. For example, the only perfect partitions of $3$ are $\{1, 1, 1\}$ and $\{1, 2\}$. $\{1, 1, 3, 4\}$ is NOT a perfect partition of $9$ because the sum $4$ can be achieved in two different ways: $4$ and $1 + 3$. How many integers $1 \le N \le 40$ each have exactly one perfect partition? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

In an urn there are n balls numbered $1, 2, \cdots, n$. They are drawn at random one by one without replacement and the numbers are recorded. What is the probability that the resulting random permutation has only one local maximum? A term in a sequence is a local maximum if it is greater than all its neighbors.

We are given a cube with edges of length $n$ cm. At our disposal is a long piece of insulating tape of width $1$ cm. It is required to stick this tape to the cube. The tape may freely cross an edge of the cube on to a different face but it must always be parallel to an edge of the cube. It may not overhang the edge of a face or cross over a vertex. How many pieces of the tape are necessary in order to completely cover the cube? (You may assume that $n$ is an integer.) (A Spivak)

2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group.

Find the number of $4 \times 4$ array whose entries are from the set $\{ 0 , 1, 2, 3 \}$ and which are such that the sum of the numbers in each of the four rows and in each of the four columns is divisible by $4$.

Prove that there exists a set $ S$ of $ n \minus{} 2$ points inside a convex polygon $ P$ with $ n$ sides, such that any triangle determined by $3$ vertices of $ P$ contains exactly one point from $ S$ inside or on the boundaries.

Each grid point of a cartesian plane is colored with one of three colors, whereby all three colors are used. Show that one can always find a right-angled triangle, whose three vertices have pairwise different colors.