An architect designed a hexagonal column with $37$ metal tubes of equal thickness. The figure shows the cross-section of this column. Is it possible to build a similar column whose number of tubes ends in $2003$? [img]https://cdn.artofproblemsolving.com/attachments/6/a/eb5714d2324aac8b78042d1f48f03b74ab0d78.png[/img]

Initially on the blackboard all the integers from $1$ to $2002$ inclusive are written in one line and in some order, without repetitions. In each step, the first and second numbers of the line are deleted and the absolute value of the subtraction of the two numbers that have just been deleted is written at the beginning of the line; the other numbers are not modified in that step, and there is a new line that has one less number than the previous step. After completing $2001$ steps, only one number remains on the board. Determine all possible values of the number left on the board by varying the order of the $2002$ numbers on the initial line (and performing the $2001$ steps).

[b]p1.[/b] A robot is at position $0$ on a number line. Each second, it randomly moves either one unit in the positive direction or one unit in the negative direction, with probability $\frac12$ of doing each. Find the probability that after $4$ seconds, the robot has returned to position $0$. [b]p2.[/b] How many positive integers $n \le 20$ are such that the greatest common divisor of $n$ and $20$ is a prime number? [b]p3.[/b] A sequence of points $A_1$, $A_2$, $A_3$, $...$, $A_7$ is shown in the diagram below, with $A_1A_2$ parallel to $A_6A_7$. We have $\angle A_2A_3A_4 = 113^o$, $\angle A_3A_4A_5 = 100^o$, and $\angle A_4A_5A_6 = 122^o$. Find the degree measure of $\angle A_1A_2A_3 + \angle A_5A_6A_7$. [center][img]https://cdn.artofproblemsolving.com/attachments/d/a/75b06a6663b2f4258e35ef0f68fcfbfaa903f7.png[/img][/center] [b]p4.[/b] Compute $$\log_3 \left( \frac{\log_3 3^{3^{3^3}}}{\log_{3^3} 3^{3^3}} \right)$$ [b]p5.[/b] In an $8\times 8$ chessboard, a pawn has been placed on the third column and fourth row, and all the other squares are empty. It is possible to place nine rooks on this board such that no two rooks attack each other. How many ways can this be done? (Recall that a rook can attack any square in its row or column provided all the squares in between are empty.) [b]p6.[/b] Suppose that $a, b$ are positive real numbers with $a > b$ and $ab = 8$. Find the minimum value of $\frac{a^2+b^2}{a-b} $. [b]p7.[/b] A cone of radius $4$ and height $7$ has $A$ as its apex and $B$ as the center of its base. A second cone of radius $3$ and height $7$ has $B$ as its apex and $A$ as the center of its base. What is the volume of the region contained in both cones? [b]p8.[/b] Let $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$ be a permutation of the numbers $1$, $2$, $3$, $4$, $5$, $6$. We say $a_i$ is visible if $a_i$ is greater than any number that comes before it; that is, $a_j < a_i$ for all $j < i$. For example, the permutation $2$, $4$, $1$, $3$, $6$, $5$ has three visible elements: $2$, $4$, $6$. How many such permutations have exactly two visible elements? [b]p9.[/b] Let $f(x) = x+2x^2 +3x^3 +4x^4 +5x^5 +6x^6$, and let $S = [f(6)]^5 +[f(10)]^3 +[f(15)]^2$. Compute the remainder when $S$ is divided by $30$. [b]p10.[/b] In triangle $ABC$, the angle bisector from $A$ and the perpendicular bisector of $BC$ meet at point $D$, the angle bisector from $B$ and the perpendicular bisector of $AC$ meet at point $E$, and the perpendicular bisectors of $BC$ and $AC$ meet at point $F$. Given that $\angle ADF = 5^o$, $\angle BEF = 10^o$, and $AC = 3$, find the length of $DF$. [img]https://cdn.artofproblemsolving.com/attachments/6/d/6bb8409678a4c44135d393b9b942f8defb198e.png[/img] [b]p11.[/b] Let $F_0 = 0$, $F_1 = 1$, and $F_n = F_{n-1} + F_{n-2}$. How many subsets $S$ of $\{1, 2,..., 2011\}$ are there such that $$F_{2012} - 1 =\sum_{i \in S}F_i?$$ [b]p12.[/b] Let $a_k$ be the number of perfect squares $m$ such that $k^3 \le m < (k + 1)^3$. For example, $a_2 = 3$ since three squares $m$ satisfy $2^3 \le m < 3^3$, namely $9$, $16$, and $25$. Compute$$ \sum^{99}_{k=0} \lfloor \sqrt{k}\rfloor a_k, $$ where $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$. [b]p13.[/b] Suppose that $a, b, c, d, e, f$ are real numbers such that $$a + b + c + d + e + f = 0,$$ $$a + 2b + 3c + 4d + 2e + 2f = 0,$$ $$a + 3b + 6c + 9d + 4e + 6f = 0,$$ $$a + 4b + 10c + 16d + 8e + 24f = 0,$$ $$a + 5b + 15c + 25d + 16e + 120f = 42.$$ Compute $a + 6b + 21c + 36d + 32e + 720f.$ [b]p14.[/b] In Cartesian space, three spheres centered at $(-2, 5, 4)$, $(2, 1, 4)$, and $(4, 7, 5)$ are all tangent to the $xy$-plane. The $xy$-plane is one of two planes tangent to all three spheres; the second plane can be written as the equation $ax + by + cz = d$ for some real numbers $a$, $b$, $c$, $d$. Find $\frac{c}{a}$ . [b]p15.[/b] Find the number of pairs of positive integers $a$, $b$, with $a \le 125$ and $b \le 100$, such that $a^b - 1$ is divisible by $125$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Some sides and diagonals of a regular $ n$-gon form a connected path that visits each vertex exactly once. A [i]parallel pair[/i] of edges is a pair of two different parallel edges of the path. Prove that (a) if $ n$ is even, there is at least one [i]parallel pair[/i]. (b) if $ n$ is odd, there can't be one single [i]parallel pair[/i].

What is the least number of moves it takes a knight to get from one corner of an $n\times n$ chessboard, where $n\ge 4$, to the diagonally opposite corner?

A tourist wants to visit each of the ten cities shown on the picture. The continuous segments on the picture denote railway lines, whereas the dashed segments denote air lines. A railway line costs $150000$ lires, and an air line costs $250000$ lires. What is the minimum possible price of a desired route? [asy] unitsize(2.5 cm); real r = 0.05; pair A, B, C, D, E, F, G, H, I, J; A = (0,0); B = dir(30); D = dir(-30); C = B + D; E = (A + B)/2; F = (B + C)/2; G = (C + D)/2; H = (D + A)/2; I = (A + B + D)/3; J = (B + C + D)/3; draw(A--B--C--D--cycle); draw(E--I--H); draw(F--J--G); draw(B--D, dashed); draw(E--H, dashed); draw(F--G, dashed); draw(I--J, dashed); filldraw(Circle(A,r),white); filldraw(Circle(B,r),white); filldraw(Circle(C,r),white); filldraw(Circle(D,r),white); filldraw(Circle(E,r),white); filldraw(Circle(F,r),white); filldraw(Circle(G,r),white); filldraw(Circle(H,r),white); filldraw(Circle(I,r),white); filldraw(Circle(J,r),white); label("$A$", A + r*dir(225), SW); [/asy]

Frodo composes a number triangle of zeroes and ones in such a way: he fills the topmost row with any $n$ digits, and in other rows he always writes $0$ under consecutive equal digits and writes $1$ under consecutive distinct digits. (An example of a triangle for $n=5$ is shown below.) In how many ways can Frodo fill the topmost row for $n=100$ so that each of $n$ rows of the triangle contains odd number of ones?\[\begin{smallmatrix}1\,0\,1\,1\,0\\1\,1\,0\,1\\0\,1\,1\\1\,0\\1\end{smallmatrix}\] (O. Rudenko and V. Brayman)

What is the minimal number of operations needed to repaint a entirely white grid $100 \times 100$ to be entirely black, if on one move we can choose $99$ cells from any row or column and change their color?

$2024$ otters live in the river. Some are friends with each other. Is it possible that, for any collection of $1012$ otters, there is exactly one additional otter that is friends with all $1012$ otters?

The $2018$ inhabitants of a city are divided in two groups: the knights(only speak the truth) and the liars(only speak the lie). The inhabitants sat in a circle and everybody spoke "My two neighbours(in the left and in the right) are liars". After this, one inhabitant got off the circle. The $2017$ inhabitants sat again in a circle(not necessarily in the same order), and everybody spoke "None of my two neighbours(in the left and in the right) is of the same group of myself" Can we determine the group of the inhabitant that got off the city?

[b]p1.[/b] Fluffy the Dog is an extremely fluffy dog. Because of his extreme fluffiness, children always love petting Fluffy anywhere. Given that Fluffy likes being petted $1/4$ of the time, out of $120$ random people who each pet Fluffy once, what is the expected number of times Fluffy will enjoy being petted? [b]p2.[/b] Andy thinks of four numbers $27$, $81$, $36$, and $41$ and whispers the numbers to his classmate Cynthia. For each number she hears, Cynthia writes down every factor of that number on the whiteboard. What is the sum of all the different numbers that are on the whiteboard? (Don't include the same number in your sum more than once) [b]p3.[/b] Charles wants to increase the area his square garden in his backyard. He increases the length of his garden by $2$ and increases the width of his garden by $3$. If the new area of his garden is $182$, then what was the original area of his garden? [b]p4.[/b] Antonio is trying to arrange his flute ensemble into an array. However, when he arranges his players into rows of $6$, there are $2$ flute players left over. When he arranges his players into rows of $13$, there are $10$ flute players left over. What is the smallest possible number of flute players in his ensemble such that this number has three prime factors? [b]p5.[/b] On the AMC $9$ (Acton Math Competition $9$), $5$ points are given for a correct answer, $2$ points are given for a blank answer and $0$ points are given for an incorrect answer. How many possible scores are there on the AMC $9$, a $15$ problem contest? [b]p6.[/b] Charlie Puth produced three albums this year in the form of CD's. One CD was circular, the second CD was in the shape of a square, and the final one was in the shape of a regular hexagon. When his producer circumscribed a circle around each shape, he noticed that each time, the circumscribed circle had a radius of $10$. The total area occupied by $1$ of each of the different types of CDs can be expressed in the form $a + b\pi + c\sqrt{d}$ where $d$ is not divisible by the square of any prime. Find $a + b + c + d$. [b]p7.[/b] You are picking blueberries and strawberries to bring home. Each bushel of blueberries earns you $10$ dollars and each bushel of strawberries earns you $8$ dollars. However your cart can only fit $24$ bushels total and has a weight limit of $100$ lbs. If a bushel of blueberries weighs $8$ lbs and each bushel of strawberries weighs $6$ lbs, what is your maximum profit. (You can only pick an integer number of bushels) [b]p8.[/b] The number $$\sqrt{2218 + 144\sqrt{35} + 176\sqrt{55} + 198\sqrt{77}}$$ can be expressed in the form $a\sqrt5 + b\sqrt7 + c\sqrt{11}$ for positive integers $a, b, c$. Find $abc$. [b]p9.[/b] Let $(x, y)$ be a point such that no circle passes through the three points $(9,15)$, $(12, 20)$, $(x, y)$, and no circle passes through the points $(0, 17)$, $(16, 19)$, $(x, y)$. Given that $x - y = -\frac{p}{q}$ for relatively prime positive integers $p$, $q$, Find $p + q$. [b]p10.[/b] How many ways can Alfred, Betty, Catherine, David, Emily and Fred sit around a $6$ person table if no more than three consecutive people can be in alphabetical order (clockwise)? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given an integer $ n\ge 2$ and a reular 2n-gon. Color all verices of the 2n-gon with n colors such that: [b](i)[/b] Each vertice is colored by exactly one color. [b](ii)[/b] Two vertices don't have the same color. Two ways of coloring, satisfying the conditions above, are called equilavent if one obtained from the other by a rotation whose center is the center of polygon. Find the total number of mutually non-equivalent ways of coloring. [i]Alternative statement:[/i] In how many ways we can color vertices of an regular 2n-polygon using n different colors such that two adjent vertices are colored by different colors. Two colorings which can be received from each other by rotation are considered as the same.

Prove that :$$\sum_{k=0}^{58}C_{2017+k}^{58-k}C_{2075-k}^{k}=\sum_{p=0}^{29}C_{4091-2p}^{58-2p}$$

For integers $a, b$, call the lattice point with coordinates $(a,b)$ [b]basic[/b] if $gcd(a,b)=1$. A graph takes the basic points as vertices and the edges are drawn in such way: There is an edge between $(a_1,b_1)$ and $(a_2,b_2)$ if and only if $2a_1=2a_2\in \{b_1-b_2, b_2-b_1\}$ or $2b_1=2b_2\in\{a_1-a_2, a_2-a_1\}$. Some of the edges will be erased, such that the remaining graph is a forest. At least how many edges must be erased to obtain this forest? At least how many trees exist in such a forest?

The six-pointed star in the figure is regular: all interior angles of the small triangles are equal. To each of the thirteen points marked are assigned a color: green or red. Prove that there will always be three points of the same color that are vertices of an equilateral triangle. [img]https://cdn.artofproblemsolving.com/attachments/b/f/c50a1f8cb81ea861f16a6a47c3b758c5993213.png[/img]

The European zoos with exactly $100$ types of species each are separated into two groups $\hat{A}$ and $\hat{B}$ in such a way that every pair of zoos $(A, B)$ $(A\in\hat{A}, B\in\hat{B})$ have some animal in common. Prove that we can colour the cages in $3$ colours (all animals of the same type live in the same cage) such that no zoo has cages of only one colour

[b]p1.[/b] Find all integer solutions of the equation $a^2 - b^2 = 2002$. [b]p2.[/b] Prove that the disks drawn on the sides of a convex quadrilateral as on diameters cover this quadrilateral. [b]p3.[/b] $30$ students from one school came to Mathematical Olympiad. In how many different ways is it possible to place them in four rooms? [b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $U=\{1,2,\ldots ,n\}$, where $n\geq 3$. A subset $S$ of $U$ is said to be [i]split[/i] by an arrangement of the elements of $U$ if an element not in $S$ occurs in the arrangement somewhere between two elements of $S$. For example, 13542 splits $\{1,2,3\}$ but not $\{3,4,5\}$. Prove that for any $n-2$ subsets of $U$, each containing at least 2 and at most $n-1$ elements, there is an arrangement of the elements of $U$ which splits all of them.

The Tower of Hanoi is a puzzle with $n$ disks of different sizes and $3$ vertical rods on it. All of the disks are initially placed on the leftmost rod, sorted by size such that the largest disk is on the bottom. On each turn, one may move the topmost disk of any nonempty rod onto any other rod, provided that it is smaller than the current topmost disk of that rod, if it exists. (For instance, if there were two disks on different rods, the smaller disk could move to either of the other two rods, but the larger disk could only move to the empty rod.) The puzzle is solved when all of the disks are moved to the rightmost rod. The specifications normally include an intelligent monk to move the disks, but instead there is a monkey making random moves (with each valid move having an equal probability of being selected). Given $64$ disks, what is the expected number of moves the monkey will have to make to solve the puzzle?

In the dog dictionary the words are any sequence of letters $A$ and $U$ for example $AA$, $UAU$ and $AUAU$. For each word, your "profundity" will be the quantity of subwords we can obtain by the removal of some letters. For each positive integer $n$, determine the largest "profundity" of word, in dog dictionary, can have with $n$ letters. Note: The word $AAUUA$ has "profundity" $14$ because your subwords are $A, U, AU, AA, UU, UA, AUU, UUA, AAU, AUA, AAA, AAUU, AAUA, AUUA$.

(a) Several identical napkins, each in the shape of a regular hexagon, are put on a table (the napkins may overlap). Each napkin has one side which is parallel to a fixed line. Is it always possible to hammer a few nails into the table so that each napkin is nailed with exactly one nail? (b) The same question for regular pentagons. (A Kanel)

Abby and Brian play the following game: They first choose a positive integer $N$. Then they write numbers on a blackboard in turn. Abby starts by writing a $1$. Thereafter, when one of them has written the number $n$, the other writes down either $n + 1$ or $2n$, provided that the number is not greater than $N$. The player who writes $N$ on the blackboard wins. (a) Determine which player has a winning strategy if $N = 2011$. (b) Find the number of positive integers $N\leqslant2011$ for which Brian has a winning strategy. (This is based on ISL 2004, Problem C5.)

Find all pairs of positive integers \((a,b)\) with the following property: there exists an integer \(N\) such that for any integers \(m\ge N\) and \(n\ge N\), every \(m\times n\) grid of unit squares may be partitioned into \(a\times b\) rectangles and fewer than \(ab\) unit squares. [i]Proposed by Holden Mui[/i]

Find the minimum value of $k$ such that there exists two sequence ${a_i},{b_i}$ for $i=1,2,\cdots ,k$ that satisfies the following conditions. (i) For all $i=1,2,\cdots ,k,$ $a_i,b_i$ is the element of $S=\{1996^n|n=0,1,2,\cdots\}.$ (ii) For all $i=1,2,\cdots, k, a_i\ne b_i.$ (iii) For all $i=1,2,\cdots, k, a_i\le a_{i+1}$ and $b_i\le b_{i+1}.$ (iv) $\sum_{i=1}^{k} a_i=\sum_{i=1}^{k} b_i.$

Consider the set $A = \{1, 2, 3, ..., 2008\}$. We say that a set is of [i]type[/i] $r, r \in \{0, 1, 2\}$, if that set is a nonempty subset of $A$ and the sum of its elements gives the remainder $r$ when divided by $3$. Denote by $X_r, r \in \{0, 1, 2\}$ the class of sets of type $r$. Determine which of the classes $X_r, r \in \{0, 1, 2\}$, is the largest.