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].

In triangle $ABC$ let $O$ and $H$ be the circumcenter and the orthocenter. The point $P$ is the reflection of $A$ with respect to $OH$. Assume that $P$ is not on the same side of $BC$ as $A$. Points $E,F$ lie on $AB,AC$ respectively such that $BE=PC \ , CF=PB$. Let $K$ be the intersection point of $AP,OH$. Prove that $\angle EKF = 90 ^{\circ}$ [i] Proposed by Iman Maghsoudi[/i]

Let $u$ be a positive number. Consider the set $S$ of all points whose rectangular coordinates $(x, y )$ satisfy all of the following conditions: $\text{(i) }\frac{a}{2}\le x\le 2a\qquad\text{(ii) }\frac{a}{2}\le y\le 2a\qquad\text{(iii) }x+y\ge a \\ \\ \qquad\text{(iv) }x+a\ge y\qquad \text{(v) }y+a\ge x$ The boundary of set $S$ is a polygon with $\textbf{(A) }3\text{ sides}\qquad\textbf{(B) }4\text{ sides}\qquad\textbf{(C) }5\text{ sides}\qquad\textbf{(D) }6\text{ sides}\qquad \textbf{(E) }7\text{ sides}$

In the square, the midpoints of the two sides were marked and the segments shown in the figure on the left were drawn. Which of the shaded quadrilaterals has the largest area? [img]https://cdn.artofproblemsolving.com/attachments/d/f/2be7bcda3fa04943687de9e043bd8baf40c98c.png[/img]

Let $a\le 1$ be a real number. Sequence $\{x_n\}$ satisfies $x_0=0, x_{n+1}= 1-a\cdot e^{x_n}$, for all $n\ge 1$, where $e$ is the natural logarithm. Prove that for any natural $n$, $x_n\ge 0$.

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 collector of rare coins has coins of denominations $1, 2,..., n$ (several coins for each denomination). He wishes to put the coins into $5$ boxes so that: (1) in each box there is at most one coin of each denomination; (2) each box has the same number of coins and the same denomination total; (3) any two boxes contain all the denominations; (4) no denomination is in all $5$ boxes. For which $n$ is this possible?

Let $n\ge4$ points on a circle be denoted by $1$ through $n$. A pair of two nonadjacent points denoted by $a$ and $b$ is called regular if all numbers on one of the arcs determined by $a$ and $b$ are less than $a$ and $b$. Prove that there are exactly $n-3$ regular pairs.

$\frac{9}{7\times 53} =$ $\text{(A)}\ \frac{.9}{.7\times 53} \qquad \text{(B)}\ \frac{.9}{.7\times .53} \qquad \text{(C)}\ \frac{.9}{.7\times 5.3} \qquad \text{(D)}\ \frac{.9}{7\times .53} \qquad \text{(E)}\ \frac{.09}{.07\times .53}$

a) Let $S$ and $P$ be the area and perimeter of some triangle. The straight lines on which its sides are located move to the outside by a distance $h$. What will be the area and perimeter of the triangle formed by the three obtained lines? b) Let $V$ and $S$ be the volume and surface area of some tetrahedron. The planes on which its faces are located are moved to the outside by a distance $h$. What will be the volume and surface area of the tetrahedron formed by the three obtained planes?

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?

Find all increasing sequences $a_1,a_2,a_3,...$ of natural numbers such that for each $i,j\in \mathbb N$, number of the divisors of $i+j$ and $a_i+a_j$ is equal. (an increasing sequence is a sequence that if $i\le j$, then $a_i\le a_j$.)

$2024$ 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?

Is there a positive integer $ n$ such that among 200th digits after decimal point in the decimal representations of $ \sqrt{n}$, $ \sqrt{n\plus{}1}$, $ \sqrt{n\plus{}2}$, $ \ldots,$ $ \sqrt{n\plus{}999}$ every digit occurs 100 times? [i]Proposed by A. Golovanov[/i]

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?

For a function ${u}$ defined on ${G \subset \Bbb{C}}$ let us denote by ${Z(u)}$ the neignborhood of unit raduis of the set of roots of ${u}$. Prove that for any compact set ${K \subset G}$ there exists a constant ${C}$ such that if ${u}$ is an arbitrary real harmonic function on ${G}$ which vanishes in a point of ${K}$ then: \[\displaystyle \sup_{z \in K} |u(z)| \leq C \sup_{Z(u)\cap G}|u(z)|.\]

Compute $\sum_{k=0}^{\infty}\int_{0}^{\frac{\pi}{3}}\sin^{2k} x \, dx$.

The base-nine representation of the number $N$ is $27{,}006{,}000{,}052_{\rm nine}$. What is the remainder when $N$ is divided by $5?$ $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$

Let $O$ be the set of odd numbers between 0 and 100. Let $T$ be the set of subsets of $O$ of size $25$. For any finite subset of integers $S$, let $P(S)$ be the product of the elements of $S$. Define $n=\textstyle{\sum_{S \in T}} P(S)$. If you divide $n$ by 17, what is the remainder?

If $x^5 - x ^3 + x = a,$ prove that $x^6 \geq 2a - 1$.

There exists a unique pair of polynomials $(P(x),Q(x))$ such that \begin{align*} P(Q(x))&= P(x)(x^2-6x+7) \\ Q(P(x))&= Q(x)(x^2-3x-2) \end{align*} Compute $P(10)+Q(-10)$. [i]Proposed by Connor Gordon[/i]

[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].