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).

A point $P$ is fixed inside the circle. $C$ is an arbitrary point of the circle, $AB$ is a chord passing through point $B$ and perpendicular to the segment $BC$. Points $X$ and $Y$ are projections of point $B$ onto lines $AC$ and $BC$. Prove that all line segments $XY$ are tangent to the same circle. (A. Zaslavsky)

Five airlines operate in a country consisting of $36$ cities. Between any pair of cities exactly one airline operates two way flights. If some airlines operates between cities $A,B$ and $B,C$ we say that the ordered triple $A,B,C$ is properly-connected. Determine the largest possible value of $k$ such that no matter how these flights are arranged there are at least $k$ properly-connected triples.

Players $A$ and $B$ play a game on a blackboard that initially contains 2020 copies of the number 1 . In every round, player $A$ erases two numbers $x$ and $y$ from the blackboard, and then player $B$ writes one of the numbers $x+y$ and $|x-y|$ on the blackboard. The game terminates as soon as, at the end of some round, one of the following holds: [list] [*] $(1)$ one of the numbers on the blackboard is larger than the sum of all other numbers; [*] $(2)$ there are only zeros on the blackboard. [/list] Player $B$ must then give as many cookies to player $A$ as there are numbers on the blackboard. Player $A$ wants to get as many cookies as possible, whereas player $B$ wants to give as few as possible. Determine the number of cookies that $A$ receives if both players play optimally.

Let $(a_1,a_2,a_3,...,a_{12})$ be a permutation of $(1,2,3,...,12)$ for which \[ a_1>a_2>a_3>a_4>a_5>a_6 \text{ and } a_6<a_7<a_8<a_9<a_{10}<a_{11}<a_{12}. \] An example of such a permutation is $(6,5,4,3,2,1,7,8,9,10,11,12)$. Find the number of such permutations.

Given a $m\times n$ grid rectangle with $m,n \ge 4$ and a closed path $P$ that is not self intersecting from inner points of the grid, let $A$ be the number of points on $P$ such that $P$ does not turn in them and let $B$ be the number of squares that $P$ goes through two non-adjacent sides of them furthermore let $C$ be the number of squares with no side in $P$. Prove that $$A=B-C+m+n-1.$$

Solve the equation $$\sqrt{x(x+7)}+\sqrt{(x+7)(x+17)}+\sqrt{(x+17)(x+24)}=12+17\sqrt2$$

Find the volume of the solid generated by a rotation of the region enclosed by the curve $y=x^3-x$ and the line $y=x$ about the line $y=x$ as the axis of rotation.

A polyomino is a connected figure constructed by joining one or more unit squares edge-to-edge. Determine, with proof, the number of non-congruent polyominoes with no holes, perimeter $180,$ and area $2024.$

$240$ students are participating a big performance show. They stand in a row and face to their coach. The coach askes them to count numbers from left to right, starting from $1$. (Of course their counts be like $1,2,3,...$)The coach askes them to remember their number and do the following action: First, if your number is divisible by $3$ then turn around. Then, if your number is divisible by $5$ then turn around. Finally, if your number is divisible by $7$ then turn around. (a) How many students are face to coach now? (b) What is the number of the $66^{\text{th}}$ student counting from left who is face to coach?

Let $a$ be a constant number such that $0<a<1$ and $V(a)$ be the volume formed by the revolution of the figure which is enclosed by the curve $y=\ln (x-a)$, the $x$-axis and two lines $x=1,x=3$ about the $x$-axis. If $a$ varies in the range of $0<a<1$, find the minimum value of $V(a)$.

Let $O$ be the circumcenter of an acute-angled triangle $ABC$, let $T$ be the circumcenter of the triangle $AOC$, and let $M$ be the midpoint of the segment $AC$. We take a point $D$ on the side $AB$ and a point $E$ on the side $BC$ that satisfy $\angle BDM = \angle BEM = \angle ABC$. Show that the straight lines $BT$ and $DE$ are perpendicular.

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

The function $ f$ satisfies \[f(x) \plus{} f(2x \plus{} y) \plus{} 5xy \equal{} f(3x \minus{} y) \plus{} 2x^2 \plus{} 1\] for all real numbers $ x$, $ y$. Determine the value of $ f(10)$.

The sequence $\{x_n\}_n$ is defined as follows: $x_1 = 0$, and for all $n \ge 1$ $$(n + 1)^3 x_{n+1} = 2n^2 (2n + 1)x_n + 2(3n + 1).$$ Prove that $\{x_n\}_n$ contains infinitely many integer numbers.

There are 2023 dice on the table. For 1 dollar, one can pick any dice and put it back on any of its four (other than top or bottom) side faces. How many dollars at a minimum will guarantee that all the dice have been repositioned to show equal number of dots on top faces? [i]Egor Bakaev[/i]

The number $2019$ is written on a blackboard. Every minute, if the number $a$ is written on the board, Evan erases it and replaces it with a number chosen from the set $$ \left\{ 0, 1, 2, \ldots, \left\lceil 2.01 a \right\rceil \right\} $$ uniformly at random. Is there an integer $N$ such that the board reads $0$ after $N$ steps with at least $99\%$ probability?

Adam has a single stack of $3 \cdot 2^n$ rocks, where $n$ is a nonnegative integer. Each move, Adam can either split an existing stack into two new stacks whose sizes differ by $0$ or $1$, or he can combine two existing stacks into one new stack. Adam keeps performing such moves until he eventually gets at least one stack with $2^n$ rocks. Find, with proof, the minimum possible number of times Adam could have combined two stacks. [i]Proposed by Anthony Wang[/i]

Solve $2a^2+3a-44=3p^n$ in positive integers where $p$ is a prime.

Let $ABC$ be an equilateral triangle and $n{}, n>1$ an integer. Let $S{}$ be the set of the $n-1$ lines parallel with $BC$ that cut $ABC$ in $n{}$ figures with equal areas and $S^{'}$ be the set of the $n-1$ lines parallel with $BC$ that cut $ABC$ in $n{}$ figures with equal perimeters. Show that $S{}$ and $S^{'}$ are disjunctive.

A sphere is centered at a point with integer coordinates and passes through the three points $(2,0,0), (0,4,0), (0,0,6),$ but not the origin $(0,0,0)$. If $r$ is the smallest possible radius of the sphere, compute $r^2$.

Let $p$ be a prime number. Find all the positive integers $n$ such that $p+n$ divides $pn$

At his usual rate a man rows $15$ miles downstream in five hours less time than it takes him to return. If he doubles his usual rate, the time downstream is only one hour less than the time upstream. In miles per hour, the rate of the stream's current is: $\text{(A)} \ 2 \qquad \text{(B)} \ \frac52 \qquad \text{(C)} \ 3 \qquad \text{(D)} \ \frac72 \qquad \text{(E)} \ 4$

Let $M$ and $N$ be two palindrome numbers, each having $9$ digits and the palindromes don't start with $0$. If $N>M$ and between $N$ and $M$ there aren't any palindromes, find all values of $N-M$.

How many three-digit numbers are not divisible by $5$, have digits that sum to less than $20$, and have the first digit equal to the third digit? $\textbf{(A) }52\qquad \textbf{(B) }60\qquad \textbf{(C) }66\qquad \textbf{(D) }68\qquad \textbf{(E) }70\qquad$