An encyclopedia consists of ${2000}$ numbered volumes. The volumes are stacked in order with number ${1}$ on top and ${2000}$ in the bottom. One may perform two operations with the stack: (i) For ${n}$ even, one may take the top ${n}$ volumes and put them in the bottom of the stack without changing the order. (ii) For ${n}$ odd, one may take the top ${n}$ volumes, turn the order around and put them on top of the stack again. How many different permutations of the volumes can be obtained by using these two operations repeatedly?

For each $i=1,2,..N$, let $a_i,b_i,c_i$ be integers such that at least one of them is odd. Show that one can find integers $x,y,z$ such that $xa_i+yb_i+zc_i$ is odd for at least $\frac{4}{7}N$ different values of $i$.

Consider all the possible subsets of the set $\{1,2,..., N\}$ which do not contain any consecutive numbers. Prove that the sum of the squares of the products of the numbers in these subsets is $(N + 1)! - 1$. (Based on idea of R.P. Stanley)

In a lake, there are 2024 leaves arranged in a row. Two frogs are positioned, one on the first leaf and the other on the second leaf. Every minute, both frogs jump simultaneously. Each time a frog jumps, it decides whether to jump to the next leaf or to the leaf that is three positions ahead. Is it possible for each leaf to be visited exactly once by exactly one of the frogs?

[b]p1.[/b] Find the number of ordered triples $(a, b, c)$ satisfying $\bullet$ $a, b, c$ are are single-digit positive integers, and $\bullet$ $a \cdot b + c = a + b \cdot c$. [b]p2.[/b] In their class Introduction to Ladders at Greendale Community College, Jan takes four tests. They realize that their test scores in chronological order form an increasing arithmetic progression with integer terms, and that the average of those scores is an integer greater than or equal to $94$. How many possible combinations of test scores could they have had? (Test scores at Greendale range between $0$ and $100$, inclusive.) [b]p3.[/b] Suppose that $a + \frac{1}{b} = 2$ and $b +\frac{1}{a} = 3$. If$ \frac{a}{b} + \frac{b}{a}$ can be expressed as $\frac{p}{q}$ in simplest terms, find $p + q$. [b]p4.[/b] Suppose that $A$ and $B$ are digits between $1$ and $9$ such that $$0.\overline{ABABAB...}+ B \cdot (0.\overline{AAA...}) = A \cdot (0.\overline{B_1B_1B_1...}) + 1$$ Find the sum of all possible values of $10A + B$. [b]p5.[/b] Let $ABC$ be an isosceles right triangle with $m\angle ABC = 90^o$. Let $D$ and $E$ lie on segments $AC$ and $BC$, respectively, such that triangles $\vartriangle ADB$ and $\vartriangle CDE$ are similar and $DE = EB$. If $\frac{AC}{AD} = 1 +\frac{\sqrt{a}}{b}$ with $a, b$ positive integers and a squarefree, then find $a + b$. [b]p6.[/b] Five bowling pins $P_1$, $P_2$,..., $P_5$ are lined up in a row. Each turn, Jemma picks a pin at random from the standing pins, and throws a bowling ball at that pin; that pin and each pin directly adjacent to it are knocked down. If the expected value of the number of turns Jemma will take to knock down all the pins is a b where a and b are relatively prime, find $a + b$. (Pins $P_i$ and $P_j$ are adjacent if and only if $|i -j| = 1$.) [b]p7.[/b] Let triangle $ABC$ have side lengths $AB = 10$, $BC = 24$, and $AC = 26$. Let $I$ be the incenter of $ABC$. If the maximum possible distance between $I$ and a point on the circumcircle of $ABC$ can be expressed as $a +\sqrt{b}$ for integers $a$ and $b$ with $b$ squarefree, find $a + b$. (The incenter of any triangle $XY Z$ is the intersection of the angle bisectors of $\angle Y XZ$, $\angle XZY$, and $\angle ZY X$.) [b]p8.[/b] How many terms in the expansion of $$(1 + x + x^2 + x^3 +... + x^{2021})(1 + x^2 + x^4 + x^6 + ... + x^{4042})$$ have coefficients equal to $1011$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A $300\times 300$ board is arbitrarily filled with $2\times 1$ dominoes with no overflow, underflow, or overlap. (Tokens can be placed vertically or horizontally.) Decide if it is possible to paint the tiles with three different colors, so that the following conditions are met: $\bullet$ Each token is painted in one and only one of the colors. $\bullet$ The same number of tiles are painted in each color. $\bullet$ No piece is a neighbor of more than two pieces of the same color. Note: Two dominoes are [i]neighbors [/i]if they share an edge.

We start with any finite list of distinct positive integers. We may replace any pair $n, n + 1$ (not necessarily adjacent in the list) by the single integer $n-2$, now allowing negatives and repeats in the list. We may also replace any pair $n, n + 4$ by $n - 1$. We may repeat these operations as many times as we wish. Either determine the most negative integer which can appear in a list, or prove that there is no such minimum.

The King decided to reduce his Council consisting of thousand wizards. He placed them in a line and placed hats with numbers from $1$ to $1001$ on their heads not necessarily in this order (one hat was hidden). Each wizard can see the numbers on the hats of all those before him but not on himself or on anyone who stayed behind him. By King's command, starting from the end of the line each wizard calls one integer from $1$ to $1001$ so that every wizard in the line can hear it. No number can be repeated twice. In the end each wizard who fails to call the number on his hat is removed from the Council. The wizards knew the conditions of testing and could work out their strategy prior to it. (a) Can the wizards work out a strategy which guarantees that more than $500$ of them remain in the Council? (b) Can the wizards work out a strategy which guarantees that at least $999$ of them remain in the Council?

$100$ betyárs stand on the Hortobágy plains. Every betyár's field of vision is a $100$ degree angle. After each of them announces the number of other betyárs they see, we compute the sum of these $100$ numbers. What is the largest value this sum can attain?

Let $T$ be the set of ordered triples $(x,y,z)$, where $x,y,z$ are integers with $0\leq x,y,z\leq9$. Players $A$ and $B$ play the following guessing game. Player $A$ chooses a triple $(x,y,z)$ in $T$, and Player $B$ has to discover $A$[i]'s[/i] triple in as few moves as possible. A [i]move[/i] consists of the following: $B$ gives $A$ a triple $(a,b,c)$ in $T$, and $A$ replies by giving $B$ the number $\left|x+y-a-b\right |+\left|y+z-b-c\right|+\left|z+x-c-a\right|$. Find the minimum number of moves that $B$ needs to be sure of determining $A$[i]'s[/i] triple.

Determine the smallest integer $n \ge 1$ such that a $n \times n$ square can be cut into square pieces of size $1 \times 1$ and $2 \times 2$ with both types occuring the same number of times.

In a regular nonagon, each vertex is colored either red or green. Three corners of the nonagon determine a triangle. Such a triangle is called [i]red [/i] or [i]green [/i] if all its vertices are red or green if all are green. Prove that for each such coloring of the nonagon there are at least two different ones , that are congruent triangles of the same color.

Let $A$ be a given finite set with some of its subsets called pretty. Let a subset be called small, if it's a subset of a pretty set. Let a subset be called big, if it has a pretty subset. (A set can be small and big simultaneously, and a set can be neither small nor big.) Let $a$ denote the number of elements of $A$, and let $p$, $s$ and $b$ denote the number of pretty, small and big sets, respectively. Prove that $2^a\cdot p\le s\cdot b$. [i]Proposed by András Imolay, Budapest[/i]

A closed non-self-intersecting polygonal chain is drawn through the centers of some squares on the $8\times 8$ chess board. Every link of the chain connects the centers of adjacent squares either horizontally, vertically or diagonally, where the two squares are adjacent if they share an edge or a corner. For the interior polygon bounded by the chain, prove that the total area of black pieces equals the total area of white pieces. (Author: D. Khramtsov)

Find all positive integers $n$ for which a square[b][i] n x n[/i][/b] can be covered with rectangles [b][i]k x 1[/i][/b] and one square [b][i]1 x 1[/i][/b] when: a) $k = 4$ b) $k = 8$

In a school there $b$ teachers and $c$ students. Suppose that a) each teacher teaches exactly $k$ students, and b)for any two (distinct) students , exactly $h$ teachers teach both of them. Prove that $\frac{b}{h}=\frac{c(c-1)}{k(k-1)}$.

Let $A{}$ be a subset formed of $16$ elements of the set $B=\{1, 2, 3, \ldots, 105, 106\}$ such that the difference between every two elements from $A$ is different from $6, 9, 12, 15, 18, 21$. Prove that there are two elements in $A{}$ whose difference is $3$.

To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y<0$, then the following operation is allowed: $x,y,z$ are replaced by $x+y,-y,z+y$ respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.

Chandler the Octopus is at a tentacle party! At this party, there is $1$ creature with $2$ tentacles, $2$ creatures with $3$ tentacles, $3$ creatures with $4$ tentacles, all the way up to $14$ creatures with $15$ tentacles. Each tentacle is distinguishable from all other tentacles. For some $2\le m < n \le 15$, a creature with m tentacles “meets” a creature with n tentacles; “meeting” another creature consists of shaking exactly 1 tentacle with each other. Find the number of ways there are to pick distinct $m < n$ between $2$ and $15$, inclusive, and then to pick a creature with $m$ tentacles to “meet” a selected creature with $n$ tentacles. [i]Proposed by Armaan Tipirneni, Richard Chen, and Denise the Octopus[/i]

Find all natural numbers $n$ for which there exists set $S$ consisting of $n$ points in the plane, satisfying the condition: For each point $A \in S$ there exist at least three points say $X, Y, Z$ from $S$ such that the segments $AX, AY$ and$ AZ$ have length $1$ (it means that $AX = AY = AZ = 1$).

Let $n\ge 2,$ be a positive integer. Numbers $\{1,2,3, ...,n\}$ are written in a row in an arbitrary order. Determine the smalles positive integer $k$ with the property: everytime it is possible to delete $k$ numbers from those written on the table, such that the remained numbers are either in an increasing or decreasing order.

Let $n$ numbers all equal to $1$ be written on a blackboard. A move consists of replacing two numbers on the board with two copies of their sum. It happens that after $h$ moves all $n$ numbers on the blackboard are equal to $m.$ Prove that $h \leq \frac{1}{2}n \log_2 m.$

A set of $1989$ numbers is given. It is known that the sum of any $10$ of them is positive. Prove that the sum of all these numbers is positive. (Folklore)

[b]p1[/b]. Points $P_1,P_2,P_3,... ,P_n$ lie on a plane such that $P_aP_b = 1$,$P_cP_d = 2$, and $P_eP_f = 2018$ for not necessarily distinct indices $a,b,c,d,e, f \in \{1, 2,... ,n\}$. Find the minimum possible value of $n$. [b]p2.[/b] Find the coefficient of the $x^2y^4$ term in the expansion of $(3x +2y)^6$. [b]p3.[/b] Find the number of positive integers $n < 1000$ such that $n$ is a multiple of $27$ and the digit sum of $n$ is a multiple of $11$. [b]p4.[/b] How many times do the minute hand and hour hand of a $ 12$-hour analog clock overlap in a $366$-day leap year? [b]p5.[/b] Find the number of ordered triples of integers $(a,b,c)$ such that $(a +b)(b +c)(c + a) = 2018$. [b]p6.[/b] Let $S$ denote the set of the first $2018$ positive integers. Call the score of a subset the sum of its maximal element and its minimal element. Find the sum of score $(x)$ over all subsets $s \in S$ [b]p7.[/b] How many ordered pairs of integers $(a,b)$ exist such that $1 \le a,b \le 20$ and $a^a$ divides $b^b$? [b]p8.[/b] Let $f$ be a function such that for every non-negative integer $p$, $f (p)$ equals the number of ordered pairs of positive integers $(a,n)$ such that $a^n = a^p \cdot n$. Find $\sum^{2018}_{p=0}f (p)$. [b]p9.[/b] A point $P$ is randomly chosen inside a regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$. What is the probability that the projections of $P$ onto the lines $\overleftrightarrow{A_i A_{i+1}}$ for $i = 1,2,... ,8$ lie on the segments $\overline{A_iA_{i+1}}$ for $i = 1,2,... ,8$ (where indices are taken $mod \,\, 8$)? [b]p10. [/b]A person keeps flipping an unfair coin until it flips $3$ tails in a row. The probability of it landing on heads is $\frac23$ and the probability it lands on tails is $\frac13$ . What is the expected value of the number of the times the coin flips? PS. You had better use hide for answers.

The Imomi archipelago consists of $n\geq 2$ islands. Between each pair of distinct islands is a unique ferry line that runs in both directions, and each ferry line is operated by one of $k$ companies. It is known that if any one of the $k$ companies closes all its ferry lines, then it becomes impossible for a traveller, no matter where the traveller starts at, to visit all the islands exactly once (in particular, not returning to the island the traveller started at). Determine the maximal possible value of $k$ in terms of $n$. [i]Anton Trygub, Ukraine[/i]