$2019$ point grasshoppers sit on a line. At each move one of the grasshoppers jumps over another one and lands at the point the same distance away from it. Jumping only to the right, the grasshoppers are able to position themselves so that some two of them are exactly $1$ mm apart. Prove that the grasshoppers can achieve the same, jumping only to the left and starting from the initial position. (Sergey Dorichenko)

Clarabelle wants to travel from $(0,0)$ to $(6,2)$ in the coordinate plane. She is able to move in one-unit steps up, down, or right, must stay between $y=0$ and $y=2$ (inclusive), and is not allowed to visit the same point twice. How many paths can she take? [i]Proposed by Connor Gordon[/i]

$100$ heaps of stones lie on a table. Two players make moves in turn. At each move, a player can remove any non-zero number of stones from the table, so that at least one heap is left untouched. The player that cannot move loses. Determine, for each initial position, which of the players, the first or the second, has a winning strategy. [i]K. Kokhas[/i] [b]EDIT.[/b] It is indeed confirmed by the sender that empty heaps are still heaps, so the third post contains the right guess of an interpretation.

For a given integer $n\ge3$, let $S_1, S_2,\ldots,S_m$ be distinct three-element subsets of the set $\{1,2,\ldots,n\}$ such that for each $1\le i,j\le m; i\neq j$ the sets $S_i\cap S_j$ contain exactly one element. Determine the maximal possible value of $m$ for each $n$.

No two of $20$ students in a class have the same scores on both written and oral examinations in mathematics. We say that student $A$ is better than $B$ if his two scores are greater than or equal to the corresponding scores of $B$. The scores are integers between $1$ and $10$. (a) Show that there exist three students $A,B,C$ such that $A$ is better than $B$ and $B$ is better than $C$. (b) Would the same be true for a class of $19$ students?

Two decompositions of a square into three rectangles are called substantially different, if reordering the rectangles does not change one into the other. How many substantially different decompositions of a $2010 \times 2010$ square in three rectangles with integer side lengths are there such that the area of one rectangle is equal to the arithmetic mean of the areas of the other rectangles?

[u]Round 1[/u] [b]p1.[/b] A circle is inscribed inside a square such that the cube of the radius of the circle is numerically equal to the perimeter of the square. What is the area of the circle? [b]p2.[/b] If the coefficient of $z^ky^k$ is $252$ in the expression $(z + y)^{2k}$, find $k$. [b]p3.[/b] Let $f(x) = \frac{4x^4-2x^3-x^2-3x-2}{x^4-x^3+x^2-x-1}$ be a function defined on the real numbers where the denominator is not zero. The graph of $f$ has a horizontal asymptote. Compute the sum of the x-coordinates of the points where the graph of $f$ intersects this horizontal asymptote. If the graph of f does not intersect the asymptote, write $0$. [u]Round 2 [/u] [b]p4.[/b] How many $5$-digit numbers have strictly increasing digits? For example, $23789$ has strictly increasing digits, but $23889$ and $23869$ do not. [b]p5.[/b] Let $$y =\frac{1}{1 +\frac{1}{9 +\frac{1}{5 +\frac{1}{9 +\frac{1}{5 +...}}}}}$$ If $y$ can be represented as $\frac{a\sqrt{b} + c}{d}$ , where $b$ is not divisible by any squares, and the greatest common divisor of $a$ and $d$ is $1$, find the sum $a + b + c + d$. [b]p6.[/b] “Counting” is defined as listing positive integers, each one greater than the previous, up to (and including) an integer $n$. In terms of $n$, write the number of ways to count to $n$. [u]Round 3 [/u] [b]p7.[/b] Suppose $p$, $q$, $2p^2 + q^2$, and $p^2 + q^2$ are all prime numbers. Find the sum of all possible values of $p$. [b]p8.[/b] Let $r(d)$ be a function that reverses the digits of the $2$-digit integer $d$. What is the smallest $2$-digit positive integer $N$ such that for some $2$-digit positive integer $n$ and $2$-digit positive integer $r(n)$, $N$ is divisible by $n$ and $r(n)$, but not by $11$? [b]p9.[/b] What is the period of the function $y = (\sin(3\theta) + 6)^2 - 10(sin(3\theta) + 7) + 13$? [u]Round 4 [/u] [b]p10.[/b] Three numbers $a, b, c$ are given by $a = 2^2 (\sum_{i=0}^2 2^i)$, $b = 2^4(\sum_{i=0}^4 2^i)$, and $c = 2^6(\sum_{i=0}^6 2^i)$ . $u, v, w$ are the sum of the divisors of a, b, c respectively, yet excluding the original number itself. What is the value of $a + b + c -u - v - w$? [b]p11.[/b] Compute $\sqrt{6- \sqrt{11}} - \sqrt{6+ \sqrt{11}}$. [b]p12.[/b] Let $a_0, a_1,..., a_n$ be such that $a_n\ne 0$ and $$(1 + x + x^3)^{341}(1 + 2x + x^2 + 2x^3 + 2x^4 + x^6)^{342} =\sum_{i=0}^n a_ix^i.$$ Find the number of odd numbers in the sequence $a_0, a_1,..., a_n$. PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2781343p24424617]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$k$ is a positive integer, $R_{n}={-k, -(k-1),..., -1, 1,..., k-1, k}$ for $n=2k$ $R_{n}={-k, -(k-1),..., -1, 0, 1,..., k-1, k}$ for $n=2k+1$. A mechanism consists of some marbles and white/red ropes that connects some marble pairs. If each one of the marbles are written on some numbers from $R_{n}$ with the property that any two connected marbles have different numbers on them, we call it [i]nice labeling[/i]. If each one of the marbles are written on some numbers from $R_{n}$ with the properties that any two connected marbles with a white rope have different numbers on them and any two connected marbles with a red rope have two numbers with sum not equal to $0$, we call it [i]precise labeling[/i]. $n\geq{3}$, if every mechanism that is labeled [i]nicely[/i] with $R_{n}$, could be labeled [i]precisely[/i] with $R_{m}$, what is the minimal value of $m$?

Zeus’s lightning is made of a copper rod of length $60$ by bending it $4$ times in alternating directions so that the angle between two adjacent parts is always $60^o$. What is the minimum value of the square of the distance between the two endpoints of the lightning? All five segments of the lightning lie in the same plane. [img]https://cdn.artofproblemsolving.com/attachments/5/1/a18206df4fde561421022c0f2b4332f5ac44a2.png[/img]

Define a [i]domino[/i] to be an ordered pair of [i]distinct[/i] positive integers. A [i]proper sequence[/i] of dominoes is a list of distinct dominoes in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i, j)$ and $(j, i)$ do not [i]both[/i] appear for any $i$ and $j$. Let $D_n$ be the set of all dominoes whose coordinates are no larger than $n$. Find the length of the longest proper sequence of dominoes that can be formed using the dominoes of $D_n$.

Let $m$ and $n$ be natural numbers with $mn$ even. Jetze is going to cover an $m \times n$ board (consisting of $m$ rows and $n$ columns) with dominoes, so that every domino covers exactly two squares, dominos do not protrude or overlap, and all squares are covered by a domino. Merlin then moves all the dominoe color red or blue on the board. Find the smallest non-negative integer $V$ (in terms of $m$ and $n$) so that Merlin can always ensure that in each row the number squares covered by a red domino and the number of squares covered by a blue one dominoes are not more than $V$, no matter how Jetze covers the board.

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]

On a blackboard, there are $11$ positive integers. Show that one can choose some (maybe all) of these numbers and place "$+$" and "$-$" in between such that the result is divisible by $2011$.

The venusian prophet Zabruberson sent to his pupils a $ 10000$-letter word, each letter being $ A$ or $ E$: the [i]Zabrubic word[/i]. Their pupils consider then that for $ 1 \leq k \leq 10000$, each word comprised of $ k$ consecutive letters of the Zabrubic word is a [i]prophetic word[/i] of length $ k$. It is known that there are at most $ 7$ prophetic words of lenght $ 3$. Find the maximum number of prophetic words of length $ 10$.

Let $A= \{a_1,a_2, \cdots, a_n \}$ and $B=\{b_1,b_2 \cdots, b_n \}$ be two positive integer sets and $|A \cap B|=1$. $C= \{ \text{all the 2-element subsets of A} \} \cup \{ \text{all the 2-element subsets of B} \}$. Function $f: A \cup B \to \{ 0, 1, 2, \cdots 2 C_n^2 \}$ is injective. For any $\{x,y\} \in C$, denote $|f(x)-f(y)|$ as the $\textsl{mark}$ of $\{x,y\}$. If $n \geq 6$, prove that at least two elements in $C$ have the same $\textsl{mark}$.

Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d\left(a\right)$: (i) move the last digit of $a$ to the first position to obtain the numb er $b$; (ii) square $b$ to obtain the number $c$; (iii) move the first digit of $c$ to the end to obtain the number $d$. (All the numbers in the problem are considered to be represented in base $10$.) For example, for $a=2003$, we get $b=3200$, $c=10240000$, and $d = 02400001 = 2400001 = d(2003)$.) Find all numbers $a$ for which $d\left( a\right) =a^2$. [i]Proposed by Zoran Sunic, USA[/i]

There is a deck of cards placed at every points $A_1, A_2, \ldots , A_n$ and $O$, where $n \geq 3$. We can do one of the following two operations at each step: $1)$ If there are more than 2 cards at some points $A_i$, we can withdraw three cards from that deck and place one each at $A_{i-1}, A_{i+1}$ and $O$. (Here $A_0=A_n$ and $A_{n+1}=A_1$); $2)$ If there are more than or equal to $n$ cards at point $O$, we can withdraw $n$ cards from that deck and place one each at $A_1, A_2, \ldots , A_n$. Show that if the total number of cards is more than or equal to $n^2+3n+1$, we can make the number of cards at every points more than or equal to $n+1$ after finitely many steps.

Let $n$ be a positive integer. A function $f : \{1, 2, \dots, 2n\} \to \{1, 2, 3, 4, 5\}$ is [i]good[/i] if $f(j+2)$ and $f(j)$ have the same parity for every $j = 1, 2, \dots, 2n-2$. Prove that the number of good functions is a perfect square.

Six decks of $n$ cards, numbered from $1$ to $n$, are given. Melanie arranges each of the decks in some order, such that for any distinct numbers $x$, $y$, and $z$ in $\{1, 2, . . . , n\}$, there is exactly one deck where card $x$ is above card $y$ and card $y$ is above card $z$. Show that there is some $n$ for which Melanie cannot arrange these six decks of cards with this property.

A rook is randomly placed on an otherwise empty $8 \times 8$ chessboard. Owen makes moves with the rook by randomly choosing $1$ of the $14$ possible moves. Find the expected value of the number of moves it takes Owen to move the rook to the top left square. Note that a rook can move any number of squares either in the horizontal or vertical direction each move.

10000 children came to a camp; every of them is friend of exactly eleven other children in the camp (friendship is mutual). Every child wears T-shirt of one of seven rainbow's colours; every two friends' colours are different. Leaders demanded that some children (at least one) wear T-shirts of other colours (from those seven colours). Survey pointed that 100 children didn't want to change their colours [translator's comment: it means that any of these 100 children (and only them) can't change his (her) colour such that still every two friends' colours will be different]. Prove that some of other children can change colours of their T-shirts such that as before every two friends' colours will be different.

The vertices of a regular polygon with $2016$ sides are colored gold or silver. Prove that there are at least $512$ different isosceles triangles whose vertices have the same color.

A cube of side $n$ is cut into $n^3$ unit cubes, and m of these cubes are marked so that the centers of any three marked cubes do not form a right-angled triangle with legs parallel to sides of the cube. Find the maximum possible value of $m$.

Adam has RM2010 in his bank account. He donates RM10 to charity every day. His first donation is on Monday. On what day will he donate his last RM10?

We call $A_{1},A_{2},A_{3}$ [i]mangool[/i] iff there is a permutation $\pi$ that $A_{\pi(2)}\not\subset A_{\pi(1)},A_{\pi(3)}\not\subset A_{\pi(1)}\cup A_{\pi(2)}$. A good family is a family of finite subsets of $\mathbb N$ like $X,A_{1},A_{2},\dots,A_{n}$. To each goo family we correspond a graph with vertices $\{A_{1},A_{2},\dots,A_{n}\}$. Connect $A_{i},A_{j}$ iff $X,A_{i},A_{j}$ are mangool sets. Find all graphs that we can find a good family corresponding to it.