Two grasshoppers sit at opposite ends of the interval $[0, 1]$. A finite number of points (greater than zero) in the interval are marked. A move is for a grasshopper to select a marked point and jump over it to the equidistant point the other side. This point must lie in the interval for the move to be allowed, but it does not have to be marked. What is the smallest $n$ such that if each grasshopper makes $n$ moves or less, then they end up with no marked points between them?

A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, $A_0,$ and the hunter's starting point, $B_0$ are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}.$ In the $n^{\text{th}}$ round of the game, three things occur in order: [list=i] [*]The rabbit moves invisibly to a point $A_n$ such that the distance between $A_{n-1}$ and $A_n$ is exactly $1.$ [*]A tracking device reports a point $P_n$ to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between $P_n$ and $A_n$ is at most $1.$ [*]The hunter moves visibly to a point $B_n$ such that the distance between $B_{n-1}$ and $B_n$ is exactly $1.$ [/list] Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after $10^9$ rounds, she can ensure that the distance between her and the rabbit is at most $100?$ [i]Proposed by Gerhard Woeginger, Austria[/i]

Keith has $10$ coins labeled $1$ through $10$, where the $i$th coin has weight $2^i$. The coins are all fair, so the probability of flipping heads on any of the coins is $\tfrac{1}{2}$. After flipping all of the coins, Keith takes all of the coins which land heads and measures their total weight, $W$. If the probability that $137\le W\le 1061$ is $\tfrac{m}{n}$ for coprime positive integers $m,n$, determine $m+n$.

The number $328$ is written on the board. Two players alternate writing positive divisors of $328$ on the board, subject to the following rules: $\bullet$ No divisor of a previously written number may be written. $\bullet$ The player who writes 328 loses. Who has a winning strategy, the first player or the second player?

Show that a unit square can be covered with three equal disks with radius less than $\frac{1}{\sqrt{2}}$. What is the smallest possible radius?

There are 100 apples on the table with total weight of 10 kg. Each apple weighs no less than 25 grams. The apples need to be cut for 100 children so that each of the children gets 100 grams. Prove that you can do it in such a way that each piece weighs no less than 25 grams.

Let $n$ be an integer bigger than $0$. Let $\mathbb{A}= ( a_1,a_2,...,a_n )$ be a set of real numbers. Find the number of functions $f:A \rightarrow A$ such that $f(f(x))-f(f(y)) \ge x-y$ for any $x,y \in \mathbb{A}$, with $x>y$.

Find the maximum number of queens you could put on $2017 \times 2017$ chess table such that each queen attacks at most $1$ other queen.

Let $n \geqslant 2$ be an integer. There are $n$ finite sets ${A_1},{A_2},\ldots,{A_n}$ which satisfy the condition \[\left| {{A_i}\Delta {A_j}} \right| = \left| {i - j} \right| \quad \forall i,j \in \left\{ {1,2,...,n} \right\}.\] Find the minimum of $\sum\limits_{i = 1}^n {\left| {{A_i}} \right|} $.

[u]Round 1[/u] [b]p1.[/b] In order to make good salad dressing, Bob needs a $0.9\%$ salt solution. If soy sauce is $15\%$ salt, how much water, in mL, does Bob need to add to $3$ mL of pure soy sauce in order to have a good salad dressing? [b]p2.[/b] Alex the Geologist is buying a canteen before he ventures into the desert. The original cost of a canteen is $\$20$, but Alex has two coupons. One coupon is $\$3$ off and the other is $10\%$ off the entire remaining cost. Alex can use the coupons in any order. What is the least amount of money he could pay for the canteen? [b]p3.[/b] Steve and Yooni have six distinct teddy bears to split between them, including exactly $1$ blue teddy bear and $1$ green teddy bear. How many ways are there for the two to divide the teddy bears, if Steve gets the blue teddy bear and Yooni gets the green teddy bear? (The two do not necessarily have to get the same number of teddy bears, but each teddy bear must go to a person.) [u]Round 2[/u] [b]p4.[/b] In the currency of Mathamania, $5$ wampas are equal to $3$ kabobs and $10$ kabobs are equal to $2$ jambas. How many jambas are equal to twenty-five wampas? [b]p5.[/b] A sphere has a volume of $81\pi$. A new sphere with the same center is constructed with a radius that is $\frac13$ the radius of the original sphere. Find the volume, in terms of $\pi$, of the region between the two spheres. [b]p6.[/b] A frog is located at the origin. It makes four hops, each of which moves it either $1$ unit to the right or $1$ unit to the left. If it also ends at the origin, how many $4$-hop paths can it take? [u]Round 3[/u] [b]p7.[/b] Nick multiplies two consecutive positive integers to get $4^5 - 2^5$ . What is the smaller of the two numbers? [b]p8.[/b] In rectangle $ABCD$, $E$ is a point on segment $CD$ such that $\angle EBC = 30^o$ and $\angle AEB = 80^o$. Find $\angle EAB$, in degrees. [b]p9.[/b] Mary’s secret garden contains clones of Homer Simpson and WALL-E. A WALL-E clone has $4$ legs. Meanwhile, Homer Simpson clones are human and therefore have $2$ legs each. A Homer Simpson clone always has $5$ donuts, while a WALL-E clone has $2$. In Mary’s secret garden, there are $184$ donuts and $128$ legs. How many WALL-E clones are there? [u]Round 4[/u] [b]p10.[/b] Including Richie, there are $6$ students in a math club. Each day, Richie hangs out with a different group of club mates, each of whom gives him a dollar when he hangs out with them. How many dollars will Richie have by the time he has hung out with every possible group of club mates? [b]p11.[/b] There are seven boxes in a line: three empty, three holding $\$10$ each, and one holding the jackpot of $\$1, 000, 000$. From the left to the right, the boxes are numbered $1, 2, 3, 4, 5, 6$ and $7$, in that order. You are told the following: $\bullet$ No two adjacent boxes hold the same contents. $\bullet$ Box $4$ is empty. $\bullet$ There is one more $\$10$ prize to the right of the jackpot than there is to the left. Which box holds the jackpot? [b]p12.[/b] Let $a$ and $b$ be real numbers such that $a + b = 8$. Let $c$ be the minimum possible value of $x^2 + ax + b$ over all real numbers $x$. Find the maximum possible value of $c$ over all such $a$ and $b$. [u]Round 5[/u] [b]p13.[/b] Let $ABCD$ be a rectangle with $AB = 10$ and $BC = 12$. Let M be the midpoint of $CD$, and $P$ be a point on $BM$ such that $BP = BC$. Find the area of $ABPD$. [b]p14.[/b] The number $19$ has the following properties: $\bullet$ It is a $2$-digit positive integer. $\bullet$ It is the two leading digits of a $4$-digit perfect square, because $1936 = 44^2$. How many numbers, including $19$, satisfy these two conditions? [b]p15.[/b] In a $3 \times 3$ grid, each unit square is colored either black or white. A coloring is considered “nice” if there is at most one white square in each row or column. What is the total number of nice colorings? Rotations and reflections of a coloring are considered distinct. (For example, in the three squares shown below, only the rightmost one has a nice coloring. [img]https://cdn.artofproblemsolving.com/attachments/e/4/e6932c822bec77aa0b07c98d1789e58416b912.png[/img] PS. You should use hide for answers. Rest rounds have been posted [url=https://artofproblemsolving.com/community/c4h2786958p24498425]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A bank has a set S of codes formed only with 0 and 1,each one with length n.Two codes are 'friends' if they are different on only one position.We know that each code has exactly k 'friends'.Prove that: 1)S has an even number of elements 2)S contains at least $2^k$ codes

What is the maximum number of possible change of directions in a path traveling on the edges of a rectangular array of $2004 \times 2004$, if the path does not cross the same place twice?.

Is it possible to cut a square with side $\sqrt{2015}$ into no more than five pieces so that these pieces can be rearranged into a rectangle with sides of integer length? (The cuts should be made using straight lines, and flipping of the pieces is disallowed.)

Seventeen football fans were planning to go to Korea to watch the World Cup football match. They selected 17 matches. The conditions of the admission tickets they booked were such that - One person should book at most one admission ticket for one match; - At most one match was same in the tickets booked by every two persons; - There was one person who booked six tickets. How many tickets did those football fans book at most?

We denote with $p_n(k)$ the number of permutations of the numbers $1,2,...,n$ that have exactly $k$ fixed points. a) Prove that $\sum_{k=0}^n kp_n (k)=n!$. b) If $s$ is an arbitrary natural number, then: $\sum_{k=0}^n k^s p_n (k)=n!\sum_{i=1}^m R(s,i)$, where with $R(s,i)$ we denote the number of partitions of the set $\{1,2,...,s\}$ into $i$ non-empty non-intersecting subsets and $m=min(s,n)$.

At the beginning of the academic year, the Olympic Center must accept a certain number of talented students for the 2024 different classes it offers. Although the admitted students are given freedom of choice in classes, there are certain rules. So, any student must take at least one class and cannot take all the classes. At the same time, there cannot be a common class that all students take, and any class must be taken by at least one student. As a final addition to the center's rules, for any student and any class that this student did not enroll in (call this type of class A), the number of students in each A must be greater than the number of classes this student enrolled. At least how many students must the center accept for these rules to be valid?

Let $A$ be a set of numbers chosen from $1,2,..., 2015$ with the property that any two distinct numbers, say $x$ and $y$, in $A$ determine a unique isosceles triangle (which is non equilateral) whose sides are of length $x$ or $y$. What is the largest possible size of $A$?

When any $11$ integers are given, prove that you can always choose $6$ integers among them so that the sum of the chosen numbers is a multiple of $6$. The $11$ integers aren't necessarily different.

All the faces of a convex polyhedron are the triangles. Prove that it is possible to paint all its edges in red and blue colour in such a way, that it is possible to move from the arbitrary vertex to every vertex along the blue edges only and along the red edges only.

On a $2023 \times 2023$ board, there are beetles on some of the cells, with at most one beetle per cell. After one minute, each beetle moves a cell to the right or to the left or to the top or to the bottom. After each further minute, the beetles continue to move to adjacent fields, but they always make a $90^\circ$ turn, i.e. when a beetle just moved to the right or to the left, it now moves to the top or to the bottom, and vice versa. What is the minimal number of beetles on the board such that no matter where they start and how they move (according to the rules), at some point two beetles will end up in the cell?

Let $ l,d,k$ be natural numbers. We want to prove that for large numbers $ n$, for each $ k$-coloring of the $ n$-dimensional cube with side length $ l$, there is a $ d$-dimensional subspace that all of its vertices have the same color. Let $ H(l,d,k)$ be the least number such that for $ n\geq H(l,d,k)$ the previus statement holds. a) Prove that: \[ H(l,d \plus{} 1,k)\leq H(l,1,k) \plus{} H(l,d,k^l)^{H(l,1,k)} \] b) Prove that \[ H(l \plus{} 1,1,k \plus{} 1)\leq H(l,1 \plus{} H(l \plus{} 1,1,k),k \plus{} 1) \] c) Prove the statement of problem. d) Prove Van der Waerden's Theorem.

There are five guys named Alan, Bob, Casey, Dan, and Eric. Each one either always tells the truth or always lies. You overhear the following discussion between them: Alan: [i]"All of us are truth-tellers."[/i] Bob: [i]"No, only Alan and I are truth-tellers."[/i] Casey: [i]"You are both liars."[/i] Dan:[i] "If Casey is a truth-teller, then Eric is too."[/i] Eric: [i]"An odd number of us are liars."[/i] Who are the liars?

On the board are written the natural numbers from $1$ to $2001$ inclusive. You have to delete some numbers so that among those that remain undeleted it is impossible to choose two different numbers such that the result of their multiplication is equal to one of the numbers that remain undeleted. What is the minimum number of numbers that must be deleted? For that amount, present an example showing which numbers are erased. Justify why, if fewer numbers are deleted, the desired property is not obtained.

There are $2017$ lines in the plane such that no three of them go through the same point. Turbo the snail sits on a point on exactly one of the lines and starts sliding along the lines in the following fashion: she moves on a given line until she reaches an intersection of two lines. At the intersection, she follows her journey on the other line turning left or right, alternating her choice at each intersection point she reaches. She can only change direction at an intersection point. Can there exist a line segment through which she passes in both directions during her journey?

Ann and Bob play a game on the edges of an infinite square grid, playing in turns. Ann plays the first move. A move consists of orienting any edge that has not yet been given an orientation. Bob wins if at any point a cycle has been created. Does Bob have a winning strategy?