Consider an alphabetized list of all the arrangements of the letters in the word BETWEEN. Then BEEENTW would be in position $1$ in the list, BEEENWT would be in position $2$ in the list, and so forth. Find the position that BETWEEN would be in the list.

Natural numbers from $1$ to $99$ (not necessarily distinct) are written on $99$ cards. It is given that the sum of the numbers on any subset of cards (including the set of all cards) is not divisible by $100$. Show that all the cards contain the same number.

Alice is at coordinate point $(0, 0)$ and wants to go to point $(11, 6)$. Similarly, Bob is at coordinate point $(5, 6)$ and wants to go to point $(16, 0)$. Both of them choose a lattice path from their current position to their target position at random (such that each lattice path has an equal probability of being chosen), where a lattice path is defined to be a path composed of unit segments with orthogonal direction (parallel to x-axis or y-axis) and of minimal length. (For instance, there are six lattice paths from $(0, 0)$ to $(2, 2)$.) If they walk with the same speed, find the probability that they meet.

Initially, all the squares of an $8\times 8$ grid are white. You start by choosing one of the squares and coloring it gray. After that, you may color additional squares gray one at a time, but you may only color a square gray if it has exactly $1$ or $3$ gray neighbors at that moment (where a neighbor is a square sharing an edge). For example, the configuration below (of a smaller $3\times 4$ grid) shows a situation where six squares have been colored gray so far. The squares that can be colored at the next step are marked with a dot. Is it possible to color all the squares gray? Justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/1/c/d50ab269f481e4e516dace06a991e6b37f2a85.png[/img]

Consider a finite sequence $a_1, a_2,...,a_n$ whose terms are natural numbers at most equal to $n$. Determine the maximum number of terms of such a sequence, if you know that every two of its neighboring terms are different and at the same time there is no quartet of terms in it such that $a_p = a_r \ne a_q = a_s$ for $p < q < r < s$.

The enemy ship has landed on a $9\times 9$ board that covers exactly $5$ squares of the board, like this: [img]https://cdn.artofproblemsolving.com/attachments/2/4/ae5aa95f5bb5e113fd5e25931a2bf8eb872dbe.png[/img] The ship is invisible. Each defensive missile covers exactly one square, and destroys the ship if it hits one of the $5$ squares that it occupies. Determine the minimum number of missiles needed to destroy the enemy ship with certainty .

Given right hexagon. The lines parallel to all the sides are drawn from all the vertices and midpoints of the sides (consider only the interior, with respect to the hexagon, parts of those lines). Thus the hexagon is divided onto $24$ triangles, and the figure has $19$ nodes. $19$ different numbers are written in those nodes. Prove that at least $7$ of $24$ triangles have the property: the numbers in its vertices increase (from the least to the greatest) counterclockwise.

Each point of the plane is coloured in one of two colours. Given an odd integer number $n \geq 3$, prove that there exist (at least) two similar triangles whose similitude ratio is $n$, each of which has a monochromatic vertex-set. [i]Vasile Pop[/i]

Let $n \ge 3$ be a positive integer. For every set $S$ with $n$ distinct positive integers, prove that there exists a bijection $f: \{1,2, \cdots n\} \rightarrow S$ which satisfies the following condition. For all $1 \le i < j < k \le n$, $f(j)^2 \neq f(i) \cdot f(k)$.

Given a set $S$ of $2024$ natural numbers satisfying the following condition: if you select any $10$ (different) numbers from $S$, then you can select another number from $S$ so that the sum of all $11$ selected numbers is divisible by $10$. Prove that one of the numbers can be thrown out of $S$ so that the resulting set $S'$ of $2023$ numbers satisfies the condition: if you choose any $9$ (different) numbers from $S'$, then you can choose another number from $S'$ so that the sum of all $10$ selected numbers is divisible by $10$. [i]K.A.Sukhov[/i]

Given a set $ M$ of points $ (x,y)$ with integral coordinates satisfying $ x^2 + y^2\leq 10^{10}$. Two players play a game. One of them marks a point on his first move. After this, on each move the moving player marks a point, which is not yet marked and joins it with the previous marked point. Players are not allowed to mark a point symmetrical to the one just chosen. So, they draw a broken line. The requirement is that lengths of edges of this broken line must strictly increase. The player, which can not make a move, loses. Who have a winning strategy?

Anett is drawing $X$-es on a $5 \times 5$ grid. For each newly drawn $X$ she gets points in the following way: She checks how many $X$-es there are in the same row (including the new one) that can be reached from the newly drawn $X$ with horizontal steps, moving only on fields that were previously marked with $X$-es. For the vertical $X$-es, she gets points the same way. a) What is the maximum number of points that she can get with drawing $25$ $X$-es? b) What is the minimum number of points that she can get with drawing $25$ $X$-es? For example, if Anett put the $X$ on the field that is marked with the circle, she would get $3$ points for the horizontal fields and $1$ point for the vertical ones. Thus, she would get $4$ points in total. [img]https://cdn.artofproblemsolving.com/attachments/5/c/662c2e4c3dea8d78e2f6397489b277daee0ad0.png[/img]

A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn. How many ways are there to put the cards in the three boxes so that the trick works?

A racing tournament has $12$ stages and $n$ participants. After each stage, all participants, depending on their place $k$, receive points $a_k$ (numbers $a_k$ are natural numbers and $a_1 > a_2 >... > a_n$). At what smallest $n$ can the organizer of the tournament choose numbers $a_1$, $...$ , $a_n$ so that after the penultimate stage for any possible distribution of places at least two participants had a chance to take first place?

Several farmers have 128 sheep. If one of them has at least half of all sheep, the rest conspire and dispossess him: everyone takes as many sheep as he already has : If two people have 64 sheep, then one of them is dispossessed. There were 7 dispossessions. Prove that all the sheep were gathered from one peasant.

Consider $9$ points in the interior of a square of side $1$. Prove that there are three of them that form a triangle with an area less than or equal to $\frac18$ .

[b]p1.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the square $ABCD$ is $14$ cm. [img]https://cdn.artofproblemsolving.com/attachments/1/1/0f80fc5f0505fa9752b5c9e1c646c49091b4ca.png[/img] [b]p2.[/b] If $a, b$, and $c$ are numbers so that $a + b + c = 0$ and $a^2 + b^2 + c^2 = 1$. Compute $a^4 + b^4 + c^4$. [b]p3.[/b] A given fraction $\frac{a}{b}$ ($a, b$ are positive integers, $a \ne b$) is transformed by the following rule: first, $1$ is added to both the numerator and the denominator, and then the numerator and the denominator of the new fraction are each divided by their greatest common divisor (in other words, the new fraction is put in simplest form). Then the same transformation is applied again and again. Show that after some number of steps the denominator and the numerator differ exactly by $1$. [b]p4.[/b] A goat uses horns to make the holes in a new $30\times 60$ cm large towel. Each time it makes two new holes. Show that after the goat repeats this $61$ times the towel will have at least two holes whose distance apart is less than $6$ cm. [b]p5.[/b] You are given $555$ weights weighing $1$ g, $2$ g, $3$ g, $...$ , $555$ g. Divide these weights into three groups whose total weights are equal. [b]p6.[/b] Draw on the regular $8\times 8$ chessboard a circle of the maximal possible radius that intersects only black squares (and does not cross white squares). Explain why no larger circle can satisfy the condition. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The integers $ 1,2,\dots,20$ are written on the blackboard. Consider the following operation as one step: [i]choose two integers $ a$ and $ b$ such that $ a\minus{}b \ge 2$ and replace them with $ a\minus{}1$ and $ b\plus{}1$[/i]. Please, determine the maximum number of steps that can be done. [i]Yudi Satria, Jakarta[/i]

$T$ is a set of all the positive integers of the form $2^k 3^l$, where $k, l$ are some non-negetive integers. Show that there exists $1998$ different elements of $T$ that satisfy the following condition. [b]Condition[/b] The sum of the $1998$ elements is again an element of $T$.

Determine the number of 2010 letter words, formed by the letters $a$, $b$, and $c$, such that at least one of the three letters is odd number of times in the word.

Given a finite string $S$ of symbols $X$ and $O$, we write $@(S)$ for the number of $X$'s in $S$ minus the number of $O$'s. (For example, $@(XOOXOOX) =-1$.) We call a string $S$ [b]balanced[/b] if every substring $T$ of (consecutive symbols) $S$ has the property $-2 \leq @(T) \leq 2$. (Thus $XOOXOOX$ is not balanced since it contains the sub-string $OOXOO$ whose $@$-value is $-3$.) Find, with proof, the number of balanced strings of length $n$.

Oliver is at a carnival. He is offered to play a game where he rolls a fair dice and receives $\$1$ if his roll is a $1$ or $2$, receives $\$2$ if his roll is a $3$ or $4$, and receives $\$3$ if his roll is a $5$ or $6$. Oliver plays the game repeatedly until he has received a total of at least $\$2$. What is the probability that he ends with $\$3$?

A $50$-card deck consists of $4$ cards labeled " i" for $i = 1, 2,..., 12$ and $2$ cards labeled "$13$". If Bob randomly chooses $2$ cards from the deck without replacement, what is the probability that his $2$ cards have the same label?

Sofia has $5$ pieces of paper on a table. He takes some of the pieces, cuts each one into $5$ little pieces, and puts them back on the table. She repeats this procedure several times until she gets tired. Could Sofia end up with $2010$ pieces on the table?

A floor is tiled with equilateral triangles of side length $1$, as shown. If you drop a needle of length $2$ somewhere on the floor , what is the largest number of triangles it could end up intersecting? (Only count the triangles whose interiors are met by the needle --- touching along edges or at corners doesn't qualify.) [img]https://cdn.artofproblemsolving.com/attachments/5/6/e7555c22ffe890b46a3ebdbda2169d23e43700.png[/img]