Given a positive integer $n$, show that the set $\{1,2,...,n\}$ can be partitioned into $m$ sets, each with the same sum, if and only if m is a divisor of $\frac{n(n + 1)}{2}$ which does not exceed $\frac{n + 1}{2}$.

Let $n$ be a positive integer. $2n+1$ tokens are in a row, each being black or white. A token is said to be [i]balanced[/i] if the number of white tokens on its left plus the number of black tokens on its right is $n$. Determine whether the number of [i]balanced[/i] tokens is even or odd.

A segment $AB$ of length $100$ is divided into $100$ little segments of length $1$ by $99$ intermediate points. Endpoint $A$ is assigned $0$ and endpoint $B$ is assigned $1$. Gustavo assigns each of the $99$ intermediate points a $0$ or a $1$, at his choice, and then color each segment of length $1$ blue or red, respecting the following rule: [i]The segments that have the same number at their ends are red, and the segments that have different numbers at their ends are blue. [/i] Determine if Gustavo can assign the $0$'s and $1$'s so as to get exactly $30$ blue segments. And $35$ blue segments? (In each case, if the answer is yes, show a distribution of $0$'s and $1$'s, and if the answer is no, explain why).

Given a convex polygon with vertices at lattice points on a plane containing origin $O$. Let $V_1$ be the set of vectors going from $O$ to the vertices of the polygon, and $V_2$ be the set of vectors going from $O$ to the lattice points that lie inside or on the boundary of the polygon (thus, $V_1$ is contained in $V_2$.) Two grasshoppers jump on the whole plane: each jump of the first grasshopper shift its position by a vector from the set $V_1$, and the second by the set $V_2$. Prove that there exists positive integer $c$ that the following statement is true: if both grasshoppers can jump from $O$ to some point $A$ and the second grasshopper needs $n$ jumps to do it, then the first grasshopper can use at most $n+c$ jumps to do so.

Prove that, for any natural number $n$, the graph of any increasing function $f : [0,1] \to [0, 1]$ can be covered by $n$ rectangles each of area whose sides are parallel to the coordinate axes. Assume that a rectangle includes both its interior and boundary points. (a) Assume that $f(x)$ is continuous on $[0,1]$. (b) Do not assume that $f(x)$ is continuous on $[0,1]$. (A Andjans, Riga) PS. (a) for O Level, (b) for A Level

For a positive integer $n$ let $t_n$ be the number of unordered triples of non-empty and pairwise disjoint subsets of a given set with $n$ elements. For example, $t_3 = 1$. Find a closed form formula for $t_n$ and determine the last digit of $t_{2022}$. (I also give here that $t_4 = 10$, for a reader to check his/her understanding of the problem statement.)

Let $ S$ be a sequence that: \[ \left\{ \begin{array}{cc} S_0\equal{}0\hfill\\ S_1\equal{}1\hfill\\ S_n\equal{}S_{n\minus{}1}\plus{}S_{n\minus{}2}\plus{}F_n& (n>1) \end{array} \right.\] such that $ F_n$ is Fibonacci sequence such that $ F_1\equal{}F_2\equal{}1$. Find $ S_n$ in terms of Fibonacci numbers.

Some squares of a $1999\times 1999$ board are occupied with pawns. Find the smallest number of pawns for which it is possible that for each empty square, the total number of pawns in the row or column of that square is at least $1999$.

There are $20$ collinear points, separated by the same distance: $$. \,\,\, . \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\, .$$ Miguel has to paint three or more of these points red, in such a way that the red points are separated by the same distance and it is impossible to paint exactly one more point red without violating the previous condition. Determine in how many ways Miguel can do his homework.

Five persons of different heights stand next to the another on numbered booths to take a picture. From how many ways can be arranged so that people in positions $ 1$ and $3$ are both taller than the person in the position $2$?

A set of $n$ points in Euclidean 3-dimensional space, no four of which are coplanar, is partitioned into two subsets $\mathcal{A}$ and $\mathcal{B}$. An $\mathcal{AB}$-tree is a configuration of $n-1$ segments, each of which has an endpoint in $\mathcal{A}$ and an endpoint in $\mathcal{B}$, and such that no segments form a closed polyline. An $\mathcal{AB}$-tree is transformed into another as follows: choose three distinct segments $A_1B_1$, $B_1A_2$, and $A_2B_2$ in the $\mathcal{AB}$-tree such that $A_1$ is in $\mathcal{A}$ and $|A_1B_1|+|A_2B_2|>|A_1B_2|+|A_2B_1|$, and remove the segment $A_1B_1$ to replace it by the segment $A_1B_2$. Given any $\mathcal{AB}$-tree, prove that every sequence of successive transformations comes to an end (no further transformation is possible) after finitely many steps.

[b]p1.[/b] There are $2024$ apples in a very large basket. First, Julie takes away half of the apples in the basket; then, Diane takes away $202$ apples from the remaining bunch. How many apples remain in the basket? [b]p2.[/b] The set of all permutations (different arrangements) of the letters in ”ABMC” are listed in alphabetical order. The first item on the list is numbered $1$, the second item is numbered $2$, and in general, the kth item on the list is numbered $k$. What number is given to ”ABMC”? [b]p3.[/b] Daniel has a water bottle that is three-quarters full. After drinking $3$ ounces of water, the water bottle is three-fifths full. The density of water is $1$ gram per milliliter, and there are around $28$ grams per ounce. How many milliliters of water could the bottle fit at full capacity? [b]p4.[/b] How many ways can four distinct $2$-by-$1$ rectangles fit on a $2$-by-$4$ board such that each rectangle is fully on the board? [b]p5.[/b] Iris and Ivy start reading a $240$ page textbook with $120$ left-hand pages and $120$ right-hand pages. Iris takes $4$ minutes to read each page, while Ivy takes $5$ minutes to read a left-hand page and $3$ minutes to read a right-hand page. Iris and Ivy move onto the next page only when both sisters have completed reading. If a sister finishes reading a page first, the other sister will start reading three times as fast until she completes the page. How many minutes after they start reading will both sisters finish the textbook? [b]p6.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length $24$. Then, let $M$ be the midpoint of $BC$. Define $P$ to be the set of all points $P$ such that $2PM = BC$. The minimum value of $AP$ can be expressed as $\sqrt{a}- b$, where $a$ and $b$ are positive integers. Find $a + b$. [b]p7.[/b] Jonathan has $10$ songs in his playlist: $4$ rap songs and $6$ pop songs. He will select three unique songs to listen to while he studies. Let $p$ be the probability that at least two songs are rap, and let $q$ be the probability that none of them are rap. Find $\frac{p}{q}$ . [b]p8.[/b] A number $K$ is called $6,8$-similar if $K$ written in base $6$ and $K$ written in base $8$ have the same number of digits. Find the number of $6,8$-similar values between $1$ and $1000$, inclusive. [b]p9.[/b] Quadrilateral $ABCD$ has $\angle ABC = 90^o$, $\angle ADC = 120^o$, $AB = 5$, $BC = 18$, and $CD = 3$. Find $AD^2$. [b]p10.[/b] Bob, Eric, and Raymond are playing a game. Each player rolls a fair $6$-sided die, and whoever has the highest roll wins. If players are tied for the highest roll, the ones that are tied reroll until one wins. At the start, Bob rolls a $4$. The probability that Eric wins the game can be expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$. [b]p11.[/b] Define the following infinite sequence $s$: $$s = \left\{\frac92,\frac{99}{2^2},\frac{999}{2^3} , ... , \overbrace{\frac{999...999}{2^k}}^{k\,\,nines}, ...\right\}$$ The sum of the first $2024$ terms in $s$, denoted $S$, can be expressed as $$S =\frac{5^a - b}{4}+\frac{1}{2^c},$$ where $a, b$, and $c$ are positive integers. Find $a + b + c$. [b]p12.[/b] Andy is adding numbers in base $5$. However, he accidentally forgets to write the units digit of each number. If he writes all the consecutive integers starting at $0$ and ending at $50$ (base $10$) and adds them together, what is the difference between Andy’s sum and the correct sum? (Express your answer in base-$10$.) [b]p13.[/b] Let $n$ be the positive real number such that the system of equations $$y =\frac{1}{\sqrt{2024 - x^2}}$$ $$y =\sqrt{x^2 - n}$$ has exactly two real solutions for $(x, y)$: $(a, b)$ and $(-a, b)$. Then, $|a|$ can be expressed as $j\sqrt{k}$, where $j$ and $k$ are integers such that $k$ is not divisible by any perfect square other than $1$. Find $j · k$. [b]p14.[/b] Nakio is playing a game with three fair $4$-sided dice. But being the cheater he is, he has secretly replaced one of the three die with his own $4$-sided die, such that there is a $1/2$ chance of rolling a $4$, and a $1/6$ chance to roll each number from $1$ to $3$. To play, a random die is chosen with equal probability and rolled. If Nakio guesses the number that is on the die, he wins. Unfortunately for him, Nakio’s friends have an anti-cheating mechanism in place: when the die is picked, they will roll it three times. If each roll lands on the same number, that die is thrown out and one of the two unused dice is chosen instead with equal probability. If Nakio always guesses $4$, the probability that he wins the game can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime. Find $m + n$. [b]p15.[/b] A particle starts in the center of a $2$m-by-$2$m square. It moves in a random direction such that the angle between its direction and a side of the square is a multiple of $30^o$. It travels in that direction at $1$ m/s, bouncing off of the walls of the square. After a minute, the position of the particle is recorded. The expected distance from this point to the start point can be written as $$\frac{1}{a}\left(b - c\sqrt{d}\right),$$ where $a$ and $b$ are relatively prime, and d is not divisible by any perfect square. Find $a + b + c + d$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

At a party every woman dances with at least one man, and no man dances with every woman. Show that there are men M and M' and women W and W' such that M dances with W, M' dances with W', but M does not dance with W', and M' does not dance with W.

We have a closed path on a vertices of a $ n$×$ n$ square which pass from each vertice exactly once . prove that we have two adjacent vertices such that if we cut the path from these points then length of each pieces is not less than quarter of total path .

On sport games there was 1991 participant from which every participant knows at least n other participants(friendship is mutual). Determine the lowest possible n for which we can be sure that there are 6 participants between which any two participants know each other.

Each positive integer should be colored red or green in such a way that the following two conditions are met: - Let $n$ be any red number. The sum of any $n$ (not necessarily different) red numbers is red. - Let $m$ be any green number. The sum of any $m$ (not necessarily different) green numbers is green. Determine all such colorings.

Let $n$ be a positive integer and define $A_n = \{1, 2, ..., n\}$. How many functions $f : A_n \to A_n$ are there such that for all $x, y \in A_n$, if $x < y$ then $f(x) \ge f(y)$?

Some paper squares of $k$ distinct colors are placed on a rectangular table, with sides parallel to the sides of the table. Suppose that for any $k$ squares of distinct colors, some two of them can be nailed on the table with only one nail. Prove that there is a color such that all squares of that color can be nailed with $2k-2$ nails.

Dene the set $F$ as the following: $F = \{(a_1,a_2,... , a_{2006}) : \forall i = 1, 2,..., 2006, a_i \in \{-1,1\}\}$ Prove that there exists a subset of $F$, called $S$ which satises the following: $|S| = 2006$ and for all $(a_1,a_2,... , a_{2006})\in F$ there exists $(b_1,b_2,... , b_{2006}) \in S$, such that $\Sigma_{i=1} ^{2006}a_ib_i = 0$.

A box whose shape is a parallelepiped can be completely filled with cubes of side $1.$ If we put in it the maximum possible number of cubes, each of volume $2$, with the sides parallel to those of the box, then exactly $40$ percent of the volume of the box is occupied. Determine the possible dimensions of the box.

In a $3 \times 25$ board, $1 \times 3$ pieces are placed (vertically or horizontally) so that they occupy entirely $3$ boxes on the board and do not have a common point. What is the maximum number of pieces that can be placed, and for that number, how many configurations are there? [hide=original formulation] Num tabuleiro 3 × 25 s˜ao colocadas pe¸cas 1 × 3 (na vertical ou na horizontal) de modo que ocupem inteiramente 3 casas do tabuleiro e n˜ao se toquem em nenhum ponto. Qual ´e o n´umero m´aximo de pe¸cas que podem ser colocadas, e para esse n´umero, quantas configura¸c˜oes existem? [url=https://www.obm.org.br/content/uploads/2018/09/Provas_OMCPLP_2018.pdf]source[/url][/hide]

In a one-round football tournament, three points were awarded for a victory. All the teams scored different numbers of points. If not three, but two points were given for a victory, then all teams would also have a different number of points, but each team's place would be different. What is the smallest number of teams for which this is possible? [i]Proposed by A. Zaslavsky[/i]

Given an integer $ n\ge 2$ and a reular 2n-gon. Color all verices of the 2n-gon with n colors such that: [b](i)[/b] Each vertice is colored by exactly one color. [b](ii)[/b] Two vertices don't have the same color. Two ways of coloring, satisfying the conditions above, are called equilavent if one obtained from the other by a rotation whose center is the center of polygon. Find the total number of mutually non-equivalent ways of coloring. [i]Alternative statement:[/i] In how many ways we can color vertices of an regular 2n-polygon using n different colors such that two adjent vertices are colored by different colors. Two colorings which can be received from each other by rotation are considered as the same.

[b]p1.[/b] The sequence $\{x_n\}$ is defined by $$x_{n+1} = \begin{cases} 2x_n - 1, \,\, if \,\, \frac12 \le x_n < 1 \\ 2x_n, \,\, if \,\, 0 \le x_n < \frac12 \end{cases}$$ where $0 \le x_0 < 1$ and $x_7 = x_0$. Find the number of sequences satisfying these conditions. [b]p2.[/b] Let $M = \{1, . . . , 2022\}$. For any nonempty set $X \subseteq M$, let $a_X$ be the sum of the maximum and the minimum number of $X$. Find the average value of $a_X$ across all nonempty subsets $X$ of $M$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.