A candy store has $100$ pieces of candy to give away. When you get to the store, there are five people in front of you, numbered from $1$ to $5$. The $i$th person in line considers the set of positive integers congruent to $i$ modulo $5$ which are at most the number of pieces of candy remaining. If this set is empty, then they take no candy. Otherwise they pick an element of this set and take that many pieces of candy. For example, the first person in line will pick an integer from the set $\{1,6,\dots,96\}$ and take that many pieces of candy. How many ways can the first five people take their share of candy so that after they are done there are at least $35$ pieces of candy remaining?

Numbers 1 and −1 are written in the cells of a board 2000×2000. It is known that the sum of all the numbers in the board is positive. Show that one can select 1000 rows and 1000 columns such that the sum of numbers written in their intersection cells is at least 1000. [i]D. Karpov[/i]

A team of geologists on a field expedition have taken with them $80$ tin cans of provisions. The $80$ cans have different weights, which are known (there is a list). After a while the names of the contents of the cans have become illegible. The cook knows what is in each can and claims that he can prove it without opening any can and only using the list and a balance which indicates the difference of weight of the objects placed on its two pans. Show that in order to do so, (a) four weight measurements will be enough, (b) three will not (AK Tolpygo)

Given any collection of $2010$ nondegenerate triangles, their sides are painted so that each triangle has one red side, one blue side, and one white side. For each color, arrange the side lengths in order: [list][*]let $b_1\le b_2\le\cdots\le b_{2011}$ denote the lengths of the blue sides; [*]let $r_1\le r_2\le\cdots\le r_{2011}$ denote the lengths of the red sides; and [*]let $w_1\le w_2\le\cdots\le w_{2011}$ denote the lengths of the white sides.[/list] Find the largest integer $k$ for which there necessarily exists at least $k$ indices $j$ such that $b_j$, $r_j$, $w_j$ are the side lengths of a nondegenerate triangle.

[b]7.1 . [/b] The area of the quadrilateral is $3$ cm$^2$ , and the lengths of its diagonals are $6$ cm and $2$ cm. Find the angle between the diagonals. [b]7.2[/b] Prove that the number $1 + 2^{3456789}$ is composite. [b]7.3[/b] $20$ people took part in the chess tournament. The participant who took clear (undivided) $19$th place scored $9.5$ points. How could they distribute points among other participants? [b]7.4[/b] The sum of the distances between the midpoints of opposite sides of a quadrilateral is equal to its semi-perimeter. Prove that this quadrilateral is a parallelogram. [b]7.5[/b] $40$ people travel on a bus without a conductor passengers carrying only coins in denominations of $10$, $15$ and $20$ kopecks. Total passengers have $ 49$ coins. Prove that passengers will not be able to pay the required amount of money to the ticket office and pay each other correctly. (Cost of a bus ticket in 1963 was 5 kopecks.) [b]7.6[/b] Some natural number $a$ is divided with a remainder by all natural numbers less than $a$. The sum of all the different (!) remainders turned out to be equal to $a$. Find $a$. [b]7.7[/b] Two squares were cut out of a chessboard. In what case is it possible and in what case not to cover the remaining squares of the board with dominoes (i.e., figures of the form $2\times 1$) without overlapping? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].

Given right hexagon $H$ with side length $1$. On the sides of $H$ points $A_1$,$A_2$,$\ldots$,$A_k$ such that at least one of them is the midpoint of some side and for every $1 \leq i \leq k$ lines $A_{i-1}A_i$ and $A_iA_{i+1}$ form equal angles with the side, that contains the point $A_i$ (let $A_0=A_k$ and $A_{k+1}=A_1$. It is known that the length of broken line $A_1A_2\ldots A_kA_1$ is a positive integer Prove that $n$ is divisible by $3$ [i]M. Zorka[/i]

Take $r$ such that $1\le r\le n$, and consider all subsets of $r$ elements of the set $\{1,2,\ldots,n\}$. Each subset has a smallest element. Let $F(n,r)$ be the arithmetic mean of these smallest elements. Prove that: \[ F(n,r)={n+1\over r+1}. \]

Numbers $\frac{49}{1}, \frac{49}{2}, ... , \frac{49}{97}$ are writen on a blackboard. Each time, we can replace two numbers (like $a, b$) with $2ab-a-b+1$. After $96$ times doing that prenominate action, one number will be left on the board. Find all the possible values fot that number.

Can ten distinct numbers $a_1, a_2, b_1, b_2, b_3, c_1, c_2, d_1, d_2, d_3$ be chosen from $\{0, 1, 2, \ldots, 14\}$, so that the $14$ differences $|a_1 - b_1|$, $|a_1 - b_2|$, $|a_1 - b_3|$, $|a_2 - b_1|$, $|a_2 - b_2|$, $|a_2 - b_3|$, $|c_1 - d_1|$, $|c_1 - d_2|$, $|c_1 - d_3|$, $|c_2 - d_1|$, $|c_2 - d_2|$, $|c_2 - d_3|$, $|a_1 - c_1|$, and $|a_2 - c_2|$ are all distinct?

Given an endless supply of white, blue and red cubes. In a circle arrange any $N$ of them. The robot, standing in any place of the circle, goes clockwise and, until one cube remains, constantly repeats this operation: destroys the two closest cubes in front of him and puts a new one behind him a cube of the same color if the destroyed ones are the same, and the third color if the destroyed two are different colors. We will call the arrangement of the cubes [i]good [/i] if the color of the cube remaining at the very end does not depends on where the robot started. We call $N$ [i]successful [/i] if for any choice of $N$ cubes all their arrangements are good. Find all successful $N$. I. Bogdanov

In an exhibition there are $100$ paintings each of which is made with exactly $k$ colors. Find the minimum possible value of $k$ if any $20$ paintings have a common color but there is no color that is used in all paintings.

In the garden of Wonderland, there are $2016$ apples, $2017$ bananas and $2018$ oranges.Two monkeys Adu and Bakar play the following game: alternatively each of them takes and eats one fruit of any kind except for the one that he took in previous turn (in the first turn, each of them can take a fruit of any kind). Who can not take a fruit is the loser. Which monkey has the winning strategy if Adu plays first?

Two players $A$ and $B$ take turns taking stones from a pile of $N$ stones. They play in the order $A$, $B$, $A$, $B$, $A$, $....$, $A$ starts the game and the one who takes out the last stone loses.$ B$ can serve on each play $1$, $2$ or 3 stones, while$ A$ can draw $2, 3, 4$ stones or $1$ stone in each turn f it is the last one in the pile. Determine for what values of $N$ does $A$ have a winning strategy, and for what values the winning strategy is $B$'s.

A [i]derangement [/i] of the letters $ABCDEF$ is a permutation of these letters so that no letter ends up in the position it began such as $BDECFA$. An [i]inversion [/i] in a permutation is a pair of letters $xy$ where $x$ appears before $y$ in the original order of the letters, but $y$ appears before $x$ in the permutation. For example, the derangement $BDECFA$ has seven inversions: $AB, AC, AD, AE, AF, CD$, and $CE$. Find the total number of inversions that appear in all the derangements of $ABCDEF$.

The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer ${n\ge 4}$, find the least number of edges of a graph that can be obtained by repeated applications of this operation from the complete graph on $n$ vertices (where each pair of vertices are joined by an edge). [i]Proposed by Norman Do, Australia[/i]

Prove that the set of positive integers $\mathbb Z^+$ can be represented as a sum of five pairwise distinct subsets with the following property: each $5$-tuple of numbers of form $(n,2n,3n,4n,5n)$, where $n\in\mathbb Z^+$, contains exactly one number from each of these five subsets.

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

There are $2022$ equally spaced points on a circular track $\gamma$ of circumference $2022$. The points are labeled $A_1, A_2, \ldots, A_{2022}$ in some order, each label used once. Initially, Bunbun the Bunny begins at $A_1$. She hops along $\gamma$ from $A_1$ to $A_2$, then from $A_2$ to $A_3$, until she reaches $A_{2022}$, after which she hops back to $A_1$. When hopping from $P$ to $Q$, she always hops along the shorter of the two arcs $\widehat{PQ}$ of $\gamma$; if $\overline{PQ}$ is a diameter of $\gamma$, she moves along either semicircle. Determine the maximal possible sum of the lengths of the $2022$ arcs which Bunbun traveled, over all possible labellings of the $2022$ points. [i]Kevin Cong[/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?

Peter has three accounts in a bank, each with an integral number of dollars. He is only allowed to transfer money from one account to another so that the amount of money in the latter is doubled. Prove that Peter can always transfer all his money into two accounts. Can Peter always transfer all his money into one account?

p1. Given $f(x)=\frac{1+x}{1-x}$ , for $x \ne 1$ . Defined $p @ q = \frac{p+q}{1+pq}$ for all positive rational numbers $p$ and $q$. Note the sequence with $a_1,a_2,a_3,...$ with $a_1=2 @3$, $a_{n}=a_{n-1}@ (n+2)$ for $n \ge 2$. Determine $f(a_{233})$ and $a_{233}$ p2. It is known that $ a$ and $ b$ are positive integers with $a > b > 2$. Is $\frac{2^a+1}{2^b-1}$ an integer? Write down your reasons. p3. Given a cube $ABCD.EFGH$ with side length $ 1$ dm. There is a square $PQRS$ on the diagonal plane $ABGH$ with points $P$ on $HG$ and $Q$ on $AH$ as shown in the figure below. Point $T$ is the center point of the square $PQRS$. The line $HT$ is extended so that it intersects the diagonal line $BG$ at $N$. Point $M$ is the projection of $N$ on $BC$. Determine the volume of the truncated prism $DCM.HGN$. [img]https://cdn.artofproblemsolving.com/attachments/f/6/22c26f2c7c66293ad7065a3c8ce3ac2ffd938b.png[/img] 4. Nine pairs of husband and wife want to take pictures in a three-line position with the background of the Palembang Ampera Bridge. There are $4$ people in the front row, $6$ people in the middle row, and $ 8$ people in the back row. They agreed that every married couple must be in the same row, and every two people next to each other must be a married couple or of the same sex. Specify the number of different possible arrangements of positions. p5. p5. A hotel provides four types of rooms with capacity, rate, and number of rooms as presented in the following table. [b] type of room, capacity of persons/ room, day / rate (Rp.), / number of rooms [/b][img]https://cdn.artofproblemsolving.com/attachments/3/c/e9e1ed86887e692f9d66349a82eaaffc730b46.jpg[/img] A group of four families wanted to stay overnight at the hotel. Each family consists of husband and wife and their unmarried children. The number of family members by gender is presented in the following table. [b]family / man / woman/ total[/b] [img]https://cdn.artofproblemsolving.com/attachments/4/6/5961b130c13723dc9fa4e34b43be30c31ee635.jpg[/img] The group leader enforces the following provisions. I. Each husband and wife must share a room and may not share a room with other married couples. II. Men and women may not share the same room unless they are from the same family. III. At least one room is occupied by all family representatives (“representative room”) IV. Each family occupies at most $3$ types of rooms. V. No rooms are occupied by more than one family except representative rooms. You are asked to arrange a room for the group so that the total cost of lodging is as low as possible. Provide two possible alternative room arrangements for each family and determine the total cost.

In a box there are $n$ balls, each colored in one of the following colors: green, red, blue or yellow. It is known that among any $28$ balls in the box at least one is green. Among any $26$ balls at least one is red. Among any $24$ balls at least one is blue. Among any $23$ balls at least one is yellow. Find the largest possible value of the number $n$.

Magicman and his helper want to do some magic trick. They have special card desk. Back of all cards is common color and face is one of $2017$ colors. Magic trick: magicman go away from scene. Then viewers should put on the table $n>1$ cards in the row face up. Helper looks at these cards, then he turn all cards face down, except one, without changing order in row. Then magicman returns on the scene, looks at cards, then show on the one card, that lays face down and names it face color. What is minimal $n$ such that magicman and his helper can has strategy to make magic trick successfully?

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

Find all natural numbers $k$ for which there exists a set $M$ of ten real numbers such that there are exactly $k$ pairwise non-congruent triangles whose side lengths are three (not necessarily distinct) elements of $M$.