Olympiad problems

2010 Dutch Mathematical Olympiad, 5

Amber and Brian are playing a game using $2010$ coins. Throughout the game, the coins are divided into a number of piles of at least 1 coin each. A move consists of choosing one or more piles and dividing each of them into two smaller piles. (So piles consisting of only $1$ coin cannot be chosen.) Initially, there is only one pile containing all $2010$ coins. Amber and Brian alternatingly take turns to make a move, starting with Amber. The winner is the one achieving the situation where all piles have only one coin. Show that Amber can win the game, no matter which moves Brian makes.

2021 Balkan MO Shortlist, C3

Tags: coin , boxes , shortlist , combinatorics

In an exotic country, the National Bank issues coins that can take any value in the interval $[0, 1]$. Find the smallest constant $c > 0$ such that the following holds, no matter the situation in that country: [i]Any citizen of the exotic country that has a finite number of coins, with a total value of no more than $1000$, can split those coins into $100$ boxes, such that the total value inside each box is at most $c$.[/i]

2021 Thailand TSTST, 3

Tags: coin , combinatorics

Let $1 \leq n \leq 2021$ be a positive integer. Jack has $2021$ coins arranged in a line where each coin has an $H$ on one side and a $T$ on the other. At the beginning, all coins show $H$ except the nth coin. Jack can repeatedly perform the following operation: he chooses a coin showing $T$, and turns over the coins next to it to the left and to the right (if any). Determine all $n$ such that Jack can make all coins show $T$ after a finite number of operations.

2005 Tournament of Towns, 5

Tags: coin , combinatorics

Among 6 coins one is counterfeit (its weight differs from that real one and neither weights is known). Using scales that show the total weight of coins placed on the cup, find the counterfeit coin in 3 weighings. [i](5 points)[/i]

2008 Hungary-Israel Binational, 2

Tags: coin , probability , combinatorics

For every natural number $ t$, $ f(t)$ is the probability that if a fair coin is tossed $ t$ times, the number of times we get heads is 2008 more than the number of tails. What is the value of $ t$ for which $ f(t)$ attains its maximum? (if there is more than one, describe all of them)

2022 Greece Team Selection Test, 4

Tags: coin , boxes , shortlist , combinatorics

In an exotic country, the National Bank issues coins that can take any value in the interval $[0, 1]$. Find the smallest constant $c > 0$ such that the following holds, no matter the situation in that country: [i]Any citizen of the exotic country that has a finite number of coins, with a total value of no more than $1000$, can split those coins into $100$ boxes, such that the total value inside each box is at most $c$.[/i]

2012 Kyiv Mathematical Festival, 2

Tags: coin , combinatorics

A hundred of silver coins are laid down in a line. A wizard can convert silver coin into golden one in $3$ seconds. Each golden coin, which is near the coin being converted, reduces this time by $1$ second. What minimal time is required for the wizard to convert all coins to gold?

2020 Iranian Combinatorics Olympiad, 7

Tags: coin , combinatorics

Seyed has 998 white coins, a red coin, and an unusual coin with one red side and one white side. He can not see the color of the coins instead he has a scanner which checks if all of the coin sides touching the scanner glass are white. Is there any algorithm to find the red coin by using the scanner at most 17 times? [i]Proposed by Seyed Reza Hosseini[/i]

2023 Indonesia TST, 2

Tags: invariant , combinatorics , monovariant , coin

Let $n > 3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbors on the right and left. Determine all initial distributions of the coins from which it is possible that, after a finite number of steps, each child has exactly one coin.

1984 Polish MO Finals, 4

Tags: probability , coin , sum , sequence , recurrence relation

A coin is tossed $n$ times, and the outcome is written in the form ($a_1,a_2,...,a_n$), where $a_i = 1$ or $2$ depending on whether the result of the $i$-th toss is the head or the tail, respectively. Set $b_j = a_1 +a_2 +...+a_j$ for $j = 1,2,...,n$, and let $p(n)$ be the probability that the sequence $b_1,b_2,...,b_n$ contains the number $n$. Express $p(n)$ in terms of $p(n-1)$ and $p(n-2)$.

2017 JBMO Shortlist, C3

Tags: coin , game strategy , combinatorics

We have two piles with $2000$ and $2017$ coins respectively. Ann and Bob take alternate turns making the following moves: The player whose turn is to move picks a pile with at least two coins, removes from that pile $t$ coins for some $2\le t \le 4$, and adds to the other pile $1$ coin. The players can choose a different $t$ at each turn, and the player who cannot make a move loses. If Ann plays first determine which player has a winning strategy.

2021 Baltic Way, 8

Tags: coin , combinatorics

We are given a collection of $2^{2^k}$ coins, where $k$ is a non-negative integer. Exactly one coin is fake. We have an unlimited number of service dogs. One dog is sick but we do not know which one. A test consists of three steps: select some coins from the collection of all coins; choose a service dog; the dog smells all of the selected coins at once. A healthy dog will bark if and only if the fake coin is amongst them. Whether the sick dog will bark or not is random. \\ Devise a strategy to find the fake coin, using at most $2^k+k+2$ tests, and prove that it works.

2021 Hong Kong TST, 3

Tags: combinatorics , coin , operation

On the table there are $20$ coins of weights $1,2,3,\ldots,15,37,38,39,40$ and $41$ grams. They all look alike but their colours are all distinct. Now Miss Adams knows the weight and colour of each coin, but Mr. Bean knows only the weights of the coins. There is also a balance on the table, and each comparison of weights of two groups of coins is called an operation. Miss Adams wants to tell Mr. Bean which coin is the $1$ gram coin by performing some operations. What is the minimum number of operations she needs to perform?

2019 Tournament Of Towns, 3

Tags: combinatorics , algorithm , coin , strategy

There are 100 visually identical coins of three types: golden, silver and copper. There is at least one coin of each type. Each golden coin weighs 3 grams, each silver coins weighs 2 grams and each copper coin weighs 1 gram. How to find the type of each coin performing no more than 101 measurements on a balance scale with no weights.

2001 Mexico National Olympiad, 6

Tags: coin , combinatorics

A collector of rare coins has coins of denominations $1, 2,..., n$ (several coins for each denomination). He wishes to put the coins into $5$ boxes so that: (1) in each box there is at most one coin of each denomination; (2) each box has the same number of coins and the same denomination total; (3) any two boxes contain all the denominations; (4) no denomination is in all $5$ boxes. For which $n$ is this possible?

2022 Azerbaijan BMO TST, C3

Tags: shortlist , coin , boxes , combinatorics

In an exotic country, the National Bank issues coins that can take any value in the interval $[0, 1]$. Find the smallest constant $c > 0$ such that the following holds, no matter the situation in that country: [i]Any citizen of the exotic country that has a finite number of coins, with a total value of no more than $1000$, can split those coins into $100$ boxes, such that the total value inside each box is at most $c$.[/i]

2014 German National Olympiad, 5

Tags: weight , coin , combinatorics

There are $9$ visually indistinguishable coins, and one of them is fake and thus lighter. We are given $3$ indistinguishable balance scales to find the fake coin; however, one of the scales is defective and shows a random result each time. Show that the fake coin can still be found with $4$ weighings.

2021 Bolivian Cono Sur TST, 1

Tags: combinatorics , coin

[b]a)[/b] Among $9$ apparently identical coins, one is false and lighter than the others. How can you discover the fake coin by making $2$ weighing in a two-course balance? [b]b)[/b] Find the least necessary number of weighing that must be done to cover a false currency between $27$ coins if all the others are true.

2023 Thailand TST, 2

Tags: combinatorics , coin , invariant , monovariant

Let $n > 3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbors on the right and left. Determine all initial distributions of the coins from which it is possible that, after a finite number of steps, each child has exactly one coin.

2024 Irish Math Olympiad, P7

Tags: coin

A game of coins is played as follows: You start with $1$ head and $1$ tail on a table. At each turn, you can perform any one of the following moves: [list=a] [*]You can turn over all the coins on the table. [*]You can triple the number of heads and tails at the table. [*]If there are at least $4$ tails on the table, you can turn over $4$ tails. [*]If there are at least $5$ tails on the table, you can turn over $3$ of the tails and discard $2$ of the tails. [/list] Knowing that at the end of the game you have $2024$ heads, what are all possible numbers of tails at the end of that game?

2022 IMO Shortlist, C4

Tags: coin , invariant , monovariant , combinatorics

Let $n > 3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbors on the right and left. Determine all initial distributions of the coins from which it is possible that, after a finite number of steps, each child has exactly one coin.

2021 Pan-American Girls' Math Olympiad, Problem 1

Tags: number theory , coin

There are $n \geq 2$ coins numbered from $1$ to $n$. These coins are placed around a circle, not necesarily in order. In each turn, if we are on the coin numbered $i$, we will jump to the one $i$ places from it, always in a clockwise order, beginning with coin number 1. For an example, see the figure below. Find all values of $n$ for which there exists an arrangement of the coins in which every coin will be visited.

2021/2022 Tournament of Towns, P3

Tags: combinatorics , coin

A pirate has five purses with 30 coins in each. He knows that one purse contains only gold coins, another one contains only silver coins, the third one contains only bronze coins, and the remaining two ones contain 10 gold, 10 silver and 10 bronze coins each. It is allowed to simultaneously take one or several coins out of any purses (only once), and examine them. What is the minimal number of taken coins that is necessary to determine for sure the content of at least one purse? [i]Mikhail Evdokimov[/i]

2013 Israel National Olympiad, 1

Tags: perimeter , coin , geometry

In the picture there are six coins, each with radius 1cm. Each coin is tangent to exactly two other coins next to it (as in the picture). Between the coins, there is an empty area whose boundary is a star-like shape. What is the perimeter of this shape? [img]https://i.imgur.com/aguQRVd.png[/img]

2002 German National Olympiad, 6

Tags: coin , distributions , combinatorics

Theo Travel, who has $5$ children, has already visited $8$ countries of the eurozone. From every country, he brought $5$ not necessarily distinct coins home. Moreover, among these $40$ coins there are exactly $5$ of every value ($1,2,5,10,20,$ and $50$ ct, $1$ and $2$ euro). He wants to give each child $8$ coins such that they are from different countries and that each child gets the same amount of money. Is this always possible?