This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 38

1996 All-Russian Olympiad Regional Round, 9.8
Tags: combinatorics , weighings
There are 8 coins, 7 of which are real, which weigh the same, and one is fake, which differs in weight from the rest. Cup scales without weights mean that if you put equal weights on their cups, then any of the cups can outweigh, but if the loads are different in mass, then the cup with a heavier load is definitely overpowered. How to definitely identify a counterfeit coin in four weighings and establish is it lighter or heavier than the others?

1990 Tournament Of Towns, (252) 6
Tags: combinatorics , weighings
We call a collection of weights (each weighing an integer value) basic if their total weight equals $200$ and each object of integer weight not greater than $200$ can be balanced exactly with a uniquely determined set of weights from the collection. (Uniquely means that we are not concerned with order or which weights of equalc value are chosen to balance against a particular object, if in fact there is a choice.) (a) Find an example of a basic collection other than the collection of $200$ weights each of value $1$. (b) How many different basic collections are there? (D. Fomin, Leningrad)

Kvant 2025, M2833
Tags: weighings , combinatorics
There are a) $26$; b) $30$ identical-looking coins in a circle. It is known that exactly two of them are fake. Real coins weigh the same, fake ones too, but they are lighter than the real ones. How can you determine in three weighings on a cup scale without weights whether there are fake coins lying nearby or not?? [i]Proposed by A. Gribalko[/i]

1997 All-Russian Olympiad Regional Round, 9.3
Tags: algebra , combinatorics , weighings
There are 300 apples, any two of which differ in weight by no more than three times. Prove that they can be arranged into bags of four apples each so that any two bags differ in weight by no more than than one and a half times.

1987 All Soviet Union Mathematical Olympiad, 442
Tags: weighings , combinatorics , algebra
It is known that, having $6$ weighs, it is possible to balance the scales with loads, which weights are successing natural numbers from $1$ to $63$. Find all such sets of weighs.

1990 Tournament Of Towns, (249) 3
Tags: combinatorics , weighings
Fifteen elephants stand in a row. Their weights are expressed by integer numbers of kilograms. The sum of the weight of each elephant (except the one on the extreme right) and the doubled weight of its right neighbour is exactly $15$ tonnes. Determine the weight of each elephant. (F.L. Nazarov)

1990 All Soviet Union Mathematical Olympiad, 534
Tags: weighings , combinatorics
Given $2n$ genuine coins and $2n$ fake coins. The fake coins look the same as genuine coins but weigh less (but all fake coins have the same weight). Show how to identify each coin as genuine or fake using a balance at most $3n$ times.

2000 Tournament Of Towns, 4
Tags: weighings , combinatorics
Among a set of $32$ coins , all identical in appearance, $30$ are real and $2$ are fake. Any two real coins have the same weight . The fake coins have the same weight , which is different from the weight of a real coin. How can one divide the coins into two groups of equal total weight by using a balance at most $4$ times? (A Shapovalov)

2003 All-Russian Olympiad Regional Round, 10.8
Tags: combinatorics , weighings
In a set of 17 externally identical coins, two are counterfeit, differing from the rest in weight. It is known that the total weight of two counterfeit coins is twice the weight of a real one.s it always possible to determine the couple of counterfeit coins, having made $5$ weighings on a cup scale without weights? (It is not necessary to determine which of the fakes is heavier.)

1989 All Soviet Union Mathematical Olympiad, 495
Tags: weighings , combinatorics
We are given $1998$ normal coins, $1$ heavy coin and $1$ light coin, which all look the same. We wish to determine whether the average weight of the two abnormal coins is less than, equal to, or greater than the weight of a normal coin. Show how to do this using a balance $4$ times or less.

1990 Tournament Of Towns, (246) 4
Tags: combinatorics , weighings
A set of $61$ coins that look alike is given. Two coins (whose weights are equal) are counterfeit. The other $59$ (genuine) coins also have the same weight, but a different weight from that of the counterfeit coins. However it is not known whether it is the genuine coins or the counterfeit coins which are heavier. How can this question be resolved by three weighings on the one balance? (It is not required to separate the counterfeit coins from the genuine ones.) (D. Fomin, Leningrad)

2000 All-Russian Olympiad Regional Round, 10.2
Tags: combinatorics , weighings
Among five outwardly identical coins, $3$ are real and two are fake, identical in weight, but it is unknown whether they are heavier or lighter than the real ones. How to find at least one real coin in the least number of weighings?

2006 All-Russian Olympiad Regional Round, 8.8
Tags: combinatorics , weighings
When making a batch of $N \ge 5$ coins, a worker mistakenly made two coins from a different material (all coins look the same). The boss knows that there are exactly two such coins, that they weigh the same, but differ in weight from the others. The employee knows what coins these are and that they are lighter than others. He needs, after carrying out two weighings on cup scales without weights, to convince his boss that the coins are counterfeit easier than real ones, and in which coins are counterfeit. Can he do it?