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: 39

1947 Moscow Mathematical Olympiad, 133
Tags: weighing , game strategy , combinatorics
Twenty cubes of the same size and appearance are made of either aluminum or of heavier duralumin. How can one find the number of duralumin cubes using not more than $11$ weighings on a balance without weights? (We assume that all cubes can be made of aluminum, but not all of duralumin.)

Kvant 2025, M2833
Tags: combinatorics , weighing
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]

2003 All-Russian Olympiad Regional Round, 10.8
Tags: combinatorics , weighing
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.)

2006 All-Russian Olympiad Regional Round, 8.8
Tags: weighing , combinatorics
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?

2011 Tournament of Towns, 3
Tags: weighing , combinatorics
A balance and a set of pairwise different weights are given. It is known that for any pair of weights from this set put on the left pan of the balance, one can counterbalance them by one or several of the remaining weights put on the right pan. Find the least possible number of weights in the set.

1949 Moscow Mathematical Olympiad, 166
Tags: weighing , combinatorics
Consider $13$ weights of integer mass (in grams). It is known that any $6$ of them may be placed onto two pans of a balance achieving equilibrium. Prove that all the weights are of equal mass.

2000 Tournament Of Towns, 5
Tags: weighing , combinatorics
A weight of $11111$ grams is placed in the left pan of a balance. Weights are added one at a time, the first weighing $1$ gram, and each subsequent one weighing twice as much as the preceding one. Each weight may be added to either pan. After a while, equilibrium is achieved. Is the $16$ gram weight placed in the left pan or the right pan? ( AV Kalinin)

1997 Tournament Of Towns, (542) 3
Tags: weighing , combinatorics
You are given $20$ weights such that any object of integer weight $m$, $1 \le m \le1997$, can be balanced by placing it on one pan of a balance and a subset of the weights on the other pan. What is the minimal value of largest of the $20$ weights if the weights are (a) all integers; (b) not necessarily integers? (M Rasin)

2013 Tournament of Towns, 3
Tags: weighing , combinatorics , number theory
Each of $11$ weights is weighing an integer number of grams. No two weights are equal. It is known that if all these weights or any group of them are placed on a balance then the side with a larger number of weights is always heavier. Prove that at least one weight is heavier than $35$ grams.

1998 Tournament Of Towns, 1
Tags: combinatorics , weighing
Nineteen weights of mass $1$ gm, $2$ gm, $3$ gm, . . . , $19$ gm are given. Nine are made of iron, nine are of bronze and one is pure gold. It is known that the total mass of all the iron weights is $90$ gm more than the total mass of all the bronze ones. Find the mass of the gold weight . (V Proizvolov)

2000 Tournament Of Towns, 4
Tags: weighing , combinatorics
Among a set of $2N$ coins, all identical in appearance, $2N - 2$ 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, if (a) $N = 16$, ( b ) $N = 11$ ? (A Shapovalov)

2013 Tournament of Towns, 4
Tags: weighing , combinatorics
Each of $100$ stones has a sticker showing its true weight. No two stones weight the same. Mischievous Greg wants to rearrange stickers so that the sum of the numbers on the stickers for any group containing from $1$ to $99$ stones is different from the true weight of this group. Is it always possible?

1990 Tournament Of Towns, (252) 6
Tags: weighing , combinatorics
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)

1996 All-Russian Olympiad Regional Round, 8.8
Tags: weighing , combinatorics
There are 4 coins, 3 of which are real, which weigh the same, and one is fake, which differs in weight from the rest. Cup scales without weights are such that if equal weights are placed 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 sure to pull. How to definitely identify a counterfeit coin in three weighings and easily establish what is it or is it heavier than the others?

1988 Tournament Of Towns, (171) 4
Tags: combinatorics , power of 2 , weighing
We have a set of weights with masses $1$ gm, $2$ gm, $4$ gm and so on, all values being powers of $2$ . Some of these weights may have equal mass. Some weights were put on both sides of a balance beam, resulting in equilibrium. It is known that on the left hand side all weights were distinct . Prove that on the right hand side there were no fewer weights than on the left hand side.

1990 Tournament Of Towns, (246) 4
Tags: weighing , combinatorics
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)

1989 All Soviet Union Mathematical Olympiad, 495
Tags: combinatorics , weighing
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.

2004 All-Russian Olympiad Regional Round, 8.2
Tags: weighing , combinatorics
There is a set of weights with the following properties: 1) It contains 5 weights, pairs of different weights. 2) For any two weights, there are two other weights of the same total weight. What is the smallest number of weights that can be in this set?

1996 All-Russian Olympiad Regional Round, 9.8
Tags: weighing , combinatorics
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?

2002 All-Russian Olympiad Regional Round, 8.8
Tags: weighing , combinatorics
Among $18$ parts placed in a row, some three in a row weigh $99 $ g each, and all the rest weigh $100$ g each. On a scale with an arrow, identify all $99$-gram parts.

2000 Tournament Of Towns, 4
Tags: weighing , 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)

1984 All Soviet Union Mathematical Olympiad, 385
Tags: weighing , combinatorics
There are scales and $(n+1)$ weights with the total weight $2n$. Each weight is an integer. We put all the weights in turn on the lighter side of the scales, starting from the heaviest one, and if the scales is in equilibrium -- on the left side. Prove that when all the weights will be put on the scales, they will be in equilibrium.

2000 All-Russian Olympiad Regional Round, 10.2
Tags: weighing , combinatorics
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?

1990 Tournament Of Towns, (258) 2
Tags: weighing , combinatorics
We call a collection of weights (each weighing an integer value) basic if their total weight equals $500$ and each object of integer weight not greater than $500$ 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 equal 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 $500$ weights each of value $1$. (b) How many different basic collections are there? (D. Fomin, Leningrad)

1997 All-Russian Olympiad Regional Round, 9.3
Tags: weighing , algebra , combinatorics
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.