Olympiad problems

1947 Moscow Mathematical Olympiad, 133

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.)

2011 Tournament of Towns, 3

Tags: weighings , 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: weighings , 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.

2004 All-Russian Olympiad Regional Round, 8.2

Tags: combinatorics , weighings

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?

2000 All-Russian Olympiad Regional Round, 9.6

Tags: combinatorics , weighings

Among $2000$ outwardly indistinguishable balls, wines - aluminum weighing 1$0$ g, and the rest - duralumin weighing $9.9$ g. It is required to select two piles of balls so that the masses of the piles are different, and the number of balls in them - the same. What is the smallest number of weighings on a cup scale without weights that can be done?

1951 Moscow Mathematical Olympiad, 192

Tags: combinatorics , algebra , weighings

a) Given a chain of $60$ links each weighing $1$ g. Find the smallest number of links that need to be broken if we want to be able to get from the obtained parts all weights $1$ g, $2$ g, . . . , $59$ g, $60$ g? A broken link also weighs $1$ g. b) Given a chain of $150$ links each weighing $1$ g. Find the smallest number of links that need to be broken if we want to be able to get from the obtained parts all weights $1$ g, $2$ g, . . . , $149$ g, $150$ g? A broken link also weighs $1$ g.

1988 Tournament Of Towns, (171) 4

Tags: weighings , combinatorics , power of 2

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.

1993 Tournament Of Towns, (388) 6

Tags: combinatorics , weighings

Construct a set of $k$ integer weights that allows you to get any total integer weight from $1$ up to $55$ grams even if some of the weights of the set are lost. Consider two versions: (a) $k = 10$, and any one of the weights may be lost; (b) $k = 12$, and any two of the weights may be lost. (D Zvonkin) (In both cases prove that the set found has the property required.)

2002 All-Russian Olympiad Regional Round, 8.8

Tags: weighings , 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.

1992 Tournament Of Towns, (328) 5

Tags: combinatorics , weighings

$50$ silver coins ordered by weight and $51$ gold coins also ordered by weight are given. All coins have different weights. You are given a balance to compare weights of any two coins. How can you find the “middle” coin (that occupying the $51$st place in weight among all $101$ coins) using $7$ weighings? (A. Andjans)

1996 All-Russian Olympiad Regional Round, 8.8

Tags: weighings , 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?

1992 Tournament Of Towns, (337) 5

Tags: weighings , combinatorics

$100$ silver coins ordered by weight and $101$ gold coins also ordered by weight are given. All coins have different weights. You are given a balance to compare weights of any two coins. How can you find the “middle” coin (that occupies the $101$-st place in weight among all $201$ coins) using the minimal number of weighings? Find this number and prove that a smaller number of weighings would be insufficient. (A. Andjans, Riga)

2000 All-Russian Olympiad Regional Round, 8.5

Tags: combinatorics , weighings

Given are $8$ weights weighing $1, 2, . . . , 8$ grams, but it is not known which one how much does it weigh. Baron Munchausen claims that he remembers which of the weights weighs how much, and to prove that he is right he is ready to conduct one weighing, as a result of which the weight of at least one of the weights will be unambiguously established. Is he cheating?

1984 All Soviet Union Mathematical Olympiad, 385

Tags: weighings , 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 Tournament Of Towns, 5

Tags: weighings , 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)

1998 Tournament Of Towns, 1

Tags: weighings , combinatorics

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)

2013 Tournament of Towns, 3

Tags: weighings , number theory , combinatorics

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.

1997 Tournament Of Towns, (535) 7

Tags: combinatorics , weighings , number theory

You are given a balance and one copy of each of ten weights of $1, 2, 4, 8, 16, 32, 64, 128, 256$ and $512$ grams. An object weighing $M$ grams, where $M$ is a positive integer, is put on one of the pans and may be balanced in different ways by placing various combinations of the given weights on either pan of the balance. (a) Prove that no object may be balanced in more than $89$ ways. (b) Find a value of $M$ such that an object weighing $M$ grams can be balanced in $89$ ways. (A Shapovalov, A Kulakov)

1985 Tournament Of Towns, (081) T2

Tags: game , game strategy , weighings , combinatorics

There are $68$ coins , each coin having a different weight than that of each other . Show how to find the heaviest and lightest coin in $100$ weighings on a balance beam. (S. Fomin, Leningrad)

2012 Tournament of Towns, 5

Tags: weighings , combinatorics

Among $239$ coins identical in appearance there are two counterfeit coins. Both counterfeit coins have the same weight different from the weight of a genuine coin. Using a simple balance, determine in three weighings whether the counterfeit coin is heavier or lighter than the genuine coin. A simple balance shows if both sides are in equilibrium or left side is heavier or lighter. It is not required to find the counterfeit coins.

1997 Tournament Of Towns, (542) 3

Tags: combinatorics , weighings

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)

1950 Moscow Mathematical Olympiad, 174

Tags: weighings , combinatorics , algebra

a) Given $555$ weights: of $1$ g, $2$ g, $3$ g, . . . , $555$ g, divide them into three piles of equal mass. b) Arrange $81$ weights of $1^2, 2^2, . . . , 81^2$ (all in grams) into three piles of equal mass.

2000 Tournament Of Towns, 4

Tags: weighings , 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: weighings , 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, (258) 2

Tags: combinatorics , weighings

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)