Olympiad problems

1996 All-Russian Olympiad Regional Round, 8.8

Tags: combinatorics , weighing

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?

2006 All-Russian Olympiad Regional Round, 8.8

Tags: combinatorics , weighing

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?

1997 All-Russian Olympiad Regional Round, 9.3

Tags: algebra , combinatorics , weighing

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.

2002 All-Russian Olympiad Regional Round, 8.8

Tags: combinatorics , weighing

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.

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)

1987 All Soviet Union Mathematical Olympiad, 442

Tags: algebra , weighing , combinatorics

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.

2013 Tournament of Towns, 4

Tags: combinatorics , weighing

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?

1950 Moscow Mathematical Olympiad, 174

Tags: combinatorics , weighing , 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.

1990 Tournament Of Towns, (258) 2

Tags: combinatorics , weighing

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)

2000 All-Russian Olympiad Regional Round, 10.2

Tags: combinatorics , weighing

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?

1984 All Soviet Union Mathematical Olympiad, 385

Tags: combinatorics , weighing

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.

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.

1974 Bundeswettbewerb Mathematik, 2

Tags: combinatorics , weighing

There are $30$ apparently equal balls, $15$ of which have the weight $a$ and the remaining $15$ have the weight $b$ with $a \ne b$. The balls are to be partitioned into two groups of $15$, according to their weight. An assistant partitioned them into two groups, and we wish to check if this partition is correct. How can we check that with as few weighings as possible?

2000 All-Russian Olympiad Regional Round, 9.6

Tags: combinatorics , weighing

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?