Found problems: 39
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?
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.
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]
$100$ 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)
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.
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?
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.
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)
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)
$50$ 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)
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.
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?
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.)
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.
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.
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.
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.
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.
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.
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)
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)
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)
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?
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)
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.)