There are 13 weights. It is known that any 12 of them could be placed in 2 scale cups with 6 weights in each cup in such a way that balance will be held.
Prove the mass of all the weights is the same, if it is known that:
a) the mass of each weight in grams is an integer;
b) the mass of each weight in grams is a rational number;
c) the mass of each weight could be any real (not negative) number.
Solve the equation \(xy = x + y\) in integers.
Author: A.K. Tolpygo
An irrational number \(\alpha\), where \(0 <\alpha <\frac 12\), is given. It defines a new number \(\alpha_1\) as the smaller of the two numbers \(2\alpha\) and \(1 - 2\alpha\). For this number, \(\alpha_2\) is determined similarly, and so on.
a) Prove that for some \(n\) the inequality \(\alpha_n <3/16\) holds.
b) Can it be that \(\alpha_n> 7/40\) for all positive integers \(n\)?
We consider a sequence of words consisting of the letters “A” and “B”. The first word in the sequence is “A”, the \(k\)-th word is obtained from the \((k-1)\)-th by the following operation: each “A” is replaced by “AAB” and each “B” by “A”. It is easy to see that each word is the beginning of the next, thus obtaining an infinite sequence of letters: AABAABAAABAABAAAB...
a) Where in this sequence will the 1000th letter “A” be?
b) Prove that this sequence is non-periodic.
Louise is confident that all her classmates have different number of friends. Is she right?
There are 100 cities all connected by roads. Each city has 6 roads coming in (or going out). How many roads do connect those cities?
There are 15 cities in a country named The Country of Fifteen Cities. The king ordered his main architect to build roads in such a way that each city was connected with other cities by exactly 5 roads, otherwise he would hang the architect. Do you think that the architect can accomplish the task or should he flee that country immediately?
The architect decided to flee The Country of 15 Cities and began to travel around the world. He arrived to a country, where every city had exactly 3 roads going to and from it. Can there be all together 100 roads in that country?
There are 9 cities named City 1, City 2, City 3, …, and City 9 in a country named The Country of the Nine Cities. Two cities are connected by a road only if the sum of the numbers made up by their names is divisible by 3. Can our travelling architect reach City 9 by starting his journey from City 1 and travelling along those roads?
Show that among any 6 people there are always either 3 people who all know each other or 3 total strangers.