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Six sacks of gold coins were found on a sunken ship of the fourteenth century. In the first four bags, there were 60, 30, 20 and 15 gold coins. When the coins were counted in the remaining two bags, someone noticed that the number of coins in the bags has a certain sequence. Having taken this into consideration, could you say how many coins are in the fifth and sixth bags?

Do you think that among the four consecutive natural numbers there will be at least one that is divisible a) by 2? b) by 3? c) by 4? d) by 5?

Does there exist a number h such that for any natural number n the number [$h \times 1969^n$] is not divisible by [$h \times 1969^{n-1}$]?

Out of the given numbers 1, 2, 3, …, 1000, find the largest number m that has this property: no matter which m of these numbers you delete, among the remaining 1000 – m numbers there are two, of which one is divisible by the other.

There are fewer than 30 people in a class. The probability that at random a selected girl is an excellent student is 3/13, and the probability that at random a chosen boy is an excellent pupil is 4/11. How many excellent students are there in the class?

Peter plays a computer game “A bunch of stones.” First in his pile of stones he has 16 stones. Players take turns taking from the pile either 1, 2, 3 or 4 stones. The one who takes the last stone wins. Peter plays this for the first time and therefore each time he takes a random number of stones, whilst not violating the rules of the game. The computer plays according to the following algorithm: on each turn, it takes the number of stones that leaves it to be in the most favorable position. The game always begins with Peter. How likely is it that Peter will win?

We are given 111 different natural numbers that do not exceed 500. Could it be that for each of these numbers, its last digit coincides with the last digit of the sum of all of the remaining numbers?

Prove that for a real positive α and a positive integer d, [α / d] = [[α] / d] is always satisfied.

From the set of numbers 1 to 2n, n + 1 numbers are chosen. Prove that among the chosen numbers there are two, one of which is divisible by another.

Write out in a row the numbers from 1 to 9 $($every number once$)$ so that every two consecutive numbers give a two-digit number that is divisible by 7 or by 13.

A country is called a Fiver if, in it, each city is connected by airlines with exactly with five other cities $($there are no international flights$)$.

a) Draw a scheme of airlines for a country that is made up of 10 cities.

b) How many airlines are there in a country of 50 cities?

c) Can there be a Fiver country, in which there are exactly 46 airlines?

A professional tennis player plays at least one match each day for training purposes. However in order to ensure he does not over-exert himself he plays no more than 12 matches a week. Prove that it is possible to find a group of consecutive days during which the player plays a total of 20 matches.

In a graph, three edges emerge from each vertex. Can there be a 1990 edges in this graph?

Let p be a prime number, and a is not divisible by p. Prove that there is a positive integer b such that ab $\equiv$ 1 $($mod p$)$.

Can there be exactly 100 roads in a state in which three roads leave each city?

You are given 11 different natural numbers that are less than or equal to 20. Prove that it is always possible to choose two numbers where one is divisible by the other.

The sequence of numbers $a_1$, $a_2$, … is given by the conditions $a_1$ = 1, $a_2$ = 143 and

for all n $≥$ 2.

Prove that all members of the sequence are integers.

Is there a sequence of natural numbers in which every natural number occurs exactly once, and for any k = 1, 2, 3, … the sum of the first k terms of the sequence is divisible by k?

Prove that for any natural number $a_1> 1$ there exists an increasing sequence of natural numbers $a_1, a_2, a_3$, …, for which $a_1^2+ a_2^2 +…+ a_k^2$ is divisible by $a_1+ a_2+…+ a_k$ for all k ≥ 1.

Which five-digit numbers are there more of: ones that are not divisible by 5 or those with neither the first nor the second digit on the left being a five?