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A numerical sequence is defined by the following conditions:

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How many complete squares are found among the first members of this sequence, not exceeding 1,000,000?

A numerical sequence is defined by the following conditions:

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Prove that among the terms of this sequence there are an infinite number of complete squares.

A group of numbers $A_1, A_2, …, A_{100}$ is created by somehow re-arranging the numbers $1, 2, …, 100$.

100 numbers are created as follows:

$$B_1=A_1, B_2=A_1+A_2, B_3=A_1+A_2+A_3, …, B_{100} = A_1+A_2+A_3…+A_{100}$$

Prove that there will always be at least 11 different remainders when dividing the numbers $B_1, B_2, …, B_{100}$ by 100.

Will the quotient or the remainder change if a divided number and the divisor are increased by 3 times?

a) Prove that within any 6 whole numbers there will be two that have a difference between them that is a multiple of 5.

b) Will this statement remain true if instead of the difference we considered the total?

Find all of the natural numbers that, when divided by 7, have the same remainder and quotient.

Prove that amongst the numbers of the form $19991999…19990…0$ – that is 1999 a number of times, followed by a number of 0s – there will be at least one divisible by 2001.

Prove that in any group of 2001 whole numbers there will be two whose difference is divisible by 2000.

All of the integers from 1 to 64 are written in an $8 \times 8$ table. Prove that in this case there are two adjacent numbers, the difference between which is not less than 5. $($Numbers that are in cells which share a common side are called adjacent$)$.

Prove that, for any integer n, among the numbers n, n + 1, n + 2, …, n + 9 there is at least one number that is mutually prime with the other nine numbers.

An infinite sequence of numbers is given. Prove that for any natural number n that is relatively prime with a number 10, you can choose a group of consecutive digits, which when written as a sequence of digits, gives a resulting number written by these digits which is divisible by n.

Prove that for any odd natural number, a, there exists a natural number, b, such that $2^b$ – 1 is divisible by a.

Prove that the 13th day of the month is more likely to occur on a Friday than on other days of the week. It is assumed that we live in the Gregorian style calendar.

Reception pupil Peter knows only the number 1. Prove that he can write a number divisible by 2001.

Upon the installation of a keypad lock, each of the 26 letters located on the lock’s keypad is assigned an arbitrary natural number known only to the owner of the lock. Different letters do not necessarily have different numbers assigned to them. After a combination of different letters, where each letter is typed once at most, is entered into the lock a summation is carried out of the corresponding numbers to the letters typed in. The lock opens only if the result of the summation is divisible by 26. Prove that for any set of numbers assigned to the 26 letters, there exists a combination that will open the lock.

10 natural numbers are written on a blackboard. Prove that it is always possible to choose some of these numbers and write “$+$” or “$-$” between them so that the resulting algebraic sum is divisible by 1001.

Prove that if $a, b, c$ are odd numbers, then at least one of the numbers $ab-1, bc-1, ca-1$ is divisible by 4.

$2n$ diplomats sit around a round table. After a break the same $2n$ diplomats sit around the same table, but this time in a different order.

Prove that there will always be two diplomats with the same number of people sitting between them, both before and after the break.

Prove that for any number d, which is not divisible by 2 or by 5, there is a number whose decimal notation contains only ones and which is divisible by d.