Can the equality \(K \times O \times T = U \times W \times E \times N \times H \times Y\) be true if the numbers from 1 to 9 are substituted for letters instead of the letters? Different letters correspond to different numbers.
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?
Prove that for any natural number \(a_1> 1\) there exists an increasing sequence of natural numbers \(a_1, a_2, a_3, \dots\), for which \(a_1^2+ a_2^2 +\dots+ a_k^2\) is divisible by \(a_1+ a_2+\dots+ a_k\) for all \(k \geq 1\).
Is there a sequence of natural numbers in which every natural number occurs exactly once, and for any \(k = 1, 2, 3, \dots\) the sum of the first \(k\) terms of the sequence is divisible by \(k\)?
The sequence of numbers \(a_1, a_2, \dots\) is given by the conditions \(a_1 = 1\), \(a_2 = 143\) and
for all \(n \geq 2\).
Prove that all members of the sequence are integers.
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.
Can there be exactly 100 roads in a state in which three roads leave each city?
Let \(p\) be a prime number, and \(a\) an integer number not divisible by \(p\). Prove that there is a positive integer \(b\) such that \(ab \equiv 1 \pmod p\).
In a graph, three edges emerge from each vertex. Can there be a 1990 edges in this graph?
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.