Let \(x_1, x_2, \dots , x_n\) be some numbers belonging to the interval \([0, 1]\). Prove that on this segment there is a number \(x\) such that \[\frac{1}{n} (|x - x_1| + |x - x_2| + \dots + |x - x_n|) = 1/2.\]
Prove there are no natural numbers \(a\) and \(b\), such as \(a^2 - 3b^2 = 8\).
Solve the equation \(3x + 5y = 7\) in integers.
Determine all the integer solutions for the equation \(21x + 48y = 6\).
Solve the equations \(x^2 = 14 + y^2\) in integers.
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 \(ab \equiv 1 \pmod p\).
Each of the 102 pupils of one school is friends with at least 68 others. Prove that among them there are four who have the same number of friends.
There were seven boxes. In some of them, seven more boxes were placed inside (not nested in each other), etc. As a result, there are 10 non-empty boxes. How many boxes are there now in total?
It is known that a certain polynomial at rational points takes rational values. Prove that all its coefficients are rational.
A scone contains raisins and sultanas. Prove that inside the scone there will always be two points 1cm apart such that either both lie inside raisins, both inside sultanas, or both lie outside of either raisins or sultanas.