There are \(n\) integers. Prove that among them either there are several numbers whose sum is divisible by \(n\) or there is one number divisible by \(n\) itself.
Prove that \(n^2 + 1\) is not divisible by \(3\) for any natural \(n\).
Solve the equation with natural numbers \(1 + m + m^2 + m^3 = 2^n\).
Is it possible to find natural numbers \(x\), \(y\) and \(z\) which satisfy the equation \(28x+30y+31z=365\)?
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 \(k\)?
Prove that there are no natural numbers \(a\) and \(b\) such that \(a^2 - 3b^2 = 8\).
Prove there are no integer solutions for the equation \(x^2=y^2+1990\).
Prove that for a real positive \(\alpha\) and a positive integer \(d\), \(\lfloor \alpha / d\rfloor = \lfloor \lfloor \alpha\rfloor / d\rfloor\) is always satisfied.
Prove that if \(p\) is a prime number and \(1 \leq k \leq p - 1\), then \(\binom{p}{k}\) is divisible by \(p\).
Prove that if \((p, q) = 1\) and \(p/q\) is a rational root of the polynomial \(P (x) = a_nx^n + \dots + a_1x + a_0\) with integer coefficients, then
a) \(a_0\) is divisible by \(p\);
b) \(a_n\) is divisible by \(q\).