Write at random a two-digit number. What is the probability that the sum of the digits of this number is 5?
There are three boxes, in each of which there are balls numbered from 0 to 9. One ball is taken from each box. What is the probability that
a) three ones were taken out;
b) three equal numbers were taken out?
A player in the card game Preferans has 4 trumps, and the other 4 are in the hands of his two opponents. What is the probability that the trump cards are distributed a) \(2: 2\); b) \(3: 1\); c) \(4: 0\)?
Determine all prime numbers \(p\) and \(q\) such that \(p^2 - 2q^2 = 1\) holds.
Suppose that there are 15 prime numbers forming an arithmetic progression with a difference of \(d\). Prove that \(d >30,000\).
Let \(a\), \(b\), \(c\) be integers; where \(a\) and \(b\) are not equal to zero.
Prove that the equation \(ax + by = c\) has integer solutions if and only if \(c\) is divisible by \(d = \mathrm{GCD} (a, b)\).
Numbers \(a, b, c\) are integers with \(a\) and \(b\) being coprime. Let us assume that integers \(x_0\) and \(y_0\) are a solution for the equation \(ax + by = c\).
Prove that every solution for this equation has the same form \(x = x_0 + kb\), \(y = y_0 - ka\), with \(k\) being a random integer.
Could it be that a) \(\sigma(n) > 3n\); b) \(\sigma(n) > 100n\)?
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.
Solve the equations in integers:
a) \(3x^2 + 5y^2 = 345\);
b) \(1 + x + x^2 + x^3 = 2^y\).