A convex polygon on the plane contains at least \(m^2+1\) points with integer coordinates. Prove that it contains \(m+1\) points with integers coordinates that lie on the same line.
Suppose a football team scores at least one goal in each of the \(20\) consecutive games. If it scores a total of \(30\) goals in those \(20\) games, prove that in some sequence of consecutive games it scores exactly \(9\) goals total.
The prime factorization of the number \(b\) is \(2 \times 5^2 \times 7 \times 13^2 \times 17\). The prime factorization of the number \(c\) is \(2^2 \times 5 \times 7^2 \times 13\). Is the first number divisible by the second one? Is the product of these two numbers, \(b \times c\), divisible by \(49000\)?
Find all the prime numbers \(p\) such that there exist natural numbers \(x\) and \(y\) for which \(p^x = y^3 + 1\).
Determine all prime numbers \(p\) such that \(5p+1\) is also prime.
Find all natural numbers \(n\) for which there exist integers \(a,b,c\) such that \(a+b+c = 0\) and the number \(a^n + b^n + c^n\) is prime.
Find all the prime numbers \(p\) such that the number \(2p^2+1\) is also prime.
A graph is called Bipartite if it is possible to split all its vertices into two groups in such a way that there are no edges connecting vertices from the same group. Find out whic of the following graphs are bipartite and which are not:
Show that a bipartite graph with \(n\) vertices cannot have more than \(\frac{n^2}{4}\) edges.
In the numeral system with base \(k\) Alex is \(50\) years old. Next year he will be \(100\) years old in the numeral system with base \(k-1\). Find the age of Alex and value of \(k\) as decimal numbers.