Tim had more hazelnuts than Tom. If Tim gave Tom as many hazelnuts as Tom already had, Tim and Tom would have the same number of hazelnuts. Instead, Tim gave Tom only a few hazelnuts (no more than five) and divided his remaining hazelnuts equally between \(3\) squirrels. How many hazelnuts did Tim give to Tom?
Prove that \(n^3 - n\) is divisible by \(24\) for any odd \(n\).
Show that if numbers \(a-b\) and \(c-d\) are divisible by \(11\), then \(ac-bd\) and \(ad - bc\) are also both divisible by \(11\).
For how many pairs of numbers \(x\) and \(y\) between \(1\) and \(100\) is the expression \(x^2 + y^2\) divisible by \(7\)?
Seven robbers are dividing a bag of coins of various denominations. It turned out that the sum could not be divided equally between them, but if any coin is set aside, the rest could be divided so that every robber would get an equal part. Prove that the bag cannot contain \(100\) coins.
Show that the equation \(x^2 +6x-1 = y^2\) has no solutions in integer \(x\) and \(y\).
On the \(xy\)-plane shown below is the graph of the function \(y=ax^2 +c\). At which points does the graph of the function \(y=cx+a\) intersect the \(x\) and \(y\) axes?
Find the largest natural number \(n\) which satisfies \(n^{200} <5^{300}\).
Does there exist a natural number which, when divided by the sum of its digits, gives a quotient and remainder both equal to the number 2011?
Solve the following inequality: \(x+y^2 +\sqrt{x-y^2-1} \leq 1\).