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 \(y\).
In a parallelogram \(ABCD\), point \(E\) belongs to the side \(CD\) and point \(F\) belongs to the side \(BC\). Show that the total red area is the same as the total blue area:
The figure below is a regular pentagram. What is larger, the black area or the blue area?
In a square, the midpoints of its sides were marked and some segments were drawn. There is another square formed in the centre. Find its area, if the side of the square has length \(10\).
In a parallelogram \(ABCD\), point \(E\) belongs to the side \(AB\), point \(F\) belongs to the side \(CD\) and point \(G\) belongs to the side \(AD\). What is more, the marked red segments \(AE\) and \(CF\) have equal lengths. Prove that the total grey area is equal to the total black area.