Problems

On an 8×8 grid (like a chessboard), an L-corner is a shape made of 3 little squares of the board that touch to make an L. You can turn the L any way you like. We place the L-corners so that none overlap. What is the fewest L-corners you must place so that no more L-corners can be added anywhere? Here is an example of how three L-corners may look like:

An airline flew exactly 10 flights each day over the course of 92 days. Each day, each plane flew no more than one flight. It is known that for any two days in this period there will be exactly one plane which flew on both those days. Prove that there is a plane that flew every day in this period.

10 children, including Billy, attended Billy’s birthday party. It turns out that any two children picked from those at the party share a grandfather. Prove that 7 of the children share a grandfather.

Three circles are constructed on a triangle, with the medians of the triangle forming the diameters of the circles. It is known that each pair of circles intersects. Let \(C_{1}\) be the point of intersection, further from the vertex \(C\), of the circles constructed from the medians \(AM_{1}\) and \(BM_{2}\). Points \(A_{1}\) and \(B_{1}\) are defined similarly. Prove that the lines \(AA_{1}\), \(BB_{1}\) and \(CC_{1}\) intersect at the same point.

The function \(f (x)\) is defined on the positive real \(x\) and takes only positive values. It is known that \(f (1) + f (2) = 10\) and \(f(a+b) = f(a) + f(b) + 2\sqrt{f(a)f(b)}\) for any \(a\) and \(b\). Find \(f (2^{2011})\).

On a chessboard, \(n\) white and \(n\) black rooks are arranged so that the rooks of different colours cannot capture one another. Find the greatest possible value of \(n\).

There are a thousand tickets with numbers 000, 001, ..., 999 and a hundred boxes with the numbers 00, 01, ..., 99. A ticket is allowed to be dropped into a box if the number of the box can be obtained from the ticket number by erasing one of the digits. Is it possible to arrange all of the tickets into 50 boxes?

2011 numbers are written on a blackboard. It turns out that the sum of any of these written numbers is also one of the written numbers. What is the minimum number of zeroes within this set of 2011 numbers?

Does there exist a real number \({\alpha}\) such that the number \(\cos {\alpha}\) is irrational, and all the numbers \(\cos 2{\alpha}\), \(\cos 3{\alpha}\), \(\cos 4{\alpha}\), \(\cos 5{\alpha}\) are rational?

The sequence of numbers \(a_1, a_2, \dots\) is given by the conditions \(a_1 = 1\), \(a_2 = 143\) and

for all \(n \geq 2\).

Prove that all members of the sequence are integers.