Problems

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.

Prove that if the numbers $x,y,z$ satisfy the following system of equations for some values of $p$ and $q$: $$\begin{array}{rl}y& =x^2+px+q,\\ z& =y^2+py+q,\\ x& =z^2+pz+q,\end{array}$$ then the inequality $x^{2}y+y^{2}z+z^{2}x\ge x^{2}z+y^{2}x+z^{2}y$ is satisfied.

On the $xy$-plane shown below is the graph of the function $y=a{x}^2+c$. 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?

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 $A{M}_1$ and $B{M}_2$. Points $A_1$ and $B_1$ are defined similarly. Prove that the lines $A{A}_1$, $B{B}_1$ and $C{C}_1$ intersect at the same point.

Solve the following inequality: $x+y^2+\sqrt{x-y^2-1}\le 1$.

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 $n$.