Problems

Below is a regular octagon. Given that its side length is \(1\), what’s the difference between the area of the red rectangle and the rest of the octagon?

\(x\), \(y\) and \(z\) are all integers. We’re told that \[\begin{align} x^3yz&=6\\ xy^3z&=24\\ xyz^3&=54. \end{align}\] What’s \(xyz\)?

In the diagram, all the small squares are of the same size. What fraction of the large square is shaded?

The letters \(O\), \(P\), \(S\) and \(T\) represent different digits from \(1\) to \(9\). The same letters correspond to the same digits, while different letters correspond to different digits.

Find \(O+P+S+T\), given that \(SPOT+POTS=15,279\).

Picasso colours every point on the circumference of a circle red or blue. Is he guaranteed to create an equilateral triangle all of whose vertices are the same colour?

Let \(A\), \(B\), \(C\), \(D\), \(E\) be five different points on the circumference of a circle in that (cyclic) order. Let \(F\) be the intersection of chords \(BD\) and \(CE\). Show that if \(AB=AE=AF\) then lines \(AF\) and \(CD\) are perpendicular.

Let \(a\), \(b\) and \(c\) be positive real numbers such that \(a+b+c=3\). Prove that \(a^a+b^b+c^c\ge3\).

David and Esther play the following game. Initially, there are three piles, each containing 1000 stones. The players take turns to make a move, with David going first. Each move consists of choosing one of the piles available, removing the unchosen pile(s) from the game, and then dividing the chosen pile into 2 or 3 non-empty piles. A player loses the game if they are unable to make a move. Prove that Esther can always win the game, no matter how David plays.

Rational numbers \(x,y,z\) are such that all the numbers \(x+y^2+z^2\), \(x^2+y+z^2\), \(x^2+y^2+z\) are integers. Prove that \(2x\) is also an integer.

A grasshopper can only make jumps exactly \(5\) inches in length. He wants to visit all \(8\) dots on the picture, where the length of the side of a unit square is one inch. Find the smallest number of jumps he will have to do if he can start and finish in any dot. It is allowed to use any point on the plane, not necessarily the ones on the picture.