Problems

\(6\) friends get together for a game of three versus three basketball. In how many ways can they be split into two teams? The order of the two teams doesn’t matter, and the order within the teams doesn’t matter.

That is, we count A,B,C vs. D,E,F as the same splitting as F,D,E vs A,C,B.

\(x\), \(y\) 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?

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.

Prove that the product of five consecutive integers is divisible by \(120\).

Prove that the vertices of a planar graph can be coloured in (at most) six different colours such that every pair of vertices joined by an edge are of different colours.

Note: a graph is planar if it can be drawn in the plane with no edges crossing. For example, three houses, each of which is connected to three utilities, is not a planar graph.
You may find it useful to use the Euler characteristic: a planar graph with \(v\) vertices, \(e\) edges and \(f\) faces satisfies \(v-e+f=2\).

Paloma wrote digits from \(0\) to \(9\) in each of the \(9\) dots below, using each digit at most once. Since there are \(9\) dots and \(10\) digits, she must have missed one digit.

In the triangles, Paloma started writing either the three digits at the corners added together (the sum), or the three digits at the corners multiplied together (the product). She gave up before finishing the final two triangles.

What numbers could Paloma have written in the interior of the red triangle? Demonstrate that you’ve found all of the possibilities.