Problems

A convex polygon \(A_1A_2...A_n\) has the following property: if we parallel push all the lines containing the sides of the polygon for a distance \(1\) outside, we will obtain another polygon, similar to the original one with the corresponding parallel sides of the same ratio. Prove that one can inscribe a circle into the original polygon \(A_1A_2...A_n\).

For a triangle \(ABC\) denote by \(R\) the radius of the circle superscribed around \(ABC\), by \(r\) the radius of circle inscribed into \(ABC\). Prove that \(R\geq 2r\) and equality holds if and only if the triangle \(ABC\) is regular. For this problem, you may want to search about the “Euler Circle” online.

Under a homothety transformation, a line \(l\) is sent to a line \(l'\) which is parallel to \(l\).

Show that a homothety is uniquely determined by where it sends any two distinct points.

Consider a homothety with center \(O\) and coefficient \(k\). Which lines are sent to themselves by this homothety? (Hint: the answer will depend on \(k\))

(IMO 1999) Two circles with centres \(A\) and \(C\) intersect at the points \(B\) and \(G\), moreover the circle with centre \(C\) goes through \(A\). A big circle is tangent to both given circles at the points \(E\) and \(F\) (see picture). The line \(BG\) intersects the big circle at the points \(H\) and \(I\). The segments \(EH\) and \(EI\) intersect with the circle with centre \(C\) at the points \(H\) and \(K\) respectively. Prove that the segment \(JK\) is tangent to the circle with center at \(A\).

On the first day Robinson Crusoe tied the goat with a single piece of rope by putting one peg into the ground. What shape did the goat graze?

An equilateral triangle is drawn on a whiteboard and a point \(P\) is drawn somewhere inside this triangle. Show that regardless of where \(P\) is drawn (as long as its inside the triangle), the sum of the distances from \(P\) to each of the sides of the triangle will always be the same.

A teacher saw the calculation \(3\times 4 = 10\) written on the whiteboard. She was about to erase it, thinking it was wrong, but then wondered whether it might have been written in a different numeral system.

Is it possible that this multiplication is correct in some base? If so, which one?

On a distant planet called Hexaris, there live two alien species: the Blipnors and the Quantoodles.

The chief alien writes on a board: “There are \(100\) aliens on this planet. Of these, \(24\) are Blipnors and \(32\) are Quantoodles.”

At first this seems confusing — the numbers do not seem to add up! Then you remember that the aliens use a different base for their numeral system.

What base are they using?