Problem of the Week

Is there is a convex quadrilateral which the diagonals divide into four triangles with areas of distinct primes?

The solution will be posted after the submission deadline.

Determine all the numbers formed by three different and non-zero digits, such that the six numbers obtained by permuting these digits leave the same remainder after division by 4.

Label the vertices of the heptagon A, B, C, D, E, F, and G. Draw a regular pentagon ABCDP so that it shares the two 108° angles.

Note that angle PDE is 168°-108°=60°. Because PD=DE, triangle PDE is equilateral. By symmetry, triangle PAG is also equilateral. So PEFG is a rhombus.

Hence, y° = angle EPG = 360°-60°-108°-60° = 132°.

Thus, y = 132.

Let H be a convex, equilateral heptagon whose angles measure 168°, 108°, 108°, 168°, x°, y°, and z° in clockwise order. Computer the number y.

The drunk passenger randomly chooses a seat. If they sit in their own seat, everyone else gets their assigned seat, so the 100th passenger gets theirs. If they choose someone else’s seat, that passenger will eventually have to choose a random empty seat. This creates a chain that eventually reaches seat 1 or seat 100. The 100th passenger has an equal possibility of ending up in each one, so the probability is 1/2.

A line of 100 passengers is waiting to board a plane with 100 assigned seats. The first passenger in line is completely drunk so sits in a random seat. Every subsequent passenger either sits in their assigned seat if it's empty, or picks a random empty seat if theirs is taken. You are the 100th (last) passenger. What is the probability that you get to sit in your actual assigned seat?