Fun problems

The following are a list of some of my favourite problems. Try to solve them before you check the solutions!!

• Which is bigger, e^{Pi} or Pi^{e} ? Do this without a calculator! solution
• If you put 3 points at random on a circle, what is the probability that the points will lie on a semicircle? solution
• Show that the sum of the lengths of the perpendiculars dropped from a random point inside an equilateral triangle is equal to the length of the perpendicular dropped from one vertex to the opposite side. solution
• Consider the picture below: We start the picture by drawing a 1 x 1 square. We then add a rectangle of area 1, placing it on the right side, to form a new rectangle. Next, we add a rectangle of area 1 to the top to form a new rectangle. We continue alternating the right and top, always adding a rectangle of area 1 to form a new rectangle. What is the limit of the ratio of the length to the height? solution

  
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• Putnam A2 1997: Players 1,2,3,...,n are seated around a table and each has a single penny. Player 1 passes a penny to Player 2, who then passes two pennies to Player 3. Player 3 then passes one penny to Player 4, who passes two pennies to Player 5, and so on, players alternatively passing one penny or two to the next player who still has some pennies. A player who runs out of pennies drops out of the game and leaves the table. Find an infinite set of numbers n for which some player ends up with all n pennies.
• Putnam A5 1997: Let N_n denote the number of ordered n-tuples of positive integers (a_1,a_2,...,a_n) such that 1/a_1 + 1/a_2 + ... + 1/a_n = 1. Determine whether N_(10) is even or odd.