3

Part of 2013 BMT Spring

Problems(5)

BMT 2013 Spring - Geometry 3

Source:

Given a regular tetrahedron

ABCD

O

\sin \angle AOB

2013 BMT Team 3

Source:

A round robin tennis tournament is played among

4

friends in which each player plays every other player only one time, resulting in either a win or a loss for each player. If overall placement is determined strictly by how many games each player won, how many possible placements are there at the end of the tournament? For example, Andy and Bob tying for first and Charlie and Derek tying for third would be one possible case.

BMT 2013 Spring - Discrete 3

Source:

Suppose we have

2013

piles of coins, with the

i

th pile containing exactly

i

coins. We wish to remove the coins in a series of steps. In each step, we are allowed to take away coins from as many piles as we wish, but we have to take the same number of coins from each pile. We cannot take away more coins than a pile actually has. What is the minimum number of steps we have to take?

BMT 2013 Spring - Analysis 3

Source:

\lim_{x\to0}\frac{\sin2x}{e^{3x}-e^{-3x}}

limitsreal analysis

2013 BMT Individual 3

Source:

Two boxes contain some number of red, yellow, and blue balls. The first box has

3

4

5

blue balls, and the second box has

6

2

7

blue balls. There are two ways to select a ball from these boxes; one could first randomly choose a box and then randomly select a ball or one could put all the balls in the same box and simply randomly select a ball from there. How much greater is the probability of drawing a red ball using the second method than the first?