Thanks everyone for trying out the problem, and congrats to those who got the problem right!
A pair of standard 6-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference.
First of all, we need to find all the possible diameters of the circle. The question states the numerical value of the area of the circle is less than the numerical value of the circle's circumference. If that is the case, then the radii will be less than 2 because
πr2 < 2πr
πr < 2π
r < 2
If a pair of standard 6-sided die is rolled once, then the diameter will have to be either 2 or 3, because the lowest value you can roll is a 2, and a 4 won't work since the radii has to be less than 2.
There is only 1 possible way to roll a 2 (1,1), and 2 possible ways to roll a 3 (1,2) (2,1).
There are 3 possible rolls out of 36 (6*6) different rolls. Thus, the answer is 1/12.
Here's the next problem:
At a competition with N players, the number of players given elite status is equal to 21+⌊log2(N-1)⌋ - N. Suppose that 19 players are given elite status. What is the sum of the two smallest possible values of N?
⌊x⌋ is the nearest integer ≤ x
Good luck! and remember, no calculators are allowed. :)