2008 Mock ARML 1 Problems
Contents
[hide]Set 1
Problem 1
Compute all real values of such that .
Problem 2
A positive integer is a yo-yo if the absolute value of the difference between any two consecutive digits of is at least . Compute the number of -digit yo-yos.
Set 2
Problem 3
In regular hexagon with side length , intersects at , and intersects at . Compute the length of .
Problem 4
There are black balls and white ball in a hat. A turn consists of picking a ball from the hat and replacing it with one of the opposite color. Compute the probability that, after a sequence of turns, there are black balls in the hat before there are white balls.
Set 3
Problem 5
The positive real numbers are in arithmetic progression in that order. They also satisfy
Compute the common difference of this arithmetic progression.
Problem 6
Square has side length . is the midpoint of , and is the midpoint of . is on such that is between and , and . Compute the length of .
Set 4
Problem 7
Compute the number of -digit base- positive integer multiples of that are also divisible by when read in base instead of base .
Problem 8
For positive real numbers ,
Compute .