Problem # 237
What is the probability that a randomly chosen x ∈ (0, 1) satisfies
blog10 4xc = blog10 xc?
{The floor function bxc is the greatest integer less than or equal to x.}
Solution:
Answer:
1
6
Proof. The probability that 0 < x1 < x < x2 < 1 is given by
Rx2
dx = x2 − x1 .
x1
Since 0 < x < 1, −n ≤ log10 x < 0 for some positive integer n. For
blog10 xc = blog10 4xc, we must have −n ≤ log10 x < log10 4x < −n + 1. This holds
1
iff 10−n ≤ x < 4x < 10−n+1 , which requires 10−n ≤ x < 10−n+1 . For each n the
4
1 −n+1
1 −n+1
3
−n
probability that 10 < x < 10
is given by 10
− 10−n = 10−n . The
4
2
4 ∞
P
3 −n
1
required probability is then given by P =
10
= .
2
6
n=1
Source: Math. Circles, Prague, 2000.