2012 USAJMO Problems/Problem 4

Revision as of 17:57, 25 April 2012 by Nsato (talk | contribs) (Created page with "== Problem == Let <math>\alpha</math> be an irrational number with <math>0 < \alpha < 1</math>, and draw a circle in the plane whose circumference has length 1. Given any integ...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Problem

Let $\alpha$ be an irrational number with $0 < \alpha < 1$, and draw a circle in the plane whose circumference has length 1. Given any integer $n \ge 3$, define a sequence of points $P_1$, $P_2$, $\dots$, $P_n$ as follows. First select any point $P_1$ on the circle, and for $2 \le k \le n$ define $P_k$ as the point on the circle for which the length of arc $P_{k - 1} P_k$ is $\alpha$, when travelling counterclockwise around the circle from $P_{k - 1}$ to $P_k$. Supose that $P_a$ and $P_b$ are the nearest adjacent points on either side of $P_n$. Prove that $a + b \le n$.

Solution

See Also

2012 USAJMO (ProblemsResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5 6
All USAJMO Problems and Solutions
Invalid username
Login to AoPS