HMMT 二月 2004 · 冲刺赛 · 第 2 题

HMMT February 2004 — Guts Round — Problem 2

[5] If the three points (1 , a, b ) ( a, 2 , b ) ( a, b, 3) are collinear (in 3-space), what is the value of a + b ?

解析

Now, in general, let p = min( { p , . . . , p } \ { 2 } ). Suppose 1 < j < p . By induction, 1 n we have a = ( j + 1) a / 2 when j is odd, and a = a / 2 when j is even. So a 6 = a for j 1 j 1 i 1 all 1 < j < p . It follows that a = a / 2 p . Then, again using induction, we get for all p 1 nonnegative integers k that a = a if k is even, and a = ( p + k + 1) a if k is odd. p + k p p + k p Clearly, a 6 = a and p + k + 1 6 = 2 p when k is odd (the left side is odd, and the right p 1 side even). It follows that a = a for no j > 1. Finally, when a = 2, we can check j 1 1 inductively that a = j + 1 for j odd and a = 1 for j even. j j So our answer is just the number of odd elements in S . There are 9 odd prime numbers 9 smaller than 30, so the answer is 2 = 512.