PUMaC 2009 · 代数（A 组） · 第 4 题

PUMaC 2009 — Algebra (Division A) — Problem 4

Find the smallest positive α (in degrees) for which all the numbers n cos α, cos 2 α, . . . , cos 2 α, . . . are negative. n 1 n

解析

Find the smallest positive α (in degrees) for which all the numbers n cos α, cos 2 α, . . . , cos 2 α, . . . are negative. Solution. 120. The answer can be guessed if one tries a few ’famous’ values (such as π/ 4 , π/ 3 etc.). The proof requires a bit more work. The cosine function is periodic, hence we can restrict our attention to the interval [0 , 2 π ]. Also, cos α < 0, so we are in the interval ( π/ 2 , 3 π/ 2). If there is a good α in the interval ( π, 3 π/ 2), then α/ 2 is also good and smaller than α . Also, α 6 = π so we obtained that if there is a smallest α , it must satisfy π/ 2 < α < π . n Generally the condition cos 2 α < 0 means that there is an integer l such that π/ 2 + 2 l π < n n n 2 α < 3 π/ 2 + 2 l π , or with the notation x = 2 α/π : n n 4 l + 1 < 2 x < 4 l + 3 n n From the inequality on α we have [ x ] = 1 (where [ . ] denotes the floor function). We prove that n in base 4, x can be written as x = 1 . 11111 . . . . The inequality 4 l + 1 < 2 x < 4 l + 3 means n n n exactly that 2 x written in base 4 has the digit 1 or 2 on the unit position. Multiplying by an appropriate power of 4, any digit in the expansion of x can be shifted to the unit position

hence every digit of x is 1 or 2. This also holds for 2 x : multiplying by an appropriate power of 2, any digit in the expansion of 2 x can be shifted to the position of the units, and so every digit of 2 x is 1 or 2. From this it follow that all digits of x are 1. Suppose the contrary, then m − 1 x = 1 . 1111 . . . 112 . . . (where there are m 1-s after the dot), and so 4 x = 11 . . . 11 . 2 . . . . Multiplying this by 2, the digit that appears on the unit position is 3, since the fractional 2 m − 1 2 m − 1 part is greater than 1 / 2. Hence 2 x = 22 . . . 223 . 0 . . . or 2 x = 22 . . . 223 . 1 . . . and this n is a contradiction with the fact that the digit on the unit position of 2 x is 1 or 2. Hence x = 1 . 11111 . . . = 4 / 3 and α = 2 π/ 3. n n 1