HMMT 二月 2023 · ALGNT 赛 · 第 3 题
HMMT February 2023 — ALGNT Round — Problem 3
题目详情
- Suppose x is a real number such that sin(1 + cos x + sin x ) = . Compute cos(1 + sin x + cos x ). 14 2
解析
- Suppose x is a real number such that sin(1 + cos x + sin x ) = . Compute cos(1 + sin x + cos x ). 14 Proposed by: Ankit Bisain, Luke Robitaille, Maxim Li, Milan Haiman, Sean Li √ 3 3 Answer: − 14 4 2 2 4 Solution: We first claim that α := 1 + cos x + sin x = 1 + sin x + cos x . Indeed, note that 4 4 2 2 2 2 2 2 sin x − cos x = (sin x + cos x )(sin x − cos x ) = sin x − cos x, 2 4 which is the desired after adding 1 + cos x + cos x to both sides. √ 13 3 3 2 Hence, since sin α = , we have cos α = ± . It remains to determine the sign. Note that α = t − t +2 14 14 2 2 where t = sin x . We have that t is between 0 and 1. In this interval, the quantity t − t +2 is maximized at t ∈ { 0 , 1 } and minimized at t = 1 / 2, so α is between 7 / 4 and 2. In particular, α ∈ ( π/ 2 , 3 π/ 2), so √ 3 3 cos α is negative. It follows that our final answer is − . 14 Remark. During the official contest, 258 contestants put the (incorrect) positive version of the answer √ 3 3 and 105 contestants answered correctly. This makes the second most submitted answer to an 14 Algebra/Number Theory problem, beat only by the correct answer to question 1. Figure : Milan. Here is how the problem was written: • Luke wanted a precalculus/trigonometry problem on the test and enlisted the help of the other problem authors, much to Sean’s dismay. 2 2 4 • Maxim proposed “given sin x + cos x , find sin x + cos x .” 2 4 4 4 2 2 • Ankit observed that sin x + cos x = cos x + sin x and proposed “given cos x + sin x , find 2 4 sin x + cos x .” • Luke suggested wrapping both expressions with an additional trig function and proposed “given √ 4 3 2 2 4 sin(cos x + sin x ) = , find cos(sin x + cos x ).” 2 • Milan suggested using the fact that fixing sin α only determines cos α up to sign and proposed √ 4 2 2 3 4 “given sin(1 + cos x + sin x ) = , find cos(1 + sin x + cos x ).” 2 4 2 • Sean noted that the function sin(1+cos x +sin x ) only achieves values in the range [sin 2 , sin 7 / 4] 4 2 and suggested to make the answer contain radicals, and proposed “given sin(1 + cos x + sin x ) = 2 13 4 , find cos(1 + sin x + cos x ).” 14 2