PUMaC 2017 · 团队赛 · 第 16 题

PUMaC 2017 — Team Round — Problem 16

(13) Robert is a robot who can move freely on the unit circle and its interior, but is attached to the origin by a retractable cord such that at any moment the cord lies in a straight line on the ground connecting Robert to the origin. Whenever his movement is counterclockwise (relative to the origin), the cord leaves a coating of black paint on the ground, and whenever his movement is clockwise, the cord leaves a coating of orange paint on the ground. The paint is dispensed regardless of whether there is already paint on the ground. The paints covers 1 2 gallon/unit , and Robert starts at (1 , 0). Each second, he moves in a straight line from the point (cos( θ ) , sin( θ )) to the point (cos( θ + α ) , sin( θ + α )), where α changes after each movement. ◦ ◦ α starts out as 253 and decreases by 2 each step. If he takes 89 steps, then the difference, in gallons, between the amount of black paint used and orange paint used can be written as √ √ a − b ◦ cot 1 , where a , b and c are positive integers and no prime divisor of c divides both a c and b twice. Find a + b + c .

解析

The black paint covers the area | sin θ | and the orange paint covers the area | sin θ | . 0 ≤ θ<π π ≤ θ< 2 π ∑ The difference is thus actually equal to sin θ , where these values of θ are the angles pivoted θ 88 ∑ ◦ ◦ by Robert. θ ( k ) = (253 − 2 k ) for 0 ≤ k < 89. Thus we wish to compute sin(253 − 2 n ) = k =0 88 m ∑ ∑ ◦ ◦ sin(88 + 164 − 2 k + 1) . Let m = 88 , n = 164 , θ = 1 . We wish to evaluate sin(( m + k =0 k =0 m ∑ 1 n − 2 k + 1) θ ) = sin(( m + n − 2 k + 1) θ ) sin θ . Use the product-to-sum rule to obtain a sin θ k =0 telescoping series; use the sum-to-product rule after collapsing the telescope to see that the sum √ √ sin(( m +1) θ ) sin(( n +1) θ ) ◦ ◦ 6 − 2 ◦ equals . Using our values, this is equal to sin 15 cot 1 = cot 1 . sin θ 4 Thus our answer is 6 + 2 + 4 = 12 . Problem written by Zack Stier and Matt Tyler