MathDB

3

Part of 2005 National High School Mathematics League

Problems(2)

Geometry

Source: 2005 National High School Mathematics League, Exam One, Problem 3

3/19/2020
△ABC\triangle ABC is inscribed to unit circle. Bisector of ∠A,∠B,∠C\angle A,\angle B,\angle C intersect the circle at A1,B1,C1A_1,B_1,C_1 respectively. The value of AA1⋅cos⁡A2+BB1⋅cos⁡B2+CC1⋅cos⁡C2sin⁡A+sin⁡B+sin⁡C\frac{\displaystyle AA_1\cdot\cos\frac{A}{2}+BB_1\cdot\cos\frac{B}{2}+CC_1\cdot\cos\frac{C}{2}}{\sin A+\sin B+\sin C} is (A)2(B)4(C)6(D)8\text{(A)}2\qquad\text{(B)}4\qquad\text{(C)}6\qquad\text{(D)}8
trigonometrygeometry
It Will Take a Lot of Calculation

Source: 2005 National High School Mathematics League, Exam Two, Problem 3

3/20/2020
For positive integer nn, define f(n)={0,if n is a perfect square[1{n}],if n is not a perfect squaref(n)=\begin{cases} 0, \text{if }n\text{ is a perfect square}\\ \displaystyle \left[\frac{1}{\{\sqrt{n}\}}\right], \text{if }n\text{ is not a perfect square}\\ \end{cases}. Find the value of ∑k=1240f(k)\sum_{k=1}^{240} f(k). Note: [x][x] is the integral part of real number xx, and {x}=x−[x]\{x\}=x-[x].
calculusintegration