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Which of the following statements are true? (A) In any triangle ABC, the minimum value of the sum [ sqrt(sinA) / (sqrt(sinB) + sqrt(sinC) - sqrt(sinA)) ] (cyclic sum) is 3. (B) If sec(A)tan(B) + tan(A)sec(B) = 30, then sec(A)sec(B) + tan(A)tan(B) = sqrt(901). (C) If sec(A)tan(B) + tan(A)sec(B) = 30, then sec(A)sec(B) + tan(A)tan(B) = sqrt(899). (D) Three lines L1: y - z - 1 = 0, x = 0; L2: z + x + 1 = 0, y = 0; L3: x - z - 1 = 0, y = 0 meet the xy-plane at points U, M, R respectively. The centroid of triangle UMR is (0, 1/3, 0).

1. Only A
2. A and B
3. Only C
4. A and D

Correct answer: A and B

Solution

A: By Nesbitt-type inequality with variables a = sqrt(sinA) etc., sum = sum a/(b+c-a) >= 3, equality at equilateral. Minimum = 3. TRUE. B/C: Let p = sec(A)tan(B)+tan(A)sec(B), q = sec(A)sec(B)+tan(A)tan(B). Then p² - q² = (secAtanB+tanAsecB)² - (secAsecB+tanAtanB)². Expanding: = sec²A*tan²B + tan²A*sec²B - sec²A*sec²B - tan²A*tan²B = sec²A(tan²B-sec²B) + tan²A(sec²B-tan²B) = -sec²A + tan²A = -(sec²A-tan²A) = -1. So q² = p² + 1 = 901, q = sqrt(901). B is TRUE, C is FALSE. D: L1 at z=0: x=0, y=1 → U=(0,1,0). L2 at z=0: x=-1, y=0 → M=(-1,0,0). L3 at z=0: x=1, y=0 → R=(1,0,0). Centroid = (0, 1/3, 0). D is TRUE. So A, B, D are true.

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