1st Mexican 1987


A1. a/b and c/d are positive fractions in their lowest terms such that a/b + c/d = 1. Show that b = d.
A2. How many positive integers divide 20! ?
A3. L and L' are parallel lines and P is a point midway between them. The variable point A lies L, and A' lies on L' so that ∠APA' = 90^o. X is the foot of the perpendicular from P to the line AA'. Find the locus of X as A varies.
A4. Let N be the product of all positive integers ≤ 100 which have exactly three positive divisors. Find N and show that it is a square.
B1. ABC is a triangle with ∠A = 90^o. M is a variable point on the side BC. P, Q are the feet of the perpendiculars from M to AB, AC. Show that the areas of BPM, MQC, AQMP cannot all be equal.
B2. Prove that (n³ - n)(5⁸ⁿ⁺⁴ + 3⁴ⁿ⁺²) is a multiple of 3804 for all positive integers n.
B3. Show that n² + n - 1 and n² + 2n have no common factor.
B4. ABCD is a tetrahedron. The plane ABC is perpendicular to the line BD. ∠ADB = ∠CDB = 45^o and ∠ABC = 90^o. Find ∠ADC. A plane through A perpendicular to DA meets the line BD at Q and the line CD at R. If AD = 1, find AQ, AR, and QR.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.