Download Contest Problem Book IV: Annual High School Examinations, by Ralph Artino, Anthony Gaglione and Niel Shell (Compiled with PDF

By Ralph Artino, Anthony Gaglione and Niel Shell (Compiled with solutions)

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Additional info for Contest Problem Book IV: Annual High School Examinations, 1973-1982 (Anneli Lax New Mathematical Library 29)

Sample text

If M is the midpoint of BC and AM = 3, what is the length of BC? (A) 2V47 (B) 2T (C) 9 (D) 4 + 2VP (E) not enough information given to solve the problem A B M C 21. Suppose f(x) is defined for all real numbers x; f(x) > 0 for all x; and f(a)f(b) = f(a + b) for all a and b. Which of the fol- lowing statements are true? I. f(O) = 1 111. f(a) = 3f(3a) for all a 11. f(-a) = I/f(a) for all a IV. f(b) > f(a) if b > a (A) III and IV only (B) 1, III and IV only (C) 1,11 and IV only (D) I, II and III only (E) All are true.

A) 7 (B) 9 (C) 11 (D) 17 (E) 19 24 THE MAA PROBLEM BOOK IV 22. Given an equilateral triangle with side of length s, consider the locus of all points P in the plane of the triangle such that the sum of the squares of the distances from P to the vertices of the triangle is a fixed number a. This locus (A) is a circle if a > 2 (B) contains only three points if a = 2S2 and is a circle if a > 2s 2 (C) is a circle with positive radius only if s2 < a < 2s 2 (D) contains only a finite number of points for any value of a (E) is none of these 23.

Then the length of PQ is (A) the average of the lengths of the internal and external common tangents (B) equal to the length of an external common tangent if and only if circles 0 and O' have equal radii ) (C) always equal to the ICnI1LIr1 VI dcli za common tangent (D) greater than the length of an external common tangent (E) the geometric mean of the lengths of the internal and external common tangents 27. If N= Vi + 2 + Hi-- 2 -F3 2r2 then N equals (A) I (B) 2V - I (C) 2 C5 (D) F5 4 (E) none of these 28.

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