18th Putnam 1958

Problem B1

Do both (1) and (2):

(1)   Given real numbers a, b, c, d with a > b, c, d, show how to construct a quadrilateral with sides a, b, c, d and the side length a parallel to that length b. What conditions must a, b, c, d satisfy?

(2)   H is the foot of the altitude from A in the acute-angled triangle ABC. D is any point on the segment AH. BD meets AC at E, and CD meets AB at F. Show that ∠AHE = ∠AHF.



(1) A necessary and sufficient condition is that we should be able to form a triangle with sides c, d, (a - b). Hence we require b + c + d ≥ a, a + d ≥ b + c, and a + c ≥ b + d. Construct two such triangles, one at each end of the segment a, then join the two tops to get the required quadrilateral.

(2) Take a line through A parallel to BC and project CF, HF, HE, BE to meet it in W, X, Y, Z. Then using similar triangles, we deduce that AY/CH = AZ/BC = BH·AW/CH (1/BC) = BH/CH AW/BC = BH/CH AX/BH = AX/CH, so AX = AY. But AH is perpendicular to XY, so the result follows.

Comment. (2) is unusually hard for a question 1, let alone half a question 1!



18th Putnam 1958

© John Scholes
25 Feb 2002