Answers to the 2006 IUPUI High School Mathematics Contest. 1 n. n 1. n + n (n 1)

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2 Answers to the 2006 IUPUI High School Mathematics Contest. The sum S n of the first n terms of the sequence a, a 2,... of positive real numbers satisfies the equation a n + a n = 2S n for n =, 2,.... Find a formula for the general term a n. Solution. Using the quadratic formula we can compute in turn a =, a 2 = 2, a 3 = 3 2 with partial sums S =, S 2 = 2, S 3 = 3. Hence we can conjecture that a n = and S n = n. We wish to show that n + = 2 n, n If we rationalize the fraction, n + n + n (n ) = 2 n 2. Let P be a point on the bisector of an angle BAC. Let l be any line passing through P. Assume that l intersects the rays AB and AC at X and Y, respectively. Show that the quantity + does not depend on the choice of l. Solution by plane geometry. Solutions using trigonometry are also possible. X B D A E Y P C Construct PD parallel to, forming isosceles triangle ADP. Similarly, construct PE parallel to, forming the congruent isosceles triangle A. As triangles Y and Y are similar, = EY DP. Also as triangles XDP and X are similar, + DP = EY + DX = DX. Adding, Because triangles ADP and A are congruent and isosceles, = DP = AE = AD. So ( + ) = EY + DX = AE + AD = 2 AE ( + )

3 Solving, we find which is independent of l. + = 2 AE + = AE 3. Find a formula for the sum }{{ }. n Solution. Let h equal the sum. Expanding each term in powers of 0 we have h = + ( + 0) + ( ) + ( n ) Each term in parentheses is a geometric series. Using we find ( k ) = 0k+ 0 = 0k+ h = n = ( n ) n = 0 ( n ) n = 0 0 n n = 0n+ 0 n 8 4. In a strange world there are n airports arranged around a giant circle, with exactly one airplane at each airport initially. Every day, exactly two of the airplanes fly, each going to one of its adjacent airports. Can the airplanes ever all gather at one airport? Solution built on that of Xingping Shen, Carmel High School. Number the airports and planes a, a 2,..., a n and p, p 2,..., p n going around the circle. We will try to gather the planes at a n. This is possible if n is odd or a multiple of 4, and impossible otherwise. Case : n is odd. As p n does not need to be moved, there are an even number of planes to move. For k n 2, p k and p n k need to move the same distance. Group each p k with p n k and each day fly one such pair one flight closer to a n. Case 2: n = 2m and m is even. Pair all of the planes except p m = p n/2 and fly them to a n as in Case. Then fly p m one step towards a n every day, paired with one other airplane that flies alternately out of and back to a n. As m is even, they arrive at a n on the same day. Case 3: If n = 2m and m is odd, the solution of Case 2 doesn t work, but we need to show there is no other solution that could work. Label airports alternately X and O around the circle. Initially the number of planes at airports marked X is m, #(X) = m, and also #(O) = m. Each day, #(X) and #(O) either each change by 2 or remain the same, depending on which airplanes fly. So if m is odd, neither #(X) nor #(O) can ever become 0.

4 2006 IUPUI/Roche Diagnostics High School Mathematics Contest Winners First Prize Winner Xingping Shen, Sophomore, Carmel High School, Teacher: Mrs. Kathie Freed Second Prize Winners Hao Yang, Junior, Carmel High School, Teacher: Mr. Matthew Wernke Tan (Tyler) Zou, Sophomore, Carmel High School, Teacher: Mrs. Kathie Freed Ruofan Xia, Freshman, Carmel High School, Teacher: Ms. Laura Diamente Tianyi Zhang, Freshman, Carmel High School, Teacher: Mrs. Sohalski Nan Lin, Senior, Ben Davis High School, Teacher: Mr. Richard Elmore Third Prize Winners Hans Zhao, Senior, Carmel High School, Teacher: Mrs. Kathie Freed Bernabe Davila, Senior, Hamilton Southeastern High School, Teacher: Mrs. Letitia McCallister Sam Tucker, Junior, North Central High School, Teacher: Mr. Paul Brown Dewei (David) Yang, Freshman, Carmel High School, Teacher: Mr. Matthew Wernke Adam Aisen, Junior, Carmel High School, Teacher: Mrs. Kathie Freed Carlin Ma, Senior, Carmel High School, Teacher: Mrs. Kathie Freed Ziwei Zhong, Sophomore, Carmel High School, Teacher: Mr. Matthew Wernke Matthew Croop, Sophomore, North Central High School, Teacher: Mr. Paul Brown Fred Pai, Junior, Hamilton Southeastern High School, Teacher: Ms. Susan Stephen Wolf, Junior, Hamilton Southeastern High School, Teacher: Ms. Susan Honorable Mention Winners Ravi Parikh, Junior, Park Tudor High School, Teacher: Mrs. Joanne Black Walter Bruen, Senior, Brebeuf Jesuit, Teacher: Mrs. Sandra Layceck Yichuan Shi, Senior, Broad Ripple High School, Teacher: Mrs. Peggy Boulden Khuchtumur Bum-Erdene, Senior, Southport High School, Teacher: Mr. Tim O Brien Paul Lee, Freshman, Carmel High School, Teacher: Mrs. Janice Mitchener Yili Shi, Sophomore, Carmel High School, Teacher: Ms. Laura Diamente Payton Lee, Senior, Carmel High School, Teacher: Mrs. Kathie Freed Jessica Ranucci, Senior, Park Tudor, Teacher: Mrs. Joanne Black Yingxue Li, Freshman, Carmel High School, Teacher: Mrs. Janice Mitchener Brian Thomas, Senior, Hamilton Southeastern High School., Teacher: Mrs. Letitia McCallister Jason Broedel, Sophomore, Brownsburg High School, Teacher: Ms. Cassie Lee Lauren Cote, Senior, North Central High School, Teacher: Mr. Paul Brown Parth Patel, Sophomore, Hamilton Southeastern High School, Teacher: Ms. Susan

5 Honorable Mention Winners Continued Derek Paul, Sophomore, Hamilton Southeastern High School, Teacher: Ms. Susan Jason Holmes, Sophomore, Hamilton Southeastern High School, Teacher: Ms. Susan Helen Yu, Junior, Carmel High School, Teacher: Mrs. Janet Mitchener Richard Fogle, Sophomore, North Central High School Anna Krayterman, Junior, Hamilton Southeastern High School, Teacher: Ms. Susan