2001 IUPUI/Roche Diagnostics High School Math Contest Winners

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2 Answers to the 998 IUPUI/TMMI Mathematics Contest. Three squares adjoin each other as in the figure. Find the sum of angles A, B and C. Answer. One plane geometry proof uses the square PQRS constructed in the figure below. From this we see that angle A + B = 45 ffi,soa + B + C = 90 ffi. A trigonometric proof is the following: so that again A + B = 45 ffi. tan(a + B) = tan A +tanb tan A tan B = = 2. The Fibonacci sequence,,2,,5,8,,2,4,55,::: is defined by setting the first two terms equal to, and thereafter by letting each term be the sum of the previous two. In other words, a n+2 (term number n +2)is defined by a n+2 = a n+ + a n for n =; 2; ;:::. Prove that if n is divisible by m, then the nth term of the sequence is divisible by the mth term. For example, 8 is divisible by 4, and the 8th term (2) is divisible by the 4th term (). Answer 2. Fix n. Let us show by mathematical induction that a n evenly divides a n, a 2n ;a n ::: a kn ::: It is clear that a n evenly divides a n. We suppose then that a n evenly divides a kn and try to deduce from this that a n evenly divides a (k+)n = a kn+n. We know so using this rule repeatedly, a kn+2 = a kn+ + a kn a kn+ =2a kn+ + a kn = a a kn+ + a 2 a kn a kn+4 =a kn+ +2a kn = a 4 a kn+ + a a kn a kn+5 =5a kn+ +a kn = a 5 a kn+ + a 4 a kn and in general a kn+p = a p a kn+ + a p a kn : When we reach p = n we have a kn+n = a n a kn+ + a n a kn :

3 By the induction assumption, a n divides both a n and a kn, so it divides the right hand side of this equation, thus it must divide a kn+n.. Points A; B and C move counterclockwise along three coplanar circles. Each point moves with the same constant angular velocity with respect to the center of its circle. How does the centroid of triangle ABC move? Answer. The centers of the three circles determine a triangle whose center of gravity is a point P. The center of gravity of triangle ABC moves with the same angular velocity around a circle with center P. To see this we first establish some notation. Let C ; C 2 ; C be the vectors from the origin to the centers of the circles, let! be the constant angular velocity, let t be time, let r ;r 2 ;r be the radii of the circles, and let ffi ;ffi 2 ;ffi be the initial angles of points A; B and C. The points on the circle with center C n are obtained from C n + r n hcos(!t); sin(!t)i or equivalently from C n + cos(!t)(r n i)+sin(!t)(r n j) () where i and j are the unit vectors h; 0i and h0; i, respectively. With the starting angles incorporated, the location of the center of gravity of the triangle is X n (C n + r n hcos(!t + ffi n ); sin(!t + ffi n )i) Note that Pn (C n) is P, the vector from the origin to P. Now we substitute the trig identities for cos(!t + ffi) and sin(!t + ffi) and factor (this is long). The result is where n is the vector P + cos(!t)n +sin(!t)n? (2) n = hr cos(ffi )+r 2 cos(ffi 2 )+r cos(ffi );r sin(ffi )+r 2 sin(ffi 2 )+r sin(ffi )i and n? is the perpendicular vector n? = h r sin(ffi ) r 2 sin(ffi 2 ) r sin(ffi );r cos(ffi )+r 2 cos(ffi 2 )+r cos(ffi )i By comparing equatons (2) and (), we see the motion is around the circle with center P and radius the length of n, and has the same angular velocity. 2

4 The solution is shorter if we introduce the complex numbers and the function cis(!t) = cos(!t) + i sin(!t). Then the nth circle can be represented as C n + r n cis(!t + ffi n )=C n + r n cis(!t) cis(ffi n ) where now C n is a complex number rather than a vector. The center of gravity is X n [C n + r n cis(ffi n ) cis(!t)] = P + [r cis(ffi )+r 2 cis(ffi 2 )+r cis(ffi )] cis(!t) : Writing [r cis(ffi )+r 2 cis(ffi 2 )+r cis(ffi )] in the form r cis(fi), we see that the center of gravity is located at P + r cis(fi) cis(!t) =P + r cis(!t + fi) : 4. For a class with two or more students, show that at least two students have the same number of friends in the class. Assume that you cannot be your own friend. Also assume that if I am your friend, then you are my friend (and vice versa). Answer 4. Suppose there are n students in the room and that everyone of them has a different number of friends. No one can have fewer than 0 friends or more than n. Since there are n people in the room and exactly n numbers from 0 through n, these numbers must correspond to the people in the room in some order. The person with n friends is friends with everyone in the room, including the person with 0 friends. This is a contradiction, as the person with 0 friends would then have as a friend the person with n friends.

5 200 IUPUI/Roche Diagnostics High School Math Contest Winners First Prize Winner Jonathan Steven Landy, Warren Central High School. Teacher: Ms. Julia Oblon Second Prize Winners Sandy Ottensmann, Brebeuf Jesuit Preparatory School, Teacher: Ms. Joan Rocap Matthew Adam Fisher, Brebeuf Jesuit Preparatory School, Teacher: Mrs. Sandra Laycock Irene Yuan Sun, Ben Davis High School, Teacher: Mr. Richard Elmore Michael Chun Chang, Hamilton Southeastern High School, Teacher: Ms. Susan Wong Kathleen Wolter, Avon High School, Teacher: Mr. Tony Record Aileen Chen, Carmel High School, Teacher: Mrs. Nancy Schulenburg Ivan Y. Dremov, North Central High School, Teacher: Mrs. Jan Wendt Karen Lai, Carmel High School, Teacher: Mrs. Nancy Schulenburg Christopher Merryman, Carmel High School, Teacher: Mrs. Laura Diamente Patrick Joseph Mihelich, Park Tudor, Teacher: Mrs. Joanne Black Caitlin Elizabeth Pauckner, Brebeuf Jesuit Preparatory School, Teacher: Ms. Joan Rocap Michael Jonathan Star, Brebeuf Jesuit Preparatory School, Teacher: Mrs. Sandra Laycock Feng Tu, Hamilton Southeastern High School, Teacher: Ms. Susan Wong Matt Willsey, Roncalli High School, Teacher: Mrs. Bonnie Ramey Jerry Wu, Carmel High School, Teacher: Mrs. Kathie Freed Honorable Mention Winners David Bauman, Roncalli High School, Teacher: Mrs. Bonnie Ramey Barbara Brill, Roncalli High School, Teacher: Sr. Anne Frederick Erika Hollins Dantzig, Brebeuf Jesuit Preparatory School, Teacher: Mr. Tim Kelaghan Scott Adam Dial, Ben Davis High School, Teacher: Mr. Wallace Mack John Durham, Warren Central High School, Teacher: Mr. John Greenlee J. Alex Emerson, Hamilton Southeastern High School, Teacher: Ms. Susan Wong Greg Hasty, Warren Central High School, Teacher: Mrs. Janis Gaerte Amy L. Hoffman, Carmel High School, Teacher: Mrs. Nancy Schulenburg Nicholas Flynn Hoover, Brebeuf Jesuit Preparatory School, Teacher: Ms. Joan Rocap Henry Jung, Ben Davis High School, Teacher: Mr. Richard Elmore Steven Linville, Franklin Community High School, Teacher: Mr. Jason Boone Gregory Ryan Martens, Brebeuf Jesuit Preparatory School, Teacher: Mr. Tim Kelaghan Bhumi Dinesh Rajkotia, North Central High School, Teacher: Mr. Paul Brown Lisa Ann Schaus, Brebeuf Jesuit Preparatory School, Teacher: Mrs. Sandra Laycock Ryan W. Tobin, Warren Central High School, Teacher: Mrs. Janis Gaerte Alexander Hilmes Toumey, Brebeuf Jesuit Preparatory School, Teacher: Mrs. Sandra Laycock Chong Yan, Carmel High School, Teacher: Mrs. Kathie Freed

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

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