Mathematics and Space Exploration

Special thanks to Kroger® and to Jorge Estrada Hernández for problem 1.

STUDENT PRIZES: Win a prize for your Mathematics Department! Prizes of $250 each will be awarded to four Mathematics Departments based on the rankings of the winning students. Thirty top entrants will receive a book on a mathematical topic. Sixteen scholarships in the amount of $2,500 per year will be awarded to winners who are directly admitted to the Purdue School of Science at IUPUI and attend full-time. This scholarship is renewable for four years, subject to certain requirements. MATHEMATICS DEPARTMENT AWARDS: The 1st place team for 2007 was Hamilton Southeastern High School. Schools awarded the 1st place trophy in the past were:

▪ Carmel, 2006 ▪ Carmel, 2005 ▪ Carmel, 2004 ▪ Hamilton Southeastern, 2003 ▪ Hamilton Southeastern, 2002 ▪ Ben Davis, 2001 ▪ Carmel, 2000 ▪ Roncalli, 1999 ▪ Brebeuf Jesuit, 1998 CEREMONY: Prizewinners will be invited to an awards ceremony at IUPUI on Friday, May 9, 2008 from 4:00 to 6:30 p.m. Parents and teachers will also be invited. The program will feature an awards presentation, refreshments and a special talk, “What Does it Take to Go to Mars (and Come Back)?” ELIGIBILITY: This contest is open to students attending high school (grades 9-12) in the 15-county area of central Indiana: Bartholomew, Boone, Brown, Clinton, Hamilton, Hancock, Hendricks, Howard, Johnson, Madison, Marion, Morgan, Putnam, Shelby and Tipton.

QUESTIONS:

1. Given any three natural numbers, show that there are two of them, say a and b, such that a3b – b3a is divisible by 30.

2. In triangle ABC, draw angle bisectors AD and CE, where D is on

BC and E is on AB. If angle B is 60 degrees, show that AC = CD + AE.

3. Let A, B, C and D be points of the plane. Let P be the disk for

which AB is a diameter, and Q the disk for which CD is a diameter. Assume that A, B do not belong to Q and C, D do not belong to P. Prove that the segments AB and CD are disjoint.

4. A tennis tournament has every participant playing every other

participant. No ties are allowed. A player A is superior if for every other player B, either A beats B or there is a third player C such that A beats C and C beats B. Prove that if there is only one superior player then he or she beats every other player.

5. Write an essay of 500 to 700 words (complete with bibliography)

on an application of mathematics to space exploration.

ENTRIES: Mail your entry by Friday, April 11, 2008 to the address below. You may obtain a copy of the questions, instructions for entering, and the cover page from your math teacher or the contest website. Solve the questions, giving your reasoning, not just the answers. Entries will be judged by professors in the IUPUI Department of Mathematical Sciences. Judging will be based on elegance of solution as well as correctness.

CONTACT INFORMATION: www.math.iupui.edu/news/contest IUPUI High School Mathematics Contest Department of Mathematical Sciences 402 North Blackford Street, LD 270 Indianapolis, IN 46202-3216 (317) 274-MATH or contest@math.iupui.edu

IUPUI 2008 High School Mathematics Contest

Presented by The IUPUI Department of Mathematical Sciences

Arp galaxies: See Image of the Day Gallery at www.NASA.gov for other great space pictures.

Starring Mittens: This artist's concept shows the view from Cassini during the star occultation that detected "Mittens," the small object to the right of the star. As Cassini watched the star pass behind Saturn's F ring

(foreground), the star blinked out when mittens blocked it, indicating it may be a solid moonlet. Image Credit: NASA/JPL/University of Colorado.

Answers to the 2008 IUPUI High School Mathematics Contest

1. Given any three natural numbers, show that there are two of them, say a and b, such that

a3b− b3a is divisible by 30.

Solution. We will use the notation a ≡ r mod n for the remainder when a is divided by n. As

examples 28 ≡ 3 mod 5 and a ≡ 0 mod 2 if and only if a is even.

Since 30 = 2·3·5, we must show that for some two of the numbers, K = a3b−b3a = ab(a+b)(a−b)

is divisible by 2, 3 and 5. Since, as we will see, K is divisible by 2 and 3 for any two numbers a and

b, we first concentrate on 5.

Out of the three numbers, choose a and b by the following procedure:

i) If any of the three numbers is equivalent to 0 mod 5, let that number be a and either of the

other two be b.

ii) Otherwise, if two of the three numbers are equivalent mod 5, let those two be a and b.

iii) When neither case i) nor ii) occurs, the three numbers have three different remainders from

the set {1, 2, 3, 4}.

- If the three remainders are {1, 2, 3} or {2, 3, 4}, let a and b be those numbers equivalent

to 2 and 3 mod 5.

- If the three remainders are {1, 3, 4} or {1, 2, 4}, let a and b be those numbers equivalent

to 1 and 4 mod 5.

In case i), a is a multiple of 5. In case ii), (a − b) is a multiple of 5. In case iii), (a + b) is a

multiple of 5. This means that from three arbitrary numbers we can always choose two, a and b, so

that K is a multiple of 5.

Now, if either a ≡ 0, or b ≡ 0 mod 2, then K is a multiple of 2 (because of the factor a or b),

while if a ≡ b ≡ 1 mod 2, then K is again a multiple of 2 (because of factor (a− b)).

Similarily, if a ≡ 0 or b ≡ 0 mod 3, K is a multiple of 3. If a ≡ b mod 3, then the factor (a − b)

makes K a multiple of 3. If a ≡ 1 mod 3 and b ≡ 2 mod 3 (or the reverse), the factor (a + b) makes

K a multiple of 3.

In all cases, K is a multiple of 2 · 3 · 5 = 30.

2. In triangle ABC, draw angle bisectors AD and CE, where D is on BC and E is on AB. If

angle B is 60 degrees, show that AC = CD + AE.

Solution by plane geometry. Solutions using trigonometry are also possible. We are given that6 B = 60◦, but all we know about angles A and C is that 6 A+ 6 C = 120◦. Let O be the point where

the angle bisectors intersect. So 6 OAC + 6 OCA = 60◦. This means 6 AOC equals 120◦. Hence we

know about its supplementary angles: 6 AOE = 6 COD = 60◦.

Now bisect 6 AOC with segment OF . The two triangles on the right, 4COF and 4COD are

congruent because they share a side and the two adjacent angles: 6 FCO = 6 DCO and 6 FOC =6 DOC = 60◦. The same is true for the two triangles on the left. So AC = CF + AF = CD + AE.

A F C

E

D

B

O

3. Let A, B,C and D be points of the plane. Let P be the disk for which AB is a diameter,

and Q the disk for which CD is a diameter. Assume that A, B do not belong to Q and C, D do not

belong to P . Prove that the segments AB and CD are disjoint.

Solution. It is logically equivalent to show that if AB and CD do intersect then either A is in Q

or B is in Q or C is in P or D is in P .

If A is in Q, we’re done. Similarly if B is in Q we’re done.

So assume A is not in Q and B is not in Q.

Let X be the midpoint of AB, the center of P and let r = BX, the radius of P .

Let M be the point of intersection of AB with CD. Without loss of generality, assume BM < AM

and DM < CM . This leads to the figure. The other cases are similar.

MA B

D

X

C

Q

Since D is on the diameter through M , it is the closest point of the circle to M . Since B is

outside the circle, d(DM) < d(MB).

So d(DX) ≤ d(MX) + d(DM) < d(MX) + d(MB) = r; in other words, D is inside P .

4. A tennis tournament has every participant playing every other participant. No ties are allowed.

A player A is superior if for every other player B, either A beats B or there is a third player C such

that A beats C and C beats B. Prove that if there is only one superior player then he or she beats

every other player.

Solution. First we show that in every such tournament, any player with a maximal number of

wins is superior. Indeed, say A has a maximal number of wins, and B is any player who beats A.

Let T be the set of players that A beats. If B beats all of the players in T then B has a larger

number of wins than A, which can’t be true as A has the maximal number of wins. So there is some

player C in T who beats B. This proves that A is superior.

Now suppose A is the only superior player. Let S be the set of players who beat A, and let Tbe the set of players that A beats. Suppose S is not empty; that is, there is a player who beats A.

Treating S as a tournament in its own right, by the result above there is a superior player B in S.

But then B is superior in the entire tournament. This contradicts the hypothesis that A is the only

superior player. So S is empty and A beats everyone.

Notes:

1. The relation “is superior to” is not transitive. Consider the case illustrated. A is superior to

B, B is superior to D by virtue of C, but A is not superior to D.

A

C D

B

2. “Superior” does not necessarily mean a winner: If A beats everybody but B and B beats only

A then B is superior, but takes the last place in the tournament.

2008 IUPUI HIGH SCHOOL MATH CONTEST

First Prize Winners Michael B. Luo, 9th Grade, Carmel High School. Teacher: Ms. Laura Diamente Reid A.Watson, 10th Grade, Carmel High School. Teacher: Ms. Janice Mitchner

Second Prize Winners Jared B.Salisbury, 11th Grade, Hamilton Southeastern. Teacher: Mrs. Susan Wong Kristin M. Shaffer, 11th Grade, Hamilton Southeastern. Teacher: Mrs. Susan Wong

Third Prize Winners Christine N. Kincaid, 11th Grade, Hamilton Southeastern. Teacher: Mrs. Susan Wong Stephanie W. Kuo, 10th Grade, Hamilton Southeastern. Teacher: Mrs. Susan Wong David K. Riggleman, 12th Grade, Perry Meridian High School. Teacher: Mr. Steve Taylor Deepa K. Singh, 10th Grade, Hamilton Southeastern. Teacher: Mrs. Susan Wong Alex B. Smith, 12th Grade, Fishers High School. Teacher: Mr. John Drozd Kiara-Chi D. Thompson, 10th Grade, Hamilton Southeastern. Teacher: Mrs. Susan Wong

Fourth Prize Winners Heba A. Elantably, 11th Grade, School of Knowledge. Teacher: Mrs. Heba Elshakmak Kathleen Hu, 10th Grade, Fishers High School. Teacher: Mrs. Kathleen Robeson Fayaaz M. Khatri, 9th Grade, Brownsburg High School. Teacher: Mrs. Micah Knobel Anna C. Roesler, 11th Grade, Hamilton Southeastern. Teacher: Mrs. Susan Wong Kaylee M. Shirrell, 10th Grade, Brownsburg High School. Teacher: Mrs. Micah Knobel David A. Zielinski, 9th Grade, Brownsburg High School. Teacher: Mrs. Micah Knobel

Honorable Mentions

Meghan S. Barry, 10th Grade, Hamilton Southeastern. Teacher: Mrs. Susan Wong Richard H. Benning, 11th Grade, Roncalli High School. Teacher: Sr. Anne Frederick Allison N. Boyd, 10th Grade, Hamilton Southeastern. Teacher: Mrs. Susan Wong Ryan B. Carr, 12th Grade, Hamilton Southeastern. Teacher: Mrs. LetitiaMcCallister Sarah L. Chang, 10th Grade, Hamilton Southeastern. Teacher: Mrs. Susan Wong Sam J. Ebling, 10th Grade, Hamilton Southeastern. Teacher: Mrs. Susan Wong Elyssa M. Goldstein, 9th Grade, Hamilton Southeastern. Teacher: Mrs. Susan Wong Matt T. Hartley, 10th Grade, Fishers High School. Teacher: Mrs. Louise Werner Christina M. Hill, 9th Grade, Brownsburg High School. Teacher: Mrs. Micah Knobel John B. Holt, 11th Grade, Hamilton Southeastern. Teacher: Mrs. Susan Wong Austin C. Hunkin, 9th Grade, Hamilton Southeastern. Teacher: Mrs. Susan Wong Kayla N. Jansen, 9th Grade, Hamilton Southeastern. Teacher: Mrs. Susan Wong Kathryn D. Kleber, 12th Grade, Fishers High School. Teacher: Mr. John Drozd Kendall C. Knoke, 10th Grade, Hamilton Southeastern. Teacher: Mrs. Susan Wong Taylor L. LaCross, 11th Grade, Hamilton Southeastern. Teacher: Mrs. Susan Wong Sawyer E. Morgan, 9th Grade, Hamilton Southeastern. Teacher: Mrs. Jo Ann Blake Emily C. Mudd, 10th Grade, Hamilton Southeastern. Teacher: Mrs. Jo Ann Blake Austin T. Mudd, 12th Grade, Hamilton Southeastern. Teacher: Mrs. Letitia McCallister Nam K. Phan, 11th Grade, Hamilton Southeastern. Teacher: Mrs. Susan Wong David J. Predajna, 11th Grade, Hamilton Southeastern. Teacher: Mrs. Susan Wong David P. Smelser, 11th Grade, Brownsburg High School. Teacher: Mrs. Dawn Fitch Hobey Tam, 12th Grade, Fishers High School. Teacher: Mr. John Drozd Julie M. Thomas, 9th Grade, Fishers High School. Teacher: Mrs. Kathleen Robeson Andrew R. Vissing, 11th Grade, Hamilton Southeastern. Teacher: Mrs. Letitia McCallister