Math League News (#1) 11/98

New Calculator Rule Calculators with algebraic manipulation capability (such as the TI-89 and TI-92) are not permitted. This is same as the American HS Mathematics Examination policy.

Contest Dates HS Contest dates are Dec. 1, Jan. 5, Feb. 2, Mar. 9, and Apr. 13. A registration form for our April Algebra Course I Contest is enclosed. Do you have a schedule conflict with our contest dates? Our rules say that, in case of vacations, special testing days, or other disruptions of the normal school day, you may give the contest on an earlier day-but you must still mail scores by Friday of the official contest week. If scores are late for due cause, please attach a brief explanation. We reserve the right to consider as unofficial late scores lacking such an explanation.

Received Your HS Contest Package Late? If you have not yet received the contests, phone 1-201-568­6328 so we can ship another set. If you just recently got the contests, please take Contest #1 as soon as possible, even if it's late!

The Score Report and the Cumulative Column Students on your score report must take the contest at the exact same time. Do not include students taking the contest during any later class period. Below is part of a score report. The Total column is for Contest 2 totals only. The Indiv. Cumulative is for student totals for the first 2 contests. This column is optional; but high scoring students not tallied here cannot be named in our newsletter. Chris Lewis got 5's on the first 2 contests and had a cumulative total of 10. Pat Harris got a 5 and had a cumulative total of 9. Team members may vary each contest-use your school's 5 best scores each time, and submit additional sheets if needed. Be sure to check the "Contest Number" box, and enter your Team Score at the top of the form.

Highest Scoring Participants
Please PRINT
Last Name, First Name
Question
1
2
3
4
5
6
Total
Indiv.
Cumulative
1. Lewis, Chris
1
1
1
1
1
0
5
10
2. Harris, Pat
1
1
1
1
1
0
5
9
3. Smith, Lee
1
1
1
0
0
0
3
4. Nelson, Jan
1
0
1
1
0
0
3
5. Sun, Ronnie
1
1
0
0
0
0
2
TEAM TOTALS
5
4
4
3
2
0
18

Completion of the "Cumulative" column is optional, but must be completed for any student who might be listed as a League high scorer.

Authentication of Scores To give credibility to our results, we authenticate scores high enough to win recognition. Awards indicate compliance with our rules. Please ask students to read the Selected Math League Rules on the back of this newsletter and sign a sheet to confirm knowledge of the rules. Keep the signed copies. Do not send them to us unless we request authentication from you.

General Comments About the Contest 675 students in Tom Armbruster's school took Contest #1. Nancy Tooker had 11 students get a 4 or 5. Pam Coryell "thought this was very good for openers. It let younger students get correct answers. They'll now take further contests. P.S. I love your cartoons." Sandia Davenport liked the first 4 problems of contest 1, but not the 5th (too susceptible to calculator solution) or #6 (too guessable). Ted Heavenrich has "some math students who were disappointed by the lack of mathematical challenge." Bryan Knight thought it "tougher than in prior years, but teachers and students enjoyed it." Susy Wong called it "challenging." Bob Smith said "Great contest to kick off the year."

Problem 1-2: A Comment From Us This idea is from The USSR Olympiad Problem Book (written before calculators).

Problem 1-3: Appeal (Denied) Two appeals for (3,0) were denied, since (3,0) is not the sought-after value of k.

Problem 1-4: Alternate Solution Dave Farber and Dennis Gwirtz said that 15+8 = 23, so an 8-15-17 Pythagorean triple works, and 15-8 = 7 is the answer. Not even algebra is needed!

Problem 1-5: Comments, Appeals (Denied), Alt. Sol'n Bryan Knight and Bob Smith sent alternate solutions. Here's Bob's: As in our solution, a3+b3+c3 = -1998-78(a+b+c). Since x = a is a root, x3+78x+666 = (x-a)´ (x2+ax-666/a) = (x-a)(x-b)(x-c). Thus, (x-b)(x-c) = x2-(b+c)x+ bc = (x2+ax-666/a). Thus, a = -b-c and a3+b3+c3 = -1998. Susy Wong, Robert Morewood, and Ted Heavenrich each had ³ 1 student use a Sharp EL-9600, an HP-48, or a TI-85 to handle complex numbers. Morewood's "students agree that the algebraic solution is much easier then the direct calculator approach, once you understand the algebra." Tim Butler used a TI-83 to locate the real root at » -5.9. The cubic became (x+5.9)(x2-5.9x+112.8-) = 0. After solving the quadratic for imaginary roots, he cubed and added the three cubes together. Some students got the real and imaginary roots and wrote their answer as an indicated sum of three cubes, rather than an actual sum. Credit was denied since all reasonable simplifications must be made where failure to make them might indicate a possible lack of knowledge. An appeal for (-1998.0000,1.074 ´ 105) was denied. Finally, Dave Farber got -1980 as an answer, but he just knew it had to be -1998. He went back and found his error!

Problem 1-6: Comments and an Alt Solution Robert Morewood "particularly liked the Friday the 13th question. It was perfect for Halloween." One student said she "wasted time trying to figure out what the guy with the dark cloud overhead had to do with 1-6." Dave Farber and Ben Wearn (who asked if there were any other all-female teams) used (mod 7) to keep track of the days in each month. The sequence of days in each month of a normal year is {31,28,31,30,31,30,31,31,30,31,30,31}. In mod 7, this becomes {3,0,3,2,3,2,3,3,2,3,2,3}. Sum the sequence to form a mod 7 shift sequence that tells you how many days of the week the 13th of the month is shifted, compared to January. For example, February is shifted 3 days. March is shifted 3+0 = 3 days. April is shifted 3+0+3 = 6 days. May is shifted 3+0+3+2 = 8 days = 1 day, etc. This sequence is {0,3,3,6,1,4,6,2,5,0,3,5}. A quick look shows that the 13th must fall on a Friday at least once (since all 7 digits appear) and at most 3 times (Feb., Mar., and Nov. are all shifted 3 days compared to Jan.). A similar analysis holds for leap years.


Statistics / Contest #1

Prob #, % Correct (top 5 each school)

1-1    97%1-4    84%
1-2    90%1-5    10%
1-3    72%1-6    50%