hu@sdcsvax.UUCP (03/22/84)
. Now, you take the AHSME (American High School Mathematics Examination). The test is in the same format as the old MAA test except that it's easier. People who score above 95 on the AHSME proceed to take the AIME (American Invitational Mathematics Examination). The AIME is a 2.5 hour test with 15 questions. The answer to each question is an integer from 0 to 999. Participants record their answers on computer cards. Top scorers from this examination are invited to participate in the USAMO. To help you judge the difficulty of these tests, here are some stats: Last year, the mean score on the AIME was 3.8 (out of 15) and the median was 3. 53 students scored a 10 or above and were invited to participate in the USAMO. Remember that the students who took this exam were already screened as being among the top math students in the country. Without further ado, and without permission from the MAA (They print and distribute for a nominal fee old tests, so I don't think they'll mind. I'll post a list of tests and an address at the end of this article.) Here are the problems. You have 2.5 hours. All answers are integers from 0 to 999. Begin 1. Find the value of a2 + a4 + a6 + ... + a98 if a1,a2,a3,... is an arithmetic progression with common difference 1, and a1 + a2 + a3+ ... + a98 = 137. [Sorry, I can't put subscripts on the terminal.] 2. The integer n is the smallest positive multiple of 15 such that every digit of n is either 0 or 8. Compute n/15. 3. A point P is chosen in the interior of triangle ABC so that when lines are drawn through P parallel to the sides of triangle ABC, the resulting smaller triangles, t1, t2, t3 in the fugure, have areas 4, 9 and 49, respectively. Find the area of triangle ABC. C. / \ / \ / \ / \ /\ /\ [Figure not drawn to scale. / \ / \ Not bad for ascii graphics!] / t2 \ / t1 \ / \ / \ /--------P--------\ / / \ \ / / \ \ / / \ \ / / t3 \ \ / / \ \ ------------------------------- A B 4. Let S be a list of positive integers--not necessarily distinct--in which the number 68 appears. The average (arithmetic mean) of the numbers in S is 56. However, if 68 is removed, the average of the remaining numbers drops to 55. What is the largest number that can appear in S? 2 2 5. Determine the value of ab if log a + log b = 5 and log b + log a = 7. 8 4 8 4 6. Three circles, each of radius 3, are drawn with centers at (14,92) (17,76) and (19,84). A line passing through (17,76) is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line? 7. The function f is defined on the set of integers and satisfies f(n) = / n-3 if n >= 1000 \ f(f(n+5)) if n < 1000 Find f(84). 8. The equation z^6 + z^3 + 1 = 0 has one complex root with argument (angle) theta between 90 degrees and 180 degrees in the complex plane. Determine the degree measure of theta. 9. In tetrahedron ABCD, edge AB has length 3 cm. The area of face ABC is 15 square cm and the area of face ABD is 12 square cm. These two faces meet each other at a 30 degree angle. Find the volume of the tetrahedron in cubic centimeters. 10. Mary told John her score on the AHSME, which was over 80. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over 80, John could not have dtermined this. What was Mary's score? (Recall that the AHSME consists of 30 multiple-choice problems and that one's score, s, is computed by the formula s=30 + 4c - 2, where c is the number of correct and w is the number of wrong answers; students are not penalized for problems left unanswered.) 11. A gardener plants three maple trees, four oak trees and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let m/n in lowest terms be the probability that no two birch trees are next to one another. Find m + n. 12. A function f is defined for all real numbers and satisfies f(2+x)=f(2-x) and f(7+x)=f(7-x) for all real x. If x=0 is a root of f(x)=0, what is the least number of roots f(x)=0 must have in the interval -1000<=x<=1000? 13. Find the value of 10cot(arccot 3 + arccot 7 + arccot 13 + arccot 21). 14. What is the largest even integer which cannot be written as the sum of two odd composite numbers? (Recall that a positive integer is said to be composite if it is divisible by at least one positive integer other than 1 and itself.) 15. Determine x^2 + y^2 + z^2 + w^2 if x^2/(2^2-1^2) + y^2/(2^2-3^2) + z^2/(2^2-5^2) + w^2/(2^2-7^2) = 1 x^2/(4^2-1^2) + y^2/(4^2-3^2) + z^2/(4^2-5^2) + w^2/(4^2-7^2) = 1 x^2/(6^2-1^2) + y^2/(6^2-3^2) + z^2/(6^2-5^2) + w^2/(6^2-7^2) = 1 x^2/(8^2-1^2) + y^2/(8^2-3^2) + z^2/(8^2-5^2) + w^2/(8^2-7^2) = 1 Whew!!!! I was pretty confident of 10 of my answers at the end of the 2.5 hour period. However, I don't have my answer sheet. Please mail me copies of your answers (together with how sure you are of them). I would be interested in knowing your solution method, too, if you want to mail it to me. Note that for the test, we were allowed ruler, compass, pencil, pen, and paper. We were not allowed to use calculators, slide rules, computers, VAXen, tutors, math professors, textbooks, notes, USENET, etc. You may order copies of old tests from: Dr. Walter E. Mientka, Executive Director American High School Mathematics Examination Department of Mathematics and Statistics University of Nebraska Lincoln, NE 68588-0322 [At least that's what it says on the back of the test.] Have fun, please send me your answers, and pardon my typos. --Alan J. Hu sdcsvax!hu
hu@sdcsvax.UUCP (03/24/84)
x<--Non-null first character Someone wrote me and told me that my first posting was truncated. This is a reposting. --Alan Hu I just took the 1984 AIME a few days ago, and I thought that you guys (sorry, that's a western dialectic colloquialism) (persons) out there might like to try them. For those of you who are wondering what this is, you might recall taking tests in high school put out by the Mathematical Association of America (MAA), the top-scorers of which would go to the United States Math Olympics (United States of America Mathematical Olympiad (USAMO)). It was a 90 minute test with 30 multiple-choice questions score on a scale from 0-150. Last year, that system was replace by a new one. Now, you take the AHSME (American High School Mathematics Examination). The test is in the same format as the old MAA test except that it's easier. People who score above 95 on the AHSME proceed to take the AIME (American Invitational Mathematics Examination). The AIME is a 2.5 hour test with 15 questions. The answer to each question is an integer from 0 to 999. Participants record their answers on computer cards. Top scorers from this examination are invited to participate in the USAMO. To help you judge the difficulty of these tests, here are some stats: Last year, the mean score on the AIME was 3.8 (out of 15) and the median was 3. 53 students scored a 10 or above and were invited to participate in the USAMO. Remember that the students who took this exam were already screened as being among the top math students in the country. Without further ado, and without permission from the MAA (They print and distribute for a nominal fee old tests, so I don't think they'll mind. I'll post a list of tests and an address at the end of this article.) Here are the problems. You have 2.5 hours. All answers are integers from 0 to 999. Begin 1. Find the value of a2 + a4 + a6 + ... + a98 if a1,a2,a3,... is an arithmetic progression with common difference 1, and a1 + a2 + a3+ ... + a98 = 137. [Sorry, I can't put subscripts on the terminal.] 2. The integer n is the smallest positive multiple of 15 such that every digit of n is either 0 or 8. Compute n/15. 3. A point P is chosen in the interior of triangle ABC so that when lines are drawn through P parallel to the sides of triangle ABC, the resulting smaller triangles, t1, t2, t3 in the fugure, have areas 4, 9 and 49, respectively. Find the area of triangle ABC. C. / \ / \ / \ / \ /\ /\ [Figure not drawn to scale. / \ / \ Not bad for ascii graphics!] / t2 \ / t1 \ / \ / \ /--------P--------\ / / \ \ / / \ \ / / \ \ / / t3 \ \ / / \ \ ------------------------------- A B 4. Let S be a list of positive integers--not necessarily distinct--in which the number 68 appears. The average (arithmetic mean) of the numbers in S is 56. However, if 68 is removed, the average of the remaining numbers drops to 55. What is the largest number that can appear in S? 2 2 5. Determine the value of ab if log a + log b = 5 and log b + log a = 7. 8 4 8 4 6. Three circles, each of radius 3, are drawn with centers at (14,92) (17,76) and (19,84). A line passing through (17,76) is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line? 7. The function f is defined on the set of integers and satisfies f(n) = / n-3 if n >= 1000 \ f(f(n+5)) if n < 1000 Find f(84). 8. The equation z^6 + z^3 + 1 = 0 has one complex root with argument (angle) theta between 90 degrees and 180 degrees in the complex plane. Determine the degree measure of theta. 9. In tetrahedron ABCD, edge AB has length 3 cm. The area of face ABC is 15 square cm and the area of face ABD is 12 square cm. These two faces meet each other at a 30 degree angle. Find the volume of the tetrahedron in cubic centimeters. 10. Mary told John her score on the AHSME, which was over 80. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over 80, John could not have determined this. What was Mary's score? (Recall that the AHSME consists of 30 multiple-choice problems and that one's score, s, is computed by the formula s=30 + 4c - 2, where c is the number of correct and w is the number of wrong answers; students are not penalized for problems left unanswered.) 11. A gardener plants three maple trees, four oak trees and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let m/n in lowest terms be the probability that no two birch trees are next to one another. Find m + n. 12. A function f is defined for all real numbers and satisfies f(2+x)=f(2-x) and f(7+x)=f(7-x) for all real x. If x=0 is a root of f(x)=0, what is the least number of roots f(x)=0 must have in the interval -1000<=x<=1000? 13. Find the value of 10cot(arccot 3 + arccot 7 + arccot 13 + arccot 21). 14. What is the largest even integer which cannot be written as the sum of two odd composite numbers? (Recall that a positive integer is said to be composite if it is divisible by at least one positive integer other than 1 and itself.) 15. Determine x^2 + y^2 + z^2 + w^2 if x^2/(2^2-1^2) + y^2/(2^2-3^2) + z^2/(2^2-5^2) + w^2/(2^2-7^2) = 1 x^2/(4^2-1^2) + y^2/(4^2-3^2) + z^2/(4^2-5^2) + w^2/(4^2-7^2) = 1 x^2/(6^2-1^2) + y^2/(6^2-3^2) + z^2/(6^2-5^2) + w^2/(6^2-7^2) = 1 x^2/(8^2-1^2) + y^2/(8^2-3^2) + z^2/(8^2-5^2) + w^2/(8^2-7^2) = 1 Whew!!!! I was pretty confident of 10 of my answers at the end of the 2.5 hour period. However, I don't have my answer sheet. Please mail me copies of your answers (together with how sure you are of them). I would be interested in knowing your solution method, too, if you want to mail it to me. Note that for the test, we were allowed ruler, compass, pencil, pen, and paper. We were not allowed to use calculators, slide rules, computers, VAXen, tutors, math professors, textbooks, notes, USENET, etc. You may order copies of old tests from: Dr. Walter E. Mientka, Executive Director American High School Mathematics Examination Department of Mathematics and Statistics University of Nebraska Lincoln, NE 68588-0322 [At least that's what it says on the back of the test.] Have fun, please send me your answers, and pardon my typos. --Alan J. Hu sdcsvax!hu