57th ANNUAL HIGH SCHOOL HONORS MATHEMATICS CONTEST

Size: px

Start display at page:

Download "57th ANNUAL HIGH SCHOOL HONORS MATHEMATICS CONTEST"

Jonas Simmons
6 years ago
Views:

1 57th ANNUAL HIGH SCHOOL HONORS MATHEMATICS CONTEST Welcome to Part II of the contest! April 19, 2014 on the campus of the University of California, San Diego PART II 4 Questions Please print your Name, School, and Contest ID number: Name First Last School 3-digit Contest ID number Please do not open the exam until told do so by the proctor. EXAMINATION DIRECTIONS: 1. Part II consists of 4 problems, each worth 25 points. These problems are essay style questions. You should put all work towards a solution in the space following the problem statement. You should show all work and justify your responses as best you can. 2. The score on Part II is added to the score on Part I to determine the highest individual scores in the contest. 3. Scoring is based on the progress you have made in understanding and solving the problem. The clarity and elegance of your response is an important part of the scoring. You may use the back side of each sheet to continue your solution, but be sure to call the reader s attention to the back side if you use it. 4. Give this entire exam to a proctor when you have completed the test to your satisfaction. Please let your coach know if you plan to attend the Awards Dinner on Wednesday, April 30, 6:00 8:pm in the UCSD Faculty Club.

2 1. Alice and ob each have a bag of 9 balls. The balls in each bag are numbered from 1 to 9. Alice and ob each remove one ball uniformly at random from their own bag. Let a be the sum of the numbers on the balls remaining in Alice s bag. Let b be the sum of the numbers on the balls remaining in ob s bag. Determine the probability that a and b differ by a multiple of 4. Part II Honors Math Contest of 4

3 2. Suppose that x Q is a rational number with the property that x 2 x Z is an integer. Prove that, in fact, x Z is an integer. Part II Honors Math Contest of 4

4 S 3. In triangle AC, A = C = 25 and AC = 30. 9The circle with diameter C intersects A at X and AC at Y. Determine the length of XY. C, A = C =25 and AC = 30. The ameter C intersects A at X and AC at the length of XY. X A Y C ce C is a diameter, then YC = 90. C, AC is isosceles and Y is an AC, then AY = YC =. θ. θ X 25 is isosceles, CA = θ. is cyclic, XY = 180 θ and so θ θ A Y C is isosceles and so XY = AY =. =. 2 YC = 90 since it is inscribed in a. isosceles, so altitude Y bisects the 2 2 Y = 25 = 20. CX = 90 since it is also inscribed in le. f AC is ( Y) = 1 ( A) ( CX) 2 X A Y C )( 20) = 1 ( 2 25) ( CX) CX = 600 = Y 20 4 we conclude that cos AY = = = A plying the Law of Cosines we get ( XY) = ( X) + ( Y) 2( X)( Y) cos XY. agoras XC), Part II Honors Math Contest of 4

5 9. The circle (x p) 2 + y 2 = r 2 has centre C and the circle x 2 +(y p) 2 = r 2 has centre D. The circles intersect at two distinct points A and, with x-coordinates a and b, respectively. (a) Prove that a + b = p and a 2 + b 2 = r 2. (b) If r is fixed and p is then found to maximize the area of quadrilateral CAD, prove that either A or is the origin. (c) If p and r are integers, determine the minimum possible distance between A and. Find 4. A school positive has integers a rowp of andn r, open each larger lockers, than numbered 1, that give this 1 through distance. n. Starting at the beginning of the row, you walk past and close every second locker until reaching the 10. end A school of the has row, a row as shown of n open in the lockers, example numbered below. 1 through Then n. youafter turnarriving around, at school walk back, and oneclose day, Josephine every second starts at locker the beginning that is of still theopen. row andyou closes continue every second in this locker manner until back and reaching forth the along end the of the row, row, until as shown onlyinone the locker exampleremains below. Then open. on her Define way f(n) back, to shebe the number closes every of the second last open lockerlocker. that is still Foropen. example, She continues if therein are this manner lockers, along then thef() row, = 11 until only one locker remains open. Define f(n) to be the number of the last open locker. as shown below. For example, if there are lockers, then f() = 11 as shown below: ! 1 2/ 3 4/ 5 6/ 7 8/ 9 10/ 11 12/ 13 14/ 1/ 3 5/ 7 9/ 11 13/! 3 7/ 11 / 3/ 11 (a) Determine f(50). (b) Prove that there is no positive integer n such that f(n) = (c) Prove that there are infinitely many positive integers n such that f(n) =f(2005). Calculate f(2014). Part II Honors Math Contest of 4

BC Exam Solutions Texas A&M High School Math Contest October 24, p(1) = b + 2 = 3 = b = 5.

BC Exam Solutions Texas A&M High School Math Contest October 24, p(1) = b + 2 = 3 = b = 5. C Exam Solutions Texas &M High School Math Contest October 4, 01 ll answers must be simplified, and If units are involved, be sure to include them. 1. p(x) = x + ax + bx + c has three roots, λ i, with