2015 CHICAGO AREA ALL-STAR MATH TEAM TRYOUTS

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Griselda Elliott
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1 Problems The average of five distinct positive integers is 85, and the average of the three largest of the integers is 100. Compute the largest possible value of the second-smallest integer. 2. In 4ABC, m\abc =60. Semicircles with diameters AB, BC, andac are constructed external to 4ABC. ThesemicirclewithdiameterAB has area 9, andthesemicirclewith diameter BC has area 16. ComputetheareaofthesemicirclewithdiameterAC. A B C ANSWER TO PROBLEM 1 ANSWER TO PROBLEM 2

2 Problems Compute the maximum possible area for a triangle with integer side lengths and perimeter At Pascal Prep, 10% of the students have athlete s foot and 15% of the students wear flipflops in the shower. After some study, it turns out that 40% of the students who wear flip-flops have athlete s foot. If a given student has athlete s foot, compute the probability that he or she wears flip-flops. ANSWER TO PROBLEM 3 ANSWER TO PROBLEM 4

3 Problems For some positive real number a, thetrianglet with vertices (0, 0), (2, 0), and (2, 4) is transformed by the mapping that sends the point (x, y) tothepoint(2x, 2x + ay), yielding T 0.IftheareaofT 0 is 34, compute a. 6. If cos(2 ) = 1 4,computethegreatestpossiblevalueofcos(3 ). ANSWER TO PROBLEM 5 ANSWER TO PROBLEM 6

4 Problems In 4ABC, M lies on AB and N lies on AC such that AM =3,MB =4,andAN/NC = 2/3. Segments BN and CM intersect at G, and AG! bisects \BAC. ComputeAC. A M G N B C 8. If 1 + r + r 2 + =17,compute1+2r +3r 2 +4r 3 +. ANSWER TO PROBLEM 7 ANSWER TO PROBLEM 8

5 Problems Xander and Aubrey agree to meet for co ee between 2 and 3pm. Each one arrives at some random time within that interval. When Xander arrives, he will wait for up to 15 minutes for Aubrey, and then he will leave if she has not yet arrived. Aubrey will only wait for up to 5 minutes for Xander before leaving. Compute the probability that the two meet up successfully. 10. Compute sin 2tan ANSWER TO PROBLEM 9 ANSWER TO PROBLEM 10

6 Problems Compute the least positive integer n such that n! isdivisibleby Fermat s flea can hop either exactly 1 unit or exactly 2 units either left or right (i.e. negative direction or positive direction) along the number line. Compute the number of possible sequences of exactly nine jumps the flea could take if the flea starts at the number 0 and ends at the number 13. ANSWER TO PROBLEM 11 ANSWER TO PROBLEM 12

7 Problems Suppose that f(x) isaquadraticpolynomialsuchthatf(1) f( 1) = 8, f(2)+f( 2) = 16, and f(1) + f( 1) = 2. Compute f(3) f( 3). 14. Cantor Cinema charges $11 per adult ticket and $7 per child ticket. One day, they take in $1800, but after an unfortunate incident with the popcorn machine, the management decides to refund each adult $2 and each child $1. Compute the minimum possible total amount of money refunded. ANSWER TO PROBLEM 13 ANSWER TO PROBLEM 14

8 Problems Compute the number of ordered pairs of positive integers (b, c), with both b and c less than 12, such that the equations x 2 + bx + c = 0 and x 2 + cx + b =0eachhavetwodistinctreal roots. 16. Avery, Toby, and Benton always get in line so that at least two of them are standing together. If the three of them get in line with Wendy, Xavier, Yusuf, and Zillah, how many arrangements are possible? ANSWER TO PROBLEM 15 ANSWER TO PROBLEM 16

9 Problems Emma, Jonah, and Helen are running a race. Jonah runs twice as fast as Helen but waits so that she has a one-minute head start, and Emma runs twice as fast as Jonah but waits so that he has a 100-yard head start. They all cross the finish line at the same instant. Compute Helen s speed in yards per second. 18. Complete the cross-number puzzle below, where each Across answer is a 4-digit number and each Down answer is a 3-digit number. No answer begins with the digit 0. NOTE: Your answer must be written in the spaces at the bottom of this page, NOT in the grid to the right of the clues. Across Down All digits are the same 1. A multiple of 11 all 5. A sum of one or more of whose digits are 5 distinct positive integral Fibonacci numbers powers of A perfect square 6 6. All digits are even 3. Four times a prime and distinct 4. Twice a prime ANSWER TO PROBLEM 17 ANSWER TO PROBLEM

10 Problems Compute the maximum possible area of a triangle in the complex plane whose vertices are 1, z + 1, and z+1,forsomecomplexnumberz. z 20. In rectangle ABCD, AB = 20 and AD = 15. Points M and N lie on BC and CD respectively, and [ABM] =[MCN]=[NDA]. Compute [AMN]. A B M D N C ANSWER TO PROBLEM 19 ANSWER TO PROBLEM 20

11 Problems Compute cos 11 cos 2 11 cos 3 11 cos 4 11 cos Compute the sum of all odd positive integers less than 200 with exactly eight positive integer divisors. ANSWER TO PROBLEM 21 ANSWER TO PROBLEM 22

12 Problems Let A, R, M, andl be positive real numbers satisfying the system below. 8 < : Compute the product ARML. log p A +log p R +logm = 2 log p R +log p M +logl = 3 log A +log p R +log p L = Compute the sum of all real values of x such that (4 x 1 ) x 3 =8 x. ANSWER TO PROBLEM 23 ANSWER TO PROBLEM 24

13 Part I Answers p , 60%, or equivalent p

14 Part II Answers p ,000,

2009 CHICAGO AREA ALL-STAR MATH TEAM TRYOUTS. Problems 1-2. Time Limit 8 Minutes

2009 CHICAGO AREA ALL-STAR MATH TEAM TRYOUTS. Problems 1-2. Time Limit 8 Minutes Problems 1-2 Time Limit 8 Minutes 1. In one week, a salesman sells p cars for q thousands of dollars each, earning a commission of $100 plus q% of the selling price each time he sells a car, where p and