Georgia Tech Mathemematical Olympiad Sample Test 1. A book has 500 pages numbered 1, 2, 3, ..., and so on. How many times does the digit 1 appear in the page numbers? a) 100 b) 150 c) 200 d) 250 e) None of the above. [Taken from UTennessee-Knoxville Fermat I 2002, Problem 13]. 2. The area of a triangle having vertices (1,0), (4,4), and (-3,2) is a) 11 square units (s.u.) b) 10 s.u. c) 4 s.u. d) 13 s.u. e) None of the above. [Taken from UTK Fermat I 2002, Problem 17]. 3. In a sequence of numbers, the sum of the first n terms is equal to 5n^2 + 6n. What is the sum of the 3rd, 4th, and 5th terms in the original sequence? a) 123 b) 155 c) 210 d) 98 e) None of the above. [Taken from the Canadian Open Mathematics Challenge, 2002, Problem 4] 4. N is a five-digit positive integer. A six-digit integer P is cons- tructed by placing a 1 at the right-hand end of N. A second six- digit integer Q is constructed by placing a 1 at the left-hand end of N. If P is three times Q, determine the value of N. a) 12345 b) 35835 c) 42457 d) 51242 e) None of the above [Taken from the Canadian Open Mathematics Challenge, 2002, Problem 7. Note that this problem is trivial since the student can use brute- force to determine the answer for the multiple-choice format. But it is a good practice problem otherwise.] 5. Suppose that M is an integer with the property that if x is randomly chosen from the set {1,2,3,..,999,1000}, the probability that x is a divisor of M is 1/100. If M <= 1000, determine the maximum possible value of M. a) 256. b) 976. c) 835. d) 1000. e) None of the above. [Taken from the Canadian Open Mathematics Challenge, 2002, Problem 8]. 6. An operation "~" is defined by a~b = a^2 + 3^b. What is the value of (2~0)~(0~1)? a) 50 b) 51 c) 52 d) 53 e) None of the above. [Taken from the Canadian Open Mathematics Challenge, 2001, Problem 1] 7. If a can be any positive integer and 2x + a = y a + y = x x + y = z determine the maximum possible value for x + y + z. a) -10. b) 0. c) 5 d) 8 e) None of the above. [Taken from the Canadian Open Mathematics Challenge, 2001, Problem 7] 8. Three problems were given to participants of a math contest. Each participant got 0, 1, 2, or 3 points for each problem. After the papers were graded it turned out that no pair of participants received matching scores for more than one problem. What is the largest possible number of participants? a) 8 b) 9 c) 12 d) 16 e) 24 [Taken from the U.Maryland H.S. Math Competition, 2002, problem 25] 9. How many positive integers less than 10,000 are of the form x^8 + y^8 for some integers x>0 and y>0? a) 5 b) 6 c) 7 d) 8 e) more than 8 [Taken from the U.Maryland H.S. Math Competition, 2001, problem 21] 10. A student writes 44*55 = 3506. The student is calculating in base b and has written a correct equation. What is b? a) 7 b) 8 c) 9 d) 12 e) 14 [Taken from the U.Maryland H.S. Math Competition, 2001, problem 22] 11. What is the largest postage that cannot be paid exactly with an unlimited supply of 6-cent and 7-cent stamps? a) 15 b) 29 c) 32 d) 41 e) 43 [Taken from the U.Maryland H.S. Math Competition, 1999, problem 18] 12. The points (-6,1), (6,10), (9,6), and (-3,-3) are the vertices of a rectangle. What is the area of the portion of this rectangle that lies above the x-axis? a) 30.5 s.u. b) 65.625 s.u. c) 52.125 s.u. d) 100 s.u. e) 46.333 s.u. [Taken from the Maritime Mathematics Competition, 2002, problem 3]. 13. Find the smallest positive integer n such that sqrt(n) - sqrt(n-1) < 0.01. a) 2500 b) 2501 c) 2502 d) 2503 e) 2504 [Taken from the Maritime Mathematics Competition, 2002, problem 4]. 14. The infinite sequence 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 ... is obtained by writing the positive integers in order. What is the 2001st digit in this sequence? a) 1 b) 2 c) 3 d) 4 e) 5 [Taken from the Maritime Mathematics Competition, 2001, problem 2]. 15. At a meeting, one mathematician remarked to another, "There are nine fewer of us here than twice the product of the two digits of our total number." How many mathematicians were at the meeting? a) 12 b) 24 c) 35 d) 47 e) 53 [Taken from the Maritime Mathematics Competition, 2000, problem 1]. 16. If N is the product of all positive integers which divides 2003^2003 exactly, what are the last 3 digits of N? (2003 is a prime) a) 027 b) 243 c) 729 d) 999 [Taken from the Catonsville Mathematics Competition 2003, problem 8]. 17. Two circles of radius 9 cm and 16 cm touch each other at a point A. A line is tangent to them at two other points B and C. The area of the triangle ABC (in cm square) is a) 138.24 b) 168.36 c) 200 d) 240 [Taken from the Catonsville Mathematics Competition 2003, problem 10] 18. If N is the product of all positive integers which divide 2002 exactly, what is the sum of the digits of N? a) 98 b) 106 c) 126 d) 202 [Taken from the Catonsville Mathematics Competition 2002, problem 7] 19. Which of the following numbers cannot be expressed as the difference of the squares of two integers? a) 314159265 b) 314159266 c) 314159267 d) 314159268 e) 314159269 [Taken from an AHSMC sample test, problem 16] 20. Two points are picked at random on the unit circle x^2 + y^2 = 1. What is the probability that the chord joining the two points has length at least 1? a) 1/4 b) 1/3 c) 1/2 d) 2/3 e) 3/4 [Taken from the 1997 N.C. State High School Math Contest, Problem 12] 21. Circle C with center P* and circle D with center P~ intersect in points Q and R such that the angle QP*R = 30 degrees, and angle QP~R = 60 degrees. What is the ratio of the area of circle C to the area of circle D? a) 2 b) 4 c) 1 / [2 - sqrt(3)] d) 7 + 4sqrt(3) e) 16. [Taken from the 1997 N.C. State High School Math Contest, Problem 14] 22. A solid regular octahedron has volume sqrt(6) cubic inches. What is the surface area of the octahedron? a) sqrt(2) / 2 sq. in. b) 2sqrt(2) / 3 sq. in. c) 2sqrt(3) sq. in. d) 4sqrt(3) sq. in. e) 6sqrt(3) sq. in. [Taken from the 1997 N.C. State High School Math Contest, Problem 19] 23. Let P be the point (3,2). Let Q be the reflection of P about the x-axis, let R be the reflection of Q about the line y = -x, and let S be the reflection of R through the origin. Then PQRS is a convex quadrilateral. What is the area of PQRS? a) 14 b) 15 c) 16 d) 17 e) 18 [Taken from the 1997 N.C. State High School Math Contest, Problem 20] 24. There are 100 US Senators, two from each of the 50 states. If 20 senators are chosen at random, what is the probability (rounded to two decimal places) that any given state is represented? a) 0.29 b) 0.31 c) 0.34 d) 0.36 e) 0.40 [Taken from the 1997 AMATYC test, Problem 18]. 25. How many positive integers are divisors (number that divides) of 2002? a) 4 b) 8 c) 16 d) 32 e) 12 [Taken from the 2002 Lehigh U. HS Math contest, problem 21].