USA-AIME-2002-美国数学邀请赛

USA AIME 2002 I 1 Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the probability that such a license plate will contain at least one palindrome (a three- letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right-to-left) is m/n, where m and n are relatively prime positive integers. Find m+ n. 2 The diagram shows twenty congruent circles arranged in three rows and enclosed in a rect- angle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as 12 (√ p− q) , where p and q are positive integers. Find p+ q. [img]6601[/img] 3 Jane is 25 years old. Dick is older than Jane. In n years, where n is a positive integer, Dick’s age and Jane’s age will both be two-digit number and will have the property that Jane’s age is obtained by interchanging the digits of Dick’s age. Let d be Dick’s present age. How many ordered pairs of positive integers (d, n) are possible? 4 Consider the sequence defined by ak = 1k2+k for k ≥ 1. Given that am + am+1 + · · ·+ an−1 = 1/29, for positive integers m and n with m < n, find m+ n. 5 Let A1, A2, A3, . . . , A12 be the vertices of a regular dodecagon. How many distinct squares in the plane of the dodecagon have at least two vertices in the set {A1, A2, A3, . . . , A12}? 6 The solutions to the system of equations log225 x+ log64 y = 4 logx 225− logy 64= 1 (0) are (x1, y1) and (x2, y2). Find log30 (x1y1x2y2). http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 1 USA AIME 2002 The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers x, y, and r with |x| > |y|, (x+ y)r = xr + rxr−1y + r(r − 1) 2 xr−2y2 + r(r − 1)(r − 2) 3! xr−3y3 + · · · What are the first three digits to the right of the decimal point in the decimal representation of( 102002 + 1 )10/7? Find the smallest integer k for which the conditions (1) a1, a2, a3, . . . is a nondecreasing sequence of positive integers (2) an = an−1 + an−2 for all n > 2 (3) a9 = k are satisfied by more than one sequence. Harold, Tanya, and Ulysses paint a very long picket fence. Harold starts with the first picket and paints every hth picket; Tanya starts with the second picket and paints everth tth picket; and Ulysses starts with the third picket and paints every uth picket. Call the positive integer 100h + 10t + u paintable when the triple (h, t, u) of positive integers results in every picket being painted exaclty once. Find the sum of all the paintable integers. In the diagram below, angle ABC is a right angle. Point D is on BC, and AD bisects angle CAB. Points E and F are on AB and AC, respectively, so that AE = 3 and AF = 10. Given that EB = 9 and FC = 27, find the integer closest to the area of quadrilateral DCFG. [img]6604[/img] Let ABCD and BCFG be two faces of a cube with AB = 12. A beam of light emanates from vertex A and reflects off face BCFG at point P, which is 7 units from BG and 5 units from BC. The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point A until it next reaches a vertex of the cube is given by m √ n, where m and n are integers and n is not divisible by the square of any prime. Find m+ n. Let F (z) = z+iz−i for all complex numbers z 6= i, and let zn = F (zn−1) for all positive integers n. Given that z0 = 1137 + i and z2002 = a+ bi, where a and b are real numbers, find a+ b. In triangle ABC the medians AD and CE have lengths 18 and 27, respectively, and AB = 24. Extend CE to intersect the circumcircle of ABC at F . The area of triangle AFB is m √ n, where m and n are positive integers and n is not divisible by the square of any prime. Find m+ n. A set S of distinct positive integers has the following property: for every integer x in S, the arithmetic mean of the set of values obtained by deleting x from S is an integer. Given that 1 belongs to S and that 2002 is the largest element of S, what is the greatet number of elements that S can have? Polyhedron ABCDEFG has six faces. Face ABCD is a square with AB = 12; face ABFG is a trapezoid with AB parallel to GF, BF = AG = 8, and GF = 6; and face CDE has CE = DE = 14. The other three faces are ADEG,BCEF, and EFG. The distance from E to face ABCD is 12. Given that EG2 = p − q√r, where p, q, and r are positive integers and r is not divisible by the square of any prime, find p+ q + r. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 2 USA AIME 2002 II 1 Given that (1) x and y are both integers between 100 and 999, inclusive; (2) y is the number formed by reversing the digits of x; and (3) z = |x− y|. (0) How many distinct values of z are possible? Three vertices of a cube are P = (7, 12, 10), Q = (8, 8, 1), and R = (11, 3, 9). What is the surface area of the cube? It is given that log6 a + log6 b + log6 c = 6, where a, b, and c are positive integers that form an increasing geometric sequence and b− a is the square of an integer. Find a+ b+ c. Patio blocks that are hexagons 1 unit on a side are used to outline a garden by placing the blocks edge to edge with n on each side. The diagram indicates the path of blocks around the garden when n = 5. [img]6613[/img] If n = 202, then the area of the garden enclosed by the path, not including the path itself, is m( √ 3/2) square units, where m is a positive integer. Find the remainder when m is divided by 1000. Find the sum of all positive integers a = 2n3m, where n and m are non-negative integers, for which a6 is not a divisor of 6a. Find the integer that is closest to 1000 ∑10000 n=3 1 n2−4 . It is known that, for all positive integers k, 12 + 22 + 32 + · · ·+ k2 = k(k + 1)(2k + 1) 6 . Find the smallest positive integer k such that 12 + 22 + 32 + · · ·+ k2 is a multiple of 200. Find the least positive integer k for which the equation b2002n c = k has no integer solutions for n. (The notation bxc means the greatest integer less than or equal to x.) http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 3 USA AIME 2002 Let S be the set {1, 2, 3, . . . , 10}. Let n be the number of sets of two non-empty disjoint subsets of S. (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when n is divided by 1000. While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of x for which the sine of x degrees is the same as the sine of x radians are mpin−pi and ppi q+pi , where m, n, p and q are positive integers. Find m+ n+ p+ q. Two distinct, real, infinite geometric series each have a sum of 1 and have the same second term. The third term of one of the series is 1/8, and the second term of both series can be written in the form √ m−n p , where m, n, and p are positive integers and m is not divisible by the square of any prime. Find 100m+ 10n+ p. A basketball player has a constant probability of .4 of making any given shot, independent of previous shots. Let an be the ratio of shots made to shots attempted after n shots. The probability that a10 = .4 and an ≤ .4 for all n such that 1 ≤ n ≤ 9 is given to be paqbr/(sc), where p, q, r, and s are primes, and a, b, and c are positive integers. Find (p+ q + r + s)(a+ b+ c). In triangle ABC, point D is on BC with CD = 2 and DB = 5, point E is on AC with CE = 1 and EA = 32, AB = 8, and AD and BE intersect at P. Points Q and R lie on AB so that PQ is parallel to CA and PR is parallel to CB. It is given that the ratio of the area of triangle PQR to the area of triangle ABC is m/n, where m and n are relatively prime positive integers. Find m+ n. The perimeter of triangle APM is 152, and the angle PAM is a right angle. A circle of radius 19 with center O on AP is drawn so that it is tangent to AM and PM. Given that OP = m/n, where m and n are relatively prime positive integers, find m+ n. Circles C1 and C2 intersect at two points, one of which is (9, 6), and the product of the radii is 68. The x-axis and the line y = mx, where m > 0, iare tangent to both circles. It is given that m can be written in the form a √ b/c, where a, b, and c are positive integers, b is not divisible by the square of any prime, and a and c are relatively prime. Find a+ b+ c. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 4