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1、2000 AIME Problems AIME Problems 2000 I March 28th 1Find the least positive integer n such that no matter how 10nis expressed as the product of any two positive integers at least one of these two integers contains the digit 0 2Let u and v be integers satisfying 0 v u Let A u v let B be the refl ecti
2、on of A across the line y x let C be the refl ection of B across the y axis let D be the refl ection of C across the x axis and let E be the refl ection of D across the y axis The area of pentagon ABCDE is 451 Find u v 3In the expansion of ax b 2000 where a and b are relatively prime positive intege
3、rs the coeffi cients of x2and x3are equal Find a b 4The diagram shows a rectangle that has been dissected into nine non overlapping squares Given that the width and the height of the rectangle are relatively prime positive integers fi nd the perimeter of the rectangle 5Each of two boxes contains bot
4、h black and white marbles and the total number of marbles in the two boxes is 25 One marble is taken out of each box randomly The probability that both marbles are black is 27 50 and the probability that both marbles are white is m n where m and n are relatively prime positive integers What is m n C
5、ontributors 4everwise white horse king88 joml88 rrusczyk 2000 AIME Problems 6For how many ordered pairs x y of integers is it true that 0 x y 0 Find m n r 14Every positive integer k has a unique factorial base expansion f1 f2 f3 fm meaning that k 1 f1 2 f2 3 f3 m fm where each fiis an integer 0 fi i
6、 and 0 fm Given that f1 f2 f3 fj is the factorial base expansion of 16 32 48 64 1968 1984 2000 fi nd the value of f1 f2 f3 f4 1 j 1fj 15Find the least positive integer n such that 1 sin45 sin46 1 sin47 sin48 1 sin133 sin134 1 sinn These problems are copyright c Mathematical Association of America ht
7、tp maa org Contributors 4everwise white horse king88 joml88 rrusczyk 2001 AIME Problems AIME Problems 2001 I March 27th 1Find the sum of all positive two digit integers that are divisible by each of their digits 2 A fi nite set S of distinct real numbers has the following properties the mean of S 1
8、is 13 less than the mean of S and the mean of S 2001 is 27 more than the mean of S Find the mean of S 3Find the sum of the roots real and non real of the equation x2001 1 2 x 2001 0 given that there are no multiple roots 4In triangle ABC angles A and B measure 60 degrees and 45 degrees respec tively
9、 The bisector of angle A intersects BC at T and AT 24 The area of triangle ABC can be written in the form a b c where a b and c are positive integers and c is not divisible by the square of any prime Find a b c 5An equilateral triangle is inscribed in the ellipse whose equation is x2 4y2 4 One verte
10、x of the triangle is 0 1 one altitude is contained in the y axis and the length of each side is pm n where m and n are relatively prime positive integers Find m n 6 A fair die is rolled four times The probability that each of the fi nal three rolls is at least as large as the roll preceding it may b
11、e expressed in the form m n where m and n are relatively prime positive integers Find m n 7Triangle ABC has AB 21 AC 22 and BC 20 Points D and E are located on AB and AC respectively such that DE is parallel to BC and contains the center of the inscribed circle of triangle ABC Then DE m n where m an
12、d n are relatively prime positive integers Find m n 8Call a positive integer N a 7 10 double if the digits of the base 7 representation of N form a base 10 number that is twice N For example 51 is a 7 10 double because its base 7 representation is 102 What is the largest 7 10 double Contributors jom
13、l88 4everwise rrusczyk 2001 AIME Problems 9In triangle ABC AB 13 BC 15 and CA 17 Point D is on AB E is on BC and F is on CA Let AD p AB BE q BC and CF r CA where p q and r are positive and satisfy p q r 2 3 and p2 q2 r2 2 5 The ratio of the area of triangle DEF to the area of triangle ABC can be wri
14、tten in the form m n where m and n are relatively prime positive integers Find m n 10Let S be the set of points whose coordinates x y and z are integers that satisfy 0 x 2 0 y 3 and 0 z 4 Two distinct points are randomly chosen from S The probability that the midpoint of the segment they determine a
15、lso belongs to S is m n where m and n are relatively prime positive integers Find m n 11In a rectangular array of points with 5 rows and N columns the points are numbered consecutively from left to right beginning with the top row Thus the top row is numbered 1 through N the second row is numbered N
16、 1 through 2N and so forth Five points P1 P2 P3 P4 and P5 are selected so that each Piis in row i Let xibe the number associated with Pi Now renumber the array consecutively from top to bottom beginning with the fi rst column Let yibe the number associated with Piafter the renumbering It is found that x1 y2 x2 y1 x3 y4 x4 y5 and x5 y3 Find the smallest possible value of N 12A sphere is inscribed in the tetrahedron whose vertices are A 6 0 0 B 0 4 0 C 0 0 2 and D 0 0 0 The radius of the sphere is
