Difference between revisions of "2000 AIME II Problems"
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The coordinates of the vertices of isosceles trapezoid <math>ABCD</math> are all integers, with <math>A=(20,100)</math> and <math>D=(21,107)</math>. The trapezoid has no horizontal or vertical sides, and <math>\overline{AB}</math> and <math>\overline{CD}</math> are the only parallel sides. The sum of the absolute values of all possible slopes for <math>\overline{AB}</math> is <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | The coordinates of the vertices of isosceles trapezoid <math>ABCD</math> are all integers, with <math>A=(20,100)</math> and <math>D=(21,107)</math>. The trapezoid has no horizontal or vertical sides, and <math>\overline{AB}</math> and <math>\overline{CD}</math> are the only parallel sides. The sum of the absolute values of all possible slopes for <math>\overline{AB}</math> is <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
− | [[2000 AIME II Problems/Problem | + | [[2000 AIME II Problems/Problem 101|Solution]] |
== Problem 12 == | == Problem 12 == |
Revision as of 19:56, 17 August 2011
2000 AIME II (Answer Key) | AoPS Contest Collections | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
The number
can be written as where and are relatively prime positive integers. Find .
Problem 2
A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola ?
Problem 3
A deck of forty cards consists of four 1's, four 2's,..., and four 10's. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let be the probability that two randomly selected cards also form a pair, where and are relatively prime positive integers. Find
Problem 4
What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors?
Problem 5
Given eight distinguishable rings, let be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of .
Problem 6
One base of a trapezoid is units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio . Let x be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area. Find the greatest integer that does not exceed .
Problem 7
Given that
find the greatest integer that is less than .
Problem 8
In trapezoid , leg is perpendicular to bases and , and diagonals and are perpendicular. Given that and , find .
Problem 9
Given that is a complex number such that , find the least integer that is greater than .
Problem 10
A circle is inscribed in quadrilateral , tangent to at and to at . Given that , , , and , find the square of the radius of the circle.
Problem 11
The coordinates of the vertices of isosceles trapezoid are all integers, with and . The trapezoid has no horizontal or vertical sides, and and are the only parallel sides. The sum of the absolute values of all possible slopes for is , where and are relatively prime positive integers. Find .
Problem 12
The points , and lie on the surface of a sphere with center and radius . It is given that , , , and that the distance from to triangle is , where , , and are positive integers, and are relatively prime, and is not divisible by the square of any prime. Find .
Problem 13
The equation has exactly two real roots, one of which is , where , and are integers, and are relatively prime, and . Find .
Problem 14
Every positive integer has a unique factorial base expansion , meaning that , where each is an integer, , and . Given that is the factorial base expansion of , find the value of .
Problem 15
Find the least positive integer such that