Difference between revisions of "2015 AIME I Problems"
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==Problem 14== | ==Problem 14== | ||
− | For each integer <math>n \ge 2</math>, let <math>A(n)</math> be the area of the region in the coordinate plane | + | For each integer <math>n \ge 2</math>, let <math>A(n)</math> be the area of the region in the coordinate plane defined by the inequalities <math>1\le x \le n</math> and <math>0\le y \le x \left\lfloor \sqrt x \right\rfloor</math>, where <math>\left\lfloor \sqrt x \right\rfloor</math> is the greatest integer not exceeding <math>\sqrt x</math>. Find the number of values of <math>n</math> with <math>2\le n \le 1000</math> for which <math>A(n)</math> is an integer. |
[[2015 AIME I Problems/Problem 14|Solution]] | [[2015 AIME I Problems/Problem 14|Solution]] |
Revision as of 17:45, 10 May 2015
2015 AIME I (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 expressions = and = are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers and .
Problem 2
The nine delegates to the Economic Cooperation Conference include officials from Mexico, officials from Canada, and officials from the United States. During the opening session, three of the delegates fall asleep. Assuming that the three sleepers were determined randomly, the probability that exactly two of the sleepers are from the same country is , where and are relatively prime positive integers. Find .
Problem 3
There is a prime number such that is the cube of a positive integer. Find .
Problem 4
Point lies on line segment with and . Points and lie on the same side of line forming equilateral triangles and . Let be the midpoint of , and be the midpoint of . The area of is . Find .
Problem 5
In a drawer Sandy has pairs of socks, each pair a different color. On Monday, Sandy selects two individual socks at random from the socks in the drawer. On Tuesday Sandy selects of the remaining socks at random, and on Wednesday two of the remaining socks at random. The probability that Wednesday is the first day Sandy selects matching socks is , where and are relatively prime positive integers. Find .
Problem 6
Point and are equally spaced on a minor arc of a circle. Points and are equally spaced on a minor arc of a second circle with center as shown in the figure below. The angle exceeds by . Find the degree measure of .
Problem 7
In the diagram below, is a square. Point is the midpoint of . Points and lie on , and and lie on and , respectively, so that is a square. Points and lie on , and and lie on and , respectively, so that is a square. The area of is 99. Find the area of .
Problem 8
For positive integer , let denote the sum of the digits of . Find the smallest positive integer satisfying .
Problem 9
Let be the set of all ordered triple of integers with . Each ordered triple in generates a sequence according to the rule for all . Find the number of such sequences for which for some .
Problem 10
Let be a third-degree polynomial with real coefficients satisfying Find .
Problem 11
Triangle has positive integer side lengths with . Let be the intersection of the bisectors of and . Suppose . Find the smallest possible perimeter of .
Problem 12
Consider all 1000-element subsets of the set . From each such subset choose the least element. The arithmetic mean of all of these least elements is , where and are relatively prime positive integers. Find .
Problem 13
With all angles measured in degrees, the product , where and are integers greater than 1. Find .
Problem 14
For each integer , let be the area of the region in the coordinate plane defined by the inequalities and , where is the greatest integer not exceeding . Find the number of values of with for which is an integer.
Problem 15
A block of wood has the shape of a right circular cylinder with radius and height , and its entire surface has been painted blue. Points and are chosen on the edge of one of the circular faces of the cylinder so that on that face measures . The block is then sliced in half along the plane that passes through point , point , and the center of the cylinder, revealing a flat, unpainted face on each half. The area of one of these unpainted faces is , where , , and are integers and is not divisible by the square of any prime. Find .
2015 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2014 AIME II |
Followed by 2015 AIME II | |
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All AIME Problems and Solutions |
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