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==Problem 10== | ==Problem 10== | ||
− | Find the sum of | + | Find the sum of all positive integers <math>n</math> such that when <math>1^3+2^3+3^3+\cdots +n^3</math> is divided by <math>n+5</math>, the remainder is <math>17</math>. |
[[2020 AIME II Problems/Problem 10 | Solution]] | [[2020 AIME II Problems/Problem 10 | Solution]] |
Revision as of 22:38, 26 November 2020
2020 AIME II (Answer Key) | AoPS Contest Collections | ||
Instructions
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Contents
Problem 1
Find the number of ordered pairs of positive integers such that .
Problem 2
Let be a point chosen uniformly at random in the interior of the unit square with vertices at , and . The probability that the slope of the line determined by and the point is greater than or equal to can be written as , where and are relatively prime positive integers. Find .
Problem 3
The value of that satisfies can be written as , where and are relatively prime positive integers. Find .
Problem 4
Triangles and lie in the coordinate plane with vertices , , , , , . A rotation of degrees clockwise around the point where , will transform to . Find .
Problem 5
For each positive integer , let be the sum of the digits in the base-four representation of and let be the sum of the digits in the base-eight representation of . For example, , and . Let be the least value of such that the base-sixteen representation of cannot be expressed using only the digits through . Find the remainder when is divided by .
Problem 6
Define a sequence recursively by , , andfor all . Then can be written as , where and are relatively prime positive integers. Find .
Problem 7
Two congruent right circular cones each with base radius and height have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance from the base of each cone. A sphere with radius lies within both cones. The maximum possible value of is , where and are relatively prime positive integers. Find .
Problem 8
Define a sequence recursively by and for integers . Find the least value of such that the sum of the zeros of exceeds .
Problem 9
While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit next to each other after the break. Find the number of possible seating orders they could have chosen after the break.
Problem 10
Find the sum of all positive integers such that when is divided by , the remainder is .
Problem 11
Let , and let and be two quadratic polynomials also with the coefficient of equal to . David computes each of the three sums , , and and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If , then , where and are relatively prime positive integers. Find .
Problem 12
Let and be odd integers greater than An rectangle is made up of unit squares where the squares in the top row are numbered left to right with the integers through , those in the second row are numbered left to right with the integers through , and so on. Square is in the top row, and square is in the bottom row. Find the number of ordered pairs of odd integers greater than with the property that, in the rectangle, the line through the centers of squares and intersects the interior of square .
Problem 13
Convex pentagon has side lengths , , and . Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of .
Problem 14
For real number let be the greatest integer less than or equal to , and define to be the fractional part of . For example, and . Define , and let be the number of real-valued solutions to the equation for . Find the remainder when is divided by .
Problem 15
Let be an acute scalene triangle with circumcircle . The tangents to at and intersect at . Let and be the projections of onto lines and , respectively. Suppose , , and . Find .
2020 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2020 AIME I Problems |
Followed by 2021 AIME I Problems | |
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