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==Problem 11== | ==Problem 11== | ||
− | + | For integers <math>a,b,c</math> and <math>d,</math> let <math>f(x)=x^2+ax+b</math> and <math>g(x)=x^2+cx+d.</math> Find the number of ordered triples <math>(a,b,c)</math> of integers with absolute values not exceeding <math>10</math> for which there is an integer <math>d</math> such that <math>g(f(2))=g(f(4))=0.</math> | |
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[[2020 AIME I Problems/Problem 11 | Solution]] | [[2020 AIME I Problems/Problem 11 | Solution]] |
Revision as of 16:54, 12 March 2020
2020 AIME I (Answer Key) | AoPS Contest Collections | ||
Instructions
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Contents
Problem 1
In with point lies strictly between and on side and point lies strictly between and on side such that The degree measure of is where and are relatively prime positive integers. Find
Problem 2
There is a unique positive real number such that the three numbers and in that order, form a geometric progression with positive common ratio. The number can be written as where and are relatively prime positive integers. Find
Problem 3
A positive integer has base-eleven representation and base-eight representation where and represent (not necessarily distinct) digits. Find the least such expressed in base ten.
Problem 4
Let be the set of positive integers with the property that the last four digits of are and when the last four digits are removed, the result is a divisor of For example, is in because is a divisor of Find the sum of all the digits of all the numbers in For example, the number contributes to this total.
Problem 5
Six cards numbered through are to be lined up in a row. Find the number of arrangements of these six cards where one of the cards can be removed leaving the remaining five cards in either ascending or descending order.
Problem 6
A flat board has a circular hole with radius and a circular hole with radius such that the distance between the centers of the two holes in Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is where and are relatively prime positive integers. Find
Problem 7
A club consisting of men and women needs to choose a committee from among its members so that the number of women on the committee is one more than the number of men on the committee. The committee could have as few as member or as many as members. Let be the number of such committees that can be formed. Find the sum of the prime numbers that divide
Problem 8
A bug walks all day and sleeps all night. On the first day, it starts at point faces east, and walks a distance of units due east. Each night the bug rotates counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to the point Then where and are relatively prime positive integers. Find
Problem 9
Let be the set of positive integer divisors of Three numbers are chosen independently and at random with replacement from the set and labeled and in the order they are chosen. The probability that both divides and divides is where and are relatively prime positive integers. Find
Problem 10
Let and be positive integers satisfying the conditions
is a multiple of and
is not a multiple of
Find the least possible value of
Problem 11
For integers and let and Find the number of ordered triples of integers with absolute values not exceeding for which there is an integer such that
Problem 12
Problem 13
Problem 14
Problem 15
Let
2020 AIME I (Problems • Answer Key • Resources) | ||
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