Difference between revisions of "2020 AIME I Problems"
(→Problem 1) |
Arnigam2007 (talk | contribs) m (→Problem 13) |
||
(45 intermediate revisions by 13 users not shown) | |||
Line 2: | Line 2: | ||
==Problem 1== | ==Problem 1== | ||
− | + | In <math>\triangle ABC</math> with <math>AB=AC,</math> point <math>D</math> lies strictly between <math>A</math> and <math>C</math> on side <math>\overline{AC},</math> and point <math>E</math> lies strictly between <math>A</math> and <math>B</math> on side <math>\overline{AB}</math> such that <math>AE=ED=DB=BC.</math> The degree measure of <math>\angle ABC</math> is <math>\tfrac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math> | |
[[2020 AIME I Problems/Problem 1 | Solution]] | [[2020 AIME I Problems/Problem 1 | Solution]] | ||
==Problem 2== | ==Problem 2== | ||
− | + | There is a unique positive real number <math>x</math> such that the three numbers <math>\log_8(2x),\log_4x,</math> and <math>\log_2x,</math> in that order, form a geometric progression with positive common ratio. The number <math>x</math> can be written as <math>\tfrac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math> | |
− | |||
[[2020 AIME I Problems/Problem 2 | Solution]] | [[2020 AIME I Problems/Problem 2 | Solution]] | ||
==Problem 3== | ==Problem 3== | ||
− | + | A positive integer <math>N</math> has base-eleven representation <math>\underline{a}\kern 0.1em\underline{b}\kern 0.1em\underline{c}</math> and base-eight representation <math>\underline1\kern 0.1em\underline{b}\kern 0.1em\underline{c}\kern 0.1em\underline{a},</math> where <math>a,b,</math> and <math>c</math> represent (not necessarily distinct) digits. Find the least such <math>N</math> expressed in base ten. | |
− | |||
[[2020 AIME I Problems/Problem 3 | Solution]] | [[2020 AIME I Problems/Problem 3 | Solution]] | ||
==Problem 4== | ==Problem 4== | ||
− | + | Let <math>S</math> be the set of positive integers <math>N</math> with the property that the last four digits of <math>N</math> are <math>2020,</math> and when the last four digits are removed, the result is a divisor of <math>N.</math> For example, <math>42{,}020</math> is in <math>S</math> because <math>4</math> is a divisor of <math>42{,}020.</math> Find the sum of all the digits of all the numbers in <math>S.</math> For example, the number <math>42{,}020</math> contributes <math>4+2+0+2+0=8</math> to this total. | |
− | |||
[[2020 AIME I Problems/Problem 4 | Solution]] | [[2020 AIME I Problems/Problem 4 | Solution]] | ||
==Problem 5== | ==Problem 5== | ||
− | + | Six cards numbered <math>1</math> through <math>6</math> 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. | |
− | |||
[[2020 AIME I Problems/Problem 5 | Solution]] | [[2020 AIME I Problems/Problem 5 | Solution]] | ||
==Problem 6== | ==Problem 6== | ||
− | + | A flat board has a circular hole with radius <math>1</math> and a circular hole with radius <math>2</math> such that the distance between the centers of the two holes is <math>7</math>. 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 <math>\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | |
− | |||
[[2020 AIME I Problems/Problem 6 | Solution]] | [[2020 AIME I Problems/Problem 6 | Solution]] | ||
==Problem 7== | ==Problem 7== | ||
− | + | A club consisting of <math>11</math> men and <math>12</math> 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 <math>1</math> member or as many as <math>23</math> members. Let <math>N</math> be the number of such committees that can be formed. Find the sum of the prime numbers that divide <math>N.</math> | |
− | |||
[[2020 AIME I Problems/Problem 7 | Solution]] | [[2020 AIME I Problems/Problem 7 | Solution]] | ||
==Problem 8== | ==Problem 8== | ||
− | + | A bug walks all day and sleeps all night. On the first day, it starts at point <math>O,</math> faces east, and walks a distance of <math>5</math> units due east. Each night the bug rotates <math>60^\circ</math> 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 <math>P.</math> Then <math>OP^2=\tfrac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math> | |
− | |||
[[2020 AIME I Problems/Problem 8 | Solution]] | [[2020 AIME I Problems/Problem 8 | Solution]] | ||
==Problem 9== | ==Problem 9== | ||
+ | Let <math>S</math> be the set of positive integer divisors of <math>20^9.</math> Three numbers are chosen independently and at random with replacement from the set <math>S</math> and labeled <math>a_1,a_2,</math> and <math>a_3</math> in the order they are chosen. The probability that both <math>a_1</math> divides <math>a_2</math> and <math>a_2</math> divides <math>a_3</math> is <math>\tfrac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m.</math> | ||
+ | [[2020 AIME I Problems/Problem 9 | Solution]] | ||
+ | ==Problem 10== | ||
+ | Let <math>m</math> and <math>n</math> be positive integers satisfying the conditions | ||
− | + | <math>\quad\bullet\ \gcd(m+n,210)=1,</math> | |
− | + | <math>\quad\bullet\ m^m</math> is a multiple of <math>n^n,</math> and | |
+ | <math>\quad\bullet\ m</math> is not a multiple of <math>n.</math> | ||
+ | Find the least possible value of <math>m+n.</math> | ||
[[2020 AIME I Problems/Problem 10 | Solution]] | [[2020 AIME I Problems/Problem 10 | Solution]] | ||
==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> | |
− | |||
[[2020 AIME I Problems/Problem 11 | Solution]] | [[2020 AIME I Problems/Problem 11 | Solution]] | ||
==Problem 12== | ==Problem 12== | ||
− | + | Let <math>n</math> be the least positive integer for which <math>149^n-2^n</math> is divisible by <math>3^3\cdot5^5\cdot7^7.</math> Find the number of positive integer divisors of <math>n.</math> | |
− | |||
[[2020 AIME I Problems/Problem 12 | Solution]] | [[2020 AIME I Problems/Problem 12 | Solution]] | ||
==Problem 13== | ==Problem 13== | ||
− | + | Point <math>D</math> lies on side <math>\overline{BC}</math> of <math>\triangle ABC</math> so that <math>\overline{AD}</math> bisects <math>\angle BAC.</math> The perpendicular bisector of <math>\overline{AD}</math> intersects the bisectors of <math>\angle ABC</math> and <math>\angle ACB</math> in points <math>E</math> and <math>F,</math> respectively. Given that <math>AB=4,BC=5,</math> and <math>CA=6,</math> the area of <math>\triangle AEF</math> can be written as <math>\tfrac{m\sqrt{n}}p,</math> where <math>m</math> and <math>p</math> are relatively prime positive integers, and <math>n</math> is a positive integer not divisible by the square of any prime. Find <math>m+n+p</math>. | |
− | |||
[[2020 AIME I Problems/Problem 13 | Solution]] | [[2020 AIME I Problems/Problem 13 | Solution]] | ||
==Problem 14== | ==Problem 14== | ||
− | + | Let <math>P(x)</math> be a quadratic polynomial with complex coefficients whose <math>x^2</math> coefficient is <math>1.</math> Suppose the equation <math>P(P(x))=0</math> has four distinct solutions, <math>x=3,4,a,b.</math> Find the sum of all possible values of <math>(a+b)^2.</math> | |
− | |||
[[2020 AIME I Problems/Problem 14 | Solution]] | [[2020 AIME I Problems/Problem 14 | Solution]] | ||
==Problem 15== | ==Problem 15== | ||
− | + | Let <math>\triangle ABC</math> be an acute triangle with circumcircle <math>\omega,</math> and let <math>H</math> be the intersection of the altitudes of <math>\triangle ABC.</math> Suppose the tangent to the circumcircle of <math>\triangle HBC</math> at <math>H</math> intersects <math>\omega</math> at points <math>X</math> and <math>Y</math> with <math>HA=3,HX=2,</math> and <math>HY=6.</math> The area of <math>\triangle ABC</math> can be written in the form <math>m\sqrt{n},</math> where <math>m</math> and <math>n</math> are positive integers, and <math>n</math> is not divisible by the square of any prime. Find <math>m+n.</math> | |
[[2020 AIME I Problems/Problem 15 | Solution]] | [[2020 AIME I Problems/Problem 15 | Solution]] | ||
− | {{AIME box|year=2020|n=I|before=[[2019 AIME II]]|after=[[2020 AIME II]]}} | + | {{AIME box|year=2020|n=I|before=[[2019 AIME II Problems]]|after=[[2020 AIME II Problems]]}} |
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 05:07, 8 March 2021
2020 AIME I (Answer Key) | AoPS Contest Collections | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
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 is . 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
Let be the least positive integer for which is divisible by Find the number of positive integer divisors of
Problem 13
Point lies on side of so that bisects The perpendicular bisector of intersects the bisectors of and in points and respectively. Given that and the area of can be written as where and are relatively prime positive integers, and is a positive integer not divisible by the square of any prime. Find .
Problem 14
Let be a quadratic polynomial with complex coefficients whose coefficient is Suppose the equation has four distinct solutions, Find the sum of all possible values of
Problem 15
Let be an acute triangle with circumcircle and let be the intersection of the altitudes of Suppose the tangent to the circumcircle of at intersects at points and with and The area of can be written in the form where and are positive integers, and is not divisible by the square of any prime. Find
2020 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2019 AIME II Problems |
Followed by 2020 AIME II Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.