Difference between revisions of "2020 AIME I Problems"
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==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]] | ||
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==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 | + | 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]] | ||
Latest revision as of 16:51, 5 February 2022
| 2020 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
|
Instructions
| ||
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
to 
, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.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) | ||
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Followed by 2020 AIME II Problems | |
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| All AIME Problems and Solutions | ||
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