Difference between revisions of "2020 AIME II Problems"
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==Problem 1== | ==Problem 1== | ||
| − | + | Find the number of ordered pairs of positive integers <math>(m,n)</math> such that <math>{m^2n = 20 ^{20}}</math>. | |
[[2020 AIME II Problems/Problem 1 | Solution]] | [[2020 AIME II Problems/Problem 1 | Solution]] | ||
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==Problem 2== | ==Problem 2== | ||
| − | + | Let <math>P</math> be a point chosen uniformly at random in the interior of the unit square with vertices at <math>(0,0), (1,0), (1,1)</math>, and <math>(0,1)</math>. The probability that the slope of the line determined by <math>P</math> and the point <math>\left(\frac58, \frac38 \right)</math> is greater than or equal to <math>\frac12</math> can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | |
[[2020 AIME II Problems/Problem 2 | Solution]] | [[2020 AIME II Problems/Problem 2 | Solution]] | ||
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==Problem 3== | ==Problem 3== | ||
| + | The value of <math>x</math> that satisfies <math>\log_{2^x} 3^{20} = \log_{2^{x+3}} 3^{2020}</math> can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
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[[2020 AIME II Problems/Problem 3 | Solution]] | [[2020 AIME II Problems/Problem 3 | Solution]] | ||
==Problem 4== | ==Problem 4== | ||
| − | + | Triangles <math>\triangle ABC</math> and <math>\triangle A'B'C'</math> lie in the coordinate plane with vertices <math>A(0,0)</math>, <math>B(0,12)</math>, <math>C(16,0)</math>, <math>A'(24,18)</math>, <math>B'(36,18)</math>, <math>C'(24,2)</math>. A rotation of <math>m</math> degrees clockwise around the point <math>(x,y)</math> where <math>0<m<180</math>, will transform <math>\triangle ABC</math> to <math>\triangle A'B'C'</math>. Find <math>m+x+y</math>. | |
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[[2020 AIME II Problems/Problem 4 | Solution]] | [[2020 AIME II Problems/Problem 4 | Solution]] | ||
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==Problem 5== | ==Problem 5== | ||
| − | + | For each positive integer <math>n</math>, let <math>f(n)</math> be the sum of the digits in the base-four representation of <math>n</math> and let <math>g(n)</math> be the sum of the digits in the base-eight representation of <math>f(n)</math>. For example, <math>f(2020) = f(133210_{\text{4}}) = 10 = 12_{\text{8}}</math>, and <math>g(2020) = \text{the digit sum of }12_{\text{8}} = 3</math>. Let <math>N</math> be the least value of <math>n</math> such that the base-sixteen representation of <math>g(n)</math> cannot be expressed using only the digits <math>0</math> through <math>9</math>. Find the remainder when <math>N</math> is divided by <math>1000</math>. | |
[[2020 AIME II Problems/Problem 5 | Solution]] | [[2020 AIME II Problems/Problem 5 | Solution]] | ||
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==Problem 6== | ==Problem 6== | ||
| − | + | Define a sequence recursively by <math>t_1 = 20</math>, <math>t_2 = 21</math>, and<cmath>t_n = \frac{5t_{n-1}+1}{25t_{n-2}}</cmath>for all <math>n \ge 3</math>. Then <math>t_{2020}</math> can be written as <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. | |
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[[2020 AIME II Problems/Problem 6 | Solution]] | [[2020 AIME II Problems/Problem 6 | Solution]] | ||
==Problem 7== | ==Problem 7== | ||
| − | + | Two congruent right circular cones each with base radius <math>3</math> and height <math>8</math> have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance <math>3</math> from the base of each cone. A sphere with radius <math>r</math> lies within both cones. The maximum possible value of <math>r^2</math> is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | |
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[[2020 AIME II Problems/Problem 7 | Solution]] | [[2020 AIME II Problems/Problem 7 | Solution]] | ||
==Problem 8== | ==Problem 8== | ||
| − | + | Define a sequence recursively by <math>f_1(x)=|x-1|</math> and <math>f_n(x)=f_{n-1}(|x-n|)</math> for integers <math>n>1</math>. Find the least value of <math>n</math> such that the sum of the zeros of <math>f_n</math> exceeds <math>500,000</math>. | |
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[[2020 AIME II Problems/Problem 8 | Solution]] | [[2020 AIME II Problems/Problem 8 | Solution]] | ||
==Problem 9== | ==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. | |
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[[2020 AIME II Problems/Problem 9 | Solution]] | [[2020 AIME II Problems/Problem 9 | Solution]] | ||
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==Problem 10== | ==Problem 10== | ||
| + | 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>. | ||
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[[2020 AIME II Problems/Problem 10 | Solution]] | [[2020 AIME II Problems/Problem 10 | Solution]] | ||
==Problem 11== | ==Problem 11== | ||
| + | Let <math>P(x) = x^2 - 3x - 7</math>, and let <math>Q(x)</math> and <math>R(x)</math> be two quadratic polynomials also with the coefficient of <math>x^2</math> equal to <math>1</math>. David computes each of the three sums <math>P + Q</math>, <math>P + R</math>, and <math>Q + R</math> and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If <math>Q(0) = 2</math>, then <math>R(0) = \frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. | ||
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[[2020 AIME II Problems/Problem 11 | Solution]] | [[2020 AIME II Problems/Problem 11 | Solution]] | ||
==Problem 12== | ==Problem 12== | ||
| − | + | Let <math>m</math> and <math>n</math> be odd integers greater than <math>1.</math> An <math>m\times n</math> rectangle is made up of unit squares where the squares in the top row are numbered left to right with the integers <math>1</math> through <math>n</math>, those in the second row are numbered left to right with the integers <math>n + 1</math> through <math>2n</math>, and so on. Square <math>200</math> is in the top row, and square <math>2000</math> is in the bottom row. Find the number of ordered pairs <math>(m,n)</math> of odd integers greater than <math>1</math> with the property that, in the <math>m\times n</math> rectangle, the line through the centers of squares <math>200</math> and <math>2000</math> intersects the interior of square <math>1099</math>. | |
[[2020 AIME II Problems/Problem 12 | Solution]] | [[2020 AIME II Problems/Problem 12 | Solution]] | ||
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==Problem 13== | ==Problem 13== | ||
| − | + | Convex pentagon <math>ABCDE</math> has side lengths <math>AB=5</math>, <math>BC=CD=DE=6</math>, and <math>EA=7</math>. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of <math>ABCDE</math>. | |
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[[2020 AIME II Problems/Problem 13 | Solution]] | [[2020 AIME II Problems/Problem 13 | Solution]] | ||
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==Problem 14== | ==Problem 14== | ||
| − | + | For real number <math>x</math> let <math>\lfloor x\rfloor</math> be the greatest integer less than or equal to <math>x</math>, and define <math>\{x\} = x - \lfloor x \rfloor</math> to be the fractional part of <math>x</math>. For example, <math>\{3\} = 0</math> and <math>\{4.56\} = 0.56</math>. Define <math>f(x)=x\{x\}</math>, and let <math>N</math> be the number of real-valued solutions to the equation <math>f(f(f(x)))=17</math> for <math>0\leq x\leq 2020</math>. Find the remainder when <math>N</math> is divided by <math>1000</math>. | |
[[2020 AIME II Problems/Problem 14 | Solution]] | [[2020 AIME II Problems/Problem 14 | Solution]] | ||
Latest revision as of 15:17, 12 March 2021
| 2020 AIME II (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
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 
, 
, and![\[t_n = \frac{5t_{n-1}+1}{25t_{n-2}}\]](http://latex.artofproblemsolving.com/0/5/5/055d678205c203b8399cd576b7812bb54597c68a.png)
for 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 | |
| 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. 
