Difference between revisions of "2020 AIME II Problems"
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− | + | {{AIME Problems|year=2020|n=II}} | |
+ | |||
+ | ==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]] | ||
+ | |||
+ | ==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]] | ||
+ | |||
+ | ==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>. | ||
+ | |||
+ | [[2020 AIME II Problems/Problem 3 | Solution]] | ||
+ | |||
+ | ==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>. | ||
+ | |||
+ | [[2020 AIME II Problems/Problem 4 | Solution]] | ||
+ | |||
+ | ==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]] | ||
+ | |||
+ | ==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>. | ||
+ | |||
+ | [[2020 AIME II Problems/Problem 6 | Solution]] | ||
+ | |||
+ | ==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>. | ||
+ | |||
+ | [[2020 AIME II Problems/Problem 7 | Solution]] | ||
+ | |||
+ | ==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>. | ||
+ | |||
+ | [[2020 AIME II Problems/Problem 8 | Solution]] | ||
+ | |||
+ | ==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. | ||
+ | |||
+ | [[2020 AIME II Problems/Problem 9 | Solution]] | ||
+ | |||
+ | ==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>. | ||
+ | |||
+ | [[2020 AIME II Problems/Problem 10 | Solution]] | ||
+ | |||
+ | ==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>. | ||
+ | |||
+ | [[2020 AIME II Problems/Problem 11 | Solution]] | ||
+ | |||
+ | ==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]] | ||
+ | |||
+ | ==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>. | ||
+ | |||
+ | [[2020 AIME II Problems/Problem 13 | Solution]] | ||
+ | |||
+ | ==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]] | ||
+ | |||
+ | ==Problem 15== | ||
+ | |||
+ | Let <math>\triangle ABC</math> be an acute scalene triangle with circumcircle <math>\omega</math>. The tangents to <math>\omega</math> at <math>B</math> and <math>C</math> intersect at <math>T</math>. Let <math>X</math> and <math>Y</math> be the projections of <math>T</math> onto lines <math>AB</math> and <math>AC</math>, respectively. Suppose <math>BT = CT = 16</math>, <math>BC = 22</math>, and <math>TX^2 + TY^2 + XY^2 = 1143</math>. Find <math>XY^2</math>. | ||
+ | |||
+ | [[2020 AIME II Problems/Problem 15 | Solution]] | ||
+ | |||
+ | {{AIME box|year=2020|n=II|before=[[2020 AIME I Problems]]|after=[[2021 AIME I Problems]]}} | ||
+ | {{MAA Notice}} |
Latest revision as of 16:17, 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 |
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
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.