Difference between revisions of "2021 AIME II Problems"
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{{AIME Problems|year=2021|n=II}} | {{AIME Problems|year=2021|n=II}} | ||
==Problem 1== | ==Problem 1== | ||
Line 9: | Line 7: | ||
==Problem 2== | ==Problem 2== | ||
Equilateral triangle <math>ABC</math> has side length <math>840</math>. Point <math>D</math> lies on the same side of line <math>BC</math> as <math>A</math> such that <math>\overline{BD} \perp \overline{BC}</math>. The line <math>\ell</math> through <math>D</math> parallel to line <math>BC</math> intersects sides <math>\overline{AB}</math> and <math>\overline{AC}</math> at points <math>E</math> and <math>F</math>, respectively. Point <math>G</math> lies on <math>\ell</math> such that <math>F</math> is between <math>E</math> and <math>G</math>, <math>\triangle AFG</math> is isosceles, and the ratio of the area of <math>\triangle AFG</math> to the area of <math>\triangle BED</math> is <math>8:9</math>. Find <math>AF</math>. | Equilateral triangle <math>ABC</math> has side length <math>840</math>. Point <math>D</math> lies on the same side of line <math>BC</math> as <math>A</math> such that <math>\overline{BD} \perp \overline{BC}</math>. The line <math>\ell</math> through <math>D</math> parallel to line <math>BC</math> intersects sides <math>\overline{AB}</math> and <math>\overline{AC}</math> at points <math>E</math> and <math>F</math>, respectively. Point <math>G</math> lies on <math>\ell</math> such that <math>F</math> is between <math>E</math> and <math>G</math>, <math>\triangle AFG</math> is isosceles, and the ratio of the area of <math>\triangle AFG</math> to the area of <math>\triangle BED</math> is <math>8:9</math>. Find <math>AF</math>. | ||
− | + | <asy> | |
− | + | pair A,B,C,D,E,F,G; | |
+ | B=origin; | ||
+ | A=5*dir(60); | ||
+ | C=(5,0); | ||
+ | E=0.6*A+0.4*B; | ||
+ | F=0.6*A+0.4*C; | ||
+ | G=rotate(240,F)*A; | ||
+ | D=extension(E,F,B,dir(90)); | ||
+ | draw(D--G--A,grey); | ||
+ | draw(B--0.5*A+rotate(60,B)*A*0.5,grey); | ||
+ | draw(A--B--C--cycle,linewidth(1.5)); | ||
+ | dot(A^^B^^C^^D^^E^^F^^G); | ||
+ | label("$A$",A,dir(90)); | ||
+ | label("$B$",B,dir(225)); | ||
+ | label("$C$",C,dir(-45)); | ||
+ | label("$D$",D,dir(180)); | ||
+ | label("$E$",E,dir(-45)); | ||
+ | label("$F$",F,dir(225)); | ||
+ | label("$G$",G,dir(0)); | ||
+ | label("$\ell$",midpoint(E--F),dir(90)); | ||
+ | </asy> | ||
[[2021 AIME II Problems/Problem 2|Solution]] | [[2021 AIME II Problems/Problem 2|Solution]] | ||
==Problem 3== | ==Problem 3== | ||
− | Find the number of permutations <math>x_1, x_2, x_3, x_4, x_5</math> of numbers <math>1, 2, 3, 4, 5</math> such that the sum of five products | + | Find the number of permutations <math>x_1, x_2, x_3, x_4, x_5</math> of numbers <math>1, 2, 3, 4, 5</math> such that the sum of five products <cmath>x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2</cmath> is divisible by <math>3</math>. |
− | <cmath>x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2</cmath>is divisible by <math>3</math>. | ||
[[2021 AIME II Problems/Problem 3|Solution]] | [[2021 AIME II Problems/Problem 3|Solution]] | ||
==Problem 4== | ==Problem 4== | ||
− | These | + | There are real numbers <math>a, b, c, </math> and <math>d</math> such that <math>-20</math> is a root of <math>x^3 + ax + b</math> and <math>-21</math> is a root of <math>x^3 + cx^2 + d.</math> These two polynomials share a complex root <math>m + \sqrt{n} \cdot i, </math> where <math>m</math> and <math>n</math> are positive integers and <math>i = \sqrt{-1}.</math> Find <math>m+n.</math> |
[[2021 AIME II Problems/Problem 4|Solution]] | [[2021 AIME II Problems/Problem 4|Solution]] | ||
==Problem 5== | ==Problem 5== | ||
− | + | For positive real numbers <math>s</math>, let <math>\tau(s)</math> denote the set of all obtuse triangles that have area <math>s</math> and two sides with lengths <math>4</math> and <math>10</math>. The set of all <math>s</math> for which <math>\tau(s)</math> is nonempty, but all triangles in <math>\tau(s)</math> are congruent, is an interval <math>[a,b)</math>. Find <math>a^2+b^2</math>. | |
[[2021 AIME II Problems/Problem 5|Solution]] | [[2021 AIME II Problems/Problem 5|Solution]] | ||
==Problem 6== | ==Problem 6== | ||
− | + | For any finite set <math>S</math>, let <math>|S|</math> denote the number of elements in <math>S</math>. Find the number of ordered pairs <math>(A,B)</math> such that <math>A</math> and <math>B</math> are (not necessarily distinct) subsets of <math>\{1,2,3,4,5\}</math> that satisfy | |
+ | <cmath>|A| \cdot |B| = |A \cap B| \cdot |A \cup B|</cmath> | ||
[[2021 AIME II Problems/Problem 6|Solution]] | [[2021 AIME II Problems/Problem 6|Solution]] | ||
==Problem 7== | ==Problem 7== | ||
− | + | Let <math>a, b, c,</math> and <math>d</math> be real numbers that satisfy the system of equations | |
+ | <cmath>\begin{align*} | ||
+ | a + b &= -3, \\ | ||
+ | ab + bc + ca &= -4, \\ | ||
+ | abc + bcd + cda + dab &= 14, \\ | ||
+ | abcd &= 30. | ||
+ | \end{align*}</cmath> | ||
+ | There exist relatively prime positive integers <math>m</math> and <math>n</math> such that | ||
+ | <cmath>a^2 + b^2 + c^2 + d^2 = \frac{m}{n}. </cmath>Find <math>m + n</math>. | ||
[[2021 AIME II Problems/Problem 7|Solution]] | [[2021 AIME II Problems/Problem 7|Solution]] | ||
==Problem 8== | ==Problem 8== | ||
− | + | An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an adjacent vertex along an edge of the cube. Initially the ant is at a vertex of the bottom face of the cube and chooses one of the three adjacent vertices to move to as its first move. For all moves after the first move, the ant does not return to its previous vertex, but chooses to move to one of the other two adjacent vertices. All choices are selected at random so that each of the possible moves is equally likely. The probability that after exactly <math>8</math> moves that ant is at a vertex of the top face on the cube is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n.</math> | |
[[2021 AIME II Problems/Problem 8|Solution]] | [[2021 AIME II Problems/Problem 8|Solution]] | ||
==Problem 9== | ==Problem 9== | ||
− | + | Find the number of ordered pairs <math>(m, n)</math> such that <math>m</math> and <math>n</math> are positive integers in the set <math>\{1, 2, ..., 30\}</math> and the greatest common divisor of <math>2^m + 1</math> and <math>2^n - 1</math> is not <math>1</math>. | |
[[2021 AIME II Problems/Problem 9|Solution]] | [[2021 AIME II Problems/Problem 9|Solution]] | ||
==Problem 10== | ==Problem 10== | ||
− | + | Two spheres with radii <math>36</math> and one sphere with radius <math>13</math> are each externally tangent to the other two spheres and to two different planes <math>\mathcal{P}</math> and <math>\mathcal{Q}</math>. The intersection of planes <math>\mathcal{P}</math> and <math>\mathcal{Q}</math> is the line <math>\ell</math>. The distance from line <math>\ell</math> to the point where the sphere with radius <math>13</math> is tangent to plane <math>\mathcal{P}</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>. | |
[[2021 AIME II Problems/Problem 10|Solution]] | [[2021 AIME II Problems/Problem 10|Solution]] | ||
==Problem 11== | ==Problem 11== | ||
− | + | A teacher was leading a class of four perfectly logical students. The teacher chose a set <math>S</math> of four integers and gave a different number in <math>S</math> to each student. Then the teacher announced to the class that the numbers in <math>S</math> were four consecutive two-digit positive integers, that some number in <math>S</math> was divisible by <math>6</math>, and a different number in <math>S</math> was divisible by <math>7</math>. The teacher then asked if any of the students could deduce what <math>S</math> is, but in unison, all of the students replied no. | |
+ | |||
+ | However, upon hearing that all four students replied no, each student was able to determine the elements of <math>S</math>. Find the sum of all possible values of the greatest element of <math>S</math>. | ||
[[2021 AIME II Problems/Problem 11|Solution]] | [[2021 AIME II Problems/Problem 11|Solution]] | ||
==Problem 12== | ==Problem 12== | ||
− | + | A convex quadrilateral has area <math>30</math> and side lengths <math>5, 6, 9,</math> and <math>7,</math> in that order. Denote by <math>\theta</math> the measure of the acute angle formed by the diagonals of the quadrilateral. Then <math>\tan \theta</math> can be written in the form <math>\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. | |
[[2021 AIME II Problems/Problem 12|Solution]] | [[2021 AIME II Problems/Problem 12|Solution]] | ||
==Problem 13== | ==Problem 13== | ||
− | + | Find the least positive integer <math>n</math> for which <math>2^n + 5^n - n</math> is a multiple of <math>1000</math>. | |
[[2021 AIME II Problems/Problem 13|Solution]] | [[2021 AIME II Problems/Problem 13|Solution]] | ||
==Problem 14== | ==Problem 14== | ||
− | + | Let <math>\Delta ABC</math> be an acute triangle with circumcenter <math>O</math> and centroid <math>G</math>. Let <math>X</math> be the intersection of the line tangent to the circumcircle of <math>\Delta ABC</math> at <math>A</math> and the line perpendicular to <math>GO</math> at <math>G</math>. Let <math>Y</math> be the intersection of lines <math>XG</math> and <math>BC</math>. Given that the measures of <math>\angle ABC, \angle BCA, </math> and <math>\angle XOY</math> are in the ratio <math>13 : 2 : 17, </math> the degree measure of <math>\angle BAC</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>. | |
[[2021 AIME II Problems/Problem 14|Solution]] | [[2021 AIME II Problems/Problem 14|Solution]] | ||
==Problem 15== | ==Problem 15== | ||
− | + | Let <math>f(n)</math> and <math>g(n)</math> be functions satisfying | |
+ | <cmath>f(n) = | ||
+ | \begin{cases} | ||
+ | \sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ | ||
+ | 1 + f(n+1) & \text{ otherwise} | ||
+ | \end{cases}</cmath> | ||
+ | and | ||
+ | <cmath>g(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ | ||
+ | 2 + g(n+2) & \text{ otherwise} | ||
+ | \end{cases}</cmath> | ||
+ | for positive integers <math>n</math>. Find the least positive integer <math>n</math> such that <math>\tfrac{f(n)}{g(n)} = \tfrac{4}{7}</math>. | ||
[[2021 AIME II Problems/Problem 15|Solution]] | [[2021 AIME II Problems/Problem 15|Solution]] |
Latest revision as of 14:31, 26 February 2024
2021 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 arithmetic mean of all the three-digit palindromes. (Recall that a palindrome is a number that reads the same forward and backward, such as or
.)
Problem 2
Equilateral triangle has side length
. Point
lies on the same side of line
as
such that
. The line
through
parallel to line
intersects sides
and
at points
and
, respectively. Point
lies on
such that
is between
and
,
is isosceles, and the ratio of the area of
to the area of
is
. Find
.
Problem 3
Find the number of permutations of numbers
such that the sum of five products
is divisible by
.
Problem 4
There are real numbers and
such that
is a root of
and
is a root of
These two polynomials share a complex root
where
and
are positive integers and
Find
Problem 5
For positive real numbers , let
denote the set of all obtuse triangles that have area
and two sides with lengths
and
. The set of all
for which
is nonempty, but all triangles in
are congruent, is an interval
. Find
.
Problem 6
For any finite set , let
denote the number of elements in
. Find the number of ordered pairs
such that
and
are (not necessarily distinct) subsets of
that satisfy
Problem 7
Let and
be real numbers that satisfy the system of equations
There exist relatively prime positive integers
and
such that
Find
.
Problem 8
An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an adjacent vertex along an edge of the cube. Initially the ant is at a vertex of the bottom face of the cube and chooses one of the three adjacent vertices to move to as its first move. For all moves after the first move, the ant does not return to its previous vertex, but chooses to move to one of the other two adjacent vertices. All choices are selected at random so that each of the possible moves is equally likely. The probability that after exactly moves that ant is at a vertex of the top face on the cube is
, where
and
are relatively prime positive integers. Find
Problem 9
Find the number of ordered pairs such that
and
are positive integers in the set
and the greatest common divisor of
and
is not
.
Problem 10
Two spheres with radii and one sphere with radius
are each externally tangent to the other two spheres and to two different planes
and
. The intersection of planes
and
is the line
. The distance from line
to the point where the sphere with radius
is tangent to plane
is
, where
and
are relatively prime positive integers. Find
.
Problem 11
A teacher was leading a class of four perfectly logical students. The teacher chose a set of four integers and gave a different number in
to each student. Then the teacher announced to the class that the numbers in
were four consecutive two-digit positive integers, that some number in
was divisible by
, and a different number in
was divisible by
. The teacher then asked if any of the students could deduce what
is, but in unison, all of the students replied no.
However, upon hearing that all four students replied no, each student was able to determine the elements of . Find the sum of all possible values of the greatest element of
.
Problem 12
A convex quadrilateral has area and side lengths
and
in that order. Denote by
the measure of the acute angle formed by the diagonals of the quadrilateral. Then
can be written in the form
, where
and
are relatively prime positive integers. Find
.
Problem 13
Find the least positive integer for which
is a multiple of
.
Problem 14
Let be an acute triangle with circumcenter
and centroid
. Let
be the intersection of the line tangent to the circumcircle of
at
and the line perpendicular to
at
. Let
be the intersection of lines
and
. Given that the measures of
and
are in the ratio
the degree measure of
can be written as
where
and
are relatively prime positive integers. Find
.
Problem 15
Let and
be functions satisfying
and
for positive integers
. Find the least positive integer
such that
.
See also
2021 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2021 AIME I |
Followed by 2022 AIME I | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.