Difference between revisions of "2021 AIME II Problems"

Latest revision as of 13:31, 26 February 2024

2021 AIME II (Answer Key) Printable version \| AoPS Contest Collections • PDF
Instructions This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$ , inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

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 $777$ or $383$ .)

Solution

Problem 2

Equilateral triangle $ABC$ has side length $840$ . Point $D$ lies on the same side of line $BC$ as $A$ such that $\overline{BD} \perp \overline{BC}$ . The line $\ell$ through $D$ parallel to line $BC$ intersects sides $\overline{AB}$ and $\overline{AC}$ at points $E$ and $F$ , respectively. Point $G$ lies on $\ell$ such that $F$ is between $E$ and $G$ , $\triangle AFG$ is isosceles, and the ratio of the area of $\triangle AFG$ to the area of $\triangle BED$ is $8:9$ . Find $AF$ . $[asy] pair A,B,C,D,E,F,G; B=origin; A=5*dir(60); C=(5,0); E=0.6*A+0.4*B; F=0.6*A+0.4*C; G=rotate(240,F)*A; D=extension(E,F,B,dir(90)); draw(D--G--A,grey); draw(B--0.5*A+rotate(60,B)*A*0.5,grey); draw(A--B--C--cycle,linewidth(1.5)); dot(A^^B^^C^^D^^E^^F^^G); label("$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$",midpoint(E--F),dir(90)); [/asy]$

Solution

Problem 3

Find the number of permutations $x_1, x_2, x_3, x_4, x_5$ of numbers $1, 2, 3, 4, 5$ such that the sum of five products $x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2$ is divisible by $3$ .

Solution

Problem 4

There are real numbers $a, b, c,$ and $d$ such that $-20$ is a root of $x^3 + ax + b$ and $-21$ is a root of $x^3 + cx^2 + d.$ These two polynomials share a complex root $m + \sqrt{n} \cdot i,$ where $m$ and $n$ are positive integers and $i = \sqrt{-1}.$ Find $m+n.$

Solution

Problem 5

For positive real numbers $s$ , let $\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$ . The set of all $s$ for which $\tau(s)$ is nonempty, but all triangles in $\tau(s)$ are congruent, is an interval $[a,b)$ . Find $a^2+b^2$ .

Solution

Problem 6

For any finite set $S$ , let $|S|$ denote the number of elements in $S$ . Find the number of ordered pairs $(A,B)$ such that $A$ and $B$ are (not necessarily distinct) subsets of $\{1,2,3,4,5\}$ that satisfy $|A| \cdot |B| = |A \cap B| \cdot |A \cup B|$

Solution

Problem 7

Let $a, b, c,$ and $d$ be real numbers that satisfy the system of equations $\begin{align*} a + b &= -3, \\ ab + bc + ca &= -4, \\ abc + bcd + cda + dab &= 14, \\ abcd &= 30. \end{align*}$ There exist relatively prime positive integers $m$ and $n$ such that $a^2 + b^2 + c^2 + d^2 = \frac{m}{n}.$ Find $m + n$ .

Solution

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 $8$ moves that ant is at a vertex of the top face on the cube is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Solution

Problem 9

Find the number of ordered pairs $(m, n)$ such that $m$ and $n$ are positive integers in the set $\{1, 2, ..., 30\}$ and the greatest common divisor of $2^m + 1$ and $2^n - 1$ is not $1$ .

Solution

Problem 10

Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\mathcal{P}$ and $\mathcal{Q}$ . The intersection of planes $\mathcal{P}$ and $\mathcal{Q}$ is the line $\ell$ . The distance from line $\ell$ to the point where the sphere with radius $13$ is tangent to plane $\mathcal{P}$ is $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

Solution

Problem 11

A teacher was leading a class of four perfectly logical students. The teacher chose a set $S$ of four integers and gave a different number in $S$ to each student. Then the teacher announced to the class that the numbers in $S$ were four consecutive two-digit positive integers, that some number in $S$ was divisible by $6$ , and a different number in $S$ was divisible by $7$ . The teacher then asked if any of the students could deduce what $S$ 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 $S$ . Find the sum of all possible values of the greatest element of $S$ .

Solution

Problem 12

A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\tan \theta$ can be written in the form $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

Solution

Problem 13

Find the least positive integer $n$ for which $2^n + 5^n - n$ is a multiple of $1000$ .

Solution

Problem 14

Let $\Delta ABC$ be an acute triangle with circumcenter $O$ and centroid $G$ . Let $X$ be the intersection of the line tangent to the circumcircle of $\Delta ABC$ at $A$ and the line perpendicular to $GO$ at $G$ . Let $Y$ be the intersection of lines $XG$ and $BC$ . Given that the measures of $\angle ABC, \angle BCA,$ and $\angle XOY$ are in the ratio $13 : 2 : 17,$ the degree measure of $\angle BAC$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Solution

Problem 15

Let $f(n)$ and $g(n)$ be functions satisfying $f(n) = \begin{cases} \sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 1 + f(n+1) & \text{ otherwise} \end{cases}$ and $g(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 2 + g(n+2) & \text{ otherwise} \end{cases}$ for positive integers $n$ . Find the least positive integer $n$ such that $\tfrac{f(n)}{g(n)} = \tfrac{4}{7}$ .

Solution

2021 AIME II (Problems • Answer Key • Resources)
Preceded by 2021 AIME I	Followed by 2022 AIME I
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Difference between revisions of "2021 AIME II Problems"

Latest revision as of 13:31, 26 February 2024

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

See also

@@ Line 1: / Line 1: @@
-These are not the problems you are looking for.
 {{AIME Problems|year=2021|n=II}}
 ==Problem 1==
-Zou and Chou are practicing their 100-meter sprints by running <math>6</math> races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is <math>\frac23</math> if they won the previous race but only <math>\frac13</math> if they lost the previous race. The probability that Zou will win exactly <math>5</math> of the <math>6</math> races is <math>\frac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
+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 <math>777</math> or <math>383</math>.)
 [[2021 AIME II Problems/Problem 1|Solution]]
 ==Problem 2==
-These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
+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]]
 ==Problem 3==
-These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
+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>.
 [[2021 AIME II Problems/Problem 3|Solution]]
 ==Problem 4==
-These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
+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]]
 ==Problem 5==
-These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
+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]]
 ==Problem 6==
-These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
+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]]
 ==Problem 7==
-These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
+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]]
 ==Problem 8==
-These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
+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]]
 ==Problem 9==
-These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
+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]]
 ==Problem 10==
-These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
+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]]
 ==Problem 11==
-These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
+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]]
 ==Problem 12==
-These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
+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]]
 ==Problem 13==
-These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
+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]]
 ==Problem 14==
-These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
+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]]
 ==Problem 15==
-These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
+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}\\
++ f(n+1) & \text{ otherwise}
+\end{cases}</cmath>
+and
+<cmath>g(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\
++ 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]]