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Difference between revisions of "2022 AIME II Problems"

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{{AIME Problems|year=2022|n=II}}
 
{{AIME Problems|year=2022|n=II}}
 
==Problem 1==
 
==Problem 1==
−
<cmath>\textbf{}</cmath>
+
Adults made up <math>\frac5{12}</math> of the crowd of people at a concert. After a bus carrying <math>50</math> more people arrived, adults made up <math>\frac{11}{25}</math> of the people at the concert. Find the minimum number of adults who could have been at the concert after the bus arrived.
 +
 +
 
 
[[2022 AIME II Problems/Problem 1|Solution]]
 
[[2022 AIME II Problems/Problem 1|Solution]]
  
 
==Problem 2==
 
==Problem 2==
−
<cmath>\textbf{Please do not post this problem until the contest is released.}</cmath>
+
Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability <math>\frac23</math>. When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability <math>\frac34</math>. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>.
 +
 
 
[[2022 AIME II Problems/Problem 2|Solution]]
 
[[2022 AIME II Problems/Problem 2|Solution]]
 +
 
==Problem 3==
 
==Problem 3==
−
<cmath>\textbf{Please do not post this problem until the contest is released.}</cmath>
+
A right square pyramid with volume <math>54</math> has a base with side length <math>6.</math> The five vertices of the pyramid all lie on a sphere with radius <math>\frac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 +
 
 
[[2022 AIME II Problems/Problem 3|Solution]]
 
[[2022 AIME II Problems/Problem 3|Solution]]
 +
 
==Problem 4==
 
==Problem 4==
−
<cmath>\textbf{Please do not post this problem until the contest is released.}</cmath>
+
There is a positive real number <math>x</math> not equal to either <math>\tfrac{1}{20}</math> or <math>\tfrac{1}{2}</math> such that<cmath>\log_{20x} (22x)=\log_{2x} (202x).</cmath>The value <math>\log_{20x} (22x)</math> can be written as <math>\log_{10} (\tfrac{m}{n})</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 +
 
 
[[2022 AIME II Problems/Problem 4|Solution]]
 
[[2022 AIME II Problems/Problem 4|Solution]]
 +
 
==Problem 5==
 
==Problem 5==
−
<cmath>\textbf{Please do not post this problem until the contest is released.}</cmath>
+
Twenty distinct points are marked on a circle and labeled <math>1</math> through <math>20</math> in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original <math>20</math> points.
 +
 
 
[[2022 AIME II Problems/Problem 5|Solution]]
 
[[2022 AIME II Problems/Problem 5|Solution]]
  
 
==Problem 6==
 
==Problem 6==
−
<cmath>\textbf{Please do not post this problem until the contest is released.}</cmath>
+
Let <math>x_1\leq x_2\leq \cdots\leq x_{100}</math> be real numbers such that <math>|x_1| + |x_2| + \cdots + |x_{100}| = 1</math> and <math>x_1 + x_2 + \cdots + x_{100} = 0</math>. Among all such <math>100</math>-tuples of numbers, the greatest value that <math>x_{76} - x_{16}</math> can achieve is <math>\tfrac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 +
 
 
[[2022 AIME II Problems/Problem 6|Solution]]
 
[[2022 AIME II Problems/Problem 6|Solution]]
 +
 
==Problem 7==
 
==Problem 7==
−
<cmath>\textbf{Please do not post this problem until the contest is released.}</cmath>
+
A circle with radius <math>6</math> is externally tangent to a circle with radius <math>24</math>. Find the area of the triangular region bounded by the three common tangent lines of these two circles.
 +
 
 
[[2022 AIME II Problems/Problem 7|Solution]]
 
[[2022 AIME II Problems/Problem 7|Solution]]
 +
 
==Problem 8==
 
==Problem 8==
−
<cmath>\textbf{Please do not post this problem until the contest is released.}</cmath>
+
Find the number of positive integers <math>n \le 600</math> whose value can be uniquely determined when the values of <math>\left\lfloor \frac n4\right\rfloor</math>, <math>\left\lfloor\frac n5\right\rfloor</math>, and <math>\left\lfloor\frac n6\right\rfloor</math> are given, where <math>\lfloor x \rfloor</math> denotes the greatest integer less than or equal to the real number <math>x</math>.
 +
 
 
[[2022 AIME II Problems/Problem 8|Solution]]
 
[[2022 AIME II Problems/Problem 8|Solution]]
 +
 
==Problem 9==
 
==Problem 9==
−
<cmath>\textbf{Please do not post this problem until the contest is released.}</cmath>
+
Let <math>\ell_A</math> and <math>\ell_B</math> be two distinct parallel lines. For positive integers <math>m</math> and <math>n</math>, distinct points <math>A_1, A_2, \allowbreak A_3, \allowbreak \ldots, \allowbreak A_m</math> lie on <math>\ell_A</math>, and distinct points <math>B_1, B_2, B_3, \ldots, B_n</math> lie on <math>\ell_B</math>. Additionally, when segments <math>\overline{A_iB_j}</math> are drawn for all <math>i=1,2,3,\ldots, m</math> and <math>j=1,\allowbreak 2,\allowbreak 3, \ldots, \allowbreak n</math>, no point strictly between <math>\ell_A</math> and <math>\ell_B</math> lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when <math>m=7</math> and <math>n=5</math>. The figure shows that there are 8 regions when <math>m=3</math> and <math>n=2</math>.
 +
<asy>
 +
import geometry;
 +
size(10cm);
 +
draw((-2,0)--(13,0));
 +
draw((0,4)--(10,4));
 +
label("$\ell_A$",(-2,0),W);
 +
label("$\ell_B$",(0,4),W);
 +
point A1=(0,0),A2=(5,0),A3=(11,0),B1=(2,4),B2=(8,4),I1=extension(B1,A2,A1,B2),I2=extension(B1,A3,A1,B2),I3=extension(B1,A3,A2,B2);
 +
draw(B1--A1--B2);
 +
draw(B1--A2--B2);
 +
draw(B1--A3--B2);
 +
label("$A_1$",A1,S);
 +
label("$A_2$",A2,S);
 +
label("$A_3$",A3,S);
 +
label("$B_1$",B1,N);
 +
label("$B_2$",B2,N);
 +
label("1",centroid(A1,B1,I1));
 +
label("2",centroid(B1,I1,I3));
 +
label("3",centroid(B1,B2,I3));
 +
label("4",centroid(A1,A2,I1));
 +
label("5",(A2+I1+I2+I3)/4);
 +
label("6",centroid(B2,I2,I3));
 +
label("7",centroid(A2,A3,I2));
 +
label("8",centroid(A3,B2,I2));
 +
dot(A1);
 +
dot(A2);
 +
dot(A3);
 +
dot(B1);
 +
dot(B2);
 +
</asy>
 +
 
 
[[2022 AIME II Problems/Problem 9|Solution]]
 
[[2022 AIME II Problems/Problem 9|Solution]]
 +
 
==Problem 10==
 
==Problem 10==
−
<cmath>\textbf{Please do not post this problem until the contest is released.}</cmath>
+
Find the remainder when<cmath>\binom{\binom{3}{2}}{2} + \binom{\binom{4}{2}}{2} + \dots +  \binom{\binom{40}{2}}{2}</cmath>is divided by <math>1000</math>.
 +
 
 
[[2022 AIME II Problems/Problem 10|Solution]]
 
[[2022 AIME II Problems/Problem 10|Solution]]
 +
 
==Problem 11==
 
==Problem 11==
−
<cmath>\textbf{Please do not post this problem until the contest is released.}</cmath>
+
Let <math>ABCD</math> be a convex quadrilateral with <math>AB=2, AD=7,</math> and <math>CD=3</math> such that the bisectors of acute angles <math>\angle{DAB}</math> and <math>\angle{ADC}</math> intersect at the midpoint of <math>\overline{BC}.</math> Find the square of the area of <math>ABCD.</math>
 +
 
 
[[2022 AIME II Problems/Problem 11|Solution]]
 
[[2022 AIME II Problems/Problem 11|Solution]]
 +
 
==Problem 12==
 
==Problem 12==
−
<cmath>\textbf{Please do not post this problem until the contest is released.}</cmath>
+
Let <math>a, b, x,</math> and <math>y</math> be real numbers with <math>a>4</math> and <math>b>1</math> such that<cmath>\frac{x^2}{a^2}+\frac{y^2}{a^2-16}=\frac{(x-20)^2}{b^2-1}+\frac{(y-11)^2}{b^2}=1.</cmath>Find the least possible value of <math>a+b.</math>
 +
 
 
[[2022 AIME II Problems/Problem 12|Solution]]
 
[[2022 AIME II Problems/Problem 12|Solution]]
 +
 
==Problem 13==
 
==Problem 13==
−
<cmath>\textbf{Please do not post this problem until the contest is released.}</cmath>
+
There is a polynomial <math>P(x)</math> with integer coefficients such that<cmath>P(x)=\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)}</cmath>holds for every <math>0<x<1.</math> Find the coefficient of <math>x^{2022}</math> in <math>P(x)</math>.
 +
 
 
[[2022 AIME II Problems/Problem 13|Solution]]
 
[[2022 AIME II Problems/Problem 13|Solution]]
 +
 
==Problem 14==
 
==Problem 14==
−
<cmath>\textbf{Please do not post this problem until the contest is released.}</cmath>
+
For positive integers <math>a</math>, <math>b</math>, and <math>c</math> with <math>a < b < c</math>, consider collections of postage stamps in denominations <math>a</math>, <math>b</math>, and <math>c</math> cents that contain at least one stamp of each denomination. If there exists such a collection that contains sub-collections worth every whole number of cents up to <math>1000</math> cents, let <math>f(a, b, c)</math> be the minimum number of stamps in such a collection. Find the sum of the three least values of <math>c</math> such that <math>f(a, b, c) = 97</math> for some choice of <math>a</math> and <math>b</math>.
 +
 
 
[[2022 AIME II Problems/Problem 14|Solution]]
 
[[2022 AIME II Problems/Problem 14|Solution]]
 +
 
==Problem 15==
 
==Problem 15==
−
<cmath>\textbf{Please do not post this problem until the contest is released.}</cmath>
+
Two externally tangent circles <math>\omega_1</math> and <math>\omega_2</math> have centers <math>O_1</math> and <math>O_2</math>, respectively. A third circle <math>\Omega</math> passing through <math>O_1</math> and <math>O_2</math> intersects <math>\omega_1</math> at <math>B</math> and <math>C</math> and <math>\omega_2</math> at <math>A</math> and <math>D</math>, as shown. Suppose that <math>AB = 2</math>, <math>O_1O_2 = 15</math>, <math>CD = 16</math>, and <math>ABO_1CDO_2</math> is a convex hexagon. Find the area of this hexagon.
 +
<asy>
 +
import geometry;
 +
size(10cm);
 +
point O1=(0,0),O2=(15,0),B=9*dir(30);
 +
circle w1=circle(O1,9),w2=circle(O2,6),o=circle(O1,O2,B);
 +
point A=intersectionpoints(o,w2)[1],D=intersectionpoints(o,w2)[0],C=intersectionpoints(o,w1)[0];
 +
filldraw(A--B--O1--C--D--O2--cycle,0.2*red+white,black);
 +
draw(w1);
 +
draw(w2);
 +
draw(O1--O2,dashed);
 +
draw(o);
 +
dot(O1);
 +
dot(O2);
 +
dot(A);
 +
dot(D);
 +
dot(C);
 +
dot(B);
 +
label("$\omega_1$",8*dir(110),SW);
 +
label("$\omega_2$",5*dir(70)+(15,0),SE);
 +
label("$O_1$",O1,W);
 +
label("$O_2$",O2,E);
 +
label("$B$",B,N+1/2*E);
 +
label("$A$",A,N+1/2*W);
 +
label("$C$",C,S+1/4*W);
 +
label("$D$",D,S+1/4*E);
 +
label("$15$",midpoint(O1--O2),N);
 +
label("$16$",midpoint(C--D),N);
 +
label("$2$",midpoint(A--B),S);
 +
label("$\Omega$",o.C+(o.r-1)*dir(270));
 +
</asy>
 +
 
 
[[2022 AIME II Problems/Problem 15|Solution]]
 
[[2022 AIME II Problems/Problem 15|Solution]]
 +
 
==See also==
 
==See also==
 
{{AIME box|year=2022|n=II|before=[[2022 AIME I Problems|2022 AIME I]]|after=[[2023 AIME I Problems|2023 AIME I]]}}
 
{{AIME box|year=2022|n=II|before=[[2022 AIME I Problems|2022 AIME I]]|after=[[2023 AIME I Problems|2023 AIME I]]}}

Latest revision as of 08:18, 15 August 2022

2022 AIME II (Answer Key)
Printable version | AoPS Contest Collections • PDF

Instructions

  1. 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.
  2. 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

Adults made up $\frac5{12}$ of the crowd of people at a concert. After a bus carrying $50$ more people arrived, adults made up $\frac{11}{25}$ of the people at the concert. Find the minimum number of adults who could have been at the concert after the bus arrived.


Solution

Problem 2

Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability $\frac23$. When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability $\frac34$. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Solution

Problem 3

A right square pyramid with volume $54$ has a base with side length $6.$ The five vertices of the pyramid all lie on a sphere with radius $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 4

There is a positive real number $x$ not equal to either $\tfrac{1}{20}$ or $\tfrac{1}{2}$ such that\[\log_{20x} (22x)=\log_{2x} (202x).\]The value $\log_{20x} (22x)$ can be written as $\log_{10} (\tfrac{m}{n})$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 5

Twenty distinct points are marked on a circle and labeled $1$ through $20$ in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original $20$ points.

Solution

Problem 6

Let $x_1\leq x_2\leq \cdots\leq x_{100}$ be real numbers such that $|x_1| + |x_2| + \cdots + |x_{100}| = 1$ and $x_1 + x_2 + \cdots + x_{100} = 0$. Among all such $100$-tuples of numbers, the greatest value that $x_{76} - x_{16}$ can achieve is $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 7

A circle with radius $6$ is externally tangent to a circle with radius $24$. Find the area of the triangular region bounded by the three common tangent lines of these two circles.

Solution

Problem 8

Find the number of positive integers $n \le 600$ whose value can be uniquely determined when the values of $\left\lfloor \frac n4\right\rfloor$, $\left\lfloor\frac n5\right\rfloor$, and $\left\lfloor\frac n6\right\rfloor$ are given, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to the real number $x$.

Solution

Problem 9

Let $\ell_A$ and $\ell_B$ be two distinct parallel lines. For positive integers $m$ and $n$, distinct points $A_1, A_2, \allowbreak A_3, \allowbreak \ldots, \allowbreak A_m$ lie on $\ell_A$, and distinct points $B_1, B_2, B_3, \ldots, B_n$ lie on $\ell_B$. Additionally, when segments $\overline{A_iB_j}$ are drawn for all $i=1,2,3,\ldots, m$ and $j=1,\allowbreak 2,\allowbreak 3, \ldots, \allowbreak n$, no point strictly between $\ell_A$ and $\ell_B$ lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when $m=7$ and $n=5$. The figure shows that there are 8 regions when $m=3$ and $n=2$. [asy] import geometry; size(10cm); draw((-2,0)--(13,0)); draw((0,4)--(10,4)); label("$\ell_A$",(-2,0),W); label("$\ell_B$",(0,4),W); point A1=(0,0),A2=(5,0),A3=(11,0),B1=(2,4),B2=(8,4),I1=extension(B1,A2,A1,B2),I2=extension(B1,A3,A1,B2),I3=extension(B1,A3,A2,B2); draw(B1--A1--B2); draw(B1--A2--B2); draw(B1--A3--B2); label("$A_1$",A1,S); label("$A_2$",A2,S); label("$A_3$",A3,S); label("$B_1$",B1,N); label("$B_2$",B2,N); label("1",centroid(A1,B1,I1)); label("2",centroid(B1,I1,I3)); label("3",centroid(B1,B2,I3)); label("4",centroid(A1,A2,I1)); label("5",(A2+I1+I2+I3)/4); label("6",centroid(B2,I2,I3)); label("7",centroid(A2,A3,I2)); label("8",centroid(A3,B2,I2)); dot(A1); dot(A2); dot(A3); dot(B1); dot(B2); [/asy]

Solution

Problem 10

Find the remainder when\[\binom{\binom{3}{2}}{2} + \binom{\binom{4}{2}}{2} + \dots + \binom{\binom{40}{2}}{2}\]is divided by $1000$.

Solution

Problem 11

Let $ABCD$ be a convex quadrilateral with $AB=2, AD=7,$ and $CD=3$ such that the bisectors of acute angles $\angle{DAB}$ and $\angle{ADC}$ intersect at the midpoint of $\overline{BC}.$ Find the square of the area of $ABCD.$

Solution

Problem 12

Let $a, b, x,$ and $y$ be real numbers with $a>4$ and $b>1$ such that\[\frac{x^2}{a^2}+\frac{y^2}{a^2-16}=\frac{(x-20)^2}{b^2-1}+\frac{(y-11)^2}{b^2}=1.\]Find the least possible value of $a+b.$

Solution

Problem 13

There is a polynomial $P(x)$ with integer coefficients such that\[P(x)=\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)}\]holds for every $0<x<1.$ Find the coefficient of $x^{2022}$ in $P(x)$.

Solution

Problem 14

For positive integers $a$, $b$, and $c$ with $a < b < c$, consider collections of postage stamps in denominations $a$, $b$, and $c$ cents that contain at least one stamp of each denomination. If there exists such a collection that contains sub-collections worth every whole number of cents up to $1000$ cents, let $f(a, b, c)$ be the minimum number of stamps in such a collection. Find the sum of the three least values of $c$ such that $f(a, b, c) = 97$ for some choice of $a$ and $b$.

Solution

Problem 15

Two externally tangent circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\Omega$ passing through $O_1$ and $O_2$ intersects $\omega_1$ at $B$ and $C$ and $\omega_2$ at $A$ and $D$, as shown. Suppose that $AB = 2$, $O_1O_2 = 15$, $CD = 16$, and $ABO_1CDO_2$ is a convex hexagon. Find the area of this hexagon. [asy] import geometry; size(10cm); point O1=(0,0),O2=(15,0),B=9*dir(30); circle w1=circle(O1,9),w2=circle(O2,6),o=circle(O1,O2,B); point A=intersectionpoints(o,w2)[1],D=intersectionpoints(o,w2)[0],C=intersectionpoints(o,w1)[0]; filldraw(A--B--O1--C--D--O2--cycle,0.2*red+white,black); draw(w1); draw(w2); draw(O1--O2,dashed); draw(o); dot(O1); dot(O2); dot(A); dot(D); dot(C); dot(B); label("$\omega_1$",8*dir(110),SW); label("$\omega_2$",5*dir(70)+(15,0),SE); label("$O_1$",O1,W); label("$O_2$",O2,E); label("$B$",B,N+1/2*E); label("$A$",A,N+1/2*W); label("$C$",C,S+1/4*W); label("$D$",D,S+1/4*E); label("$15$",midpoint(O1--O2),N); label("$16$",midpoint(C--D),N); label("$2$",midpoint(A--B),S); label("$\Omega$",o.C+(o.r-1)*dir(270)); [/asy]

Solution

See also

2022 AIME II (Problems • Answer Key • Resources)
Preceded by
2022 AIME I 		Followed by
2023 AIME I
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. AMC logo.png

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