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== | ||
− | 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. | + | 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]] | ||
Line 21: | Line 22: | ||
==Problem 5== | ==Problem 5== | ||
− | < | + | 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== | ||
− | < | + | 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== | ||
− | < | + | 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== | ||
− | < | + | 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== | ||
− | < | + | 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>\ | + | 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== | ||
− | < | + | 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>\ | + | 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>\ | + | 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== | ||
− | < | + | 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== | ||
− | < | + | 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 09:18, 15 August 2022
2022 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
Adults made up of the crowd of people at a concert. After a bus carrying
more people arrived, adults made up
of the people at the concert. Find the minimum number of adults who could have been at the concert after the bus arrived.
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 . When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability
. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is
, where
and
are relatively prime positive integers. Find
.
Problem 3
A right square pyramid with volume has a base with side length
The five vertices of the pyramid all lie on a sphere with radius
, where
and
are relatively prime positive integers. Find
.
Problem 4
There is a positive real number not equal to either
or
such that
The value
can be written as
, where
and
are relatively prime positive integers. Find
.
Problem 5
Twenty distinct points are marked on a circle and labeled through
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
points.
Problem 6
Let be real numbers such that
and
. Among all such
-tuples of numbers, the greatest value that
can achieve is
, where
and
are relatively prime positive integers. Find
.
Problem 7
A circle with radius is externally tangent to a circle with radius
. Find the area of the triangular region bounded by the three common tangent lines of these two circles.
Problem 8
Find the number of positive integers whose value can be uniquely determined when the values of
,
, and
are given, where
denotes the greatest integer less than or equal to the real number
.
Problem 9
Let and
be two distinct parallel lines. For positive integers
and
, distinct points
lie on
, and distinct points
lie on
. Additionally, when segments
are drawn for all
and
, no point strictly between
and
lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when
and
. The figure shows that there are 8 regions when
and
.
Problem 10
Find the remainder whenis divided by
.
Problem 11
Let be a convex quadrilateral with
and
such that the bisectors of acute angles
and
intersect at the midpoint of
Find the square of the area of
Problem 12
Let and
be real numbers with
and
such that
Find the least possible value of
Problem 13
There is a polynomial with integer coefficients such that
holds for every
Find the coefficient of
in
.
Problem 14
For positive integers ,
, and
with
, consider collections of postage stamps in denominations
,
, and
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
cents, let
be the minimum number of stamps in such a collection. Find the sum of the three least values of
such that
for some choice of
and
.
Problem 15
Two externally tangent circles and
have centers
and
, respectively. A third circle
passing through
and
intersects
at
and
and
at
and
, as shown. Suppose that
,
,
, and
is a convex hexagon. Find the area of this hexagon.
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 |
- 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.