Difference between revisions of "2022 AIME I Problems"
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− | The | + | {{AIME Problems|year=2022|n=I}} |
+ | |||
+ | ==Problem 1== | ||
+ | |||
+ | Quadratic polynomials <math>P(x)</math> and <math>Q(x)</math> have leading coefficients <math>2</math> and <math>-2,</math> respectively. The graphs of both polynomials pass through the two points <math>(16,54)</math> and <math>(20,53).</math> Find <math>P(0) + Q(0).</math> | ||
+ | |||
+ | [[2022 AIME I Problems/Problem 1|Solution]] | ||
+ | |||
+ | ==Problem 2== | ||
+ | |||
+ | Find the three-digit positive integer <math>\underline{a}\,\underline{b}\,\underline{c}</math> whose representation in base nine is <math>\underline{b}\,\underline{c}\,\underline{a}_{\,\text{nine}},</math> where <math>a,</math> <math>b,</math> and <math>c</math> are (not necessarily distinct) digits. | ||
+ | |||
+ | [[2022 AIME I Problems/Problem 2|Solution]] | ||
+ | |||
+ | ==Problem 3== | ||
+ | |||
+ | In isosceles trapezoid <math>ABCD,</math> parallel bases <math>\overline{AB}</math> and <math>\overline{CD}</math> have lengths <math>500</math> and <math>650,</math> respectively, and <math>AD=BC=333.</math> The angle bisectors of <math>\angle A</math> and <math>\angle D</math> meet at <math>P,</math> and the angle bisectors of <math>\angle B</math> and <math>\angle C</math> meet at <math>Q.</math> Find <math>PQ.</math> | ||
+ | |||
+ | [[2022 AIME I Problems/Problem 3|Solution]] | ||
+ | |||
+ | ==Problem 4== | ||
+ | |||
+ | Let <math>w = \dfrac{\sqrt{3} + i}{2}</math> and <math>z = \dfrac{-1 + i\sqrt{3}}{2},</math> where <math>i = \sqrt{-1}.</math> Find the number of ordered pairs <math>(r,s)</math> of positive integers not exceeding <math>100</math> that satisfy the equation <math>i \cdot w^r = z^s.</math> | ||
+ | |||
+ | [[2022 AIME I Problems/Problem 4|Solution]] | ||
+ | |||
+ | ==Problem 5== | ||
+ | |||
+ | A straight river that is <math>264</math> meters wide flows from west to east at a rate of <math>14</math> meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of <math>D</math> meters downstream from Sherry. Relative to the water, Melanie swims at <math>80</math> meters per minute, and Sherry swims at <math>60</math> meters per minute. At the same time, Melanie and Sherry begin swimming in straight lines to a point on the north bank of the river that is equidistant from their starting positions. The two women arrive at this point simultaneously. Find <math>D.</math> | ||
+ | |||
+ | [[2022 AIME I Problems/Problem 5|Solution]] | ||
+ | |||
+ | ==Problem 6== | ||
+ | |||
+ | Find the number of ordered pairs of integers <math>(a,b)</math> such that the sequence <cmath>3,4,5,a,b,30,40,50</cmath> is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression. | ||
+ | |||
+ | [[2022 AIME I Problems/Problem 6|Solution]] | ||
+ | |||
+ | ==Problem 7== | ||
+ | |||
+ | Let <math>a,b,c,d,e,f,g,h,i</math> be distinct integers from <math>1</math> to <math>9.</math> The minimum possible positive value of <cmath>\dfrac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i}</cmath> 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> | ||
+ | |||
+ | [[2022 AIME I Problems/Problem 7|Solution]] | ||
+ | |||
+ | ==Problem 8== | ||
+ | |||
+ | Equilateral triangle <math>\triangle ABC</math> is inscribed in circle <math>\omega</math> with radius <math>18.</math> Circle <math>\omega_A</math> is tangent to sides <math>\overline{AB}</math> and <math>\overline{AC}</math> and is internally tangent to <math>\omega.</math> Circles <math>\omega_B</math> and <math>\omega_C</math> are defined analogously. Circles <math>\omega_A,</math> <math>\omega_B,</math> and <math>\omega_C</math> meet in six points---two points for each pair of circles. The three intersection points closest to the vertices of <math>\triangle ABC</math> are the vertices of a large equilateral triangle in the interior of <math>\triangle ABC,</math> and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of <math>\triangle ABC.</math> The side length of the smaller equilateral triangle can be written as <math>\sqrt{a} - \sqrt{b},</math> where <math>a</math> and <math>b</math> are positive integers. Find <math>a+b.</math> | ||
+ | |||
+ | [[2022 AIME I Problems/Problem 8|Solution]] | ||
+ | |||
+ | ==Problem 9== | ||
+ | |||
+ | Ellina has twelve blocks, two each of red (<math>\textbf{R}</math>), blue (<math>\textbf{B}</math>), yellow (<math>\textbf{Y}</math>), green (<math>\textbf{G}</math>), orange (<math>\textbf{O}</math>), and purple (<math>\textbf{P}</math>). Call an arrangement of blocks <math>\textit{even}</math> if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement | ||
+ | <cmath>\textbf{R B B Y G G Y R O P P O}</cmath> | ||
+ | is even. Ellina arranges her blocks in a row in random order. The probability that her arrangement is even is <math>\frac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math> | ||
+ | |||
+ | [[2022 AIME I Problems/Problem 9|Solution]] | ||
+ | |||
+ | ==Problem 10== | ||
+ | |||
+ | Three spheres with radii <math>11,</math> <math>13,</math> and <math>19</math> are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at <math>A,</math> <math>B,</math> and <math>C,</math> respectively, and the centers of the spheres all lie on the same side of this plane. Suppose that <math>AB^2 = 560.</math> Find <math>AC^2.</math> | ||
+ | |||
+ | [[2022 AIME I Problems/Problem 10|Solution]] | ||
+ | |||
+ | ==Problem 11== | ||
+ | |||
+ | Let <math>ABCD</math> be a parallelogram with <math>\angle BAD < 90^\circ.</math> A circle tangent to sides <math>\overline{DA},</math> <math>\overline{AB},</math> and <math>\overline{BC}</math> intersects diagonal <math>\overline{AC}</math> at points <math>P</math> and <math>Q</math> with <math>AP < AQ,</math> as shown. Suppose that <math>AP=3,</math> <math>PQ=9,</math> and <math>QC=16.</math> Then the area of <math>ABCD</math> can be expressed in the form <math>m\sqrt{n},</math> where <math>m</math> and <math>n</math> are positive integers, and <math>n</math> is not divisible by the square of any prime. Find <math>m+n.</math> | ||
+ | |||
+ | <asy> | ||
+ | defaultpen(linewidth(0.6)+fontsize(11)); | ||
+ | size(8cm); | ||
+ | pair A,B,C,D,P,Q; | ||
+ | A=(0,0); | ||
+ | label("$A$", A, SW); | ||
+ | B=(6,15); | ||
+ | label("$B$", B, NW); | ||
+ | C=(30,15); | ||
+ | label("$C$", C, NE); | ||
+ | D=(24,0); | ||
+ | label("$D$", D, SE); | ||
+ | P=(5.2,2.6); | ||
+ | label("$P$", (5.8,2.6), N); | ||
+ | Q=(18.3,9.1); | ||
+ | label("$Q$", (18.1,9.7), W); | ||
+ | draw(A--B--C--D--cycle); | ||
+ | draw(C--A); | ||
+ | draw(Circle((10.95,7.45), 7.45)); | ||
+ | dot(A^^B^^C^^D^^P^^Q); | ||
+ | </asy> | ||
+ | |||
+ | [[2022 AIME I Problems/Problem 11|Solution]] | ||
+ | |||
+ | ==Problem 12== | ||
+ | |||
+ | For any finite set <math>X,</math> let <math>|X|</math> denote the number of elements in <math>X.</math> Define <cmath>S_n = \sum |A \cap B|,</cmath> where the sum is taken over all ordered pairs <math>(A,B)</math> such that <math>A</math> and <math>B</math> are subsets of <math>\{1,2,3,\ldots,n\}</math> with <math>|A|=|B|.</math> For example, <math>S_2 = 4</math> because the sum is taken over the pairs of subsets <cmath>(A,B) \in \left\{(\emptyset,\emptyset),(\{1\},\{1\}),(\{1\},\{2\}),(\{2\},\{1\}),(\{2\},\{2\}),(\{1,2\},\{1,2\})\right\},</cmath> giving <math>S_2 = 0+1+0+0+1+2=4.</math> Let <math>\frac{S_{2022}}{S_{2021}} = \frac{p}{q},</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. Find the remainder when <math>p+q</math> is divided by <math>1000.</math> | ||
+ | |||
+ | [[2022 AIME I Problems/Problem 12|Solution]] | ||
+ | |||
+ | ==Problem 13== | ||
+ | |||
+ | Let <math>S</math> be the set of all rational numbers that can be expressed as a repeating decimal in the form <math>0.\overline{abcd},</math> where at least one of the digits <math>a,</math> <math>b,</math> <math>c,</math> or <math>d</math> is nonzero. Let <math>N</math> be the number of distinct numerators obtained when numbers in <math>S</math> are written as fractions in lowest terms. For example, both <math>4</math> and <math>410</math> are counted among the distinct numerators for numbers in <math>S</math> because <math>0.\overline{3636} = \frac{4}{11}</math> and <math>0.\overline{1230} = \frac{410}{3333}.</math> Find the remainder when <math>N</math> is divided by <math>1000.</math> | ||
+ | |||
+ | [[2022 AIME I Problems/Problem 13|Solution]] | ||
+ | |||
+ | ==Problem 14== | ||
+ | |||
+ | Given <math>\triangle ABC</math> and a point <math>P</math> on one of its sides, call line <math>\ell</math> the <math>\textit{splitting line}</math> of <math>\triangle ABC</math> through <math>P</math> if <math>\ell</math> passes through <math>P</math> and divides <math>\triangle ABC</math> into two polygons of equal perimeter. Let <math>\triangle ABC</math> be a triangle where <math>BC = 219</math> and <math>AB</math> and <math>AC</math> are positive integers. Let <math>M</math> and <math>N</math> be the midpoints of <math>\overline{AB}</math> and <math>\overline{AC},</math> respectively, and suppose that the splitting lines of <math>\triangle ABC</math> through <math>M</math> and <math>N</math> intersect at <math>30^\circ.</math> Find the perimeter of <math>\triangle ABC.</math> | ||
+ | |||
+ | [[2022 AIME I Problems/Problem 14|Solution]] | ||
+ | |||
+ | ==Problem 15== | ||
+ | |||
+ | Let <math>x,</math> <math>y,</math> and <math>z</math> be positive real numbers satisfying the system of equations: | ||
+ | <cmath>\begin{align*} | ||
+ | \sqrt{2x-xy} + \sqrt{2y-xy} &= 1 \ | ||
+ | \sqrt{2y-yz} + \sqrt{2z-yz} &= \sqrt2 \ | ||
+ | \sqrt{2z-zx} + \sqrt{2x-zx} &= \sqrt3. | ||
+ | \end{align*}</cmath> | ||
+ | Then <math>\left[ (1-x)(1-y)(1-z) \right]^2</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> | ||
+ | |||
+ | [[2022 AIME I Problems/Problem 15|Solution]] | ||
+ | |||
+ | ==See Also== | ||
+ | {{AIME box|year=2022|n=I|before=[[2021 AIME II Problems|2021 AIME II]]|after=[[2022 AIME II Problems|2022 AIME II]]}} | ||
+ | * [[American Invitational Mathematics Examination]] | ||
+ | * [[AIME Problems and Solutions]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 18:06, 2 January 2023
2022 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
[hide]Problem 1
Quadratic polynomials and
have leading coefficients
and
respectively. The graphs of both polynomials pass through the two points
and
Find
Problem 2
Find the three-digit positive integer whose representation in base nine is
where
and
are (not necessarily distinct) digits.
Problem 3
In isosceles trapezoid parallel bases
and
have lengths
and
respectively, and
The angle bisectors of
and
meet at
and the angle bisectors of
and
meet at
Find
Problem 4
Let and
where
Find the number of ordered pairs
of positive integers not exceeding
that satisfy the equation
Problem 5
A straight river that is meters wide flows from west to east at a rate of
meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of
meters downstream from Sherry. Relative to the water, Melanie swims at
meters per minute, and Sherry swims at
meters per minute. At the same time, Melanie and Sherry begin swimming in straight lines to a point on the north bank of the river that is equidistant from their starting positions. The two women arrive at this point simultaneously. Find
Problem 6
Find the number of ordered pairs of integers such that the sequence
is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression.
Problem 7
Let be distinct integers from
to
The minimum possible positive value of
can be written as
where
and
are relatively prime positive integers. Find
Problem 8
Equilateral triangle is inscribed in circle
with radius
Circle
is tangent to sides
and
and is internally tangent to
Circles
and
are defined analogously. Circles
and
meet in six points---two points for each pair of circles. The three intersection points closest to the vertices of
are the vertices of a large equilateral triangle in the interior of
and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of
The side length of the smaller equilateral triangle can be written as
where
and
are positive integers. Find
Problem 9
Ellina has twelve blocks, two each of red (), blue (
), yellow (
), green (
), orange (
), and purple (
). Call an arrangement of blocks
if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement
is even. Ellina arranges her blocks in a row in random order. The probability that her arrangement is even is
where
and
are relatively prime positive integers. Find
Problem 10
Three spheres with radii
and
are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at
and
respectively, and the centers of the spheres all lie on the same side of this plane. Suppose that
Find
Problem 11
Let be a parallelogram with
A circle tangent to sides
and
intersects diagonal
at points
and
with
as shown. Suppose that
and
Then the area of
can be expressed in the form
where
and
are positive integers, and
is not divisible by the square of any prime. Find
Problem 12
For any finite set let
denote the number of elements in
Define
where the sum is taken over all ordered pairs
such that
and
are subsets of
with
For example,
because the sum is taken over the pairs of subsets
giving
Let
where
and
are relatively prime positive integers. Find the remainder when
is divided by
Problem 13
Let be the set of all rational numbers that can be expressed as a repeating decimal in the form
where at least one of the digits
or
is nonzero. Let
be the number of distinct numerators obtained when numbers in
are written as fractions in lowest terms. For example, both
and
are counted among the distinct numerators for numbers in
because
and
Find the remainder when
is divided by
Problem 14
Given and a point
on one of its sides, call line
the
of
through
if
passes through
and divides
into two polygons of equal perimeter. Let
be a triangle where
and
and
are positive integers. Let
and
be the midpoints of
and
respectively, and suppose that the splitting lines of
through
and
intersect at
Find the perimeter of
Problem 15
Let
and
be positive real numbers satisfying the system of equations:
Then
can be written as
where
and
are relatively prime positive integers. Find
See Also
2022 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2021 AIME II |
Followed by 2022 AIME II | |
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.