Difference between revisions of "2024 AIME II Problems"
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==Problem 1== | ==Problem 1== | ||
+ | Among the <math>900</math> residents of Aimeville, there are <math>195</math> who own a diamond ring, <math>367</math> who own a set of golf clubs, and <math>562</math> who own a garden spade. In addition, each of the <math>900</math> residents owns a bag of candy hearts. There are <math>437</math> residents who own exactly two of these things, and <math>234</math> residents who own exactly three of these things. Find the number of residents of Aimeville who own all four of these things. | ||
[[2024 AIME II Problems/Problem 1|Solution]] | [[2024 AIME II Problems/Problem 1|Solution]] | ||
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==Problem 2== | ==Problem 2== | ||
+ | A list of positive integers has the following properties: | ||
+ | |||
+ | <math>\bullet</math> The sum of the items in the list is <math>30</math>. | ||
+ | |||
+ | <math>\bullet</math> The unique mode of the list is <math>9</math>. | ||
+ | |||
+ | <math>\bullet</math> The median of the list is a positive integer that does not appear in the list itself. | ||
+ | |||
+ | Find the sum of the squares of all the items in the list. | ||
[[2024 AIME II Problems/Problem 2|Solution]] | [[2024 AIME II Problems/Problem 2|Solution]] | ||
Line 12: | Line 22: | ||
==Problem 3== | ==Problem 3== | ||
+ | Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right is <math>999</math>, and the sum of the three numbers formed by reading top to bottom is <math>99</math>. The grid below is an example of such an arrangement because <math>8+991=999</math> and <math>9+9+81=99</math>. | ||
+ | |||
+ | <cmath> | ||
+ | \begin{array}{|c|c|c|} \hline | ||
+ | 0 & 0 & 8 \\ \hline | ||
+ | 9 & 9 & 1 \\ \hline | ||
+ | \end{array} | ||
+ | </cmath> | ||
[[2024 AIME II Problems/Problem 3|Solution]] | [[2024 AIME II Problems/Problem 3|Solution]] | ||
==Problem 4== | ==Problem 4== | ||
+ | |||
+ | Let <math>x,y</math> and <math>z</math> be positive real numbers that satisfy the following system of equations: | ||
+ | <cmath>\log_2\left({x \over yz}\right) = {1 \over 2}</cmath><cmath>\log_2\left({y \over xz}\right) = {1 \over 3}</cmath><cmath>\log_2\left({z \over xy}\right) = {1 \over 4}</cmath> | ||
+ | Then the value of <math>\left|\log_2(x^4y^3z^2)\right|</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>. | ||
Line 21: | Line 43: | ||
==Problem 5== | ==Problem 5== | ||
− | + | Let <math>ABCDEF</math> be a convex equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are extensions of segments <math>\overline{AB}</math>, <math>\overline{CD}</math>, and <math>\overline{EF}</math> has side lengths <math>200, 240,</math> and <math>300</math>. Find the side length of the hexagon. | |
[[2024 AIME II Problems/Problem 5|Solution]] | [[2024 AIME II Problems/Problem 5|Solution]] | ||
==Problem 6== | ==Problem 6== | ||
− | + | Alice chooses a set <math>A</math> of positive integers. Then Bob lists all finite nonempty sets <math>B</math> of positive integers with the property that the maximum element of <math>B</math> belongs to <math>A</math>. Bob's list has <math>2024</math> sets. Find the sum of the elements of <math>A</math>. | |
− | |||
[[2024 AIME II Problems/Problem 6|Solution]] | [[2024 AIME II Problems/Problem 6|Solution]] | ||
Line 33: | Line 54: | ||
==Problem 7== | ==Problem 7== | ||
− | + | Let <math>N</math> be the greatest four-digit integer with the property that whenever one of its digits is changed to <math>1</math>, the resulting number is divisible by <math>7</math>. Let <math>Q</math> and <math>R</math> be the quotient and remainder, respectively, when <math>N</math> is divided by <math>1000</math>. Find <math>Q+R</math>. | |
[[2024 AIME II Problems/Problem 7|Solution]] | [[2024 AIME II Problems/Problem 7|Solution]] | ||
Line 39: | Line 60: | ||
==Problem 8== | ==Problem 8== | ||
+ | Torus <math>T</math> is the surface produced by revolving a circle with radius 3 around an axis in the plane of the circle that is a distance 6 from the center of the circle (so like a donut). Let <math>S</math> be a sphere with a radius 11. When <math>T</math> rests on the inside of <math>S</math>, it is internally tangent to <math>S</math> along a circle with radius <math>r_i</math>, and when <math>T</math> rests on the outside of <math>S</math>, it is externally tangent to <math>S</math> along a circle with radius <math>r_o</math>. The difference <math>r_i-r_o</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>. | ||
+ | <asy> | ||
+ | unitsize(0.3 inch); | ||
+ | draw(ellipse((0,0), 3, 1.75)); | ||
+ | draw((-1.2,0.1)..(-0.8,-0.03)..(-0.4,-0.11)..(0,-0.15)..(0.4,-0.11)..(0.8,-0.03)..(1.2,0.1)); | ||
+ | draw((-1,0.04)..(-0.5,0.12)..(0,0.16)..(0.5,0.12)..(1,0.04)); | ||
+ | draw((0,2.4)--(0,-0.15)); | ||
+ | draw((0,-0.15)--(0,-1.75), dashed); | ||
+ | draw((0,-1.75)--(0,-2.25)); | ||
+ | draw(ellipse((2,0), 1, 0.9)); | ||
+ | draw((2.03,-0.02)--(2.9,-0.4)); | ||
+ | </asy> | ||
[[2024 AIME II Problems/Problem 8|Solution]] | [[2024 AIME II Problems/Problem 8|Solution]] | ||
==Problem 9== | ==Problem 9== | ||
+ | There are <math>25</math> indistinguishable white chips and <math>25</math> indistinguishable black chips. Find the number of ways to place some of these chips in a <math>5 \times 5</math> grid such that | ||
+ | * each cell contains at most one chip | ||
+ | |||
+ | * all chips in the same row and all chips in the same column have the same colour | ||
+ | |||
+ | * any additional chip placed on the grid would violate one or more of the previous two conditions. | ||
[[2024 AIME II Problems/Problem 9|Solution]] | [[2024 AIME II Problems/Problem 9|Solution]] | ||
==Problem 10== | ==Problem 10== | ||
− | + | Let <math>\triangle</math><math>ABC</math> have incenter <math>I</math> and circumcenter <math>O</math> with <math>\overline{IA} \perp \overline{OI}</math>, circumradius <math>13</math>, and inradius <math>6</math>. Find <math>AB \cdot AC</math>. | |
− | |||
[[2024 AIME II Problems/Problem 10|Solution]] | [[2024 AIME II Problems/Problem 10|Solution]] | ||
Line 56: | Line 94: | ||
==Problem 11== | ==Problem 11== | ||
+ | Find the number of triples of nonnegative integers <math>(a, b, c)</math> satisfying <math>a + b + c = 300</math> and | ||
+ | <cmath>a^2 b + a^2 c + b^2 a + b^2 c + c^2 a + c^2 b = 6,000,000.</cmath> | ||
[[2024 AIME II Problems/Problem 11|Solution]] | [[2024 AIME II Problems/Problem 11|Solution]] | ||
==Problem 12== | ==Problem 12== | ||
− | + | Let <math>O(0,0),A(\tfrac{1}{2},0),</math> and <math>B(0,\tfrac{\sqrt{3}}{2})</math> be points in the coordinate plane. Let <math>\mathcal{F}</math> be the family of segments <math>\overline{PQ}</math> of unit length lying in the first quadrant with <math>P</math> on the <math>x</math>-axis and <math>Q</math> on the <math>y</math>-axis. There is a unique point <math>C</math> on <math>\overline{AB},</math> distinct from <math>A</math> and <math>B,</math> that does not belong to any segment from <math>\mathcal{F}</math> other than <math>\overline{AB}</math>. Then <math>OC^2=\tfrac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. | |
− | |||
[[2024 AIME II Problems/Problem 12|Solution]] | [[2024 AIME II Problems/Problem 12|Solution]] | ||
==Problem 13== | ==Problem 13== | ||
− | + | Let <math>\omega\neq 1</math> be a 13th root of unity. Find the remainder when | |
− | + | <cmath>\prod_{k=0}^{12}(2-2\omega^k+\omega^{2k})</cmath> | |
+ | is divided by 1000. | ||
[[2024 AIME II Problems/Problem 13|Solution]] | [[2024 AIME II Problems/Problem 13|Solution]] | ||
==Problem 14== | ==Problem 14== | ||
− | + | Let <math>b \geq 2</math> be an integer. Call a positive integer <math>n</math> <math>b\textit{-eautiful}</math> if it has exactly two digits when expressed in base <math>b</math>, and these two digits sum to <math>\sqrt{n}</math>. For example, <math>81</math> is <math>13</math>-eautiful because <math>81=\underline{6}</math><math>\underline{3}_{13}</math> and <math>6+3=\sqrt{81}</math>. Find the least integer <math>b\geq 2</math> for which there are more than ten <math>b</math>-eautiful integers. | |
− | |||
[[2024 AIME II Problems/Problem 14|Solution]] | [[2024 AIME II Problems/Problem 14|Solution]] | ||
==Problem 15== | ==Problem 15== | ||
+ | Find the number of rectangles that can be formed inside a fixed regular dodecagon (<math>12</math>-gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. The diagram below shows three of those rectangles. | ||
− | + | <asy> | |
+ | unitsize(0.6 inch); | ||
+ | for(int i=0; i<360; i+=30) { | ||
+ | dot(dir(i), 4+black); | ||
+ | draw(dir(i)--dir(i+30)); | ||
+ | } | ||
+ | draw(dir(120)--dir(330)); | ||
+ | filldraw(dir(210)--dir(240)--dir(30)--dir(60)--cycle, mediumgray, linewidth(1.5)); | ||
+ | draw((0,0.366)--(0.366,0), linewidth(1.5)); | ||
+ | </asy> | ||
[[2024 AIME II Problems/Problem 15|Solution]] | [[2024 AIME II Problems/Problem 15|Solution]] |
Latest revision as of 09:59, 29 July 2024
2024 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
Among the residents of Aimeville, there are
who own a diamond ring,
who own a set of golf clubs, and
who own a garden spade. In addition, each of the
residents owns a bag of candy hearts. There are
residents who own exactly two of these things, and
residents who own exactly three of these things. Find the number of residents of Aimeville who own all four of these things.
Problem 2
A list of positive integers has the following properties:
The sum of the items in the list is
.
The unique mode of the list is
.
The median of the list is a positive integer that does not appear in the list itself.
Find the sum of the squares of all the items in the list.
Problem 3
Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right is , and the sum of the three numbers formed by reading top to bottom is
. The grid below is an example of such an arrangement because
and
.
Problem 4
Let and
be positive real numbers that satisfy the following system of equations:
Then the value of
is
where
and
are relatively prime positive integers. Find
.
Problem 5
Let be a convex equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are extensions of segments
,
, and
has side lengths
and
. Find the side length of the hexagon.
Problem 6
Alice chooses a set of positive integers. Then Bob lists all finite nonempty sets
of positive integers with the property that the maximum element of
belongs to
. Bob's list has
sets. Find the sum of the elements of
.
Problem 7
Let be the greatest four-digit integer with the property that whenever one of its digits is changed to
, the resulting number is divisible by
. Let
and
be the quotient and remainder, respectively, when
is divided by
. Find
.
Problem 8
Torus is the surface produced by revolving a circle with radius 3 around an axis in the plane of the circle that is a distance 6 from the center of the circle (so like a donut). Let
be a sphere with a radius 11. When
rests on the inside of
, it is internally tangent to
along a circle with radius
, and when
rests on the outside of
, it is externally tangent to
along a circle with radius
. The difference
can be written as
, where
and
are relatively prime positive integers. Find
.
Problem 9
There are indistinguishable white chips and
indistinguishable black chips. Find the number of ways to place some of these chips in a
grid such that
- each cell contains at most one chip
- all chips in the same row and all chips in the same column have the same colour
- any additional chip placed on the grid would violate one or more of the previous two conditions.
Problem 10
Let have incenter
and circumcenter
with
, circumradius
, and inradius
. Find
.
Problem 11
Find the number of triples of nonnegative integers satisfying
and
Problem 12
Let and
be points in the coordinate plane. Let
be the family of segments
of unit length lying in the first quadrant with
on the
-axis and
on the
-axis. There is a unique point
on
distinct from
and
that does not belong to any segment from
other than
. Then
, where
and
are relatively prime positive integers. Find
.
Problem 13
Let be a 13th root of unity. Find the remainder when
is divided by 1000.
Problem 14
Let be an integer. Call a positive integer
if it has exactly two digits when expressed in base
, and these two digits sum to
. For example,
is
-eautiful because
and
. Find the least integer
for which there are more than ten
-eautiful integers.
Problem 15
Find the number of rectangles that can be formed inside a fixed regular dodecagon (-gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. The diagram below shows three of those rectangles.
See also
2024 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2024 AIME I |
Followed by 2025 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.