Difference between revisions of "2024 AIME I Problems"
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− | The | + | {{AIME Problems|year=2024|n=I}} |
+ | |||
+ | ==Problem 1== | ||
+ | Every morning, Aya does a <math>9</math> kilometer walk, and then finishes at the coffee shop. One day, she walks at <math>s</math> kilometers per hour, and the walk takes <math>4</math> hours, including <math>t</math> minutes at the coffee shop. Another morning, she walks at <math>s+2</math> kilometers per hour, and the walk takes <math>2</math> hours and <math>24</math> minutes, including <math>t</math> minutes at the coffee shop. This morning, if she walks at <math>s+\frac12</math> kilometers per hour, how many minutes will the walk take, including the <math>t</math> minutes at the coffee shop? | ||
+ | |||
+ | [[2024 AIME I Problems/Problem 1|Solution]] | ||
+ | |||
+ | ==Problem 2== | ||
+ | Real numbers <math>x</math> and <math>y</math> with <math>x,y>1</math> satisfy <math>\log_x(y^x)=\log_y(x^{4y})=10.</math> What is the value of <math>xy</math>? | ||
+ | |||
+ | [[2024 AIME I Problems/Problem 2|Solution]] | ||
+ | |||
+ | ==Problem 3== | ||
+ | Alice and Bob play the following game. A stack of <math>n</math> tokens lies before them. The players take turns with Alice going first. On each turn, the player removes <math>1</math> token or <math>4</math> tokens from the stack. The player who removes the last token wins. Find the number of positive integers <math>n</math> less than or equal to <math>2024</math> such that there is a strategy that guarantees that Bob wins, regardless of Alice’s moves. | ||
+ | |||
+ | [[2024 AIME I Problems/Problem 3|Solution]] | ||
+ | |||
+ | ==Problem 4== | ||
+ | Jen enters a lottery by picking <math>4</math> distinct numbers from <math>S=\{1,2,3,\cdots,9,10\}.</math> <math>4</math> numbers are randomly chosen from <math>S.</math> She wins a prize if at least two of her numbers were <math>2</math> of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of her winning the grand prize given that she won a prize is <math>\tfrac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
+ | |||
+ | [[2024 AIME I Problems/Problem 4|Solution]] | ||
+ | |||
+ | ==Problem 5== | ||
+ | Rectangles <math>ABCD</math> and <math>EFGH</math> are drawn such that <math>D,E,C,F</math> are collinear. Also, <math>A,D,H,G</math> all lie on a circle. If <math>BC=16,</math> <math>AB=107,</math> <math>FG=17,</math> and <math>EF=184,</math> what is the length of <math>CE</math>? | ||
+ | |||
+ | <asy> | ||
+ | import graph; | ||
+ | unitsize(0.1cm); | ||
+ | |||
+ | pair A = (0,0);pair B = (70,0);pair C = (70,16);pair D = (0,16);pair E = (3,16);pair F = (90,16);pair G = (90,33);pair H = (3,33); | ||
+ | dot(A^^B^^C^^D^^E^^F^^G^^H); | ||
+ | label("$A$", A, S);label("$B$", B, S);label("$C$", C, N);label("$D$", D, N);label("$E$", E, S);label("$F$", F, S);label("$G$", G, N);label("$H$", H, N); | ||
+ | draw(E--D--A--B--C--E--H--G--F--C); | ||
+ | </asy> | ||
+ | |||
+ | [[2024 AIME I Problems/Problem 5|Solution]] | ||
+ | |||
+ | ==Problem 6== | ||
+ | Consider the paths of length <math>16</math> that follow the lines from the lower left corner to the upper right corner on an <math>8\times 8</math> grid. Find the number of such paths that change direction exactly four times, like in the examples shown below. | ||
+ | |||
+ | <asy> | ||
+ | size(7.5cm); | ||
+ | usepackage("tikz");label("\begin{tikzpicture}[scale=.4]\draw(0,0)grid(8,8);\draw[line width=2,red](0,0)--(2,0)--(2,3)--(5,3)--(5,8)--(8,8);\end{tikzpicture}",origin); | ||
+ | label("\begin{tikzpicture}[scale=.4]\draw(0,0)grid(8,8);\draw[line width=2,red](0,0)--(0,3)--(3,3)--(3,5)--(8,5)--(8,8);\end{tikzpicture}",E); | ||
+ | </asy> | ||
+ | |||
+ | [[2024 AIME I Problems/Problem 6|Solution]] | ||
+ | |||
+ | ==Problem 7== | ||
+ | Find the largest possible real part of <cmath>(75+117i)z+\frac{96+144i}{z}</cmath>where <math>z</math> is a complex number with <math>|z|=4</math>. | ||
+ | |||
+ | [[2024 AIME I Problems/Problem 7|Solution]] | ||
+ | |||
+ | ==Problem 8== | ||
+ | Eight circles of radius <math>34</math> can be placed tangent to <math>\overline{BC}</math> of <math>\triangle ABC</math> so that the circles are sequentially tangent to each other, with the first circle being tangent to <math>\overline{AB}</math> and the last circle being tangent to <math>\overline{AC}</math>, as shown. Similarly, <math>2024</math> circles of radius <math>1</math> can be placed tangent to <math>\overline{BC}</math> in the same manner. The inradius of <math>\triangle ABC</math> can be expressed 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> | ||
+ | pair A = (2,1); pair B = (0,0); pair C = (3,0); | ||
+ | dot(A^^B^^C); | ||
+ | label("$A$", A, N); label("$B$", B, S); label("$C$", C, S); | ||
+ | draw(A--B--C--cycle); | ||
+ | for(real i=0.62; i<2.7; i+=0.29){ | ||
+ | draw(circle((i,0.145), 0.145)); | ||
+ | } | ||
+ | </asy> | ||
+ | |||
+ | [[2024 AIME I Problems/Problem 8|Solution]] | ||
+ | |||
+ | ==Problem 9== | ||
+ | Let <math>ABCD</math> be a rhombus whose vertices all lie on the hyperbola <math>\tfrac{x^2}{20}-\tfrac{y^2}{24}=1</math> and are in that order. If its diagonals intersect at the origin, find the largest number less than <math>BD^2</math> for all rhombuses <math>ABCD</math>. | ||
+ | |||
+ | [[2024 AIME I Problems/Problem 9|Solution]] | ||
+ | |||
+ | ==Problem 10== | ||
+ | Let <math>ABC</math> be a triangle inscribed in circle <math>\omega</math>. Let the tangents to <math>\omega</math> at <math>B</math> and <math>C</math> intersect at point <math>D</math>, and let <math>\overline{AD}</math> intersect <math>\omega</math> at <math>P</math>. If <math>AB=5</math>, <math>BC=9</math>, and <math>AC=10</math>, <math>AP</math> can be written as the form <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime integers. Find <math>m + n</math>. | ||
+ | |||
+ | [[2024 AIME I Problems/Problem 10|Solution]] | ||
+ | |||
+ | ==Problem 11== | ||
+ | Each vertex of a regular octagon is independently colored either red or blue with equal probability. The probability that the octagon can then be rotated so that all of the blue vertices end up at positions where there were originally red vertices is <math>\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m+n</math>? | ||
+ | |||
+ | [[2024 AIME I Problems/Problem 11|Solution]] | ||
+ | |||
+ | ==Problem 12== | ||
+ | Define <math>f(x)=|| x|-\tfrac{1}{2}|</math> and <math>g(x)=|| x|-\tfrac{1}{4}|</math>. Find the number of intersections of the graphs of <cmath>y=4 g(f(\sin (2 \pi x))) \quad\text{ and }\quad x=4 g(f(\cos (3 \pi y))).</cmath> | ||
+ | |||
+ | [[2024 AIME I Problems/Problem 12|Solution]] | ||
+ | |||
+ | ==Problem 13== | ||
+ | Let <math>p</math> be the least prime number for which there exists a positive integer <math>n</math> such that <math>n^{4}+1</math> is divisible by <math>p^{2}</math>. Find the least positive integer <math>m</math> such that <math>m^{4}+1</math> is divisible by <math>p^{2}</math>. | ||
+ | |||
+ | [[2024 AIME I Problems/Problem 13|Solution]] | ||
+ | |||
+ | ==Problem 14== | ||
+ | Let <math>ABCD</math> be a tetrahedron such that <math>AB = CD = \sqrt{41}</math>, <math>AC = BD = \sqrt{80}</math>, and <math>BC = AD = \sqrt{89}</math>. There exists a point <math>I</math> inside the tetrahedron such that the distances from <math>I</math> to each of the faces of the tetrahedron are all equal. This distance can be written in the form <math>\frac{m \sqrt{n}}{p}</math>, when <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, <math>m</math> and <math>p</math> are relatively prime, and <math>n</math> is not divisible by the square of any prime. Find <math>m+n+p</math>. | ||
+ | |||
+ | [[2024 AIME I Problems/Problem 14|Solution]] | ||
+ | |||
+ | ==Problem 15== | ||
+ | Let <math>\mathcal{B}</math> be the set of rectangular boxes with surface area <math>54</math> and volume <math>23</math>. Let <math>r</math> be the radius of the smallest sphere that can contain each of the rectangular boxes that are elements of <math>\mathcal{B}</math>. The value of <math>r^2</math> can be written as <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. | ||
+ | |||
+ | [[2024 AIME I Problems/Problem 15|Solution]] | ||
+ | |||
+ | ==See also== | ||
+ | {{AIME box|year=2024|n=I|before=[[2023 AIME II Problems|2023 AIME II]]|after=[[2024 AIME II Problems|2024 AIME II]]}} | ||
+ | * [[American Invitational Mathematics Examination]] | ||
+ | * [[AIME Problems and Solutions]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 20:00, 6 February 2024
2024 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
Problem 1
Every morning, Aya does a kilometer walk, and then finishes at the coffee shop. One day, she walks at
kilometers per hour, and the walk takes
hours, including
minutes at the coffee shop. Another morning, she walks at
kilometers per hour, and the walk takes
hours and
minutes, including
minutes at the coffee shop. This morning, if she walks at
kilometers per hour, how many minutes will the walk take, including the
minutes at the coffee shop?
Problem 2
Real numbers and
with
satisfy
What is the value of
?
Problem 3
Alice and Bob play the following game. A stack of tokens lies before them. The players take turns with Alice going first. On each turn, the player removes
token or
tokens from the stack. The player who removes the last token wins. Find the number of positive integers
less than or equal to
such that there is a strategy that guarantees that Bob wins, regardless of Alice’s moves.
Problem 4
Jen enters a lottery by picking distinct numbers from
numbers are randomly chosen from
She wins a prize if at least two of her numbers were
of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of her winning the grand prize given that she won a prize is
where
and
are relatively prime positive integers. Find
.
Problem 5
Rectangles and
are drawn such that
are collinear. Also,
all lie on a circle. If
and
what is the length of
?
Problem 6
Consider the paths of length that follow the lines from the lower left corner to the upper right corner on an
grid. Find the number of such paths that change direction exactly four times, like in the examples shown below.
Problem 7
Find the largest possible real part of where
is a complex number with
.
Problem 8
Eight circles of radius can be placed tangent to
of
so that the circles are sequentially tangent to each other, with the first circle being tangent to
and the last circle being tangent to
, as shown. Similarly,
circles of radius
can be placed tangent to
in the same manner. The inradius of
can be expressed as
, where
and
are relatively prime positive integers. Find
.
Problem 9
Let be a rhombus whose vertices all lie on the hyperbola
and are in that order. If its diagonals intersect at the origin, find the largest number less than
for all rhombuses
.
Problem 10
Let be a triangle inscribed in circle
. Let the tangents to
at
and
intersect at point
, and let
intersect
at
. If
,
, and
,
can be written as the form
, where
and
are relatively prime integers. Find
.
Problem 11
Each vertex of a regular octagon is independently colored either red or blue with equal probability. The probability that the octagon can then be rotated so that all of the blue vertices end up at positions where there were originally red vertices is , where
and
are relatively prime positive integers. What is
?
Problem 12
Define and
. Find the number of intersections of the graphs of
Problem 13
Let be the least prime number for which there exists a positive integer
such that
is divisible by
. Find the least positive integer
such that
is divisible by
.
Problem 14
Let be a tetrahedron such that
,
, and
. There exists a point
inside the tetrahedron such that the distances from
to each of the faces of the tetrahedron are all equal. This distance can be written in the form
, when
,
, and
are positive integers,
and
are relatively prime, and
is not divisible by the square of any prime. Find
.
Problem 15
Let be the set of rectangular boxes with surface area
and volume
. Let
be the radius of the smallest sphere that can contain each of the rectangular boxes that are elements of
. The value of
can be written as
, where
and
are relatively prime positive integers. Find
.
See also
2024 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2023 AIME II |
Followed by 2024 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.