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AIME Problems
2024 AIME
2024 AIME
Part of
AIME Problems
Subcontests
(15)
12
2
Hide problems
Silly Extra Intersection
Define
f
(
x
)
=
∣
∣
x
∣
−
1
2
∣
f(x)=|| x|-\tfrac{1}{2}|
f
(
x
)
=
∣∣
x
∣
−
2
1
∣
and
g
(
x
)
=
∣
∣
x
∣
−
1
4
∣
g(x)=|| x|-\tfrac{1}{4}|
g
(
x
)
=
∣∣
x
∣
−
4
1
∣
. Find the number of intersections of the graphs of y=4 g(f(\sin (2 \pi x))) \text{ and } x=4 g(f(\cos (3 \pi y))).
Family of segments
Let
O
(
0
,
0
)
O(0,0)
O
(
0
,
0
)
,
A
(
1
2
,
0
)
A(\tfrac{1}{2},0)
A
(
2
1
,
0
)
, and
B
(
0
,
3
2
)
B(0, \tfrac{\sqrt{3}}{2})
B
(
0
,
2
3
)
be points in the coordinate plane. Let
F
\mathcal{F}
F
be the family of segments
P
Q
‾
\overline{PQ}
PQ
of unit length lying in the first quadrant with
P
P
P
on the
x
x
x
-axis and
Q
Q
Q
on the
y
y
y
-axis. There is a unique point
C
C
C
on
A
B
‾
\overline{AB}
A
B
, distinct from
A
A
A
and
B
B
B
, that does not belong to any segment from
F
\mathcal{F}
F
other than
A
B
‾
\overline{AB}
A
B
. Then
O
C
2
=
p
q
OC^2 = \tfrac{p}{q}
O
C
2
=
q
p
where
p
p
p
and
q
q
q
are relatively prime positive integers. Find
p
+
q
p + q
p
+
q
.
14
2
Hide problems
DUM Bashing
Let
A
B
C
D
ABCD
A
BC
D
be a tetrahedron such that
A
B
=
C
D
=
41
AB = CD = \sqrt{41}
A
B
=
C
D
=
41
,
A
C
=
B
D
=
80
AC = BD = \sqrt{80}
A
C
=
B
D
=
80
, and
B
C
=
A
D
=
89
BC = AD = \sqrt{89}
BC
=
A
D
=
89
. There exists a point
I
I
I
inside the tetrahedron such that the distances from
I
I
I
to each of the faces of the tetrahedron are all equal. This distance can be written in the form
m
n
p
\frac{m \sqrt{n}}{p}
p
m
n
, when
m
m
m
,
n
n
n
, and
p
p
p
are positive integers,
m
m
m
and
p
p
p
are relatively prime, and
n
n
n
is not divisible by the square of any prime. Find
m
+
n
+
p
m+n+p
m
+
n
+
p
.
Number Theory is Beautiful <3
Let
b
≥
2
b\ge 2
b
≥
2
be an integer. Call a positive integer
n
n
n
b
b
b
-eautiful if it has exactly two digits when expressed in base
b
b
b
and these two digits sum to
n
\sqrt{n}
n
. For example,
81
81
81
is
13
13
13
-eautiful because
81
=
6
‾
3
‾
13
81 = \underline{6} \ \underline{3}_{13}
81
=
6
3
13
and
6
+
3
=
81
6 + 3 = \sqrt{81}
6
+
3
=
81
. Find the least integer
b
≥
2
b\ge 2
b
≥
2
for which there are more than ten
b
b
b
-eautiful integers.
15
2
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Box Inequalities
Let
B
\mathcal{B}
B
be the set of rectangular boxes that have volume
23
23
23
and surface area
54
54
54
. Suppose
r
r
r
is the least possible radius of a sphere that can fit any element of
B
\mathcal{B}
B
inside it. Then
r
2
r^{2}
r
2
can be expressed as
p
q
\tfrac{p}{q}
q
p
, where
p
p
p
and
q
q
q
are relatively prime positive integers. Find
p
+
q
p+q
p
+
q
.
2002 I #5: Level 2
Find the number of rectangles that can be formed inside a fixed regular dodecagon (
12
12
12
-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(40); real r = pi/6; pair A1 = (cos(r),sin(r)); pair A2 = (cos(2r),sin(2r)); pair A3 = (cos(3r),sin(3r)); pair A4 = (cos(4r),sin(4r)); pair A5 = (cos(5r),sin(5r)); pair A6 = (cos(6r),sin(6r)); pair A7 = (cos(7r),sin(7r)); pair A8 = (cos(8r),sin(8r)); pair A9 = (cos(9r),sin(9r)); pair A10 = (cos(10r),sin(10r)); pair A11 = (cos(11r),sin(11r)); pair A12 = (cos(12r),sin(12r)); draw(A1--A2--A3--A4--A5--A6--A7--A8--A9--A10--A11--A12--cycle); filldraw(A2--A1--A8--A7--cycle, mediumgray, linewidth(1.2)); draw(A4--A11); draw(0.365*A3--0.365*A12, linewidth(1.2)); dot(A1); dot(A2); dot(A3); dot(A4); dot(A5); dot(A6); dot(A7); dot(A8); dot(A9); dot(A10); dot(A11); dot(A12); [/asy]
6
2
Hide problems
Direction-Changing Paths
Consider the paths of length
16
16
16
that go from the lower left corner to the upper right corner of an
8
×
8
8\times 8
8
×
8
grid. Find the number of such paths that change direction exactly
4
4
4
times.
Normal AB MAA names with sets of integers
Alice chooses a set
A
A
A
of positive integers. Then Bob lists all finite nonempty sets
B
B
B
of positive integers with the property that the maximum element of
B
B
B
belongs to
A
A
A
. Bob's list has
2024
2024
2024
sets. Find the sum of the elements of
A
A
A
13
2
Hide problems
Prime Squared
Let
p
p
p
be the least prime number for which there exists a positive integer
n
n
n
such that
n
4
+
1
n^{4}+1
n
4
+
1
is divisible by
p
2
p^{2}
p
2
. Find the least positive integer
m
m
m
such that
m
4
+
1
m^{4}+1
m
4
+
1
is divisible by
p
2
p^{2}
p
2
.
2023 AIME II P8 all over again
Let
ω
≠
1
\omega \ne 1
ω
=
1
be a
13
13
13
th root of unity. Find the remainder when
∏
k
=
0
12
(
2
−
2
ω
k
+
ω
2
k
)
\prod_{k=0}^{12} \left(2 - 2\omega^k + \omega^{2k} \right)
k
=
0
∏
12
(
2
−
2
ω
k
+
ω
2
k
)
is divided by
1000
1000
1000
.
10
2
Hide problems
Olympiad Geometry on AIME yay
Let
△
A
B
C
\triangle ABC
△
A
BC
have side lengths
A
B
=
5
,
B
C
=
9
,
AB = 5, BC = 9,
A
B
=
5
,
BC
=
9
,
and
C
A
=
10.
CA = 10.
C
A
=
10.
The tangents to the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
at
B
B
B
and
C
C
C
intersect at point
D
,
D,
D
,
and
A
D
‾
\overline{AD}
A
D
intersects the circumcircle at
P
≠
A
.
P \ne A.
P
=
A
.
The length of
A
P
‾
\overline{AP}
A
P
is equal to
m
n
,
\frac{m}{n},
n
m
,
where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
.
m + n.
m
+
n
.
2019 CIME I/14 moment
Let
△
A
B
C
\triangle ABC
△
A
BC
have circumcenter
O
O
O
and incenter
I
I
I
with
I
A
‾
⊥
O
I
‾
\overline{IA}\perp\overline{OI}
I
A
⊥
O
I
, circumradius
13
13
13
, and inradius
6
6
6
. Find
A
B
⋅
A
C
AB\cdot AC
A
B
⋅
A
C
.
4
2
Hide problems
Jen, Subset of 10
Jen randomly picks
4
4
4
distinct elements from
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
}
\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
}
. The lottery machine also picks
4
4
4
distinct elements. If the lottery machine picks at least
2
2
2
of Jen’s numbers, Jen wins a prize. If the lottery machine’s numbers are all
4
4
4
of Jen’s, Jen wins the Grand Prize. Given that Jen wins a prize, what is the probability she wins a Grand Prize?
Another early logarithm
Let
x
,
y
x,y
x
,
y
and
z
z
z
be positive real numbers that satisfy the following system of equations:
log
2
(
x
y
z
)
=
1
2
\log_2\left({x \over yz}\right) = {1 \over 2}
lo
g
2
(
yz
x
)
=
2
1
log
2
(
y
x
z
)
=
1
3
\log_2\left({y \over xz}\right) = {1 \over 3}
lo
g
2
(
x
z
y
)
=
3
1
log
2
(
z
x
y
)
=
1
4
\log_2\left({z \over xy}\right) = {1 \over 4}
lo
g
2
(
x
y
z
)
=
4
1
Then the value of
∣
log
2
(
x
4
y
3
z
2
)
∣
\left|\log_2(x^4y^3z^2)\right|
lo
g
2
(
x
4
y
3
z
2
)
is
m
n
{m \over n}
n
m
where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
m+n
m
+
n
11
2
Hide problems
Octagons
Each vertex of a regular octagon is coloured either red or blue with equal probability. The probability that the octagon can then be rotated in such a way that all of the blue vertices end up at points that were originally red is
m
n
\tfrac{m}{n}
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers. What is
m
+
n
m+n
m
+
n
?
Nerfed 2023 JMO 1
Find the number of triples of nonnegative integers
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
satisfying
a
+
b
+
c
=
300
a + b + c = 300
a
+
b
+
c
=
300
and
a
2
b
+
a
2
c
+
b
2
a
+
b
2
c
+
c
2
a
+
c
2
b
=
6
,
000
,
000.
a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6{,}000{,}000.
a
2
b
+
a
2
c
+
b
2
a
+
b
2
c
+
c
2
a
+
c
2
b
=
6
,
000
,
000.
9
2
Hide problems
Conic geo returns
Let
A
A
A
,
B
B
B
,
C
C
C
, and
D
D
D
be points in the coordinate plane on the hyperbola
x
2
20
−
y
2
24
=
1
\tfrac{x^{2}}{20}-\tfrac{y^{2}}{24}=1
20
x
2
−
24
y
2
=
1
such that
A
B
C
D
ABCD
A
BC
D
is a rhombus whose diagonals intersect at the origin. Find the greatest real number that is less than
B
D
2
BD^{2}
B
D
2
for all such rhombi.
Another beautiful grid problem
There is a collection of
25
25
25
indistinguishable black chips and
25
25
25
indistinguishable white chips. Find the number of ways to place some of these chips in
25
25
25
unit cells of a
5
×
5
5 \times 5
5
×
5
grid so that:[*]each cell contains at most one chip, [*]all chips in the same row and all chips in the same column have the same color, [*]any additional chip placed on the grid would violate one or more of the previous two conditions.
1
2
Hide problems
the numbers sum to 2024
Among the
900
900
900
residents of Aimeville, there are
195
195
195
who own a diamond ring,
367
367
367
who own a set of golf clubs, and
562
562
562
who own a garden spade. In addition, each of the
900
900
900
residents owns a bag of candy hearts. There are
437
437
437
residents who own exactly two of these things, and
234
234
234
residents who own exactly three of these things. Find the number of residents of Aimeville who own all four of these things.
Reread again and again and again and again…
Every morning, Aya does a
9
9
9
kilometer walk, and then finishes at the coffee shop. One day, she walks at
s
s
s
kilometers per hour, and the walk takes
4
4
4
hours, including
t
t
t
minutes at the coffee shop. Another morning, she walks at
s
+
2
s+2
s
+
2
kilometers per hour, and the walk takes
2
2
2
hours and
24
24
24
minutes, including
t
t
t
minutes at the coffee shop. This morning, if she walks at
s
+
1
2
s+\frac12
s
+
2
1
kilometers per hour, how many minutes will the walk take, including the
t
t
t
minutes at the coffee shop?
3
2
Hide problems
How to Win the Game
Alice and Bob play the following game. A stack of
n
n
n
tokens lies before them. The players take turns with Alice going first. On each turn, the player removes
1
1
1
token or
4
4
4
tokens from the stack. The player who removes the last token wins. Find the number of positive integers
n
n
n
less than or equal to
2024
2024
2024
such that there is a strategy that guarantees that Bob wins, regardless of Alice’s moves.
Beautiful grid problem
Find the number of ways to place a digit in each cell of a
2
×
3
2 \times 3
2
×
3
grid so that the sum of the two numbers formed by reading left to right is
999
999
999
, and the sum of the three numbers formed by reading top to bottom is
99
99
99
. The grid below is an example of such an arrangement because
8
+
991
=
999
8+991 = 999
8
+
991
=
999
and
9
+
9
+
81
=
99
9+9+81 = 99
9
+
9
+
81
=
99
. [asy] unitsize(0.7cm); draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((0,2)--(3,2)); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,0)--(2,2)); draw((3,0)--(3,2)); label("
9
9
9
", (0.5,0.5)); label("
9
9
9
", (1.5,0.5)); label("
1
1
1
", (2.5,0.5)); label("
0
0
0
", (0.5,1.5)); label("
0
0
0
", (1.5,1.5)); label("
8
8
8
", (2.5,1.5)); [/asy]
8
2
Hide problems
Incircles
Eight circles of radius
34
34
34
can be placed tangent to side
B
C
‾
\overline{BC}
BC
of
△
A
B
C
\triangle ABC
△
A
BC
such that the first circle is tangent to
A
B
‾
\overline{AB}
A
B
, subsequent circles are externally tangent to each other, and the last is tangent to
A
C
‾
\overline{AC}
A
C
. Similarly,
2024
2024
2024
circles of radius
1
1
1
can also be placed along
B
C
‾
\overline{BC}
BC
in this manner. The inradius of
△
A
B
C
\triangle ABC
△
A
BC
is
m
n
\tfrac{m}{n}
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
m+n
m
+
n
.
Torus Geo Showed Up on AIME
Torus
T
\mathcal T
T
is the surface produced by revolving a circle with radius
3
3
3
around an axis in the plane a distance
6
6
6
from the center of the circle. When a sphere of radius
11
11
11
rests inside
T
\mathcal T
T
, it is internally tangent to
T
\mathcal T
T
along a circle with radius
r
i
r_{i}
r
i
, and when it rests outside
T
\mathcal T
T
, it is externally tangent along a circle with radius
r
o
r_{o}
r
o
. The difference
r
i
−
r
o
=
m
n
r_{i}-r_{o}=\tfrac{m}{n}
r
i
−
r
o
=
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
m+n
m
+
n
.
2
2
Hide problems
EZ logarithm V2
Real numbers
x
x
x
and
y
y
y
with
x
,
y
>
1
x,y>1
x
,
y
>
1
satisfy
log
x
(
y
x
)
=
log
y
(
x
4
y
)
=
10.
\log_x(y^x)=\log_y(x^{4y})=10.
lo
g
x
(
y
x
)
=
lo
g
y
(
x
4
y
)
=
10.
What is the value of
x
y
xy
x
y
?
List of positive integers
A list of positive integers has the following properties: - The sum of the items in the list is
30
30
30
. - The unique mode of the list is
9
9
9
. - 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.
5
2
Hide problems
PoP goes the weasel
Rectangles
A
B
C
D
ABCD
A
BC
D
and
E
F
G
H
EFGH
EFG
H
are drawn such that
D
,
E
,
C
,
F
D,E,C,F
D
,
E
,
C
,
F
are collinear. Also,
A
,
D
,
H
,
G
A,D,H,G
A
,
D
,
H
,
G
all lie on a circle. If
B
C
=
16
,
BC=16,
BC
=
16
,
A
B
=
107
,
AB=107,
A
B
=
107
,
F
G
=
17
,
FG=17,
FG
=
17
,
and
E
F
=
184
,
EF=184,
EF
=
184
,
what is the length of
C
E
CE
CE
? [asy] unitsize(1 cm);pair A, B, C, D, E, F, G, H;A = (0,0); B = (5,0); C = (5,1.5); D = (0,1.5); E = (1,1.5); F = (8,1.5); G = (8,3.5); H = (1,3.5);draw(A--B--C--D--cycle); draw(E--F--G--H--cycle);dot("A", A, SW); dot("B", B, SE); dot("C", C, SE); dot("D", D, NW); dot("E", E, NW); dot("F", F, SE); dot("G", G, NE); dot("H", H, NW);[/asy]
2024 AIME II P5
Let ABCDEF be an equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are the extensions of AB, CD and EF has side lengths 200, 240 and 300 respectively. Find the side length of the hexagon.
7
2
Hide problems
Complex Expression
Find the largest possible real part of
(
75
+
117
i
)
z
+
96
+
144
i
z
(75+117i)z+\frac{96+144i}{z}
(
75
+
117
i
)
z
+
z
96
+
144
i
where
z
z
z
is a complex number with
∣
z
∣
=
4
|z|=4
∣
z
∣
=
4
.
Digit Replacements
Let
N
N
N
be the greatest four-digit integer with the property that whenever one of its digits is changed to
1
1
1
, the resulting number is divisible by
7
7
7
. Let
Q
Q
Q
and
R
R
R
be the quotient and remainder, respectively, when
N
N
N
is divided by
1000
1000
1000
. Find
Q
+
R
Q+R
Q
+
R
.