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AIME Problems
2003 AIME Problems
2003 AIME Problems
Part of
AIME Problems
Subcontests
(15)
15
2
Hide problems
A difficult geometry problem
In
△
A
B
C
\triangle ABC
△
A
BC
,
A
B
=
360
AB = 360
A
B
=
360
,
B
C
=
507
BC = 507
BC
=
507
, and
C
A
=
780
CA = 780
C
A
=
780
. Let
M
M
M
be the midpoint of
C
A
‾
\overline{CA}
C
A
, and let
D
D
D
be the point on
C
A
‾
\overline{CA}
C
A
such that
B
D
‾
\overline{BD}
B
D
bisects angle
A
B
C
ABC
A
BC
. Let
F
F
F
be the point on
B
C
‾
\overline{BC}
BC
such that
D
F
‾
⊥
B
D
‾
\overline{DF} \perp \overline{BD}
D
F
⊥
B
D
. Suppose that
D
F
‾
\overline{DF}
D
F
meets
B
M
‾
\overline{BM}
BM
at
E
E
E
. The ratio
D
E
:
E
F
DE: EF
D
E
:
EF
can be written in the form
m
/
n
m/n
m
/
n
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
m + n
m
+
n
.
Beastly Polynomial
Let
P
(
x
)
=
24
x
24
+
∑
j
=
1
23
(
24
−
j
)
(
x
24
−
j
+
x
24
+
j
)
.
P(x)=24x^{24}+\sum_{j=1}^{23}(24-j)(x^{24-j}+x^{24+j}).
P
(
x
)
=
24
x
24
+
j
=
1
∑
23
(
24
−
j
)
(
x
24
−
j
+
x
24
+
j
)
.
Let
z
1
,
z
2
,
…
,
z
r
z_{1},z_{2},\ldots,z_{r}
z
1
,
z
2
,
…
,
z
r
be the distinct zeros of
P
(
x
)
,
P(x),
P
(
x
)
,
and let
z
k
2
=
a
k
+
b
k
i
z_{k}^{2}=a_{k}+b_{k}i
z
k
2
=
a
k
+
b
k
i
for
k
=
1
,
2
,
…
,
r
,
k=1,2,\ldots,r,
k
=
1
,
2
,
…
,
r
,
where
i
=
−
1
,
i=\sqrt{-1},
i
=
−
1
,
and
a
k
a_{k}
a
k
and
b
k
b_{k}
b
k
are real numbers. Let
∑
k
=
1
r
∣
b
k
∣
=
m
+
n
p
,
\sum_{k=1}^{r}|b_{k}|=m+n\sqrt{p},
k
=
1
∑
r
∣
b
k
∣
=
m
+
n
p
,
where
m
,
m,
m
,
n
,
n,
n
,
and
p
p
p
are integers and
p
p
p
is not divisible by the square of any prime. Find
m
+
n
+
p
.
m+n+p.
m
+
n
+
p
.
14
2
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Digits 2, 5, and 1
The decimal representation of
m
/
n
m/n
m
/
n
, where
m
m
m
and
n
n
n
are relatively prime positive integers and
m
<
n
m < n
m
<
n
, contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of
n
n
n
for which this is possible.
Area of Hexagon
Let
A
=
(
0
,
0
)
A=(0,0)
A
=
(
0
,
0
)
and
B
=
(
b
,
2
)
B=(b,2)
B
=
(
b
,
2
)
be points on the coordinate plane. Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a convex equilateral hexagon such that
∠
F
A
B
=
12
0
∘
,
\angle FAB=120^\circ,
∠
F
A
B
=
12
0
∘
,
A
B
‾
∥
D
E
‾
,
\overline{AB}\parallel \overline{DE},
A
B
∥
D
E
,
B
C
‾
∥
E
F
,
‾
\overline{BC}\parallel \overline{EF,}
BC
∥
EF
,
C
D
‾
∥
F
A
‾
,
\overline{CD}\parallel \overline{FA},
C
D
∥
F
A
,
and the y-coordinates of its vertices are distinct elements of the set
{
0
,
2
,
4
,
6
,
8
,
10
}
.
\{0,2,4,6,8,10\}.
{
0
,
2
,
4
,
6
,
8
,
10
}
.
The area of the hexagon can be written in the form
m
n
,
m\sqrt{n},
m
n
,
where
m
m
m
and
n
n
n
are positive integers and n is not divisible by the square of any prime. Find
m
+
n
.
m+n.
m
+
n
.
13
2
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Binary representations
Let
N
N
N
be the number of positive integers that are less than or equal to 2003 and whose base-2 representation has more 1's than 0's. Find the remainder when
N
N
N
is divided by 1000.
Another Random Bug
A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is
m
/
n
,
m/n,
m
/
n
,
where
m
m
m
and
n
n
n
are relatively prime positive integers, find
m
+
n
.
m+n.
m
+
n
.
12
2
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A convex quadrilateral
In convex quadrilateral
A
B
C
D
ABCD
A
BC
D
,
∠
A
≅
∠
C
\angle A \cong \angle C
∠
A
≅
∠
C
,
A
B
=
C
D
=
180
AB = CD = 180
A
B
=
C
D
=
180
, and
A
D
≠
B
C
AD \neq BC
A
D
=
BC
. The perimeter of
A
B
C
D
ABCD
A
BC
D
is 640. Find
⌊
1000
cos
A
⌋
\lfloor 1000 \cos A \rfloor
⌊
1000
cos
A
⌋
. (The notation
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
means the greatest integer that is less than or equal to
x
x
x
.)
Members of a Committee
The members of a distinguished committee were choosing a president, and each member gave one vote to one of the
27
27
27
candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least
1
1
1
than the number of votes for that candidate. What is the smallest possible number of members of the committee?
11
2
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Sides of a triangle
An angle
x
x
x
is chosen at random from the interval
0
∘
<
x
<
9
0
∘
0^\circ < x < 90^\circ
0
∘
<
x
<
9
0
∘
. Let
p
p
p
be the probability that the numbers
sin
2
x
\sin^2 x
sin
2
x
,
cos
2
x
\cos^2 x
cos
2
x
, and
sin
x
cos
x
\sin x \cos x
sin
x
cos
x
are not the lengths of the sides of a triangle. Given that
p
=
d
/
n
p = d/n
p
=
d
/
n
, where
d
d
d
is the number of degrees in
arctan
m
\arctan m
arctan
m
and
m
m
m
and
n
n
n
are positive integers with
m
+
n
<
1000
m + n < 1000
m
+
n
<
1000
, find
m
+
n
m + n
m
+
n
.
Area of a Subtriangle
Triangle
A
B
C
ABC
A
BC
is a right triangle with
A
C
=
7
,
AC=7,
A
C
=
7
,
B
C
=
24
,
BC=24,
BC
=
24
,
and right angle at
C
.
C.
C
.
Point
M
M
M
is the midpoint of
A
B
,
AB,
A
B
,
and
D
D
D
is on the same side of line
A
B
AB
A
B
as
C
C
C
so that
A
D
=
B
D
=
15.
AD=BD=15.
A
D
=
B
D
=
15.
Given that the area of triangle
C
D
M
CDM
C
D
M
may be expressed as
m
n
p
,
\frac{m\sqrt{n}}{p},
p
m
n
,
where
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
.
10
2
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Isosceles triangle
Triangle
A
B
C
ABC
A
BC
is isosceles with
A
C
=
B
C
AC = BC
A
C
=
BC
and
∠
A
C
B
=
10
6
∘
\angle ACB = 106^\circ
∠
A
CB
=
10
6
∘
. Point
M
M
M
is in the interior of the triangle so that
∠
M
A
C
=
7
∘
\angle MAC = 7^\circ
∠
M
A
C
=
7
∘
and
∠
M
C
A
=
2
3
∘
\angle MCA = 23^\circ
∠
MC
A
=
2
3
∘
. Find the number of degrees in
∠
C
M
B
\angle CMB
∠
CMB
.
Optimizing a Sum
Two positive integers differ by
60.
60.
60.
The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers?
9
2
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Balanced integers
An integer between 1000 and 9999, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?
Two Polynomials
Consider the polynomials
P
(
x
)
=
x
6
−
x
5
−
x
3
−
x
2
−
x
P(x)=x^{6}-x^{5}-x^{3}-x^{2}-x
P
(
x
)
=
x
6
−
x
5
−
x
3
−
x
2
−
x
and
Q
(
x
)
=
x
4
−
x
3
−
x
2
−
1.
Q(x)=x^{4}-x^{3}-x^{2}-1.
Q
(
x
)
=
x
4
−
x
3
−
x
2
−
1.
Given that
z
1
,
z
2
,
z
3
,
z_{1},z_{2},z_{3},
z
1
,
z
2
,
z
3
,
and
z
4
z_{4}
z
4
are the roots of
Q
(
x
)
=
0
,
Q(x)=0,
Q
(
x
)
=
0
,
find
P
(
z
1
)
+
P
(
z
2
)
+
P
(
z
3
)
+
P
(
z
4
)
.
P(z_{1})+P(z_{2})+P(z_{3})+P(z_{4}).
P
(
z
1
)
+
P
(
z
2
)
+
P
(
z
3
)
+
P
(
z
4
)
.
8
2
Hide problems
An AP and a GP
In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by 30. Find the sum of the four terms.
Strange Sequence
Find the eighth term of the sequence
1440
,
1440,
1440
,
1716
,
1716,
1716
,
1848
,
…
,
1848,\ldots,
1848
,
…
,
whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.
7
2
Hide problems
Possible perimeters
Point
B
B
B
is on
A
C
‾
\overline{AC}
A
C
with
A
B
=
9
AB = 9
A
B
=
9
and
B
C
=
21
BC = 21
BC
=
21
. Point
D
D
D
is not on
A
C
‾
\overline{AC}
A
C
so that
A
D
=
C
D
AD = CD
A
D
=
C
D
, and
A
D
AD
A
D
and
B
D
BD
B
D
are integers. Let
s
s
s
be the sum of all possible perimeters of
△
A
C
D
\triangle ACD
△
A
C
D
. Find
s
s
s
.
Area of a Rhombus
Find the area of rhombus
A
B
C
D
ABCD
A
BC
D
given that the radii of the circles circumscribed around triangles
A
B
D
ABD
A
B
D
and
A
C
D
ACD
A
C
D
are
12.5
12.5
12.5
and
25
25
25
, respectively.
6
2
Hide problems
Triangles in a Cube
The sum of the areas of all triangles whose vertices are also vertices of a
1
×
1
×
1
1\times 1 \times 1
1
×
1
×
1
cube is
m
+
n
+
p
m+\sqrt{n}+\sqrt{p}
m
+
n
+
p
, where
m
m
m
,
n
n
n
, and
p
p
p
are integers. Find
m
+
n
+
p
m+n+p
m
+
n
+
p
.
Rotation about a Centroid
In triangle
A
B
C
,
ABC,
A
BC
,
A
B
=
13
,
AB=13,
A
B
=
13
,
B
C
=
14
,
BC=14,
BC
=
14
,
A
C
=
15
,
AC=15,
A
C
=
15
,
and point
G
G
G
is the intersection of the medians. Points
A
′
,
A',
A
′
,
B
′
,
B',
B
′
,
and
C
′
,
C',
C
′
,
are the images of
A
,
A,
A
,
B
,
B,
B
,
and
C
,
C,
C
,
respectively, after a
18
0
∘
180^\circ
18
0
∘
rotation about
G
.
G.
G
.
What is the area if the union of the two regions enclosed by the triangles
A
B
C
ABC
A
BC
and
A
′
B
′
C
′
?
A'B'C'?
A
′
B
′
C
′
?
5
2
Hide problems
Neighbourhood of a box
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is
(
m
+
n
π
)
/
p
(m + n \pi)/p
(
m
+
nπ
)
/
p
, where
m
m
m
,
n
n
n
, and
p
p
p
are positive integers, and
n
n
n
and
p
p
p
are relatively prime, find
m
+
n
+
p
m + n + p
m
+
n
+
p
.
Cutting a Cylindrical Log
A cylindrical log has diameter
12
12
12
inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a
4
5
∘
45^\circ
4
5
∘
angle with the plane of the first cut. The intersection of these two planes has exactly one point in common with the log. The number of cubic inches in the wedge can be expressed as
n
π
,
n\pi,
nπ
,
where
n
n
n
is a positive integer. Find
n
.
n.
n
.
4
2
Hide problems
Logs and Trig
Given that
log
10
sin
x
+
log
10
cos
x
=
−
1
\log_{10} \sin x + \log_{10} \cos x = -1
lo
g
10
sin
x
+
lo
g
10
cos
x
=
−
1
and that
log
10
(
sin
x
+
cos
x
)
=
1
2
(
log
10
n
−
1
)
\log_{10} (\sin x + \cos x) = \textstyle \frac{1}{2} (\log_{10} n - 1)
lo
g
10
(
sin
x
+
cos
x
)
=
2
1
(
lo
g
10
n
−
1
)
, find
n
n
n
.
Tetrahedron Volume Ratios
In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is
m
/
n
m/n
m
/
n
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
.
m+n.
m
+
n
.
3
2
Hide problems
A sum of maximums
Let the set
S
=
{
8
,
5
,
1
,
13
,
34
,
3
,
21
,
2
}
\mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}
S
=
{
8
,
5
,
1
,
13
,
34
,
3
,
21
,
2
}
. Susan makes a list as follows: for each two-element subset of
S
\mathcal{S}
S
, she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.
Counting "Good Words"
Define a
g
o
o
d
w
o
r
d
good~word
g
oo
d
w
or
d
as a sequence of letters that consists only of the letters
A
,
A,
A
,
B
,
B,
B
,
and
C
C
C
−
-
−
some of these letters may not appear in the sequence
−
-
−
and in which
A
A
A
is never immediately followed by
B
,
B,
B
,
B
B
B
is never immediately followed by
C
,
C,
C
,
and
C
C
C
is never immediately followed by
A
.
A.
A
.
How many seven-letter good words are there?
2
2
Hide problems
100 Circles
One hundred concentric circles with radii
1
,
2
,
3
,
…
,
100
1, 2, 3, \dots, 100
1
,
2
,
3
,
…
,
100
are drawn in a plane. The interior of the circle of radius 1 is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green regions to the area of the circle of radius 100 can be expressed as
m
/
n
m/n
m
/
n
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
m + n
m
+
n
.
Multiples of 8
Let
N
N
N
be the greatest integer multiple of
8
,
8,
8
,
no two of whose digits are the same. What is the remainder when
N
N
N
is divided by
1000
?
1000?
1000
?
1
2
Hide problems
Multiple Factorials
Given that
(
(
3
!
)
!
)
!
3
!
=
k
⋅
n
!
,
\frac{((3!)!)!}{3!} = k \cdot n!,
3
!
((
3
!)!)!
=
k
⋅
n
!
,
where
k
k
k
and
n
n
n
are positive integers and
n
n
n
is as large as possible, find
k
+
n
k + n
k
+
n
.
Possible Products N
The product
N
N
N
of three positive integers is
6
6
6
times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of
N
.
N.
N
.