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National and Regional Contests
USA Contests
MAA AMC
AIME Problems
1989 AIME Problems
1989 AIME Problems
Part of
AIME Problems
Subcontests
(15)
15
1
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Triangle Cevians
Point
P
P
P
is inside
△
A
B
C
\triangle ABC
△
A
BC
. Line segments
A
P
D
APD
A
P
D
,
B
P
E
BPE
BPE
, and
C
P
F
CPF
CPF
are drawn with
D
D
D
on
B
C
BC
BC
,
E
E
E
on
A
C
AC
A
C
, and
F
F
F
on
A
B
AB
A
B
(see the figure at right). Given that
A
P
=
6
AP=6
A
P
=
6
,
B
P
=
9
BP=9
BP
=
9
,
P
D
=
6
PD=6
P
D
=
6
,
P
E
=
3
PE=3
PE
=
3
, and
C
F
=
20
CF=20
CF
=
20
, find the area of
△
A
B
C
\triangle ABC
△
A
BC
.[asy] size(200); pair A=origin, B=(7,0), C=(3.2,15), D=midpoint(B--C), F=(3,0), P=intersectionpoint(C--F, A--D), ex=B+40*dir(B--P), E=intersectionpoint(B--ex, A--C); draw(A--B--C--A--D^^C--F^^B--E); pair point=P; label("
A
A
A
", A, dir(point--A)); label("
B
B
B
", B, dir(point--B)); label("
C
C
C
", C, dir(point--C)); label("
D
D
D
", D, dir(point--D)); label("
E
E
E
", E, dir(point--E)); label("
F
F
F
", F, dir(point--F)); label("
P
P
P
", P, dir(0));[/asy]
14
1
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Complex Numbers
Given a positive integer
n
n
n
, it can be shown that every complex number of the form
r
+
s
i
r+si
r
+
s
i
, where
r
r
r
and
s
s
s
are integers, can be uniquely expressed in the base
−
n
+
i
-n+i
−
n
+
i
using the integers
1
,
2
,
…
,
n
2
1,2,\ldots,n^2
1
,
2
,
…
,
n
2
as digits. That is, the equation
r
+
s
i
=
a
m
(
−
n
+
i
)
m
+
a
m
−
1
(
−
n
+
i
)
m
−
1
+
⋯
+
a
1
(
−
n
+
i
)
+
a
0
r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0
r
+
s
i
=
a
m
(
−
n
+
i
)
m
+
a
m
−
1
(
−
n
+
i
)
m
−
1
+
⋯
+
a
1
(
−
n
+
i
)
+
a
0
is true for a unique choice of non-negative integer
m
m
m
and digits
a
0
,
a
1
,
…
,
a
m
a_0,a_1,\ldots,a_m
a
0
,
a
1
,
…
,
a
m
chosen from the set
{
0
,
1
,
2
,
…
,
n
2
}
\{0,1,2,\ldots,n^2\}
{
0
,
1
,
2
,
…
,
n
2
}
, with
a
m
≠
0
a_m\ne 0
a
m
=
0
. We write
r
+
s
i
=
(
a
m
a
m
−
1
…
a
1
a
0
)
−
n
+
i
r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}
r
+
s
i
=
(
a
m
a
m
−
1
…
a
1
a
0
)
−
n
+
i
to denote the base
−
n
+
i
-n+i
−
n
+
i
expansion of
r
+
s
i
r+si
r
+
s
i
. There are only finitely many integers
k
+
0
i
k+0i
k
+
0
i
that have four-digit expansions
k
=
(
a
3
a
2
a
1
a
0
)
−
3
+
i
a
3
≠
0.
k=(a_3a_2a_1a_0)_{-3+i}~~~~a_3\ne 0.
k
=
(
a
3
a
2
a
1
a
0
)
−
3
+
i
a
3
=
0.
Find the sum of all such
k
k
k
.
13
1
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Subset
Let
S
S
S
be a subset of
{
1
,
2
,
3
,
…
,
1989
}
\{1,2,3,\ldots,1989\}
{
1
,
2
,
3
,
…
,
1989
}
such that no two members of
S
S
S
differ by
4
4
4
or
7
7
7
. What is the largest number of elements
S
S
S
can have?
12
1
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Tetrahedron
Let
A
B
C
D
ABCD
A
BC
D
be a tetrahedron with
A
B
=
41
AB=41
A
B
=
41
,
A
C
=
7
AC=7
A
C
=
7
,
A
D
=
18
AD=18
A
D
=
18
,
B
C
=
36
BC=36
BC
=
36
,
B
D
=
27
BD=27
B
D
=
27
, and
C
D
=
13
CD=13
C
D
=
13
, as shown in the figure. Let
d
d
d
be the distance between the midpoints of edges
A
B
AB
A
B
and
C
D
CD
C
D
. Find
d
2
d^{2}
d
2
.[asy] pair C=origin, D=(4,11), A=(8,-5), B=(16,0); draw(A--B--C--D--B^^D--A--C); draw(midpoint(A--B)--midpoint(C--D), dashed); label("27", B--D, NE); label("41", A--B, SE); label("7", A--C, SW); label("
d
d
d
", midpoint(A--B)--midpoint(C--D), NE); label("18", (7,8), SW); label("13", (3,9), SW); pair point=(7,0); label("
A
A
A
", A, dir(point--A)); label("
B
B
B
", B, dir(point--B)); label("
C
C
C
", C, dir(point--C)); label("
D
D
D
", D, dir(point--D));[/asy]
11
1
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Integer Sample
A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let
D
D
D
be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of
⌊
D
⌋
\lfloor D\rfloor
⌊
D
⌋
? (For real
x
x
x
,
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
is the greatest integer less than or equal to
x
x
x
.)
10
1
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Triangle Trig
Let
a
a
a
,
b
b
b
,
c
c
c
be the three sides of a triangle, and let
α
\alpha
α
,
β
\beta
β
,
γ
\gamma
γ
, be the angles opposite them. If
a
2
+
b
2
=
1989
c
2
a^2+b^2=1989c^2
a
2
+
b
2
=
1989
c
2
, find
cot
γ
cot
α
+
cot
β
.
\frac{\cot \gamma}{\cot \alpha+\cot \beta}.
cot
α
+
cot
β
cot
γ
.
9
1
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Got what it takes to disprove Euler?
One of Euler's conjectures was disproved in then 1960s by three American mathematicians when they showed there was a positive integer
n
n
n
such that 133^5 \plus{} 110^5 \plus{} 84^5 \plus{} 27^5 \equal{} n^5. Find the value of
n
n
n
.
8
1
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Perfect Square Sequence
Assume that
x
1
,
x
2
,
…
,
x
7
x_1,x_2,\ldots,x_7
x
1
,
x
2
,
…
,
x
7
are real numbers such that
x
1
+
4
x
2
+
9
x
3
+
16
x
4
+
25
x
5
+
36
x
6
+
49
x
7
=
1
4
x
1
+
9
x
2
+
16
x
3
+
25
x
4
+
36
x
5
+
49
x
6
+
64
x
7
=
12
9
x
1
+
16
x
2
+
25
x
3
+
36
x
4
+
49
x
5
+
64
x
6
+
81
x
7
=
123.
\begin{array}{r} x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7=1\,\,\,\,\,\,\,\, \\ 4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7=12\,\,\,\,\, \\ 9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7=123. \\ \end{array}
x
1
+
4
x
2
+
9
x
3
+
16
x
4
+
25
x
5
+
36
x
6
+
49
x
7
=
1
4
x
1
+
9
x
2
+
16
x
3
+
25
x
4
+
36
x
5
+
49
x
6
+
64
x
7
=
12
9
x
1
+
16
x
2
+
25
x
3
+
36
x
4
+
49
x
5
+
64
x
6
+
81
x
7
=
123.
Find the value of
16
x
1
+
25
x
2
+
36
x
3
+
49
x
4
+
64
x
5
+
81
x
6
+
100
x
7
.
16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7.
16
x
1
+
25
x
2
+
36
x
3
+
49
x
4
+
64
x
5
+
81
x
6
+
100
x
7
.
7
1
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Arithmetic Progression
If the integer
k
k
k
is added to each of the numbers
36
36
36
,
300
300
300
, and
596
596
596
, one obtains the squares of three consecutive terms of an arithmetic series. Find
k
k
k
.
6
1
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Physicsish Aime Problem
Two skaters, Allie and Billie, are at points
A
A
A
and
B
B
B
, respectively, on a flat, frozen lake. The distance between
A
A
A
and
B
B
B
is
100
100
100
meters. Allie leaves
A
A
A
and skates at a speed of
8
8
8
meters per second on a straight line that makes a
6
0
∘
60^\circ
6
0
∘
angle with
A
B
AB
A
B
. At the same time Allie leaves
A
A
A
, Billie leaves
B
B
B
at a speed of
7
7
7
meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?[asy] defaultpen(linewidth(0.8)); draw((100,0)--origin--60*dir(60), EndArrow(5)); label("
A
A
A
", origin, SW); label("
B
B
B
", (100,0), SE); label("
100
100
100
", (50,0), S); label("
6
0
∘
60^\circ
6
0
∘
", (15,0), N);[/asy]
5
1
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Another Biased Coin...
When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to
0
0
0
and is the same as that of getting heads exactly twice. Let
i
j
\frac ij
j
i
, in lowest terms, be the probability that the coin comes up heads in exactly
3
3
3
out of
5
5
5
flips. Find
i
+
j
i+j
i
+
j
.
4
1
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Arithmetic Series Practice
If
a
<
b
<
c
<
d
<
e
a<b<c<d<e
a
<
b
<
c
<
d
<
e
are consecutive positive integers such that
b
+
c
+
d
b+c+d
b
+
c
+
d
is a perfect square and
a
+
b
+
c
+
d
+
e
a+b+c+d+e
a
+
b
+
c
+
d
+
e
is a perfect cube, what is the smallest possible value of
c
c
c
?
3
1
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Decimal Fun
Suppose
n
n
n
is a positive integer and
d
d
d
is a single digit in base 10. Find
n
n
n
if
n
810
=
0.
d
25
d
25
d
25
…
\frac{n}{810}=0.d25d25d25\ldots
810
n
=
0.
d
25
d
25
d
25
…
2
1
Hide problems
Number of Polygons
Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices?
1
1
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Compute This
Compute
(
31
)
(
30
)
(
29
)
(
28
)
+
1
\sqrt{(31)(30)(29)(28)+1}
(
31
)
(
30
)
(
29
)
(
28
)
+
1
.