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Contests
National and Regional Contests
USA Contests
MAA AMC
AIME Problems
1992 AIME Problems
1992 AIME Problems
Part of
AIME Problems
Subcontests
(15)
15
1
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Factorial Tails
Define a positive integer
n
n
n
to be a factorial tail if there is some positive integer
m
m
m
such that the decimal representation of
m
!
m!
m
!
ends with exactly
n
n
n
zeroes. How many positive integers less than
1992
1992
1992
are not factorial tails?
14
1
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Triangle Problem
In triangle
A
B
C
ABC
A
BC
,
A
′
A'
A
′
,
B
′
B'
B
′
, and
C
′
C'
C
′
are on the sides
B
C
BC
BC
,
A
C
AC
A
C
, and
A
B
AB
A
B
, respectively. Given that
A
A
′
AA'
A
A
′
,
B
B
′
BB'
B
B
′
, and
C
C
′
CC'
C
C
′
are concurrent at the point
O
O
O
, and that
A
O
O
A
′
+
B
O
O
B
′
+
C
O
O
C
′
=
92
,
\frac{AO}{OA'}+\frac{BO}{OB'}+\frac{CO}{OC'}=92,
O
A
′
A
O
+
O
B
′
BO
+
O
C
′
CO
=
92
,
find
A
O
O
A
′
⋅
B
O
O
B
′
⋅
C
O
O
C
′
.
\frac{AO}{OA'}\cdot \frac{BO}{OB'}\cdot \frac{CO}{OC'}.
O
A
′
A
O
⋅
O
B
′
BO
⋅
O
C
′
CO
.
12
1
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Chomp
In a game of Chomp, two players alternately take bites from a 5-by-7 grid of unit squares. To take a bite, a player chooses one of the remaining squares, then removes ("eats'') all squares in the quadrant defined by the left edge (extended upward) and the lower edge (extended rightward) of the chosen square. For example, the bite determined by the shaded square in the diagram would remove the shaded square and the four squares marked by
×
.
\times.
×
.
(The squares with two or more dotted edges have been removed form the original board in previous moves.) [asy] defaultpen(linewidth(0.7)); fill((2,2)--(2,3)--(3,3)--(3,2)--cycle, mediumgray); int[] array={5, 5, 5, 4, 2, 2, 2, 0}; pair[] ex = {(2,3), (2,4), (3,2), (3,3)}; draw((3,5)--(7,5)^^(4,4)--(7,4)^^(4,3)--(7,3), linetype("3 3")); draw((4,4)--(4,5)^^(5,2)--(5,5)^^(6,2)--(6,5)^^(7,2)--(7,5), linetype("3 3")); int i, j; for(i=0; i<7; i=i+1) { for(j=0; j
array[i+1]) { draw((i+1,array)--(i+1,array[i+1])); }} for(i=0; i<4; i=i+1) { draw(ex--(ex.x+1, ex.y+1), linewidth(1.2)); draw((ex.x+1, ex.y)--(ex.x, ex.y+1), linewidth(1.2)); }[/asy]The object of the game is to make one's opponent take the last bite. The diagram shows one of the many subsets of the set of 35 unit squares that can occur during the game of Chomp. How many different subsets are there in all? Include the full board and empty board in your count.
11
1
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Lines and Transformations
Lines
l
1
l_1
l
1
and
l
2
l_2
l
2
both pass through the origin and make first-quadrant angles of
π
70
\frac{\pi}{70}
70
π
and
π
54
\frac{\pi}{54}
54
π
radians, respectively, with the positive x-axis. For any line
l
l
l
, the transformation
R
(
l
)
R(l)
R
(
l
)
produces another line as follows:
l
l
l
is reflected in
l
1
l_1
l
1
, and the resulting line is reflected in
l
2
l_2
l
2
. Let
R
(
1
)
(
l
)
=
R
(
l
)
R^{(1)}(l)=R(l)
R
(
1
)
(
l
)
=
R
(
l
)
and
R
(
n
)
(
l
)
=
R
(
R
(
n
−
1
)
(
l
)
)
R^{(n)}(l)=R\left(R^{(n-1)}(l)\right)
R
(
n
)
(
l
)
=
R
(
R
(
n
−
1
)
(
l
)
)
. Given that
l
l
l
is the line
y
=
19
92
x
y=\frac{19}{92}x
y
=
92
19
x
, find the smallest positive integer
m
m
m
for which
R
(
m
)
(
l
)
=
l
R^{(m)}(l)=l
R
(
m
)
(
l
)
=
l
.
13
1
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Max Area Without Calculus
Triangle
A
B
C
ABC
A
BC
has
A
B
=
9
AB=9
A
B
=
9
and
B
C
:
A
C
=
40
:
41
BC: AC=40: 41
BC
:
A
C
=
40
:
41
. What's the largest area that this triangle can have?
10
1
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Area on the Complex Plane
Consider the region
A
A
A
in the complex plane that consists of all points
z
z
z
such that both
z
40
\frac{z}{40}
40
z
and
40
z
‾
\frac{40}{\overline{z}}
z
40
have real and imaginary parts between
0
0
0
and
1
1
1
, inclusive. What is the integer that is nearest the area of
A
A
A
?
9
1
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Nice Trapezoid Problem
Trapezoid
A
B
C
D
ABCD
A
BC
D
has sides
A
B
=
92
AB=92
A
B
=
92
,
B
C
=
50
BC=50
BC
=
50
,
C
D
=
19
CD=19
C
D
=
19
, and
A
D
=
70
AD=70
A
D
=
70
, with
A
B
AB
A
B
parallel to
C
D
CD
C
D
. A circle with center
P
P
P
on
A
B
AB
A
B
is drawn tangent to
B
C
BC
BC
and
A
D
AD
A
D
. Given that
A
P
=
m
n
AP=\frac mn
A
P
=
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers, find
m
+
n
m+n
m
+
n
.
8
1
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Sequence of Numbers
For any sequence of real numbers
A
=
(
a
1
,
a
2
,
a
3
,
…
)
A=(a_1,a_2,a_3,\ldots)
A
=
(
a
1
,
a
2
,
a
3
,
…
)
, define
Δ
A
\Delta A
Δ
A
to be the sequence
(
a
2
−
a
1
,
a
3
−
a
2
,
a
4
−
a
3
,
…
)
(a_2-a_1,a_3-a_2,a_4-a_3,\ldots)
(
a
2
−
a
1
,
a
3
−
a
2
,
a
4
−
a
3
,
…
)
, whose
n
th
n^\text{th}
n
th
term is
a
n
+
1
−
a
n
a_{n+1}-a_n
a
n
+
1
−
a
n
. Suppose that all of the terms of the sequence
Δ
(
Δ
A
)
\Delta(\Delta A)
Δ
(
Δ
A
)
are
1
1
1
, and that
a
19
=
a
92
=
0
a_{19}=a_{92}=0
a
19
=
a
92
=
0
. Find
a
1
a_1
a
1
.
7
1
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Tetrahedron Volume
Faces
A
B
C
ABC
A
BC
and
B
C
D
BCD
BC
D
of tetrahedron
A
B
C
D
ABCD
A
BC
D
meet at an angle of
3
0
∘
30^\circ
3
0
∘
. The area of face
A
B
C
ABC
A
BC
is
120
120
120
, the area of face
B
C
D
BCD
BC
D
is
80
80
80
, and
B
C
=
10
BC=10
BC
=
10
. Find the volume of the tetrahedron.
6
1
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Adding Integers
For how many pairs of consecutive integers in
{
1000
,
1001
,
1002
,
…
,
2000
}
\{1000,1001,1002,\ldots,2000\}
{
1000
,
1001
,
1002
,
…
,
2000
}
is no carrying required when the two integers are added?
4
1
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Pascal's Triangle
In Pascal's Triangle, each entry is the sum of the two entries above it. The first few rows of the triangle are shown below.\begin{array}{c@{\hspace{8em}} c@{\hspace{6pt}}c@{\hspace{6pt}}c@{\hspace{6pt}}c@{\hspace{4pt}}c@{\hspace{2pt}} c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{3pt}}c@{\hspace{6pt}} c@{\hspace{6pt}}c@{\hspace{6pt}}c} \vspace{4pt} \text{Row 0: } & & & & & & & 1 & & & & & & \\\vspace{4pt} \text{Row 1: } & & & & & & 1 & & 1 & & & & & \\\vspace{4pt} \text{Row 2: } & & & & & 1 & & 2 & & 1 & & & & \\\vspace{4pt} \text{Row 3: } & & & & 1 & & 3 & & 3 & & 1 & & & \\\vspace{4pt} \text{Row 4: } & & & 1 & & 4 & & 6 & & 4 & & 1 & & \\\vspace{4pt} \text{Row 5: } & & 1 & & 5 & &10& &10 & & 5 & & 1 & \\\vspace{4pt} \text{Row 6: } & 1 & & 6 & &15& &20& &15 & & 6 & & 1 \end{array}In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio
3
:
4
:
5
3: 4: 5
3
:
4
:
5
?
5
1
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Decimal Expansion
Let
S
S
S
be the set of all rational numbers
r
r
r
,
0
<
r
<
1
0<r<1
0
<
r
<
1
, that have a repeating decimal expansion in the form
0.
a
b
c
a
b
c
a
b
c
…
=
0.
a
b
c
‾
,
0.abcabcabc\ldots=0.\overline{abc},
0.
ab
c
ab
c
ab
c
…
=
0.
ab
c
,
where the digits
a
a
a
,
b
b
b
, and
c
c
c
are not necessarily distinct. To write the elements of
S
S
S
as fractions in lowest terms, how many different numerators are required?
3
1
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Win Ratio
A tennis player computes her win ratio by dividing the number of matches she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly
.
500
.500
.500
. During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than
.
503
.503
.503
. What's the largest number of matches she could've won before the weekend began?
2
1
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Ascending Numbers
A positive integer is called ascending if, in its decimal representation, there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are there?
1
1
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Sum of Fractions
Find the sum of all positive rational numbers that are less than
10
10
10
and that have denominator
30
30
30
when written in lowest terms.