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AIME Problems
1983 AIME Problems
1983 AIME Problems
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AIME Problems
Subcontests
(15)
14
1
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Chord Through Two Circles
In the adjoining figure, two circles of radii 6 and 8 are drawn with their centers 12 units apart. At
P
P
P
, one of the points of intersection, a line is drawn in such a way that the chords
Q
P
QP
QP
and
P
R
PR
PR
have equal length. Find the square of the length of
Q
P
QP
QP
.[asy]unitsize(2.5mm); defaultpen(linewidth(.8pt)+fontsize(12pt)); dotfactor=3;pair O1=(0,0), O2=(12,0); path C1=Circle(O1,8), C2=Circle(O2,6); pair P=intersectionpoints(C1,C2)[0]; path C3=Circle(P,sqrt(130)); pair Q=intersectionpoints(C3,C1)[0]; pair R=intersectionpoints(C3,C2)[1];draw(C1); draw(C2); //draw(O2--O1); //dot(O1); //dot(O2); draw(Q--R);label("
Q
Q
Q
",Q,N); label("
P
P
P
",P,dir(80)); label("
R
R
R
",R,E); //label("12",waypoint(O1--O2,0.4),S);[/asy]
15
1
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Two Intersecting Chords
The adjoining figure shows two intersecting chords in a circle, with
B
B
B
on minor arc
A
D
AD
A
D
. Suppose that the radius of the circle is 5, that
B
C
=
6
BC = 6
BC
=
6
, and that
A
D
AD
A
D
is bisected by
B
C
BC
BC
. Suppose further that
A
D
AD
A
D
is the only chord starting at
A
A
A
which is bisected by
B
C
BC
BC
. It follows that the sine of the minor arc
A
B
AB
A
B
is a rational number. If this fraction is expressed as a fraction
m
/
n
m/n
m
/
n
in lowest terms, what is the product
m
n
mn
mn
? [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=dir(200), D=dir(95), M=midpoint(A--D), C=dir(30), BB=C+2*dir(C--M), B=intersectionpoint(M--BB, Circle(origin, 1)); draw(Circle(origin, 1)^^A--D^^B--C); real r=0.05; pair M1=midpoint(M--D), M2=midpoint(M--A); draw((M1+0.1*dir(90)*dir(A--D))--(M1+0.1*dir(-90)*dir(A--D))); draw((M2+0.1*dir(90)*dir(A--D))--(M2+0.1*dir(-90)*dir(A--D))); pair point=origin; 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]
4
1
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Cutting Tool
A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is
50
\sqrt{50}
50
cm, the length of
A
B
AB
A
B
is 6 cm, and that of
B
C
BC
BC
is 2 cm. The angle
A
B
C
ABC
A
BC
is a right angle. Find the square of the distance (in centimeters) from
B
B
B
to the center of the circle. [asy] size(150); defaultpen(linewidth(0.65)+fontsize(11)); real r=10; pair O=(0,0),A=r*dir(45),B=(A.x,A.y-r),C; path P=circle(O,r); C=intersectionpoint(B--(B.x+r,B.y),P); draw(Arc(O, r, 45, 360-17.0312)); draw(A--B--C);dot(A); dot(B); dot(C); label("
A
A
A
",A,NE); label("
B
B
B
",B,SW); label("
C
C
C
",C,SE); [/asy]
12
1
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Length of Diameter
Diameter
A
B
AB
A
B
of a circle has length a 2-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord
C
D
CD
C
D
. The distance from their intersection point
H
H
H
to the center
O
O
O
is a positive rational number. Determine the length of
A
B
AB
A
B
.
10
1
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Certain Palindromes
The numbers 1447, 1005, and 1231 have something in common: each is a four-digit number beginning with 1 that has exactly two identical digits. How many such numbers are there?
13
1
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Find the sum of some sums of a set
For
{
1
,
2
,
3
,
…
,
n
}
\{1, 2, 3, \dots, n\}
{
1
,
2
,
3
,
…
,
n
}
and each of its nonempty subsets a unique alternating sum is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. (For example, the alternating sum for
{
1
,
2
,
4
,
6
,
9
}
\{1, 2, 4, 6, 9\}
{
1
,
2
,
4
,
6
,
9
}
is
9
−
6
+
4
−
2
+
1
=
6
9 - 6 + 4 - 2 + 1 = 6
9
−
6
+
4
−
2
+
1
=
6
and for
{
5
}
\{5\}
{
5
}
it is simply 5.) Find the sum of all such alternating sums for
n
=
7
n = 7
n
=
7
.
6
1
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Remainders
Let
a
n
=
6
n
+
8
n
a_n = 6^n + 8^n
a
n
=
6
n
+
8
n
. Determine the remainder on dividing
a
83
a_{83}
a
83
by 49.
5
1
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Playing with Complex Numbers
Suppose that the sum of the squares of two complex numbers
x
x
x
and
y
y
y
is 7 and the sum of the cubes is 10. What is the largest real value that
x
+
y
x + y
x
+
y
can have?
3
1
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Product of Roots
What is the product of the real roots of the equation
x
2
+
18
x
+
30
=
2
x
2
+
18
x
+
45
?
x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}\,\,?
x
2
+
18
x
+
30
=
2
x
2
+
18
x
+
45
?
2
1
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Absolute Value
Let
f
(
x
)
=
∣
x
−
p
∣
+
∣
x
−
15
∣
+
∣
x
−
p
−
15
∣
f(x) = |x - p| + |x - 15| + |x - p - 15|
f
(
x
)
=
∣
x
−
p
∣
+
∣
x
−
15∣
+
∣
x
−
p
−
15∣
, where
0
<
p
<
15
0 < p < 15
0
<
p
<
15
. Determine the minimum value taken by
f
(
x
)
f(x)
f
(
x
)
for
x
x
x
in the interval
p
≤
x
≤
15
p \le x \le 15
p
≤
x
≤
15
.
1
1
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Logarithmic Manipulation
Let
x
x
x
,
y
y
y
, and
z
z
z
all exceed 1 and let
w
w
w
be a positive number such that \log_x w = 24, \log_y w = 40 \text{and} \log_{xyz} w = 12. Find
log
z
w
\log_z w
lo
g
z
w
.
9
1
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Minimum value of trig expression
Find the minimum value of
9
x
2
sin
2
x
+
4
x
sin
x
\frac{9x^2 \sin^2 x + 4}{x \sin x}
x
sin
x
9
x
2
sin
2
x
+
4
for
0
<
x
<
π
0 < x < \pi
0
<
x
<
π
.
8
1
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Binomial Coeff
What is the largest 2-digit prime factor of the integer
n
=
(
200
100
)
n = \binom{200}{100}
n
=
(
100
200
)
?
11
1
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Find the Volume!
The solid shown has a square base of side length
s
s
s
. The upper edge is parallel to the base and has length
2
s
2s
2
s
. All other edges have length
s
s
s
. Given that
s
=
6
2
s = 6 \sqrt{2}
s
=
6
2
, what is the volume of the solid? [asy] import three; size(170); pathpen = black+linewidth(0.65); pointpen = black; currentprojection = perspective(30,-20,10); real s = 6 * 2^.5; triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6); draw(F--B--C--F--E--A--B); draw(A--D--E, dashed); draw(D--C, dashed); label("
2
s
2s
2
s
", (s/2, s/2, 6), N); label("
s
s
s
", (s/2, 0, 0), SW); [/asy]
7
1
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Knights at the Round Table
Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices of three being equally likely - and are sent off to slay a troublesome dragon. Let
P
P
P
be the probability that at least two of the three had been sitting next to each other. If
P
P
P
is written as a fraction in lowest terms, what is the sum of the numerator and denominator?