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Duke Math Meet (DMM)
1998 Duke Math Meet
1998 Duke Math Meet
Part of
Duke Math Meet (DMM)
Subcontests
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1998 DMM Individual Round - Duke Math Meet
p1. Find the greatest integer
n
n
n
such that
n
log
10
4
n \log_{10} 4
n
lo
g
10
4
does not exceed
log
10
1998
\log_{10} 1998
lo
g
10
1998
. p2. Rectangle
A
B
C
D
ABCD
A
BC
D
has sides
A
B
=
C
D
=
12
/
5
AB = CD = 12/5
A
B
=
C
D
=
12/5
,
B
C
=
D
A
=
5
BC = DA = 5
BC
=
D
A
=
5
. Point
P
P
P
is on
A
D
AD
A
D
with
∠
B
P
C
=
9
0
o
\angle BPC = 90^o
∠
BPC
=
9
0
o
. Compute
B
P
+
P
C
BP + PC
BP
+
PC
. p3. Compute the number of sequences of four decimal digits
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
(each between
0
0
0
and
9
9
9
inclusive) containing no adjacent repeated digits. (That is, each digit is distinct from the digits directly before and directly after it.) p4. Solve for
t
t
t
,
−
π
/
4
≤
t
≤
π
/
4
-\pi/4 \le t \le \pi/4
−
π
/4
≤
t
≤
π
/4
:
sin
3
t
+
sin
2
t
cos
t
+
sin
t
cos
2
t
+
cos
3
t
=
6
2
\sin^3 t + \sin^2 t \cos t + \sin t \cos^2 t + \cos^3 t =\frac{\sqrt6}{2}
sin
3
t
+
sin
2
t
cos
t
+
sin
t
cos
2
t
+
cos
3
t
=
2
6
p5. Find all integers
n
n
n
such that
n
−
3
n - 3
n
−
3
divides
n
2
+
2
n^2 + 2
n
2
+
2
. p6. Find the maximum number of bishops that can occupy an
8
×
8
8 \times 8
8
×
8
chessboard so that no two of the bishops attack each other. (Bishops can attack an arbitrary number of squares in any diagonal direction.) p7. Points
A
,
B
,
C
A, B, C
A
,
B
,
C
, and
D
D
D
are on a Cartesian coordinate system with
A
=
(
0
,
1
)
A = (0, 1)
A
=
(
0
,
1
)
,
B
=
(
1
,
1
)
B = (1, 1)
B
=
(
1
,
1
)
,
C
=
(
1
,
−
1
)
C = (1,-1)
C
=
(
1
,
−
1
)
, and
D
=
(
−
1
,
0
)
D = (-1, 0)
D
=
(
−
1
,
0
)
. Compute the minimum possible value of
P
A
+
P
B
+
P
C
+
P
D
PA + PB + PC + PD
P
A
+
PB
+
PC
+
P
D
over all points
P
P
P
. p8. Find the number of distinct real values of
x
x
x
which satisfy
(
x
−
1
)
(
x
−
2
)
(
x
−
3
)
(
x
−
4
)
(
x
−
5
)
(
x
−
6
)
(
x
−
7
)
(
x
−
8
)
(
x
−
9
)
(
x
−
10
)
+
(
1
2
⋅
3
2
⋅
5
2
⋅
7
2
⋅
9
2
)
/
2
10
=
0.
(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)+(1^2 \cdot 3^2\cdot 5^2\cdot 7^2\cdot 9^2)/2^{10} = 0.
(
x
−
1
)
(
x
−
2
)
(
x
−
3
)
(
x
−
4
)
(
x
−
5
)
(
x
−
6
)
(
x
−
7
)
(
x
−
8
)
(
x
−
9
)
(
x
−
10
)
+
(
1
2
⋅
3
2
⋅
5
2
⋅
7
2
⋅
9
2
)
/
2
10
=
0.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
1998 DMM Team Round - Duke Math Meet
p1. Find the perimeter of a regular hexagon with apothem
3
3
3
. p2. Concentric circles of radius
1
1
1
and r are drawn on a circular dartboard of radius
5
5
5
. The probability that a randomly thrown dart lands between the two circles is
0.12
0.12
0.12
. Find
r
r
r
. p3. Find all ordered pairs of integers
(
x
,
y
)
(x, y)
(
x
,
y
)
with
0
≤
x
≤
100
0 \le x \le 100
0
≤
x
≤
100
,
0
≤
y
≤
100
0 \le y \le 100
0
≤
y
≤
100
satisfying
x
y
=
(
x
−
22
)
(
y
+
15
)
.
xy = (x - 22) (y + 15) .
x
y
=
(
x
−
22
)
(
y
+
15
)
.
p4. Points
A
1
A_1
A
1
,
A
2
A_2
A
2
,
.
.
.
...
...
,
A
12
A_{12}
A
12
are evenly spaced around a circle of radius
1
1
1
, but not necessarily in order. Given that chords
A
1
A
2
A_1A_2
A
1
A
2
,
A
3
A
4
A_3A_4
A
3
A
4
, and
A
5
A
6
A_5A_6
A
5
A
6
have length
2
2
2
and chords
A
7
A
8
A_7A_8
A
7
A
8
and
A
9
A
10
A_9A_{10}
A
9
A
10
have length
2
s
i
n
(
π
/
12
)
2 sin (\pi / 12)
2
s
in
(
π
/12
)
, find all possible lengths for chord
A
11
A
12
A_{11}A_{12}
A
11
A
12
. p5. Let
a
a
a
be the number of digits of
2
1998
2^{1998}
2
1998
, and let
b
b
b
be the number of digits in
5
1998
5^{1998}
5
1998
. Find
a
+
b
a + b
a
+
b
. p6. Find the volume of the solid in
R
3
R^3
R
3
defined by the equations
x
2
+
y
2
≤
2
x^2 + y^2 \le 2
x
2
+
y
2
≤
2
x
+
y
+
∣
z
∣
≤
3.
x + y + |z| \le 3.
x
+
y
+
∣
z
∣
≤
3.
p7. Positive integer
n
n
n
is such that
3
n
3n
3
n
has
28
28
28
positive divisors and
4
n
4n
4
n
has
36
36
36
positive divisors. Find the number of positive divisors of
n
n
n
. p8. Define functions
f
f
f
and
g
g
g
by
f
(
x
)
=
x
+
x
f (x) = x +\sqrt{x}
f
(
x
)
=
x
+
x
and
g
(
x
)
=
x
+
1
/
4
g (x) = x + 1/4
g
(
x
)
=
x
+
1/4
. Compute
g
(
f
(
g
(
f
(
g
(
f
(
g
(
f
(
3
)
)
)
)
)
)
)
)
.
g(f(g(f(g(f(g(f(3)))))))).
g
(
f
(
g
(
f
(
g
(
f
(
g
(
f
(
3
))))))))
.
(Your answer must be in the form
a
+
b
c
a + b \sqrt{ c}
a
+
b
c
where
a
a
a
,
b
b
b
, and
c
c
c
are rational.) p9. Sequence
(
a
1
,
a
2
,
.
.
.
)
(a_1, a_2,...)
(
a
1
,
a
2
,
...
)
is defined recursively by
a
1
=
0
a_1 = 0
a
1
=
0
,
a
2
=
100
a_2 = 100
a
2
=
100
, and
a
n
=
2
a
n
−
1
−
a
n
−
2
−
3
a_n = 2a_{n-1}-a_{n-2}-3
a
n
=
2
a
n
−
1
−
a
n
−
2
−
3
. Find the greatest term in the sequence
(
a
1
,
a
2
,
.
.
.
)
(a_1, a_2,...)
(
a
1
,
a
2
,
...
)
. p10. Points
X
=
(
3
/
5
,
0
)
X = (3/5, 0)
X
=
(
3/5
,
0
)
and
Y
=
(
0
,
4
/
5
)
Y = (0, 4/5)
Y
=
(
0
,
4/5
)
are located on a Cartesian coordinate system. Consider all line segments which (like
X
Y
‾
\overline{XY}
X
Y
) are of length 1 and have one endpoint on each axis. Find the coordinates of the unique point
P
P
P
on
X
Y
‾
\overline{XY}
X
Y
such that none of these line segments (except
X
Y
‾
\overline{XY}
X
Y
itself) pass through
P
P
P
. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
1998 DMM Tiebreaker Round - Duke Math Meet
p1A Positive reals
x
x
x
,
y
y
y
, and
z
z
z
are such that
x
/
y
+
y
/
x
=
7
x/y +y/x = 7
x
/
y
+
y
/
x
=
7
and
y
/
z
+
z
/
y
=
7
y/z +z/y = 7
y
/
z
+
z
/
y
=
7
. There are two possible values for
z
/
x
+
x
/
z
;
z/x + x/z;
z
/
x
+
x
/
z
;
find the greater value. p1B Real values
x
x
x
and
y
y
y
are such that
x
+
y
=
2
x+y = 2
x
+
y
=
2
and
x
3
+
y
3
=
3
x^3+y^3 = 3
x
3
+
y
3
=
3
. Find
x
2
+
y
2
x^2+y^2
x
2
+
y
2
. p2 Set
A
=
{
5
,
6
,
8
,
13
,
20
,
22
,
33
,
42
}
A = \{5, 6, 8, 13, 20, 22, 33, 42\}
A
=
{
5
,
6
,
8
,
13
,
20
,
22
,
33
,
42
}
. Let
∑
S
\sum S
∑
S
denote the sum of the members of
S
S
S
; then
∑
A
=
149
\sum A = 149
∑
A
=
149
. Find the number of (not necessarily proper) subsets
B
B
B
of
A
A
A
for which
∑
B
≥
75
\sum B \ge 75
∑
B
≥
75
. p3
99
99
99
dots are evenly spaced around a circle. Call two of these dots ”close” if they have
0
0
0
,
1
1
1
, or
2
2
2
dots between them on the circle. We wish to color all
99
99
99
dots so that any two dots which are close are colored differently. How many such colorings are possible using no more than
4
4
4
different colors? p4 Given a
9
×
9
9 \times 9
9
×
9
grid of points, count the number of nondegenerate squares that can be drawn whose vertices are in the grid and whose center is the middle point of the grid. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.