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Math Prize For Girls Problems
2011 Math Prize For Girls Problems
2011 Math Prize For Girls Problems
Part of
Math Prize For Girls Problems
Subcontests
(20)
20
1
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Math Prize 2011 Problem 20
Let
A
B
C
ABC
A
BC
be an equilateral triangle with each side of length 1. Let
X
X
X
be a point chosen uniformly at random on side
A
B
‾
\overline{AB}
A
B
. Let
Y
Y
Y
be a point chosen uniformly at random on side
A
C
‾
\overline{AC}
A
C
. (Points
X
X
X
and
Y
Y
Y
are chosen independently.) Let
p
p
p
be the probability that the distance
X
Y
XY
X
Y
is at most
1
3
4
\dfrac{1}{\sqrt[4]{3}}\,
4
3
1
. What is the value of
900
p
900p
900
p
, rounded to the nearest integer?
19
1
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Math Prize 2011 Problem 19
If
−
1
<
x
<
1
-1 < x < 1
−
1
<
x
<
1
and
−
1
<
y
<
1
-1 < y < 1
−
1
<
y
<
1
, define the "relativistic sum''
x
⊕
y
x \oplus y
x
⊕
y
to be
x
⊕
y
=
x
+
y
1
+
x
y
.
x \oplus y = \frac{x + y}{1 + xy} \, .
x
⊕
y
=
1
+
x
y
x
+
y
.
The operation
⊕
\oplus
⊕
is commutative and associative. Let
v
v
v
be the number
v
=
17
7
−
1
17
7
+
1
.
v = \frac{\sqrt[7]{17} - 1}{\sqrt[7]{17} + 1} \, .
v
=
7
17
+
1
7
17
−
1
.
What is the value of
v
⊕
v
⊕
v
⊕
v
⊕
v
⊕
v
⊕
v
⊕
v
⊕
v
⊕
v
⊕
v
⊕
v
⊕
v
⊕
v
?
v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \, ?
v
⊕
v
⊕
v
⊕
v
⊕
v
⊕
v
⊕
v
⊕
v
⊕
v
⊕
v
⊕
v
⊕
v
⊕
v
⊕
v
?
(In this expression,
⊕
\oplus
⊕
appears 13 times.)
18
1
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Math Prize 2011 Problem 18
The polynomial
P
P
P
is a quadratic with integer coefficients. For every positive integer
n
n
n
, the integers
P
(
n
)
P(n)
P
(
n
)
and
P
(
P
(
n
)
)
P(P(n))
P
(
P
(
n
))
are relatively prime to
n
n
n
. If
P
(
3
)
=
89
P(3) = 89
P
(
3
)
=
89
, what is the value of
P
(
10
)
P(10)
P
(
10
)
?
17
1
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Math Prize 2011 Problem 17
There is a polynomial
P
P
P
such that for every real number
x
x
x
,
x
512
+
x
256
+
1
=
(
x
2
+
x
+
1
)
P
(
x
)
.
x^{512} + x^{256} + 1 = (x^2 + x + 1) P(x).
x
512
+
x
256
+
1
=
(
x
2
+
x
+
1
)
P
(
x
)
.
When
P
P
P
is written in standard polynomial form, how many of its coefficients are nonzero?
16
1
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Math Prize 2011 Problem 16
Let
N
N
N
be the number of ordered pairs of integers
(
x
,
y
)
(x, y)
(
x
,
y
)
such that
4
x
2
+
9
y
2
≤
1000000000.
4x^2 + 9y^2 \le 1000000000.
4
x
2
+
9
y
2
≤
1000000000.
Let
a
a
a
be the first digit of
N
N
N
(from the left) and let
b
b
b
be the second digit of
N
N
N
. What is the value of
10
a
+
b
10a + b
10
a
+
b
?
15
1
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Math Prize 2011 Problem 15
The game of backgammon has a "doubling" cube, which is like a standard 6-faced die except that its faces are inscribed with the numbers 2, 4, 8, 16, 32, and 64, respectively. After rolling the doubling cube four times at random, we let
a
a
a
be the value of the first roll,
b
b
b
be the value of the second roll,
c
c
c
be the value of the third roll, and
d
d
d
be the value of the fourth roll. What is the probability that
a
+
b
c
+
d
\frac{a + b}{c + d}
c
+
d
a
+
b
is the average of
a
c
\frac{a}{c}
c
a
and
b
d
\frac{b}{d}
d
b
?
14
1
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Math Prize 2011 Problem 14
If
0
≤
p
≤
1
0 \le p \le 1
0
≤
p
≤
1
and
0
≤
q
≤
1
0 \le q \le 1
0
≤
q
≤
1
, define
F
(
p
,
q
)
F(p, q)
F
(
p
,
q
)
by
F
(
p
,
q
)
=
−
2
p
q
+
3
p
(
1
−
q
)
+
3
(
1
−
p
)
q
−
4
(
1
−
p
)
(
1
−
q
)
.
F(p, q) = -2pq + 3p(1-q) + 3(1-p)q - 4(1-p)(1-q).
F
(
p
,
q
)
=
−
2
pq
+
3
p
(
1
−
q
)
+
3
(
1
−
p
)
q
−
4
(
1
−
p
)
(
1
−
q
)
.
Define
G
(
p
)
G(p)
G
(
p
)
to be the maximum of
F
(
p
,
q
)
F(p, q)
F
(
p
,
q
)
over all
q
q
q
(in the interval
0
≤
q
≤
1
0 \le q \le 1
0
≤
q
≤
1
). What is the value of
p
p
p
(in the interval
0
≤
p
≤
1
0 \le p \le 1
0
≤
p
≤
1
) that minimizes
G
(
p
)
G(p)
G
(
p
)
?
13
1
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Math Prize 2011 Problem 13
The number 104,060,465 is divisible by a five-digit prime number. What is that prime number?
12
1
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Math Prize 2011 Problem 12
If
x
x
x
is a real number, let
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
be the greatest integer that is less than or equal to
x
x
x
. If
n
n
n
is a positive integer, let
S
(
n
)
S(n)
S
(
n
)
be defined by
S
(
n
)
=
⌊
n
1
0
⌊
log
n
⌋
⌋
+
10
(
n
−
1
0
⌊
log
n
⌋
⋅
⌊
n
1
0
⌊
log
n
⌋
⌋
)
.
S(n) = \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor + 10 \left( n - 10^{\lfloor \log n \rfloor} \cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor \right) \, .
S
(
n
)
=
⌊
1
0
⌊
l
o
g
n
⌋
n
⌋
+
10
(
n
−
1
0
⌊
l
o
g
n
⌋
⋅
⌊
1
0
⌊
l
o
g
n
⌋
n
⌋
)
.
(All the logarithms are base 10.) How many integers
n
n
n
from 1 to 2011 (inclusive) satisfy
S
(
S
(
n
)
)
=
n
S(S(n)) = n
S
(
S
(
n
))
=
n
?
11
1
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Math Prize 2011 Problem 11
The sequence
a
0
a_0
a
0
,
a
1
a_1
a
1
,
a
2
a_2
a
2
,
…
\ldots\,
…
satisfies the recurrence equation
a
n
=
2
a
n
−
1
−
2
a
n
−
2
+
a
n
−
3
a_n = 2 a_{n-1} - 2 a_{n - 2} + a_{n - 3}
a
n
=
2
a
n
−
1
−
2
a
n
−
2
+
a
n
−
3
for every integer
n
≥
3
n \ge 3
n
≥
3
. If
a
20
=
1
a_{20} = 1
a
20
=
1
,
a
25
=
10
a_{25} = 10
a
25
=
10
, and
a
30
=
100
a_{30} = 100
a
30
=
100
, what is the value of
a
1331
a_{1331}
a
1331
?
10
1
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Math Prize 2011 Problem 10
There are real numbers
a
a
a
and
b
b
b
such that for every positive number
x
x
x
, we have the identity
tan
−
1
(
1
x
−
x
8
)
+
tan
−
1
(
a
x
)
+
tan
−
1
(
b
x
)
=
π
2
.
\tan^{-1} \bigl( \frac{1}{x} - \frac{x}{8} \bigr) + \tan^{-1}(ax) + \tan^{-1}(bx) = \frac{\pi}{2} \, .
tan
−
1
(
x
1
−
8
x
)
+
tan
−
1
(
a
x
)
+
tan
−
1
(
b
x
)
=
2
π
.
(Throughout this equation,
tan
−
1
\tan^{-1}
tan
−
1
means the inverse tangent function, sometimes written
arctan
\arctan
arctan
.) What is the value of
a
2
+
b
2
a^2 + b^2
a
2
+
b
2
?
9
1
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Math Prize 2011 Problem 9
Let
A
B
C
ABC
A
BC
be a triangle. Let
D
D
D
be the midpoint of
B
C
‾
\overline{BC}
BC
, let
E
E
E
be the midpoint of
A
D
‾
\overline{AD}
A
D
, and let
F
F
F
be the midpoint of
B
E
‾
\overline{BE}
BE
. Let
G
G
G
be the point where the lines
A
B
AB
A
B
and
C
F
CF
CF
intersect. What is the value of
A
G
A
B
\frac{AG}{AB}
A
B
A
G
?
8
1
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Math Prize 2011 Problem 8
In the figure below, points
A
A
A
,
B
B
B
, and
C
C
C
are distance 6 from each other. Say that a point
X
X
X
is reachable if there is a path (not necessarily straight) connecting
A
A
A
and
X
X
X
of length at most 8 that does not intersect the interior of
B
C
‾
\overline{BC}
BC
. (Both
X
X
X
and the path must lie on the plane containing
A
A
A
,
B
B
B
, and
C
C
C
.) Let
R
R
R
be the set of reachable points. What is the area of
R
R
R
? [asy] unitsize(40); pair A = dir(90); pair B = dir(210); pair C = dir(330); dot(A); dot(B); dot(C); draw(B -- C); label("
A
A
A
", A, N); label("
B
B
B
", B, W); label("
C
C
C
", C, E); [/asy]
7
1
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Math Prize 2011 Problem 7
If
z
z
z
is a complex number such that
z
+
z
−
1
=
3
,
z + z^{-1} = \sqrt{3},
z
+
z
−
1
=
3
,
what is the value of
z
2010
+
z
−
2010
?
z^{2010} + z^{-2010} \, ?
z
2010
+
z
−
2010
?
6
1
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Math Prize 2011 Problem 6
Two circles each have radius 1. No point is inside both circles. The circles are contained in a square. What is the area of the smallest such square?
5
1
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Math Prize 2011 Problem 5
Let
△
A
B
C
\triangle ABC
△
A
BC
be a triangle with
A
B
=
3
AB = 3
A
B
=
3
,
B
C
=
4
BC = 4
BC
=
4
, and
A
C
=
5
AC = 5
A
C
=
5
. Let
I
I
I
be the center of the circle inscribed in
△
A
B
C
\triangle ABC
△
A
BC
. What is the product of
A
I
AI
A
I
,
B
I
BI
B
I
, and
C
I
CI
C
I
?
4
1
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Math Prize 2011 Problem 4
If
x
>
10
x > 10
x
>
10
, what is the greatest possible value of the expression
(
log
x
)
log
log
log
x
−
(
log
log
x
)
log
log
x
?
{( \log x )}^{\log \log \log x} - {(\log \log x)}^{\log \log x} ?
(
lo
g
x
)
l
o
g
l
o
g
l
o
g
x
−
(
lo
g
lo
g
x
)
l
o
g
l
o
g
x
?
All the logarithms are base 10.
3
1
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Math Prize 2011 Problem 3
The figure below shows a triangle
A
B
C
ABC
A
BC
with a semicircle on each of its three sides. [asy] unitsize(5); pair A = (0, 20 * 21) / 29.0; pair B = (-20^2, 0) / 29.0; pair C = (21^2, 0) / 29.0; draw(A -- B -- C -- cycle); label("
A
A
A
", A, S); label("
B
B
B
", B, S); label("
C
C
C
", C, S); filldraw(arc((A + C)/2, C, A)--cycle, gray); filldraw(arc((B + C)/2, C, A)--cycle, white); filldraw(arc((A + B)/2, A, B)--cycle, gray); filldraw(arc((B + C)/2, A, B)--cycle, white); [/asy] If
A
B
=
20
AB = 20
A
B
=
20
,
A
C
=
21
AC = 21
A
C
=
21
, and
B
C
=
29
BC = 29
BC
=
29
, what is the area of the shaded region?
2
1
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Math Prize 2011 Problem 2
Express
2
+
3
\sqrt{2 + \sqrt{3}}
2
+
3
in the form
a
+
b
c
\frac{a + \sqrt{b}}{\sqrt{c}}
c
a
+
b
, where
a
a
a
is a positive integer and
b
b
b
and
c
c
c
are square-free positive integers.
1
1
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Math Prize 2011 Problem 1
If
m
m
m
and
n
n
n
are integers such that
3
m
+
4
n
=
100
3m + 4n = 100
3
m
+
4
n
=
100
, what is the smallest possible value of
∣
m
−
n
∣
\left| m - n \right|
∣
m
−
n
∣
?