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2002 Stanford Mathematics Tournament
2002 Stanford Mathematics Tournament
Part of
Stanford Mathematics Tournament
Subcontests
(7)
1
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2002 SMT Team Round - Stanford Math Tournament
p1. Evaluate
i
\sqrt{i}
i
in the form
a
+
b
i
a + bi
a
+
bi
with
a
>
0
a > 0
a
>
0
, where
a
a
a
and
b
b
b
are real numbers and
i
i
i
is
−
1
\sqrt{-1}
−
1
. p2. Let
A
A
A
be the matrix
[
3
2
1
2
0
4
0
6
3
]
\begin{bmatrix} 3 & 2 & 1 \\ 2 & 0 & 4 \\ 0 & 6 & 3 \end{bmatrix}
3
2
0
2
0
6
1
4
3
.What is
d
e
t
(
A
−
1
)
det(A^{-1})
d
e
t
(
A
−
1
)
? p3. How many positive integers divide the number of positive integers that divide
200
2
2002
2002^{2002}
200
2
2002
? p4. In base
6
6
6
, how many
(
2
n
+
1
)
(2n + 1)
(
2
n
+
1
)
-digit numbers are palindromes? p5. In the diagram below, what is the sum of five angles
θ
i
\theta_i
θ
i
numbered,
∑
n
=
i
5
θ
n
\sum_{n=i}^5 \theta_n
∑
n
=
i
5
θ
n
? https://cdn.artofproblemsolving.com/attachments/5/2/21d2302852680db9d2c8b31d3f01c9d0b67b56.png p6. Let
x
x
x
be the smallest number such that
x
x
x
written out in English (i.c.
1
,
647
1,647
1
,
647
is one thousand six hundred forty seven) has exactly
300
300
300
letters. What is the most common digit
(
0
−
9
)
(0-9)
(
0
−
9
)
in
x
x
x
? p7. Define
g
(
x
)
=
∫
x
x
+
1
2
t
d
t
g(x) = \int_{x}^{x+1} 2^t dt
g
(
x
)
=
∫
x
x
+
1
2
t
d
t
, and
g
′
(
x
)
=
d
d
x
g
(
x
)
g'(x) =\frac{d}{dx}g(x)
g
′
(
x
)
=
d
x
d
g
(
x
)
. Compute
g
′
(
10
)
g'(10)
g
′
(
10
)
. p8. If
x
y
=
24
xy = 24
x
y
=
24
with
x
x
x
and
y
y
y
real, what is the minimum value that
x
2
+
4
y
2
x^2 + 4y^2
x
2
+
4
y
2
can attain? p9. Find the cubic polynomial
f
(
x
)
f(x)
f
(
x
)
such that
f
(
1
)
=
1
f(1) = 1
f
(
1
)
=
1
,
f
(
2
)
=
5
f(2) = 5
f
(
2
)
=
5
,
f
(
3
)
=
14
f(3) = 14
f
(
3
)
=
14
, and
f
(
4
)
=
30
f(4) = 30
f
(
4
)
=
30
. p10. What is
1
2
+
2
2
+
3
2
+
.
.
.
+
200
2
2
1^2 + 2^2 + 3^2 + ... + 2002^2
1
2
+
2
2
+
3
2
+
...
+
200
2
2
? p11. The
r
r
r
-th power mean
P
r
P_r
P
r
of
n
n
n
numbers
x
1
,
.
.
.
,
x
n
x_1,...,x_n
x
1
,
...
,
x
n
is defined as
P
r
(
x
1
,
.
.
.
,
x
n
)
=
(
x
1
r
+
.
.
.
+
x
n
r
n
)
1
/
r
P_r(x_1,...,x_n) = \left(\frac{x_1^r+...+x_n^r}{n} \right)^{1/r}
P
r
(
x
1
,
...
,
x
n
)
=
(
n
x
1
r
+
...
+
x
n
r
)
1/
r
for
r
≠
0
r\ne 0
r
=
0
, and
P
0
=
(
x
1
x
2
.
.
.
x
n
)
1
/
n
P_0 = (x_1x_2 ... x_n)^{1/n}
P
0
=
(
x
1
x
2
...
x
n
)
1/
n
The Power Mean Inequality says that if
r
>
s
r > s
r
>
s
, then
P
r
≥
P
s
P_r \ge P_s
P
r
≥
P
s
. Using this fact, find out how many ordered pairs of positive integers
(
x
,
y
)
(x, y)
(
x
,
y
)
satisfy
48
x
y
−
x
2
−
y
2
≥
289
48\sqrt{xy}-x^2 - y^2 \ge 289
48
x
y
−
x
2
−
y
2
≥
289
. p12. After meeting him in the afterlife, Gauss challenges Fermat to a boxing match. Each mathematician is wearing glasses, and Gauss has a
1
/
3
1/3
1/3
probability of knocking off Fermat’s glasses during the match, whereas Fermat has a
1
/
5
1/5
1/5
chance of knocking off Gauss’s glasses. Each mathematician has a
1
/
2
1/2
1/2
chance of losing without his glasses and a
1
/
5
1/5
1/5
chance of losing anyway with his. Note that it is possible for both Fermat and Gauss to lose (simultaneous knockout) or for neither to lose (the match is a draw). Given that Gauss wins the match (and Fermat loses), what is the probability that Gauss has lost his glasses? p13. Evaluate
1
−
1
+
1
−
1
+
1
−
1
+
.
.
.
\frac{1}{-1+\frac{1}{-1+\frac{1}{-1+...}}}
−
1
+
−
1
+
−
1
+
...
1
1
1
p14. What is the smallest positive integer
x
x
x
such that
x
2
+
x
+
41
x^2 + x + 41
x
2
+
x
+
41
is not prime? p15. Let
A
(
t
)
A(t)
A
(
t
)
be an
n
×
n
n \times n
n
×
n
square matrix whose entries are all functions of
t
t
t
, and suppose that
d
e
t
A
(
t
)
≠
0
det A(t)\ne 0
d
e
t
A
(
t
)
=
0
for all
t
t
t
. Then
d
A
d
t
=
A
′
\frac{dA}{dt}= A'
d
t
d
A
=
A
′
is simply the matrix formed by differentiating each entry of
A
(
t
)
A(t)
A
(
t
)
with respect to
t
t
t
. Write
d
d
t
(
A
−
1
(
t
)
)
\frac{d}{dt}(A{-1} (t))
d
t
d
(
A
−
1
(
t
))
in terms of
A
(
t
)
A(t)
A
(
t
)
and
A
′
A'
A
′
, where the only differentiation occurs in
A
′
A'
A
′
itself. PS. You had better use hide for answers .
6
1
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SMT 2002 Algebra #6
How many integers
x
x
x
, from
10
10
10
to
99
99
99
inclusive, have the property that the remainder of
x
2
x^2
x
2
divided by
100
100
100
is equal to the square of the units digit of
x
x
x
?
5
1
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SMT 2002 Algebra #5
Solve for
a
,
b
,
c
a, b, c
a
,
b
,
c
given that
a
≤
b
≤
c
a \le b \le c
a
≤
b
≤
c
, and
a
+
b
+
c
=
−
1
a+b+c=-1
a
+
b
+
c
=
−
1
a
b
+
b
c
+
a
c
=
−
4
ab+bc+ac=-4
ab
+
b
c
+
a
c
=
−
4
a
b
c
=
−
2
abc=-2
ab
c
=
−
2
4
1
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SMT 2002 Algebra #4
Suppose that
n
2
−
2
m
2
=
m
(
n
+
3
)
−
3
n^2-2m^2=m(n+3)-3
n
2
−
2
m
2
=
m
(
n
+
3
)
−
3
. Find all integers
m
m
m
such that all corresponding solutions for
n
n
n
will not be real.
3
2
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SMT 2002 Algebra #3
A clockmaker wants to design a clock such that the area swept by each hand (second, minute, and hour) in one minute is the same (all hands move continuously). What is the length of the hour hand divided by the length of the second hand?
SMT 2002 Geometry #3
An equilateral triangle has has sides
1
1
1
inch long. An ant walks around the triangle maintaining a distance of
1
1
1
inch from the triangle at all times. How far does the ant walk?
2
2
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SMT 2002 Algebra #2
Solve for all real
x
x
x
that satisfy the equation
4
x
=
2
x
+
6
4^x=2^x+6
4
x
=
2
x
+
6
SMT 2002 Geometry #2
Upon cutting a certain rectangle in half, you obtain two rectangles that are scaled down versions of the original. What is the ratio of the longer side length to the shorter side length?
1
1
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SMT 2002 Algebra #1
Completely factor the polynomial
x
4
−
x
3
−
5
x
2
+
3
x
+
6
x^4-x^3-5x^2+3x+6
x
4
−
x
3
−
5
x
2
+
3
x
+
6