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1984 MMPC
1984 MMPC
Part of
Michigan Mathematics Prize Competition
Subcontests
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1
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1984 MMPC , Part 2 = Michigan Mathematics Prize Competition
p1. For what integers
n
n
n
is
2
6
+
2
9
+
2
n
2^6 + 2^9 + 2^n
2
6
+
2
9
+
2
n
the square of an integer? p2. Two integers are chosen at random (independently, with repetition allowed) from the set
{
1
,
2
,
3
,
.
.
.
,
N
}
\{1,2,3,...,N\}
{
1
,
2
,
3
,
...
,
N
}
. Show that the probability that the sum of the two integers is even is not less than the probability that the sum is odd. p3. Let
X
X
X
be a point in the second quadrant of the plane and let
Y
Y
Y
be a point in the first quadrant. Locate the point
M
M
M
on the
x
x
x
-axis such that the angle
X
M
XM
XM
makes with the negative end of the
x
x
x
-axis is twice the angle
Y
M
YM
Y
M
makes with the positive end of the
x
x
x
-axis. p4. Let
a
,
b
a,b
a
,
b
be positive integers such that
a
≥
b
3
a \ge b \sqrt3
a
≥
b
3
. Let
α
n
=
(
a
+
b
3
)
n
=
a
n
+
b
n
3
\alpha^n = (a + b\sqrt3)^n = a_n + b_n\sqrt3
α
n
=
(
a
+
b
3
)
n
=
a
n
+
b
n
3
for
n
=
1
,
2
,
3
,
.
.
.
n = 1,2,3,...
n
=
1
,
2
,
3
,
...
. i. Prove that
lim
n
→
+
∞
a
n
b
n
\lim_{n \to + \infty} \frac{a_n}{b_n}
lim
n
→
+
∞
b
n
a
n
exists. ii. Evaluate this limit. p5. Suppose
m
m
m
and
n
n
n
are the hypotenuses are of Pythagorean triangles, i.e,, there are positive integers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
, so that
m
2
=
a
2
+
b
2
m^2 = a^2 + b^2
m
2
=
a
2
+
b
2
and
n
2
=
c
2
+
d
2
n^2= c^2 + d^2
n
2
=
c
2
+
d
2
. Show than
m
n
mn
mn
is the hypotenuse of at least two distinct Pythagorean triangles.Hint: you may not assume that the pair
(
a
,
b
)
(a,b)
(
a
,
b
)
is different from the pair
(
c
,
d
)
(c,d)
(
c
,
d
)
. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.