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Michigan Mathematics Prize Competition
1961 MMPC
1961 MMPC
Part of
Michigan Mathematics Prize Competition
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1
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1961 MMPC , Part 2 = Michigan Mathematics Prize Competition
p1.
x
,
y
,
z
x,y,z
x
,
y
,
z
are required to be non-negative whole numbers, find all solutions to the pair of equations
x
+
y
+
z
=
40
x+y+z=40
x
+
y
+
z
=
40
2
x
+
4
y
+
17
z
=
301.
2x + 4y + 17z = 301.
2
x
+
4
y
+
17
z
=
301.
p2. Let
P
P
P
be a point lying between the sides of an acute angle whose vertex is
O
O
O
. Let
A
,
B
A,B
A
,
B
be the intersections of a line passing through
P
P
P
with the sides of the angle. Prove that the triangle
A
O
B
AOB
A
OB
has minimum area when
P
P
P
bisects the line segment
A
B
AB
A
B
. p3. Find all values of
x
x
x
for which
∣
3
x
−
2
∣
+
∣
3
x
+
1
∣
=
3
|3x-2|+|3x+1|=3
∣3
x
−
2∣
+
∣3
x
+
1∣
=
3
. p4. Prove that
x
2
+
y
2
+
z
2
x^2+y^2+z^2
x
2
+
y
2
+
z
2
cannot be factored in the form
(
a
x
+
b
y
+
c
z
)
(
d
x
+
e
y
+
f
z
)
,
(ax + by + cz) (dx + ey + fz),
(
a
x
+
b
y
+
cz
)
(
d
x
+
ey
+
f
z
)
,
a
,
b
,
c
,
d
,
e
,
f
a, b, c, d, e, f
a
,
b
,
c
,
d
,
e
,
f
real. p5. Let
f
(
x
)
f(x)
f
(
x
)
be a continuous function for all real values of
x
x
x
such that
f
(
a
)
≤
f
(
b
)
f(a)\le f(b)
f
(
a
)
≤
f
(
b
)
whenever
a
≤
b
a\le b
a
≤
b
. Prove that, for every real number
r
r
r
, the equation
x
+
f
(
x
)
=
r
x + f(x) = r
x
+
f
(
x
)
=
r
has exactly one solution. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.