MathDB
Problems
Contests
National and Regional Contests
USA Contests
USA - Other Middle and High School Contests
Michigan Mathematics Prize Competition
1976 MMPC
1976 MMPC
Part of
Michigan Mathematics Prize Competition
Subcontests
(1)
1
Hide problems
1976 MMPC , Part 2 = Michigan Mathematics Prize Competition
p1. The total cost of
1
1
1
football,
3
3
3
tennis balls and
7
7
7
golf balls is
$
14
\$14
$14
, while that of
1
1
1
football,
4
4
4
tennis balls and
10
10
10
golf balls is
$
17
\$17
$17
.If one has
$
20
\$20
$20
to spend, is this sufficient to buy a)
3
3
3
footballs and
2
2
2
tennis balls? b)
2
2
2
footballs and
3
3
3
tennis balls? p2. Let
A
B
‾
\overline{AB}
A
B
and
C
D
‾
\overline{CD}
C
D
be two chords in a circle intersecting at a point
P
P
P
(inside the circle). a) Prove that
A
P
⋅
P
B
=
C
P
⋅
P
D
AP \cdot PB = CP\cdot PD
A
P
⋅
PB
=
CP
⋅
P
D
. b) If
A
B
‾
\overline{AB}
A
B
is perpendicular to
C
D
‾
\overline{CD}
C
D
and the length of
A
P
‾
\overline{AP}
A
P
is
2
2
2
, the length of
P
B
‾
\overline{PB}
PB
is
6
6
6
, and the length of
P
D
‾
\overline{PD}
P
D
is
3
3
3
, find the radius of the circle. p3. A polynomial
P
(
x
)
P(x)
P
(
x
)
of degree greater than one has the remainder
2
2
2
when divided by
x
−
2
x-2
x
−
2
and the remainder
3
3
3
when divided by
x
−
3
x-3
x
−
3
. Find the remainder when
P
(
x
)
P(x)
P
(
x
)
is divided by
x
2
−
5
x
+
6
x^2-5x+6
x
2
−
5
x
+
6
. p4. Let
x
1
=
2
x_1= 2
x
1
=
2
and
x
n
+
1
=
x
n
+
(
3
n
+
2
)
x_{n+1}=x_n+ (3n+2)
x
n
+
1
=
x
n
+
(
3
n
+
2
)
for all
n
n
n
greater than or equal to one. a) Find a formula expressing
x
n
x_n
x
n
as a function of
n
n
n
. b) Prove your result. p5. The point
M
M
M
is the midpoint of side
B
C
‾
\overline{BC}
BC
of a triangle
A
B
C
ABC
A
BC
. a) Prove that
A
M
≤
1
2
A
B
+
1
2
A
C
AM \le \frac12 AB + \frac12 AC
A
M
≤
2
1
A
B
+
2
1
A
C
.b) A fly takes off from a certain point and flies a total distance of
4
4
4
meters, returning to the starting point. Explain why the fly never gets outside of some sphere with a radius of one meter. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.