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Michigan Mathematics Prize Competition
1970 MMPC
1970 MMPC
Part of
Michigan Mathematics Prize Competition
Subcontests
(1)
1
Hide problems
1970 MMPC , Part 2 = Michigan Mathematics Prize Competition
p1. Show that the
n
×
n
n \times n
n
×
n
determinant
∣
1
+
x
1
1
.
.
.
1
1
1
+
x
1
.
.
.
1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
1
.
.
.
.
1
+
x
∣
\begin{vmatrix} 1+x & 1 & 1 & . & . & . & 1 \\ 1 & 1+x & 1 & . & . & . & 1 \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ 1 & 1 & . & . & . & . & 1+x \\ \end{vmatrix}
1
+
x
1
.
.
1
1
1
+
x
.
.
1
1
1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
1
.
.
1
+
x
has the value zero when
x
=
−
n
x = -n
x
=
−
n
p2. Let
c
>
a
≥
b
c > a \ge b
c
>
a
≥
b
be the lengths of the sides of an obtuse triangle. Prove that
c
n
=
a
n
+
b
n
c^n = a^n + b^n
c
n
=
a
n
+
b
n
for no positive integer
n
n
n
. p3. Suppose that
p
1
=
p
2
2
+
p
3
2
+
p
4
2
p_1 = p_2^2+ p_3^2 + p_4^2
p
1
=
p
2
2
+
p
3
2
+
p
4
2
, where
p
1
p_1
p
1
,
p
2
p_2
p
2
,
p
3
p_3
p
3
, and
p
4
p_4
p
4
are primes. Prove that at least one of
p
2
p_2
p
2
,
p
3
p_3
p
3
,
p
4
p_4
p
4
is equal to
3
3
3
. p4. Suppose
X
X
X
and
Y
Y
Y
are points on tJhe boundary of the triangular region
A
B
C
ABC
A
BC
such that the segment
X
Y
XY
X
Y
divides the region into two parts of equal area. If
X
Y
XY
X
Y
is the shortest such segment and
A
B
=
5
AB = 5
A
B
=
5
,
B
C
=
4
BC = 4
BC
=
4
,
A
C
=
3
AC = 3
A
C
=
3
calculate the length of
X
Y
XY
X
Y
.Hint: Of all triangles having the same area and same vertex angle the one with the shortest base is isosceles. Clearly justify all claims. p5. Find all solutions of the following system of simultaneous equations
x
+
y
+
z
=
7
,
x
2
+
y
2
+
z
2
=
31
,
x
3
+
y
3
+
z
3
=
154
x + y + z = 7\,\, , \,\, x^2 + y^2 + z^2 = 31\,\,, \,\,x^3 + y^3 + z^3 = 154
x
+
y
+
z
=
7
,
x
2
+
y
2
+
z
2
=
31
,
x
3
+
y
3
+
z
3
=
154
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.