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Eotvos Mathematical Competition (Hungary)
1908 Eotvos Mathematical Competition
1908 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
3
1
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difference in the sides of 2 regular decagons is equal to the circumradius
A regular polygon of 10 sides (a regular decagon) may be inscribed in a circle in the following two distinct ways: Divide the circumference into
10
10
10
equal arcs and (1) join each division point to the next by straight line segments, (2) join each division point to the next but two by straight line segments. (See figures). Prove that the difference in the side lengths of these two decagons is equal to the radius of their circumscribed circle. https://cdn.artofproblemsolving.com/attachments/7/9/41c38d08f4f89e07852942a493df17eaaf7498.png
2
1
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a^n > b^n + c^n if a^2=b^2+c^2
Let
n
n
n
be an integer greater than
2
2
2
. Prove that the
n
n
n
th power of the length of the hypotenuse of a right triangle is greater than the sum of the
n
n
n
th powers of the lengths of the legs.
1
1
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2^n | a^3 -b^3 iff 2^n | a-b
Given two odd integers
a
a
a
and
b
b
b
; prove that
a
3
−
b
3
a^3 -b^3
a
3
−
b
3
is divisible by
2
n
2^n
2
n
if and only if
a
−
b
a-b
a
−
b
is divisible by
2
n
2^n
2
n
.