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Bosnia Herzegovina Contests
Bosnia And Herzegovina - Regional Olympiad
2008 Bosnia And Herzegovina - Regional Olympiad
2008 Bosnia And Herzegovina - Regional Olympiad
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Bosnia And Herzegovina - Regional Olympiad
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If (b^{n}-1)(b-1) is perfect square then 8 divides b
Let
b
b
b
be an even positive integer. Assume that there exist integer
n
>
1
n > 1
n
>
1
such that \frac {b^{n} \minus{} 1}{b \minus{} 1} is perfect square. Prove that
b
b
b
is divisible by 8.
Well known p^{4}+q^{4}=r^{4}
Prove that equation p^{4}\plus{}q^{4}\equal{}r^{4} does not have solution in set of prime numbers.
Simple and easy
Find all positive integers
a
a
a
and
b
b
b
such that \frac{a^{4}\plus{}a^{3}\plus{}1}{a^{2}b^{2}\plus{}ab^{2}\plus{}1} is an integer.
2
3
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Nice inequality
For arbitrary reals
x
x
x
,
y
y
y
and
z
z
z
prove the following inequality: x^{2} \plus{} y^{2} \plus{} z^{2} \minus{} xy \minus{} yz \minus{} zx \geq \max \{\frac {3(x \minus{} y)^{2}}{4} , \frac {3(y \minus{} z)^{2}}{4} , \frac {3(y \minus{} z)^{2}}{4} \}
Easy inequality
IF
a
a
a
,
b
b
b
and
c
c
c
are positive reals such that a^{2}\plus{}b^{2}\plus{}c^{2}\equal{}1 prove the inequality: \frac{a^{5}\plus{}b^{5}}{ab(a\plus{}b)}\plus{} \frac {b^{5}\plus{}c^{5}}{bc(b\plus{}c)}\plus{}\frac {c^{5}\plus{}a^{5}}{ca(a\plus{}b)}\geq 3(ab\plus{}bc\plus{}ca)\minus{}2.
Nice and easy
If
a
a
a
,
b
b
b
and
c
c
c
are positive reals prove inequality: \left(1\plus{}\frac{4a}{b\plus{}c}\right)\left(1\plus{}\frac{4b}{a\plus{}c}\right)\left(1\plus{}\frac{4c}{a\plus{}b}\right) > 25.
1
4
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