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Contests
National and Regional Contests
Kyrgyzstan Contests
Kyrgyzstan Regional Olympiad
2017 Kyrgyzstan Regional Olympiad
2017 Kyrgyzstan Regional Olympiad
Part of
Kyrgyzstan Regional Olympiad
Subcontests
(4)
4
1
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number theory
Prove that for all n=3,4,5.... there excist odd x,y such
2
n
=
x
2
+
7
y
2
2^n=x^2 + 7y^2
2
n
=
x
2
+
7
y
2
.
3
1
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inequality algebra
If
∣
x
∣
<
1
{|x|}<{1}
∣
x
∣
<
1
and
∣
y
∣
<
1
{|y|}<1
∣
y
∣
<
1
then prove that
∣
x
−
y
1
−
x
y
∣
<
1
|\frac{x-y}{1-xy}|<1
∣
1
−
x
y
x
−
y
∣
<
1
2
1
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equation
x
2
+
2
y
2
=
1
x^2 + 2y^2 = 1
x
2
+
2
y
2
=
1
solve in integers.
1
1
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inequality
a
3
+
b
3
+
3
a
b
c
≥
c
3
a^3 + b^3 + 3abc \ge\ c^3
a
3
+
b
3
+
3
ab
c
≥
c
3
prove that where a,b and c are sides of triangle.