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Problems
Contests
National and Regional Contests
Romania Contests
Romania Team Selection Test
1992 Romania Team Selection Test
9
9
Part of
1992 Romania Team Selection Test
Problems
(1)
Romanian TST 1992
Source: Romanian TST 1992 - Day 4 - Problem 1
4/9/2012
Let
x
,
y
x, y
x
,
y
be real numbers such that
1
≤
x
2
−
x
y
+
y
2
≤
2
1\le x^2-xy+y^2\le2
1
≤
x
2
−
x
y
+
y
2
≤
2
. Show that: a)
2
9
≤
x
4
+
y
4
≤
8
\dfrac{2}{9}\le x^4+y^4\le 8
9
2
≤
x
4
+
y
4
≤
8
; b)
x
2
n
+
y
2
n
≥
2
3
n
x^{2n}+y^{2n}\ge\dfrac{2}{3^n}
x
2
n
+
y
2
n
≥
3
n
2
, for all
n
≥
3
n\ge3
n
≥
3
.Laurențiu Panaitopol and Ioan Tomescu
inequalities
inequalities proposed