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National and Regional Contests
Romania Contests
Romania - Local Contests
Grigore Moisil Intercounty
2005 Grigore Moisil Urziceni
1
1
Part of
2005 Grigore Moisil Urziceni
Problems
(2)
Easy inequality
Source:
11/28/2019
Prove that
5
x
+
6
x
≤
4
x
+
8
x
,
5^x+6^x\le 4^x+8^x,
5
x
+
6
x
≤
4
x
+
8
x
,
for any nonegative real numbers
x
.
x.
x
.
inequalities
monotony
System of four variables
Source:
11/28/2019
Find the nonnegative real numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
that satisfy the following system:
{
a
3
+
2
a
b
c
+
b
c
d
−
6
=
a
a
2
b
+
b
2
c
+
a
b
d
+
b
d
2
=
b
a
2
b
+
a
2
c
+
b
c
2
+
c
d
2
=
c
d
3
+
a
b
2
+
a
b
c
+
b
c
d
−
6
=
d
\left\{ \begin{matrix} a^3+2abc+bcd-6&=&a \\a^2b+b^2c+abd+bd^2&=&b\\a^2b+a^2c+bc^2+cd^2&=&c\\d^3+ab^2+abc+bcd-6&=&d \end{matrix} \right.
⎩
⎨
⎧
a
3
+
2
ab
c
+
b
c
d
−
6
a
2
b
+
b
2
c
+
ab
d
+
b
d
2
a
2
b
+
a
2
c
+
b
c
2
+
c
d
2
d
3
+
a
b
2
+
ab
c
+
b
c
d
−
6
=
=
=
=
a
b
c
d
equations
system of equations
algebra