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1979 Romania Team Selection Tests
3.
Inequality in system of equations
Inequality in system of equations
Source: Romanian TST 1979 day 2 P3
April 15, 2023
inequalities
simultaneous equation
algebra
system of equations
Problem Statement
Let
a
,
b
,
c
∈
R
a,b,c\in \mathbb{R}
a
,
b
,
c
∈
R
with
a
2
+
b
2
+
c
2
=
1
a^2+b^2+c^2=1
a
2
+
b
2
+
c
2
=
1
and
λ
∈
R
>
0
∖
{
1
}
\lambda\in \mathbb{R}_{>0}\setminus\{1\}
λ
∈
R
>
0
∖
{
1
}
. Then for each solution
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of the system of equations:
{
x
−
λ
y
=
a
,
y
−
λ
z
=
b
,
z
−
λ
x
=
c
.
\begin{cases} x-\lambda y=a,\\ y-\lambda z=b,\\ z-\lambda x=c. \end{cases}
⎩
⎨
⎧
x
−
λ
y
=
a
,
y
−
λ
z
=
b
,
z
−
λ
x
=
c
.
we have
x
2
+
y
2
+
z
2
⩽
1
(
λ
−
1
)
2
\displaystyle x^2+y^2+z^2\leqslant \frac1{(\lambda-1)^2}
x
2
+
y
2
+
z
2
⩽
(
λ
−
1
)
2
1
.Radu Gologan
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