MathDB
Problems
Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad Regional Round
2009 All-Russian Olympiad Regional Round
10.7
x_k^2 - x_k x_{k+1}+x^2_{n+1}=c
x_k^2 - x_k x_{k+1}+x^2_{n+1}=c
Source: All-Russian MO 2009 Regional 10.7
August 27, 2024
algebra
system of equations
Problem Statement
Positive numbers
x
1
,
x
2
,
.
.
.
,
x
2009
x_1, x_2, . . ., x_{2009}
x
1
,
x
2
,
...
,
x
2009
satisfy the equalities
x
1
2
−
x
1
x
2
+
x
2
2
=
x
2
2
−
x
2
x
3
+
x
3
2
=
x
3
2
−
x
3
x
4
+
x
4
2
=
.
.
.
=
x
2008
2
−
x
2008
x
2009
+
x
2009
2
=
x
2009
2
−
x
2009
x
1
+
x
1
2
x^2_1 - x_1x_2 +x^2_2 =x^2_2 -x_2x_3+x^2_3=x^2_3 -x_3x_4+x^2_4= ...= x^2_{2008}- x_{2008}x_{2009}+x^2_{2009}= x^2_{2009}-x_{2009}x_1+x^2_1
x
1
2
−
x
1
x
2
+
x
2
2
=
x
2
2
−
x
2
x
3
+
x
3
2
=
x
3
2
−
x
3
x
4
+
x
4
2
=
...
=
x
2008
2
−
x
2008
x
2009
+
x
2009
2
=
x
2009
2
−
x
2009
x
1
+
x
1
2
. Prove that the numbers
x
1
,
x
2
,
.
.
.
,
x
2009
x_1, x_2, . . ., x_{2009}
x
1
,
x
2
,
...
,
x
2009
are equal.
Back to Problems
View on AoPS