MathDB
Problems
Contests
International Contests
Czech-Polish-Slovak Match
2023 Czech-Polish-Slovak Match
2
n-variable inequality
n-variable inequality
Source: CAPS Match 2023 P2
June 30, 2023
inequalities
Problem Statement
Let
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
be reals such that for all
k
=
1
,
2
,
…
,
n
k=1,2, \ldots, n
k
=
1
,
2
,
…
,
n
,
n
a
k
≥
a
1
2
+
a
2
2
+
…
+
a
k
2
na_k \geq a_1^2+a_2^2+ \ldots+a_k^2
n
a
k
≥
a
1
2
+
a
2
2
+
…
+
a
k
2
. Prove that there exist at least
n
10
\frac{n} {10}
10
n
indices
k
k
k
, such that
a
k
≤
1000
a_k \leq 1000
a
k
≤
1000
.
Back to Problems
View on AoPS