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National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
2018 Ukraine Team Selection Test
12
12
Part of
2018 Ukraine Team Selection Test
Problems
(1)
Sum of f on a template is zero
Source: Ukraine TST for IMO 2018 P12
3/7/2019
Let
n
n
n
be a positive integer and
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots,a_n
a
1
,
a
2
,
…
,
a
n
be integers. Function
f
:
Z
→
R
f: \mathbb{Z} \rightarrow \mathbb{R}
f
:
Z
→
R
is such that for all integers
k
k
k
and
l
l
l
,
l
≠
0
l \neq 0
l
=
0
,
∑
i
=
1
n
f
(
k
+
a
i
l
)
=
0.
\sum_{i=1}^n f(k+a_il)=0.
i
=
1
∑
n
f
(
k
+
a
i
l
)
=
0.
Prove that
f
≡
0
f \equiv 0
f
≡
0
.
algebra
function
combinatorics