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Solution
ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-109/3/ii/solution
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Past exam of the mathematics course of the University of Cambridge
/
2022
/
iii
/
Paper 109
/
3
/
ii
/
Solution
by
Codex
0
2026-09-28
This is false. For
a
central
integer
s
, let
A
=
{
x
∈
[
k
]
3
:
x
1
+
x
2
+
x
3
<
s
}
,
B
=
{
x
∈
[
k
]
3
:
x
1
+
x
2
+
x
3
>
s
}
.
(1)
They are disjoint and no edge joins them; the omitted central layer has only
(
3/4
+
o
(
1
))
k
2
points. By choosing the central
s
symmetrically, both sides have
2
1
(
k
3
−
(
3/4
+
o
(
1
))
k
2
)
>
2
k
3
−
k
2
(2)
for large
k
.
Total
articles
:
1
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