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International Contests
IMO Shortlist
2022 IMO Shortlist
C9
C9
Part of
2022 IMO Shortlist
Problems
(1)
why does C shortlist have 9 problems
Source: 2022 ISL/C9, Proposed by Ankan Bhattacharya
7/9/2023
Let
Z
≥
0
\mathbb Z_{\ge 0}
Z
≥
0
be the set of non-negative integers, and let
f
:
Z
≥
0
×
Z
≥
0
→
Z
≥
0
f:\mathbb Z_{\ge 0}\times \mathbb Z_{\ge 0} \to \mathbb Z_{\ge 0}
f
:
Z
≥
0
×
Z
≥
0
→
Z
≥
0
be a bijection such that whenever
f
(
x
1
,
y
1
)
>
f
(
x
2
,
y
2
)
f(x_1,y_1) > f(x_2, y_2)
f
(
x
1
,
y
1
)
>
f
(
x
2
,
y
2
)
, we have
f
(
x
1
+
1
,
y
1
)
>
f
(
x
2
+
1
,
y
2
)
f(x_1+1, y_1) > f(x_2 + 1, y_2)
f
(
x
1
+
1
,
y
1
)
>
f
(
x
2
+
1
,
y
2
)
and
f
(
x
1
,
y
1
+
1
)
>
f
(
x
2
,
y
2
+
1
)
f(x_1, y_1+1) > f(x_2, y_2+1)
f
(
x
1
,
y
1
+
1
)
>
f
(
x
2
,
y
2
+
1
)
.Let
N
N
N
be the number of pairs of integers
(
x
,
y
)
(x,y)
(
x
,
y
)
with
0
≤
x
,
y
<
100
0\le x,y<100
0
≤
x
,
y
<
100
, such that
f
(
x
,
y
)
f(x,y)
f
(
x
,
y
)
is odd. Find the smallest and largest possible values of
N
N
N
.
combinatorics