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Simon Marais Mathematical Competition
2024 Simon Marais Mathematical Competition
A3
A3
Part of
2024 Simon Marais Mathematical Competition
Problems
(1)
m(a,b) is largest integer <= (1+na)/(1+nb) for all n>=1
Source: SMMC 2024 A3
10/12/2024
Let
W
W
W
be a fixed positive integer. Let
S
S
S
be the set of all pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of positive integers such that
a
≠
b
a \neq b
a
=
b
. For each
(
a
,
b
)
∈
S
(a, b) \in S
(
a
,
b
)
∈
S
, let
m
(
a
,
b
)
m(a,b)
m
(
a
,
b
)
be the largest integer satisfying
m
(
a
,
b
)
≤
1
+
n
a
1
+
n
b
m(a, b) \leq \frac{1 + na}{1 + nb}
m
(
a
,
b
)
≤
1
+
nb
1
+
na
for all integers
n
≥
1
n \geq 1
n
≥
1
.(a) For each
(
a
,
b
)
∈
S
(a, b) \in S
(
a
,
b
)
∈
S
, prove that there exists a positive integer
k
k
k
such that
m
(
a
,
b
)
≤
1
+
n
a
W
+
n
b
m(a,b) \leq \frac{1 + na}{W + nb}
m
(
a
,
b
)
≤
W
+
nb
1
+
na
for all
n
≥
k
n \geq k
n
≥
k
.(b) For each
(
a
,
b
)
∈
S
(a, b) \in S
(
a
,
b
)
∈
S
, let
k
(
a
,
b
)
k(a,b)
k
(
a
,
b
)
be the smallest value of
k
k
k
that satisfies the condition of part (a). Determine
max
{
k
(
a
,
b
)
∣
(
a
,
b
)
∈
S
}
\max \{k(a,b) \mid (a,b) \in S \}
max
{
k
(
a
,
b
)
∣
(
a
,
b
)
∈
S
}
or prove that it does not exist.
algebra