MathDB
Problems
Contests
National and Regional Contests
Malaysia Contests
Malaysian IMO Training Camp
BIMO 2020
BIMO 2020
Part of
Malaysian IMO Training Camp
Subcontests
(2)
1
1
Hide problems
f(x^2+f(x+y))=y+xf(x+1)
Find all functions
f
:
R
→
R
f : \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
such that for all reals
x
,
y
x, y
x
,
y
,
f
(
x
2
+
f
(
x
+
y
)
)
=
y
+
x
f
(
x
+
1
)
f(x^2+f(x+y))=y+xf(x+1)
f
(
x
2
+
f
(
x
+
y
))
=
y
+
x
f
(
x
+
1
)
2
1
Hide problems
Partitioning $\mathbb{N}$
Let
a
1
,
a
2
,
⋯
a_1,a_2,\cdots
a
1
,
a
2
,
⋯
be a strictly increasing sequence on positive integers.Is it always possible to partition the set of natural numbers
N
\mathbb{N}
N
into infinitely many subsets with infinite cardinality
A
1
,
A
2
,
⋯
A_1,A_2,\cdots
A
1
,
A
2
,
⋯
, so that for every subset
A
i
A_i
A
i
, if we denote
b
1
<
b
2
<
⋯
b_1<b_2<\cdots
b
1
<
b
2
<
⋯
be the elements of
A
i
A_i
A
i
, then for every
k
∈
N
k\in \mathbb{N}
k
∈
N
and for every
1
≤
i
≤
a
k
1\le i\le a_k
1
≤
i
≤
a
k
, it satisfies
b
i
+
1
−
b
i
≤
k
b_{i+1}-b_{i}\le k
b
i
+
1
−
b
i
≤
k
?