MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2018 India National Olympiad
2018 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
3
1
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INMO 2018 -- Problem #3
Let
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
be two circles with respective centres
O
1
O_1
O
1
and
O
2
O_2
O
2
intersecting in two distinct points
A
A
A
and
B
B
B
such that
∠
O
1
A
O
2
\angle{O_1AO_2}
∠
O
1
A
O
2
is an obtuse angle. Let the circumcircle of
Δ
O
1
A
O
2
\Delta{O_1AO_2}
Δ
O
1
A
O
2
intersect
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
respectively in points
C
(
≠
A
)
C (\neq A)
C
(
=
A
)
and
D
(
≠
A
)
D (\neq A)
D
(
=
A
)
. Let the line
C
B
CB
CB
intersect
Γ
2
\Gamma_2
Γ
2
in
E
E
E
; let the line
D
B
DB
D
B
intersect
Γ
1
\Gamma_1
Γ
1
in
F
F
F
. Prove that, the points
C
,
D
,
E
,
F
C, D, E, F
C
,
D
,
E
,
F
are concyclic.
6
1
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INMO 2018 -- Problem #6
Let
N
\mathbb N
N
denote set of all natural numbers and let
f
:
N
→
N
f:\mathbb{N}\to\mathbb{N}
f
:
N
→
N
be a function such that
(a)
f
(
m
n
)
=
f
(
m
)
.
f
(
n
)
\text{(a)} f(mn)=f(m).f(n)
(a)
f
(
mn
)
=
f
(
m
)
.
f
(
n
)
for all
m
,
n
∈
N
m,n \in\mathbb{N}
m
,
n
∈
N
;
(b)
m
+
n
\text{(b)} m+n
(b)
m
+
n
divides
f
(
m
)
+
f
(
n
)
f(m)+f(n)
f
(
m
)
+
f
(
n
)
for all
m
,
n
∈
N
m,n\in \mathbb N
m
,
n
∈
N
.Prove that, there exists an odd natural number
k
k
k
such that
f
(
n
)
=
n
k
f(n)= n^k
f
(
n
)
=
n
k
for all
n
n
n
in
N
\mathbb{N}
N
.
2
1
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INMO 2018 -- Problem #2
For any natural number
n
n
n
, consider a
1
×
n
1\times n
1
×
n
rectangular board made up of
n
n
n
unit squares. This is covered by
3
3
3
types of tiles :
1
×
1
1\times 1
1
×
1
red tile,
1
×
1
1\times 1
1
×
1
green tile and
1
×
2
1\times 2
1
×
2
domino. (For example, we can have
5
5
5
types of tiling when
n
=
2
n=2
n
=
2
: red-red ; red-green ; green-red ; green-green ; and blue.) Let
t
n
t_n
t
n
denote the number of ways of covering
1
×
n
1\times n
1
×
n
rectangular board by these
3
3
3
types of tiles. Prove that,
t
n
t_n
t
n
divides
t
2
n
+
1
t_{2n+1}
t
2
n
+
1
.
5
1
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INMO 2018 -- Problem #5
There are
n
≥
3
n\ge 3
n
≥
3
girls in a class sitting around a circular table, each having some apples with her. Every time the teacher notices a girl having more apples than both of her neighbours combined, the teacher takes away one apple from that girl and gives one apple each to her neighbours. Prove that, this process stops after a finite number of steps. (Assume that, the teacher has an abundant supply of apples.)
4
1
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INMO 2018 -- Problem #4
Find all polynomials with real coefficients
P
(
x
)
P(x)
P
(
x
)
such that
P
(
x
2
+
x
+
1
)
P(x^2+x+1)
P
(
x
2
+
x
+
1
)
divides
P
(
x
3
−
1
)
P(x^3-1)
P
(
x
3
−
1
)
.
1
1
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INMO 2018 -- Problem #1
Let
A
B
C
ABC
A
BC
be a non-equilateral triangle with integer sides. Let
D
D
D
and
E
E
E
be respectively the mid-points of
B
C
BC
BC
and
C
A
CA
C
A
; let
G
G
G
be the centroid of
Δ
A
B
C
\Delta{ABC}
Δ
A
BC
. Suppose,
D
D
D
,
C
C
C
,
E
E
E
,
G
G
G
are concyclic. Find the least possible perimeter of
Δ
A
B
C
\Delta{ABC}
Δ
A
BC
.