MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2022 India National Olympiad
2022 India National Olympiad
Part of
India National Olympiad
Subcontests
(3)
3
1
Hide problems
INMO 2022
For a positive integer
N
N
N
, let
T
(
N
)
T(N)
T
(
N
)
denote the number of arrangements of the integers
1
,
2
,
⋯
N
1, 2, \cdots N
1
,
2
,
⋯
N
into a sequence
a
1
,
a
2
,
⋯
a
N
a_1, a_2, \cdots a_N
a
1
,
a
2
,
⋯
a
N
such that
a
i
>
a
2
i
a_i > a_{2i}
a
i
>
a
2
i
for all
i
i
i
,
1
≤
i
<
2
i
≤
N
1 \le i < 2i \le N
1
≤
i
<
2
i
≤
N
and
a
i
>
a
2
i
+
1
a_i > a_{2i+1}
a
i
>
a
2
i
+
1
for all
i
i
i
,
1
≤
i
<
2
i
+
1
≤
N
1 \le i < 2i+1 \le N
1
≤
i
<
2
i
+
1
≤
N
. For example,
T
(
3
)
T(3)
T
(
3
)
is
2
2
2
, since the possible arrangements are
321
321
321
and
312
312
312
(a) Find
T
(
7
)
T(7)
T
(
7
)
(b) If
K
K
K
is the largest non-negative integer so that
2
K
2^K
2
K
divides
T
(
2
n
−
1
)
T(2^n - 1)
T
(
2
n
−
1
)
, show that
K
=
2
n
−
n
−
1
K = 2^n - n - 1
K
=
2
n
−
n
−
1
. (c) Find the largest non-negative integer
K
K
K
so that
2
K
2^K
2
K
divides
T
(
2
n
+
1
)
T(2^n + 1)
T
(
2
n
+
1
)
1
1
Hide problems
INMO 2022
Let
D
D
D
be an interior point on the side
B
C
BC
BC
of an acute-angled triangle
A
B
C
ABC
A
BC
. Let the circumcircle of triangle
A
D
B
ADB
A
D
B
intersect
A
C
AC
A
C
again at
E
(
≠
A
)
E(\ne A)
E
(
=
A
)
and the circumcircle of triangle
A
D
C
ADC
A
D
C
intersect
A
B
AB
A
B
again at
F
(
≠
A
)
F(\ne A)
F
(
=
A
)
. Let
A
D
AD
A
D
,
B
E
BE
BE
, and
C
F
CF
CF
intersect the circumcircle of triangle
A
B
C
ABC
A
BC
again at
D
1
(
≠
A
)
D_1(\ne A)
D
1
(
=
A
)
,
E
1
(
≠
B
)
E_1(\ne B)
E
1
(
=
B
)
and
F
1
(
≠
C
)
F_1(\ne C)
F
1
(
=
C
)
, respectively. Let
I
I
I
and
I
1
I_1
I
1
be the incentres of triangles
D
E
F
DEF
D
EF
and
D
1
E
1
F
1
D_1E_1F_1
D
1
E
1
F
1
, respectively. Prove that
E
,
F
,
I
,
I
1
E,F, I, I_1
E
,
F
,
I
,
I
1
are concyclic.
2
1
Hide problems
Permutation times exponential zero
Find all natural numbers
n
n
n
for which there is a permutation
σ
\sigma
σ
of
{
1
,
2
,
…
,
n
}
\{1,2,\ldots, n\}
{
1
,
2
,
…
,
n
}
that satisfies:
∑
i
=
1
n
σ
(
i
)
(
−
2
)
i
−
1
=
0
\sum_{i=1}^n \sigma(i)(-2)^{i-1}=0
i
=
1
∑
n
σ
(
i
)
(
−
2
)
i
−
1
=
0