MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2024 India National Olympiad
2024 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
5
1
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Maximal radius of rotated triangle
Let points
A
1
A_1
A
1
,
A
2
A_2
A
2
and
A
3
A_3
A
3
lie on the circle
Γ
\Gamma
Γ
in a counter-clockwise order, and let
P
P
P
be a point in the same plane. For
i
∈
{
1
,
2
,
3
}
i \in \{1,2,3\}
i
∈
{
1
,
2
,
3
}
, let
τ
i
\tau_i
τ
i
denote the counter-clockwise rotation of the plane centred at
A
i
A_i
A
i
, where the angle of rotation is equial to the angle at vertex
A
i
A_i
A
i
in
△
A
1
A
2
A
3
\triangle A_1A_2A_3
△
A
1
A
2
A
3
. Further, define
P
i
P_i
P
i
to be the point
τ
i
+
2
(
τ
i
(
τ
i
+
1
(
P
)
)
)
\tau_{i+2}(\tau_{i}(\tau_{i+1}(P)))
τ
i
+
2
(
τ
i
(
τ
i
+
1
(
P
)))
, where the indices are taken modulo
3
3
3
(i.e.,
τ
4
=
τ
1
\tau_4 = \tau_1
τ
4
=
τ
1
and
τ
5
=
τ
2
\tau_5 = \tau_2
τ
5
=
τ
2
).Prove that the radius of the circumcircle of
△
P
1
P
2
P
3
\triangle P_1P_2P_3
△
P
1
P
2
P
3
is at most the radius of
Γ
\Gamma
Γ
. Proposed by Anant Mudgal
4
1
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Weird Function
A finite set
S
\mathcal{S}
S
of positive integers is called cardinal if
S
\mathcal{S}
S
contains the integer
∣
S
∣
|\mathcal{S}|
∣
S
∣
where
∣
S
∣
|\mathcal{S}|
∣
S
∣
denotes the number of distinct elements in
S
\mathcal{S}
S
. Let
f
f
f
be a function from the set of positive integers to itself such that for any cardinal set
S
\mathcal{S}
S
, the set
f
(
S
)
f(\mathcal{S})
f
(
S
)
is also cardinal. Here
f
(
S
)
f(\mathcal{S})
f
(
S
)
denotes the set of all integers that can be expressed as
f
(
a
)
f(a)
f
(
a
)
where
a
∈
S
a \in \mathcal{S}
a
∈
S
. Find all possible values of
f
(
2024
)
f(2024)
f
(
2024
)
Proposed by Sutanay Bhattacharya
6
1
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floor and square roots
For each positive integer
n
≥
3
n \ge 3
n
≥
3
, define
A
n
A_n
A
n
and
B
n
B_n
B
n
as
A
n
=
n
2
+
1
+
n
2
+
3
+
⋯
+
n
2
+
2
n
−
1
A_n = \sqrt{n^2 + 1} + \sqrt{n^2 + 3} + \cdots + \sqrt{n^2+2n-1}
A
n
=
n
2
+
1
+
n
2
+
3
+
⋯
+
n
2
+
2
n
−
1
B
n
=
n
2
+
2
+
n
2
+
4
+
⋯
+
n
2
+
2
n
.
B_n = \sqrt{n^2 + 2} + \sqrt{n^2 + 4} + \cdots + \sqrt{n^2 + 2n}.
B
n
=
n
2
+
2
+
n
2
+
4
+
⋯
+
n
2
+
2
n
.
Determine all positive integers
n
≥
3
n\ge 3
n
≥
3
for which
⌊
A
n
⌋
=
⌊
B
n
⌋
\lfloor A_n \rfloor = \lfloor B_n \rfloor
⌊
A
n
⌋
=
⌊
B
n
⌋
. Note. For any real number
x
x
x
,
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
denotes the largest integer
N
≤
x
N\le x
N
≤
x
.Anant Mudgal and Navilarekallu Tejaswi
2
1
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Red Alert!
All the squares of a
2024
×
2024
2024 \times 2024
2024
×
2024
board are coloured white. In one move, Mohit can select one row or column whose every square is white, choose exactly
1000
1000
1000
squares in that row or column, and colour all of them red. Find maximum number of squares Mohit can colour in a finite number of moves. Proposed by Pranjal Srivastava
3
1
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Easy Number Theory
Let
p
p
p
be an odd prime and
a
,
b
,
c
a,b,c
a
,
b
,
c
be integers so that the integers a^{2023}+b^{2023}, b^{2024}+c^{2024}, a^{2025}+c^{2025} are divisible by
p
p
p
. Prove that
p
p
p
divides each of
a
,
b
,
c
a,b,c
a
,
b
,
c
. Proposed by Navilarekallu Tejaswi
1
1
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Center lies on circumcircle of other
In triangle
A
B
C
ABC
A
BC
with
C
A
=
C
B
CA=CB
C
A
=
CB
, point
E
E
E
lies on the circumcircle of
A
B
C
ABC
A
BC
such that
∠
E
C
B
=
9
0
∘
\angle ECB=90^{\circ}
∠
ECB
=
9
0
∘
. The line through
E
E
E
parallel to
C
B
CB
CB
intersects
C
A
CA
C
A
in
F
F
F
and
A
B
AB
A
B
in
G
G
G
. Prove that the center of the circumcircle of triangle
E
G
B
EGB
EGB
lies on the circumcircle of triangle
E
C
F
ECF
ECF
.Proposed by Prithwijit De