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Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
2023 India Regional Mathematical Olympiad
2023 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(6)
6
2
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Ahh... Grids!
Consider a set of
16
16
16
points arranged in
4
×
4
4 \times 4
4
×
4
square grid formation. Prove that if any
7
7
7
of these points are coloured blue, then there exists an isosceles right-angled triangle whose vertices are all blue.
RMO KV P-6
The diagonals
A
C
AC
A
C
and
B
D
BD
B
D
of a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
meet at
P
P
P
. The point
Q
Q
Q
is chosen on the segment
B
C
BC
BC
so that
P
Q
PQ
PQ
is perpendicular to
A
C
AC
A
C
. Prove that the line joining the centres of the circumcircles of triangles
A
P
D
APD
A
P
D
and
B
Q
D
BQD
BQ
D
is parallel to
A
D
AD
A
D
.
5
2
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RMO 2023 P-5
Let
n
>
k
>
1
n>k>1
n
>
k
>
1
be positive integers. Determine all positive real numbers
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
which satisfy
∑
i
=
1
n
k
a
i
k
(
k
−
1
)
a
i
k
+
1
=
∑
i
=
1
n
a
i
=
n
.
\sum_{i=1}^n \sqrt{\frac{k a_i^k}{(k-1) a_i^k+1}}=\sum_{i=1}^n a_i=n .
i
=
1
∑
n
(
k
−
1
)
a
i
k
+
1
k
a
i
k
=
i
=
1
∑
n
a
i
=
n
.
RMO KV P-5
The side lengths
a
,
b
,
c
a,b,c
a
,
b
,
c
of a triangle
A
B
C
ABC
A
BC
are positive integers. Let:\\
T
n
=
(
a
+
b
+
c
)
2
n
−
(
a
−
b
+
c
)
2
n
−
(
a
+
b
−
c
)
2
n
+
(
a
−
b
−
c
)
2
n
T_{n}=(a+b+c)^{2n}-(a-b+c)^{2n}-(a+b-c)^{2n}+(a-b-c)^{2n}
T
n
=
(
a
+
b
+
c
)
2
n
−
(
a
−
b
+
c
)
2
n
−
(
a
+
b
−
c
)
2
n
+
(
a
−
b
−
c
)
2
n
for any positive integer
n
n
n
. If
T
2
2
T
1
=
2023
\frac{T_{2}}{2T_{1}}=2023
2
T
1
T
2
=
2023
and
a
>
b
>
c
a>b>c
a
>
b
>
c
, determine all possible perimeters of the triangle
A
B
C
ABC
A
BC
.
4
2
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RMO 2023 P4
Let
Ω
1
,
Ω
2
\Omega_1, \Omega_2
Ω
1
,
Ω
2
be two intersecting circles with centres
O
1
,
O
2
O_1, O_2
O
1
,
O
2
respectively. Let
l
l
l
be a line that intersects
Ω
1
\Omega_1
Ω
1
at points
A
,
C
A, C
A
,
C
and
Ω
2
\Omega_2
Ω
2
at points
B
,
D
B, D
B
,
D
such that
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
are collinear in that order. Let the perpendicular bisector of segment
A
B
A B
A
B
intersect
Ω
1
\Omega_1
Ω
1
at points
P
,
Q
P, Q
P
,
Q
; and the perpendicular bisector of segment
C
D
C D
C
D
intersect
Ω
2
\Omega_2
Ω
2
at points
R
,
S
R, S
R
,
S
such that
P
,
R
P, R
P
,
R
are on the same side of
l
l
l
. Prove that the midpoints of
P
R
,
Q
S
P R, Q S
PR
,
QS
and
O
1
O
2
O_1 O_2
O
1
O
2
are collinear.
RMO KV P-4
The set
X
X
X
of
N
N
N
four-digit numbers formed from the digits
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
1,2,3,4,5,6,7,8
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
satisfies the following condition: for any two different digits from
1
,
2
,
3
,
4
,
,
6
,
7
,
8
1,2,3,4,,6,7,8
1
,
2
,
3
,
4
,,
6
,
7
,
8
there exists a number in
X
X
X
which contains both of them. \\Determine the smallest possible value of
N
N
N
.
3
2
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RMO 2023 P3
For any natural number
n
n
n
, expressed in base 10 , let
s
(
n
)
s(n)
s
(
n
)
denote the sum of all its digits. Find all natural numbers
m
m
m
and
n
n
n
such that
m
<
n
m<n
m
<
n
and
(
s
(
n
)
)
2
=
m
and
(
s
(
m
)
)
2
=
n
.
(s(n))^2=m \text { and }(s(m))^2=n .
(
s
(
n
)
)
2
=
m
and
(
s
(
m
)
)
2
=
n
.
RMO KV region P-3
Let
f
(
x
)
f(x)
f
(
x
)
be a polynomial with real coefficients of degree 2. Suppose that for some pairwise distinct real numbers ,
a
,
b
,
c
a,b,c
a
,
b
,
c
we have:\\
f
(
a
)
=
b
c
,
f
(
b
)
=
a
c
,
f
(
c
)
=
a
b
f(a)=bc , f(b)=ac, f(c)=ab
f
(
a
)
=
b
c
,
f
(
b
)
=
a
c
,
f
(
c
)
=
ab
Dertermine
f
(
a
+
b
+
c
)
f(a+b+c)
f
(
a
+
b
+
c
)
in terms of
a
,
b
,
c
a,b,c
a
,
b
,
c
.
2
2
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RMO 2023 P2
Let
ω
\omega
ω
be a semicircle with
A
B
A B
A
B
as the bounding diameter and let
C
D
C D
C
D
be a variable chord of the semicircle of constant length such that
C
,
D
C, D
C
,
D
lie in the interior of the arc
A
B
A B
A
B
. Let
E
E
E
be a point on the diameter
A
B
A B
A
B
such that
C
E
C E
CE
and
D
E
D E
D
E
are equally inclined to the line
A
B
A B
A
B
. Prove that (a) the measure of
∠
C
E
D
\angle C E D
∠
CE
D
is a constant; (b) the circumcircle of triangle
C
E
D
C E D
CE
D
passes through a fixed point.
RMO KV 2023 P2
Given a prime number
p
p
p
such that
2
p
2p
2
p
is equal to the sum of the squares of some four consecutive positive integers. Prove that
p
−
7
p-7
p
−
7
is divisible by 36.
1
2
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RMO 2023 P1
Let
N
\mathbb{N}
N
be the set of all positive integers and
S
=
{
(
a
,
b
,
c
,
d
)
∈
N
4
:
a
2
+
b
2
+
c
2
=
d
2
}
S=\left\{(a, b, c, d) \in \mathbb{N}^4: a^2+b^2+c^2=d^2\right\}
S
=
{
(
a
,
b
,
c
,
d
)
∈
N
4
:
a
2
+
b
2
+
c
2
=
d
2
}
. Find the largest positive integer
m
m
m
such that
m
m
m
divides abcd for all
(
a
,
b
,
c
,
d
)
∈
S
(a, b, c, d) \in S
(
a
,
b
,
c
,
d
)
∈
S
.
RMO KV 2023 P1
Given a triangle
A
B
C
ABC
A
BC
with
∠
A
C
B
=
12
0
∘
.
\angle ACB = 120^{\circ}.
∠
A
CB
=
12
0
∘
.
A point
L
L
L
is marked in the side
A
B
AB
A
B
such that
C
L
CL
C
L
bisects
∠
A
C
B
.
\angle ACB.
∠
A
CB
.
Points
N
N
N
and
K
K
K
are chosen in the sides
A
C
AC
A
C
and
B
C
BC
BC
such that
C
K
+
C
N
=
C
L
.
CK+CN=CL.
C
K
+
CN
=
C
L
.
Prove that the triangle
K
L
N
KLN
K
L
N
is equilateral.