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Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
2024 India Regional Mathematical Olympiad
2024 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(6)
6
2
Hide problems
RMO 2024 Q6
Let
n
≥
2
n \geq 2
n
≥
2
be a positive integer. Call a sequence
a
1
,
a
2
,
⋯
,
a
k
a_1, a_2, \cdots , a_k
a
1
,
a
2
,
⋯
,
a
k
of integers an
n
n
n
-chain if
1
=
a
2
<
a
2
<
⋯
<
a
k
=
n
1 = a_2 < a_ 2 < \cdots < a_k =n
1
=
a
2
<
a
2
<
⋯
<
a
k
=
n
,
a
i
a_i
a
i
divides
a
i
+
1
a_{i+1}
a
i
+
1
for all
i
i
i
,
1
≤
i
≤
k
−
1
1 \leq i \leq k-1
1
≤
i
≤
k
−
1
. Let
f
(
n
)
f(n)
f
(
n
)
be the number of
n
n
n
-chains where
n
≥
2
n \geq 2
n
≥
2
. For example,
f
(
4
)
=
2
f(4) = 2
f
(
4
)
=
2
corresponds to the
4
4
4
-chains
{
1
,
4
}
\{1,4\}
{
1
,
4
}
and
{
1
,
2
,
4
}
\{1,2,4\}
{
1
,
2
,
4
}
. Prove that
f
(
2
m
⋅
3
)
=
2
m
−
1
(
m
+
2
)
f(2^m \cdot 3) = 2^{m-1} (m+2)
f
(
2
m
⋅
3
)
=
2
m
−
1
(
m
+
2
)
for every positive integer
m
m
m
.
RMO KV 2024 Q6
Let
X
X
X
be a set of
11
11
11
integers. Prove that one can find a nonempty subset
{
a
1
,
a
2
,
⋯
,
a
k
}
\{a_1, a_2, \cdots , a_k \}
{
a
1
,
a
2
,
⋯
,
a
k
}
of
X
X
X
such that
3
3
3
divides
k
k
k
and
9
9
9
divides the sum
∑
i
=
1
k
4
i
a
i
\sum_{i=1}^{k} 4^i a_i
∑
i
=
1
k
4
i
a
i
.
4
2
Hide problems
RMO 2024 Q4
Let
a
1
,
a
2
,
a
3
,
a
4
a_1,a_2,a_3,a_4
a
1
,
a
2
,
a
3
,
a
4
be real numbers such that
a
1
2
+
a
2
2
+
a
3
2
+
a
4
2
=
1
a_1^2 + a_2^2 + a_3^2 + a_4^2 = 1
a
1
2
+
a
2
2
+
a
3
2
+
a
4
2
=
1
. Show that there exist
i
,
j
i,j
i
,
j
with
1
≤
i
<
j
≤
4
1 \leq i < j \leq 4
1
≤
i
<
j
≤
4
, such that
(
a
i
−
a
j
)
2
≤
1
5
(a_i - a_j)^2 \leq \frac{1}{5}
(
a
i
−
a
j
)
2
≤
5
1
.
RMO KV 2024 Q4
Let
n
>
1
n>1
n
>
1
be a positive integer. Call a rearrangement
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2, \cdots , a_n
a
1
,
a
2
,
⋯
,
a
n
of
1
,
2
,
⋯
,
n
1,2, \cdots , n
1
,
2
,
⋯
,
n
nice if for every
k
=
2
,
3
,
⋯
,
n
k = 2 ,3, \cdots , n
k
=
2
,
3
,
⋯
,
n
, we have that
a
1
2
+
a
2
2
+
⋯
+
a
k
2
a_1^2 + a_2^2 + \cdots + a_k^2
a
1
2
+
a
2
2
+
⋯
+
a
k
2
is not divisible by
k
k
k
. Determine which positive integers
n
>
1
n>1
n
>
1
have a nice arrangement.
5
2
Hide problems
Some Length Equality
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral such that
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
. Let
O
O
O
be the circumcenter of
A
B
C
D
ABCD
A
BC
D
and
L
L
L
be the point on
A
D
AD
A
D
such that
O
L
OL
O
L
is perpendicular to
A
D
AD
A
D
. Prove that
O
B
⋅
(
A
B
+
C
D
)
=
O
L
⋅
(
A
C
+
B
D
)
.
OB\cdot(AB+CD) = OL\cdot(AC + BD).
OB
⋅
(
A
B
+
C
D
)
=
O
L
⋅
(
A
C
+
B
D
)
.
RMO KV 2024 Q5
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
B
C
=
2
0
∘
\angle ABC = 20^{\circ}
∠
A
BC
=
2
0
∘
and
∠
A
C
B
=
4
0
∘
\angle ACB = 40^{\circ}
∠
A
CB
=
4
0
∘
. Let
D
D
D
be a point on
B
C
BC
BC
such that
∠
B
A
D
=
∠
D
A
C
\angle BAD = \angle DAC
∠
B
A
D
=
∠
D
A
C
. Let the incircle of triangle
A
B
C
ABC
A
BC
touch
B
C
BC
BC
at
E
E
E
. Prove that
B
D
=
2
⋅
C
E
BD = 2 \cdot CE
B
D
=
2
⋅
CE
.
3
2
Hide problems
RMO 2024 Q3
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
. Let
D
D
D
be the point on
B
C
BC
BC
such that
A
D
AD
A
D
is perpendicular to
B
C
BC
BC
. Let
O
,
H
,
G
O,H,G
O
,
H
,
G
be the circumcenter, orthocenter and centroid of triangle
A
B
C
ABC
A
BC
respectively. Suppose that
2
⋅
O
D
=
23
⋅
H
D
2 \cdot OD = 23 \cdot HD
2
⋅
O
D
=
23
⋅
HD
. Prove that
G
G
G
lies on the incircle of triangle
A
B
C
ABC
A
BC
.
RMO KV 2024 Q3
Let
A
B
C
ABC
A
BC
be an equilateral triangle. Suppose
D
D
D
is the point on
B
C
BC
BC
such that
B
D
+
D
C
=
1
:
3
BD+DC = 1:3
B
D
+
D
C
=
1
:
3
. Let the perpendicular bisector of
A
D
AD
A
D
intersect
A
B
,
A
C
AB,AC
A
B
,
A
C
at
E
,
F
E,F
E
,
F
respectively. Prove that
49
⋅
[
B
D
E
]
=
25
⋅
[
C
D
F
]
49 \cdot [BDE] = 25 \cdot [CDF]
49
⋅
[
B
D
E
]
=
25
⋅
[
C
D
F
]
, where
[
X
Y
Z
]
[XYZ]
[
X
Y
Z
]
denotes the area of the triangle
X
Y
Z
XYZ
X
Y
Z
.
2
2
Hide problems
RMO 2024 Q2
For a positive integer
n
n
n
, let
R
(
n
)
R(n)
R
(
n
)
be the sum of the remainders when
n
n
n
is divided by
1
,
2
,
⋯
,
n
1,2, \cdots , n
1
,
2
,
⋯
,
n
. For example,
R
(
4
)
=
0
+
0
+
1
+
0
=
1
,
R(4) = 0 + 0 + 1 + 0 = 1,
R
(
4
)
=
0
+
0
+
1
+
0
=
1
,
R
(
7
)
=
0
+
1
+
1
+
3
+
2
+
1
+
0
=
8
R(7) = 0 + 1 + 1 + 3 + 2 + 1 + 0 = 8
R
(
7
)
=
0
+
1
+
1
+
3
+
2
+
1
+
0
=
8
. Find all positive integers such that
R
(
n
)
=
n
−
1
R(n) = n-1
R
(
n
)
=
n
−
1
.
RMO KV 2024 Q2
Show that there do not exist non-zero real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
such that the following statements hold simultaneously:
∙
\bullet
∙
the equation
a
x
2
+
b
x
+
c
=
0
ax^2 + bx + c = 0
a
x
2
+
b
x
+
c
=
0
has two distinct roots
x
1
,
x
2
x_1,x_2
x
1
,
x
2
;
∙
\bullet
∙
the equation
b
x
2
+
c
x
+
a
=
0
bx^2 + cx + a = 0
b
x
2
+
c
x
+
a
=
0
has two distinct roots
x
2
,
x
3
x_2,x_3
x
2
,
x
3
;
∙
\bullet
∙
the equation
c
x
2
+
a
x
+
b
=
0
cx^2 + ax + b = 0
c
x
2
+
a
x
+
b
=
0
has two distinct roots
x
3
,
x
1
x_3,x_1
x
3
,
x
1
. (Note that
x
1
,
x
2
,
x
3
x_1,x_2,x_3
x
1
,
x
2
,
x
3
may be real or complex numbers.)
1
2
Hide problems
RMO 2024 Q1
Let
n
>
1
n>1
n
>
1
be a positive integer. Call a rearrangement
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2, \cdots , a_n
a
1
,
a
2
,
⋯
,
a
n
of
1
,
2
,
⋯
,
n
1,2, \cdots , n
1
,
2
,
⋯
,
n
nice if for every
k
=
2
,
3
,
⋯
,
n
k = 2,3, \cdots , n
k
=
2
,
3
,
⋯
,
n
, we have that
a
1
+
a
2
+
⋯
+
a
k
a_1 + a_2 + \cdots + a_k
a
1
+
a
2
+
⋯
+
a
k
is not divisible by
k
k
k
. (a) If
n
>
1
n>1
n
>
1
is odd, prove that there is no nice arrangement of
1
,
2
,
⋯
,
n
1,2, \cdots , n
1
,
2
,
⋯
,
n
. (b) If
n
n
n
is even, find a nice arrangement of
1
,
2
,
⋯
,
n
1,2, \cdots , n
1
,
2
,
⋯
,
n
.
RMO KV 2024 Q1
Find all positive integers
x
,
y
x,y
x
,
y
such that
202
x
+
4
x
2
=
y
2
202x + 4x^2 = y^2
202
x
+
4
x
2
=
y
2
.