MathDB
Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
2019 India Regional Mathematical Olympiad
2019 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(6)
6
2
Hide problems
Collinarity
Suppose
91
91
91
distinct positive integers greater than
1
1
1
are given such that there are at least
456
456
456
pairs among them which are relatively prime. Show that one can find four integers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
among them such that
gcd
(
a
,
b
)
=
gcd
(
b
,
c
)
=
gcd
(
c
,
d
)
=
gcd
(
d
,
a
)
=
1.
\gcd(a,b)=\gcd(b,c)=\gcd(c,d)=\gcd(d,a)=1.
g
cd
(
a
,
b
)
=
g
cd
(
b
,
c
)
=
g
cd
(
c
,
d
)
=
g
cd
(
d
,
a
)
=
1.
Circles with centers in a set
Let
k
k
k
be a positive real number. In the
X
−
Y
X-Y
X
−
Y
coordinate plane, let
S
S
S
be the set of all points of the form
(
x
,
x
2
+
k
)
(x,x^2+k)
(
x
,
x
2
+
k
)
where
x
∈
R
x\in\mathbb{R}
x
∈
R
. Let
C
C
C
be the set of all circles whose center lies in
S
S
S
, and which are tangent to
X
X
X
-axis. Find the minimum value of
k
k
k
such that any two circles in
C
C
C
have at least one point of intersection.
4
2
Hide problems
Array compatibility
Consider the following
3
×
2
3\times 2
3
×
2
array formed by using the numbers
1
,
2
,
3
,
4
,
5
,
6
1,2,3,4,5,6
1
,
2
,
3
,
4
,
5
,
6
,
(
a
11
a
12
a
21
a
22
a
31
a
32
)
=
(
1
6
2
5
3
4
)
.
\begin{pmatrix} a_{11}& a_{12}\\a_{21}& a_{22}\\ a_{31}& a_{32}\end{pmatrix}=\begin{pmatrix}1& 6\\2& 5\\ 3& 4\end{pmatrix}.
a
11
a
21
a
31
a
12
a
22
a
32
=
1
2
3
6
5
4
.
Observe that all row sums are equal, but the sum of the square of the squares is not the same for each row. Extend the above array to a
3
×
k
3\times k
3
×
k
array
(
a
i
j
)
3
×
k
(a_{ij})_{3\times k}
(
a
ij
)
3
×
k
for a suitable
k
k
k
, adding more columns, using the numbers
7
,
8
,
9
,
…
,
3
k
7,8,9,\dots ,3k
7
,
8
,
9
,
…
,
3
k
such that
∑
j
=
1
k
a
1
j
=
∑
j
=
1
k
a
2
j
=
∑
j
=
1
k
a
3
j
and
∑
j
=
1
k
(
a
1
j
)
2
=
∑
j
=
1
k
(
a
2
j
)
2
=
∑
j
=
1
k
(
a
3
j
)
2
\sum_{j=1}^k a_{1j}=\sum_{j=1}^k a_{2j}=\sum_{j=1}^k a_{3j}~~\text{and}~~\sum_{j=1}^k (a_{1j})^2=\sum_{j=1}^k (a_{2j})^2=\sum_{j=1}^k (a_{3j})^2
j
=
1
∑
k
a
1
j
=
j
=
1
∑
k
a
2
j
=
j
=
1
∑
k
a
3
j
and
j
=
1
∑
k
(
a
1
j
)
2
=
j
=
1
∑
k
(
a
2
j
)
2
=
j
=
1
∑
k
(
a
3
j
)
2
Two numbers leaving same remainder
Let
a
1
,
a
2
,
⋯
,
a
6
,
a
7
a_1,a_2,\cdots,a_6,a_7
a
1
,
a
2
,
⋯
,
a
6
,
a
7
be seven positive integers. Let
S
S
S
be the set of all numbers of the form
a
i
2
+
a
j
2
a_i^2+a_j^2
a
i
2
+
a
j
2
where
1
≤
i
<
j
≤
7
1\leq i<j\leq 7
1
≤
i
<
j
≤
7
. Prove that there exist two elements of
S
S
S
which have the same remainder on dividing by
36
36
36
.
5
2
Hide problems
Perpendicular lines
In an acute angled triangle
A
B
C
ABC
A
BC
, let
H
H
H
be the orthocenter, and let
D
,
E
,
F
D,E,F
D
,
E
,
F
be the feet of altitudes from
A
,
B
,
C
A,B,C
A
,
B
,
C
to the opposite sides, respectively. Let
L
,
M
,
N
L,M,N
L
,
M
,
N
be the midpoints of the segments
A
H
,
E
F
,
B
C
AH, EF, BC
A
H
,
EF
,
BC
respectively. Let
X
,
Y
X,Y
X
,
Y
be the feet of altitudes from
L
,
N
L,N
L
,
N
on to the line
D
F
DF
D
F
respectively. Prove that
X
M
XM
XM
is perpendicular to
M
Y
MY
M
Y
.
Counting problem with three types of objects
There is a pack of 27 distinct cards, and each card has three values on it. The first value is a shape from
{
Δ
,
□
,
⊙
}
\{\Delta,\square,\odot\}
{
Δ
,
□
,
⊙
}
; the second value is a letter from
{
A
,
B
,
C
}
\{A,B,C\}
{
A
,
B
,
C
}
; and the third value is a number from
{
1
,
2
,
3
}
\{1,2,3\}
{
1
,
2
,
3
}
. In how many ways can we choose an unordered set of 3 cards from the pack, so that no two of the chosen cards have two matching values. For example we can chose
{
Δ
A
1
,
Δ
B
2
,
⊙
C
3
}
\{\Delta A1,\Delta B2,\odot C3\}
{
Δ
A
1
,
Δ
B
2
,
⊙
C
3
}
But we cannot choose
{
Δ
A
1
,
□
B
2
,
Δ
C
1
}
\{\Delta A1,\square B2,\Delta C1\}
{
Δ
A
1
,
□
B
2
,
Δ
C
1
}
2
2
Hide problems
Equilateral pair
Let
A
B
C
ABC
A
BC
be a triangle with circumcircle
Ω
\Omega
Ω
and let
G
G
G
be the centroid of triangle
A
B
C
ABC
A
BC
. Extend
A
G
,
B
G
AG, BG
A
G
,
BG
and
C
G
CG
CG
to meet the circle
Ω
\Omega
Ω
again in
A
1
,
B
1
A_1, B_1
A
1
,
B
1
and
C
1
C_1
C
1
. Suppose
∠
B
A
C
=
∠
A
1
B
1
C
1
,
∠
A
B
C
=
∠
A
1
C
1
B
1
\angle BAC = \angle A_1 B_1 C_1, \angle ABC = \angle A_1 C_1 B_1
∠
B
A
C
=
∠
A
1
B
1
C
1
,
∠
A
BC
=
∠
A
1
C
1
B
1
and
∠
A
C
B
=
B
1
A
1
C
1
\angle ACB = B_1 A_1 C_1
∠
A
CB
=
B
1
A
1
C
1
. Prove that
A
B
C
ABC
A
BC
and
A
1
B
1
C
1
A_1 B_1 C_1
A
1
B
1
C
1
are equilateral triangles.
Construct circle ...
Given a circle
τ
\tau
τ
, let
P
P
P
be a point in its interior, and let
l
l
l
be a line through
P
P
P
. Construct with proof using ruler and compass, all circles which pass through
P
P
P
, are tangent to
τ
\tau
τ
and whose center lies on line
l
l
l
.
3
2
Hide problems
Cyclic Non-homogeneous in 3 variables
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Prove that
a
a
2
+
b
3
+
c
3
+
b
b
2
+
a
3
+
c
3
+
c
c
2
+
a
3
+
b
3
≤
1
5
a
b
c
\frac{a}{a^2+b^3+c^3}+\frac{b}{b^2+a^3+c^3}+\frac{c}{c^2+a^3+b^3}\leq\frac{1}{5abc}
a
2
+
b
3
+
c
3
a
+
b
2
+
a
3
+
c
3
b
+
c
2
+
a
3
+
b
3
c
≤
5
ab
c
1
System of Equations
Find all triples of non-negative real numbers
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
which satisfy the following set of equations
a
2
+
a
b
=
c
a^2+ab=c
a
2
+
ab
=
c
b
2
+
b
c
=
a
b^2+bc=a
b
2
+
b
c
=
a
c
2
+
c
a
=
b
c^2+ca=b
c
2
+
c
a
=
b
1
2
Hide problems
Rationally yours!
Suppose
x
x
x
is a non zero real number such that both
x
5
x^5
x
5
and
20
x
+
19
x
20x+\frac{19}{x}
20
x
+
x
19
are rational numbers. Prove that
x
x
x
is a rational number.
Sum of gcd of n and 2019-n.
For each
n
∈
N
n\in\mathbb{N}
n
∈
N
let
d
n
d_n
d
n
denote the gcd of
n
n
n
and
(
2019
−
n
)
(2019-n)
(
2019
−
n
)
. Find value of
d
1
+
d
2
+
⋯
d
2018
+
d
2019
d_1+d_2+\cdots d_{2018}+d_{2019}
d
1
+
d
2
+
⋯
d
2018
+
d
2019