MathDB
Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
2017 India Regional Mathematical Olympiad
2017 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(6)
6
1
Hide problems
RMO 2017 P6
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be real numbers, each greater than
1
1
1
. Prove that
x
+
1
y
+
1
+
y
+
1
z
+
1
+
z
+
1
x
+
1
≤
x
−
1
y
−
1
+
y
−
1
z
−
1
+
z
−
1
x
−
1
\dfrac{x+1}{y+1}+\dfrac{y+1}{z+1}+\dfrac{z+1}{x+1} \leq \dfrac{x-1}{y-1}+\dfrac{y-1}{z-1}+\dfrac{z-1}{x-1}
y
+
1
x
+
1
+
z
+
1
y
+
1
+
x
+
1
z
+
1
≤
y
−
1
x
−
1
+
z
−
1
y
−
1
+
x
−
1
z
−
1
.
5
1
Hide problems
RMO 2017 P5
Let
Ω
\Omega
Ω
be a circle with a chord
A
B
AB
A
B
which is not a diameter.
Γ
1
\Gamma_{1}
Γ
1
be a circle on one side of
A
B
AB
A
B
such that it is tangent to
A
B
AB
A
B
at
C
C
C
and internally tangent to
Ω
\Omega
Ω
at
D
D
D
. Likewise, let
Γ
2
\Gamma_{2}
Γ
2
be a circle on the other side of
A
B
AB
A
B
such that it is tangent to
A
B
AB
A
B
at
E
E
E
and internally tangent to
Ω
\Omega
Ω
at
F
F
F
. Suppose the line
D
C
DC
D
C
intersects
Ω
\Omega
Ω
at
X
≠
D
X \neq D
X
=
D
and the line
F
E
FE
FE
intersects
Ω
\Omega
Ω
at
Y
≠
F
Y \neq F
Y
=
F
. Prove that
X
Y
XY
X
Y
is a diameter of
Ω
\Omega
Ω
.
4
1
Hide problems
RMO 2017 P4
Consider
n
2
n^2
n
2
unit squares in the
x
y
xy
x
y
plane centered at point
(
i
,
j
)
(i,j)
(
i
,
j
)
with integer coordinates,
1
≤
i
≤
n
1 \leq i \leq n
1
≤
i
≤
n
,
1
≤
j
≤
n
1 \leq j \leq n
1
≤
j
≤
n
. It is required to colour each unit square in such a way that whenever
1
≤
i
<
j
≤
n
1 \leq i < j \leq n
1
≤
i
<
j
≤
n
and
1
≤
k
<
l
≤
n
1 \leq k < l \leq n
1
≤
k
<
l
≤
n
, the three squares with centres at
(
i
,
k
)
,
(
j
,
k
)
,
(
j
,
l
)
(i,k),(j,k),(j,l)
(
i
,
k
)
,
(
j
,
k
)
,
(
j
,
l
)
have distinct colours. What is the least possible number of colours needed?
3
1
Hide problems
RMO 2017 P3
Let
P
(
x
)
=
x
2
+
x
2
+
b
P(x)=x^2+\dfrac x 2 +b
P
(
x
)
=
x
2
+
2
x
+
b
and
Q
(
x
)
=
x
2
+
c
x
+
d
Q(x)=x^2+cx+d
Q
(
x
)
=
x
2
+
c
x
+
d
be two polynomials with real coefficients such that
P
(
x
)
Q
(
x
)
=
Q
(
P
(
x
)
)
P(x)Q(x)=Q(P(x))
P
(
x
)
Q
(
x
)
=
Q
(
P
(
x
))
for all real
x
x
x
. Find all real roots of
P
(
Q
(
x
)
)
=
0
P(Q(x))=0
P
(
Q
(
x
))
=
0
.
2
1
Hide problems
RMO 2017 P2
Show that the equation
a
3
+
(
a
+
1
)
3
+
…
+
(
a
+
6
)
3
=
b
4
+
(
b
+
1
)
4
a^3+(a+1)^3+\ldots+(a+6)^3=b^4+(b+1)^4
a
3
+
(
a
+
1
)
3
+
…
+
(
a
+
6
)
3
=
b
4
+
(
b
+
1
)
4
has no solutions in integers
a
,
b
a,b
a
,
b
.
1
1
Hide problems
RMO 2017 P1
Let
A
O
B
AOB
A
OB
be a given angle less than
18
0
∘
180^{\circ}
18
0
∘
and let
P
P
P
be an interior point of the angular region determined by
∠
A
O
B
\angle AOB
∠
A
OB
. Show, with proof, how to construct, using only ruler and compass, a line segment
C
D
CD
C
D
passing through
P
P
P
such that
C
C
C
lies on the way
O
A
OA
O
A
and
D
D
D
lies on the ray
O
B
OB
OB
, and
C
P
:
P
D
=
1
:
2
CP:PD=1:2
CP
:
P
D
=
1
:
2
.