MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2011 India National Olympiad
2011 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
4
1
Hide problems
Five vertices are chosen from a regular nonagon [INMO 2011]
Suppose five of the nine vertices of a regular nine-sided polygon are arbitrarily chosen. Show that one can select four among these five such that they are the vertices of a trapezium.
6
1
Hide problems
Functional equation from R to R-[INMO 2011]
Find all functions
f
:
R
→
R
f:\mathbb{R}\to \mathbb R
f
:
R
→
R
satisfying
f
(
x
+
y
)
f
(
x
−
y
)
=
(
f
(
x
)
+
f
(
y
)
)
2
−
4
x
2
f
(
y
)
,
f(x+y)f(x-y)=\left(f(x)+f(y)\right)^2-4x^2f(y),
f
(
x
+
y
)
f
(
x
−
y
)
=
(
f
(
x
)
+
f
(
y
)
)
2
−
4
x
2
f
(
y
)
,
For all
x
,
y
∈
R
x,y\in\mathbb R
x
,
y
∈
R
.
3
1
Hide problems
Polynomials P(x), Q(x) with a_n-b_n as prime [INMO 2011]
Let
P
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
0
P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0
P
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
0
and
Q
(
x
)
=
b
n
x
n
+
b
n
−
1
x
n
−
1
+
⋯
+
b
0
Q(x)=b_nx^n+b_{n-1}x^{n-1}+\cdots+b_0
Q
(
x
)
=
b
n
x
n
+
b
n
−
1
x
n
−
1
+
⋯
+
b
0
be two polynomials with integral coefficients such that
a
n
−
b
n
a_n-b_n
a
n
−
b
n
is a prime and
a
n
b
0
−
a
0
b
n
≠
0
,
a_nb_0-a_0b_n\neq 0,
a
n
b
0
−
a
0
b
n
=
0
,
and
a
n
−
1
=
b
n
−
1
.
a_{n-1}=b_{n-1}.
a
n
−
1
=
b
n
−
1
.
Suppose that there exists a rational number
r
r
r
such that
P
(
r
)
=
Q
(
r
)
=
0.
P(r)=Q(r)=0.
P
(
r
)
=
Q
(
r
)
=
0.
Prove that
r
∈
Z
.
r\in\mathbb Z.
r
∈
Z
.
5
1
Hide problems
Cyclic quadrilateral with midpoints of arcs;-[INMO 2011]
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral inscribed in a circle
Γ
.
\Gamma.
Γ.
Let
E
,
F
,
G
,
H
E,F,G,H
E
,
F
,
G
,
H
be the midpoints of arcs
A
B
,
B
C
,
C
D
,
A
D
AB,BC,CD,AD
A
B
,
BC
,
C
D
,
A
D
of
Γ
,
\Gamma,
Γ
,
respectively. Suppose that
A
C
⋅
B
D
=
E
G
⋅
F
H
.
AC\cdot BD=EG\cdot FH.
A
C
⋅
B
D
=
EG
⋅
F
H
.
Show that
A
C
,
B
D
,
E
G
,
F
H
AC,BD,EG,FH
A
C
,
B
D
,
EG
,
F
H
are all concurrent.
1
1
Hide problems
D on BC, E on CA, F on AB such that BDF=CED=AFE -[INMO 2011]
Let
D
,
E
,
F
D,E,F
D
,
E
,
F
be points on the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively of a triangle
A
B
C
ABC
A
BC
such that
B
D
=
C
E
=
A
F
BD=CE=AF
B
D
=
CE
=
A
F
and
∠
B
D
F
=
∠
C
E
D
=
∠
A
F
E
.
\angle BDF=\angle CED=\angle AFE.
∠
B
D
F
=
∠
CE
D
=
∠
A
FE
.
Show that
△
A
B
C
\triangle ABC
△
A
BC
is equilateral.
2
1
Hide problems
A number n is faithful if n=a+b+c, a|b, b|c-[INMO 2011]
Call a natural number
n
n
n
faithful if there exist natural numbers
a
<
b
<
c
a<b<c
a
<
b
<
c
such that
a
∣
b
,
a|b,
a
∣
b
,
and
b
∣
c
b|c
b
∣
c
and
n
=
a
+
b
+
c
.
n=a+b+c.
n
=
a
+
b
+
c
.
(
i
)
(i)
(
i
)
Show that all but a finite number of natural numbers are faithful.
(
i
i
)
(ii)
(
ii
)
Find the sum of all natural numbers which are not faithful.