MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2013 India National Olympiad
2013 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
2
1
Hide problems
m,n natural number, p prime: m(4m^2+m+12)=3(p^n-1)
Find all
m
,
n
∈
N
m,n\in\mathbb N
m
,
n
∈
N
and primes
p
≥
5
p\geq 5
p
≥
5
satisfying
m
(
4
m
2
+
m
+
12
)
=
3
(
p
n
−
1
)
.
m(4m^2+m+12)=3(p^n-1).
m
(
4
m
2
+
m
+
12
)
=
3
(
p
n
−
1
)
.
5
1
Hide problems
[ODC]=[HEA]=[GFB], find all values of C
In an acute triangle
A
B
C
,
ABC,
A
BC
,
let
O
,
G
,
H
O,G,H
O
,
G
,
H
be its circumcentre, centroid and orthocenter. Let
D
∈
B
C
,
E
∈
C
A
D\in BC, E\in CA
D
∈
BC
,
E
∈
C
A
and
O
D
⊥
B
C
,
H
E
⊥
C
A
.
OD\perp BC, HE\perp CA.
O
D
⊥
BC
,
H
E
⊥
C
A
.
Let
F
F
F
be the midpoint of
A
B
.
AB.
A
B
.
If the triangles
O
D
C
,
H
E
A
,
G
F
B
ODC, HEA, GFB
O
D
C
,
H
E
A
,
GFB
have the same area, find all the possible values of
∠
C
.
\angle C.
∠
C
.
6
1
Hide problems
a,b,c,x,y,z>0; a+b+c=x+y+z, abc=xyz
Let
a
,
b
,
c
,
x
,
y
,
z
a,b,c,x,y,z
a
,
b
,
c
,
x
,
y
,
z
be six positive real numbers satisfying
x
+
y
+
z
=
a
+
b
+
c
x+y+z=a+b+c
x
+
y
+
z
=
a
+
b
+
c
and
x
y
z
=
a
b
c
.
xyz=abc.
x
yz
=
ab
c
.
Further, suppose that
a
≤
x
<
y
<
z
≤
c
a\leq x<y<z\leq c
a
≤
x
<
y
<
z
≤
c
and
a
<
b
<
c
.
a<b<c.
a
<
b
<
c
.
Prove that
a
=
x
,
b
=
y
a=x,b=y
a
=
x
,
b
=
y
and
c
=
z
.
c=z.
c
=
z
.
1
1
Hide problems
O_1P, O_2Q tangents to (O_1), (O_2) [INMO 2013 P1]
Let
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
be two circles touching each other externally at
R
.
R.
R
.
Let
O
1
O_1
O
1
and
O
2
O_2
O
2
be the centres of
Γ
1
\Gamma_1
Γ
1
and
Γ
2
,
\Gamma_2,
Γ
2
,
respectively. Let
ℓ
1
\ell_1
ℓ
1
be a line which is tangent to
Γ
2
\Gamma_2
Γ
2
at
P
P
P
and passing through
O
1
,
O_1,
O
1
,
and let
ℓ
2
\ell_2
ℓ
2
be the line tangent to
Γ
1
\Gamma_1
Γ
1
at
Q
Q
Q
and passing through
O
2
.
O_2.
O
2
.
Let
K
=
ℓ
1
∩
ℓ
2
.
K=\ell_1\cap \ell_2.
K
=
ℓ
1
∩
ℓ
2
.
If
K
P
=
K
Q
KP=KQ
K
P
=
K
Q
then prove that the triangle
P
Q
R
PQR
PQR
is equilateral.
4
1
Hide problems
T_n - n is even
Let
N
N
N
be an integer greater than
1
1
1
and let
T
n
T_n
T
n
be the number of non empty subsets
S
S
S
of
{
1
,
2
,
.
.
.
.
.
,
n
}
\{1,2,.....,n\}
{
1
,
2
,
.....
,
n
}
with the property that the average of the elements of
S
S
S
is an integer.Prove that
T
n
−
n
T_n - n
T
n
−
n
is always even.
3
1
Hide problems
Equation has no integer solution.
Let
a
,
b
,
c
,
d
∈
N
a,b,c,d \in \mathbb{N}
a
,
b
,
c
,
d
∈
N
such that
a
≥
b
≥
c
≥
d
a \ge b \ge c \ge d
a
≥
b
≥
c
≥
d
. Show that the equation
x
4
−
a
x
3
−
b
x
2
−
c
x
−
d
=
0
x^4 - ax^3 - bx^2 - cx -d = 0
x
4
−
a
x
3
−
b
x
2
−
c
x
−
d
=
0
has no integer solution.