MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2015 India National Olympiad
2015 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
1
1
Hide problems
Inmo 2015
Let
A
B
C
ABC
A
BC
be a right-angled triangle with
∠
B
=
9
0
∘
\angle{B}=90^{\circ}
∠
B
=
9
0
∘
. Let
B
D
BD
B
D
is the altitude from
B
B
B
on
A
C
AC
A
C
. Let
P
,
Q
P,Q
P
,
Q
and
I
I
I
be the incenters of triangles
A
B
D
,
C
B
D
ABD,CBD
A
B
D
,
CB
D
and
A
B
C
ABC
A
BC
respectively.Show that circumcenter of triangle
P
I
Q
PIQ
P
I
Q
lie on the hypotenuse
A
C
AC
A
C
.
5
1
Hide problems
Inmo 2015
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral.Let diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect at
P
P
P
. Let
P
E
,
P
F
,
P
G
PE,PF,PG
PE
,
PF
,
PG
and
P
H
PH
P
H
are altitudes from
P
P
P
on the side
A
B
,
B
C
,
C
D
AB,BC,CD
A
B
,
BC
,
C
D
and
D
A
DA
D
A
respectively. Show that
A
B
C
D
ABCD
A
BC
D
has a incircle if and only if
1
P
E
+
1
P
G
=
1
P
F
+
1
P
H
.
\frac{1}{PE}+\frac{1}{PG}=\frac{1}{PF}+\frac{1}{PH}.
PE
1
+
PG
1
=
PF
1
+
P
H
1
.
4
1
Hide problems
Inmo 2015
There are four basketball players
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
. Initially the ball is with
A
A
A
. The ball is always passed from one person to a different person. In how many ways can the ball come back to
A
A
A
after
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
s
e
v
e
n
<
/
s
p
a
n
>
<span class='latex-bold'>seven</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
se
v
e
n
<
/
s
p
an
>
moves? (for example
A
→
C
→
B
→
D
→
A
→
B
→
C
→
A
A\rightarrow C\rightarrow B\rightarrow D\rightarrow A\rightarrow B\rightarrow C\rightarrow A
A
→
C
→
B
→
D
→
A
→
B
→
C
→
A
, or
A
→
D
→
A
→
D
→
C
→
A
→
B
→
A
)
A\rightarrow D\rightarrow A\rightarrow D\rightarrow C\rightarrow A\rightarrow B\rightarrow A)
A
→
D
→
A
→
D
→
C
→
A
→
B
→
A
)
.
6
1
Hide problems
Inmo 2015
Show that from a set of
11
11
11
square integers one can select six numbers
a
2
,
b
2
,
c
2
,
d
2
,
e
2
,
f
2
a^2,b^2,c^2,d^2,e^2,f^2
a
2
,
b
2
,
c
2
,
d
2
,
e
2
,
f
2
such that
a
2
+
b
2
+
c
2
≡
d
2
+
e
2
+
f
2
(
m
o
d
12
)
a^2+b^2+c^2 \equiv d^2+e^2+f^2\pmod{12}
a
2
+
b
2
+
c
2
≡
d
2
+
e
2
+
f
2
(
mod
12
)
.
2
1
Hide problems
Inmo 2015
For any natural number
n
>
1
n > 1
n
>
1
write the finite decimal expansion of
1
n
\frac{1}{n}
n
1
(for example we write
1
2
=
0.4
9
‾
\frac{1}{2}=0.4\overline{9}
2
1
=
0.4
9
as its infinite decimal expansion not
0.5
)
0.5)
0.5
)
. Determine the length of non-periodic part of the (infinite) decimal expansion of
1
n
\frac{1}{n}
n
1
.
3
1
Hide problems
INMO 2015
Find all real functions
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
f
(
x
2
+
y
f
(
x
)
)
=
x
f
(
x
+
y
)
f(x^2+yf(x))=xf(x+y)
f
(
x
2
+
y
f
(
x
))
=
x
f
(
x
+
y
)
.