MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2016 India National Olympiad
2016 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
P6
1
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A non-constant AP with square of terms also as terms of AP
Consider a nonconstant arithmetic progression
a
1
,
a
2
,
⋯
,
a
n
,
⋯
a_1, a_2,\cdots, a_n,\cdots
a
1
,
a
2
,
⋯
,
a
n
,
⋯
. Suppose there exist relatively prime positive integers
p
>
1
p>1
p
>
1
and
q
>
1
q>1
q
>
1
such that
a
1
2
,
a
p
+
1
2
a_1^2, a_{p+1}^2
a
1
2
,
a
p
+
1
2
and
a
q
+
1
2
a_{q+1}^2
a
q
+
1
2
are also the terms of the same arithmetic progression. Prove that the terms of the arithmetic progression are all integers.
P5
1
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Point on hypotenuse making the two inradii equal
Let
A
B
C
ABC
A
BC
be a right-angle triangle with
∠
B
=
9
0
∘
\angle B=90^{\circ}
∠
B
=
9
0
∘
. Let
D
D
D
be a point on
A
C
AC
A
C
such that the inradii of the triangles
A
B
D
ABD
A
B
D
and
C
B
D
CBD
CB
D
are equal. If this common value is
r
′
r^{\prime}
r
′
and if
r
r
r
is the inradius of triangle
A
B
C
ABC
A
BC
, prove that
1
r
′
=
1
r
+
1
B
D
.
\cfrac{1}{r'}=\cfrac{1}{r}+\cfrac{1}{BD}.
r
′
1
=
r
1
+
B
D
1
.
P4
1
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A regular polygon obtained on a circle from blue & red pts
Suppose
2016
2016
2016
points of the circumference of a circle are colored red and the remaining points are colored blue . Given any natural number
n
≥
3
n\ge 3
n
≥
3
, prove that there is a regular
n
n
n
-sided polygon all of whose vertices are blue.
P3
1
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Everything comes to 1 eventually
Let
N
\mathbb{N}
N
denote the set of natural numbers. Define a function
T
:
N
→
N
T:\mathbb{N}\rightarrow\mathbb{N}
T
:
N
→
N
by
T
(
2
k
)
=
k
T(2k)=k
T
(
2
k
)
=
k
and
T
(
2
k
+
1
)
=
2
k
+
2
T(2k+1)=2k+2
T
(
2
k
+
1
)
=
2
k
+
2
. We write
T
2
(
n
)
=
T
(
T
(
n
)
)
T^2(n)=T(T(n))
T
2
(
n
)
=
T
(
T
(
n
))
and in general
T
k
(
n
)
=
T
k
−
1
(
T
(
n
)
)
T^k(n)=T^{k-1}(T(n))
T
k
(
n
)
=
T
k
−
1
(
T
(
n
))
for any
k
>
1
k>1
k
>
1
.(i) Show that for each
n
∈
N
n\in\mathbb{N}
n
∈
N
, there exists
k
k
k
such that
T
k
(
n
)
=
1
T^k(n)=1
T
k
(
n
)
=
1
.(ii) For
k
∈
N
k\in\mathbb{N}
k
∈
N
, let
c
k
c_k
c
k
denote the number of elements in the set
{
n
:
T
k
(
n
)
=
1
}
\{n: T^k(n)=1\}
{
n
:
T
k
(
n
)
=
1
}
. Prove that
c
k
+
2
=
c
k
+
1
+
c
k
c_{k+2}=c_{k+1}+c_k
c
k
+
2
=
c
k
+
1
+
c
k
, for
k
≥
1
k\ge 1
k
≥
1
.
P2
1
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Which system of equalities implies $a=b=c$
For positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
which of the following statements necessarily implies
a
=
b
=
c
a=b=c
a
=
b
=
c
: (I)
a
(
b
3
+
c
3
)
=
b
(
c
3
+
a
3
)
=
c
(
a
3
+
b
3
)
a(b^3+c^3)=b(c^3+a^3)=c(a^3+b^3)
a
(
b
3
+
c
3
)
=
b
(
c
3
+
a
3
)
=
c
(
a
3
+
b
3
)
, (II)
a
(
a
3
+
b
3
)
=
b
(
b
3
+
c
3
)
=
c
(
c
3
+
a
3
)
a(a^3+b^3)=b(b^3+c^3)=c(c^3+a^3)
a
(
a
3
+
b
3
)
=
b
(
b
3
+
c
3
)
=
c
(
c
3
+
a
3
)
? Justify your answer.
P1
1
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Orthocenter lies on incircle in isosceles ∆
Let
A
B
C
ABC
A
BC
be a triangle in which
A
B
=
A
C
AB=AC
A
B
=
A
C
. Suppose the orthocentre of the triangle lies on the incircle. Find the ratio
A
B
B
C
\frac{AB}{BC}
BC
A
B
.