MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2017 India National Olympiad
2017 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
6
1
Hide problems
INMO 2017 Question 6
Let
n
≥
1
n\ge 1
n
≥
1
be an integer and consider the sum
x
=
∑
k
≥
0
(
n
2
k
)
2
n
−
2
k
3
k
=
(
n
0
)
2
n
+
(
n
2
)
2
n
−
2
⋅
3
+
(
n
4
)
2
n
−
k
⋅
3
2
+
⋯
.
x=\sum_{k\ge 0} \dbinom{n}{2k} 2^{n-2k}3^k=\dbinom{n}{0}2^n+\dbinom{n}{2}2^{n-2}\cdot{}3+\dbinom{n}{4}2^{n-k}\cdot{}3^2 + \cdots{}.
x
=
k
≥
0
∑
(
2
k
n
)
2
n
−
2
k
3
k
=
(
0
n
)
2
n
+
(
2
n
)
2
n
−
2
⋅
3
+
(
4
n
)
2
n
−
k
⋅
3
2
+
⋯
.
Show that
2
x
−
1
,
2
x
,
2
x
+
1
2x-1,2x,2x+1
2
x
−
1
,
2
x
,
2
x
+
1
form the sides of a triangle whose area and inradius are also integers.
2
1
Hide problems
INMO 2017 Question 2
Suppose
n
≥
0
n \ge 0
n
≥
0
is an integer and all the roots of
x
3
+
α
x
+
4
−
(
2
×
201
6
n
)
=
0
x^3 + \alpha x + 4 - ( 2 \times 2016^n) = 0
x
3
+
αx
+
4
−
(
2
×
201
6
n
)
=
0
are integers. Find all possible values of
α
\alpha
α
.
1
1
Hide problems
INMO 2017 Question 1
In the given figure,
A
B
C
D
ABCD
A
BC
D
is a square sheet of paper. It is folded along
E
F
EF
EF
such that
A
A
A
goes to a point
A
′
A'
A
′
different from
B
B
B
and
C
C
C
, on the side
B
C
BC
BC
and
D
D
D
goes to
D
′
D'
D
′
. The line
A
′
D
′
A'D'
A
′
D
′
cuts
C
D
CD
C
D
in
G
G
G
. Show that the inradius of the triangle
G
C
A
′
GCA'
GC
A
′
is the sum of the inradii of the triangles
G
D
′
F
GD'F
G
D
′
F
and
A
′
B
E
A'BE
A
′
BE
.[asy] size(5cm); pair A=(0,0),B=(1,0),C=(1,1),D=(0,1),Ap=(1,0.333),Dp,Ee,F,G; Ee=extension(A,B,(A+Ap)/2,bisectorpoint(A,Ap)); F=extension(C,D,(A+Ap)/2,bisectorpoint(A,Ap)); Dp=reflect(Ee,F)*D; G=extension(C,D,Ap,Dp); D(MP("A",A,W)--MP("E",Ee,S)--MP("B",B,E)--MP("A^{\prime}",Ap,E)--MP("C",C,E)--MP("G",G,NE)--MP("D^{\prime}",Dp,N)--MP("F",F,NNW)--MP("D",D,W)--cycle,black); draw(Ee--Ap--G--F); dot(A);dot(B);dot(C);dot(D);dot(Ap);dot(Dp);dot(Ee);dot(F);dot(G); draw(Ee--F,dashed); [/asy]
5
1
Hide problems
AI perpendicular to PQ and AI=PQ
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
=
9
0
∘
\angle{A}=90^{\circ}
∠
A
=
9
0
∘
and
A
B
<
A
C
AB<AC
A
B
<
A
C
. Let
A
D
AD
A
D
be the altitude from
A
A
A
on to
B
C
BC
BC
, Let
P
,
Q
P,Q
P
,
Q
and
I
I
I
denote respectively the incentres of triangle
A
B
D
,
A
C
D
ABD,ACD
A
B
D
,
A
C
D
and
A
B
C
ABC
A
BC
. Prove that
A
I
AI
A
I
is perpendicular to
P
Q
PQ
PQ
and
A
I
=
P
Q
AI=PQ
A
I
=
PQ
.
4
1
Hide problems
Convex pentagon with consecutive integer lengths
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a convex pentagon in which
∠
A
=
∠
B
=
∠
C
=
∠
D
=
12
0
∘
\angle{A}=\angle{B}=\angle{C}=\angle{D}=120^{\circ}
∠
A
=
∠
B
=
∠
C
=
∠
D
=
12
0
∘
and the side lengths are five consecutive integers in some order. Find all possible values of
A
B
+
B
C
+
C
D
AB+BC+CD
A
B
+
BC
+
C
D
.
3
1
Hide problems
Number of triples (x,a,b)
Find the number of triples
(
x
,
a
,
b
)
(x,a,b)
(
x
,
a
,
b
)
where
x
x
x
is a real number and
a
,
b
a,b
a
,
b
belong to the set
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
\{1,2,3,4,5,6,7,8,9\}
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
such that
x
2
−
a
{
x
}
+
b
=
0.
x^2-a\{x\}+b=0.
x
2
−
a
{
x
}
+
b
=
0.
where
{
x
}
\{x\}
{
x
}
denotes the fractional part of the real number
x
x
x
.