MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2019 India National OIympiad
2019 India National OIympiad
Part of
India National Olympiad
Subcontests
(6)
6
1
Hide problems
INMO 2019 P6
Let
f
f
f
be a function defined from
(
(
x
,
y
)
:
x
,
y
((x,y) : x,y
((
x
,
y
)
:
x
,
y
real,
x
y
≠
0
)
xy\ne 0)
x
y
=
0
)
to the set of all positive real numbers such that
(
i
)
f
(
x
y
,
z
)
=
f
(
x
,
z
)
⋅
f
(
y
,
z
)
(i) f(xy,z)= f(x,z)\cdot f(y,z)
(
i
)
f
(
x
y
,
z
)
=
f
(
x
,
z
)
⋅
f
(
y
,
z
)
for all
x
,
y
≠
0
x,y \ne 0
x
,
y
=
0
(
i
i
)
f
(
x
,
y
z
)
=
f
(
x
,
y
)
⋅
f
(
x
,
z
)
(ii) f(x,yz)= f(x,y)\cdot f(x,z)
(
ii
)
f
(
x
,
yz
)
=
f
(
x
,
y
)
⋅
f
(
x
,
z
)
for all
x
,
y
≠
0
x,y \ne 0
x
,
y
=
0
(
i
i
i
)
f
(
x
,
1
−
x
)
=
1
(iii) f(x,1-x) = 1
(
iii
)
f
(
x
,
1
−
x
)
=
1
for all
x
≠
0
,
1
x \ne 0,1
x
=
0
,
1
Prove that
(
a
)
f
(
x
,
x
)
=
f
(
x
,
−
x
)
=
1
(a) f(x,x) = f(x,-x) = 1
(
a
)
f
(
x
,
x
)
=
f
(
x
,
−
x
)
=
1
for all
x
≠
0
x \ne 0
x
=
0
(
b
)
f
(
x
,
y
)
⋅
f
(
y
,
x
)
=
1
(b) f(x,y)\cdot f(y,x) = 1
(
b
)
f
(
x
,
y
)
⋅
f
(
y
,
x
)
=
1
for all
x
,
y
≠
0
x,y \ne 0
x
,
y
=
0
The condition (ii) was left out in the paper leading to an incomplete problem during contest.
5
1
Hide problems
INMO 2019 P5
Let
A
B
AB
A
B
be the diameter of a circle
Γ
\Gamma
Γ
and let
C
C
C
be a point on
Γ
\Gamma
Γ
different from
A
A
A
and
B
B
B
. Let
D
D
D
be the foot of perpendicular from
C
C
C
on to
A
B
AB
A
B
.Let
K
K
K
be a point on the segment
C
D
CD
C
D
such that
A
C
AC
A
C
is equal to the semi perimeter of
A
D
K
ADK
A
DK
.Show that the excircle of
A
D
K
ADK
A
DK
opposite
A
A
A
is tangent to
Γ
\Gamma
Γ
.
4
1
Hide problems
INMO 2019 P4
Let
n
n
n
and
M
M
M
be positive integers such that
M
>
n
n
−
1
M>n^{n-1}
M
>
n
n
−
1
. Prove that there are
n
n
n
distinct primes
p
1
,
p
2
,
p
3
⋯
,
p
n
p_1,p_2,p_3 \cdots ,p_n
p
1
,
p
2
,
p
3
⋯
,
p
n
such that
p
j
p_j
p
j
divides
M
+
j
M + j
M
+
j
for all
1
≤
j
≤
n
1 \le j \le n
1
≤
j
≤
n
.
3
1
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INMO 2019 P3
Let
m
,
n
m,n
m
,
n
be distinct positive integers. Prove that
g
c
d
(
m
,
n
)
+
g
c
d
(
m
+
1
,
n
+
1
)
+
g
c
d
(
m
+
2
,
n
+
2
)
≤
2
∣
m
−
n
∣
+
1.
gcd(m,n) + gcd(m+1,n+1) + gcd(m+2,n+2) \le 2|m-n| + 1.
g
c
d
(
m
,
n
)
+
g
c
d
(
m
+
1
,
n
+
1
)
+
g
c
d
(
m
+
2
,
n
+
2
)
≤
2∣
m
−
n
∣
+
1.
Further, determine when equality holds.
2
1
Hide problems
INMO 2019 P2
Let
A
1
B
1
C
1
D
1
E
1
A_1B_1C_1D_1E_1
A
1
B
1
C
1
D
1
E
1
be a regular pentagon.For
2
≤
n
≤
11
2 \le n \le 11
2
≤
n
≤
11
, let
A
n
B
n
C
n
D
n
E
n
A_nB_nC_nD_nE_n
A
n
B
n
C
n
D
n
E
n
be the pentagon whose vertices are the midpoint of the sides
A
n
−
1
B
n
−
1
C
n
−
1
D
n
−
1
E
n
−
1
A_{n-1}B_{n-1}C_{n-1}D_{n-1}E_{n-1}
A
n
−
1
B
n
−
1
C
n
−
1
D
n
−
1
E
n
−
1
. All the
5
5
5
vertices of each of the
11
11
11
pentagons are arbitrarily coloured red or blue. Prove that four points among these
55
55
55
points have the same colour and form the vertices of a cyclic quadrilateral.
1
1
Hide problems
INMO 2019 P1
Let
A
B
C
ABC
A
BC
be a triangle with
∠
B
A
C
>
90
\angle{BAC} > 90
∠
B
A
C
>
90
. Let
D
D
D
be a point on the segment
B
C
BC
BC
and
E
E
E
be a point on line
A
D
AD
A
D
such that
A
B
AB
A
B
is tangent to the circumcircle of triangle
A
C
D
ACD
A
C
D
at
A
A
A
and
B
E
BE
BE
is perpendicular to
A
D
AD
A
D
. Given that
C
A
=
C
D
CA=CD
C
A
=
C
D
and
A
E
=
C
E
AE=CE
A
E
=
CE
. Determine
∠
B
C
A
\angle{BCA}
∠
BC
A
in degrees.