MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2003 India National Olympiad
2003 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
6
1
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What color is ticket $123123123
Each lottery ticket has a 9-digit numbers, which uses only the digits
1
1
1
,
2
2
2
,
3
3
3
. Each ticket is colored red, blue or green. If two tickets have numbers which differ in all nine places, then the tickets have different colors. Ticket
122222222
122222222
122222222
is red, and ticket
222222222
222222222
222222222
is green. What color is ticket
123123123
123123123
123123123
?
5
1
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Area is at least .....
Let a, b, c be the sidelengths and S the area of a triangle ABC. Denote
x
=
a
+
b
2
x=a+\frac{b}{2}
x
=
a
+
2
b
,
y
=
b
+
c
2
y=b+\frac{c}{2}
y
=
b
+
2
c
and
z
=
c
+
a
2
z=c+\frac{a}{2}
z
=
c
+
2
a
. Prove that there exists a triangle with sidelengths x, y, z, and the area of this triangle is
≥
9
4
S
\geq\frac94 S
≥
4
9
S
.
4
1
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Find all 7-digit numbers
Find all
7
7
7
-digit numbers which use only the digits
5
5
5
and
7
7
7
and are divisible by
35
35
35
.
3
1
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Find the sum of the non-real roots
Show that
8
x
4
−
16
x
3
+
16
x
2
−
8
x
+
k
=
0
8x^4 - 16x^3 + 16x^2 - 8x + k = 0
8
x
4
−
16
x
3
+
16
x
2
−
8
x
+
k
=
0
has at least one real root for all real
k
k
k
. Find the sum of the non-real roots.
1
1
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Show that kd bisects
Let
P
P
P
be an interior point of an acute-angled triangle
A
B
C
ABC
A
BC
. The line
B
P
BP
BP
meets the line
A
C
AC
A
C
at
E
E
E
, and the line
C
P
CP
CP
meets the line
A
B
AB
A
B
at
F
F
F
. The lines
A
P
AP
A
P
and
E
F
EF
EF
intersect each other at
D
D
D
. Let
K
K
K
be the foot of the perpendicular from the point
D
D
D
to the line
B
C
BC
BC
. Show that the line
K
D
KD
KD
bisects the angle
∠
E
K
F
\angle EKF
∠
E
K
F
.
2
1
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Find all primes
Find all primes
p
,
q
p,q
p
,
q
and even
n
>
2
n>2
n
>
2
such that
p
n
+
p
n
−
1
+
.
.
.
+
1
=
q
2
+
q
+
1
p^n+p^{n-1}+...+1=q^2+q+1
p
n
+
p
n
−
1
+
...
+
1
=
q
2
+
q
+
1
.