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Contests
National and Regional Contests
India Contests
India National Olympiad
1987 India National Olympiad
1987 India National Olympiad
Part of
India National Olympiad
Subcontests
(9)
9
1
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Indian Mathematical Olympiad 1987 - Problem 9
Prove that any triangle having two equal internal angle bisectors (each measured from a vertex to the opposite side) is isosceles.
8
1
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Indian Mathematical Olympiad 1987 - Problem 8
Three congruent circles have a common point
O
O
O
and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incentre and the circumcentre of the triangle and the common point
O
O
O
are collinear.
7
1
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Indian Mathematical Olympiad 1987 - Problem 7
Construct the
△
A
B
C
\triangle ABC
△
A
BC
, given
h
a
h_a
h
a
,
h
b
h_b
h
b
(the altitudes from
A
A
A
and
B
B
B
) and
m
a
m_a
m
a
, the median from the vertex
A
A
A
.
6
1
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Indian Mathematical Olympiad 1987 - Problem 6
Prove that if coefficients of the quadratic equation ax^2\plus{}bx\plus{}c\equal{}0 are odd integers, then the roots of the equation cannot be rational numbers.
5
1
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Indian Mathematical Olympiad 1987 - Problem 5
Find a finite sequence of 16 numbers such that: (a) it reads same from left to right as from right to left. (b) the sum of any 7 consecutive terms is \minus{}1, (c) the sum of any 11 consecutive terms is \plus{}1.
4
1
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Indian Mathematical Olympiad 1987 - Problem 4
If
x
x
x
,
y
y
y
,
z
z
z
, and
n
n
n
are natural numbers, and
n
≥
z
n\geq z
n
≥
z
then prove that the relation x^n \plus{} y^n \equal{} z^n does not hold.
3
1
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Indian Mathematical Olympiad 1987 - Problem 3
Let
T
T
T
be the set of all triplets
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of integers such that
1
≤
a
<
b
<
c
≤
6
1 \leq a < b < c \leq 6
1
≤
a
<
b
<
c
≤
6
For each triplet
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
in
T
T
T
, take number
a
⋅
b
⋅
c
a\cdot b \cdot c
a
⋅
b
⋅
c
. Add all these numbers corresponding to all the triplets in
T
T
T
. Prove that the answer is divisible by 7.
2
1
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Indian Mathematical Olympiad 1987 - Problem 2
Determine the largest number in the infinite sequence
1
,
2
2
,
3
3
,
4
4
,
…
,
n
n
,
…
1, \sqrt[2]{2},\sqrt[3]{3},\sqrt[4]{4}, \dots, \sqrt[n]{n},\dots
1
,
2
2
,
3
3
,
4
4
,
…
,
n
n
,
…
1
1
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Indian Mathematical Olympiad 1987 - Problem 1
Given
m
m
m
and
n
n
n
as relatively prime positive integers greater than one, show that
log
10
m
log
10
n
\frac{\log_{10} m}{\log_{10} n}
lo
g
10
n
lo
g
10
m
is not a rational number.