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Contests
National and Regional Contests
India Contests
India IMO Training Camp
2011 India IMO Training Camp
2011 India IMO Training Camp
Part of
India IMO Training Camp
Subcontests
(3)
3
4
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1
3
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Prove that the radius of the circle..
Let
A
B
C
ABC
A
BC
be a triangle each of whose angles is greater than
3
0
∘
30^{\circ}
3
0
∘
. Suppose a circle centered with
P
P
P
cuts segments
B
C
BC
BC
in
T
,
Q
;
C
A
T,Q; CA
T
,
Q
;
C
A
in
K
,
L
K,L
K
,
L
and
A
B
AB
A
B
in
M
,
N
M,N
M
,
N
such that they are on a circle in counterclockwise direction in that order.Suppose further
P
Q
K
,
P
L
M
,
P
N
T
PQK,PLM,PNT
PQ
K
,
P
L
M
,
PNT
are equilateral. Prove that:
a
)
a)
a
)
The radius of the circle is
2
a
b
c
a
2
+
b
2
+
c
2
+
4
3
S
\frac{2abc}{a^2+b^2+c^2+4\sqrt{3}S}
a
2
+
b
2
+
c
2
+
4
3
S
2
ab
c
where
S
S
S
is area.
b
)
a
⋅
A
P
=
b
⋅
B
P
=
c
⋅
P
C
.
b) a\cdot AP=b\cdot BP=c\cdot PC.
b
)
a
⋅
A
P
=
b
⋅
BP
=
c
⋅
PC
.
Indian TST Number theory
Find all positive integer
n
n
n
satisfying the conditions
a
)
n
2
=
(
a
+
1
)
3
−
a
3
a)n^2=(a+1)^3-a^3
a
)
n
2
=
(
a
+
1
)
3
−
a
3
b
)
2
n
+
119
b)2n+119
b
)
2
n
+
119
is a perfect square.
ABC be an acute-angled
Let
A
B
C
ABC
A
BC
be an acute-angled triangle. Let
A
D
,
B
E
,
C
F
AD,BE,CF
A
D
,
BE
,
CF
be internal bisectors with
D
,
E
,
F
D, E, F
D
,
E
,
F
on
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
respectively. Prove that
E
F
B
C
+
F
D
C
A
+
D
E
A
B
≥
1
+
r
R
\frac{EF}{BC}+\frac{FD}{CA}+\frac{DE}{AB}\geq 1+\frac{r}{R}
BC
EF
+
C
A
F
D
+
A
B
D
E
≥
1
+
R
r
2
2
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Indian TST Algebra
Suppose
a
1
,
…
,
a
n
a_1,\ldots,a_n
a
1
,
…
,
a
n
are non-integral real numbers for
n
≥
2
n\geq 2
n
≥
2
such that
a
1
k
+
…
+
a
n
k
{a_1}^k+\ldots+{a_n}^k
a
1
k
+
…
+
a
n
k
is an integer for all integers
1
≤
k
≤
n
1\leq k\leq n
1
≤
k
≤
n
. Prove that none of
a
1
,
…
,
a
n
a_1,\ldots,a_n
a
1
,
…
,
a
n
is rational.
No integer of form n^7 + 7 is a perfect square
Prove that for no integer
n
n
n
is n^7 \plus{} 7 a perfect square.