MathDB
Problems
Contests
National and Regional Contests
India Contests
India IMO Training Camp
2010 India IMO Training Camp
2010 India IMO Training Camp
Part of
India IMO Training Camp
Subcontests
(12)
12
1
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Indian Team Selection Test 2010 ST4 P3
Prove that there are infinitely many positive integers
m
m
m
for which there exists consecutive odd positive integers
p
m
<
q
m
p_m<q_m
p
m
<
q
m
such that
p
m
2
+
p
m
q
m
+
q
m
2
p_m^2+p_mq_m+q_m^2
p
m
2
+
p
m
q
m
+
q
m
2
and
p
m
2
+
m
⋅
p
m
q
m
+
q
m
2
p_m^2+m\cdot p_mq_m+q_m^2
p
m
2
+
m
⋅
p
m
q
m
+
q
m
2
are both perfect squares. If
m
1
,
m
2
m_1, m_2
m
1
,
m
2
are two positive integers satisfying this condition, then we have
p
m
1
≠
p
m
2
p_{m_1}\neq p_{m_2}
p
m
1
=
p
m
2
11
1
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Indian Team Selection Test 2010 ST4 P2
Find all functions
f
:
R
⟶
R
f:\mathbb{R}\longrightarrow\mathbb{R}
f
:
R
⟶
R
such that
f
(
x
+
y
)
+
x
y
=
f
(
x
)
f
(
y
)
f(x+y)+xy=f(x)f(y)
f
(
x
+
y
)
+
x
y
=
f
(
x
)
f
(
y
)
for all reals
x
,
y
x, y
x
,
y
10
1
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Indian Team Selection Test 2010 ST4 P1
Let
A
B
C
ABC
A
BC
be a triangle. Let
Ω
\Omega
Ω
be the brocard point. Prove that
(
A
Ω
B
C
)
2
+
(
B
Ω
A
C
)
2
+
(
C
Ω
A
B
)
2
≥
1
\left(\frac{A\Omega}{BC}\right)^2+\left(\frac{B\Omega}{AC}\right)^2+\left(\frac{C\Omega}{AB}\right)^2\ge 1
(
BC
A
Ω
)
2
+
(
A
C
B
Ω
)
2
+
(
A
B
C
Ω
)
2
≥
1
5
1
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Indian Team Selection Test 2010 ST2 P2
Given an integer
k
>
1
k>1
k
>
1
, show that there exist an integer an
n
>
1
n>1
n
>
1
and distinct positive integers
a
1
,
a
2
,
⋯
a
n
a_1,a_2,\cdots a_n
a
1
,
a
2
,
⋯
a
n
, all greater than
1
1
1
, such that the sums
∑
j
=
1
n
a
j
\sum_{j=1}^n a_j
∑
j
=
1
n
a
j
and
∑
j
=
1
n
ϕ
(
a
j
)
\sum_{j=1}^n \phi (a_j)
∑
j
=
1
n
ϕ
(
a
j
)
are both
k
k
k
-th powers of some integers. (Here
ϕ
(
m
)
\phi (m)
ϕ
(
m
)
denotes the number of positive integers less than
m
m
m
and relatively prime to
m
m
m
.)
1
1
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Perpendicular
Let
A
B
C
ABC
A
BC
be a triangle in which
B
C
<
A
C
BC<AC
BC
<
A
C
. Let
M
M
M
be the mid-point of
A
B
AB
A
B
,
A
P
AP
A
P
be the altitude from
A
A
A
on
B
C
BC
BC
, and
B
Q
BQ
BQ
be the altitude from
B
B
B
on to
A
C
AC
A
C
. Suppose that
Q
P
QP
QP
produced meets
A
B
AB
A
B
(extended) at
T
T
T
. If
H
H
H
is the orthocenter of
A
B
C
ABC
A
BC
, prove that
T
H
TH
T
H
is perpendicular to
C
M
CM
CM
.
3
1
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Indian Team Selection Test 2010 ST1 P3
For any integer
n
≥
2
n\ge 2
n
≥
2
, let
N
(
n
)
N(n)
N
(
n
)
be the maximum number of triples
(
a
j
,
b
j
,
c
j
)
,
j
=
1
,
2
,
3
,
⋯
,
N
(
n
)
,
(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),
(
a
j
,
b
j
,
c
j
)
,
j
=
1
,
2
,
3
,
⋯
,
N
(
n
)
,
consisting of non-negative integers
a
j
,
b
j
,
c
j
a_j,b_j,c_j
a
j
,
b
j
,
c
j
(not necessarily distinct) such that the following two conditions are satisfied:(a)
a
j
+
b
j
+
c
j
=
n
,
a_j+b_j+c_j=n,
a
j
+
b
j
+
c
j
=
n
,
for all
j
=
1
,
2
,
3
,
⋯
N
(
n
)
j=1,2,3,\cdots N(n)
j
=
1
,
2
,
3
,
⋯
N
(
n
)
; (b)
j
≠
k
j\neq k
j
=
k
, then
a
j
≠
a
k
a_j\neq a_k
a
j
=
a
k
,
b
j
≠
b
k
b_j\neq b_k
b
j
=
b
k
and
c
j
≠
c
k
c_j\neq c_k
c
j
=
c
k
.Determine
N
(
n
)
N(n)
N
(
n
)
for all
n
≥
2
n\ge 2
n
≥
2
.
2
1
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Indian Team Selection Test 2010 ST1 P2
Two polynomials
P
(
x
)
=
x
4
+
a
x
3
+
b
x
2
+
c
x
+
d
P(x)=x^4+ax^3+bx^2+cx+d
P
(
x
)
=
x
4
+
a
x
3
+
b
x
2
+
c
x
+
d
and
Q
(
x
)
=
x
2
+
p
x
+
q
Q(x)=x^2+px+q
Q
(
x
)
=
x
2
+
p
x
+
q
have real coefficients, and
I
I
I
is an interval on the real line of length greater than
2
2
2
. Suppose
P
(
x
)
P(x)
P
(
x
)
and
Q
(
x
)
Q(x)
Q
(
x
)
take negative values on
I
I
I
, and they take non-negative values outside
I
I
I
. Prove that there exists a real number
x
0
x_0
x
0
such that
P
(
x
0
)
<
Q
(
x
0
)
P(x_0)<Q(x_0)
P
(
x
0
)
<
Q
(
x
0
)
.
7
1
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Indian Team Selection Test 2010 ST3 P1
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilaterla and let
E
E
E
be the point of intersection of its diagonals
A
C
AC
A
C
and
B
D
BD
B
D
. Suppose
A
D
AD
A
D
and
B
C
BC
BC
meet in
F
F
F
. Let the midpoints of
A
B
AB
A
B
and
C
D
CD
C
D
be
G
G
G
and
H
H
H
respectively. If
Γ
\Gamma
Γ
is the circumcircle of triangle
E
G
H
EGH
EG
H
, prove that
F
E
FE
FE
is tangent to
Γ
\Gamma
Γ
.
8
1
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Indian Team Selection Test 2010 ST3 P2
Call a positive integer good if either
N
=
1
N=1
N
=
1
or
N
N
N
can be written as product of even number of prime numbers, not necessarily distinct. Let
P
(
x
)
=
(
x
−
a
)
(
x
−
b
)
,
P(x)=(x-a)(x-b),
P
(
x
)
=
(
x
−
a
)
(
x
−
b
)
,
where
a
,
b
a,b
a
,
b
are positive integers.(a) Show that there exist distinct positive integers
a
,
b
a,b
a
,
b
such that
P
(
1
)
,
P
(
2
)
,
⋯
,
P
(
2010
)
P(1),P(2),\cdots ,P(2010)
P
(
1
)
,
P
(
2
)
,
⋯
,
P
(
2010
)
are all good numbers. (b) Suppose
a
,
b
a,b
a
,
b
are such that
P
(
n
)
P(n)
P
(
n
)
is a good number for all positive integers
n
n
n
. Prove that
a
=
b
a=b
a
=
b
.
4
1
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Indian Team Selection Test 2010 ST2 P1
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
a
b
+
b
c
+
c
a
≤
3
a
b
c
ab+bc+ca\le 3abc
ab
+
b
c
+
c
a
≤
3
ab
c
. Prove that
a
2
+
b
2
a
+
b
+
b
2
+
c
2
b
+
c
+
c
2
+
a
2
c
+
a
+
3
≤
2
(
a
+
b
+
b
+
c
+
c
+
a
)
\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\le \sqrt{2} (\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})
a
+
b
a
2
+
b
2
+
b
+
c
b
2
+
c
2
+
c
+
a
c
2
+
a
2
+
3
≤
2
(
a
+
b
+
b
+
c
+
c
+
a
)
6
1
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Indian Team Selection Test 2010 ST2 P3
Let
n
≥
2
n\ge 2
n
≥
2
be a given integer. Show that the number of strings of length
n
n
n
consisting of
0
′
0'
0
′
s and
1
′
1'
1
′
s such that there are equal number of
00
00
00
and
11
11
11
blocks in each string is equal to
2
(
n
−
2
⌊
n
−
2
2
⌋
)
2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}
2
(
⌊
2
n
−
2
⌋
n
−
2
)
9
1
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Indian Team Selection Test 2010 ST3 P3
Let
A
=
(
a
j
k
)
A=(a_{jk})
A
=
(
a
jk
)
be a
10
×
10
10\times 10
10
×
10
array of positive real numbers such that the sum of numbers in row as well as in each column is
1
1
1
. Show that there exists
j
<
k
j<k
j
<
k
and
l
<
m
l<m
l
<
m
such that
a
j
l
a
k
m
+
a
j
m
a
k
l
≥
1
50
a_{jl}a_{km}+a_{jm}a_{kl}\ge \frac{1}{50}
a
j
l
a
km
+
a
jm
a
k
l
≥
50
1