MathDB
Problems
Contests
National and Regional Contests
India Contests
India IMO Training Camp
2006 India IMO Training Camp
2006 India IMO Training Camp
Part of
India IMO Training Camp
Subcontests
(3)
3
3
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Union of the triangular regions BAD,CBE,ACF covers ABC
Let
A
B
C
ABC
A
BC
be an equilateral triangle, and let
D
,
E
D,E
D
,
E
and
F
F
F
be points on
B
C
,
B
A
BC,BA
BC
,
B
A
and
A
B
AB
A
B
respectively. Let
∠
B
A
D
=
α
,
∠
C
B
E
=
β
\angle BAD= \alpha, \angle CBE=\beta
∠
B
A
D
=
α
,
∠
CBE
=
β
and
∠
A
C
F
=
γ
\angle ACF =\gamma
∠
A
CF
=
γ
. Prove that if
α
+
β
+
γ
≥
12
0
∘
\alpha+\beta+\gamma \geq 120^\circ
α
+
β
+
γ
≥
12
0
∘
, then the union of the triangular regions
B
A
D
,
C
B
E
,
A
C
F
BAD,CBE,ACF
B
A
D
,
CBE
,
A
CF
covers the triangle
A
B
C
ABC
A
BC
.
n arithmetic progressions of integers each of k terms
Let
A
1
,
A
2
,
⋯
,
A
n
A_1,A_2,\cdots , A_n
A
1
,
A
2
,
⋯
,
A
n
be arithmetic progressions of integers, each of
k
k
k
terms, such that any two of these arithmetic progressions have at least two common elements. Suppose
b
b
b
of these arithmetic progressions have common difference
d
1
d_1
d
1
and the remaining arithmetic progressions have common difference
d
2
d_2
d
2
where
0
<
b
<
n
0<b<n
0
<
b
<
n
. Prove that
b
≤
2
(
k
−
d
2
g
c
d
(
d
1
,
d
2
)
)
−
1.
b \le 2\left(k-\frac{d_2}{gcd(d_1,d_2)}\right)-1.
b
≤
2
(
k
−
g
c
d
(
d
1
,
d
2
)
d
2
)
−
1.
Set: Prove that intersection of all A_i's is not a null set
Let
A
1
,
A
2
,
…
,
A
n
A_1,A_2,\ldots,A_n
A
1
,
A
2
,
…
,
A
n
be subsets of a finite set
S
S
S
such that
∣
A
j
∣
=
8
|A_j|=8
∣
A
j
∣
=
8
for each
j
j
j
. For a subset
B
B
B
of
S
S
S
let
F
(
B
)
=
{
j
∣
1
≤
j
≤
n
and
A
j
⊂
B
}
F(B)=\{j \mid 1\le j\le n \ \ \text{and} \ A_j \subset B\}
F
(
B
)
=
{
j
∣
1
≤
j
≤
n
and
A
j
⊂
B
}
. Suppose for each subset
B
B
B
of
S
S
S
at least one of the following conditions holds(a)
∣
B
∣
>
25
|B| > 25
∣
B
∣
>
25
,(b) F(B)={\O},(c) \bigcap_{j\in F(B)} A_j \neq {\O}.Prove that A_1\cap A_2 \cap \cdots \cap A_n \neq {\O}.
2
3
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d_7^2+d_{15}^2=d_{16}^2, d_{17}=?
the positive divisors
d
1
,
d
2
,
⋯
,
d
k
d_1,d_2,\cdots,d_k
d
1
,
d
2
,
⋯
,
d
k
of a positive integer
n
n
n
are ordered
1
=
d
1
<
d
2
<
⋯
<
d
k
=
n
1=d_1<d_2<\cdots<d_k=n
1
=
d
1
<
d
2
<
⋯
<
d
k
=
n
Suppose
d
7
2
+
d
15
2
=
d
16
2
d_7^2+d_{15}^2=d_{16}^2
d
7
2
+
d
15
2
=
d
16
2
. Find all possible values of
d
17
d_{17}
d
17
.
Collection of pairwise mutually disjoint p-element subsets
Let
p
p
p
be a prime number and let
X
X
X
be a finite set containing at least
p
p
p
elements. A collection of pairwise mutually disjoint
p
p
p
-element subsets of
X
X
X
is called a
p
p
p
-family. (In particular, the empty collection is a
p
p
p
-family.) Let
A
A
A
(respectively,
B
B
B
) denote the number of
p
p
p
-families having an even (respectively, odd) number of
p
p
p
-element subsets of
X
X
X
. Prove that
A
A
A
and
B
B
B
differ by a multiple of
p
p
p
.
Prove that there exists 3 integers satisfying the conditions
Let
u
j
k
u_{jk}
u
jk
be a real number for each
j
=
1
,
2
,
3
j=1,2,3
j
=
1
,
2
,
3
and each
k
=
1
,
2
k=1,2
k
=
1
,
2
and let
N
N
N
be an integer such that
max
1
≤
k
≤
2
∑
j
=
1
3
∣
u
j
k
∣
≤
N
\max_{1\le k \le 2} \sum_{j=1}^3 |u_{jk}| \leq N
1
≤
k
≤
2
max
j
=
1
∑
3
∣
u
jk
∣
≤
N
Let
M
M
M
and
l
l
l
be positive integers such that
l
2
<
(
M
+
1
)
3
l^2 <(M+1)^3
l
2
<
(
M
+
1
)
3
. Prove that there exist integers
ξ
1
,
ξ
2
,
ξ
3
\xi_1,\xi_2,\xi_3
ξ
1
,
ξ
2
,
ξ
3
not all zero, such that
max
1
≤
j
≤
3
ξ
j
≤
M
and
∣
∑
j
=
1
3
u
j
k
ξ
k
∣
≤
M
N
l
for k=1,2
\max_{1\le j \le 3}\xi_j \le M\ \ \ \ \text{and} \ \ \ \left|\sum_{j=1}^3 u_{jk}\xi_k\right| \le \frac{MN}{l} \ \ \ \ \text{for k=1,2}
1
≤
j
≤
3
max
ξ
j
≤
M
and
j
=
1
∑
3
u
jk
ξ
k
≤
l
MN
for k=1,2
1
4
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