MathDB
Problems
Contests
National and Regional Contests
India Contests
India IMO Training Camp
2007 India IMO Training Camp
2007 India IMO Training Camp
Part of
India IMO Training Camp
Subcontests
(3)
2
2
Hide problems
a+b<=c+1, b+c<=a+1, c+a<=b+1 implies \sum a^2<=2abc+1
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be non-negative real numbers such that
a
+
b
≤
c
+
1
,
b
+
c
≤
a
+
1
a+b\leq c+1, b+c\leq a+1
a
+
b
≤
c
+
1
,
b
+
c
≤
a
+
1
and
c
+
a
≤
b
+
1.
c+a\leq b+1.
c
+
a
≤
b
+
1.
Show that
a
2
+
b
2
+
c
2
≤
2
a
b
c
+
1.
a^2+b^2+c^2\leq 2abc+1.
a
2
+
b
2
+
c
2
≤
2
ab
c
+
1.
Find all integer solutions to y^2=x^3-p^2x, p is a prime
Find all integer solutions
(
x
,
y
)
(x,y)
(
x
,
y
)
of the equation
y
2
=
x
3
−
p
2
x
,
y^2=x^3-p^2x,
y
2
=
x
3
−
p
2
x
,
where
p
p
p
is a prime such that
p
≡
3
m
o
d
4.
p\equiv 3 \mod 4.
p
≡
3
mod
4.
3
3
Hide problems
Find the number of "balanced" n-strings
Given a finite string
S
S
S
of symbols
X
X
X
and
O
O
O
, we denote
Δ
(
s
)
\Delta(s)
Δ
(
s
)
as the number of
X
′
X'
X
′
s in
S
S
S
minus the number of
O
′
O'
O
′
s (For example,
Δ
(
X
O
O
X
O
O
X
)
=
−
1
\Delta(XOOXOOX)=-1
Δ
(
XOOXOOX
)
=
−
1
). We call a string
S
S
S
balanced if every sub-string
T
T
T
of (consecutive symbols)
S
S
S
has the property
−
1
≤
Δ
(
T
)
≤
2.
-1\leq \Delta(T)\leq 2.
−
1
≤
Δ
(
T
)
≤
2.
(Thus
X
O
O
X
O
O
X
XOOXOOX
XOOXOOX
is not balanced, since it contains the sub-string
O
O
X
O
O
OOXOO
OOXOO
whose
Δ
\Delta
Δ
value is
−
3.
-3.
−
3.
Find, with proof, the number of balanced strings of length
n
n
n
.
S={1,2,...,n}; T_{f}(j)=1,0; Determine sum(sum(T_f(j))
Let
X
\mathbb X
X
be the set of all bijective functions from the set
S
=
{
1
,
2
,
⋯
,
n
}
S=\{1,2,\cdots, n\}
S
=
{
1
,
2
,
⋯
,
n
}
to itself. For each
f
∈
X
,
f\in \mathbb X,
f
∈
X
,
define
T
f
(
j
)
=
{
1
,
if
f
(
12
)
(
j
)
=
j
,
0
,
otherwise
T_f(j)=\left\{\begin{aligned} 1, \ \ \ & \text{if} \ \ f^{(12)}(j)=j,\\ 0, \ \ \ & \text{otherwise}\end{aligned}\right.
T
f
(
j
)
=
{
1
,
0
,
if
f
(
12
)
(
j
)
=
j
,
otherwise
Determine
∑
f
∈
X
∑
j
=
1
n
T
f
(
j
)
.
\sum_{f\in\mathbb X}\sum_{j=1}^nT_{f}(j).
∑
f
∈
X
∑
j
=
1
n
T
f
(
j
)
.
(Here
f
(
k
)
(
x
)
=
f
(
f
(
k
−
1
)
(
x
)
)
f^{(k)}(x)=f(f^{(k-1)}(x))
f
(
k
)
(
x
)
=
f
(
f
(
k
−
1
)
(
x
))
for all
k
≥
2.
k\geq 2.
k
≥
2.
)
Find all functional equations satisfying f(x+y)+f(x)f(y)=...
Find all function(s)
f
:
R
→
R
f:\mathbb R\to\mathbb R
f
:
R
→
R
satisfying the equation
f
(
x
+
y
)
+
f
(
x
)
f
(
y
)
=
(
1
+
y
)
f
(
x
)
+
(
1
+
x
)
f
(
y
)
+
f
(
x
y
)
;
f(x+y)+f(x)f(y)=(1+y)f(x)+(1+x)f(y)+f(xy);
f
(
x
+
y
)
+
f
(
x
)
f
(
y
)
=
(
1
+
y
)
f
(
x
)
+
(
1
+
x
)
f
(
y
)
+
f
(
x
y
)
;
For all
x
,
y
∈
R
.
x,y\in\mathbb R.
x
,
y
∈
R
.
1
1
Hide problems
The common tangent to (I), (N) is parallel to Euler line
Show that in a non-equilateral triangle, the following statements are equivalent:
(
a
)
(a)
(
a
)
The angles of the triangle are in arithmetic progression.
(
b
)
(b)
(
b
)
The common tangent to the Nine-point circle and the Incircle is parallel to the Euler Line.