MathDB
Problems
Contests
National and Regional Contests
India Contests
India IMO Training Camp
2005 India IMO Training Camp
2005 India IMO Training Camp
Part of
India IMO Training Camp
Subcontests
(3)
3
2
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Ascent and descent
A merida path of order
2
n
2n
2
n
is a lattice path in the first quadrant of
x
y
xy
x
y
- plane joining
(
0
,
0
)
(0,0)
(
0
,
0
)
to
(
2
n
,
0
)
(2n,0)
(
2
n
,
0
)
using three kinds of steps
U
=
(
1
,
1
)
U=(1,1)
U
=
(
1
,
1
)
,
D
=
(
1
,
−
1
)
D= (1,-1)
D
=
(
1
,
−
1
)
and
L
=
(
2
,
0
)
L= (2,0)
L
=
(
2
,
0
)
, i.e.
U
U
U
joins
x
,
y
)
x,y)
x
,
y
)
to
(
x
+
1
,
y
+
1
)
(x+1,y+1)
(
x
+
1
,
y
+
1
)
etc... An ascent in a merida path is a maximal string of consecutive steps of the form
U
U
U
. If
S
(
n
,
k
)
S(n,k)
S
(
n
,
k
)
denotes the number of merdia paths of order
2
n
2n
2
n
with exactly
k
k
k
ascents, compute
S
(
n
,
1
)
S(n,1)
S
(
n
,
1
)
and
S
(
n
,
n
−
1
)
S(n,n-1)
S
(
n
,
n
−
1
)
.
A trignometric eqn
For real numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
not all equal to
0
0
0
, define a real function
f
(
x
)
=
a
+
b
cos
2
x
+
c
sin
5
x
+
d
cos
8
x
f(x) = a +b\cos{2x} + c\sin{5x} +d \cos{8x}
f
(
x
)
=
a
+
b
cos
2
x
+
c
sin
5
x
+
d
cos
8
x
. Suppose
f
(
t
)
=
4
a
f(t) = 4a
f
(
t
)
=
4
a
for some real
t
t
t
. prove that there exist a real number
s
s
s
s.t.
f
(
s
)
<
0
f(s)<0
f
(
s
)
<
0
2
3
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Prove that one can find...
Prove that one can find a
n
0
∈
N
n_{0} \in \mathbb{N}
n
0
∈
N
such that
∀
m
≥
n
0
\forall m \geq n_{0}
∀
m
≥
n
0
, there exist three positive integers
a
a
a
,
b
b
b
,
c
c
c
such that (i)
m
3
<
a
<
b
<
c
<
(
m
+
1
)
3
m^3 < a < b < c < (m+1)^3
m
3
<
a
<
b
<
c
<
(
m
+
1
)
3
; (ii)
a
b
c
abc
ab
c
is the cube of an integer.
Prove that there exist integers st....
Given real numbers
a
,
α
,
β
,
σ
a
n
d
ϱ
a,\alpha,\beta, \sigma \ and \ \varrho
a
,
α
,
β
,
σ
an
d
ϱ
s.t.
σ
,
ϱ
>
0
\sigma, \varrho > 0
σ
,
ϱ
>
0
and
σ
ϱ
=
1
16
\sigma \varrho = \frac{1}{16}
σ
ϱ
=
16
1
, prove that there exist integers
x
x
x
and
y
y
y
s.t.
−
σ
≤
(
x
+
α
(
a
x
+
y
+
β
)
≤
ϱ
- \sigma \leq (x+\alpha_(ax + y + \beta ) \leq \varrho
−
σ
≤
(
x
+
α
(
a
x
+
y
+
β
)
≤
ϱ
Phi(n)
Determine all positive integers
n
>
2
n > 2
n
>
2
, such that
1
2
φ
(
n
)
≡
1
(
m
o
d
6
)
\frac{1}{2} \varphi(n) \equiv 1 ( \bmod 6)
2
1
φ
(
n
)
≡
1
(
mod
6
)
1
4
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