MathDB
Problems
Contests
National and Regional Contests
India Contests
India IMO Training Camp
2002 India IMO Training Camp
2002 India IMO Training Camp
Part of
India IMO Training Camp
Subcontests
(14)
18
1
Hide problems
Number of paths from A to C which cross AC from below
Consider the square grid with
A
=
(
0
,
0
)
A=(0,0)
A
=
(
0
,
0
)
and
C
=
(
n
,
n
)
C=(n,n)
C
=
(
n
,
n
)
at its diagonal ends. Paths from
A
A
A
to
C
C
C
are composed of moves one unit to the right or one unit up. Let
C
n
C_n
C
n
(n-th catalan number) be the number of paths from
A
A
A
to
C
C
C
which stay on or below the diagonal
A
C
AC
A
C
. Show that the number of paths from
A
A
A
to
C
C
C
which cross
A
C
AC
A
C
from below at most twice is equal to
C
n
+
2
−
2
C
n
+
1
+
C
n
C_{n+2}-2C_{n+1}+C_n
C
n
+
2
−
2
C
n
+
1
+
C
n
14
1
Hide problems
(x+y+z)^2 \equiv axyz mod p
Let
p
p
p
be an odd prime and let
a
a
a
be an integer not divisible by
p
p
p
. Show that there are
p
2
+
1
p^2+1
p
2
+
1
triples of integers
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
with
0
≤
x
,
y
,
z
<
p
0 \le x,y,z < p
0
≤
x
,
y
,
z
<
p
and such that
(
x
+
y
+
z
)
2
≡
a
x
y
z
(
m
o
d
p
)
(x+y+z)^2 \equiv axyz \pmod p
(
x
+
y
+
z
)
2
≡
a
x
yz
(
mod
p
)
12
1
Hide problems
N_0 is the union of all disjoint olympic sets
Let
a
,
b
a,b
a
,
b
be integers with
0
<
a
<
b
0<a<b
0
<
a
<
b
. A set
{
x
,
y
,
z
}
\{x,y,z\}
{
x
,
y
,
z
}
of non-negative integers is olympic if
x
<
y
<
z
x<y<z
x
<
y
<
z
and if
{
z
−
y
,
y
−
x
}
=
{
a
,
b
}
\{z-y,y-x\}=\{a,b\}
{
z
−
y
,
y
−
x
}
=
{
a
,
b
}
. Show that the set of all non-negative integers is the union of pairwise disjoint olympic sets.
9
1
Hide problems
On how many days did they eat the same kind of fruit?
On each day of their tour of the West Indies, Sourav and Srinath have either an apple or an orange for breakfast. Sourav has oranges for the first
m
m
m
days, apples for the next
m
m
m
days, followed by oranges for the next
m
m
m
days, and so on. Srinath has oranges for the first
n
n
n
days, apples for the next
n
n
n
days, followed by oranges for the next
n
n
n
days, and so on. If
gcd
(
m
,
n
)
=
1
\gcd(m,n)=1
g
cd
(
m
,
n
)
=
1
, and if the tour lasted for
m
n
mn
mn
days, on how many days did they eat the same kind of fruit?
8
1
Hide problems
Well known property of sigma(n)
Let
σ
(
n
)
=
∑
d
∣
n
d
\sigma(n)=\sum_{d|n} d
σ
(
n
)
=
∑
d
∣
n
d
, the sum of positive divisors of an integer
n
>
0
n>0
n
>
0
.(a) Show that
σ
(
m
n
)
=
σ
(
m
)
σ
(
n
)
\sigma(mn)=\sigma(m)\sigma(n)
σ
(
mn
)
=
σ
(
m
)
σ
(
n
)
for positive integers
m
m
m
and
n
n
n
with
g
c
d
(
m
,
n
)
=
1
gcd(m,n)=1
g
c
d
(
m
,
n
)
=
1
(b) Find all positive integers
n
n
n
such that
σ
(
n
)
\sigma(n)
σ
(
n
)
is a power of
2
2
2
.
6
1
Hide problems
Determine the number of n-tuples of integers
Determine the number of
n
n
n
-tuples of integers
(
x
1
,
x
2
,
⋯
,
x
n
)
(x_1,x_2,\cdots ,x_n)
(
x
1
,
x
2
,
⋯
,
x
n
)
such that
∣
x
i
∣
≤
10
|x_i| \le 10
∣
x
i
∣
≤
10
for each
1
≤
i
≤
n
1\le i \le n
1
≤
i
≤
n
and
∣
x
i
−
x
j
∣
≤
10
|x_i-x_j| \le 10
∣
x
i
−
x
j
∣
≤
10
for
1
≤
i
,
j
≤
n
1 \le i,j \le n
1
≤
i
,
j
≤
n
.
3
1
Hide problems
Quadratics of the form ax^2+2bx+c
Let
X
=
{
2
m
3
n
∣
0
≤
m
,
n
≤
9
}
X=\{2^m3^n|0 \le m, \ n \le 9 \}
X
=
{
2
m
3
n
∣0
≤
m
,
n
≤
9
}
. How many quadratics are there of the form
a
x
2
+
2
b
x
+
c
ax^2+2bx+c
a
x
2
+
2
b
x
+
c
, with equal roots, and such that
a
,
b
,
c
a,b,c
a
,
b
,
c
are distinct elements of
X
X
X
?
17
1
Hide problems
(1+iT)^n=f(T)+ig(T) where i is the square root of -1
Let
n
n
n
be a positive integer and let
(
1
+
i
T
)
n
=
f
(
T
)
+
i
g
(
T
)
(1+iT)^n=f(T)+ig(T)
(
1
+
i
T
)
n
=
f
(
T
)
+
i
g
(
T
)
where
i
i
i
is the square root of
−
1
-1
−
1
, and
f
f
f
and
g
g
g
are polynomials with real coefficients. Show that for any real number
k
k
k
the equation
f
(
T
)
+
k
g
(
T
)
=
0
f(T)+kg(T)=0
f
(
T
)
+
k
g
(
T
)
=
0
has only real roots.
7
1
Hide problems
Construct triangle, incircle and ninepoint circle given
Given two distinct circles touching each other internally, show how to construct a triangle with the inner circle as its incircle and the outer circle as its nine point circle.
2
1
Hide problems
2002 consecutive postve intgrs containing exactly 150 primes
Show that there is a set of
2002
2002
2002
consecutive positive integers containing exactly
150
150
150
primes. (You may use the fact that there are
168
168
168
primes less than
1000
1000
1000
)
20
1
Hide problems
an easy cyclic inequality
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers. Prove that
a
b
+
b
c
+
c
a
≥
c
+
a
c
+
b
+
a
+
b
a
+
c
+
b
+
c
b
+
a
\frac{a}b+\frac{b}c+\frac{c}a \geq \frac{c+a}{c+b}+\frac{a+b}{a+c}+\frac{b+c}{b+a}
b
a
+
c
b
+
a
c
≥
c
+
b
c
+
a
+
a
+
c
a
+
b
+
b
+
a
b
+
c
21
1
Hide problems
p^n has 2002 consecutive zeros
Given a prime
p
p
p
, show that there exists a positive integer
n
n
n
such that the decimal representation of
p
n
p^n
p
n
has a block of
2002
2002
2002
consecutive zeros.
1
1
Hide problems
gamma1,2,3 are semicircles show that AB=DE
Let
A
,
B
A,B
A
,
B
and
C
C
C
be three points on a line with
B
B
B
between
A
A
A
and
C
C
C
. Let
Γ
1
,
Γ
2
,
Γ
3
\Gamma_1,\Gamma_2, \Gamma_3
Γ
1
,
Γ
2
,
Γ
3
be semicircles, all on the same side of
A
C
AC
A
C
and with
A
C
,
A
B
,
B
C
AC,AB,BC
A
C
,
A
B
,
BC
as diameters, respectively. Let
l
l
l
be the line perpendicular to
A
C
AC
A
C
through
B
B
B
. Let
Γ
\Gamma
Γ
be the circle which is tangent to the line
l
l
l
, tangent to
Γ
1
\Gamma_1
Γ
1
internally, and tangent to
Γ
3
\Gamma_3
Γ
3
externally. Let
D
D
D
be the point of contact of
Γ
\Gamma
Γ
and
Γ
3
\Gamma_3
Γ
3
. The diameter of
Γ
\Gamma
Γ
through
D
D
D
meets
l
l
l
in
E
E
E
. Show that
A
B
=
D
E
AB=DE
A
B
=
D
E
.
5
1
Hide problems
easy inequality with a^2+b^2+c^2=3abc
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive reals such that
a
2
+
b
2
+
c
2
=
3
a
b
c
a^2+b^2+c^2=3abc
a
2
+
b
2
+
c
2
=
3
ab
c
. Prove that
a
b
2
c
2
+
b
c
2
a
2
+
c
a
2
b
2
≥
9
a
+
b
+
c
\frac{a}{b^2c^2}+\frac{b}{c^2a^2}+\frac{c}{a^2b^2} \geq \frac{9}{a+b+c}
b
2
c
2
a
+
c
2
a
2
b
+
a
2
b
2
c
≥
a
+
b
+
c
9