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Problems
Contests
National and Regional Contests
India Contests
ISI Entrance Examination
2013 ISI Entrance Examination
2013 ISI Entrance Examination
Part of
ISI Entrance Examination
Subcontests
(8)
3
1
Hide problems
|f(x+y)-f(x-y)-y|<=y^2 for all x,y
Let
f
:
R
→
R
f:\mathbb R\to\mathbb R
f
:
R
→
R
satisfy
∣
f
(
x
+
y
)
−
f
(
x
−
y
)
−
y
∣
≤
y
2
|f(x+y)-f(x-y)-y|\leq y^2
∣
f
(
x
+
y
)
−
f
(
x
−
y
)
−
y
∣
≤
y
2
For all
(
x
,
y
)
∈
R
2
.
(x,y)\in\mathbb R^2.
(
x
,
y
)
∈
R
2
.
Show that
f
(
x
)
=
x
2
+
c
f(x)=\frac x2+c
f
(
x
)
=
2
x
+
c
where
c
c
c
is a constant.
8
1
Hide problems
ABCD square, AB along y=x+8, C,D on y=x^2 [ISI 2013/8]
Let
A
B
C
D
ABCD
A
BC
D
be a square such that
A
B
AB
A
B
lies along the line
y
=
x
+
8
,
y=x+8,
y
=
x
+
8
,
and
C
C
C
and
D
D
D
lie on the parabola
y
=
x
2
.
y=x^2.
y
=
x
2
.
Find all possible values of sidelength of the square.
5
1
Hide problems
AB=BC=r/2, AD diameter, find CD [ISI Entrance 2013/5]
Let
A
D
AD
A
D
be a diameter of a circle of radius
r
,
r,
r
,
and let
B
,
C
B,C
B
,
C
be points on the circle such that
A
B
=
B
C
=
r
2
AB=BC=\frac r2
A
B
=
BC
=
2
r
and
A
≠
C
.
A\neq C.
A
=
C
.
Find the ratio
C
D
r
.
\frac{CD}{r}.
r
C
D
.
6
1
Hide problems
polynomials p,q: p(x)^3-q(x)^3=p(x^3)-q(x^3)
Let
p
(
x
)
p(x)
p
(
x
)
and
q
(
x
)
q(x)
q
(
x
)
be two polynomials, both of which have their sum of coefficients equal to
s
.
s.
s
.
Let
p
,
q
p,q
p
,
q
satisfy
p
(
x
)
3
−
q
(
x
)
3
=
p
(
x
3
)
−
q
(
x
3
)
.
p(x)^3-q(x)^3=p(x^3)-q(x^3).
p
(
x
)
3
−
q
(
x
)
3
=
p
(
x
3
)
−
q
(
x
3
)
.
Show that (i) There exists an integer
a
≥
1
a\geq1
a
≥
1
and a polynomial
r
(
x
)
r(x)
r
(
x
)
with
r
(
1
)
≠
0
r(1)\neq0
r
(
1
)
=
0
such that
p
(
x
)
−
q
(
x
)
=
(
x
−
1
)
a
r
(
x
)
.
p(x)-q(x)=(x-1)^ar(x).
p
(
x
)
−
q
(
x
)
=
(
x
−
1
)
a
r
(
x
)
.
(ii) Show that
s
2
=
3
a
−
1
,
s^2=3^{a-1},
s
2
=
3
a
−
1
,
where
a
a
a
is described as above.
7
1
Hide problems
N(N-101) is a perfect square, ISI Entrance Exam 2013/7
Find all natural numbers
N
N
N
for which
N
(
N
−
101
)
N(N-101)
N
(
N
−
101
)
is a perfect square.
4
1
Hide problems
Some player writes all others' names
In a badminton tournament, each of
n
n
n
players play all the other
n
−
1
n-1
n
−
1
players. Each game results in either a win, or a loss. The players then write down the names of those whom they defeated, and also of those who they defeated. For example, if
A
A
A
beats
B
B
B
and
B
B
B
beats
C
,
C,
C
,
then
A
A
A
writes the names of both
B
B
B
and
C
C
C
. Show that there will be one person, who has written down the names of all the other
n
−
1
n-1
n
−
1
players. [hide="Clarification"] Consider a game between
A
,
B
,
C
,
D
,
E
,
F
,
G
A,B,C,D,E,F,G
A
,
B
,
C
,
D
,
E
,
F
,
G
where
A
A
A
defeats
B
B
B
and
C
C
C
and
B
B
B
defeats
E
,
F
E,F
E
,
F
,
C
C
C
defeats
E
.
E.
E
.
Then
A
A
A
's list will have
(
B
,
C
,
E
,
F
)
(B,C,E,F)
(
B
,
C
,
E
,
F
)
, and will not include
G
.
G.
G
.
2
1
Hide problems
Range of f(x), ISI 2013, 2
For
x
≥
0
x\ge 0
x
≥
0
, define
f
(
x
)
=
1
x
+
2
cos
x
f(x)=\frac1{x+2\cos x}
f
(
x
)
=
x
+
2
cos
x
1
Find the set
{
y
∈
R
:
y
=
f
(
x
)
,
x
≥
0
}
\{ y \in \mathbb{R}: y=f(x), x\ge 0\}
{
y
∈
R
:
y
=
f
(
x
)
,
x
≥
0
}
1
1
Hide problems
minimum value of S, ISI 2013
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real number greater than
1
1
1
. Let
S
=
log
a
b
c
+
log
b
c
a
+
log
c
a
b
S=\log_a {bc}+\log_b {ca}+\log_c {ab}
S
=
lo
g
a
b
c
+
lo
g
b
c
a
+
lo
g
c
ab
Find the minimum possible value of
S
S
S
.