MathDB
Problems
Contests
National and Regional Contests
India Contests
ISI Entrance Examination
2014 ISI Entrance Examination
2014 ISI Entrance Examination
Part of
ISI Entrance Examination
Subcontests
(8)
8
1
Hide problems
Lotus leafs and a frog
n
(
>
1
)
n(>1)
n
(
>
1
)
lotus leaves are arranged in a circle. A frog jumps from a particular leaf from another under the following rule: [*]It always moves clockwise. [*]From starting it skips one leaf and then jumps to the next. After that it skips two leaves and jumps to the following. And the process continues. (Remember the frog might come back on a leaf twice or more.) Given that it reaches all leaves at least once. Show
n
n
n
cannot be odd.
7
1
Hide problems
Integral inequality
Let
f
:
[
0
,
∞
)
→
R
f: [0,\infty)\to \mathbb{R}
f
:
[
0
,
∞
)
→
R
a non-decreasing function. Then show this inequality holds for all
x
,
y
,
z
x,y,z
x
,
y
,
z
such that
0
≤
x
<
y
<
z
0\le x<y<z
0
≤
x
<
y
<
z
. \begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge (z-y)\int_{x}^{z}f(u)\,\mathrm{du} \end{align*}
6
1
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Longest line contained in the region
Define
A
=
{
(
x
,
y
)
∣
x
=
u
+
v
,
y
=
v
,
u
2
+
v
2
≤
1
}
\mathcal{A}=\{(x,y)|x=u+v,y=v, u^2+v^2\le 1\}
A
=
{(
x
,
y
)
∣
x
=
u
+
v
,
y
=
v
,
u
2
+
v
2
≤
1
}
. Find the length of the longest segment that is contained in
A
\mathcal{A}
A
.
5
1
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Bad NT
Prove that sum of
12
12
12
consecutive integers cannot be a square. Give an example of
11
11
11
consecutive integers whose sum is a perfect square.
4
1
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Identically zero
Let
f
,
g
f,g
f
,
g
are defined in
(
a
,
b
)
(a,b)
(
a
,
b
)
such that
f
(
x
)
,
g
(
x
)
∈
C
2
f(x),g(x)\in\mathcal{C}^2
f
(
x
)
,
g
(
x
)
∈
C
2
and non-decreasing in an interval
(
a
,
b
)
(a,b)
(
a
,
b
)
. Also suppose
f
′
′
(
x
)
=
g
(
x
)
,
g
′
′
(
x
)
=
f
(
x
)
f^{\prime \prime}(x)=g(x),g^{\prime \prime}(x)=f(x)
f
′′
(
x
)
=
g
(
x
)
,
g
′′
(
x
)
=
f
(
x
)
. Also it is given that
f
(
x
)
g
(
x
)
f(x)g(x)
f
(
x
)
g
(
x
)
is linear in
(
a
,
b
)
(a,b)
(
a
,
b
)
. Show that
f
≡
0
and
g
≡
0
f\equiv 0 \text{ and } g\equiv 0
f
≡
0
and
g
≡
0
in
(
a
,
b
)
(a,b)
(
a
,
b
)
.
2
1
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Co-ordinate bash? Maybe not..
Let us consider a triangle
Δ
P
Q
R
\Delta{PQR}
Δ
PQR
in the co-ordinate plane. Show for every function
f
:
R
2
→
R
,
f
(
X
)
=
a
x
+
b
y
+
c
f: \mathbb{R}^2\to \mathbb{R}\;,f(X)=ax+by+c
f
:
R
2
→
R
,
f
(
X
)
=
a
x
+
b
y
+
c
where
X
≡
(
x
,
y
)
and
a
,
b
,
c
∈
R
X\equiv (x,y) \text{ and } a,b,c\in\mathbb{R}
X
≡
(
x
,
y
)
and
a
,
b
,
c
∈
R
and every point
A
A
A
on
Δ
P
Q
R
\Delta PQR
Δ
PQR
or inside the triangle we have the inequality: \begin{align*} & f(A)\le \text{max}\{f(P),f(Q),f(R)\} \end{align*}
3
1
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Local extrema or distinct roots?
Consider
f
(
x
)
=
x
4
+
a
x
3
+
b
x
2
+
c
x
+
d
(
a
,
b
,
c
,
d
∈
R
)
f(x)=x^4+ax^3+bx^2+cx+d\; (a,b,c,d\in\mathbb{R})
f
(
x
)
=
x
4
+
a
x
3
+
b
x
2
+
c
x
+
d
(
a
,
b
,
c
,
d
∈
R
)
. It is known that
f
f
f
intersects X-axis in at least
3
3
3
(distinct) points. Show either
f
f
f
has
4
4
4
d
i
s
t
i
n
c
t
\mathbf{distinct}
distinct
real roots or it has
3
3
3
d
i
s
t
i
n
c
t
\mathbf{distinct}
distinct
real roots and one of them is a point of local maxima or minima.
1
1
Hide problems
Counting friends in two ways
Suppose a class contains
100
100
100
students. Let, for
1
≤
i
≤
100
1\le i\le 100
1
≤
i
≤
100
, the
i
th
i^{\text{th}}
i
th
student have
a
i
a_i
a
i
many friends. For
0
≤
j
≤
99
0\le j\le 99
0
≤
j
≤
99
let us define
c
j
c_j
c
j
to be the number of students who have strictly more than
j
j
j
friends. Show that \begin{align*} & \sum_{i=1}^{100}a_i=\sum_{j=0}^{99}c_j \end{align*}