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Problems
Contests
National and Regional Contests
India Contests
ISI Entrance Examination
2021 ISI Entrance Examination
2021 ISI Entrance Examination
Part of
ISI Entrance Examination
Subcontests
(8)
8
1
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Rising water level of a pond (ISI 2021)
A pond has been dug at the Indian Statistical Institute as an inverted truncated pyramid with a square base (see figure below). The depth of the pond is 6m. The square at the bottom has side length 2m and the top square has side length 8m. Water is filled in at a rate of
19
3
\tfrac{19}{3}
3
19
cubic meters per hour. At what rate is the water level rising exactly
1
1
1
hour after the water started to fill the pond?https://cdn.artofproblemsolving.com/attachments/0/9/ff8cac4bb4596ec6c030813da7e827e9a09dfd.png
2
1
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Nice Functional Equations (ISI 2021)
Let
f
:
Z
→
Z
f : \mathbb{Z} \to \mathbb{Z}
f
:
Z
→
Z
be a function satisfying
f
(
0
)
≠
0
=
f
(
1
)
f(0) \neq 0 = f(1)
f
(
0
)
=
0
=
f
(
1
)
. Assume also that
f
f
f
satisfies equations (A) and (B) below. \begin{eqnarray*}f(xy) = f(x) + f(y) -f(x) f(y)\qquad\mathbf{(A)}\\ f(x-y) f(x) f(y) = f(0) f(x) f(y)\qquad\mathbf{(B)} \end{eqnarray*} for all integers
x
,
y
x,y
x
,
y
.(i) Determine explicitly the set
{
f
(
a
)
:
a
∈
Z
}
\big\{f(a)~:~a\in\mathbb{Z}\big\}
{
f
(
a
)
:
a
∈
Z
}
. (ii) Assuming that there is a non-zero integer
a
a
a
such that
f
(
a
)
≠
0
f(a) \neq 0
f
(
a
)
=
0
, prove that the set
{
b
:
f
(
b
)
≠
0
}
\big\{b~:~f(b) \neq 0\big\}
{
b
:
f
(
b
)
=
0
}
is infinite.
6
1
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Equilateral Triangle inside Equilateral Triangles.
If a given equilateral triangle
Δ
\Delta
Δ
of side length
a
a
a
lies in the union of five equilateral triangles of side length
b
b
b
, show that there exist four equilateral triangles of side length
b
b
b
whose union contains
Δ
\Delta
Δ
.
4
1
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A Differentiable Involution.
Let
g
:
(
0
,
∞
)
→
(
0
,
∞
)
g:(0,\infty) \rightarrow (0,\infty)
g
:
(
0
,
∞
)
→
(
0
,
∞
)
be a differentiable function whose derivative is continuous, and such that
g
(
g
(
x
)
)
=
x
g(g(x)) = x
g
(
g
(
x
))
=
x
for all
x
>
0
x> 0
x
>
0
. If
g
g
g
is not the identity function, prove that
g
g
g
must be strictly decreasing.
7
1
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Finding Bounds on Roots
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be three real numbers which are roots of a cubic polynomial, and satisfy
a
+
b
+
c
=
6
a+b+c=6
a
+
b
+
c
=
6
and
a
b
+
b
c
+
c
a
=
9
ab+bc+ca=9
ab
+
b
c
+
c
a
=
9
. Suppose
a
<
b
<
c
a<b<c
a
<
b
<
c
. Show that
0
<
a
<
1
<
b
<
3
<
c
<
4.
0<a<1<b<3<c<4.
0
<
a
<
1
<
b
<
3
<
c
<
4.
5
1
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A Polynomial Limit
Let
a
0
,
a
1
,
…
,
a
19
∈
R
a_0, a_1,\dots, a_{19} \in \mathbb{R}
a
0
,
a
1
,
…
,
a
19
∈
R
and
P
(
x
)
=
x
20
+
∑
i
=
0
19
a
i
x
i
,
x
∈
R
.
P(x) = x^{20} + \sum_{i=0}^{19}a_ix^i, x \in \mathbb{R}.
P
(
x
)
=
x
20
+
i
=
0
∑
19
a
i
x
i
,
x
∈
R
.
If
P
(
x
)
=
P
(
−
x
)
P(x)=P(-x)
P
(
x
)
=
P
(
−
x
)
for all
x
∈
R
x \in \mathbb{R}
x
∈
R
, and
P
(
k
)
=
k
2
,
P(k)=k^2,
P
(
k
)
=
k
2
,
for
k
=
0
,
1
,
2
,
…
,
9
k=0, 1, 2, \dots, 9
k
=
0
,
1
,
2
,
…
,
9
then find
lim
x
→
0
P
(
x
)
sin
2
x
.
\lim_{x\rightarrow 0} \frac{P(x)}{\sin^2x}.
x
→
0
lim
sin
2
x
P
(
x
)
.
1
1
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A Tale of Three Cities
There are three cities each of which has exactly the same number of citizens, say
n
n
n
. Every citizen in each city has exactly a total of
(
n
+
1
)
(n+1)
(
n
+
1
)
friends in the other two cities. Show that there exist three people, one from each city, such that they are friends. We assume that friendship is mutual (that is, a symmetric relation).
3
1
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Unique Rational Number Representation
Prove that every positive rational number can be expressed uniquely as a finite sum of the form
a
1
+
a
2
2
!
+
a
3
3
!
+
⋯
+
a
n
n
!
,
a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\dots+\frac{a_n}{n!},
a
1
+
2
!
a
2
+
3
!
a
3
+
⋯
+
n
!
a
n
,
where
a
n
a_n
a
n
are integers such that
0
≤
a
n
≤
n
−
1
0 \leq a_n \leq n-1
0
≤
a
n
≤
n
−
1
for all
n
>
1
n > 1
n
>
1
.