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Problems
Contests
National and Regional Contests
India Contests
ISI Entrance Examination
2024 ISI Entrance UGB
2024 ISI Entrance UGB
Part of
ISI Entrance Examination
Subcontests
(8)
P7
1
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Revolutionary Parabola and and my dizzy head
Consider a container of the shape obtained by revolving a segment of parabola
x
=
1
+
y
2
x = 1 + y^2
x
=
1
+
y
2
around the
y
y
y
-axis as shown below. The container is initially empty. Water is poured at a constant rate of
1
cm
3
1\, \text{cm}^3
1
cm
3
into the container. Let
h
(
t
)
h(t)
h
(
t
)
be the height of water inside container at time
t
t
t
. Find the time
t
t
t
when the rate of change of
h
(
t
)
h(t)
h
(
t
)
is maximum.
P3
1
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ISI 2024 P3
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral with all the internal angles
<
π
< \pi
<
π
. Squares are drawn on each side as shown in the picture below. Let
Δ
1
,
Δ
2
,
Δ
3
,
Δ
4
\Delta_1 , \Delta_2 , \Delta_3 , \Delta_4
Δ
1
,
Δ
2
,
Δ
3
,
Δ
4
denote the areas of the shaded triangles as shown. Prove that
Δ
1
−
Δ
2
+
Δ
3
−
Δ
4
=
0.
\Delta_1 - \Delta_2 + \Delta_3 - \Delta_4 = 0.
Δ
1
−
Δ
2
+
Δ
3
−
Δ
4
=
0.
[asy] //made from sweat and hardwork by SatisfiedMagma import olympiad; import geometry;size(250);pair A = (-3,0); pair B = (0,2); pair C = (5.88,0.44); pair D = (0.96, -1.86);pair H = B + rotate(90)*(C-B); pair G = C + rotate(270)*(B-C);pair J = C + rotate(90)*(D-C); pair I = D + rotate(270)*(C-D);pair L = D + rotate(90)*(A-D); pair K = A + rotate(270)*(D-A);pair F = A + rotate(90)*(B-A); pair E = B + rotate(270)*(A-B); draw(B--H--G--C--B, blue); draw(C--J--I--D--C, red); draw(B--E--F--A--B, orange); draw(D--L--K--A--D, magenta); draw(L--I, fuchsia); draw(J--G, fuchsia); draw(E--H, fuchsia); draw(F--K, fuchsia);pen lightFuchsia = deepgreen + 0.5*white;fill(D--L--I--cycle, lightFuchsia); fill(A--K--F--cycle, lightFuchsia); fill(E--B--H--cycle, lightFuchsia); fill(C--J--G--cycle, lightFuchsia);label("
△
2
\triangle_2
△
2
", (E+B+H)/3); label("
△
4
\triangle_4
△
4
", (D+L+I)/3); label("
△
3
\triangle_3
△
3
", (C+G+J)/3); label("
△
1
\triangle_1
△
1
", (A+F+K)/3);dot("
A
A
A
", A, S); dot("
B
B
B
", B, S); dot("
C
C
C
", C, S); dot("
D
D
D
", D, N); dot("
H
H
H
", H, dir(H)); dot("
G
G
G
", G, dir(G)); dot("
J
J
J
", J, dir(J)); dot("
I
I
I
", I, dir(I)); dot("
L
L
L
", L, dir(L)); dot("
K
K
K
", K, dir(K)); dot("
F
F
F
", F, dir(F)); dot("
E
E
E
", E, dir(E)); [/asy]
P4
1
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ISI 2024 P4
Let
f
:
R
→
R
f: \mathbb R \to \mathbb R
f
:
R
→
R
be a function which is differentiable at
0
0
0
. Define another function
g
:
R
→
R
g: \mathbb R \to \mathbb R
g
:
R
→
R
as follows:
g
(
x
)
=
{
f
(
x
)
sin
(
1
x
)
if
x
≠
0
<
/
b
r
>
0
if
x
=
0.
g(x) = \begin{cases} f(x)\sin\left(\frac 1x\right) ~ &\text{if} ~ x \neq 0 \\</br>0 &\text{if} ~ x = 0. \end{cases}
g
(
x
)
=
{
f
(
x
)
sin
(
x
1
)
<
/
b
r
>
0
if
x
=
0
if
x
=
0.
Suppose that
g
g
g
is also differentiable at
0
0
0
. Prove that
g
′
(
0
)
=
f
′
(
0
)
=
f
(
0
)
=
g
(
0
)
=
0.
g'(0) = f'(0) = f(0) = g(0) = 0.
g
′
(
0
)
=
f
′
(
0
)
=
f
(
0
)
=
g
(
0
)
=
0.
P8
1
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ISI 2024 P8
In a sports tournament involving
N
N
N
teams, each team plays every other team exactly one. At the end of every match, the winning team gets
1
1
1
point and losing team gets
0
0
0
points. At the end of the tournament, the total points received by the individual teams are arranged in decreasing order as follows:
x
1
≥
x
2
≥
⋯
≥
x
N
.
x_1 \ge x_2 \ge \cdots \ge x_N .
x
1
≥
x
2
≥
⋯
≥
x
N
.
Prove that for any
1
≤
k
≤
N
1\le k \le N
1
≤
k
≤
N
,
N
−
k
2
≤
x
k
≤
N
−
k
+
1
2
\frac{N - k}{2} \le x_k \le N - \frac{k+1}{2}
2
N
−
k
≤
x
k
≤
N
−
2
k
+
1
P6
1
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ISI 2024 P6
Let
x
1
,
…
,
x
2024
x_1 , \dots , x_{2024}
x
1
,
…
,
x
2024
be non negative real numbers with
∑
i
=
1
2024
x
i
=
1
\displaystyle{\sum_{i=1}^{2024}}x_i = 1
i
=
1
∑
2024
x
i
=
1
. Find, with proof, the minimum and maximum possible values of the following expression
∑
i
=
1
1012
x
i
+
∑
i
=
1013
2024
x
i
2
.
\sum_{i=1}^{1012} x_i + \sum_{i=1013}^{2024} x_i^2 .
i
=
1
∑
1012
x
i
+
i
=
1013
∑
2024
x
i
2
.
P5
1
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ISI 2024 P5
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial with real coefficients. Let
α
1
,
…
,
α
k
\alpha_1 , \dots , \alpha_k
α
1
,
…
,
α
k
be the distinct real roots of
P
(
x
)
=
0
P(x)=0
P
(
x
)
=
0
. If
P
′
P'
P
′
is the derivative of
P
P
P
, show that for each
i
=
1
,
…
,
k
i=1,\dots , k
i
=
1
,
…
,
k
lim
x
→
α
i
(
x
−
α
i
)
P
′
(
x
)
P
(
x
)
=
r
i
,
\lim_{x\to \alpha_i} \frac{(x-\alpha_i)P'(x)}{P(x)} = r_i,
x
→
α
i
lim
P
(
x
)
(
x
−
α
i
)
P
′
(
x
)
=
r
i
,
for some positive integer
r
i
r_i
r
i
.
P2
1
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ISI 2024 P2
Suppose
n
≥
2
n\ge 2
n
≥
2
. Consider the polynomial
Q
n
(
x
)
=
1
−
x
n
−
(
1
−
x
)
n
.
Q_n(x) = 1-x^n - (1-x)^n .
Q
n
(
x
)
=
1
−
x
n
−
(
1
−
x
)
n
.
Show that the equation
Q
n
(
x
)
=
0
Q_n(x) = 0
Q
n
(
x
)
=
0
has only two real roots, namely
0
0
0
and
1
1
1
.
P1
1
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ISI 2024 P1
Find, with proof, all possible values of
t
t
t
such that
lim
n
→
∞
(
1
+
2
1
/
3
+
3
1
/
3
+
⋯
+
n
1
/
3
n
t
)
=
c
\lim_{n \to \infty} \left( \frac{1 + 2^{1/3} + 3^{1/3} + \dots + n^{1/3}}{n^t} \right ) = c
n
→
∞
lim
(
n
t
1
+
2
1/3
+
3
1/3
+
⋯
+
n
1/3
)
=
c
for some real
c
>
0
c>0
c
>
0
. Also find the corresponding values of
c
c
c
.