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National and Regional Contests
India Contests
ISI B.Math Entrance Exam
2005 ISI B.Math Entrance Exam
2005 ISI B.Math Entrance Exam
Part of
ISI B.Math Entrance Exam
Subcontests
(8)
8
1
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B.Math - matrix
In how many ways can one fill an
n
∗
n
n*n
n
∗
n
matrix with
+
1
+1
+
1
and
−
1
-1
−
1
so that the product of the entries in each row and each column equals
−
1
-1
−
1
?
7
1
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B.Math - Area locus
Let
M
M
M
be a point in the triangle
A
B
C
ABC
A
BC
such that
area
(
A
B
M
)
=
2
⋅
area
(
A
C
M
)
\text{area}(ABM)=2 \cdot \text{area}(ACM)
area
(
A
BM
)
=
2
⋅
area
(
A
CM
)
Show that the locus of all such points is a straight line.
6
1
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B.Math - interval
Let
a
0
=
0
<
a
1
<
a
2
<
.
.
.
<
a
n
a_0=0<a_1<a_2<...<a_n
a
0
=
0
<
a
1
<
a
2
<
...
<
a
n
be real numbers . Supppose
p
(
t
)
p(t)
p
(
t
)
is a real valued polynomial of degree
n
n
n
such that
∫
a
j
a
j
+
1
p
(
t
)
d
t
=
0
∀
0
≤
j
≤
n
−
1
\int_{a_j}^{a_{j+1}} p(t)\,dt = 0\ \ \forall \ 0\le j\le n-1
∫
a
j
a
j
+
1
p
(
t
)
d
t
=
0
∀
0
≤
j
≤
n
−
1
Show that , for
0
≤
j
≤
n
−
1
0\le j\le n-1
0
≤
j
≤
n
−
1
, the polynomial
p
(
t
)
p(t)
p
(
t
)
has exactly one root in the interval
(
a
j
,
a
j
+
1
)
(a_j,a_{j+1})
(
a
j
,
a
j
+
1
)
5
1
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B.Math - maximum
Find the point in the closed unit disc
D
=
{
(
x
,
y
)
∣
x
2
+
y
2
≤
1
}
D=\{ (x,y) | x^2+y^2\le 1 \}
D
=
{(
x
,
y
)
∣
x
2
+
y
2
≤
1
}
at which the function
f
(
x
,
y
)
=
x
+
y
f(x,y)=x+y
f
(
x
,
y
)
=
x
+
y
attains its maximum .
3
1
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B.Math - quadrilateral
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral such that the sum of a pair of opposite sides equals the sum of other pair of opposite sides
(
A
B
+
C
D
=
A
D
+
B
C
)
(AB+CD=AD+BC)
(
A
B
+
C
D
=
A
D
+
BC
)
. Prove that the circles inscribed in triangles
A
B
C
ABC
A
BC
and
A
C
D
ACD
A
C
D
are tangent to each other.
2
1
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B.Math - limit
Let
a
1
=
1
a_1=1
a
1
=
1
and
a
n
=
n
(
a
n
−
1
+
1
)
a_n=n(a_{n-1}+1)
a
n
=
n
(
a
n
−
1
+
1
)
for all
n
≥
2
n\ge 2
n
≥
2
. Define :
P
n
=
(
1
+
1
a
1
)
.
.
.
(
1
+
1
a
n
)
P_n=\left(1+\frac{1}{a_1}\right)...\left(1+\frac{1}{a_n}\right)
P
n
=
(
1
+
a
1
1
)
...
(
1
+
a
n
1
)
Compute
lim
n
→
∞
P
n
\lim_{n\to \infty} P_n
lim
n
→
∞
P
n
1
1
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B.Math - integer part
For any
k
∈
Z
+
k\in\mathbb{Z}^+
k
∈
Z
+
, prove that:-
2
(
k
+
1
−
k
)
<
1
k
<
2
(
k
−
k
−
1
)
2(\sqrt{k+1}-\sqrt{k})<\frac{1}{\sqrt{k}}<2(\sqrt{k}-\sqrt{k-1})
2
(
k
+
1
−
k
)
<
k
1
<
2
(
k
−
k
−
1
)
Also compute integral part of
1
2
+
1
3
+
.
.
.
+
1
10000
\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{10000}}
2
1
+
3
1
+
...
+
10000
1
.
4
1
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interesting cardinality
For a set
S
S
S
we denote its cardinality by
∣
S
∣
|S|
∣
S
∣
. Let
e
1
,
e
2
,
…
,
e
k
e_1,e_2,\ldots,e_k
e
1
,
e
2
,
…
,
e
k
be non-negative integers. Let
A
k
A_k
A
k
(respectively
B
k
B_k
B
k
) be the set of all
k
k
k
-tuples
(
f
1
,
f
2
,
…
,
f
k
)
(f_1,f_2,\ldots,f_k)
(
f
1
,
f
2
,
…
,
f
k
)
of integers such that
0
≤
f
i
≤
e
i
0\leq f_i\leq e_i
0
≤
f
i
≤
e
i
for all
i
i
i
and
∑
i
=
1
k
f
i
\sum_{i=1}^k f_i
∑
i
=
1
k
f
i
is even (respectively odd). Show that
∣
A
k
∣
−
∣
B
k
∣
=
0
or
1
|A_k|-|B_k|=0 \textrm{ or } 1
∣
A
k
∣
−
∣
B
k
∣
=
0
or
1
.