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National and Regional Contests
India Contests
ISI B.Math Entrance Exam
2009 ISI B.Math Entrance Exam
2009 ISI B.Math Entrance Exam
Part of
ISI B.Math Entrance Exam
Subcontests
(10)
4
1
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find min(x^2+y^2)
Find the values of
x
,
y
x,y
x
,
y
for which
x
2
+
y
2
x^2+y^2
x
2
+
y
2
takes the minimum value where
(
x
+
5
)
2
+
(
y
−
12
)
2
=
14
(x+5)^2+(y-12)^2=14
(
x
+
5
)
2
+
(
y
−
12
)
2
=
14
.
10
1
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B.Math(Hons.) Admission Test 2009 problem 10
Given odd integers
a
,
b
,
c
a,b,c
a
,
b
,
c
prove that the equation
a
x
2
+
b
x
+
c
=
0
ax^2+bx+c=0
a
x
2
+
b
x
+
c
=
0
cannot have a solution
x
x
x
which is a rational number.
9
1
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B.Math(Hons.) Admission Test 2009 problem 9
Let
f
(
x
)
=
a
x
2
+
b
x
+
c
f(x)=ax^2+bx+c
f
(
x
)
=
a
x
2
+
b
x
+
c
where
a
,
b
,
c
a,b,c
a
,
b
,
c
are real numbers. Suppose
f
(
−
1
)
,
f
(
0
)
,
f
(
1
)
∈
[
−
1
,
1
]
f(-1),f(0),f(1) \in [-1,1]
f
(
−
1
)
,
f
(
0
)
,
f
(
1
)
∈
[
−
1
,
1
]
. Prove that
∣
f
(
x
)
∣
≤
3
2
|f(x)|\le \frac{3}{2}
∣
f
(
x
)
∣
≤
2
3
for all
x
∈
[
−
1
,
1
]
x \in [-1,1]
x
∈
[
−
1
,
1
]
.
6
1
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B.Math(Hons.) Admission Test 2009 problem 6
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be integers such that
a
d
−
b
c
ad-bc
a
d
−
b
c
is non zero. Suppose
b
1
,
b
2
b_1,b_2
b
1
,
b
2
are integers both of which are multiples of
a
d
−
b
c
ad-bc
a
d
−
b
c
. Prove that there exist integers simultaneously satisfying both the equalities
a
x
+
b
y
=
b
1
,
c
x
+
d
y
=
b
2
ax+by=b_1, cx+dy=b_2
a
x
+
b
y
=
b
1
,
c
x
+
d
y
=
b
2
.
5
1
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B.Math(Hons.) Admission Test 2009 problem 5
Let
p
p
p
be a prime number bigger than
5
5
5
. Suppose, the decimal expansion of
1
p
\frac{1}{p}
p
1
looks like
0.
a
1
a
2
⋯
a
r
‾
0.\overline{a_1a_2\cdots a_r}
0.
a
1
a
2
⋯
a
r
where the line denotes a recurring decimal. Prove that
1
0
r
10^r
1
0
r
leaves a remainder of
1
1
1
on dividing by
p
p
p
.
8
1
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B.Math(Hons.) Admission Test 2009 Problem 8
Suppose you are given six colours and, are asked to colour each face of a cube by a different colour. Determine the different number of colouring possible.
3
1
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ISI B.Math(Hons.) Admission Test 2009 Problem 3
Let
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
11
,
12
,
⋯
1,2,3,4,5,6,7,8,9,11,12,\cdots
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
11
,
12
,
⋯
be the sequence of all positive integers which do not contain the digit zero. Write
{
a
n
}
\{a_n\}
{
a
n
}
for this sequence. By comparing with a geometric series, show that
∑
k
=
1
n
1
a
k
<
90
\sum_{k=1}^n \frac{1}{a_k} < 90
∑
k
=
1
n
a
k
1
<
90
.
2
1
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ISI B.Math(Hons.) Admission Test 2009 Problem 2
Let
c
c
c
be a fixed real number. Show that a root of the equation
x
(
x
+
1
)
(
x
+
2
)
⋯
(
x
+
2009
)
=
c
x(x+1)(x+2)\cdots(x+2009)=c
x
(
x
+
1
)
(
x
+
2
)
⋯
(
x
+
2009
)
=
c
can have multiplicity at most
2
2
2
. Determine the number of values of
c
c
c
for which the equation has a root of multiplicity
2
2
2
.
1
1
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ISI B.Math(Hons.) Admission Test 2009 Problem 1
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be non-zero real numbers. Suppose
α
,
β
,
γ
\alpha, \beta, \gamma
α
,
β
,
γ
are complex numbers such that
∣
α
∣
=
∣
β
∣
=
∣
γ
∣
=
1
|\alpha|=|\beta|=|\gamma|=1
∣
α
∣
=
∣
β
∣
=
∣
γ
∣
=
1
. If
x
+
y
+
z
=
0
=
α
x
+
β
y
+
γ
z
x+y+z=0=\alpha x+\beta y+\gamma z
x
+
y
+
z
=
0
=
αx
+
β
y
+
γ
z
, then prove that
α
=
β
=
γ
\alpha =\beta =\gamma
α
=
β
=
γ
.
7
1
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max area, rectangle in triangle
Compute the maximum area of a rectangle which can be inscribed in a triangle of area
M
M
M
.