MathDB
Problems
Contests
National and Regional Contests
India Contests
ISI Entrance Examination
2015 ISI Entrance Examination
2015 ISI Entrance Examination
Part of
ISI Entrance Examination
Subcontests
(8)
1
1
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Harmonic series of positive integers
Let
m
1
<
m
2
<
…
m
k
−
1
<
m
k
m_1< m_2 < \ldots m_{k-1}< m_k
m
1
<
m
2
<
…
m
k
−
1
<
m
k
be
k
k
k
distinct positive integers such that their reciprocals are in arithmetic progression.1.Show that
k
<
m
1
+
2
k< m_1 + 2
k
<
m
1
+
2
. 2. Give an example of such a sequence of length
k
k
k
for any positive integer
k
k
k
.
2
1
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Cyclic quad on a parabola!
Let
y
=
x
2
+
a
x
+
b
y = x^2 + ax + b
y
=
x
2
+
a
x
+
b
be a parabola that cuts the coordinate axes at three distinct points. Show that the circle passing through these three points also passes through
(
0
,
1
)
(0,1)
(
0
,
1
)
.
3
1
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Sum of max element of subsets
Consider the set
S
=
1
,
2
,
3
,
…
,
j
S = {1,2,3,\ldots , j}
S
=
1
,
2
,
3
,
…
,
j
. Let
m
(
A
)
m(A)
m
(
A
)
denote the maximum element of
A
A
A
. Prove that
∑
A
⊆
S
m
(
A
)
=
(
j
−
1
)
2
j
+
1
\sum_ {A\subseteq S} m(A) = (j-1)2^j +1
A
⊆
S
∑
m
(
A
)
=
(
j
−
1
)
2
j
+
1
4
1
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Show that a negative number r exists such that $p(r)=q(r)$
Let
p
(
x
)
=
x
7
+
x
6
+
b
5
x
5
+
⋯
+
b
0
p(x) = x^7 +x^6 + b_5 x^5 + \cdots +b_0
p
(
x
)
=
x
7
+
x
6
+
b
5
x
5
+
⋯
+
b
0
and
q
(
x
)
=
x
5
+
c
4
x
4
+
⋯
+
c
0
q(x) = x^5 + c_4 x^4 + \cdots +c_0
q
(
x
)
=
x
5
+
c
4
x
4
+
⋯
+
c
0
. If
p
(
i
)
=
q
(
i
)
p(i)=q(i)
p
(
i
)
=
q
(
i
)
for
i
=
1
,
2
,
3
,
⋯
,
6
i=1,2,3,\cdots,6
i
=
1
,
2
,
3
,
⋯
,
6
. Show that there exists a negative integer r such that
p
(
r
)
=
q
(
r
)
p(r)=q(r)
p
(
r
)
=
q
(
r
)
.
6
1
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Find all n such that $7 | 5^n +1$
Find all
n
∈
N
n\in \mathbb{N}
n
∈
N
so that 7 divides
5
n
+
1
5^n + 1
5
n
+
1
7
1
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What are the sidelengths of the triangle ?
Let
γ
1
,
γ
2
,
γ
3
\gamma_1, \gamma_2,\gamma_3
γ
1
,
γ
2
,
γ
3
be three circles of unit radius which touch each other externally. The common tangent to each pair of circles are drawn (and extended so that they intersect) and let the triangle formed by the common tangents be
△
X
Y
Z
\triangle XYZ
△
X
Y
Z
. Find the length of each side of
△
X
Y
Z
\triangle XYZ
△
X
Y
Z
8
1
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Find all functions with the following property
Find all the functions
f
:
R
→
R
f:\mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
such that
∣
f
(
x
)
−
f
(
y
)
∣
=
2
∣
x
−
y
∣
|f(x)-f(y)| = 2 |x - y|
∣
f
(
x
)
−
f
(
y
)
∣
=
2∣
x
−
y
∣
5
1
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Show it has *exactly* n roots
If
0
<
a
1
<
⋯
<
a
n
0<a_1< \cdots < a_n
0
<
a
1
<
⋯
<
a
n
, show that the following equation has exactly
n
n
n
roots.
a
1
a
1
−
x
+
a
2
a
2
−
x
+
a
3
a
3
−
x
+
⋯
+
a
n
a
n
−
x
=
2015
\frac{a_1}{a_1-x}+\frac{a_2}{a_2-x}+ \frac{a_3}{a_3-x}+ \cdots + \frac {a_n}{a_n - x} = 2015
a
1
−
x
a
1
+
a
2
−
x
a
2
+
a
3
−
x
a
3
+
⋯
+
a
n
−
x
a
n
=
2015