MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2014 India National Olympiad
2014 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
4
1
Hide problems
quadratic will have integer roots
Written on a blackboard is the polynomial
x
2
+
x
+
2014
x^2+x+2014
x
2
+
x
+
2014
. Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of
x
x
x
by
1
1
1
. And at this turn, Hobbes should either increase or decrease the constant coefficient by
1
1
1
. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.
6
1
Hide problems
Maximal size of set of subsets
Let
n
>
1
n>1
n
>
1
be a natural number. Let
U
=
{
1
,
2
,
.
.
.
,
n
}
U=\{1,2,...,n\}
U
=
{
1
,
2
,
...
,
n
}
, and define
A
Δ
B
A\Delta B
A
Δ
B
to be the set of all those elements of
U
U
U
which belong to exactly one of
A
A
A
and
B
B
B
. Show that
∣
F
∣
≤
2
n
−
1
|\mathcal{F}|\le 2^{n-1}
∣
F
∣
≤
2
n
−
1
, where
F
\mathcal{F}
F
is a collection of subsets of
U
U
U
such that for any two distinct elements of
A
,
B
A,B
A
,
B
of
F
\mathcal{F}
F
we have
∣
A
Δ
B
∣
≥
2
|A\Delta B|\ge 2
∣
A
Δ
B
∣
≥
2
. Also find all such collections
F
\mathcal{F}
F
for which the maximum is attained.
3
1
Hide problems
upper bound of quotient
Let
a
,
b
a,b
a
,
b
be natural numbers with
a
b
>
2
ab>2
ab
>
2
. Suppose that the sum of their greatest common divisor and least common multiple is divisble by
a
+
b
a+b
a
+
b
. Prove that the quotient is at most
a
+
b
4
\frac{a+b}{4}
4
a
+
b
. When is this quotient exactly equal to
a
+
b
4
\frac{a+b}{4}
4
a
+
b
2
1
Hide problems
Prove that expression is always even.
Let
n
n
n
be a natural number. Prove that,
⌊
n
1
⌋
+
⌊
n
2
⌋
+
⋯
+
⌊
n
n
⌋
+
⌊
n
⌋
\left\lfloor \frac{n}{1} \right\rfloor+ \left\lfloor \frac{n}{2} \right\rfloor + \cdots + \left\lfloor \frac{n}{n} \right\rfloor + \left\lfloor \sqrt{n} \right\rfloor
⌊
1
n
⌋
+
⌊
2
n
⌋
+
⋯
+
⌊
n
n
⌋
+
⌊
n
⌋
is even.
5
1
Hide problems
Circum centre of triangle and orthocentre Parallel
In a acute-angled triangle
A
B
C
ABC
A
BC
, a point
D
D
D
lies on the segment
B
C
BC
BC
. Let
O
1
,
O
2
O_1,O_2
O
1
,
O
2
denote the circumcentres of triangles
A
B
D
ABD
A
B
D
and
A
C
D
ACD
A
C
D
respectively. Prove that the line joining the circumcentre of triangle
A
B
C
ABC
A
BC
and the orthocentre of triangle
O
1
O
2
D
O_1O_2D
O
1
O
2
D
is parallel to
B
C
BC
BC
.
1
1
Hide problems
Prove that triangle is isoceles
In a triangle
A
B
C
ABC
A
BC
, let
D
D
D
be the point on the segment
B
C
BC
BC
such that
A
B
+
B
D
=
A
C
+
C
D
AB+BD=AC+CD
A
B
+
B
D
=
A
C
+
C
D
. Suppose that the points
B
B
B
,
C
C
C
and the centroids of triangles
A
B
D
ABD
A
B
D
and
A
C
D
ACD
A
C
D
lie on a circle. Prove that
A
B
=
A
C
AB=AC
A
B
=
A
C
.