MathDB
Problems
Contests
National and Regional Contests
India Contests
India Pre-Regional Mathematical Olympiad
2016 India PRMO
2016 India PRMO
Part of
India Pre-Regional Mathematical Olympiad
Subcontests
(16)
16
1
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2016 preRMO p16, average of arithmetic and geometric mean, is perfect square
For positive real numbers
x
x
x
and
y
y
y
, define their special mean to be average of their arithmetic and geometric means. Find the total number of pairs of integers
(
x
,
y
)
(x, y)
(
x
,
y
)
, with
x
≤
y
x \le y
x
≤
y
, from the set of numbers
{
1
,
2
,
.
.
.
,
2016
}
\{1,2,...,2016\}
{
1
,
2
,
...
,
2016
}
, such that the special mean of
x
x
x
and
y
y
y
is a perfect square.
15
1
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2016 preRMO p15, least common multiple of m and n equals 600
Find the number of pairs of positive integers
(
m
;
n
)
(m; n)
(
m
;
n
)
, with
m
≤
n
m \le n
m
≤
n
, such that the ‘least common multiple’ (LCM) of
m
m
m
and
n
n
n
equals
600
600
600
.
14
1
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2016 preRMO p14, |a - b|<= 3, a,b in m element subset of {1,2,...,2016}
Find the minimum value of
m
m
m
such that any
m
m
m
-element subset of the set of integers
{
1
,
2
,
.
.
.
,
2016
}
\{1,2,...,2016\}
{
1
,
2
,
...
,
2016
}
contains at least two distinct numbers
a
a
a
and
b
b
b
which satisfy
∣
a
−
b
∣
≤
3
|a - b|\le 3
∣
a
−
b
∣
≤
3
.
13
1
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2016 preRMO p13, how many times 2 appears in {1,2,..,1000}
Find the total number of times the digit ‘
2
2
2
’ appears in the set of integers
{
1
,
2
,
.
.
,
1000
}
\{1,2,..,1000\}
{
1
,
2
,
..
,
1000
}
. For example, the digit ’
2
2
2
’ appears twice in the integer
229
229
229
.
12
1
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2016 preRMO p12, 1 + 1/\sqrt2+1/\sqrt3+..+ 1/\sqrt{99}+ 1/\sqrt{100}
Let
S
=
1
+
1
2
+
1
3
+
1
4
+
.
.
.
+
1
99
+
1
100
S = 1 + \frac{1}{\sqrt2}+ \frac{1}{\sqrt3}+\frac{1}{\sqrt4}+...+ \frac{1}{\sqrt{99}}+ \frac{1}{\sqrt{100}}
S
=
1
+
2
1
+
3
1
+
4
1
+
...
+
99
1
+
100
1
. Find
[
S
]
[S]
[
S
]
.You may use the fact that
n
<
1
2
(
n
+
n
+
1
)
<
n
+
1
\sqrt{n} < \frac12 (\sqrt{n} +\sqrt{n+1}) <\sqrt{n+1}
n
<
2
1
(
n
+
n
+
1
)
<
n
+
1
for all integers
n
≥
1
n \ge 1
n
≥
1
.
11
1
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2016 preRMO p11, M is max of (x^4y+x^3y+x^2y+xy+xy^2+xy^3+xy^4) if x+y=3
For real numbers
x
x
x
and
y
y
y
, let
M
M
M
be the maximum value of the expression
x
4
y
+
x
3
y
+
x
2
y
+
x
y
+
x
y
2
+
x
y
3
+
x
y
4
x^4y + x^3y + x^2y + xy + xy^2 + xy^3 + xy^4
x
4
y
+
x
3
y
+
x
2
y
+
x
y
+
x
y
2
+
x
y
3
+
x
y
4
, subject to
x
+
y
=
3
x + y = 3
x
+
y
=
3
. Find
[
M
]
[M]
[
M
]
.
8
1
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2016 preRMO p8, [x/100 [x/100]]=5
Find the number of integer solutions of
[
x
100
[
x
100
]
]
=
5
\left[\frac{x}{100} \left[\frac{x}{100}\right]\right]= 5
[
100
x
[
100
x
]
]
=
5
10
1
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2016 preRMO p10, [M], M is max of (6x-3y-8z) when 2x^2+3y^2+4z^2 = 1
Let
M
M
M
be the maximum value of
(
6
x
−
3
y
−
8
z
)
(6x-3y-8z)
(
6
x
−
3
y
−
8
z
)
, subject to
2
x
2
+
3
y
2
+
4
z
2
=
1
2x^2+3y^2+4z^2 = 1
2
x
2
+
3
y
2
+
4
z
2
=
1
. Find
[
M
]
[M]
[
M
]
.
9
1
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2016 preRMO p9, 4 b^2 -a^3 where a,b roots of x^2 + x - 3 = 0
Let
a
a
a
and
b
b
b
be the roots of the equation
x
2
+
x
−
3
=
0
x^2 + x - 3 = 0
x
2
+
x
−
3
=
0
. Find the value of the expression 4 b^2 -a^3.
7
1
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2016 preRMO p7, coefficient of a^5b^5c^5d^6 in (bcd + acd + abd + abc)^7
Find the coefficient of
a
5
b
5
c
5
d
6
a^5b^5c^5d^6
a
5
b
5
c
5
d
6
in the expansion of the following expression
(
b
c
d
+
a
c
d
+
a
b
d
+
a
b
c
)
7
(bcd +acd +abd +abc)^7
(
b
c
d
+
a
c
d
+
ab
d
+
ab
c
)
7
6
1
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2016 preRMO p6, ray of light reflects on circumference
Suppose a circle
C
C
C
of radius
2
\sqrt2
2
touches the
Y
Y
Y
-axis at the origin
(
0
,
0
)
(0, 0)
(
0
,
0
)
. A ray of light
L
L
L
, parallel to the
X
X
X
-axis, reflects on a point
P
P
P
on the circumference of
C
C
C
, and after reflection, the reflected ray
L
′
L'
L
′
becomes parallel to the
Y
Y
Y
-axis. Find the distance between the ray
L
L
L
and the
X
X
X
-axis.
5
1
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2016 preRMO p5, line perpendicular to BC bisects triangle's area
Consider a triangle
A
B
C
ABC
A
BC
with
A
B
=
13
,
B
C
=
14
,
C
A
=
15
AB = 13, BC = 14, CA = 15
A
B
=
13
,
BC
=
14
,
C
A
=
15
. A line perpendicular to
B
C
BC
BC
divides the interior of
△
B
C
\vartriangle BC
△
BC
into two regions of equal area. Suppose that the aforesaid perpendicular cuts
B
C
BC
BC
at
D
D
D
, and cuts
△
A
B
C
\vartriangle ABC
△
A
BC
again at
E
E
E
. If
L
L
L
is the length of the line segment
D
E
DE
D
E
, find
L
2
L^2
L
2
.
4
1
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2016 preRMO p4, <C=90^o, AD = DE = EF = FB, CD^2 +CE^2 +CF^2 = 350
Consider a right-angled triangle
A
B
C
ABC
A
BC
with
∠
C
=
9
0
o
\angle C = 90^o
∠
C
=
9
0
o
. Suppose that the hypotenuse
A
B
AB
A
B
is divided into four equal parts by the points
D
,
E
,
F
D,E,F
D
,
E
,
F
, such that
A
D
=
D
E
=
E
F
=
F
B
AD = DE = EF = FB
A
D
=
D
E
=
EF
=
FB
. If
C
D
2
+
C
E
2
+
C
F
2
=
350
CD^2 +CE^2 +CF^2 = 350
C
D
2
+
C
E
2
+
C
F
2
=
350
, find the length of
A
B
AB
A
B
.
3
1
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2016 preRMO p3, adding digits of N and so on until you get single digit
Suppose
N
N
N
is any positive integer. Add the digits of
N
N
N
to obtain a smaller integer. Repeat this process of digit-addition till you get a single digit numbem. Find the number of positive integers
N
≤
1000
N \le 1000
N
≤
1000
, such that the final single-digit number
n
n
n
is equal to
5
5
5
. Example:
N
=
563
→
(
5
+
6
+
3
)
=
14
→
(
1
+
4
)
=
5
N = 563\to (5 + 6 + 3) = 14 \to(1 + 4) = 5
N
=
563
→
(
5
+
6
+
3
)
=
14
→
(
1
+
4
)
=
5
will be counted as one such integer.
2
1
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2016 preRMO p2, x^{2016}+(2016!+1!)x^{2015}+(2015!+2!)x^{2014}+...+(1!+2016!)=0
Find the number of integer solutions of the equation
x
2016
+
(
2016
!
+
1
!
)
x
2015
+
(
2015
!
+
2
!
)
x
2014
+
.
.
.
+
(
1
!
+
2016
!
)
=
0
x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + ... + (1! + 2016!) = 0
x
2016
+
(
2016
!
+
1
!)
x
2015
+
(
2015
!
+
2
!)
x
2014
+
...
+
(
1
!
+
2016
!)
=
0
1
1
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2016 preRMO p1 (5 \cdot 3^m) + 4 = n^2
Consider all possible integers
n
≥
0
n \ge 0
n
≥
0
such that (5 \cdot 3^m) + 4 = n^2 holds for some corresponding integer
m
≥
0
m \ge 0
m
≥
0
. Find the sum of all such
n
n
n
.