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Problems
Contests
National and Regional Contests
India Contests
India Pre-Regional Mathematical Olympiad
2018 India PRMO
2018 India PRMO
Part of
India Pre-Regional Mathematical Olympiad
Subcontests
(30)
29
1
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2018 preRMO p29, angle chasing candidate, 2 incentres given
Let
D
D
D
be an interior point of the side
B
C
BC
BC
of a triangle
A
B
C
ABC
A
BC
. Let
I
1
I_1
I
1
and
I
2
I_2
I
2
be the incentres of triangles
A
B
D
ABD
A
B
D
and
A
C
D
ACD
A
C
D
respectively. Let
A
I
1
AI_1
A
I
1
and
A
I
2
AI_2
A
I
2
meet
B
C
BC
BC
in
E
E
E
and
F
F
F
respectively. If
∠
B
I
1
E
=
6
0
o
\angle BI_1E = 60^o
∠
B
I
1
E
=
6
0
o
, what is the measure of
∠
C
I
2
F
\angle CI_2F
∠
C
I
2
F
in degrees?
28
1
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2018 preRMO p28, 8 chocolates of different brands among 3 kids
Let
N
N
N
be the number of ways of distributing
8
8
8
chocolates of different brands among
3
3
3
children such that each child gets at least one chocolate, and no two children get the same number of chocolates. Find the sum of the digits of
N
N
N
.
26
1
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2018 preRMO p26, choose 60 unit squares from a 11 x 11 chessboard
What is the number of ways in which one can choose
60
60
60
unit squares from a
11
×
11
11 \times 11
11
×
11
chessboard such that no two chosen squares have a side in common?
22
1
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2018 preRMO p22, k=sum of no in partition of {1, 2, ..., 20}, good integer
A positive integer
k
k
k
is said to be good if there exists a partition of
{
1
,
2
,
3
,
.
.
.
,
20
}
\{1, 2, 3,..., 20\}
{
1
,
2
,
3
,
...
,
20
}
into disjoint proper subsets such that the sum of the numbers in each subset of the partition is
k
k
k
. How many good numbers are there?
24
1
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2018 preRMO p24, triangles with integer angles, not similar, N/100 ?
If
N
N
N
is the number of triangles of different shapes (i.e., not similar) whose angles are all integers (in degrees), what is
N
100
\frac{N}{100}
100
N
?
17
1
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2018 preRMO p17 \A=\D, AB=DE=17, BC=EF=10,AC−DF=12, AC + DF ?
Triangles
A
B
C
ABC
A
BC
and
D
E
F
DEF
D
EF
are such that
∠
A
=
∠
D
,
A
B
=
D
E
=
17
,
B
C
=
E
F
=
10
\angle A = \angle D, AB = DE = 17, BC = EF = 10
∠
A
=
∠
D
,
A
B
=
D
E
=
17
,
BC
=
EF
=
10
and
A
C
−
D
F
=
12
AC - DF = 12
A
C
−
D
F
=
12
. What is
A
C
+
D
F
AC + DF
A
C
+
D
F
?
12
1
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2018 preRMO p12, 8tuples wanted, multiple of 3, in {-1,1 }
Determine the number of
8
8
8
-tuples
(
ϵ
1
,
ϵ
2
,
.
.
.
,
ϵ
8
)
(\epsilon_1, \epsilon_2,...,\epsilon_8)
(
ϵ
1
,
ϵ
2
,
...
,
ϵ
8
)
such that
ϵ
1
,
ϵ
2
,
.
.
.
,
8
∈
{
1
,
−
1
}
\epsilon_1, \epsilon_2, ..., 8 \in \{1,-1\}
ϵ
1
,
ϵ
2
,
...
,
8
∈
{
1
,
−
1
}
and
ϵ
1
+
2
ϵ
2
+
3
ϵ
3
+
.
.
.
+
8
ϵ
8
\epsilon_1 + 2\epsilon_2 + 3\epsilon_3 +...+ 8\epsilon_8
ϵ
1
+
2
ϵ
2
+
3
ϵ
3
+
...
+
8
ϵ
8
is a multiple of
3
3
3
.
14
1
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2018 preRMO p14, x=cos1cos2cos3..·cos89, y=cos2cos6cos10...cos86
If
x
=
c
o
s
1
o
c
o
s
2
o
c
o
s
3
o
.
.
.
c
o
s
8
9
o
x = cos 1^o cos 2^o cos 3^o...cos 89^o
x
=
cos
1
o
cos
2
o
cos
3
o
...
cos
8
9
o
and
y
=
c
o
s
2
o
c
o
s
6
o
c
o
s
1
0
o
.
.
.
c
o
s
8
6
o
y = cos 2^o cos 6^o cos 10^o...cos 86^o
y
=
cos
2
o
cos
6
o
cos
1
0
o
...
cos
8
6
o
, then what is the integer nearest to
2
7
log
2
y
x
\frac27 \log_2 \frac{y}{x}
7
2
lo
g
2
x
y
?
10
1
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2018 preRMO p10, (BC^2 + CA^2 + AB^2)/100, perp. medians
In a triangle
A
B
C
ABC
A
BC
, the median from
B
B
B
to
C
A
CA
C
A
is perpendicular to the median from
C
C
C
to
A
B
AB
A
B
. If the median from
A
A
A
to
B
C
BC
BC
is
30
30
30
, determine
B
C
2
+
C
A
2
+
A
B
2
100
\frac{BC^2 + CA^2 + AB^2}{100}
100
B
C
2
+
C
A
2
+
A
B
2
.
9
1
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2018 preRMO p9, a+b is root of x^2 +ax+b = 0, a,b integers
Suppose
a
,
b
a, b
a
,
b
are integers and
a
+
b
a+b
a
+
b
is a root of
x
2
+
a
x
+
b
=
0
x^2 +ax+b = 0
x
2
+
a
x
+
b
=
0
. What is the maximum possible value of
b
2
b^2
b
2
?
8
1
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2018 preRMO p8, angle chasing with a chord
Let
A
B
AB
A
B
be a chord of a circle with centre
O
O
O
. Let
C
C
C
be a point on the circle such that
∠
A
B
C
=
3
0
o
\angle ABC =30^o
∠
A
BC
=
3
0
o
and
O
O
O
lies inside the triangle
A
B
C
ABC
A
BC
. Let
D
D
D
be a point on
A
B
AB
A
B
such that
∠
D
C
O
=
∠
O
C
B
=
2
0
o
\angle DCO = \angle OCB = 20^o
∠
D
CO
=
∠
OCB
=
2
0
o
. Find the measure of
∠
C
D
O
\angle CDO
∠
C
D
O
in degrees.
7
1
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2018 preRMO p7 distances from consecutive vertices of regular hexagon
A point
P
P
P
in the interior of a regular hexagon is at distances
8
,
8
,
16
8,8,16
8
,
8
,
16
units from three consecutive vertices of the hexagon, respectively. If
r
r
r
is radius of the circumscribed circle of the hexagon, what is the integer closest to
r
r
r
?
11
1
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2018 preRMO p11, teacups in the kitchen, some wiht handles
There are several teacups in the kitchen, some with handles and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly
1200
1200
1200
. What is the maximum possible number of cups in the kitchen?
30
1
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2018 preRMO p30 integer P(x), P(1) = 4 and P(5) = 136, P(3) ?
Let
P
(
x
)
P(x)
P
(
x
)
=
a
0
+
a
1
x
+
a
2
x
2
+
⋯
+
a
n
x
n
a_0+a_1x+a_2x^2+\cdots +a_nx^n
a
0
+
a
1
x
+
a
2
x
2
+
⋯
+
a
n
x
n
be a polynomial in which
a
i
a_i
a
i
is non-negative integer for each
i
∈
i \in
i
∈
{
0
,
1
,
2
,
3
,
.
.
.
.
,
n
0,1,2,3,....,n
0
,
1
,
2
,
3
,
....
,
n
} . If
P
(
1
)
=
4
P(1) = 4
P
(
1
)
=
4
and
P
(
5
)
=
136
P(5) = 136
P
(
5
)
=
136
, what is the value of
P
(
3
)
P(3)
P
(
3
)
?
25
1
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2018 preRMO p25 remainders divides by 11,13,15
Let
T
T
T
be the smallest positive integers which, when divided by
11
,
13
,
15
11,13,15
11
,
13
,
15
leaves remainders in the sets {
7
,
8
,
9
7,8,9
7
,
8
,
9
}, {
1
,
2
,
3
1,2,3
1
,
2
,
3
}, {
4
,
5
,
6
4,5,6
4
,
5
,
6
} respectively. What is the sum of the squares of the digits of
T
T
T
?
20
1
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2018 preRMO p20, product of whose digits equals n^2 -15n -27
Determine the sum of all possible positive integers
n
,
n,
n
,
the product of whose digits equals
n
2
−
15
n
−
27
n^2 -15n -27
n
2
−
15
n
−
27
.
18
1
Hide problems
2018 preRMO p18, diopantine 4abc = (a+3)(b+3)(c+3), a+b+c =?
If
a
,
b
,
c
≥
4
a, b, c \ge 4
a
,
b
,
c
≥
4
are integers, not all equal, and
4
a
b
c
=
(
a
+
3
)
(
b
+
3
)
(
c
+
3
)
4abc = (a+3)(b+3)(c+3)
4
ab
c
=
(
a
+
3
)
(
b
+
3
)
(
c
+
3
)
then what is the value of
a
+
b
+
c
a+b+c
a
+
b
+
c
?
15
1
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2018 preRMO p15, 2a-b, a-2b and a+b all distinct squares
Let
a
a
a
and
b
b
b
be natural numbers such that
2
a
−
b
2a-b
2
a
−
b
,
a
−
2
b
a-2b
a
−
2
b
and
a
+
b
a+b
a
+
b
are all distinct squares. What is the smallest possible value of
b
b
b
?
13
1
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2018 preRMO p13 computational, median in right triangle
In a triangle
A
B
C
ABC
A
BC
, right angled at
A
A
A
, the altitude through
A
A
A
and the internal bisector of
∠
A
\angle A
∠
A
have lengths
3
3
3
and
4
4
4
, respectively. Find the length of the median through
A
A
A
.
6
1
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2018 preRMO p6 , a+b-c=1 and a^2+b^2-c^2=-1, sum of a^2+b^2+c^2 ?
Integers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfy
a
+
b
−
c
=
1
a+b-c=1
a
+
b
−
c
=
1
and
a
2
+
b
2
−
c
2
=
−
1
a^2+b^2-c^2=-1
a
2
+
b
2
−
c
2
=
−
1
. What is the sum of all possible values of
a
2
+
b
2
+
c
2
a^2+b^2+c^2
a
2
+
b
2
+
c
2
?
5
1
Hide problems
2018 preRMO p5 computational with a tangential trapezium
Let
A
B
C
D
ABCD
A
BC
D
be a trapezium in which
A
B
/
/
C
D
AB //CD
A
B
//
C
D
and
A
D
⊥
A
B
AD \perp AB
A
D
⊥
A
B
. Suppose
A
B
C
D
ABCD
A
BC
D
has an incircle which touches
A
B
AB
A
B
at
Q
Q
Q
and
C
D
CD
C
D
at
P
P
P
. Given that
P
C
=
36
PC = 36
PC
=
36
and
Q
B
=
49
QB = 49
QB
=
49
, find
P
Q
PQ
PQ
.
4
1
Hide problems
2018 preRMO p4 equation 166 x 56 = 8590 is valid in some base b >=10
The equation
166
×
56
=
8590
166\times 56 = 8590
166
×
56
=
8590
is valid in some base
b
≥
10
b \ge 10
b
≥
10
(that is,
1
,
6
,
5
,
8
,
9
,
0
1, 6, 5, 8, 9, 0
1
,
6
,
5
,
8
,
9
,
0
are digits in base
b
b
b
in the above equation). Find the sum of all possible values of
b
≥
10
b \ge 10
b
≥
10
satisfying the equation.
3
1
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2018 preRMO p3 - 6-digit numbers of the form abccba , b odd, divisible by 7
Consider all
6
6
6
-digit numbers of the form
a
b
c
c
b
a
abccba
ab
cc
ba
where
b
b
b
is odd. Determine the number of all such
6
6
6
-digit numbers that are divisible by
7
7
7
.
1
1
Hide problems
2018 preRMO p1 - book is published in three volumes
A book is published in three volumes, the pages being numbered from
1
1
1
onwards. The page numbers are continued from the first volume to the second volume to the third. The number of pages in the second volume is
50
50
50
more than that in the first volume, and the number pages in the third volume is one and a half times that in the second. The sum of the page numbers on the first pages of the three volumes is
1709
1709
1709
. If
n
n
n
is the last page number, what is the largest prime factor of
n
n
n
?
16
1
Hide problems
Interesting Summation
What is the value of
∑
1
≤
i
<
j
≤
10
(
i
+
j
)
i
+
j
=
o
d
d
{ \sum_{1 \le i< j \le 10}(i+j)}_{i+j=odd}
∑
1
≤
i
<
j
≤
10
(
i
+
j
)
i
+
j
=
o
dd
−
∑
1
≤
i
<
j
≤
10
(
i
+
j
)
i
+
j
=
e
v
e
n
- { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=even}
−
∑
1
≤
i
<
j
≤
10
(
i
+
j
)
i
+
j
=
e
v
e
n
23
1
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Simple inequality
What is the largest positive integer
n
n
n
such that
a
2
b
29
+
c
31
+
b
2
c
29
+
a
31
+
c
2
a
29
+
b
31
≥
n
(
a
+
b
+
c
)
\frac{a^2}{\frac{b}{29} + \frac{c}{31}}+\frac{b^2}{\frac{c}{29} + \frac{a}{31}}+\frac{c^2}{\frac{a}{29} + \frac{b}{31}} \ge n(a+b+c)
29
b
+
31
c
a
2
+
29
c
+
31
a
b
2
+
29
a
+
31
b
c
2
≥
n
(
a
+
b
+
c
)
holds for all positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
.
2
1
Hide problems
Find the closest integer
In a quadrilateral ABCD, it is given that AB = AD = 13, BC = CD = 20, BD = 24. If r is the radius of the circle inscribable in the quadrilateral, then what is the integer closest to r?
21
1
Hide problems
PRMO 2018
Let
Δ
A
B
C
\Delta ABC
Δ
A
BC
be an acute-angled triangle and let
H
H
H
be its orthocentre. Let
G
1
,
G
2
G_1, G_2
G
1
,
G
2
and
G
3
G_3
G
3
be the centroids of the triangles
Δ
H
B
C
,
Δ
H
C
A
\Delta HBC , \Delta HCA
Δ
H
BC
,
Δ
H
C
A
and
Δ
H
A
B
\Delta HAB
Δ
H
A
B
respectively. If the area of
Δ
G
1
G
2
G
3
\Delta G_1G_2G_3
Δ
G
1
G
2
G
3
is
7
7
7
units, what is the area of
Δ
A
B
C
\Delta ABC
Δ
A
BC
?
27
1
Hide problems
let's do some combinatorics
What is the number of ways in which one can color the squares of a
4
×
4
4\times 4
4
×
4
chessboard with colors red and blue such that each row as well as each column has exactly two red squares and two blue squares?
19
1
Hide problems
Infinite 6's
Let
N
=
6
+
66
+
666
+
.
.
.
.
+
666..66
N=6+66+666+....+666..66
N
=
6
+
66
+
666
+
....
+
666..66
, where there are hundred
6
′
s
6's
6
′
s
in the last term in the sum. How many times does the digit
7
7
7
occur in the number
N
N
N