MathDB
Problems
Contests
National and Regional Contests
India Contests
India Pre-Regional Mathematical Olympiad
2019 India PRMO
2019 India PRMO
Part of
India Pre-Regional Mathematical Olympiad
Subcontests
(31)
21 incorrect
1
Hide problems
PRMO 2019 Leg 2 P21 the incorrect problem, show why
Consider the set
E
E
E
of all positive integers
n
n
n
such that when divided by
9
,
10
,
11
9,10,11
9
,
10
,
11
respectively, the remainders(in that order) are all
>
1
>1
>
1
and form a non constant geometric progression. If
N
N
N
is the largest element of
E
E
E
, find the sum of digits of
E
E
E
30
2
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Set / Equal Sums
Let
E
E
E
denote the set of all natural numbers
n
n
n
such that
3
<
n
<
100
3 < n < 100
3
<
n
<
100
and the set
{
1
,
2
,
3
,
…
,
n
}
\{ 1, 2, 3, \ldots , n\}
{
1
,
2
,
3
,
…
,
n
}
can be partitioned in to
3
3
3
subsets with equal sums. Find the number of elements of
E
E
E
.
Floors and Fracs
For any real number
x
x
x
, let
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
denote the integer part of
x
x
x
;
{
x
}
\{ x \}
{
x
}
be the fractional part of
x
x
x
(
{
x
}
\{x\}
{
x
}
=
=
=
x
−
x-
x
−
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
). Let
A
A
A
denote the set of all real numbers
x
x
x
satisfying
{
x
}
=
x
+
⌊
x
⌋
+
⌊
x
+
(
1
/
2
)
⌋
20
\{x\} =\frac{x+\lfloor x \rfloor +\lfloor x + (1/2) \rfloor }{20}
{
x
}
=
20
x
+
⌊
x
⌋
+
⌊
x
+
(
1/2
)⌋
If
S
S
S
is the sume of all numbers in
A
A
A
, find
⌊
S
⌋
\lfloor S \rfloor
⌊
S
⌋
29
2
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Geometry / Median / Angle Bisector
In a triangle
A
B
C
ABC
A
BC
, the median
A
D
AD
A
D
(with
D
D
D
on
B
C
BC
BC
) and the angle bisector
B
E
BE
BE
(with
E
E
E
on
A
C
AC
A
C
) are perpedicular to each other. If
A
D
=
7
AD = 7
A
D
=
7
and
B
E
=
9
BE = 9
BE
=
9
, find the integer nearest to the area of triangle
A
B
C
ABC
A
BC
.
Last Geo
Let
A
B
C
ABC
A
BC
be an acute angled triangle with
A
B
=
15
AB=15
A
B
=
15
and
B
C
=
8
BC=8
BC
=
8
. Let
D
D
D
be a point on
A
B
AB
A
B
such that
B
D
=
B
C
BD=BC
B
D
=
BC
. Consider points
E
E
E
on
A
C
AC
A
C
such that
∠
D
E
B
=
∠
B
E
C
\angle DEB=\angle BEC
∠
D
EB
=
∠
BEC
. If
α
\alpha
α
denotes the product of all possible values of
A
E
AE
A
E
, find
⌊
α
⌋
\lfloor \alpha \rfloor
⌊
α
⌋
the integer part of
α
\alpha
α
.
28
2
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Geometry / Incicles and Incenters
Let
A
B
C
ABC
A
BC
be a triangle with sides
51
,
52
,
53
51, 52, 53
51
,
52
,
53
. Let
Ω
\Omega
Ω
denote the incircle of
△
A
B
C
\bigtriangleup ABC
△
A
BC
. Draw tangents to
Ω
\Omega
Ω
which are parallel to the sides of
A
B
C
ABC
A
BC
. Let
r
1
,
r
2
,
r
3
r_1, r_2, r_3
r
1
,
r
2
,
r
3
be the inradii of the three corener triangles so formed, Find the largest integer that does not exceed
r
1
+
r
2
+
r
3
r_1 + r_2 + r_3
r
1
+
r
2
+
r
3
.
Computation Geo
In a triangle
A
B
C
ABC
A
BC
, it is known that
∠
A
=
10
0
∘
\angle A=100^{\circ}
∠
A
=
10
0
∘
and
A
B
=
A
C
AB=AC
A
B
=
A
C
. The internal angle bisector
B
D
BD
B
D
has length
20
20
20
units. Find the length of
B
C
BC
BC
to the nearest integer, given that
sin
1
0
∘
≈
0.174
\sin 10^{\circ} \approx 0.174
sin
1
0
∘
≈
0.174
27
2
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Combinatorics Question
We will say that a rearrangement of the letters of a word has no fixed letters if, when the rearrangement is placed directly below the word, no column has the same letter repeated. For instance
H
B
R
A
T
A
HBRATA
H
BR
A
T
A
is a rearragnement with no fixed letter of
B
H
A
R
A
T
BHARAT
B
H
A
R
A
T
. How many distinguishable rearrangements with no fixed letters does
B
H
A
R
A
T
BHARAT
B
H
A
R
A
T
have? (The two
A
A
A
s are considered identical.)
Honey and The Ant
A conical glass is in the form of a right circular cone. The slant height is
21
21
21
and the radius of the top rim of the glass is
14
14
14
. An ant at the mid point of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. If
d
d
d
the shortest distance it should crawl to reach the honey drop, what is the integer part of
d
d
d
?https://i.imgur.com/T1Y3zwR.png
26
2
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Computing minimum value of an equation given a particular relation
Positive integers
x
,
y
,
z
x, y, z
x
,
y
,
z
satisfy
x
y
+
z
=
160
xy + z = 160
x
y
+
z
=
160
. Compute the smallest possible value of
x
+
y
z
x + yz
x
+
yz
.
Bouncy Ball
A friction-less board has the shape of an equilateral triangle of side length
1
1
1
meter with bouncing walls along the sides. A tiny super bouncy ball is fired from vertex
A
A
A
towards the side
B
C
BC
BC
. The ball bounces off the walls of the board nine times before it hits a vertex for the first time. The bounces are such that the angle of incidence equals the angle of reflection. The distance travelled by the ball in meters is of the form
N
\sqrt{N}
N
, where
N
N
N
is an integer. What is the value of
N
N
N
?
25
2
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Simple Geometry word problem involving circles and tangents
A village has a circular wall around it, and the wall has four gates pointing north, south, east and west. A tree stands outside the village,
16
m
16 \, \mathrm{m}
16
m
north of the north gate, and it can be just seen appearing on the horizon from a point
48
m
48 \, \mathrm{m}
48
m
east of the south gate. What is the diamter in meters, of the wall that surrounds the village?
Easy Geo Again
Let
A
B
C
ABC
A
BC
be an isosceles triangle with
A
B
=
B
C
AB=BC
A
B
=
BC
. A trisector of
∠
B
\angle B
∠
B
meets
A
C
AC
A
C
at
D
D
D
. If
A
B
,
A
C
AB,AC
A
B
,
A
C
and
B
D
BD
B
D
are integers and
A
B
−
B
D
AB-BD
A
B
−
B
D
=
=
=
3
3
3
, find
A
C
AC
A
C
.
24
2
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Combinatorics / Counting squares
A
1
×
n
1 \times n
1
×
n
rectangle (
n
≥
1
n \geq 1
n
≥
1
) is divided into
n
n
n
unit (
1
×
1
1 \times 1
1
×
1
) squares. Each square of this rectangle is colored red, blue or green. Let
f
(
n
)
f(n)
f
(
n
)
be the number of colourings of the rectangle in which there are an even number of red squares. What is the largest prime factor of
f
(
9
)
/
f
(
3
)
f(9)/f(3)
f
(
9
)
/
f
(
3
)
? (The number of red squares can be zero.)
9's in a Number
For
n
≥
1
n \geq 1
n
≥
1
, let
a
n
a_n
a
n
be the number beginning with
n
n
n
9
9
9
's followed by
744
744
744
; eg.,
a
4
=
9999744
a_4=9999744
a
4
=
9999744
. Define
f
(
n
)
=
max
{
m
∈
N
∣
2
m
divides
a
n
}
f(n)=\text{max}\{m\in \mathbb{N} \mid2^m ~ \text{divides} ~ a_n \}
f
(
n
)
=
max
{
m
∈
N
∣
2
m
divides
a
n
}
, for
n
≥
1
n\geq 1
n
≥
1
. Find
f
(
1
)
+
f
(
2
)
+
f
(
3
)
+
⋯
+
f
(
10
)
f(1)+f(2)+f(3)+ \cdots + f(10)
f
(
1
)
+
f
(
2
)
+
f
(
3
)
+
⋯
+
f
(
10
)
.
23
2
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Geometry / Cyclic Quadilateral
Let
A
B
C
D
ABCD
A
BC
D
be a convex cyclic quadilateral. Suppose
P
P
P
is a point in the plane of the quadilateral such that the sum of its distances from the vertices of
A
B
C
D
ABCD
A
BC
D
is the least. If
{
P
C
,
P
B
,
P
C
,
P
D
}
=
{
3
,
4
,
6
,
8
}
\{PC, PB, PC, PD\} = \{3, 4, 6, 8\}
{
PC
,
PB
,
PC
,
P
D
}
=
{
3
,
4
,
6
,
8
}
, what is the maxumum possible area of
A
B
C
D
ABCD
A
BC
D
?
Polynomial With Trigo Roots
Let
t
t
t
be the area of a regular pentagon with each side equal to
1
1
1
. Let
P
(
x
)
=
0
P(x)=0
P
(
x
)
=
0
be the polynomial equation with least degree, having integer coefficients, satisfied by
x
=
t
x=t
x
=
t
and the
gcd
\gcd
g
cd
of all the coefficients equal to
1
1
1
. If
M
M
M
is the sum of the absolute values of the coefficients of
P
(
x
)
P(x)
P
(
x
)
, What is the integer closest to
M
\sqrt{M}
M
? (
sin
1
8
∘
=
(
5
−
1
)
/
2
\sin 18^{\circ}=(\sqrt{5}-1)/2
sin
1
8
∘
=
(
5
−
1
)
/2
)
22
2
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Telescoping Sum
What is the greatest integer not exceeding the sum
∑
n
=
1
1599
1
n
\sum^{1599}_{n=1} \dfrac{1}{\sqrt{n}}
∑
n
=
1
1599
n
1
?
Trig Geometry
In parallelogram
A
B
C
D
ABCD
A
BC
D
,
A
C
=
10
AC=10
A
C
=
10
and
B
D
=
28
BD=28
B
D
=
28
. The points
K
K
K
and
L
L
L
in the plane of
A
B
C
D
ABCD
A
BC
D
move in such a way that
A
K
=
B
D
AK=BD
A
K
=
B
D
and
B
L
=
A
C
BL=AC
B
L
=
A
C
. Let
M
M
M
and
N
N
N
be the midpoints of
C
K
CK
C
K
and
D
L
DL
D
L
, respectively. What is the maximum walue of
cot
2
(
∠
B
M
D
2
)
+
tan
2
(
∠
A
N
C
2
)
\cot^2 (\tfrac{\angle BMD}{2})+\tan^2(\tfrac{\angle ANC}{2})
cot
2
(
2
∠
BM
D
)
+
tan
2
(
2
∠
A
NC
)
?
20
2
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Number Theory / Sets
Consider the set
E
E
E
of all natural numbers
n
n
n
such that whenn divided by
11
,
12
,
13
11, 12, 13
11
,
12
,
13
, respectively, the remainders, int that order, are distinct prime numbers in an arithmetic progression. If
N
N
N
is the largest number in
E
E
E
, find the sum of digits of
N
N
N
.
Combi With Digits
How many
4
−
4-
4
−
digit numbers
a
b
c
d
‾
\overline{abcd}
ab
c
d
are there such that
a
<
b
<
c
<
d
a<b<c<d
a
<
b
<
c
<
d
and
b
−
a
<
c
−
b
<
d
−
c
b-a<c-b<d-c
b
−
a
<
c
−
b
<
d
−
c
?
21
1
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Primes / Sets / Number Theory
Consider the set
E
=
{
5
,
6
,
7
,
8
,
9
}
E = \{5, 6, 7, 8, 9\}
E
=
{
5
,
6
,
7
,
8
,
9
}
. For any partition
A
,
B
{A, B}
A
,
B
of
E
E
E
, with both
A
A
A
and
B
B
B
non-empty, consider the number obtained by adding the product of elements of
A
A
A
to the product of elements of
B
B
B
. Let
N
N
N
be the largest prime number amonh these numbers. Find the sum of the digits of
N
N
N
.
17
2
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Combinatorics / Multinomial
How many ordered triplets
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of positive integers such that
30
a
+
50
b
+
70
c
≤
343
30a + 50b + 70c \leq 343
30
a
+
50
b
+
70
c
≤
343
.
More Squares
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be distinct positive integers such that
b
+
c
−
a
b+c-a
b
+
c
−
a
,
c
+
a
−
b
c+a-b
c
+
a
−
b
and
a
+
b
−
c
a+b-c
a
+
b
−
c
are all perfect squares. What is the largest possible value of
a
+
b
+
c
a+b+c
a
+
b
+
c
smaller than
100
100
100
?
19
2
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Geometry Problem
Let
A
B
AB
A
B
be a diameter of a circle and let
C
C
C
be a point on the segement
A
B
AB
A
B
such that
A
C
:
C
B
=
6
:
7
AC : CB = 6 : 7
A
C
:
CB
=
6
:
7
. Let
D
D
D
be a point on the circle such that
D
C
DC
D
C
is perpendicular to
A
B
AB
A
B
. Let
D
E
DE
D
E
be the diameter through
D
D
D
. If
[
X
Y
Z
]
[XYZ]
[
X
Y
Z
]
denotes the area of the triangle
X
Y
Z
XYZ
X
Y
Z
, find
[
A
B
D
]
/
[
C
D
E
]
[ABD]/[CDE]
[
A
B
D
]
/
[
C
D
E
]
to the nearest integer.
Easy Geometry
If
15
15
15
and
9
9
9
are lengths of two medians of a triangle, what is the maximum possible area of the triangle to the nearest integer ?
18
2
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Combinatorics / Multinomial
Find the number of ordered triples
(
a
,
b
)
(a, b)
(
a
,
b
)
of positive integers with
a
<
b
a < b
a
<
b
and
100
≤
a
,
b
≤
1000
100 \leq a, b \leq 1000
100
≤
a
,
b
≤
1000
satisfy
gcd
(
a
,
b
)
:
l
c
m
(
a
,
b
)
=
1
:
495
\gcd(a, b) : \mathrm{lcm}(a, b) = 1 : 495
g
cd
(
a
,
b
)
:
lcm
(
a
,
b
)
=
1
:
495
?
Factors Of A Prime
What is the smallest prime number
p
p
p
such that
p
3
+
4
p
2
+
4
p
p^3+4p^2+4p
p
3
+
4
p
2
+
4
p
has exactly
30
30
30
positive divisors ?
16
2
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Minimizing the difference between variables in a linear equation
A pen costs
R
s
.
13
\mathrm{Rs.}\, 13
Rs.
13
and a note book costs
R
s
.
17
\mathrm{Rs.}\, 17
Rs.
17
. A school spends exactly
R
s
.
10000
\mathrm{Rs.}\, 10000
Rs.
10000
in the year
2017
−
18
2017-18
2017
−
18
to buy
x
x
x
pens and
y
y
y
note books such that
x
x
x
and
y
y
y
are as close as possible (i.e.,
∣
x
−
y
∣
|x-y|
∣
x
−
y
∣
is minimum). Next year, in
2018
−
19
2018-19
2018
−
19
, the school spends a little more than
R
s
.
10000
\mathrm{Rs.}\, 10000
Rs.
10000
and buys
y
y
y
pens and
x
x
x
note books. How much more did the school pay?
Simple Divisibility
Let
N
N
N
denote the number of all natural numbers
n
n
n
such that
n
n
n
is divisible by a prime
p
>
n
p> \sqrt{n}
p
>
n
and
p
<
20
p<20
p
<
20
. What is the value of
N
N
N
?
15
2
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Parallel Diagonals of a 10-gon
In how many ways can a pair of parallel diagonals of a regular polygon of
10
10
10
sides be selected?
Negative Bases
In base-
2
2
2
notation, digits are
0
0
0
and
1
1
1
only and the places go up in powers of
−
2
-2
−
2
. For example,
11011
11011
11011
stands for
(
−
2
)
4
+
(
−
2
)
3
+
(
−
2
)
1
+
(
−
2
)
0
(-2)^4+(-2)^3+(-2)^1+(-2)^0
(
−
2
)
4
+
(
−
2
)
3
+
(
−
2
)
1
+
(
−
2
)
0
and equals number
7
7
7
in base
10
10
10
. If the decimal number
2019
2019
2019
is expressed in base
−
2
-2
−
2
how many non-zero digits does it contain ?
14
2
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Smallest Prime and Perfect Square of a smallest relation
Find the smallest positive integer
n
≥
10
n \geq 10
n
≥
10
such that
n
+
6
n + 6
n
+
6
is a prime and
9
n
+
7
9n + 7
9
n
+
7
is a perfect square.
Circle in Real Plane
Let
R
\mathcal{R}
R
denote the circular region in the
x
y
−
xy-
x
y
−
plane bounded by the circle
x
2
+
y
2
=
36
x^2+y^2=36
x
2
+
y
2
=
36
. The lines
x
=
4
x=4
x
=
4
and
y
=
3
y=3
y
=
3
divide
R
\mathcal{R}
R
into four regions
R
i
,
i
=
1
,
2
,
3
,
4
\mathcal{R}_i ~ , ~i=1,2,3,4
R
i
,
i
=
1
,
2
,
3
,
4
. If
∣
R
i
∣
\mid \mathcal{R}_i \mid
∣
R
i
∣
denotes the area of the region
R
i
\mathcal{R}_i
R
i
and if
∣
R
1
∣
>
\mid \mathcal{R}_1 \mid >
∣
R
1
∣>
∣
R
2
∣
>
\mid \mathcal{R}_2 \mid >
∣
R
2
∣>
∣
R
3
∣
>
\mid \mathcal{R}_3 \mid >
∣
R
3
∣>
∣
R
4
∣
\mid \mathcal{R}_4 \mid
∣
R
4
∣
, determine
∣
R
1
∣
\mid \mathcal{R}_1 \mid
∣
R
1
∣
−
-
−
∣
R
2
∣
\mid \mathcal{R}_2 \mid
∣
R
2
∣
−
-
−
∣
R
3
∣
\mid \mathcal{R}_3 \mid
∣
R
3
∣
+
+
+
∣
R
4
∣
\mid \mathcal{R}_4 \mid
∣
R
4
∣
.
13
2
Hide problems
Simple Summation Problem
Each of the numbers
x
1
,
x
2
,
…
,
x
101
x_1, x_2, \ldots, x_{101}
x
1
,
x
2
,
…
,
x
101
is
±
1
\pm 1
±
1
. What is the smallest positive value of
∑
1
≤
i
<
j
≤
101
x
i
x
j
\sum_{1\leq i < j \leq 101} x_i x_j
∑
1
≤
i
<
j
≤
101
x
i
x
j
?
Sum of distinct powers of 7
Consider the sequence
1
,
7
,
8
,
49
,
50
,
56
,
57
,
343
…
1,7,8,49,50,56,57,343\ldots
1
,
7
,
8
,
49
,
50
,
56
,
57
,
343
…
which consists of sums of distinct powers of
7
7
7
, that is,
7
0
7^0
7
0
,
7
1
7^1
7
1
,
7
0
+
7
1
7^0+7^1
7
0
+
7
1
,
7
2
7^2
7
2
,
…
\ldots
…
in increasing order. At what position will
16856
16856
16856
occur in this sequence?
12
2
Hide problems
Find sum of natural numbers satisfying the following condition
A natural number
k
>
1
k > 1
k
>
1
is called good if there exist natural numbers
a
1
<
a
2
<
⋯
<
a
k
a_1 < a_2 < \cdots < a_k
a
1
<
a
2
<
⋯
<
a
k
such that
1
a
1
+
1
a
2
+
⋯
+
1
a
k
=
1
\dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots + \dfrac{1}{\sqrt{a_k}} = 1
a
1
1
+
a
2
1
+
⋯
+
a
k
1
=
1
.Let
f
(
n
)
f(n)
f
(
n
)
be the sum of the first
n
n
n
[good numbers,
n
≥
n \geq
n
≥
1. Find the sum of all values of
n
n
n
for which
f
(
n
+
5
)
/
f
(
n
)
f(n+5)/f(n)
f
(
n
+
5
)
/
f
(
n
)
is an integer.
Easy counting
Let
N
N
N
be the number of ways of choosing a subset of
5
5
5
distinct numbers from the set
10
a
+
b
:
1
≤
a
≤
5
,
1
≤
b
≤
5
{10a+b:1\leq a\leq 5, 1\leq b\leq 5}
10
a
+
b
:
1
≤
a
≤
5
,
1
≤
b
≤
5
where
a
,
b
a,b
a
,
b
are integers, such that no two of the selected numbers have the same units digits and no two have the same tens digit. What is the remainder when
N
N
N
is divided by
73
73
73
?
11
2
Hide problems
Trigonometry and Triangles
How many distinct triangles
A
B
C
ABC
A
BC
are tjere, up to simplilarity, such that the magnitudes of the angles
A
,
B
A, B
A
,
B
and
C
C
C
in degrees are positive integers and satisfy
cos
A
cos
B
+
sin
A
sin
B
sin
k
C
=
1
\cos{A}\cos{B} + \sin{A}\sin{B}\sin{kC} = 1
cos
A
cos
B
+
sin
A
sin
B
sin
k
C
=
1
for some positive integer
k
k
k
, where
k
C
kC
k
C
does not exceet
36
0
∘
360^{\circ}
36
0
∘
?
Easy SFFT
Find the largest value of
a
b
a^b
a
b
such that the positive integers
a
,
b
>
1
a,b>1
a
,
b
>
1
satisfy
a
b
b
a
+
a
b
+
b
a
=
5329
a^bb^a+a^b+b^a=5329
a
b
b
a
+
a
b
+
b
a
=
5329
10
2
Hide problems
Geometry Angle Bisectors Problem
Let
A
B
C
ABC
A
BC
be a triangle and let
Ω
\Omega
Ω
be its circumcircle. The internal bisectors of angles
A
,
B
A, B
A
,
B
and
C
C
C
intersect
Ω
\Omega
Ω
at
A
1
,
B
1
A_1, B_1
A
1
,
B
1
and
C
1
C_1
C
1
, respectively, and the internal bisectors of angles
A
1
,
B
1
A_1, B_1
A
1
,
B
1
and
C
1
C_1
C
1
of the triangles
A
1
A
2
A
3
A_1 A_2 A_ 3
A
1
A
2
A
3
intersect
Ω
\Omega
Ω
at
A
2
,
B
2
A_2, B_2
A
2
,
B
2
and
C
2
C_2
C
2
, respectively. If the smallest angle of the triangle
A
B
C
ABC
A
BC
is
4
0
∘
40^{\circ}
4
0
∘
, what is the magnitude of the smallest angle of the triangle
A
2
B
2
C
2
A_2 B_2 C_2
A
2
B
2
C
2
in degrees?
A weird walk timing
One day I went for a walk in the morning at
x
x
x
minutes past
5
′
O
5'O
5
′
O
clock, where
x
x
x
is a 2 digit number. When I returned, it was
y
y
y
minutes past
6
′
O
6'O
6
′
O
clock, and I noticed that (i) I walked for exactly
x
x
x
minutes and (ii)
y
y
y
was a 2 digit number obtained by reversing the digits of
x
x
x
. How many minutes did I walk?
9
2
Hide problems
Simple Fractions
Let the rational number
p
/
q
p/q
p
/
q
be closest to but not equal to
22
/
7
22/7
22/7
among all rational numbers with denominator
<
100
< 100
<
100
. What is the value of
p
−
3
q
p - 3q
p
−
3
q
?
Circumcentre on circumcircle
The centre of the circle passing through the midpoints of the sides of am isosceles triangle
A
B
C
ABC
A
BC
lies on the circumcircle of triangle
A
B
C
ABC
A
BC
. If the larger angle of triangle
A
B
C
ABC
A
BC
is
α
∘
\alpha^{\circ}
α
∘
and the smaller one
β
∘
\beta^{\circ}
β
∘
then what is the value of
α
−
β
\alpha-\beta
α
−
β
?
8
2
Hide problems
Algebra Problem
How many positive integers
n
n
n
are there such that
3
≤
n
≤
100
3 \leq n \leq 100
3
≤
n
≤
100
and
x
2
n
+
x
+
1
x^{2^{n}} + x + 1
x
2
n
+
x
+
1
is divisible by
x
2
+
x
+
1
x^2 + x + 1
x
2
+
x
+
1
?
(a+b)^k-a^k-b^k
Let
F
k
(
a
,
b
)
=
(
a
+
b
)
k
−
a
k
−
b
k
F_k(a,b)=(a+b)^k-a^k-b^k
F
k
(
a
,
b
)
=
(
a
+
b
)
k
−
a
k
−
b
k
and let
S
=
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
S={1,2,3,4,5,6,7,8,9,10}
S
=
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
. For how many ordered pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
with
a
,
b
∈
S
a,b\in S
a
,
b
∈
S
and
a
≤
b
a\leq b
a
≤
b
is
F
5
(
a
,
b
)
F
3
(
a
,
b
)
\frac{F_5(a,b)}{F_3(a,b)}
F
3
(
a
,
b
)
F
5
(
a
,
b
)
an integer?
7
2
Hide problems
Hands of clock at 90 degrees
On a clock, there are two instants between
12
12
12
noon and
1
P
M
1 \,\mathrm{PM}
1
PM
, when the hour hand and the minute hannd are at right angles. The difference in minutes between these two instants is written as
a
+
b
c
a + \dfrac{b}{c}
a
+
c
b
, where
a
,
b
,
c
a, b, c
a
,
b
,
c
are positive integers, with
b
<
c
b < c
b
<
c
and
b
/
c
b/c
b
/
c
in the reduced form. What is the value of
a
+
b
+
c
a+b+c
a
+
b
+
c
?
Sum of digits
Let
s
(
n
)
s(n)
s
(
n
)
denote the sum of digits of a positive integer
n
n
n
in base
10
10
10
. If
s
(
m
)
=
20
s(m)=20
s
(
m
)
=
20
and
s
(
33
m
)
=
120
s(33m)=120
s
(
33
m
)
=
120
, what is the value of
s
(
3
m
)
s(3m)
s
(
3
m
)
?
6
2
Hide problems
Simple NT Problem
Let
a
b
c
‾
\overline{abc}
ab
c
be a three digit number with nonzero digits such that
a
2
+
b
2
=
c
2
a^2 + b^2 = c^2
a
2
+
b
2
=
c
2
. What is the largest possible prime factor of
a
b
c
‾
\overline{abc}
ab
c
An easy Geo
Let
A
B
C
ABC
A
BC
be a triangle such that
A
B
=
A
C
AB=AC
A
B
=
A
C
. Suppose the tangent to the circumcircle of ABC at B is perpendicular to AC. Find angle ABC measured in degrees
4
2
Hide problems
Geometry / Computation
An ant leaves the anthill for its morning exercise. It walks
4
4
4
feet east and then makes a
16
0
∘
160^\circ
16
0
∘
turn to the right and walks
4
4
4
more feet. If the ant continues this patterns until it reaches the anthill again, what is the distance in feet it would have walked?
Recursive sequence and divisibility
Let
a
1
=
24
a_1=24
a
1
=
24
and form the sequence
a
n
a_n
a
n
,
n
≥
2
n\geq 2
n
≥
2
by
a
n
=
100
a
n
−
1
+
134
a_n=100a_{n-1}+134
a
n
=
100
a
n
−
1
+
134
. The first few terms are
24
,
2534
,
253534
,
25353534
,
…
24,2534,253534,25353534,\ldots
24
,
2534
,
253534
,
25353534
,
…
What is the least value of
n
n
n
for which
a
n
a_n
a
n
is divisible by
99
99
99
?
5
2
Hide problems
Simple Combinatorics on seating in a circle
Five persons wearing badges with numbers
1
,
2
,
3
,
4
,
5
1, 2, 3, 4, 5
1
,
2
,
3
,
4
,
5
are seated on
5
5
5
chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different)
Easy Division problem
Let
N
N
N
be the smallest positive integer such that
N
+
2
N
+
3
N
+
…
+
9
N
N+2N+3N+\ldots +9N
N
+
2
N
+
3
N
+
…
+
9
N
is a number all of whose digits are equal. What is the sum of digits of
N
N
N
?
3
2
Hide problems
Simple Recurrence solving
Let
x
1
x_{1}
x
1
be a positive real number and for every integer
n
≥
1
n \geq 1
n
≥
1
let
x
n
+
1
=
1
+
x
1
x
2
…
x
n
−
1
x
n
x_{n+1} = 1 + x_{1}x_{2}\ldots x_{n-1}x_{n}
x
n
+
1
=
1
+
x
1
x
2
…
x
n
−
1
x
n
. If
x
5
=
43
x_{5} = 43
x
5
=
43
, what is the sum of digits of the largest prime factors of
x
6
x_{6}
x
6
?
Difference of squares
Find the number of positive integers less than 101 that can not be written as the difference of two squares of integers.
2
2
Hide problems
Simple Function Solving
Ket
f
(
x
)
=
x
2
+
a
x
+
b
f(x) = x^{2} +ax + b
f
(
x
)
=
x
2
+
a
x
+
b
. If for all nonzero real
x
x
x
f
(
x
+
1
x
)
=
f
(
x
)
+
f
(
1
x
)
f\left(x + \dfrac{1}{x}\right) = f\left(x\right) + f\left(\dfrac{1}{x}\right)
f
(
x
+
x
1
)
=
f
(
x
)
+
f
(
x
1
)
and the roots of
f
(
x
)
=
0
f(x) = 0
f
(
x
)
=
0
are integers, what is the value of
a
2
+
b
2
a^{2}+b^{2}
a
2
+
b
2
?
Minimal Polynomial
If
x
=
2
+
3
+
6
x=\sqrt2+\sqrt3+\sqrt6
x
=
2
+
3
+
6
is a root of
x
4
+
a
x
3
+
b
x
2
+
c
x
+
d
=
0
x^4+ax^3+bx^2+cx+d=0
x
4
+
a
x
3
+
b
x
2
+
c
x
+
d
=
0
where
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are integers, what is the value of
∣
a
+
b
+
c
+
d
∣
|a+b+c+d|
∣
a
+
b
+
c
+
d
∣
?
1
2
Hide problems
Removed coreners from a square
Form a square with sides of length
5
5
5
, triangular pieces from the four coreners are removed to form a regular octagonn. Find the area removed to the nearest integer.
A tough P1
Consider the sequence of numbers
[
n
+
2
n
+
1
2
]
\left[n+\sqrt{2n}+\frac12\right]
[
n
+
2
n
+
2
1
]
, where
[
x
]
[x]
[
x
]
denotes the greatest integer not exceeding
x
x
x
. If the missing integers in the sequence are
n
1
<
n
2
<
n
3
<
…
n_1<n_2<n_3<\ldots
n
1
<
n
2
<
n
3
<
…
find
n
12
n_{12}
n
12