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Problems
Contests
National and Regional Contests
India Contests
India IOQM
2022 IOQM India
2022 IOQM India
Part of
India IOQM
Subcontests
(12)
11
1
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IOQM 2022 P11
In how many ways can four married couples sit in a merry-go-round with identical seats such that men and women occupy alternate seats and no husband seats next to his wife?
10
1
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IOQM 2022 P10
Suppose that
P
P
P
is the polynomial of least degree with integer coefficients such that
P
(
7
+
5
)
=
2
(
7
−
5
)
P(\sqrt{7} + \sqrt{5}) = 2(\sqrt{7} - \sqrt{5})
P
(
7
+
5
)
=
2
(
7
−
5
)
Find
P
(
2
)
P(2)
P
(
2
)
.
9
1
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IOQM 2022 P9
Let
P
0
=
(
3
,
1
)
P_0 = (3,1)
P
0
=
(
3
,
1
)
and define
P
n
+
1
=
(
x
n
,
y
n
)
P_{n+1} = (x_n, y_n)
P
n
+
1
=
(
x
n
,
y
n
)
for
n
≥
0
n \ge 0
n
≥
0
by
x
n
+
1
=
−
3
x
n
−
y
n
2
,
y
n
+
1
=
−
x
n
+
y
n
2
x_{n+1} = - \frac{3x_n - y_n}{2}, y_{n+1} = - \frac{x_n + y_n}{2}
x
n
+
1
=
−
2
3
x
n
−
y
n
,
y
n
+
1
=
−
2
x
n
+
y
n
Find the area of the quadrilateral formed by the points
P
96
,
P
97
,
P
98
,
P
99
P_{96}, P_{97}, P_{98}, P_{99}
P
96
,
P
97
,
P
98
,
P
99
.
12
1
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IOQM 2022 P12
A
12
×
12
12 \times 12
12
×
12
board is divided into
144
144
144
unit squares by drawing lines parallel to the sides. Two rooks placed on two unit squares are said to be non-attacking if they are not in the same column or same row. Find the least number
N
N
N
such that if
N
N
N
rooks are placed on the unit squares, one rook per square, we can always find
7
7
7
rooks such that no two are attacking each other.
8
1
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IOQM 2022 P8
For any real number
t
t
t
, let
⌊
t
⌋
\lfloor t \rfloor
⌊
t
⌋
denote the largest integer
≤
t
\le t
≤
t
. Suppose that
N
N
N
is the greatest integer such that
⌊
⌊
⌊
N
⌋
⌋
⌋
=
4
\left \lfloor \sqrt{\left \lfloor \sqrt{\left \lfloor \sqrt{N} \right \rfloor}\right \rfloor}\right \rfloor = 4
⌊
⌊
N
⌋
⌋
=
4
Find the sum of digits of
N
N
N
.
7
1
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IOQM 2022 P7
Find the number of maps
f
:
{
1
,
2
,
3
}
→
{
1
,
2
,
3
,
4
,
5
}
f: \{1,2,3\} \rightarrow \{1,2,3,4,5\}
f
:
{
1
,
2
,
3
}
→
{
1
,
2
,
3
,
4
,
5
}
such that
f
(
i
)
≤
f
(
j
)
f(i) \le f(j)
f
(
i
)
≤
f
(
j
)
whenever
i
<
j
i < j
i
<
j
.
6
1
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IOQM 2022 P6
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive real numbers such that
x
2
+
y
2
=
49
,
y
2
+
y
z
+
z
2
=
36
x^2 + y^2 = 49, y^2 + yz + z^2 = 36
x
2
+
y
2
=
49
,
y
2
+
yz
+
z
2
=
36
and
x
2
+
3
x
z
+
z
2
=
25
x^2 + \sqrt{3}xz + z^2 = 25
x
2
+
3
x
z
+
z
2
=
25
. If the value of
2
x
y
+
3
y
z
+
z
x
2xy + \sqrt{3}yz + zx
2
x
y
+
3
yz
+
z
x
can be written as
p
q
p \sqrt{q}
p
q
where
p
,
q
∈
Z
p,q \in \mathbb{Z}
p
,
q
∈
Z
and
q
q
q
is squarefree, find
p
+
q
p+q
p
+
q
.
5
1
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IOQM 2022 P5
In parallelogram
A
B
C
D
ABCD
A
BC
D
, the longer side is twice the shorter side. Let
X
Y
Z
W
XYZW
X
Y
Z
W
be the quadrilateral formed by the internal bisectors of the angles of
A
B
C
D
ABCD
A
BC
D
. If the area of
X
Y
Z
W
XYZW
X
Y
Z
W
is
10
10
10
, find the area of
A
B
C
D
ABCD
A
BC
D
4
1
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IOQM 2022 P4
Consider the set of all 6-digit numbers consisting of only three digits,
a
,
b
,
c
a,b,c
a
,
b
,
c
where
a
,
b
,
c
a,b,c
a
,
b
,
c
are distinct. Suppose the sum of all these numbers is
593999406
593999406
593999406
. What is the largest remainder when the three digit number
a
b
c
abc
ab
c
is divided by
100
100
100
?
3
1
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IOQM 2022 P3
Consider the set
T
\mathcal{T}
T
of all triangles whose sides are distinct prime numbers which are also in arithmetic progression. Let
△
∈
T
\triangle \in \mathcal{T}
△
∈
T
be the triangle with least perimeter. If
a
∘
a^{\circ}
a
∘
is the largest angle of
△
\triangle
△
and
L
L
L
is its perimeter, determine the value of
a
L
\frac{a}{L}
L
a
.
2
1
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IOQM 2022 P2
Ria writes down the numbers
1
,
2
,
⋯
,
101
1,2,\cdots, 101
1
,
2
,
⋯
,
101
in red and blue pens. The largest blue number is equal to the number of numbers written in blue and the smallest red number is equal to half the number of numbers in red. How many numbers did Ria write with red pen?
1
1
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IOQM 2022 P1
Three parallel lines
L
1
,
L
2
,
L
2
L_1, L_2, L_2
L
1
,
L
2
,
L
2
are drawn in the plane such that the perpendicular distance between
L
1
L_1
L
1
and
L
2
L_2
L
2
is
3
3
3
and the perpendicular distance between lines
L
2
L_2
L
2
and
L
3
L_3
L
3
is also
3
3
3
. A square
A
B
C
D
ABCD
A
BC
D
is constructed such that
A
A
A
lies on
L
1
L_1
L
1
,
B
B
B
lies on
L
3
L_3
L
3
and
C
C
C
lies on
L
2
L_2
L
2
. Find the area of the square.