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Assam Mathematics Olympiad
2023 Assam Mathematics Olympiad
2023 Assam Mathematics Olympiad
Part of
Assam Mathematics Olympiad
Subcontests
(18)
18
1
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Assam Mathematics Olympiad 2023 Category III Q18
A circle of radius
2
2
2
is inscribed in an isosceles trapezoid with the area of
28
28
28
. Find the length of the side of the trapezoid.
17
1
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Assam Mathematics Olympiad 2023 Category III Q17
If in
△
A
B
C
\bigtriangleup ABC
△
A
BC
,
A
D
AD
A
D
is the altitude and
A
E
AE
A
E
is the diameter of the circumcircle through
A
A
A
, then prove that
A
B
⋅
A
C
=
A
D
⋅
A
E
AB\cdot AC = AD \cdot AE
A
B
⋅
A
C
=
A
D
⋅
A
E
. Use this result to show that if
A
B
C
D
ABCD
A
BC
D
is a cyclic quadrilateral then show that
A
C
⋅
(
A
B
⋅
B
C
+
C
D
⋅
D
A
)
=
B
D
⋅
(
D
A
⋅
A
B
+
B
C
⋅
C
D
)
AC \cdot (AB \cdot BC + CD \cdot DA) = BD\cdot (DA\cdot AB + BC \cdot CD)
A
C
⋅
(
A
B
⋅
BC
+
C
D
⋅
D
A
)
=
B
D
⋅
(
D
A
⋅
A
B
+
BC
⋅
C
D
)
.
16
1
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Assam Mathematics Olympiad 2023 Category III Q16
n
n
n
is a positive integer such that the product of all its positive divisors is
n
3
n^3
n
3
. Find all such
n
n
n
less than
100
100
100
.
15
1
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Assam Mathematics Olympiad 2023 Category III Q15
Let
f
(
x
)
f(x)
f
(
x
)
be a polynomial of degree
3
3
3
with real coefficients satisfying
∣
f
(
x
)
∣
=
12
|f(x)| = 12
∣
f
(
x
)
∣
=
12
for
x
=
1
,
2
,
3
,
5
,
6
,
7
x = 1, 2, 3, 5, 6, 7
x
=
1
,
2
,
3
,
5
,
6
,
7
. Find
∣
f
(
0
)
∣
|f(0)|
∣
f
(
0
)
∣
.
14
1
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Assam Mathematics Olympiad 2023 Category III Q14
Find all possible triples of integers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfying
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
.
13
1
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Assam Mathematics Olympiad 2023 Category III Q13
Let
S
(
r
)
S(r)
S
(
r
)
denote the sum of the infinite geometric series
17
+
17
r
+
17
r
2
+
17
r
3
+
.
.
.
17 + 17r + 17r^2 +17r^3 + . . .
17
+
17
r
+
17
r
2
+
17
r
3
+
...
for
−
1
<
r
<
1
-1 < r < 1
−
1
<
r
<
1
. If
S
(
a
)
×
S
(
−
a
)
=
2023
S(a) \times S(-a) = 2023
S
(
a
)
×
S
(
−
a
)
=
2023
, find
S
(
a
)
+
S
(
−
a
)
S(a) + S(-a)
S
(
a
)
+
S
(
−
a
)
.
12
1
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Assam Mathematics Olympiad 2023 Category III Q12
In quadrilateral
A
B
C
D
ABCD
A
BC
D
,
A
D
∣
∣
B
C
AD || BC
A
D
∣∣
BC
, diagonals
A
C
AC
A
C
and
B
D
BD
B
D
are perpendicular to each other,
X
X
X
and
Y
Y
Y
are mid-points of
A
B
AB
A
B
and
C
D
CD
C
D
respectively. Prove that
A
B
+
C
D
≥
A
D
+
B
C
AB + CD \geq AD + BC
A
B
+
C
D
≥
A
D
+
BC
.
11
1
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Assam Mathematics Olympiad 2023 Category III Q11
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial of degree
10
10
10
with non-negative integer coefficients. The remainder when
P
(
x
)
P(x)
P
(
x
)
is divided by
(
x
−
1
)
(x - 1)
(
x
−
1
)
is
3
3
3
. How many such polynomials are there ?
10
1
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Assam Mathematics Olympiad 2023 Category III Q10
If
a
,
b
,
c
≠
0
a,b,c \neq 0
a
,
b
,
c
=
0
, prove that
a
2
+
b
2
c
2
+
b
2
+
c
2
a
2
+
c
2
+
a
2
b
2
≥
6
\frac{a^2+b^2}{c^2} +\frac{b^2+c^2}{a^2} +\frac{c^2+a^2}{b^2} \geq 6
c
2
a
2
+
b
2
+
a
2
b
2
+
c
2
+
b
2
c
2
+
a
2
≥
6
.
9
1
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Assam Mathematics Olympiad 2023 Category III Q9
What is the smallest positive integer having
24
24
24
positive divisors?
8
1
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Assam Mathematics Olympiad 2023 Category III Q8
If
n
n
n
is a positive even number, find the last two digits of
(
2
6
n
+
26
)
−
(
6
2
n
−
62
)
(2^{6n}+26)-(6^{2n}-62)
(
2
6
n
+
26
)
−
(
6
2
n
−
62
)
.
7
1
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Assam Mathematics Olympiad 2023 Category III Q7
If
x
y
z
=
1
xyz=1
x
yz
=
1
find the value of
(
1
1
+
x
+
1
y
+
1
1
+
y
+
1
z
+
1
1
+
z
+
1
x
)
2
\left(\frac{1}{1+x+\frac{1}{y}}+\frac{1}{1+y+\frac{1}{z}}+\frac{1}{1+z+\frac{1}{x}}\right)^2
(
1
+
x
+
y
1
1
+
1
+
y
+
z
1
1
+
1
+
z
+
x
1
1
)
2
.
6
1
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Assam Mathematics Olympiad 2023 Category III Q6
What is the remainder when
12
8
2023
128^{2023}
12
8
2023
is divided by
126
126
126
?
5
1
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Assam Mathematics Olympiad 2023 Category III Q5
What is the least possible value of
x
2
+
y
2
−
x
−
y
−
x
y
x^2 + y^2 - x - y - xy
x
2
+
y
2
−
x
−
y
−
x
y
where
x
,
y
x, y
x
,
y
are real numbers ?
4
1
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Assam Mathematics Olympiad 2023 Category III Q4
Real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfy
(
2
b
−
a
)
2
+
(
2
b
−
c
)
2
=
2
(
2
b
2
−
a
c
)
(2b - a)^2 + (2b - c)^2 = 2(2b^2 - ac)
(
2
b
−
a
)
2
+
(
2
b
−
c
)
2
=
2
(
2
b
2
−
a
c
)
. Prove that
a
+
c
=
2
b
a + c = 2b
a
+
c
=
2
b
.
3
1
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Assam Mathematics Olympiad 2023 Category III Q3
Find the number of integer solutions of
∣
∣
x
∣
−
2023
∣
<
2020
||x| - 2023| < 2020
∣∣
x
∣
−
2023∣
<
2020
.
2
1
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Assam Mathematics Olympiad 2023 Category III Q2
An umbrella seller has umbrellas of
7
7
7
different colours. He has a total of
2023
2023
2023
umbrellas in stock but because of the plastic packaging, the colours are not visible. What is the minimum number of umbrellas that one must buy in order to ensure that at least
23
23
23
umbrellas are of the same colour ?
1
1
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Assam Mathematics Olympiad 2023 Category III Q1
What is the
288
288
288
th term of the sequence
a
,
b
,
b
,
c
,
c
,
c
,
d
,
d
,
d
,
d
,
e
,
e
,
e
,
e
,
e
,
f
,
f
,
f
,
f
,
f
,
f
,
.
.
.
?
a,b,b,c,c,c,d,d,d,d,e,e,e,e,e,f,f,f,f,f,f,...?
a
,
b
,
b
,
c
,
c
,
c
,
d
,
d
,
d
,
d
,
e
,
e
,
e
,
e
,
e
,
f
,
f
,
f
,
f
,
f
,
f
,
...
?