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National and Regional Contests
Malaysia Contests
Malaysia IMONST
2020 Malaysia IMONST 2
2020 Malaysia IMONST 2
Part of
Malaysia IMONST
Subcontests
(6)
6
1
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Malaysia IMONST 2 Senior Problem 6
Consider the following one-person game: A player starts with score
0
0
0
and writes the number
20
20
20
on an empty whiteboard. At each step, she may erase any one integer (call it a) and writes two positive integers (call them
b
b
b
and
c
c
c
) such that
b
+
c
=
a
b + c = a
b
+
c
=
a
. The player then adds
b
×
c
b\times c
b
×
c
to her score. She repeats the step several times until she ends up with all
1
1
1
's on the whiteboard. Then the game is over, and the final score is calculated. Let
M
,
m
M, m
M
,
m
be the maximum and minimum final score that can be possibly obtained respectively. Find
M
−
m
M-m
M
−
m
.
5
1
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Malaysia IMONST 2 Senior Problem 5
Let
p
p
p
and
q
q
q
be real numbers such that the quadratic equation
x
2
+
p
x
+
q
=
0
x^2 + px + q = 0
x
2
+
p
x
+
q
=
0
has two distinct real solutions
x
1
x_1
x
1
and
x
2
x_2
x
2
. Suppose
∣
x
1
−
x
2
∣
=
1
|x_1-x_2|=1
∣
x
1
−
x
2
∣
=
1
,
∣
p
−
q
∣
=
1
|p-q|=1
∣
p
−
q
∣
=
1
. Prove that
p
,
q
,
x
1
,
x
2
p, q, x_1, x_2
p
,
q
,
x
1
,
x
2
are all integers.
4
1
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Malaysia IMONST 2 Senior Problem 4
Given are four circles
Γ
1
,
Γ
2
,
Γ
3
,
Γ
4
\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4
Γ
1
,
Γ
2
,
Γ
3
,
Γ
4
. Circles
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
are externally tangent at point
A
A
A
. Circles
Γ
2
\Gamma_2
Γ
2
and
Γ
3
\Gamma_3
Γ
3
are externally tangent at point
B
B
B
. Circles
Γ
3
\Gamma_3
Γ
3
and
Γ
4
\Gamma_4
Γ
4
are externally tangent at point
C
C
C
. Circles
Γ
4
\Gamma_4
Γ
4
and
Γ
1
\Gamma_1
Γ
1
are externally tangent at point
D
D
D
. Prove that
A
B
C
D
ABCD
A
BC
D
is cyclic.
3
2
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Malaysia IMONST 2 Senior Problem 3
Find all possible integer values of
n
n
n
such that
12
n
2
+
12
n
+
11
12n^2 + 12n + 11
12
n
2
+
12
n
+
11
is a
4
4
4
-digit number with equal digits.
a and b are both divisible by 11 when a^2+b^2 is divisible by 11
Given integers
a
a
a
and
b
b
b
such that
a
2
+
b
2
a^2+b^2
a
2
+
b
2
is divisible by
11
11
11
. Prove that
a
a
a
and
b
b
b
are both divisible by
11
11
11
.
2
2
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Malaysia IMONST 2 Senior Problem 2
Prove that
1
−
1
2
+
1
3
−
1
4
+
⋯
+
1
2019
−
1
2020
=
1
1011
+
1
1012
+
⋯
+
1
2020
1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots +\frac{1}{2019}-\frac{1}{2020}=\frac{1}{1011}+\frac{1}{1012}+\cdots +\frac{1}{2020}
1
−
2
1
+
3
1
−
4
1
+
⋯
+
2019
1
−
2020
1
=
1011
1
+
1012
1
+
⋯
+
2020
1
every equilateral triangle can be divided into n>=6 equilateral triangles.
Prove that for any integer
n
≥
6
n\ge 6
n
≥
6
we can divide an equilateral triangle completely into
n
n
n
smaller equilateral triangles.
1
2
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Malaysia IMONST 2 Senior Problem 1
Given a trapezium with two parallel sides of lengths
m
m
m
and
n
n
n
, where
m
m
m
,
n
n
n
are integers, prove that it is possible to divide the trapezium into several congruent triangles.
inradius r=1/2 (a + b - c) in right triangle (2016 Izumrud Olympiad 8-9 p5)
Prove that if
a
a
a
and
b
b
b
are legs,
c
c
c
is the hypotenuse of a right triangle, then the radius of a circle inscribed in this triangle can be found by the formula
r
=
1
2
(
a
+
b
−
c
)
r = \frac12 (a + b - c)
r
=
2
1
(
a
+
b
−
c
)
.