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Problems
Contests
National and Regional Contests
Malaysia Contests
Malaysia National Olympiad
2019 Malaysia National Olympiad
2019 Malaysia National Olympiad
Part of
Malaysia National Olympiad
Subcontests
(8)
6
1
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Prime factors
It is known that
2018
(
201
9
39
+
201
9
37
+
.
.
.
+
2019
)
+
1
2018(2019^{39}+2019^{37}+...+2019)+1
2018
(
201
9
39
+
201
9
37
+
...
+
2019
)
+
1
is prime. How many positive factors does
201
9
41
+
1
2019^{41}+1
201
9
41
+
1
have?
5
1
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Triangle geometry
In a triangle
A
B
C
,
ABC,
A
BC
,
point
D
D
D
lies on
A
B
AB
A
B
. It is given that
A
D
=
25
,
B
D
=
24
,
B
C
=
28
,
C
D
=
20.
A
C
=
?
AD=25, BD=24, BC=28, CD=20. AC=?
A
D
=
25
,
B
D
=
24
,
BC
=
28
,
C
D
=
20.
A
C
=
?
4
1
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Counting subset sizes
Let
A
=
{
1
,
2
,
.
.
.
,
100
}
A=\{1,2,...,100\}
A
=
{
1
,
2
,
...
,
100
}
and
f
(
k
)
,
k
∈
N
f(k), k\in N
f
(
k
)
,
k
∈
N
be the size of the largest subset of
A
A
A
such that no two elements differ by
k
k
k
. How many solutions are there to
f
(
k
)
=
50
f(k)=50
f
(
k
)
=
50
?
3
1
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Number equal to digits' factorials
A factorian is defined to be a number such that it is equal to the sum of it's digits' factorials. What is the smallest three digit factorian?
1
1
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Sum of logarithms
Evaluate the following sum
1
log
2
1
7
+
1
log
3
1
7
+
1
log
4
1
7
+
1
log
5
1
7
+
1
log
6
1
7
−
1
log
7
1
7
−
1
log
8
1
7
−
1
log
9
1
7
−
1
log
10
1
7
\frac{1}{\log_2{\frac{1}{7}}}+\frac{1}{\log_3{\frac{1}{7}}}+\frac{1}{\log_4{\frac{1}{7}}}+\frac{1}{\log_5{\frac{1}{7}}}+\frac{1}{\log_6{\frac{1}{7}}}-\frac{1}{\log_7{\frac{1}{7}}}-\frac{1}{\log_8{\frac{1}{7}}}-\frac{1}{\log_9{\frac{1}{7}}}-\frac{1}{\log_{10}{\frac{1}{7}}}
lo
g
2
7
1
1
+
lo
g
3
7
1
1
+
lo
g
4
7
1
1
+
lo
g
5
7
1
1
+
lo
g
6
7
1
1
−
lo
g
7
7
1
1
−
lo
g
8
7
1
1
−
lo
g
9
7
1
1
−
lo
g
10
7
1
1
B3
1
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Problem B3
An arithmetic sequence of five terms is considered
g
o
o
d
good
g
oo
d
if it contains 19 and 20. For example,
18.5
,
19.0
,
19.5
,
20.0
,
20.5
18.5,19.0,19.5,20.0,20.5
18.5
,
19.0
,
19.5
,
20.0
,
20.5
is a
g
o
o
d
good
g
oo
d
sequence. For every
g
o
o
d
good
g
oo
d
sequence, the sum of its terms is totalled. What is the total sum of all
g
o
o
d
good
g
oo
d
sequences?
B2
1
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Problem B2
Given a parallelogram
A
B
C
D
ABCD
A
BC
D
, a point M is chosen such that
∠
D
A
C
=
∠
M
A
C
\angle DAC=\angle MAC
∠
D
A
C
=
∠
M
A
C
and
∠
C
A
B
=
∠
M
A
B
.
\angle CAB=\angle MAB.
∠
C
A
B
=
∠
M
A
B
.
Prove
A
M
B
M
=
(
A
C
B
D
)
2
\frac{AM}{BM}=\left(\frac{AC}{BD}\right)^2
BM
A
M
=
(
B
D
A
C
)
2
B1
1
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Problem B1
Given three nonzero real numbers
a
,
b
,
c
,
a,b,c,
a
,
b
,
c
,
such that
a
>
b
>
c
a>b>c
a
>
b
>
c
, prove the equation has at least one real root.
1
x
+
a
+
1
x
+
b
+
1
x
+
c
−
3
x
=
0
\frac{1}{x+a}+\frac{1}{x+b}+\frac{1}{x+c}-\frac{3}{x}=0
x
+
a
1
+
x
+
b
1
+
x
+
c
1
−
x
3
=
0
@below sorry, I believe I fixed it with the added constraint.