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National and Regional Contests
Malaysia Contests
Malaysia National Olympiad
2010 Malaysia National Olympiad
2010 Malaysia National Olympiad
Part of
Malaysia National Olympiad
Subcontests
(9)
4
3
Hide problems
angle wanted, equal ones, 68^o, 72^o- Malaysia 2010 OMK Bongsu / Juniors p4
In the diagram,
∠
A
O
B
=
∠
B
O
C
\angle AOB = \angle BOC
∠
A
OB
=
∠
BOC
and
∠
C
O
D
=
∠
D
O
E
=
∠
E
O
F
\angle COD = \angle DOE = \angle EOF
∠
CO
D
=
∠
D
OE
=
∠
EOF
. Given that
∠
A
O
D
=
8
2
o
\angle AOD = 82^o
∠
A
O
D
=
8
2
o
and
∠
B
O
E
=
6
8
o
\angle BOE = 68^o
∠
BOE
=
6
8
o
. Find
∠
A
O
F
\angle AOF
∠
A
OF
. https://cdn.artofproblemsolving.com/attachments/b/2/deba6cd740adbf033ad884fff8e13cd21d9c5a.png
2 squares and 1 circle - Malaysia 2010 OMK Muda / Intermediate p4
A square
A
B
C
D
ABCD
A
BC
D
has side length
1
1
1
. A circle passes through the vertices of the square. Let
P
,
Q
,
R
,
S
P, Q, R, S
P
,
Q
,
R
,
S
be the midpoints of the arcs which are symmetrical to the arcs
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
,
D
A
DA
D
A
when reflected on sides
A
B
AB
A
B
,
B
B
B
C,
C
D
CD
C
D
,
D
A
DA
D
A
, respectively. The area of square
P
Q
R
S
PQRS
PQRS
is
a
+
b
2
a+b\sqrt2
a
+
b
2
, where
a
a
a
and
b
b
b
are integers. Find the value of
a
+
b
a+b
a
+
b
. https://cdn.artofproblemsolving.com/attachments/4/3/fc9e1bd71b26cfd9ff076db7aa0a396ae64e72.png
OMK 2010
A semicircle has diameter
X
Y
XY
X
Y
. A square
P
Q
R
S
PQRS
PQRS
with side length 12 is inscribed in the semicircle with
P
P
P
and
S
S
S
on the diameter. Square
S
T
U
V
STUV
ST
U
V
has
T
T
T
on
R
S
RS
RS
,
U
U
U
on the semicircle, and
V
V
V
on
X
Y
XY
X
Y
. What is the area of
S
T
U
V
STUV
ST
U
V
?
1
3
Hide problems
areas related to 2 squares - Malaysia 2010 OMK Bongsu / Juniors p1
A square with side length
2
2
2
cm is placed next to a square with side length
6
6
6
cm, as shown in the diagram. Find the shaded area, in cm
2
^2
2
. https://cdn.artofproblemsolving.com/attachments/5/7/ceb4912a6e73ca751113b2b5c92cbfdbb6e0d1.png
OMK 2010
Triangles
O
A
B
,
O
B
C
,
O
C
D
OAB,OBC,OCD
O
A
B
,
OBC
,
OC
D
are isoceles triangles with
∠
O
A
B
=
∠
O
B
C
=
∠
O
C
D
=
∠
9
0
o
\angle OAB=\angle OBC=\angle OCD=\angle 90^o
∠
O
A
B
=
∠
OBC
=
∠
OC
D
=
∠9
0
o
. Find the area of the triangle
O
A
B
OAB
O
A
B
if the area of the triangle
O
C
D
OCD
OC
D
is 12.
area among rectangles - Malaysia 2010 OMK Sulung / Seniors p1
In the diagram, congruent rectangles
A
B
C
D
ABCD
A
BC
D
and
D
E
F
G
DEFG
D
EFG
have a common vertex
D
D
D
. Sides
B
C
BC
BC
and
E
F
EF
EF
meet at
H
H
H
. Given that
D
A
=
D
E
=
8
DA = DE = 8
D
A
=
D
E
=
8
,
A
B
=
E
F
=
12
AB = EF = 12
A
B
=
EF
=
12
, and
B
H
=
7
BH = 7
B
H
=
7
. Find the area of
A
B
H
E
D
ABHED
A
B
H
E
D
. https://cdn.artofproblemsolving.com/attachments/f/b/7225fa89097e7b20ea246b3aa920d2464080a5.png
9
3
Hide problems
OMK 2010
A number of runners competed in a race. When Ammar finished, there were half as many runners who had finished before him compared to the number who finished behind him. Julia was the 10th runner to finish behind Ammar. There were twice as many runners who had finished before Julia compared to the number who finished behind her. How many runners were there in the race?
OMK 2010
Let
m
m
m
and
n
n
n
be positive integers such that
2
n
+
3
m
2^n+3^m
2
n
+
3
m
is divisible by
5
5
5
. Prove that
2
m
+
3
n
2^m+3^n
2
m
+
3
n
is divisible by
5
5
5
.
OMK 2010
Show that there exist integers
m
m
m
and
n
n
n
such that
m
n
=
50
+
7
3
−
50
−
7
3
.
\dfrac{m}{n}=\sqrt[3]{\sqrt{50}+7}-\sqrt[3]{\sqrt{50}-7}.
n
m
=
3
50
+
7
−
3
50
−
7
.
8
3
Hide problems
OMK 2010
Find the last digit of
7
1
×
7
2
×
7
3
×
⋯
×
7
2009
×
7
2010
.
7^1\times 7^2\times 7^3\times \cdots \times 7^{2009}\times 7^{2010}.
7
1
×
7
2
×
7
3
×
⋯
×
7
2009
×
7
2010
.
OMK 2010
For any number
x
x
x
, let
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
denotes the greatest integer less than or equal to
x
x
x
. A sequence
a
1
,
a
2
,
⋯
a_1,a_2,\cdots
a
1
,
a
2
,
⋯
is given, where
a
n
=
⌊
2
n
+
1
2
⌋
.
a_n=\left\lfloor{\sqrt{2n}+\dfrac{1}{2}}\right\rfloor.
a
n
=
⌊
2
n
+
2
1
⌋
.
How many values of
k
k
k
are there such that
a
k
=
2010
a_k=2010
a
k
=
2010
?
OMK 2010
Show that
log
a
b
c
+
log
b
c
a
+
log
c
a
b
≥
4
(
log
a
b
c
+
log
b
c
a
+
log
c
a
b
)
\log_{a}bc+\log_bca+\log_cab \ge 4(\log_{ab}c+\log_{bc}a+\log_{ca}b)
lo
g
a
b
c
+
lo
g
b
c
a
+
lo
g
c
ab
≥
4
(
lo
g
ab
c
+
lo
g
b
c
a
+
lo
g
c
a
b
)
for all
a
,
b
,
c
a,b,c
a
,
b
,
c
greater than 1.
7
3
Hide problems
OMK 2010
Let
A
B
C
ABC
A
BC
be a triangle in which
A
B
=
A
C
AB=AC
A
B
=
A
C
. A point
I
I
I
lies inside the triangle such that
∠
A
B
I
=
∠
C
B
I
\angle ABI=\angle CBI
∠
A
B
I
=
∠
CB
I
and
∠
B
A
I
=
∠
C
A
I
\angle BAI=\angle CAI
∠
B
A
I
=
∠
C
A
I
. Prove that
∠
B
I
A
=
9
0
o
+
∠
C
2
\angle BIA=90^o+\dfrac{\angle C}{2}
∠
B
I
A
=
9
0
o
+
2
∠
C
OMK 2010
Let
A
B
C
ABC
A
BC
be a triangle in which
A
B
=
A
C
AB=AC
A
B
=
A
C
and let
I
I
I
be its incenter. It is known that
B
C
=
A
B
+
A
I
BC=AB+AI
BC
=
A
B
+
A
I
. Let
D
D
D
be a point on line
B
A
BA
B
A
extended beyond
A
A
A
such that
A
D
=
A
I
AD=AI
A
D
=
A
I
. Prove that
D
A
I
C
DAIC
D
A
I
C
is a cyclic quadrilateral.
OMK 2010
A line segment of length 1 is given on the plane. Show that a line segment of length
2010
\sqrt{2010}
2010
can be constructed using only a straightedge and a compass.
3
3
Hide problems
OMK 2010
Adam has RM2010 in his bank account. He donates RM10 to charity every day. His first donation is on Monday. On what day will he donate his last RM10?
OMK 2010
Let
γ
=
α
×
β
\gamma=\alpha \times \beta
γ
=
α
×
β
where
α
=
999
⋯
9
\alpha=999 \cdots 9
α
=
999
⋯
9
(2010 '9') and
β
=
444
⋯
4
\beta=444 \cdots 4
β
=
444
⋯
4
(2010 '4') Find the sum of digits of
γ
\gamma
γ
.
OMK 2010
Let
N
=
a
b
c
‾
N=\overline{abc}
N
=
ab
c
be a three-digit number. It is known that we can construct an isoceles triangle with
a
,
b
a,b
a
,
b
and
c
c
c
as the length of sides. Determine how many possible three-digit number
N
N
N
there are.(
N
=
a
b
c
‾
N=\overline{abc}
N
=
ab
c
means that
a
,
b
a,b
a
,
b
and
c
c
c
are digits of
N
N
N
, and not
N
=
a
×
b
×
c
N=a\times b\times c
N
=
a
×
b
×
c
.)
2
3
Hide problems
OMK 2010
A student wrote down the following sequence of numbers : the first number is 1, the second number is 2, and after that, each number is obtained by adding together all the previous numbers. Determine the 12th number in the sequence.
OMK 2010
A meeting is held at a round table. It is known that 7 women have a woman on their right side, and 12 women have a man on their right side. It is also known that 75% of the men have a woman on their right side. How many people are sitting at the round table?
OMK 2010
Find
x
x
x
such that
201
0
log
10
x
=
1
1
log
10
(
1
+
3
+
5
+
⋯
+
4019
)
.
2010^{\log_{10}x}=11^{\log_{10}(1+3+5+\cdots +4019).}
201
0
l
o
g
10
x
=
1
1
l
o
g
10
(
1
+
3
+
5
+
⋯
+
4019
)
.
6
3
Hide problems
OMK 2010
Find the number of different pairs of positive integers
(
a
,
b
)
(a,b)
(
a
,
b
)
for which
a
+
b
≤
100
a+b\le100
a
+
b
≤
100
and
a
+
1
b
1
a
+
b
=
10
\dfrac{a+\frac{1}{b}}{\frac{1}{a}+b}=10
a
1
+
b
a
+
b
1
=
10
OMK 2010
A two-digit integer is divided by the sum of its digits. Find the largest remainder that can occur.
Smallest integer
Find the smallest integer
k
≥
3
k\ge3
k
≥
3
with the property that it is possible to choose two of the number
1
,
2
,
.
.
.
,
k
1,2,...,k
1
,
2
,
...
,
k
in such a way that their product is equal to the sum of the remaining
k
−
2
k-2
k
−
2
numbers.
5
3
Hide problems
ratio of areas, square, circle - Malaysia 2010 OMK Bongsu / Juniors p5
A circle and a square overlap such that the overlapping area is
50
%
50\%
50%
of the area of the circle, and is
25
%
25\%
25%
of the area of the square, as shown in the figure. Find the ratio of the area of the square outside the circle to the area of the whole figure. https://cdn.artofproblemsolving.com/attachments/e/2/c209a95f457dbf3c46f66f82c0a45cc4b5c1c8.png
OMK 2010
Find the number of triples of nonnegative integers
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
such that
x
2
+
2
x
y
+
y
2
−
z
2
=
9.
x^2+2xy+y^2-z^2=9.
x
2
+
2
x
y
+
y
2
−
z
2
=
9.
All Odd Numbers
Let
n
n
n
be an integer greater than 1. If all digits of
97
n
97n
97
n
are odd, find the smallest possible value of
n
n
n
.