MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
Taiwan APMO Prelininary
2018 Taiwan APMO Preliminary
2018 Taiwan APMO Preliminary
Part of
Taiwan APMO Prelininary
Subcontests
(7)
7
1
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Turning around
240
240
240
students are participating a big performance show. They stand in a row and face to their coach. The coach askes them to count numbers from left to right, starting from
1
1
1
. (Of course their counts be like
1
,
2
,
3
,
.
.
.
1,2,3,...
1
,
2
,
3
,
...
)The coach askes them to remember their number and do the following action: First, if your number is divisible by
3
3
3
then turn around. Then, if your number is divisible by
5
5
5
then turn around. Finally, if your number is divisible by
7
7
7
then turn around. (a) How many students are face to coach now? (b) What is the number of the
6
6
th
66^{\text{th}}
6
6
th
student counting from left who is face to coach?
6
1
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PQRE area wanted
Let
A
B
C
D
ABCD
A
BC
D
be an unit aquare.
E
,
F
E,F
E
,
F
be the midpoints of
C
D
,
B
C
CD,BC
C
D
,
BC
respectively.
A
E
AE
A
E
intersects the diagonal
B
D
BD
B
D
at
P
P
P
.
A
F
AF
A
F
intersects
B
D
,
B
E
BD,BE
B
D
,
BE
at
Q
,
R
Q,R
Q
,
R
respectively. Find the area of quadrilateral
P
Q
R
E
PQRE
PQRE
.
5
1
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A.P and G.P
Let (i)
a
1
,
a
2
,
a
3
a_1,a_2,a_3
a
1
,
a
2
,
a
3
is an arithmetic progression and
a
1
+
a
2
+
a
3
=
18
a_1+a_2+a_3=18
a
1
+
a
2
+
a
3
=
18
(ii)
b
1
,
b
2
,
b
3
b_1,b_2,b_3
b
1
,
b
2
,
b
3
is a geometric progression and
b
1
b
2
b
3
=
64
b_1b_2b_3=64
b
1
b
2
b
3
=
64
If
a
1
+
b
1
,
a
2
+
b
2
,
a
3
+
b
3
a_1+b_1,a_2+b_2,a_3+b_3
a
1
+
b
1
,
a
2
+
b
2
,
a
3
+
b
3
are all positive integers and it is a ageometric progression, then find the maximum value of
a
3
a_3
a
3
.
4
1
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Possibility of all rows and colums sum are even
If we fill
1
∼
16
1\sim 16
1
∼
16
into
4
×
4
4\times4
4
×
4
chessboard randomly. What is the possibility of the sum of each rows and columns are all even?
3
1
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Positive integer solution
Let
a
,
b
a,b
a
,
b
be positive integers satisfying
a
b
2
b
2
−
a
=
a
+
2
b
4
b
\sqrt{\dfrac{ab}{2b^2-a}}=\dfrac{a+2b}{4b}
2
b
2
−
a
ab
=
4
b
a
+
2
b
. Find
∣
10
(
a
−
5
)
(
b
−
15
)
∣
+
8
|10(a-5)(b-15)|+8
∣10
(
a
−
5
)
(
b
−
15
)
∣
+
8
.
2
1
Hide problems
Gcd, quotient
Let
k
,
x
,
y
k,x,y
k
,
x
,
y
be postive integers. The quotients of
k
k
k
divided by
x
2
,
y
2
x^2, y^2
x
2
,
y
2
are
n
,
n
+
148
n,n+148
n
,
n
+
148
respectively.(
k
k
k
is divisible by
x
2
x^2
x
2
and
y
2
y^2
y
2
) (a) If
gcd
(
x
,
y
)
=
1
\gcd(x,y)=1
g
cd
(
x
,
y
)
=
1
, then find
k
k
k
. (b) If
gcd
(
x
,
y
)
=
4
\gcd(x,y)=4
g
cd
(
x
,
y
)
=
4
, then find
k
k
k
.
1
1
Hide problems
Trapezoid, DF//BC
Let trapezoid
A
B
C
D
ABCD
A
BC
D
inscribed in a circle
O
O
O
,
A
B
∣
∣
C
D
AB||CD
A
B
∣∣
C
D
. Tangent at
D
D
D
wrt
O
O
O
intersects line
A
C
AC
A
C
at
F
F
F
,
D
F
∣
∣
B
C
DF||BC
D
F
∣∣
BC
. If
C
A
=
5
,
B
C
=
4
CA=5, BC=4
C
A
=
5
,
BC
=
4
, then find
A
F
AF
A
F
.