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Problems
Contests
National and Regional Contests
Vietnam Contests
Hanoi Open Mathematics Competition
2009 Hanoi Open Mathematics Competitions
2009 Hanoi Open Mathematics Competitions
Part of
Hanoi Open Mathematics Competition
Subcontests
(11)
11
1
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subset of {1,2,..,100} with 48 elements has 11/(a+b) (HOMC 2009 J Q11)
Let
A
=
{
1
,
2
,
.
.
.
,
100
}
A = \{1,2,..., 100\}
A
=
{
1
,
2
,
...
,
100
}
and
B
B
B
is a subset of
A
A
A
having
48
48
48
elements. Show that
B
B
B
has two distint elements
x
x
x
and
y
y
y
whose sum is divisible by
11
11
11
.
8
1
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diophantine ab+1/2a+1/3b <1004/3 (HOMC 2009 J Q8)
Find all the pairs of the positive integers such that the product of the numbers of any pair plus the half of one of the numbers plus one third of the other number is three times less than
1004
1004
1004
.
6
1
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M = ab + cd when a^2+b^2=4, c^2+d^2=4, ac+bd = 2 (HOMC 2009 J Q6)
Suppose that
4
4
4
real numbers
a
,
b
,
c
,
d
a, b,c,d
a
,
b
,
c
,
d
satisfy the conditions
{
a
2
+
b
2
=
4
c
2
+
d
2
=
4
a
c
+
b
d
=
2
\begin{cases} a^2 + b^2 = 4\\ c^2 + d^2 = 4 \\ ac + bd = 2 \end{cases}
⎩
⎨
⎧
a
2
+
b
2
=
4
c
2
+
d
2
=
4
a
c
+
b
d
=
2
Find the set of all possible values the number
M
=
a
b
+
c
d
M = ab + cd
M
=
ab
+
c
d
can take.
5
1
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m^7- m is divisible by 42 (HOMC 2009 J Q5)
Prove that
m
7
−
m
m^7- m
m
7
−
m
is divisible by
42
42
42
for any positive integer
m
m
m
.
3
1
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1/a+1/b=1/c => a+b square HOMC 2009 J Q3
Let
a
,
b
,
c
a, b,c
a
,
b
,
c
be positive integers with no common factor and satisfy the conditions
1
a
+
1
b
=
1
c
\frac1a +\frac1b=\frac1c
a
1
+
b
1
=
c
1
Prove that
a
+
b
a + b
a
+
b
is a square.
1
1
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7 / abc + (7 - a)(7 - b)(7 - c) when a,b,c \in {1,...,6} (HOMC 2009 J Q1)
Let
a
,
b
,
c
a,b, c
a
,
b
,
c
be
3
3
3
distinct numbers from
{
1
,
2
,
3
,
4
,
5
,
6
}
\{1, 2,3, 4, 5, 6\}
{
1
,
2
,
3
,
4
,
5
,
6
}
Show that
7
7
7
divides
a
b
c
+
(
7
−
a
)
(
7
−
b
)
(
7
−
c
)
abc + (7 - a)(7 - b)(7 - c)
ab
c
+
(
7
−
a
)
(
7
−
b
)
(
7
−
c
)
10
2
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computational with altitudes and midpoints (2009 HOMC Junior Q10)
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
A
B
=
4
AB =4
A
B
=
4
and
C
D
CD
C
D
be the altitude through
C
C
C
with
C
D
=
3
CD = 3
C
D
=
3
. Find the distance between the midpoints of
A
D
AD
A
D
and
B
C
BC
BC
d^2+(a-b)^2<c^2 where d is diameter of incricle (HOMC 2009 Q10)
Prove that
d
2
+
(
a
−
b
)
2
<
c
2
d^2+(a-b)^2<c^2
d
2
+
(
a
−
b
)
2
<
c
2
,where
d
d
d
is diameter of the inscribed circle of
△
A
B
C
\vartriangle ABC
△
A
BC
9
2
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another area computational, BR = RS = SC (2009 HOMC Junior Q9)
Let be given
△
A
B
C
\vartriangle ABC
△
A
BC
with area
(
△
A
B
C
)
=
60
(\vartriangle ABC) = 60
(
△
A
BC
)
=
60
cm
2
^2
2
. Let
R
,
S
R,S
R
,
S
lie in
B
C
BC
BC
such that
B
R
=
R
S
=
S
C
BR = RS = SC
BR
=
RS
=
SC
and
P
,
Q
P,Q
P
,
Q
be midpoints of
A
B
AB
A
B
and
A
C
AC
A
C
, respectively. Suppose that
P
S
PS
PS
intersects
Q
R
QR
QR
at
T
T
T
. Evaluate area
(
△
P
Q
T
)
(\vartriangle PQT)
(
△
PQT
)
.
T= (BCA')^2+ (BCA')^2+ (ABC')^2, sum of areas, (HOMC 2009 Q9)
Give an acute-angled triangle
A
B
C
ABC
A
BC
with area
S
S
S
, let points
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
be located as follows:
A
′
A'
A
′
is the point where altitude from
A
A
A
on
B
C
BC
BC
meets the outwards facing semicirle drawn on
B
X
BX
BX
as diameter.Points
B
′
,
C
′
B',C'
B
′
,
C
′
are located similarly. Evaluate the sum
T
=
(
T=(
T
=
(
area
△
B
C
A
′
)
2
+
(
\vartriangle BCA')^2+(
△
BC
A
′
)
2
+
(
area
△
C
A
B
′
)
2
+
(
\vartriangle CAB')^2+(
△
C
A
B
′
)
2
+
(
area
△
A
B
C
′
)
2
\vartriangle ABC')^2
△
A
B
C
′
)
2
.
2
1
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Hanoi Open Mathematics Competition 2009
Show that there is a natural number
n
n
n
such that the number
a
=
n
!
a = n!
a
=
n
!
ends exactly in
2009
2009
2009
zeros.
4
1
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Prove that $a$ is divisible by 23
Suppose that
a
=
2
b
a=2^b
a
=
2
b
, where
b
=
2
10
n
+
1
b=2^{10n+1}
b
=
2
10
n
+
1
. Prove that
a
a
a
is divisible by 23 for any positive integer
n
n
n
7
1
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Hanoi Open ....
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be positive integers such that
a
+
b
+
c
+
d
=
99
a+b+c+d=99
a
+
b
+
c
+
d
=
99
. Find the maximum and minimum of product
a
b
c
d
abcd
ab
c
d