MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Hanoi Open Mathematics Competition
2008 Hanoi Open Mathematics Competitions
2008 Hanoi Open Mathematics Competitions
Part of
Hanoi Open Mathematics Competition
Subcontests
(10)
9
1
Hide problems
find point, equal angles, parallelogram related (HOMC 2008 Q9)
Consider a triangle
A
B
C
ABC
A
BC
. For every point M
∈
B
C
\in BC
∈
BC
,we define
N
∈
C
A
N \in CA
N
∈
C
A
and
P
∈
A
B
P \in AB
P
∈
A
B
such that
A
P
M
N
APMN
A
PMN
is a parallelogram. Let
O
O
O
be the intersection of
B
N
BN
BN
and
C
P
CP
CP
. Find
M
∈
B
C
M \in BC
M
∈
BC
such that
∠
P
M
O
=
∠
O
M
N
\angle PMO=\angle OMN
∠
PMO
=
∠
OMN
5
2
Hide problems
4x4 (absolute) inequalites imply a 5th inequality (HOMC 2008 J Q5)
Suppose
x
,
y
,
z
,
t
x, y, z, t
x
,
y
,
z
,
t
are real numbers such that
{
∣
x
+
y
+
z
−
t
∣
≤
1
∣
y
+
z
+
t
−
x
∣
≤
1
∣
z
+
t
+
x
−
y
∣
≤
1
∣
t
+
x
+
y
−
z
∣
≤
1
\begin{cases} |x + y + z -t |\le 1 \\ |y + z + t - x|\le 1 \\ |z + t + x - y|\le 1 \\ |t + x + y - z|\le 1 \end{cases}
⎩
⎨
⎧
∣
x
+
y
+
z
−
t
∣
≤
1
∣
y
+
z
+
t
−
x
∣
≤
1
∣
z
+
t
+
x
−
y
∣
≤
1
∣
t
+
x
+
y
−
z
∣
≤
1
Prove that
x
2
+
y
2
+
z
2
+
t
2
≤
1
x^2 + y^2 + z^2 + t^2 \le 1
x
2
+
y
2
+
z
2
+
t
2
≤
1
.
max P(x)- min P(x)= b-a, for all x\in [a,b] (HOMC 2008 Q5)
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
of degree
1
1
1
such that
m
a
x
a
≤
x
≤
b
P
(
x
)
−
m
i
n
a
≤
x
≤
b
P
(
x
)
=
b
−
a
\underset {a\le x\le b}{max} P(x) - \underset {a\le x\le b}{min} P(x) =b-a
a
≤
x
≤
b
ma
x
P
(
x
)
−
a
≤
x
≤
b
min
P
(
x
)
=
b
−
a
,
∀
a
,
b
∈
R
\forall a,b\in R
∀
a
,
b
∈
R
where
a
<
b
a < b
a
<
b
10
1
Hide problems
max{a, b, c}>= 2, a+b+c=5, min{a^2+b^2+c^2} (HOMC 2008 J Q10)
Let
a
,
b
,
c
∈
[
1
,
3
]
a,b,c \in [1, 3]
a
,
b
,
c
∈
[
1
,
3
]
and satisfy the following conditions:
m
a
x
{
a
,
b
,
c
}
≥
2
max \{a, b, c\}\ge 2
ma
x
{
a
,
b
,
c
}
≥
2
and
a
+
b
+
c
=
5
a + b + c = 5
a
+
b
+
c
=
5
What is the smallest possible value of
a
2
+
b
2
+
c
2
a^2 + b^2 + c^2
a
2
+
b
2
+
c
2
?
6
1
Hide problems
P(x^2-1)=x^4-3x^2+3, P(x^2+1) = ? (HOMC 2008 J Q6)
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial such that
P
(
x
2
−
1
)
=
x
4
−
3
x
2
+
3
P(x^2 - 1) = x^4 - 3x^2 + 3
P
(
x
2
−
1
)
=
x
4
−
3
x
2
+
3
. Find
P
(
x
2
+
1
)
P(x^2 + 1)
P
(
x
2
+
1
)
.
3
2
Hide problems
coefficient of x into (1+x)(1-2x)(1+3x)(1-4x) ... (1-2008x) (HOMC 2008 J Q3)
Find the coefficient of
x
x
x
in the expansion of
(
1
+
x
)
(
1
−
2
x
)
(
1
+
3
x
)
(
1
−
4
x
)
.
.
.
(
1
−
2008
x
)
(1 + x)(1 - 2x)(1 + 3x)(1 - 4x) ...(1 - 2008x)
(
1
+
x
)
(
1
−
2
x
)
(
1
+
3
x
)
(
1
−
4
x
)
...
(
1
−
2008
x
)
.
x^2+8z=3+2y^2, impossible diophantine in N* (HOMC 2008 Q3)
Show that the equation
x
2
+
8
z
=
3
+
2
y
2
x^2 + 8z = 3 + 2y^2
x
2
+
8
z
=
3
+
2
y
2
has no solutions of positive integers
x
,
y
x, y
x
,
y
and
z
z
z
.
2
2
Hide problems
how many integers belong to (a,2008a), where a>0 given (HOMC 2008 J Q2)
How many integers belong to (
a
,
2008
a
a,2008a
a
,
2008
a
), where
a
a
a
(
a
>
0
a > 0
a
>
0
) is given.
diophantine m^2 + 2n^2 = 3(m + 2n) (HOMC 2008 Q2)
Find all pairs
(
m
,
n
)
(m, n)
(
m
,
n
)
of positive integers such that
m
2
+
2
n
2
=
3
(
m
+
2
n
)
m^2 + 2n^2 = 3(m + 2n)
m
2
+
2
n
2
=
3
(
m
+
2
n
)
1
2
Hide problems
1 to 2008 have the sum of their digits divisible by 5 (HOMC 2008 J Q1)
How many integers from
1
1
1
to
2008
2008
2008
have the sum of their digits divisible by
5
5
5
?
How many integers are in (b,2008b], where b > 0 given (HOMC 2008 Q1)
How many integers are there in
(
b
,
2008
b
]
(b,2008b]
(
b
,
2008
b
]
, where
b
b
b
(
b
>
0
b > 0
b
>
0
) is given.
7
2
Hide problems
<DAC+<EBD+<ACE+<BDA+<CEB in covex pentagon (2008 HOMC Junior Q7)
The figure
A
B
C
D
E
ABCDE
A
BC
D
E
is a convex pentagon. Find the sum
∠
D
A
C
+
∠
E
B
D
+
∠
A
C
E
+
∠
B
D
A
+
∠
C
E
B
\angle DAC + \angle EBD +\angle ACE +\angle BDA + \angle CEB
∠
D
A
C
+
∠
EB
D
+
∠
A
CE
+
∠
B
D
A
+
∠
CEB
?
a,b,c consecutive odd, a^2+b^2+c^2 = 4 equal digits (HOMC 2008 Q7)
Find all triples
(
a
,
b
,
c
)
(a, b,c)
(
a
,
b
,
c
)
of consecutive odd positive integers such that
a
<
b
<
c
a < b < c
a
<
b
<
c
and
a
2
+
b
2
+
c
2
a^2 + b^2 + c^2
a
2
+
b
2
+
c
2
is a four digit number with all digits equal.
4
2
Hide problems
diophantine m^2 + n^2 = 3(m + n) (HOMC 2008 J Q4)
Find all pairs
(
m
,
n
)
(m,n)
(
m
,
n
)
of positive integers such that
m
2
+
n
2
=
3
(
m
+
n
)
m^2 + n^2 = 3(m + n)
m
2
+
n
2
=
3
(
m
+
n
)
.
HOMC 2008 P4
Prove that there exists an infinite number of relatively prime pairs
(
m
,
n
)
(m, n)
(
m
,
n
)
of positive integers such that the equation
x
3
−
n
x
+
m
n
=
0
x^3-nx+mn=0
x
3
−
n
x
+
mn
=
0
has three distint integer roots.
8
2
Hide problems
shorter diagonal of a rhombus (2008 HOMC Junior Q8)
The sides of a rhombus have length
a
a
a
and the area is
S
S
S
. What is the length of the shorter diagonal?
Convex quadrilateral
Consider a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. Let
O
O
O
be the intersection of
A
C
AC
A
C
and
B
D
BD
B
D
;
M
,
N
M, N
M
,
N
be the centroid of
Δ
A
O
B
\Delta AOB
Δ
A
OB
and
Δ
C
O
D
\Delta COD
Δ
CO
D
and
P
,
Q
P, Q
P
,
Q
be orthocenter of
Δ
B
O
C
\Delta BOC
Δ
BOC
and
Δ
D
O
A
\Delta DOA
Δ
D
O
A
, respectively. Prove that
M
N
⊥
P
Q
MN\bot PQ
MN
⊥
PQ
.