MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Hanoi Open Mathematics Competition
2007 Hanoi Open Mathematics Competitions
2007 Hanoi Open Mathematics Competitions
Part of
Hanoi Open Mathematics Competition
Subcontests
(15)
7
2
Hide problems
9 points in equilateral with area <=\sqrt3 (HOMC 2007 Junior Q7)
Nine points, no three of which lie on the same straight line, are located inside an equilateral triangle of side
4
4
4
. Prove that some three of these points are vertices of a triangle whose area is not greater than
3
\sqrt3
3
.
ij divides x_i+x_j for any distinct positive integer i,j (HOMC 2007 Q7)
Find all sequences of integer
x
1
,
x
2
,
.
.
,
x
n
,
.
.
.
x_1,x_2,..,x_n,...
x
1
,
x
2
,
..
,
x
n
,
...
such that
i
j
ij
ij
divides
x
i
+
x
j
x_i+x_j
x
i
+
x
j
for any distinct positive integer
i
i
i
,
j
j
j
.
13
2
Hide problems
locus , same area inside a triangle (2007 HOMC Junior Q13)
Let be given triangle
A
B
C
ABC
A
BC
. Find all points
M
M
M
such that area of
△
M
A
B
\vartriangle MAB
△
M
A
B
= area of
△
M
A
C
\vartriangle MAC
△
M
A
C
HOMC(Vietnam)
Let ABC be an acute-angle triangle with BC > CA. Let O, H and F be the circumcenter, orthocentre and the foot of its altitude CH, respectively. Suppose that the perpendicular to OF at F meet the side CA at P. Prove FHP = BAC.
6
2
Hide problems
90-60-30 triangle, angle bisector computational (HOMC 2007 J Q6)
In triangle
A
B
C
,
∠
B
A
C
=
6
0
o
,
∠
A
C
B
=
9
0
o
ABC, \angle BAC = 60^o, \angle ACB = 90^o
A
BC
,
∠
B
A
C
=
6
0
o
,
∠
A
CB
=
9
0
o
and
D
D
D
is on
B
C
BC
BC
. If
A
D
AD
A
D
bisects
∠
B
A
C
\angle BAC
∠
B
A
C
and
C
D
=
3
CD = 3
C
D
=
3
cm, calculate
D
B
DB
D
B
.
HOMC (Vietnam)
Let
P
(
x
)
=
x
3
+
a
x
2
+
b
x
+
1
P(x) = x^3 + ax^2 + bx + 1
P
(
x
)
=
x
3
+
a
x
2
+
b
x
+
1
and
∣
P
(
x
)
∣
≤
1
|P(x)| \leq 1
∣
P
(
x
)
∣
≤
1
for all x such that
∣
x
∣
≤
1
|x| \leq 1
∣
x
∣
≤
1
. Prove that
∣
a
∣
+
∣
b
∣
≤
5
|a| + |b| \leq 5
∣
a
∣
+
∣
b
∣
≤
5
.
8
2
Hide problems
HOMC (Vietnam)
Let a; b; c be positive integers. Prove that
(
b
+
c
−
a
)
2
(
b
+
c
)
2
+
a
2
+
(
c
+
a
−
b
)
2
(
c
+
a
)
2
+
b
2
+
(
a
+
b
−
c
)
2
(
a
+
b
)
2
+
c
2
≥
3
5
\frac{(b+c-a)^2}{(b+c)^2+a^2} + \frac{(c+a-b)^2}{(c+a)^2+b^2} + \frac{(a+b-c)^2}{(a+b)^2+c^2} \geq \frac{3}{5}
(
b
+
c
)
2
+
a
2
(
b
+
c
−
a
)
2
+
(
c
+
a
)
2
+
b
2
(
c
+
a
−
b
)
2
+
(
a
+
b
)
2
+
c
2
(
a
+
b
−
c
)
2
≥
5
3
locus inside an equilateral triangle (2007 HOMC Senior Q8)
Let
A
B
C
ABC
A
BC
be an equilateral triangle. For a point
M
M
M
inside
△
A
B
C
\vartriangle ABC
△
A
BC
, let
D
,
E
,
F
D,E,F
D
,
E
,
F
be the feet of the perpendiculars from
M
M
M
onto
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
, respectively. Find the locus of all such points
M
M
M
for which
∠
F
D
E
\angle FDE
∠
F
D
E
is a right angle.
5
2
Hide problems
HOMC(Vietnam)
Let be given an open interval ( \alpha; \beta) with
α
−
β
=
1
27
\alpha - \beta = \frac{1}{27}
α
−
β
=
27
1
. Determine the maximum number of irreducible fractions
a
b
\frac{a}{b}
b
a
in ( \alpha; \beta) with
1
≤
b
≤
2007
1 \leq b \leq 2007
1
≤
b
≤
2007
?
AE - EB=\sqrt{3}, integer computational with circle (2007 HOMC Senior Q5)
Suppose that
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
are points on a circle,
A
B
AB
A
B
is the diameter,
C
D
CD
C
D
is perpendicular to
A
B
AB
A
B
and meets
A
B
AB
A
B
and meets
A
B
AB
A
B
at
E
,
A
B
E , AB
E
,
A
B
and
C
D
CD
C
D
are integers and
A
E
−
E
B
=
3
AE - EB=\sqrt{3}
A
E
−
EB
=
3
. Find
A
E
AE
A
E
?
4
2
Hide problems
the number of digits
[help me] Let m and n denote the number of digits in
2
2007
2^{2007}
2
2007
and
5
2007
5^{2007}
5
2007
when expressed in base 10. What is the sum m + n?
HOMC (Vietnam)
List the numbers
2
;
3
3
;
4
4
;
5
5
;
6
6
.
\sqrt{2}; \sqrt[3]{3}; \sqrt[4]{4}; \sqrt[5]{5}; \sqrt[6]{6}.
2
;
3
3
;
4
4
;
5
5
;
6
6
.
in order from greatest to least.
3
2
Hide problems
possible number of diagonals of a convex polygon (HOMC 2007 J Q3)
Which of the following is a possible number of diagonals of a convex polygon?(A)
02
02
02
(B)
21
21
21
(C)
32
32
32
(D)
54
54
54
(E)
63
63
63
HOMC (Vietnam)
Find the number of di erent positive integer triples (x; y; z) satsfying the equations x+y-z=1 and
x
2
+
y
2
−
z
2
=
1
x^2+y^2-z^2=1
x
2
+
y
2
−
z
2
=
1
.
10
2
Hide problems
HOMC (Vietnam)
Let a; b; c be positive real numbers such that
1
b
c
+
1
c
a
+
1
a
b
≥
1
\frac{1}{bc}+\frac{1}{ca}+\frac{1}{ab} \geq 1
b
c
1
+
c
a
1
+
ab
1
≥
1
. Prove that
a
b
c
+
b
c
a
+
c
a
b
≥
1
\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab} \geq 1
b
c
a
+
c
a
b
+
ab
c
≥
1
.
smallest possible value of x^2+2y^2-x-2y-xy (HOMC 2007 Q10)
What is the smallest possible value of
x
2
+
2
y
2
−
x
−
2
y
−
x
y
x^2+2y^2-x-2y-xy
x
2
+
2
y
2
−
x
−
2
y
−
x
y
?
9
2
Hide problems
HOMC (Hanoi Open Mathematical Competition)
A triangle is said to be the Heron triangle if it has integer sides and integer area. In a Heron triangle, the sides a; b; c satisfy the equation b=a(a-c). Prove that the triangle is isosceles.
Prove that $ a_k\in[2006;2007]$
Let
a
1
,
a
2
,
.
.
.
,
a
2007
a_1,a_2,...,a_{2007}
a
1
,
a
2
,
...
,
a
2007
be real number such that
a
1
+
a
2
+
.
.
.
+
a
2007
≥
200
7
2
a_1+a_2+...+a_{2007}\geq 2007^{2}
a
1
+
a
2
+
...
+
a
2007
≥
200
7
2
and
a
1
2
+
a
2
2
+
.
.
.
+
a
2007
2
≤
200
7
3
−
1
a_1^{2}+a_2^{2}+...+a_{2007}^{2}\leq 2007^{3}-1
a
1
2
+
a
2
2
+
...
+
a
2007
2
≤
200
7
3
−
1
. Prove that
a
k
∈
[
2006
;
2008
]
a_k\in[2006;2008]
a
k
∈
[
2006
;
2008
]
for all
k
∈
{
1
,
2
,
.
.
.
,
2007
}
k\in\left \{ 1,2,...,2007 \right \}
k
∈
{
1
,
2
,
...
,
2007
}
12
2
Hide problems
Calculate the sum
Calculate the sum
5
2.7
+
5
7.12
+
.
.
.
+
5
2002.2007
\frac{5}{2.7}+\frac{5}{7.12}+...+\frac{5}{2002.2007}
2.7
5
+
7.12
5
+
...
+
2002.2007
5
HOMC (Vietnam)
Calculate the sum
1
2.7.12
+
1
7.12.17
+
.
.
.
+
1
1997.2002.2007
\frac{1}{2.7.12} + \frac{1}{7.12.17} + ... + \frac{1}{1997.2002.2007}
2.7.12
1
+
7.12.17
1
+
...
+
1997.2002.2007
1
.
14
2
Hide problems
How many ordered pairs of integers
How many ordered pairs of integers (x; y) satisfy the equation: 2
x
2
x^2
x
2
+
y
2
y^2
y
2
+ xy = 2(x + y)?
HOMC(Vietnam)
How many ordered pairs of integers (x; y) satisfy the equation
x
2
+
y
2
+
x
y
=
4
(
x
+
y
)
?
x^2 + y^2 + xy = 4(x + y)?
x
2
+
y
2
+
x
y
=
4
(
x
+
y
)?
.
11
2
Hide problems
How many possible values
How many possible values are there for the sum a + b + c + d if a; b; c; d are positive integers and abcd = 2007:
HOMC(Vietnam)
Find all polynomials P(x) satisfying the equation
(
2
x
−
1
)
P
(
x
)
=
(
x
−
1
)
P
(
2
x
)
,
∀
x
.
(2x-1)P(x) = (x-1)P(2x), \forall x.
(
2
x
−
1
)
P
(
x
)
=
(
x
−
1
)
P
(
2
x
)
,
∀
x
.
2
2
Hide problems
What is largest positive integer
What is largest positive integer n satisfying the following inequality:
n
2006
n^{2006}
n
2006
<
7
2007
7^{2007}
7
2007
?
HOMC (Vietnam)
Which is largest positive integer n satisfying the following inequality:
n
2007
>
(
2007
)
n
n^{2007} > (2007)^n
n
2007
>
(
2007
)
n
.
1
2
Hide problems
What is the last two digits of the number
What is the last two digits of the number (3 + 7 + 11 + ... + 2007)^2?
HOMC (Vietnam)
What is the last two digits of the number (11^2 + 15^2 + 19^2 + ... + 2007^2)^2?
15
2
Hide problems
HOMC(Vietnam)
Let
p
=
a
b
c
‾
p = \overline{abc}
p
=
ab
c
be the 3-digit prime number. Prove that the equation
a
x
2
+
b
x
+
c
=
0
ax^2 + bx + c = 0
a
x
2
+
b
x
+
c
=
0
has no rational roots.
Quadratic equation
Let
p
=
a
b
c
d
‾
p = \overline{abcd}
p
=
ab
c
d
be a
4
4
4
-digit prime number. Prove that the equation
a
x
3
+
b
x
2
+
c
x
+
d
=
0
ax^3+bx^2+cx+d=0
a
x
3
+
b
x
2
+
c
x
+
d
=
0
has no rational roots.