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Problems
Contests
National and Regional Contests
Vietnam Contests
Hanoi Open Mathematics Competition
2006 Hanoi Open Mathematics Competitions
2006 Hanoi Open Mathematics Competitions
Part of
Hanoi Open Mathematics Competition
Subcontests
(9)
3
2
Hide problems
dioplantine, x^2 + y -z = 100 and x + y^2 - z = 124 (HOMC 2006 J Q3)
Find the number of different positive integer triples
(
x
,
y
,
z
)
(x, y,z)
(
x
,
y
,
z
)
satisfying the equations
x
2
+
y
−
z
=
100
x^2 + y -z = 100
x
2
+
y
−
z
=
100
and
x
+
y
2
−
z
=
124
x + y^2 - z = 124
x
+
y
2
−
z
=
124
:
c^{log_{b}a}+a^{log_{c}b} (HOMC 2006 Q3)
Suppose that
a
log
b
c
+
b
log
c
a
=
m
a^{\log_{b}c}+b^{\log_{c}a}=m
a
l
o
g
b
c
+
b
l
o
g
c
a
=
m
. Find the value of
c
log
b
a
+
a
log
c
b
c^{\log_{b}a}+a^{\log_{c}b}
c
l
o
g
b
a
+
a
l
o
g
c
b
.
6
2
Hide problems
locus of equal areas inside a regular hexagon (HOMC 2006 Juniors Q6)
The figure
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
is a regular hexagon. Find all points
M
M
M
belonging to the hexagon such that Area of triangle
M
A
C
=
MAC =
M
A
C
=
Area of triangle
M
C
D
MCD
MC
D
.
another circle computational (2006 HOMC Senior Q7)
On the circle of radius
30
30
30
cm are given
2
2
2
points A,B with
A
B
=
16
AB = 16
A
B
=
16
cm and
C
C
C
is a midpoint of
A
B
AB
A
B
. What is the perpendicular distance from
C
C
C
to the circle?
7
1
Hide problems
computational with a circle (2006 HOMC junior Q7)
On the circle
(
O
)
(O)
(
O
)
of radius
15
15
15
cm are given
2
2
2
points
A
,
B
A, B
A
,
B
. The altitude
O
H
OH
O
H
of the triangle
O
A
B
OAB
O
A
B
intersect
(
O
)
(O)
(
O
)
at
C
C
C
. What is
A
C
AC
A
C
if
A
B
=
16
AB = 16
A
B
=
16
cm?
9
2
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smallest possible value x^2 + y^2 - x -y - xy (HOMC 2006 J Q9)
What is the smallest possible value of
x
2
+
y
2
−
x
−
y
−
x
y
x^2 + y^2 - x -y - xy
x
2
+
y
2
−
x
−
y
−
x
y
?
Inequalities
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be real numbers such that
x
2
+
y
2
+
z
2
=
1
x^2+y^2+z^2=1
x
2
+
y
2
+
z
2
=
1
.Find the largest posible value of
∣
x
3
+
y
3
+
z
3
−
x
y
z
∣
|x^3+y^3+z^3-xyz|
∣
x
3
+
y
3
+
z
3
−
x
yz
∣
4
2
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HOMC Junior
For any real numbers
x
,
y
x,y
x
,
y
that satisfies the equation
x
+
y
−
x
y
=
155
x+y-xy=155
x
+
y
−
x
y
=
155
and
x
2
+
y
2
=
325
x^2+y^2=325
x
2
+
y
2
=
325
, Find
∣
x
3
−
y
3
∣
|x^3-y^3|
∣
x
3
−
y
3
∣
Which is large
Which is larger:
2
2
2^{\sqrt{2}}
2
2
;
2
1
+
1
2
2^{1+\frac{1}{\sqrt{2}}}
2
1
+
2
1
and 3 ?
2
1
Hide problems
Find the last three digits of the sum
Find the last three digits of the sum
200
5
11
2005^{11}
200
5
11
+
200
5
12
2005^{12}
200
5
12
+ ... +
200
5
2006
2005^{2006}
200
5
2006
1
2
Hide problems
last 2 digits of the (11 + 12 + 13 + ... + 2006)^2 (HOMC 2006 J Q1)
What is the last two digits of the number
(
11
+
12
+
13
+
.
.
.
+
2006
)
2
(11 + 12 + 13 + ... + 2006)^2
(
11
+
12
+
13
+
...
+
2006
)
2
?
What is the last three digits of the sum
What is the last three digits of the sum 11! + 12! + 13! + + 2006!
8
2
Hide problems
computational without circles (2006 HOMC Junior Q8)
In
△
A
B
C
,
P
Q
/
/
B
C
\vartriangle ABC, PQ // BC
△
A
BC
,
PQ
//
BC
where
P
P
P
and
Q
Q
Q
are points on
A
B
AB
A
B
and
A
C
AC
A
C
respectively. The lines
P
C
PC
PC
and
Q
B
QB
QB
intersect at
G
G
G
. It is also given
E
F
/
/
B
C
EF//BC
EF
//
BC
, where
G
∈
E
F
,
E
∈
A
B
G \in EF, E \in AB
G
∈
EF
,
E
∈
A
B
and
F
∈
A
C
F\in AC
F
∈
A
C
with
P
Q
=
a
PQ = a
PQ
=
a
and
E
F
=
b
EF = b
EF
=
b
. Find value of
B
C
BC
BC
.
test HOMC 2006
Find all polynomials P(x) such that P(x)+P(1/x)=x+1/x
5
1
Hide problems
Largest product
Suppose
n
n
n
is a positive integer and 3 arbitrary numbers numbers are chosen from the set
1
,
2
,
3
,
.
.
.
,
3
n
+
1
1,2,3,...,3n+1
1
,
2
,
3
,
...
,
3
n
+
1
with their sum equal to
3
n
+
1
3n+1
3
n
+
1
. What is the largest possible product of those 3 numbers?