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Problems
Contests
National and Regional Contests
Vietnam Contests
Hanoi Open Mathematics Competition
2010 Hanoi Open Mathematics Competitions
2010 Hanoi Open Mathematics Competitions
Part of
Hanoi Open Mathematics Competition
Subcontests
(10)
1
1
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compare 888...888 x333..333, 444...444x666...667 (HOMC 2010 J Q1)
Compare the numbers: P = 888...888 \times 333 .. 333 (
2010
2010
2010
digits of
8
8
8
and
2010
2010
2010
digits of
3
3
3
) and
Q
=
444...444
×
666...6667
Q = 444...444\times 666...6667
Q
=
444...444
×
666...6667
(
2010
2010
2010
digits of
4
4
4
and
2009
2009
2009
digits of
6
6
6
)(A):
P
=
Q
P = Q
P
=
Q
, (B):
P
>
Q
P > Q
P
>
Q
, (C):
P
<
Q
P < Q
P
<
Q
.
4
1
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a \in (1,9) such (a - 1/a) is an integer (HOMC 2010 J Q4)
How many real numbers
a
∈
(
1
,
9
)
a \in (1,9)
a
∈
(
1
,
9
)
such that the corresponding number
a
−
1
a
a- \frac1a
a
−
a
1
is an integer?(A):
0
0
0
, (B):
1
1
1
, (C):
8
8
8
, (D):
9
9
9
, (E) None of the above.
3
1
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5 last digits of 5^{2010} (HOMC 2010 J Q3)
Find
5
5
5
last digits of the number
M
=
5
2010
M = 5^{2010}
M
=
5
2010
.(A):
65625
65625
65625
, (B):
45625
45625
45625
, (C):
25625
25625
25625
, (D):
15625
15625
15625
, (E) None of the above.
2
1
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2^{2n} + 2^n + 5 divisible by 5 (HOMC 2010 J Q2)
Find the number of integer
n
n
n
from the set
{
2000
,
2001
,
.
.
.
,
2010
}
\{2000,2001,...,2010\}
{
2000
,
2001
,
...
,
2010
}
such that
2
2
n
+
2
n
+
5
2^{2n} + 2^n + 5
2
2
n
+
2
n
+
5
is divisible by
7
7
7
(A):
0
0
0
, (B):
1
1
1
, (C):
2
2
2
, (D):
3
3
3
, (E) None of the above.
6
2
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greatest integer <= (2 +\sqrt3)^5 (HOMC 2010 J Q6)
Find the greatest integer less than
(
2
+
3
)
5
(2 +\sqrt3)^5
(
2
+
3
)
5
.(A):
721
721
721
(B):
722
722
722
(C):
723
723
723
(D):
724
724
724
(E) None of the above.
2(abc+bcd+cda+dab)/(p^2+q^2+r^2+s^2) \in Z (HOMC 2010 Q6)
Let
a
,
b
a,b
a
,
b
be the roots of the equation
x
2
−
p
x
+
q
=
0
x^2-px+q = 0
x
2
−
p
x
+
q
=
0
and let
c
,
d
c, d
c
,
d
be the roots of the equation
x
2
−
r
x
+
s
=
0
x^2 - rx + s = 0
x
2
−
r
x
+
s
=
0
, where
p
,
q
,
r
,
s
p, q, r,s
p
,
q
,
r
,
s
are some positive real numbers. Suppose that
M
=
2
(
a
b
c
+
b
c
d
+
c
d
a
+
d
a
b
)
p
2
+
q
2
+
r
2
+
s
2
M =\frac{2(abc + bcd + cda + dab)}{p^2 + q^2 + r^2 + s^2}
M
=
p
2
+
q
2
+
r
2
+
s
2
2
(
ab
c
+
b
c
d
+
c
d
a
+
d
ab
)
is an integer. Determine
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
.
5
1
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different colorings of bw in a 2x2 table (HOMC 2010 J Q5)
Each box in a
2
x
2
2x2
2
x
2
table can be colored black or white. How many different colorings of the table are there?(A):
4
4
4
, (B):
8
8
8
, (C):
16
16
16
, (D):
32
32
32
, (E) None of the above.
7
1
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integer a such 2x^2-30x+a=0 has prime roots (HOMC 2010 J Q7)
Determine all positive integer
a
a
a
such that the equation
2
x
2
−
30
x
+
a
=
0
2x^2 - 30x + a = 0
2
x
2
−
30
x
+
a
=
0
has two prime roots, i.e. both roots are prime numbers.
8
1
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n and (n^3+2n^2+2n+4) both perfect squares (HOMC 2010 J Q8)
If
n
n
n
and
n
3
+
2
n
2
+
2
n
+
4
n^3+2n^2+2n+4
n
3
+
2
n
2
+
2
n
+
4
are both perfect squares, fi nd
n
n
n
.
10
1
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max of x/(2x + y) +y/(2y + z)+z/(2z + x) in R^+ (HOMC 2010 J Q10)
Find the maximum value of
M
=
x
2
x
+
y
+
y
2
y
+
z
+
z
2
z
+
x
M =\frac{x}{2x + y} +\frac{y}{2y + z}+\frac{z}{2z + x}
M
=
2
x
+
y
x
+
2
y
+
z
y
+
2
z
+
x
z
,
x
,
y
,
z
>
0
x,y, z > 0
x
,
y
,
z
>
0
9
2
Hide problems
area computational (2010 HOMC Junior Q9)
Let be given a triangle
A
B
C
ABC
A
BC
and points
D
,
M
,
N
D,M,N
D
,
M
,
N
belong to
B
C
,
A
B
,
A
C
BC,AB,AC
BC
,
A
B
,
A
C
, respectively. Suppose that
M
D
MD
M
D
is parallel to
A
C
AC
A
C
and
N
D
ND
N
D
is parallel to
A
B
AB
A
B
. If
S
△
B
M
D
=
9
S_{\vartriangle BMD} = 9
S
△
BM
D
=
9
cm
2
,
S
△
D
N
C
=
25
^2, S_{\vartriangle DNC} = 25
2
,
S
△
D
NC
=
25
cm
2
^2
2
, compute
S
△
A
M
N
S_{\vartriangle AMN}
S
△
A
MN
?
x,y\in N*, 3x^2+x=4y^2+y => x-y perfect integer (HOMC 2010 Q9)
Let
x
,
y
x,y
x
,
y
be the positive integers such that
3
x
2
+
x
=
4
y
2
+
y
3x^2 +x = 4y^2 + y
3
x
2
+
x
=
4
y
2
+
y
. Prove that
x
−
y
x - y
x
−
y
is a perfect (square).