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Problems
Contests
National and Regional Contests
Vietnam Contests
Hanoi Open Mathematics Competition
2011 Hanoi Open Mathematics Competitions
2011 Hanoi Open Mathematics Competitions
Part of
Hanoi Open Mathematics Competition
Subcontests
(12)
3
1
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\sqrt[3]{(2011)^3 + 3x(2011)^2 + 4x 2011+ 5} (HOMC 2011 Q3)
What is the largest integer less than to \sqrt[3]{(2011)^3 + 3 \times (2011)^2 + 4 \times 2011+ 5} ?(A)
2010
2010
2010
, (B)
2011
2011
2011
, (C)
2012
2012
2012
, (D)
2013
2013
2013
, (E) None of the above.
1
2
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total no of intersections of 3 lines in plane (HOMC 2011 J Q1)
Three lines are drawn in a plane. Which of the following could NOT be the total number of points of intersections?(A)
0
0
0
(B)
1
1
1
(C)
2
2
2
(D)
3
3
3
(E) They all could.
divisible by 8 or at least one of its digits is 8 (HOMC 2011 Q1)
An integer is called "octal" if it is divisible by
8
8
8
or if at least one of its digits is
8
8
8
. How many integers between
1
1
1
and
100
100
100
are octal?(A):
22
22
22
, (B):
24
24
24
, (C):
27
27
27
, (D):
30
30
30
, (E):
33
33
33
4
2
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4 statements on real, how many correct? (HOMC 2011 J Q4)
Among the five statements on real numbers below, how many of them are correct? "If
a
<
b
<
0
a < b < 0
a
<
b
<
0
then
a
<
b
2
a < b^2
a
<
b
2
" , "If
0
<
a
<
b
0 < a < b
0
<
a
<
b
then
a
<
b
2
a < b^2
a
<
b
2
", "If
a
3
<
b
3
a^3 < b^3
a
3
<
b
3
then
a
<
b
a < b
a
<
b
", "If
a
2
<
b
2
a^2 < b^2
a
2
<
b
2
then
a
<
b
a < b
a
<
b
", "If
∣
a
∣
<
∣
b
∣
|a| < |b|
∣
a
∣
<
∣
b
∣
then
a
<
b
a < b
a
<
b
",(A)
0
0
0
, (B)
1
1
1
, (C)
2
2
2
, (D)
3
3
3
, (E)
4
4
4
1 + x + x^2 + x^3 + ...+ x^{2011} >=0 for x>=-1 (HOMC 2011 Q4)
Prove that
1
+
x
+
x
2
+
x
3
+
.
.
.
+
x
2011
≥
0
1 + x + x^2 + x^3 + ...+ x^{2011} \ge 0
1
+
x
+
x
2
+
x
3
+
...
+
x
2011
≥
0
for every
x
≥
−
1
x \ge - 1
x
≥
−
1
.
7
2
Hide problems
4a^2 + 3a + 5 is divisible by 6 (HOMC 2011 Q7)
How many positive integers a less than
100
100
100
such that
4
a
2
+
3
a
+
5
4a^2 + 3a + 5
4
a
2
+
3
a
+
5
is divisible by
6
6
6
.
system x + y = 3 and x^4 - y^4 = 8x - y (HOMC 2011 J Q7)
Find all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of real numbers satisfying the system :
{
x
+
y
=
3
x
4
−
y
4
=
8
x
−
y
\begin{cases} x + y = 3 \\ x^4 - y^4 = 8x - y \end{cases}
{
x
+
y
=
3
x
4
−
y
4
=
8
x
−
y
8
1
Hide problems
min of |x + 1| + |x + 5|+ |x + 14| + |x + 97| + |x + 1920| (HOMC 2011 J Q8)
Find the minimum value of
S
=
∣
x
+
1
∣
+
∣
x
+
5
∣
+
∣
x
+
14
∣
+
∣
x
+
97
∣
+
∣
x
+
1920
∣
S = |x + 1| + |x + 5|+ |x + 14| + |x + 97| + |x + 1920|
S
=
∣
x
+
1∣
+
∣
x
+
5∣
+
∣
x
+
14∣
+
∣
x
+
97∣
+
∣
x
+
1920∣
.
9
2
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1 + x + x^2 + x^3 + ... + x^{2011} = 0 (HOMC 2011 J Q9)
Solve the equation
1
+
x
+
x
2
+
x
3
+
.
.
.
+
x
2011
=
0
1 + x + x^2 + x^3 + ... + x^{2011} = 0
1
+
x
+
x
2
+
x
3
+
...
+
x
2011
=
0
.
function in N*xN*, f(x+1, y) = y[f(x, y)+f(x, y -1)], f(5, 5) (HOMC 2011 Q9)
For every pair of positive integers
(
x
,
y
)
(x, y)
(
x
,
y
)
we de fine
f
(
x
,
y
)
f(x,y)
f
(
x
,
y
)
as follows:
f
(
x
,
1
)
=
x
f(x,1) = x
f
(
x
,
1
)
=
x
f
(
x
,
y
)
=
0
f(x,y) = 0
f
(
x
,
y
)
=
0
if
y
>
x
y > x
y
>
x
f
(
x
+
1
,
y
)
=
y
[
f
(
x
,
y
)
+
f
(
x
,
y
−
1
)
]
f(x +1,y) = y[f(x,y)+ f(x, y-1)]
f
(
x
+
1
,
y
)
=
y
[
f
(
x
,
y
)
+
f
(
x
,
y
−
1
)]
Evaluate
f
(
5
,
5
)
f(5, 5)
f
(
5
,
5
)
.
6
2
Hide problems
diophantine m^2 + n^2 + 3 = 4(m + n) (HOMC 2011 J Q6)
Find all positive integers
(
m
,
n
)
(m,n)
(
m
,
n
)
such that
m
2
+
n
2
+
3
=
4
(
m
+
n
)
m^2 + n^2 + 3 = 4(m + n)
m
2
+
n
2
+
3
=
4
(
m
+
n
)
system x + y = 2 and x^4 - y^4 = 5x - 3y (HOMC 2011 Q6)
Find all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of real numbers satisfying the system :
{
x
+
y
=
2
x
4
−
y
4
=
5
x
−
3
y
\begin{cases} x + y = 2 \\ x^4 - y^4 = 5x - 3y \end{cases}
{
x
+
y
=
2
x
4
−
y
4
=
5
x
−
3
y
11
1
Hide problems
area of a quadrilateral (2011 HOMC Junior Q11)
Given a quadrilateral
A
B
C
D
ABCD
A
BC
D
with
A
B
=
B
C
=
3
AB = BC =3
A
B
=
BC
=
3
cm,
C
D
=
4
CD = 4
C
D
=
4
cm,
D
A
=
8
DA = 8
D
A
=
8
cm and
∠
D
A
B
+
∠
A
B
C
=
18
0
o
\angle DAB + \angle ABC = 180^o
∠
D
A
B
+
∠
A
BC
=
18
0
o
. Calculate the area of the quadrilateral.
5
2
Hide problems
The perfect squares
Let M = 7!.8!.9!.10!.11!.12!. How many factors of M are perfect squares ?
max of abc if a+2b+3c=100, a,b,c\in N* (HOMC 2011 Q5)
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive integers such that
a
+
2
b
+
3
c
=
100
a + 2b +3c = 100
a
+
2
b
+
3
c
=
100
. Find the greatest value of
M
=
a
b
c
M = abc
M
=
ab
c
2
2
Hide problems
The last digit of the number
The last digit of the number A =
7
2011
7^{2011}
7
2011
is ?
compare expontentials with sqrt (HOMC 2011 Q2)
What is the smallest number ?(A)
3
3
3
(B)
2
2
2^{\sqrt2}
2
2
(C)
2
1
+
1
2
2^{1+\frac{1}{\sqrt2}}
2
1
+
2
1
(D)
2
1
2
+
2
2
3
2^{\frac12} + 2^{\frac23}
2
2
1
+
2
3
2
(E)
2
5
3
2^{\frac53}
2
3
5
10
2
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another sum PQ+PE+QF inside a right triangle (2011 HOMC Junior Q10)
Consider a right -angle triangle
A
B
C
ABC
A
BC
with
A
=
9
0
o
A=90^{o}
A
=
9
0
o
,
A
B
=
c
AB=c
A
B
=
c
and
A
C
=
b
AC=b
A
C
=
b
. Let
P
∈
A
C
P\in AC
P
∈
A
C
and
Q
∈
A
B
Q\in AB
Q
∈
A
B
such that
∠
A
P
Q
=
∠
A
B
C
\angle APQ=\angle ABC
∠
A
PQ
=
∠
A
BC
and
∠
A
Q
P
=
∠
A
C
B
\angle AQP = \angle ACB
∠
A
QP
=
∠
A
CB
. Calculate
P
Q
+
P
E
+
Q
F
PQ+PE+QF
PQ
+
PE
+
QF
, where
E
E
E
and
F
F
F
are the projections of
B
B
B
and
Q
Q
Q
onto
B
C
BC
BC
, respectively.
Prove that equality
Two bisectors BD and CE of the triangle ABC intersect at O. Suppose that BD.CE = 2BO.OC. Denote by H the point in BC such that OH perpendicular BC. Prove that AB.AC = 2HB.HC.
12
2
Hide problems
Determine the minimum value
Suppose that
a
>
0
;
b
>
0
a > 0; b > 0
a
>
0
;
b
>
0
and
a
+
b
≤
1
a + b \leq 1
a
+
b
≤
1
. Determine the minimum value of
M
=
1
a
b
+
1
a
2
+
a
b
+
1
a
b
+
b
2
+
1
a
2
+
b
2
M=\frac{1}{ab} +\frac{1}{a^2+ab}+\frac{1}{ab+b^2}+\frac{1}{a^2+b^2}
M
=
ab
1
+
a
2
+
ab
1
+
ab
+
b
2
1
+
a
2
+
b
2
1
.
Help me[homc2011]
Suppose that
∣
a
x
2
+
b
x
+
c
∣
≥
∣
x
2
−
1
∣
|ax^2+bx+c| \geq |x^2-1|
∣
a
x
2
+
b
x
+
c
∣
≥
∣
x
2
−
1∣
for all real numbers x. Prove that
∣
b
2
−
4
a
c
∣
≥
4
|b^2-4ac|\geq 4
∣
b
2
−
4
a
c
∣
≥
4
.