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Problems
Contests
National and Regional Contests
Vietnam Contests
Hanoi Open Mathematics Competition
2013 Hanoi Open Mathematics Competitions
2013 Hanoi Open Mathematics Competitions
Part of
Hanoi Open Mathematics Competition
Subcontests
(15)
15
2
Hide problems
(ax + b)/x=(Ax + B)/Cx \in Q (HOMC 2013 J Q15)
Denote by
Q
Q
Q
and
N
∗
N^*
N
∗
the set of all rational and positive integer numbers, respectively. Suppose that
a
x
+
b
x
∈
Q
\frac{ax + b}{x} \in Q
x
a
x
+
b
∈
Q
for every
x
∈
N
∗
x \in N^*
x
∈
N
∗
: Prove that there exist integers
A
,
B
,
C
A,B,C
A
,
B
,
C
such that
a
x
+
b
x
=
A
x
+
B
C
x
\frac{ax + b}{x}= \frac{Ax + B}{Cx}
x
a
x
+
b
=
C
x
A
x
+
B
for all
x
∈
N
∗
x \in N^*
x
∈
N
∗
(ax + b)/(cx+d)=(Ax + B)/(Cx+D) \in Q (HOMC 2013 Q15)
Denote by
Q
Q
Q
and
N
∗
N^*
N
∗
the set of all rational and positive integer numbers, respectively. Suppose that
a
x
+
b
c
x
+
d
∈
Q
\frac{ax + b}{cx + d} \in Q
c
x
+
d
a
x
+
b
∈
Q
for every
x
∈
N
∗
x \in N^*
x
∈
N
∗
: Prove that there exist integers
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
such that
a
x
+
b
c
x
+
d
=
A
x
+
B
C
x
+
D
\frac{ax + b}{cx + d}= \frac{Ax + B}{Cx+D}
c
x
+
d
a
x
+
b
=
C
x
+
D
A
x
+
B
for all
x
∈
N
∗
x \in N^*
x
∈
N
∗
14
2
Hide problems
system x^3+y=x^2+1, 2y^3+z=2y^2+1, 3z^3+x=3z^2+1 (HOMC 2014 J Q14)
Solve the system of equations
{
x
3
+
y
=
x
2
+
1
2
y
3
+
z
=
2
y
2
+
1
3
z
3
+
x
=
3
z
2
+
1
\begin{cases} x^3+y = x^2+1\\ 2y^3+z=2y^2+1 \\ 3z^3+x=3z^2+1 \end{cases}
⎩
⎨
⎧
x
3
+
y
=
x
2
+
1
2
y
3
+
z
=
2
y
2
+
1
3
z
3
+
x
=
3
z
2
+
1
x^3+1/3 y=x^2+x-4/3 and y^3+1/4 z=y^2+y -5/4 and ... (HOMC 2013 Q14)
Solve the system of equations
{
x
3
+
1
3
y
=
x
2
+
x
−
4
3
y
3
+
1
4
z
=
y
2
+
y
−
5
4
z
3
+
1
5
x
=
z
2
+
z
−
6
5
\begin{cases} x^3+\frac13 y=x^2+x -\frac43 \\ y^3+\frac14 z=y^2+y -\frac54 \\ z^3+\frac15 x=z^2+z -\frac65 \end{cases}
⎩
⎨
⎧
x
3
+
3
1
y
=
x
2
+
x
−
3
4
y
3
+
4
1
z
=
y
2
+
y
−
4
5
z
3
+
5
1
x
=
z
2
+
z
−
5
6
13
2
Hide problems
1/x+1/y=1/6 and 3/x+2/y=5/6 (HOMC 2013 J Q13)
Solve the system of equations
{
1
x
+
1
y
=
1
6
3
x
+
2
y
=
5
6
\begin{cases} \frac{1}{x}+\frac{1}{y}=\frac{1}{6} \\ \frac{3}{x}+\frac{2}{y}=\frac{5}{6} \end{cases}
{
x
1
+
y
1
=
6
1
x
3
+
y
2
=
6
5
system xy=1 and x/(x^4+y^2)+y/(x^2+y^4)=1 (HOMC 2013 Q13)
Solve the system of equations
{
x
y
=
1
x
x
4
+
y
2
+
y
x
2
+
y
4
=
1
\begin{cases} xy=1 \\ \frac{x}{x^4+y^2}+\frac{y}{x^2+y^4}=1\end{cases}
{
x
y
=
1
x
4
+
y
2
x
+
x
2
+
y
4
y
=
1
12
2
Hide problems
|ax^2+bx+c|<1, x\in [-1,1]=> f(x) = 2x^2 - 1 has 2 real roots (HOMC 2013 J Q12 )
If
f
(
x
)
=
a
x
2
+
b
x
+
c
f(x) = ax^2 + bx + c
f
(
x
)
=
a
x
2
+
b
x
+
c
satisfies the condition
∣
f
(
x
)
∣
<
1
;
∀
x
∈
[
−
1
,
1
]
|f(x)| < 1; \forall x \in [-1, 1]
∣
f
(
x
)
∣
<
1
;
∀
x
∈
[
−
1
,
1
]
, prove that the equation
f
(
x
)
=
2
x
2
−
1
f(x) = 2x^2 - 1
f
(
x
)
=
2
x
2
−
1
has two real roots.
trinomial, f(\sqrt2)=3, |f(x)| < 1 for all x \in [-1, 1] (HOMC 2013 Q12)
The function
f
(
x
)
=
a
x
2
+
b
x
+
c
f(x) = ax^2 + bx + c
f
(
x
)
=
a
x
2
+
b
x
+
c
satisfies the following conditions:
f
(
2
)
=
3
f(\sqrt2)=3
f
(
2
)
=
3
and
∣
f
(
x
)
∣
≤
1
|f(x)| \le 1
∣
f
(
x
)
∣
≤
1
for all
x
∈
[
−
1
,
1
]
x \in [-1, 1]
x
∈
[
−
1
,
1
]
. Evaluate the value of
f
(
2013
)
f(\sqrt{2013})
f
(
2013
)
11
2
Hide problems
a,b,c,d,e>0 min d , (x + a)(x + b)(x + c)=x^3+3dx^2+3x+e^3 (HOMC 2013 J Q11)
The positive numbers
a
,
b
,
c
,
d
,
e
a, b, c,d,e
a
,
b
,
c
,
d
,
e
are such that the following identity hold for all real number
x
x
x
:
(
x
+
a
)
(
x
+
b
)
(
x
+
c
)
=
x
3
+
3
d
x
2
+
3
x
+
e
3
(x + a)(x + b)(x + c) = x^3 + 3dx^2 + 3x + e^3
(
x
+
a
)
(
x
+
b
)
(
x
+
c
)
=
x
3
+
3
d
x
2
+
3
x
+
e
3
. Find the smallest value of
d
d
d
.
(x+a)(x+b)(x+c)(x+d) = x^4+4px^3+6x^2+4qx+1 (HOMC 2013 Q11)
The positive numbers
a
,
b
,
c
,
d
,
p
,
q
a, b,c, d, p, q
a
,
b
,
c
,
d
,
p
,
q
are such that
(
x
+
a
)
(
x
+
b
)
(
x
+
c
)
(
x
+
d
)
=
x
4
+
4
p
x
3
+
6
x
2
+
4
q
x
+
1
(x+a)(x+b)(x+c)(x+d) = x^4+4px^3+6x^2+4qx+1
(
x
+
a
)
(
x
+
b
)
(
x
+
c
)
(
x
+
d
)
=
x
4
+
4
p
x
3
+
6
x
2
+
4
q
x
+
1
holds for all real numbers
x
x
x
. Find the smallest value of
p
p
p
or the largest value of
q
q
q
.
10
2
Hide problems
max of S/(2S+p+2), S,p area, perimeter of a rectangle fixed S (HOMC 2013 J Q10 )
Consider the set of all rectangles with a given area
S
S
S
. Find the largest value o
M
=
S
2
S
+
p
+
2
M = \frac{S}{2S+p + 2}
M
=
2
S
+
p
+
2
S
where
p
p
p
is the perimeter of the rectangle.
max of (16-p)/(p^2+2p), S,p area, perimeter, fixed s (HOMC 2013 Q10)
Consider the set of all rectangles with a given area
S
S
S
. Find the largest value o
M
=
16
−
p
p
2
+
2
p
M = \frac{16-p}{p^2+2p}
M
=
p
2
+
2
p
16
−
p
where
p
p
p
is the perimeter of the rectangle.
9
2
Hide problems
x+y<= 1 and 2/xy +1/(x^2+y^2)=10 and x,y>0 (HOMC 2013 J Q9)
Solve the following system in positive numbers
{
x
+
y
≤
1
2
x
y
+
1
x
2
+
y
2
=
10
\begin{cases} x+y\le 1 \\ \frac{2}{xy} +\frac{1}{x^2+y^2}=10\end{cases}
{
x
+
y
≤
1
x
y
2
+
x
2
+
y
2
1
=
10
P(t) = t^3 + at^2 + bt + c, P(2013) >1/64 (HOMC 2013 Q9)
A given polynomial
P
(
t
)
=
t
3
+
a
t
2
+
b
t
+
c
P(t) = t^3 + at^2 + bt + c
P
(
t
)
=
t
3
+
a
t
2
+
b
t
+
c
has
3
3
3
distinct real roots. If the equation
(
x
2
+
x
+
2013
)
3
+
a
(
x
2
+
x
+
2013
)
2
+
b
(
x
2
+
x
+
2013
)
+
c
=
0
(x^2 +x+2013)^3 +a(x^2 +x+2013)^2 + b(x^2 + x + 2013) + c = 0
(
x
2
+
x
+
2013
)
3
+
a
(
x
2
+
x
+
2013
)
2
+
b
(
x
2
+
x
+
2013
)
+
c
=
0
has no real roots, prove that
P
(
2013
)
>
1
64
P(2013) >\frac{1}{64}
P
(
2013
)
>
64
1
3
2
Hide problems
largest integer [(n+1)a]-[na], a=\sqrt{2013}/\sqrt{2014} (HOMC 2013 J Q3)
The largest integer not exceeding
[
(
n
+
1
)
a
]
−
[
n
a
]
[(n+1)a]-[na]
[(
n
+
1
)
a
]
−
[
na
]
where
n
n
n
is a natural number,
a
=
2013
2014
a=\frac{\sqrt{2013}}{\sqrt{2014}}
a
=
2014
2013
is:(A):
1
1
1
, (B):
2
2
2
, (C):
3
3
3
, (D):
4
4
4
, (E) None of the above.
8x^3 +6x - 1, x=1/2(\sqrt[3]{2+\sqrt5}+\sqrt[3]{2-\sqrt5}) (HOMC 2013 Q3)
What is the largest integer not exceeding
8
x
3
+
6
x
−
1
8x^3 +6x - 1
8
x
3
+
6
x
−
1
, where
x
=
1
2
(
2
+
5
3
+
2
−
5
3
)
x =\frac12 \left(\sqrt[3]{2+\sqrt5} + \sqrt[3]{2-\sqrt5}\right)
x
=
2
1
(
3
2
+
5
+
3
2
−
5
)
?(A):
1
1
1
, (B):
2
2
2
, (C):
3
3
3
, (D):
4
4
4
, (E) None of the above.
5
2
Hide problems
diophantine (12x -1)(6x-1)(4x -1)(3x -1) = 330 (HOMC 2013 J Q5)
The number of integer solutions
x
x
x
of the equation below
(
12
x
−
1
)
(
6
x
−
1
)
(
4
x
−
1
)
(
3
x
−
1
)
=
330
(12x -1)(6x - 1)(4x -1)(3x - 1) = 330
(
12
x
−
1
)
(
6
x
−
1
)
(
4
x
−
1
)
(
3
x
−
1
)
=
330
is(A):
0
0
0
, (B):
1
1
1
, (C):
2
2
2
, (D):
3
3
3
, (E): None of the above.
2013 as a sum of m composite numbers, max m (HOMC 2013 Q5)
The number
n
n
n
is called a composite number if it can be written in the form
n
=
a
×
b
n = a\times b
n
=
a
×
b
, where
a
,
b
a, b
a
,
b
are positive integers greater than
1
1
1
. Write number
2013
2013
2013
in a sum of
m
m
m
composite numbers. What is the largest value of
m
m
m
?(A):
500
500
500
, (B):
501
501
501
, (C):
502
502
502
, (D):
503
503
503
, (E): None of the above.
4
2
Hide problems
A even not divisible by 10, last 2 digits of A^20 (HOMC 2013 J Q4)
Let
A
A
A
be an even number but not divisible by
10
10
10
. The last two digits of
A
20
A^{20}
A
20
are:(A):
46
46
46
, (B):
56
56
56
, (C):
66
66
66
, (D):
76
76
76
, (E): None of the above.
x_n=[(n+1)a]-[na], a=\sqrt{2013/2014} (HOMC 2013 Q4)
Let
x
0
=
[
a
]
,
x
1
=
[
2
a
]
−
[
a
]
,
x
2
=
[
3
a
]
−
[
2
a
]
,
x
3
=
[
3
a
]
−
[
4
a
]
,
x
4
=
[
5
a
]
−
[
4
a
]
,
x
5
=
[
6
a
]
−
[
5
a
]
,
.
.
.
,
x_0 = [a], x_1 = [2a] - [a], x_2 = [3a] - [2a], x_3 = [3a] - [4a],x_4 = [5a] - [4a],x_5 = [6a] - [5a], . . . ,
x
0
=
[
a
]
,
x
1
=
[
2
a
]
−
[
a
]
,
x
2
=
[
3
a
]
−
[
2
a
]
,
x
3
=
[
3
a
]
−
[
4
a
]
,
x
4
=
[
5
a
]
−
[
4
a
]
,
x
5
=
[
6
a
]
−
[
5
a
]
,
...
,
where
a
=
2013
2014
a=\frac{\sqrt{2013}}{\sqrt{2014}}
a
=
2014
2013
.The value of
x
9
x_9
x
9
is: (A):
2
2
2
(B):
3
3
3
(C):
4
4
4
(D):
5
5
5
(E): None of the above.
2
2
Hide problems
n^2 + 2014 is a perfect square (HOMC 2013 J Q2)
How many natural numbers
n
n
n
are there so that
n
2
+
2014
n^2 + 2014
n
2
+
2014
is a perfect square?(A):
1
1
1
, (B):
2
2
2
, (C):
3
3
3
, (D):
4
4
4
, (E) None of the above.
min of f(x) =|x| +|(1 - 2013x)/(2013 - x)| when x \in [-1, 1] (HOMC 2013 Q2)
The smallest value of the function
f
(
x
)
=
∣
x
∣
+
∣
1
−
2013
x
2013
−
x
∣
f(x) =|x| +\left|\frac{1 - 2013x}{2013 - x}\right|
f
(
x
)
=
∣
x
∣
+
2013
−
x
1
−
2013
x
where
x
∈
[
−
1
,
1
]
x \in [-1, 1]
x
∈
[
−
1
,
1
]
is:(A):
1
2012
\frac{1}{2012}
2012
1
, (B):
1
2013
\frac{1}{2013}
2013
1
, (C):
1
2014
\frac{1}{2014}
2014
1
, (D):
1
2015
\frac{1}{2015}
2015
1
, (E): None of the above.
1
2
Hide problems
2013 as a sum of m primes, smallest m (HOMC 2013 J Q1)
Write
2013
2013
2013
as a sum of
m
m
m
prime numbers. The smallest value of
m
m
m
is:(A):
2
2
2
, (B):
3
3
3
, (C):
4
4
4
, (D):
1
1
1
, (E): None of the above.
3-digit perfect, increase by 1 each digit and remain perfect (HOMC 2013 Q1)
How many three-digit perfect squares are there such that if each digit is increased by one, the resulting number is also a perfect square?(A):
1
1
1
, (B):
2
2
2
, (C):
4
4
4
, (D):
8
8
8
, (E) None of the above.
7
2
Hide problems
angle chasing with equilaterals (2013 HOMC Senior Q7)
Let
A
B
C
ABC
A
BC
be an equilateral triangle and a point M inside the triangle such that
M
A
2
=
M
B
2
+
M
C
2
MA^2 = MB^2 +MC^2
M
A
2
=
M
B
2
+
M
C
2
. Draw an equilateral triangle
A
C
D
ACD
A
C
D
where
D
≠
B
D \ne B
D
=
B
. Let the point
N
N
N
inside
△
A
C
D
\vartriangle ACD
△
A
C
D
such that
A
M
N
AMN
A
MN
is an equilateral triangle. Determine
∠
B
M
C
\angle BMC
∠
BMC
.
equilaterals outside a 90-60-30 triangle (2013 HOMC Junior Q7)
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
=
9
0
o
,
∠
B
=
6
0
o
\angle A = 90^o, \angle B = 60^o
∠
A
=
9
0
o
,
∠
B
=
6
0
o
and
B
C
=
1
BC = 1
BC
=
1
cm. Draw outside of
△
A
B
C
\vartriangle ABC
△
A
BC
three equilateral triangles
A
B
D
,
A
C
E
ABD,ACE
A
B
D
,
A
CE
and
B
C
F
BCF
BCF
. Determine the area of
△
D
E
F
\vartriangle DEF
△
D
EF
.
6
2
Hide problems
scent of a classic area inequality (2013 HOMC Junior Q6)
Let
A
B
C
ABC
A
BC
be a triangle with area
1
1
1
(cm
2
^2
2
). Points
D
,
E
D,E
D
,
E
and
F
F
F
lie on the sides
A
B
,
B
C
AB, BC
A
B
,
BC
and CA, respectively. Prove that
m
i
n
{
min\{
min
{
area of
△
A
D
F
,
\vartriangle ADF,
△
A
D
F
,
area of
△
B
E
D
,
\vartriangle BED,
△
BE
D
,
area of
△
C
E
F
}
≤
1
4
\vartriangle CEF\} \le \frac14
△
CEF
}
≤
4
1
(cm
2
^2
2
).
C_n^{2013} > C_a^{2013}, combinations (HOMC 2013 Q6)
Let be given
a
∈
{
0
,
1
,
2
,
3
,
.
.
.
,
100
}
.
a\in\{0,1,2, 3,..., 100\}.
a
∈
{
0
,
1
,
2
,
3
,
...
,
100
}
.
Find all
n
∈
{
1
,
2
,
3
,
.
.
.
,
2013
}
n \in\{1,2, 3,..., 2013\}
n
∈
{
1
,
2
,
3
,
...
,
2013
}
such that
C
n
2013
>
C
a
2013
C_n^{2013} > C_a^{2013}
C
n
2013
>
C
a
2013
, where
C
k
m
=
m
!
k
!
(
m
−
k
)
!
C_k^m=\frac{m!}{k!(m -k)!}
C
k
m
=
k
!
(
m
−
k
)!
m
!
.
8
1
Hide problems
equal areas inside a pentagon (2013 HOMC Junior Q8 Senior Q8)
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a convex pentagon and area of
△
A
B
C
=
\vartriangle ABC =
△
A
BC
=
area of
△
B
C
D
=
\vartriangle BCD =
△
BC
D
=
area of
△
C
D
E
=
\vartriangle CDE=
△
C
D
E
=
area of
△
D
E
A
=
\vartriangle DEA =
△
D
E
A
=
area of
△
E
A
B
\vartriangle EAB
△
E
A
B
. Given that area of
△
A
B
C
D
E
=
2
\vartriangle ABCDE = 2
△
A
BC
D
E
=
2
. Evaluate the area of area of
△
A
B
C
\vartriangle ABC
△
A
BC
.