MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Hanoi Open Mathematics Competition
2015 Hanoi Open Mathematics Competitions
2015 Hanoi Open Mathematics Competitions
Part of
Hanoi Open Mathematics Competition
Subcontests
(15)
13
2
Hide problems
(x^2 + y^2 - 2) (x + y)^2 +(xy + 1)^2 = 0, rational (HOMC 2015 J Q13)
Give rational numbers
x
,
y
x, y
x
,
y
such that
(
x
2
+
y
2
−
2
)
(
x
+
y
)
2
+
(
x
y
+
1
)
2
=
0
(x^2 + y^2 - 2) (x + y)^2 + (xy + 1)^2 = 0
(
x
2
+
y
2
−
2
)
(
x
+
y
)
2
+
(
x
y
+
1
)
2
=
0
Prove that
1
+
x
y
\sqrt{1 + xy}
1
+
x
y
is a rational number.
c=a^{2014}+b^{2014} ,d=a^{2015}+b^{2015}, rel. prime (HOMC 15 Q13)
Let
m
m
m
be given odd number, and let
a
,
b
a, b
a
,
b
denote the roots of equation
x
2
+
m
x
−
1
=
0
x^2 + mx - 1 = 0
x
2
+
m
x
−
1
=
0
and
c
=
a
2014
+
b
2014
c = a^{2014} + b^{2014}
c
=
a
2014
+
b
2014
,
d
=
a
2015
+
b
2015
d =a^{2015} + b^{2015}
d
=
a
2015
+
b
2015
. Prove that
c
c
c
and
d
d
d
are relatively prime numbers.
15
2
Hide problems
max of 4(Σ a_i^3 )-(Σa_i^4) when Σa_i^2 <=8 (HOMC 2015 J Q15)
Let the numbers
a
,
b
,
c
a, b,c
a
,
b
,
c
satisfy the relation
a
2
+
b
2
+
c
2
≤
8
a^2+b^2+c^2 \le 8
a
2
+
b
2
+
c
2
≤
8
. Determine the maximum value of
M
=
4
(
a
3
+
b
3
+
c
3
)
−
(
a
4
+
b
4
+
c
4
)
M = 4(a^3 + b^3 + c^3) - (a^4 + b^4 + c^4)
M
=
4
(
a
3
+
b
3
+
c
3
)
−
(
a
4
+
b
4
+
c
4
)
max of 4(sum a_i^3 )-(sum a_i^4) when sum a_i^2 <=12 (HOMC 2015 S Q15)
Let the numbers
a
,
b
,
c
a, b,c
a
,
b
,
c
satisfy the relation
a
2
+
b
2
+
c
2
+
d
2
≤
12
a^2+b^2+c^2+d^2 \le 12
a
2
+
b
2
+
c
2
+
d
2
≤
12
. Determine the maximum value of
M
=
4
(
a
3
+
b
3
+
c
3
+
d
3
)
−
(
a
4
+
b
4
+
c
4
+
d
4
)
M = 4(a^3 + b^3 + c^3+d^3) - (a^4 + b^4 + c^4+d^4)
M
=
4
(
a
3
+
b
3
+
c
3
+
d
3
)
−
(
a
4
+
b
4
+
c
4
+
d
4
)
14
1
Hide problems
diophantine 2xy^2 + x + y + 1 = x^2 + 2y^2 + xy (HOMC 2015 J Q14)
Determine all pairs of integers
(
x
,
y
)
(x, y)
(
x
,
y
)
such that
2
x
y
2
+
x
+
y
+
1
=
x
2
+
2
y
2
+
x
y
2xy^2 + x + y + 1 = x^2 + 2y^2 + xy
2
x
y
2
+
x
+
y
+
1
=
x
2
+
2
y
2
+
x
y
.
9
1
Hide problems
a^3+b^3+c^3+2[(ab)^3+(bc)^3+(ca)^3] >= 3(a^2b+b^2c+c^2a) (HOMCJ'15-9)
Let
a
,
b
,
c
a, b,c
a
,
b
,
c
be positive numbers with
a
b
c
=
1
abc = 1
ab
c
=
1
. Prove that
a
3
+
b
3
+
c
3
+
2
[
(
a
b
)
3
+
(
b
c
)
3
+
(
c
a
)
3
]
≥
3
(
a
2
b
+
b
2
c
+
c
2
a
)
a^3 + b^3 + c^3 + 2[(ab)^3 + (bc)^3 + (ca)^3] \ge 3(a^2b + b^2c + c^2a)
a
3
+
b
3
+
c
3
+
2
[(
ab
)
3
+
(
b
c
)
3
+
(
c
a
)
3
]
≥
3
(
a
2
b
+
b
2
c
+
c
2
a
)
.
8
2
Hide problems
(2015x - 2014)^3 = 8(x - 1)^3 + (2013x - 2012)^3 (HOMC 2015 J Q8)
Solve the equation
(
2015
x
−
2014
)
3
=
8
(
x
−
1
)
3
+
(
2013
x
−
2012
)
3
(2015x -2014)^3 = 8(x-1)^3 + (2013x -2012)^3
(
2015
x
−
2014
)
3
=
8
(
x
−
1
)
3
+
(
2013
x
−
2012
)
3
(x + 1)^3(x - 2)^3 + (x -1)^3(x + 2)^3 = 8(x^2 -2)^3 (HOMC 2015 Q8)
Solve the equation
(
x
+
1
)
3
(
x
−
2
)
3
+
(
x
−
1
)
3
(
x
+
2
)
3
=
8
(
x
2
−
2
)
3
.
(x + 1)^3(x - 2)^3 + (x -1)^3(x + 2)^3 = 8(x^2 -2)^3.
(
x
+
1
)
3
(
x
−
2
)
3
+
(
x
−
1
)
3
(
x
+
2
)
3
=
8
(
x
2
−
2
)
3
.
7
1
Hide problems
x^4 = 2x^2 + [x] (HOMC 2015 J Q7)
Solve equation
x
4
=
2
x
2
+
⌊
x
⌋
x^4 = 2x^2 + \lfloor x \rfloor
x
4
=
2
x
2
+
⌊
x
⌋
, where
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
is an integral part of
x
x
x
.
6
1
Hide problems
1+2abc>=a^2+b^2+c^2 => 1+2(abc)^2>=a^4+b^4+c^4 (HOMC 2015 J Q6)
Let
a
,
b
,
c
∈
[
−
1
,
1
]
a, b, c \in [-1, 1]
a
,
b
,
c
∈
[
−
1
,
1
]
such that
1
+
2
a
b
c
≥
a
2
+
b
2
+
c
2
1 + 2abc \ge a^2 + b^2 + c^2
1
+
2
ab
c
≥
a
2
+
b
2
+
c
2
. Prove that
1
+
2
a
2
b
2
c
2
≥
a
4
+
b
4
+
c
4
1 + 2a^2b^2c^2 \ge a^4 + b^4 + c^4
1
+
2
a
2
b
2
c
2
≥
a
4
+
b
4
+
c
4
.
5
1
Hide problems
a+b+c= (a-b)(b-c)(c-a) = m (mod 27) (HOMC 2015 J Q5)
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
and
m
m
m
(
0
≤
m
≤
26
0 \le m \le 26
0
≤
m
≤
26
) be integers such that
a
+
b
+
c
=
(
a
−
b
)
(
b
−
c
)
(
c
−
a
)
=
m
a + b + c = (a - b)(b- c)(c - a) = m
a
+
b
+
c
=
(
a
−
b
)
(
b
−
c
)
(
c
−
a
)
=
m
(mod
27
27
27
) then
m
m
m
is(A):
0
0
0
, (B):
1
1
1
, (C):
25
25
25
, (D):
26
26
26
(E): None of the above.
4
1
Hide problems
regular hexagon + equilateral equal perimeter (HOMC 2015 J Q4)
A regular hexagon and an equilateral triangle have equal perimeter. If the area of the triangle is
4
3
4\sqrt3
4
3
square units, the area of the hexagon is(A):
5
3
5\sqrt3
5
3
, (B):
6
3
6\sqrt3
6
3
, (C):
7
3
7\sqrt3
7
3
, (D):
8
3
8\sqrt3
8
3
, (E): None of the above.
3
2
Hide problems
sum of even positive <100, nod divis. by 3 (HOMC 2015 J Q3)
The sum of all even positive integers less than
100
100
100
those are not divisible by
3
3
3
is(A):
938
938
938
, (B):
940
940
940
, (C):
1634
1634
1634
, (D):
1638
1638
1638
, (E): None of the above.
(x-a)(x-b)/(c-a)(c-b)+(x-b)(x-c)/(a-b)(a-c)}+(x-c)(x-a)/(b-c)(b-a)= x (HOMC15-3)
Suppose that
a
>
b
>
c
>
1
a > b > c > 1
a
>
b
>
c
>
1
. One of solutions of the equation
(
x
−
a
)
(
x
−
b
)
(
c
−
a
)
(
c
−
b
)
+
(
x
−
b
)
(
x
−
c
)
(
a
−
b
)
(
a
−
c
)
+
(
x
−
c
)
(
x
−
a
)
(
b
−
c
)
(
b
−
a
)
=
x
\frac{(x - a)(x - b)}{(c - a)(c - b)}+\frac{(x - b)(x - c)}{(a - b)(a - c)}+\frac{(x - c)(x - a)}{(b - c)(b - a)}= x
(
c
−
a
)
(
c
−
b
)
(
x
−
a
)
(
x
−
b
)
+
(
a
−
b
)
(
a
−
c
)
(
x
−
b
)
(
x
−
c
)
+
(
b
−
c
)
(
b
−
a
)
(
x
−
c
)
(
x
−
a
)
=
x
is(A):
−
1
-1
−
1
, (B):
−
2
-2
−
2
, (C):
0
0
0
, (D):
1
1
1
, (E): None of the above.
2
1
Hide problems
last digit of 2017^{2017} - 2013^{2015} (HOMC 2015 J Q2)
The last digit of number
201
7
2017
−
201
3
2015
2017^{2017} - 2013^{2015}
201
7
2017
−
201
3
2015
is(A):
2
2
2
, (B):
4
4
4
, (C):
6
6
6
, (D):
8
8
8
, (E): None of the above.
1
1
Hide problems
7th term of sequence -1, 4, -2, 3,-3, 2,... (HOMC 2015 J Q1)
What is the
7
7
7
th term of the sequence
{
−
1
,
4
,
−
2
,
3
,
−
3
,
2
,
.
.
.
}
\{-1, 4,-2, 3,-3, 2,...\}
{
−
1
,
4
,
−
2
,
3
,
−
3
,
2
,
...
}
?(A)
−
1
-1
−
1
(B)
−
2
-2
−
2
(C)
−
3
-3
−
3
(D)
−
4
-4
−
4
(E) None of the above
12
2
Hide problems
integer altitudes in triangle, 3,7 and d (2015 HOMC Junior Q12)
Give a triangle
A
B
C
ABC
A
BC
with heights
h
a
=
3
h_a = 3
h
a
=
3
cm,
h
b
=
7
h_b = 7
h
b
=
7
cm and
h
c
=
d
h_c = d
h
c
=
d
cm, where
d
d
d
is an integer. Determine
d
d
d
.
perpendiculars wanted, isosceles related (2015 HOMC Senior Q12)
Give an isosceles triangle
A
B
C
ABC
A
BC
at
A
A
A
. Draw ray
C
x
Cx
C
x
being perpendicular to
C
A
,
B
E
CA, BE
C
A
,
BE
perpendicular to
C
x
Cx
C
x
(
E
∈
C
x
E \in Cx
E
∈
C
x
).Let
M
M
M
be the midpoint of
B
E
BE
BE
, and
D
D
D
be the intersection point of
A
M
AM
A
M
and
C
x
Cx
C
x
. Prove that
B
D
⊥
B
C
BD \perp BC
B
D
⊥
BC
.
10
1
Hide problems
concyclic centers of squares out right triangle (2015 HOMC Junior-Senior Q10 )
A right-angled triangle has property that, when a square is drawn externally on each side of the triangle, the six vertices of the squares that are not vertices of the triangle are concyclic. Assume that the area of the triangle is
9
9
9
cm
2
^2
2
. Determine the length of sides of the triangle.
11
1
Hide problems
AD x BI x CH <= AC x BD x OK (2015 HOMC Junior - Senior Q11)
Given a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. Let
O
O
O
be the intersection point of diagonals
A
C
AC
A
C
and
B
D
BD
B
D
and let
I
,
K
,
H
I , K , H
I
,
K
,
H
be feet of perpendiculars from
B
,
O
,
C
B , O , C
B
,
O
,
C
to
A
D
AD
A
D
, respectively. Prove that
A
D
×
B
I
×
C
H
≤
A
C
×
B
D
×
O
K
AD \times BI \times CH \le AC \times BD \times OK
A
D
×
B
I
×
C
H
≤
A
C
×
B
D
×
O
K
.