MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Hanoi Open Mathematics Competition
2016 Hanoi Open Mathematics Competitions
2016 Hanoi Open Mathematics Competitions
Part of
Hanoi Open Mathematics Competition
Subcontests
(15)
5
1
Hide problems
diophantine 3x^2 + x = 4y^2 + y (HOMC 2016 J Q5)
There are positive integers
x
,
y
x, y
x
,
y
such that
3
x
2
+
x
=
4
y
2
+
y
3x^2 + x = 4y^2 + y
3
x
2
+
x
=
4
y
2
+
y
, and
(
x
−
y
)
(x - y)
(
x
−
y
)
is equal to(A):
2013
2013
2013
(B):
2014
2014
2014
(C):
2015
2015
2015
(D):
2016
2016
2016
(E): None of the above.
9
2
Hide problems
3x4 system of inequalities given, max |x|+|y|+|z| (HOMC 2016 J Q9)
Let
x
,
y
,
z
x, y,z
x
,
y
,
z
satisfy the following inequalities
{
∣
x
+
2
y
−
3
z
∣
≤
6
∣
x
−
2
y
+
3
z
∣
≤
6
∣
x
−
2
y
−
3
z
∣
≤
6
∣
x
+
2
y
+
3
z
∣
≤
6
\begin{cases} | x + 2y - 3z| \le 6 \\ | x - 2y + 3z| \le 6 \\ | x - 2y - 3z| \le 6 \\ | x + 2y + 3z| \le 6 \end{cases}
⎩
⎨
⎧
∣
x
+
2
y
−
3
z
∣
≤
6
∣
x
−
2
y
+
3
z
∣
≤
6
∣
x
−
2
y
−
3
z
∣
≤
6
∣
x
+
2
y
+
3
z
∣
≤
6
Determine the greatest value of
M
=
∣
x
∣
+
∣
y
∣
+
∣
z
∣
M = |x| + |y| + |z|
M
=
∣
x
∣
+
∣
y
∣
+
∣
z
∣
.
a+b+c=a^2+b^2+c^2 \in Z, abc = m^2/n^3 (HOMC 2016 S Q9)
Let rational numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfy the conditions
a
+
b
+
c
=
a
2
+
b
2
+
c
2
∈
Z
a + b + c = a^2 + b^2 + c^2 \in Z
a
+
b
+
c
=
a
2
+
b
2
+
c
2
∈
Z
. Prove that there exist two relative prime numbers
m
,
n
m, n
m
,
n
such that
a
b
c
=
m
2
n
3
abc =\frac{m^2}{n^3}
ab
c
=
n
3
m
2
.
8
2
Hide problems
diophantine x^3 - (x + y + z)^2 = (y + z)^3 + 34 (HOMC 2016 J Q8)
Find all positive integers
x
,
y
,
z
x,y,z
x
,
y
,
z
such that
x
3
−
(
x
+
y
+
z
)
2
=
(
y
+
z
)
3
+
34
x^3 - (x + y + z)^2 = (y + z)^3 + 34
x
3
−
(
x
+
y
+
z
)
2
=
(
y
+
z
)
3
+
34
3-digit no which equal to cube of sum of all digits (HOMC 2016 S Q8)
Determine all
3
3
3
-digit numbers which are equal to cube of the sum of all its digits.
7
1
Hide problems
no of triangles in a 3x3 grid (by 9 points) (HOMC 2016 J Q7)
Nine points form a grid of size
3
×
3
3\times 3
3
×
3
. How many triangles are there with
3
3
3
vertices at these points?
6
2
Hide problems
min a>0 such 2016 integers belong (a, 2016a], (HOMC 2016 J Q6)
Determine the smallest positive number
a
a
a
such that the number of all integers belonging to
(
a
,
2016
a
]
(a, 2016a]
(
a
,
2016
a
]
is
2016
2016
2016
.
16 elements of {1, 2, 3,..., 106} with difference ... (HOMC 2016 S Q6)
Let
A
A
A
consist of
16
16
16
elements of the set
{
1
,
2
,
3
,
.
.
.
,
106
}
\{1, 2, 3,..., 106\}
{
1
,
2
,
3
,
...
,
106
}
, so that the difference of two arbitrary elements in
A
A
A
are different from
6
,
9
,
12
,
15
,
18
,
21
6, 9, 12, 15, 18, 21
6
,
9
,
12
,
15
,
18
,
21
. Prove that there are two elements of
A
A
A
for which their difference equals to
3
3
3
.
14
2
Hide problems
2015a^2+a = 2016b^2+b, \sqrt{a-b} is natural (HOMC 2016 J Q14)
Given natural numbers
a
,
b
a,b
a
,
b
such that
2015
a
2
+
a
=
2016
b
2
+
b
2015a^2+a = 2016b^2+b
2015
a
2
+
a
=
2016
b
2
+
b
. Prove that
a
−
b
\sqrt{a-b}
a
−
b
is a natural number.
f(m)=f(2015) f(2016), f (x)=x^2 +px +q (HOMC 2016 S Q14)
Let
f
(
x
)
=
x
2
+
p
x
+
q
f (x) = x^2 + px + q
f
(
x
)
=
x
2
+
p
x
+
q
, where
p
,
q
p, q
p
,
q
are integers. Prove that there is an integer
m
m
m
such that
f
(
m
)
=
f
(
2015
)
⋅
f
(
2016
)
f (m) = f (2015) \cdot f (2016)
f
(
m
)
=
f
(
2015
)
⋅
f
(
2016
)
.
15
2
Hide problems
f(2014)=2015, f(2015)=2016, f(2013)- f(2016) prime (HOMC 2016 J Q15)
Find all polynomials of degree
3
3
3
with integer coeffcients such that
f
(
2014
)
=
2015
,
f
(
2015
)
=
2016
f(2014) = 2015, f(2015) = 2016
f
(
2014
)
=
2015
,
f
(
2015
)
=
2016
and
f
(
2013
)
−
f
(
2016
)
f(2013) - f(2016)
f
(
2013
)
−
f
(
2016
)
is a prime number.
min T=42a^2+34b^2+43c^2 when 18ab+9ca+29bc=1 (HOMC 2016 S Q15)
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be real numbers satisfying the condition
18
a
b
+
9
c
a
+
29
b
c
=
1
18ab + 9ca + 29bc = 1
18
ab
+
9
c
a
+
29
b
c
=
1
. Find the minimum value of the expression
T
=
42
a
2
+
34
b
2
+
43
c
2
T = 42a^2 + 34b^2 + 43c^2
T
=
42
a
2
+
34
b
2
+
43
c
2
.
4
1
Hide problems
monkey in Zoo is lucky when eats 3 diff. fruits (HOMC 2016 J Q4)
A monkey in Zoo becomes lucky if he eats three different fruits. What is the largest number of monkeys one can make lucky, by having
20
20
20
oranges,
30
30
30
bananas,
40
40
40
peaches and
50
50
50
tangerines? Justify your answer.(A):
30
30
30
(B):
35
35
35
(C):
40
40
40
(D):
45
45
45
(E): None of the above.
3
1
Hide problems
max of M=a^2+b^2- ab when a^3 +b^3 = a^5 +b^5 (HOMC 2016 J Q3)
Given two positive numbers
a
,
b
a,b
a
,
b
such that
a
3
+
b
3
=
a
5
+
b
5
a^3 +b^3 = a^5 +b^5
a
3
+
b
3
=
a
5
+
b
5
, then the greatest value of
M
=
a
2
+
b
2
−
a
b
M = a^2 + b^2 - ab
M
=
a
2
+
b
2
−
ab
is(A):
1
4
\frac14
4
1
(B):
1
2
\frac12
2
1
(C):
2
2
2
(D):
1
1
1
(E): None of the above.
2
2
Hide problems
n + s(n) = 2016, sum of digits (HOMC 2016 J Q2)
The number of all positive integers
n
n
n
such that
n
+
s
(
n
)
=
2016
n + s(n) = 2016
n
+
s
(
n
)
=
2016
, where
s
(
n
)
s(n)
s
(
n
)
is the sum of all digits of
n
n
n
is(A):
1
1
1
(B):
2
2
2
(C):
3
3
3
(D):
4
4
4
(E): None of the above.
a,b,c replaced by 1/3 min(a,b,c) (HOMC 2016 Q S2)
Given an array of numbers
A
=
(
672
,
673
,
674
,
.
.
.
,
2016
)
A = (672, 673, 674, ..., 2016)
A
=
(
672
,
673
,
674
,
...
,
2016
)
on table. Three arbitrary numbers
a
,
b
,
c
∈
A
a,b,c \in A
a
,
b
,
c
∈
A
are step by step replaced by number
1
3
m
i
n
(
a
,
b
,
c
)
\frac13 min(a,b,c)
3
1
min
(
a
,
b
,
c
)
. After
672
672
672
times, on the table there is only one number
m
m
m
, such that(A):
0
<
m
<
1
0 < m < 1
0
<
m
<
1
(B):
m
=
1
m = 1
m
=
1
(C):
1
<
m
<
2
1 < m < 2
1
<
m
<
2
(D):
m
=
2
m = 2
m
=
2
(E): None of the above.
1
2
Hide problems
2016 = 2^5 + 2^6 + ...+ 2^m (HOMC 2016 J Q1)
If
2016
=
2
5
+
2
6
+
.
.
.
+
2
m
2016 = 2^5 + 2^6 + ...+ 2^m
2016
=
2
5
+
2
6
+
...
+
2
m
then
m
m
m
is equal to(A):
8
8
8
(B):
9
9
9
(C):
10
10
10
(D):
11
11
11
(E): None of the above.
10-digit numbers by digits 1, 2, 3 (HOMC 2016 S Q1)
How many are there
10
10
10
-digit numbers composed from the digits
1
,
2
,
3
1, 2, 3
1
,
2
,
3
only and in which, two neighbouring digits differ by
1
1
1
:(A):
48
48
48
(B):
64
64
64
(C):
72
72
72
(D):
128
128
128
(E): None of the above.
12
1
Hide problems
fixed line related to random circle (2016 HOMC Q12)
Let
A
A
A
be a point inside the acute angle
x
O
y
xOy
x
O
y
. An arbitrary circle
ω
\omega
ω
passes through
O
,
A
O, A
O
,
A
, intersecting
O
x
Ox
O
x
and
O
y
Oy
O
y
at the second intersection
B
B
B
and
C
C
C
, respectively. Let
M
M
M
be the midpoint of
B
C
BC
BC
. Prove that
M
M
M
is always on a fixed line (when
ω
\omega
ω
changes, but always goes through
O
O
O
and
A
A
A
).
13
2
Hide problems
H midpoint of DE iff F midpoint of BC (2016 HOMC Junior Q13)
Let
H
H
H
be orthocenter of the triangle
A
B
C
ABC
A
BC
. Let
d
1
,
d
2
d_1, d_2
d
1
,
d
2
be lines perpendicular to each-another at
H
H
H
. The line
d
1
d_1
d
1
intersects
A
B
,
A
C
AB, AC
A
B
,
A
C
at
D
,
E
D, E
D
,
E
and the line d_2 intersects
B
C
B C
BC
at
F
F
F
. Prove that
H
H
H
is the midpoint of segment
D
E
DE
D
E
if and only if
F
F
F
is the midpoint of segment
B
C
BC
BC
.
|2a + b| >=4, |ax^2 + bx + c| <=1 for x \in [-1, 1] (HOMC 2016 S Q13)
Find all triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of real numbers such that
∣
2
a
+
b
∣
≥
4
|2a + b| \ge 4
∣2
a
+
b
∣
≥
4
and
∣
a
x
2
+
b
x
+
c
∣
≤
1
|ax^2 + bx + c| \le 1
∣
a
x
2
+
b
x
+
c
∣
≤
1
∀
x
∈
[
−
1
,
1
]
\forall x \in [-1, 1]
∀
x
∈
[
−
1
,
1
]
.
11
2
Hide problems
equal segments wanted , 2 lines, midpoint (2016 HOMC Junior Q11)
Let be given a triangle
A
B
C
ABC
A
BC
, and let
I
I
I
be the midpoint of
B
C
BC
BC
. The straight line
d
d
d
passing
I
I
I
intersects
A
B
,
A
C
AB,AC
A
B
,
A
C
at
M
,
N
M,N
M
,
N
, respectively. The straight line
d
′
d'
d
′
(
≠
d
\ne d
=
d
) passing
I
I
I
intersects
A
B
,
A
C
AB, AC
A
B
,
A
C
at
Q
,
P
Q, P
Q
,
P
, respectively. Suppose
M
,
P
M, P
M
,
P
are on the same side of
B
C
BC
BC
and
M
P
,
N
Q
MP , NQ
MP
,
NQ
intersect
B
C
BC
BC
at
E
E
E
and
F
F
F
, respectively. Prove that
I
E
=
I
F
IE = I F
I
E
=
I
F
.
TF \cdot AD = ID \cdot AT (2016 HOMC Q11 )
Let
I
I
I
be the incenter of triangle
A
B
C
ABC
A
BC
and
ω
\omega
ω
be its circumcircle. Let the line
A
I
AI
A
I
intersect
ω
\omega
ω
at point
D
≠
A
D \ne A
D
=
A
. Let
F
F
F
and
E
E
E
be points on side
B
C
BC
BC
and arc
B
D
C
BDC
B
D
C
respectively such that
∠
B
A
F
=
∠
C
A
E
<
1
2
∠
B
A
C
\angle BAF = \angle CAE < \frac12 \angle BAC
∠
B
A
F
=
∠
C
A
E
<
2
1
∠
B
A
C
. Let
X
X
X
be the second point of intersection of line
E
I
EI
E
I
with
ω
\omega
ω
and
T
T
T
be the point of intersection of segment
D
X
DX
D
X
with line
A
F
AF
A
F
. Prove that
T
F
⋅
A
D
=
I
D
⋅
A
T
TF \cdot AD = ID \cdot AT
TF
⋅
A
D
=
I
D
⋅
A
T
.
10
1
Hide problems
h_a + 4h_b + 9h_c > 36r, altitudes (HOMC 2016 Junior Q10)
Let
h
a
,
h
b
,
h
c
h_a, h_b, h_c
h
a
,
h
b
,
h
c
and
r
r
r
be the lengths of altitudes and radius of the inscribed circle of
△
A
B
C
\vartriangle ABC
△
A
BC
, respectively. Prove that
h
a
+
4
h
b
+
9
h
c
>
36
r
h_a + 4h_b + 9h_c > 36r
h
a
+
4
h
b
+
9
h
c
>
36
r
.