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Problems
Contests
National and Regional Contests
Vietnam Contests
Hanoi Open Mathematics Competition
2017 Hanoi Open Mathematics Competitions
2017 Hanoi Open Mathematics Competitions
Part of
Hanoi Open Mathematics Competition
Subcontests
(15)
10
2
Hide problems
x^2-2ax+b=0,x^2- 2bx+c=0, x^2-2cx+a=0, integer roots (HOMC 2017 J Q10)
Find all non-negative integers
a
,
b
,
c
a, b, c
a
,
b
,
c
such that the roots of equations:
{
x
2
−
2
a
x
+
b
=
0
x
2
−
2
b
x
+
c
=
0
x
2
−
2
c
x
+
a
=
0
\begin{cases}x^2 - 2ax + b = 0 \\ x^2- 2bx + c = 0 \\ x^2 - 2cx + a = 0 \end{cases}
⎩
⎨
⎧
x
2
−
2
a
x
+
b
=
0
x
2
−
2
b
x
+
c
=
0
x
2
−
2
c
x
+
a
=
0
are non-negative integers.
8-letter words from {C ,H,M, O}, word sequence (HOMC 2017 S Q10)
Consider all words constituted by eight letters from
{
C
,
H
,
M
,
O
}
\{C ,H,M, O\}
{
C
,
H
,
M
,
O
}
. We arrange the words in an alphabet sequence. Precisely, the first word is
C
C
C
C
C
C
C
C
CCCCCCCC
CCCCCCCC
, the second one is
C
C
C
C
C
C
C
H
CCCCCCCH
CCCCCCC
H
, the third is
C
C
C
C
C
C
C
M
CCCCCCCM
CCCCCCCM
, the fourth one is
C
C
C
C
C
C
C
O
,
.
.
.
,
CCCCCCCO, ...,
CCCCCCCO
,
...
,
and the last word is
O
O
O
O
O
O
O
O
OOOOOOOO
OOOOOOOO
. a) Determine the
2017
2017
2017
th word of the sequence? b) What is the position of the word
H
O
M
C
H
O
M
C
HOMCHOMC
H
OMC
H
OMC
in the sequence?
8
1
Hide problems
3x3 system x^3-3x =4-y, 2y^3-6y=6-z, 3z^3-9z=8 -x (HOMC 2017 J Q8)
Determine all real solutions
x
,
y
,
z
x, y, z
x
,
y
,
z
of the following system of equations:
{
x
3
−
3
x
=
4
−
y
2
y
3
−
6
y
=
6
−
z
3
z
3
−
9
z
=
8
−
x
\begin{cases} x^3 - 3x = 4 - y \\ 2y^3 - 6y = 6 - z \\ 3z^3 - 9z = 8 - x\end{cases}
⎩
⎨
⎧
x
3
−
3
x
=
4
−
y
2
y
3
−
6
y
=
6
−
z
3
z
3
−
9
z
=
8
−
x
7
2
Hide problems
two last digits of 2^{2017} + 2017^2 (HOMC 2017 J Q7)
Determine two last digits of number
Q
=
2
2017
+
201
7
2
Q = 2^{2017} + 2017^2
Q
=
2
2017
+
201
7
2
min of T = x^3 + y^3 when 44 | x^2 + y^2 (HOMC 2017 S Q7 )
Let two positive integers
x
,
y
x, y
x
,
y
satisfy the condition
44
/
(
x
2
+
y
2
)
44 /( x^2 + y^2)
44/
(
x
2
+
y
2
)
. Determine the smallest value of
T
=
x
3
+
y
3
T = x^3 + y^3
T
=
x
3
+
y
3
.
6
2
Hide problems
diophantine with prime p, 2^mp^2 + 27 = q^3 (HOMC 2017 J Q6)
Find all triples of positive integers
(
m
,
p
,
q
)
(m,p,q)
(
m
,
p
,
q
)
such that
2
m
p
2
+
27
=
q
3
2^mp^2 + 27 = q^3
2
m
p
2
+
27
=
q
3
and
p
p
p
is a prime.
2-parameter diophantine system (HOMC 2017 S Q6)
Find all pairs of integers
a
,
b
a, b
a
,
b
such that the following system of equations has a unique integral solution
(
x
,
y
,
z
)
(x , y , z )
(
x
,
y
,
z
)
:
{
x
+
y
=
a
−
1
x
(
y
+
1
)
−
z
2
=
b
\begin{cases}x + y = a - 1 \\ x(y + 1) - z^2 = b \end{cases}
{
x
+
y
=
a
−
1
x
(
y
+
1
)
−
z
2
=
b
5
2
Hide problems
a, b, c are 2-digit, 3-digit, and 4-digit numbers (HOMC 2017 J Q5)
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be two-digit, three-digit, and four-digit numbers, respectively. Assume that the sum of all digits of number
a
+
b
a+b
a
+
b
, and the sum of all digits of
b
+
c
b + c
b
+
c
are all equal to
2
2
2
. The largest value of
a
+
b
+
c
a + b + c
a
+
b
+
c
is(A):
1099
1099
1099
(B):
2099
2099
2099
(C):
1199
1199
1199
(D):
2199
2199
2199
(E): None of the above.
replace x,y from 2017 numbers with x+7xy+y, last (HOMC 2017 S Q5)
Write
2017
2017
2017
following numbers on the blackboard:
−
1008
1008
,
−
1007
1008
,
.
.
.
,
−
1
1008
,
0
,
1
1008
,
2
1008
,
.
.
.
,
1007
1008
,
1008
1008
-\frac{1008}{1008}, -\frac{1007}{1008}, ..., -\frac{1}{1008}, 0,\frac{1}{1008},\frac{2}{1008}, ... ,\frac{1007}{1008},\frac{1008}{1008}
−
1008
1008
,
−
1008
1007
,
...
,
−
1008
1
,
0
,
1008
1
,
1008
2
,
...
,
1008
1007
,
1008
1008
. One processes some steps as: erase two arbitrary numbers
x
,
y
x, y
x
,
y
on the blackboard and then write on it the number
x
+
7
x
y
+
y
x + 7xy + y
x
+
7
x
y
+
y
. After
2016
2016
2016
steps, there is only one number. The last one on the blackboard is(A):
−
1
1008
-\frac{1}{1008}
−
1008
1
(B):
0
0
0
(C):
1
1008
\frac{1}{1008}
1008
1
(D):
−
144
1008
-\frac{144}{1008}
−
1008
144
(E): None of the above
4
2
Hide problems
last digit 2^1+3^5+4^9+5^{13}+ ... +505^{2013}+506^{2017} (HOMC 2017 J Q4)
Put
S
=
2
1
+
3
5
+
4
9
+
5
13
+
.
.
.
+
50
5
2013
+
50
6
2017
S = 2^1 + 3^5 + 4^9 + 5^{13} + ... + 505^{2013} + 506^{2017}
S
=
2
1
+
3
5
+
4
9
+
5
13
+
...
+
50
5
2013
+
50
6
2017
. The last digit of
S
S
S
is(A):
1
1
1
(B):
3
3
3
(C):
5
5
5
(D):
7
7
7
(E): None of the above.
c(x-a)(x-b)/(c-a)(cb)+a(x-b)(x-c)/(a-b)(a-c)+b(x-c)(x-a)/(b-c)(b-a)+1 HOMC17.4
Let a,b,c be three distinct positive numbers. Consider the quadratic polynomial
P
(
x
)
=
c
(
x
−
a
)
(
x
−
b
)
(
c
−
a
)
(
c
−
b
)
+
a
(
x
−
b
)
(
x
−
c
)
(
a
−
b
)
(
a
−
c
)
+
b
(
x
−
c
)
(
x
−
a
)
(
b
−
c
)
(
b
−
a
)
+
1
P (x) =\frac{c(x - a)(x - b)}{(c -a)(c -b)}+\frac{a(x - b)(x - c)}{(a - b)(a - c)}+\frac{b(x -c)(x - a)}{(b - c)(b - a)}+ 1
P
(
x
)
=
(
c
−
a
)
(
c
−
b
)
c
(
x
−
a
)
(
x
−
b
)
+
(
a
−
b
)
(
a
−
c
)
a
(
x
−
b
)
(
x
−
c
)
+
(
b
−
c
)
(
b
−
a
)
b
(
x
−
c
)
(
x
−
a
)
+
1
. The value of
P
(
2017
)
P (2017)
P
(
2017
)
is(A):
2015
2015
2015
(B):
2016
2016
2016
(C):
2017
2017
2017
(D):
2018
2018
2018
(E): None of the above.
3
2
Hide problems
n^2 + 4n + 25 is perfect square (HOMC 2017 J Q3)
Suppose
n
2
+
4
n
+
25
n^2 + 4n + 25
n
2
+
4
n
+
25
is a perfect square. How many such non-negative integers
n
n
n
's are there?(A):
1
1
1
(B):
2
2
2
(C):
4
4
4
(D):
6
6
6
(E): None of the above.
diophantine x^4 + 4y^4 + z^4 + 4 = 8xyz (HOMC 2017 S Q3)
The number of real triples
(
x
,
y
,
z
)
(x , y , z )
(
x
,
y
,
z
)
that satisfy the equation
x
4
+
4
y
4
+
z
4
+
4
=
8
x
y
z
x^4 + 4y^4 + z^4 + 4 = 8xyz
x
4
+
4
y
4
+
z
4
+
4
=
8
x
yz
is (A):
0
0
0
, (B):
1
1
1
, (C):
2
2
2
, (D):
8
8
8
, (E): None of the above.
2
2
Hide problems
diophantine 2^x - y^2 = 1 (HOMC 2017 J Q2)
How many pairs of positive integers
(
x
,
y
)
(x, y)
(
x
,
y
)
are there, those satisfy the identity
2
x
−
y
2
=
1
2^x - y^2 = 1
2
x
−
y
2
=
1
?(A):
1
1
1
(B):
2
2
2
(C):
3
3
3
(D):
4
4
4
(E): None of the above.
diophantine 2^x - y^2 = 4 (HOMC 2017 S Q2)
How many pairs of positive integers
(
x
,
y
)
(x, y)
(
x
,
y
)
are there, those satisfy the identity
2
x
−
y
2
=
4
2^x - y^2 = 4
2
x
−
y
2
=
4
? (A):
1
1
1
(B):
2
2
2
(C):
3
3
3
(D):
4
4
4
(E): None of the above.
1
2
Hide problems
|x_1|+|x_2|+|x_3| , P(x)=x^3 -6x^2+5x +12 (HOMC 2017 J Q1)
Suppose
x
1
,
x
2
,
x
3
x_1, x_2, x_3
x
1
,
x
2
,
x
3
are the roots of polynomial
P
(
x
)
=
x
3
−
6
x
2
+
5
x
+
12
P(x) = x^3 - 6x^2 + 5x + 12
P
(
x
)
=
x
3
−
6
x
2
+
5
x
+
12
The sum
∣
x
1
∣
+
∣
x
2
∣
+
∣
x
3
∣
|x_1| + |x_2| + |x_3|
∣
x
1
∣
+
∣
x
2
∣
+
∣
x
3
∣
is(A):
4
4
4
(B):
6
6
6
(C):
8
8
8
(D):
14
14
14
(E): None of the above.
|x_1|+|x_2|+|x_3| , P(x)=x^3 -4x^2 -3x +2 (HOMC 2017 S Q1)
Suppose
x
1
,
x
2
,
x
3
x_1, x_2, x_3
x
1
,
x
2
,
x
3
are the roots of polynomial
P
(
x
)
=
x
3
−
4
x
2
−
3
x
+
2
P(x) = x^3 - 4x^2 -3x + 2
P
(
x
)
=
x
3
−
4
x
2
−
3
x
+
2
. The sum
∣
x
1
∣
+
∣
x
2
∣
+
∣
x
3
∣
|x_1| + |x_2| + |x_3|
∣
x
1
∣
+
∣
x
2
∣
+
∣
x
3
∣
is (A):
4
4
4
(B):
6
6
6
(C):
8
8
8
(D):
10
10
10
(E): None of the above.
13
2
Hide problems
min of a/(b+c-a)+4b/(c+a-b)+ 9c/(a+b-c), if a+b+c=12 (HOMC 2017 J Q13)
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be the side-lengths of triangle
A
B
C
ABC
A
BC
with
a
+
b
+
c
=
12
a+b+c = 12
a
+
b
+
c
=
12
. Determine the smallest value of
M
=
a
b
+
c
−
a
+
4
b
c
+
a
−
b
+
9
c
a
+
b
−
c
M =\frac{a}{b + c - a}+\frac{4b}{c + a - b}+\frac{9c}{a + b - c}
M
=
b
+
c
−
a
a
+
c
+
a
−
b
4
b
+
a
+
b
−
c
9
c
.
|AM | -|PM| >= d fon any M \in BC (HOMC 2017 Q13)
Let
A
B
C
ABC
A
BC
be a triangle. For some
d
>
0
d>0
d
>
0
let
P
P
P
stand for a point inside the triangle such that
∣
A
B
∣
−
∣
P
B
∣
≥
d
|AB| - |P B| \ge d
∣
A
B
∣
−
∣
PB
∣
≥
d
, and
∣
A
C
∣
−
∣
P
C
∣
≥
d
|AC | - |P C | \ge d
∣
A
C
∣
−
∣
PC
∣
≥
d
. Is the following inequality true
∣
A
M
∣
−
∣
P
M
∣
≥
d
|AM | - |P M | \ge d
∣
A
M
∣
−
∣
PM
∣
≥
d
, for any position of
M
∈
B
C
M \in BC
M
∈
BC
?
12
2
Hide problems
2017 consecutive integers with 17 primes (HOMC 2017 J Q12)
Does there exist a sequence of
2017
2017
2017
consecutive integers which contains exactly
17
17
17
primes?
length of NM depends on position of C? (HOMC 2017 Q12)
Let
(
O
)
(O)
(
O
)
denote a circle with a chord
A
B
AB
A
B
, and let
W
W
W
be the midpoint of the minor arc
A
B
AB
A
B
. Let
C
C
C
stand for an arbitrary point on the major arc
A
B
AB
A
B
. The tangent to the circle
(
O
)
(O)
(
O
)
at
C
C
C
meets the tangents at
A
A
A
and
B
B
B
at points
X
X
X
and
Y
Y
Y
, respectively. The lines
W
X
W X
W
X
and
W
Y
W Y
WY
meet
A
B
AB
A
B
at points
N
N
N
and
M
M
M
, respectively. Does the length of segment
N
M
NM
NM
depend on position of
C
C
C
?
11
2
Hide problems
square covered by 8 squares (2017 HOMC Junior Q11)
Let
S
S
S
denote a square of the side-length
7
7
7
, and let eight squares of the side-length
3
3
3
be given. Show that
S
S
S
can be covered by those eight small squares.
| <PAB - <PAC| >= |<PBC - <PCB| in equilateral (HOMC 2017 Q11)
Let
A
B
C
ABC
A
BC
be an equilateral triangle, and let
P
P
P
stand for an arbitrary point inside the triangle. Is it true that
∣
∠
P
A
B
−
∠
P
A
C
∣
≥
∣
∠
P
B
C
−
∠
P
C
B
∣
| \angle PAB - \angle PAC| \ge | \angle PBC - \angle PCB|
∣∠
P
A
B
−
∠
P
A
C
∣
≥
∣∠
PBC
−
∠
PCB
∣
?
9
2
Hide problems
equilateral, can be covered by 5 equilaterals (2017 HOMC Junior Q9)
Prove that the equilateral triangle of area
1
1
1
can be covered by five arbitrary equilateral triangles having the total area
2
2
2
.
2018 pieces by cutting 2017 times a square, numbers 0,1,2 (HOMC 2017 S Q9)
Cut off a square carton by a straight line into two pieces, then cut one of two pieces into two small pieces by a straight line, ect. By cutting
2017
2017
2017
times we obtain
2018
2018
2018
pieces. We write number
2
2
2
in every triangle, number 1 in every quadrilateral, and
0
0
0
in the polygons. Is the sum of all inserted numbers always greater than
2017
2017
2017
?
15
2
Hide problems
any quadrilateral can be divided into 9 isosceles (2017 HOMC Junior Q15)
Show that an arbitrary quadrilateral can be divided into nine isosceles triangles.
cover a square with 8 squares, impossible (HOMC 2017 Q15)
Let
S
S
S
denote a square of side-length
7
7
7
, and let eight squares with side-length
3
3
3
be given. Show that it is impossible to cover
S
S
S
by those eight small squares with the condition: an arbitrary side of those (eight) squares is either coincided, parallel, or perpendicular to others of
S
S
S
.
14
2
Hide problems
trapezoid, 4 perpendiculars, collinear ? (2017 HOMC Junior Q14)
Given trapezoid
A
B
C
D
ABCD
A
BC
D
with bases
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
(
A
B
<
C
D
AB < CD
A
B
<
C
D
). Let
O
O
O
be the intersection of
A
C
AC
A
C
and
B
D
BD
B
D
. Two straight lines from
D
D
D
and
C
C
C
are perpendicular to
A
C
AC
A
C
and
B
D
BD
B
D
intersect at
E
E
E
, i.e.
C
E
⊥
B
D
CE \perp BD
CE
⊥
B
D
and
D
E
⊥
A
C
DE \perp AC
D
E
⊥
A
C
. By analogy,
A
F
⊥
B
D
AF \perp BD
A
F
⊥
B
D
and
B
F
⊥
A
C
BF \perp AC
BF
⊥
A
C
. Are three points
E
,
O
,
F
E , O, F
E
,
O
,
F
located on the same line?
P=m^{2003}n^{2017}-m^{2017}n^{2003} divisible by 24,7 (HOMC 2017 S Q14)
Put
P
=
m
2003
n
2017
−
m
2017
n
2003
P = m^{2003}n^{2017} - m^{2017}n^{2003}
P
=
m
2003
n
2017
−
m
2017
n
2003
, where
m
,
n
∈
N
m, n \in N
m
,
n
∈
N
. a) Is
P
P
P
divisible by
24
24
24
? b) Do there exist
m
,
n
∈
N
m, n \in N
m
,
n
∈
N
such that
P
P
P
is not divisible by
7
7
7
?