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Problems
Contests
National and Regional Contests
Vietnam Contests
Hanoi Open Mathematics Competition
2019 Hanoi Open Mathematics Competitions
2019 Hanoi Open Mathematics Competitions
Part of
Hanoi Open Mathematics Competition
Subcontests
(15)
15
1
Hide problems
HANOI letters in a 2x5 rectangle (HOMC 2019 JI-15)
Given a
2
×
5
2\times 5
2
×
5
rectangle is divided into unit squares as figure below. https://cdn.artofproblemsolving.com/attachments/6/a/9432bbf40f6d89ee1cbb507e1a3f65326c6a13.png How many ways are there to write the letters
H
,
A
,
N
,
O
,
I
H, A, N, O, I
H
,
A
,
N
,
O
,
I
into all of the unit squares, such that two neighbor squares (the squares with a common side) do not contain the same letters? (Each unit square is filled by only one letter and each letter may be used several times or not used as well.)
14
1
Hide problems
max of M =3(ab+bc+ca)-2abc if a+b+c=3, a,b,c>=0 (HOMC 2019 JI-14)
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be nonnegative real numbers satisfying
a
+
b
+
c
=
3
a + b + c =3
a
+
b
+
c
=
3
. a) If
c
>
3
2
c > \frac32
c
>
2
3
, prove that
3
(
a
b
+
b
c
+
c
a
)
−
2
a
b
c
<
7
3(ab + bc + ca) - 2abc < 7
3
(
ab
+
b
c
+
c
a
)
−
2
ab
c
<
7
. b) Find the greatest possible value of
M
=
3
(
a
b
+
b
c
+
c
a
)
−
2
a
b
c
M =3(ab + bc + ca) - 2abc
M
=
3
(
ab
+
b
c
+
c
a
)
−
2
ab
c
.
13
1
Hide problems
locus of points in a equilateral with distances sidelengths (HOMC 2019 JI-13)
Find all points inside a given equilateral triangle such that the distances from it to three sides of the given triangle are the side lengths of a triangle.
12
1
Hide problems
x^2 + ax + b with a,b integers, add 1 or - 1 to a or b (HOMC 2019 JI-12)
Given an expression
x
2
+
a
x
+
b
x^2 + ax + b
x
2
+
a
x
+
b
where
a
,
b
a,b
a
,
b
are integer coefficients. At any step, one can change the expression by adding either
1
1
1
or
−
1
-1
−
1
to only one of the two coefficients
a
,
b
a, b
a
,
b
. a) Suppose that the initial expression has
a
=
−
7
a =-7
a
=
−
7
and
b
=
19
b = 19
b
=
19
. Show your modification steps to obtain a new expression that has zero value at some integer value of
x
x
x
. b) Starting from the initial expression as above, one gets the expression
x
2
−
17
x
+
9
x^2 - 17x + 9
x
2
−
17
x
+
9
after
m
m
m
modification steps. Prove that at a certain step
k
k
k
with
k
<
m
k < m
k
<
m
, the obtained expression has zero value at some integer value of
x
x
x
.
11
1
Hide problems
x^2 - 2xy + 5y^2 + 2x - 6y - 3 = 0 diophantine (HOMC 2019 JI-11)
Find all integers
x
x
x
and
y
y
y
satisfying the following equation
x
2
−
2
x
y
+
5
y
2
+
2
x
−
6
y
−
3
=
0
x^2 - 2xy + 5y^2 + 2x - 6y - 3 = 0
x
2
−
2
x
y
+
5
y
2
+
2
x
−
6
y
−
3
=
0
.
10
1
Hide problems
sum of greatest odd divisors of n (HOMC 2019 JI-10)
For any positive integer
n
n
n
, let
r
n
r_n
r
n
denote the greatest odd divisor of
n
n
n
. Compute
T
=
r
100
+
r
101
+
r
102
+
.
.
.
+
r
200
T =r_{100}+ r_{101} + r_{102}+...+r_{200}
T
=
r
100
+
r
101
+
r
102
+
...
+
r
200
9
1
Hide problems
min of S = 2a +3 +32/(a - b) (2b +3)^2, for a>b>0 (HOMC 2019 JI-9)
Let
a
a
a
and
b
b
b
be positive real numbers with
a
>
b
a > b
a
>
b
. Find the smallest possible values of
S
=
2
a
+
3
+
32
(
a
−
b
)
(
2
b
+
3
)
2
S = 2a +3 +\frac{32}{(a - b)(2b +3)^2}
S
=
2
a
+
3
+
(
a
−
b
)
(
2
b
+
3
)
2
32
8
1
Hide problems
area chasing, ratios in a triangle related (HOMC 2019 JI-8)
Let
A
B
C
ABC
A
BC
be a triangle, and
M
M
M
be the midpoint of
B
C
BC
BC
, Let
N
N
N
be the point on the segment
A
M
AM
A
M
with
A
N
=
3
N
M
AN = 3NM
A
N
=
3
NM
, and
P
P
P
be the intersection point of the lines
B
N
BN
BN
and
A
C
AC
A
C
. What is the area in cm
2
^2
2
of the triangle
A
N
P
ANP
A
NP
if the area of the triangle
A
B
C
ABC
A
BC
is
40
40
40
cm
2
^2
2
?
7
1
Hide problems
p+q in terms of n when pq-1= n^3 for odd primes p,q (HOMC 2019 JI-7)
Let
p
p
p
and
q
q
q
be odd prime numbers. Assume that there exists a positive integer
n
n
n
such that
p
q
−
1
=
n
3
pq-1= n^3
pq
−
1
=
n
3
. Express
p
+
q
p+q
p
+
q
in terms of
n
n
n
6
1
Hide problems
max n so that 10x11x12 x...x50 divisible by 10^n (HOMC 2019 JI-6)
What is the largest positive integer
n
n
n
such that
10
×
11
×
12
×
.
.
.
×
50
10 \times 11 \times 12 \times ... \times 50
10
×
11
×
12
×
...
×
50
is divisible by
1
0
n
10^n
1
0
n
?
5
1
Hide problems
computational with bisector, area wanted (HOMC 2019 JI-5)
Let
A
B
C
ABC
A
BC
be a triangle and
A
D
AD
A
D
be the bisector of the triangle (
D
∈
(
B
C
)
D \in (BC)
D
∈
(
BC
)
) Assume that
A
B
=
14
AB =14
A
B
=
14
cm,
A
C
=
35
AC = 35
A
C
=
35
cm and
A
D
=
12
AD = 12
A
D
=
12
cm; which of the following is the area of triangle
A
B
C
ABC
A
BC
in cm
2
^2
2
?A.
1176
5
\frac{1176}{5}
5
1176
B.
1167
5
\frac{1167}{5}
5
1167
C.
234
234
234
D.
1176
7
\frac{1176}{7}
7
1176
E.
236
236
236
4
1
Hide problems
no of connected subsequences of 1,2,...,100 (HOMC 2019 JI-4)
How many connected subsequences (i.e, consisting of one element or consecutive elements) of the following sequence are there:
1
,
2
,
.
.
.
,
100
1,2,...,100
1
,
2
,
...
,
100
?A.
1010
1010
1010
B.
2020
2020
2020
C.
3030
3030
3030
D.
4040
4040
4040
E.
5050
5050
5050
3
1
Hide problems
P(5)- P(0)=? if P(x) =ax + b, P(2)- P(1)= 3 (HOMC 2019 JI-3)
Let
a
a
a
and
b
b
b
be real numbers, and the polynomial
P
(
x
)
=
a
x
+
b
P(x) =ax + b
P
(
x
)
=
a
x
+
b
such that
P
(
2
)
−
P
(
1
)
=
3
P(2)- P(1)= 3
P
(
2
)
−
P
(
1
)
=
3
: Compute the value of
P
(
5
)
−
P
(
0
)
P(5)- P(0)
P
(
5
)
−
P
(
0
)
.A.
11
11
11
B.
13
13
13
C.
15
15
15
D.
17
17
17
E.
19
19
19
2
1
Hide problems
last digit of $4^{3^{2019}} (HOMC 2019 JI-2)
What is the last digit of
4
3
2019
4^{3^{2019}}
4
3
2019
?A.
0
0
0
B.
2
2
2
C.
4
4
4
D.
6
6
6
E.
8
8
8
1
1
Hide problems
larger expression than both a and y for x,y>0 (HOMC 2019 JI-1)
Let
x
x
x
and
y
y
y
be positive real numbers. Which of the following expressions is larger than both
x
x
x
and
y
y
y
?A.
x
y
+
1
xy + 1
x
y
+
1
B.
(
x
+
y
)
2
(x + y)^2
(
x
+
y
)
2
C.
x
2
+
y
x^2 + y
x
2
+
y
D.
x
(
x
+
y
)
x(x + y)
x
(
x
+
y
)
E.
(
x
+
y
+
1
)
2
(x + y + 1)^2
(
x
+
y
+
1
)
2