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Contests
National and Regional Contests
Vietnam Contests
Hanoi Open Mathematics Competition
2018 Hanoi Open Mathematics Competitions
2018 Hanoi Open Mathematics Competitions
Part of
Hanoi Open Mathematics Competition
Subcontests
(15)
12
2
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right angle inside a rectangle wanted (HOMC 2018 Ind. p12)
Let ABCD be a rectangle with
4
5
o
<
∠
A
D
B
<
6
0
o
45^o < \angle ADB < 60^o
4
5
o
<
∠
A
D
B
<
6
0
o
. The diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect at
O
O
O
. A line passing through
O
O
O
and perpendicular to
B
D
BD
B
D
meets
A
D
AD
A
D
and
C
D
CD
C
D
at
M
M
M
and
N
N
N
respectively. Let
K
K
K
be a point on side
B
C
BC
BC
such that
M
K
∥
A
C
MK \parallel AC
M
K
∥
A
C
. Show that
∠
M
K
N
=
9
0
o
\angle MKN = 90^o
∠
M
K
N
=
9
0
o
. https://cdn.artofproblemsolving.com/attachments/4/1/1d37b96cebaea3409ade7ce6711ac2d3fc2ef9.png
AC is tangent to circumcircle of DEF (HOMC 2018 ind. sen12)
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
<
A
C
AB < AC
A
B
<
A
C
, and let
B
E
BE
BE
and
C
F
CF
CF
be the altitudes. Let the median
A
M
AM
A
M
intersect
B
E
BE
BE
at point
P
P
P
, and let line
C
P
CP
CP
intersect
A
B
AB
A
B
at point
D
D
D
(see Figure 2). Prove that
D
E
∥
B
C
DE \parallel BC
D
E
∥
BC
, and
A
C
AC
A
C
is tangent to the circumcircle of
△
D
E
F
\vartriangle DEF
△
D
EF
. https://cdn.artofproblemsolving.com/attachments/f/7/bbad9f6019a77c6aa46c3a821857f06233cb93.png
15
2
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pq/(p + q)=(m^2 + 6)/(m + 1) diophantine (HOMC 2018 Ind. p15)
Find all pairs of prime numbers
(
p
,
q
)
(p,q)
(
p
,
q
)
such that for each pair
(
p
,
q
)
(p,q)
(
p
,
q
)
, there is a positive integer m satisfying
p
q
p
+
q
=
m
2
+
6
m
+
1
\frac{pq}{p + q}=\frac{m^2 + 6}{m + 1}
p
+
q
pq
=
m
+
1
m
2
+
6
.
n lines on plane such every line intersects 12 (HOMC 2018 ind. sen15)
There are
n
n
n
distinct straight lines on a plane such that every line intersects exactly
12
12
12
others. Determine all the possible values of
n
n
n
.
14
2
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polynomial P(k) =k/(k + 1), k=0,...,2017, p(2018)=? (HOMC 2018 Ind. p14)
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial with degree
2017
2017
2017
such that
P
(
k
)
=
k
k
+
1
P(k) =\frac{k}{k + 1}
P
(
k
)
=
k
+
1
k
,
∀
k
=
0
,
1
,
2
,
.
.
.
,
2017
\forall k = 0, 1, 2, ..., 2017
∀
k
=
0
,
1
,
2
,
...
,
2017
. Calculate
P
(
2018
)
P(2018)
P
(
2018
)
.
max T= (a-b)^{2018}+(b-c)^{2018}+(c-a)^{2018} (HOMC 2018 ind. sen14)
Let
a
,
b
,
c
a,b, c
a
,
b
,
c
denote the real numbers such that
1
≤
a
,
b
,
c
≤
2
1 \le a, b, c\le 2
1
≤
a
,
b
,
c
≤
2
. Consider
T
=
(
a
−
b
)
2018
+
(
b
−
c
)
2018
+
(
c
−
a
)
2018
T = (a - b)^{2018} + (b - c)^{2018} + (c - a)^{2018}
T
=
(
a
−
b
)
2018
+
(
b
−
c
)
2018
+
(
c
−
a
)
2018
. Determine the largest possible value of
T
T
T
.
13
2
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27 handshakes of m x n students (HOMC 2018 Ind. p13)
A competition room of HOMC has
m
×
n
m \times n
m
×
n
students where
m
,
n
m, n
m
,
n
are integers larger than
2
2
2
. Their seats are arranged in
m
m
m
rows and
n
n
n
columns. Before starting the test, every student takes a handshake with each of his/her adjacent students (in the same row or in the same column). It is known that there are totally
27
27
27
handshakes. Find the number of students in the room.
n = S(n) + P(n), sum , product digits (HOMC 2018 ind. sen13)
For a positive integer
n
n
n
, let
S
(
n
)
,
P
(
n
)
S(n), P(n)
S
(
n
)
,
P
(
n
)
denote the sum and the product of all the digits of
n
n
n
respectively. 1) Find all values of n such that
n
=
P
(
n
)
n = P(n)
n
=
P
(
n
)
: 2) Determine all values of n such that
n
=
S
(
n
)
+
P
(
n
)
n = S(n) + P(n)
n
=
S
(
n
)
+
P
(
n
)
.
11
2
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(xy + 2)^2 = x^2 + y^2 diophantine (HOMC 2018 Ind. p11)
Find all pairs of nonnegative integers
(
x
,
y
)
(x, y)
(
x
,
y
)
for which
(
x
y
+
2
)
2
=
x
2
+
y
2
(xy + 2)^2 = x^2 + y^2
(
x
y
+
2
)
2
=
x
2
+
y
2
.
2^n + 11 is divisible by 2^k - 1 (HOMC 2018 ind. sen11)
Find all positive integers
k
k
k
such that there exists a positive integer
n
n
n
, for which
2
n
+
11
2^n + 11
2
n
+
11
is divisible by
2
k
−
1
2^k - 1
2
k
−
1
.
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4
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4
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4
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