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Subcontests
(4)
4
2
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a^3+b^3+c^3= 2p and 3ab=2c^2
Find all the positive integers
a
,
b
,
c
a,b,c
a
,
b
,
c
such that
3
a
b
=
2
c
2
3ab= 2c^2
3
ab
=
2
c
2
and
a
3
+
b
3
+
c
3
a^3+b^3+c^3
a
3
+
b
3
+
c
3
is the double of a prime number.
find all a such that n\mid phi(n)!+a for infinitely many n
Find all integers
a
a
a
such that there are infinitely many positive integers
n
n
n
such that
n
n
n
divides
ϕ
(
n
)
!
+
a
\phi(n)!+a
ϕ
(
n
)!
+
a
.
3
2
Hide problems
simple geometry
In a triangle scalene
A
B
C
ABC
A
BC
, let
I
I
I
be its incenter and
D
D
D
the intersection of
A
I
AI
A
I
and
B
C
BC
BC
. Let
M
M
M
and
N
N
N
points where the incircle touches
A
B
AB
A
B
and
A
C
AC
A
C
, respectively. Let
F
F
F
be the second intersection of the circumcircle
(
A
M
N
)
(AMN)
(
A
MN
)
with the circumcircle
(
A
B
C
)
(ABC)
(
A
BC
)
. Let
T
T
T
the intersection of
A
F
AF
A
F
and
B
C
BC
BC
. Let
J
J
J
be the intersection of
T
I
TI
T
I
with the line parallel of
F
I
FI
F
I
that passes through
D
D
D
. Prove that the line
A
J
AJ
A
J
is perpendicular to
B
C
BC
BC
.
set of points with integer coordinates
Let
C
C
C
be the set of points
(
x
,
y
)
(x,y)
(
x
,
y
)
with integer coordinates in the plane where
1
≤
x
≤
900
1\leq x\leq 900
1
≤
x
≤
900
and
1
≤
y
≤
1000
1\leq y\leq 1000
1
≤
y
≤
1000
. A polygon
P
P
P
with vertices in
C
C
C
is called emerald if
P
P
P
has exactly zero or two vertices in each row and each column and all the internal angles of
P
P
P
are
9
0
∘
90^\circ
9
0
∘
or
27
0
∘
270^\circ
27
0
∘
. Find the greatest value of
k
k
k
such that we can color
k
k
k
points in
C
C
C
such that any subset of these
k
k
k
points is not the set of vertices of an emerald polygon.https://cdn.discordapp.com/attachments/954427908359876608/1299737432010395678/image.png?ex=671e4a4f&is=671cf8cf&hm=ce008541975226a0e9ea53a93592a7469d8569baca945c1c207d4a722126bb60&On the left, an example of an emerald polygon; on the right, an example of a non-emerald polygon.
2
1
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x^2= 5^y+3^z, (x,y,z) positive integers
Show that there are no triples of positive integers
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
satisfying the equation
x
2
=
5
y
+
3
z
x^2= 5^y+3^z
x
2
=
5
y
+
3
z
1
2
Hide problems
words that contain some palindrome
A word is a sequence of capital letters of our alphabet (that is, there are 26 possible letters). A word is called palindrome if has at least two letters and is spelled the same forward and backward. For example, the words "ARARA" e "NOON" are palindromes, but the words "ESMERALDA" and "A" are not palindromes. We say that a word
x
x
x
contains a word
y
y
y
if there are consecutive letters of
x
x
x
that together form the word
y
y
y
. For example, the word "ARARA" contains the word "RARA" and also the word "ARARA", but doesn't contain the word "ARRA". Compute the number of words of 14-letter that contain some palindrome.
a^2-bc=b^2-ac=c^2-ab=a^3+b^3+c^3; possible values of (a+b+c)
The nonzero real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
are such that:
a
2
−
b
c
=
b
2
−
a
c
=
c
2
−
a
b
=
a
3
+
b
3
+
c
3
a^2-bc= b^2-ac= c^2-ab= a^3+b^3+c^3
a
2
−
b
c
=
b
2
−
a
c
=
c
2
−
ab
=
a
3
+
b
3
+
c
3
. Compute the possible values of
a
+
b
+
c
a+b+c
a
+
b
+
c
.