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Problems
Contests
National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2022 Greece Junior Math Olympiad
2022 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
4
1
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When is it a common divisor?
Find all couples of non-zero integers
(
x
,
y
)
(x,y)
(
x
,
y
)
such that,
x
2
+
y
2
x^2+y^2
x
2
+
y
2
is a common divisor of
x
5
+
y
x^5+y
x
5
+
y
and
y
5
+
x
y^5+x
y
5
+
x
.
3
1
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Numbers on the board
On the board we write a series of
n
n
n
numbers, where
n
≥
40
n \geq 40
n
≥
40
, and each one of them is equal to either
1
1
1
or
−
1
-1
−
1
, such that the following conditions both hold:(i) The sum of every
40
40
40
consecutive numbers is equal to
0
0
0
. (ii) The sum of every
42
42
42
consecutive numbers is not equal to
0
0
0
.We denote by
S
n
S_n
S
n
the sum of the
n
n
n
numbers of the board. Find the maximum possible value of
S
n
S_n
S
n
for all possible values of
n
n
n
.
2
1
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Let's play with angles
Let
A
B
C
ABC
A
BC
be an isosceles triangle, and point
D
D
D
in its interior such that
D
B
^
C
=
3
0
∘
,
D
B
^
A
=
5
0
∘
,
D
C
^
B
=
5
5
∘
D \hat{B} C=30^\circ, D \hat{B}A=50^\circ, D \hat{C}B=55^\circ
D
B
^
C
=
3
0
∘
,
D
B
^
A
=
5
0
∘
,
D
C
^
B
=
5
5
∘
(a) Prove that
B
^
=
C
^
=
8
0
∘
\hat B=\hat C=80^\circ
B
^
=
C
^
=
8
0
∘
. (b) Find the measure of the angle
D
A
^
C
D \hat{A} C
D
A
^
C
.
1
1
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Polynomial + inequality?
(a) Find the value of the real number
k
k
k
, for which the polynomial
P
(
x
)
=
x
3
−
k
x
+
2
P(x)=x^3-kx+2
P
(
x
)
=
x
3
−
k
x
+
2
has the number
2
2
2
as a root. In addition, for the value of
k
k
k
you will find, write this polynomial as the product of two polynomials with integer coefficients. (b) If the positive real numbers
a
,
b
a,b
a
,
b
satisfy the equation
2
a
+
b
+
4
a
b
=
10
,
2a+b+\frac{4}{ab}=10,
2
a
+
b
+
ab
4
=
10
,
find the maximum possible value of
a
a
a
.