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Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2023 Greece National Olympiad
2023 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
4
1
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Find maximal N such that no two are pairs
A class consists of 26 students with two students sitting on each desk. Suddenly, the students decide to change seats, such that every two students that were previously sitting together are now apart. Find the maximum value of positive integer
N
N
N
such that, regardless of the students' sitting positions, at the end there is a set
S
S
S
consisting of
N
N
N
students satisfying the following property: every two of them have never been sitting together.
3
1
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Another "triangle BIC" configuration
A triangle
A
B
C
ABC
A
BC
with
A
B
>
A
C
AB>AC
A
B
>
A
C
is given,
A
D
AD
A
D
is the A-angle bisector with point
D
D
D
on
B
C
BC
BC
and point
I
I
I
is the incenter of triangle
A
B
C
ABC
A
BC
. Point M is the midpoint of segment
A
D
AD
A
D
and point
F
F
F
is the second intersection of
M
B
MB
MB
with the circumcirle of triangle
B
I
C
BIC
B
I
C
. Prove that
A
F
⊥
F
C
AF\bot FC
A
F
⊥
FC
.
2
1
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A classic NT but it can be generalised
Find all positive integers
N
N
N
that are perfect squares and their decimal representation consists of
n
n
n
digits equal to 2 and one digit equal to 5, where
n
n
n
takes positive integer values.
1
1
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A nice system of equations
Find all quadruplets (x, y, z, w) of positive real numbers that satisfy the following system:
{
x
y
z
+
1
x
+
1
=
y
z
w
+
1
y
+
1
=
z
w
x
+
1
z
+
1
=
w
x
y
+
1
w
+
1
x
+
y
+
z
+
w
=
48
\begin{cases} \frac{xyz+1}{x+1}= \frac{yzw+1}{y+1}= \frac{zwx+1}{z+1}= \frac{wxy+1}{w+1}\\ x+y+z+w= 48 \end{cases}
{
x
+
1
x
yz
+
1
=
y
+
1
yz
w
+
1
=
z
+
1
z
w
x
+
1
=
w
+
1
w
x
y
+
1
x
+
y
+
z
+
w
=
48