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Problems
Contests
National and Regional Contests
Kosovo Contests
Kosovo Team Selection Test
2024 Kosovo Team Selection Test
2024 Kosovo Team Selection Test
Part of
Kosovo Team Selection Test
Subcontests
(3)
P3
1
Hide problems
almost symmetric beautiful functional equation
Find all functions
f
:
R
→
R
f:\mathbb R\to\mathbb R
f
:
R
→
R
such that
(
x
−
y
)
f
(
x
+
y
)
−
(
x
+
y
)
f
(
x
−
y
)
=
2
y
(
f
(
x
)
−
f
(
y
)
−
1
)
(x-y)f(x+y) - (x+y)f(x-y) = 2y(f(x)-f(y) - 1)
(
x
−
y
)
f
(
x
+
y
)
−
(
x
+
y
)
f
(
x
−
y
)
=
2
y
(
f
(
x
)
−
f
(
y
)
−
1
)
for all
x
,
y
∈
R
x, y\in\mathbb R
x
,
y
∈
R
.
P2
1
Hide problems
Three concurrent chords in a circle
Let
ω
\omega
ω
be a circle and let
A
A
A
be a point lying outside of
ω
\omega
ω
. The tangents from
A
A
A
to
ω
\omega
ω
touch
ω
\omega
ω
at points
B
B
B
and
C
C
C
. Let
M
M
M
be the midpoint of
B
C
BC
BC
and let
D
D
D
a point on the side
B
C
BC
BC
different from
M
M
M
. The circle with diameter
A
D
AD
A
D
intersects
ω
\omega
ω
at points
X
X
X
and
Y
Y
Y
and the circumcircle of
△
A
B
C
\bigtriangleup ABC
△
A
BC
again at
E
E
E
. Prove that
A
D
AD
A
D
,
E
M
EM
EM
, and
X
Y
XY
X
Y
are concurrent.
P1
1
Hide problems
p^q +5q -2 being a prime
Find all prime numbers
p
p
p
and
q
q
q
such that
p
q
+
5
q
−
2
p^q + 5q - 2
p
q
+
5
q
−
2
is also a prime number.