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Problems
Contests
National and Regional Contests
Israel Contests
Israel Olympic Revenge
2023 Israel Olympic Revenge
2023 Israel Olympic Revenge
Part of
Israel Olympic Revenge
Subcontests
(4)
P4
1
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Integer averages implies constant
Let
c
c
c
be a positive real and
a
1
,
a
2
,
…
a_1, a_2, \dots
a
1
,
a
2
,
…
be a sequence of nonnegative integers satisfying the following conditions for every positive integer
n
n
n
:(i)
2
a
1
+
2
a
2
+
⋯
+
2
a
n
n
\frac{2^{a_1}+2^{a_2}+\cdots+2^{a_n}}{n}
n
2
a
1
+
2
a
2
+
⋯
+
2
a
n
is an integer;(ii)\textbullet 2^{a_n}\leq cn.Prove that the sequence
a
1
,
a
2
,
…
a_1, a_2, \dots
a
1
,
a
2
,
…
is eventually constant.
P3
1
Hide problems
Same equation with different condition
Find all (weakly) increasing
f
:
R
→
R
f\colon \mathbb{R}\to \mathbb{R}
f
:
R
→
R
for which
f
(
f
(
x
)
+
y
)
=
f
(
f
(
y
)
+
x
)
f(f(x)+y)=f(f(y)+x)
f
(
f
(
x
)
+
y
)
=
f
(
f
(
y
)
+
x
)
holds for all
x
,
y
∈
R
x, y\in \mathbb{R}
x
,
y
∈
R
.
P2
1
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UV is parallel to BC, weird mixtilinear condition
Triangle
Δ
A
B
C
\Delta ABC
Δ
A
BC
is inscribed in circle
Ω
\Omega
Ω
. The tangency point of
Ω
\Omega
Ω
and the
A
A
A
-mixtilinear circle of
Δ
A
B
C
\Delta ABC
Δ
A
BC
is
T
T
T
. Points
E
E
E
,
F
F
F
were chosen on
A
C
AC
A
C
,
A
B
AB
A
B
respectively so that
E
F
∥
B
C
EF\parallel BC
EF
∥
BC
and
(
T
E
F
)
(TEF)
(
TEF
)
is tangent to
Ω
\Omega
Ω
. Let
ω
\omega
ω
denote the
A
A
A
-excircle of
Δ
A
E
F
\Delta AEF
Δ
A
EF
, which is tangent to sides
E
F
EF
EF
,
A
E
AE
A
E
,
A
F
AF
A
F
at
K
K
K
,
Y
Y
Y
,
Z
Z
Z
respectively. Line
A
T
AT
A
T
intersects
ω
\omega
ω
at two points
P
P
P
,
Q
Q
Q
with
P
P
P
between
A
A
A
and
Q
Q
Q
. Let
Q
K
QK
Q
K
and
Y
Z
YZ
Y
Z
intersect at
V
V
V
, and let the tangent to
ω
\omega
ω
at
P
P
P
and the tangent to
Ω
\Omega
Ω
at
T
T
T
intersect at
U
U
U
. Prove that
U
V
∥
B
C
UV\parallel BC
U
V
∥
BC
.
P1
1
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Apples on a tree
Armadillo and Badger are playing a game. Armadillo chooses a nonempty tree (a simple acyclic graph) and places apples at some of its vertices (there may be several apples on a single vertex). First, Badger picks a vertex
v
0
v_0
v
0
and eats all its apples. Next, Armadillo and Badger take turns alternatingly, with Armadillo starting. On the
i
i
i
-th turn the animal whose turn it is picks a vertex
v
i
v_i
v
i
adjacent to
v
i
−
1
v_{i-1}
v
i
−
1
that hasn't been picked before and eats all its apples. The game ends when there is no such vertex
v
i
v_i
v
i
.Armadillo's goal is to have eaten more apples than Badger once the game ends. Can she fulfill her wish?