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Problems
Contests
National and Regional Contests
Israel Contests
Israel Olympic Revenge
2020 Israel Olympic Revenge
2020 Israel Olympic Revenge
Part of
Israel Olympic Revenge
Subcontests
(6)
P3
1
Hide problems
Bounds on degree of polynomials
For each positive integer
n
n
n
, define
f
(
n
)
f(n)
f
(
n
)
to be the least positive integer for which the following holds:For any partition of
{
1
,
2
,
…
,
n
}
\{1,2,\dots, n\}
{
1
,
2
,
…
,
n
}
into
k
>
1
k>1
k
>
1
disjoint subsets
A
1
,
…
,
A
k
A_1, \dots, A_k
A
1
,
…
,
A
k
, all of the same size, let
P
i
(
x
)
=
∏
a
∈
A
i
(
x
−
a
)
P_i(x)=\prod_{a\in A_i}(x-a)
P
i
(
x
)
=
∏
a
∈
A
i
(
x
−
a
)
. Then there exist
i
≠
j
i\neq j
i
=
j
for which
deg
(
P
i
(
x
)
−
P
j
(
x
)
)
≥
n
k
−
f
(
n
)
\deg(P_i(x)-P_j(x))\geq \frac{n}{k}-f(n)
de
g
(
P
i
(
x
)
−
P
j
(
x
))
≥
k
n
−
f
(
n
)
a) Prove that there is a constant
c
c
c
so that
f
(
n
)
≤
c
⋅
n
f(n)\le c\cdot \sqrt{n}
f
(
n
)
≤
c
⋅
n
for all
n
n
n
.b) Prove that for infinitely many
n
n
n
, one has
f
(
n
)
≥
ln
(
n
)
f(n)\ge \ln(n)
f
(
n
)
≥
ln
(
n
)
.
P2
1
Hide problems
Mutually repulsive sets
Let
A
,
B
⊂
Z
A, B\subset \mathbb{Z}
A
,
B
⊂
Z
be two sets of integers. We say that
A
,
B
A,B
A
,
B
are mutually repulsive if there exist positive integers
m
,
n
m,n
m
,
n
and two sequences of integers
α
1
,
α
2
,
…
,
α
n
\alpha_1, \alpha_2, \dots, \alpha_n
α
1
,
α
2
,
…
,
α
n
and
β
1
,
β
2
,
…
,
β
m
\beta_1, \beta_2, \dots, \beta_m
β
1
,
β
2
,
…
,
β
m
, for which there is a unique integer
x
x
x
such that the number of its appearances in the sequence of sets
A
+
α
1
,
A
+
α
2
,
…
,
A
+
α
n
A+\alpha_1, A+\alpha_2, \dots, A+\alpha_n
A
+
α
1
,
A
+
α
2
,
…
,
A
+
α
n
is different than the number of its appearances in the sequence of sets
B
+
β
1
,
…
,
B
+
β
m
B+\beta_1, \dots, B+\beta_m
B
+
β
1
,
…
,
B
+
β
m
.For a given quadruple of positive integers
(
n
1
,
d
1
,
n
2
,
d
2
)
(n_1,d_1, n_2, d_2)
(
n
1
,
d
1
,
n
2
,
d
2
)
, determine whether the sets
A
=
{
d
1
,
2
d
1
,
…
,
n
1
d
1
}
A=\{d_1, 2d_1, \dots, n_1d_1\}
A
=
{
d
1
,
2
d
1
,
…
,
n
1
d
1
}
B
=
{
d
2
,
2
d
2
,
…
,
n
2
d
2
}
B=\{d_2, 2d_2, \dots, n_2d_2\}
B
=
{
d
2
,
2
d
2
,
…
,
n
2
d
2
}
are mutually repulsive.For a set
X
⊂
Z
X\subset \mathbb{Z}
X
⊂
Z
and
c
∈
Z
c\in \mathbb{Z}
c
∈
Z
, we define
X
+
c
=
{
x
+
c
∣
x
∈
X
}
X+c=\{x+c\mid x\in X\}
X
+
c
=
{
x
+
c
∣
x
∈
X
}
.
P1
1
Hide problems
Finite preimages, symmetry
Find all functions
f
:
R
→
R
f:\mathbb{R}\to \mathbb{R}
f
:
R
→
R
such that for all
x
,
y
∈
R
x,y\in \mathbb{R}
x
,
y
∈
R
one has
f
(
f
(
x
)
+
y
)
=
f
(
x
+
f
(
y
)
)
f(f(x)+y)=f(x+f(y))
f
(
f
(
x
)
+
y
)
=
f
(
x
+
f
(
y
))
and in addition the set
f
−
1
(
a
)
=
{
b
∈
R
∣
f
(
b
)
=
a
}
f^{-1}(a)=\{b\in \mathbb{R}\mid f(b)=a\}
f
−
1
(
a
)
=
{
b
∈
R
∣
f
(
b
)
=
a
}
is a finite set for all
a
∈
R
a\in \mathbb{R}
a
∈
R
.
N
1
Hide problems
Divisibility sequence equals identity infinitely many times.
Let
a
1
,
a
2
,
a
3
,
.
.
.
a_1,a_2,a_3,...
a
1
,
a
2
,
a
3
,
...
be an infinite sequence of positive integers. Suppose that a sequence
a
1
,
a
2
,
…
a_1,a_2,\ldots
a
1
,
a
2
,
…
of positive integers satisfies
a
1
=
1
a_1=1
a
1
=
1
and
a
n
=
∑
n
≠
d
∣
n
a
d
a_{n}=\sum_{n\neq d|n}a_d
a
n
=
n
=
d
∣
n
∑
a
d
for every integer
n
>
1
n>1
n
>
1
. Prove that the exist infinitely many integers
k
k
k
such that
a
k
=
k
a_k=k
a
k
=
k
.
G
1
Hide problems
Condition on right angle involving isodynamic points.
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
≠
A
C
AB\neq AC
A
B
=
A
C
. The angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
intersects with
B
C
BC
BC
at a point
D
D
D
.
B
E
,
C
F
BE,CF
BE
,
CF
are the altitudes of the triangle and
A
p
1
,
A
p
2
Ap_1,Ap_2
A
p
1
,
A
p
2
are the isodynamic points of triangle
A
B
C
ABC
A
BC
.Let the
A
A
A
-median of
A
B
C
ABC
A
BC
intersect
E
F
EF
EF
at
T
T
T
. Show that the line connecting
T
T
T
with the nine-point center of
A
B
C
ABC
A
BC
is perpendicular to
B
C
BC
BC
if and only if
∠
A
p
1
D
A
p
2
=
9
0
∘
\angle Ap_1DAp_2=90^\circ
∠
A
p
1
D
A
p
2
=
9
0
∘
.
P4
1
Hide problems
Many Feuerbach points and similar quads
Original post by shalomrav, but for some reason the mods locked the problem without any solves :noo:Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral inscribed in circle
Ω
\Omega
Ω
. Let
F
A
F_A
F
A
be the (associated with
Ω
\Omega
Ω
) Feuerbach point of the triangle formed by the tangents to
Ω
\Omega
Ω
at
B
,
C
,
D
B,C,D
B
,
C
,
D
, that is, the point of tangency of
Ω
\Omega
Ω
and the nine-point circle of that triangle. Define
F
B
,
F
C
,
F
D
F_B, F_C, F_D
F
B
,
F
C
,
F
D
similarly. Let
A
′
A'
A
′
be the intersection of the tangents to
Ω
\Omega
Ω
at
A
A
A
and
F
A
F_A
F
A
. Define
B
′
,
C
′
,
D
′
B', C', D'
B
′
,
C
′
,
D
′
similarly. Prove that quadrilaterals
A
B
C
D
ABCD
A
BC
D
and
A
′
B
′
C
′
D
′
A'B'C'D'
A
′
B
′
C
′
D
′
are similar