MathDB
Problems
Contests
National and Regional Contests
Israel Contests
Israel Olympic Revenge
2019 Israel Olympic Revenge
2019 Israel Olympic Revenge
Part of
Israel Olympic Revenge
Subcontests
(5)
P2
1
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5779 dimensional polytopes
A
5779
5779
5779
-dimensional polytope is call a
k
k
k
-tope if it has exactly
k
k
k
5778
5778
5778
-dimensional faces. Find all sequences
b
5780
,
b
5781
,
…
,
b
11558
b_{5780}, b_{5781}, \dots, b_{11558}
b
5780
,
b
5781
,
…
,
b
11558
of nonnegative integers, not all
0
0
0
, such that the following condition holds:It is possible to tesselate every
5779
5779
5779
-dimensional polytope with convex
5779
5779
5779
-dimensional polytopes, such that the number of
k
k
k
-topes in the tessellation is proportional to
b
k
b_k
b
k
, while there are no
k
k
k
-topes in the tessellation if
k
∉
{
5780
,
5781
,
…
,
11558
}
k\notin \{5780, 5781, \dots, 11558\}
k
∈
/
{
5780
,
5781
,
…
,
11558
}
.
P1
1
Hide problems
Number of bijective polynomials,
A polynomial
P
P
P
in
n
n
n
variables and real coefficients is called magical if
P
(
N
n
)
⊂
N
P(\mathbb{N}^n)\subset \mathbb{N}
P
(
N
n
)
⊂
N
, and moreover the map
P
:
N
n
→
N
P: \mathbb{N}^n \to \mathbb{N}
P
:
N
n
→
N
is a bijection. Prove that for all positive integers
n
n
n
, there are at least
n
!
⋅
(
C
(
n
)
−
C
(
n
−
1
)
)
n!\cdot (C(n)-C(n-1))
n
!
⋅
(
C
(
n
)
−
C
(
n
−
1
))
magical polynomials, where
C
(
n
)
C(n)
C
(
n
)
is the
n
n
n
-th Catalan number.Here
N
=
{
0
,
1
,
2
,
…
}
\mathbb{N}=\{0,1,2,\dots\}
N
=
{
0
,
1
,
2
,
…
}
.
P4
1
Hide problems
Rugged functions.
Call a function
Z
>
0
→
Z
>
0
\mathbb Z_{>0}\rightarrow \mathbb Z_{>0}
Z
>
0
→
Z
>
0
\emph{M-rugged} if it is unbounded and satisfies the following two conditions:
(
1
)
(1)
(
1
)
If
f
(
n
)
∣
f
(
m
)
f(n)|f(m)
f
(
n
)
∣
f
(
m
)
and
f
(
n
)
<
f
(
m
)
f(n)<f(m)
f
(
n
)
<
f
(
m
)
then
n
∣
m
n|m
n
∣
m
.
(
2
)
(2)
(
2
)
∣
f
(
n
+
1
)
−
f
(
n
)
∣
≤
M
|f(n+1)-f(n)|\leq M
∣
f
(
n
+
1
)
−
f
(
n
)
∣
≤
M
. a. Find all
1
−
r
u
g
g
e
d
1-rugged
1
−
r
ugg
e
d
functions. b. Determine if the number of
2
−
r
u
g
g
e
d
2-rugged
2
−
r
ugg
e
d
functions is smaller than
2019
2019
2019
.
P3
1
Hide problems
Circumcenters form a circumscribed quadrilateral.
Let
A
B
C
D
ABCD
A
BC
D
be a circumscribed quadrilateral, assume
A
B
C
D
ABCD
A
BC
D
is not a kite. Denote the circumcenters of triangle
A
B
C
,
B
C
D
,
C
D
A
,
D
A
B
ABC,BCD,CDA,DAB
A
BC
,
BC
D
,
C
D
A
,
D
A
B
by
O
D
,
O
A
,
O
B
,
O
C
O_D,O_A,O_B,O_C
O
D
,
O
A
,
O
B
,
O
C
respectively. a. Prove that
O
A
O
B
O
C
O
D
O_AO_BO_CO_D
O
A
O
B
O
C
O
D
is circumscribed. b. Let the angle bisector of
∠
B
A
D
\angle BAD
∠
B
A
D
intersect the angle bisector of
∠
O
B
O
A
O
D
\angle O_BO_AO_D
∠
O
B
O
A
O
D
in
X
X
X
. Similarly define the points
Y
,
Z
,
W
Y,Z,W
Y
,
Z
,
W
. Denote the incenters of
A
B
C
D
,
O
A
O
B
O
C
O
D
ABCD, O_AO_BO_CO_D
A
BC
D
,
O
A
O
B
O
C
O
D
by
I
,
J
I,J
I
,
J
respectively. Express the angles
∠
Z
Y
J
,
∠
X
Y
I
\angle ZYJ,\angle XYI
∠
Z
Y
J
,
∠
X
Y
I
in terms of angles of quadrilateral
A
B
C
D
ABCD
A
BC
D
.
G
1
Hide problems
Tangent from midpoint to excircle.
Let
ω
\omega
ω
be the
A
A
A
-excircle of triangle
A
B
C
ABC
A
BC
and
M
M
M
the midpoint of side
B
C
BC
BC
.
G
G
G
is the pole of
A
M
AM
A
M
w.r.t
ω
\omega
ω
and
H
H
H
is the midpoint of segment
A
G
AG
A
G
. Prove that
M
H
MH
M
H
is tangent to
ω
\omega
ω
.