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Problems
Contests
National and Regional Contests
Israel Contests
Israel Olympic Revenge
2022 Israel Olympic Revenge
2022 Israel Olympic Revenge
Part of
Israel Olympic Revenge
Subcontests
(4)
4
1
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Ellipsoid problem
A (not necessarily regular) tetrahedron
A
1
A
2
A
3
A
4
A_1A_2A_3A_4
A
1
A
2
A
3
A
4
is given in space. For each pair of indices
1
≤
i
<
j
≤
4
1\leq i<j\leq 4
1
≤
i
<
j
≤
4
, an ellipsoid with foci
A
i
,
A
j
A_i,A_j
A
i
,
A
j
and string length
ℓ
i
j
\ell_{ij}
ℓ
ij
, for positive numbers
ℓ
i
j
\ell_{ij}
ℓ
ij
, is given (in all 6 ellipsoids were built).For each
i
=
1
,
2
i=1,2
i
=
1
,
2
, a pair of points
X
i
≠
X
i
′
X_i\neq X'_i
X
i
=
X
i
′
was chosen so that
X
i
,
X
i
′
X_i, X'_i
X
i
,
X
i
′
both belong to all three ellipsoids with
A
i
A_i
A
i
as one of their foci. Prove that the lines
X
1
X
1
′
,
X
2
X
2
′
X_1X'_1, X_2X'_2
X
1
X
1
′
,
X
2
X
2
′
share a point in space if and only if
ℓ
13
+
ℓ
24
=
ℓ
14
+
ℓ
23
\ell_{13}+\ell_{24}=\ell_{14}+\ell_{23}
ℓ
13
+
ℓ
24
=
ℓ
14
+
ℓ
23
Remark: An ellipsoid with foci
P
,
Q
P,Q
P
,
Q
and string length
ℓ
>
∣
P
Q
∣
\ell>|PQ|
ℓ
>
∣
PQ
∣
is defined here as the set of points
X
X
X
in space for which
∣
X
Q
∣
+
∣
X
P
∣
=
ℓ
|XQ|+|XP|=\ell
∣
XQ
∣
+
∣
XP
∣
=
ℓ
.
3
1
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Approximating with Egyptian fractions
Determine if there exist positive real numbers
x
,
α
x, \alpha
x
,
α
, so that for any non-empty finite set of positive integers
S
S
S
, the inequality
∣
x
−
∑
s
∈
S
1
s
∣
>
1
max
(
S
)
α
\left|x-\sum_{s\in S}\frac{1}{s}\right|>\frac{1}{\max(S)^\alpha}
x
−
s
∈
S
∑
s
1
>
max
(
S
)
α
1
holds, where
max
(
S
)
\max(S)
max
(
S
)
is defined as the maximum element of the finite set
S
S
S
.
2
1
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Strong triples
A triple
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of positive integers is called strong if the following holds: for each integer
m
>
1
m>1
m
>
1
, the number
a
+
b
+
c
a+b+c
a
+
b
+
c
does not divide
a
m
+
b
m
+
c
m
a^m+b^m+c^m
a
m
+
b
m
+
c
m
. The sum of a strong triple
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
is defined as
a
+
b
+
c
a+b+c
a
+
b
+
c
.Prove that there exists an infinite collection of strong triples, the sums of which are all pairwise coprime.
1
1
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Dissecting square into n+1 rectangles
For each positive integer
n
n
n
, decide whether it is possible to tile a square with exactly
n
+
1
n+1
n
+
1
similar rectangles, each with a positive area and aspect ratio
1
:
n
1:n
1
:
n
.