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Problems
Contests
National and Regional Contests
Israel Contests
Grosman Mathematical Olympiad
1995 Grosman Memorial Mathematical Olympiad
1995 Grosman Memorial Mathematical Olympiad
Part of
Grosman Mathematical Olympiad
Subcontests
(7)
6
1
Hide problems
f(q)= 1 + f(q/(1-2q))
(a) Prove that there is a unique function
f
:
Q
→
Q
f : Q \to Q
f
:
Q
→
Q
satisfying: (i)
f
(
q
)
=
1
+
f
(
q
1
−
2
q
)
f(q)= 1 + f\left(\frac{q}{1-2q}\right)
f
(
q
)
=
1
+
f
(
1
−
2
q
q
)
for
0
<
q
<
1
2
0<q< \frac12
0
<
q
<
2
1
(ii)
f
(
q
)
=
1
+
f
(
q
−
1
)
f(q)= 1 + f(q-1)
f
(
q
)
=
1
+
f
(
q
−
1
)
for
1
<
q
≤
2
1<q\le 2
1
<
q
≤
2
(iii)
f
(
q
)
f
(
1
q
)
=
1
f(q)f\left(\frac{1}{q}\right)=1
f
(
q
)
f
(
q
1
)
=
1
for all
q
∈
Q
+
q\in Q^+
q
∈
Q
+
(b) For this function
f
f
f
, find all
r
∈
Q
+
r\in Q^+
r
∈
Q
+
such that
f
(
r
)
=
r
f(r) = r
f
(
r
)
=
r
7
1
Hide problems
good real functions, related to lattice points
For a given positive integer
n
n
n
, let
A
n
A_n
A
n
be the set of all points
(
x
,
y
)
(x,y)
(
x
,
y
)
in the coordinate plane with
x
,
y
∈
{
0
,
1
,
.
.
.
,
n
}
x,y \in \{0,1,...,n\}
x
,
y
∈
{
0
,
1
,
...
,
n
}
. A point
(
i
,
j
)
(i, j)
(
i
,
j
)
is called internal if
0
<
i
,
j
<
n
0 < i, j < n
0
<
i
,
j
<
n
. A real function
f
f
f
, defined on
A
n
A_n
A
n
, is called good if it has the following property: For every internal point
x
x
x
, the value of
f
(
x
)
f(x)
f
(
x
)
is the arithmetic mean of its values on the four neighboring points (i.e. the points at the distance
1
1
1
from
x
x
x
). Prove that if
f
f
f
and
g
g
g
are good functions that coincide at the non-internal points of
A
n
A_n
A
n
, then
f
≡
g
f \equiv g
f
≡
g
.
5
1
Hide problems
equalizing planes in space
For non-coplanar points are given in space. A plane
π
\pi
π
is called equalizing if all four points have the same distance from
π
\pi
π
. Find the number of equilizing planes.
4
1
Hide problems
locus of centers of orthogonal circles to intersecting circles
Two given circles
α
\alpha
α
and
β
\beta
β
intersect each other at two points. Find the locus of the centers of all circles that are orthogonal to both
α
\alpha
α
and
β
\beta
β
.
3
1
Hide problems
2 thieves stole an open chain with 2k white and 2m black beads
Two thieves stole an open chain with
2
k
2k
2
k
white beads and
2
m
2m
2
m
black beads. They want to share the loot equally, by cutting the chain to pieces in such a way that each one gets
k
k
k
white beads and
m
m
m
black beads. What is the minimal number of cuts that is always sufficient?
2
1
Hide problems
player II can prevent player I from winning in a game on infinite square board
Two players play a game on an infinite board that consists of unit squares. Player
I
I
I
chooses a square and marks it with
O
O
O
. Then player
I
I
II
II
chooses another square and marks it with
X
X
X
. They play until one of the players marks a whole row or a whole column of five consecutive squares, and this player wins the game. If no player can achieve this, the result of the game is a tie. Show that player
I
I
II
II
can prevent player
I
I
I
from winning.
1
1
Hide problems
exist d_i and d_j among divisors of 1995 such that
Positive integers
d
1
,
d
2
,
.
.
.
,
d
n
d_1,d_2,...,d_n
d
1
,
d
2
,
...
,
d
n
are divisors of
1995
1995
1995
. Prove that there exist
d
i
d_i
d
i
and
d
j
d_j
d
j
among them, such the denominator of the reduced fraction
d
i
/
d
j
d_i/d_j
d
i
/
d
j
is at least
n
n
n