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Problems
Contests
International Contests
Hungary-Israel Binational
1998 Hungary-Israel Binational
1998 Hungary-Israel Binational
Part of
Hungary-Israel Binational
Subcontests
(3)
3
2
Hide problems
trinomial and consecutive natural numbers
Let
a
,
b
,
c
,
m
,
n
a, b, c, m, n
a
,
b
,
c
,
m
,
n
be positive integers. Consider the trinomial
f
(
x
)
=
a
x
2
+
b
x
+
c
f (x) = ax^{2}+bx+c
f
(
x
)
=
a
x
2
+
b
x
+
c
. Show that there exist
n
n
n
consecutive natural numbers
a
1
,
a
2
,
.
.
.
,
a
n
a_{1}, a_{2}, . . . , a_{n}
a
1
,
a
2
,
...
,
a
n
such that each of the numbers
f
(
a
1
)
,
f
(
a
2
)
,
.
.
.
,
f
(
a
n
)
f (a_{1}), f (a_{2}), . . . , f (a_{n})
f
(
a
1
)
,
f
(
a
2
)
,
...
,
f
(
a
n
)
has at least
m
m
m
different prime factors.
partitions of n into a sum of positive integers
Let
n
n
n
be a positive integer. We consider the set
P
P
P
of all partitions of
n
n
n
into a sum of positive integers (the order is irrelevant). For every partition
α
\alpha
α
, let
a
k
(
α
)
a_{k}(\alpha)
a
k
(
α
)
be the number of summands in
α
\alpha
α
that are equal to
k
,
k
=
1
,
2
,
.
.
.
,
n
.
k, k = 1,2,...,n.
k
,
k
=
1
,
2
,
...
,
n
.
Prove that
∑
α
∈
P
1
1
a
1
(
α
)
a
1
(
α
)
!
⋅
2
a
2
(
α
)
a
2
(
α
)
!
.
.
.
n
a
n
(
α
)
a
n
(
α
)
!
=
1.
\sum_{\alpha\in P}\frac{1}{1^{a_{1}(\alpha)}a_{1}(\alpha)!\cdot 2^{a_{2}(\alpha)}a_{2}(\alpha)!...n^{a_{n}(\alpha)}a_{n}(\alpha)!}=1.
∑
α
∈
P
1
a
1
(
α
)
a
1
(
α
)!
⋅
2
a
2
(
α
)
a
2
(
α
)!
...
n
a
n
(
α
)
a
n
(
α
)!
1
=
1.
2
2
Hide problems
inequality on radius
A triangle ABC is inscribed in a circle with center
O
O
O
and radius
R
R
R
. If the inradii of the triangles
O
B
C
,
O
C
A
,
O
A
B
OBC, OCA, OAB
OBC
,
OC
A
,
O
A
B
are
r
1
,
r
2
,
r
3
r_{1}, r_{2}, r_{3}
r
1
,
r
2
,
r
3
, respectively, prove that
1
r
1
+
1
r
2
+
1
r
3
≥
4
3
+
6
R
.
\frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}\geq\frac{4\sqrt{3}+6}{R}.
r
1
1
+
r
2
1
+
r
3
1
≥
R
4
3
+
6
.
triangles are constructd in exterior of a convex hexagon
On the sides of a convex hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
, equilateral triangles are constructd in its exterior. Prove that the third vertices of these six triangles are vertices of a regular hexagon if and only if the initial hexagon is affine regular. (A hexagon is called affine regular if it is centrally symmetric and any two opposite sides are parallel to the diagonal determine by the remaining two vertices.)
1
2
Hide problems
solve 5^x - 3^y = 16
Find all positive integers
x
x
x
and
y
y
y
such that
5
x
−
3
y
=
16
5^{x}-3^{y}= 16
5
x
−
3
y
=
16
.
a game
A player is playing the following game. In each turn he flips a coin and guesses the outcome. If his guess is correct, he gains
1
1
1
point; otherwise he loses all his points. Initially the player has no points, and plays the game until he has
2
2
2
points. (a) Find the probability
p
n
p_{n}
p
n
that the game ends after exactly
n
n
n
flips. (b) What is the expected number of flips needed to finish the game?