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Contests
International Contests
Czech-Polish-Slovak Match
1995 Czech and Slovak Match
1995 Czech and Slovak Match
Part of
Czech-Polish-Slovak Match
Subcontests
(6)
2
1
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f ,g : Z -> Z, f (g(x)+y) = g( f (y)+x) , g 1-1
Find all pairs of functions
f
,
g
:
Z
→
Z
f ,g : Z \rightarrow Z
f
,
g
:
Z
→
Z
that satisfy
f
(
g
(
x
)
+
y
)
=
g
(
f
(
y
)
+
x
)
f (g(x)+y) = g( f (y)+x)
f
(
g
(
x
)
+
y
)
=
g
(
f
(
y
)
+
x
)
for all integers
x
,
y
x,y
x
,
y
and such that
g
(
x
)
=
g
(
y
)
g(x) = g(y)
g
(
x
)
=
g
(
y
)
only if
x
=
y
x = y
x
=
y
.
3
1
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maximum ratio in triangle with lattice points as vertices
Consider all triangles
A
B
C
ABC
A
BC
in the cartesian plane whose vertices are at lattice points (i.e. with integer coordinates) and which contain exactly one lattice point (to be denoted
P
P
P
) in its interior. Let the line
A
P
AP
A
P
meet
B
C
BC
BC
at
E
E
E
. Determine the maximum possible value of the ratio
A
P
P
E
\frac{AP}{PE}
PE
A
P
.
4
1
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minimum (x+y) when (x+\sqrt{1+x^2})(y+\sqrt{1+y^2}) = p
For each real number
p
>
1
p > 1
p
>
1
, find the minimum possible value of the sum
x
+
y
x+y
x
+
y
, where the numbers
x
x
x
and
y
y
y
satisfy the equation
(
x
+
1
+
x
2
)
(
y
+
1
+
y
2
)
=
p
(x+\sqrt{1+x^2})(y+\sqrt{1+y^2}) = p
(
x
+
1
+
x
2
)
(
y
+
1
+
y
2
)
=
p
.
5
1
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reflections of intersection wrt sides ABCD are concyclic
The diagonals of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
are orthogonal and intersect at point
E
E
E
. Prove that the reflections of
E
E
E
in the sides of quadrilateral
A
B
C
D
ABCD
A
BC
D
lie on a circle.
6
1
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Hard Algebra
Find all triples
(
x
;
y
;
p
)
(x; y; p)
(
x
;
y
;
p
)
of two non-negative integers
x
,
y
x, y
x
,
y
and a prime number p such that
p
x
−
y
p
=
1
p^x-y^p=1
p
x
−
y
p
=
1
1
1
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Czech-Slovak 1995!
Let a_1\equal{}2, a_2\equal{}5 and a_{n\plus{}2}\equal{}(2\minus{}n^2)a_{n\plus{}1}\plus{} (2\plus{}n^2)a_n for
n
≥
1
n\geq 1
n
≥
1
. Do there exist
p
,
q
,
r
p,q,r
p
,
q
,
r
so that a_pa_q \equal{}a_r?