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Contests
International Contests
Czech-Polish-Slovak Match
2004 Czech-Polish-Slovak Match
2004 Czech-Polish-Slovak Match
Part of
Czech-Polish-Slovak Match
Subcontests
(6)
6
1
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The number of stones in a heap is a square
On the table there are
k
≥
3
k \ge 3
k
≥
3
heaps of
1
,
2
,
…
,
k
1, 2, \dots , k
1
,
2
,
…
,
k
stones. In the first step, we choose any three of the heaps, merge them into a single new heap, and remove
1
1
1
stone from this new heap. Thereafter, in the
i
i
i
-th step (
i
≥
2
i \ge 2
i
≥
2
) we merge some three heaps containing more than
i
i
i
stones in total and remove
i
i
i
stones from the new heap. Assume that after a number of steps a single heap of
p
p
p
stones remains on the table. Show that the number
p
p
p
is a perfect square if and only if so are both
2
k
+
2
2k + 2
2
k
+
2
and
3
k
+
1
3k + 1
3
k
+
1
. Find the least
k
k
k
with this property.
5
1
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Triangles that share an orthocentre
Points
K
,
L
,
M
K,L,M
K
,
L
,
M
on the sides
A
B
,
B
C
,
C
A
AB,BC,CA
A
B
,
BC
,
C
A
respectively of a triangle
A
B
C
ABC
A
BC
satisfy
A
K
K
B
=
B
L
L
C
=
C
M
M
A
\frac{AK}{KB} = \frac{BL}{LC} = \frac{CM}{MA}
K
B
A
K
=
L
C
B
L
=
M
A
CM
. Show that the triangles
A
B
C
ABC
A
BC
and
K
L
M
KLM
K
L
M
have a common orthocenter if and only if
△
A
B
C
\triangle ABC
△
A
BC
is equilateral.
2
1
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Triples of primes with a divisibility property
Show that for each natural number
k
k
k
there exist only finitely many triples
(
p
,
q
,
r
)
(p, q, r)
(
p
,
q
,
r
)
of distinct primes for which
p
p
p
divides
q
r
−
k
qr-k
q
r
−
k
,
q
q
q
divides
p
r
−
k
pr-k
p
r
−
k
, and
r
r
r
divides
p
q
−
k
pq - k
pq
−
k
.
1
1
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An Equation Satisfied by the Coefficients of Polynomials
Show that real numbers,
p
,
q
,
r
p, q, r
p
,
q
,
r
satisfy the condition
p
4
(
q
−
r
)
2
+
2
p
2
(
q
+
r
)
+
1
=
p
4
p^4(q-r)^2 + 2p^2(q+r) + 1 = p^4
p
4
(
q
−
r
)
2
+
2
p
2
(
q
+
r
)
+
1
=
p
4
if and only if the quadratic equations
x
2
+
p
x
+
q
=
0
x^2 + px + q = 0
x
2
+
p
x
+
q
=
0
and
y
2
−
p
y
+
r
=
0
y^2 - py + r = 0
y
2
−
p
y
+
r
=
0
have real roots (not necessarily distinct) which can be labeled by
x
1
,
x
2
x_1,x_2
x
1
,
x
2
and
y
1
,
y
2
y_1,y_2
y
1
,
y
2
, respectively, in such a way that
x
1
y
1
−
x
2
y
2
=
1
x_1y_1 - x_2y_2 = 1
x
1
y
1
−
x
2
y
2
=
1
.
3
1
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Czech-Polish-Slovak 2004
A point P in the interior of a cyclic quadrilateral ABCD satisfies ∠BPC = ∠BAP + ∠PDC. Denote by E, F and G the feet of the perpendiculars from P to the lines AB, AD and DC, respectively. Show that the triangles FEG and PBC are similar.
4
1
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Solve System
Solve in real numbers the system of equations: \begin{align*} \frac{1}{xy}&=\frac{x}{z}+1 \\ \frac{1}{yz}&=\frac{y}{x}+1 \\ \frac{1}{zx}&=\frac{z}{y}+1 \\ \end{align*}