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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1952 Czech and Slovak Olympiad III A
1952 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(4)
2
1
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Triangular table
Consider a triangular table of positive integers \begin{matrix} & & & a_{11} & a_{12} & a_{13} & & & \\ & & a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & & \\ & a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} & a_{37} & \\ \iddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{matrix} The first row consists of odd numbers only. For
i
>
1
,
j
≥
1
i>1,j\ge1
i
>
1
,
j
≥
1
we have
a
i
j
=
a
i
−
1
,
j
−
2
+
a
i
−
1
,
j
−
1
+
a
i
−
1
,
j
.
a_{ij}=a_{i-1,j-2}+a_{i-1,j-1}+a_{i-1,j}.
a
ij
=
a
i
−
1
,
j
−
2
+
a
i
−
1
,
j
−
1
+
a
i
−
1
,
j
.
If we get out of range with the second index, we consider such
a
a
a
to be zero (eg.
a
22
=
0
+
a
11
+
a
12
a_{22}=0+a_{11}+a_{12}
a
22
=
0
+
a
11
+
a
12
and
a
37
=
a
25
+
0
+
0
a_{37}=a_{25}+0+0
a
37
=
a
25
+
0
+
0
). Show that for every
i
>
1
i>1
i
>
1
there is
j
∈
{
1
,
…
,
2
i
+
1
}
j\in\{1,\ldots,2i+1\}
j
∈
{
1
,
…
,
2
i
+
1
}
such that
a
i
j
a_{ij}
a
ij
is even.
4
1
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Intersection counting
Let
p
,
q
p,q
p
,
q
be positive integers. Consider a rectangle
A
B
C
D
ABCD
A
BC
D
with lengths of sides
p
p
p
and
q
q
q
that consists of
p
q
pq
pq
unital squares. How many of these squares are crossed by diagonal
A
C
AC
A
C
?
3
1
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Parallel rays in a quadrilateral
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with
A
B
=
C
D
AB=CD
A
B
=
C
D
. Let
R
,
S
R,S
R
,
S
be midpoints of sides
A
D
,
B
C
AD,BC
A
D
,
BC
respectively. Consider rays
A
U
,
D
V
AU, DV
A
U
,
D
V
parallel with ray
R
S
RS
RS
and all of them point in the same direction. Show that
∠
B
A
U
=
∠
C
D
V
\angle BAU=\angle CDV
∠
B
A
U
=
∠
C
D
V
.
1
1
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Rational square roots
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive rational numbers such that
a
+
b
=
c
\sqrt a+\sqrt b=c
a
+
b
=
c
. Show that
a
\sqrt a
a
and
b
\sqrt b
b
are also rational.