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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2017 Czech And Slovak Olympiad III A
2017 Czech And Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
1
1
Hide problems
100 diamonds, 50 genuine and 50 false, expert tells a number in every 2
There are
100
100
100
diamonds on the pile,
50
50
50
of which are genuine and
50
50
50
false. We invited a peculiar expert who alone can recognize which are which. Every time we show him some three diamonds, he would pick two and tell (truthfully) how many of them are genuine . Decide whether we can surely detect all genuine diamonds regardless how the expert chooses the pairs to be considered.
6
1
Hide problems
k =\frac{x^2 - xy + 2y^2}{x + y} has odd no of ordered integer pairs when 7/k
Given is a nonzero integer
k
k
k
. Prove that equation
k
=
x
2
−
x
y
+
2
y
2
x
+
y
k =\frac{x^2 - xy + 2y^2}{x + y}
k
=
x
+
y
x
2
−
x
y
+
2
y
2
has an odd number of ordered integer pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
just when
k
k
k
is divisible by seven.
5
1
Hide problems
circumcircle passed through arc midpoint of other circumcircle
Given is the acute triangle
A
B
C
ABC
A
BC
with the intersection of altitudes
H
H
H
. The angle bisector of angle
B
H
C
BHC
B
H
C
intersects side
B
C
BC
BC
at point
D
D
D
. Mark
E
E
E
and
F
F
F
the symmetrics of the point
D
D
D
wrt lines
A
B
AB
A
B
and
A
C
AC
A
C
. Prove that the circle circumscribed around the triangle
A
E
F
AEF
A
EF
passes through the midpoint of the arc
B
A
C
BAC
B
A
C
4
1
Hide problems
number sections of a number with only 0s and 1s, total sum wanted
For each sequence of
n
n
n
zeros and
n
n
n
units, we assign a number that is a number sections of the same digits in it. (For example, sequence
00111001
00111001
00111001
has
4
4
4
such sections
00
,
111
,
00
,
1
00, 111,00, 1
00
,
111
,
00
,
1
.) For a given
n
n
n
we sum up all the numbers assigned to each such sequence. Prove that the sum total is equal to
(
n
+
1
)
(
2
n
n
)
(n+1){2n \choose n}
(
n
+
1
)
(
n
2
n
)
3
1
Hide problems
f(y - xy) = f(x)y + (x - 1)^2 f(y)
Find all functions
f
:
R
→
R
f: R \to R
f
:
R
→
R
such that for all real numbers
x
,
y
x, y
x
,
y
holds
f
(
y
−
x
y
)
=
f
(
x
)
y
+
(
x
−
1
)
2
f
(
y
)
f(y - xy) = f(x)y + (x - 1)^2 f(y)
f
(
y
−
x
y
)
=
f
(
x
)
y
+
(
x
−
1
)
2
f
(
y
)
2
1
Hide problems
find k,l such that ka^2 + lb^2> c^2 for every sidelengths a,b,c
Find all pairs of real numbers
k
,
l
k, l
k
,
l
such that inequality
k
a
2
+
l
b
2
>
c
2
ka^2 + lb^2> c^2
k
a
2
+
l
b
2
>
c
2
applies to the lengths of sides
a
,
b
,
c
a, b, c
a
,
b
,
c
of any triangle.