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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1979 Czech And Slovak Olympiad IIIA
1979 Czech And Slovak Olympiad IIIA
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
Hide problems
if natural number m, 1 < m < n, not divisible by n, then m is prime.
Find all natural numbers
n
n
n
,
n
<
1
0
7
n < 10^7
n
<
1
0
7
, for which: If natural number
m
m
m
,
1
<
m
<
n
1 < m < n
1
<
m
<
n
, is not divisible by
n
n
n
, then
m
m
m
is prime.
5
1
Hide problems
max line segment that bisects perimeter of triangle
Given a triangle
A
B
C
ABC
A
BC
with side sizes
a
≥
b
≥
c
a \ge b \ge c
a
≥
b
≥
c
. Among all pairs of points
X
,
Y
X, Y
X
,
Y
on the boundary of triangle
A
B
C
ABC
A
BC
, which this boundary divides into two parts of equal length, find all such for which the distance is
X
Y
X Y
X
Y
maximum.
4
1
Hide problems
\(sum x_i )^2 <= n sum_ x_i x_{n-i+1}
Let
n
n
n
be any natural number. Find all
n
n
n
-tuples of real numbers
x
1
≤
x
2
≤
.
.
.
≤
x
n
x_1\le x_2\le ... \le x_n
x
1
≤
x
2
≤
...
≤
x
n
, for which holds
(
∑
i
=
1
n
x
i
)
2
≤
n
∑
i
=
1
n
x
i
x
n
−
i
+
1
.
\left(\sum_{i=1}^n x_i\right)^2 \le n \sum_{i=1}^n x_i x_{n-i+1}.
(
i
=
1
∑
n
x
i
)
2
≤
n
i
=
1
∑
n
x
i
x
n
−
i
+
1
.
3
1
Hide problems
cyclic ABCD inscibed in unit circle is square if |AB| | x BC| x |CD| x |DA| >=4
If in a quadrilateral
A
B
C
D
ABCD
A
BC
D
whose vertices lie on a circle of radius
1
1
1
, holds
∣
A
B
∣
⋅
∣
B
C
∣
⋅
∣
C
D
∣
⋅
∣
D
A
∣
≥
4
|AB| \cdot |BC| \cdot |CD|\cdot |DA| \ge 4
∣
A
B
∣
⋅
∣
BC
∣
⋅
∣
C
D
∣
⋅
∣
D
A
∣
≥
4
, then
A
B
C
D
ABCD
A
BC
D
is a square. Prove it. [hide=Hint given in contest] You can use Ptolemy's formula
∣
A
B
∣
⋅
∣
C
D
∣
+
∣
B
C
∣
⋅
∣
A
D
∣
=
∣
A
C
∣
⋅
∣
B
D
∣
|AB| \cdot |CD| + |BC|\cdot |AD|= |AC| \cdot|BD|
∣
A
B
∣
⋅
∣
C
D
∣
+
∣
BC
∣
⋅
∣
A
D
∣
=
∣
A
C
∣
⋅
∣
B
D
∣
2
1
Hide problems
cube has common center with cuboid, min, common volumes
Given a cuboid
Q
Q
Q
with dimensions
a
,
b
,
c
a, b, c
a
,
b
,
c
,
a
<
b
<
c
a < b < c
a
<
b
<
c
. Find the length of the edge of a cube
K
K
K
, which has parallel faces and a common center with the given cuboid so that the volume of the difference of the sets
Q
∪
K
Q \cup K
Q
∪
K
and
Q
∩
K
Q \cap K
Q
∩
K
is minimal.
1
1
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x + 2y + 5z=10n diophantine
Let
n
n
n
be a given natural number. Determine the number of all orderer triples
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
of non-negative integers
x
,
y
,
z
x, y, z
x
,
y
,
z
that satisfy the equation
x
+
2
y
+
5
z
=
10
n
.
x + 2y + 5z=10n.
x
+
2
y
+
5
z
=
10
n
.