MathDB
Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2024 Czech and Slovak Olympiad III A
2024 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
5
1
Hide problems
a_{k+1} = 3a_k -[ 2a_k]- [ a_k ]
Let
(
a
k
)
k
=
0
∞
(a_k)^{\infty}_{k=0}
(
a
k
)
k
=
0
∞
be a sequence of real numbers such that if
k
k
k
is a non-negative integer, then
a
k
+
1
=
3
a
k
−
⌊
2
a
k
⌋
−
⌊
a
k
⌋
.
a_{k+1} = 3a_k - \lfloor 2a_k \rfloor - \lfloor a_k \rfloor.
a
k
+
1
=
3
a
k
−
⌊
2
a
k
⌋
−
⌊
a
k
⌋
.
Definitely all positive integers
n
n
n
such that if
a
0
=
1
/
n
a_0 = 1/n
a
0
=
1/
n
, then this sequence is constant after a certain term.
6
1
Hide problems
integer triangles wanted, with 2 congruent triangles inscribed with prime r
Find all right triangles with integer side lengths in which two congruent circles with prime radius can be inscribed such that they are externally tangent, both touch the hypotenuse, and each is tangent to another leg of the right triangle.
4
1
Hide problems
10 boys and 10 girls at the party
There were
10
10
10
boys and
10
10
10
girls at the party. Every boy likes a different 'positive' number of girls. Every girl likes a different positive number of boys. Define the largest non-negative integer
n
n
n
such that it is always possible to form at least
n
n
n
disjoint pairs in which both like the other.
3
1
Hide problems
tetraminoes in 20x20 grid
Find the largest natural number
n
n
n
such that any set of
n
n
n
tetraminoes, each of which is one of the four shapes in the picture, can be placed without overlapping in a
20
×
20
20 \times 20
20
×
20
table (no tetramino extends beyond the borders of the table), such that each tetramino covers exactly 4 cells of the 20x20 table. An individual tetramino is allowed to turn and flip at will. https://cdn.artofproblemsolving.com/attachments/b/9/0dddb25c2aa07536b711ded8363679e47972d6.png
2
1
Hide problems
OA=OB if <PAD = <ADP=< CBP =< PCB =< CPD
Let the interior point
P
P
P
of the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
be such that
∣
∠
P
A
D
∣
=
∣
∠
A
D
P
∣
=
∣
∠
C
B
P
∣
=
∣
∠
P
C
B
∣
=
∣
∠
C
P
D
∣
.
|\angle PAD| = |\angle ADP| = |\angle CBP| = |\angle PCB| = |\angle CPD|.
∣∠
P
A
D
∣
=
∣∠
A
D
P
∣
=
∣∠
CBP
∣
=
∣∠
PCB
∣
=
∣∠
CP
D
∣.
Let
O
O
O
be the center of the circumcircle of the triangle
C
P
D
CPD
CP
D
. Prove that
∣
O
A
∣
=
∣
O
B
∣
|OA| = |OB|
∣
O
A
∣
=
∣
OB
∣
.
1
1
Hide problems
gcd(a,b) x lcm(b,c), gcd(b,c)x lcm(c,a), ,gcd(c,a) x lcm(a,b)
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive integers such that one of the values
g
c
d
(
a
,
b
)
⋅
l
c
m
(
b
,
c
)
,
g
c
d
(
b
,
c
)
⋅
l
c
m
(
c
,
a
)
,
g
c
d
(
c
,
a
)
−
⋅
l
c
m
(
a
,
b
)
gcd(a,b) \cdot lcm(b,c), \,\,\,\, gcd(b,c)\cdot lcm(c,a), \,\,\,\, gcd(c,a)-\cdot lcm(a,b)
g
c
d
(
a
,
b
)
⋅
l
c
m
(
b
,
c
)
,
g
c
d
(
b
,
c
)
⋅
l
c
m
(
c
,
a
)
,
g
c
d
(
c
,
a
)
−
⋅
l
c
m
(
a
,
b
)
is equal to the product of the remaining two. Prove that one of the numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
is a multiple of another of them.