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Problems
Contests
International Contests
Czech-Polish-Slovak Match
2024 CAPS Match
2024 CAPS Match
Part of
Czech-Polish-Slovak Match
Subcontests
(6)
6
1
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every prime p divides floor(a+b*sqrt(2024)^p)-c
Determine whether there exist infinitely many triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of positive integers such that every prime
p
p
p
divides
⌊
(
a
+
b
2024
)
p
⌋
−
c
.
\left\lfloor\left(a+b\sqrt{2024}\right)^p\right\rfloor-c.
⌊
(
a
+
b
2024
)
p
⌋
−
c
.
5
1
Hide problems
Parametrized functional equation
Let
α
≠
0
\alpha\neq0
α
=
0
be a real number. Determine all functions
f
:
R
→
R
f:\mathbb R\to\mathbb R
f
:
R
→
R
such that
f
(
x
2
+
y
2
)
=
f
(
x
−
y
)
f
(
x
+
y
)
+
α
y
f
(
y
)
f\left(x^2+y^2\right)=f(x-y)f(x+y)+\alpha yf(y)
f
(
x
2
+
y
2
)
=
f
(
x
−
y
)
f
(
x
+
y
)
+
α
y
f
(
y
)
holds for all
x
,
y
∈
R
.
x, y\in\mathbb R.
x
,
y
∈
R
.
4
1
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Quadrilateral with tree same sides
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral, such that
A
B
=
B
C
=
C
D
.
AB = BC = CD.
A
B
=
BC
=
C
D
.
There are points
X
,
Y
X, Y
X
,
Y
on rays
C
A
,
B
D
,
CA, BD,
C
A
,
B
D
,
respectively, such that
B
X
=
C
Y
.
BX = CY.
BX
=
C
Y
.
Let
P
,
Q
,
R
,
S
P, Q, R, S
P
,
Q
,
R
,
S
be the midpoints of segments
B
X
,
C
Y
,
BX, CY ,
BX
,
C
Y
,
X
D
,
Y
A
,
XD, YA,
X
D
,
Y
A
,
respectively. Prove that points
P
,
Q
,
R
,
S
P, Q, R, S
P
,
Q
,
R
,
S
lie on a circle.
3
1
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Incenter is a point of disc given by circumcircle
Let
A
B
C
ABC
A
BC
be a triangle and
D
D
D
a point on its side
B
C
.
BC.
BC
.
Points
E
,
F
E, F
E
,
F
lie on the lines
A
B
,
A
C
AB, AC
A
B
,
A
C
beyond vertices
B
,
C
,
B, C,
B
,
C
,
respectively, such that
B
E
=
B
D
BE = BD
BE
=
B
D
and
C
F
=
C
D
.
CF = CD.
CF
=
C
D
.
Let
P
P
P
be a point such that
D
D
D
is the incenter of triangle
P
E
F
.
P EF.
PEF
.
Prove that
P
P
P
lies inside the circumcircle
Ω
\Omega
Ω
of triangle
A
B
C
ABC
A
BC
or on it.
2
1
Hide problems
Sweet n-configurations
For a positive integer
n
n
n
, an
n
n
n
-configuration is a family of sets
⟨
A
i
,
j
⟩
1
≤
i
,
j
≤
n
.
\left\langle A_{i,j}\right\rangle_{1\le i,j\le n}.
⟨
A
i
,
j
⟩
1
≤
i
,
j
≤
n
.
An
n
n
n
-configuration is called sweet if for every pair of indices
(
i
,
j
)
(i, j)
(
i
,
j
)
with
1
≤
i
≤
n
−
1
1\le i\le n -1
1
≤
i
≤
n
−
1
and
1
≤
j
≤
n
1\le j\le n
1
≤
j
≤
n
we have
A
i
,
j
⊆
A
i
+
1
,
j
A_{i,j}\subseteq A_{i+1,j}
A
i
,
j
⊆
A
i
+
1
,
j
and
A
j
,
i
⊆
A
j
,
i
+
1
.
A_{j,i}\subseteq A_{j,i+1}.
A
j
,
i
⊆
A
j
,
i
+
1
.
Let
f
(
n
,
k
)
f(n, k)
f
(
n
,
k
)
denote the number of sweet
n
n
n
-configurations such that
A
n
,
n
⊆
{
1
,
2
,
…
,
k
}
A_{n,n}\subseteq \{1, 2,\ldots , k\}
A
n
,
n
⊆
{
1
,
2
,
…
,
k
}
. Determine which number is larger:
f
(
2024
,
202
4
2
)
f\left(2024, 2024^2\right)
f
(
2024
,
202
4
2
)
or
f
(
202
4
2
,
2024
)
.
f\left(2024^2, 2024\right).
f
(
202
4
2
,
2024
)
.
1
1
Hide problems
Set of fractions
Determine whether there exist 2024 distinct positive integers satisfying the following: if we consider every possible ratio between two distinct numbers (we include both
a
/
b
a/b
a
/
b
and
b
/
a
b/a
b
/
a
), we will obtain numbers with finite decimal expansions (after the decimal point) of mutually distinct non-zero lengths.