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International Contests
Czech-Polish-Slovak Match
2021 Czech-Austrian-Polish-Slovak Match
2021 Czech-Austrian-Polish-Slovak Match
Part of
Czech-Polish-Slovak Match
Subcontests
(6)
6
1
Hide problems
So many circles
Let
A
B
C
ABC
A
BC
be an acute triangle and suppose points
A
,
A
b
,
B
a
,
B
,
B
c
,
C
b
,
C
,
C
a
,
A, A_b, B_a, B, B_c, C_b, C, C_a,
A
,
A
b
,
B
a
,
B
,
B
c
,
C
b
,
C
,
C
a
,
and
A
c
A_c
A
c
lie on its perimeter in this order. Let
A
1
≠
A
A_1 \neq A
A
1
=
A
be the second intersection point of the circumcircles of triangles
A
A
b
C
a
AA_bC_a
A
A
b
C
a
and
A
A
c
B
a
AA_cB_a
A
A
c
B
a
. Analogously,
B
1
≠
B
B_1 \neq B
B
1
=
B
is the second intersection point of the circumcircles of triangles
B
B
c
A
b
BB_cA_b
B
B
c
A
b
and
B
B
a
C
b
BB_aC_b
B
B
a
C
b
, and
C
1
≠
C
C_1 \neq C
C
1
=
C
is the second intersection point of the circumcircles of triangles
C
C
a
B
c
CC_aB_c
C
C
a
B
c
and
C
C
b
A
c
CC_bA_c
C
C
b
A
c
. Suppose that the points
A
1
,
B
1
,
A_1, B_1,
A
1
,
B
1
,
and
C
1
C_1
C
1
are all distinct, lie inside the triangle
A
B
C
ABC
A
BC
, and do not lie on a single line. Prove that lines
A
A
1
,
B
B
1
,
C
C
1
,
AA_1, BB_1, CC_1,
A
A
1
,
B
B
1
,
C
C
1
,
and the circumcircle of triangle
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
all pass through a common point.Josef Tkadlec (Czech Republic), Patrik Bak (Slovakia)
5
1
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Interesting sequence with \sqrt{2}
The sequence
a
1
,
a
2
,
a
3
,
…
a_1, a_2, a_3, \ldots
a
1
,
a
2
,
a
3
,
…
satisfies
a
1
=
1
a_1=1
a
1
=
1
, and for all
n
≥
2
n \ge 2
n
≥
2
, it holds that
a
n
=
{
a
n
−
1
+
3
if
n
−
1
∈
{
a
1
,
a
2
,
…
,
,
a
n
−
1
}
;
a
n
−
1
+
2
otherwise
.
a_n= \begin{cases} a_{n-1}+3 ~~ \text{if} ~ n-1 \in \{ a_1,a_2,\ldots,,a_{n-1} \} ; \\ a_{n-1}+2 ~~ \text{otherwise}. \end{cases}
a
n
=
{
a
n
−
1
+
3
if
n
−
1
∈
{
a
1
,
a
2
,
…
,,
a
n
−
1
}
;
a
n
−
1
+
2
otherwise
.
Prove that for all positive integers n, we have
a
n
<
n
⋅
(
1
+
2
)
.
a_n < n \cdot (1 + \sqrt{2}).
a
n
<
n
⋅
(
1
+
2
)
.
Dominik Burek (Poland) (also known as [url=https://artofproblemsolving.com/community/user/100466]Burii)
4
1
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Number of sequences
Determine the number of
2021
2021
2021
-tuples of positive integers such that the number
3
3
3
is an element of the tuple and consecutive elements of the tuple differ by at most
1
1
1
.Walther Janous (Austria)
3
1
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Interlaced polygons
For any two convex polygons
P
1
P_1
P
1
and
P
2
P_2
P
2
with mutually distinct vertices, denote by
f
(
P
1
,
P
2
)
f(P_1, P_2)
f
(
P
1
,
P
2
)
the total number of their vertices that lie on a side of the other polygon. For each positive integer
n
≥
4
n \ge 4
n
≥
4
, determine
max
{
f
(
P
1
,
P
2
)
∣
P
1
and
P
2
are convex
n
-gons
}
.
\max \{ f(P_1, P_2) ~ | ~ P_1 ~ \text{and} ~ P_2 ~ \text{are convex} ~ n \text{-gons} \}.
max
{
f
(
P
1
,
P
2
)
∣
P
1
and
P
2
are convex
n
-gons
}
.
(We say that a polygon is convex if all its internal angles are strictly less than
18
0
∘
180^\circ
18
0
∘
.)Josef Tkadlec (Czech Republic)
2
1
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Classic configuration revisited
In an acute triangle
A
B
C
ABC
A
BC
, the incircle
ω
\omega
ω
touches
B
C
BC
BC
at
D
D
D
. Let
I
a
I_a
I
a
be the excenter of
A
B
C
ABC
A
BC
opposite to
A
A
A
, and let
M
M
M
be the midpoint of
D
I
a
DI_a
D
I
a
. Prove that the circumcircle of triangle
B
M
C
BMC
BMC
is tangent to
ω
\omega
ω
.Patrik Bak (Slovakia)
1
1
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Routine number theory
Find all quadruples
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
of positive integers satisfying
gcd
(
a
,
b
,
c
,
d
)
=
1
\gcd(a, b, c, d) = 1
g
cd
(
a
,
b
,
c
,
d
)
=
1
and
a
∣
b
+
c
,
b
∣
c
+
d
,
c
∣
d
+
a
,
d
∣
a
+
b
.
a | b + c, ~ b | c + d, ~ c | d + a, ~ d | a + b.
a
∣
b
+
c
,
b
∣
c
+
d
,
c
∣
d
+
a
,
d
∣
a
+
b
.
Vítězslav Kala (Czech Republic)