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KoMaL A Problems
KoMaL A Problems 2024/2025
KoMaL A Problems 2024/2025
Part of
KoMaL A Problems
Subcontests
(3)
A. 886
1
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Erasing progressions from blackboard
Let
k
k
k
and
n
n
n
be two given distinct positive integers greater than
1
1
1
. There are finitely many (not necessarily distinct) integers written on the blackboard. Kázmér is allowed to erase
k
k
k
consecutive elements of an arithmetic sequence with a difference not divisible by
k
k
k
. Similarly, Nándor is allowed to erase
n
n
n
consecutive elements of an arithmetic sequence with a difference that is not divisible by
n
n
n
. The initial numbers on the blackboard have the property that both Kázmér and Nándor can erase all of them (independently from each other) in a finite number of steps. Prove that the difference of biggest and the smallest number on the blackboard is at least
φ
(
n
)
+
φ
(
k
)
\varphi(n)+\varphi(k)
φ
(
n
)
+
φ
(
k
)
.Proposed by Boldizsár Varga, Budapest
A. 885
1
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Incenter of small triangle lies on incircle
Let triangle
A
B
C
ABC
A
BC
be a given acute scalene triangle with altitudes
B
E
BE
BE
and
C
F
CF
CF
. Let
D
D
D
be the point where the incircle of
△
A
B
C
\,\triangle ABC
△
A
BC
touches side
B
C
BC
BC
. The circumcircle of
△
B
D
E
\triangle BDE
△
B
D
E
meets line
A
B
AB
A
B
again at point
K
K
K
, the circumcircle of
△
C
D
F
\triangle CDF
△
C
D
F
meets line
A
C
AC
A
C
again at point
L
L
L
. The circumcircle of
△
B
D
E
\triangle BDE
△
B
D
E
and
△
C
D
F
\triangle CDF
△
C
D
F
meet line
K
L
KL
K
L
again at
X
X
X
and
Y
Y
Y
, respectively. Prove that the incenter of
△
D
X
Y
\triangle DXY
△
D
X
Y
lies on the incircle of
△
A
B
C
\,\triangle ABC
△
A
BC
.Proposed by Luu Dong, Vietnam
A. 884
1
Hide problems
Bound for sum of negatives implies positiive config
We fill in an
n
×
n
n\times n
n
×
n
table with real numbers such that the sum of the numbers in each row and each coloumn equals
1
1
1
. For which values of
K
K
K
is the following statement true: if the sum of the absolute values of the negative entries in the table is at most
K
K
K
, then it's always possible to choose
n
n
n
positive entries of the table such that each row and each coloumn contains exactly one of the chosen entries.Proposed by Dávid Bencsik, Budapest