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National and Regional Contests
North Macedonia Contests
Memorial "Aleksandar Blazhevski-Cane"
2020 Memorial "Aleksandar Blazhevski-Cane"
2020 Memorial "Aleksandar Blazhevski-Cane"
Part of
Memorial "Aleksandar Blazhevski-Cane"
Subcontests
(3)
3
1
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finite solutions of F_{n,k}(x,y)=x!+n^k+n+1-y^k=0
For given integers
n
>
0
n>0
n
>
0
and
k
>
1
k> 1
k
>
1
, let
F
n
,
k
(
x
,
y
)
=
x
!
+
n
k
+
n
+
1
−
y
k
F_{n,k}(x,y)=x!+n^k+n+1-y^k
F
n
,
k
(
x
,
y
)
=
x
!
+
n
k
+
n
+
1
−
y
k
. Prove that there are only finite couples
(
a
,
b
)
(a,b)
(
a
,
b
)
of positive integers such that
F
n
,
k
(
a
,
b
)
=
0
F_{n,k}(a,b)=0
F
n
,
k
(
a
,
b
)
=
0
2
1
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numbers in mxn board, 2 operations available, can all numbers become -1?
One positive integer is written in each
1
×
1
1 \times 1
1
×
1
square of the
m
×
n
m \times n
m
×
n
board. The following operations are allowed : (1) In an arbitrarily selected row of the board, all numbers should be reduced by
1
1
1
. (2) In an arbitrarily selected column of the board, double all the numbers. Is it always possible, after a final number of steps, for all the numbers written on the board to be equal to
−
1
-1
−
1
? (Explain the answer.)
1
1
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2 angle bisectors interesect on a diagonal of ABCD, wanted and given
A convex quadrilateral
A
B
C
D
ABCD
A
BC
D
is given in which the bisectors of the interior angles
∠
A
B
C
\angle ABC
∠
A
BC
and
∠
A
D
C
\angle ADC
∠
A
D
C
have a common point on the diagonal
A
C
AC
A
C
. Prove that the bisectors of the interior angles
∠
B
A
D
\angle BAD
∠
B
A
D
and
∠
B
C
D
\angle BCD
∠
BC
D
have a common point on the diagonal
B
D
BD
B
D
.