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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1905 Eotvos Mathematical Competition
1905 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
3
1
Hide problems
1/ AA1 + 1/ BB1 = 1/ CC1
Let
C
1
C_1
C
1
be any point on side
A
B
AB
A
B
of a triangle
A
B
C
ABC
A
BC
, and draw
C
1
C
C_1C
C
1
C
. Let
A
1
A_1
A
1
be the intersection of
B
C
BC
BC
extended and the line through
A
A
A
parallel to
C
C
1
CC_1
C
C
1
, similarly let
B
1
B_1
B
1
be the intersection of
A
C
AC
A
C
extended and the line through
B
B
B
parallel to
C
C
1
CC_1
C
C
1
. Prove that
1
A
A
1
+
1
B
B
1
=
1
C
C
1
.
\frac{1}{AA_1}+\frac{1}{BB_1}=\frac{1}{CC_1}.
A
A
1
1
+
B
B
1
1
=
C
C
1
1
.
2
1
Hide problems
sum of the areas of the removed squares from 9 squares of unit square
Divide the unit square into
9
9
9
equal squares by means of two pairs of lines parallel to the sides (see figure). Now remove the central square. Treat the remaining
8
8
8
squares the same way, and repeat the process
n
n
n
times.(a) How many squares of side length
1
/
3
n
1/3^n
1/
3
n
remain? (b) What is the sum of the areas of the removed squares as
n
n
n
becomes infinite? https://cdn.artofproblemsolving.com/attachments/7/d/3e6e68559919583c24d4457f946bc4cef3922f.png
1
1
Hide problems
x + py = n , x + y = p^2 , NT system
For given positive integers
n
n
n
and
p
p
p
, find neaessary and sufficient conditions for the system of equations
x
+
p
y
=
n
,
x
+
y
=
p
2
x + py = n , \\ x + y = p^2
x
+
p
y
=
n
,
x
+
y
=
p
2
to have a solution
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
of positive integers. Prove also that there is at most one such solution.