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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1909 Eotvos Mathematical Competition
1909 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
3
1
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circles related to orthoc triangle
Let
A
1
,
B
1
,
C
1
A_1, B_1, C_1
A
1
,
B
1
,
C
1
, be the feet of the altitudes of
△
A
B
C
\vartriangle ABC
△
A
BC
drawn from the vertices
A
,
B
,
C
A, B, C
A
,
B
,
C
respectively, and let
M
M
M
be the orthocenter (point of intersection of altitudes) of
△
A
B
C
\vartriangle ABC
△
A
BC
. Assume that the orthic triangle (i.e. the triangle whose vertices are the feet of the altitudes of the original triangle)
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
exists. Prove that each of the points
M
M
M
,
A
A
A
,
B
B
B
, and
C
C
C
is the center of a circle tangent to all three sides (extended if necessary) of
△
A
1
B
1
C
1
\vartriangle A_1B_1C_1
△
A
1
B
1
C
1
. What is the difference in the behavior of acute and obtuse triangles
A
B
C
ABC
A
BC
?
2
1
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a< 1/2 (sin a + tan a)
Show that the radian measure of an acute angle is less than the arithmetic mean of its sine and its tangent.
1
1
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sum of perfect cubes is not a perfect cube, for 3 consecutives
Consider any three consecutive natural numbers. Prove that the cube of the largest cannot be the sum of the cubes of the other two.