MathDB
Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1971 Kurschak Competition
1971 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
1
1
Hide problems
sen B_1A_1C / sen C_2A_2B =B_2C_2 / B_1C_1
A straight line cuts the side
A
B
AB
A
B
of the triangle
A
B
C
ABC
A
BC
at
C
1
C_1
C
1
, the side
A
C
AC
A
C
at
B
1
B_1
B
1
and the line
B
C
BC
BC
at
A
1
A_1
A
1
.
C
2
C_2
C
2
is the reflection of
C
1
C_1
C
1
in the midpoint of
A
B
AB
A
B
, and
B
2
B_2
B
2
is the reflection of
B
1
B_1
B
1
in the midpoint of
A
C
AC
A
C
. The lines
B
2
C
2
B_2C_2
B
2
C
2
and
B
C
BC
BC
intersect at
A
2
A_2
A
2
. Prove that
s
e
n
B
1
A
1
C
s
e
n
C
2
A
2
B
=
B
2
C
2
B
1
C
1
\frac{sen \, \, B_1A_1C}{sen\, \, C_2A_2B} = \frac{B_2C_2}{B_1C_1}
se
n
C
2
A
2
B
se
n
B
1
A
1
C
=
B
1
C
1
B
2
C
2
https://cdn.artofproblemsolving.com/attachments/3/8/774da81495df0a0f7f2f660ae9f516cf70df06.png
3
1
Hide problems
30 boxes each with a unique key.
There are
30
30
30
boxes each with a unique key. The keys are randomly arranged in the boxes, so that each box contains just one key and the boxes are locked. Two boxes are broken open, thus releasing two keys. What is the probability that the remaining boxes can be opened without forcing them?
2
1
Hide problems
11 pairs from 22 points in plane
Given any
22
22
22
points in the plane, no three collinear. Show that the points can be divided into
11
11
11
pairs, so that the
11
11
11
line segments defined by the pairs have at least five different intersections