第9届（2013）国际Zhautykov数学奥林匹克试题（英文）IX International Zhautykov Olympiad in Mathematics
Almaty, 2013
15 January, 2013, 9.00–13.30
First day
(Each problem is worth 7 points)
1. Given a trapezoid
(
) with
. Point
is chosen on the lateral side
. Let
and
be the circumcenters of the tria...
IX International Zhautykov Olympiad in Mathematics
Almaty, 2013
15 January, 2013, 9.00–13.30
First day
(Each problem is worth 7 points)
1. Given a trapezoid
(
) with
. Point
is chosen on the lateral side
. Let
and
be the circumcenters of the triangles
and
respectively. The circumcircles of the triangles
and
meet again at the point
. Prove that the line
passes through the point
.
2. Find all odd positive integers
such that there is a permutation
of the numbers
where
divides one of the numbers
and
for each
(we assume
).
3. Let
and
. Prove that
.
IX International Zhautykov Olympiad in Mathematics
Almaty, 2013
16 January, 2013, 9.00–13.30
Second day
(Each problem is worth 7 points)
4. A quadratic trinomial
with real coefficients is given. Prove that there is a positive integer n such that the equation
has no rational roots.
5. Given convex hexagon
with
,
,
. The distance between the lines
and
is equal to the distance between the lines
and
and to the distance between the lines
and
. Prove that the sum
does not exceed the perimeter of hexagon
.
6. A
table consists of 100 unit cells. A block is a
square consisting of 4 unit cells of the table. A set C of n blocks covers the table (i.e. each cell of the table is covered by some block of C ) but no
blocks of C cover the table. Find the largest possible value of n.
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