Ημέρα 1η - 30 Απριλίου 2013
1. In triangle 
, points 
lie on sides 
respectively. Let 
, 
, 
denote the circumcircles of triangles 
, 
, 
, respectively. Given the fact that segment 
intersects , , again at 
respectively, prove that
, points lie on sides respectively. Let , , denote the circumcircles of triangles , , , respectively. Given the fact that segment intersects , , again at respectively, prove that
. 
plot 
equally spaced points around a circle. Label one of them 
, and place a marker at . One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of 
distinct moves available; two from each point. Let 
count the number of ways to advance around the circle exactly twice, beginning and ending at , without repeating a move. Prove that 
for all 
.
5. Given postive integers 
and , prove that there is a positive integer 
such that the numbers 
and 
have the same number of occurrences of each non-zero digit when written in base ten.
.
3. Let be a positive integer. There are 
marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing marks. Initially, each mark has the black side up. An operation is to choose a line parallel to the sides of the triangle, and flipping all the marks on that line. A configuration is called admissible if it can be obtained from the initial configuration by performing a finite number of operations. For each admissible configuration 
, let 
denote the smallest number of operations required to obtain from the initial configuration. Find the maximum value of , where varies over all admissible configurations.
be a positive integer. There are marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing marks. Initially, each mark has the black side up. An operation is to choose a line parallel to the sides of the triangle, and flipping all the marks on that line. A configuration is called admissible if it can be obtained from the initial configuration by performing a finite number of operations. For each admissible configuration , let denote the smallest number of operations required to obtain from the initial configuration. Find the maximum value of , where varies over all admissible configurations.
Ημέρα 2η - 1 Μαΐου 2013
4. Find all real numbers 
satisfying
satisfying and , prove that there is a positive integer such that the numbers and have the same number of occurrences of each non-zero digit when written in base ten.
6. Let be a triangle. Find all points 
on segment 
satisfying the following property: If 
and 
are the intersections of line 
with the common external tangent lines of the circumcircles of triangles 
and 
, then
be a triangle. Find all points on segment satisfying the following property: If and are the intersections of line with the common external tangent lines of the circumcircles of triangles and , then
Πηγή: artofproblemsolving
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