Given positive integers and , prove that there exists a positive integer such that the numbers and have the same occurrences of each non-zero digit in base .
Let be a triangle with . Points and lie on sides and , respectively, such that and . Segments and meet at . Determine whether or not it is possible for segments to all have integer side lengths.
Find all functions such that for all integers that satisfy , the following equality holds:
A circle is divided into congruent arcs by points. The points are colored with four colors such that some points are colored each of Red, Blue, Green, and Yellow. Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.
Find all positive integers for which there exists non-negative integers such that
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