1977 IMO Problems
Problems of the 19th IMO 1977 in Yugoslavia.
In the interior of a square we construct the equilateral triangles Prove that the midpoints of the four segments and the midpoints of the eight segments are the 12 vertices of a regular dodecagon.
In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.
Let be a given number greater than 2. We consider the set of all the integers of the form with A number from is called indecomposable in if there are not two numbers and from so that Prove that there exist a number that can be expressed as the product of elements indecomposable in in more than one way. (Expressions which differ only in order of the elements of will be considered the same.)
Let be given reals. We consider the function defined byProve that if for any real number we have then and
Let be two natural numbers. When we divide by , we the the remainder and the quotient Determine all pairs for which
Let be the set of positive integers. Let be a function defined on , which satisfies the inequality for all . Prove that for any we have
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