1978 IMO Problems
Problems of the 20th IMO 1978 in Romania.
Contents
Day I
Problem 1
Let and be positive integers such that . In their decimal representations, the last three digits of are equal, respectively, to the last three digits of . Find and such that has its least value.
Problem 2
We consider a fixed point in the interior of a fixed sphere We construct three segments , perpendicular two by two with the vertexes on the sphere We consider the vertex which is opposite to in the parallelepiped (with right angles) with as edges Find the locus of the point when take all the positions compatible with our problem.
Problem 3
Let a sequence with all its terms positive The positive integer which doesn't belong to the sequence is Find
Day II
Problem 4
In a triangle we have A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides in the points respectively Prove that the midpoint of is the center of the inscribed circle of the triangle
Problem 5
Let be an injective function from in itself. Prove that for any we have:
Problem 6
An international society has its members from six different countries. The list of members contain names, numbered . Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.
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1978 IMO (Problems) • Resources | ||
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