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11.5

Part of IV Soros Olympiad 1997 - 98 (Russia)

Problems(3)

log_{2n-2} (n^2 + 2) is rational (IV Soros Olympiad 1997-98 R2 11.5)

Source:

6/1/2024
Find all integers nn for which log⁡2n−2(n2+2)\log_{2n-2} (n^2 + 2) is a rational number.
number theorylogarithms
sides of parallelogram are diagonals of 4 squares

Source: IV Soros Olympiad 1997-98 Corr. 11.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/1/2024
The sides of the parallelogram serve as the diagonals of the four squares. The vertices of the squares lying in the part of the plane external to the parallelogram (the sides of the squares emerging from these vertices do not have common points with the parallelogram) serve as the vertices of a quadrilateral of area aa, the four vertices opposite to them form a quadrilateral of area bb. Find the area of the parallelogram.
geometryparallelogramareas
KM _|_ AD, cyclic ABCD, BK=CK, <KBC+< AMB= 90^o

Source: IV Soros Olympiad 1997-98 R3 11.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/2/2024
Let MM be the point of intersection of the diagonals of the inscribed quadrilateral ABCDABCD, and let the angle ∠AMB\angle AMB be an acute angle. On the side BCBC, as a base, an isosceles triangle BCKBCK is constructed in the direction external to the quadrilateral such that ∠KBC+∠AMB=90o\angle KBC+\angle AMB= 90^o. Prove that line KMKM is perpendicular to ADAD.
geometryperpendicular