MathDB

11.7

Part of IV Soros Olympiad 1997 - 98 (Russia)

Problems(2)

4 collinear points, 3 circles, computational

Source: IV Soros Olympiad 1997-98 Corr. 11.7 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/1/2024
On straight line ℓ\ell there are points AA, BB, CC and DD, following in the indicated order: AB=aAB = a, BC=bBC = b, CD=cCD = c. Segments ADAD and BCBC serve as chords of two circles, and the sum of the angular values of the arcs of these circles located on one side of ℓ\ell is equal to 360o360^o. A third circle passes through AA and BB, intersecting the first two at points KK and MM. The straight line KMKM intersects ℓ\ell at point EE. Find AEAE.
geometrycircles
log_{\frac12} x\ ge 16^x (IV Soros Olympiad 1997-98 R3 11.7)

Source:

6/2/2024
Solve the inequality log⁡12x≥16x\log_{\frac12} x\ge 16^x
algebrainequalitieslogarithms