MathDB

4

Part of 2010 Mathcenter Contest

Problems(2)

f :P -> P, ABCD convex , f(A),f(B),f(C),f (D) vertices of concave

Source: Mathcenter Contest / Oly - Thai Forum 2010 (R1) p4 https://artofproblemsolving.com/community/c3196914_mathcenter_contes

11/15/2022
Let PP be a plane. Prove that there is no function f:P→Pf :P\rightarrow P where, for any convex quadrilateral ABCDABCD, the points f(A)f(A), f(B)f(B), f(C)f(C), f(D)f (D) are the vertices of a concave quadrilateral.
(tatari/nightmare)
geometryfunctionalconvex
point construction, non intersecting chords related, PE/EQ=1/2

Source: Mathcenter Contest / Oly - Thai Forum 2010 (R2) p4 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

11/15/2022
In a circle, two non-intersecting chords AB,CDAB,CD are drawn.On the chord ABAB,a point EE (different from AA,BB) is taken Consider the arc ABAB that does not contain the points C,DC,D. With a compass and a straighthedge, find all possible point FF on that arc such that PEEQ=12\dfrac{PE}{EQ}=\dfrac{1}{2}, where PP and QQ are the points in which the chord ABAB meets the segment FCFC and FDFD.
(tatari/nightmare)
geometryconstruction