MathDB

3

Part of 2001 Tuymaada Olympiad

Problems(3)

A convex quadrilateral, bisectors, circumcircles..

Source: Tuymaada 2001, day 2, problem 3.

4/30/2007
ABCDABCD is a convex quadrilateral; half-lines DADA and CBCB meet at point QQ; half-lines BABA and CDCD meet at point PP. It is known that ∠AQB=∠APD\angle AQB=\angle APD. The bisector of angle ∠AQB\angle AQB meets the sides ABAB and CDCD of the quadrilateral at points XX and YY, respectively; the bisector of angle ∠APD\angle APD meets the sides ADAD and BCBC at points ZZ and TT, respectively. The circumcircles of triangles ZQTZQT and XPYXPY meet at point KK inside the quadrilateral. Prove that KK lies on the diagonal ACAC.
Proposed by S. Berlov
geometrycircumcirclepower of a pointradical axisgeometry proposed
Wanted : 3 quadratic polynomials such that P(x)+Q(y)=R(z).

Source: Tuymaada 2001, day 1, problem 3.

4/30/2007
Do there exist quadratic trinomials P,  Q,  RP, \ \ Q, \ \ R such that for every integers xx and yy an integer zz exists satisfying P(x)+Q(y)=R(z)?P(x)+Q(y)=R(z)?
Proposed by A. Golovanov
quadraticsalgebrapolynomialfunctionalgebra proposed
concyclic points, Tuymaada juniors 2001

Source: Tuymaada Junior 2001 p3

4/30/2019
Let ABC be an acute isosceles triangle (AB=BCAB=BC) inscribed in a circle with center OO . The line through the midpoint of the chord ABAB and point OO intersects the line ACAC at LL and the circle at the point PP. Let the bisector of angle BACBAC intersects the circle at point KK. Lines ABAB and PKPK intersect at point DD. Prove that the points L,B,DL,B,D and PP lie on the same circle.
Concyclicgeometryangle bisectorcircumcircle