MathDB

1

Part of 2007 Indonesia TST

Problems(5)

trigonometry trivia

Source: Indonesia IMO 2007 TST, Stage 2, Test 1, Problem 1

11/15/2009
Let P P be a point in triangle ABC ABC, and define α,β,γ \alpha,\beta,\gamma as follows: \alpha\equal{}\angle BPC\minus{}\angle BAC,   \beta\equal{}\angle CPA\minus{}\angle \angle CBA,   \gamma\equal{}\angle APB\minus{}\angle ACB. Prove that PA\dfrac{\sin \angle BAC}{\sin \alpha}\equal{}PB\dfrac{\sin \angle CBA}{\sin \beta}\equal{}PC\dfrac{\sin \angle ACB}{\sin \gamma}.
trigonometrygeometrycircumcirclegeometry proposed
lattice points

Source: unknown

9/24/2014
Call an nn-gon to be lattice if its vertices are lattice points. Prove that inside every lattice convex pentagon there exists a lattice point.
geometryparallelogramcombinatorics unsolvedcombinatorics
locus and similarity

Source: Indonesia IMO 2007 TST, Stage 2, Test 2, Problem 1

11/15/2009
Given triangle ABC ABC and its circumcircle Γ \Gamma, let M M and N N be the midpoints of arcs BC BC (that does not contain A A) and CA CA (that does not contain B B), repsectively. Let X X be a variable point on arc AB AB that does not contain C C. Let O1 O_1 and O2 O_2 be the incenter of triangle XAC XAC and XBC XBC, respectively. Let the circumcircle of triangle XO1O2 XO_1O_2 meets Γ \Gamma at Q Q. (a) Prove that QNO1 QNO_1 and QMO2 QMO_2 are similar. (b) Find the locus of Q Q as X X varies.
geometrycircumcircleincentergeometry proposed
inequality

Source: Indonesia IMO 2007 TST, Stage 2, Test 4, Problem 1

11/15/2009
Let a,b,c a,b,c be real numbers. Prove that (ab\plus{}bc\plus{}ca\minus{}1)^2 \le (a^2\plus{}1)(b^2\plus{}1)(c^2\plus{}1).
inequalitiesinequalities proposed
incircle of a quadrilateral

Source: Indonesia IMO 2007 TST, Stage 2, Test 5, Problem 1

11/15/2009
Let ABCD ABCD be a cyclic quadrilateral and O O be the intersection of diagonal AC AC and BD BD. The circumcircles of triangle ABO ABO and the triangle CDO CDO intersect at K K. Let L L be a point such that the triangle BLC BLC is similar to AKD AKD (in that order). Prove that if BLCK BLCK is a convex quadrilateral, then it has an incircle.
geometrycircumcirclecyclic quadrilateralangle bisectorgeometry proposed