MathDB

3

Part of 2011 Bosnia Herzegovina Team Selection Test

Problems(2)

Old and easy sums

Source: Bosnia and Herzegovina 2011

5/16/2011
Numbers 1,2,...,2n1,2, ..., 2n are partitioned into two sequences a1<a2<...<ana_1<a_2<...<a_n and b1>b2>...>bnb_1>b_2>...>b_n. Prove that number W=∣a1−b1∣+∣a2−b2∣+...+∣an−bn∣W= |a_1-b_1|+|a_2-b_2|+...+|a_n-b_n| is a perfect square.
absolute valuealgebra proposedalgebra
AD/BC=AF/FB=DG/GC

Source: Bosnia and Herzegovina 2011

5/16/2011
In quadrilateral ABCDABCD sides ADAD and BCBC aren't parallel. Diagonals ACAC and BDBD intersect in EE. FF and GG are points on sides ABAB and DCDC such AFFB=DGGC=ADBC\frac{AF}{FB}=\frac{DG}{GC}=\frac{AD}{BC} Prove that if E,F,GE, F, G are collinear then ABCDABCD is cyclic.
trigonometrygeometrygeometric transformationhomothetygeometry proposed