MathDB

1

Part of 2015 Danube Mathematical Competition

Problems(2)

Trunk numbers

Source: Danube Mathematical Competition,Juniors #1

10/31/2015
Consider a positive integer n=a1a2...ak‾,k≥2n=\overline{a_1a_2...a_k},k\ge 2.A trunk of nn is a number of the form a1a2...at‾,1≤t≤k−1\overline{a_1a_2...a_t},1\le t\le k-1.(For example,the number 2323 is a trunk of 23512351.) By T(n)T(n) we denote the sum of all trunk of nn and let S(n)=a1+a2+...+akS(n)=a_1+a_2+...+a_k.Prove that n=S(n)+9⋅T(n)n=S(n)+9\cdot T(n).
number theory
cyclic ABCD, diagonals intersection, 2 incenters, isosceles wanted

Source: Danube 2015 Seniors P1

9/8/2018
Let ABCDABCD be a cyclic quadrangle, let the diagonals ACAC and BDBD cross at OO, and let II and JJ be the incentres of the triangles ABCABC and ABDABD, respectively. The line IJIJ crosses the segments OAOA and OBOB at MM and NN, respectively. Prove that the triangle OMNOMN is isosceles.
geometryisoscelescyclic quadrilateralincenter