MathDB

4

Part of 2007 Postal Coaching

Problems(2)

circumradius of MIN is twice that of ABC

Source: Indian Postal Coaching 2007 set 3 p4

5/25/2020
Let BEBE and CFCF be the bisectors of ∠B\angle B and ∠C\angle C of a triangle ABCABC whose incentre is II. Suppose EFEF, extended, meets the circumcircle of ABCABC in M,NM,N. Show that the circumradius of MINMIN is twice that of ABCABC.
geometrycircumradiuscircumcircleangle bisector
A_1 \cup A_2 \cup ... \cup A_n may be colored with 2 colors

Source: Indian Postal Coaching 2007 set 6 p4

5/25/2020
Let A1,A2,...,AnA_1,A_2,...,A_n be nn finite subsets of a set X,n≥2X, n \ge 2, such that (i) ∣Ai∣≥2,1≤i≤n|A_i| \ge 2, 1 \le i \le n, (ii) ∣Ai∩Aj∣≠1,j≤i<j≤n |A_i \cap A_j | \ne 1, j \le i < j \le n. Prove that the elements of A1∪A2∪...∪AnA_1 \cup A_2 \cup ... \cup A_n may be colored with 22 colors so that no AiA_i is colored by the same color.
ColoringcombinatoricsSubsets