Subcontests
(4)coloring 1-8 with black or white, given rules, find each number's color
We color one of the numbers 1,...,8 with white or black according to the following rules:
i) number 4 gets colored white and one at lest of the following numbers gets colored black
ii) if two numbers a,b are colored in a different color and a+b≤8, then number a+b gets colored black.
iii) if two numbers a,b are colored in a different color and a⋅b≤8, then number a⋅b gets colored white.
If by those rules, all numbers get colored, find the color of each number. 3 circumcircles have a common chord, concyclic points
Given an acute and scalene triangle ABC with AB<AC and random line (e) that passes throuh the center of the circumscribed circles c(O,R). Line (e), intersects sides BC,AC,AB at points A1,B1,C1 respectively (point C1 lies on the extension of AB towards B). Perpendicular from A on line (e) and AA1 intersect circumscribed circle c(O,R) at points M and A2 respectively. Prove that
a) points O,A1,A2,M are consyclic
b) if (c2) is the circumcircle of triangle (OBC1) and (c3) is the circumcircle of triangle (OCB1), then circles (c1),(c2) and (c3) have a common chord
given last 2-digit part of 9 numbers, find last 2-digit part of sum of squares
Nine positive integers a1,a2,...,a9 have their last 2-digit part equal to 11,12,13,14,15,16,17,18 and 19 respectively. Find the last 2-digit part of the sum of their squares.