Subcontests
(5)Greece JBMO TST 2017
[url=https://artofproblemsolving.com/community/c675547]Greece JBMO TST 2017[url=http://artofproblemsolving.com/community/c6h1663730p10567608]Problem 1. Positive real numbers a,b,c satisfy a+b+c=1. Prove that
(a+1)2a(1−a)+(b+1)2b(1−b)+(c+1)2c(1−c)≥8(ab+bc+ca).
Also, find the values of a,b,c for which the equality happens.[url=http://artofproblemsolving.com/community/c6h1663731p10567619]Problem 2. Let ABC be an acute-angled triangle inscribed in a circle C(O,R) and F a point on the side AB such that AF<AB/2. The circle c1(F,FA) intersects the line OA at the point A′ and the circle C at K. Prove that the quadrilateral BKFA′ is cyclic and its circumcircle contains point O.[url=http://artofproblemsolving.com/community/c6h1663732p10567627]Problem 3. Prove that for every positive integer n, the number An=72n−48n−1 is a multiple of 9.[url=http://artofproblemsolving.com/community/c6h1663734p10567640]Problem 4. Let ABC be an equilateral triangle of side length a, and consider D, E and F the midpoints of the sides (AB),(BC), and (CA), respectively. Let H be the the symmetrical of D with respect to the line BC. Color the points A,B,C,D,E,F,H with one of the two colors, red and blue.[*] How many equilateral triangles with all the vertices in the set {A,B,C,D,E,F,H} are there?
[*] Prove that if points B and E are painted with the same color, then for any coloring of the remaining points there is an equilateral triangle with vertices in the set {A,B,C,D,E,F,H} and having the same color.
[*] Does the conclusion of the second part remain valid if B is blue and E is red?
Coloring and equilateral triangles
Let ABC be an equilateral triangle of side length a, and consider D, E and F the midpoints of the sides (AB),(BC), and (CA), respectively. Let H be the the symmetrical of D with respect to the line BC. Color the points A,B,C,D,E,F,H with one of the two colors, red and blue.[*] How many equilateral triangles with all the vertices in the set {A,B,C,D,E,F,H} are there?
[*] Prove that if points B and E are painted with the same color, then for any coloring of the remaining points there is an equilateral triangle with vertices in the set {A,B,C,D,E,F,H} and having the same color.
[*] Does the conclusion of the second part remain valid if B is blue and E is red?
cyc sum (a+1)\sqrt{2a(1-a)} \geq 8(ab+bc+ca)
Positive real numbers a,b,c satisfy a+b+c=1. Prove that
(a+1)2a(1−a)+(b+1)2b(1−b)+(c+1)2c(1−c)≥8(ab+bc+ca).
Also, find the values of a,b,c for which the equality happens.