Subcontests
(6)Interesting conditional geo in orthocentre configuration
Let ABC be an acute angled triangle with orthocentre H. Let E=BH∩AC and F=CH∩AB. Let D,M,N denote the midpoints of segments AH,BD,CD respectively, and T=FM∩EN. Suppose D,E,T,F are concylic. Prove that DT passes through the circumcentre of ABC.Proposed by Pranjal Srivastava Everyone loves tiling
1. Can a 7×7 square be tiled with the two types of tiles shown in the figure? (Tiles can be rotated and reflected but cannot overlap or be broken)2. Find the least number N of tiles of type A that must be used in the tiling of a 1011×1011 square. Give an example of a tiling that contains exactly N tiles of type A.
[asy]
size(4cm, 0);
pair a = (-10,0), b = (0, 0), c = (10, 0), d = (20, 0), e = (20, 10), f = (10, 10), g = (0, 10), h = (0, 20), ii = (-10, 20), j = (-10, 10);
draw(a--b--c--f--g--h--ii--cycle);
draw(g--b);
draw(j--g);
draw(f--c);
draw((30, 0)--(30, 20)--(50,20)--(50,0)--cycle);
draw((40,20)--(40,0));
draw((30,10)--(50,10));
label((0,0), "(A)", S);
label((40,0), "(B)", S);
[/asy]
Proposed by Muralidharan Somasundaran What are those floors doing here?
Given that a1,a2,…,a10 are positive real numbers, determine the smallest possible value of i=1∑10⌊ai+ai+17ai⌋ where we define a11=a1.Proposed by Sutanay Bhattacharya