MathDB

2

Part of 2009 Cuba MO

Problems(2)

2008^2 hydras in the Hidro planet

Source: Cuba, Concurso Nacional 2009

2/5/2014
In Hidro planet were living 200822008^2 hydras some time ago. One of them had 1 tentacle, other 2 and so on to the last with 200822008^2 tentacles. Let t(H)t(H) be the number of tentacles of hydra HH. The pairing of H1H_1 and H2H_2 (where t(H1)<t(H2)t(H_1) < t(H_2)) is a new hydra with t(H2)−t(H1)+8t(H_2)-t(H_1)+8 tentacles, in case of t(H1)=t(H2)t(H_1)=t(H_2) they die. An expedition found in Hidro the last hydra with 23 tentacles. That could be true ?
combinatorics proposedcombinatorics
ABC is equilateral if circles (XYZ) and (ABC) are equal

Source: 2009 Cuba 2.72

8/27/2024
Let II be the incenter of an acute riangle ABCABC. Let CA(A,AI)C_A(A, AI) be the circle with center AA and radius AIAI. Circles CB(B,BI)C_B(B, BI), CC(C,CI)C_C(C, CI) are defined in an analogous way. Let X,Y,ZX, Y, Z be the intersection points of CBC_B with CCC_C, CCC_C with CAC_A, CAC_A with CBC_B respectively (different than II) . Show that if the radius of the circle that passes through the points X,Y,ZX, Y, Z is equal to the radius of the circle that passes through points AA, BB and CC then triangle ABCABC is equilateral.
Equilateralincenterequal circlesgeometry