Subcontests
(6)Finally an actual geometry problem!
Let ABCD be a circumscribed quadrilateral and T=AC∩BD. Let I1, I2, I3, I4 the incenters of ΔTAB, ΔTBC, TCD, TDA, respectively, and J1, J2, J3, J4 the incenters of ΔABC, ΔBCD, ΔCDA, ΔDAB. Show that I1I2I3I4 is a cyclic quadrilateral and its center is J1J3∩J2J4 Does Ceva work in 3D too?
Define a \emph{big circle} in a sphere as a circle that has two diametrically oposite points of the sphere in it. Suppose (AB) as the big circle that passes through A and B. Also, let a \emph{Spheric Triangle} be 3 connected by big circles. The angle between two circles that intersect is defined by the angle between the two tangent lines from the intersection point through the two circles in their respective planes. Define also ∠XYZ the angle between (XY) and (YZ). Two circles are tangent if the angle between them is 0. All the points in the following problem are in a sphere S.
Let ΔABC be a spheric triangle with all its angles <90∘ such that there is a circle ω tangent to (BC),(CA),(AB) in D,E,F. Show that there is P∈S with ∠PAB=∠DAC, ∠PCA=∠FCB, ∠PBA=∠EBC. "Arrows problems are too easy"
Find all f:R→R continuous functions such that limx→∞f(x)=∞ and ∀x,y∈R,∣x−y∣>φ,∃n<φ2023,n∈N such that
fn(x)+fn(y)=x+y