Problems proposed for olympiads & contests

Prove that in every given triangle ABC with circumradius R inradius r and semiperimeter s we have that (\frac{4R+r}{s})^2+\frac{9r}{4R+r} \geq 4.

Let ABC be a triangle, let E, F be the tangency points of the incircle \Gamma(I) to the sides AC, respectively AB, and let M be the midpoint of the side BC. Let N = AM \cap EF, let \gamma(M) be the circle of diameter BC, and let X, Y be the other (than B, C) intersection points of BI, respectively CI, with \gamma. Prove that NX/NY= AC/AB.

Consider \rho a semicircle of diameter AB. A line parallel to AB cuts the semicircle at points C, D such that AD separates B, C. The line parallel to AD through C intersects the semicircle for the second time at E. Let F be the intersection point of the lines BE and CD. The line parallel through F at AD cuts AB in P. Prove that PC is tangent to \rho.

Let \rho(O) be a circle and A a point outside of it. Denote by B, C the points where the tangents from A with respect to \rho meet the circle, by D the point on \rho for which O \in (AD), by X the foot of the perpendicular from B to CD, by Y the midpoint of the segment BX and by finally, by Z the second intersection of the line DY with \rho. Prove that the lines ZA and ZC are perpendicular.

Let d be a line outside of a given circle \mathcal{C}(O). Denote by A the orthogonal projection of O on the line d and let M be an arbitrary point on the circle \mathcal{C}. Denote by X, Y the intersection points of the circle \Gamma of diameter AM and \mathcal{C}, and d, respectively. Prove that the line XY passes through a fixed point, not depending of the position of M.

Given an acute-angled triangle ABC, let AA_1, BB_1, CC_1 be its altitudes. Some circle passes through the points A_1 and B_1 and touches the arc AB of the circumcircle of ABC, not containing the vertex C, at a point C_2. The points A_2 and B_2 are defined similarly. Prove that the lines A_1A_2, B_1B_2, C_1C_2 are concurrent on the Euler line of triangle ABC.

Let ABC be a triangle with \angle BAC < \angle ACB. Let D, E be points on the sides AC and AB, such that the angles ACB and BED are congruent. If F lies in the interior of the quadrilateral BCDE such that the circumcircle of triangle BCF is tangent to the circumcircle of DEF and the circumcircle of BEF is tangent to the circumcircle of CDF, prove that the points A, C, E, F are concyclic.

Let ABC be a given triangle with the incenter I. Denote by X, Y, Z the intersections of the lines AI, BI, CI with the sides BC, CA, and AB, respectively. Consider \mathcal{K}_a the circle tangent simultanously to the sidelines AB, AC, and internally to the circumcircle \mathcal{C}(O) of ABC, and let A’ be the tangency point of \mathcal{K}_a with \mathcal{C}. Similarly, define B’, and C’. Prove that the circumcircles of triangles AXA’, BYB’, and CZC’ are coaxal, having two common (distinct) points.

Let  A’ be an arbitrary point on the side BC of a triangle ABC. Denote by \mathcal{T}_{A}^b, \mathcal{T}_{A}^c the circles simultanously tangent to AA’, A’B, \Gamma and AA’, A’C, \Gamma, respectively, where \Gamma  is the circumcircle of ABC.
(1) Prove that \mathcal{T}_{A}^b,  \mathcal{T}_{A}^c are congruent if and only if AA’ passes through the Nagel point of triangle ABC.
(2) Prove that the radii of \mathcal{T}_{A}^b, \mathcal{T}_{A}^c are in arithmetical progression with the inradius of ABC if and only if latex AA’ passes through the orthocenter of triangle ABC.

Let ABC be a given triangle with the incenter I. Denote by X, Y, Z the intersections of the lines AI, BI, CI with the sides BC, CA, and AB, respectively. Consider \mathcal{K}_a the circle tangent simultanously to the sidelines AB, AC, and internally to the circumcircle \mathcal{C}(O) of ABC, and let A’ be the tangency point of \mathcal{K}_a with \mathcal{C}. Similarly, define B’, and C’. Prove that the circumcircles of triangles AXA’, BYB’, and CZC’ are coaxal, having two common (distinct) points.

Given a scalene acute triangle ABC with AC > BC let F be the foot of the altitude from C. Let P be a point on AB, different from A so that AF = PF. Let H, O, M be the orthocenter, circumcenter and midpoint of [AC], respectively. Let X be the intersection point of BC and HP. Let Y be the intersection point of OM and FX and let OF intersect AC at Z. Prove that  the points F, M, Y, Z are concyclic.