Problems proposed for journals

Let \mathcal{C}(O) be a given circle and let P be a point outside of it. The tangents from P intersect the circle at A and B. Let M be the midpoint of AP and let N be the second intersection of BM with \mathcal{C}. Prove that PN=2MN.

Let ABC be a triangle and let D, E, F be the tangency points of its incircle with the sides BC, CA and AB, respectively. Prove that the centroid of triangle ABC and the centroid of DEF are isogonally conjugated with respect to triangle ABC if and only if the triangle ABC is equilateral.

Let O and I be the circumcenter and the incenter of a given triangle ABC, respectively. Denote by D the tangency point of the incircle of triangle ABC with the side BC and let E, F be the second intersection points of the circumcircle of ABC with the lines AI and AO, respectively. Let S be the intersection of FI and ED, M the intersection of SC and BE, and N the intersection of AC and BF. Prove that the points M, I and N are collinear.

Let a, b, c be the lengths of the sides of a given triangle ABC. Prove that (a+b)(b+c)(c+a) + (-a+b+c)(a-b+c)(a+b-c) \geq 9abc.

Let ABC be a given triangle with incenter I. Denote by M, N, P the midpoints of the sides BC, CA, AB, respectively. Prove that IM^2+IN^2+IP^2 \geq r(R+r). Does the inequality remain true if I is replaced by an arbitrary point P lying in the plane of ABC?

Let ABC be a triangle and let E, F be the feet of the internal angle bisectors of B and C, respectively. Denote by O the circumcenter of triangle ABC and by I_a the center of the excircle corresponding to the vertex A. Prove that OI_a \perp EF.

Let ABC be a triangle and let D, E, F be the tangency points of its incircle \omega with the sides BC, CA and AB, respectively. Let T \in AD \cap \omega, M \in BT \cap \omega, and N \in CT \cap \omega. Consider \rho_1 a circle tangent to \omega at T, and \rho_2 a circle tangent to \omega at D, such that \rho_1, \rho_2 intersect on the chord XY. Prove that the points X, Y, M, N are concyclic.

Let ABC be a given triangle and let (I_a), (I_b), (I_c) be its corresponding excircles. Let P be a point in the interior of the triangle ABC with cevian triangle A_1B_1C_1. Denote by X, Y, Z the intersection points of the tangents from A_1, B_1, C_1 to (I_a), (I_b), (I_c) with the corresponding excircles. Prove that the lines AX, BY, CZ are concurrent.

Let ABC be a triangle and let D, E, F be the tangency points of its incircle \omega with the sides BC, CA and AB, respectively. Let X_1, X_2 be the intersections of the line EF with the circumcircle of triangle ABC. Similarly, define Y_1, Y_2 and Z_1, Z_2. Prove that the radical center of the circumcircles of triangles DX_1X_2, EY_1Y_2 and FZ_1Z_2 lies on the line OI.

Consider ABC an arbitrary triangle and P a point in its plane. Let D, E, and F be three points on the lines through P perpendicular to the lines BC, CA, and AB, respectively. Prove that if triangle DEF is equilateral, and if the point P lies on the Euler line of triangle ABC, then the center of triangle DEF also lies on the Euler line of triangle ABC.

Let x, y, z be three positive real numbers. Prove that \sum{\frac{y+z}{x}}-4\sum{\frac{x}{y+z}} \geq 1 - \frac{8xyz}{(y+z)(z+x)(x+y)}.

Let ABC be a triangle and let M, N, P be the midpoints of the sides BC, CA, AB, respectively. Denote by X, Y, Z the midpoints of the altitudes emerging from vertices A, B, C, respectively. Prove that the radical center of the circles AMX, BNY, CPZ is the center of the nine-point circle of triangle ABC.

Let a, b, c be three positive real numbers. Prove that
\sum_{cyc}{\sqrt{\frac{b+c}{a}}}\geq 2\sum_{cyc}{\sqrt{\frac{a}{b+c}}}\cdot\sqrt{1+\frac{(a+b)(b+c)(c+a)-8abc}{4\sum{a(a+b)(a+c)}}}.

Find all complex numbers x, y, z of modulus 1, satisfying \frac{y^2+z^2}{x}+\frac{z^2+x^2}{y}+\frac{x^2+y^2}{z} = 2(x+y+z).

Let P be an arbitrary point in the plane of a given triangle ABC and let P’ be its isogonal conjugate with respect to this triangle. Let I be the incenter of ABC and let A_1B_1C_1 be the cevian triangle of P. Denote by A_2, B_2, C_2 be the midpoints of the segments IA_1, IB_1, and IC_1, respectively. Prove that the lines XA_2, YB_2, ZC_2 are concurrent on the line IP’.

Let M be a point on the circumcircle of triangle ABC and lying on the arc BC that does not contain A. Let I be the incenter of ABC, and let E and F be the feet of the perpendiculars from I to lines MB and MC, respectively. Prove that the value of (IE+IF)/AM does not depend of the position of M.

Let a, b, c be positive real numbers. Prove that
\sum_{cyc}{\frac{a^{2}+bc}{b+c}} \geq \sum_{cyc}{\frac{a(b^{2}+c^{2})}{a^{2}+bc}}.

In triangle ABC, let M and Q be points on segment AB, and similarly let N and R be points on AC, and P and S be points on BC. Let d_1 be the line through M, N, d_2 the line through P, Q, and d_3 the line through R, S. Let \rho(X,Y,Z) denote the ratio of the length of XZ to that of XY. Let m=\rho(M,A,B), n=\rho(N,A,C), p=\rho(P,B,C), q=\rho(Q,B,A), r=\rho(R,C,A), and s=\rho(S,C,B). Prove that the lines d_1, d_2, d_3 are concurrent if and only if mpr+nqs+mq+nr+ps=1.

Let P be an arbitrary point on the circumcircle \Gamma(O) of a given triangle ABC. Let \rho(I) be the incenter of the triangle and let X, Y be the intersection points of the sideline BC with the tangents from P to the incircle \rho. Prove that the second intersection of the circumcircle of triangle PXY with \Gamma is the tangency point of the mixtilinear incircle given in angle A with \Gamma. (The mixtilinear incircle given in angle A is the circle simultanously tangent to the sidelines AB, AC and internally to \Gamma.)