MathDB

2

Part of 2005 Croatia National Olympiad

Problems(4)

lines join incenter and vertices of a triangle

Source: Croatian NMC 2005, 1st Grade

5/8/2007
The lines joining the incenter of a triangle to the vertices divide the triangle into three triangles. If one of these triangles is similar to the initial one,determine the angles of the triangle.
geometryincenter
four lines have a common point

Source: Croatian NMC 2005, 3rd Grade

5/8/2007
The incircle of a triangle ABCABC touches AC,BCAC, BC , and ABAB at M,NM , N, and RR, respectively. Let SS be a point on the smaller arc MNMN and tt be the tangent to this arc at SS . The line tt meets NCNC at PP and MCMC at QQ. Prove that the lines AP,BQ,SR,MNAP, BQ, SR, MN have a common point.
geometrygeometry proposed
circumcenters of two triangles

Source: Croatian NMC 2005, 2nd Grade

5/8/2007
Let UU be the incenter of a triangle ABCABC and O1,O2,O3O_{1}, O_{2}, O_{3} be the circumcenters of the triangles BCU,CAU,ABUBCU, CAU, ABU , respectively. Prove that the circumcircles of the triangles ABCABC and O1O2O3O_{1}O_{2}O_{3} have the same center.
geometrycircumcircleincentergeometric transformationhomothetygeometry proposed
P(x) \geq (x + 1)^{n}

Source: Croatian NMC 2005, 4 th Grade

5/7/2007
Let P(x)P(x) be a monic polynomial of degree nn with nonnegative coefficients and the free term equal to 11. Prove that if all the roots of P(x)P(x) are real, then P(x)≥(x+1)nP(x) \geq (x+1)^{n} holds for every x≥0x \geq 0.
algebrapolynomiallimitinequalitiesalgebra unsolved