MathDB

5

Part of 2017 Saint Petersburg Mathematical Olympiad

Problems(3)

St. Petersburg MO 2017 Grade 9 P5

Source: St. Petersburg MO 2017 Grade 9 P5

5/3/2018
Given a scalene triangle ABCABC with ∠B=130∘\angle B=130^{\circ}. Let HH be the foot of altitude from BB. DD and EE are points on the sides ABAB and BCBC, respectively, such that DH=EHDH=EH and ADECADEC is a cyclic quadrilateral. Find ∠DHE\angle{DHE}.
geometry
Simple inequality

Source: St. Petersburg 2017

8/18/2017
Let x,y,z>0x,y,z>0 and xyz=xy+yz+zx\sqrt{xyz}=xy+yz+zx. Prove thatx+y+z≤13.x+y+z\leq \frac{1}{3}.
inequalities
St. Petersburg MO 2017 Grade 11 P5

Source: St. Petersburg MO 2017 Grade 11 P5

5/3/2018
Given a tetrahedron PABCPABC, draw the height PHPH from vertex PP to ABCABC. From point HH, draw perpendiculars HA’,HB’,HC’HA’,HB’,HC’ to the lines PA,PB,PCPA,PB,PC. Suppose the planes ABCABC and A’B’C’A’B’C’ intersects at line ℓ\ell. Let OO be the circumcenter of triangle ABCABC. Prove that OH⊥ℓOH\perp \ell.
geometry