MathDB

1.1

Part of 2016 Saudi Arabia Pre-TST

Problems(2)

x + y + z + 1/2x yz >= 4 when x^2 +y^2 + z^2 = 2(x y + yz + z x)

Source: 2016 Saudi Arabia Pre-TST Level 4 1.1

9/13/2020
Let x,y,zx, y, z be positive real numbers satisfy the condition x2+y2+z2=2(xy+yz+zx)x^2 +y^2 + z^2 = 2(x y + yz + z x). Prove that x+y+z+12xyz≥4x + y + z + \frac{1}{2x yz} \ge 4
algebrainequalities
tangent circle to 3 given circles internally also tangent to a 4th circle

Source: 2016 Saudi Arabia Pre-TST Level 4+ 1.1

9/13/2020
Let ABCABC be an acute, non isosceles triangle, AX,BY,CZAX, BY, CZ are the altitudes with X,Y,ZX, Y, Z belong to BC,CA,ABBC, CA,AB respectively. Respectively denote (O1),(O2),(O3)(O_1), (O_2), (O_3) as the circumcircles of triangles AYZ,BZX,CXYAY Z, BZX, CX Y . Suppose that (K)(K) is a circle that internal tangent to (O1),(O2),(O3)(O_1), (O_2), (O_3). Prove that (K)(K) is tangent to circumcircle of triangle ABCABC.
geometrytangent circlesaltitudes