MathDB

7

Part of 1999 All-Russian Olympiad

Problems(2)

Prove that BFO is right if AEDC is cyclic

Source: All-Russian MO 1999

12/31/2012
A circle through vertices AA and BB of triangle ABCABC meets side BCBC again at DD. A circle through BB and CC meets side ABAB at EE and the first circle again at FF. Prove that if points AA, EE, DD, CC lie on a circle with center OO then ∠BFO\angle BFO is right.
radical axisgeometry
x^2+y^3>=x^3+y^4 implies x^3+y^3<=2

Source: All-Russian MO 1999

12/31/2012
Positive numbers x,yx,y satisfy x2+y3≥x3+y4x^2+y^3 \ge x^3+y^4. Prove that x3+y3≤2x^3+y^3 \le 2.
RussiaHigh school olympiadalgebraInequality