MathDB

7

Part of 2009 Saint Petersburg Mathematical Olympiad

Problems(3)

Special sequence

Source: St Peterburg Olympiad 2009, Grade 11, P7

8/30/2017
f(x)=x2+xf(x)=x^2+x b1,...,b10000>0b_1,...,b_{10000}>0 and ∣bn+1−f(bn)∣≤11000|b_{n+1}-f(b_n)|\leq \frac{1}{1000} for n=1,...,9999n=1,...,9999 Prove, that there is such a1>0a_1>0 that an+1=f(an);n=1,...,9999a_{n+1}=f(a_n);n=1,...,9999 and ∣an−bn∣<1100|a_n-b_n|<\frac{1}{100}
algebra
Another geometry

Source: St Petersburg Olympiad 2009, Grade 10, P7

8/30/2017
Points Y,XY,X lies on AB,BCAB,BC of △ABC\triangle ABC and X,Y,A,CX,Y,A,C are concyclic. AXAX and CYCY intersect in OO. Points M,NM,N are midpoints of ACAC and XYXY. Prove, that BOBO is tangent to circumcircle of △MON\triangle MON
geometrycircumcircle
Discriminants

Source: St Petersburg Olympiad 2009, Grade 9, P7

8/30/2017
Discriminants of square trinomials f(x),g(x),h(x),f(x)+g(x),f(x)+h(x),g(x)+h(x)f(x),g(x),h(x),f(x)+g(x),f(x)+h(x),g(x)+h(x) equals 11. Prove that f(x)+h(x)+g(x)≡0f(x)+h(x)+g(x) \equiv 0
algebra