MathDB

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Part of 2012 Saint Petersburg Mathematical Olympiad

Problems(3)

Common roots

Source: St Petersburg Olympiad 2012, Grade 11, P1

9/29/2017
a,b,ca,b,c are reals, such that every pair of equations of x3−ax2+b=0,x3−bx2+c=0,x3−cx2+a=0x^3-ax^2+b=0,x^3-bx^2+c=0,x^3-cx^2+a=0 has common root. Prove a=b=ca=b=c
algebra
system of cubic

Source: St Petersburg Olympiad 2012, Grade 10, P1

9/29/2017
{x3−ax2+b3=0x3−bx2+c3=0x3−cx2+a3=0\begin{cases} x^3-ax^2+b^3=0 \\x^3-bx^2+c^3=0 \\ x^3-cx^2+a^3=0 \end{cases} Prove that system hasn`t solutions if a,b,ca,b,c are different.
algebra
Number of roots

Source: St Petersburg Olympiad 2012, Grade 9, P1

9/29/2017
Find all integer bb such that [x2]−2012x+b=0[x^2]-2012x+b=0 has odd number of roots.
algebranumber theory