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IMO training

  1. Given that \(a,b,c> 0\) and \(a+b+c=ab+bc+ca\).Prove that: \[\sum \frac{a^2}{a^2-a+1}\leq 3\]

  2. Let \(x;y;z >0\) . Prove that \(xyz+2(x^2+y^2+z^2)+8\geq 5(x+y+z)\)
    Revised version,find all the constant \(k\) such that \(xyz+k(x^2+y^2+z^2)+8\geq (k+3)(x+y+z)\) for all \(x;y;z >0\)

  3. Let \(x;y;z >0\) Prove that \((x+y+z)^{2}(x^2+y^2)(y^2+z^2)(z^2+x^2)\geq 8(x^2y^2+y^2z^2+z^2x^2)^{2}\)

  4. Let \(0\leq a;b;c\leq 1\) . Put \(x=1-a+ab;y=1-b+bc;z=1-c+ca\)
    a) Prove that \(x+y+z\geq 2\)
    b)Prove that \(x^2+y^2+z^2\geq \frac{3}{2}\)
    c)Prove or disprove \[x^3+y^3+z^3\geq \frac{5}{4}\]

  5. Let be given positive integer \(n\). Find the least real number \(k\) such that \((xy)^{k}(x^n+y^n)\leq 2\) for all positive real numbers \(x;y\) satisfying the condition \(x+y=2\)

These problems has been taken from Vietnam TST practice for Hong Kong IMO 2016.

Note by Ms Ht
5 months ago

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