1967 IMO SL #4

Suppose medians $m_a$ and $m_b$ of a triangle are orthogonal. Prove that 

a) Using medians of that triangle it is possible to construct a rectangular triangle. 

b) The following inequality:\[5(a^2+b^2-c^2) \geq 8ab,\]is valid, where $a,b$ and $c$ are side length of the given triangle.



a) Let $ABCD$ be a parallelogram and $K,L$ the midpoints of segments $BC, CD$. The sides of $\triangle{AKL}$ are equal and parallel to the medians of triangle $ABC$. This proves the desired. $\square$

b) Using the formulas $4m_a^2=2b^2+2c^2-a^2$, etc., we can see that $$m_a^2+m_b^2=m_c^2 \implies a^2+b^2=5c^2.$$  

Then, we have $$5(a^2+b^2-c^2)=4(a^2+b^2) \geq 8ab,$$ as desired. $\square$


Comments

  1. Anonymous1/23/2022

    nice solution

    ReplyDelete
  2. Anonymous1/24/2022

    part a is very elegant

    great job

    ReplyDelete
  3. Anonymous1/25/2022

    hi

    your part a was rlly elegant as @anonymous (comment#2) said

    waiting for the next blog post :)

    ReplyDelete
  4. schukkayapally1/26/2022

    yeah nice sol

    and also thanks for making this blog :D
    i could get a decent score on the AMC 10
    but im literally failing aime mocks (like actually i get 5s and 6s)
    i prolly wont qual for jmo this year
    but if i keep reading ur blog then i might next year :)

    ReplyDelete

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