Recent content by anemone

Let ##x,\,y>0## and ##x+y \leq 1##. Prove that ##(1-\frac{1}{x^2})(1-\frac{1}{y^2})\geq 9##.

Find all integers n such that ##\dfrac{n^2+3}{2n+4}## is an integer as well.

Find the real solutions for ##\sqrt{z-y^2-6x-26}+x^2+6y+z-8=0##.

Sorry for the late reply to this POTW's thread! I have been real busy with work, and I apologize for MIA for almost two weeks too. Here is another bright solution of other that I wish to share with the community:

Let a and b be positive real numbers such that ##a^5+b^3 \le a^2+b^2##. Prove that ##a(a+b) \le 2##.

Let ##a,\,b,\,c## and ##d## be any four real numbers but not all equal to zero. Prove that the roots of the polynomial ##f(x)=x^6+ax^3+bx^2+cx+d## cannot all be real.

Thanks everyone for the overwhelming participation. I want to encourage for those who have "partially" solved the question right but cannot prove those are the unique answers to have further attempt and I wish you success in finding the proof. However, here is one brilliant solution of someone...

Find all triples (a, b, c) of positive integers such that ##a^3+b^3+c^3=(abc)^2##.

Thanks for so many participation to this POTW! I am delighted to see all the correct solutions! Yes, challenging math problems will be used for high school POTW section because it is our goal to post tricky problems that could stimulate students' creativity, curiosity, self-confidence as well...

I am so sorry for the late reply to this POTW's thread as I have been so extremely busy with work. Here is the solution of other and I hope that readers will appreciate the beauty of the solution as much as I do.

If a and b are two real numbers such that ##a^3-3a^2+5a=1## and ##b^3-3b^2+5b=5##, evaluate a + b.

Congratulations to @malawi_glenn for your correct solution! And thanks for your participation!

Thanks @I like Serena for the reply!

Of course! I don't see why not, and as a matter of fact, thanks for showing interest in the old POTW!

Yes!