min perimeter, triangle with integer sidelengths, <B = 2 <А, <C > 90^o

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min perimeter, triangle with integer sidelengths, <B = 2 <А, <C > 90^o

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Source: 2007 All-Ukrainian Correspondence MO of magazine ''In the World of Mathematics'', grades 5-11 p9

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#1 h Apr 28, 2021, 4:39 PM • 4 Y

Y by HWenslawski, Mango247, Mango247, Mango247

In triangle $ABC$ , the lengths of all sides are integers, $\angle B=2 \angle A$ and $\angle C> 90^o$ . Find the smallest possible perimeter of this triangle.

This post has been edited 2 times. Last edited by parmenides51, Apr 29, 2021, 12:33 PM
Reason: source typo

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#2 h Apr 28, 2021, 9:17 PM

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We have $b^2=a(a+c)$ and $a^2+b^2<c^2$
$a|b^2$ so $a=xy^2,b=xyz$ for some integers $x,y,z$
$c=x(z^2-y^2)$ and we have $c<a+b \to z^2-y^2<y^2+yz \to z<2y$ and
$c^2>a^2+b^2 \to (z^2-y^2)^2>y^4+y^2z^2 \to z^2>3y^2$ so $(2y-1)^2 \geq z^2>3y^2 \to y \geq 4 \to z\geq 7$
And $P=a+b+c=x(z^2+yz) \geq 77$ (for $(a,b,c)=(16,28,33)$ )

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