Newsletter #47 - Solution

Problem

Find all real solutions of the system

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.16em" columnalign="left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><msup><mo stretchy="false">)</mo><mn>3</mn></msup><mo>=</mo><mi>z</mi><mo separator="true">,</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><mi>y</mi><mo>+</mo><mi>z</mi><msup><mo stretchy="false">)</mo><mn>3</mn></msup><mo>=</mo><mi>x</mi><mo separator="true">,</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><mi>z</mi><mo>+</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>3</mn></msup><mo>=</mo><mi>y</mi><mi mathvariant="normal">.</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{array}{l} (x+y)^3 = z, \\ (y+z)^3 = x, \\ (z+x)^3 = y. \end{array}</annotation></semantics></math>(x+y)3=z,(y+z)3=x,(z+x)3=y.

Solution - by Dashamir Hoxha (Albania)

We want to solve the system:

Subtract the second equation from the third:

Factor the right-hand side:

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>y</mi><mo>−</mo><mi>x</mi><mo>=</mo><mo stretchy="false">(</mo><mi>z</mi><mo>+</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo>−</mo><mi>z</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>z</mi><mo>+</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mi>z</mi><mo>+</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mi>y</mi><mo>+</mo><mi>z</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">y - x = (z+x-y-z)\big((z+x)^2 + (z+x)(y+z) + (y+z)^2\big).</annotation></semantics></math>y−x=(z+x−y−z)((z+x)2+(z+x)(y+z)+(y+z)2).

Thus,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>−</mo><mi>y</mi><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="true" minsize="1.8em" maxsize="1.8em">(</mo><mo stretchy="false">(</mo><mi>y</mi><mo>+</mo><mi>z</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>z</mi><mo>+</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mi>z</mi><mo>+</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo fence="false" stretchy="true" minsize="1.8em" maxsize="1.8em">)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">x-y = (x-y)\Big((y+z)^2+(y+z)(z+x)+(z+x)^2\Big).</annotation></semantics></math>x−y=(x−y)((y+z)2+(y+z)(z+x)+(z+x)2).

Let

Since

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>R</mi><mo>=</mo><msup><mrow><mo fence="true">(</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mi>z</mi><mo>+</mo><mi>x</mi><mo stretchy="false">)</mo><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>+</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mo stretchy="false">(</mo><mi>z</mi><mo>+</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>≥</mo><mn>0</mn><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">R = \left(y+z+\frac{1}{2}(z+x)\right)^2 + \frac{3}{4}(z+x)^2 \geq 0,</annotation></semantics></math>R=(y+z+21(z+x))2+43(z+x)2≥0,

we conclude

By symmetry, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>y</mi><mo>=</mo><mi>z</mi><mo>=</mo><mi>t</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">x=y=z=t.</annotation></semantics></math>x=y=z=t.

Substitute into the first equation:

Hence,

<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn><mtext> or </mtext><mi>t</mi><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow></mfrac></mrow><annotation encoding="application/x-tex">t = 0 \text{ or } t = \pm\frac{1}{2\sqrt{2}}</annotation></semantics></math>t=0 or t=±22<svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/></svg>1

Therefore, the real solutions are:

<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow></mfrac><mo separator="true">,</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow></mfrac><mo separator="true">,</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow></mfrac><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow></mfrac><mo separator="true">,</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow></mfrac><mo separator="true">,</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow></mfrac><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">(x,y,z) \in \{(0,0,0), (\frac{1}{2\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{2\sqrt{2}}), (-\frac{1}{2\sqrt{2}}, -\frac{1}{2\sqrt{2}}, -\frac{1}{2\sqrt{2}})\}</annotation></semantics></math>(x,y,z)∈{(0,0,0),(22<svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/></svg>1,22<svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/></svg>1,22<svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/></svg>1),(−22<svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/></svg>1,−22<svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/></svg>1,−22<svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 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Thank you!

Thanks to Advait Nene (Pittsburgh, PA), Michael David (Canada), CK (California), Ritoprovo Roy (Prague), Dashamir Hoxha (Albania), JW (The Hague), jwenpaq (Singapore), Marek Grabarski (Poland), jc (New York), David J. Webb (Hawaii), Madhur Agrawal (Mumbai), Nikolaj Peters (Denmark), and Ankit (Bay Area) for submitting solutions to this week's challenge problem.