RU-ALLRUSSIAN 1982 Problem 21 Do there exist polynomials p(x),q(x),r(x)\mathrm{p}(\mathrm{x}), \mathrm{q}(\mathrm{x}), \mathrm{r}(\mathrm{x})p(x),q(x),r(x) such that p(x−y+z)3+q(y−z−1)3+r(z−2x+1)3=1\mathrm{p}(\mathrm{x} - \mathrm{y} + \mathrm{z})^{3} + \mathrm{q}(\mathrm{y} - \mathrm{z} - 1)^{3} + \mathrm{r}(\mathrm{z} - 2\mathrm{x} + 1)^{3} = 1p(x−y+z)3+q(y−z−1)3+r(z−2x+1)3=1 for all x,y,z\mathrm{x}, \mathrm{y}, \mathrm{z}x,y,z ? Do there exist polynomials p(x),q(x),r(x)\mathrm{p}(\mathrm{x}), \mathrm{q}(\mathrm{x}), \mathrm{r}(\mathrm{x})p(x),q(x),r(x) such that p(x−y+z)3+q(y−z−1)3+r(z−x+1)3=1\mathrm{p}(\mathrm{x} - \mathrm{y} + \mathrm{z})^{3} + \mathrm{q}(\mathrm{y} - \mathrm{z} - 1)^{3} + \mathrm{r}(\mathrm{z} - \mathrm{x} + 1)^{3} = 1p(x−y+z)3+q(y−z−1)3+r(z−x+1)3=1 for all x,y,z\mathrm{x}, \mathrm{y}, \mathrm{z}x,y,z ? Dataset text NuminaMath olympiads. Quoted with source. Solutions No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label. Add a solution PreviousProblem 20 NextProblem 22 Locator ru-allrussian/1982/p21 · Tag 4SD9NBMW8H · Edit this page on GitHub · Keys: j next, k previous