Home Competitions AMO-Bench 2025 Problem 10
AMO-Bench 2025 Problem 10
Let x , y , z x,y,z x , y , z x y + x z + y z ≠ 1 {xy} + {xz} + {yz} \neq 1 x y + x z + y z = 1 ( x 2 − 1 ) ( y 2 − 1 ) x y + ( x 2 − 1 ) ( z 2 − 1 ) x z + ( z 2 − 1 ) ( y 2 − 1 ) z y = 4 \frac{\left( {{x}^{2} - 1}\right) \left( {{y}^{2} - 1}\right) }{xy} + \frac{\left( {{x}^{2} - 1}\right) \left( {{z}^{2} - 1}\right) }{xz} + \frac{\left( {{z}^{2} - 1}\right) \left( {{y}^{2} - 1}\right) }{zy} = 4 x y ( x 2 − 1 ) ( y 2 − 1 ) + x z ( x 2 − 1 ) ( z 2 − 1 ) + z y ( z 2 − 1 ) ( y 2 − 1 ) = 4 9 x 2 + 13689 y 2 z 2 − 27378 y z + 28561 ( y + z ) 2 9{x}^{2} + {13689}{y}^{2}{z}^{2} - {27378yz} + \frac{28561}{{\left( y + z\right) }^{2}} 9 x 2 + 13689 y 2 z 2 − 27378 y z + ( y + z ) 2 28561
Dataset text AMO-Bench . Open licence.
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