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B2

Part of 1967 Putnam

Problems(1)

Putnam 1967 B2

Source: Putnam 1967

5/14/2022
Let 0≤p,r≤10\leq p,r\leq 1 and consider the identities a)  (px+(1−p)y)2=ax2+bxy+cy2,       b)  (px+(1−p)y)(rx+(1−r)y)=αx2+βxy+γy2.a)\; (px+(1-p)y)^{2}=a x^2 +bxy +c y^2, \;\;\;\, b)\; (px+(1-p)y)(rx+(1-r)y) =\alpha x^2 + \beta xy + \gamma y^2. Show that a)  max⁡(a,b,c)≥49,        b)  max⁡(α,β,γ)≥49. a)\; \max(a,b,c) \geq \frac{4}{9}, \;\;\;\; b)\; \max( \alpha, \beta , \gamma) \geq \frac{4}{9}.
Putnaminequalities