tag:blogger.com,1999:blog-2400513859305780710.post6617077137480733638..comments2019-07-11T22:41:53.239-07:00Comments on Mathematical Food For Thought: Fishy Triangular Number. Topic: Inequalities. Level: AIME.Jeffrey Wanghttp://www.blogger.com/profile/11114458640271201663noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-2400513859305780710.post-13221513848627533642007-04-12T13:38:36.000-07:002007-04-12T13:38:36.000-07:00Man, I was about to post a problem like that.Anywa...Man, I was about to post a problem like that.<br><br>Anyway, consider the largest and smallest of the five variables; WLOG the largest is a and the smallest is e. If a is greater than 1/2, then either b or c must be greater than a, contradiction; hence a ≤ 1/2. Similarly, if e is less than 1/2, then either a or b must be less than e, contradiction; hence e ≥ 1/2. Together, this implies a = b = c = d = e = 1/2.t0rajir0unoreply@blogger.comtag:blogger.com,1999:blog-2400513859305780710.post-74459803918590909522007-04-12T14:53:40.000-07:002007-04-12T14:53:40.000-07:00Hmm, I'm not sure you aren't losing any ge...Hmm, I'm not sure you aren't losing any generality with that WLOG. Since the set of equations is cyclic, picking one to be the largest and another to be the smallest (especially ones that share an equation) should change the problem...<br><br>There is a way to avoid that, in any case, because we can say WLOG $ a $ is the maximum, but then $ e^2 = a^3+b^3 \ge c^3+b^3 = a^2 $ so $ e \ge a $ and thus $ e = a $. We can make similar arguments for the remaining variables.paladin8noreply@blogger.comtag:blogger.com,1999:blog-2400513859305780710.post-5268261408270798382007-04-12T15:47:28.000-07:002007-04-12T15:47:28.000-07:00Sorry, the WLOG should be made separately for the ...Sorry, the WLOG should be made separately for the largest and the smallest. That is, do the argument for the largest first, then the argument for the smallest.t0rajir0unoreply@blogger.comtag:blogger.com,1999:blog-2400513859305780710.post-31072032763138940202007-04-12T15:58:11.000-07:002007-04-12T15:58:11.000-07:00Oh, I see. That makes sense.Oh, I see. That makes sense.paladin8noreply@blogger.com