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Question Number 74246 by TawaTawa last updated on 20/Nov/19

Answered by MJS last updated on 20/Nov/19

$$1249$$$$...$$$$12526$$$$24\mathrm{triples}$$$$2347$$$$...$$$$22426$$$$22\mathrm{triples}$$$$3445$$$$...$$$$32425$$$$21\mathrm{triples}$$$$$$$$\mathrm{starting}\mathrm{with}4:19$$$$\mathrm{starting}\mathrm{with}5:18$$$$6:16$$$$7:15$$$$8:13$$$$9:12$$$$10:10$$$$11:9$$$$12:7$$$$13:6$$$$14:4$$$$15:3$$$$16:1\left[161719\right]$$$$$$$$\Rightarrow$$$$200\mathrm{triples}$$

Commented by mind is power last updated on 21/Nov/19

$$Nice$$

Commented by TawaTawa last updated on 21/Nov/19

$$\mathrm{God}\mathrm{bless}\mathrm{your}\mathrm{sir}$$

Answered by mind is power last updated on 21/Nov/19

$$r+s+t⩾3+t\Rightarrow t⩽49$$$$r+s+t⩽t-2+t-1+t\Rightarrow t⩾\frac{55}{3}\Rightarrow t⩾19$$$$r+s+t⩾1+s+t$$$$\Rightarrow s⩽51-t$$$$r+s+t⩽s-1+s+t\Rightarrow 52⩽2s-1+t$$$$\Rightarrow s⩾\frac{53-t}{2}$$$$\Rightarrow \frac{53-t}{2}⩽s⩽51-t$$$$s⩽t-1\Rightarrow s⩽min(51-t,t-1)$$$$t-1⩾51-t\Rightarrow t⩾26$$$$t⩾26\frac{53-t}{2}⩽s⩽51-t$$$$t⩽26\frac{53-t}{2}⩽s⩽t-1$$$$\underset{t=19}{\sum ^{26}}\sum _{\frac{53-t}{2}⩽s⩽t-1}+\underset{27}{\sum ^{49}}\sum _{\frac{53-t}{2}⩽s⩽51-t}$$$$\underset{9}{\sum ^{12}}\sum _{26-n⩽s⩽2n}1+\underset{10}{\sum ^{13}}\sum _{27-n⩽s⩽2n-1}1+\underset{14}{\sum ^{24}}\sum _{27-n⩽s⩽51-2n}1$$$$+\underset{13}{\sum ^{24}}\sum _{26-n⩽s⩽50-2n}$$$$=\underset{9}{\sum ^{12}}(3n-25)+\underset{10}{\sum ^{13}}(3n-27)+\underset{14}{\sum ^{24}}(-n+25)$$$$+\underset{13}{\sum ^{24}}(-n+25)$$$$=2+5+8+11+3+6+9+12$$$$$$$$=$$$$+2.\underset{14}{\sum ^{24}}(-n+25)+12$$$$=(11+1).11+12+56=132+12+56=200$$

Commented by TawaTawa last updated on 21/Nov/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

Commented by mind is power last updated on 21/Nov/19

$$y^{\prime }rewelcom$$

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