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$$1!\times2!\times3!\times...\times100! =1^{100}\times2^{99}\times3^{98}\times...\times100^1$$ $$=\left[1^{100}\times3^{98}\times...\times99^2\right]\times\left[2^{99}\times4^{97}\times...\times100^1\right]$$ $$=\left[1^{50}\times3^{49}\times...\times99^1\right]^2\times\left[2^{99}\times4^{97}\times...\times100^1\right]$$ $$=\left[1^{50}\times3^{49}\times...\times99^1\right]^2\times\left[(2\times1)^{99}\times(2\times2)^{97}\times...\times(2\times50)^1\right]$$ $$=\left[1^{50}\times3^{49}\times...\times99^1\right]^2\times2^{99+97+...+1}\left[1^{99}\times2^{97}\times...\times50^1\right]$$ $$=\left[1^{50}\times3^{49}\times...\times99^1\right]^2\times2^{2500}\left[1^{99}\times2^{97}\times...\times50^1\right]$$ $$\frac{1!\times2!\times3!\times...\times100! }{50!}=\left[1^{50}\times3^{49}\times...\times99^1\right]^2\times2^{2500}\left[1^{98}\times2^{96}\times...\times50^0\right]$$ $$=\left[1^{50}\times3^{49}\times...\times99^1\times2^{1250}\times1^{49}\times2^{48}\times...\times50^0\right]^2$$

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$$111...1=(10n\pm1)^2$$$$=100n^2\pm20n+1$$ $$=10(10n^2\pm2n)+1$$

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$$(a^2+b^2)(c^2+d^2)$$ $$a^2c^2+b^2c^2+a^2d^2+b^2d^2$$ $$(bd-ac)^2=b^2d^2+a^2c^2-2abcd$$ $$(bc+ad)^2=b^2c^2+a^2d^2+2abcd$$ $$(a^2+b^2)(c^2+d^2)=(bd-ac)^2+(bc+ad)^2$$

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$$n^2=(10A+B)^2\\=100A+20AB+B^2$$

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$$n^2 \leq T_a$$ $$n^2 \leq \frac{1}{2}a(a+1)$$ $$2n^2 \leq a^2+a$$ $$2n^2 \leq 2a^2$$ $$n^2 \leq a^2$$ $$n \leq a$$ $$T_{a+2}=\frac{1}{2}(a+2)(a+3)$$ $$=\frac{1}{2}(a^2+5a+6)$$ $$=\frac{1}{2}(a^2+a)+\frac{1}{2}(4a+6)$$ $$=\frac{1}{2}a(a+1)+2a+3$$ $$=T_a+2a+3$$ $$T_a+2a+3 \geq n^2+2n+3$$ $$>n^2+2n+1 = (n+1)^2$$

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$$2(a^2 +b^2 ) = 2a^2 +2b^2$$ $$= a^2 + 2ab + b^2 + a^2 - 2ab + b^2$$ $$= (a+b)^2 +(a-b)^2$$

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