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# Some questions of UKMO

($$1$$) If $$a,b$$ and $$c$$ are real numbers such that $$a<b<c$$, $$a+b+c=6$$ and $$ab+bc+ca=9$$.

Prove that $$0<a<1<b<3<c<4$$.

($$2$$) Find all triples ($$a,b,c$$) such that

$\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\left(1+\frac{1}{c}\right)=2$

($$3$$) How many perfect squares are there $$\pmod{ 2^n}$$?