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Part of 2012 Kosovo National Mathematical Olympiad

Problems(4)

Kosovo MO 2012 Grade 9, Problem 1

Source: Kosovo MO 2012 Grade 9, Problem 1

Find the value of

(1+2)(1+2^2)(1+2^4)(1+2^8)...(1+2^{2048})

Kosovo MO 2012 Grade 10, Problem 1

Source: Kosovo MO 2012 Grade 10, Problem 1

Find the two last digits of

2012^{2012}

Kosovo MO 2012 Grade 11, Problem 1

Source: Kosovo MO 2012 Grade 11, Problem 1

(x^2-x-1)^n=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x+a_0

a_i,i\in\{0,1,2,..,2n\}

a_1+a_3+...+a_{2n-1}

a_0+a_2+a_4+...+a_{2n}

Kosovo MO 2012 Grade 12, Problem 1

Source: Kosovo MO 2012 Grade 12, Problem 1

Prove that for all

n\in\mathbb{N}

\sum_{k=0}^n\dbinom {n}{k}^2=\dbinom {2n}{n}