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Problem 1 Let for 𝑛 - natural number we define 𝑎𝑛 = 1 + 22 + 33 + . . . + 𝑛𝑛 . Show that there exist infinitely many 𝑛 such that 𝑎𝑛 is composite number. Problem 2 Solve in natural numbers ∙ 𝑥3 + 𝑦 4 = 22003 ∙ 𝑥3 + 𝑦 4 = 22005 Problem 3 Denote by (2𝑘 + 1)!! = 1 · 3 · 5 · . . . · (2𝑘 − 1) · (2𝑘 + 1). Show that for odd n we can state that 2𝑛! − 1 is a multiple of 𝑛!! Problem 4 Find all natural 𝑛, so that 2𝑛 + 𝑛2 004 is prime. Problem 5 What is the hundred digit of number 22006 + 22007 + 22008 + 22009 + 22010 Problem 6 𝑛𝑛 𝑛 Show that for any 𝑛 ≥ 3 we have that 𝑛𝑛 − 𝑛𝑛 is a multiple of 1989. Problem 7 𝑛 Show that for any 𝑛 we have that 23 + 1 is a multiple of 3𝑘+1 but not a multiple of 3𝑘+2 . Problem 8 Find all natural k, for which ∙ 3𝑘 − 1 is a multiple of 13 ∙ 3𝑘 + 1 is a multiple of 13 1