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
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