1

Part of 2005 Italy TST

Problems(2)

Two fixed points of a function

Source:

f:\{1, 2,\ldots ,1600\}\rightarrow\{1, 2,\ldots ,1600\}

f(1)=1

and f^{2005}(x)=x \text{for}\ x=1,2,\ldots ,1600.

(a)

f

has a fixed point different from

1

(b)

n>1600

f:\{1,\ldots ,n\}\rightarrow\{1,\ldots ,n\}

satisfying the above condition has at least two fixed points.

functionalgebra unsolvedalgebra

Least number of conspiring students

Source:

A stage course is attended by

n \ge 4

students. The day before the final exam, each group of three students conspire against another student to throw him/her out of the exam. Prove that there is a student against whom there are at least

\sqrt[3]{(n-1)(n- 2)}

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