MathDB

4

Part of 2017 239 Open Mathematical Olympiad

Problems(2)

f^k(a)-a has no root

Source: 239 2017 S4

5/22/2020
A polynomial f(x)f(x) with integer coefficients is given. We define d(a,k)=∣fk(a)−a∣.d(a,k)=|f^k(a)-a|. It is known that for each integer aa and natural number kk, d(a,k)d(a,k) is positive. Prove that for all such a,ka,k, d(a,k)≥k3.d(a,k) \geq \frac{k}{3}. (fk(x)=f(fk−1(x)),f0(x)=x.f^k(x)=f(f^{k-1}(x)), f^0(x)=x.)
algebrapolynomial
A cannon shooting at an invisible tank in 100x100 table

Source: 239 2017 J4

6/3/2020
An invisible tank is on a 100×100100 \times 100 table. A cannon can fire at any 6060 cells of the board after that the tank will move to one of the adjacent cells (by side). Then the progress is repeated. Can the cannon grantee to shoot the tank?
combinatorics