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## International Mathematics Competition for University Students (IMC 2020)

Maple

This year, the International Mathematics Competition for University Students  (IMC) took place online (due to Coronavirus), https://www.imc-math.org.uk/?year=2020

One of the sponsors was Maplesoft.

Here is a Maple solution for one of the most difficult problems.

Problem 4, Day 1.

A polynomial  with real coeffcients satisfies the equation

, for all real .

Prove that  for   .

A Maple solution.

Obviously, the degree of the polynomial must be 101.

We shall find effectively p(x).

 > restart;
 > n:=100;
 (1)
 > p:= x -> add(a[k]*x^k, k=0..n+1):
 > collect(expand( p(x+1) - p(x) - x^n ), x):
 > S:=solve([coeffs(%,x)]):
 > f:=unapply(expand(eval(p(1-x)-p(x), S)), x);
 (2)
 > plot(f, 0..1); # Visual check: f(x)>0 for 0
 > f(0), f(1/4), f(1/2);
 (3)
 > sturm(f(x), x, 0, 1/2);
 (4)

So, the polynomial f has a unique zero in the interval (0, 1/2]. Since f(1/2) = 0  and f(1/4) > 0, it results that  f > 0 in the interval  (0, 1/2). Q.E.D.