Equations in polynomials (Table of contents)
# Equations in Polynomials: Problems and Solutions

**Probem 1** Find all polynomials \(P\) such that \(P(x)^2+P(\frac1x)^2=
P(x^2)P(\frac1{x^2})\).
**Probem 2** Do there exist non-linear polynomials \(P\) and \(Q\) such that
\(P(Q(x))=(x-1)(x-2)\cdots(x-15)\)?
**Probem 3** Determine all polynomials \(P\) for which \(P(x)^2-2=
2P(2x^2-1)\).
**Probem 4** Determine all polynomials \(P\) for which \(P(x)^2-1=
4P(x^2-4x+1)\).
**Probem 5** For which real values of \(a\) does there exist a rational
function \(f(x)\) that satisfies \(f(x^2)=f(x)^2-a\)?
**Probem 6** Find all polynomials \(P\) satisfying \(P(x^2+1)=P(x)^2+1\) for
all \(x\).
**Probem 7** If a polynomial \(P\) with real coefficients satisfies for all
\(x\) \[P(\cos x)=P(\sin x),\] prove that there exists a polynomial
\(Q\) such that for all \(x\), \(P(x)=Q(x^4-x^2).\)
**Probem 8** Find all quadruples of polynomials \((P_1,P_2,P_3,P_4)\) such
that, whenever natural numbers \(x,y,z,t\) satisfy \(xy-zt=1\), it
holds that \(P_1(x)P_2(y)-P_3(z)P_4(t)=1\).
**Probem 9 (IMO
2004.2)** Find all polynomials \(P(x)\) with real coefficients that
satisfy the equality \[P(a-b)+P(b-c)+P(c-a)=2P(a+b+c)\] for all
triples \(a,b,c\) of real numbers such that \(ab+bc+ca=0\).
**Probem 10**
**Probem 11** Prove that if the polynomials \(P\) and \(Q\) have a real root
each and \[P(1+x+Q(x)^2)=Q(1+x+P(x)^2),\] then \(P\equiv Q\).
**Probem 12** If \(P\) and \(Q\) are monic polynomials with \(P(P(x))=Q(Q(x))\),
prove that \(P\equiv Q\).
**Probem 13** Assume that there exist complex polynomials \(P,Q,R\) such
that \[P^a+Q^b=R^c,\] where \(a,b,c\) are natural numbers. Show
that \(\frac1a+\frac1b+\frac1c > 1\).
**Probem 14** The lateral surface of a cylinder is divided by \(n-1\)
planes parallel to the base and \(m\) meridians into \(mn\) cells
(\(n\geq1\), \(m\geq3\)). Two cells are called neighbors if they have
a common side. Prove that it is possible to write real numbers
in the cells, not all zero, so that the number in each cell
equals the sum of the numbers in the neighboring cells, if and
only if there exist \(k,l\) with \(n+1\nmid k\) such that \[\cos
\frac{2l\pi}m+\cos\frac{k\pi}{n+1}=\frac12.\]

- (a) If a real polynomial \(P(x)\) satisfies \(P(x)\geq0\) for all
\(x\), show that there exist real polynomials \(A(x)\) and \(B(x)\) such
that \(P(x)=A(x)^2+B(x)^2\).

- (b) If a real polynomial \(P(x)\) satisfies \(P(x)\geq0\) for all \(x\geq0\), show that there exist real polynomials \(A(x)\) and \(B(x)\) such that \(P(x)=A(x)^2+xB(x)^2\).