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
 
  • (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 \( P(x)=A(x)^2+xB(x)^2 \).

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 \[ \cos \frac{2l\pi}m+\cos\frac{k\pi}{n+1}=\frac12.\]




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