Problems

Problem 1
 

A monic polynomial \( f(x) \) of fourth degree satisfies \( f(1)=10 \), \( f(2)=20 \) and \( f(3)=30 \). Determine \( f(12)+f(-8) \).

Problem 2
 

Consider complex polynomials \( P(x)=x^n+a_1x^{n-1}+\cdots +a_n \) with the zeros \( x_1,\dots,x_n \), and \( Q(x)=x^n+b_1x^{n-1}+\cdots+ b_n \) with the zeros \( x_1^2,\dots,x_n^2 \). Prove that if \( a_1+a_3+a_5+\cdots \) and \( a_2+a_4+a_6+\cdots \) are real numbers, then \( b_1+b_2+\dots+b_n \) is also real.

Problem 3
 

If a polynomial \( P \) with real coefficients satisfies for all \( x \) \[ P(\cos x)=P(\sin x),\] show that there exists a polynomial \( Q \) such that \( P(x)=Q(x^4-x^2) \) for each \( x \).

Problem 4
 

  • (a) Prove that for each \( n\in\mathbb{N} \) there is a polynomial \( T_n \) with integer coefficients and the leading coefficient \( 2^{n-1} \) such that \( T_n(\cos x)=\cos nx \) for all \( x \).

  • (b) Prove that the polynomials \( T_n \) satisfy \( T_{m+n}+T_{m-n}= 2T_mT_n \) for all \( m,n\in\mathbb{N} \), \( m\geq n \).

  • (c) Prove that the polynomial \( U_n \) given by \( U_n(2x)=2T_n(x) \) also has integer coefficients and satisfies \( U_n(x+x^{-1})=x^n+x^{-n} \).

The polynomials \( T_n(x) \) are known as the Chebyshev polynomials.

Problem 5
 

Prove that if \( \cos\frac pq\pi=a \) is a rational number for some \( p,q\in\mathbb{Z} \), then \( a\in\{0,\pm\frac12,\pm1\} \).

Problem 6
 

Prove that the maximum in absolute value of any monic real polynomial of \( n \)-th degree on \( [-1,1] \) is not less than \( \frac1{2^{n-1}} \).

Problem 7
 

The polynomial \( P \) of \( n \)-th degree is such that, for each \( i=0,1,\dots,n \), \( P(i) \) equals the remainder of \( i \) modulo 2. Evaluate \( P(n+1) \).

Problem 8
 

A polynomial \( P(x) \) of \( n \)-th degree satisfies \( P(i)=\frac1i \) for \( i=1,2,\dots,n+1 \). Find \( P(n+2) \).

Problem 9
 

Let \( P(x) \) be a real polynomial.

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

Problem 10
 

Prove that if the equation \( Q(x)=ax^2+(c-b)x+(e-d)=0 \) has real roots greater than 1, where \( a,b,c,d,e\in\mathbb{R} \), then the equation \( P(x)=ax^4+bx^3+cx^2+dx+e=0 \) has at least one real root.

Problem 11
 

A monic polynomial \( P \) with real coefficients satisfies \( |P(i)|< 1 \). Prove that there is a root \( z=a+bi \) of \( P \) such that \( (a^2+b^2+1)^2< 4b^2+1 \).

Problem 12
 

For what real values of \( a \) does there exist a rational function \( f(x) \) that satisfies \( f(x^2)=f(x)^2-a \)? (A rational function is a quotient of two polynomials.)

Problem 13
 

Find all polynomials \( P \) satisfying \( P(x^2+1)= P(x)^2+1 \) for all \( x \).

Problem 14
 

Find all \( P \) for which \( P(x)^2-2=2P(2x^2-1) \).

Problem 15
 

If the polynomials \( P \) and \( Q \) each have a real root and \[ P(1+x+Q(x)^2)=Q(1+x+P(x)^2),\] prove that \( P\equiv Q \).

Problem 16 (IMO04.2)
 

Find all polynomials \( P(x) \) with real coefficients satisfying 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 \( ab+bc+ca=0 \).

Problem 17
 

A sequence of integers \( (a_n)_{n=1}^{\infty} \) has the property that \( m-n\mid a_m-a_n \) for any distinct \( m,n\in\mathbb{N} \). Suppose that there is a polynomial \( P(x) \) such that \( |a_n|< P(n) \) for all \( n \). Show that there exists a polynomial \( Q(x) \) such that \( a_n= Q(n) \) for all \( n \).

Problem 18 (IMO06.5)
 

Let \( P(x) \) be a polynomial of degree \( n> 1 \) with integer coefficients and let \( k \) be a natural number. Consider the polynomial \( Q(x)=P(P(\dots P(P(x))\dots)) \), where \( P \) is applied \( k \) times. Prove that there exist at most \( n \) integers \( t \) such that \( Q(t)=t \).

Problem 19
 

If \( P \) and \( Q \) are monic polynomials such that \( P(P(x))=Q(Q(x)) \), prove that \( P\equiv Q \).

Problem 20
 

Let \( m,n \) and \( a \) be natural numbers and \( p< a-1 \) a prime number. Prove that the polynomial \( f(x)=x^m(x-a)^n+p \) is irreducible.

Problem 21
 

Prove that the polynomial \( F(x)=(x^2+x)^{2^n}+1 \) is irreducible for all \( n\in\mathbb{N} \).

Problem 22
 

A polynomial \( P(x) \) has the property that for every \( y\in \mathbb{Q} \) there exists \( x\in\mathbb{Q} \) such that \( P(x)=y \). Prove that \( P \) is a linear polynomial.

Problem 23
 

Let \( P(x) \) be a monic polynomial of degree \( n \) whose zeros are \( i-1,i-2,\dots,i-n \) (where \( i^2=-1 \)) and let \( R(x) \) and \( S(x) \) be the real polynomials such that \( P(x)=R(x)+iS(x) \). Prove that the polynomial \( R(x) \) has \( n \) real zeros.

Problem 24
 

Let \( a,b,c \) be natural numbers. Prove that if there exist coprime polynomials \( P,Q,R \) with complex coefficients such that \[ P^a+Q^b=R^c,\] then \( \frac1a+\frac1b+\frac1c> 1 \).

Corollary: The Last Fermat Theorem for polynomials.

Problem 25
 

Suppose that all zeros of a monic polynomial \( P(x) \) with integer coefficients are of module 1. Prove that there are only finitely many such polynomials of any given degree; hence show that all its zeros are actually roots of unity, i.e. \( P(x)\mid(x^n-1)^k \) for some natural \( n,k \).


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