# 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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