Olympiad problems

2007 Nicolae Păun, 1

Let be nine nonzero decimal digits $ a_1,a_2,a_3,b_1,b_2,b_3,c_1,c_2,c_3 $ chosen such that the polynom $$ \left( 100a_1+10a_2+a_3 \right) X^2 +\left( 100b_1+10b_2+b_3 \right) X +100c_1+10c_2+c_3 $$ admits at least a real solution. Prove that at least one of the polynoms $ a_iX^2+b_iX+c_i\quad (i\in\{1,2,3\}) $ admits at least a real solution. [i]Nicolae Mușuroia[/i]

2011 N.N. Mihăileanu Individual, 2

Tags: find all functions , polynom

Let $ 0 $ be a root for a polynom $ f\in\mathbb{R}[X] $ that has the property that $ f(X^2-X+1) =f^2(X)-f(X)+1. $ Determine this polynom. [i]Nelu Chichirim[/i]

2011 N.N. Mihăileanu Individual, 1

Tags: algebra , quadratic , polynom

Let be a quadratic polynom that has the property that the modulus of the sum between the leading and the free coefficient is smaller than the modulus of the middle coefficient. Prove that this polynom admits two distinct real roots, one belonging to the interval $ (-1,1) , $ and the other belonging outside of the interval $ (-1,1). $

2011 Laurențiu Duican, 3

Tags: cauchy schwarz , polynom , algebra

Let $ n\ge 2 $ be a perfect square and let be $ n $ natural numbers $ m_1,m_2,\ldots ,m_n. $ Prove that if the polynom $$ X^2-\left( 1+ m_1^2+m_2^2+\cdots +m_n^2 \right) X+m_1m_2+m_2m_3+\cdots +m_{n-1}m_n +m_nm_1\in \mathbb{N} [X] $$ is reducible, then its two roots are perfect squares.

2010 Laurențiu Panaitopol, Tulcea, 3

Tags: polynom , algebra

Let be two polynoms $ P,Q\in\mathbb{R} [X] $ having the property that $$ \left| \{ n\in\{ 0\}\cup\mathbb{N} | P(n)\le Q(n) \} \right| =\left| \{ n\in\{ 0\}\cup\mathbb{N} | P(n)\ge Q(n) \} \right| =\infty .$$ Show that $ P=Q. $ [i]Laurențiu Panaitopol[/i]

2007 Nicolae Păun, 3

Tags: polynom , function , algebra

Let $ a,b,c,d $ be four real numbers such that $ |ax^3+bx^2+cx+d|\le 1,\forall x\in [0,1] . $ Prove that $ |dx^2+cx^2+bx+a|\le 9/2,\forall x\in [0,1] . $ [i]Lavinia Savu[/i]

2011 Laurențiu Duican, 4

Tags: ring theory , field , finite field , polynom , irreducibility over field , algebra

Consider a finite field $ K. $ [b]a)[/b] Prove that there is an element $ k $ in $ K $ having the property that the polynom $ X^3+k $ is irreducible in $ K[X], $ if $ \text{ord} (K)\equiv 1\pmod {12}. $ [b]b)[/b] Is [b]a)[/b] still true if, intead, $ \text{ord} (K) \equiv -1\pmod{12} ? $ [i]Dorel Miheț[/i]

1986 Traian Lălescu, 1.3

Tags: algebra , real analysis , polynom , supremum , polynomial function , extremal

Let be four real numbers. Find the polynom of least degree such that two of these numbers are some locally extreme values, and the other two are the respective points of local extrema.

2008 Grigore Moisil Intercounty, 2

Tags: complex number , polynom , linear algebra

Let be a polynom $ P $ of grade at least $ 2 $ and let be two $ 2\times 2 $ complex matrices such that $$ AB-BA\neq 0=P(AB)-P(BA). $$ Prove that there is a complex number $ \alpha $ having the property that $ P(AB)=\alpha I_2. $ [i]Titu Andreescu[/i] and [i]Dorin Andrica[/i]