Found problems: 9
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). $
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]
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.
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]
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]
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.
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]
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]
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]