Found problems: 109
Determine the quadratic functions $f(x) = ax^2 + bx + c$ for which there exists an interval $(h, k)$ such that for all $x \in (h, k)$ it holds that $f(x)f(x + 1) < 0$ and $f(x)f(x -1) < 0$.
The square polynomial $x^2+ax+b+1$ has natural roots. Prove that $(a^2+b^2)$ is a composite number.
Cris has the equation $-2x^2 + bx + c = 0$, and Cristian increases the coefficients of the Cris equation by $1$, obtaining the equation $-x^2 + (b + 1) x + (c + 1) = 0$. Mariloli notices that the real solutions of the Cristian's equation are the squares of the real solutions of the Cris equation. Find all possible values that can take the coefficients $b$ and $c$.
Determine all the integers $a$ and $b$, such that $\sqrt{2010 + 2 \sqrt{2009}}$ be a solution of the equation $x^2 + ax + b = 0$. Prove that for such $a$ and $b$ the number$\sqrt{2010 - 2 \sqrt{2009}}$ is not a solution to the given equation.
Integers $a,b,c$ are such that the values of the trinomials $bx^2+cx+a$ and $cx^2+ax+b$ at $x=1234$ coincide. Can the first trinomial at $x = 1$ take the value $2009$?
The absolute values of all roots of the quadratic equation $x^2+Ax+B = 0$ and $x^2+Cx+D = 0$ are less then $1$. Prove that so are absolute values of the roots of the quadratic equation $x^2 + \frac{A + C}{2} x + \frac{B + D}{2} = 0$.
Find all pairs of real numbers $(x, y)$ that satisfy the equation $(x + y)^2 = (x + 3) (y - 3)$.
Prove that a quadratic trinomial $x^2 + ax + b (a, b \in R)$ cannot attain at ten consecutive integral points values equal to powers of $2$ with non-negative integral exponent.
[i](F. Petrov )[/i]
Consider the trinomial $f(x) = x^2 + 2bx + c$ with integer coefficients $b$ and $c$.
Prove that if $f(n) \ge 0$ for all integers $n$, then $f(x) \ge 0$ even for all rational numbers $x$.
Prove that if equations $$x^2 + mx + n = 0
\,\,\,\, and\,\, \,\,
x^2 + px + q = 0$$
have a common root, there is a relationship between the coefficients of these equations
$$ (n - q)^2 - (m - p) (np - mq) = 0.$$
A $6\times6$ square is given, and a quadratic trinomial with a positive leading coefficient is placed in each of its cells. There are $108$ coefficents in total, and these coefficents are chosen from the set $[-66;47]$, and each coefficient is different from each other. Prove that there exists at least one column such that the polynomial you get by summing the six trinomials in that column has a real root.
* The quadratic expression $ax^2 + bx + c$ is a square (of an integer) for any integer $x$. Prove that $ax^2 + bx + c = (dx + e)^2$ for some integers d and e.
$a, b, c, d$ are integers. Show that $x^2 + ax + b = y^2 + cy + d$ has infinitely many integer solutions iff $a^2 - 4b = c^2 - 4d$.
What is the smallest possible natural number $n$ for which the equation $x^2 -nx + 2014 = 0$ has integer roots?
Given integers $a$, $ b$ and $c$, $c\ne b$. It is known that the square trinomials $ax^2 + bx + c$ and $(c-b)x^2 + (c- a)x + (a + b)$ have a common root (not necessarily integer). Prove that $a+b+2c$ is divisible by $3$.
(a) Let $u$, $v$, and $w$ be the real solutions to the equation $x^3 - 7x + 7 = 0$. Show that there exists a quadratic polynomial $f$ with rational coefficients such that $u = f(v)$, $v = f(w)$, and $w = f(u)$.
(b) Let $u$, $v$, and $w$ be the real solutions to the equation $x^3 -7x+4 = 0$. Show that there does not exist a quadratic polynomial $f $with rational coefficients such that $u = f(v)$, $v = f(w)$, and $w = f(u)$.
Product of square trinomials $x^2 - a_1x + b_1$, $x^2 - a_2x + b_2$, $...$, $x^2-a_nx + b_n$ is equal to the polynomial $P(x) = x^{2n} +c_1x^{2n-1} +c_2x^{2n-2} +...+ c_{2n-1}x + c_{2n}$, where the coefficients are $c_1$, $c_2$, $...$ , $c_{2n}$ are positive. Show that for some $k$ ($1\le k \le n$) the coefficients $a_k$ and $b_k$ are positive.
Show that, for each integer $z \ge 3$, there exist two two-digit numbers $A$ and $B$ in base $z$, one equal to the other one read in reverse order, such that the equation $x^2 -Ax+B$ has one double root. Prove that this pair is unique for a given $z$. For instance, in base $10$ these numbers are $A = 18, B = 81$.
Calculate all real numbers $r $ with the following properties:
If real numbers $a, b, c$ satisfy the inequality$ | ax^2 + bx + c | \le 1$ for each $x \in [ - 1, 1]$, then they also satisfy the inequality $| cx^2 + bx + a | \le r$ for each $ x \in [- 1, 1]$.
The quadratic trinomials $ f $, $ g $ and $ h $ are such that for every real $ x $ the numbers $ f (x) $, $ g (x) $ and $ h (x) $ are the lengths of the sides of some triangles, and the numbers $ f (x) -1 $, $ g (x) -1 $ and $ h (x) -1 $ are not the lengths of the sides of the triangle. Prove that at least of the polynomials $ f + g-h $, $ f + h-g $, $ g + h-f $ is constant.
Prove that if the roots of the equation $x^2 + px + q = 0$ are real, then for any real number $a$ the roots of the equation $$x^2 + px + q + (x + a) (2x + p) = 0$$ are also real.
Let $p,q, r, s$ be real numbers with $q \ne -1$ and $s \ne -1$. Prove that the quadratic equations $x^2 + px+q = 0$ and $x^2 +rx+s = 0$ have a common root, while their other roots are inverse of each other, if and only if $pr = (q+1)(s+1)$ and $p(q+1)s = r(s+1)q$.
(A double root is counted twice.)
Find the integers $a \ne 0, b$ and $c$ such that $x = 2 +\sqrt3$ would be a solution of the quadratic equation $ax^2 + bx + c = 0$.
A quadratic trinomial with integer coefficients is called [i]admissible [/i] if its leading coeffi cient is $1$, its roots are integers and the absolute values of coefficients do not exceed $2013$. Basil has summed up all admissible quadratic trinomials. Prove that the resulting trinomial has no real roots.
Does there exist a quadratic trinomial $p(x)$ with integer coefficients such that, for every natural number $n$ whose decimal representation consists of digits $1$, $p(n)$ also consists only of digits $1$?