Found problems: 109
Find the smallest positive integer $a$ such that for some integers $b$, $c$ the polynomial $ax^2 - bx + c$ has two distinct zeros in the interval $(0,1)$.
The families of second degree functions $f_m, g_m: R\to R, $ are considered , $f_m (x) = (m^2 + 1) x^2 + 3mx + m^2 - 1$, $g_m (x) = m^2x^2 + mx - 1$, where $m$ is a real nonzero parameter.
Show that, for any function $h$ of the second degree with the property that $g_m (x) \le h (x) \le f_m (x)$ for any real $x$, there exists $\lambda \in [0, 1]$ which verifies the condition $h (x) = \lambda f_m (x) + (1- \lambda) g_m (x)$, whatever real $x$ is.
Let $a$ and $b$ be the roots of the equation $x^2 + x - 3 = 0$. Find the value of the expression $4 b^2 -a^3$.
Given the equations
(1) $ax^2 + bx + c = 0$
(2)$ -ax^2 + bx + c = 0$
prove that if $x_1$ and $x_2$ are some roots of equations (1) and (2), respectively, then there is a root $x_3$ of the equation $$\frac{a}{2}x^2 + bx + c = 0$$ such that either $x_1 \le x_3 \le x_2$ or $x_1 \ge x_3 \ge x_2$.
The teacher wrote on a blackboard: $$x^2 + 10x + 20$$ Then all the pupils in the class came up in turn and either decreased or increased by $1$ either the free coefficient or the coefficient at $x$, but not both. Finally they have obtained: $$x^2 + 20x + 10$$ Is it true that some time during the process there was written the square polynomial with the integer roots?
For any real number$m$, the equation $$x^2+(m-2)x- (m+3)=0$$ has two solutions, denoted $x_1 $and $ x_2$. Determine $m$ such that $x_1^2+x_2^2$ is the minimum possible.
Suppose that $a, b$ are natural numbers so that all the roots of $x^2 + ax - b$ and $x^2 - ax + b$ are integers. Show that exists a right triangle with integer sides, with $a$ the length of the hypotenuse and $b$ the area .
The monic quadratic trinomial $f(x)$ has $2$ different roots. Could it be that the equation $f(f(x)) = 0$ has $3$ different root, and the equation $f(f(f(x))) = 0$ has $7$ different roots?
The equation $(x+a) (x+b) = 9$ has a root $a+b$. Prove that $ab\le 1$.