Found problems: 8
Find all triples of integers ${(x, y, z)}$ satisfying ${x^3 + y^3 + z^3 - 3xyz = 2003}$
[b]a)[/b] Prove that, for any integer $ k, $ the equation $ x^3-24x+k=0 $ has at most an integer solution.
[b]b)[/b] Show that the equation $ x^3+24x-2016=0 $ has exactly one integer solution.
A given polynomial $P(t) = t^3 + at^2 + bt + c$ has $3$ distinct real roots.
If the equation $(x^2 +x+2013)^3 +a(x^2 +x+2013)^2 + b(x^2 + x + 2013) + c = 0$ has no real roots,
prove that $P(2013) >\frac{1}{64}$
Find all pairs of cubic equations $x^3+ax^2+bx+c=0$ and $x^3+bx^2+ax+c=0$ where $a, b$ are positive integers, $c\neq 0$ is an integer, such that each equation has three integer roots and exactly one of these three roots is common to both the equations.
Determine all $6$-tuples $(x,y,z,u,v,w)$ of integers satisfying the equation
\[x^3+7y^3+49z^3=2u^3+14v^3+98w^3.\]
A cubic trinomial $x^3 + px + q$ with integer coefficients $p$ and $q$ is said to be [i]irrational [/i] if it has three pairwise distinct real irrational roots $a_1,a_2, a_3$
Find all irrational cubic trinomials for which the value of $|a_1| + [a_2| + |a_3|$ is the minimal possible.
(E. Barabanov)
The equation $x^3-3x^2+1=0$ has three real solutions $x_1<x_2<x_3$. Show that for any positive integer $n$, the number $\left\lceil x_3^n\right\rceil$ is a multiple of $3$.
Real constants $ a, b, c$ are such that there is exactly one square all of whose vertices lie on the cubic curve $ y \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c.$ Prove that the square has sides of length $ \sqrt[4]{72}.$