Found problems: 26
Given a prime $p$, show that there exist two integers $a, b$ which satisfies the following.
For all integers $m$, $m^3+ 2017am+b$ is not a multiple of $p$.
a) Let $ax^3 + bx^2 + cx + d$ be divisible by $5$ for given positive integers $a, b, c, d$ and any integer $x$. Prove that $a, b, c$ and $d$ are all divisible by $5$.
b) Let $ax^4 + bx^3 + cx^2 + dx + e$ be divisible by $7$ for given positive integers $a, b, c, d, e$ and all integers $x$. Prove that $a, b, c, d$ and $e$ are all divisible by $7$.
Let $ p(x)$ be a cubic polynomial with rational coefficients. $ q_1$, $ q_2$, $ q_3$, ... is a sequence of rationals such that $ q_n \equal{} p(q_{n \plus{} 1})$ for all positive $ n$. Show that for some $ k$, we have $ q_{n \plus{} k} \equal{} q_n$ for all positive $ n$.
If you have an algorithm for finding all the real zeros of any cubic polynomial, how do you find the real solutions to $x = p(y), y = p(x)$, where $p$ is a cubic polynomial?
Find all pairs of integers $(a,b)$ so that each of the two cubic polynomials $$x^3+ax+b \, \, \text{and} \, \, x^3+bx+a$$ has all the roots to be integers.
[i]Proposed by Prithwijit De and Sutanay Bhattacharya[/i]
Let $a,b,c$ be the three different solutions of $x^3-x-1 = 0$. Compute $a^4+b^5+c^6-c$.
Determine all real numbers $a, b, c, d$ such that the polynomial $f(x) = ax^3 +bx^2 + cx + d$ satis fies simultaneously the folloving conditions $\begin {cases} |f(x)| \le 1 \,for \, |x| \le 1 \\ f(2) = 26 \end {cases}$
Let $ p(x)$ be a cubic polynomial with rational coefficients. $ q_1$, $ q_2$, $ q_3$, ... is a sequence of rationals such that $ q_n \equal{} p(q_{n \plus{} 1})$ for all positive $ n$. Show that for some $ k$, we have $ q_{n \plus{} k} \equal{} q_n$ for all positive $ n$.
A game is played in three moves. The first player picks any real number, then the second player makes it the coefficient of a cubic, except that the coefficient of $x^3$ is already fixed at $1$. Can the first player make his choices so that the final cubic has three distinct integer roots?
Let $f(x) = ax^3 + bx^2 + cx + d$ be a polynomial with real coefficients. Given that $f(x)$ has three real positive roots and that $f(0) < 0$, prove that $2b^3+ 9a^2 d - 7abc \le 0$.
Prove that if two cubic polynomials with integer coefficients have an irrational root in common, then they have another common irrational root.
Let $u, v, w$ be the roots of $x^3 -5x^2 + 4x-3 = 0$. Find a cubic polynomial having $u^3, v^3, w^3$ as roots.
Let $p(x)$ be a cubic polynomial with integer coefficients with leading coefficient $1$ and with one of its roots equal to the product of the other two. Show that $2p(-1)$ is a multiple of $p(1)+p(-1)-2(1+p(0)).$
Let $p(x)$ be a cubic polynomial with integer coefficients with leading coefficient $1$ and with one of its roots equal to the product of the other two. Show that $2p(-1)$ is a multiple of $p(1)+p(-1)-2(1+p(0)).$
Let $p_1,p_2,p_3,p_4$ be four distinct primes. Prove that there is no polynomial $Q(x) = ax^3 + bx^2 + cx + d$ with integer coefficients such that $|Q(p_1)| =|Q(p_2)| = |Q(p_3)|= |Q(p_4 )| = 3$.
Let $p(x) = (x- x_1)(x- x_2)(x- x_3)$, where $x_1, x_2$ and $x_3$ are real. Show that $p(x) p''(x) \le p'(x)^2$ for all $x$.
Let $a, b, c$ be the roots of the equation $x^3-9x^2+11x-1 = 0$, and define $s =\sqrt{a}+\sqrt{b}+\sqrt{c}$.
Compute $s^4 -18s^2 - 8s$ .
$p(x)$ is the cubic $x^3 - 3x^2 + 5x$. If $h$ is a real root of $p(x) = 1$ and $k$ is a real root of $p(x) = 5$, find $h + k$.
If $p (x) = c_0 + c_1x + c_2x^2 + c_3x^3$ is a polynomial with integer coefficients with $a, b,c$ integers and different from each other, prove that it cannot happen simultaneously that $p (a) = b$, $p (b) = c$ and $p (c) = a$.
Let $S$ be the set of all real polynomials $f(x) = ax^3 + bx^2 + cx + d$ such that $|f(x)| \le 1$ for all $ -1 \le x \le 1$. Show that the set of possible $|a|$ for $f$ in $S$ is bounded above and find the smallest upper bound.
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)
Prove that the numbers $\tan^2 20^o, \tan^2 40^o,\tan^2 80^o$ are the roots of the equation $x^3 - 33x^2 + 27x - 33 = 0$.
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}.$
Solve the equation: $x^3-2ax^2+(a^2-2\sqrt2 a -6)x + 2\sqrt2 a^2+ 8a + 4\sqrt2 =0$
For all valid values of $a$ and $b$, solve the equation
$$\frac{x^3}{(x-a) (x-b)} +\frac{a^3}{(a-b) (a-x)} + \frac{b^3}{ (b-x) (b-a)}= x^2 + a + b$$