This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 26

2012 Belarus Team Selection Test, 1
Tags: sum , min , irrational , cubic , cubic equation , polynomial , algebra
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)

1982 IMO Shortlist, 7
Tags: algebra , root , cubic , polynomial
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 $p(1)+p(-1)-2(1+p(0)).$

1990 IMO Shortlist, 26
Tags: number theory , algebra , polynomial , cubic , iteration
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$.

1974 Chisinau City MO, 74
Tags: parameter , cubic , algebra
Solve the equation: $x^3-2ax^2+(a^2-2\sqrt2 a -6)x + 2\sqrt2 a^2+ 8a + 4\sqrt2 =0$

1957 Moscow Mathematical Olympiad, 347
Tags: cubic , divisible , polynomial , algebra
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 $7$.

2006 Thailand Mathematical Olympiad, 8
Tags: radical , algebra , cubic
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 $s^4 -18s^2 - 8s$ .

1992 All Soviet Union Mathematical Olympiad, 576
Tags: cubic , algebra , polynomial
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?

1982 IMO Longlists, 16
Tags: algebra , root , cubic , polynomial
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 $p(1)+p(-1)-2(1+p(0)).$

1997 Austrian-Polish Competition, 5
Tags: polynomial , prime , integer polynomial , cubic
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 $|Q(p_1)| =|Q(p_2)| = |Q(p_3)|= |Q(p_4 )| = 3$.

1991 All Soviet Union Mathematical Olympiad, 552
Tags: cubic , algebra , polynomial
$p(x)$ 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 $h + k$.

VII Soros Olympiad 2000 - 01, 9.5
Tags: equation , algebra , parameterization , cubic
For all valid values ​​of $a$ and $b$, solve the equation $$\frac{x^3}{(x-a) (x-b)} +\frac{a^3}{(a-b) (a-x)} + \frac{b^3}{ (b-x) (b-a)}= x^2 + a + b$$

2021 India National Olympiad, 2
Tags: algebra , cubic , integer root , vieta
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]

1990 All Soviet Union Mathematical Olympiad, 533
Tags: polynomial , game , algebra , game strategy , integer root , cubic
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?

1975 Chisinau City MO, 117
Tags: trigonometry , cubic
Prove that the numbers $\tan^2 20^o, \tan^2 40^o,\tan^2 80^o$ are the roots of the equation $x^3 - 33x^2 + 27x - 33 = 0$.

1966 Poland - Second Round, 2
Tags: polynomial , algebra , cubic
Prove that if two cubic polynomials with integer coefficients have an irrational root in common, then they have another common irrational root.

1969 All Soviet Union Mathematical Olympiad, 125
Tags: algebra , cubic
Given an equation $$x^3 + ?x^2 + ?x + ? = 0$$ First player substitutes an integer on the place of one of the interrogative marks, than the same do the second with one of the two remained marks, and, finally, the first puts the integer instead of the last mark. Explain how can the first provide the existence of three integer roots in the obtained equation. (The roots may coincide.)

1965 Swedish Mathematical Competition, 5
Tags: algebra , polynomial , cubic , bounded
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.

2004 Thailand Mathematical Olympiad, 3
Tags: polynomial , algebra , cubic
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.

1990 IMO Longlists, 95
Tags: polynomial , number theory , cubic , iteration , algebra
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$.

1995 Chile National Olympiad, 3
Tags: algebra , polynomial , cubic
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 $p (c) = a$.

1970 Swedish Mathematical Competition, 4
Tags: polynomial , algebra , cubic , analysis
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 $x$.

2014 Hanoi Open Mathematics Competitions, 11
Tags: algebra , cubic , inequalities
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 $\begin {cases} |f(x)| \le 1 \,for \, |x| \le 1 \\ f(2) = 26 \end {cases}$

1991 IMO Shortlist, 22
Tags: algebra , polynomial , square , cubic , cubic equation
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}.$

2017 Korea National Olympiad, problem 5
Tags: polynomial , cubic , divisibility , number theory
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 $p$.

2011 QEDMO 9th, 2
Tags: algebra , polynomial , cubic
Let $a,b,c$ be the three different solutions of $x^3-x-1 = 0$. Compute $a^4+b^5+c^6-c$.