Olympiad problems

2018 Ecuador NMO (OMEC), 4

Let $k$ be a real number. Show that the polynomial $p (x) = x^3-24x + k$ has at most an integer root.

1990 All Soviet Union Mathematical Olympiad, 533

Tags: integer roots , Cubic , polynomial , game , game strategy , algebra

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?

2021 India National Olympiad, 2

Tags: algebra , Cubics , integer roots , Vieta , INMO , v_2

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]

1984 All Soviet Union Mathematical Olympiad, 383

Tags: integer roots , trinomial , polynomial , game , combinatorics

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?

1941 Moscow Mathematical Olympiad, 077

Tags: integer roots , Integer Polynomial , polynomial , algebra

A polynomial $P(x)$ with integer coefficients takes odd values at $x = 0$ and $x = 1$. Prove that $P(x)$ has no integer roots.

2013 Tournament of Towns, 5

Tags: algebra , trinomial , integer roots

A quadratic trinomial with integer coefficients is called [i]admissible [/i] if its leading coefficient 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.