Olympiad problems

2010 Dutch IMO TST, 5

The polynomial $A(x) = x^2 + ax + b$ with integer coefficients has the following property: for each prime $p$ there is an integer $k$ such that $A(k)$ and $A(k + 1)$ are both divisible by $p$. Proof that there is an integer $m$ such that $A(m) = A(m + 1) = 0$.

2019 Azerbaijan Junior NMO, 1

Tags: cell , quadratic , trinomial , quadratic trinomial

A $6\times6$ square is given, and a quadratic trinomial with a positive leading coefficient is placed in each of its cells. There are $108$ coefficents in total, and these coefficents are chosen from the set $[-66;47]$, and each coefficient is different from each other. Prove that there exists at least one column such that the polynomial you get by summing the six trinomials in that column has a real root.

1973 All Soviet Union Mathematical Olympiad, 180

Tags: trinomial , algebra

The square polynomial $f(x)=ax^2+bx+c$ is of such a sort, that the equation $f(x)=x$ does not have real roots. Prove that the equation $f(f(x))=0$ does not have real roots also.

2006 Thailand Mathematical Olympiad, 15

Tags: trinomial , number theory

How many positive integers $n < 2549$ are there such that $x^2 + x - n$ has an integer rooot?

1902 Eotvos Mathematical Competition, 1

Tags: algebra , trinomial

Prove that any quadratic expression $$Q(x) = Ax^2 + Bx + C$$ (a) can be put into the form $$Q(x) = k \frac{x(x- 1)}{1 \cdot 2} + \ell x + m$$ where $k, \ell, m$ depend on the coefficients $A,B,C$ and (b) $Q(x)$ takes on integral values for every integer $x$ if and only if $k, \ell, m$ are integers.

2019 Dutch IMO TST, 1

Tags: trinomial , quadratic trinomial , algebra , polynomial

Let $P(x)$ be a quadratic polynomial with two distinct real roots. For all real numbers $a$ and $b$ satisfying $|a|,|b| \ge 2017$, we have $P(a^2+b^2) \ge P(2ab)$. Show that at least one of the roots of $P$ is negative.

2005 Estonia National Olympiad, 4

Tags: equation , algebra , trinomial

Find all pairs of real numbers $(x, y)$ that satisfy the equation $(x + y)^2 = (x + 3) (y - 3)$.

1979 Chisinau City MO, 171

Tags: trinomial , algebra

Are there numbers $a, b$ such that $| a -b |\le 1979$ and the equation $ax^2 + (a + b) x + b = x$ has no roots?

1997 Estonia National Olympiad, 2

Tags: trinomial , algebra , quadratic polynomial

Find the integers $a \ne 0, b$ and $c$ such that $x = 2 +\sqrt3$ would be a solution of the quadratic equation $ax^2 + bx + c = 0$.