Search problems

33 problems match “quadratic equations”. Clear

Counting z with integer parts of f(z) ●●●●○ 100 ptsLet $f : \mathbb{C} \to \mathbb{C}$ be defined by $f(z) = z^2 + iz + 1$ . How many complex numbers $z$ are there such that $\text{Im}(z) > 0$ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $10$ ?
Participants in a chess tournament ●●●●○ 100 ptsEach participant in a chess tournament played one game against each of the other participants. A win earns one point, a draw earns half a point, and a loss earns no points. At the end of the tournament, it was determined that each participant had earned exactly half of their points in games against the three competitors who finished in the last three places. How many participants were there in the tournament?
Longest chain of quadratic equations ●●●●○ 100 ptsQuadratic equations are solved one after another. Each equation has leading coefficient 1, of the form $x^2+p_jx+q_j=0$ . Whenever an equation has two distinct real roots, the next equation is formed as follows: its constant term is the larger root of the current equation, and its linear coefficient is the smaller root of the current equation. The process stops at the first equation that does not have two distinct real roots. Find the maximum possible number of quadratic equations in such a sequence.