ITOT 44 Autumn 2022 · THPT · A (Advanced)

Let $a,b,c$ be positive integers not exceeding 100. Find the greatest rational root that the equation $ax^2+bx+c=0$ can have.

Let $p,q\ge2$ be coprime integers with $|p-q|>1$. Prove that some positive integer $n$ satisfies $\operatorname{lcm}(p+n,q+n)\lt \operatorname{lcm}(p,q)$.

Two circles $\omega,\Omega$ are concentric. The chord $AD$ of $\Omega$ is tangent to $\omega$. Choose $P$ in the interior of the smaller circular segment bounded by $AD$ and $\Omega$. The tangents from $P$ to $\omega$ meet the major arc $AD$ of $\Omega$ at $B,C$. Segments $BD,AC$ meet at $Q$. Prove that segment $PQ$ passes through the midpoint of $AD$.

A finite square board is divided into unit cells. Between every pair of side-adjacent cells there is initially a closed door. A beetle starts in one cell. When it crosses a closed door, it opens it in its direction of travel and leaves it open; an open door can subsequently be crossed only in that same direction. Prove that at every moment the beetle can return to its starting cell.

An infinite arithmetic progression of positive integers contains two terms with the same digit sum. Must it contain another term with that same digit sum?

Among some banknotes with pairwise distinct positive-integer denominations, exactly $N$ are counterfeit. In one test a detector reports the sum of the denominations of the genuine notes in any chosen subset. Prove that all counterfeit notes can be identified in $N$ tests if (a) $N=2$; (b) $N=3$. Original weights: (a) 2 points, (b) 7 points.

A round pizza is to be shared by $N$ friends. Can it be cut using at most 100 straight cuts, without moving pieces before cutting is finished, and then all resulting pieces distributed so that each friend receives the same total area, when (a) $N=201$; (b) $N=400$? The Vietnamese collection permits removal of already separated pieces; the official original’s stricter no-moving wording is also addressed in the editorial.