A polynomial $P(x,y)$ satisfies, for every integer $n\ge0$, that each of $P(n,y),P(x,n)$ is either the zero polynomial or has degree at most n. Can $P(x,x)$ have odd degree? Both the book and official original include n=0. The book says “a nonzero constant” where the official original says “the zero polynomial”; the effect of this wording is discussed below.
In an acute triangle ABC, interior side points A′∈BC,B′∈CA,C′∈AB give cevians AA′,BB′,CC′ concurrent at an interior point P. In each circle having one cevian as diameter, draw the chord through P perpendicular to that diameter. The three chords have equal length. Prove P is the orthocenter of ABC.
There are 100 visually identical coins: gold 3 g, silver 2 g and bronze 1 g, with at least one of each type. Using a balance scale with no external weights, determine every coin’s type in at most 101 weighings.
A strictly increasing positive sequence $(a_i)_{i\in\mathbb Z}$ extends infinitely in both directions. For each positive k, let b_k be the smallest integer bounding above the ratio of the sum of any k consecutive terms to the largest of those terms. Prove that either b_k=k for all k, or the sequence b_k is eventually constant.
An interior point M of a convex quadrilateral ABCD is equidistant from lines AB,CD and also equidistant from lines BC,AD. Its area equals $MA\cdot MC+MB\cdot MD$. Prove that ABCD (a) is cyclic; (b) has an incircle.
A cube of $(2N)^3$ unit cubes is pierced by axis-parallel needles, each passing through exactly 2 N unit cubes. Every unit cube is pierced by at least one needle. A set of needles is regular if no two pierce the same unit cube. (a) Prove there is a regular set of 2 N² needles using at most two directions. (b) Determine the largest guaranteed size of a regular set. The official original says “at most two directions”; the book’s phrasing is less precise.
Some integers from 1 through n are red. Whenever red a,b,c (not necessarily distinct) satisfy $n\mid a(b-c)$, one has b=c. Prove that at most $\varphi(n)$ integers are red, where $\varphi(n)$ counts positive integers at most n coprime to n. The official definition uses “at most n”; the book says “less than n”, which differs at n=1.