4

Part of ICMC 4

Problems(2)

Unique rational sum

Source: ICMC 2021 Round 1 P4

Does there exist a set

\mathcal{R}

of positive rational numbers such that every positive rational number is the sum of the elements of a unique finite subset of

\mathcal{R}

?
Proposed by Tony Wang

combinatoricspositive rationalscollege contestsICMC

Almost inversion

Source: ICMC 2021 Round 2 P4

\mathbb R^2

denote the Euclidean plane. A continuous function

f : \mathbb R^2 \to \mathbb R^2

maps circles to circles. (A point is not a circle.) Prove that it maps lines to lines.
Proposed by Tony Wang

analysisICMCcollege contestsgeometry