 
Summary: ON INJECTIVITY OF COMBINATORIAL
RADON TRANSFORM OF ORDER FIVE
Tewodros Amdeberhan and Melkamu Zeleke
Abstract. In the present work, we give a proof of the injectivity of the combinatorial radon
transform of order five.
The problem of determining members of a set by their sums of a fixed order was posed by
Leo Moser and partially settled by Ewell, Fraenkel, Gordon, Selfridge, and Straus. Following
the notation of [BL], the general problem can be stated in the following way.
For any given (k; n) 2 Z \Theta Z, with 2 Ÿ k Ÿ n, we choose arbitrarily an nset Xn =
fx 1 ; x 2 ; :::; xng then form the set W k
n (Xn ) = foe i g of all sums of k distinct elements of Xn and
ask:
Does there exist an nset X 0
n different from Xn giving rise to the same set of sums as does
Xn ? More formally, we can describe the problem as follows:
Define a mapping W k
n from the set fXng of all nsets to the set of all
\Gamma n
k
\Delta sets by the rule:
