By Andreescu T., Feng Z.

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10] G. BLIND & R. BLIND: Convex polytopes without triangular faces, Israel J. Math. 71 (1990), 129-134. [11] A. BOCKMAYR, F. EISENBRAND, M. HARTMANN & A. S. , to appear. [12] B. BOLLOBAS: Random Graphs, Academic Press, New York 1985. [13] C. CHAN & D. P. , to appear. [14] T. html [15] V. CHVATAL: Edmonds polytopes and a hierarchy of combinatorial problems, Discrete Math. 4 (1973), 305-337. [16] J. H. E. COHN: On determinants with elements ±1, II, Bulletin London Math. Soc. 21 (1989), 36-42. [17] J.

For the upper bound x(n) :::; Pn-I we use that the entries of A-I can be written as b.. _ lJ - (_ )i+jdet(Aij) I det(A) , where the cofactors A ij E {O, I}(n-I)x(n-I) satisfy Idet(Aij)1 :::; Pn-I by definition, and the invertible matrix A satisfies Idet(A)1 ~ I since it is integral. (2) Super-multiplicativity. For the lower bound it is sufficient to construct "bad" matrices of size 2m X 2m , because of the following simple construction, which establishes x(nl + n2) x(nl)' x(n2)' Take "bad" invertible O/I-matrices A and B of sizes nl x nl and n2 x n2, such that X(A) = Idet Anl,n) det AI and X(B) = Idet B ll / det BI.

E(det(C)2) = nL Hint, by Bernd Gartner: Use det(C) = L~=l (-l)i-l cli det(Cli ), and analyze the expected values of the summands in n det(C)2 = L(det(C1i ))2 + i=l L( -l)i+jcliClj det(Cli ) det(C1j ). i#j 10. What is the largest absolute value of the determinant of an n x n matrix with coefficients in {-I, 0, I}? With coefficients in the interval [0, I]? With coefficients in [-1, I]? (It is reported that this is a question that was asked by L. Collatz at an international conference in 1961, and answered a year later by Ehlich & Zeller [22].

### 102 Combinatorial problems from the training of USA IMO team by Andreescu T., Feng Z.

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