# Latvian Olympiad in Informatics Lessons Learned

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8 Latvian Olympiad in Informatics Lessons Learned 85 Table 1 Examples Input data Output data Comment The following booklets were created (exam paper sets are underlined): (1 2 1)(212)(121) (2 1 2) The following booklets were created (exam paper sets are underlined): (12341)(23412)(34123)(41234)(1234) Despite the fact that the last booklet contains only four pages, it still contains full set of exam paper pages and therefore is valid. The only remaining case is K<N 2K 2. According to pigeonhole principle there are one or two first pages in each full group. a booklet is valid if there are two first pages or group ends with page number K (only one of these two situations can take place). The number { of first pages located in full groups is M, if{ KM N FP = N } <K, M 1, if{ KM. N } K. The number of full groups containing two first pages can be calculated as N FP [ KM N ]. The number of full groups ending with page number K can be calculated as KM [ LCM(K,N) ], where LCM(K, N) is the least common multiple of K and N. Using the formula KN = LCM(K, N) GCD(K, N) where GCD(K, N) is the greatest common divisor of K and N, the previous formula can be transformed as follows: [ ] [ ] KM M GCD(K, N) = = LCM(K, N) N [ M N GCD(K,N) ]. This transformation may be helpful to limit the intermediate results. An uncompleted group can still be valid booklet if { KM N } K. Summing up all these calculations we obtain a surprisingly simple formula: [ ] KM M + N [ 4.2. Task Benefit M N GCD(K,N) ] Task Description Given N cards, having one side coloured in green and the other side in a red colour. On each side some integer is written. When any two cards are chosen (lets denote them by

9 86 M. Opmanis Table 2 Example Input data Output data Comment The maximum difference of benefits is 9 1 between the fourth and the second card: ( 6) = A and B), it is possible to calculate the benefit of A against B. This benefit is calculated as the product of the numbers written on the green side of card A and on the red side of card B. For example, if 10 is written on the green side of the card A, and 3 on its red side, and 7 is written on the green side of the card B, and ( 2) on the red side, then the benefit of A against B is 10 ( 2) = 20 and the benefit of B against a is 7 3=21. When both benefits are calculated, it is possible to find difference between these benefits. In the given example it is 41. If the same cards would be chosen in the reverse order, the difference of benefits would be 41. Write a program, which, for a given description of N (N ) cards finds the maximum possible difference of benefits. It is known that all the numbers written on cards are between and See Table Task Solution (by Sergejs Me Iņiks) We begin with some observations that reveal the geometric meaning of the task. Let us denote by s ij the benefit of card with index i against the card with index j, and by x i and y i the numbers written on the green and red sides of the card with index i respectively. We can observe that: 1) s ij = s ji, i.e., the maximum cannot be negative, because for any negative s ij we can choose s ji = s ij >s ij Therefore, it suffices to find the maximum value of s ij = x i y j x j y i. 2) Replacing x i by x i and y i by y i does not change the value of s ij = x i y j x j y i = (x i y j x j y i ) = ( x i )y j x j ( y i ). 3) Consider a triangle OAB in the coordinate plane, where O the point of origin having coordinates (0,0), A the point with coordinates(x i,y i ) and B the point with coordinates (x j,y j ), then x i y j x j y i equals to doubled area of the triangle OAB. Therefore, the original task can be reduced to the following one: Let us construct the set of points P = {P 0,P 1,...,P N 1,P N }, where the coordinates of P 0 are (0,0), and for all i =1,...,N the coordinates of P i correspond to the

10 Latvian Olympiad in Informatics Lessons Learned 87 card (x i,y i ) as follows: { ( xi, y i ) if y i < 0 or (y i =0and x i < 0), (x i,y i ) otherwise. The triangle must be found with the largest area from all triangles such that one vertex is the point P 0 and two other vertices are placed in the two different points P i and P j. Let us denote the convex hull of P by conv(p ). By construction, P 0 belongs to conv(p ) because P 0 is the leftmost of all the lowest points of P. By purely geometrical means we can prove that all vertices of the desired triangle with maximum area are located in the vertices of conv(p ). Let the points A, B, C be located in the interior of a convex polygon M. We want to construct a triangle ABV such that V is the vertex of M, andtheareaof ABV S ABV >S ABC. The ray AC crosses the boundary of the polygon M at some point D. Obviously, S ABD >S ABC. If D is a vertex of M, wesetv = D. Otherwise, D is located on the some side UW of M (see Fig.1). If UW is parallel to AB, then S ABU = S ABW = S ABD >S ABC, otherwise one of the points U and W (let this be point W, for definiteness) is located closer to AB than the point D, and the other vertex (point U) is located further from the AB than point D. Then, choosing V = Uwe obtain that S ABV >S ABC. Now we can extend the ray VBtill the intersection with the perimeter of the polygon Mand use the same reasoning as above. Afterwards, in the same way we can extend the ray VA. To construct conv(p ) let s sort all points P i except P 0 by the polar angle and then apply the Graham scan. Note that all necessary comparisons can be easily implemented in integer arithmetic. The time complexity of constructing the convex hull is O(N log N) with required memory is linear in N. Let conv(p ) to be the polygon B 0 B 1...B K, where B 0 is the point of origin (0, 0). Fig. 1. Triangle with maximum area within polygon.

12 Latvian Olympiad in Informatics Lessons Learned 89 M. Opmanis is researcher at the Institute of Mathematics and Computer Science of University of Latvia. Since the very first Latvian Olympiads in Informatics in 1986 he is involved in its organization at various positions (last years head of jury). He is deputy or team leader of Latvian IOI teams since 1996 and team leader of Latvian team at Baltic Olympiads in Informatics since the first Olympiad in 1995 (except 2008 and 2010). M. Opmanis was head of jury of Baltic Olympiad in Informatics at BOI 1996, 1999, 2004 and Since 2012 he is member of IOI International Committee.

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