Experiment with simple cases.

In some cases, working backward from the desired result is helpful. Even if you can solve a problem, do read the solutions. They may contain some ideas that did not occur in your solutions, and they viii Introduction may discuss strategic and tactical approaches that can be used else- where. The formal solutions are also models of elegant presenta- tion that you should emulate, but they often obscure the torturous process of investigation, false starts, inspiration and attention to detail that led to them.

When you read the solutions, try to re- construct the thinking that went into them. Many of the problems have multiple solutions, but not all are outlined here. All terms in boldface are defined in the Glossary. Use the glossary and the reading list to further your mathematical education. Meaningful problem solving takes practice. Don't get discouraged if you have trouble at first. New Jersey: Princeton University Press, Every two sets have exactly one element in common. Solved by Jayson Dwight S. Note that here, some of the groups may be empty.

Partial credit for Jayson Dwight S. The figure for the problem is given below. Suppose such a triangle exists. Hence, no triangle exists. Catindig [Ateneo de Manila HS]. Austrian-Polish Mathematics Competition, Suppose such a partitioning exists. Hence, no partitioning exists. In either case, the "fractional part" of u satisfies the estimate Note also that 1 1. Clearly, a is close to 2 if m is large enough.

Problem 14, Solut ion 1 The sum of four integers is divisible by 3 if and only. Enumerate these patterns 1 through 5, in the order as they are listed above. The argument, however, breaks down when k is divisible by 3. P roblem 1 5 For what natural numbers n is it possible to tile the with 2 x 2 and 3 x 3-squares?

## KUCZMA - Definition and synonyms of kuczma in the Polish dictionary

Thus let n be odd and suppose the chessboard has been tiled as described. In each 3 x 3-tile, colour blue the three cells unit squares adj acent to its left edge, colour red the three cells adjacent to its right edge, and colour green the three cells in the middle; the 2 x 2-tiles remain uncoloured. Enumerate the columns vertical lines of the board 1 through n. This in view of conditions 5 shows that n must be divisible by 3. In conclusion, if the board can be tiled as required, then n is divisible by 2 or 3. The converse implication is obvious.

## Australian Mathematics Trust Polish and Austrian Mathematical Olympiads, 1981-1995

Problem 1 5 , S o lution 2 A colouring argument can be used in a yet smarter manner. Colour all the columns of the board black and white alternately, in a "zebra" fashion.

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For n odd, the two outer columns are coloured alike - say, black; so there are n black cells more than white ones. Suppose the tiling is possible. Each 2 x 2-tile covers two white cells and two black cells. Each 3 x 3-tile covers three white cells and six black cells, or conversely.

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The difference between the number of black cells and the number of white cells covered by a single tile equals 3, -3 or 0. The total difference between the numbers of black and white cells in the whole board equals n. Thus n is the sum of some threes, some minus-threes, and some zeros - hence, it is a number divisible by 3. Conclusion as in Solution 1: a tiling in question is possible if and only if n is divisible by 2 or by 3. Remark The easy "if" part results from "uniform" tilings, using tiles of only one of the two kinds.

Problem 1 6 A triangular prism is a pentahedron whose two parallel faces "top base" and "bottom base" are congruent triangles and the remaining three faces are parallelograms. Link A rithmetic a nd Combinatorics 47 the fourth point with one of the vertices of that base 3 possibilities ; the connecting segment will be a side edge of the prism , which is thereby fully determined. The points A, B are supposed to lie in one base of the prism under construction, and C, D in the other. Hence, if the four given points are to be vertices of a triangular prism, two of them call them P, Q must be the endpoints of a side edge.

The other two points R and S cannot be joined by an edge, since they are not coplanar with P and Q. This done, we can attach each of R, S to either P or Q. This defines four possible cases. We claim that in each case the prism is uniquely determined. In each one of the first two cases, we have already one base triangle and the side edge P Q ; translate that triangle by the corresponding vector PQ or QP to get the other base.

R ; the points R' and S' are the remaining two vertices of the prism. The fourth case is analogous.

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We see that, on the total, there exist 6 4 24 triangular prisms with four vertices in the given points. S uppose S is a set of squares such that every two squares in S can be connected by a path consisting of consecutively adjacent squares. Two squares are adj acent if they have a common 48 Solutions edge. Show that there are at least white squares in S. Problem 17, S o lution 1 If every two squares in a certain set can be linked within that set by a path consisting of consecutively adjacent squares, we will say that the set is connected.

Take any connected set S composed o f n squares. Define the distance between two squares as the minimum number of edges one has to cross while going from one square to the other, along an admissible path within S.

Choose and fix a pair of squares A, B E S whose distance is a maximum; denote their distance by m. Thus there exists a path CoC 1. Remove from S square Cm - 1 together with those squares adj acent to C m - 1 whose distance from A is exactly m. Denote by S' the set that remains.

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Note that square Cm - 2 has not been removed its distance from A is m - 2 and not m. We now show that S' is connected. Choose a square D E S'. There exists a path EoE 1. Squares Eo, E 1. S uppose not. This is The theorem formulated at the beginning is now proved. To see that is optimal, consider a horizontal 3 x rectangle, with white squares removed from the upper row and from the lower row but not from the middle one. This is a connected set of squares, out of which exactly are white.

Assume this is true for all positive integers smaller than a fixed integer 1. Let S be a connected set of squares. Choose a white square W E S. From any other square in S a path contained in S leads to W , and this path necessarily passes through one of the four squares adjacent to W. Label these sets St ,. Suppose consists of n i squares, of them being white. The claim i s proved. Problem 18, Solution 1 Consider words of length n that begin with an a and satisfy condition 2. Consider Once guessed, this equality is easily proved by induction.

Note that a n is the number we had to calculate. Its value is given by formula 4.

A rithmetic a nd Com binatorics 51 Remark The explicit formula 4 could be derived from 3 without guessing, by the usual method of solving linear recursions compare Problem 10, Solution 3, for instance. P roblem 1 8 , Solut ion 2 Assume n 3. Imagine a row of n empty cells. They have to be filled-in with symbols a, b, c, observing conditions 1 and 2. In the two outer cells, as must be placed; this is prescribed.

Consider k to be fixed, for the while. There remain n - 2 - k cells to be filled with other symbols.