MathDB

1997 May Olympiad

Part of May Olympiad

Subcontests

(5)
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graceful distribution of numbers in circles

In the figures, the vertices are marked with a circle. The segments that join vertices are called paths. Non-negative integers are distributed to the vertices and, to the paths, the differences between the numbers at their ends. https://cdn.artofproblemsolving.com/attachments/d/6/e6fce93719a5b35dbf34d58652b01a8631de57.gif
We will say that a distribution of numbers is graceful if all the numbers from 11 to nn appear in the paths, where nn is the number of paths.
The following is an example of graceful distribution: https://cdn.artofproblemsolving.com/attachments/1/1/a8c2b4fde673ca902b655804c4f5321f9666e9.gif Give -if possible- a graceful distribution for the following figures. If you can't do it, show why.

5-digits numbers of tickets

Joaquín and his brother Andrés go to class every day on the 6262 bus. Joaquín always pays for the tickets. Each ticket has a 55-digit number printed on it. One day, Joaquín observes that the numbers on his tickets - his and his brother's - as well as being consecutive, are such that the sum of the ten digits is precisely 6262. Andrés asks him if the sum of the digits of any of the tickets is 3535 and, knowing the answer, he can directly say the number of each ticket. What were those numbers?
5
2

areas of a hexagon with all angles equal and sides 1,2,3,4,5,6

What are the possible areas of a hexagon with all angles equal and sides 1,2,3,4,51, 2, 3, 4, 5, and 66, in some order?

43 friends fror birthday, cake of 15-gon

When Pablo turns 1515, he throws a party inviting 4343 friends. He presents them with a cake n the form of a regular 1515-sided polygon and on it 1515 candles. The candles are arranged so that between candles and vertices there are never three aligned (any three candles are not aligned, nor are any two candles with a vertex of the polygon, nor are any two vertices of the polygon with a candle). Then Pablo divides the cake into triangular pieces, by means of cuts that join candles to each other or candles and vertices, but also do not intersect with others already made. Why, by doing this, Paul was able to distribute a piece to each of his guests but he was left without eating?
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1
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no of 7-digit multiples of 388 that end in 388

How many seven-digit numbers are multiples of 388388 and end in 388388?

numbers 0-9 to 9 squares

On a square board with 99 squares (three by three), nine elements of the set S={0,1,2,3,4,5,6,7,8,9}S=\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} must be placed, different from each other, so that each one is in a box and the following conditions are met: ∙\bullet The sums of the numbers in the second and third rows are, respectively, double and triple the sum of the numbers in the first row. ∙\bullet The sum of the numbers in the second and third columns are, respectively, double and triple the sum of the numbers in the first column. Show all the possible ways to place elements of SS on the board, fulfilling the indicated conditions.
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