ITOT 43 Autumn 2021 - THCS O - additive practice

The autumn Tournament of Towns is held once each year. In 2021 the calendar year is divisible by the tournament number: $2021\div43=47$. How many more times will this happen in the future? (4 points)



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Practice scoring adds all whole-question scores (maximum 24). Official ITOT scoring adds only the three highest whole-question scores (maximum 15), after summing subparts within each question. The importer does not enforce best-three scoring. Duration: 300 minutes from historical organizer general rules, not an individually printed duration on this paper.

A cube is cut into eight rectangular boxes by three planes, each parallel to a face of the cube. Each box is painted entirely white or grey so that boxes sharing a face have different colours, as illustrated. The grey-box volumes are $1,6,8,12$. Determine the white-box volumes. (5 points)

A pirate has five purses, each holding 30 coins. One contains only gold, another only silver, a third only bronze; each of the other two contains 10 gold,10 silver and 10 bronze coins. In each turn, one or more coins may be taken at random from a purse and examined. What is the minimum total number of coins needed to guarantee identifying the remaining contents of at least one purse? (5 points)

Source wording note: the supplied Vietnamese text permits turns as phrased above. The official original instead permits a single simultaneous choice of how many coins to take from each purse, only once. The answer is the same for both readings; the distinction is explained in the solution.



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A convex polygon with $n>4$ vertices has the property that whenever a diagonal cuts off a triangle, that triangle is isosceles. Prove that among any four sides of the polygon at least two have equal length. (5 points)



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Twenty players enter a chess tournament. Each plays each other player twice, once with White and once with Black. Player $X$ is “no weaker than” player $Y$ if $X$ has at least as many wins with White as $Y$, and at least as many wins with Black as $Y$. Must there be distinct players $A,B$ such that $A$ is no weaker than $B$? (5 points)



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