ITOT 43 Spring 2022 - THCS A - additive practice

Find the largest positive integer $n$ such that for every prime $p$ with $2\lt p\lt n$, the number $n-p$ is prime. (4 points)



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Practice scoring adds all whole-question scores (maximum 60). Official ITOT scoring adds only the three highest whole-question scores (maximum 32), 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.

Prove that every convex quadrilateral can contain three non-overlapping copies of itself, each with all lengths reduced by a factor of 2. Copies may share boundary points but not interiors. (7 points)



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What is the smallest possible number of different leading decimal digits among $n,2n,3n,\ldots,9n$, where $n$ is a positive integer? (7 points)



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Some cells of a $100\times100$ board are coloured black. Every row and column that contains black cells contains an odd number of them. The middle black cells of different rows lie in different columns; the middle black cells of different columns lie in different rows. (a) Prove that some cell is simultaneously the middle black cell of its row and of its column. (5 points) (b) Must every middle black cell of a row also be the middle black cell of its column? (5 points)



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Two triangles overlap as shown, forming a six-pointed star. The incircles of the six outer small triangles all have equal radii. Prove that the incircles of the two original triangles have equal radii. (10 points)

There are 21 medals:7 gold,7 silver and 7 bronze. Exactly one medal is defective. A defective gold medal is lighter than a genuine gold medal; a defective bronze medal is heavier than a genuine bronze medal; a defective silver medal may be lighter or heavier than a genuine silver medal. Genuine medals of the same metal have equal mass. Find the defective medal in three weighings on a balance scale without weights. (10 points)



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A convex polygon can be tiled by unit squares and unit equilateral triangles in exactly $p$ different ways, where $p$ is prime. Prove that the polygon has a side of length $p-1$. (12 points)



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