ITOT 43 Autumn 2021 - THPT O - additive practice

A positive integer $k$ is called interesting if the product of the first $k$ prime numbers is divisible by $k$. For example,2 is interesting because $2\cdot3=6$ is divisible by 2. What is the largest possible number of consecutive positive integers that are all interesting? (3 points)



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Practice scoring adds all whole-question scores (maximum 25). Official ITOT scoring adds only the three highest whole-question scores (maximum 18), 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. (4 points)

All cells of a $2021\times2021$ board are white. Ivan chooses two cells and colours them black. On each subsequent turn, every cell sharing an edge with at least one black cell becomes black simultaneously. With an optimal initial choice, what is the minimum number of turns needed for the entire board to become black? (6 points)



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A segment $AB$ is fixed in space. Points $X,Y,Z$ satisfy that $ABX$ is an equilateral triangle and $ABYZ$ is a square. Prove that the orthocentre of triangle $XYZ$ lies on a fixed circle. (6 points)



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Start with the interval $[0,1]$. At each step choose one of the current intervals, cut it into two smaller intervals, and write down the product of their lengths. Prove that the sum of all numbers written down cannot exceed $1/2$. (6 points)



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