ITOT 43 Spring 2022 - THPT A - additive practice

What is the smallest possible number of different leading decimal digits among $n,2n,3n,\ldots,9n$, where $n$ is a positive integer? (5 points)



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Practice scoring adds all whole-question scores (maximum 57). Official ITOT scoring adds only the three highest whole-question scores (maximum 28), 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.

On a blank sheet of paper, perpendicular coordinate axes $x,y$ with equal scales were drawn and the graph $y=f(x)$ was plotted. The marks on the $x$-axis and the entire $y$-axis were then erased. Using only pencil, straightedge and compass, reconstruct the $y$-axis when (a) $f(x)=3^x$; (b) $f(x)=\log_a x$, where the unknown constant satisfies $a>1$. (4+4 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. (8 points)

A rook traverses an $n\times n$ board, each move to an edge-adjacent cell, visiting each cell exactly once. Bob writes the integers 1 through $n^2$ in the order of the visits. Let $M$ be the largest absolute difference between numbers in any two edge-adjacent cells. What is the smallest possible value of $M$? (8 points)



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What is the largest possible number of roots in $(0,1)$ of a monic polynomial of degree 2022 with integer coefficients? (8 points)



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There are 300 sages and 25 possible hat colours. Each sage wears one hat and sees all the other hats but not their own. They know that the numbers of hats of the 25 colours are pairwise different. They must simultaneously guess their own colours. Can they agree in advance on a strategy guaranteeing at least 150 correct guesses? (8 points)



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A spacecraft is at distance $a>0$ from an unknown plane separating two half-spaces. The crew knows $a$ but not the plane's direction. The craft may follow any path in space, records its travelled distance, and has a sensor signalling contact with the plane. Can the crew guarantee contact with total travel at most (a) $14a$; (b) $13a$? (6+6 points)



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