ITOT 43 Spring 2022 - THPT O - additive practice

Peter starts with a positive integer and multiplies it by 5, then multiplies the result by 5, and continues for a total of $k$ multiplications. Neither the initial integer nor any result contains the decimal digit 7. Prove that there is a positive integer which can be multiplied successively by 2 a total of $k$ times with no initial or resulting number containing 7. (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 16), 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.

Eight visually identical coins consist of five genuine and three counterfeit coins. Genuine coins all have the same weight; counterfeit coins all have the same smaller weight. The fox gives Pinocchio three coins. Can he determine whether all three are genuine using at most two weighings on an equal-arm balance without weights? (4 points)



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For a positive integer $n$, call $(a_1,\ldots,a_n)$ interesting if $a_i=i$ or $i+1$ for every $i$. Call it even or odd according to the parity of $a_1+\cdots+a_n$. Alice records the product of the entries of every odd interesting sequence; Bob does the same for every even one. Whose recorded numbers have the greater sum, and by how much, as a function of $n$? (5 points)



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Alice and Bob independently tile a $20\times21$ board with $1\times3$ or $3\times1$ trominoes. They then compare their tilings, and Alice pays Bob one dollar for each tromino occupying exactly the same cells in both tilings. What is the largest amount Bob can guarantee? (5 points)



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A quadrilateral $ABCD$ is inscribed in a circle $\omega$ with centre $O$. The circumcircle of triangle $AOC$ meets the lines $AB,BC,CD,DA$ for the second time at $M,N,K,L$, respectively. Prove that the lines $MN,KL$ and the tangents to $\omega$ at $A,C$ are all tangent to one circle. (6 points)



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