ITOT 43 Spring 2022 - THCS O - additive practice

Two people walk towards one another along a straight line at different constant speeds. At the same instant each releases a dog to run forwards. The dogs have the same constant speed. Each dog runs to the other person, then returns to its own owner. Which dog returns first: the faster walker’s dog or the slower walker’s dog? (3 points)



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Practice scoring adds all whole-question scores (maximum 23). 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.

Petya chooses a positive integer and repeatedly multiplies by 5. Is it true that from some point onwards every resulting number contains the digit 5 in its decimal representation? (4 points)



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There are 11 indistinguishable-looking coins:7 genuine and 4 counterfeit. All genuine coins have equal mass; all counterfeits have equal smaller mass. Fox gives Pinocchio 4 coins. Can he determine in only two balance-scale weighings whether all four are genuine? (5 points)



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In square $ABCD$, point $P$ lies on diagonal $AC$. Let $H$ be the orthocentre of triangle $APD$, and let $M,N$ be the midpoints of $AD,CD$. Prove $PN\perp MH$. (5 points)

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? (6 points)



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