ITOT 43 Autumn 2021 - THPT A - additive practice

Twelve cards numbered 1 through 12 are dealt to four wizards $A,B,C,D$, three each. Each sees only their own cards. They speak in order: A: “I have card 8.” B: “All three of my numbers are prime.” C: “I have three composite numbers with a common prime divisor.” D: “Now I know everyone's cards.” All statements are true. Which cards does A have? (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 31), 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 rook starts on some cell of a $10\times10$ board. Each move goes to an edge-adjacent cell, and its path visits every cell exactly once. Prove that for each main diagonal of the board, there are two consecutive moves in which it first leaves that diagonal and then returns to it. (7 points)



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Kolya stands in the plane together with 2020 other grasshoppers. He jumps over each of them once in an arbitrary order. At each jump the grasshopper jumped over is the midpoint of Kolya's starting and finishing positions. A point is a possible finishing point if some order puts Kolya there after 2020 jumps. What is the largest possible number $N$ of finishing points, over all initial placements? (7 points)



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Find the smallest real number $k$ such that among any three nonzero real numbers there are two, $a,b$, with
$$|a-b|\le k\quad\text{or}\quad\left|\frac1a-\frac1b\right|\le k.$$
(7 points)



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Point $P$ lies inside parallelogram $ABCD$ and satisfies $\angle PDA=\angle PBA$. Let $\omega_1$ be the excircle opposite $A$ of triangle $PAB$, and $\omega_2$ the incircle of triangle $PCD$. Prove that these circles have a common tangent parallel to $AD$. (9 points)



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Twenty savoury cakes and twenty sweet cakes are arranged in a row in an arbitrary order. Alice and Bob alternate taking one cake from either end, with Alice first. Alice wants exactly ten of each kind, while Bob tries to prevent this. Can Alice always achieve her aim? (10 points)



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A $2\times2$ square board is covered by two triangular sheets. Must it be true that (a) at least one of its four unit cells is entirely covered by one sheet? (6 points) (b) some square of side 1, allowed to have any orientation, fits entirely inside one sheet? (6 points)



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