ITOT 42 Spring 2021 - THPT O - additive practice

(a) A convex pentagon is divided into three triangles by non-crossing diagonals. Can the three triangle centroids be collinear? (2 points) (b) Answer the same question for a non-convex pentagon. (2 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.

Maria has four indistinguishable-looking weights of masses 1000,1002,1004,1005 grams. Each next weighing may depend on previous results. (a) With an ordinary two-pan balance, can she certainly determine the mass of every weight in four weighings? (2 points) (b) The same question when the balance also indicates equality when the total mass on the left pan is exactly 1 gram lighter than that on the right pan, as stated in the supplied Vietnamese source. (2 points)

Source discrepancy: the official English original uses the opposite calibration: its left pan itself is 1 gram lighter, so the loaded masses balance when the LEFT load is 1 gram HEAVIER. Both readings are analyzed separately below; the answer to each is unchanged.



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For which positive integers $n$ can the product of $n$ consecutive positive integers equal the sum of $n$ consecutive positive integers? The two consecutive sequences need not be the same. (5 points)



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An ordinary quadratic equation has at most two real roots. Can an equation $\lfloor x^2\rfloor+px+q=0$, with $p\ne0$, have more than 100 real roots? Here $\lfloor x^2\rfloor$ is the greatest integer not exceeding $x^2$. (5 points)



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Let $O$ be the circumcentre of triangle $ABC$, and let $M$ be the midpoint of $AC$. The line $BO$ meets the altitudes $AA_1,CC_1$ at $H_a,H_c$, respectively. The circumcircles of triangles $BH_aA$ and $BH_cC$ meet again at $K$. Prove that $K$ lies on $BM$. (6 points)

Source clarification: the official original specifies an acute triangle; the supplied Vietnamese text omits that adjective. The stated lines, nondegenerate circles and distinct second intersection are understood to exist. The proof below treats the official acute configuration and gives an algebraic extension whenever these objects are defined.



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