ITOT 42 Spring 2021 - THPT A - additive practice

Several children are in a room with 1000 sweets in a bag. In some order, each child divides the current number of sweets by the current number of children in the room, rounds if necessary, takes that many sweets and leaves. Boys always round up and girls always round down. Prove that the total number of sweets taken by the boys is independent of the order. (4 points)



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Practice scoring adds all whole-question scores (maximum 53). 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.

Does there exist a positive integer $n$ such that for every pair of real numbers $x,y$ there are nonzero real numbers $a_1,\ldots,a_n$ satisfying
$$x=a_1+\cdots+a_n,\qquad y=\frac1{a_1}+\cdots+\frac1{a_n}?$$
(5 points)



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Let $M$ be the midpoint of side $BC$ of triangle $ABC$. A circle $\omega$ through $A$ touches $BC$ at $M$ and meets $AB,AC$ again at $D,E$, respectively. Let $X,Y$ be the midpoints of $BE,CD$, respectively. Prove that the circumcircle of $MXY$ is tangent to $\omega$. (5 points)



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There is a row of $100N$ ham sandwiches. In one action a boy eats the first sandwich from either end; in one action his cat either eats the ham from one remaining sandwich or does nothing. The boy performs 100 actions per turn, the cat one action per turn, and they alternate, with the boy starting. The boy wins if the last sandwich he eats contains ham. Can he guarantee a win for every positive integer $N$, regardless of the cat's play? (8 points)



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100 tourists arrive at a hotel with single rooms numbered $1,2,\ldots,n$. Exactly $k$ rooms, with unknown numbers, are under repair; the others are vacant. Each tourist checks rooms in a chosen order, possibly different for different tourists, and takes the first room not under repair. They cannot see occupancy before checking a room and must never check a room already occupied by another tourist. They coordinate a strategy beforehand. For each $k$, find the smallest $n$ guaranteeing success. (8 points)



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Find at least one real number $A$ such that for every positive integer $n$, the distance from $\lceil A^n\rceil$ to the nearest perfect square of an integer is exactly 2. Here $\lceil x\rceil$ is the smallest integer not less than $x$. (10 points)



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Let $n>2$ be an integer. Peter wants to draw $n$ pairwise disjoint arcs of length $\alpha$ on great circles of the unit sphere. Prove: (a) this is possible for every $\alpha\lt \pi+2\pi/n$; (6 points) (b) this is impossible for every $\alpha>\pi+2\pi/n$. (7 points) Arc lengths are positive; endpoints count as part of an arc.



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