The number $2021=43\cdot47$ is composite. Prove that inserting any number of digits 8 between 20 and 21 gives another composite number. (4 points)
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Practice scoring adds all whole-question scores (maximum 52). Official ITOT scoring adds only the three highest whole-question scores (maximum 30), 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 room contains several children and a bag of 1000 sweets. In some order each child divides the current number of sweets by the number of children still in the room, rounds the result if needed, takes that many sweets and leaves. Boys always round upwards and girls downwards. The process continues until the room is empty. Prove that the total sweets received by the boys does not depend on the order. (5 points)
In equilateral triangle $ABC$, point $E$ lies on $AB$, point $F$ on $AC$, and point $K$ on the line $AB$ outside segment $AB$, with $AE=CF=BK$. Let $P$ be the midpoint of $EF$. Prove that $\angle KPC$ is a right angle. (6 points)
An island has 50 natives standing in a circle. Each announces first the age of the neighbour on the left, then the age of the neighbour on the right. A truth-teller gives both ages correctly; a liar decreases one age by 1 and increases the other by 1, choosing which. From all the announcements, can a visitor always determine the type of every native? (7 points)
At the centre of every cell of a rectangular grid $M$ is a point-sized light bulb, initially off. In one turn choose a line passing through no bulb such that all bulbs on one side are off, and switch on all bulbs on that side. At least one bulb must be switched on each turn. Find the largest possible number of turns to switch on all bulbs for:
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. (10 points)
Let $p,q$ be coprime positive integers. A frog starts at 0 on the integer line and makes a nonempty sequence of hops, each either $p$ units right or $q$ units left, eventually returning to 0. Prove that for every positive integer $d<p+q$, two positions visited by the frog differ by exactly $d$. (12 points)
