Compete · Serbia
Opštinsko takmičenje (Municipal round) 2018
1For a natural number \(n\), let \(f(n)\) be the number written with the same digits taken in the opposite order (that is, read from right to left) whenever \(n\) is not divisible by \(10\); if \(10 \mid n\), …2Let \(K\) be the midpoint of the side \(CD\) of a rectangle \(ABCD\). The lines \(BK\) and \(AC\) are perpendicular to each other and meet at the point \(H\), and \(G\) denotes the foot of the perpendicular …3A mathematical commission has \(2n\) members, where \(n \geqslant 3\). Every member of the commission is in a quarrel with exactly one other member (the relation is symmetric). In how many ways can the …4A clock has three hands, each turning at its own constant speed: the second hand completes a full circle in one minute, the minute hand in one hour, and the hour hand in twelve hours. At midnight all three …5Baron Munchausen lives in a country \(Z\) which has \(2018\) cities, some pairs of them joined by roads (every road can be travelled in both directions). The Baron has established that there is a city …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.