Solution-1

Let Kārlis’s speed be \(k\), Sandris’s speed \(s\), Vilnis’s speed \(v\), Mārtiņš’s speed \(m\), and the track length \(\ell\).

From the fact that Kārlis meets Vilnis every \(12\) minutes, it follows that \(k+v=\frac{\ell}{12}\).

From the fact that Sandris overtakes Kārlis every 20 minutes, it follows that \(s-k=\frac{\ell}{20}\).

From the fact that Mārtiņš overtakes Vilnis every 5 minutes, it follows that \(m-v=\frac{\ell}{5}\).

Since \(m+s=(m-v)+(s-k)+(k+v)=\frac{\ell}{5}+\frac{\ell}{20}+\frac{\ell}{12}=\frac{\ell}{3}\), we conclude that Mārtiņš meets Sandris every \(3\) minutes.

Solution 2

Assume that Kārlis and Sandris move to the right, but Vilnis and Kārlis — to the left. Also assume that they all start moving at the same time from the same point, and determine how many laps and in which direction in \(60\) minutes the others make relative to Kārlis (\(60\) is chosen as the least common multiple of \(5\), \(12\), and \(20\) minutes). From the fact that Kārlis meets Vilnis every \(12\) minutes, it follows that Vilnis has completed \(60:12=5\) laps to the left relative to Kārlis.

From the fact that Mārtiņš overtakes Vilnis every \(5\) minutes, it follows that Mārtiņš has completed \(60:5=12\) laps to the left relative to Vilnis, hence he has completed \(12+5=17\) laps to the left relative to Kārlis.

From the fact that Sandris overtakes Kārlis every \(20\) minutes, it follows that Sandris has completed \(60:20=3\) laps to the right relative to Kārlis.

Since Mārtiņš has completed \(17\) laps to the left relative to Kārlis, while Sandris has completed \(3\) laps to the right, they have met \(17+3=20\) times in these \(60\) minutes. Thus they meet every \(60:20=3\) minutes.

Atrisinājums-2

Pieņemsim, ka Kārlis un Sandris pārvietojas pa labi, bet Vilnis un Kārlis - pa kreisi. Pieņemsim arī, ka viņi visi sāk pārvietoties vienā laikā no viena punkta un noskaidrosim, cik apļus un kurā virzienā \(60\) minūtēs Kārli apdzen pārējie (\(60\) izvēlēts, kā \(5\), \(12\) un \(20\) minūšu mazākais kopīgais dalāmais). No tā, ka Kārlis satiek Vilni ik pa \(12\) minūtēm izriet, ka Vilnis ir veicis \(60: 12=5\) apļus pa kreisi attiecībā pret Kārli.

No tā, ka Mārtiņš apdzen Vilni ik pa \(5\) minūtēm izriet, ka Mārtiņš ir veicis \(60:5=12\) apļus pa kreisi attiecībā pret Vilni, tātad viņš veicis \(12+5=17\) apļus pa kreisi attiecībā pret Kārli.

No tā, ka Sandris apdzen Kārli ik pa 20 minūtēm izriet, ka Sandris ir veicis \(60:20=3\) apļus pa labi attiecībā pret Kārli.

Tā kā Mārtiņš ir veicis attiecībā pret Kārli \(17\) apļus pa kreisi, bet Sandris veicis 3 apļus pa labi, tad viņi šajās \(60\) minūtēs ir satikušies \(17+3=20\) reizes. Tātad viņi satiekas ik pēc \(60:20=3\) minūtēm.