by Ben Monreal
• COTENANCY
• COTTONY
• DARKEST PERU
• EMBOLI
• ERSTWHILE
• ANTHEMIC
• BYWORD
• METRIC SPACE
• MONONOKE
• XENOPHOBIC

The answers can be paired such that placing a Holi answer before a Patriots' Day answer reveals a color:

 COTENANCY ANTHEMIC COTTONY XENOPHOBIC DARKESTPERU BYWORD EMBOLI METRICSPACE ERSTWHILE MONONOKE

These strings are all sixteen letters long, so they can each have their ends connected to form five rings. Five rings, the five colors, and the note in the flavortext about preparing for a marathon next year in Tokyo should all suggest the Olympics, and in fact these five rings can be linked together to form the Olympic rings, with intersecting letters matching.

 A N T H E O P H O B R U B Y W Y M N I E O C H I L E M I C S P A R N W C X O R O T C D A N E T O Y N O T T S E K R A S O E M R E K O N M I L O B

Once the answers are assembled into rings, the marathon route describes a path on the rings themselves. (There are several ways to assemble 16-letter rings to form the Olympic logo, but the marathon route reveals the number of stops between some intersections, which should help with the construction.) Here’s a rundown of the letters the route takes you to:

Start on the RED route, heading southeast. Check in at your starting place.
The red route is the red ring, and the next step reveals that there will be an intersection seven places later and then three after that. This fixes the starting line as the W in BYWORD.
Continue forward seven places, and you’ll reach an intersection; go straight through that intersection and turn right at the intersection three places later. After turning, continue forward one place and check in.
Turn right at the P in METRIC SPACE / DARKEST PERU, and check in at the A in METRIC SPACE.
Continue forward six places (going straight through an intersection along the way) and check in.
Check in at the L in EMBOLI.
Continue forward and turn left at the second intersection you encounter. Then turn left at the next intersection you encounter. After turning, continue forward seven places (going straight through an intersection along the way) and check in.
Turn left at the C in XENOPHOBIC / METRIC SPACE, turn left at the E in XENOPHOBIC / ERSTWHILE, and check in at the K in MONONOKE.
Continue forward and turn left at the second intersection you encounter. After turning, continue forward seven places and check in.
Turn left at the I in ANTHEMIC / ERSTWHILE, and check in at the Y in COTENANCY.
Continue forward six places and turn left at the intersection. After turning, continue forward one place and check in.
Turn left at the T in COTENANCY / ERSTWHILE, and check in at the W in ERSTWHILE.
Continue forward and turn left at the third intersection you encounter. After turning, continue forward one place and check in.
Turn left at the N in MONONOKE / COTTONY, and check in at the O in COTTONY.
Turn right at the next intersection you encounter. After turning, continue forward ten places (going straight through an intersection along the way) and check in.
Turn right at the T in COTTONY / METRIC SPACE, and check in at the A in METRIC SPACE (again).
Turn right at the next intersection you encounter. After turning, continue forward nine places (passing your starting point) and check in at the finish line! Congratulations!
Turn right at the P in METRIC SPACE / DARKEST PERU, and check in at the D in BYWORD.
The letters at the nine “check-in” points spell the answer WALKY WOAD, which is a special dye the Holi Town revelers have prepared for marathon walkers.