PC2026-P1 Best solutions and final results

Problem 1

In total, 329 solutions were submitted for problem 1, of which 182 were correct. Congratulations to everyone who successfully solved it!

There were 119 correct solutions submitted before the first hint was published, and only those were considered as candidates for the best solution awards. Additionally, since there were so many well-written solutions to choose from, the handwritten solutions were not considered this time.

The award winners are as follows.

Zhuoheng Wu (bonus: 0.25 pts) uses the sum and difference of the relative speeds to obtain a very economical solution. Recognising the transverse velocity as the altitude of the velocity triangle turns the displacement calculation into a single straightforward integral.
Alon Hart (bonus: 0.15 pts) gives an illuminating symmetry argument for the conserved speed difference and identifies its geometric meaning: the trajectory in velocity space is a hyperbola. This makes the key idea particularly accessible.
Jakub Kliment (bonus: 0.15 pts) exploits the invariant through a simple relation between the horizontal velocity and the speed. This gives a compact route to both the terminal velocity and the required displacement.
Thach Nguyen (bonus: 0.15 pts) chooses the direction of the velocity as the integration variable. The resulting cancellation simplifies the displacement integral and provides an elegant alternative to working with velocity components.
Muntean Rares (bonus: 0.1 pts) recognises that elliptic coordinates fit the geometry of the problem: the sum and difference of the relative speeds take simple forms, and one coordinate remains constant throughout the motion.
Jerzy Gamdzyk (bonus: 0.1 pts) interprets the motion as gradient descent in velocity space and explains how its qualitative behaviour guides the calculation, particularly the use of the monotonically decreasing transverse velocity as a variable.
Jonathan Burger (bonus: 0.1 pts) brings together the gradient-flow interpretation and the hyperbolic trajectory in velocity space, showing how this geometric structure allows time to be eliminated from the displacement calculation.

Final results for Problem 1

Pre-university students:

University students: