2016
Shadows
- Round
- Theory
- Points
- 10
- Level
- Bronze
- Taught in
- Section 1.7
Adapted from IOAA 2016 (Bhubaneswar), theory problem 4. We have restated it in our own words and kept the original data. The solution was last checked on 15 October 2026.
An observer in the northern hemisphere finds that on a certain day the shortest shadow of a vertical stick 1.000 m tall is 1.732 m long, and the longest shadow of the same stick is 5.671 m long. Find the observer's latitude and the Sun's declination on that day. Treat the Sun as a point and ignore refraction.
When is a shadow longest? What does a longest shadow of finite length say about the Sun?
- The Sun's altitude follows from
- The longest shadow is finite, so the Sun never sets. It is circumpolar, and its least altitude is at lower culmination. By (1.12) and (1.13),
- With
AnswerThe observer is at latitude 80° N, and the Sun's declination is +20°.
This problem also appears among the olympiad problems of Section 1.7, Problems for Chapter 1, which teaches the ideas it uses. Equation and figure numbers refer to the course.