Maths Olympiad Prep

Library / HMMT November /330 of 672

HMMT November

Algebra Difficulty 5.0 AIME Find the answer

Find the value of sin⁡2B+sin⁡2C−sin⁡2Asin⁡Bsin⁡C\frac{\sin^{2}B+\sin^{2}C-\sin^{2}A}{\sin B \sin C} given that sin⁡Bsin⁡C=ACAB\frac{\sin B}{\sin C}=\frac{AC}{AB}, sin⁡Csin⁡B=ABAC\frac{\sin C}{\sin B}=\frac{AB}{AC}, and sin⁡Asin⁡Bsin⁡C=BCAC⋅AB\frac{\sin A}{\sin B \sin C}=\frac{BC}{AC \cdot AB}.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Using the Law of Sines, we have sin⁡2B+sin⁡2C−sin⁡2Asin⁡Bsin⁡C=sin⁡Bsin⁡C+sin⁡Csin⁡B−sin⁡Asin⁡Bsin⁡Asin⁡C=ACAB+ABAC−BCACBCAB=8380\frac{\sin^{2}B+\sin^{2}C-\sin^{2}A}{\sin B \sin C}=\frac{\sin B}{\sin C}+\frac{\sin C}{\sin B}-\frac{\sin A}{\sin B} \frac{\sin A}{\sin C}=\frac{AC}{AB}+\frac{AB}{AC}-\frac{BC}{AC} \frac{BC}{AB}=\frac{83}{80}

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.