MathDB

5

Part of 2019 Caucasus Mathematical Olympiad

Problems(2)

a_ia_j+a_ka_l all equal

Source: IV Caucasus Mathematic Olympiad

4/7/2019
Vasya has a numeric expression □⋅□+□⋅□ \Box \cdot \Box +\Box \cdot \Box and 4 cards with numbers that can be put on 4 free places in the expression. Vasya tried to put cards in all possible ways and all the time obtained the same value as a result. Prove that equal numbers are written on three of his cards.
algebra
Different relation in a triangle

Source: IV Caucasus Mathematic Olympiad

4/7/2019
Given a triangle ABCABC with BC=aBC=a, CA=bCA=b, AB=cAB=c, ∠BAC=α\angle BAC = \alpha, ∠CBA=β\angle CBA = \beta, ∠ACB=γ\angle ACB = \gamma. Prove that asin⁡(β−γ)+bsin⁡(γ−α)+csin⁡(α−β)=0. a \sin(\beta-\gamma) + b \sin(\gamma-\alpha) +c\sin(\alpha-\beta) = 0.
trigonometryLaw of Sinesgeometry