Monthly Archives: May 2023

Trigonometry Challenge – 1 ⭐⭐

Posted on May 13, 2023 by kevin

Prove that $\ \ \ \ \ \ \ \dfrac{sin 20^\circ}{cos 20^\circ-2\cdot sin 10^\circ}=tan 30^\circ\ \ \ \ \ \ \ \ \ $ Solution Proof: Because $$ sin 10^\circ-sin 10^\circ=0,\ \ \ sin^2 20^\circ+cos^2 20^\circ=1,\ \ \ sin 30^\circ … Continue reading →

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Cyclic System of Equations – 3

Posted on May 1, 2023 by kevin

Find all real solutions of the following equations: $$a+b=c^2$$ $$b+c=d^2$$ $$c+d=e^2$$ $$d+e=a^2$$ $$e+a=b^2$$ Solution: If $a=b=c=d=e$, then we have two real solutions as $$a=b=c=d=e=0$$ and $$a=b=c=d=e=2$$ Without loss of generality, assume $a\le b \le c \le d \le e$ and … Continue reading →

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