Category Archives: Algebra

Find real solutions for the following equations: $$a+bcd = 2$$ $$b+cda=2$$ $$c+dab=2$$ $$d + abc=2$$ Solution: Because $a+bcd=2$, $b+cda=2$, we have $a+bcd=b+cda$. Factorizing it, we have $$(a-b)(cd-1)=0$$ Therefore either $a=b$ or $cd=1$. Case 1: If $a=b$, we have $$a+ac^2=2$$ $$c+da^2=2$$ … Continue reading →

Let $a > 1$ and $x > 1$ satisfy $$\log_a(\log_a(\log_a 2) + \log_a 24 – 128) = 128$$ and $\log_a(\log_a x) = 256$. Find the remainder when $x$ is divided by $1000$. 🔑 Solution: Let $a=2^n$, we have $$log_{2^n}(\log_{2^n}(\log_{2^n}2)+\log_{2^n}24-128)=128$$ $$log_{2^n}(\log_{2^n}2)+\log_{2^n}24-128=2^{128n}$$ … Continue reading →

One day sometime before 12 noon, the snow started to fall. A snow plower started to remove snow from the streets at 12 o’clock. In the first hour, it advanced 6 miles; in the second hour, it advanced 3 miles. … Continue reading →