Category Archives: Algebra

Algebra Challenge – 2 ⭐⭐

Posted on July 4, 2023 by kevin

On the $X$-$Y$ plane, two circles centered at $(0,0)$ with radius $1$ and $2$ respectively. Let point $A=(-1,0)$, $B=(1,0)$, and $C$ is a point on the bigger circle. Find the locus of the orthocenter $P$ of $\triangle{ABC}$. Click here for … Continue reading →

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Algebra Challenge – 1 ⭐

Posted on June 26, 2023 by kevin

If $x=\sqrt[3]{9}+\sqrt[3]{3}+1$, find the value of $(\dfrac{2}{x}+1\big{)}^3$. Click here for the solution. Solution: Let $y=\sqrt[3]{3}$, we have $$x=y^2+y+1=\dfrac{y^3-1}{y-1}=\dfrac{(\sqrt[3]{3})^3-1}{\sqrt[3]{3}-1}=\dfrac{2}{\sqrt[3]{3}-1}$$ Therefore $$(\dfrac{2}{x}+1)^3=(2\cdot \dfrac{\sqrt[3]{3}-1}{2}+1)^3=(\sqrt[3]{3})^3=\boxed{3}$$

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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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Geometry Challenge – 13

Posted on April 18, 2023 by kevin

$ABCD$ is a square, P is an inner point such that $PA:PB:PC=1:2:3$. Find $\angle{APB}$ in degrees. A B C D P Click here for the solution. Solution 1: As shown in the diagram at the right, link $AC$. Without loss … Continue reading →

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Cyclic System of Equations – 2

Posted on March 28, 2023 by kevin

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 →