Monthly Archives: December 2025

Algebra Challenge – 12/28/2025

Posted on December 28, 2025 by kevin

Find all real values $x$ such that $4^x+6^x=9^x$.🔑 Solutions: Dividing $6^x$ on both side of the equation, we have: $$\Big{(}\dfrac{2}{3}\Big{)}^x+1=\Big{(}\dfrac{3}{2}\Big{)}^x$$ Let $y=\Big{(}\dfrac{2}{3}\Big{)}^x$, we have $$y+1=\dfrac{1}{y}$$ i.e. $$y^2+y-1=0$$ Solving the above equation, we have $$y=\dfrac{-1\pm\sqrt{5}}{2}$$ Because $x$ is real, and $y=\Big{(}\dfrac{2}{3}\Big{)}^x>0$, … Continue reading →

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Number Theory Challenge – 12/21/2025

Posted on December 21, 2025 by kevin

Prove that for every positive integer $n$, there is a positive integer $m$ so that $2^n | (19^m-97)$.🔑 Proof: We prove it by induction. Base Case: For $n=1,2,3$, we have $$n=1, m=1 \Longrightarrow 2^1|(19^1-97)$$ $$n=2, m=2 \Longrightarrow 2^2|(19^2-97)$$ $$n=3,m=2 \Longrightarrow … Continue reading →

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Number Theory Challenge – 12/20/2025

Posted on December 20, 2025 by kevin

Prove that for all positive integer $n$, $19^{2^n}=1+m\cdot 2^{n+2}$, where $m$ is a positive odd integer.🔑 Proof: We prove it by induction. Base Case: When $n=1$, we have $$19^{2^n}=361=1+45\cdot 2^3=1+45\cdot 2^{n+2}$$ Therefore we prove the case when $n=1$, is true … Continue reading →

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Number Theory Challenge – 12/12/2025

Posted on December 12, 2025 by kevin

Prove that for integers $a$,$b$,$c$, if $9|(a^3+b^3+c^3)$, then at least of them is divisible by $3$.🔑 Proof: Let $a=3A+x$, $b=3B+y$, $c=3C+z$, where $A$, $B$, $C$ are integers, $a\pmod 3=x$, $b\pmod 3=y$, $c\pmod 3=z$. Therefore $0\le x,y,z\le 2$. Without loss of … Continue reading →

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