Monthly Archives: January 2023

AIME 2017 I – Problem 14

Posted on January 13, 2023 by kevin

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 →

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Combination Challenge – 2023/01/06

Posted on January 6, 2023 by kevin

Prove $$\sum_{k=1}^n\binom{n}{k}\binom{n-1}{k-1}=\binom{2n-1}{n}$$ 🔑 Proof: Rewrite the left side as: $$\sum_{k=1}^n\binom{n}{k}\binom{n-1}{n-k}$$ The above can be interpreted as the number of ways to choose $n$ balls from $2n-1$ distinct balls, with balls divided into two groups, one group with $k$ distinct balls, … Continue reading →

Posted in Combinatorics, Daily Problems | Comments Off on Combination Challenge – 2023/01/06