Author Archives: kevin

Algebra Challenge – 2025/05/03

Posted on May 3, 2025 by kevin

Let $n$ be the real number. Define $[n]$ be the integer part of $n$, and $\{n\}$ be the decimal part of $n$. Solve the following equations: $$\begin{array}{ccccccc} \{x\} & + & [y] & + & \{z\} & = 2.9\\ \{y\} … Continue reading →

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Combination Challenge – 2025/04/29

Posted on April 29, 2025 by kevin

Let $f(n)$ be the number of binary strings with length as $n$ that they do not contain a sub-string as “$010$” nor “$101$”. Prove that for $n\ge 3$: $f(n)=f(n-1)+f(n-2)$ 🔑 Proof: For $n=1$, we have $2$ possible binary strings, as … Continue reading →

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Combination Challenge – 2025/04/21

Posted on April 21, 2025 by kevin

Find the number of ways to write $24$ as the sum of at least three positive integer multiples of $3$. For example, count $3+18+3$, $18+3+3$, and $3+6+3+9+3$, but not $18+6$ or $24$. (PurpleComet 2023, Middle School, Problem 8) Click here … Continue reading →

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Combination Challenge – 2025/03/05

Posted on March 5, 2025 by kevin

A soccer team has $8$ players. They need to form a starting lineup with one goalkeeper, one captain, one vice-captain, three unique field players, and two bench players. However, two specific players do not want to be captain, vice-captain, or … Continue reading →

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Geometry Challenge – 18

Posted on February 9, 2025 by kevin

Two squares $ABCD$ and $DEFG$ are inscribed inside a unit semi-circle, as shown in the following diagram, with $CD$ and $DE$ on the same line, $A$, $D$, $G$ on the diameter of the semi-circle, and $B$ and $F$ on the … Continue reading →

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