Author Archives: kevin

MathCounts Geometry Exercise – 1

Posted on October 28, 2022 by kevin

25 20 30 ________ One rectangle is divided into four smaller rectangles. The areas of three smaller rectangles are $25\ cm^2$, $20\ cm^2$, and $30\ cm^2$ respectively, as shown in the diagram. Find the area of the shaded region (unit: … Continue reading →

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Distance between the incenter and circumcenter

Posted on October 7, 2022 by kevin

The distance between the incenter and circumcenter of a triangle, is calculated by Euler’s theorem in geometry: $$d=\sqrt{R(R-2r)}$$ which also implies $$R\ge2r$$ For a Bicentric quadrilateral, the distance between the incenter and circumcenter can be calculated by Fuss’s theorem or … Continue reading →

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Trigonometry Challenge 2022/09/26

Posted on September 26, 2022 by kevin

Let $\alpha$, $\beta$, and $\gamma$ be the interior angles of $\triangle{ABC}$. Find all solutions so that $$\cos\alpha\cdot\cos\beta+\sin\alpha\cdot\sin\beta\cdot\sin\gamma=1$$ 🔑 Solution: Since $\alpha$, $\beta$, and $\gamma$ are the interior angles of $\triangle{ABC}$, we have $$0\lt\alpha,\ \beta,\ \gamma\lt\pi$$ Therefore $$0\lt\sin\alpha,\ \sin\beta,\ \sin\gamma\le 1$$ … Continue reading →

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Probability Challenge 2022/08/29

Posted on August 29, 2022 by kevin

A frog can randomly jump exactly 1 yard away in any random direction constantly. (1) What is the probability that the frog is within 1 yard away from its starting point after 2 jumps? (2) What is the probability that … Continue reading →

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Algebra Challenge 2022/07/25

Posted on July 25, 2022 by kevin

Let $x$ and $a$ are real numbers, and $a$ is a constant with $a>=0$, and $x^2=a(x-\lfloor x \rfloor)$. Find the number of solutions for $x$, in terms of $a$. Click here for the solution. Solution: Let $y=x-\lfloor x \rfloor$ and … Continue reading →

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