Category Archives: Circles in a Square

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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Circles in a Square – Part 12

Posted on May 21, 2024 by kevin

As shown in the figure, $ABCD$ is a square, $E$ is the mid-point of $AB$. The circle with its center at $H$ is tangent with $AD$, $AE$ and $DE$. The circle with its center at $F$ is tangent with $BC$, … Continue reading →

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MATHCOUNTS Exercises – 16

Posted on January 26, 2020 by kevin

A semicircle and a circle are placed inside a square with sides of length 4 cm as shown. The circle is tangent to two adjacent sides of the square and to the semicircle. The diameter of the semicircle is a … Continue reading →

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Circles in a Square – Part 11

Posted on January 11, 2020 by kevin

8 semi-circles are drawn along the side lines inside of a unit square as shown in the following figure, with another circle centered at the center of the square and tangent to all of 8 semi-circles. What is the area of … Continue reading →

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Circles in a Square – Part 10

Posted on December 17, 2019 by kevin

In this 10th post of this series, consider the following figure with two co-tangent semi-circles drawn, centered at point $E$ on $\overline{AB}$ and $F$ on $\overline{BC}$. A full circle centered at point G, tangent with both semi-circles, line $\overline{AD}$ and $\overline{CD}$ in a unit square $ABCD$. Find … Continue reading →