Category Archives: Algebra

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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Geometry Challenge – 9

Posted on February 21, 2022 by kevin

In the diagram below, two tangent semi-circles centered at A and B are inscribed in a large semi-circle center at O, and circle C is tangent to semi-circle A, B and O. Point A, O, and B are on the … Continue reading →

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Storing Cans on Shelf

Posted on November 8, 2021 by kevin

If cans can be placed on top of the one another straight up, how many cylindrical cans $4$ inches in diameter and $6$ inches high can be stored on a shelf $2$ feet wide and $6$ feet long if the … Continue reading →

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Geometry Challenge – 8 ⭐

Posted on October 16, 2021 by kevin

In unit square $ABCD$, point $E$ and $F$ are located on edge $CD$, with $E$ closer to $D$ and $F$ closer to $C$. Line $BE$ and $AF$ intersect at $G$, forming two triangles: $\triangle{ABG}$ and $\triangle{EFG}$. Find the minimum value … Continue reading →

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Geometry Challenge – 6 ⭐⭐⭐⭐⭐

Posted on August 4, 2021 by kevin

In parallelogram $ABCD$, diagonal $AC$ tangents the incircle of $\triangle{ABC}$ at $P$. Let $r_1$ and $r_2$ be the radii of incircles of $\triangle{ADP}$ and $\triangle{DCP}$ respectively. 1. Prove that $\dfrac{r_1}{r_2}=\dfrac{AP}{CP}$ 2. If $AD=PD$, and $\dfrac{AD+CD}{AC}=p$, where $p>1$, prove that $\dfrac{r_1}{r_2}=1+\dfrac{1}{p}$. … Continue reading →

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