Author Archives: kevin

Algebra/Geometry Challenge – 2

Posted on December 8, 2024 by kevin

Cyclic quadrilateral $ABCD$ has lengths $BC=CD=3$, $AB=5$, $AD=8$. What is the length of the shorter diagonal of $ABCD$? Click here for the solution. Solution: Let $AC$ and $BD$ intersect at $E$, $x=BD$, $y=AE$, $z=CE$, $AC=y+z$. Because $BC=CD$, $AC$ is angle … Continue reading →

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Stengel’s Theorem – Extended Cross Ladders Theorem

Posted on November 7, 2024 by kevin

If you remember, the Classic Cross Ladders Theorem was used in one of the solutions to this MathCounts problem. The Extended Cross Ladders Theorem, also known as Stengel’s Theorem, applies to cross ladders in a triangle. Specifically, in $\triangle{ABC}$ as … Continue reading →

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AMC 10 Exercise – 2

Posted on October 20, 2024 by kevin

What is the tens digit of $2019^{2020}-2021$? All the numbers $2$,$3$,$4$,$5$ are assigned to the four faces of a tetrahedron. For each of the four vertices of the tetrahedron, a product of three numbers are computed, where the three numbers … Continue reading →

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Geometry Challenge – Area of Irregular Pentagon

Posted on October 15, 2024 by kevin

As shown in the diagram below, a star sign consists of five straight lines. It produces five triangles and a pentagon. If areas of five triangles are 3, 10, 7, 15, and 8 square unit respectively. Find the area of … Continue reading →

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AMC 10 Exercise – 1

Posted on September 20, 2024 by kevin

Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2:1$? Two right circular cylinders … Continue reading →