Author Archives: kevin

MATHCOUNTS Exercises

Posted on August 27, 2023 by kevin

Two dices are thrown simultaneously. The probability that the sum of the two numbers on the top faces of the dices is a prime number is __________. Suppose $ABCD$ is a regular triangular pyramid, with each face as a unit … Continue reading →

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MathCounts Training – Number Theory – 2

Posted on August 25, 2023 by kevin

________ In base $b$, $441_b$ is equal to $n^2$ in base 10, and $351_b$ is equal to $(n-2)^2$. What is the value of $b$, expressed in base 10? ________ The base-three representation of $0.\overline{12}$ is equivalent to what base-ten common … Continue reading →

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MathCounts Training – Number Theory – 1

Posted on August 18, 2023 by kevin

________ Some positive integers have exactly four positive factors. For example, 35 has only 1, 5, 7, and 35 as its factors. What is the sum of the smallest five positive integers that have exactly four factors. ________ What is … Continue reading →

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Algebra Challenge – 2 ⭐⭐

Posted on July 4, 2023 by kevin

On the $X$-$Y$ plane, two circles centered at $(0,0)$ with radius $1$ and $2$ respectively. Let point $A=(-1,0)$, $B=(1,0)$, and $C$ is a point on the bigger circle. Find the locus of the orthocenter $P$ of $\triangle{ABC}$. Click here for … Continue reading →

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Algebra Challenge – 1 ⭐

Posted on June 26, 2023 by kevin

If $x=\sqrt[3]{9}+\sqrt[3]{3}+1$, find the value of $(\dfrac{2}{x}+1\big{)}^3$. Click here for the solution. Solution: Let $y=\sqrt[3]{3}$, we have $$x=y^2+y+1=\dfrac{y^3-1}{y-1}=\dfrac{(\sqrt[3]{3})^3-1}{\sqrt[3]{3}-1}=\dfrac{2}{\sqrt[3]{3}-1}$$ Therefore $$(\dfrac{2}{x}+1)^3=(2\cdot \dfrac{\sqrt[3]{3}-1}{2}+1)^3=(\sqrt[3]{3})^3=\boxed{3}$$