Author Archives: kevin

MATHCOUNTS Exercise – Convolution of Non-zero Squares

Posted on March 1, 2021 by kevin

A four by four grid of unit squares contains squares of various sizes (1 by 1 through 4 by 4), each of which are formed entirely from squares in the grid. In each of the 16 unit squares, write the … Continue reading →

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MATHCOUNTS Exercise – 22

Posted on January 1, 2021 by kevin

Point $E$ and $F$ are inside the square $ABCD$, with $DE=10$, $EF=6$, $BF=4$, and $\angle{DEF}=\angle{BFE}=90^\circ$. Find the area of the square $ABCD$. Click here for the solutions. Solution 1: Draw line $DB$, intersecting $EF$ at $G$. We have $\triangle{DEG}\sim\triangle{BFG}$. $\therefore … Continue reading →

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Sum and Product of All Factors

Posted on December 18, 2020 by kevin

For a nature number $$n=p_{1}^{q_{1}}\times p_{2}^{q_{2}}\times \cdots \times p_{m}^{q_{m}}$$ where $p_{i} (1 \le i \le m) $ are unique prime numbers. The total number of positive factors of $n$ is $$f=(q_{1}+1)\times(q_{2}+1)\times\cdots\times(q_{m}+1)$$ The sum of all positive factors of $n$ is … Continue reading →

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Cheat Sheet for Distributing $k$ Balls into $n$ Boxes

Posted on November 19, 2020 by kevin

Distribution of Restriction $k\text{ Balls}$ $n\text{ Boxes}$ None $\le 1$ $\ge 1$ $=1$ $distinct$ $distinct$ $$n^k$$ $$(n)_k$$ $$n!S(k,n)$$ $$n!\text{ or }0$$ $identical$ $distinct$ $${n+k-1}\choose k$$ $$n\choose k$$ $${k-1}\choose{n-1}$$ $$1\text{ or }0$$ $distinct$ $identical$ $$\sum_{i=1}^{n}S(k,i)$$ $$1\text{ or }0$$ $$S(k,n)$$ $$1\text{ or … Continue reading →

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Geometry Fold Problems

Posted on October 16, 2020 by kevin

A unit equilateral $\triangle{ABC}$ is folded over line $DE$, forming a quadrilateral $BCDE$, with $A$ touching $BC$ at $A’$, and $\triangle{BA’E}$ is a right triangle. The area of $BCDE$ is __________. Answer: $\dfrac{17\sqrt{3}-27}{8}\approx 0.305608$ A quarter-circle $ABC$, with its center … Continue reading →

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