Category Archives: Daily Problems

Trigonometry Challenge 2022/09/26

Posted on September 26, 2022 by kevin

Let $\alpha$, $\beta$, and $\gamma$ be the interior angles of $\triangle{ABC}$. Find all solutions so that $$\cos\alpha\cdot\cos\beta+\sin\alpha\cdot\sin\beta\cdot\sin\gamma=1$$ 🔑 Solution: Since $\alpha$, $\beta$, and $\gamma$ are the interior angles of $\triangle{ABC}$, we have $$0\lt\alpha,\ \beta,\ \gamma\lt\pi$$ Therefore $$0\lt\sin\alpha,\ \sin\beta,\ \sin\gamma\le 1$$ … Continue reading →

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MATHCOUNTS Exercise – 01/24/2022

Posted on January 24, 2022 by kevin

In unit square $ABCD$, $E$ and $F$ are midpoints of $CD$ and $AD$ respectively. Line $AE$ and $CF$ intersect at $G$. $M$ and $N$ are incenters of $\triangle{AFG}$ and $\triangle{CEG}$. Find $MN$.

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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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Mass Points Exercises

Posted on July 22, 2021 by kevin

(AIME 2011) In triangle $ABC, AB=\dfrac{20}{11}AC.$ The angle bisector of $\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of intersection of $AC$ and $BM$. Find $\dfrac{CP}{PA}$. SolutionWithout loss … Continue reading →

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