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2023 CMO Problems/Problem 5

Problem

In an acute triangle $\triangle A B C, K$ is a point on the extension of $B C$. Through $K$, draw lines parallel to $A B$ and $A C$, denoted as $K P$ and $K Q$ respectively, such that $B K=B P$ and $C K=C Q$. Let the circumcircle of $\triangle K P Q$ intersect $A K$ at point $T$. Prove:

(1) $\angle B T C+\angle A P B=\angle C Q A$;

(2) $A P \cdot B T \cdot C Q=A Q \cdot C T \cdot B P$.

Solution 1

Proof for (1): Let the side lengths of $\triangle A B C$ be $a, b, c$. We have $K A=K^{\prime} T-2 K B-K C$. Let $B C=a, C A=b, A B=c$. \[\angle A=\angle B=\angle K^{\prime} \Rightarrow \frac{A B}{\sin C}=\frac{B C}{\sin A}=\frac{C A}{\sin B}\]

Assume $K A=K^{\prime} T-2 K B-K C$ : \[\begin{gathered} 2 K A \cos (\angle K-\angle A-\theta)=2 a \cos \theta \\ \Rightarrow \cos (\angle \theta)=2 a \cos (\angle C+\theta) \end{gathered}\]

Assume $K T \sin A=K P \sin \theta+C \sin (\angle A+\theta)$ : \[\begin{gathered} \Rightarrow \angle B K C+\angle C=\theta, \quad \angle A^{\prime}=180^{\circ}-\angle C^{\prime}-(\angle A+B)=\angle K+A \\ \Rightarrow \frac{\sin (\angle A)}{\sin (\angle A+\theta)}=\frac{\cos B}{\cos C} \\ \Rightarrow \cos (\angle A-B) \leq \cos \theta \Rightarrow \cos (\angle A-B-C) \Rightarrow \cos (\angle A-B-C) \end{gathered}\]

Assume $\frac{a}{\sin B}=\frac{b}{\sin A}$ \[A P: B T: C T=A B: C T: B P\] Proof: (2) \[\begin{gathered} A P=A R, A P=A S, C Q=C K, B P=B K \Rightarrow \frac{B T}{A R} \cdot \frac{A R}{C T} \cdot \frac{C A}{B P}=1 \Rightarrow(*) \\ \frac{B T}{A R}=\frac{\sin \angle B A T}{\sin \angle A B C}=\frac{\sin \angle A T}{\sin B} \Rightarrow \frac{A S}{C T}=\frac{\sin C}{\sin \theta} \end{gathered}\] \[CK=b \sin \theta \ (\sin (c-\theta), BK=c \sin (A+\theta) / \sin (c-\theta)\] \[(*) \Leftrightarrow \frac{\sin (A+\theta)}{\sin B} \cdot \frac{\sin c}{\sin \theta} \cdot \frac{6 \sin \theta}{c \sin (A+\theta)}=1 \Leftrightarrow \frac{b}{c}=\frac{\sin B}{\sin C}\]

~xiaohuangya|szm

See Also

2023 CMO(CHINA) (Problems • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
1 • 2 • 3 • 4 • 5 • 6
All CMO(CHINA) Problems and Solutions
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