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Problem 2

Part of 2000 Mongolian Mathematical Olympiad

Problems(2)

externally tangent circles, prove collinearity of intersection points

Source: Mongolia MO 2000 Grade 10 P2

4/22/2021
Circles ω1,ω2,ω3\omega_1,\omega_2,\omega_3 with centers O1,O2,O3O_1,O_2,O_3, respectively, are externally tangent to each other. The circle ω1\omega_1 touches ω2\omega_2 at P1P_1 and ω3\omega_3 at P2P_2. For any point AA on ω1\omega_1, A1A_1 denotes the point symmetric to AA with respect to O1O_1. Show that the intersection points of AP2AP_2 with ω3\omega_3, A1P3A_1P_3 with ω2\omega_2, and AP3AP_3 with A1P2A_1P_2 lie on a line.
geometrycircles
inequality in vectors

Source: Mongolia MO 2000 Teachers P2

4/22/2021
Let n≥2n\ge2. For any two nn-vectors x⃗=(x1,…,xn)\vec x=(x_1,\ldots,x_n) and y⃗=(y1,…,yn)\vec y=(y_1,\ldots,y_n), we define f(x⃗,y⃗)=x1y1‾−∑i=2nxiyi‾.f\left(\vec x,\vec y\right)=x_1\overline{y_1}-\sum_{i=2}^nx_i\overline{y_i}.Prove that if f(x⃗,x⃗)≥0f\left(\vec x,\vec x\right)\ge0, and f(y⃗,y⃗)≥0f\left(\vec y,\vec y\right)\ge0, then ∣f(x⃗,y⃗)∣2≥f(x⃗,x⃗)f(y⃗,y⃗)\left|f\left(\vec x,\vec y\right)\right|^2\ge f\left(\vec x,\vec x\right)f\left(\vec y,\vec y\right).
inequalitiesgeometryVectorsvector