answer:1. **Initial Setup:** - Consider triangle ABC with sides AB, BC, and CA. - Points D and E are placed on sides AB and BC respectively, such that (AD:DB = CE:EB = 2:1). 2. **Assumption for Contradiction:** - Assume that line segment DE is **not** parallel to side AC. - Draw a line through point D parallel to AC. Let this line intersect side BC at point E'. 3. **Use of Previous Problem's Insight:** - By our assumption and the relevant geometric principles (particularly considering the properties of similar triangles and parallel lines), point E' divides BC in the same ratio as the original point E. 4. **Ratio Comparison:** - Given (CE:EB = 2:1), ensure point E' divides side BC in a similar manner. - Triangles (ADE') and (ACD) being similar (due to (DE' parallel AC)), and using the ratio (AD:DB = 2:1), means that point E' must divide BC in the ratio (CE':E'B = 2:1). 5. **Conclusion from Ratio Identity:** - Since points E and E' both divide BC into the ratio (2:1), they must coincide. - Therefore, (E' = E), indicating that DE **is** parallel to AC. Thus, by contradiction, we have established that: [ DE parallel AC ] boxed{DE parallel AC}

answer:By Vieta's Formulas, we know that c is the sum of the three roots of the polynomial x^3 - cx^2 + dx - 3990. Also, Vieta's Formulas indicate that 3990 is the product of the three roots. Factoring 3990 into its prime components, we have 3990 = 2 cdot 3 cdot 5 cdot 7 cdot 19. We need to combine these factors to form three roots to minimize c. Let's consider combining the smallest factors to minimize the sum: - Combine 2, 3, and 5 to form one root: 2 cdot 3 cdot 5 = 30. - The second root can be 7, and the third root can be 19. Thus, the roots are 30, 7, and 19. Summing these, we get c = 30 + 7 + 19 = boxed{56}.

answer:(1) From the question, the variable x satisfies frac{1+x}{1-x} > 0, ...(2 points) The above equation is equivalent to (1+x)(1-x) > 0, ...(3 points) That is, (x+1)(x-1) < 0, ...(4 points) So, -1 < x < 1, ...(6 points) (2) Since the domain of the function is symmetric about the origin, ...(7 points) And, f(-x) = lg frac{1+(-x)}{1-(-x)} = lg frac{1-x}{1+x} = lg (frac{1+x}{1-x})^{-1} = -lg frac{1+x}{1-x} = -f(x). Therefore, f(x) is an odd function, ...(12 points) The final answers are: (1) The domain of the function is boxed{-1 < x < 1}. (2) The function f(x) is boxed{text{an odd function}}.

answer:The line l making an angle theta = tan^{-1}left(frac{1}{3}right) with the x-axis is reflected across l_1, which makes an angle alpha = frac{pi}{60}. The reflected line l' makes an angle 2alpha - theta = frac{pi}{30} - theta with the x-axis. Next, l' is reflected across l_2, which makes an angle beta = frac{pi}{45}. The resulting line l'' makes an angle 2beta - (frac{pi}{30} - theta) = frac{pi}{15} - theta with the x-axis. The line R^{(n)}(l) then makes an angle of theta + left(frac{pi}{15} - thetaright) cdot n = frac{pi n}{15} + theta(1-n) with the x-axis. For R^{(n)}(l) to coincide with the original line l, frac{pi n}{15} must be an integer multiple of 2pi, i.e., frac{pi n}{15} = 2pi k for some integer k. This simplifies to n = 30k. The smallest positive integer k is 1, thus n = 30k = 30. The smallest positive integer m for which R^{(m)}(l)=l is boxed{30}.