answer:We use the same approach as in the previous solution but start with b_1 = 2. We note that: [ b_{2^1} = b_2 = 1 cdot b_1 = 2, ] [ b_{2^2} = b_4 = 2 cdot b_2 = 2 cdot 2 = 2^2, ] [ b_{2^3} = b_8 = 4 cdot b_4 = 4 cdot 2^2 = 2^3, ] [ b_{2^4} = b_{16} = 8 cdot b_8 = 8 cdot 2^3 = 2^4, ] and in general, it seems that b_{2^n} = 2^n. This pattern suggests that: [ b_{2^{100}} = 2^{100}. ] Conclusion: The value of b_{2^{100}} in this modified sequence is boxed{2^{100}}.

answer:Start by simplifying each term separately: 1. **Simplify (625)^frac{1}{4}**: - Since 625 = 5^4, the fourth root of 625 is 5^{frac{4}{4}}=5. 2. **Simplify (343)^frac{1}{3}**: - Since 343 = 7^3, the cube root of 343 is 7^{frac{3}{3}}=7. 3. **Multiply the results**: - Multiply 5 and 7 together to get the final answer: 5 times 7 = 35. So, the simplified value of (625)^frac{1}{4}(343)^frac{1}{3} is boxed{35}. Conclusion: The new problem was simplified by factoring the numbers to their prime bases and applying the appropriate roots. The multiplication of the results gives a clear and correct answer.

answer:**Analysis** This problem examines the application of the sum of two angles in sine functions, trigonometric identities, the periodic formula of trigonometric functions, and the monotonicity of the sine function, testing simplification and calculation abilities. Given that f(x)=sin(omega x+varphi)+cos(omega x+varphi)=sqrt{2}sinleft(omega x+varphi+frac{pi}{4}right), since the function f(x) (omega > 0, 0 < varphi < pi) is an odd function, **Solution** From the given, f(x)=sin(omega x+varphi)+cos(omega x+varphi)=sqrt{2}sinleft(omega x+varphi+frac{pi}{4}right) Since the function f(x) (omega > 0, 0 < varphi < pi) is an odd function, varphi+frac{pi}{4}=kpi (kinmathbb{Z}), varphi=frac{3pi}{4}, f(x)=-sqrt{2}sin(omega x), The absolute difference between the x-coordinates of two adjacent intersection points of y=sqrt{2} and f(x) is frac{pi}{2}, Thus, omega=4, f(x)=-sqrt{2}sin(4x), For xinleft(frac{pi}{8}, frac{3pi}{8}right), 4xinleft(frac{pi}{2}, frac{3pi}{2}right), thus f(x) is increasing on left(frac{pi}{8}, frac{3pi}{8}right), Therefore, the correct choice is boxed{text{D}}.

answer:**I. megoldás:** 1. **Labeling Points and Angles:** Let the vertices of the desired hexagon be labeled as follows: - Points D and E on side AB. - Points F and G on side BC. - Points H and J on side CA. Denote the angles of the triangle ABC as usual: - alpha at vertex A - beta at vertex B - gamma at vertex C 2. **Calculating Hexagon Angles:** The hexagon's angles can be easily calculated since its sides are parallel to the sides of the triangle ABC. Hence, [ begin{aligned} angle DEF &= angle GHJ = 180^circ - alpha, angle EFG &= angle HJD = 180^circ - gamma, angle FGH &= angle JDE = 180^circ - beta. end{aligned} ] 3. **Constructing a Similar Hexagon:** Because the sides of the hexagon are equal and its angles are known, we can construct a similar hexagon. Consider a hexagon labeled D', E', F', G', H', J'. 4. **Extending Sides:** By extending every second side of the hexagon, we obtain a triangle A'B'C' similar to the original triangle ABC. The resulting figure can be scaled from a suitable center to match the required dimensions of our desired hexagon. 5. **Parallel Construction:** For simplicity, consider constructing the hexagon such that: - Side J'D', D'E', and E'F' are taken parallel to BC, AB, and AC respectively. This ensures that triangle A'B'C' is similarly positioned as ABC and the scaling process is simplified for ease of construction. **II. megoldás:** 1. **Identifying Similar Triangles:** The triangles formed by the segments cut off by the hexagon sides, namely, - HGC, - EBF, - ADJ, are similar to triangle ABC and thus to each other, with angles alpha, beta, and gamma respectively. 2. **Hexagon Sides in Similar Triangles:** Each triangle includes one side of the hexagon: - Each side is positioned opposite to a different angle of the triangle ABC. Moreover, sides AJ, JH, and HC must jointly add up to AC: [ AJ + JH + HC = AB. ] 3. **Choosing Segment Length:** Select a segment of any chosen length and consider it as one side of our hexagon. Using the conditions defined: - Construct similar triangles H'G'C' and A'D'J' corresponding to HGC and ADJ. 4. **Dividing Original Triangle Side:** Using the obtained divisions A'J', J'H', H'C': - Partition side AC in the triangle ABC appropriately. 5. **Determining Hexagon Side:** This process provides one side of the hexagon directly. 6. **Calculation of Side Length:** By denoting the side of hexagon as d and the sides of the triangle a, b, c, we can express: [ AC = frac{d}{a} cdot b + d + frac{d}{c} cdot b = b, ] Therefore, [ d = frac{abc}{ab + bc + ca}, quad text{or equivalently,} quad frac{1}{d} = frac{1}{a} + frac{1}{b} + frac{1}{c}. ] **Conclusion:** Therefore, the side length of the hexagon is given by: [ boxed{frac{abc}{ab + bc + ca}} ]