answer:Since point P(1,m) is outside the ellipse frac{x^{2}}{4}+y^{2}=1, we have that m^{2} > frac{3}{4}. Now consider the circle x^{2}+y^{2}=1 with center at (0,0). The distance d from the center of the circle to the line y-2mx-sqrt{3}=0 can be calculated using the formula for the distance from a point to a line: d = frac{left| -2m cdot 0 + 1 cdot 0 - sqrt{3} right|}{sqrt{(2m)^{2} + 1^{2}}} = frac{sqrt{3}}{sqrt{4m^2 + 1}}. Since we have m^{2} > frac{3}{4}, let's simplify the inequality for the denominator under the square root: 4m^2 + 1 > 4 cdot frac{3}{4} + 1 = 4 + 1 = 5. Now, we rewrite the distance as: d < frac{sqrt{3}}{sqrt{5}} < frac{sqrt{3}}{sqrt{4}} = frac{sqrt{3}}{2} < 1. Because the distance d from the line to the center of the circle is less than the radius of the circle, the line y=2mx+ sqrt{3} and the circle x^{2}+y^{2}=1 must intersect. Hence, the positional relationship between the line and the circle is intersection, so the correct answer is: boxed{B}.

answer:1. **Convert Elena's hourly wage to cents**: Elena earns 25 dollars per hour. Since there are 100 cents in a dollar, her hourly wage in cents is: [ 25 text{ dollars} times 100 text{ cents/dollar} = 2500 text{ cents} ] 2. **Calculate the tax deduction in cents**: The state tax rate is 2.4%. To find the amount deducted for taxes in cents, we calculate 2.4% of 2500 cents: [ 2.4% text{ of } 2500 text{ cents} = 0.024 times 2500 = 60 text{ cents} ] 3. **Conclusion**: Therefore, 60 cents per hour of Elena's wages are used to pay state taxes. [ 60 ] The final answer is boxed{C}

answer:1. **Symmedian and Reflection Properties**: - Given that (AD) is a symmedian of triangle (ABC), we know that (D) lies on the circumcircle of (triangle ABC). - Reflect (D) about (BC) to get point (E). By properties of reflection, (E) will also lie on the circumcircle of (triangle ABC). 2. **Reflections about Sides**: - Reflect (E) about (AB) to get point (C_0). - Reflect (E) about (AC) to get point (B_0). 3. **Concurrent Lines**: - We need to prove that lines (AD), (BB_0), and (CC_0) are concurrent if and only if (angle BAC = 60^circ). 4. **Special Properties in a (60^circ) Triangle**: - In a triangle where (angle BAC = 60^circ), the Dumpty point coincides with the First Fermat point. - The First Fermat point is known to be the point such that the total distance from the vertices of the triangle is minimized. - Since the Dumpty point and the First Fermat point coincide, and the Neuberg cubic passes through these points, it also passes through the (HM)-point, which is point (E). 5. **Conclusion**: - Since (E) lies on the Neuberg cubic and the lines (AD), (BB_0), and (CC_0) are concurrent, this configuration is only possible if (angle BAC = 60^circ). (blacksquare)

answer:- A regular pentagon can be divided into equal isosceles triangles by drawing lines from the center to each vertex. - In a regular pentagon with side length s, the area can be calculated by using the formula text{Area} = frac{1}{4}sqrt{5(5+2sqrt{5})}s^2. For ABCDE, if each side is equal to half the side length of PAD (since AD in PAD corresponds to AD in the pentagon), then s = 5 and the pentagon's area is frac{1}{4}sqrt{5(5+2sqrt{5})}5^2 = frac{25}{4}sqrt{5(5+2sqrt{5})}. - Like in the hexagon case, the altitude of the pyramid can be considered. Since PAD is an equilateral triangle and P is the apex of the pyramid right above AD, PO (the altitude) can be found knowing that in an equilateral triangle with side length 10, the altitude from any vertex to the midpoint of the opposite side splits the triangle into two 30-60-90 triangles. Therefore, PO = 5sqrt{3} (altitude formula for an equilateral triangle). - Using the volume formula V = frac{1}{3} text{Base Area} times text{Height}, the volume of the pyramid is V = frac{1}{3} times frac{25}{4}sqrt{5(5+2sqrt{5})} times 5sqrt{3} = frac{125sqrt{15(5+2sqrt{5})}}{12}. Conclusion: boxed{frac{125sqrt{15(5+2sqrt{5})}}{12}}