answer:The sum of two prime numbers is greater than 2, so if this sum is to be prime, it must be odd. Therefore, one of the primes in the sum must be 2, and the other must be odd. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Of the odd primes, 3, 5, 11, 17, and 23 added to 2 provide a prime number. Therefore, there are 5 possible pairs whose sum is prime: (2 + 3), (2 + 5), (2 + 11), (2 + 17), and (2 + 23). The total number of pairs that can be formed from ten numbers is computed using the combination formula for choosing two elements from ten: binom{10}{2} = 45. The probability is therefore frac{5}{45} = boxed{frac{1}{9}}.

answer:1. **Initial Setup**: Vasya has a collection of squares and triangles. Using all the pieces, Vasya can form a polyhedron where these shapes serve as the faces of the polyhedron. 2. **Example with Original Set**: Assume Vasya initially has: - 1 square and - 4 equilateral triangles. The sides of these triangles are equal to the sides of the square. 3. **Polyhedron from Original Set**: With 1 square and 4 equilateral triangles, Vasya can construct a square pyramid. The square acts as the base, and the 4 equilateral triangles form the lateral faces of the pyramid. 4. **Replacement by Petya**: Petya replaces 2 triangles with 2 squares of the same side length. Now, the set consists of: - 3 squares and - 2 equilateral triangles. 5. **Construction with New Set**: Vasya can construct a prism with the new set: - The base is a triangle formed by the 2 equilateral triangles. - The lateral faces are formed by the 3 squares, with each square attached to a side of the triangular base, creating a triangular prism. # Conclusion Vasya can form a polyhedron (a prism) using the new set that includes: - 3 squares - 2 equilateral triangles. boxed{text{Prism}}

answer:Since f(x) is an even function defined for every x in mathbb{R} and satisfies f(x+3)=-frac{1}{f(x)}, we can infer that: f(x+6) = -frac{1}{f(x+3)} = f(x), which means the period of function f(x) is 6. Therefore, we have: f(107.5) = f(18 times 6 - 0.5) = f(-0.5). Applying the property of the even function f(-x) = f(x) and the functional equation given, we get: f(107.5) = -frac{1}{f(-0.5+3)} = -frac{1}{f(2.5)}. Given that f(x) = 4x when x in (-infty, 0), we can say that this also applies to x in (0, +infty) since f(x) is an even function. Hence, for f(2.5), we have: f(107.5) = -frac{1}{f(2.5)} = -frac{1}{f(-2.5)} = -frac{1}{-4 times 2.5} = boxed{frac{1}{10}}. Hence, the correct option is B.

answer:Solution: From aoverrightarrow{MA} + boverrightarrow{MB} + frac{sqrt{3}}{3}coverrightarrow{MC} = aoverrightarrow{MA} + boverrightarrow{MB} + frac{sqrt{3}}{3}c(-overrightarrow{MA} - overrightarrow{MB}) = (a- frac{sqrt{3}}{3}c)overrightarrow{MA} + (b- frac{sqrt{3}}{3}c)overrightarrow{MB} = overrightarrow{0}, Since overrightarrow{MA} and overrightarrow{MB} are not collinear, it follows that a- frac{sqrt{3}}{3}c = b- frac{sqrt{3}}{3}c = 0, Therefore, a = b = frac{sqrt{3}}{3}c. In triangle ABC, by using the cosine rule, we can find that cos A = frac{sqrt{3}}{2}, therefore A = frac{pi}{6}. If a = 3, then b = 3, c = 3sqrt{3}, the area of triangle ABC, S = frac{1}{2}bcsin A = frac{1}{2} times 3 times 3sqrt{3} times frac{1}{2} = frac{9sqrt{3}}{4}, Hence, the answers are boxed{frac{pi}{6}} for the size of the internal angle A and boxed{frac{9sqrt{3}}{4}} for the area of triangle ABC.