answer:1. **Define the Given Circle**: - Let the given circle be (k) with center (O) and radius (r). 2. **Define Triangle Properties**: - Let (A, B, C) be the vertices of the triangle. - The angle at vertex (A) is (alpha). - The median from vertex (B) to the midpoint (B_1) of side (AC) has a given length (s_b). 3. **Using Inscribed and Central Angles**: - According to the relationship between inscribed and central angles, the central angle (BOC) subtended by the arc (BC) is (2alpha). 4. **Constructing Points B and C**: - Choose any radius (OF) of the circle (k). Measure an angle (2alpha) centered at (O), ensuring the arms of this angle intersect the circle (k) at points (B) and (C). - Denote the segment (BAC). Point (A) must lie on the arc (BC) that does not include the point (F). This arc is part of the circle (i). 5. **Locating Point (B_1)**: - Point (B_1) should be the midpoint of side (AC). This implies (B_1) is a scaled-down image of (A) relative to the center (C). - Thus, (B_1) must lie on the image circle of the arc, denoted as (i_1), which is the Thales' circle using the diameter (OC). 6. **Intersection of Circles**: - To further locate point (B_1), consider the circle (k_b) centered at (B) with radius (s_b). This circle represents all possible points (B_1) such that the segment (BB_1) is (s_b). - The intersection points of (i_1) and (k_b) provide the possible positions for (B_1). 7. **Determining Point A**: - Finally, point (A) can be found as the intersection of the circle (k) with the line (CB_1). - This process defines the points of triangle (A, B, C). 8. **Number of Solutions**: - The number of possible solutions depends on the intersections of circles (i_1) and (k_b). There can be: - 2 solutions if there are two intersection points. - 1 solution if there is exactly one intersection point. - 0 solutions if there are no intersection points. # Conclusion: The solution to constructing a triangle with the given conditions can result in 2, 1, or 0 triangles depending on the intersections of the circles (i_1) and (k_b). The correctness of the construction is evident from geometric properties. (blacksquare)

answer:To solve for f(3) given f(x) = 3x + 1, we substitute x with 3: [ begin{align*} f(3) &= 3(3) + 1 &= 9 + 1 &= boxed{10}. end{align*} ]

answer:Choose the geometric sequence 1, 2, 4, and let n=1 to get X=1, Y=3, Z=7. Substitute these values to verify, only option D satisfies the condition. Therefore, the correct answer is boxed{text{D}}.

answer:To solve this problem, we need to distribute 11 balls into three boxes such that each box contains a different number of balls, and then find both the maximum and minimum number of balls in any single box. 1. **Identify the constraints**: - We have 11 balls in total. - Each box must contain a different number of balls. - There are three boxes. 2. **Assign the minimum possible number of balls to the first two boxes**: - Let's assign 1 ball to the first box. - Let's assign 2 balls to the second box. 3. **Calculate the remaining balls for the third box**: - Remaining balls ( = 11 - 1 - 2 ). [ text{Remaining balls} = 11 - 1 - 2 = 8 ] 4. **Verification**: - Now we have assigned 1 ball to the first box, 2 balls to the second box, and 8 balls to the third box. - These assignments meet the required condition: (1 neq 2 neq 8 neq 1). Conclusion: [ boxed{8} ] 5. **Minimum possible number of balls in the box with the most balls**: - To find the minimum such maximum, we should distribute the balls as evenly as possible while maintaining that each box has a different number of balls. - We test distributing balls in closely related counts, like (2, 3, 4): [ 2 + 3 + 4 = 9 ] - Since 9 is not 11, this arrangement does not work. We need to test all remaining logical distributions: (1 + 5 + 5 = 11 quad text{(fails distinct requirement)}) (1 + 4 + 6 = 11 quad text{(works)}) - This means the new valid distribution is (1, 4, 6). Conclusion: [ boxed{5} ] Final conclusion: The most a box can contain at least of 5 balls while making it different should contain maximum 8 balls. [ boxed{8, 5} ]