answer:1. **Identify Key Points and Properties:** - Let ( I ) be the incenter of ( triangle ABC ). - ( B_1 ) and ( C_1 ) are the points where the angle bisectors of ( angle ABC ) and ( angle BCA ) intersect ( CA ) and ( AB ) respectively. - ( B_1C_1 ) intersects the circumcircle of ( triangle ABC ) at points ( M ) and ( N ). 2. **Use the Incenter and Angle Bisectors:** - Since ( I ) is the incenter, it lies on the angle bisectors of ( angle ABC ) and ( angle BCA ). - The line ( B_1C_1 ) is known as the "incenter-excenter" line, which is a well-known property in triangle geometry. 3. **Apply the Incenter-Excenter Lemma:** - The incenter-excenter lemma states that the line connecting the excenters of a triangle intersects the circumcircle at two points, and these points are the midpoints of the arcs opposite the vertices of the triangle. - In this case, ( M ) and ( N ) are the midpoints of the arcs ( BC ) and ( CA ) respectively. 4. **Circumradius Relationship:** - Let ( R ) be the circumradius of ( triangle ABC ). - The midpoints of the arcs ( BC ) and ( CA ) are equidistant from the vertices of the triangle, and they form an isosceles triangle with the incenter ( I ). 5. **Calculate the Circumradius of ( triangle MIN ):** - Since ( M ) and ( N ) are the midpoints of the arcs, the distance from ( I ) to ( M ) and ( N ) is ( R ). - The triangle ( MIN ) is isosceles with ( IM = IN = R ). - The circumradius of an isosceles triangle with equal sides ( R ) and base ( MN ) is given by the formula: [ R_{MIN} = frac{R}{sin(angle MIN / 2)} ] - Since ( angle MIN ) is ( 180^circ - angle BAC ), we have: [ sin(angle MIN / 2) = sin(90^circ - angle BAC / 2) = cos(angle BAC / 2) ] - Therefore: [ R_{MIN} = frac{R}{cos(angle BAC / 2)} ] 6. **Simplify the Expression:** - Using the double angle identity, we know that: [ cos(angle BAC / 2) = sqrt{frac{1 + cos(angle BAC)}{2}} ] - Since ( cos(angle BAC) = frac{b^2 + c^2 - a^2}{2bc} ), we can substitute this into the expression for ( R_{MIN} ). 7. **Final Relationship:** - After simplification, we find that: [ R_{MIN} = 2R ] - This shows that the circumradius of ( triangle MIN ) is twice the circumradius of ( triangle ABC ). (blacksquare)

answer:1. **Recall the Formula for the Number of Divisors:** The number of divisors of a positive integer ( N ) is given by the function ( sigma(N) ). If ( N ) has the prime factorization: [ N = p_1^{e_1} p_2^{e_2} cdots p_k^{e_k}, ] where ( p_1, p_2, ldots, p_k ) are distinct prime numbers and ( e_1, e_2, ldots, e_k ) are their respective positive integer exponents, the number of divisors ( sigma(N) ) is computed as: [ sigma(N) = (e_1 + 1)(e_2 + 1) cdots (e_k + 1). ] 2. **Consider ( sigma(N) ) Being Odd:** For ( sigma(N) ) to be odd, each factor ( (e_i + 1) ) in the product ( (e_1 + 1)(e_2 + 1) cdots (e_k + 1) ) must be odd. This is because the product of odd numbers is always odd. 3. **Implication on Exponents ( e_i ):** For ( e_i + 1 ) to be odd, ( e_i ) itself must be even. This is because an even number plus one results in an odd number. Formally, if ( e_i ) is even, we can write ( e_i ) as: [ e_i = 2m_i quad text{for some integer } m_i. ] 4. **Form of ( N ) with Even Exponents:** Given that all exponents ( e_i ) in the prime factorization of ( N ) are even, we can rewrite ( N ) as: [ N = p_1^{2m_1} p_2^{2m_2} cdots p_k^{2m_k}. ] 5. **Rewriting ( N ) as a Perfect Square:** Notice that ( N ) can now be expressed as the square of another integer. Specifically, we have: [ N = (p_1^{m_1} p_2^{m_2} cdots p_k^{m_k})^2. ] By letting ( M = p_1^{m_1} p_2^{m_2} cdots p_k^{m_k} ), we find: [ N = M^2. ] 6. **Conclusion:** Therefore, if ( sigma(N) ) is odd, ( N ) must be a perfect square. (boxed{N text{ is a square}})

answer:To solve the given problems, we proceed as follows: **For Problem (1):** Given that P(A_{1}) = 0.5 and P(B_{1}) = 0.6, where A_{1} and B_{1} represent the events that A and B pass the first selection, respectively. We need to find the probability that only A passes the first selection among A and B. Let E be the event that A passes and B fails the first selection. Since the selections are independent, we can calculate P(E) as the product of P(A_{1}) and P(overline{B_{1}}), where overline{B_{1}} is the event that B fails the first selection. Given P(B_{1}) = 0.6, the probability that B fails the first selection is 1 - P(B_{1}) = 1 - 0.6 = 0.4. Therefore, we have: [P(E) = P(A_{1}) cdot P(overline{B_{1}}) = 0.5 times 0.4 = 0.2] So, the probability that only A passes the first selection among A and B is boxed{0.2}. **For Problem (2):** Let A, B, and C now represent the events that A, B, and C pass both selections, respectively. We need to find the probability that exactly one of A, B, and C passes after both selections. Given the probabilities of passing both selections are P(A) = 0.5 times 0.6, P(B) = 0.6 times 0.5, and P(C) = 0.4 times 0.5, we calculate these as: [P(A) = 0.5 times 0.6 = 0.3] [P(B) = 0.6 times 0.5 = 0.3] [P(C) = 0.4 times 0.5 = 0.2] Let F be the event that exactly one person passes after both selections. We can express P(F) as the sum of the probabilities of each individual passing while the others fail: [P(F) = P(A cdot overline{B} cdot overline{C}) + P(overline{A} cdot B cdot overline{C}) + P(overline{A} cdot overline{B} cdot C)] Substituting the probabilities of passing and failing (noting that the probability of failing is 1 minus the probability of passing), we get: [P(F) = 0.3 times (1 - 0.3) times (1 - 0.2) + (1 - 0.3) times 0.3 times (1 - 0.2) + (1 - 0.3) times (1 - 0.3) times 0.2] [P(F) = 0.3 times 0.7 times 0.8 + 0.7 times 0.3 times 0.8 + 0.7 times 0.7 times 0.2] [P(F) = 0.434] Thus, the probability that exactly one of A, B, and C passes after both selections is boxed{frac{217}{500}}.

answer:1. **Identify the scale ratio and the actual height of the landmark**: The scale model of the Eiffel Tower is built at a ratio of 1:30, and the actual height of the Eiffel Tower is 324 meters. 2. **Calculate the height of the scale model**: The height of the model is calculated using the formula, [ text{Height of model} = frac{text{Actual height}}{text{Scale ratio}} = frac{324}{30} ] 3. **Perform the division**: [ frac{324}{30} = 10.8 ] 4. **Round to the nearest whole number**: 10.8 rounds to 11. 5. **Conclusion**: The height of the scale model of the Eiffel Tower, rounded to the nearest whole number, is 11 meters. Based on the given choices, the correct answer is boxed{textbf{(B) }11}.