answer:Let [ g(n) = leftlceil frac{119n}{120} rightrceil - leftlfloor frac{120n}{121} rightrfloor. ] We investigate changes in g(n) as n increases by 120*121 = 14520 (the least common multiple of the denominators). Note: [ g(n + 14520) = leftlceil frac{119(n + 14520)}{120} rightrceil - leftlfloor frac{120(n + 14520)}{121} rightrfloor ] [ = leftlceil frac{119n}{120} + 119 rightrceil - leftlfloor frac{120n}{121} + 120 rightrfloor ] [ = leftlceil frac{119n}{120} rightrceil + 119 - leftlfloor frac{120n}{121} rightrfloor - 120 ] [ = leftlceil frac{119n}{120} rightrceil - leftlfloor frac{120n}{121} rightrfloor - 1 ] [ = g(n) - 1. ] Thus, each residue class modulo 14520 leads to a decrease in the function value by 1 per period. Therefore, for each residue class r modulo 14520, there is a unique integer n such that g(n) = 1 and n equiv r pmod{14520}. Hence, the total number of integers solving the given equation is boxed{14520}.

answer:Given the inverse proportion function y=frac{k}{x}, and it passes through the point left(3,-5right), we aim to determine in which quadrant the graph of this function should be. Step 1: Substitute the given point into the function to find the constant k. [ y=frac{k}{x} Rightarrow -5=frac{k}{3} ] Step 2: Solve for k. [ k = 3 times (-5) = -15 ] Step 3: Analyze the implication of k < 0 for the graph of the function. Since k = -15 < 0, the function y=frac{-15}{x} will produce positive y values when x is negative and negative y values when x is positive. This behavior indicates that the graph of the function will be located in the quadrants where x and y have opposite signs. Step 4: Determine the quadrants. The quadrants where x and y have opposite signs are the second quadrant (where x is negative and y is positive) and the fourth quadrant (where x is positive and y is negative). Therefore, the graph of the function should be in the second and fourth quadrants. Final Answer: boxed{C}.

answer:1. We first extend the non-parallel sides AD and BC of the trapezoid ABCD until they intersect at a point K. 2. By similarity of triangles triangle KDC and triangle KAB, with the similarity ratio given by [ frac{CD}{AB} = frac{2}{3}, ] we find that [ frac{KD}{KA} = frac{2}{3}. ] 3. Let KD = 2x and KA = 3x. Then, [ AD = AK - KD = 3x - 2x = x. ] 4. Given AP: PD = 3: 2, we can determine [ PD = frac{2}{5} AD = frac{2}{5} x. ] 5. Therefore, [ KP = KD + PD = 2x + frac{2}{5}x = frac{10x}{5} + frac{2x}{5} = frac{12x}{5}. ] 6. Assuming the area of the quadrilateral PDCQ is twice the area of APQB, we start with the area relationship [ S_{PDCQ} = 2 S_{APQB}. ] Let S_{triangle KDC} = S. Then, since the trapezoid ABCD shares the area between KDC and KAB, [ frac{S_{triangle KDC}}{S_{triangle KAB}} = left(frac{2}{3}right)^2 = frac{4}{9}, ] leading to [ S_{triangle KAB} = frac{9}{4} S_{triangle KDC} = frac{9}{4} S. ] 7. Considering S_{ABCD} = S_{triangle KAB} - S_{triangle KDC}, [ S_{ABCD} = frac{9}{4} S - S = frac{5}{4} S. ] 8. Given the area proportion, we calculate S_{PDCQ}: [ S_{PDCQ} = frac{2}{3} S_{ABCD} = frac{2}{3} cdot frac{5}{4} S = frac{5}{6} S. ] 9. Similarly for the triangle area: [ S_{triangle KPD} = S + S_{PDCQ} = S + frac{5}{6} S = frac{11}{6} S. ] 10. We take the ratio of areas formed by triangle KDC and triangle KPQ: [ frac{S_{triangle KDC}}{S_{triangle KPQ}} = frac{S}{ frac{11}{6} S } = frac{6}{11}. ] 11. Since also: [ frac{S_{triangle KDC}}{S_{triangle KPQ}} = frac{KD}{KP} cdot frac{KC}{KQ}, ] substituting known values gives: [ frac{2}{frac{12}{5}} cdot frac{KC}{KQ} = frac{5}{6} cdot frac{KC}{KQ} = frac{6}{11}. ] 12. Solving yields: [ frac{KC}{KQ} = frac{36}{55}. ] 13. Thus, since the point lies such that frac{KC}{KB} must be checked, and noting the inequality: [ frac{2}{3} > frac{36}{55}, ] confirming Q lies on the line extending beyond point B. 14. Testing the hypothesis that area A of APQB to be the larger part leads similarly to: [ S_{PDCQ} = frac{1}{3} S_{ABCD} = frac{5}{12} S, ] and verifying areas: [ S_{triangle KPD} = frac{17}{12} S, ] and the ratio: [ frac{S_{triangle KDC}}{S_{triangle KPQ}} = frac{S}{frac{17}{12} S} = frac{12}{17}, ] yielding: [ frac{KC}{KQ} = frac{72}{95}. ] Verifying leads to the correct proportion: # Conclusion: [ boxed{23: 13} ]

answer:1. **Understand the Problem Context**: - The accused on a small island is allowed to make only one statement in defense. - It is known that the accused was born and raised on a neighboring island of knights (who always tell the truth) and liars (who always lie). - The statement made by the accused is: "The person who indeed committed the crime that I am accused of is a liar." 2. **Analyze the Statement for Logical Consistency**: - The goal is to determine if the statement helps the accused or not. 3. **Case 1: Assume the Accused is a Knight**: - **Sub-assumption**: The accused tells the truth (since knights always tell the truth). - **Implication**: - The phrase "The person who indeed committed the crime that I am accused of is a liar" is true. - Therefore, the real criminal must be a liar. - **Conclusion**: - If the real criminal is a liar, and the accused (being a knight) cannot be a liar, then the accused cannot be the real criminal. - Hence, the accused is innocent. 4. **Case 2: Assume the Accused is a Liar**: - **Sub-assumption**: The accused lies (since liars always lie). - **Implication**: - The phrase "The person who indeed committed the crime that I am accused of is a liar" is false (because liars lie). - Therefore, the real criminal cannot be a liar (as the statement must be false), so the real criminal must be a knight. - **Conclusion**: - If the real criminal is a knight, and the accused (being a liar) cannot be one, then the accused cannot be the real criminal. - Hence, the accused is innocent. 5. **General Conclusion**: - In both cases (whether the accused is a knight or a liar), the analysis concludes that the accused cannot be the real criminal based on the truth or falsehood of the statement made. - Therefore, making this statement was reasonable for the accused, as it helped in removing any suspicion from themselves. boxed{text{The statement helped the accused to remove suspicions and assert their innocence.}}