| {"input": "A1 Terms that contain the same letters, and the exponents of these letters are also the same, are called like terms.", "target": "Let's solve this step by step.\n**Answer:** According to the concept of like terms, which are terms that contain the same letters and the exponents of these letters are also the same, we can fill in the blank as follows:\n\nThe blank should be filled with text{exponents}.\n#### text{exponents"} | |
| {"input": "A2 Prove the negation of the conclusion \"There is at most one obtuse angle in a triangle\" using proof by contradiction.", "target": "Let's solve this step by step.\n**Analysis**\n\nThe problem tests our understanding of the negation of statements. The key to solving this problem is to master the rules for writing the negation of statements, particularly those involving quantifiers. The statement contains the quantifier \"at most,\" which corresponds to \"at least\" when writing its negation. According to the rules of proof by contradiction, we only need to write the negation of the statement \"There is at most one obtuse angle in a triangle\" to find the negation of its conclusion.\n\n**Solution**\n\nThe negation of the statement \"There is at most one obtuse angle in a triangle\" is \"There are at least two obtuse angles.\"\n\n**Step-by-step reasoning:**\n1. Identify the statement: \"There is at most one obtuse angle in a triangle.\"\n2. Determine the corresponding quantifier for \"at most\" when writing its negation, which is \"at least.\"\n3. Write the negation of the statement: \"There are at least two obtuse angles.\"\n\nThus, the answer is text{There are at least two obtuse angles.}\n#### text{There are at least two obtuse angles."} | |
| {"input": "A3 a , band c can do a piece of work in 11 days , 20 days and 55 days respectively , working alone . how soon can the work be done if a is assisted by band c on alternate days ?", "target": "Let's solve this step by step.\nTo solve this problem, we need to determine the amount of work each band can complete in one day and then calculate how much work can be done when A is assisted by B and C on alternate days.\n\nLet's denote the total work as W.\n\nBand A can complete the work in 11 days, so the work done by A in one day is W/11.\nBand B can complete the work in 20 days, so the work done by B in one day is W/20.\nBand C can complete the work in 55 days, so the work done by C in one day is W/55.\n\nNow, let's consider the work done when A is assisted by B and C on alternate days.\n\nOn the first day, A and B work together, so the work done is (W/11 + W/20).\nOn the second day, A and C work together, so the work done is (W/11 + W/55).\n\nLet's calculate the work done in two days when A is assisted by B on the first day and by C on the second day:\n\nWork done in two days = (W/11 + W/20) + (W/11 + W/55)\n= 2W/11 + W/20 + W/55\n\nTo find a common denominator, we can use the least common multiple (LCM) of 11, 20, and 55, which is 220.\n\nWork done in two days = (2W/11 * 20/20) + (W/20 * 11/11) + (W/55 * 4/4)\n= (40W/220) + (11W/220) + (4W/220)\n= (40W + 11W + 4W) / 220\n= 55W/220\n= W/4\n\nSo, in two days, A, B, and C together can complete 1/4 of the work.\n\nNow, we need to find out how many such two-day cycles are required to complete the entire work W.\n\nSince 1/4 of the work is done in two days, the entire work will be done in 2 * 4 = 8 days.\n\nTherefore, the work can be done in 8 days if A is assisted by B and C on alternate days.\n#### 8"} | |
| {"input": "A4 A polygon has each of its exterior angles equal to 10\u00b0. How many diagonals does it have?", "target": "Let's solve this step by step.\nLet the polygon have n sides. Then, the sum of its n exterior angles is 360\u00b0, which means n * 10\u00b0 = 360\u00b0, so n = 36.\nFor this 36-sided polygon, each vertex can connect to 33 other vertices (excluding the 2 adjacent vertices) with a diagonal.\nSince a diagonal connects 2 vertices, the number of diagonals is 18 * 33 = 594.\nTherefore, the answer is 594.\n#### 594"} | |
| {"input": "A5 Changes in interest rates will have a certain impact on stock prices. According to analysis, the probability of a stock price increase when interest rates are lowered is 0.7, the probability of a stock price increase when interest rates remain unchanged is 0.2, and the probability of a stock price increase when interest rates are raised is 0.1. Assuming the probability of interest rate reduction is 0.6 and the probability of interest rate remaining unchanged is 0.3, then the probability of a stock price increase for this stock is ______.", "target": "Let's solve this step by step.\nTo solve this problem, we'll apply the law of total probability. Let's denote the events as follows:\n\n- A: Interest rate reduction\n- B: Interest rate remains unchanged\n- C: Interest rate increase\n- D: Stock price increase\n\nGiven probabilities are:\n\n- P(A) = 0.6\n- P(B) = 0.3\n- P(C) = 1 - P(A) - P(B) = 0.1 (since the probabilities must sum up to 1)\n- P(D|A) = 0.7 (probability of D given A)\n- P(D|B) = 0.2 (probability of D given B)\n- P(D|C) = 0.1 (probability of D given C)\n\nUsing the law of total probability, we can calculate the total probability of D (stock price increase) as follows:\n\nP(D) = P(A)P(D|A) + P(B)P(D|B) + P(C)P(D|C)\n\nSubstituting the given values:\n\nP(D) = (0.6)(0.7) + (0.3)(0.2) + (0.1)(0.1)\n\nP(D) = 0.42 + 0.06 + 0.01\n\nP(D) = 0.49\n\nTherefore, the probability of a stock price increase for this stock is 0.49.\n#### 0.49"} | |