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question:5. Unlocking the communicator is done by entering a 4-digit numerical code on the touch screen. The arrangement of digits on the keyboard changes after entering the code depending on a random prime number k from 7 to 2017, and the digit i is displayed as a_{i}, which is the last digit of the number ik. The user enters digits from the left column with the left hand, and the rest with the right hand. Restore the lock code if it is known that when entering the code, the user entered the digits as follows:when a_{3}=3 - right, right, left, right;when a_{3}=9 - left, left, left, left;when a_{3}=1 - right, right, right, right;when a_{3}=7 - left, right, right, right.In your answer, specify the obtained code.| mathrm{a}_{1} | mathrm{a}_{2} | mathrm{a}_{3} | | || :--- | :--- | :--- | :---: | :---: || mathrm{a}_{4} | mathrm{a}_{5} | mathrm{a}_{6} | | || mathrm{a}_{7} | mathrm{a}_{8} | mathrm{a}_{9} | | || | | | | mathrm{a}_{0} || | | | | || | | | | |
answer:5. Unlocking the communicator is done by entering a 4-digit numerical code on the touch screen. The layout of the digits on the keyboard changes after entering the code based on a random prime number k from 7 to 2017, and the digit i is displayed as a_{i}, which is the last digit of the number i k. The user enters digits from the left column with the left hand, and the rest with the right hand. Restore the lock code if it is known that when entering the code, the user entered the digits as follows:when a_{3}=3 - right, right, left, right;when a_{3}=9 - left, left, left, left;when a_{3}=1 - right, right, right, right;when a_{3}=7 - left, right, right, right.In the answer, specify the obtained code.| 1 | 2 | 3 || :--- | :--- | :--- || 4 | 5 | 6 || 7 | 8 | 9 || 0 | | || | | || a_{1} | a_{2} | mathrm{a}_{3} || :---: | :---: | :---: || mathrm{a}_{4} | a_{5} | mathrm{a}_{6} || a_{7} | a_{8} | mathrm{a}_{9} || | mathrm{a}_{0} | |Answer: 3212
question:In triangle ABC, let a, b, and c be the lengths of the sides opposite to the interior angles A, B, and C, respectively, and acos C+left(2b+cright)cos A=0. (1) Find the value of angle A. (2) If D is the midpoint of segment BC and AD=frac{7}{2}, AC=3, find the area of triangle ABC.
answer: Step-by-Step Solution# Part (1): Finding the Value of Angle AGiven the equation acos C+left(2b+cright)cos A=0, we can manipulate it using trigonometric identities:1. Replace a and b with their respective expressions in terms of sin using the Law of Sines, a = frac{sin A}{sin B} and b = frac{sin B}{sin A}, and simplify: [ sin Acos C + left(2sin B + sin Cright)cos A = 0 ]2. Apply the sum-to-product identities: [ sin(A+C) + sin(A-C) = -2sin Bcos A ]3. Since A + C + B = pi, we know sin(A+C) = sin(pi - B) = sin B, simplifying the equation to: [ sin B + sin(A-C) = -2sin Bcos A ]4. Given that sin B > 0, we can isolate cos A: [ cos A = -frac{1}{2} ]5. Considering the range of A within (0, pi), we find: [ A = frac{2pi}{3} ]Thus, the value of angle A is boxed{frac{2pi}{3}}.# Part (2): Finding the Area of Triangle ABCGiven that D is the midpoint of BC and AD = frac{7}{2}, AC = 3, we use the median formula:1. Apply the formula for the length of the median: [ AD^2 = frac{1}{4}(2AB^2 + 2AC^2 - BC^2) ] Simplifying with given values: [ frac{49}{4} = frac{1}{4}(2c^2 + 18 - c^2) ]2. Solve for c^2: [ c^2 - 3c - 40 = 0 ]3. Solving the quadratic equation yields c = 8 (since c > 0).Finally, to find the area of triangle ABC, we use the formula S = frac{1}{2}bcsin A:1. Substitute the known values: [ S = frac{1}{2} times 3 times 8 times frac{sqrt{3}}{2} ]2. Simplify to find the area: [ S = 6sqrt{3} ]Therefore, the area of triangle ABC is boxed{6sqrt{3}}.
question:(-8)^{frac{1}{3}} cdot frac{(sqrt{ab^{-1}})^3}{(0.2)^{-2}(a^3b^{-3})^{frac{1}{2}}} = ______.
answer:The original expression can be simplified to -2 cdot frac{a^{frac{3}{2}}b^{-frac{3}{2}}}{25a^{frac{3}{2}}b^{-frac{3}{2}}} = -frac{2}{25}.Therefore, the answer is: -frac{2}{25}.This can be derived using the properties of exponentiation.This question tests the properties of exponentiation, reasoning, and computational skills, and is considered a basic question.So, the final answer is boxed{-frac{2}{25}}.
question:16. A polygon is said to be friendly if it is regular and it also has angles that when measured in degrees are either integers or half-integers (i.e. have a decimal part of exactly 0.5 ). How many different friendly polygons are there?
answer:SolUTION028The interior angle will be an integer or half-integer precisely when the exterior angle is an integer or a half-integer respectively. A regular polygon of n sides has exterior angle frac{360}{n}. This is either an integer or a half-integer if, and only if, frac{720}{n} is an integer. Therefore, n must be a factor of 720 . The factors of 720 are 720,360,240,180,144,120,90,80,72,60,48,45,40, 36,30,24,20,18,16,15,12,10,9,8,6,5,4 and 3 (ignoring 2 and 1 neither of which is a valid number of sides for a polygon). There are 28 numbers in this list, so there are 28 different friendly polygons.
question:Read the following material:We define: If the solution to the one-variable linear equation ax=b involving x is x=b+a, then we call this equation a "sum solution equation." For example, the solution to the equation 2x=-4 is x=-2, and since -2=-4+2, the equation 2x=-4 is a "sum solution equation." Please answer the following questions based on the above rules:(1) The following one-variable linear equations involving x are "sum solution equations":①frac{2}{3}x=-frac{2}{3}②-3x=frac{9}{4}③5x=-2(2) If the one-variable linear equation involving x is 3x=2a-10 and is a "sum solution equation," find the value of a.
answer: Step-by-Step Solution# Part 1: Identifying "Sum Solution Equations"Equation ①: frac{2}{3}x=-frac{2}{3}- Solution: x = -1- Check if it's a "sum solution equation": -1 neq frac{2}{3} + (-frac{2}{3}) = 0- Conclusion: Equation ① is not a "sum solution equation".Equation ②: -3x=frac{9}{4}- Solution: x = -frac{3}{4}- Check if it's a "sum solution equation": -frac{3}{4} = -3 + frac{9}{4}- Conclusion: Equation ② is a "sum solution equation".Equation ③: 5x=-2- Solution: x = -frac{2}{5}- Check if it's a "sum solution equation": -frac{2}{5} neq -2 + 5 = 3- Conclusion: Equation ③ is not a "sum solution equation".Therefore, the answer to part 1 is: boxed{text{②}}.# Part 2: Finding the Value of aGiven: 3x=2a-10 is a "sum solution equation".- Solve for x: x = frac{2a-10}{3}- Since it's a "sum solution equation", we have: 3 + (2a - 10) = frac{2a - 10}{3}- Solving for a: 9 + 2a - 10 = 2a - 10- Simplify and solve for a: a = frac{11}{4}Therefore, the value of a is: boxed{frac{11}{4}}.
question:8. Arrange all positive integers whose sum of digits is 10 in ascending order to form a sequence left{a_{n}right}, if a_{n}=2017, then n= qquad
answer:a_{1}=19, a_{2}=28, cdots, a_{9}=91, a_{10}=109, a_{11}=118, cdots, there are 9 two-digit numbers;for three-digit numbers, when the first digit is 1, there are 10, when the first digit is 2, there are 9, ..., when the first digit is 9, there are 2, that is, there are 10+9+cdots+2=54 three-digit numbers;for four-digit numbers, when the first digit is 1 and the hundred's place is 0, there are 10, when the hundred's place is 1, there are 9, ..., when the hundred's place is 9, there is 1, at this time, there are 10+9+cdots+1=55 four-digit numbers;for four-digit numbers, when the first digit is 2, there are 2008, 2017. So n=9+54+55+2=120.