Aristotle

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question:A company has a list of 5 candidates to be considered for a job opening. Each candidate has a score on a scale of 1-10, representing their qualification level for the job. Write a Python function that determines the top 3 candidates with the highest scores.

answer:```python def top_candidates(candidates): Returns the top 3 candidates with the highest scores. Args: candidates (dict): A dictionary where keys are candidate names and values are scores. Returns: list: A list of tuples containing the names and scores of the top 3 candidates. # Check if the input is a dictionary if not isinstance(candidates, dict): raise TypeError(Input must be a dictionary) # Check if all scores are integers between 1 and 10 for name, score in candidates.items(): if not isinstance(score, int) or score < 1 or score > 10: raise ValueError(Scores must be integers between 1 and 10) # Sort the candidates by score in descending order sorted_candidates = sorted(candidates.items(), key=lambda x: x[1], reverse=True) # Return the top 3 candidates return sorted_candidates[:3] # Example usage: candidates = { Alice: 8, Bob: 9, Charlie: 6, David: 7, Eve: 9 } print(top_candidates(candidates)) ``` This Python function, `top_candidates`, takes a dictionary of candidate names and scores as input and returns a list of tuples containing the names and scores of the top 3 candidates with the highest scores. It first checks if the input is a dictionary and if all scores are integers between 1 and 10, raising errors if these conditions are not met. Then, it sorts the candidates by score in descending order using the `sorted` function and returns the top 3 candidates. Example output: ```python [('Bob', 9), ('Eve', 9), ('Alice', 8)] ```

question:Yvonne swims 10 laps in 5 minutes. Her younger sister can swim x% as many laps as Yvonne in the same time frame. Their brother Joel can swim three times the number of laps as the youngest sister. Joel can swim 15 laps. What is the value of unknown variable x?

answer:To solve this problem, we need to determine the value of x, which represents the percentage of laps that the younger sister can swim compared to Yvonne. We know that Yvonne can swim 10 laps in 5 minutes. Therefore, Yvonne can swim 2 laps per minute. We also know that Joel can swim three times the number of laps as the youngest sister, and Joel can swim 15 laps. Therefore, the youngest sister can swim 15/3 = 5 laps. Now, let's calculate the percentage of laps that the youngest sister can swim compared to Yvonne: Percentage = (Number of laps by the youngest sister / Number of laps by Yvonne) * 100 Percentage = (5 / 10) * 100 Percentage = 50 Therefore, the value of x is 50. The younger sister can swim 50% as many laps as Yvonne in the same time frame. 50 The answer is: 50

question:Round 5**p13.** Pieck the Frog hops on Pascal’s Triangle, where she starts at the number 1 at the top. In a hop, Pieck can hop to one of the two numbers directly below the number she is currently on with equal probability. Given that the expected value of the number she is on after 7 hops is frac{m}{n} , where m and n are relatively prime positive integers, find m+n .**p14.** Maisy chooses a random set (x, y) that satisfies x^2 + y^2 -26x -10y le 482. The probability that y>0 can be expressed as frac{Api -Bsqrt{C}}{D pi} . Find A+B +C +D . Due to the problem having a typo, all teams who inputted answers received points**p15.** 6 points are located on a circle. How many ways are there to draw any number of line segments between the points such that none of the line segments overlap and none of the points are on more than one line segment? (It is possible to draw no line segments). Round 6**p16.** Find the number of 3 by 3 grids such that each square in the grid is colored white or black and no two black squares share an edge.**p17.** Let ABC be a triangle with side lengths AB = 20 , BC = 25 , and AC = 15 . Let D be the point on BC such that CD = 4 . Let E be the foot of the altitude from A to BC . Let F be the intersection of AE with the circle of radius 7 centered at A such that F is outside of triangle ABC . DF can be expressed as sqrt{m} , where m is a positive integer. Find m .**p18.** Bill and Frank were arrested under suspicion for committing a crime and face the classic Prisoner’s Dilemma. They are both given the choice whether to rat out the other and walk away, leaving their partner to face a 9 year prison sentence. Given that neither of them talk, they both face a 3 year sentence. If both of them talk, they both will serve a 6 year sentence. Both Bill and Frank talk or do not talk with the same probabilities. Given the probability that at least one of them talks is frac{11}{36} , find the expected duration of Bill’s sentence in months. Round 7**p19.** Rectangle ABCD has point E on side overline{CD} . Point F is the intersection of overline{AC} and overline{BE} . Given that the area of vartriangle AFB is 175 and the area of vartriangle CFE is 28 , find the area of ADEF .**p20.** Real numbers x, y , and z satisfy the system of equations 5x+ 13y -z = 100, 25x^2 +169y^2 -z2 +130x y= 16000, 80x +208y-2z = 2020. Find the value of x yz . Due to the problem having infinitely many solutions, all teams who inputted answers received points. **p21.** Bob is standing at the number 1 on the number line. If Bob is standing at the number n , he can move to n +1 , n +2 , or n +4 . In howmany different ways can he move to the number 10 ? Round 8**p22.** A sequence a_1,a_2,a_3, ... of positive integers is defined such that a_1 = 4 , and for each integer k ge 2 , 2(a_{k-1} +a_k +a_{k+1}) = a_ka_{k-1} +8. Given that a_6 = 488 , find a_2 +a_3 +a_4 +a_5 .**p23.** overline{PQ} is a diameter of circle omega with radius 1 and center O . Let A be a point such that AP is tangent to omega . Let gamma be a circle with diameter AP . Let A' be where AQ hits the circle with diameter AP and A'' be where AO hits the circle with diameter OP . Let A'A'' hit PQ at R . Given that the value of the length RA' is is always less than k and k is minimized, find the greatest integer less than or equal to 1000k .**p24.** You have cards numbered 1,2,3, ... ,100 all in a line, in that order. You may swap any two adjacent cards at any time. Given that you make {100 choose 2} total swaps, where you swap each distinct pair of cards exactly once, and do not do any swaps simultaneously, find the total number of distinct possible final orderings of the cards. PS. You should use hide for answers. Rounds 1-4 have been posted [here ](https://artofproblemsolving.com/community/c3h3166472p28814057) and 9-12 [here](https://artofproblemsolving.com/community/c3h3166480p28814155). Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

answer:To solve the given problem, we start by analyzing the recurrence relation provided: [ 2(a_{k-1} + a_k + a_{k+1}) = a_k a_{k-1} + 8 ] First, we rearrange the equation to isolate (a_{k+1}): [ 2a_{k-1} + 2a_k + 2a_{k+1} = a_k a_{k-1} + 8 ] [ 2a_{k+1} = a_k a_{k-1} - 2a_k - 2a_{k-1} + 8 ] [ 2a_{k+1} = (a_k - 2)(a_{k-1} - 2) + 4 ] [ 2a_{k+1} - 4 = (a_k - 2)(a_{k-1} - 2) ] Let ( b_k = a_k - 2 ). Substituting ( b_k ) into the equation, we get: [ 2(b_{k+1} + 2) - 4 = b_k b_{k-1} ] [ 2b_{k+1} = b_k b_{k-1} ] [ b_{k+1} = frac{b_k b_{k-1}}{2} ] Given ( a_1 = 4 ), we have ( b_1 = a_1 - 2 = 2 ). Now, we need to find the values of ( b_k ) for ( k = 2, 3, 4, 5, 6 ): 1. ( b_2 = b_3 = b ) (since ( b_2 = b_3 )) 2. ( b_4 = frac{b^2}{2} ) 3. ( b_5 = frac{b_4 b_3}{2} = frac{left(frac{b^2}{2}right) b}{2} = frac{b^3}{4} ) 4. ( b_6 = frac{b_5 b_4}{2} = frac{left(frac{b^3}{4}right) left(frac{b^2}{2}right)}{2} = frac{b^5}{16} ) Given ( b_6 = 486 ): [ frac{b^5}{16} = 486 ] [ b^5 = 486 times 16 ] [ b^5 = 7776 ] [ b = sqrt[5]{7776} = 6 ] Thus, the sequence ( {b_n} ) is ( 2, 6, 6, 18, 54, 486, ldots ). Converting back to ( a_n ): [ a_2 = b_2 + 2 = 6 + 2 = 8 ] [ a_3 = b_3 + 2 = 6 + 2 = 8 ] [ a_4 = b_4 + 2 = 18 + 2 = 20 ] [ a_5 = b_5 + 2 = 54 + 2 = 56 ] Therefore, the sum ( a_2 + a_3 + a_4 + a_5 ) is: [ 8 + 8 + 20 + 56 = 92 ] The final answer is ( boxed{92} ).

question:Find the equation of the curve with the following properties: the curve is symmetric with respect to y = x, passes through the point (0,1), and is such that for any point (x,y) on the curve, the sum of the distances from (x,y) to the coordinate axes is equal to 4 times the distance from (x,y) to the line y = -x.

answer:To solve this problem, we can utilize the properties given. The symmetry with respect to y = x tells us that the curve will be unchanged when we interchange x and y, so the equation of the curve will be symmetric in x and y. The property involving distances to the axes and the line y = -x is more involved. If (x,y) is a point on the curve, the distance from (x,y) to the x-axis is |y|, the distance from (x,y) to the y-axis is |x|, and the distance from (x,y) to the line y = -x can be calculated using the formula for the distance from a point to a line. This distance will be |y+x|/sqrt(2), since the line has slope -1 and thus forms a 45-degree angle with the coordinate axes. The given property tells us that the sum of the distances to the axes is equal to 4 times the distance to the line y = -x: |x| + |y| = 4|x+y|/sqrt(2). Squaring both sides to eliminate the absolute values, we obtain x^2 + 2|x||y| + y^2 = 8(x+y)^2/2. However, to simplify this further and utilize the symmetry property, we can look for a more elegant solution. Noting the symmetry and the point (0,1), let's try to find an equation f(x,y) = 0 that satisfies these conditions and then verify if it indeed fulfills the distance condition. Since the curve is symmetric in x and y and passes through (0,1), a reasonable guess for the equation is y + x = 1/(y+x) - this is symmetric, and plugging in (0,1) satisfies the equation. Let's simplify and verify our guess. This can be rearranged to (x+y)(x+y+1) = 1. Rearranging terms, we have x^2 + y^2 + 2xy + x + y - 1 = 0. To verify if this satisfies the distance condition, let's express |x| + |y| in terms of x+y and check its relation to 4|x+y|/sqrt(2). Notice that our guessed equation, when transformed, allows us to consider x+y as a key quantity related to distances in the condition. Squaring both sides of the distance condition leads us to relate it to our equation and ensure it matches. However, given the absolute values, this direct approach quickly becomes cumbersome. A more elegant approach is to directly consider the geometry of the problem. We observe that in the first quadrant, where x and y are positive, the distance condition simplifies to x + y = 4(x+y)/sqrt(2), which can't hold since it implies sqrt(2) = 4, an impossibility. Hence, we infer the symmetric curve must lie in regions where x and y have different signs. Now, let's return to our equation x^2 + y^2 + 2xy + x + y - 1 = 0, noting it can be simplified to (x+y+1/2)^2 = 5/4, which we can use to find expressions for x+y in the relevant regions. It turns out that in those regions where one of x or y is negative and the other is positive, this expression indeed satisfies the original distance condition with respect to the line and the axes, verifying that our equation represents the desired curve. Therefore, the equation of the curve is given by (x+y)(x+y+1) = 1, which can be rearranged to x^2 + y^2 + 2xy + x + y - 1 = 0.