answer:To solve for the largest constant ( K ) such that the inequality [sqrt{frac{ab}{c}}+sqrt{frac{bc}{a}}+sqrt{frac{ac}{b}} geq K sqrt{a + b + c}]holds for all positive real numbers ( a, b, ) and ( c ), we can proceed as follows:1. Simplify and Understand the Inequality: The inequality involves square roots and proportions of ( a, b, ) and ( c ). To find the largest ( K ), we need to consider specific values of ( a, b, ) and ( c ) and check the behavior of the inequality.2. Use SymPy to Symbolically Manipulate the Inequality:Let's use the AM-GM inequality (Arithmetic Mean-Geometric Mean inequality) to derive some insights, and then we can use Python and SymPy to check our findings.The AM-GM inequality states that for non-negative real numbers:[ frac{x_1 + x_2 + ldots + x_n}{n} geq sqrt[n]{x_1 x_2 cdots x_n} ]Applying AM-GM to the individual terms:[sqrt{frac{ab}{c}} geq frac{2sqrt{ab}}{2sqrt{c}} quad text{and similarly for other terms}]Using symmetric properties and simplifications, we can hypothesize a value for ( K ) and then verify it using SymPy.```pythonimport sympy as sp# Define the variablesa, b, c, K = sp.symbols('a b c K', positive=True)# Define the LHS of the inequalitylhs = sp.sqrt(a * b / c) + sp.sqrt(b * c / a) + sp.sqrt(c * a / b)# Define the RHS of the inequalityrhs = K * sp.sqrt(a + b + c)# Form the inequalityinequality = sp.Ge(lhs, rhs)# To find the largest K, we can try to solve the inequality by finding the minimum value of lhs/rhslhs_simplified = lhs / sp.sqrt(a + b + c)# We want to find the largest constant K such that lhs_simplified >= K for all positive a, b, c# Let's calculate the minimum value of lhs_simplifiedK_value = sp.Min(lhs_simplified)# Print the simplified form of K_valueprint(K_value)``````output(sqrt(a)*sqrt(b)/sqrt(c) + sqrt(a)*sqrt(c)/sqrt(b) + sqrt(b)*sqrt(c)/sqrt(a))/sqrt(a + b + c)```To find the largest constant ( K ) such that[sqrt{frac{ab}{c}} + sqrt{frac{bc}{a}} + sqrt{frac{ac}{b}} geq K sqrt{a + b + c}]holds for all positive real numbers ( a, b, ) and ( c ), we need to analyze the expression[frac{sqrt{frac{ab}{c}} + sqrt{frac{bc}{a}} + sqrt{frac{ac}{b}}}{sqrt{a + b + c}}]and find its minimum value.Let's take a geometric approach. By symmetry and considering cases where ( a = b = c ), we can gain insights. Let's start by evaluating the expression when ( a = b = c ).```pythonimport sympy as sp# Define the variablesa, b, c = sp.symbols('a b c', positive=True)# Define the expression (lhs / sqrt(a + b + c))lhs_simplified = (sp.sqrt(a * b / c) + sp.sqrt(b * c / a) + sp.sqrt(c * a / b)) / sp.sqrt(a + b + c)# Evaluate the expression when a = b = ca_value, b_value, c_value = sp.symbols('a_value b_value c_value', positive=True)expr_at_equal_values = lhs_simplified.subs({a: a_value, b: a_value, c: a_value})# Simplify the expressionexpr_simplified = sp.simplify(expr_at_equal_values)# Print the simplified form of the expression when a = b = cprint(expr_simplified)# To find the largest K, we need to check the general case (for generic a, b, c)# Let's find the minimum value of lhs_simplifiedK_value = sp.Min(lhs_simplified)# Print the simplified form of K_valueprint(K_value)``````outputsqrt(3)(sqrt(a)*sqrt(b)/sqrt(c) + sqrt(a)*sqrt(c)/sqrt(b) + sqrt(b)*sqrt(c)/sqrt(a))/sqrt(a + b + c)```From the output, it's clear that when ( a = b = c ), the expression simplifies to:[frac{3sqrt{frac{a^2}{a}}}{sqrt{3a}} = frac{3sqrt{a}}{sqrt{3a}} = sqrt{3}.]Let's analyze this further to infer that the minimum value of the expression [frac{sqrt{frac{ab}{c}} + sqrt{frac{bc}{a}} + sqrt{frac{ac}{b}}}{sqrt{a + b + c}}]is indeed (sqrt{3}). This implies that the largest constant ( K ) such that [sqrt{frac{ab}{c}} + sqrt{frac{bc}{a}} + sqrt{frac{ac}{b}} geq K sqrt{a + b + c}]holds for all positive real numbers ( a, b, ) and ( c ) is ( K = sqrt{3} ).Thus, we can conclude that[K = boxed{sqrt{3}}.]

answer:To solve the problem of finding the number of positive integers less than 1000 that cannot be divided evenly by 5 or 7, we can use the principle of inclusion and exclusion. Steps to Solve:1. Count the Total Numbers: The total number of positive integers less than 1000 is (999) (from 1 to 999).2. Count Numbers Divisible by 5: Use the floor division to count the multiples of 5 below 1000.3. Count Numbers Divisible by 7: Use the floor division to count the multiples of 7 below 1000.4. Count Numbers Divisible by Both 5 and 7 (i.e., 35): Use the floor division to count the multiples of 35 below 1000.5. Use Inclusion-Exclusion Principle: The number of integers divisible by either 5 or 7 is given by: [ |A cup B| = |A| + |B| - |A cap B| ] Where (A) is the set of multiples of 5 and (B) is the set of multiples of 7.6. Subtract from Total: The number of integers that cannot be divided evenly by 5 or 7 is: [ 999 - |A cup B| ]Let's implement this in Python to get the exact count.```python# Total numbers less than 1000total_numbers = 999# Count numbers divisible by 5count_5 = total_numbers // 5# Count numbers divisible by 7count_7 = total_numbers // 7# Count numbers divisible by 35 (both 5 and 7)count_35 = total_numbers // 35# Using the inclusion-exclusion principle to find the count of numbers divisible by 5 or 7count_5_or_7 = count_5 + count_7 - count_35# Numbers that cannot be divided evenly by 5 or 7result = total_numbers - count_5_or_7print(result)``````output686```The number of positive integers less than 1000 that cannot be divided evenly by 5 or 7 is (boxed{686}).

answer:To find the distance from a point ( M_{0} ) to a plane passing through three points ( M_{1}, M_{2}, M_{3} ), we need to:1. Determine the equation of the plane passing through the points ( M_{1}, M_{2}, M_{3} ).2. Use the formula for the distance from a point to a plane to find the distance.Let's break down the steps:1. Equation of the Plane: The general equation of a plane is given by ( ax + by + cz = d ). To find (a, b, c, ) and ( d ), we use the cross product of vectors formed by the points on the plane. Given points: [ M_{1}(1, 2, 0) ] [ M_{2}(3, 0, -3) ] [ M_{3}(5, 2, 6) ] Form vectors: [ vec{M_{1}M_{2}} = langle 3-1, 0-2, -3-0 rangle = langle 2, -2, -3 rangle ] [ vec{M_{1}M_{3}} = langle 5-1, 2-2, 6-0 rangle = langle 4, 0, 6 rangle ] Compute the cross product of these vectors to get the normal vector to the plane: [ vec{n} = vec{M_{1}M_{2}} times vec{M_{1}M_{3}} ]2. Distance from Point to Plane: The distance ( D ) from a point ( M_{0}(x_{0}, y_{0}, z_{0}) ) to the plane ( ax + by + cz = d ) is given by: [ D = frac{|ax_{0} + by_{0} + cz_{0} - d|}{sqrt{a^2 + b^2 + c^2}} ]Let's implement this in Python using SymPy:```pythonimport sympy as sp# Define symbolsx, y, z = sp.symbols('x y z')# PointsM1 = sp.Point3D(1, 2, 0)M2 = sp.Point3D(3, 0, -3)M3 = sp.Point3D(5, 2, 6)M0 = sp.Point3D(-13, -8, 16)# Vectors M1M2 and M1M3M1M2 = sp.Matrix(M2 - M1)M1M3 = sp.Matrix(M3 - M1)# Cross product to find the normal vector of the planenormal_vector = M1M2.cross(M1M3)# Equation of the plane: ax + by + cz = da, b, c = normal_vectord = normal_vector.dot(sp.Matrix(M1))# Plane equationplane_eq = a*x + b*y + c*z - d# Distance from M0 to the planex0, y0, z0 = M0distance = sp.Abs(a*x0 + b*y0 + c*z0 - d) / sp.sqrt(a2 + b2 + c2)# Simplify and print the distancedistance_simplified = sp.simplify(distance)print(distance_simplified)``````output134/7```The distance from the point ( M_{0}(-13, -8, 16) ) to the plane passing through the points ( M_{1}(1, 2, 0) ), ( M_{2}(3, 0, -3) ), and ( M_{3}(5, 2, 6) ) is (boxed{frac{134}{7}}).

answer:To determine the smallest integer value of ( m ) for which the quadratic equation (x^2 - 2x - m = 0) has two distinct real roots, we need to analyze the discriminant of the quadratic equation. For a quadratic equation of the form (ax^2 + bx + c = 0), the discriminant (Delta) is given by:[Delta = b^2 - 4ac]The roots of the quadratic equation are real and distinct if and only if the discriminant is positive ((Delta > 0)).Given the quadratic equation (x^2 - 2x - m = 0), we have:- (a = 1)- (b = -2)- (c = -m)The discriminant for our quadratic is:[Delta = (-2)^2 - 4 cdot 1 cdot (-m) = 4 + 4m]We need the discriminant to be positive for the equation to have two distinct real roots:[4 + 4m > 0]Solving this inequality will give us the required condition on (m):[4 + 4m > 0 implies 4m > -4 implies m > -1]The smallest integer value of (m) that satisfies this inequality is (m = 0).Let's verify this using Python and SymPy to ensure our reasoning is correct.```pythonimport sympy as sp# Define the variablem = sp.symbols('m')# Define the discriminant of the quadratic equationdiscriminant = 4 + 4 * m# Solve the inequality for the discriminant to be greater than zerosolution = sp.solve(discriminant > 0, m)print(solution)``````output(-1 < m) & (m < oo)```The output indicates that ( m ) must be greater than (-1). Therefore, the smallest integer value of ( m ) that satisfies this condition is:[boxed{0}]Thus, the smallest integer value of ( m ) for which the quadratic equation ( x^2 - 2x - m = 0 ) has two distinct real roots is ( 0 ).

answer:To solve the problem, we need to find which of the given expressions is equal to (sqrt{t^4 + t^2}) for all real values of (t).Let's analyze the given expression step-by-step and compare it with the given options to identify the correct one.1. Analyze the Expression: [ sqrt{t^4 + t^2} ] Factor out (t^2) from inside the square root: [ sqrt{t^4 + t^2} = sqrt{t^2(t^2 + 1)} = sqrt{t^2} cdot sqrt{t^2 + 1} ] Since (sqrt{t^2} = |t|): [ sqrt{t^4 + t^2} = |t| cdot sqrt{t^2 + 1} ]2. Compare with Given Options: Now let's compare (|t| cdot sqrt{t^2 + 1}) with each given option. - (textbf{(A)} t^3): This is not equivalent. - (textbf{(B)} t^2 + t): This is not equivalent. - (textbf{(C)} |t^2 + t|): This is not equivalent. - (textbf{(D)} t sqrt{t^2 + 1}): This is only equivalent for ( t geq 0 ). - (textbf{(E)} |t| sqrt{1 + t^2}): This is equivalent for all real values of (t).Thus, the correct answer is (textbf{(E)}).Let's verify our reasoning using Python and SymPy.```pythonimport sympy as sp# Define the variablet = sp.symbols('t')# Define the given expressiongiven_expr = sp.sqrt(t4 + t2)# Define the option expressionsoption_A = t3option_B = t2 + toption_C = sp.Abs(t2 + t)option_D = t * sp.sqrt(t2 + 1)option_E = sp.Abs(t) * sp.sqrt(1 + t2)# Check if given expression equals any of the optionsresults = { 'A': sp.simplify(given_expr - option_A) == 0, 'B': sp.simplify(given_expr - option_B) == 0, 'C': sp.simplify(given_expr - option_C) == 0, 'D': sp.simplify(given_expr - option_D) == 0, 'E': sp.simplify(given_expr - option_E) == 0}print(results)``````output{'A': False, 'B': False, 'C': False, 'D': False, 'E': False}```It appears that the output indicates none of the options exactly match the given expression directly. However, this could be due to the simplification process not accounting for the equivalence in a general form. Let's take a closer look at the expressions and manually compare them more carefully.We already reasoned that:[sqrt{t^4 + t^2} = |t| cdot sqrt{t^2 + 1}]To confirm our reasoning, let's examine the expression manually. Specifically, let's re-evaluate the final simplification process and check the input more carefully.We'll re-verify by checking the equivalence using a different approach: substituting specific values for (t) and comparing the results.Let's modify and simplify the code by checking for a range of values for (t):```pythonimport sympy as sp# Define the variablet = sp.symbols('t')# Define the given expressiongiven_expr = sp.sqrt(t4 + t2)# Define the option expressionsoption_E = sp.Abs(t) * sp.sqrt(1 + t2)# Check equivalence for a range of values of tt_values = [-10, -1, -0.5, 0, 0.5, 1, 10]results = []for value in t_values: given_value = given_expr.subs(t, value) option_value = option_E.subs(t, value) results.append((value, given_value, option_value, sp.simplify(given_value - option_value) == 0))print(results)``````output[(-10, 10*sqrt(101), 10*sqrt(101), True), (-1, sqrt(2), sqrt(2), True), (-0.5, 0.559016994374947, 0.559016994374947, True), (0, 0, 0, True), (0.5, 0.559016994374947, 0.559016994374947, True), (1, sqrt ... (output truncated)```The output confirms that (sqrt{t^4 + t^2}) is indeed equal to (|t| sqrt{t^2 + 1}) for various values of (t).Given this confirmation, we can conclude that the correct answer is:[boxed{|t| sqrt{1 + t^2}}]This corresponds to option (textbf{(E)}).

answer:To solve the problem of finding the probability of drawing a full house from a standard deck of 52 cards, we need to follow a systematic approach. Here's how:1. Calculate the total number of 5-card hands possible from a 52-card deck: This can be found using the combination formula ( C(n, k) = frac{n!}{k!(n-k)!} ), where ( n ) is the total number of items to choose from, and ( k ) is the number of items to choose.2. Calculate the number of favorable outcomes: A full house consists of a 3-of-a-kind of one rank and a 2-of-a-kind of another rank. We need to: - Choose 1 rank out of the 13 for the 3-of-a-kind. - Choose 3 cards out of the 4 for that rank. - Choose 1 rank out of the remaining 12 for the 2-of-a-kind. - Choose 2 cards out of the 4 for that rank.3. Compute the probability: The probability ( P ) of drawing a full house is given by the ratio of the number of favorable outcomes to the total number of 5-card hands.Let's compute these steps using Python code.First, we calculate the total number of 5-card hands possible:[ text{Total 5-card hands} = binom{52}{5} = frac{52!}{5!(52-5)!} ]Then, calculate the number of ways to draw a full house:[ text{Number of full houses} = left( binom{13}{1} times binom{4}{3} right) times left( binom{12}{1} times binom{4}{2} right) = (13 times 4) times (12 times 6)]Finally, compute the probability of drawing a full house:[ text{Probability} = frac{text{Number of full houses}}{text{Total 5-card hands}} ]Now, let's implement these calculations using Python:```pythonimport math# Total number of 5-card handstotal_5_card_hands = math.comb(52, 5)# Number of ways to choose the rank for 3-of-a-kindways_to_choose_rank_3 = math.comb(13, 1)# Number of ways to choose 3 out of 4 cards for the chosen rankways_to_choose_cards_3 = math.comb(4, 3)# Number of ways to choose the rank for 2-of-a-kind (excluding the rank already chosen for 3-of-a-kind)ways_to_choose_rank_2 = math.comb(12, 1)# Number of ways to choose 2 out of 4 cards for the chosen rankways_to_choose_cards_2 = math.comb(4, 2)# Total number of full house combinationsnumber_of_full_houses = (ways_to_choose_rank_3 * ways_to_choose_cards_3) * (ways_to_choose_rank_2 * ways_to_choose_cards_2)# Probability of drawing a full houseprobability_full_house = number_of_full_houses / total_5_card_handsprint((total_5_card_hands, number_of_full_houses, probability_full_house))``````output(2598960, 3744, 0.0014405762304921968)```The calculations provide the following results:- The total number of 5-card hands possible from a 52-card deck is ( 2,598,960 ).- The number of full house combinations is ( 3,744 ).- The probability of drawing a full house is approximately ( 0.0014405762304921968 ), or about ( 0.1441% ).Thus, the probability that five cards chosen at random form a full house is (boxed{0.0014405762304921968}).