answer:We know that Madeline got 2 mistakes, which is half as many mistakes as Leo. This means Leo got 2 * 2 = 4 mistakes. Brent scored 25 and has x more mistakes than Leo, which means Brent got 4 + x mistakes. To find Madeline's score, we subtract the number of mistakes she made from the total possible score: score = total possible score - number of mistakes. If Madeline got 28, her score would be 28. We can set up the equation: 28 = total possible score - 2. Simplifying, we get: total possible score = 30. Since Brent scored 25 and has x more mistakes than Leo, we can write: 25 = total possible score - (4 + x). Substituting the value of total possible score, we get: 25 = 30 - (4 + x). Simplifying, we get: 25 = 26 - x. Subtracting 26 from both sides, we get: -1 = -x. Multiplying both sides by -1, we get: x = 1. The value of x is 1. 1 The answer is: 1

answer:We know that the centroid of a triangle divides each median into segments with lengths in the ratio 2:1. Let M_a, M_b, and M_c be the midpoints of overline{BC}, overline{CA}, overline{AB}, respectively. Then G is the average of A, B, and C, so [GA = frac{2}{3} GM_a,]and similarly for GB and GC. Then begin{align*} k cdot PG^2 + GA^2 + GB^2 + GC^2 &= k cdot left( frac{2}{3} GM_a right)^2 + left( frac{2}{3} GM_a right)^2 + left( frac{2}{3} GM_b right)^2 + left( frac{2}{3} GM_c right)^2 &= frac{4}{9} left( k + 1 + 1 + 1 right) GM_a^2 &= frac{4}{9} (k + 3) GM_a^2. end{align*}By the Triangle Inequality, PA + AP ge AM_a, so PA^2 + 2 cdot PA cdot AP + AP^2 ge AM_a^2. Then [PA^2 + PB^2 + PC^2 ge AM_a^2 + BM_b^2 + CM_c^2.]From the Apollonian Circle Theorem, [AM_a^2 + BM_b^2 + CM_c^2 = 3 left( GA^2 + GB^2 + GC^2 right).]Hence, [PA^2 + PB^2 + PC^2 ge 3 left( GA^2 + GB^2 + GC^2 right).]Thus, k must be boxed{3}. The answer is: 3

answer:Dorothy needs to pay 18% of 60000, which is 0.18 x 60000 = 10800 in taxes. After she pays the taxes, she will have 60000 - 10800 = 49200 left. 49200 The answer is: 49200

answer:We want to find the value of X in the given situation. To find the remainder when r^{13} + 1 is divided by r - X, we can use the Remainder Theorem. According to the Remainder Theorem, if we divide a polynomial f(r) by r - X, the remainder is equal to f(X). In this case, the polynomial is r^{13} + 1. So, the remainder is f(X) = X^{13} + 1. Given that the remainder is 2, we can write: X^{13} + 1 = 2 Subtracting 1 from both sides of the equation, we find: X^{13} = 1 Taking the 13th root of both sides of the equation, we get: X = 1 The value of X is 1. The answer is: 1