answer:1. **Total Point Combinations**: Calculate the total number of ways to choose two points from a 4-by-4 grid, binom{16}{2} = 120. 2. **Lines through Three Points**: Count lines that go through three points. Horizontal, vertical, and two main diagonals (each direction) that include four points, and additional diagonals that include three points. There are 2 main horizontal lines, 2 main diagonals, and 4 sets of three-point diagonals. - Horizontal and vertical by row/column: 4 + 4 = 8 (excluding vertical lines). - Diagonal lines (through 3 or more points): 2 main diagonals (four points each), and 8 (each with three points). - Total collinear lines through three or more points: 8 + 10 = 18. 3. **Adjust for Overcounts**: Adjust for triple-counted lines. Subtract twice the number of lines through three points, 120 - 2 times 18 = 84. Final answer: boxed{84} lines.

answer:Since the vertical asymptotes are at x = -1 and x = 2, the denominator of g(x) can be factored as A(x+1)(x-2). The horizontal asymptote is given by y = 1/A. From the condition that g(x) > 0.3 for x > 3, we have: [0.3 le frac{1}{A} < 1.] As A is an integer, it follows that A = 3. The denominator of the function is Ax^2 + Bx + C = 3(x+1)(x-2) = 3x^2 - 3x - 6. Thus, A+B+C = 3 - 3 - 6 = boxed{-6}.

answer:First, we consider k = lfloor sqrt{n} rfloor, meaning [ frac{n+1500}{80} = k ] Rearrange to find n: [ n = 80k - 1500 ] Since n must be non-negative, 80k - 1500 geq 0, simplifying to k geq frac{1500}{80}, approximately k geq 18.75. Since k is an integer, k geq 19. Next, establish the upper boundary: n = (k+1)^2 - 1 must hold, with k < sqrt{n+1} < k+1, which gives: [ n leq (k+1)^2 - 1 ] Plugging in n = 80k - 1500 shows: [ 80k - 1500 leq (k+1)^2 - 1 ] [ 80k - 1500 leq k^2 + 2k + 1 - 1 ] [ k^2 - 78k + 1499 geq 0 ] We find integer solutions k where both inequalities hold. Solving k^2 - 78k + 1499 geq 0 provides an upper bound for k. Finding the roots of k^2 - 78k + 1499: [ k = frac{78 pm sqrt{78^2 - 4 cdot 1499}}{2} ] [ k = frac{78 pm sqrt{6084 - 5996}}{2} ] [ k = frac{78 pm sqrt{88}}{2} ] Finding exact values for k isn't possible here, but we consider it close to k = 37, as it should be an integer close to 78/2. Testing for k geq 19 and approximately k leq 37, and calculating n = 80k - 1500, we check if k is equal to lfloor sqrt{n} rfloor for each value of k in this range. Conclusion: Considering each integer k from 19 to 37, after recalculations, we find 18 viable k values that hold the equality. Thus the answer is boxed{18}.

answer:1. **Apply the difference of squares formula**: Given frac{980^2}{210^2 - 206^2}, use the difference of squares, a^2 - b^2 = (a-b)(a+b): - Here, a = 210 and b = 206. [ 210^2 - 206^2 = (210 - 206)(210 + 206) = 4 times 416 = 1664. ] 2. **Simplify the entire expression**: Substitute the simplified denominator back into the original expression: [ frac{980^2}{210^2 - 206^2} = frac{980^2}{1664}. ] 3. **Compute the square of 980 and simplify further**: Calculate 980^2 and simplify the expression: - 980^2 = 960400. [ frac{960400}{1664}. ] 4. **Final calculation**: Divide 960400 by 1664: [ frac{960400}{1664} = 577.5. ] 5. **Conclusion with the boxed answer**: The value of the expression is 577.5. The final answer is boxed{A}.