answer: 103


answer: 3375


answer: \frac{1}{576}


answer: -984


answer: 890


answer: \frac{1311}{2017}


answer: \frac{9 \sqrt{23}}{23}


answer: 1-\frac{2}{\pi}


answer: 1037


answer: \frac{-1+\sqrt{17}}{2}, \frac{-1-\sqrt{17}}{2}


answer: 56


answer: 29


answer: 105


answer: 2304


answer: 200


answer: 6300


answer: 2^{25} \cdot 26!


answer: \frac{2025}{101}


answer: \frac{4}{9}


answer: \frac{448}{3}

pred: Thus after a step, we can go from (k=4) to (k=3) or (k=2). If X=2, merges happen on two nonadjacent gaps, merging two pairs of snakes simultaneously: that would combine the 4 snakes


answer: 26


answer: 63


answer: 8\sqrt{10}


answer: 20


answer: \sqrt{23}-2 \sqrt{3}


answer: 9\sqrt{15}


answer: \frac{7}{18}


answer: \sqrt{6}


answer: 14+4\sqrt{37}

pred: Check triangle AXD sides: AD=24, AX = sqrt(a^2 + s^2) = sqrt(16.04^2 +18.91^2) = sqrt(257.3 + 357.6) = sqrt(614.9) ≈24.79. DX = sqrt(a^2 + (s-24)^2) = sqrt(257.3 + ( -5.09)^2 ≈257.3 +25.91=283.21) => sqrt≈16.83. So sides: AD=24, AX≈24.79, DX≈16.83. Largest side is AX≈24.79 (actually slightly > AD=24). Check angle at D opposite AX? Actually largest side is AX ≈24.79. Check if 24.79^2 >24^2 +16.83^2? 24.79^2≈614.9; 24^2 +16.83^2 =576+283.2≈859.2. So 614.9<859.2, so angle opposite AX is acute. Check also if 24^2 > 24.79^2 +16.83^2? No; 576 <859, so angle at X opposite AD? AD is not the largest side now, but check angle at X opposite side AD=24. So check AD^2 =576 vs other sides sum: 24.79^2 +16.83^2 ≈614.9+283.2=898.1 >576, so angle at X is acute. Check angle at A opposite side DX: DX^2=283.2 vs sum of squares of other two sides: 2


answer: \sqrt{\frac{95}{24}}

pred: Thus compute α = (S - W)/2, β = (W+Y)/2, γ = (S - Y)/2. These