32. [17] Let ABC be an acute triangle and D be the foot of altitude from A to BC . Let X and Y be points on the segment BC such that ∠ BAX = ∠ Y AC , BX = 2, XY = 6, and Y C = 3. Given that AD = 12, compute BD . Proposed by: Sarunyu Thongjarast √ √ Answer: 12 2 − 16 = 288 − 16 Solution 1: A P C B X Y Let the line tangent to ⊙ ( ABC ) at A intersect line BC at P . Claim 1. Line P A is tangent to ⊙ ( AXY ). Proof. Note that ∠ P AX = ∠ P AB + ∠ BAX = ∠ P CA + ∠ CAY = ∠ P Y A. This proves the desired tangency. Now, by power of point, we have, 2 P B · P C = P X · P Y = P A P B ( P B + 11) = ( P B + 2)( P B + 8) P B = 16 2 P A = P B · P C = 16 · 27 √ P A = 12 3 2 2 2 2 2 2 By Pythagorean theorem, AD + DP = P A . Therefore, P D = P A − AD = 432 − 144 = 288, so √ √ P D = 12 2. Finally, BD = P D − P B = 12 2 − 16 . Solution 2: Since ∠ BAX = ∠ CAY , by Steiner ratio theorem, we get that 2 AB BX BY 2 8 16 = · = · = . 2 AC CX CY 9 3 27 2 2 2 2 Thus, if BD = x , then by Pythagorean theorem, we get that AB = 144+ x and AC = 144+(11 − x ) . Combining with the displayed equations gives 2 2 27(144 + x ) = 16(144 + (11 − x ) ) 2 2 27(144 + x ) = 16(144 + x ) + 16( − 22 x + 121) 2 11(144 + x ) = 16 · 11 · ( − 2 x + 11) 2 x + 144 + 16(2 x − 11) = 0 2 x + 32 x − 32 = 0 √ x = 12 2 − 16 , (where we only take the positive solution since x > 0). Solution 3: Suppose that BD = c . From the length conditions, we get that c 2 − c 11 − c 8 − c tan ∠ BAD = , tan ∠ DAX = , tan ∠ DAC = , tan ∠ DAY = 12 12 12 12 Thus, using tangent addition formula, we get that c 2 − c 2

24 12 12 12 tan ∠ BAX = tan( ∠ BAD + ∠ DAX ) = = = 2 c 2 − c 144 − 2 c + c 2 144 − 2 c + c 1 − · 12 12 144 11 − c 8 − c 3 − 36 12 12 12 tan ∠ Y AC = tan( ∠ DAC − ∠ DAY ) = = = 2 11 − c 8 − c 2 c − 19 c +232 c − 19 c + 232 1 + · 12 12 144 Hence, the condition ∠ BAX = ∠ CAY translates to 24 36

2 2 144 − 2 c + c c − 19 c + 232 2 2 36( c − 2 c + 144) − 24( c − 19 c + 232) = 0 2 2 3( c − 2 c + 144) − 2( c − 19 c + 232) = 0 2 c + 32 c − 32 = 0 √ c = − 16 ± 12 2 √ Since c is positive, the answer is BD = 12 2 − 16 .