11/4 Meeting 1: Splitting Circles Equally by Area and Perimeter Solution: 2: Logic Questions about Knaves and Jokers Solution: 3: AMC Question https://artofproblemsolving.com/wiki/index.php/2017_AMC_12B_Problems/Problem_18 Solution: 4: Solve for the Perimeter of 5 conjoined semicircles Solution: Solving for Diameter: https://www.youtube.com/watch?v=Xrx8oa8NeXI Once Diameter is Known, Perimeter can be calculated as such: 2(22) + 2(12) + 16 = 84→ side lengths 5(36π/2) = 5(18π) = 80π → semicircle arcs ∴ Perimeter is 84 + 80π 11/12 Meeting 1: Law of Sines Problem Solution: https://www.youtube.com/watch?v=JX9IiaOxG8 Essentially: 1. a/sin(∠CBD) = e/sin(∠BDC) = d/sin(∠ACB) (using law of sines in triangles ABC and BCD) 2. ∠ABD + ∠ E = 180 degrees (opposite angles in inscribed quadrilateral (ABDE) sums to 180) 3. ∠ABD = 180∠E, ∠ACD = 180∠E, therefore ∠ABD = ∠ACD (same opposite angles theorem) 4. can rewrite equation in step 1 to be a/sin(∠CBD) = d/sin(∠ACB) CBD = ∠ 5. As ∠ B  ∠ ABD, ∠ CBD = ∠ B  (180∠E) ⇒ ∠ CBD = ∠ B+∠E180 ⇒ ∠ CBD + 180 = ∠ B+∠E 6. thus, sin(∠CBD + 180) = sin(∠B+∠E) = sin(∠CBD) (as sin(x+pi) = sin(x)) 7. As ∠ ACB = ∠ C ∠ ACD, ∠ ACB = ∠ C  (∠180∠E), and ∠ ACB + 180 = ∠ C+∠E 8. thus, sin(∠ACB+180) = sin(∠C+∠E) = sin(∠ACB) 9. equation in step 4 ⇒ a/sin(∠B+∠E) = d/sin(∠C+∠E), negatives cancel out to get answer 2: Colliding Ants Solution: 3: 3 angles in a star problem Solution: 1. Method 1: a. The sum of the interior angles of a star is equal to 180 degrees i. Proof can be found here: https://www.youtube.com/watch?v=K3wYAGNmWJI b. thus A + B + C + 50 + 30 = 180, then A + B + C = 100 4: Logic question about tiles Solution: Hints: 1. We know that Vesta needs to have 2 even tiles, limiting her to {2,4}, {2,6}, or {4,6} 2. We know that Ysabel’s 2 tiles have to contain at least 1 prime number, limiting her to {1,3}, {1,5}, or {1,7} 3. As Alain has the greatest amount of tiles he has 3, while the others have two 4. Alain’s consecutive condition requires him to go over at least one even number; as the number 2 is present in at least 2 of Vesta’s possible numbers, Alain’s range cannot include two, limiting it to {3,4,5}, {4,5,6}, and {5,6,7} 5. From the given ranges the correct solutions can be logiced out
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