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2018 MMATHS Mathathon Rounds 5-7 Math Majors of America Tournament for HS

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February 20, 2022
algebrageometrycombinatoricsnumber theoryMMATHS

Problem Statement

Round 5
p13. Circles ω1\omega_1, ω2\omega_2, and ω3\omega_3 have radii 88, 55, and 55, respectively, and each is externally tangent to the other two. Circle ω4\omega_4 is internally tangent to ω1\omega_1, ω2\omega_2, and ω3\omega_3, and circle ω5\omega_5 is externally tangent to the same three circles. Find the product of the radii of ω4\omega_4 and ω5\omega_5.
p14. Pythagoras has a regular pentagon with area 11. He connects each pair of non-adjacent vertices with a line segment, which divides the pentagon into ten triangular regions and one pentagonal region. He colors in all of the obtuse triangles. He then repeats this process using the smaller pentagon. If he continues this process an infinite number of times, what is the total area that he colors in? Please rationalize the denominator of your answer.
p15. Maisy arranges 6161 ordinary yellow tennis balls and 33 special purple tennis balls into a 4×4×44 \times 4 \times 4 cube. (All tennis balls are the same size.) If she chooses the tennis balls’ positions in the cube randomly, what is the probability that no two purple tennis balls are touching?
Round 6
p16. Points A,B,CA, B, C, and DD lie on a line (in that order), and △BCE\vartriangle BCE is isosceles with BE‾=CE‾\overline{BE} = \overline{CE}. Furthermore, FF lies on BE‾\overline{BE} and GG lies on CE‾\overline{CE} such that △BFD\vartriangle BFD and △CGA\vartriangle CGA are both congruent to △BCE\vartriangle BCE. Let HH be the intersection of DF‾\overline{DF} and AG‾\overline{AG}, and let II be the intersection of BE‾\overline{BE} and AG‾\overline{AG}. If m∠BCE=arcsin(1213)m \angle BCE = arcsin \left( \frac{12}{13} \right), what is HI‾FI‾\frac{\overline{HI}}{\overline{FI}} ?
p17. Three states are said to form a tri-state area if each state borders the other two. What is the maximum possible number of tri-state areas in a country with fifty states? Note that states must be contiguous and that states touching only at “corners” do not count as bordering.
p18. Let a,b,c,da, b, c, d, and ee be integers satisfying 2(23)2+23a+2b+(23)2c+23d+e=02(\sqrt[3]{2})^2 + \sqrt[3]{2}a + 2b + (\sqrt[3]{2})^2c +\sqrt[3]{2}d + e = 0 and 255i+25a−55ib−5c+5id+e=025\sqrt5 i + 25a - 5\sqrt5 ib - 5c + \sqrt5 id + e = 0 where i=−1i =\sqrt{-1}. Find ∣a+b+c+d+e∣|a + b + c + d + e|.
Round 7
p19. What is the greatest number of regions that 100100 ellipses can divide the plane into? Include the unbounded region.
p20. All of the faces of the convex polyhedron PP are congruent isosceles (but NOT equilateral) triangles that meet in such a way that each vertex of the polyhedron is the meeting point of either ten base angles of the faces or three vertex angles of the faces. (An isosceles triangle has two base angles and one vertex angle.) Find the sum of the numbers of faces, edges, and vertices of PP.
p21. Find the number of ordered 20182018-tuples of integers (x1,x2,....x2018)(x_1, x_2, .... x_{2018}), where each integer is between −20182-2018^2 and 201822018^2 (inclusive), satisfying 6(1x1+2x2+...⋅+2018x2018)2≥(2018)(2019)(4037)(x12+x22+...+x20182).6(1x_1 + 2x_2 +...· + 2018x_{2018})^2 \ge (2018)(2019)(4037)(x^2_1 + x^2_2 + ... + x^2_{2018}).
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2784936p24472982]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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