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2022 CMWMC Guts Round 8/8 - Carnegie Mellon University Womens' Competition

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August 12, 2023
CMWMCalgebrageometrycombinatoricsnumber theory

Problem Statement

Set 8
p22. For monic quadratic polynomials P=x2+ax+bP = x^2 + ax + b and Q=x2+cx+dQ = x^2 + cx + d, where 1≤a,b,c,d≤101 \le a, b, c, d \le 10 are integers, we say that PP and QQ are friends if there exists an integer 1≤n≤101 \le n \le 10 such that P(n)=Q(n)P(n) = Q(n). Find the total number of ordered pairs (P,Q)(P, Q) of such quadratic polynomials that are friends.
p23. A three-dimensional solid has six vertices and eight faces. Two of these faces are parallel equilateral triangles with side length 11, △A1A2A3\vartriangle A_1A_2A_3 and △B1B2B3\vartriangle B_1B_2B_3. The other six faces are isosceles right triangles — △A1B2A3\vartriangle A_1B_2A_3, △A2B3A1\vartriangle A_2B_3A_1, △A3B1A2\vartriangle A_3B_1A_2, △B1A2B3\vartriangle B_1A_2B_3, △B2A3B1\vartriangle B_2A_3B_1, △B3A1B2\vartriangle B_3A_1B_2 — each with a right angle at the second vertex listed (so for instace △A1B2A3\vartriangle A_1B_2A_3 has a right angle at B2B_2). Find the volume of this solid.
p24. The digits 0,1,2,3,4,5,6,7,8,90, 1, 2, 3, 4, 5, 6, 7, 8, 9 are each colored red, blue, or green. Find the number of colorings such that any integer n≥2 n \ge 2 has that (a) If nn is prime, then at least one digit of nn is not blue. (b) If nn is composite, then at least one digit of nn is not green.
PS. You should use hide for answers.
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