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Harvard-MIT November Tournament #0j0q

Harvard-MIT November TournamentHard

From module Newton's Sums

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Understand the scale
Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.
Polynomial interpolation: Newton, Lagrange

Problem

Problem:

w,x,y,zw, x, y, z are real numbers such that

w+x+y+z=52w+4x+8y+16z=73w+9x+27y+81z=114w+16x+64y+256z=1\begin{aligned} w+x+y+z & =5 \\ 2 w+4 x+8 y+16 z & =7 \\ 3 w+9 x+27 y+81 z & =11 \\ 4 w+16 x+64 y+256 z & =1 \end{aligned}

What is the value of 5w+25x+125y+625z?5 w+25 x+125 y+625 z ?

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