Maths Olympiad Prep

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Algebra Difficulty 5.5 AIME, harder Find the answer

10. There are 50 RMB notes with denominations of 1 yuan, 2 yuan, and 5 yuan. To make their total value 100 yuan, how can the number of notes of these denominations be chosen?

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

10. x,y,zx, y, z represent the number of 1 yuan, 2 yuan, and 5 yuan notes respectively. x+y+z=50,x+2y+5z=100x+y+z=50, x+2 y+5 z=100.
x=36+3t,y=2−4t,z=12+t,t=0,−1,−2,⋯ ,−12. x=36+3 t, y=2-4 t, z=12+t, t=0,-1,-2, \cdots,-12 \text {. }

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.