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222-9+11+12:2*14+14 = ? ( )

Toán Lớp 7: Cho dãy tỉ số bằng nhau: $\frac{x}{y+z+t}$ =$\frac{y}{z+t+x}$ =$\frac{z}{x+t+y}$ =$\frac{t}{x+y+z}$ . CMR: A=$\frac{x+y}{z+t}$ +$\fra

Toán Lớp 7: Cho dãy tỉ số bằng nhau:
$\frac{x}{y+z+t}$ =$\frac{y}{z+t+x}$ =$\frac{z}{x+t+y}$ =$\frac{t}{x+y+z}$ .
CMR: A=$\frac{x+y}{z+t}$ +$\frac{y+z}{x+t}$ +$\frac{z+t}{x+y}$ +$\frac{t+x}{y+z}$ là 1 số nguyên.

Comments ( 1 )

  1. Ta có: x/(y+z+t)=y/(x+z+t)=z/(x+y+t)=t/(x+y+z)

    =>(y+z+t)/x=(x+z+t)/y=(x+y+t)/z=(x+y+z)/t

    =>(y+z+t)/x +1=(x+z+t)/y +1=(x+y+t)/z +1=(x+y+z)/t +1

    =>(y+z+t)/x + x/x=(x+z+t)/y + y/y=(x+y+t)/z + z/z=(x+y+z)/t + t/t

    =>(x+y+z+t)/x=(x+y+z+t)/y=(x+y+z+t)/z=(x+y+z+t)/t 

    =>x=y=z=t , thay vào biểu thức A,ta có:

    A=(x+x)/(x+x) +(x+x)/(x+x) +(x+x)/(x+x) +(x+x)/(x+x)

    A=1+1+1+1

    A=4

    Vì 4\in ZZ

    =>A\in ZZ

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222-9+11+12:2*14+14 = ? ( )

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