Challenging Systems of Equations
Μια συλλογή με προχωρημένα προβλήματα συστημάτων εξισώσεων. Δοκιμάστε τα όλα!
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$$\begin{cases}
3\!\left(x+\dfrac1x\right)=4\!\left(y+\dfrac1y\right)=5\!\left(z+\dfrac1z\right)\\[4pt]
xy+yz+zx=1\\
(x,y,z\in\mathbb R^{*})
\end{cases}$$
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$$\begin{cases}
y+z+yz=11\\
z+x+zx=7\\
x+y+xy=5\\
(x,y,z\in\mathbb R)
\end{cases}$$
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$$\begin{cases}
x+2y=\sqrt3\\
2x\sqrt{1-4y^2}+4y\sqrt{1-x^2}=\sqrt3\\
(x,y\in\mathbb R)
\end{cases}$$
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$$\begin{cases}
x^2+y^2=z+3\\
y^2+z^2=x+3\\
z^2+x^2=y+3\\
(x,y,z\in\mathbb R)
\end{cases}$$
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$$\begin{cases}
x+y+z=1\\
\sqrt{x+yz}+\sqrt{y+zx}+\sqrt{z+xy}=2\\
(x,y,z\ge0)
\end{cases}$$
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$$\begin{cases}
x^2+y^2=1\\
z^2+t^2=1\\
xz+yt=0\\
(2x+z)(2y+t)=2\\
(x,y,z,t\in\mathbb R)
\end{cases}$$
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$$\begin{cases}
x\sqrt{1-y^2}+y\sqrt{1-x^2}=1\\
(1-x)(1+y)=2\\
(x,y\in\mathbb R)
\end{cases}$$
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$$\begin{cases}
x^2-y^2-w=0\\
w^2-t^2-x=0\\
t+2xy=0\\
y+2wt=0\\
(x,y,w,t\in\mathbb R)
\end{cases}$$
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$$\begin{cases}
x+y+z=xy+yz+zx+1\\
x^{2013}+y^{2013}+z^{2013}=2\\
(x,y,z\in[0,1])
\end{cases}$$
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$$\begin{cases}
x(x+1)=y\\
y(y+1)=z\\
z(z+1)=x\\
(x,y,z\in\mathbb R)
\end{cases}$$
-
$$\begin{cases}
(x+1)(x+4)=y\\
(y+1)(y+4)=z\\
(z+1)(z+4)=x\\
(x,y,z\in\mathbb R)
\end{cases}$$
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$$\begin{cases}
x^{2013}+y^{2013}=x^3+y^3\\
x^{2014}+y^{2014}=x^4+y^4\\
(x,y\in\mathbb R)
\end{cases}$$
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$$\begin{cases}
x^2-y+z=1\\
x+y^2+z^2=3\\
xy+yz-zx=1\\
(x,y,z\in\mathbb Q)
\end{cases}$$
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$$\begin{cases}
x+y+z=5\\
xy+yz+zx=x+7\\
(x,y,z\in\mathbb R)
\end{cases}$$
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$$\begin{cases}
x=\sqrt{2y+3}\\
y=\sqrt{2z+3}\\
z=\sqrt{2x+3}\\
(x,y,z\in\mathbb R)
\end{cases}$$
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$$\begin{cases}
\dfrac{2}{1+x}+2\sqrt{\dfrac{2y}{1+y}}=3\\[6pt]
\dfrac{2}{1+y}+2\sqrt{\dfrac{2z}{1+z}}=3\\[6pt]
\dfrac{2}{1+z}+2\sqrt{\dfrac{2x}{1+x}}=3\\[6pt]
(x,y,z\in(0,+\infty))
\end{cases}$$
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$$\begin{cases}
z\,(2^x+3^y)=5\\[2pt]
\dfrac{2^y+3^x}{z}=5\\[6pt]
x+y+2z=z^2+3\\
(x,y,z\in\mathbb R)
\end{cases}$$
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$$\begin{cases}
\dfrac1x+\dfrac{1}{y+z}=\dfrac65\\[6pt]
\dfrac1y+\dfrac{1}{z+x}=\dfrac34\\[6pt]
\dfrac1z+\dfrac{1}{x+y}=\dfrac23\\[6pt]
(x,y,z\in\mathbb R)
\end{cases}$$
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$$\begin{cases}
3x+y+z=4\\
xy+yz+zx=-1\\
x^2+y^2+z^2=6\\
(x,y,z\in\mathbb R)
\end{cases}$$
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$$\begin{cases}
2^x+3^y=5\\
2^y+3^x=5\\
(x,y>0)
\end{cases}$$
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$$\begin{cases}
\dfrac{y}{x}-9xy=2\\[6pt]
\dfrac{z}{y}-9yz=6\\[6pt]
\dfrac{3x}{z}-3zx=2\\[6pt]
(x,y,z\in\mathbb R)
\end{cases}$$
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$$\begin{cases}
x+y+z=3\\
x^3+y^3+z^3=3\\
(x,y,z\in\mathbb Z)
\end{cases}$$
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$$\frac{x(x+y)}{y+z}+y=\frac{z(z+x)}{x+y}+x=\frac{y(y+z)}{z+x}+z,\qquad (x,y,z>0)$$
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$$\begin{cases}
36x^2y-27y^3=8\\
4x^3-27xy^2=4\\
(x,y\in\mathbb R)
\end{cases}$$
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$$\begin{cases}
x+y+z=\dfrac{1}{3}\\[6pt]
2(1-x)(1-y)(1-z)=(1+x)(1+y)(1+z)\\
(x,y,z\in[0,+\infty))
\end{cases}$$
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$$\begin{cases}
x^2+5x+3=y\\
y^2+5y+3=z\\
z^2+5z+3=x\\
(x,y,z\in\mathbb R)
\end{cases}$$
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$$\begin{cases}
x^3=x+3y+12\\
y^3=-y+4z+6\\
z^3=9z+2x-32\\
(x,y,z\in\mathbb R)
\end{cases}$$
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$$\begin{cases}
2^x=2y\\
2^y=2x\\
(x,y\in\mathbb R)
\end{cases}$$
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$$\begin{cases}
3^{\,x-y}=\dfrac{6x+3}{x+y+2}\\[8pt]
\bigl((x^3+x)^3\bigr)+4(y+1)^3=10x+2y+2\\
(x,y\in[0,+\infty))
\end{cases}$$
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$$\begin{cases}
\sqrt{6x^2+3y^2}+\sqrt{6y^2+3z^2}+\sqrt{6z^2+3x^2}=6039\\
x+y+z=2013\\
(x,y,z\in\mathbb R)
\end{cases}$$
- $$\begin{cases} x_1+x_2+\cdots+x_n=0\\ 3^{x_1}+3^{x_2}+\cdots+3^{x_n}=3\\ (x_1,\ldots,x_n\in\mathbb R) \end{cases}$$
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