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10.4

Part of IV Soros Olympiad 1997 - 98 (Russia)

Problems(3)

2x4 radical system (IV Soros Olympiad 1997-98 Correspondence 10.4)

Source:

6/1/2024
Solve the system of equations {x+y+z+t=61−x2+4−y2+9−z2+16−t2=8\begin{cases} x+y+z+t=6 \\ \sqrt{1-x^2}+\sqrt{4-y^2}+\sqrt{9-z^2}+\sqrt{16-t^2}=8 \end{cases}
system of equationsalgebra
sin^2 x+psin x+q=0 (IV Soros Olympiad 1997-98 R2 10.4)

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6/1/2024
Draw on the plane (p,q)(p, q) all points with coordinates (p,q)(p,q), for which the equation sin⁡2x+psin⁡x+q=0\sin^2x+p\sin x+q=0 has solutions and all its positive solutions form an arithmetic progression.
trigonometryalgebra
\sqrt{\sqrt{2x^2+x-3}+2x^2-3}=x (IV Soros Olympiad 1997-98 R3 10.4)

Source:

6/2/2024
Solve the equation 2x2+x−3+2x2−3=x. \sqrt{\sqrt{2x^2+x-3}+2x^2-3}=x.
algebraRadicals