MathDB

9.6

Part of II Soros Olympiad 1995 - 96 (Russia)

Problems(3)

\sqrt[3]{5\sqrt{13}+18}-... (II Soros Olympiad 1995-96 R1 9.6)

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6/2/2024
Without using a calculator (especially a computer), find out what is more: 513+183−213+53   or   1\sqrt[3]{5\sqrt{13}+18}- \sqrt[3]{2\sqrt{13}+5} \,\,\, or \,\,\, 1
algebraRadicals
3 distances , equilateral (II Soros Olympiad 1995-96 R2 9.6)

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6/3/2024
There is a point inside a regular triangle located at distances 55, 66 and 77 from its vertices. Find the area of this regular triangle.
geometryEquilateral
f(x)=x^2-6x+5, f(y)<=f(x)<= - f(y) (II Soros Olympiad 1995-96 R3 9.4)

Source:

6/4/2024
Let f(x)=x2−6x+5f(x)=x^2-6x+5. On the plane (x,y)(x, y) draw a set of points M(x,y)M(x, y) whose coordinates satisfy the inequalities {f(x)+f(y)≤0f(x)−f(y)≥0\begin{cases} f(x)+f(y)\le 0 \\ f(x)-f(y)\ge 0 \end{cases}
system of equationsinequalitiesalgebraanalytic geometry