MathDB

10.2

Part of IV Soros Olympiad 1997 - 98 (Russia)

Problems(3)

sum k!/((x+k)..(x+1))=1 (IV Soros Olympiad 1997-98 Correspondence 10.2)

Source:

6/1/2024
Solve the equation 10x+10+10⋅9(x+10)(x+9)+10⋅9⋅8(x+10)(x+9)(x+8)+...+10⋅9⋅...⋅2⋅1(x+10)(x+9)⋅...⋅(x+1)=11\frac{10}{x+10}+\frac{10\cdot 9}{(x+10)(x+9)}+\frac{10\cdot 9\cdot 8}{(x+10)(x+9)(x+8)}+ ...+\frac{10\cdot 9\cdot ... \cdot 2 \cdot 1}{(x+10)(x+9)\cdot ... \cdot(x+1)}=11
algebra
line passes through arc midpoint

Source: IV Soros Olympiad 1997-98 R3 10.2 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/2/2024
Let MM be the point of intersection of the diagonals of the inscribed quadrilateral ABCDABCD. Prove that if AB=AM,AB = AM, then a line passing through MM perpendicular to ADAD passes through the midpoint of the arc BCBC.
geometryarc midpoint
\sqrt[3]{x^3+6x^2-6x-1}=\sqrt{x^2+4x+1} (IV Soros Olympiad 1997-98 R2 10.2)

Source:

6/1/2024
Solve the equation x3+6x2−6x−13=x2+4x+1\sqrt[3]{x^3+6x^2-6x-1}=\sqrt{x^2+4x+1}
algebraRadicals