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9.2

Part of IV Soros Olympiad 1997 - 98 (Russia)

Problems(3)

nested radicals (IV Soros Olympiad 1997-98 Correspondence 9.2)

Source:

6/1/2024
Solve the equation 21+x1+(x+1)1+(x+2)1+(x+3)(x+5)=x2\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+(x+3)(x+5)}}}}=x
algebraRadicals
7-digit number by 3 terms of A.P. (IV Soros Olympiad 1997-98 R3 9.2)

Source:

6/1/2024
The student wrote on the board three natural numbers that are consecutive members of one arithmetic progression. Then he erased the commas separating the numbers, resulting in a seven-digit number. What is the largest number that could result?
number theoryalgebraArithmetic Progression
x^2+5\sqrt2 x+a<0 (IV Soros Olympiad 1997-98 R2 9.2)

Source:

6/1/2024
Find all values of the parameter aa for which there exist exactly two integer values of xx that satisfy the inequality x2+52x+a<0.x^2+5\sqrt2 x+a<0.
algebrainequalitiesparameter