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11.1

Part of I Soros Olympiad 1994-95 (Rus + Ukr)

Problems(3)

(2^x +3^x)/(3^x +4^x)<=5/7 if x>=1 (I Soros Olympiad 1994-95 R1 11.1)

Source:

7/31/2021
Prove that for real x≥1x\ge 1, holds the inequality 2x+3x3x+4x≤57\frac{2^x +3^x }{3^x +4^x} \le \frac57
algebrainequalities
2^{1995} > 5^{856} (I Soros Olympiad 1994-99 Round 2 11.1)

Source:

5/26/2024
Without using a calculator, prove that 21995>58562^{1995} > 5^{856}
algebrainequalities
f(x +y) = f(x) +f(y), f(1/x)=f(x)/x^2 (I Soros Olympiad 1994-95 Ukraine R2 11.1

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6/6/2024
Let the function f:R→Rf:R \to R satisfies the following conditions: 1) for all x,y∈Rx, y\in R, f(x+y)=f(x)+f(y) f(x +y) = f(x) +f(y) 2)f(1)=1 f(1)=1 3) for all x≠0x \ne 0 , f(1/x)=f(x)x2 f(1/x) =\frac{f(x)}{x^2} Prove that for all x∈Rx \in R, f(x)=xf(x) = x.
algebrafunctional