MathDB

11.5

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(3)

D=6G, D=25R, D=120T, G=4R, G=21T, R=5T- VI Soros Olympiad 1999-00 Round 1 11.5

Source:

5/21/2024
At the currency exchange of the island of Luck they sell dinars (D), guilders (G), reals (R) and thalers (T). Stock brokers have the right to make a purchase and sale transaction with any pair of currencies no more than once per day. The exchange rates are as follows: D=6GD = 6G, D=25RD = 25R, D=120TD = 120T, G=4RG = 4R, G=21TG = 21T, R=5TR = 5T. For example, the entry D=6GD = 6G means that 11 dinar can be bought for 66 guilders (or 66 guilders can be sold for 11 dinar). In the morning the broker had 8080 dinars, 100100 guilders, 100100 reals and 50,40050,400 thalers. In the evening he had the same number of dinars and thalers. What is the maximum value of this number?
number theory
(1 + 2x)P(2x) = (1 + 2^{1999}x)P(x) .

Source: VI Soros Olympiad 1990-00 R2 11.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/28/2024
Find all polynomials P(x)P(x) with real coefficients such that for all real xx holds the equality (1+2x)P(2x)=(1+21999x)P(x).(1 + 2x)P(2x) = (1 + 2^{1999}x)P(x) .
algebrapolynomial
sum (\sqrt{x_k^2-1}}{x_{k+1}} <= 1/sqrt2 n

Source: VI Soros Olympiad 1990-00 R3 11.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/29/2024
Let n≥2 n \ge 2 and x1x_1, x2x_2, ......, xnx_n be real numbers from the segment [1,2][1,\sqrt2]. Prove that holds the inequality x12−1x2+x22−1x3+...+xn2−1x1≤22n.\frac{\sqrt{x_1^2-1}}{x_2}+\frac{\sqrt{x_2^2-1}}{x_3}+...+\frac{\sqrt{x_n^2-1}}{x_1} \le \frac{\sqrt2}{2} n.
algebrainequalities