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11.3

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

Problems(2)

min root of x^3 + ax - a^3 - 29 = 0 (I Soros Olympiad 1994-99 Round 2 11.3)

Source:

5/26/2024
For each non-negative aa, consider the equation x3+ax−a3−29=0.x^3 + ax - a^3 - 29 = 0. Let xox_o be the positive root of this equation. Prove that for all a>0a > 0 such a root exists. What is the smallest value of xox_o?
algebrapolynomial
4(BC + CA)< 5AB if 2 < BAC + 3 <ABC= 180^o

Source: I Soros Olympiad 1994-95 Ukraine R2 11.3 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/6/2024
It is known that in the triangle ABCABC, 2∠BAC+3∠ABC=180o 2 \angle BAC + 3 \angle ABC= 180^o. Prove that 4(BC+CA)<5AB4(BC + CA)< 5AB.
geometryanglesgeometric inequality