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Problem 777

AIME late

Algebra Difficulty 5.4 Find the answer

34. De Morgan and Others. Mathematician Augustus De Morgan, who died in 1871, used to say that he was $x$ years old in the year $x^{2}$ . Jasper Jenkins, wishing to outdo him, told me in 1925 that he was $\mathrm{a}^{2}+b^{2}$ years old in the year $\mathrm{a}^{4}+b^{4}$ , that his age was $2 m$ in the year $2 m^{2}$ , and finally, that he was $3 n$ years old in the year $3 n^{4}$ .

In what year were De Morgan and Jenkins born?

Your answer

Official solution

34. De Morgan was born in 1806. When he was 43 years old, the current year equaled the square of his age - 1849. Jenkins was born in 1860. He was $5^{2}+6^{2}(61)$ years old in the $5^{4}+6^{4}(1921)$ year. In the $2 \times 31^{2}$ (1922) year, he turned $2 \times 31$ (62) years old. Finally, his age was $3 \times 5$ (15) years in the $3 \times 5^{4}$ (1875) year.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.