HMMT February, 2018

Algebra Difficulty 4.8 AIME Find the answer United States

Problem:

Randall proposes a new temperature system called Felsius temperature with the following conversion between Felsius $\{ \}^{\circ} E$ , Celsius $\{ \}^{\circ} C$ , and Fahrenheit $\{ \}^{\circ} F$ :
$\{ \}^{\circ} E = \frac{7 \times \{ \}^{\circ} C}{5} + 16 = \frac{7 \times \{ \}^{\circ} F - 80}{9}.$
For example, $0^{\circ} C = 16^{\circ} E$ . Let $x, y, z$ be real numbers such that $x^{\circ} C = x^{\circ} E$ , $y^{\circ} E = y^{\circ} F$ , $z^{\circ} C = z^{\circ} F$ . Find $x + y + z$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

Notice that $(5 k)^{\circ} C = (7 k + 16)^{\circ} E = (9 k + 32)^{\circ} F$ , so Felsius is an exact average of Celsius and Fahrenheit at the same temperature. Therefore we conclude that $x = y = z$ , and it is not difficult to compute that they are all equal to $-40$ .

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