The wording of the problem and underlying assumptions make this problem challenging on many levels. I find that this is a common theme with “excellent” problems; one has to assume certain truths that will make the problem solvable, otherwise it would be intractable.

Does LaTeX work in your blog? I’m testing here: $latex frac{5}{8}$

]]>If f(x) <= x and f(x + y) <= f(x) + f(y) for all real x and y, then consider x = y = 0. Because f(0) = 0. But we know that f(0) <= 0, so f(0) = 0. Now consider real numbers a and -a. We know that f(0) <= f(a) + f(-a) so 0 <= f(a) + f(-a). But f(a) <= a and f(-a) <= -a, so f(a) + f(-a) <= 0, which means that f(a) + f(-a) = 0, and we can conclude that the ONLY function that satisfies the requirements is f(x) = x ]]>

Let the total milk consumed be m and the total coffee consumed be c

m/4 + c/6 = 1 cup

Total number of cups consumed = m+c = t

t – c/3 = 4

t + m/2 = 6

Since t is an integer, and t > 4 and t < 6

We get t = 5. Total number of people in the family = 5 ðŸ™‚

Nice problem. Thanks

]]>This is for 8th graders after all.

]]>I’m kinda new here, should I be solving these in the comments, or will people complain about spoilers if I do that?

]]>What I meant to say was that the first implies f(0) <= 0 and the second implies f(0) >= 0.

Did that come out right?

]]>