This is problem 18 on the grade 11 and 12, 2025 Math Kangaroo Contest. A student drew the graphs of two linear functions in a coordinate system as shown. Observe the picture. It is two lines in, you know, R2. The expression AB plus CD minus in parentheses AC plus BD, close parentheses, is always A, negative, B, non-positive, C, positive, D, zero, or E, none of the above is always true. Take your time. Pause the video. Okay. So, we don't really know anything about A, B, C, or D exactly. Like, maybe we could get, maybe we could try to draw some stuff here and try to get values. We don't know really anything. Like, these are all kind of mysteries to us. But what we can do is sort of judge whether or not they're negative or positive, and then see if we can make any judgments from there. So, what we do have is that, well, let's take a look at this, like, first line highlighted in cyan. We know that the slope is positive, and therefore, you know, of the line equation, that means that A is positive. And we also know that it intersects the y-axis at a positive point. So, when x equals zero, we get that y equals positive B, or equals B, which is positive. So, then we know that B is also positive. Okay. Let's do the same thing for the other line, now highlighted in green. We do the same thing. The slope is negative, so we get that C is negative. And similarly, we get, you know, D is below the x-axis when it intersects the y-axis. So, we get that D is also negative. And I'm going to just, like, represent these two as minuses, and the positives as pluses. And then you can do some arithmetic with these pluses and minuses. So, let's plug each thing into this equation. We get that positive times positive, plus a negative times a negative, for A, B, plus C, D. And then we get minus, in parentheses, a positive times a negative, plus a positive times a negative. And I apologize if my notation is maybe a little bit unclear. But, yeah, it should just sort of be like that. So, then, if we judge what each of these evaluates to be, well, positive times positive, that's positive. Negative times negative, that's positive. So, we have a positive plus a positive, minus, in parentheses, a positive times a negative. Well, that's a negative. And then, plus a positive times a negative. Again, another negative. But we get that this, eventually, the negative plus negative turns out to be negative. But then a negative times a negative, that turns into a positive. So, we can just cross out this entire thing and evaluate this to be positive. So, we end up with positive, plus positive, plus positive, which is always necessarily positive. And, therefore, our answer should be C.

linear functions

expression evaluation

positive slope

negative slope

y-intercept