I actually wasn’t going to do a weekend challenge this week because it was a super busy week, culminating in today’s flight to Minnesota with my whole family for a wedding we’re attending tomorrow, but this question came to me on the plane and I think it’s good enough to post from the hotel.

Prize this week: It won’t raaaeeaaaaiiiiin on your wedding day.
Note: figure not drawn to scale.

If the legs of a right triangle are x + m and x + n and the hypotenuse is √ax² + 16x + c as shown above for all non-negative values of x, and a, c, m and n are positive integer constants, what is the greatest possible value of ac?

The triangle is green because weddings are festive.

Post your answers in the comments; I’ll post the solution Monday.

UPDATE: nobody got this one. As usual, this challenge question is a level or two harder than you’d see on the SAT, so don’t sweat it too hard. However, I did draw my inspiration here from a real question from a real SAT, and you can see it in the Blue Book: test 3 section 5 #8.

Also, here’s a mea culpa: because I initially posted this from a hotel in Minnesota between wedding festivities, I didn’t originally word the question as carefully as I should have. If you saw this right when it was posted, I apologize. Careful wording is important in math questions, and I hold myself to a higher standard than that.

Solution below the cut.

Alright, so let’s take the obvious first step and pythagorize:

(x + m)2 + (x + n)2 = (√ax² + 16x + c)2
x2 + 2mx + m2 + x2 + 2nx + n2ax² + 16x + c
Now do a little rearranging:

2x2 + 2mx + 2nx + m2 + n2ax² + 16x + c
2x2 + (2m + 2n)x + m2 + n2 = ax² + 16x + c

The important thing here is that this needs to be true for ALL non-negative values of x, which means that the coefficients on the left and right of this equation have to be equal to each other*. In other words, 2 = a2m + 2n = 16, and m2 + n2 = c.

Now you start to see where this is going. Simplify 2m + 2n = 16 to m + n = 8, and start looking at positive integer possibilities for m and n that will add up to 8, and then sum their squares to find values:

7 + 1 = 8; 72 + 12 = 50
6 + 2 = 8; 62 + 22 = 40
5 + 3 = 8; 52 + 32 = 34
4 + 4 = 8; 42 + 42 = 32
Greatest value for c is 50. Since we saw before that the value of a is 2, the greatest possible value of ac is 100. Phew.
* I’m not going to do it rigorously here, but I know this is a sticking point for a lot of people, so you can prove this to yourself by plugging in different values for x and seeing what happens. For example: if you say x = 1, then your left side will be 2 + 2m + 2n + m2 + n2 and your right side will be a + 16 + c. Then you could say m = 2 and n = 3, and your left side would be 2 + 4 + 6 + 4 + 9 = 25. You could then set a = 5 and c = 4 to make the right side = 25 as well. Those same numbers will NOT work when you set x = 2. If the equation has to work for all values of x, then the coefficients of the terms containing x have to be equal on either side of the equation.

Comments (3)

Well, that explains it. I set x=1 and saw that the other integers could be as high as I wanted them. I figured I was just missing something…

Yeah. This is one of the most difficult things to wrap your head around that the SAT will occasionally toss into a test. Usually (as in the Blue Book question I referenced above) it’ll be multiple choice, and thus susceptible to backsolve, but for the Weekend Challenge I wanted to take it up a couple notches. 🙂

It isn’t too hard to explain the seemingly mysterious fact that if two quadratics are equal for all values of x, their coefficients must match (though, as you say, it is obviously beyond the scope of most SAT reasoning, in drawing on a deep idea from precalculus). Suppose we have ax^2+bx+c=dx^2+ex+f. Collecting terms on the left, we have (a-d)x^2+(b-e)x+(c-f)=0, and this must be true for all values of x — i.e. the function (a-d)x^2+(b-e)x+(c-f) must have infinitely many roots. If the coefficients weren’t all 0, the left hand side would be a nontrivial polynomial of degree 2, which by the fundamental theorem of algebra can’t have more than 2 roots. This is a contradiction, so the coefficients must all be 0, i.e. a-d=0, b-e=0, and c-f=0. And this shows a=d, b=e, and c=f. I think this is accessible to most high school students who have had a course that covers roots of polynomials.

Of course, there’s also a less elegant algebraic, rather than precalculus, perspective here. Plugging in different values of x gives a system of equations. When x=0, we have


which means c=f.

When x=1, we have


which means a+b=d+e.

When x=2, we have


But 4a+2b-4d-2e=2a+2(a+b)-2d-2(d+e) and since a+b=d+e from above, this third equation just says 2a-2d=0, or a=d. Back substituting into the second equation gives b=e.

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