1) Two of the sides of a triangle are 18 and 25. The length of the third side is also a positive integer. How many different possible values are there for the third side length? (Assume that the triangle is non-degenerate.)

2) Triangle ABC has altitudes $\overline{AD}$, $\overline{BE}$, and $\overline{CF}.$ If AD = 12, BE = 14, and CF is a positive integer, then find the largest possible value of CF.

3) Let n be a positive integer. In triangle ABC, AB = 3n, AC = 2n + 15, BC = n +30, and angle A > angle B > angle C. How many possible values of n are there?

I've been having some trouble with these last few questions. All responses welcome as they are due wednesday night. I've tried a few equations but to no avail...

Paradox Jun 22, 2020

#1**0 **

What have you done so far?

In the future, please post the AoPS question you are most stuck with.

Funny I was crunching on these earlier today...

For the first... I'll give a hint

By triangle inequality we have

\(\begin{align*} n + 18 &> 25, \\ n + 25 &> 18, \\ 18 + 25 &> n, \end{align*}\)

Also, maybe search these up.

Typically AoPS questions are asked very often, and are answered.

BTW are you in 2387? The one with Hokaj?

hugomimihu Jun 23, 2020

#2**0 **

Yes, I am in 2387.

I managed to solve number 1 and number 3 myself last night. It was a silly mistake I made and miscounted my answer. I didn't even need to come to here for a solution...opss. The only one I'm still stuck on is the one about altitudes (number 2). I'll make sure to only post a single question when I'm absolutly stuck. Thanks for the quick feedback btw!

Paradox
Jun 23, 2020