A USAMTS Year 34 Round 3 & Overall Review
What’s USAMTS?
USAMTS is a free, high school oriented, proof based contest that is funded by NSA. The contest consists of three rounds, each with 5 problems and worth 5 points. The contest is fully remote and is a month long since the problems are hard enough to not get immediately. The contest is a great way to think about problems constantly and sharpen your math skills.
Problem 1
So this puzzle was a bit different from the others because there was no givens in the sense that there was no X’s or Y’s given to us.
Although we were given the hearts and stars which was helpful. There are some essential things to note here which would be
that the row of stars on the top needed just one X in front of it since each star is already in a row with one other X and Y.
It’s the same case for the rightmost column with stars. The first “given” that I found was that below the star in the second to bottommost row, that needs a Y below it.
From there you know that the Y in the second to right column needs to be in the bottom 3 squares.
From there it’s a whole chain of “givens” that provide you the answer.
Problem 2
This problem was pretty straightforward, at first I was about to go with the wrong answer that there is no function because I was simply thinking that only polynomials could only work although there are many functions such as piecewise functions. I chose a function with mod 5 because I though that could cycle in 4 times so luckily that worked. I spent a lot of fruitless time researching cyclic functional equations and basically trying to manipulate $\frac{1-x}{1+x}$ since that was a solution, but it just didn’t output integer solutions. So sometimes just thinking about the problems for a little bit of time can get you the answer.
Problem 4
This was a problem that I spent a lot of time on but I still got it wrong. The answer had something to do with ellipses and noticing that O and H could be the foci of the ellipse. I still don’t know how I misunderstood it, but I thought that the area would be the whole circle because when I made a system of equations and graphed the soutions, it seems that it filled the entire area. I am still unsure of what I misunderstood.
Anyway I created a system of equations with the centroid of the triangle and the equation of a circle with a radius 10. I got 5 equations with 6 variables and I put it into Wolfram Alpha and got many solutions that could work so I’m assuming there was just some disconnect with the problem.
Problem 5
This was quite the tricky problem, and I spent a lot of time on this one too. The strategy that I took was finding configurations for smaller values instead of 2022 outright. For $n = 2$, we notice that a octogon works out. Although, finding $n = 3$ was really hard for some reason. I tried just drawing an octagon and drawing three dots on each side and putting dots on the interior of the octogon. Although I saw that you had to have the points all inside some “enclosed” region. For example, when I drew an octogon the other points that I draw should be in the inside and not on the outside otherwise one of the lines with a slope of -1 or 1 wouldn’t work. Like how a square’s corners wouldn’t work if we wanted to cover 2 points. When I looked at the solution it was one of those where I don’t think I would’ve been able to get it but it didn’t feel that unfamiliar either.
Final Thoughts
This round was significantly harder than the other two to me at least because the proofs felt more involved especially for 3, 4, and 5. While the problems that I got right were probably more minimal compared to the ones that I got in the other rounds, the problems were hard and nice to think about. Overall, USAMTS is a good contest to solidfy your proof skills, because usually if you do AMC or AIME you have enough knowledge to do some of the proofs. The puzzle also gives everyone an easy 5 points since you just need to be persistent.
Good news is that I’m on the leaderboard for rounds 1 and 2 of the contest so far, so let’s hope that my score isn’t totally demolished by this round.
Now that USAMTS is done, I guess it’s time to work on the other upcoming olympiads. Maybe I’ll make a post after the f=ma when discussion period is allowed.