2023 AMC Proposals
by djmathman, Nov 15, 2023, 6:52 PM
I had quite a few this year! Most of them were on the easy end, though. That's part of the natural progression of things, I guess; I've been less interested in making hard problems and more focused on the easier end of the exam. Though this year it looks like the problems on the hard end were not great; I have some thoughts about a few of them....
12A 6 (March 2023). Points
and 
lie on the graph of 
. The midpoint of 
is 
. What is the positive difference between the 
-coordinates of and ?
10A 12 (March 2023). How many three-digit positive integers
satisfy the following properties?
12A 16 (March 2023). Consider the set of complex numbers
satisfying 
. The maximum value of the imaginary part of can be written in the form 
, where 
and 
are relatively prime positive integers. What is 
?
10B 1/12B 1 (March 2023). A mom is pouring orange juice for her 4 kids into 4 identical glasses. She fills the first 3 full, but only fills one third of the glass for the last one. How much does she need to pour from the 3 full glasses to fill all of the glasses to an equal amount?
10B 5 (March 2023?). Maddy and Lara see a list of numbers written on a blackboard. Maddy adds
to each number in the list and finds that the sum of her new numbers is 
. Lara multiplies each number in the list by and finds that the sum of her new numbers is also . How many numbers are written on the blackboard?
10B 7 (March 2023). Square
is rotated 
clockwise about its center to obtain square 
, as shown below. What is the degree measure of 
?
![[asy] size(170); defaultpen(linewidth(0.6)); real r = 25; draw(dir(135)--dir(45)--dir(315)--dir(225)--cycle); draw(dir(135-r)--dir(45-r)--dir(315-r)--dir(225-r)--cycle); label("$A$",dir(135),NW); label("$B$",dir(45),NE); label("$C$",dir(315),SE); label("$D$",dir(225),SW); label("$E$",dir(135-r),N); label("$F$",dir(45-r),E); label("$G$",dir(315-r),S); label("$H$",dir(225-r),W); [/asy]](//latex.artofproblemsolving.com/4/0/4/404883b79d8ef1c910971e24f1655976ce3afbae.png)
12B 12 (December 2020). For complex numbers
and 
, define the binary operation 
by ![\[w\otimes z=ac+bd\,i.\]](//latex.artofproblemsolving.com/c/8/e/c8ecf215ad3e92354fba57bc8ffdd4526021f183.png)
Suppose is a complex number such that 
. What is 
?
You'll notice a lot of March 2023 problems here. Some of you may also remember that I wrote the following last year:
It turns out that right before the AMC constructor meetings last year I had a sudden interest in constructing some problems for the test. The AMC heads requested some complex numbers and logarithm problems, so I focused my efforts on those, with a few extra problems thrown in there for good measure. This was the result of about two weeks' worth of thought, putting research to the side for a bit. There's nothing super flashy here, but I'm decently happy with this set. Favorite is almost certainly 12A #16, though 10B #7 is up there as one of my favorites, too.
12A 6 (March 2023). Points
and 
lie on the graph of 
. The midpoint of 
is 
. What is the positive difference between the 
-coordinates of and ?10A 12 (March 2023). How many three-digit positive integers
satisfy the following properties?
- The number is divisible by
. - The number formed by reversing the digits of is divisible by
.
.
.12A 16 (March 2023). Consider the set of complex numbers
satisfying 
. The maximum value of the imaginary part of can be written in the form 
, where 
and 
are relatively prime positive integers. What is 
?10B 1/12B 1 (March 2023). A mom is pouring orange juice for her 4 kids into 4 identical glasses. She fills the first 3 full, but only fills one third of the glass for the last one. How much does she need to pour from the 3 full glasses to fill all of the glasses to an equal amount?
10B 5 (March 2023?). Maddy and Lara see a list of numbers written on a blackboard. Maddy adds
to each number in the list and finds that the sum of her new numbers is 
. Lara multiplies each number in the list by and finds that the sum of her new numbers is also . How many numbers are written on the blackboard?10B 7 (March 2023). Square
is rotated 
clockwise about its center to obtain square 
, as shown below. What is the degree measure of 
?![[asy] size(170); defaultpen(linewidth(0.6)); real r = 25; draw(dir(135)--dir(45)--dir(315)--dir(225)--cycle); draw(dir(135-r)--dir(45-r)--dir(315-r)--dir(225-r)--cycle); label("$A$",dir(135),NW); label("$B$",dir(45),NE); label("$C$",dir(315),SE); label("$D$",dir(225),SW); label("$E$",dir(135-r),N); label("$F$",dir(45-r),E); label("$G$",dir(315-r),S); label("$H$",dir(225-r),W); [/asy]](http://latex.artofproblemsolving.com/4/0/4/404883b79d8ef1c910971e24f1655976ce3afbae.png)
12B 12 (December 2020). For complex numbers
and 
, define the binary operation 
by ![\[w\otimes z=ac+bd\,i.\]](http://latex.artofproblemsolving.com/c/8/e/c8ecf215ad3e92354fba57bc8ffdd4526021f183.png)
Suppose is a complex number such that 
. What is 
?
You'll notice a lot of March 2023 problems here. Some of you may also remember that I wrote the following last year:
Quote:
In any case, that'll be it for a while. As I mentioned before, I didn't submit anything for the 2023-2024 series of contests, so you probably won't see any dj problems there. I probably won't be completely dead, but I certainly don't feel the same passion and fire for problem construction as I used to. (Right now, I'm more focused on improving my mental well-being anyway.) 
This post has been edited 1 time. Last edited by djmathman, Nov 15, 2023, 6:55 PM
April 2025
November 2024