MathDB

10

Part of 2023 AIME

Problems(2)

did U silly this?

Source: 2023 AIME I #10

2/8/2023
There exists a unique positive integer aa for which the sum U=∑n=12023⌊n2−na5⌋U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor is an integer strictly between −1000-1000 and 10001000. For that unique aa, find a+Ua+U.
(Note that ⌊x⌋\lfloor x\rfloor denotes the greatest integer that is less than or equal to xx.)
AMCAIMEAIME I
Integers in a Grid

Source: 2023 AIME II/10

2/16/2023
Let NN be the number of ways to place the integers 11 through 1212 in the 1212 cells of a 2×62\times 6 grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by 33. One way to do this is shown below. Find the number of positive integer divisors of NN.
[asy] size(160); defaultpen(linewidth(0.6)); for(int j=0;j<=6;j=j+1) { draw((j,0)--(j,2)); } for(int i=0;i<=2;i=i+1) { draw((0,i)--(6,i)); } for(int k=1;k<=12;k=k+1) { label(""+((string)k)+""+((string) k)+"",(floor((k-1)/2)+0.5,k%2+0.5)); } [/asy]
AMCAIME