MathDB

8

Part of 2021 Durer Math Competition Finals

Problems(2)

g(x) = (x -1)^2(p + 2x)^2

Source: 2021 Dürer Math Competition Finals Day2 E8 https://artofproblemsolving.com/community/c2749870_

1/8/2022
John found all real numbers pp such that in the polynomial g(x)=(x−1)2(p+2x)2g(x) = (x -1)^2(p + 2x)^2 , the quadratic term has coefficient 20212021. What is the sum of all of these values pp?
algebrapolynomial
no of true statements about 300 concerning disibility

Source: 2021 Dürer Math Competition Finals Day2 E+8 https://artofproblemsolving.com/community/c2749870_

1/9/2022
Benedek wrote the following 300300 statements on a piece of paper. 2∣1!2 | 1! 3∣1!   3∣2!3 | 1! \,\,\, 3 | 2! 4∣1!   4∣2!   4∣3!4 | 1! \,\,\, 4 | 2! \,\,\, 4 | 3! 5∣1!   5∣2!   5∣3!   5∣4!5 | 1! \,\,\, 5 | 2! \,\,\, 5 | 3! \,\,\, 5 | 4! ...... 24∣1!   24∣2!   24∣3!   24∣4!   ⋅⋅⋅   24∣23!24 | 1! \,\,\, 24 | 2! \,\,\, 24 | 3! \,\,\, 24 | 4! \,\,\, · · · \,\,\, 24 | 23! 25∣1!   25∣2!   25∣3!   25∣4!   ⋅⋅⋅   25∣23!   25∣24!25 | 1! \,\,\, 25 | 2! \,\,\, 25 | 3! \,\,\, 25 | 4! \,\,\, · · · \,\,\, 25 | 23! \,\,\, 25 | 24! How many true statements did Benedek write down? The symbol | denotes divisibility, e.g. 6∣4!6 | 4! means that 66 is a divisor of number 4!4!.
number theorydividesdivisible