MathDB

3

Part of 2021 Harvard-MIT Mathematics Tournament.

Problems(4)

2021 Algebra/NT #3: Polynomial with two known points

Source:

5/30/2021
Among all polynomials P(x)P(x) with integer coefficients for which P(−10)=145P(-10) = 145 and P(9)=164P(9) = 164, compute the smallest possible value of ∣P(0)∣.|P(0)|.
algebrapolynomialnumber theory
2021 Combo #3: Kylo Ren plays games

Source:

5/30/2021
Let NN be a positive integer. Brothers Michael and Kylo each select a positive integer less than or equal to NN, independently and uniformly at random. Let pNp_N denote the probability that the product of these two integers has a units digit of 00. The maximum possible value of pNp_N over all possible choices of NN can be written as ab,\tfrac ab, where aa and bb are relatively prime positive integers. Compute 100a+b100a + b.
Combo
2021 Geo #3: Midpoint of a triangle

Source:

5/30/2021
Triangle ABCABC has a right angle at CC, and DD is the foot of the altitude from CC to ABAB. Points L,M,L, M, and NN are the midpoints of segments AD,DC,AD, DC, and CA,CA, respectively. If CL=7CL = 7 and BM=12,BM = 12, compute BN2BN^2.
geometry
2021 Team #3

Source:

6/27/2021
Let mm be a positive integer. Show that there exists a positive integer nn such that each of the 2m+12m+1 integers 2n−m,2n−(m−1),…,2n+(m−1),2n+m 2^{n}-m,2^{n}-(m-1),\ldots,2^{n}+(m-1),2^{n}+m is positive and composite.
number theoryComposite