MathDB

R2

Part of 2022 CMWMC

Problems(2)

2022 CMWMC Relay Round 2/4 - Carnegie Mellon University Womens' Competition

Source:

8/12/2023
Set 2
2.1 What is the last digit of 2022+20222022+2022(20222022)2022 + 2022^{2022} + 2022^{(2022^{2022})}?
2.2 Let TT be the answer to the previous problem. CMIMC executive members are trying to arrange desks for CMWMC. If they arrange the desks into rows of 55 desks, they end up with 11 left over. If they instead arrange the desks into rows of 77 desks, they also end up with 11 left over. If they instead arrange the desks into rows of 1111 desks, they end up with TT left over. What is the smallest possible (non-negative) number of desks they could have?
2.3 Let TT be the answer to the previous problem. Compute the largest value of kk such that 11k11^k divides T!=T(T−1)(T−2)...(2)(1).T! = T(T - 1)(T - 2)...(2)(1).
PS. You should use hide for answers.
CMWMCnumber theory
2022 CMWMC Guts Round 2/8 - Carnegie Mellon University Womens' Competition

Source:

8/12/2023
Set 2
p4. △ABC\vartriangle ABC is an isosceles triangle with AB=BCAB = BC. Additionally, there is DD on BCBC with AC=DA=BD=1AC = DA = BD = 1. Find the perimeter of △ABC\vartriangle ABC.
p5. Let rr be the positive solution to the equation 100r2+2r−1=0100r^2 + 2r - 1 = 0. For an appropriate AA, the infinite series Ar+Ar2+Ar3+Ar4+...Ar + Ar^2 + Ar^3 + Ar^4 +... has sum 11. Find AA.
p6. Let N(k)N(k) denote the number of real solutions to the equation x4−x2=kx^4 -x^2 = k. As kk ranges from −∞-\infty to ∞\infty, the value of N(k)N(k) changes only a finite number of times. Write the sequence of values of N(k)N(k) as an ordered tuple (i.e. if N(k)N(k) went from 11 to 33 to 22, you would write (1,3,2)(1, 3, 2)).
PS. You should use hide for answers.
CMWMCalgebrageometrycombinatoricsnumber theory