MathDB

16

Part of 2021 AMC 12/AHSME Fall

Problems(2)

Linus Tech Tips

Source: AMC 12A #16

11/11/2021
An organization has 3030 employees, 2020 of whom have a brand A computer while the other 1010 have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if their computers are directly connected by a cable or by relaying messages through a series of connected computers. Initially, no computer is connected to any other. A technician arbitrarily selects one computer of each brand and installs a cable between them, provided there is not already a cable between that pair. The technician stops once every employee can communicate with each other. What is the maximum possible number of cables used?
<spanclass=′latex−bold′>(A)</span> 190<spanclass=′latex−bold′>(B)</span> 191<spanclass=′latex−bold′>(C)</span> 192<spanclass=′latex−bold′>(D)</span> 195<spanclass=′latex−bold′>(E)</span> 196<span class='latex-bold'>(A)</span>\ 190 \qquad<span class='latex-bold'>(B)</span>\ 191 \qquad<span class='latex-bold'>(C)</span>\ 192 \qquad<span class='latex-bold'>(D)</span>\ 195 \qquad<span class='latex-bold'>(E)</span>\ 196
GCD Condition

Source: 2022 AMC 12B #16

11/17/2021
Let a,b,a, b, and cc be positive integers such that a+b+c=23a+b+c=23 and gcd⁡(a,b)+gcd⁡(b,c)+gcd⁡(c,a)=9.\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9. What is the sum of all possible distinct values of a2+b2+c2a^{2}+b^{2}+c^{2}?
<spanclass=′latex−bold′>(A)</span> 259<spanclass=′latex−bold′>(B)</span> 438<spanclass=′latex−bold′>(C)</span> 516<spanclass=′latex−bold′>(D)</span> 625<spanclass=′latex−bold′>(E)</span> 687<span class='latex-bold'>(A)</span> ~259\qquad<span class='latex-bold'>(B)</span> ~438\qquad<span class='latex-bold'>(C)</span> ~516\qquad<span class='latex-bold'>(D)</span> ~625\qquad<span class='latex-bold'>(E)</span> ~687
Proposed by djmathman
number theorygreatest common divisorAMCAMC 12AMC 12 B