MathDB

25

Part of 2022 MIG

Problems(2)

2022 A Individual #25

Source:

4/28/2022
For all positive integers a>1a > 1, there are divisors of 2021a2021a that are not divisors of 20212021. If there are twelve unshared divisors, including 2021a2021a, which of the following answer choices could be a possible value of aa?
<spanclass=latexbold>(A)</span>9<spanclass=latexbold>(B)</span>10<spanclass=latexbold>(C)</span>16<spanclass=latexbold>(D)</span>18<spanclass=latexbold>(E)</span>19<span class='latex-bold'>(A) </span>9\qquad<span class='latex-bold'>(B) </span>10\qquad<span class='latex-bold'>(C) </span>16\qquad<span class='latex-bold'>(D) </span>18\qquad<span class='latex-bold'>(E) </span>19
2022 A Individual
2022 B Individual #25

Source:

4/30/2022
In the below diagram, the three rectangles are similar. Find the area of rectangle ABCDABCD.
[asy] size(6cm); draw((0,0)--(10,0)--(10,8.16496580928)--(0,8.16496580928)--(0,0)); draw((2,0)--(6,3.26598632371)--(2,8.16496580928)--(-2,4.89897949)--(2,0)); draw((10,3.26598632371)--(6,3.26598632371)--(6,0)); label("AA",(0,8.16496580928), NW); label("DD",(0,0), SW); label("CC",(10,0), SE); label("BB",(10,8.16496580928), NE); label("EE",(2,8.16496580928), N); label("FF",(2,0), S); label("33",(1,0),S); label("33",(1,8.16496580928),N); label("1212",(6,8.16496580928),N); [/asy]
<spanclass=latexbold>(A)</span>753<spanclass=latexbold>(B)</span>1202<spanclass=latexbold>(C)</span>1003<spanclass=latexbold>(D)</span>180<spanclass=latexbold>(E)</span>756<span class='latex-bold'>(A) </span>75\sqrt{3}\qquad<span class='latex-bold'>(B) </span>120\sqrt{2}\qquad<span class='latex-bold'>(C) </span>100\sqrt{3}\qquad<span class='latex-bold'>(D) </span>180\qquad<span class='latex-bold'>(E) </span>75\sqrt{6}
2022 B Individual