MathDB

12

Part of 2021 MOAA

Problems(2)

Team Round P12

Source:

10/16/2021
Let △ABC\triangle ABC have AB=9AB=9 and AC=10AC=10. A semicircle is inscribed in △ABC\triangle ABC with its center on segment BCBC such that it is tangent ABAB at point DD and ACAC at point EE. If AD=2DBAD=2DB and rr is the radius of the semicircle, r2r^2 can be expressed as mn\frac{m}{n} for relatively prime positive integers mm and nn. Compute m+nm+n.
Proposed by Andy Xu
MOAA 2021team
Gunga P12

Source:

10/16/2021
Andy wishes to open an electronic lock with a keypad containing all digits from 00 to 99. He knows that the password registered in the system is 24692469. Unfortunately, he is also aware that exactly two different buttons (but he does not know which ones) a‾\underline{a} and b‾\underline{b} on the keypad are broken −- when a‾\underline{a} is pressed the digit bb is registered in the system, and when b‾\underline{b} is pressed the digit aa is registered in the system. Find the least number of attempts Andy needs to surely be able to open the lock.
Proposed by Andrew Wen
MOAA 2021Gunga