Olympiad problems

2021 MOAA, 4

How many of the following capital English letters look the same when rotated $180^\circ$ about their center? [center]A B C D E F G H I J K L M N O P Q R S T U V W X Y Z[/center] [i]Proposed by William Yue[/i]

MOAA Gunga Bowls, 2021.12

Tags: MOAA 2021 , Gunga

Andy wishes to open an electronic lock with a keypad containing all digits from $0$ to $9$. He knows that the password registered in the system is $2469$. Unfortunately, he is also aware that exactly two different buttons (but he does not know which ones) $\underline{a}$ and $\underline{b}$ on the keypad are broken $-$ when $\underline{a}$ is pressed the digit $b$ is registered in the system, and when $\underline{b}$ is pressed the digit $a$ is registered in the system. Find the least number of attempts Andy needs to surely be able to open the lock. [i]Proposed by Andrew Wen[/i]

2021 MOAA, 13

Tags: MOAA 2021 , Gunga

Determine the greatest power of $2$ that is a factor of $3^{15}+3^{11}+3^{6}+1$. [i]Proposed by Nathan Xiong[/i]