[u]Set 1[/u]

[b]G1.[/b] The Daily Challenge office has a machine that outputs the number $2.75$ when operated. If it is operated $12$ times, then what is the sum of all $12$ of the machine outputs?


[b]G2.[/b] A car traveling at a constant velocity $v$ takes $30$ minutes to travel a distance of $d$. How long does it take, in minutes, for it travel $10d$ with a constant velocity of $2.5v$?


[b]G3.[/b]  Andy originally has $3$ times as many jelly beans as Andrew. After Andrew steals 15 of Andy’s jelly beans, Andy now only has $2$ times as many jelly beans as Andrew. Find the number of jelly beans Andy originally had.

[u]Set 2[/u]


[b]G4.[/b] A coin is weighted so that it is $3$ times more likely to come up as heads than tails. How many times more likely is it for the coin to come up heads twice consecutively than tails twice consecutively?


[b]G5.[/b] There are $n$ students in an Areteem class. When 1 student is absent, the students can be evenly divided into groups of $5$. When $8$ students are absent, the students can evenly be divided into groups of $7$. Find the minimum possible value of $n$.


[b]G6.[/b] Trapezoid $ABCD$ has $AB \parallel CD$ such that $AB = 5$, $BC = 4$ and $DA = 2$. If there exists a point $M$ on $CD$ such that $AM = AD$ and $BM = BC$, find $CD$.


[u]Set 3[/u]


[b]G7.[/b] Angeline has $10$ coins (either pennies, nickels, or dimes) in her pocket. She has twice as many nickels as pennies. If she has $62$ cents in total, then how many dimes does she have?


[b]G8.[/b] Equilateral triangle $ABC$ has side length $6$. There exists point $D$ on side $BC$ such that the area of $ABD$ is twice the area of $ACD$. There also exists point $E$ on segment $AD$ such that the area of $ABE$ is twice the area of $BDE$. If $k$ is the area of triangle $ACE$, then find $k^2$.


[b]G9.[/b] A number $n$ can be represented in base $ 6$ as $\underline{aba}_6$ and base $15$ as $\underline{ba}_{15}$, where $a$ and $b$ are not necessarily distinct digits. Find $n$.


PS. You should use hide for answers. Sets 4-6 have been posted [url=https://artofproblemsolving.com/community/c3h3131305p28367080]here[/url] and 7-9 [url=https://artofproblemsolving.com/community/c3h3131308p28367095]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
9. A number \(n\) can be represented in base \(6\) as \(\underline{aba}_6\) and base \(15\) as \(\underline{ba}_{15}\), where \(a\) and \(b\) are not necessarily distinct digits. Find \(n\).

We have:

\[
n = 6^2 \cdot a + 6 \cdot b + a = 36a + 6b + a = 37a + 6b
\]

\[
n = 15 \cdot b + a
\]

Equating the two expressions for \(n\):

\[
37a + 6b = 15b + a
\]

Solving for \(a\) and \(b\):

\[
37a - a = 15b - 6b \\
36a = 9b \\
4a = b
\]

Since \(1 \leq a, b \leq 5\), the only possible values are \(a = 1\) and \(b = 4\). Thus:

\[
n = 6^2 \cdot 1 + 6 \cdot 4 + 1 = 36 + 24 + 1 = 61
\]

The final answer is \(\boxed{61}\)