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Hey Math Circle Students!
Here are the advanced problems for the next $3$ days! These are more difficult than AMC 8 Problems 1520 but they fit the range of $2025$ and some are even AMC 10 level. Each of these problems are worth $10$ points for your leaderboard score, but the challenge problem is worth 15 points!
If you don’t understand one of the questions, feel free to post in the questions forum here
1. John is trying to group marbles. If he groups them in sets of 5, 2 are left over. If he groups them in sets of 7, 4 are left over. If he groups them in sets of 11, 8 are left over. What is the minimum number of marbles John must be grouping?
2. Find the maximum value of $k$ such that $28^k$ divides $100!$ where the exclamation mark represents the factorial function (e.g. $5!=5*4*3*2*1$ and $4!=4*3*2*1$)
3. Find the number of positive integer $x$ less than $7$ to $x^2 \equiv 2 \bmod 7$. The existence of a solution is surprising because $2$ is not a perfect square. What about the solutions less than $15$ to $x^2 \equiv 1 \bmod 15$? Why is this interesting as well?
4. How many values of $n$ less than $100$ exist such that $1+2+…+n$ is divisible by $n$. What about the number of values of $n$ such that $1^2+2^2+….+n^2$ is divisible by $n$. (NOTE: $1+2+..+n=\frac{n(n+1)}{2}$ and $1^2+2^2+..+n^2=\frac{n(n+1)(2n+1)}{6}$)
CHALLENGE PROBLEM:
Let be a strictly increasing sequence of positive integers such that What is the remainder when is divided by ?
Remember that you can post your solutions below to these problems and I will keep a running leaderboards for the top 10 students who participate the most! I especially appreciate the same problem being solved in different ways!

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