[u]Round 1[/u]


[b]p1.[/b] What is the remainder when $2021$ is divided by $102$?


[b]p2.[/b] Brian has $2$ left shoes and $2$ right shoes. Given that he randomly picks $2$ of the $4$ shoes, the probability he will get a left shoe and a right shoe is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Compute $p + q$.


[b]p3.[/b] In how many ways can $59$ be written as a sum of two perfect squares? (The order of the two perfect squares does not matter.)


[u]Round 2 [/u]


[b]p4.[/b] Two positive integers have a sum of $60$. Their least common multiple is $273$. What is the positive diffeerence between the two numbers?


[b]p5.[/b] How many ways are there to distribute $13$ identical apples among $4$ identical boxes so that no two boxes receive the same number of apples? A box may receive zero apples.


[b]p6.[/b] In square $ABCD$ with side length $5$, $P$ lies on segment $AB$ so that $AP = 3$ and $Q$ lies on segment $AD$ so that $AQ = 4$. Given that the area of triangle $CPQ$ is $x$, compute $2x$.


[u]Round 3 [/u]


[b]p7.[/b] Find the number of ordered triples $(a, b, c)$ of nonnegative integers such that $2a+3b+5c = 15$.


[b]p8.[/b] What is the greatest integer $n \le 15$ such that $n + 1$ and $n^2 + 3$ are both prime?


[b]p9.[/b] For positive integers $a, b$, and $c$, suppose that $gcd \,\,(a, b) = 21$, $gcd \,\,(a, c) = 10$, and $gcd \,\,(b,c) = 11$. Find $\frac{abc}{lcm \,\,(a,b,c)}$ . (Note: $gcd$ is the greatest common divisor function and $lcm$ is the least common multiple function.)


[u]Round 4[/u]


[b]p10.[/b] The vertices of a square in the coordinate plane are at $(0, 0)$, $(0, 6)$, $(6, 0)$, and $(6, 6)$. Line $\ell$ intersects the square at exactly two lattice points (that is, points with integer coordinates). How many such lines $\ell$ are there that divide the square into two regions, one of them having an area of $12$?


[b]p11.[/b] Let $f(n)$ be defined as follows for positive integers $n$: $f(1) = 0$, $f(n) = 1$ if $n$ is prime, and $f(n) = f(n - 1) + 1$ otherwise. What is the maximum value of $f(n)$ for $n \le 120$?


[b]p12.[/b] The graph of the equation $y = x^3 + ax^2 + bx + c$ passes through the points $(2,4)$, $(3, 9)$, and $(4, 16)$. What is $b$?


PS. You should use hide for answers. Rounds 5- 8 have been posted [url=https://artofproblemsolving.com/community/c3h2949415p26408227]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
### Problem 1
1. We need to find the remainder when \(2021\) is divided by \(102\).
2. Perform the division: 
   \[
   2021 \div 102 \approx 19.814
   \]
3. The integer part of the division is \(19\), so:
   \[
   2021 = 102 \times 19 + r
   \]
4. Calculate the remainder \(r\):
   \[
   r = 2021 - 102 \times 19 = 2021 - 1938 = 83
   \]

The final answer is \(\boxed{8}\).