[u]Round 1[/u]


[b]p1.[/b] If $2m = 200 cm$ and $m \ne 0$, find $c$.


[b]p2.[/b] A right triangle has two sides of lengths $3$ and $4$. Find the smallest possible length of the third side.


[b]p3.[/b] Given that $20(x + 17) = 17(x + 20)$, determine the value of $x$.


[u]Round 2[/u]


[b]p4.[/b] According to the Egyptian Metropolitan Culinary Community, food service is delayed on $\frac23$ of flights departing from Cairo airport. On average, if  flights with delayed food service have twice as many passengers per flight as those without, what is the probability that a passenger departing from Cairo airport experiences delayed food service?


[b]p5.[/b] In a positive geometric sequence $\{a_n\}$, $a_1 = 9$, $a_9 = 25$. Find the integer $k$ such that $a_k = 15$


[b]p6.[/b] In the Delicate, Elegant, and Exotic Music Organization, pianist Hans is selling two types of  owers with different prices (per  ower): magnolias and myosotis. His friend Alice originally plans to buy a bunch containing $50\%$ more magnolias than myosotis for $\$50$, but then she realizes that if she buys $50\%$ less magnolias and $50\%$ more myosotis than her original plan, she would still need to pay the same amount of money. If instead she buys $50\%$ more magnolias and $50\%$ less myosotis than her original plan, then how much, in dollars, would she need to pay?


[u]Round 3[/u]


[b]p7.[/b] In square $ABCD$, point $P$ lies on side $AB$ such that $AP = 3$,$BP = 7$. Points $Q,R, S$ lie on sides $BC,CD,DA$ respectively such that $PQ = PR = PS = AB$. Find the area of quadrilateral $PQRS$.


[b]p8.[/b] Kristy is thinking of a number $n < 10^4$ and she says that $143$ is one of its divisors. What is the smallest number greater than $143$ that could divide $n$?


[b]p9.[/b] A positive integer $n$ is called [i]special [/i] if the product of the $n$ smallest prime numbers is divisible by the sum of the $n$ smallest prime numbers. Find the sum of the three smallest special numbers.


[u]Round 4[/u]


[b]p10.[/b]  In the diagram below, all adjacent points connected with a segment are unit distance apart. Find the number of squares whose vertices are among the points in the diagram and whose sides coincide with the drawn segments.
[img]https://cdn.artofproblemsolving.com/attachments/b/a/923e4d2d44e436ccec90661648967908306fea.png[/img]


[b]p11.[/b] Geyang tells Junze that he is thinking of a positive integer. Geyang gives Junze the following clues:
$\bullet$ My number has three distinct odd digits.
$\bullet$ It is divisible by each of its three digits, as well as their sum.
What is the sum of all possible values of Geyang's number?


[b]p12.[/b] Regular octagon $ABCDEFGH$ has center $O$ and side length $2$. A circle passes through $A,B$, and $O$. What is the area of the part of the circle that lies outside of the octagon?


PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2936505p26278645]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. **Identify the number of $1 \times 1$ squares:**

   The diagram is a $7 \times 7$ grid of points, which means there are $6 \times 6 = 36$ $1 \times 1$ squares. However, the diagram also includes additional points around the eyes and other areas, which contribute to more $1 \times 1$ squares. By carefully counting all the $1 \times 1$ squares, we find there are $52$ such squares.

2. **Identify the number of $2 \times 2$ squares:**

   A $2 \times 2$ square requires a $3 \times 3$ grid of points. In a $7 \times 7$ grid, there are $(7-2) \times (7-2) = 5 \times 5 = 25$ possible $2 \times 2$ squares. However, due to the specific arrangement of points in the diagram, only $7$ of these $2 \times 2$ squares are valid.

3. **Check for larger squares:**

   Larger squares such as $3 \times 3$ or bigger would require a larger grid of points. Given the constraints of the diagram, there are no valid larger squares.

4. **Sum the total number of squares:**

   Adding the number of $1 \times 1$ and $2 \times 2$ squares, we get:
   \[
   52 + 7 = 59
   \]

The final answer is $\boxed{59}$