[b]p1.[/b] What is the smallest possible value for the product of two real numbers that differ by ten?


[b]p2.[/b] Determine the number of positive integers $n$ with $1 \le n \le  400$ that satisfy the following:
$\bullet$ $n$ is a square number.
$\bullet$ $n$ is one more than a multiple of $5$.
$\bullet$ $n$ is even.


[b]p3.[/b] How many positive integers less than $2019$ are either a perfect cube or a perfect square but not both?


[b]p4.[/b] Felicia draws the heart-shaped figure $GOAT$ that is made of two semicircles of equal area and an equilateral triangle, as shown below. If $GO = 2$, what is the area of the figure?
[img]https://cdn.artofproblemsolving.com/attachments/3/c/388daa657351100f408ab3f1185f9ab32fcca5.png[/img]


[b]p5.[/b] For distinct digits $A, B$, and $ C$:
$$\begin{tabular}{cccc}
  & A & A \\
&   B & B \\
+ &   C & C \\
\hline
  A & B & C \\
\end{tabular}$$ Compute $A \cdot B \cdot C$.


[b]p6 [/b] What is the difference between the largest and smallest value for $lcm(a,b,c)$, where $a,b$, and $c$ are distinct positive integers between $1$ and $10$, inclusive?


[b]p7.[/b] Let $A$ and $B$ be points on the circumference of a circle with center $O$ such that $\angle AOB = 100^o$. If $X$ is the midpoint of minor arc $AB$ and $Y$ is on the circumference of the circle such that $XY\perp  AO$, find the measure of $\angle OBY$ .


[b]p8. [/b]When Ben works at twice his normal rate and Sammy works at his normal rate, they can finish a project together in $6$ hours. When Ben works at his normal rate and Sammy works as three times his normal rate, they can finish the same project together in $4$ hours. How many hours does it take Ben and Sammy to finish that project if they each work together at their normal rates?


[b][b]p9.[/b][/b] How many positive integer divisors $n$ of $20000$ are there such that when $20000$ is divided by $n$, the quotient is divisible by a square number greater than $ 1$?


[b]p10.[/b] What’s the maximum number of Friday the $13$th’s that can occur in a year?


[b]p11.[/b] Let circle $\omega$ pass through points $B$ and $C$ of triangle $ABC$. Suppose $\omega$ intersects segment $AB$ at a point $D \ne B$ and intersects segment $AC$ at a point $E \ne C$. If $AD = DC = 12$, $DB = 3$, and $EC = 8$, determine the length of $EB$.


[b]p12.[/b] Let $a,b$ be integers that satisfy the equation $2a^2 - b^2 + ab = 18$. Find the ordered pair $(a,b)$.


[b]p13.[/b] Let $a,b,c$ be nonzero complex numbers such that $a -\frac{1}{b}= 8, b -\frac{1}{c}= 10, c -\frac{1}{a}= 12.$
Find $abc -\frac{1}{abc}$ .


[b]p14.[/b] Let $\vartriangle ABC$ be an equilateral triangle of side length $1$. Let $\omega_0$ be the incircle of  $\vartriangle ABC$, and for $n > 0$, define the infinite progression of circles $\omega_n$ as follows:
$\bullet$ $\omega_n$ is tangent to $AB$ and $AC$ and externally tangent to $\omega_{n-1}$. 
$\bullet$ The area of  $\omega_n$ is strictly less than the area of $\omega_{n-1}$.
Determine the total area enclosed by all $\omega_i$ for $i \ge 0$.


[b]p15.[/b] Determine the remainder when $13^{2020} +11^{2020}$ is divided by $144$.


[b]p16.[/b] Let $x$ be a solution to $x +\frac{1}{x}= 1$. Compute $x^{2019} +\frac{1}{x^{2019}}$ .


[b]p17. [/b]The positive integers are colored black and white such that if $n$ is one color, then $2n$ is the other color. If all of the odd numbers are colored black, then how many numbers between $100$ and $200$ inclusive are colored white?


[b]p18.[/b] What is the expected number of rolls it will take to get all six values of a six-sided die face-up at least once?


[b]p19.[/b] Let $\vartriangle ABC$ have side lengths $AB = 19$, $BC = 2019$, and $AC = 2020$. Let $D,E$ be the feet of the angle bisectors drawn from $A$ and $B$, and let $X,Y$ to be the feet of the altitudes from $C$ to $AD$ and $C$ to $BE$, respectively. Determine the length of $XY$ .


[b]p20.[/b] Suppose I have $5$ unit cubes of cheese that I want to divide evenly amongst $3$ hungry mice. I can cut the cheese into smaller blocks, but cannot combine blocks into a bigger block. Over all possible choices of cuts in the cheese, what’s the largest possible volume of the smallest block of cheese?


PS. You had better use hide for answers.
**p3.** How many positive integers less than \(2019\) are either a perfect cube or a perfect square but not both?

To solve this problem, we will use the Principle of Inclusion/Exclusion (PIE).

1. **Count the perfect squares less than \(2019\):**
   \[
   \lfloor \sqrt{2019} \rfloor = \lfloor 44.94 \rfloor = 44
   \]
   So, there are \(44\) perfect squares less than \(2019\).

2. **Count the perfect cubes less than \(2019\):**
   \[
   \lfloor \sqrt[3]{2019} \rfloor = \lfloor 12.63 \rfloor = 12
   \]
   So, there are \(12\) perfect cubes less than \(2019\).

3. **Count the numbers that are both perfect squares and perfect cubes (i.e., perfect sixth powers) less than \(2019\):**
   \[
   \lfloor \sqrt[6]{2019} \rfloor = \lfloor 3.65 \rfloor = 3
   \]
   So, there are \(3\) perfect sixth powers less than \(2019\).

4. **Apply the Principle of Inclusion/Exclusion:**
   \[
   \text{Total} = (\text{Perfect squares}) + (\text{Perfect cubes}) - (\text{Perfect sixth powers})
   \]
   \[
   \text{Total} = 44 + 12 - 3 = 53
   \]

5. **Count the numbers that are either perfect squares or perfect cubes but not both:**
   \[
   \text{Result} = \text{Total} - 2 \times (\text{Perfect sixth powers})
   \]
   \[
   \text{Result} = 53 - 2 \times 3 = 53 - 6 = 47
   \]

The final answer is \(\boxed{47}\)