[b]p1.[/b] What is $20\times 19 + 20 \div (2 - 7)$?


[b]p2.[/b] Will has three spinners. The first has three equally sized sections numbered $1$, $2$, $3$; the second has four equally sized sections numbered $1$, $2$, $3$, $4$; and the third has five equally sized sections numbered $1$, $2$, $3$, $4$, $5$. When Will spins all three spinners, the probability that the same number appears on all three spinners is $p$. Compute $\frac{1}{p}$.



[b]p3.[/b] Three girls and five boys are seated randomly in a row of eight desks. Let $p$ be the probability that the students at the ends of the row are both boys. If $p$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m + n$.


[b]p4.[/b] Jaron either hits a home run or strikes out every time he bats. Last week, his batting average was $.300$. (Jaron's batting average is the number of home runs he has hit divided by the number of times he has batted.) After hitting $10$ home runs and striking out zero times in the last week, Jaron has now raised his batting average to $.310$. How many home runs has Jaron now hit?


[b]p5.[/b] Suppose that the sum $$\frac{1}{1 \cdot 4} +\frac{1}{4 \cdot 7}+ ...+\frac{1}{97 \cdot 100}$$ is expressible as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$.


[b]p6.[/b] Let $ABCD$ be a unit square with center $O$, and $\vartriangle OEF$ be an equilateral triangle with center $A$. Suppose that $M$ is the area of the region inside the square but outside the triangle and $N$ is the area of the region inside the triangle but outside the square, and let $x = |M -N|$ be the positive difference between $M$ and $N$. If $$x =\frac1 8(p -\sqrt{q})$$ for positive integers $p$ and $q$, find $p + q$.


[b]p7.[/b] Find the number of seven-digit numbers such that the sum of any two consecutive digits is divisible by $3$. For example, the number $1212121$ satisfies this property.


[b]p8.[/b] There is a unique positive integer $x$ such that $x^x$ has $703$ positive factors. What is $x$?


[b]p9.[/b] Let $x$ be the number of digits in $2^{2019}$ and let $y$ be the number of digits in $5^{2019}$. Compute $x + y$.


[b]p10.[/b] Let $ABC$ be an isosceles triangle with $AB = AC = 13$ and $BC = 10$. Consider the set of all points $D$ in three-dimensional space such that $BCD$ is an equilateral triangle. This set of points forms a circle $\omega$. Let $E$ and $F$ be points on $\omega$ such that $AE$ and $AF$ are tangent to $\omega$. If $EF^2$ can be expressed in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, determine $m + n$.


PS. You should use hide for answers.  Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To solve the problem, we need to find a positive integer \( x \) such that \( x^x \) has exactly 703 positive factors. 

1. **Understanding the number of factors:**
   The number of factors of a number \( n \) can be determined if we know its prime factorization. If \( n \) has the prime factorization \( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \), then the number of factors of \( n \) is given by:
   \[
   (e_1 + 1)(e_2 + 1) \cdots (e_k + 1)
   \]

2. **Prime factorization of \( x^x \):**
   Suppose \( x = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k} \). Then:
   \[
   x^x = (p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k})^x = p_1^{a_1 x} p_2^{a_2 x} \cdots p_k^{a_k x}
   \]
   The exponents in the prime factorization of \( x^x \) are \( a_1 x, a_2 x, \ldots, a_k x \).

3. **Number of factors of \( x^x \):**
   The number of factors of \( x^x \) is:
   \[
   (a_1 x + 1)(a_2 x + 1) \cdots (a_k x + 1)
   \]
   We need this product to equal 703.

4. **Factorizing 703:**
   We factorize 703:
   \[
   703 = 19 \times 37
   \]
   Since 19 and 37 are both prime, we have two factors.

5. **Matching the form:**
   We need to match the form \( (a_1 x + 1)(a_2 x + 1) = 703 \). Given that 703 has two prime factors, we can assume \( x \) has the form \( p_1^{a_1} \) where \( p_1 \) is a prime number and \( a_1 = 1 \).

6. **Solving for \( x \):**
   Let \( x = p_1 \). Then:
   \[
   (x + 1)(x + 1) = 703
   \]
   Solving for \( x \):
   \[
   x + 1 = 19 \quad \text{or} \quad x + 1 = 37
   \]
   Thus:
   \[
   x = 18 \quad \text{or} \quad x = 36
   \]

7. **Verification:**
   - For \( x = 18 \):
     \[
     18^{18} = (2 \cdot 3^2)^{18} = 2^{18} \cdot 3^{36}
     \]
     The number of factors is:
     \[
     (18 + 1)(36 + 1) = 19 \cdot 37 = 703
     \]
   - For \( x = 36 \):
     \[
     36^{36} = (2^2 \cdot 3^2)^{36} = 2^{72} \cdot 3^{72}
     \]
     The number of factors is:
     \[
     (72 + 1)(72 + 1) = 73 \cdot 73 = 5329 \quad (\text{not equal to 703})
     \]

Thus, the only valid solution is \( x = 18 \).

The final answer is \( \boxed{18} \).