[b]p1.[/b] If $x$ is a real number that satisfies $\frac{48}{x} = 16$, find the value of $x$.


[b]p2.[/b] If $ABC$ is a right triangle with hypotenuse $BC$ such that $\angle ABC = 35^o$, what is $\angle BCA$ in degrees?
[img]https://cdn.artofproblemsolving.com/attachments/a/b/0f83dc34fb7934281e0e3f988ac34f653cc3f1.png[/img]

[b]p3.[/b] If $a\vartriangle b = a + b - ab$, find $4\vartriangle 9$.


[b]p4.[/b] Grizzly is $6$ feet tall. He measures his shadow to be $4$ feet long. At the same time, his friend Panda helps him measure the shadow of a nearby lamp post, and it is $6$ feet long. How tall is the lamp post in feet?


[b]p5.[/b] Jerry is currently twice as old as Tom was $7$ years ago. Tom is $6$ years younger than Jerry. How many years old is Tom?


[b]p6.[/b] Out of the $10, 000$ possible four-digit passcodes on a phone, how many of them contain only prime digits?


[b]p7.[/b] It started snowing, which means Moor needs to buy snow shoes for his $6$ cows and $7$ sky bison. A cow has $4$ legs, and a sky bison has $6$ legs. If Moor has 36 snow shoes already, how many more shoes does he need to buy? Assume cows and sky bison wear the same type of shoe and each leg gets one shoe.


[b]p8.[/b] How many integers $n$ with $1 \le n \le 100$ have exactly $3$ positive divisors?


[b]p9.[/b] James has three $3$ candies and $3$ green candies. $3$ people come in and each randomly take $2$ candies. What is the probability that no one got $2$ candies of the same color? Express your answer as a decimal or a fraction in lowest terms.


[b]p10.[/b] When Box flips a strange coin, the coin can land heads, tails, or on the side. It has a $\frac{1}{10}$probability of landing on the side, and the probability of landing heads equals the probability of landing tails. If Box flips a strange coin $3$ times, what is the probability that the number of heads flipped is equal to the number of tails flipped? Express your answer as a decimal or a fraction in lowest terms.


[b]p11.[/b] James is travelling on a river. His canoe goes $4$ miles per hour upstream and $6$ miles per hour  downstream. He travels $8$ miles upstream and then $8$ miles downstream (to where he started). What is his average speed, in miles per hour? Express your answer as a decimal or a fraction in lowest terms.


[b]p12.[/b] Four boxes of cookies and one bag of chips cost exactly $1000$ jelly beans. Five bags of chips and one box of cookies cost less than $1000$ jelly beans. If both chips and cookies cost a whole number of jelly beans, what is the maximum possible cost of a bag of chips?


[b]p13.[/b] June is making a pumpkin pie, which takes the shape of a truncated cone, as shown below. The pie tin is $18$ inches wide at the top, $16$ inches wide at the bottom, and $1$ inch high. How many cubic inches of pumpkin filling are needed to fill the pie?
[img]https://cdn.artofproblemsolving.com/attachments/7/0/22c38dd6bc42d15ad9352817b25143f0e4729b.png[/img]

[b]p14.[/b] For two real numbers $a$ and $b$, let $a\# b = ab - 2a - 2b + 6$. Find a positive real number $x$ such that $(x\#7) \#x = 82$.


[b]p15.[/b] Find the sum of all positive integers $n$ such that $\frac{n^2 + 20n + 51}{n^2 + 4n + 3}$ is an integer.


[b]p16.[/b] Let $ABC$ be a right triangle with hypotenuse $AB$ such that $AC = 36$ and $BC = 15$. A semicircle is inscribed in $ABC$ as shown, such that the diameter $XC$ of the semicircle lies on side $AC$ and that the semicircle is tangent to $AB$. What is the radius of the semicircle?
[img]https://cdn.artofproblemsolving.com/attachments/4/2/714f7dfd09f6da1d61a8f910b5052e60dcd2fb.png[/img]

[b]p17.[/b] Let $a$ and $b$ be relatively prime positive integers such that the product $ab$ is equal to the least common multiple of $16500$ and $990$. If $\frac{16500}{a}$ and $\frac{990}{b}$ are both integers, what is the minimum value of $a + b$?


[b]p18.[/b] Let $x$ be a positive real number so that $x - \frac{1}{x} = 1$. Compute $x^8 - \frac{1}{x^8}$  .


[b]p19.[/b] Six people sit around a round table. Each person rolls a standard $6$-sided die. If no two people sitting next to each other rolled the same number, we will say that the roll is valid. How many dierent rolls are valid?


[b]p20.[/b] Given that $\frac{1}{31} = 0.\overline{a_1a_2a_3a_4a_5... a_n}$ (that is, $\frac{1}{31}$ can be written as the repeating decimal expansion $0.a_1a_2... a_na_1a_2... a_na_1a_2...$ ), what is the minimum value of $n$?


PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To solve the given problem, we need to find a positive real number \( x \) such that \((x \# 7) \# x = 82\), where the operation \(\#\) is defined as \(a \# b = ab - 2a - 2b + 6\).

1. First, we need to compute \( x \# 7 \):
   \[
   x \# 7 = x \cdot 7 - 2x - 2 \cdot 7 + 6 = 7x - 2x - 14 + 6 = 5x - 8
   \]

2. Next, we need to compute \((5x - 8) \# x\):
   \[
   (5x - 8) \# x = (5x - 8) \cdot x - 2(5x - 8) - 2x + 6
   \]
   Simplify the expression step-by-step:
   \[
   (5x - 8) \cdot x = 5x^2 - 8x
   \]
   \[
   -2(5x - 8) = -10x + 16
   \]
   \[
   -2x = -2x
   \]
   \[
   \text{Combining all terms: } 5x^2 - 8x - 10x + 16 - 2x + 6 = 5x^2 - 20x + 22
   \]

3. Set the expression equal to 82 and solve for \( x \):
   \[
   5x^2 - 20x + 22 = 82
   \]
   \[
   5x^2 - 20x + 22 - 82 = 0
   \]
   \[
   5x^2 - 20x - 60 = 0
   \]

4. Solve the quadratic equation \( 5x^2 - 20x - 60 = 0 \):
   \[
   x^2 - 4x - 12 = 0 \quad \text{(dividing the entire equation by 5)}
   \]
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   Here, \( a = 1 \), \( b = -4 \), and \( c = -12 \):
   \[
   x = \frac{4 \pm \sqrt{16 + 48}}{2} = \frac{4 \pm \sqrt{64}}{2} = \frac{4 \pm 8}{2}
   \]
   \[
   x = \frac{4 + 8}{2} = 6 \quad \text{or} \quad x = \frac{4 - 8}{2} = -2
   \]

5. Since we need a positive real number \( x \):
   \[
   x = 6
   \]

The final answer is \( \boxed{6} \).