[u]Round 1[/u]


[b]p1.[/b] A circle is inscribed inside a square such that the cube of the radius of the circle is numerically equal to the perimeter of the square. What is the area of the circle?


[b]p2.[/b] If the coefficient of $z^ky^k$ is $252$ in the expression $(z + y)^{2k}$, find $k$.


[b]p3.[/b] Let $f(x) = \frac{4x^4-2x^3-x^2-3x-2}{x^4-x^3+x^2-x-1}$ be a function defined on the real numbers where the denominator is not zero. The graph of $f$ has a horizontal asymptote. Compute the sum of the x-coordinates of the points where the graph of $f$ intersects this horizontal asymptote. If the graph of f does not intersect the asymptote, write $0$.


[u]Round 2 [/u]


[b]p4.[/b] How many $5$-digit numbers have strictly increasing digits? For example, $23789$ has strictly increasing digits, but $23889$ and $23869$ do not.

[b]p5.[/b] Let
$$y =\frac{1}{1 +\frac{1}{9 +\frac{1}{5 +\frac{1}{9 +\frac{1}{5 +...}}}}}$$ If $y$ can be represented as $\frac{a\sqrt{b} + c}{d}$ , where $b$ is not divisible by any squares, and the greatest common divisor of $a$ and $d$ is $1$, find the sum $a + b + c + d$.

[b]p6.[/b] “Counting” is defined as listing positive integers, each one greater than the previous, up to (and including) an integer $n$. In terms of $n$, write the number of ways to count to $n$.


[u]Round 3 [/u]


[b]p7.[/b] Suppose $p$, $q$, $2p^2 + q^2$, and $p^2 + q^2$ are all prime numbers. Find the sum of all possible values of $p$.


[b]p8.[/b] Let $r(d)$ be a function that reverses the digits of the $2$-digit integer $d$. What is the smallest $2$-digit positive integer $N$ such that for some $2$-digit positive integer $n$ and $2$-digit positive integer $r(n)$, $N$ is divisible by $n$ and $r(n)$, but not by $11$?


[b]p9.[/b] What is the period of the function $y = (\sin(3\theta) + 6)^2 - 10(sin(3\theta) + 7) + 13$?


[u]Round 4 [/u]


[b]p10.[/b] Three numbers $a, b, c$ are given by $a = 2^2 (\sum_{i=0}^2 2^i)$, $b = 2^4(\sum_{i=0}^4  2^i)$, and $c  = 2^6(\sum_{i=0}^6  2^i)$ . $u, v, w$ are the sum of the divisors of a, b, c respectively, yet excluding the original number itself. What is the value of $a + b + c -u - v - w$?


[b]p11.[/b] Compute $\sqrt{6- \sqrt{11}} - \sqrt{6+ \sqrt{11}}$.


[b]p12.[/b] Let $a_0, a_1,..., a_n$ be such that $a_n\ne 0$ and $$(1 + x + x^3)^{341}(1 + 2x + x^2 + 2x^3 + 2x^4 + x^6)^{342} =\sum_{i=0}^n a_ix^i.$$ Find the number of odd numbers in the sequence $a_0, a_1,..., a_n$.



PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2781343p24424617]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To find the number of 5-digit numbers with strictly increasing digits, we need to choose 5 distinct digits from the set \(\{1, 2, 3, 4, 5, 6, 7, 8, 9\}\). Note that 0 cannot be included because it cannot be a leading digit in a 5-digit number.

1. **Choose 5 digits from 9 possible digits:**
   We need to select 5 digits out of the 9 available digits. The number of ways to choose 5 digits from 9 is given by the binomial coefficient \(\binom{9}{5}\).

2. **Calculate the binomial coefficient:**
   \[
   \binom{9}{5} = \frac{9!}{5!(9-5)!} = \frac{9!}{5!4!}
   \]
   Simplifying the factorials:
   \[
   \binom{9}{5} = \frac{9 \times 8 \times 7 \times 6 \times 5!}{5! \times 4 \times 3 \times 2 \times 1} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = \frac{3024}{24} = 126
   \]

3. **Conclusion:**
   Since the digits are chosen from 1 to 9 and must be in strictly increasing order, each selection of 5 digits will automatically form a valid 5-digit number with strictly increasing digits.

\[
\boxed{126}
\]