[u]Round 9[/u]


[b]p25.[/b] Let $S$ be the set of the first $2017$ positive integers. Find the number of elements $n \in S$ such that $\sum^n_{i=1} \left\lfloor \frac{n}{i} \right\rfloor$ is even.


[b]p26.[/b] Let $\{x_n\}_{n \ge 0}$ be a sequence with $x_0 = 0$,$x_1 = \frac{1}{20}$ ,$x_2 = \frac{1}{17}$ ,$x_3 = \frac{1}{10}$ , and $x_n = \frac12 ((x_{n-2} +x_{n-4})$ for $n\ge  4$. Compute $$ \left\lfloor \frac{1}{x_{2017!} -x_{2017!-1}} \right\rfloor.$$


[b]p27.[/b] Let $ABCDE$ be be a cyclic pentagon. Given that $\angle CEB = 17^o$, find $\angle CDE  + \angle EAB$, in degrees.


[u]Round 10[/u]


[b]p28.[/b] Let $S = \{1,2,4, ... ,2^{2016},2^{2017}\}$. For each $0 \le  i \le 2017$, let $x_i$ be chosen uniformly at random from the subset of $S$ consisting of the divisors of $2^i$ . What is the expected number of distinct values in the set $\{x_0,x_1,x_2,... ,x_{2016},x_{2017}\}$?


[b]p29.[/b] For positive real numbers $a$ and $b$, the points $(a, 0)$, $(20,17)$ and $(0,b)$ are collinear. Find the minimum possible value of $a+b$.


[b]p30.[/b] Find the sum of the distinct prime factors of $2^{36}-1$.

 
[u]Round 11[/u]


[b]p31.[/b] There exist two angle bisectors of the lines $y = 20x$ and $y = 17x$ with slopes $m_1$ and $m_2$. Find the unordered pair $(m_1,m_2)$.


[b]p32.[/b] Triangle 4ABC has sidelengths $AB = 13$, $BC = 14$, $C A =15$ and orthocenter $H$. Let $\Omega_1$ be the circle through $B$ and $H$, tangent to $BC$, and let $\Omega_2$ be the circle through $C$ and $H$, tangent to $BC$. Finally, let $R \ne H$ denote the second intersection of $\Omega_1$ and $\Omega_2$. Find the length $AR$.


[b]p33.[/b] For a positive integer $n$, let $S_n = \{1,2,3, ...,n\}$ be the set of positive integers less than or equal to $n$. Additionally, let $$f (n) = |\{x \in S_n : x^{2017}\equiv x \,\, (mod \,\, n)\}|.$$ Find $f (2016)- f (2015)+ f (2014)- f (2013)$.



[u]Round 12[/u]


[b]p34. [/b] Estimate the value of $\sum^{2017}_{n=1} \phi (n)$, where $\phi (n)$ is the number of numbers less than or equal $n$ that are relatively prime to n. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be max $\max \left(0,\lfloor  15 - 75 \frac{|A-E|}{A} \rceil \right).$


[b]p35.[/b] An up-down permutation of order $n$ is a permutation $\sigma$ of $(1,2,3, ..., n)$ such that $\sigma(i ) <\sigma (i +1)$ if and only if $i$ is odd. Denote by $P_n$ the number of up-down permutations of order $n$. Estimate the value of $P_{20} +P_{17}$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, 16 -\lceil  \max \left(\frac{E}{A}, 2- \frac{E}{A}\right) \rceil \right).$


[b]p36.[/b] For positive integers $n$, superfactorial of $n$, denoted $n\$ $, is defined as the product of the first $n$ factorials. In other words, we have $n\$ = \prod^n_{i=1}(i !)$. Estimate the number of digits in the product $(20\$)\cdot (17\$)$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be  $\max \left(0, \lfloor 15 -\frac12 |A-E| \rfloor \right).$


PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158491p28715220]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3158514p28715373]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To solve the problem, we need to find the sum of the angles $\angle CDE$ and $\angle EAB$ in a cyclic pentagon $ABCDE$ given that $\angle CEB = 17^\circ$.

1. **Identify the key property of cyclic polygons:**
   In a cyclic polygon, the sum of the opposite angles is $180^\circ$. This property will help us determine the relationships between the angles in the pentagon.

2. **Label the angles:**
   Let $\angle EAB = x$, $\angle ABC = y$, $\angle BCD = z$, $\angle CDE = w$, and $\angle DEA = v$. We are given that $\angle CEB = 17^\circ$.

3. **Use the property of cyclic polygons:**
   Since $ABCDE$ is a cyclic pentagon, the sum of the interior angles is $(5-2) \times 180^\circ = 540^\circ$.

4. **Express the sum of the angles:**
   The sum of the interior angles of the pentagon is:
   \[
   x + y + z + w + v = 540^\circ
   \]

5. **Use the given angle $\angle CEB = 17^\circ$:**
   Since $\angle CEB = 17^\circ$, and angles subtended by the same arc in a cyclic polygon are equal, we can use this information to find the relationships between the other angles.

6. **Determine the sum of the angles $\angle CDE$ and $\angle EAB$:**
   We need to find $\angle CDE + \angle EAB$. Using the property of cyclic polygons, we know that:
   \[
   \angle CDE + \angle EAB = 180^\circ - \angle CEB
   \]
   Substituting the given value:
   \[
   \angle CDE + \angle EAB = 180^\circ - 17^\circ = 163^\circ
   \]

The final answer is $\boxed{163^\circ}$.