[i]Round 5[/i]


[b]p13.[/b] Kelvin Amphibian, a not-frog who lives on the coordinate plane, likes jumping around. Each step, he jumps either to the spot that is $1$ unit to the right and 2 units up, or the spot that is $2$ units to the right and $1$ unit up, from his current location. He chooses randomly among these two choices with equal probability. He starts at the origin and jumps for a long time. What is the probability that he lands on $(10, 8)$ at some time in his journey?


[b]p14.[/b] Points $A, B, C$, and $D$ are randomly chosen on the circumference of a unit circle. What is the probability that line segments $AB$ and $CD$ intersect inside the circle?


[b]p15.[/b] Let $P(x)$ be a quadratic polynomial with two consecutive integer roots. If it is also known that $\frac{P(2017)}
{P(2016)} = \frac{2016}{2017}$ , find the larger root of $P(x)$.


[u]Round 6[/u]


[b]p16.[/b] Let $S_n$ be the sum of reciprocals of the integers between $1$ and $n$ inclusive. Find a triple $(a, b, c)$ of positive integers such that $S_{2017} \cdot  S_{2017} - S_{2016} \cdot S_{2018} = \frac{S_a+S_b}{c}$ .


[b]p17.[/b] Suppose that $m$ and $n$ are both positive integers. Alec has $m$ standard $6$-sided dice, each labelled $1$ to $6$ inclusive on the sides, while James has $n$ standard $12$-sided dice, each labelled $1$ to $12$ inclusive on the sides. They decide to play a game with their dice. They each toss all their dice simultaneously and then compute the sum of the numbers that come up on their dice. Whoever has a higher sum wins (if the sums are equal, they tie). Given that both players have an equal chance of winning, determine the minimum possible value of mn.


[b]p18.[/b]  Overlapping rectangles $ABCD$ and $BEDF$ are congruent to each other and both have area $1$. Given that $A,C,E, F$ are the vertices of a square, find the area of the square.


[u]Round 7[/u]


[b]p19.[/b] Find the number of solutions to the equation $$||| ... |||||x| + 1| - 2| + 3| - 4| +... - 98| + 99| - 100| = 0$$


[b]p20.[/b] A split of a positive integer in base $10$ is the separation of the integer into two nonnegative integers, allowing leading zeroes. For example, $2017$ can be split into $2$ and $017$ (or $17$), $20$ and $17$, or $201$ and $7$. A split is called squarish if both integers are nonzero perfect squares. $49$ and $169$ are the two smallest  perfect squares that have a squarish split ($4$ and $9$, $16$ and $9$ respectively). Determine all other perfect squares less than $2017$ with at least one squarish split.


[b]p21.[/b] Polynomial $f(x) = 2x^3 + 7x^2 - 3x + 5$ has zeroes $a, b$ and $c$. Cubic polynomial $g(x)$ with $x^3$-coefficient $1$ has zeroes $a^2$, $b^2$ and $c2$. Find the sum of coefficients of $g(x)$.


[u]Round 8[/u]


[b]p22.[/b] Two congruent circles, $\omega_1$ and $\omega_2$, intersect at points $A$ and $B$. The centers of $\omega_1$ and $\omega_2$ are $O_1$ and $O_2$ respectively. The arc $AB$ of $\omega_1$ that lies inside $\omega_2$ is trisected by points $P$ and $Q$, with the points lying in the order $A, P, Q,B$. Similarly, the arc $AB$ of $\omega_2$ that lies inside $\omega_1$ is trisected by points $R$ and $S$, with the points lying in the order $A,R, S,B$. Given that $PQ = 1$ and $PR =\sqrt2$, find the measure of $\angle AO_1B$ in degrees.


[b]p23.[/b] How many ordered triples of $(a, b, c)$ of integers between $-10$ and $10$ inclusive satisfy the equation $-abc = (a + b)(b + c)(c + a)$?


[b]p24.[/b] For positive integers $n$ and $b$ where $b > 1$, define $s_b(n)$ as the sum of digits in the base-$b$ representation of $n$. A positive integer $p$ is said to dominate another positive integer $q$ if for all positive integers $n$, $s_p(n)$ is greater than or equal to $s_q(n)$. Find the number of ordered pairs $(p, q)$ of distinct positive integers between $2$ and $100$ inclusive such that $p$ dominates $q$.


PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h2936487p26278546]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To find the sum of the coefficients of the polynomial \( g(x) \) whose roots are \( \alpha^2, \beta^2, \gamma^2 \), we start by using Vieta's formulas for the polynomial \( f(x) = 2x^3 + 7x^2 - 3x + 5 \).

1. **Identify the roots and their sums/products using Vieta's formulas:**
   Given \( f(x) = 2x^3 + 7x^2 - 3x + 5 \), let the roots be \( \alpha, \beta, \gamma \). By Vieta's formulas, we have:
   \[
   \alpha + \beta + \gamma = -\frac{7}{2}
   \]
   \[
   \alpha\beta + \beta\gamma + \gamma\alpha = -\frac{3}{2}
   \]
   \[
   \alpha\beta\gamma = -\frac{5}{2}
   \]

2. **Calculate the sums and products of the squares of the roots:**
   We need to find the polynomial \( g(x) \) whose roots are \( \alpha^2, \beta^2, \gamma^2 \). First, we calculate:
   \[
   \alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha)
   \]
   Substituting the values from Vieta's formulas:
   \[
   \alpha^2 + \beta^2 + \gamma^2 = \left(-\frac{7}{2}\right)^2 - 2\left(-\frac{3}{2}\right)
   \]
   \[
   \alpha^2 + \beta^2 + \gamma^2 = \frac{49}{4} + 3 = \frac{49}{4} + \frac{12}{4} = \frac{61}{4}
   \]

3. **Calculate the product of the squares of the roots:**
   \[
   \alpha^2 \beta^2 + \beta^2 \gamma^2 + \gamma^2 \alpha^2 = (\alpha\beta)^2 + (\beta\gamma)^2 + (\gamma\alpha)^2
   \]
   Using the identity for the sum of squares of products:
   \[
   (\alpha\beta + \beta\gamma + \gamma\alpha)^2 = \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 + 2\alpha\beta\gamma(\alpha + \beta + \gamma)
   \]
   Substituting the known values:
   \[
   \left(-\frac{3}{2}\right)^2 = \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 + 2\left(-\frac{5}{2}\right)\left(-\frac{7}{2}\right)
   \]
   \[
   \frac{9}{4} = \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 + \frac{35}{2}
   \]
   \[
   \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 = \frac{9}{4} - \frac{70}{4} = \frac{9 - 70}{4} = -\frac{61}{4}
   \]

4. **Calculate the product of all the squares of the roots:**
   \[
   \alpha^2 \beta^2 \gamma^2 = (\alpha \beta \gamma)^2 = \left(-\frac{5}{2}\right)^2 = \frac{25}{4}
   \]

5. **Form the polynomial \( g(x) \):**
   The polynomial \( g(x) \) with roots \( \alpha^2, \beta^2, \gamma^2 \) is:
   \[
   g(x) = x^3 - (\alpha^2 + \beta^2 + \gamma^2)x^2 + (\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2)x - \alpha^2\beta^2\gamma^2
   \]
   Substituting the calculated values:
   \[
   g(x) = x^3 - \frac{61}{4}x^2 - \frac{61}{4}x - \frac{25}{4}
   \]
   To find the sum of the coefficients, we evaluate \( g(1) \):
   \[
   g(1) = 1^3 - \frac{61}{4} \cdot 1^2 - \frac{61}{4} \cdot 1 - \frac{25}{4}
   \]
   \[
   g(1) = 1 - \frac{61}{4} - \frac{61}{4} - \frac{25}{4} = 1 - \frac{147}{4} = 1 - 36.75 = -35.75
   \]

The final answer is \(\boxed{-35.75}\).