[u]Round 5[/u]


[b]p13.[/b] Initially, the three numbers $20$, $201$, and $2016$ are written on a blackboard. Each minute, Zhuo selects two of the numbers on the board and adds $1$ to each. Find the minimum $n$ for which Zhuo can make all three numbers equal to $n$.


[b]p14.[/b] Call a three-letter string rearrangeable if, when the first letter is moved to the end, the resulting string comes later alphabetically than the original string. For example, $AAA$ and $BAA$ are not rearrangeable, while $ABB$ is rearrangeable. How many three-letters strings with (not necessarily distinct) uppercase letters are rearrangeable?


[b]p15.[/b] Triangle $ABC$ is an isosceles right triangle with $\angle C = 90^o$ and $AC = 1$. Points $D$, $E$ and $F$ are chosen on sides $BC$,$CA$ and $AB$, respectively, such that $AEF$, $BFD$, $CDE$, and $DEF$ are isosceles right triangles. Find the sum of all distinct possible lengths of segment $DE$.


[u]Round 6[/u]


[b]p16.[/b] Let $p, q$, and $r$ be prime numbers such that $pqr = 17(p + q + r)$. Find the value of the product $pqr$.


[b]p17.[/b] A cylindrical cup containing some water is tilted $45$ degrees from the vertical. The point on the surface of the water closest to the bottom of the cup is $6$ units away. The point on the surface of the water farthest from the bottom of the cup is $10$ units away. Compute the volume of the water in the cup.


[b]p18.[/b] Each dot in an equilateral triangular grid with $63$ rows and $2016 = \frac12 \cdot 63 \cdot 64$ dots is colored black or white. Every unit equilateral triangle with three dots has the property that exactly one of its vertices is colored black. Find all possible values of the number of black dots in the grid.


[u]Round 7[/u]


[b]p19.[/b] Tomasz starts with the number $2$. Each minute, he either adds $2$ to his number, subtracts $2$ from his number, multiplies his number by $2$, or divides his number by $2$. Find the minimum number of minutes he will need in order to make his number equal $2016$.


[b]p20.[/b] The edges of a regular octahedron $ABCDEF$ are painted with $3$ distinct colors such that no two edges with the same color lie on the same face. In how many ways can the octahedron be painted? Colorings are considered different under rotation or reflection.


[b]p21.[/b] Jacob is trapped inside an equilateral triangle $ABC$ and must visit each edge of triangle $ABC$ at least once. (Visiting an edge means reaching a point on the edge.) His distances to sides $AB$, $BC$, and $CA$ are currently $3$, $4$, and $5$, respectively. If he does not need to return to his starting point, compute the least possible distance that Jacob must travel.


[u]Round 8[/u]


[b]p22.[/b] Four integers $a, b, c$, and $d$ with a $\le b \le c \le d$ satisfy the property that the product of any two of them is equal to the sum of the other two. Given that the four numbers are not all equal, determine the $4$-tuple $(a, b, c, d)$.


[b]p23.[/b] In equilateral triangle $ABC$, points $D$,$E$, and $F$ lie on sides $BC$,$CA$ and $AB$, respectively, such that $BD = 4$ and $CD = 5$. If $DEF$ is an isosceles right triangle with right angle at $D$, compute $EA + FA$.


[b]p24.[/b] On each edge of a regular tetrahedron, four points that separate the edge into five equal segments are marked. There are sixteen planes that are parallel to a face of the tetrahedron and pass through exactly three of the marked points. When the tetrahedron is cut along each of these sixteen planes, how many new tetrahedrons are produced?


PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2934049p26256220]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To solve the problem, we need to determine the minimum value \( n \) such that the three numbers \( 20 \), \( 201 \), and \( 2016 \) can all be made equal by repeatedly selecting two of the numbers and adding 1 to each.

1. **Initial Setup**:
   - The numbers on the board are \( 20 \), \( 201 \), and \( 2016 \).

2. **Understanding the Operation**:
   - Each operation increases two of the numbers by 1. This means the sum of the three numbers increases by 2 with each operation.

3. **Calculate the Total Sum**:
   - Initially, the sum of the numbers is \( 20 + 201 + 2016 = 2237 \).

4. **Determine the Target Sum**:
   - If all three numbers are to be equal to \( n \), then the sum of the three numbers must be \( 3n \).

5. **Set Up the Equation**:
   - We need \( 3n \) to be the sum after some number of operations. Since each operation increases the sum by 2, we can write:
     \[
     3n = 2237 + 2k
     \]
     where \( k \) is the number of operations performed.

6. **Solve for \( n \)**:
   - Rearrange the equation to solve for \( n \):
     \[
     n = \frac{2237 + 2k}{3}
     \]
   - For \( n \) to be an integer, \( 2237 + 2k \) must be divisible by 3.

7. **Check Divisibility**:
   - Calculate \( 2237 \mod 3 \):
     \[
     2237 \div 3 = 745 \text{ remainder } 2 \quad \Rightarrow \quad 2237 \equiv 2 \pmod{3}
     \]
   - For \( 2237 + 2k \) to be divisible by 3, \( 2k \) must be congruent to 1 modulo 3:
     \[
     2k \equiv 1 \pmod{3}
     \]
   - The smallest \( k \) that satisfies this is \( k = 2 \):
     \[
     2 \times 2 = 4 \quad \Rightarrow \quad 4 \equiv 1 \pmod{3}
     \]

8. **Calculate \( n \)**:
   - Substitute \( k = 2 \) into the equation for \( n \):
     \[
     n = \frac{2237 + 2 \times 2}{3} = \frac{2237 + 4}{3} = \frac{2241}{3} = 747
     \]

Thus, the minimum \( n \) for which Zhuo can make all three numbers equal is \( 747 \).

The final answer is \( \boxed{747} \).