Kana and Dragon Quest game
Contest: Codeforces Round 635 (Div. 2)
Difficulty rating: 900
Tags: greedy, implementation, math

Kana was just an ordinary high school girl before a talent scout discovered her. Then, she became an idol. But different from the stereotype, she is also a gameholic.

One day Kana gets interested in a new adventure game called Dragon Quest. In this game, her quest is to beat a dragon.

The dragon has a hit point of $$$x$$$ initially. When its hit point goes to $$$0$$$ or under $$$0$$$, it will be defeated. In order to defeat the dragon, Kana can cast the two following types of spells.

- Void Absorption Assume that the dragon's current hit point is $$$h$$$, after casting this spell its hit point will become $$$\left\lfloor \frac{h}{2} \right\rfloor + 10$$$. Here $$$\left\lfloor \frac{h}{2} \right\rfloor$$$ denotes $$$h$$$ divided by two, rounded down.
- Lightning Strike This spell will decrease the dragon's hit point by $$$10$$$. Assume that the dragon's current hit point is $$$h$$$, after casting this spell its hit point will be lowered to $$$h-10$$$.

Due to some reasons Kana can only cast no more than $$$n$$$ Void Absorptions and $$$m$$$ Lightning Strikes. She can cast the spells in any order and doesn't have to cast all the spells. Kana isn't good at math, so you are going to help her to find out whether it is possible to defeat the dragon.

Input format:
The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$)  — the number of test cases.

The next $$$t$$$ lines describe test cases. For each test case the only line contains three integers $$$x$$$, $$$n$$$, $$$m$$$ ($$$1\le x \le 10^5$$$, $$$0\le n,m\le30$$$)  — the dragon's intitial hit point, the maximum number of Void Absorptions and Lightning Strikes Kana can cast respectively.

Output format:
If it is possible to defeat the dragon, print "YES" (without quotes). Otherwise, print "NO" (without quotes).

You can print each letter in any case (upper or lower).

Constraints:
Time limit: 1.0 s
Memory limit: 256.0 MB

Note:
One possible casting sequence of the first test case is shown below:

- Void Absorption $$$\left\lfloor \frac{100}{2} \right\rfloor + 10=60$$$.
- Lightning Strike $$$60-10=50$$$.
- Void Absorption $$$\left\lfloor \frac{50}{2} \right\rfloor + 10=35$$$.
- Void Absorption $$$\left\lfloor \frac{35}{2} \right\rfloor + 10=27$$$.
- Lightning Strike $$$27-10=17$$$.
- Lightning Strike $$$17-10=7$$$.
- Lightning Strike $$$7-10=-3$$$.

Example input:
7
100 3 4
189 3 4
64 2 3
63 2 3
30 27 7
10 9 1
69117 21 2

Example output:
YES
NO
NO
YES
YES
YES
YES

Editorial:
First, it's better not to cast Void Absorptions after a Lightning Strike.
Otherwise, there will be a Void Absorption right after a Lightning Strike. Supposing the hit point was $$$x$$$ before casting these two spells, if you cast Lightning Strike first, after these two spells the hit point will be $$$\left\lfloor \frac{x-10}{2} \right\rfloor +10 = \left\lfloor \frac{x}{2} \right\rfloor + 5$$$, but if you cast Void Absorption first, after these two spells the hit point will be $$$\left\lfloor \frac{x}{2} \right\rfloor$$$, which is smaller.
So the solution is to cast Void Absorptions until the dragon's hit point won't decrease with more casts or you can't cast more Void Absorptions, then cast all Lightning Strikes you have.
We can also use DP to solve this problem. Let $$$f_{i,j,k} = 0/1$$$ be possibility of defeating the dragon when the dragon's hit point is at $$$i$$$, you can cast $$$j$$$ more Void Absorptions and $$$k$$$ more Lightning Strikes. This is a slow solution but is enough to get accepted. You may have to use boolean arrays instead of integer arrays to save memory.