095. Techno Pop

Codeforces

IDCF10269095

Time1000ms

Memory256MB

Difficulty—

English · Original

Formal · Original

Techno Pop After listening to the Kraftwerk song Techno Pop, you decide that you want to pop a balloon. You have a spherical balloon that will pop when it reaches a volume of $v 1$ in^3, and you blow up the balloon inside of a cube-shaped box with a volume of $v 2$ in^3. Given these values, figure out whether or not it is possible to pop the balloon in the cube-shaped box that you have. Recall that the formula for the volume of a sphere is $frac(4, 3) pi r^3$, where $r$ represents the radius of the sphere, and the formula for the volume of a cube (or a cube-shaped box) is $s^3$, where $s$ represents the side length of the cube. The only line of input contains two space-separated integers: $v 1$ and $v 2$, representing the volume that the spherical balloon needs to be greater than or equal to in order to pop, and the volume of the cube-shaped box, respectively. Both values are given in $i n^3$. Output "YES" (no quotes) if the balloon can pop inside of the box, and "NO" (no quotes) otherwise. In the first example case, the balloon has room to inflate to at most 14.1 $i n^3$, so the balloon can inflate to 10 $i n^3$ and pop. In the second example case, the balloon only has room to inflate to 4.2 $i n^3$, so the balloon cannot pop inside of the cube-shaped box. ## Input The only line of input contains two space-separated integers: $v 1$ and $v 2$, representing the volume that the spherical balloon needs to be greater than or equal to in order to pop, and the volume of the cube-shaped box, respectively. Both values are given in $i n^3$. ## Output Output "YES" (no quotes) if the balloon can pop inside of the box, and "NO" (no quotes) otherwise. [samples] ## Note In the first example case, the balloon has room to inflate to at most 14.1 $i n^3$, so the balloon can inflate to 10 $i n^3$ and pop.In the second example case, the balloon only has room to inflate to 4.2 $i n^3$, so the balloon cannot pop inside of the cube-shaped box.

**Definitions** Let $ v_1 \in \mathbb{R}^+ $ be the minimum volume required for the balloon to pop. Let $ v_2 \in \mathbb{R}^+ $ be the volume of the cube-shaped box. **Constraints** The balloon is spherical and must fit entirely within the cube. The maximum radius $ r_{\text{max}} $ of the balloon is constrained by the cube’s side length $ s = \sqrt[3]{v_2} $: $$ r_{\text{max}} = \frac{s}{2} = \frac{\sqrt[3]{v_2}}{2} $$ **Objective** Compute the maximum possible volume $ V_{\text{max}} $ of the balloon inside the cube: $$ V_{\text{max}} = \frac{4}{3} \pi r_{\text{max}}^3 = \frac{4}{3} \pi \left( \frac{\sqrt[3]{v_2}}{2} \right)^3 = \frac{\pi}{6} v_2 $$ Determine whether: $$ V_{\text{max}} \geq v_1 $$ Output "YES" if $ \frac{\pi}{6} v_2 \geq v_1 $, otherwise "NO".

API Response (JSON)

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    "name": "095. Techno Pop",
    "description": {
      "content": "Techno Pop After listening to the Kraftwerk song Techno Pop, you decide that you want to pop a balloon. You have a spherical balloon that will pop when it reaches a volume of $v 1$ in^3, and you blow",
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    "platform": "Codeforces",
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