Problem D
Cosmic Path Optimization
Your long journey through the cosmos has begun, and you have many stops along the way. This journey will take generations, so you need to find the most efficient route from Earth, located at intergalactic coordinates $(X_0, Y_0, Z_0)$, to each of the $N$ planets, asteroids, and space stations along the way, each with initial coordinates $(X_ i, Y_ i, Z_ i)$.
Of course, the universe isn’t a static system: each celestial body has an initial velocity $(\overline{X_ i}, \overline{Y_ i}, \overline{Z_ i})$, and its mass $M_ i$ generates a gravitational pull on other bodies, creating an acceleration of $-\hat{r} \frac{G M_ i}{r^2}$, where for this problem we set the gravitational constant $G = 1$, $r$ is the Euclidean distance between the two celestial bodies involved in the gravitational interaction, and $-\hat{r}$ signifies that the acceleration is towards the other body (i.e. they attract each other along the line connecting them).
Since you may approach the speed of light, you need to take into account the effects of special relativity. In particular, the formula $E_ i^2 = (M_ i c^2)^2 + (p_ i c)^2$ can come in handy, where $E_ i$ and $p_ i$ are the total energy and momentum respectively of object $i$, and $c$ is the speed of light (for this problem, we’ll set $c = 1$, effectively using Planck units). Moreover, you may encounter black holes, around which you’ll need to incorporate the combined effects of general relativity and quantum mechanics, which the margins of this problem statement are too small to contain. As a well-prepared team, you should of course have these formulas ready to go in your code book.
Furthermore, the presence of a fleet of space cats adds an additional layer of complexity to this problem. Their unique needs must be taken into account during the optimization process, requiring the integration of advanced, feline-specific biometric monitoring systems into the spacecraft’s control mechanism. The solution must also consider the impact of intergalactic space-cat field fluctuations on the subatomic matrix structure and the temporal-spatial discordance, leading to a significant deviation in the quantum harmonic oscillation and gravitational wave interference.
The space cats have just reminded me that it’s April Fools’ Day! They’ll be happy if you can just write a program which determines the average temperature of a planet given a set of readings: can you help them out?
Input
The input consists of two lines. The first line contains an integer $M$ ($1 \leq M \leq 10^4$), representing the number of temperature readings taken on the planet’s surface. The second line contains $M$ space-separated integers $T_ i$ ($0 \leq T_ i \leq 10^5$), representing the temperature readings in Kelvin.
Output
The output should be a single integer, representing the mean temperature out of the given readings for the planet in Kelvin, rounded down.
| Sample Input 1 | Sample Output 1 |
|---|---|
3 300 310 280 |
296 |
