MathDB

R9

Part of 2023 Stanford Mathematics Tournament

Problems(1)

2023 SMT Guts Round 9 p25-27 - Stanford Math Tournament

Source:

8/31/2023
p25. You are given that 1000!1000! has 25682568 decimal digits. Call a permutation π\pi of length 10001000 good if π(2i)>π(2i−1)\pi(2i) > \pi (2i - 1) for all 1≤i≤5001 \le i \le 500 and π(2i)>π(2i+1)\pi (2i) > \pi (2i + 1) for all 1≤i≤4991 \le i \le 499. Let NN be the number of good permutations. Estimate DD, the number of decimal digits in NN. You will get max⁡(0,25−⌈∣D−X∣10⌉)\max \left( 0, 25 - \left\lceil \frac{|D-X|}{10} \right\rceil \right) points, where XX is the true answer.
p26. A year is said to be interesting if it is the product of 33, not necessarily distinct, primes (for example 22⋅52^2 \cdot 5 is interesting, but 22⋅3⋅52^2 \cdot 3 \cdot 5 is not). How many interesting years are there between 5000 5000 and 1000010000, inclusive? For an estimate of EE, you will get max⁡(0,25−⌈∣E−X∣10⌉)\max \left( 0, 25 - \left\lceil \frac{|E-X|}{10} \right\rceil \right) points, where XX is the true answer.
p27. Sam chooses 10001000 random lattice points (x,y)(x, y) with 1≤x,y≤10001 \le x, y \le 1000 such that all pairs (x,y)(x, y) are distinct. Let NN be the expected size of the maximum collinear set among them. Estimate ⌊100N⌋\lfloor 100N \rfloor. Let SS be the answer you provide and XX be the true value of ⌊100N⌋\lfloor 100N \rfloor. You will get max⁡(0,25−⌈∣S−X∣10⌉)\max \left( 0, 25 - \left\lceil \frac{|S-X|}{10} \right\rceil \right) points for your estimate.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
Stanford Math Tournamentnumber theorycombinatorics