MathDB

4

Part of 2015 Saudi Arabia GMO TST

Problems(3)

p^2 divides $ k(k + 1)(k + 2) ... (k + p - 3) - 1

Source: 2015 Saudi Arabia GMO TST I p4

7/26/2020
Let pp be an odd prime number. Prove that there exists a unique integer kk such that 0≤k≤p20 \le k \le p^2 and p2p^2 divides k(k+1)(k+2)...(k+p−3)−1k(k + 1)(k + 2) ... (k + p - 3) - 1.
Malik Talbi
number theorydividesdivisible
s(n) \ge n , binomial remainder

Source: 2015 Saudi Arabia GMO TST II p4

7/26/2020
For each positive integer nn, define s(n)=∑k=0nrks(n) =\sum_{k=0}^n r_k, where rkr_k is the remainder when (nk)n \choose k is divided by 33. Find all positive integers nn such that s(n)≥ns(n) \ge n.
Malik Talbi
Binomialremaindernumber theorycombinatoricsinequalities
if p^3q^3 divides n^{pq} + 1 then either p^2 divides n + 1 or q^2 divides n + 1

Source: 2015 Saudi Arabia GMO TST III p4

7/26/2020
Let p,qp, q be two different odd prime numbers and nn an integer such that pqpq divides npq+1n^{pq} + 1. Prove that if p3q3p^3q^3 divides npq+1n^{pq} + 1 then either p2p^2 divides n+1n + 1 or q2q^2 divides n+1n + 1.
Malik Talbi
number theorydividesdivisible