Maths Olympiad Daily Problems

try ioqm 3 or 5 marker

#PUMaC #Combinatorics

bhai nmtc inter ka paper daal do koi

NMTC ka Paper Upload krdo Guysss

BEST OF LUCK EVERYONE FOR THEIR NMTC EXAM ~ GO AND ROCK YOUR PAPER (Mera toh Adv ka tst h🤡)

good 2-3 marker

Qn 96 is an upgraded daddy version of the 2025 F(n) = remainder left when n^n is divided by 7 qn

Basically a^p(p-1) congruent 1 mod p² So by concept of order where d is the order then d divides p=> d=p or d | p-1 => a^d congrunt 1 mod p² => (a^d)^k congruent 1 mod p² => a^(dk) congrunt 1 mod p² => a^(p-1) congruent 1 mod p² => a^p congrunt a mod p² meaning it's 1/a mod p² If d>p then d=p(some prime factor of p-1). = pm for some m If m = Odd: a^p congruent 1 mod p If m = Even: a^p congrunt ±1 mod p Ideally we would need congruent -1 mod p to get the least p Goal: Parameterise p s.t. we can produce maximum -1 while satisfying the conditions in the qn This was my approach You can have a fresh approach or try to parameterise something from here to get the n

Attempt 💀💀

For any positive integer n, let τ (n) denote the number of positive divisors of n. If n is a positive integer such that τ(n^2)/τ(n)= 3, compute τ(n^7)/τ(n).

87 is a good qn asks your basics pretty well

Ari repeatedly rolls a standard, fair, six-sided die. Let R(n) be the n th number rolled, and let Q(n) = R(1)· R(2)· . . . · R(n). Let the probability that there exists an n such that Q(n) = 100 and for all m < n, Q(m) is not a perfect square be p/q where p, q are relatively prime positive integers. Find the largest prime factor of p + q.