MathDB

N

Part of 2022 Indonesia TST

Problems(4)

Polynomic Prime Product Generator

Source: Indonesian Stage 1 TST for IMO 2022, Test 2 (Number Theory), Canadian MO Qualification Repechage 2018 No 7

12/23/2021
Let nn be a natural number, with the prime factorisation n=p1e1p2e2⋯prer n = p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r} where p1,…,prp_1, \ldots, p_r are distinct primes, and eie_i is a natural number. Define rad(n)=p1p2⋯pr rad(n) = p_1p_2 \cdots p_r to be the product of all distinct prime factors of nn. Determine all polynomials P(x)P(x) with rational coefficients such that there exists infinitely many naturals nn satisfying P(n)=rad(n)P(n) = rad(n).
polynomialnumber theoryprimeProductinfinite
Relatively Prime Construction

Source: Indonesian Stage 1 TST for IMO 2022, Test 1 (Number Theory)

12/11/2021
Prove that there exists a set X⊆NX \subseteq \mathbb{N} which contains exactly 2022 elements such that for every distinct a,b,c∈Xa, b, c \in X the following equality: gcd⁡(an+bn,c)=1 \gcd(a^n+b^n, c) = 1 is satisfied for every positive integer nn.
number theoryrelatively prime
An Order Identity

Source: Indonesian Stage 1 TST for IMO 2022, Test 3 (Number Theory)

12/25/2021
Given positive odd integers mm and nn where the set of all prime factors of mm is the same as the set of all prime factors nn, and n∣mn \vert m. Let aa be an arbitrary integer which is relatively prime to mm and nn. Prove that: om(a)=on(a)×mgcd⁡(m,aon(a)−1) o_m(a) = o_n(a) \times \frac{m}{\gcd(m, a^{o_n(a)}-1)} where ok(a)o_k(a) denotes the smallest positive integer such that aok(a)≡1a^{o_k(a)} \equiv 1 (mod kk) holds for some natural number k>1k > 1.
Ordernumber theoryidentityDivisibility
GCD of the Solutions to a Quadratic Diophantine

Source: Indonesian Stage 1 TST for IMO 2022, Test 4 (Number Theory)

12/25/2021
For each natural number nn, let f(n)f(n) denote the number of ordered integer pairs (x,y)(x,y) satisfying the following equation: x2−xy+y2=n. x^2 - xy + y^2 = n. a) Determine f(2022)f(2022). b) Determine the largest natural number mm such that mm divides f(n)f(n) for every natural number nn.
quadraticsgreatest common divisornumber theoryIntegersdiophantine