MathDB

3

Part of 2023 Germany Team Selection Test

Problems(2)

On elements of a finite set with sum 0

Source: German TST 2023 AIMO 3, Problem 3

11/2/2023
Let AA be a non-empty set of integers with the following property: For each a∈Aa \in A, there exist not necessarily distinct integers b,c∈Ab,c \in A so that a=b+ca=b+c.
(a) Proof that there are examples of sets AA fulfilling above property that do not contain 00 as element.
(b) Proof that there exist a1,…,ar∈Aa_1,\ldots,a_r \in A with r≥1r \ge 1 and a1+⋯+ar=0a_1+\cdots+a_r=0.
(c) Proof that there exist pairwise distinct a1,…,ara_1,\ldots,a_r with r≥1r \ge 1 and a1+⋯+ar=0a_1+\cdots+a_r=0.
SetsSumelements of setcombinatorics
Control prime powers dividing product of polynomial values

Source: German TST 2023, Test 4, Problem 3

7/15/2023
Let f(x)f(x) be a monic polynomial of degree 20232023 with positive integer coefficients. Show that for any sufficiently large integer NN and any prime number p>2023Np>2023N, the product f(1)f(2)…f(N)f(1)f(2)\dots f(N) is at most (20232)\binom{2023}{2} times divisible by pp. Proposed by Ashwin Sah
algebrapolynomialnumber theoryprime numbersalgebra proposed