MathDB

4

Part of 2018 Middle European Mathematical Olympiad

Problems(2)

A representation

Source: MEMO 2018 T4

9/2/2018
Let nn be a positive integer and u1,u2,⋯ ,unu_1,u_2,\cdots ,u_n be positive integers not larger than 2k,2^k, for some integer k≥3.k\geq 3. A representation of a non-negative integer tt is a sequence of non-negative integers a1,a2,⋯ ,ana_1,a_2,\cdots ,a_n such that t=a1u1+a2u2+⋯+anun.t=a_1u_1+a_2u_2+\cdots +a_nu_n. Prove that if a non-negative integer tt has a representation,then it also has a representation where less than 2k2k of numbers a1,a2,⋯ ,ana_1,a_2,\cdots ,a_n are non-zero.
number theoryalgebra
p(2018)= p(2019)

Source: MEMO 2018 I4

9/7/2018
(a) Prove that for every positive integer mm there exists an integer n≥mn\ge m such that ⌊n1⌋⋅⌊n2⌋⋯⌊nm⌋=(nm) (∗)\left \lfloor \frac{n}{1} \right \rfloor \cdot \left \lfloor \frac{n}{2} \right \rfloor \cdots \left \lfloor \frac{n}{m} \right \rfloor =\binom{n}{m} \\\\\\\\\\\\\\\ (*) (b) Denote by p(m)p(m) the smallest integer n≥mn \geq m such that the equation (∗) (*) holds. Prove that p(2018)=p(2019).p(2018) = p(2019). Remark: For a real number x,x, we denote by ⌊x⌋\left \lfloor x \right \rfloor the largest integer not larger than x.x.
number theoryalgebrafloor function