MathDB

1

Part of 2019 Romania Team Selection Test

Problems(5)

product - sum is a perfect sqaure

Source: Vietnam TST 2002 for the 43th IMO, problem 6

6/26/2005
Prove that there exists an integer nn, n≥2002n\geq 2002, and nn distinct positive integers a1,a2,…,ana_1,a_2,\ldots,a_n such that the number N=a12a22⋯an2−4(a12+a22+⋯+an2)N= a_1^2a_2^2\cdots a_n^2 - 4(a_1^2+a_2^2+\cdots + a_n^2) is a perfect square.
number theory unsolvednumber theory
number theory

Source:

2/2/2017
Let k≥2k\geq 2,n1,n2,⋯ ,nk∈N+n_1,n_2,\cdots ,n_k\in \mathbb{N}_+,satisfied n2∣2n1−1,n3∣2n2−1,⋯ ,nk∣2nk−1−1,n1∣2nk−1n_2|2^{n_1}-1,n_3|2^{n_2}-1,\cdots ,n_k|2^{n_{k-1}}-1,n_1|2^{n_k}-1. Prove:n1=n2=⋯=nk=1n_ 1=n_ 2=\cdots=n_k=1.
number theoryalgebra
Minimum of sum of n variables under constraint

Source: Romanian TST for 2019 IMO

10/1/2019
Let be a natural number n≥3. n\ge 3. Find inf⁡1=P(x1,x2,…,xn)x1,x2,…,xn∈R>0∑i=1n(1xi−xi), \inf_{\stackrel{ x_1,x_2,\ldots ,x_n\in\mathbb{R}_{>0}}{1=P\left( x_1,x_2,\ldots ,x_n\right)}}\sum_{i=1}^n\left( \frac{1}{x_i} -x_i \right) , where P(x1,x2,…,xn):=∑i=1n1xi+n−1, P\left( x_1,x_2,\ldots ,x_n\right) :=\sum_{i=1}^n \frac{1}{x_i+n-1} , and find in which circumstances this infimum is attained.
inequalitiesminimumalgebra
Nice geometry proposed by Pavel Kozhevnikov

Source:

10/1/2019
Let I,O I,O denote the incenter, respectively, the circumcenter of a triangle ABC. ABC. The A-excircle A\text{-excircle} touches the lines AB,AC,BC AB,AC,BC at K,L, K,L, respectively, M. M. The midpoint of KL KL lies on the circumcircle of ABC. ABC. Show that the points I,M,O I,M,O are collinear.
Павел Кожевников
geometryincentercircumcircle
Another extrema of sum (now happy, rmtf1111?)

Source: Romanian TST for 2019 IMO

10/1/2019
Determine the largest value the expression ∑1≤i<j≤4(xi+xj)xixj \sum_{1\le i<j\le 4} \left( x_i+x_j \right)\sqrt{x_ix_j} may achieve, as x1,x2,x3,x4 x_1,x_2,x_3,x_4 run through the non-negative real numbers, and add up to 1. 1. Find also the specific values of this numbers that make the above sum achieve the asked maximum.
inequalitiesalgebra