MathDB

Problem 2

Part of 2023 Kyiv City MO Round 1

Problems(5)

Equal signs needed easy

Source: Kyiv City MO 2023 Round 1, Problem 7.2

12/16/2023
You are given n≥3n \geq 3 distinct real numbers. Prove that one can choose either 33 numbers with positive sum, or 22 numbers with negative sum.
Proposed by Mykhailo Shtandenko
algebra
Easy algebra for 8-graders

Source: Kyiv City MO 2023 8.2

5/14/2023
Positive integers kk and nn are given such that 3≤k≤n3 \le k \le n.Prove that among any nn pairwise distinct real numbers one can choose either kk numbers with positive sum, or k−1k-1 numbers with negative sum. Proposed by Mykhailo Shtandenko
algebra
Equal signs needed

Source: Kyiv City MO 2023 9.2

5/14/2023
Non-zero real numbers a,ba, b and cc are given such that ab+bc+ac=0ab+bc+ac=0. Prove that numbers a+b+ca+b+c and 1a+b+1b+c+1a+c\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{a+c} are either both positive or both negative. Proposed by Mykhailo Shtandenko
algebra
System error

Source: Kyiv City MO 2023 Round 1, Problem 10.2

12/16/2023
For any given real a,b,ca, b, c solve the following system of equations: {ax3+by=cz5,az3+bx=cy5,ay3+bz=cx5.\left\{\begin{array}{l}ax^3+by=cz^5,\\az^3+bx=cy^5,\\ay^3+bz=cx^5.\end{array}\right. Proposed by Oleksiy Masalitin, Bogdan Rublov
algebrasystem of equations
Pairs with close sums

Source: Kyiv City MO 2023 Round 1, Problem 11.2

12/16/2023
You are given n≥4n\geq 4 positive real numbers. Consider all n(n−1)2\frac{n(n-1)}{2} pairwise sums of these numbers. Show that some two of these sums differ in at most 2n−2\sqrt[n-2]{2} times.
Proposed by Anton Trygub
algebra