MathDB

Problem 1

Part of 2024 Kyiv City MO Round 2

Problems(4)

At least one nonnegative

Source: Kyiv City MO 2024 Round 2, Problem 7.1

2/4/2024
Prove that for any real numbers x,y,zx, y, z at least one of numbers x2+y+14,y2+z+14,z2+x+14x^2 + y + \frac{1}{4}, y^2 + z + \frac{1}{4}, z^2 + x + \frac{1}{4} is nonnegative.
Proposed by Oleksii Masalitin
algebrainequalities
System of equations: easy

Source: Kyiv City MO 2024 Round 2, Problem 9.1

2/4/2024
Solve the following system of equations in real numbers: {x2=y2+z2,x2023=y2023+z2023,x2025=y2025+z2025.\left\{\begin{array}{l}x^2=y^2+z^2,\\x^{2023}=y^{2023}+z^{2023},\\x^{2025}=y^{2025}+z^{2025}.\end{array}\right. Proposed by Mykhailo Shtandenko, Anton Trygub
algebrasystem of equations
Concatenating powers

Source: Kyiv City MO 2024 Round 2, Problem 10.1

2/4/2024
For some positive integer nn, Katya wrote on the board next to each other numbers 2n2^n and 14n14^n (in this order), thus forming a new number AA. Can the number A−1A - 1 be prime?
Proposed by Oleksii Masalitin
number theoryDigits
System of equations: hard

Source: Kyiv City MO 2024 Round 2, Problem 11.1

2/4/2024
Solve the following system of equations in real numbers: {x2=y2+z2,x2024=y2024+z2024,x2025=y2025+z2025.\left\{\begin{array}{l}x^2=y^2+z^2,\\x^{2024}=y^{2024}+z^{2024},\\x^{2025}=y^{2025}+z^{2025}.\end{array}\right. Proposed by Mykhailo Shtandenko, Anton Trygub, Bogdan Rublov
algebrasystem of equations