Home Competitions CZ-SK-MO 1953
CZ-SK-MO 1953
Czech-Slovak Mathematical Olympiad 1953.
Print the paper
Other problems
Problem 1
Find the locus of all numbers z ∈ C z\in\mathbb C z ∈ C z + z ˉ = a ⋅ ∣ z ∣ , z+\bar z=a\cdot|z|, z + z ˉ = a ⋅ ∣ z ∣ , a ∈ R a\in\mathbb R a ∈ R
Problem 2
Let α , β , γ \alpha,\beta,\gamma α , β , γ φ \varphi φ α = φ + π 4 , β = π − 3 φ . \alpha=\varphi+\frac\pi4, \beta=\pi-3\varphi. α = φ + 4 π , β = π − 3 φ . α > γ . \alpha>\gamma. α > γ .
Problem 3
Prove that the inequality ( a 1 + ⋯ + a n ) ( 1 a 1 + ⋯ + 1 a n ) ≥ n 2 \left(a_1+\cdots+a_n\right)\left(\frac{1}{a_1}+\cdots+\frac{1}{a_n}\right)\ge n^2 ( a 1 + ⋯ + a n ) ( a 1 1 + ⋯ + a n 1 ) ≥ n 2 a 1 , … , a n a_1,\ldots,a_n a 1 , … , a n
Problem 4
Consider skew lines a , b a,b a , b ρ \rho ρ X ∈ a , Y ∈ b X\in a,Y\in b X ∈ a , Y ∈ b X Y ∥ ρ . XY\parallel\rho. X Y ∥ ρ . M M M X Y , XY, X Y , X X X a a a