1959 IMO Problems

Problems of the 1st IMO 1959 Romania.

Day I

Problem 1

Prove that $\displaystyle\frac{21n+4}{14n+3}$ is irreducible for every natural number $\displaystyle n$ .

Problem 2

For what real values of $\displaystyle x$ is

$\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A,$

given (a) $A = \sqrt{2}$ , (b) $\displaystyle A=1$ , (c) $\displaystyle A=2$ , where only non-negative real numbers are admitted for square roots?

Solution

Problem 3

Let $\displaystyle a,b,c$ be real numbers. Consider the quadratic equation in $\displaystyle \cos{x}$ :

$\displaystyle a\cos ^{2}x + b\cos{x} + c = 0.$

Using the numbers $\displaystyle a,b,c$ , form a quadratic equation in $\displaystyle \cos{2x}$ , whose roots are the same as those of the original equation. Compare the equations in $\displaystyle \cos{x}$ and $\displaystyle \cos{2x}$ for $\displaystyle a=4, b=2, c=-1$ .

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1959 IMO Problems

Contents

Day I

Problem 1

Problem 2

Problem 3

Day II

Problem 4

Problem 5

Problem 6

Resources