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1959 IMO Problems

From AoPSWiki

Problems of the 1st IMO 1959 Romania.

Contents

1 Day I
2 Day II
3 Resources

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?

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 $\displaystyle a=4, b=2, c=-1$ .

Day II

Problem 4

Construct a right triangle with a given hypotenuse $\displaystyle c$ such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.

Problem 5

An arbitrary point $\displaystyle M$ is selected in the interior of the segment $\displaystyle AB$ . The squares $\displaystyle AMCD$ and $\displaystyle MBEF$ are constructed on the same side of $\displaystyle AB$ , with the segments $\displaystyle AM$ and $\displaystyle MB$ as their respective bases. The circles about these squares, with respective centers $\displaystyle P$ and $\displaystyle Q$ , intersect at $\displaystyle M$ and also at another point $\displaystyle N$ . Let $\displaystyle N'$ denote the point of intersection of the straight lines $\displaystyle AF$ and $\displaystyle BC$ .

(a) Prove that the points $\displaystyle N$ and $\displaystyle N'$ coincide.

(b) Prove that the straight lines $\displaystyle MN$ pass through a fixed point $\displaystyle S$ independent of the choice of $\displaystyle M$ .

(c) Find the locus of the midpoints of the segments $\displaystyle PQ$ as $\displaystyle M$ varies between $\displaystyle A$ and $\displaystyle B$ .

Problem 6

Two planes, $\displaystyle P$ and $\displaystyle Q$ , intersect along the line $\displaystyle p$ . The point $\displaystyle A$ is in the plane $\displaystyle P$ , and the point $\displaystyle {C}$ is in the plane $\displaystyle Q$ ; neither of these points lies on the straight line $\displaystyle p$ . Construct an isosceles trapezoid $\displaystyle ABCD$ (with $\displaystyle AB$ parallel to $\displaystyle DC$ ) in which a circle can be constructed, and with vertices $\displaystyle B$ and $\displaystyle D$ lying in the planes $\displaystyle P$ and $\displaystyle Q$ , respectively.

Resources

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