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Difference between revisions of "1969 IMO Problems"

(Created page with 'Problems of the 11th IMO 1969 in Romania. ==Problem 1== Prove that there are infinitely many natural numbers <math>a</math> with the following property: the number <math>z…')
 
 
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* [[1969 IMO]]
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* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1969 IMO 1969 Problems on the Resources page]
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* [[IMO Problems and Solutions, with authors]]
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* [[Mathematics competition resources]] {{IMO box|year=1969|before=[[1968 IMO]]|after=[[1970 IMO]]}}

Latest revision as of 12:34, 29 January 2021

Problems of the 11th IMO 1969 in Romania.

Problem 1

Prove that there are infinitely many natural numbers $a$ with the following property: the number $z = n^4 + a$ is not prime for any natural number $n$.

Solution

Problem 2

Let $a_1, a_2, \cdots, a_n$ be real constants, $x$ a real variable, and \[f(x) = \cos(a_1 + x) + \frac{1}{2}\cos(a_2 + x) + \frac{1}{4}\cos(a_3 + x) + \cdots + \frac{1}{2^{n - 1}}\cos(a_n + x).\] Given that $f(x_1) = f(x_2) = 0$, prove that $x_2 - x_1 = m\pi$ for some integer $m$.

Solution

Problem 3

For each value of $k = 1, 2, 3, 4, 5$, find necessary and sufficient conditions on the number $a > 0$ so that there exists a tetrahedron with k edges of length $a$, and the remaining $6 - k$ edges of length 1.

Solution

Problem 4

A semicircular arc $\gamma$ is drawn on $AB$ as diameter. $C$ is a point on $\gamma$ other than $A$ and $B$, and $D$ is the foot of the perpendicular from $C$ to $AB$. We consider three circles, $\gamma_1, \gamma_2, \gamma_3$, all tangent to the line $AB$. Of these, $\gamma_1$ is inscribed in $\Delta ABC$, while $\gamma_2$ and $\gamma_3$ are both tangent to $CD$ and to $\gamma$, one on each side of $CD$. Prove that $\gamma_1$, $\gamma_2$ and $\gamma_3$ have a second tangent in common.

Solution

Problem 5

Given $n > 4$ points in the plane such that no three are collinear. Prove that there are at least $\tbinom{n - 3}{2}$ convex quadrilaterals whose vertices are four of the given points.

Solution

Problem 6

Prove that for all real numbers $x_1, x_2, y_1, y_2, z_1, z_2$, with $x_1 > 0, x_2 > 0, x_1y_1 - z_1^2 > 0, x_2y_2 - z_2^2 > 0$, the inequality \[\frac{8}{(x_1 + x_2)(y_1 + y_2) - (z_1 + z_2)^2} \leq \frac{1}{x_1y_1 - z_1^2} + \frac{1}{x_2y_2 - z_2^2}\] is satisfied. Give necessary and sufficient conditions for equality.

Solution

1969 IMO (Problems) • Resources
Preceded by
1968 IMO 		1 • 2 • 3 • 4 • 5 • 6 Followed by
1970 IMO
All IMO Problems and Solutions
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