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Difference between revisions of "1968 IMO Problems"

(Created page with 'Problems of the 10th IMO 1968 in USSR. ==Problem 1== Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is …')
 
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Let <math>f</math> be a real-valued function defined for all real numbers <math>x</math> such that, for some positive constant <math>a</math>, the equation
 
Let <math>f</math> be a real-valued function defined for all real numbers <math>x</math> such that, for some positive constant <math>a</math>, the equation
−
<cmath>f(x + a) = \frac{1}{2} + \sqrt{f(x) - [f(x)]^2}</cmath>
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<cmath>f(x + a) = \frac{1}{2} + \sqrt{f(x) - (f(x))^2}</cmath>
 
holds for all <math>x</math>.
 
holds for all <math>x</math>.
  

Revision as of 14:08, 26 April 2012

Problems of the 10th IMO 1968 in USSR.

Problem 1

Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another.

Solution

Problem 2

Find all natural numbers $x$ such that the product of their digits (in decimal notation) is equal to $x^2 - 10x - 22$.

Solution

Problem 3

Consider the system of equations \[ax_1^2 + bx_1 + c = x_2\] \[ax_2^2 + bx_2 + c = x_3\] \[\cdots\] \[ax_{n-1}^2 + bx_{n-1} + c = x_n\] \[ax_n^2 + bx_n + c = x_1\] with unknowns $x_1, x_2, \cdots, x_n$ where $a, b, c$ are real and $a \neq 0$. Let $\Delta = (b - 1)^2 - 4ac$. Prove that for this system

(a) if $\Delta < 0$, there is no solution,

(b) if $\Delta = 0$, there is exactly one solution,

(c) if $\Delta > 0$, there is more than one solution.

Solution

Problem 4

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle.

Solution

Problem 5

Let $f$ be a real-valued function defined for all real numbers $x$ such that, for some positive constant $a$, the equation \[f(x + a) = \frac{1}{2} + \sqrt{f(x) - (f(x))^2}\] holds for all $x$.

(a) Prove that the function $f$ is periodic (i.e., there exists a positive number $b$ such that $f(x + b) = f(x)$ for all $x$).

(b) For $a = 1$, give an example of a non-constant function with the required properties.

Solution

Problem 6

For every natural number $n$, evaluate the sum \[\sum_{k = 0}^\infty\bigg[\frac{n + 2^k}{2^{k + 1}}\bigg] = \Big[\frac{n + 1}{2}\Big] + \Big[\frac{n + 2}{4}\Big] + \cdots + \bigg[\frac{n + 2^k}{2^{k + 1}}\bigg] + \cdots\] (The symbol $[x]$ denotes the greatest integer not exceeding $x$.)

Solution

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