Difference between revisions of "1972 IMO Problems"
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Find all solutions <math>(x_1, x_2, x_3, x_4, x_5)</math> of the system of inequalities | Find all solutions <math>(x_1, x_2, x_3, x_4, x_5)</math> of the system of inequalities | ||
<cmath>(x_1^2 - x_3x_5)(x_2^2 - x_3x_5) \leq 0 \\ | <cmath>(x_1^2 - x_3x_5)(x_2^2 - x_3x_5) \leq 0 \\ | ||
| − | (x_2^2 - x_4x_1)(x_3^2 - x_4x_1) \leq 0 \\ | + | ,(x_2^2 - x_4x_1)(x_3^2 - x_4x_1) \leq 0 \\ |
| − | (x_3^2 - x_5x_2)(x_4^2 - x_5x_2) \leq 0 \\ | + | ,(x_3^2 - x_5x_2)(x_4^2 - x_5x_2) \leq 0 \\ |
| − | (x_4^2 - x_1x_3)(x_5^2 - x_1x_3) \leq 0 \\ | + | ,(x_4^2 - x_1x_3)(x_5^2 - x_1x_3) \leq 0 \\ |
| − | (x_5^2 - x_2x_4)(x_1^2 - x_2x_4) \leq 0</cmath> | + | ,(x_5^2 - x_2x_4)(x_1^2 - x_2x_4) \leq 0</cmath> |
where <math>x_1, x_2, x_3, x_4, x_5</math> are positive real numbers. | where <math>x_1, x_2, x_3, x_4, x_5</math> are positive real numbers. | ||
Revision as of 08:49, 7 January 2018
Problems of the 14th IMO 1972 in Poland.
Problem 1
Prove that from a set of ten distinct two-digit numbers (in the decimal system), it is possible to select two disjoint subsets whose members have the same sum.
Problem 2
Prove that if 
, every quadrilateral that can be inscribed in a circle can be dissected into 
quadrilaterals each of which is inscribable in a circle.
Problem 3
Let 
and be arbitrary non-negative integers. Prove that
![\[\frac{(2m)!(2n)!}{m!n!(m+n)!}\]](http://latex.artofproblemsolving.com/3/2/3/323dce2539f7ac15a9beb292b765590b9cf1f5ad.png)
is an integer. (
.)
Problem 4
Find all solutions 
of the system of inequalities
![\[(x_1^2 - x_3x_5)(x_2^2 - x_3x_5) \leq 0 \\ ,(x_2^2 - x_4x_1)(x_3^2 - x_4x_1) \leq 0 \\ ,(x_3^2 - x_5x_2)(x_4^2 - x_5x_2) \leq 0 \\ ,(x_4^2 - x_1x_3)(x_5^2 - x_1x_3) \leq 0 \\ ,(x_5^2 - x_2x_4)(x_1^2 - x_2x_4) \leq 0\]](http://latex.artofproblemsolving.com/4/6/a/46adddcbcf6b21578ba380eff7906f6ce5169f22.png)
where 
are positive real numbers.
Problem 5
Let 
and 
be real-valued functions defined for all real values of 
and 
, and satisfying the equation
![\[f(x + y) + f(x - y) = 2f(x)g(y)\]](http://latex.artofproblemsolving.com/9/7/f/97f8d013a410592034cbfed478e6a77b6573100d.png)
for all 
. Prove that if 
is not identically zero, and if 
for all , then 
for all .
Problem 6
Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.
