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. | ||
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[[1972 IMO Problems/Problem 6|Solution]] | [[1972 IMO Problems/Problem 6|Solution]] | ||
+ | |||
+ | * [[1962 IMO]] | ||
+ | * [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1962 IMO 1962 Problems on the Resources page] | ||
+ | * [[IMO Problems and Solutions, with authors]] | ||
+ | * [[Mathematics competition resources]] {{IMO box|year=1972|before=[[1971 IMO]]|after=[[1973 IMO]]}} |
Latest revision as of 14:15, 29 January 2021
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
is an integer. (
.)
Problem 4
Find all solutions of the system of inequalities
where
are positive real numbers.
Problem 5
Let and
be real-valued functions defined for all real values of
and
, and satisfying the equation
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.
- 1962 IMO
- IMO 1962 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1972 IMO (Problems) • Resources | ||
Preceded by 1971 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1973 IMO |
All IMO Problems and Solutions |