Difference between revisions of "1972 IMO Problems"

(Created page with '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 disjoi…')
 
 
(2 intermediate revisions by 2 users not shown)
Line 16: Line 16:
  
 
Let <math>m</math> and <math>n</math> be arbitrary non-negative integers. Prove that
 
Let <math>m</math> and <math>n</math> be arbitrary non-negative integers. Prove that
<cmath>\frac{(2m)!(2n)!}{(m/n)!(m+n)!}</cmath>
+
<cmath>\frac{(2m)!(2n)!}{m!n!(m+n)!}</cmath>
 
is an integer. (<math>0! = 1</math>.)
 
is an integer. (<math>0! = 1</math>.)
  
Line 25: Line 25:
 
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.
  
Line 46: Line 46:
  
 
[[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.

Solution

Problem 2

Prove that if $n \geq 4$, every quadrilateral that can be inscribed in a circle can be dissected into $n$ quadrilaterals each of which is inscribable in a circle.

Solution

Problem 3

Let $m$ and $n$ be arbitrary non-negative integers. Prove that \[\frac{(2m)!(2n)!}{m!n!(m+n)!}\] is an integer. ($0! = 1$.)

Solution

Problem 4

Find all solutions $(x_1, x_2, x_3, x_4, x_5)$ of the system of inequalities \[(x_1^2 - x_3x_5)(x_2^2 - x_3x_5) \leq 0 \\ ,(x_2^2 - x_4x_1)(x_3^2 - x_4x_1) \leq 0 \\ ,(x_3^2 - x_5x_2)(x_4^2 - x_5x_2) \leq 0 \\ ,(x_4^2 - x_1x_3)(x_5^2 - x_1x_3) \leq 0 \\ ,(x_5^2 - x_2x_4)(x_1^2 - x_2x_4) \leq 0\] where $x_1, x_2, x_3, x_4, x_5$ are positive real numbers.

Solution

Problem 5

Let $f$ and $g$ be real-valued functions defined for all real values of $x$ and $y$, and satisfying the equation \[f(x + y) + f(x - y) = 2f(x)g(y)\] for all $x, y$. Prove that if $f(x)$ is not identically zero, and if $|f(x)| \leq 1$ for all $x$, then $|g(y)| \leq 1$ for all $y$.

Solution

Problem 6

Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.

Solution

1972 IMO (Problems) • Resources
Preceded by
1971 IMO
1 2 3 4 5 6 Followed by
1973 IMO
All IMO Problems and Solutions