Difference between revisions of "1972 Canadian MO"

Latest revision as of 17:37, 19 September 2024

\textbf{1972 Canadian MO Problems and Solutions}

Problem 1:

Given three distinct unit circles, each of which is tangent to the other two, find the radii of the circles which are tangent to all three circles.

Solution:

Problem 2:

Let $a_1, a_2, ... , a_n$ be non-negative numbers. Define $M$ to be the sum of all of products of pairs $a_ia_j (i>j)$ , i.e.

$M = a_1(a_2 + a_3 + ... + a_n) + a_2(a_3 + a_4 + ... a_n) + ... a_{n-1}a_n.$

Prove that the sqaure of at least one of the numbers a_1, a_2, ... a_n does not exceet $frac{2M}{n(n-1)}$ .

@@ Line 11: / Line 11: @@
 Problem 2:
-Let <math>a_1, a_2, ... , a_n</math> be non-negative numbers. Define <math>M</math> to be the sum of all of products of pairs <math>a_ia_j</math>
+Let <math>a_1, a_2, ... , a_n</math> be non-negative numbers. Define <math>M</math> to be the sum of all of products of pairs <math>a_ia_j (i>j)</math>, i.e.
+<cmath>M = a_1(a_2 + a_3 + ... + a_n) + a_2(a_3 + a_4 + ... a_n) + ... a_{n-1}a_n.</cmath>
+Prove that the sqaure of at least one of the numbers a_1, a_2, ... a_n does not exceet <math>frac{2M}{n(n-1)}</math>.