Difference between revisions of "1966 IMO Problems/Problem 1"
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==Solution== | ==Solution== | ||
− | { | + | Let us draw a Venn Diagram. |
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+ | PLEASE PROVIDE A DIAGRAM. | ||
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+ | Let <math>a</math> be the number of students solving both B and C. Then for some positive integer <math>x</math>, <math>2x - a</math> students solved B only, and <math>x - a</math> students solved C only. Let <math>2y - 1</math> be the number of students solving A; then <math>y</math> is the number of students solving A only. We have by given <cmath>2y - 1 + 3x - a = 24</cmath> and <cmath>y = 3x - 2a.</cmath> Substituting for y into the first equation gives <cmath>9x - 5a = 25.</cmath> Thus, because <math>x</math> and <math>a</math> are positive integers with <math>x-a \ge 0</math>, we have <math>x = 5</math> and <math>a = 4</math>. (Note that <math>x = 10</math> and <math>a = 13</math> does not work.) Hence, the number of students solving B only is <math>2x - a = 10 - 4 = \boxed{6}.</math> | ||
==See Also== | ==See Also== | ||
{{IMO box|year=1966|before=First Question|num-a=2}} | {{IMO box|year=1966|before=First Question|num-a=2}} |
Revision as of 23:45, 28 April 2014
In a mathematical contest, three problems, , , and were posed. Among the participants there were students who solved at least one problem each. Of all the contestants who did not solve problem , the number who solved was twice the number who solved . The number of students who solved only problem was one more than the number of students who solved and at least one other problem. Of all students who solved just one problem, half did not solve problem . How many students solved only problem ?
Solution
Let us draw a Venn Diagram.
PLEASE PROVIDE A DIAGRAM.
Let be the number of students solving both B and C. Then for some positive integer , students solved B only, and students solved C only. Let be the number of students solving A; then is the number of students solving A only. We have by given and Substituting for y into the first equation gives Thus, because and are positive integers with , we have and . (Note that and does not work.) Hence, the number of students solving B only is
See Also
1966 IMO (Problems) • Resources | ||
Preceded by First Question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |