Difference between revisions of "2024 USAJMO Problems/Problem 1"
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Let <math>ABCD</math> be a cyclic quadrilateral with <math>AB=7</math> and <math>CD=8</math>. Points <math>P</math> and <math>Q</math> are selected on line segment <math>AB</math> so that <math>AP=BQ=3</math>. Points <math>R</math> and <math>S</math> are selected on line segment <math>CD</math> so that <math>CR=DS=2</math>. Prove that <math>PQRS</math> is a quadrilateral. | Let <math>ABCD</math> be a cyclic quadrilateral with <math>AB=7</math> and <math>CD=8</math>. Points <math>P</math> and <math>Q</math> are selected on line segment <math>AB</math> so that <math>AP=BQ=3</math>. Points <math>R</math> and <math>S</math> are selected on line segment <math>CD</math> so that <math>CR=DS=2</math>. Prove that <math>PQRS</math> is a quadrilateral. | ||
| − | == Solution 1 == | + | == Solution 1 (One-liner) == |
| + | <math>OP=OQ=\sqrt{R^2-3.5^2+0.5^2}=\sqrt{R^2-12}=\sqrt{R^2-4^2+2^2}=OR=OS</math> | ||
==See Also== | ==See Also== | ||
{{USAJMO newbox|year=2024|before=First Question|num-a=2}} | {{USAJMO newbox|year=2024|before=First Question|num-a=2}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 20:52, 19 March 2024
Contents
Problem
Let 
be a cyclic quadrilateral with 
and 
. Points 
and 
are selected on line segment 
so that 
. Points 
and 
are selected on line segment 
so that 
. Prove that 
is a quadrilateral.
Solution 1 (One-liner)
See Also
| 2024 USAJMO (Problems • Resources) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAJMO Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
