Difference between revisions of "2022 USAJMO Problems/Problem 5"

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===Problem===
 
===Problem===
Let <math>ABCD</math> be a rhombus, and let <math>K</math> and <math>L</math> be points such that <math>K</math> lies inside the rhombus, <math>L</math> lies outside the rhombus, and <math>KA=KB=LC=LD</math>. Prove that there exist points <math>X</math> and <math>Y</math> on lines <math>AC</math> and <math>BD</math> such that <math>KXLY</math> is also a rhombus.
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Find all pairs of primes <math>(p,q)</math> for which <math>p-q</math> and <math>pq-q</math> are both perfect squares.
  
 
==Solution==
 
==Solution==

Revision as of 20:36, 19 April 2022

Problem

Find all pairs of primes $(p,q)$ for which $p-q$ and $pq-q$ are both perfect squares.

Solution