Consulta de contagem lenta com cláusula where

Você pode ter várias condições em suas joins usando ON condition1 AND condition2 etc.

SELECT COUNT(appeloffre0_.ID_APPEL_OFFRE) as size  
FROM  ao.appel_offre appeloffre0_  
JOIN ao.acheteur acheteur1_ ON appeloffre0_.ID_ACHETEUR=acheteur1_.ID_ACHETEUR 
JOIN ao.lot lot2_ ON appeloffre0_.ID_APPEL_OFFRE=lot2_.ID_APPEL_OFFRE AND lot2_.ESTIMATION_COUT>=1 
JOIN ao.lieu_execution lieuexecut3_ ON appeloffre0_.ID_APPEL_OFFRE=lieuexecut3_.ID_APPEL_OFFRE AND lieuexecut3_.ID_ACTIVITE=1
JOIN ao.ao_activite aoactivite4_ ON appeloffre0_.ID_APPEL_OFFRE=aoactivite4_.ID_APPEL_OFFRE AND aoactivite4_.ID_ACTIVITE=1
WHERE appeloffre0_.DATE_OUVERTURE_PLIS>'2015-01-01' 
AND (appeloffre0_.CATEGORIE='fournitures' OR appeloffre0_.CATEGORIE='travaux' OR appeloffre0_.CATEGORIE='services') 
AND acheteur1_.ID_ENTITE_MERE=2;