I can not join in all the computers to the project because "Project communication failed". Reset the project does not help. Only the first 2 machines perform tasks.

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I see that 3 out of 4 your computers work fine and already gained credits. The 4th computer has 4 aborted tasks, so it at least started them.

If you get "Project communication failed", check your network connection and proxy settings first.

First paragraph is true.

But rest of the computers display a massages:

834	https://sech.me/boinc/Amicable/	2017-02-15 13:19:15	update requested by user	
835	https://sech.me/boinc/Amicable/	2017-02-15 13:19:17	Fetching scheduler list	
836	                                2017-02-15 13:19:19	Project communication failed: attempting access to reference site	
837	                                2017-02-15 13:19:21	Internet access OK - project servers may be temporarily down.	

All other BOINC projects are working without problems.

Maybe there are some limitations connections from a single IP address?

After resetting project:

842	https://sech.me/boinc/Amicable/	2017-02-15 13:26:49	Resetting project	
843	https://sech.me/boinc/Amicable/	2017-02-15 13:26:53	Fetching scheduler list	
844               			2017-02-15 13:26:55	Project communication failed: attempting access to reference site	
845			                2017-02-15 13:26:57	Internet access OK - project servers may be temporarily down.	

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It's something on your machines that prevents them from connecting. Enable http debug log:
1) Create cc_config.xml in BOINC directory
2) Add flag like this:

<cc_config>
   <log_flags>
       <http_debug>1</http_debug>
   </log_flags>
</cc_config>

More info on cc_config: https://boinc.berkeley.edu/wiki/Client_configuration

The client will log all http errors.

The problem is the same as in the project BURP.

It is about the certificate ca-bundle.crt located in a directory Program Files\BOINC\. Server Amicable verify this certificate. If the certificate is invalid, it denies the connection.

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So you need to update BOINC client to the newest version there. Server doesn't deny the connection. It's the client which doesn't know server's certificate authority (Let's Encrypt Authority X3) and doesn't connect because of that.

P.S. BURP uses a certificate from StartCom Certification Authority which also has a new certificate now.