iMac does not wake up

Thanks for your suggestions.

Yesterday I set CCC to skip all backups for 24 hours.
Today morning the screen woke up correctly.
I will reset NVRAM and SMC as you suggested and see what happens tomorrow.

Last issues I already had pressed power button for a short time but nothing happened.
Tomorrow I'll post news.
Thank you

"For all systems the hibernatefile should be "/var/vm/sleepimage" by default, and for laptops the hibernatemode should have a value of 3 and for desktops it should be 0. If these values are different, then try resetting your computer's SMC"

After resetting my hibernatemode is 0 (iMac)

Yesterday I reset SMC and Carbon Copy Cloner made its job during the night.

Today morning the display didn't wake up. Pressing repeatedly the space bar I hear the system sound when you are doing something wrong.
I already have checked HD integrity and it was okay.

Now I am resetting SMC and NVRAM.

These are the steps and results:

• Shut down
• Unplug power cable 15"
• Plug power cable and wait 10"
• Power on
• Press immediately after the sound CMD-Option P and R and keep them pressed
  Black screen with Apple logo and progress bar -> Grey backround -> Select language (mouse does not respond) -> OS X Utilities (mouse responds after a while)
  Step 5 is not accomplished
  

Resetting NVRAM

1. Shut down your Mac.
2. Locate the following keys on the keyboard: Command (⌘), Option, P, and R.
3. Turn on your Mac.
4. Press and hold the Command-Option-P-R keys immediately after you hear the startup sound.
5. Hold these keys until the computer restarts and you hear the startup sound for a second time.
6. Release the keys.

What can I do?

Back to my desk after 9 hours away I cannot get my display on.

Surely system is on, I see a blinking light in the iMac port of my router.
I also have access from another computer in my network, even I can play a video stored in an external drive attached to my iMac.

I have a CCC task which is executed every 3 hours, so I don't rule CCC on this subject.

As I said before I couldn't end steps to reset NVRAM

I tried changing step 4

Press and hold the Command-Option-P-R keys immediately after you hear the startup sound.

by

Press and hold the Command-Option-P-R keys immediately before you hear the startup sound.

The same chords but sounding differently, poorer, I heard after a while, then I released the 4 keys and another startup sound was played (same as the last one).
Finally the system started. Only time zone was changed.
Now I have to wait until tomorrow

Edit: I didn't have to wait.
After 90 min or so I cannot wake up the display. There was a CCC backup performed during this time
Power off

Power on - I see the router light corresponding to the iMac on immediately but startup sound delays more than usual (around 10 seconds)

I'm updating CCC.

Next test: manually ran a CCC task and put system to sleep.

After a couple of minutes display didn't wake up again.

I've asked CCC support to read this thread

Digging...

I ran another CCC task, put the system to sleep and immediately moved the mouse and pressed some keys: black screen, didn't wake up

Thanks to Mike Bombich from CCC I believe I found the problem.

He asked me to disable Logmein and Teamviewer.

I completely removed Logmein I don't use anymore and turned off Teamviewer.
Put system to sleep and it woke up correctly with or without CCC running a task.
So I am contacting Teamviewer team

I uninstalled Teamweaver, put to sleep and again display do not wake up.

It is something random.

I don't know what to think.

I am going to disconnect all external disks to see what happens.
Anyway if it is random... I don't know
Someone in Mac stuff to help me?

I spent thousands of dollars!!!!

In a week I have pressed the power button more times than the last ten years!!

Disconnected all wires but power.

Power on.
Sleep
Display still off!!

Thank you Eric. I will post that info later.

Last night I was looking system logs from Console

I extracted the lines after going to Sleep:

31/8/16 18:21:03,581 identityservicesd[301]: <IMMacNotificationCenterManager: 0x7fa148e45930>: notification observer: com.apple.iChat notification: __CFNotification 0x7fa148c05180 {name = _NSDoNotDisturbEnabledNotification}

31/8/16 18:21:03,719 identityservicesd[301]: <IMMacNotificationCenterManager: 0x7fa148e45930>: NC Disabled: NO

31/8/16 18:21:03,723 identityservicesd[301]: <IMMacNotificationCenterManager: 0x7fa148e45930>: DND Enabled: YES

31/8/16 18:21:03,723 identityservicesd[301]: <IMMacNotificationCenterManager: 0x7fa148e45930>: Updating enabled: NO (Topics: (

))

31/8/16 18:21:04,000 kernel[0]: PM response took 2179 ms (248, netbiosd)

31/8/16 18:21:05,163 WindowServer[223]: device_generate_lock_screen_screenshot: authw 0x0(0)[inf, inf, 0, 0] shield 0x7fbebb056c00(2001), dev [2560,1440]

31/8/16 18:21:08,935 com.apple.xpc.launchd[1]: (com.logmein.hamachi[1058]) Service could not initialize: Unable to set current working directory. error = 2: No such file or directory, path = /Library/Application Support/LogMeIn Hamachi/: 15G31: xpcproxy + 12068 [1343][F7717708-ACF7-307D-B04E-998DFC36598F]: 0x2

31/8/16 18:21:08,935 com.apple.xpc.launchd[1]: (com.logmein.hamachi) Service only ran for 0 seconds. Pushing respawn out by 10 seconds.

31/8/16 18:21:13,293 RunOuc[1060]: WARNING: The Gestalt selector gestaltSystemVersion is returning 10.9.6 instead of 10.11.6. This is not a bug in Gestalt -- it is a documented limitation. Use NSProcessInfo's operatingSystemVersion property to get correct system version number.

Call location:

I don't understand most of that but I highlighted in bold what it took my attention.

I don't use Logmein for 2 years. It is a program very difficult to erase completely.

I note that this is a 2 months old 27" iMac 2015. I migrated all my stuff from an old iMac using Migration Apple tools.

Yesterday I installed Logmein again for the purpose to uninstall it and consequently remove that service and other stuff that could be still installed.

I have to say that all Logmein ways to uninstall the program they recommend in their site are wrong.
No way to uninstall it.

The matter is that after I installed Logmein my display always turns on after setting system to sleep (auto or manually)

tygb, I already had performed those reset steps. I always have WiFi off.
Restore Defaults: I don't want to put hard disks to sleep when possible and I want to startup after a power failure