On the map, Cyprus seems not to far away from Lebanon. Approximately 100 kilometers. Is it possible to see parts of Lebanon from anywhere in Cyprus?
I started trying to solve this by finding photos from Cyprus. I found some, but they're not very conclusive.
Then I realised we could do this mathematically!
The approximate distance to the horizon from an observer close to the Earth's surface is given by:
where d is in kilometres and h is height above sea level in metres.
As you've pointed out, it's about 100km - I plugged it into GMap-Pedometer and reckon it's about 105km.
So solving for h, we find we need to be at a height of 742m above sea level at the closest point.
Unfortunately nowhere even comes close on the eastern side.
So we have one shot - Mount Olympus, which is 1952m.
Solving for d, we can get d = ~170km.
However the distance from the top of Mount Olympus to the nearest point in Lebanon is well over 250km, meaning that NO, you cannot see Lebanon from any point in Cyprus.
The highest point in Lebanon is Qurnat As Sawda. So perhaps you can see from Olympus to here? Let's find out:
From the top of Olympus to the top of Qurnat As Sawda is about 302km as the crow flies. If we sum their heights, we get 5040m. To be able to see 302km, we solve for h, and work that out as a mountain which would need to be 6134m high. So again, no, but it is a bit closer :)
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It doesn't look like it's possible, if you are standing on the beach and not taking photos from some elevated vantage point. The eastern part of Cyprus is fairly flat, with elevation of less than 50 m, especially on the side you'd be able to see Lebanon.
The easiest way to confirm is to look for photos taken in this region, and see whether you'd be able to see something there. I looked through some photos in Panoramio and wasn't able to find a single one on which the Lebanese coast is visible. Admittedly, the photos are not of highest quality, and the meteorological conditions are not perfect, but it seems that it's not possible.
For comparison, in my current residence in Jönköping, Sweden, the island of Visingsö is around 30 km north from the city in the Vättern lake, and is barely visible most of the days -- you can only see the mirrored image of the island "hanging" above the water, even in near-perfect conditions.
[I just saw Mark's excellent scientific answer, but if mathematics are not your thing, this sort of reasoning could be of use in similar circumstances.]
I've been on the Karpaz peninsula, the part of Cyprus that is closest to Lebanon and no, you can't see anything but sea, even from the very tip - though admittedly I didn't have binoculars or tried to find the highest point. But there aren't really any mountains in that area, just some minor hills.
Here's a picture from that area (and even pointing in the right direction, I think):
I found this on the net. Syrian mountains (maybe Lebanese) seen from Cape Greco early in the morning.