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I think that it should be F2 because F is smaller than N and O thus having a small bond length. And small bond length means greater energy (my teacher told me).

But the answer is N2.

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Your textbook probably contains the data in a table such as this:

$$ \begin{array}{|c|c|c|} \hline \mathrm{bond} &\mathrm{length(pm)} & \mathrm{energy(kJ/mol)} \\ \hline \ce{N-N} & 145 & 170 \\ \hline \ce{O-O} & 148 & 145 \\ \hline \ce{F-F} & 142 & 158 \\ \hline \hline \ce{N#N} & 110 & 945 \\ \hline \ce{O=O}& 121 & 498 \\ \hline \end{array} $$

So if you consider that dinitrogen has a triple bond and dioxygen a double bond, you see that breaking dinitrogen into atoms requires a much higher bond dissociation energy. Also, you see that the bond length in dinitrogen is the shortest, so here your teacher's rule of thumb (shorter bond, stronger bond) works nicely.

F is smaller than N and O thus having a small bond length

That rule does not work, even if you compare just the single bonds. It probably works if you stay in the same group (comparing F, Cl, Br, I), but in the end, you have to look at the experimental data or, if there is no data available, measure it yourself.

And small bond length means greater energy (my teacher told me)

That rule often works, but again not for the single bonds N-N, O-O and F-F. The differences in bond length and energy are subtle, and that's when these rules often break down.

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