If this is for a gaming rig, then you may want to consider AMD's FX-6300 / FX-6350 CPUs which have 6 cores and are within your budget. The two downsides of AMD CPUs are:
1. Most importantly less powerful cores compared to Intel. Therefore, games using only 1 or 2 cores will definitely run better on Intel CPUs.
2. Less of a concern for most people is that they consume more power than Intel CPUs. Under stress a FX-6xxx CPU consumes 2x more power than Core i3 CPUs.
While on a per core basis AMD CPUs are weaker than Intel CPUs, the extra cores can help make up the difference against the Core i3 in games that can take advantage of more than 2 cores.
In general, the Core i3 is capable of providing good performance in many games, or at least average. However, Crysis 3 is the 1st game where the Core i3 does very poorly. Compared to a quad core i5 CPU of similar speed benchmarks have shown around a 40% lower performance. Let me just be clear about this. Crysis 3 is an exception not the rule. The FX-6xxx would provide better performance in Crysis 3 compared to the Core i3, but it would be less than a Core i5.
The FX-6300 can be bought for $110 using the rebate code while the FX-6350 sells for $130 with the rebate code. Links are below.
http://www.newegg.com/Product/Product.aspx?Item=N82E16819113286
http://www.newegg.com/Product/Product.aspx?Item=N82E16819113327