3 challenging AMC 10 problems

My older son and took a look at three challenging problem from the 2006 AMC 10 a this morning. The last one gave him quite a bit of trouble. I find it difficult to know ahead of time what problems are going to be particularly challenging, but even on the challenging problems the ideas about how to approach the problem are fascinating to me.

The three problems are #17, #18, and #19 here:

Problems #17, #18, and #19 from the 2006 AMC 10 B

Here’s the first – a probability problem:

 

The second – a problem about a specific sequence:

 

and the third (in three parts) – a pretty tough geometry problem:

 

 

 

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