MathDB

22

Part of 2012 AMC 10

Problems(2)

Sum of Odd and Even Positive Integers

Source: 2012 AMC10A Problem #22

2/8/2012
The sum of the first mm positive odd integers is 212212 more than the sum of the first nn positive even integers. What is the sum of all possible values of nn?
<spanclass=′latex−bold′>(A)</span> 255<spanclass=′latex−bold′>(B)</span> 256<spanclass=′latex−bold′>(C)</span> 257<spanclass=′latex−bold′>(D)</span> 258<spanclass=′latex−bold′>(E)</span> 259 <span class='latex-bold'>(A)</span>\ 255 \qquad<span class='latex-bold'>(B)</span>\ 256 \qquad<span class='latex-bold'>(C)</span>\ 257 \qquad<span class='latex-bold'>(D)</span>\ 258 \qquad<span class='latex-bold'>(E)</span>\ 259
modular arithmeticnumber theoryDiophantine equationalgebrasystem of equationsAMC
Ways to List Integers

Source: 2012 AMC 10B Problem #22

2/23/2012
Let (a1,a2,…,a10)(a_1,a_2, \dots ,a_{10}) be a list of the first 1010 positive integers such that for each 2≤i≤102 \le i \le 10 either ai+1a_i+1 or ai−1a_i-1 or both appear somewhere before aia_i in the list. How many such lists are there?
<spanclass=′latex−bold′>(A)</span> 120<spanclass=′latex−bold′>(B)</span> 512<spanclass=′latex−bold′>(C)</span> 1024<spanclass=′latex−bold′>(D)</span> 181,440<spanclass=′latex−bold′>(E)</span> 362,880 <span class='latex-bold'>(A)</span>\ 120\qquad<span class='latex-bold'>(B)</span>\ 512\qquad<span class='latex-bold'>(C)</span>\ 1024\qquad<span class='latex-bold'>(D)</span>\ 181,440\qquad<span class='latex-bold'>(E)</span>\ 362,880
AMC