MathDB

17

Part of 2008 AMC 12/AHSME

Problems(2)

Even/Odd Sequence

Source: AMC 12 2008A Problem 17

2/17/2008
Let a1,a2,… a_1,a_2,\dots be a sequence of integers determined by the rule a_n\equal{}a_{n\minus{}1}/2 if a_{n\minus{}1} is even and a_n\equal{}3a_{n\minus{}1}\plus{}1 if a_{n\minus{}1} is odd. For how many positive integers a1≤2008 a_1 \le 2008 is it true that a1 a_1 is less than each of a2 a_2, a3 a_3, and a4 a_4? <spanclass=′latex−bold′>(A)</span> 250<spanclass=′latex−bold′>(B)</span> 251<spanclass=′latex−bold′>(C)</span> 501<spanclass=′latex−bold′>(D)</span> 502<spanclass=′latex−bold′>(E)</span> 1004 <span class='latex-bold'>(A)</span>\ 250 \qquad <span class='latex-bold'>(B)</span>\ 251 \qquad <span class='latex-bold'>(C)</span>\ 501 \qquad <span class='latex-bold'>(D)</span>\ 502 \qquad <span class='latex-bold'>(E)</span>\ 1004
modular arithmeticAMC
Right Triangle on Parabola

Source: AMC 12 2008B Problem 17

2/29/2008
Let A A, B B, and C C be three distinct points on the graph of y\equal{}x^2 such that line AB AB is parallel to the x x-axis and △ABC \triangle{ABC} is a right triangle with area 2008 2008. What is the sum of the digits of the y y-coordinate of C C? <spanclass=′latex−bold′>(A)</span> 16<spanclass=′latex−bold′>(B)</span> 17<spanclass=′latex−bold′>(C)</span> 18<spanclass=′latex−bold′>(D)</span> 19<spanclass=′latex−bold′>(E)</span> 20 <span class='latex-bold'>(A)</span>\ 16 \qquad <span class='latex-bold'>(B)</span>\ 17 \qquad <span class='latex-bold'>(C)</span>\ 18 \qquad <span class='latex-bold'>(D)</span>\ 19 \qquad <span class='latex-bold'>(E)</span>\ 20
conicsparabolageometryanalytic geometrycalculusgraphing linesslope