Difficulty:MediumNCTM Standard:Algebra Standard: write equivalent forms of equations, inequalities, and systemsof equations and solve them with fluency–mentally or with paper and pencil in simple cases and usingtechnology in all cases.Mathworld.com Classification:Number Theory>Arithmetic>Fractions>Percent
Suppose that2x3-x6is an integer. Which of the following statements mustbe true aboutx?2008 AMC 10 A, Problem #9—2008 AMC 12 A, Problem #5—“2x3-x6=x2.”SolutionAnswer (B):Because2x3-x6=x2is an integer,xmust be even.The casex= 4shows thatxis notnecessarily a multiple of 3 and that none of the other statements must betrue.Difficulty:Medium-hardNCTM Standard:Number and Operations Standard: understand numbers, ways of representingnumbers, relationships among numbers, and number systems.Mathworld.com Classification:Number Theory>Integers
Each of the sides of a squareS1with area 16 isbisected, and a smaller squareS2is constructed usingthe bisection points as vertices. The same process iscarried out onS2to construct an even smaller squareS3. What is the area ofS3?Difficulty:MediumNCTM Standard:Geometry Standard:apply transformations and use symmetry to analyzemathematical situations.Mathworld.com Classification:Geometry>Plane Geometry>Squares
While Steve and LeRoy are fishing 1 mile from shore,their boat springs a leak, and water comes in at aconstant rate of 10 gallons per minute. The boat willsink if it takes in more than 30 gallons of water. Stevestarts rowing toward the shore at a constant rate of4 miles per hour while LeRoy bails water out of theboat. What is the slowest rate, in gallons per minute,at which LeRoy can bail if they are to reach the shorewithout sinking?