Prove that if 51 integers are selected from the first hundred positive integers, (1 : s s 100), there must be a pair of these integers with a sum equal to 101.

If you shift the linear parent function, f(x)=x, up 13 units, what is the equation of the new function?

Acollection of coins has value of 64 cents. there are two more nickels than dimes and three times as many pennies as dimes in this collection. how many of each kind of coin are in the collection?

Amira is painting a rectangular banner 2 1/4 yards wide. the banner will have a blue background. amira has enough paint to cover 1 1/2 square yards of the wall. the school colors are blue and yellow, so amira wants to add yellow rectangles on the left and right sides of the blue rectangle. the yellow rectangles will each be 3/4 yards wide and the same height as the blue rectangle. what will be the total area of the two yellow rectangles?