Unanswered Questions

Determine the smallest positive integer y for which there is a positive integer x satisfying the equation \(2^{13}+2^{10}+2^x=y^2\)

A fraction, expressed in its lowest terms \(\dfrac{a}{b}\) , can also be written in the form  \(\dfrac{2}{n}+\dfrac{1}{n^2}\) , where n is a positive integer. If a + b = 1024, what is the value of a? 

A selection of 3 whatsits, 7 doovers and 1 thingy cost a total of $329. A selection of 4 whatsits, 10 doovers and 1 thingy cost a total of $441. What is the total cost, in dollars, of 1 whatsit, 1 doover and 1 thingy?

A circle is inscribed in a hexagon ABCDEF so that each side of the hexagon is tangent to the circle. Find the perimeter of the hexagon if AB = 6, CD = 7, and EF = 8.

A number written in base a is 123_a. The same number written in base b is 146_b. What is the minimum value of a + b?

27) Given x, y, z \(\in Z\) and

a = x² - yz                             c = z² - xy 

b = y² - xz

Prove that: a³ + b³ + c³ - 3abc is a square number.

26) Find x, y, z such that:

\(\left\{{}\begin{matrix}x+y+xy=8\\y+z+yz=15\\z+x+zx=35\end{matrix}\right.\)

Three containers A, B, and C contain a total of 48 apples. First, 6 apples are taken from A and are put into B. Second, 9 apples are taken from B and are put into C. Now, each container has the same number of apples. What is the original number of apples in container A? 

Ali has 5 consecutive numbers while Ben has 7 consecutive numbers, none of the Ali’s number is in the group of Ben’s numbers. If the second number of Ali’s number is 5 and the sum of Ali’s and Ben’s numbers are 128. What is the largest number of Ben’s number? 

Sam, Tom and Una are three chefs of a restaurant. One day, they cooked 320 plates of spaghetti and in this day, Sam cooked for 6 hours, Tom cooked for 8 hours and Una cooked for 5 hours. They also cook spaghetti at different speeds, with Sam cooking 5 plates for every 3 plates Tom cooks and every 2 plates Una cooks. How many plates of spaghetti did Sam cook this day?