Rewrite the following expression as an integer:

Side AC of right triangle ABC is divided into 8 equal parts, and 7 interior segments parallel to BC are drawn.  If segment BC=10, determine the sum of the 7 segment lengths.  Don’t add in the 10 for BC.  (If the answer is a whole number, report the whole number.  If it is a fraction, round to the nearest whole number.)

How many different combinations of integers side lengths can be used to create an isosceles triangle with perimeter 2021?  

A rectangle is cut up into 4 pieces as shown.  If the pieces are rearranged to form a square, determine the perimeter of that square.  

Suppose there are 8 points on a circle, evenly spaced.  How many right triangles can be made using three of the points as vertices?

The number 19 appears in the 3rd row and 4th column. In which ROW does 2021 appear?  

Calculate the following expression:

Let A be a 2-digit integer and let B be a 2-digit integer with the same digits as A but in the opposite order.  Find A so that A^2-B^2 is a positive perfect square.

In the mythical land of Smullyana, trolls always speak lies before they reach the age of 10, but once 10 or older, they always tell the truth. Also, it is customary to only discuss ages using whole numbers of years (nobody would ever claim to be 12 and a half).   While walking in a forest in Smullyana, you meet four trolls.  Poppy says "I am not yet 8 years old. Cooper is the same age as either Branch or me.”  Branch says “My age is at most two years more than Poppy’s age.”  Cooper says “If I am older than Poppy, I am the same age as Branch.”  Biggie says, “Either I am less than 10, or else I am as old as Poppy, Branch, and Cooper combined.”  How old is Biggie?

There is exactly one way to fill in the circles with three distinct positive integers so that the sum of each pair is a perfect square and the sum of all three numbers is less than 68.  Find the sum of the three numbers in this case.

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Use a numeral. For example, if you are in 10th grade, write 10. Do NOT write "tenth" and do not include a "th." Write just the number. Thanks.

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