G6 Math Olympiad Part K

Chloe and Zoe are both students in Ms. Demeanor's math class. Last night, they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only 80% of the problems she solved alone, but overall 88% of her answers were correct. Zoe had correct answers to 90% of the problems she solved alone. What was Zoe's overall percentage of correct answers?

In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path only allows moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture.

(a)  

In the figure below, choose point D on $$\overline{BC}$$ so that △ACD and △ABD have equal perimeters. What is the area of △ABD?

Starting with some gold coins and some empty treasure chests, I tried to put 9 gold coins in each treasure chest, but that left 2 treasure chests empty. So instead I put 6 gold coins in each treasure chest, but then I had 3 gold coins left over. How many gold coins did I have?

(a)  

In the non-convex quadrilateral ABCD shown below, $$\angle BCD$$ is a right angle, AB=12, BC=4, CD=3, and AD=13. What is the area of quadrilateral ABCD ?

For any positive integer M , the notation M! denotes the product of the integers 1 through M. What is the largest integer n for which $$5^n$$ is a factor of the sum 98! + 99! + 100! ?

(a)  

An integer between 1000 and 9999, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct?

Suppose a, b, and c are nonzero real numbers, and a + b + c = 0. What are the possible value(s) for $$\frac{a}{\left|a\right|}+\frac{b}{\left|b\right|}+\frac{c}{\left|c\right|}+\frac{abc}{\left|abc\right|}$$ ?

(a)  

In the right triangle ABC, AC=12, BC=5, and angle C is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?

Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by 5 minutes over the preceding day. Each of the four days, her distance traveled was also an integer number of miles. What was the total number of miles for the four trips?

(a)