2013-14 Problems

7th and 8th graders

February, 2014

 
C.953. For the positive whole numbers x, y and z,
 
x + 2y + 2z = 2013 and 2x + 2y + z = 2014
What is the greatest possible value of x + y + z?

C.954. Prove that there is at least one Friday the 13th every year.

C.955. A company sells its detergent in three different size boxes: small, medium and large. The medium size is 50% more expensive than the small size, and contains 20% less detergent than the large size. The large size is 50% heavier than the small size, and 25% more expensive than the medium size. Which box is the most economical?

C.956. Let points X and Y divide the diagonal BD of parallelogram ABCD into 3 congruent (equally long) sections. Prove that quadrilateral AXCY is also a parallelogram. What is the ration of the areas of ABCD and AXCY?
C.957. At a pizzeria you may buy circular pizzas with diameters 20 cm and 25 cm. They deliver, also: the prices of the two different size pizzas with ham are $5.70 and $6.80. The smaller size pizza ala "Frutti de mare" (with all kinds of seafood on top) costs $6.80. (These prices include delivery charges.) Suppose that the delivery charge does not depend on the kind and size of the pizza, and that the prices of the pizzas, without the delivery charges, are proportional to the areas of the circular pizzas. How much does a larger pizza ala "Frutti de mare" cost with delivery charges? (Use 3.14 for pi.)

C.958. Oleg drew the following diagram in his new notebook. His older sister, Kate, kept on looking at the diagram and then finally said that it is impossible to draw such a pentagon into a circle. Find Oleg's mistake. (Side 5 is perpendicular to side 12. Which number or numbers on the diagram are incorrect?)
C.959. There is a 6m by 8m sized rectangular shaped barn standing in the middle of a meadow. They tied a dog to one of the corners of the barn on a leash that is as log as the diagonal of the barn. What is the area of the territory of the dog?

C.960. Prove that the problem below is divisible by 186. 
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