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题目： If n is an integer greater than 50, then the expression $$(n^{2} – 2n)(n^{2} – 1)$$ MUST be divisible by which of the following?I. 8II. 12III. 18

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题目： If x and y are positive integers, and 1 is the greatest common divisor of x and y, what is the greatest common divisor of 2x and 3y?

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题目： A popular website requires users to create a password consisting of digits only. If no digit may be repeated and each password must be at least 9 digits long, how many passwords are possible?

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题目： In how many ways can 16 different gifts be divided among four children such that each child receives exactly four gifts?

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题目： A marketer bought N crates of empty cardboard gift boxes. Each crate held Q individual gift boxes, and the lot of N crates was purchases at a wholesale price of W dollars. This marketer will sell collections of J cardboard gift boxes to retailers, at a price of P dollars for each collection. (Note: J is a divisor of Q.) The marketer knows that, when he has sold all the cardboard gift boxes this way, he wants to net a total profit of Z dollars on the entire transaction. What price P for these collections must he charge, to net this profit? Express P in terms of N, Q, W, J, and Z.

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题目： In how many different ways can 3 identical green shirts and 3 identical red shirts be distributed among 6 children such that each child receives a shirt?

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题目： In the grid of points above, each point has equal vertical & horizontal spacing from the others. A small 45-45-90 triangle is drawn. Counting this triangle, how many triangles congruent to this one, of any orientation, can be constructed from points in this grid?

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题目： Which of the following equations is true for all positive values of x and y?

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题目： How many four digit numbers have no repeat digits, do not contain zero, and have a sum of digits equal to 28?

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题目： The above 11 x 11 grid of dots is evenly spaced: each dot is separated by one unit, vertically or horizontally, from its nearest neighbors. Drawn in the diagram is a single right triangle with legs of length 2 & 1. How many right triangles, of this particular size and shape, in any orientation, can be formed by three dots from this grid?

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题目： $$(30^{30}) * (29^{29}) * (28^{28}) * . . . * (3^{3}) * (2^{2}) * (1^{1}) = N$$What is the highest value of K, such that $$\frac{N}{(125^{K})}$$ is an integer?

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题目： This year, a woman has a lucrative one-year position. During this year, she will give a fraction f of her salary to her husband, a private investor, to invest and they will live this year on the remainder. Through investments, her husband can turn each dollar she gives him into 1 + r, which will be deposited in a bank account. Their goal is to save & invest enough money so they can live off this money for two years following the end of the wife's position. Toward this end, they want to choose f such that the amount in the account at the end of the year is twice what they lived off this year. In terms of r, what should f be?

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题目： Pump A can empty a pool in A minutes, and pump B can empty the same pool in B minutes. Pump A begins emptying the pool for 1 minute before pump B joins. Beginning from the time pump A starts, how many minutes will it take to empty the pool?

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题目： Given that $$n = 10^{a} + 10^{b} + 10^{c}$$, where a, b, and c are distinct positive integers, how many different positive values of n result if n is less than 1 billion (1,000,000,000) ?

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题目： In the diagram above, angle C = 90º and AC = BC. Point M is the midpoint of AB. Arc AXB has its center at C, and passes through A and B. Arc AYB has its center at M and passes through A and B. The shaded region between the two arcs is called a "lune." What is the ratio of the area of the lune to the area of triangle ABC?