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美国奥数训练题(国家奥数队领队出题)3


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Last month a pet store sold three times as many cats as dogs. If the store had sold the same number of cats but eight more dogs, it would have sold twice as many cats as dogs. Ho

w many cats did the pet store sell last month? What is the greatest three-digit divisor of ?

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The diagram below shows a large square divided into nine congruent smaller squares. There are circles inscribed in five of the smaller squares. The total area covered by all the five circles is . Find the area of the large square.

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The following diagram shows an equilateral triangle and two squares that share common edges. The area of each square is . Find the distance from point to point .

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Find the sum of the squares of the values that

satisfy

.

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Find the least positive integer so that both and have prime factorizations with exactly four (not necessarily distinct) prime factors. Two convex polygons have a total of 33 sides and 243 diagonals. Find the number of diagonals in the polygon with the greater number of sides. In the tribe of Zimmer, being able to hike long distances and knowing the roads through the forest are both extremely important, so a boy who reaches the age of manhood is not designated as a man by the tribe until he completes an interesting rite of passage. The man must go on a sequence of hikes. The first hike is a kilometer hike down the main road. The second hike is a kilometer hike down a secondary road. Each hike goes down a different road and is a quarter kilometer longer than the previous hike. The rite of passage is completed at the end of the hike where the cumulative distance walked by the man on all his hikes exceeds kilometers. So in the tribe of Zimmer, how many roads must a man walk down, before you call him a man? Find the value of that satisfies

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Consider a sequence of eleven squares that have side lengths . Eleven copies of a single square each with area have the same total area as the total area of the eleven squares of the sequence. Find .

11

Define .

and suppose that . Find

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Ted flips seven fair coins. there are relatively prime positive integers and so that is the probability that Ted flips at least two heads given that he flips at least three tails. Find . Find the smallest positive integer such that the decimal representation of has its last digits all equal to . A circle in the first quadrant with center on the curve is tangent to the -axis and the line . The radius of the circle is where and are relatively prime positive integers. Find . Let be a positive integer whose digits add up to . What is the greatest possible product the digits of can have? Let , , and be non-zero real number such that , , and . There are relatively prime positive integers and so that . Find .

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How many positive integer solutions are there to where and ? Find the number of three-digit numbers such that its first two digits are each divisible by its third

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digit.

19

Find the remainder when by .

is divided

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Square has side length . Let be the midpoint of segment , and let be the point on segment a distance from point . Point is on segment so that is perpendicular to segment . The length of segment can be written as where and are positive integers, and is not divisible by the square of any prime. Find . Each time you click a toggle switch, the switch either turns from off to on or from on to off. Suppose that you start with three toggle sswitches with one of them on and two of them off. On each move you randomly select one of the three switches and click it. Let and be relatively prime positive integers so that is the probability that after four such clicks, one switch will be on and two of them will be off. Find . The diagram below shows circles radius and externally tangent to each other and internally tangent to a circle radius . There are relatively prime positive integers and so that a circle radius is internally tangent to the circle radius and externally tangent to the other two circles as shown. Find .

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Find the greatest seven-digit integer divisible by whose digits, in order, are where , , and are single digits. Let and be positive integers such that is a solution to the equation . Find .

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Find the largest prime that divides

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A paper cup has a base that is a circle with radius , a top that is a circle with radius , and sides that connect the two circles with straight line segments as shown below. This cup has height and volume . A second cup that is exactly the same shape as the first is held upright inside the first cup so that its base is a distance of from the base of the first cup. The volume of liquid that will t inside the first cup and outside the second cup can be written where and are relatively prime positive integers. Find .

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You have some white one-by-one tiles and some black and white two-bye-one tiles as shown below. There are four different color patterns that can be generated when using these tiles to cover a threeby-one rectangoe by laying these tiles side by side (WWW, BWW, WBW, WWB). How many different color patterns can be generated when using these tiles to cover a ten-by-one rectangle?

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A bag contains green candies and red candies. You randomly select one candy at a time to eat. If you eat five candies, there are relatively prime positive integers and so that is the probability that you do not eat a green candy after you eat a red candy. Find .

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Let and . Let be a randomly chosen function from the set into itself. There are relatively prime positive integers and such that is the probablity that is a one-to-one function on given that it maps one-to-one into and it maps one-to-one into . Find . The diagram below shows four regular hexagons each with side length meter attached to the sides

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of a square. This figure is drawn onto a thin sheet of metal and cut out. The hexagons are then bent upward along the sides of the square so that meets , meets , meets , and meets . If the resulting dish is filled with water, the water will rise to the height of the corner where the and meet. there are relatively prime positive integers and so that the number of cubic meters of water the dish will hold is . Find .

Middle School 1 Evaluate .

2 The diagram below shows rectangle , and is the midpoint of side area of the shaded region.

where . If

is the midpoint of side and , find the

3 While Peter was driving from home to work, he noticed that after driving 21 miles, the distance he had left to drive was 30 percent of the total distance from home to work. How many miles was his complete trip home to work? 4 How many two-digit positive integers contain at least one digit equal to 5?

5 Meredith drives 5 miles to the northeast, then 15 miles to the southeast, then 25 miles to the southwest, then 35 miles to the northwest, and finally 20 miles to the northeast. How many miles is Meredith from where she started? 6 Volume equals one fourth of the sum of the volumes and , while volume equals one sixth of the sum of the volumes and . There are relatively prime positive integers and so that the ratio of volume to the sum of the other two volumes is . Find .

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A snail crawls centimeters in minutes. At this rate, how many centimeters can the snail crawl is 85 minutes?

8 Seven boys and three girls are playing basketball. I how many different ways can they make two teams of five players so that both teams have at least one girl?

9 Points

and

lie inside rectangle . If .

with and , find the area of

the quadrilateral

10 Find the least positive multiple of 999 that does not have a 9 as a digit.

11 For some integers and the function and . Find

has the properties that .

consists of a square and an equilateral triangle 12 Pentagon that share the side . A circle centered at has area 24. The intersection of the circle and the pentagon has half the area of the pentagon. Find the area of the pentagon.

13 Find the least positive integer digits add to 23.

which is both a multiple of 19 and whose

14 At the 4 PM show, all the seats in the theater were taken, and 65 percent of the audience was children. At the 6 PM show, again, all the seats were taken, but this time only 50 percent of the audience was children. Of all the people who attended either of the shows, 57 percent were children although there were 12 adults and 28 children who attended both shows. How many people does the theater seat? 15 The top and bottom of a rectangular tank each has area 48 square inches. The four vertical sides of the tank is 13 inches. Find the sum of the height, the

width, and the length of this tank in inches. 16 The following sequence lists all the positive rational numbers that do not exceed by first listing the fraction with denominator 2, followed by the one with denominator 3, followed by the two fractions with denominator 4 in increasing order, and so forth so that the sequence is Let and be relatively prime positive integers so that the . Find . fraction in

the list is equal to

17 The diagram below shows nine points on a circle where . Given that and is perpendicular to , there are relatively prime positive integers and so that the degree measure of is . Find .

18 Find the smallest positive integer whose remainder when divided by is , when divided by is , and when divided by is . 19 A teacher suggests four possible books for students to read. Each of six

students selects one of the four books. How many ways can these selections be made if each of the books is read by at least one student? 20 In the following addition, different letters represent different non-zero digits. What is the 5-digit number ?


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