1. Prove that 1/64 < (1/2)(3/4)(5/6)... (2009/2010) < 1/44 

2. Find the smallest positive integer N such that N! is a multiple of 10²⁰⁰⁹.

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1. A man had a 10-gallon keg of wine and a jug.  One day, he drew off a jugful of wine and filled up the keg with water.  Later on, when the wine and water had got thoroughly mixed, he drew off another jugful and again filled up the keg with water.  The keg then contained equal quantities of wine and water.  What was the capacity of the jug?

Source: unknown

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1. Unique Solution: For which real numbers, a, does the equation

a 3^x + 3^-x = 3

have a unique solution?

Source: Crux Mathematica, April 2002, from a Finnish High School math contest.

2. Two Solutions: For what values of m does sqrt(x-5)=mx+2 have two solutions?

Source: unknown

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Prove the trigonometric identity tan(x - y) + tan(y - z) + tan(z - x) = tan(x - y) *tan(y -z)*tan(z - x)

Source: unknown

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Prove that if any set of nine distinct integers has sum greater than 200, then there is a subset of four of the integers whose sum is greater than 100.

Source: unknown

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Let S be a set of rational numbers with the following properties:
1) 1/2 is an element of S
2) If x is an element of S, then both 1/(x+1) is an element of S and x/(x+1) is an element of S
Prove that S contains all rational numbers in the interval 0<x<1.

Source: unknown

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1. X is a set with n elements. Find the number of triples (A, B, C), where A, B, C are subsets of X, such that A is a subset of B and B is a subset of C. 

2. Let m and n be integers greater than 1. Consider an m*n rectangular grid of points in the plane. Some k of these points are colored red in such a way that no three red points are the vertices of a right-angled triangle, two of whose sides are parallel to the sides of the grid. Determine the greatest possible value of k for any given values of m,n > 1.

Source: unknown

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1. When I sum five numbers in every possible pair combination, I get the values: 0,1,2,4,7,8,9,10,11,12.  What are the original 5 numbers?

2. When I sum a different set of five numbers in every possible group of 3, I get the values: 0,3,4,8,9,10,11,12,14,19.  What are the original 5 numbers?

3. Is it possible to find a set of 5 numbers in either case above which results in the sums 1-10?  Find an example or prove it impossible.

4. If the above problem is not possible, what is the longest series of sequential sums you can find? For example, problems 1 and 2 have six and five sequential sums, (7-12) and (8-12) respectively.

Source: unknown

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What is the area of triangle whose sides are sqrt(61), sqrt(153), sqrt(388)

You get partial credit for solving it with Heron's Formula, and full (or even extra) credit for using the following hint: each of the numbers 61, 153, and 388 is the product of a square and one or more primes of the form 4k+1.

Source: Henry Dudeney

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Charlie has 7 cards numbered 1, 2, 3, 4, 5, 6 and 7 and randomly deals 3 of them to Alice and 3 to Bob. All three people look at the cards that they hold.  Can Alice and Bob communicate with each other, in the presence of Charlie, so that after the communication Alice knows which cards Bob has, and Bob knows which cards Alice has, but, for any card except the one he has, Charlie does not know whether Alice or Bob has it? 

Source: unknown

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Let F(x) be a non-decreasing function for all x in [0 1], such that

2F(x/3)=F(x) and F(x)+F(1-x)=1

1. Find F(173/1993) and F(1/13).

2. If possible give a general algorithm to find F(x) for any x.

Other questions:

3. Is F(x) uniquely determined(in [0,1])?

4. Is F(x) continuous(in [0,1])?

Source: unknown

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