Roll a standard pair of six-sided dice, and note the sum. There is one way of obtaining a 2, two ways of obtaining a 3, and so on, up to one way of obtaining a 12. Find all other pairs of six-sided dice such that:
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Let {an} be a strictly increasing sequence of positive integers such that:
Show that an = n, for every positive integer, n.
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If x and y are positive real numbers, show that xy + yx > 1.
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Show that the area of a regular polygon with 2n sides and unit perimeter is 
, where there are n − 1 twos under both sets of nested radical signs.
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Given any sequence of n integers, show that there exists a consecutive subsequence the sum of whose elements is a multiple of n.
For example, in sequence {1,5,1,2} a consecutive subsequence with this property is the last three elements; in {1,−3,−7} it is simply the second element.
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Find the smallest natural number greater than 1 billion (109) that has exactly 1000 positive divisors. (The term divisor includes 1 and the number itself. So, for example, 9 has three positive divisors.)
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Suppose xy = yx, where x and y are positive real numbers, with x < y. Show that x = 2, y = 4 is the only integer solution. Are there further rational solutions? (That is, with x and y rational.) For what values of x do real solutions exist?
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Let p(x) be a polynomial with integer coefficients. Show that, if the constant term is odd, and the sum of all the coefficients is odd, then p has no integer roots. (That is, if p(x) = a0 + a1x + ... + anxn, a0 is odd, and a0 + a1 + ... + an is odd, then there is no integer k such that p(k) = 0.)
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Suppose n fair 6-sided dice are rolled simultaneously. What is the expected value of the score on the highest valued die?
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