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The Second Canadian Mathematical Olympiads
1970年第二届加拿大数学奥林匹克
1. Find all triples of real numbers such that the product of any two of the numbers plus the third is 2. The triangle ABC has angle A > 90o. The altitude from A is AD and the altitude from B is BE. Show that BC + AD ≥ AC + BE. When do we have equality ?
3. Every ball in a collection is one of two colors and one of two weights. There is at least one of each color and at least one of each weight. Show that there are two balls with different color and different weight.
4. Find all positive integers whose first digit is 6 and such that the effect of deleting the first digit is to divide the number by 25. Show that there is no positive integer such that the deletion of its first digit divides it by 35.
5. A quadrilateral has one vertex on each side of a square side 1. Show that the sum of the squares of its sides is at least 2 and at most 4.
6. Given three non-collinear points O, A, B show how to construct a circle center O such that the tangents from A and B are parallel.
7. Given any sequence of five integers, show that three terms have sum divisible by 3.
8. P lies on the line y = x and Q lies on the line y = 2x. Find the locus for the midpoint of PQ, if |PQ| = 4.
9. Let a1 = 0, a2n+1 = a2n = n. Let s(n) = a1 + a2 + ... + an. Find a formula for s(n) and show that s(m + n) = mn + s(m - n) for m > n.
10. A monic polynomial p(x) with integer coefficients takes the value 5 at four distinct integer values of x. Show that it does not take the value 8 at any integer value of x.
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