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The Fifth Canadian Mathematical Olympiads
1973年第五届加拿大数学奥林匹克
1. (1) For what x do we have x < 0 and x < 1/(4x) ? (2) What is the greatest integer n such that 4n + 13 < 0 and n(n+3) > 16? (3) Give an example of a rational number between 11/24 and 6/13. (4) Express 100000 as a product of two integers which are not divisible by 10. (5) Find 1/log236 + 1/log336.
2. Find all real numbers x such that x + 1 = |x + 3| - |x - 1|.
3. Show that if p and p+2 are primes then p = 3 or 6 divides p+1.
4. Let P0, P1, ... , P8 be a convex 9-gon. Draw the diagonals P0P3, P0P6, P0P7, P1P3, P4P6, thus dividing the 9-gon into seven triangles. How many ways can we label these triangles from 1 to 7, so that Pn belongs to triangle n for n = 1, 2, ... , 7.
5. Let s(n) = 1 + 1/2 + 1/3 + ... + 1/n. Show that s(1) + s(2) + ... + s(n-1) = n s(n) - n.
6. C is a circle with chord AB (not a diameter). XY is any diameter. Find the locus of the intersection of the lines AX and BY.
7. Let an = 1/(n(n+1) ). (1) Show that 1/n = 1/(n+1) + an. (2) Show that for any integer n > 1 there are positive integers r < s such that 1/n = ar + ar+1 + ... + as.
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