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1. 1983 cities are served by ten airlines. All services are both ways. There is a direct service between any two cities. Show that at least one of the airlines can offer a round trip with an odd number of landings.
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2. Let s(n) be the sum of the positive divisors of n. For example, s(6) = 1 + 2 + 3 + 6 = 12. Show that there are infinitely many n such that s(n)/n > s(m)/m for all m < n.
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4. ABC is a triangle with AB not equal to AC. P is taken on the opposite side of AB to C such that PA = PB. Q is taken on the opposite side of AC to B such that QA = QC and ∠Q = ∠P. R is taken on the same side of BC as A such that RB = RC and ∠R = ∠P. Show that APRQ is a parallelogram.
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5. Let Sn be the set of all strictly decreasing sequences of n positive integers such that no term divides any other term. Given two sequences A = (ai) and B = (bj) we say that A < B if for some k, ak < bk and ai = bi for i < k. For example, (7, 5, 3) < (9, 7, 2) < (9, 8, 7). Find the sequence A in Sn such that A < B for any other B in Sn.
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6. The n positive integers a1, a2, ... , an have sum 2n+2. Show that we can find an integer r such that:
ar+1 ≤ 3
ar+1 + ar+2 ≤ 5
...
ar+1 + ar+2 + ... + an + a1 + ... + ar-1 ≤ 2n-1
Note that ar does not appear in the inequalities. Show that there are either one or two r for which the inequalities hold, and that if there is only one, then all the inequalities are strict.
For example, suppose n = 3, a1 = 1, a2 = 6, a3 = 1. Then r = 1 and r = 3 do not work, but r = 2 does work: a3 = 1 < 3, a3 + a1 = 2 < 5. So r is unique and the inequalities are strict.
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7. Let N be a positive integer. Define the sequence a0, a1, a2, ... by a0 = 0, an+1 = N(an + 1) + (N + 1)an + 2( N(N+1)an(an+1) )1/2. Show that all terms are positive integers.
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8. 3n students are sitting in 3 rows of n. The students leave one at a time. All leaving orders are equally likely. Find the probability that there are never two rows where the number of students remaining differs by 2 or more.
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9. Let m be any integer and n any positive integer. Show that there is a polynomial p(x) with integral coefficients such that | p(x) - m/n | < 1/n2 for all x in some interval of length 1/n.
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10. c is a positive real constant and b = (1 + c)/(2 + c). f is a real-valued function defined on the interval [0, 1] such that f(2x) = f(x)/b for 0 ≤ x ≤ 1/2 and f(x) = b - (1 - b) f(2x - 1) for 1/2 ≤ x ≤ 1. Show that 0 < f(x) - x < c for all 0 < x < 1.
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12. Let S be the set of lattice points (a, b, c) in space such that 0 ≤ a, b, c ≤ 1982. Find the number of ways of coloring each point red or blue so that the number of red vertices of each rectangular parallelepiped (with sides parallel to the axes) is 0, 4 or 8.
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14. Does there exist a set S of positive integers such that given any integer n > 1 we can find a, b in S such that a + b = n, and if a, b, c, d are all in S and all < 10 and a + b = c + d, then a = c or d.
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15. Let S be the set of polynomials anxn + an-1xn-1 + ... + a0 with non-negative real coefficients such that a0 = an ≤ a1 = an-1 ≤ a2 = an-2 ≤ ... . For example, x3 + 2.1 x2 + 2.1 x + 1 or 0.1 x2 + 15 x + 0.1. Show that the product of any two members of S belongs to S.
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16. Given n distinct points in the plane, let D be the greatest distance between two points and d the shortest distance between two points. Show that D ≥ d √3 (√n - 1)/2.
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18. Let F1 = 1, F2 = 1, Fn+2 = Fn+1 + Fn be the Fibonacci sequence. Let p(x) be a polynomial of degree 990 such that p(k) = Fk for k = 992, 993, ... , 1982. Show that p(1983) = F1983 - 1.
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19. k is a positive real. Solve the equations:
x1 | x1 | = x2 | x2| + (x1 - k) | x1 - k |
x2 | x2 | = x3 | x3| + (x2 - k) | x2 - k |
...
xn | xn | = x1 | x1| + (xn - k) | xn - k |.
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20. Find the greatest integer not exceeding 1 + 1/2k + 1/3k + ... + 1/Nk, where k = 1982/1983 and N = 21983.
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21. Let n be a positive integer which is not a prime power. Show that there is a permutation a1, a2, ... , an of 1, 2, ... , n such that cos(2πa1/n) + 2 cos(2πa2/n) + 3 cos(2πa3/n) + ... + n cos(2πan/n) = 0.
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24. dn is the last non-zero digit of n! . Show that there is no k such that dn = dn+k for all sufficiently large n.
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Note: problems 3, 11, 13, 17, 22 and 23 were used in the Olympiad and are not shown here.
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