1. The circle C has center O and radius r and contains the points A and B. The circle C' touches the rays OA and OB and has center O' and radius r'. Find the area of the quadrilateral OAO'B.
2. The sequence an is defined by a1 = 1, a2 = 2, an+2 = an+1 + an. The sequence bn is defined by b1 = 2, b2 = 1, bn+2 = bn+1 + bn. How many integers belong to both sequences?
3. N is a sum of n powers of 2. If N is divisible by 2m - 1, prove that n ≥ m. Does there exist a number divisible by 11...1 (m 1s) which has the sum of its digits less than m?
4. A non-negative real is written at each vertex of a cube. The sum of the eight numbers is 1. Two players choose faces of the cube alternately. A player cannot choose a face already chosen or the one opposite, so the first player plays twice, the second player plays once. Can the first player arrange that the vertex common to all three chosen faces is ≤ 1/6?
5. A library is open every day except Wednesday. One day three boys, A, B, C visit the library together for the first time. Thereafter they visit the library many times. A always makes his next visit two days after the previous visit, unless the library is closed on that day, in which case he goes the following day. B always makes his next visit three days after the previous visit (or four if the library is closed). C always makes his next visit four days after the previous visit (or five if the library is closed). For example, if A went first on Monday, his next visit would be Thursday, then Saturday. If B went first on Monday, his next visit would be on Thursday. All three boys are subsequently in the library on a Monday. What day of the week was their first visit?
6. ABCD is a parallelogram and AB is not equal to BC. M is chosen so that (1) ∠MAC = ∠DAC and M is on the opposite side of AC to D, and (2) ∠MBD = ∠CBD and M is on the opposite side of BD to C. Find AM/BM in terms of k = AC/BD.
7. 3n points divide a circle into 3n arcs. One third of the arcs have length 1, one third have length 2 and one third have length 3. Show that two of the points are at opposite ends of a diameter.
8. M is a point inside a regular tetrahedron. Show that we can find two vertices A, B of the tetrahedron such that cos AMB ≤ -1/3.
9. 0 < x, y, z < π/2. We have cos x = x, sin(cos y) = y, cos(sin z) = z. Which of x, y, z is the largest and which the smallest?
10. P is a polygon with 2n+1 sides. A new polygon is derived by taking as its vertices the midpoints of the sides of P. This process is repeated. Show that we must eventually reach a polygon which is homothetic to P.
11. a1, a2, ... , a1982 is a permutation of 1, 2, ... , 1982. If a1 > a2, we swap a1 and a2. Then if (the new) a2 > a3 we swap a2 and a3. And so on. After 1981 potential swaps we have a new permutation b1, b2, ... , b1982. We then compare b1982 and b1981. If b1981 > b1982, we swap them. We then compare b1980 and (the new) b1981. So we arrive finally at c1, c2, ... , c1982. We find that a100 = c100. What value is a100?
12. Cucumber River has parallel banks a distance 1 meter apart. It has some islands with total perimeter 8 meters. It is claimed that it is always possible to cross the river (starting from an arbitrary point) by boat in at most 3 meters. Is the claim always true for any arrangement of islands? [Neglect the current.]
13. The parabola y = x2 is drawn and then the axes are deleted. Can you restore them using ruler and compasses?
14. An integer is put in each cell of an n x n array. The difference between the integers in cells which share a side is 0 or 1. Show that some integer occurs at least n times.
15. x is a positive integer. Put a = x1/12, b = x1/4, c = x1/6. Show that 2a + 2b ≥ 21+c.
16. What is the largest subset of {1, 2, ... , 1982} with the property that no element is the product of two other distinct elements.
17. A real number is assigned to each unit square in an infinite sheet of squared paper. Show that some cell contains a number that is less than or equal to at least four of its eight neighbors.
18. Given a real sequence a1, a2, ... , an, show that it is always possible to choose a subsequence such that (1) for each i ≤ n-2 at least one and at most two of ai, ai+1, ai+2 are chosen and (2) the sum of the absolute values of the numbers in the subsequence is at least 1/6 ∑ 1n |ai|.
19. An n x n array has a cross in n - 1 cells. A move consists of moving a row to a new position or moving a column to a new position. For example, one might move row 2 to row 5, so that row 1 remained in the same position, row 3 became row 2, row 4 became row 3, row 5 became row 4, row 2 became row 5 and the remaining rows remained in the same position. Show that by a series of moves one can end up with all the crosses below the main diagonal.
20. Let {a} denote the difference between a and the nearest integer. For example {3.8} = 0.2, {-5.4} = 0.4. Show that |a| |a-1| |a-2| ... |a-n| >= {a} n!/2n.
21. Do there exist polynomials p(x), q(x), r(x) such that p(x-y+z)3 + q(y-z-1)3 + r(z-2x+1)3 = 1 for all x, y, z? Do there exist polynomials p(x), q(x), r(x) such that p(x-y+z)3 + q(y-z-1)3 + r(z-x+1)3 = 1 for all x, y, z?
22. A tetrahedron T' has all its vertices inside the tetrahedron T. Show that the sum of the edge lengths of T' is less than 4/3 times the corresponding sum for T.
Solutions
The sequence an is defined by a1 = 1, a2 = 2, an+2 = an+1 + an. The sequence bn is defined by b1 = 2, b2 = 1, bn+2 = bn+1 + bn. How many integers belong to both sequences?
Answer
1,2,3 only
Solution
The first few terms are:
n 1 2 3 4 5 6 7Note that for n = 4, 5 we have an-1 < bn < an. So by a trivial induction, the inequality holds for all n ≥ 4.
an 1 2 3 5 8 13 21
bn 2 1 3 4 7 11 18
Problem 15
x is a positive integer. Put a = x1/12, b = x1/4, c = x1/6. Show that 2a + 2b ≥ 21+c.
Solution
Put x = r12. Since x is a positive integer, we have r ≥ 1. We have to show that (2r + 2r3)/2 ≥ 2r2. But this follows immediately from AM/GM.
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