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IMO预选题1998

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IMO预选题1998 Problems Submitted to the IMO Problem Committee 39th International Mathematical Olympiad, Taiwan 1998 1. A convex quadrilateral ABCD has perpendicular diagonals. The perpendicular bisectors of AB and CD meet at a unique point P inside ABCD. Prove that ...

IMO预选题1998
Problems Submitted to the IMO Problem Committee 39th International Mathematical Olympiad, Taiwan 1998 1. A convex quadrilateral ABCD has perpendicular diagonals. The perpendicular bisectors of AB and CD meet at a unique point P inside ABCD. Prove that ABCD is cyclic if and only if the triangles ABP and CDP have equal areas. 2. Let ABCD be a circumscribed quadrilateral. Given that AE/EB = CF/FD and PE/PF = AB/CD prove that area(APD)/area(BPC) is constant with respect to the choice of E and F. 3. Let I be the incenter of a triangle ABC. Let K, L and M be the points of tangency of the incircle of ABC with AB, BC and CA. The line t passes through B and is parallel to KL. The lines MK and ML intersect t at the points R and S. Prove that angle RIS is acute. 4. Let ABC be a triangle and M and N two internal isogonic points. Prove that: 1= ⋅ ⋅ � sym ACAB ANAM 5. Let ABC be a triangle, H its orthocenter, O its circumcenter and R its circumradius. Let D be the reflection of A across BC, E be that of B across CA and F that of C across AB. Prove that D, E and F are collinear if and only if OH = 2R. 6. Let ABCDEF be a convex hexagon. If B + D + F = 360 and 1= FA EF DE CD BC AB then 1= BD FD EF AE CA BC . 7. Let ABC be a triangle such that angle ACB is twice angle ABC. Let D be the point on the side BC such that CD = 2BD. The segment AD is extended to E so that AD = DE. Prove that ECB + 180 = 2EBC. 8. Let L be a circle and ABC an inscribed triangle so that BC is a diameter. BC intersects the tangent through A to L in D. E is the reflection of A across BC, X the foot of the perpendicular from A to BE, and Y the midpoint of AX. Let the line BY meet L again at Z. Prove that BD is tangent to the circumcircle of ADZ. 9. Prove that if a1, …, an are positive real numbers with sum less than 1 then the inequality holds: 1 2121 2121 1 )1)...(1)(1)(...( )...1(... + ≤ −−−+++ −−−− n nn nn naaaaaa aaaaaa . 10. Prove that if r1, r2, …, rn ≥ 1 then: n n n i i rrr n r ...11 1 211 + ≥ + � = . 11. Prove that if xyz = 1 then: 4 3 )1)(1( 3 ≥ ++ � zy x . 12. Define the numbers c(n, k), k ≤ n this way : c(n, 0) = c(n, n) = 1, n nonnegative integer. c(n+1, k) = 2kc(n, k) + c(n, k-1), k ≤ n. Prove that c(n, k) = c(n-k, k). 13. Find the least possible value of f(1998) where f is a function defined from the set of nonnegative integers to the same set so that for all nonnegative integers m, n we have f(n2f(m)) = m[f(n)]2. 14. Find all pairs of whole numbers (x, y) so that x2y + x + y divides xy2 + y + 7. 15. Find all pairs of real numbers (a, b) so that a[bn] = b[an] for all whole positive n. 16. Find the smallest n ≥ 4 so that for any n integer numbers one can find among these four: a, b, c and d so that a + b = c + d (mod 20). 17. The sequence ai is defined this way: an+1 is the smallest whole number greater than an so that ai + aj is different from 3ak (i, j, k not necessarily distinct). Find a1998. 18. Determine all positive integers n for which there exists an integer m so that 2n - 1 divides m2 + 9. 19. For any positive integer n let d(n) denote the number of n's positive divisors (including 1 and itself). Determine all positive integers m for which there exists a positive integer n so that d(n2)/d(n) = m. 20. Prove that for any n positive integer there is an n digit number, with none of its digits 0, so that it is divisible by the sum of its digits. 21. The numbers ai are some positive integer numbers so that any positive integer n can be uniquely written as n = ai+2aj+4ak (i, j, k not necessarily distinct). Find a1998. 22. A matrix with real entries has the property that the sum of the elements on each row and column is integer. Prove that all the elements can be changed into either their lower or upper integer part so that the sum of elements on each row or column remains unchanged. 23. Let n be an integer greater than 1. A number is called attainable if is equal to 1 or can be obtained from 1 using a finite sequence of the following operations: the operation is either + or *. + and * alternate. + is done either with 2 or with n. is done either with 2 or with n. Prove that if n ≥ 9 then there is an infinite number of numbers that are not attainable. If n = 3 then all positive integers except 7 are attainable. 24. The numbers from 1 to 9 are written randomly in a row. In a move one may choose any block of consecutive numbers that are either in an increasing or a decreasing order and reverse the order of the numbers in it. For example 916532748 may be changed into 913562748 (block 653). Prove that in at most 12 moves one can arrange the numbers increasingly. 25. U is the set of all whole positive numbers less than or equal to n. S, a subset of U, is called split by a set of elements if a number not in S appears in the set between two elements of S. (e.g. 1 3 5 4 2 splits 1,2,3 but does not split 3,4,5). Prove that for any n - 2 subsets of U with at least 2 and at most n-1 elements there exists a set from U that splits all of them. 26. In a contest there are m candidates and n judges, where n ≥ 3 is odd. Each candidate is evaluated by each judge with either pass or fail. Suppose that each pair of judges agrees on at most k candidates. Prove that n n m k 2 1−≥ . 27. Let K be a 10 complete graph. Each side is colored in one of k colors so that for any k of the 10 vertices there are k sides with ends in the k vertices that are colored distinctly. Find k for which this is possible. (Prove that the least k for which it is possible is 5). 28. On an m x n board there are mn cards with one side white and the other black. At first all cards have the white side upwards except for one card in the upper left corner. In a move one can take out a card with black side up and turn all adjacent cards upside-down. Find all m and n for which all cards may be removed from the board.
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