MathDB

1998 IMO Shortlist

Part of IMO Shortlist

Subcontests

(8)
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IMO ShortList 1998, number theory problem 3

Determine the smallest integer n≥4n\geq 4 for which one can choose four different numbers a,b,ca,b,c and dd from any nn distinct integers such that a+b−c−da+b-c-d is divisible by 2020.

IMO ShortList 1998, algebra problem 3

Let x,yx,y and zz be positive real numbers such that xyz=1xyz=1. Prove that x3(1+y)(1+z)+y3(1+z)(1+x)+z3(1+x)(1+y)≥34. \frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)} \geq \frac{3}{4}.

IMO ShortList 1998, combinatorics theory problem 3

Cards numbered 1 to 9 are arranged at random in a row. In a move, one may choose any block of consecutive cards whose numbers are in ascending or descending order, and switch the block around. For example, 9 1 6 5 3‾\underline{6\ 5\ 3} 2 7 4 82\ 7\ 4\ 8 may be changed to 919 1 3 5 6‾\underline{3\ 5\ 6} 2 7 4 82\ 7\ 4\ 8. Prove that in at most 12 moves, one can arrange the 9 cards so that their numbers are in ascending or descending order.
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IMO ShortList 1998, algebra problem 1

Let a1,a2,…,ana_{1},a_{2},\ldots ,a_{n} be positive real numbers such that a1+a2+⋯+an<1a_{1}+a_{2}+\cdots +a_{n}<1. Prove that a1a2⋯an[1−(a1+a2+⋯+an)](a1+a2+⋯+an)(1−a1)(1−a2)⋯(1−an)≤1nn+1. \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}.

IMO ShortList 1998, combinatorics theory problem 1

A rectangular array of numbers is given. In each row and each column, the sum of all numbers is an integer. Prove that each nonintegral number xx in the array can be changed into either ⌈x⌉\lceil x\rceil or ⌊x⌋\lfloor x\rfloor so that the row-sums and column-sums remain unchanged. (Note that ⌈x⌉\lceil x\rceil is the least integer greater than or equal to xx, while ⌊x⌋\lfloor x\rfloor is the greatest integer less than or equal to xx.)
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IMO ShortList 1998, number theory problem 8

Let a0,a1,a2,…a_{0},a_{1},a_{2},\ldots be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form ai+2aj+4aka_{i}+2a_{j}+4a_{k}, where i,ji,j and kk are not necessarily distinct. Determine a1998a_{1998}.

A difficult problem [tangent circles in right triangles]

Let ABCABC be a triangle such that ∠A=90∘\angle A=90^{\circ } and ∠B<∠C\angle B<\angle C. The tangent at AA to the circumcircle ω\omega of triangle ABCABC meets the line BCBC at DD. Let EE be the reflection of AA in the line BCBC, let XX be the foot of the perpendicular from AA to BEBE, and let YY be the midpoint of the segment AXAX. Let the line BYBY intersect the circle ω\omega again at ZZ. Prove that the line BDBD is tangent to the circumcircle of triangle ADZADZ. [hide="comment"] Edited by Orl.
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calculation !

Let ABCABC be a triangle such that ∠ACB=2∠ABC\angle ACB=2\angle ABC. Let DD be the point on the side BCBC such that CD=2BDCD=2BD. The segment ADAD is extended to EE so that AD=DEAD=DE. Prove that ∠ECB+180∘=2∠EBC. \angle ECB+180^{\circ }=2\angle EBC.

IMO ShortList 1998, number theory problem 7

Prove that for each positive integer nn, there exists a positive integer with the following properties: It has exactly nn digits. None of the digits is 0. It is divisible by the sum of its digits.

IMO ShortList 1998, combinatorics theory problem 7

A solitaire game is played on an m×nm\times n rectangular board, using mnmn markers which are white on one side and black on the other. Initially, each square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one may take away one marker with its black side up, but must then turn over all markers which are in squares having an edge in common with the square of the removed marker. Determine all pairs (m,n)(m,n) of positive integers such that all markers can be removed from the board.
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geO 98 [convex hexagon ABCDEF with B + D + F = 360°]

Let ABCDEFABCDEF be a convex hexagon such that ∠B+∠D+∠F=360∘\angle B+\angle D+\angle F=360^{\circ } and ABBC⋅CDDE⋅EFFA=1. \frac{AB}{BC} \cdot \frac{CD}{DE} \cdot \frac{EF}{FA} = 1. Prove that BCCA⋅AEEF⋅FDDB=1. \frac{BC}{CA} \cdot \frac{AE}{EF} \cdot \frac{FD}{DB} = 1.

IMO ShortList 1998, combinatorics theory problem 6

Ten points are marked in the plane so that no three of them lie on a line. Each pair of points is connected with a segment. Each of these segments is painted with one of kk colors, in such a way that for any kk of the ten points, there are kk segments each joining two of them and no two being painted with the same color. Determine all integers kk, 1≤k≤101\leq k\leq 10, for which this is possible.