MathDB

2001 IMO Shortlist

Part of IMO Shortlist

Subcontests

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IMO ShortList 2001, geometry problem 7

Let OO be an interior point of acute triangle ABCABC. Let A1A_1 lie on BCBC with OA1OA_1 perpendicular to BCBC. Define B1B_1 on CACA and C1C_1 on ABAB similarly. Prove that OO is the circumcenter of ABCABC if and only if the perimeter of A1B1C1A_1B_1C_1 is not less than any one of the perimeters of AB1C1,BC1A1AB_1C_1, BC_1A_1, and CA1B1CA_1B_1.

IMO ShortList 2001, combinatorics problem 7

A pile of nn pebbles is placed in a vertical column. This configuration is modified according to the following rules. A pebble can be moved if it is at the top of a column which contains at least two more pebbles than the column immediately to its right. (If there are no pebbles to the right, think of this as a column with 0 pebbles.) At each stage, choose a pebble from among those that can be moved (if there are any) and place it at the top of the column to its right. If no pebbles can be moved, the configuration is called a final configuration. For each nn, show that, no matter what choices are made at each stage, the final configuration obtained is unique. Describe that configuration in terms of nn. [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=119189]IMO ShortList 2001, combinatorics problem 7, alternative
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IMO ShortList 2001, geometry problem 6

Let ABCABC be a triangle and PP an exterior point in the plane of the triangle. Suppose the lines APAP, BPBP, CPCP meet the sides BCBC, CACA, ABAB (or extensions thereof) in DD, EE, FF, respectively. Suppose further that the areas of triangles PBDPBD, PCEPCE, PAFPAF are all equal. Prove that each of these areas is equal to the area of triangle ABCABC itself.

IMO ShortList 2001, number theory problem 6

Is it possible to find 100100 positive integers not exceeding 25,00025,000, such that all pairwise sums of them are different?

IMO ShortList 2001, combinatorics problem 6

For a positive integer nn define a sequence of zeros and ones to be balanced if it contains nn zeros and nn ones. Two balanced sequences aa and bb are neighbors if you can move one of the 2n2n symbols of aa to another position to form bb. For instance, when n=4n = 4, the balanced sequences 0110100101101001 and 0011010100110101 are neighbors because the third (or fourth) zero in the first sequence can be moved to the first or second position to form the second sequence. Prove that there is a set SS of at most 1n+1(2nn)\frac{1}{n+1} \binom{2n}{n} balanced sequences such that every balanced sequence is equal to or is a neighbor of at least one sequence in SS.
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IMO ShortList 2001, geometry problem 5

Let ABCABC be an acute triangle. Let DAC,EABDAC,EAB, and FBCFBC be isosceles triangles exterior to ABCABC, with DA=DC,EA=EBDA=DC, EA=EB, and FB=FCFB=FC, such that \angle ADC = 2\angle BAC,   \angle BEA= 2 \angle ABC,   \angle CFB = 2 \angle ACB. Let D′D' be the intersection of lines DBDB and EFEF, let E′E' be the intersection of ECEC and DFDF, and let F′F' be the intersection of FAFA and DEDE. Find, with proof, the value of the sum DBDD′+ECEE′+FAFF′. \frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}.

IMO ShortList 2001, algebra problem 5

Find all positive integers a1,a2,…,ana_1, a_2, \ldots, a_n such that 99100=a0a1+a1a2+⋯+an−1an, \frac{99}{100} = \frac{a_0}{a_1} + \frac{a_1}{a_2} + \cdots + \frac{a_{n-1}}{a_n}, where a0=1a_0 = 1 and (ak+1−1)ak−1≥ak2(ak−1)(a_{k+1}-1)a_{k-1} \geq a_k^2(a_k - 1) for k=1,2,…,n−1k = 1,2,\ldots,n-1.

IMO ShortList 2001, combinatorics problem 5

Find all finite sequences (x0,x1,…,xn)(x_0, x_1, \ldots,x_n) such that for every jj, 0≤j≤n0 \leq j \leq n, xjx_j equals the number of times jj appears in the sequence.
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