MathDB

2

Part of 1967 IMO Shortlist

Problems(11)

IMO LongList 1967, Bulgaria 2

Source: IMO LongList 1967, Bulgaria 2

11/14/2004
Prove that 13n2+12n+16≥(n!)2n,\frac{1}{3}n^2 + \frac{1}{2}n + \frac{1}{6} \geq (n!)^{\frac{2}{n}}, and let n≥1n \geq 1 be an integer. Prove that this inequality is only possible in the case n=1.n = 1.
factorialInequalityIMO ShortlistIMO Longlist
IMO LongList 1967, Great Britain 2

Source:

12/16/2004
If xx is a positive rational number show that xx can be uniquely expressed in the form x=∑k=1nakk!x = \sum^n_{k=1} \frac{a_k}{k!} where a1,a2,…a_1, a_2, \ldots are integers, 0≤an≤n−10 \leq a_n \leq n - 1, for n>1,n > 1, and the series terminates. Show that xx can be expressed as the sum of reciprocals of different integers, each of which is greater than 106.10^6.
inductionalgebraseries summationrational numberIMO ShortlistIMO Longlist
IMO LongList 1967, Hungary 2

Source: IMO LongList 1967, Hungary 2

12/16/2004
In the space n≥3n \geq 3 points are given. Every pair of points determines some distance. Suppose all distances are different. Connect every point with the nearest point. Prove that it is impossible to obtain (closed) polygonal line in such a way.
geometry3D geometryeuclidean distanceCyclicpoint setIMO ShortlistIMO Longlist
IMO LongList 1967, Mongolia 2

Source: IMO LongList 1967, Mongolia 2

12/16/2004
An urn contains balls of kk different colors; there are nin_i balls of i−thi-th color. Balls are selected at random from the urn, one by one, without replacement, until among the selected balls mm balls of the same color appear. Find the greatest number of selections.
combinatoricscounting
IMO LongList 1967, Italy 2

Source: IMO LongList 1967, Italy 2

12/16/2004
Let ABCDABCD be a regular tetrahedron. To an arbitrary point MM on one edge, say CDCD, corresponds the point P=P(M)P = P(M) which is the intersection of two lines AHAH and BKBK, drawn from AA orthogonally to BMBM and from BB orthogonally to AMAM. What is the locus of PP when MM varies ?
geometry3D geometrytetrahedronperpendicular bisectorIMO ShortlistIMO Longlist
IMO LongList 1967, Poland 2

Source: IMO LongList 1967, Poland 2

12/16/2004
Prove this proposition: Center the sphere circumscribed around a tetrahedron which coincides with the center of a sphere inscribed in that tetrahedron if and only if the skew edges of the tetrahedron are equal.
geometry3D geometryspheretetrahedroncircumcircleIMO ShortlistIMO Longlist
IMO LongList 1967, Romania 2

Source: IMO LongList 1967, Romania 2

12/16/2004
The equation x5+5λx4−x3+(λα−4)x2−(8λ+3)x+λα−2=0x^5 + 5 \lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8 \lambda + 3)x + \lambda \alpha - 2 = 0 is given. Determine α\alpha so that the given equation has exactly (i) one root or (ii) two roots, respectively, independent from λ.\lambda.
algebrapolynomialDiophantine equationparametric equationrootsIMO ShortlistIMO Longlist
IMO LongList 1967, Socialists Republic Of Czechoslovakia 2

Source: IMO LongList 1967, Socialists Republic Of Czechoslovakia 2

12/16/2004
Find all real solutions of the system of equations: ∑k=1nxki=ai\sum^n_{k=1} x^i_k = a^i for i=1,2,…,n.i = 1,2, \ldots, n.
algebrasystem of equationspolynomialIMO ShortlistIMO Longlist
IMO LongList 1967, Sweden 2

Source: IMO LongList 1967, Sweden 2

12/16/2004
Let nn and kk be positive integers such that 1≤n≤N+11 \leq n \leq N+1, 1≤k≤N+11 \leq k \leq N+1. Show that: min⁡n≠k∣sin⁡n−sin⁡k∣<2N. \min_{n \neq k} |\sin n - \sin k| < \frac{2}{N}.
trigonometryInequalityTrigonometric inequalityminimumIMO ShortlistIMO Longlist
IMO LongList 1967, The Democratic Republic Of Germany 2

Source: IMO LongList 1967, The Democratic Republic Of Germany 2

12/16/2004
Which fractions pq, \dfrac{p}{q}, where p,qp,q are positive integers <100< 100, is closest to 2?\sqrt{2} ? Find all digits after the point in decimal representation of that fraction which coincide with digits in decimal representation of 2\sqrt{2} (without using any table).
number theoryapproximationdecimal representationrational numberirrational numberIMO ShortlistIMO Longlist