MathDB

1

Part of 1967 IMO Shortlist

Problems(5)

All numbers are exact cubes

Source: IMO Longlist 1967, Bulgaria 1

10/14/2005
Prove that all numbers of the sequence \frac{107811}{3},   \frac{110778111}{3}, \frac{111077781111}{3},   \ldots are exact cubes.
algebranumber theorydecimal representationperfect cubeIMO ShortlistIMO Longlist
IMO LongList 1967, Mongolia 1

Source: IMO LongList 1967, Mongolia 1

12/16/2004
Given m+nm+n numbers: ai,a_i, i=1,2,…,m,i = 1,2, \ldots, m, bjb_j, j=1,2,…,n,j = 1,2, \ldots, n, determine the number of pairs (ai,bj)(a_i,b_j) for which ∣i−j∣≥k,|i-j| \geq k, where kk is a non-negative integer.
combinatoricscountingIMO ShortlistIMO Longlist
IMO LongList 1967, Romania 1

Source: IMO LongList 1967, Romania 1

12/16/2004
Decompose the expression into real factors: E=1−sin⁡5(x)−cos⁡5(x).E = 1 - \sin^5(x) - \cos^5(x).
trigonometryalgebraTrigonometric EquationsfactorizationIMO ShortlistIMO Longlist
IMO LongList 1967, Sweden 1

Source: IMO LongList 1967, Sweden 1

12/16/2004
Determine all positive roots of the equation xx=12. x^x = \frac{1}{\sqrt{2}}.
algebraequationrootsIMO ShortlistIMO LonglistExponential equation
IMO LongList 1967, The Democratic Republic Of Germany 1

Source: IMO LongList 1967, The Democratic Republic Of Germany 1

12/16/2004
Find whether among all quadrilaterals, whose interiors lie inside a semi-circle of radius rr, there exist one (or more) with maximum area. If so, determine their shape and area.
geometryquadrilateralareamaximizationIMO ShortlistIMO Longlist