MathDB

3

Part of 1967 IMO Shortlist

Problems(11)

IMO LongList 1967, Bulgaria 3

Source: IMO LongList 1967, Bulgaria 3

11/14/2004
Prove the trigonometric inequality cos⁡x<1−x22+x416,\cos x < 1 - \frac{x^2}{2} + \frac{x^4}{16}, when x∈(0,π2).x \in \left(0, \frac{\pi}{2} \right).
trigonometrycalculusTaylor seriesInequalityTrigonometric inequalityIMO ShortlistIMO Longlist
IMO LongList 1967, Great Britain 3

Source: IMO LongList 1967, Great Britain 3

12/16/2004
The nn points P1,P2,…,PnP_1,P_2, \ldots, P_n are placed inside or on the boundary of a disk of radius 1 in such a way that the minimum distance DnD_n between any two of these points has its largest possible value Dn.D_n. Calculate DnD_n for n=2n = 2 to 7. and justify your answer.
geometrypoint seteuclidean distancemaximizationIMO ShortlistIMO Longlist
IMO LongList 1967, Hungary 3

Source: IMO LongList 1967, Hungary 3

12/16/2004
Without using tables, find the exact value of the product: P=∏k=17cos⁡(kπ15).P = \prod^7_{k=1} \cos \left(\frac{k \pi}{15} \right).
trigonometryalgebraProductCalculateIMO ShortlistIMO Longlistcomplex numbers
IMO LongList 1967, Mongolia 3

Source: IMO LongList 1967, Mongolia 3

12/16/2004
Determine the volume of the body obtained by cutting the ball of radius RR by the trihedron with vertex in the center of that ball, it its dihedral angles are α,β,γ.\alpha, \beta, \gamma.
geometry3D geometrysphereVolumeIMO ShortlistIMO Longlist
IMO LongList 1967, Italy 3

Source: IMO LongList 1967, Italy 3

12/16/2004
Which regular polygon can be obtained (and how) by cutting a cube with a plane ?
geometry3D geometrypolygoncubeIntersectionIMO ShortlistIMO Longlist
IMO LongList 1967, Poland 3

Source: IMO LongList 1967, Poland 3

12/16/2004
Prove that for arbitrary positive numbers the following inequality holds 1a+1b+1c≤a8+b8+c8a3b3c3.\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq \frac{a^8 + b^8 + c^8}{a^3b^3c^3}.
Inequalitythree variable inequalityMuirheadIMO ShortlistIMO Longlistalgebrainequalities proposed
IMO LongList 1967, Romania 3

Source: IMO LongList 1967, Romania 3

12/16/2004
Suppose that pp and qq are two different positive integers and xx is a real number. Form the product (x+p)(x+q).(x+p)(x+q). Find the sum S(x,n)=∑(x+p)(x+q),S(x,n) = \sum (x+p)(x+q), where pp and qq take values from 1 to n.n. Does there exist integer values of xx for which S(x,n)=0.S(x,n) = 0.
algebrapolynomialSummationequationDiophantine equationIMO ShortlistIMO Longlist
IMO LongList 1967, Socialists Republic Of Czechoslovakia 3

Source: IMO LongList 1967, Socialists Republic Of Czechoslovakia 3

12/16/2004
Circle kk and its diameter ABAB are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on ABAB and two other vertices on k.k.
geometryincenterangle bisectorLocusLocus problemsIMO ShortlistIMO Longlist
IMO LongList 1967, Sweden 3

Source: IMO LongList 1967, Sweden 3

12/16/2004
The function φ(x,y,z)\varphi(x,y,z) defined for all triples (x,y,z)(x,y,z) of real numbers, is such that there are two functions ff and gg defined for all pairs of real numbers, such that φ(x,y,z)=f(x+y,z)=g(x,y+z)\varphi(x,y,z) = f(x+y,z) = g(x,y+z) for all real numbers x,yx,y and z.z. Show that there is a function hh of one real variable, such that φ(x,y,z)=h(x+y+z)\varphi(x,y,z) = h(x+y+z) for all real numbers x,yx,y and z.z.
functionalgebrafunctional equationIMO ShortlistIMO Longlist
IMO LongList 1967, The Democratic Republic Of Germany 3

Source: IMO LongList 1967, The Democratic Republic Of Germany 3

12/16/2004
Suppose tan⁡α=pq\tan \alpha = \dfrac{p}{q}, where pp and qq are integers and q≠0q \neq 0. Prove that the number tan⁡β\tan \beta for which tan⁡2β=tan⁡3α\tan {2 \beta} = \tan {3 \alpha} is rational only when p2+q2p^2 + q^2 is the square of an integer.
trigonometrynumber theoryTrigonometric EquationsDiophantine equationIMO ShortlistIMO Longlist