MathDB

Problem 5

Part of 1999 Mongolian Mathematical Olympiad

Problems(5)

inequality

Source:

2/23/2017
Given a;b;ca;b;c satisfying a2+b2+c2=2a^{2}+b^{2}+c^{2}=2 . Prove that: a) ∣a+b+c−abc∣⩽2\left | a+b+c-abc \right |\leqslant 2 . b) ∣a3+b3+c3−3abc∣⩽22\left | a^{3}+b^{3}+c^{3}-3abc \right |\leqslant 2\sqrt{2}
Inequalityinequalities
find locus when circle through two points varies

Source: Mongolia 1999 Grade 9 P5

5/4/2021
Let DD be a point in the angle ABCABC. A circle γ\gamma passing through BB and DD intersects the lines ABAB and BCBC at MM and NN respectively. Find the locus of the midpoint of MNMN when circle γ\gamma varies.
geometry
existence of subset with conditions

Source: Mongolia 1999 Grade 10 P5

5/5/2021
Let A1,…,AmA_1,\ldots,A_m be three-element subsets of an nn-element set XX such that ∣Ai∪Aj∣≤1|A_i\cup A_j|\le1 whenever i≠ji\ne j. Prove that there exists a subset AA of XX with ∣A∣≥2n|A|\ge2\sqrt n such that it does not contain any of the AiA_i.
combinatorics
# of polynomials divisible by x^3+x^2+x+1, coefficients in [1999], degree=6

Source: Mongolia 1999 Teachers elementary level P5

5/6/2021
Find the number of polynomials P(x)P(x) of degree 66 whose coefficients are in the set {1,2,…,1999}\{1,2,\ldots,1999\} and which are divisible by x3+x2+x+1x^3+x^2+x+1.
algebrapolynomial
Mongolia 1999

Source:

10/30/2008
The edge lengths of a tetrahedron are a, b, c, d, e, f, the areas of its faces are S1, S2, S3, S4, and its volume is V . Prove that 2 S1 S2 S3 S4 > 3V abcdef this problem comes from: http://www.imomath.com/othercomp/jkasfvgkusa/MonMO99.pdf I was just wondering if someone could write it in LATEX form. _____________________________________ EDIT by moderator: If you type The edge lengths of a tetrahedron are a,b,c,d,e,f,a, b, c, d, e, f, the areas of its faces are S1,S2,S3,S4,S_1, S_2, S_3, S_4, and its volume is V.V. Prove that
2S1S2S3S4>3Vabcdef62 \sqrt{S_1 S_2 S_3 S_4} > 3V \sqrt[6]{abcdef} it shows up as: The edge lengths of a tetrahedron are a,b,c,d,e,f, a, b, c, d, e, f, the areas of its faces are S1,S2,S3,S4, S_1, S_2, S_3, S_4, and its volume is V. V. Prove that 2S1S2S3S4>3Vabcdef6 2 \sqrt{S_1 S_2 S_3 S_4} > 3V \sqrt[6]{abcdef}
geometry3D geometrytetrahedronLaTeXgeometry unsolved