MathDB

4

Part of 1996 Romania National Olympiad

Problems(4)

d(I_1, AD)+ d(I_2, AD)=1/4 BC, bisectors in right ABC (1996 Romania NMO VII p4)

Source:

8/13/2024
In the right triangle ABCABC (m(∠A)=90om ( \angle A) = 90^o) DD is the foot of the altitude from AA. The bisectors of the angles ABDABD and ADBADB intersect in I1I_1 and the bisectors of the angles ACDACD and ADCADC in I2I_2. Find the angles of the triangle if the sum of distances from I1I_1 and I2I_2 to ADAD is equal to 14\frac14 of the length of BCBC.
geometryright triangle
loci of 5 points wanted if GH//BC or BCHG is inscriptible

Source: 1996 Romania NMO IX p4

8/13/2024
In the triangle ABCABC the incircle JJ touches the sides BCBC, CACA, ABAB in DD, EE, FF, respectively. The segments (BE)(BE) and (CF)(CF) intersect JJ in G,HG,H. If BB and CC are fixed points, find the loci of points A,D,E,F,G,HA, D, E, F, G, H if GH∥BCGH \parallel BC and the loci of the same points if BCHGBCHG is an inscriptible quadrilateral.
geometryLocus
3d geo helps provin 6-var ineq xyz+uv(1- x)+(1-y)(1-v)t +(1- z)(1- w)(1- t)<1

Source: 1996 Romania NMO VIII p4

8/13/2024
a) Let ABCDAB CD be a regular tetrahedron. On the sides ABAB, ACAC and ADAD, the points MM, NN and PP, are considered. Determine the volume of the tetrahedron AMNPAMNP in terms of x,y,zx, y, z, where x=AMx=AM, y=ANy=AN, z=APz=AP.
b) Show that for any real numbers x,y,z,t,u,v∈(0,1)x, y, z, t, u, v \in (0, 1) : xyz+uv(1−x)+(1−y)(1−v)t+(1−z)(1−w)(1−t)<1.xyz + uv(1- x) + (1- y)(1- v)t + (1- z)(1- w)(1- t) < 1.
inequalitiesalgebrageometric inequalityGeometric Inequalities3D geometrygeometry
not hard but a cute

Source: Romani 1996

8/23/2005
Let a,b,c∈Za,b,c\in Z and aa be the even number and bb be the odd number. Show that for every integer nn there exist one positive integer xx such that 2n∣ax2+bx+c2^n\mid ax^2+bx+c
number theory