MathDB

Problem 3

Part of 2004 Croatia National Olympiad

Problems(4)

absolute value inequality (Croatian MO 2004 1st Grade P3)

Source:

4/8/2021
Prove that for any three real numbers x,y,zx,y,z the following inequality holds: ∣x∣+∣y∣+∣z∣−∣x+y∣−∣y+z∣−∣z+x∣+∣x+y+z∣≥0.|x|+|y|+|z|-|x+y|-|y+z|-|z+x|+|x+y+z|\ge0.
absolute valueinequalities
sequence of largest prime factor of product of previous terms

Source: Croatian MO 2004 2nd Grade P3

4/8/2021
The sequence (pn)n∈N(p_n)_{n\in\mathbb N} is defined by p1=2p_1=2 and, for n≥2n\ge2, pnp_n is the largest prime factor of p1p2⋯pn−1+1p_1p_2\cdots p_{n-1}+1. Show that pn≠5p_n\ne5 for all nn.
number theory
Prove consphericity in a tetrahedron

Source: Croatian MO 2004 3rd Grade P3

4/9/2021
The altitudes of a tetrahedron meet at a single point. Prove that this point, the centroid of one face of the tetrahedron, the foot of the altitude on that face, and the three points dividing the other three altitudes in ratio 2:12:1 (closer to the feet) all lie on a sphere.
geometry3D geometrytetrahedron
three sequences

Source: Croatian MO 2004 4th Grade P3

4/9/2021
The sequences (xn),(yn),(zn),n∈N(x_n),(y_n),(z_n),n\in\mathbb N, are defined by the relations xn+1=2xnxn2−1,yn+1=2ynyn2−1,zn+1=2znzn2−1,x_{n+1}=\frac{2x_n}{x_n^2-1},\qquad y_{n+1}=\frac{2y_n}{y_n^2-1},\qquad z_{n+1}=\frac{2z_n}{z_n^2-1},where x1=2x_1=2, y1=4y_1=4, and x1y1z1=x1+y1+z1x_1y_1z_1=x_1+y_1+z_1. (a) Show that xn2≠1x_n^2\ne1, yn2≠1y_n^2\ne1, zn2≠1z_n^2\ne1 for all nn; (b) Does there exist a k∈Nk\in\mathbb N for which xk+yk+zk=0x_k+y_k+z_k=0?
algebraSequences