MathDB

Problem 3

Part of 2003 Croatia National Olympiad

Problems(4)

find angles in isosceles triangle (Croatia MO 2003 1st Grade P3)

Source:

4/9/2021
In an isosceles triangle with base aa, lateral side bb, and height to the base vv, it holds that a2+v≥b2\frac a2+v\ge b\sqrt2. Find the angles of the triangle. Compute its area if b=82b=8\sqrt2.
geometry
inequality on sequence (Croatia MO 2003 2nd Grade P3)

Source:

4/10/2021
For positive numbers a1,a2,…,ana_1,a_2,\ldots,a_n (n≥2n\ge2) denote s=a1+…+ans=a_1+\ldots+a_n. Prove that a1s−a1+…+ans−an≥nn−1.\frac{a_1}{s-a_1}+\ldots+\frac{a_n}{s-a_n}\ge\frac n{n-1}.
inequalities
angles in a tetrahedron

Source: Croatia MO 2003 3rd Grade P3

4/10/2021
In a tetrahedron ABCDABCD, all angles at vertex DD are equal to α\alpha and all dihedral angles between faces having DD as a vertex are equal to ϕ\phi. Prove that there exists a unique α\alpha for which ϕ=2α\phi=2\alpha.
geometry3D geometrytetrahedron
reversing order of first k natural numbers

Source: Croatia MO 2003 4th Grade P3

4/10/2021
The natural numbers 11 through 20032003 are arranged in a sequence. We repeatedly perform the following operation: If the first number in the sequence is kk, the order of the first kk terms is reversed. Prove that after several operations number 11 will occur on the first place.
game