MathDB

Problem 2

Part of 1999 Croatia National Olympiad

Problems(4)

How to prove this inequality?

Source: Romanian Mathematical Contest

1/19/2020
How do I prove that, for every a,b,ca, b, c positive real numbers such that a+b+c=1a+b+c = 1 the following inequality holds: a3a2+b2+b3b2+c2+c3c2+a2≥12\frac{a^3}{a^2+b^2} +\frac{b^3}{b^2+c^2} +\frac {c^3}{c^2+a^2} \geq \frac{1}{2}?
inequalitiesinequalities unsolved
x^4-2ax^2+x+a^2-a=0

Source: Croatia 1999 2nd Grade P2

5/17/2021
For a real parameter aa, solve the equation x4−2ax2+x+a2−a=0x^4-2ax^2+x+a^2-a=0. Find all aa for which all solutions are real.
algebra
pyramid configuration

Source: Croatia 1999 3rd Grade P2

5/17/2021
The base of a pyramid ABCDVABCDV is a rectangle ABCDABCD with the sides AB=aAB=a and BC=bBC=b, and all lateral edges of the pyramid have length cc. Find the area of the intersection of the pyramid with a plane that contains the diagonal BDBD and is parallel to VAVA.
geometry3D geometrypyramid
number of permutations, barely increasing condition

Source: Croatia 1999 4th Grade P2

5/17/2021
Let n>1n>1 be an integer. Find the number of permutations (a1,a2,…,an)(a_1,a_2,\ldots,a_n) of the numbers 1,2,…,n1,2,\ldots,n such that ai>ai+1a_i>a_{i+1} holds for exactly one i∈{1,2,…,n−1}i\in\{1,2,\ldots,n-1\}.
combinatorics