MathDB

Problem 2

Part of 2004 Croatia National Olympiad

Problems(4)

orthogonal medians (Croatian MO 2004 1st Grade P2)

Source:

4/8/2021
Prove that the medians from the vertices AA and BB of a triangle ABCABC are orthogonal if and only if BC2+AC2=5AB2BC^2+AC^2=5AB^2.
geometry
inequality, sum of a^2/(a+b)(a+c) (Croatian MO 2004 2nd Grade P2)

Source:

4/8/2021
If a,b,ca,b,c are positive numbers, prove the inequality a2(a+b)(a+c)+b2(b+c)(b+a)+c2(c+a)(c+b)≥34.\frac{a^2}{(a+b)(a+c)}+\frac{b^2}{(b+c)(b+a)}+\frac{c^2}{(c+a)(c+b)}\ge\frac34.
inequalities
inequality in triangle

Source: Croatian MO 2004 3rd Grade P2

4/8/2021
If a,b,ca,b,c are the sides and α,β,γ\alpha,\beta,\gamma the corresponding angles of a triangle, prove the inequality cos⁡αa3+cos⁡βb3+cos⁡γc3≥32abc.\frac{\cos\alpha}{a^3}+\frac{\cos\beta}{b^3}+\frac{\cos\gamma}{c^3}\ge\frac3{2abc}.
inequalitiesgeometric inequality
given angle relationships in triangle, prove equality

Source: Croatian MO 2004 4th Grade P2

4/9/2021
Points PP and QQ inside a triangle ABCABC with sides a,b,ca,b,c and the corresponding angle α,β,γ\alpha,\beta,\gamma satisfy ∠BPC=∠CPA=∠APB=120∘\angle BPC=\angle CPA=\angle APB=120^\circ and ∠BQC=60∘+α\angle BQC=60^\circ+\alpha, ∠CQA=60∘+β\angle CQA=60^\circ+\beta, ∠AQB=60∘+γ\angle AQB=60^\circ+\gamma. Prove the equality (AP+BP+CP)3⋅AQ⋅BQ⋅CQ=(abc)2.(AP+BP+CP)^3\cdot AQ\cdot BQ\cdot CQ=(abc)^2.
geometry