Here are some of my composed problems which are exposed to public:

I am a member of the stackexchange and AoPS communities and occasionally solve problems there. Here are some of my selected solutions to problems in these forums:

- For reals \(a, b, c\) holds \((1+a^2)(b-c)^2+(1+b^2)(c-a)^2+(1+c^2)(a-b)^2\ge 2\sqrt{3}|(a-b)(b-c)(c-a)|\)
- For positive reals \(x, y, z\) holds \(5xyz+\left[ \sqrt{\frac{xy+yz+zx}{3}} +\frac{3(x+y)(y+z)(z+x)}{4(xy+yz+zx)} \right]^3 \ge 4(x+y)(y+z)(z+x)\)
- For \(a, b, c\in[0, 4]\) with \(ab+bc+ca=4\) holds \(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\le 3+\sqrt 5\)
- For \(a\ge b\ge c\ge d\ge 0\) with \(a^4+b^4+c^4+d^4=4\) hold \(\frac{1-abcd}{(ac-bd)^2}\ge \frac 1{\sqrt 2}\) and \(\frac{1-abcd}{(a^2-b^2)(c^2-d^2)}\ge 1\)
- For \(a\ge b\ge c\ge 0\) with \(a^2+b^2+c^2=3\) holds \(1-\sqrt[3]{abc}\ge \frac 2{3}(a-b)(b-c)\)
- For \(a\ge b\ge c\ge d\ge 0\) with \(a^2+b^2+c^2+d^2=4\) holds \(1-\sqrt[5]{abcd}\ge \frac 2{5}(a-b)(c-d)\)
- For positive reals \(a, b, c\) with \(a+b+c=1\) maximize \(abc(a-b)(b-c)(c-a)\)
- For reals \(\alpha, \beta, \gamma\) with \(\alpha^3+\beta^3+\gamma^3=0\) holds \(\left[(\alpha-\beta)^2+(\beta-\gamma)^2+(\gamma-\alpha)^2\right]\left[\alpha^4+\beta^4+\gamma^4\right]\geq\left[\alpha^2+\beta^2+\gamma^2\right]^3\)

- Tetrahedron's volume is not less than the third of its biheights' product
- For point \(M\) in face \(ABC\) of tetrahedron \(SABC\) and points \(X\in SA\) with \(SBC\parallel MX\) etc. holds \(27[MXYZ]\le 2[SABC]\)
- For bicentric \(ABCD\) with incenter \(I\) holds \((IA^2 + IC^2)(IB^2 + ID^2) \ge AB \cdot BC \cdot CD \cdot DA\)
- For point \(M\) inside tetrahedron \(A_1A_2A_3A_4\) with \(A_{ij}=A_iA_j\cap MA_kA_l\), \(V_i=[A_iA_{ij}A_{ik}A_{il}]\), \(V=[A_1A_2A_3A_4]\), \(V_0=[A_{12}A_{13}A_{14}A_{23}A_{24}A_{34}]\) holds \(VV_0\geq32\sqrt{V_1V_2V_3V_4}\)
- Doubled distance between the intersections of adjacent angles' bisectors in convex quadrilateral is not less than its sides' alternating sum
- For \(A_1, B_1, C_1\) on the sides of \(ABC\) with \(AB_1A_1C_1\) cyclic holds \(\frac{4S_{A_1B_1C_1}}{S_{ABC}}\le \left(\frac{B_1C_1}{AA_1}\right)^2\)
- For \(A_1\in BC, B_1\in AC\) and \(K\) the midpoint of \(AB\) holds \(9[KA_1B_1]\ge 2[ABC]\)
- The perimeter of the bisectral triangle does not exceed the semiperimeter of the triangle
- For the tetrahedron \(ABCD\) with angles \(\alpha, \beta, \gamma\) between the opposite edges and in- and circumradii \(r, R\) holds \(\left(\frac{3r}{R}\right)^3\leq \sin \frac{\alpha +\beta +\gamma}{3}\)
- For convex body \(A\) there is a plane on which the projection area of \(A\) is not less than surface area of \(A\)
- For point \(P\) in tetrahedron \(ABCD\) with circumradius \(R\), circumcenter \(O\) and \(x=OP\) holds \((R + x) (R - x)^3 \le PA\cdot PB\cdot PC\cdot PD \le (R + x)^3 (R - x)\)
- Construct a plane through the centroid of tetrahedron minimizing sum of volumes of some tetrahedra
- For \(M, N, P\) on the sides of \(ABC\) dividing the sides in triples of equal cyclic sums holds \(4[MNP]\le [ABC]\)

- If some two of the three corner triangles formed by inscribed equilateral triangle are equidistant from the respective point then the second external tangents of the three corner incircles concur
- Find \(AB/CD\) in tetrahedron \(ABCD\) with dihedrons \(BC, CD, DA\) of measure \(\alpha\), and the others of measure \(\beta\)
- Is there a polyhedron whose area ratio of any two faces is at least \(2\)?
- Rigid flat triangle \(T\) with area \(<4\), may be inserted in a triangular hole of area \(3\)
- For two points of a circle joined by a broken line shorter than the diameter exists diameter not intersecting the broken line
- At least \(n-1\) of the angles \(\angle A_iOA_j\) for point \(O\) inside convex polyhedron \(A_1A_2\dots A_n\) are not acute
- Find the probability that for point inside an \(n-1\)-simplex the volumes of the \(n\) smaller simplices are sides of a polygon

- Chris van Tienhoven's Encyclopedia of quadrilateral objects
- PlanetMath
- Encyclopedia of Mathematics

- Cut the Knot - Alexander Bogomolny's website