Composed problems
Here are some of my composed problems which are exposed to public:
Selected solutions
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
\(\left[(\alpha-\beta)^2+(\beta-\gamma)^2+(\gamma-\alpha)^2\right]\left[\alpha^4+\beta^4+\gamma^4\right]\geq\left[\alpha^2+\beta^2+\gamma^2\right]^3\)
- Minimize \({\left( {\frac{x}{{2x - y - z}}} \right)^2} + {\left( {\frac{y}{{2y - z - x}}} \right)^2} + {\left( {\frac{z}{{2z - x - y}}} \right)^2}\)
- Find \(x, y>0\) with \(\frac{1}{x+y^2}+\frac{1}{x^2+y}+\frac{1}{xy+1}=\frac{1}{2}(\frac{1}{x}+\frac{1}{y}+\frac{1}{xy})\)
- 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 \(4[MNP]\le
[ABC]\)
- The shortest altitude of a triangle is not longer than the doubled longest altitude of an inscribed acute triangle
- Sum of distances to the faces of an \(n\)-hedron from a point inside it within unit distance to each vertex does not exceed \(n-2\)
- The sum of squares of the face areas of a polyhedron inside unit cube does not exceed \(6\)
- If the vertex angles of trihedron \(OABC\) are \(60^\circ\) then \(AB \cdot BC + BC \cdot CA + CA \cdot AB \ge OA^2 + OB^2 + OC^2\)
- If \(M, N\) are isogonal conjugates inside a triangle \(ABC\) of inradius \(r\) and \(d_a=\textrm{dist}(M,BC), d_a'=\textrm{dist}(N,BC)\) etc. then \(d_ad_bd_cd_a'd_b'd_c' \le r^6\)
- If \(T\) is the Toricelii point of \(ABC\) then \(AB^2 \cdot BC^2 \cdot CA^2 \geq 3(TA^2\cdot TB + TB^2 \cdot TC + TC^2 \cdot TA)(TA\cdot TB^2 + TB \cdot TC^2 + TC \cdot TA^2)\)
- \(\forall \Delta ABC, \forall x, y, z > 0, \forall M\) holds \(x\cdot MA^2+y\cdot MB^2+z\cdot MC^2\geq \frac{xyz}{x+y+z} (\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z})\)
- If no two adjacent faces of a circumscribed polyhedron are painted then the painted area does not exceed the not painted area
Collected problems
Here are chosen problems collected by me that stand out with their beauty:
Favorite books
References
Encyclopedias
Other resources