MathDB

P3

Part of Russian TST 2016

Problems(6)

Incircle of curved triangle?

Source: Russian TST 2016, Day 8 P3 (Groups A & B)

4/19/2023
Two circles ω1\omega_1 and ω2\omega_2 intersecting at points XX{} and YY{} are inside the circle Ω\Omega and touch it at points AA{} and BB{}, respectively; the segments ABAB and XYXY intersect. The line ABAB intersects the circles ω1\omega_1 and ω2\omega_2 again at points CC{} and DD{}, respectively. The circle inscribed in the curved triangle CDXCDX touches the side CDCD at the point ZZ{}. Prove that XZXZ is a bisector of ∠AXB\angle AXB{}.
geometrycircles
Beautiful geometrical inequality

Source: Russian TST 2016, Day 8 P3 (Group NG)

4/19/2023
Prove that for any points A,B,C,DA,B,C,D in the plane, the following inequality holds ABDA+DB+BCDB+DC⩾ACDA+DC.\frac{AB}{DA+DB}+\frac{BC}{DB+DC}\geqslant\frac{AC}{DA+DC}.
geometryInequality
Inequality (idk what to say)

Source: Russian TST 2016, Day 10 P3 (Group A), P4 (Group B)

4/19/2023
Let a,b,ca,b,c be positive real numbers such that a2+b2+c2⩾3a^2+b^2+c^2\geqslant 3. Prove that a2a+b2+b2b+c2+c2c+a2⩾32.\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geqslant\frac{3}{2}.
algebrainequalities
Nice graph theory

Source: Russian TST 2016, Day 11 P3 (Group NG), P4 (Group A)

4/19/2023
A simple graph has NN{} vertices and less than 3(N−1)/23(N-1)/2 edges. Prove that its vertices can be divided into two non-empty groups so that each vertex has at most one neighbor in the group it doesn't belong to.
combinatoricsgraph theory
Very similar to ELMO 2016/6

Source: Russian TST 2016, Day 12 P3

4/20/2023
The scalene triangle ABCABC has incenter II{} and circumcenter OO{}. The points BAB_A and CAC_A are the projections of the points BB{} and CC{} onto the line AIAI. A circle with a diameter BACAB_AC_A intersects the line BCBC at the points KAK_A and LAL_A.
[*]Prove that the circumcircle of the triangle AKALAAK_AL_A touches the incircle of the triangle ABCABC at some point TAT_A. [*]Define the points TBT_B and TCT_C analogously. Prove that the lines ATA,BTBAT_A,BT_B and CTCCT_C intersect on the line OIOI.
geometryincircle
Product of four rational numbers with sum zero

Source: Russian TST 2016, Day 13 P3

4/20/2023
Prove that any rational number can be represented as a product of four rational numbers whose sum is zero.
number theoryrational number