MathDB

3

Part of 2021 Israel TST

Problems(5)

Tangent to incircles.

Source: ISR 2021 TST1 p.3

5/4/2022
Let ABCABC be an acute triangle with orthocenter HH. Prove that there is a line ll which is parallel to BCBC and tangent to the incircles of ABHABH and ACHACH.
geometryincircleorthocentercommon tangent
Bars is hungry!

Source: ISR 2021 TST2 p.3

5/4/2022
A game is played on a n×nn \times n chessboard. In the beginning Bars the cat occupies any cell according to his choice. The dd sparrows land on certain cells according to their choice (several sparrows may land in the same cell). Bars and the sparrows play in turns. In each turn of Bars, he moves to a cell adjacent by a side or a vertex (like a king in chess). In each turn of the sparrows, precisely one of the sparrows flies from its current cell to any other cell of his choice. The goal of Bars is to get to a cell containing a sparrow. Can Bars achieve his goal a) if d=⌊3⋅n225⌋d=\lfloor \frac{3\cdot n^2}{25}\rfloor, assuming nn is large enough? b) if d=⌊3⋅n219⌋d=\lfloor \frac{3\cdot n^2}{19}\rfloor, assuming nn is large enough? c) if d=⌊3⋅n214⌋d=\lfloor \frac{3\cdot n^2}{14}\rfloor, assuming nn is large enough?
Combinatorial gamescombinatorics
Congruent triangles inscribed in ABC

Source: ISR 2021 February TST p.3

5/4/2022
Consider a triangle ABCABC and two congruent triangles A1B1C1A_1B_1C_1 and A2B2C2A_2B_2C_2 which are respectively similar to ABCABC and inscribed in it: Ai,Bi,CiA_i,B_i,C_i are located on the sides of ABCABC in such a way that the points AiA_i are on the side opposite to AA, the points BiB_i are on the side opposite to BB, and the points CiC_i are on the side opposite to CC (and the angle at A are equal to angles at AiA_i etc.). The circumcircles of A1B1C1A_1B_1C_1 and A2B2C2A_2B_2C_2 intersect at points PP and QQ. Prove that the line PQPQ passes through the orthocenter of ABCABC.
geometrycongruent triangles
8 real variables, minimal k

Source: 2021 Israel TST 8 P3

5/31/2022
What is the smallest value of kk for which the inequality \begin{align*} ad-bc+yz&-xt+(a+c)(y+t)-(b+d)(x+z)\leq \\ &\leq k\left(\sqrt{a^2+b^2}+\sqrt{c^2+d^2}+\sqrt{x^2+y^2}+\sqrt{z^2+t^2}\right)^2 \end{align*} holds for any 88 real numbers a,b,c,d,x,y,z,ta,b,c,d,x,y,z,t?
Edit: Fixed a mistake! Thanks @below.
inequalities
Incenters of incenters

Source: 2021 Israel TST Test 6 P3

7/25/2022
In an inscribed quadrilateral ABCDABCD, we have BC=CDBC=CD but AB≠ADAB\neq AD. Points II and JJ are the incenters of triangles ABCABC and ACDACD respectively. Point KK was chosen on segment ACAC so that IK=JKIK=JK. Points MM and NN are the incenters of triangles AIKAIK and AJKAJK. Prove that the perpendicular to CDCD at DD and the perpendicular to KIKI at II intersect on the circumcircle of MANMAN.
geometryincentercircumcircle