MathDB

3

Part of 2022 Israel TST

Problems(3)

n kinds of fruit, friends in a class

Source: 2022 Israel TST 3 P3

5/22/2022
A class has 30 students. To celebrate 'Tu BiShvat' each student chose some dried fruits out of nn different kinds. Say two students are friends if they both chose from the same type of fruit. Find the minimal nn so that it is possible that each student has exactly 66 friends.
combinatoricscombinatorics unsolved
Circumcenter on BC, Angle Bisectors

Source: 2022 Israel TST 8 P3

5/21/2022
In triangle ABCABC, the angle bisectors are BEBE and CFCF (where E,FE, F are on the sides of the triangle), and their intersection point is II. Point NN lies on the circumcircle of AEFAEF, and the angle ∠IAN\angle IAN is right. The circumcircle of AEFAEF meets the line NINI a second time at the point LL. Show that the circumcenter of AILAIL lies on line BCBC.
geometrycircumcircleIMO 2022 tstangle bisectorcircles
AX and HT meet on circumcircle

Source: 2022 Israel TST test 10 P3

7/18/2022
Scalene triangle ABCABC has incenter II and circumcircle Ω\Omega with center OO. HH is the orthocenter of triangle BICBIC, and TT is a point on Ω\Omega for which ∠ATI=90∘\angle ATI=90^\circ. Circle (AIO)(AIO) intersects line IHIH again at XX. Show that the lines AX,HTAX, HT intersect on Ω\Omega.
geometryTSTIsraelcircumcircleincenter