MathDB

3

Part of 2022 Thailand TSTST

Problems(3)

XY bisects BC

Source: Thailand TSTST 2021, test 1, P3

8/16/2022
An acute scalene triangle ABCABC with circumcircle Ω\Omega is given. The altitude from BB intersects side ACAC at B1B_1 and circle Ω\Omega at B2B_2. The circle with diameter B1B2B_1B_2 intersects circle Ω\Omega again at B3B_3. Similarly, the altitude from CC intersects side ABAB at C1C_1 and circle Ω\Omega at C2C_2. The circle with diameter C1C2C_1C_2 intersects circle Ω\Omega again at C3C_3. Let XX be the intersection of lines B1B3B_1B_3 and C1C3C_1C_3, and let YY be the intersection of lines B3CB_3C and C3BC_3B. Prove that line XYXY bisects side BCBC.
geometryThailandcircumcircle
Weird function

Source: Thailand TSTST 2021, test 2, P3

8/16/2022
Let SS be the set of the positive integers greater than 11, and let nn be from SS. Does there exist a function ff from SS to itself such that for all pairwise distinct positive integers a1,a2,...,ana_1, a_2,...,a_n from SS, we have f(a1)f(a2)...f(an)=f(a1na2n...ann)f(a_1)f(a_2)...f(a_n)=f(a_1^na_2^n...a_n^n)?
functionalgebra
n sums are positive

Source: Thailand TSTST 2021, test 3, P3

8/16/2022
An odd positive integer nn is called pretty if there exists at least one permutation a1,a2,...,ana_1, a_2,..., a_n, of 1,2,...,n1,2,...,n, such that all nn sums a1−a2+a3−...+ana_1-a_2+a_3-...+a_n, a2−a3+a4−...+a1a_2-a_3+a_4-...+a_1,..., an−a1+a2−...+an−1a_n-a_1+a_2-...+a_{n-1} are positive. Find all pretty integers.
combinatorics