MathDB

6

Part of 2023 Iran Team Selection Test

Problems(2)

Point P in triangle BHC

Source: Iran TST 2023 ; Exam 1 Problem 6

3/16/2023
ABCABC is an acute triangle with orthocenter HH. Point PP is in triangle BHCBHC that ∠HPC=3∠HBC\angle HPC = 3 \angle HBC and ∠HPB=3∠HCB\angle HPB =3 \angle HCB . Reflection of point PP through BH,CHBH,CH is X,YX,Y. if SS is the center of circumcircle of AXYAXY , Prove that: ∠BAS=∠CAP\angle BAS = \angle CAP
Proposed by Pouria Mahmoudkhan Shirazi
geometrycircumcircle
2n subset from {0,1,2,...,2n-1}

Source: Iran TST 2023 ; Exam 2 Problem 6

3/16/2023
Suppose that we have 2n2n non-empty subset of {0,1,2,...,2n−1} \big\{0,1,2,...,2n-1\big\} that sum of the elements of these subsets is (2n+12) \binom{2n+1}{2} . Prove that we can choose one element from every subset that some of them is (2n2) \binom{2n}{2}
Proposed by Morteza Saghafian and Afrouz Jabalameli
Subsetscombinatorics