MathDB

2

Part of 2020 Iran MO (3rd Round)

Problems(4)

verry original of Iran to propose this.

Source: Iranian Third Round 2020 Algebra exam Problem2

11/20/2020
let a1,a2,...,ana_1,a_2,...,a_n,b1,b2,...,bnb_1,b_2,...,b_n,c1,c2,...,cnc_1,c_2,...,c_n be real numbers. prove that ∑cyc∑i∈{1,...,n}(3ai−bi−ci)2≥∑cyc∑i∈{1,2,...,n}ai2 \sum_{cyc}{ \sqrt{\sum_{i \in \{1,...,n\} }{ (3a_i-b_i-c_i)^2}}} \ge \sum_{cyc}{\sqrt{\sum_{i \in \{1,2,...,n\}}{a_i^2}}}
inequalities
Nice geometry in Iran.

Source: Iranian Third Round 2020 Geometry exam Problem2

11/18/2020
Triangle ABCABC with it's circumcircle Γ\Gamma is given. Points DD and EE are chosen on segment BCBC such that ∠BAD=∠CAE\angle BAD=\angle CAE. The circle ω\omega is tangent to ADAD at AA with it's circumcenter lies on Γ\Gamma. Reflection of AA through BCBC is A′A'. If the line A′EA'E meet ω\omega at LL and KK. Then prove either BLBL and CKCK or BKBK and CLCL meet on Γ\Gamma.
geometry
a C7 problem with a C1 hardness

Source: Iranian Third Round 2020 Combinatorics exam Problem2

11/18/2020
For each nn find the number of ways one can put the numbers {1,2,3,...,n}\{1,2,3,...,n\} numbers on the circle, such that if for any 44 numbers a,b,c,da,b,c,d where n∣a+b−c−dn|a+b-c-d. The segments joining a,ba,b and c,dc,d do not meet inside the circle. (Two ways are said to be identical , if one can be obtained from rotaiting the other)
combinatoricscirclemoduloidentical
polynomial with integer roots.

Source: Iranian Third Round 2020 Number Theory exam Problem2

11/21/2020
Find all polynomials PP with integer coefficients such that all the roots of Pn(x)P^n(x) are integers. (here Pn(x)P^n(x) means P(P(...(P(x))...))P(P(...(P(x))...)) where PP is repeated nn times)
number theorypolynomialinteger root