MathDB

7

Part of 2008 Mathcenter Contest

Problems(3)

(DEF) <=\frac{EF^2}{4 AD^2} if AFDE is cyclic and (ABC)=1

Source: Mathcenter Contest / Oly - Thai Forum 2008 R1 p7 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

11/10/2022
ABCABC is a triangle with an area of 11 square meter. Given the point DD on BCBC, point EE on CACA, point FF on ABAB, such that quadrilateral AFDEAFDE is cyclic. Prove that the area of DEF≤EF24AD2DEF \le \frac{EF^2}{4 AD^2}.
(holmes)
geometryarea of a triangleareasgeometric inequality
\sigma(p^2)=\sigma(q^b)

Source: Mathcenter Contest / Oly - Thai Forum 2008 R2 p7 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

11/11/2022
For every positive integer nn, σ(n)\sigma(n) is equal to the sum of all the positive divisors of nn (for example, σ(6)=1+2+3+6=12\sigma(6)=1+2+3+6=12) . Find the solution of the equation σ(p2)=σ(qb)\sigma(p^2)=\sigma(q^b) where pp and qq are primes where p>q and bb are positive integers.
(gools)
number theorysum of divisors
arithmetic sequence a_n with a_1 prime factor >=i

Source: Mathcenter Contest / Oly - Thai Forum 2008 R3 p7 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

11/9/2022
Let n,dn,d be natural numbers. Prove that there is an arithmetic sequence of positive integers. a1,a2,...,ana_1,a_2,...,a_n with common difference of dd and aia_i with prime factor greater than or equal to ii for all values i=1,2,...,ni=1,2,...,n.
(nooonuii)
arithmetic sequencenumber theory