MathDB

2

Part of Oliforum Contest I 2008

Problems(4)

AMC right angle

Source: oliforum contest, round 1, problem 2

9/21/2008
Let ABCD ABCD be a cyclic quadrilateral with AB>CD AB>CD and BC>AD BC>AD. Take points X X and Y Y on the sides AB AB and BC BC, respectively, so that AX\equal{}CD and AD\equal{}CY. Let M M be the midpoint of XY XY. Prove that AMC AMC is a right angle.
vectorgeometrycircumcirclegeometric transformationreflectionparallelogramtrapezoid
strange sequence with floor function

Source: Oliforum Contest I 2008 2.2 https://artofproblemsolving.com/community/c2487525_oliforum_contes

9/28/2021
Let {an}n∈N0 \{a_n\}_{n \in \mathbb{N}_0} be a sequence defined as follows: a1=0 a_1=0, an=a[n2]+(−1)n(n+1)/2 a_n=a_{[\frac{n}{2}]}+(-1)^{n(n+1)/2}, where [x] [x] denotes the floor function. For every k≥0 k \ge 0, find the number n(k) n(k) of positive integers n n such that 2k≤n<2k+1 2^k \le n < 2^{k+1} and an=0 a_n=0.
floor functionalgebrafunction
Oliforum contest- final round - problem2

Source:

12/8/2008
Find all non-negative integers x,y,z x,y,z such that 5^x \plus{} 7^y \equal{} 2^z. :lol: (Daniel Kohen, University of Buenos Aires - Buenos Aires,Argentina)
number theorygreatest common divisornumber theory unsolved
sum 1/{a_j(a_j+a_{j+1})(a_j+a_{j+1}+a_{j+2})...(a_j+a_{j+1}+...+a_{j+n-2})}}

Source: Oliforum Contest I 2008 unused https://artofproblemsolving.com/community/c2487525_oliforum_contes

9/28/2021
Let a1,a2,...,an a_1,a_2,...,a_n with arithmetic mean equals zero; what is the value of: ∑j=1n1aj(aj+aj+1)(aj+aj+1+aj+2)...(aj+aj+1+...+aj+n−2) \sum_{j=1}^n{\frac{1}{a_j(a_j+a_{j+1})(a_j+a_{j+1}+a_{j+2})...(a_j+a_{j+1}+...+a_{j+n-2})}} , where an+k=ak a_{n+k}=a_k ?
algebra