MathDB

7

Part of 2013 BMT Spring

Problems(5)

BMT 2013 Spring - Geometry 7

Source:

12/29/2021
Let ABCABC be a triangle with BC=5BC = 5, CA=3CA = 3, and AB=4AB = 4. Variable points P,QP, Q are on segments ABAB, ACAC, respectively such that the area of APQAPQ is half of the area of ABCABC. Let xx and yy be the lengths of perpendiculars drawn from the midpoint of PQPQ to sides ABAB and ACAC, respectively. Find the range of values of 2y+3x2y + 3x.
geometry
2013 BMT Team 7

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1/5/2022
Consider the infinite polynomial G(x)=F1x+F2x2+F3x3+...G(x) = F_1x+F_2x^2 +F_3x^3 +... defined for 0<x<5−120 < x <\frac{\sqrt5 -1}{2} where Fk is the kkth term of the Fibonacci sequence defined to be Fk=Fk−1+Fk−2F_k = F_{k-1} + F_{k-2} with F1=1F_1 = 1, F2=1F_2 = 1. Determine the value a such that G(a)=2G(a) = 2.
number theory
BMT 2013 Spring - Discrete 7

Source:

1/6/2022
Denote by S(a,b)S(a,b) the set of integers kk that can be represented as k=a⋅m+b⋅nk=a\cdot m+b\cdot n, for some non-negative integers mm and nn. So, for example, S(2,4)={0,2,4,6,…}S(2,4)=\{0,2,4,6,\ldots\}. Then, find the sum of all possible positive integer values of xx such that S(18,32)S(18,32) is a subset of S(3,x)S(3,x).
number theory
BMT 2013 Spring - Analysis 7

Source:

1/6/2022
If x,yx,y are positive real numbers satisfying x3−xy+1=y3x^3-xy+1=y^3, find the minimum possible value of yy.
inequalitiesalgebra
2013 BMT Individual 7

Source:

1/18/2022
Given real numbers a,b,ca, b, c such that a+b−c=ab−bc−ca=abc=8a + b - c = ab- bc - ca = abc = 8. Find all possible values of aa.
algebra