MathDB

8

Part of 2020 BMT Fall

Problems(5)

BMT Algebra #8 - Inequality

Source:

10/11/2020
Compute the smallest value CC such that the inequality x2(1+y)+y2(1+x)≤(x4+4)(y4+4)+Cx^2(1+y)+y^2(1+x)\le \sqrt{(x^4+4)(y^4+4)}+C holds for all real xx and yy.
Bmtalgebrainequalities
BMT 2020 Fall - Geometry 8

Source:

12/30/2021
Let triangle △ABC \vartriangle ABC have AB=17AB = 17, BC=14BC = 14, CA=12CA = 12. Let MAM_A, MBM_B, MCM_C be midpoints of BC‾\overline{BC}, AC‾\overline{AC}, and AB‾\overline{AB} respectively. Let the angle bisectors of A A, B B, and CC intersect BC‾\overline{BC}, AC‾\overline{AC}, and AB‾\overline{AB} at PP, QQ, and RR, respectively. Reflect MAM_A about AP‾\overline{AP}, MBM_B about BQ‾\overline{BQ}, and MCM_C about CR‾\overline{CR} to obtain MA′M'_A, MB′M'_B, MC′M'_C, respectively. The lines AMA′AM'_A, BMB′BM'_B, and CMC′CM'_C will then intersect BC‾\overline{BC}, AC‾\overline{AC}, and AB‾\overline{AB} at DD, EE, and FF, respectively. Given that AD‾\overline{AD}, BE‾\overline{BE}, and CF‾\overline{CF} concur at a point KK inside the triangle, in simplest form, the ratio [KAB]:[KBC]:[KCA][KAB] : [KBC] : [KCA] can be written in the form p:q:rp : q : r, where pp, qq and r r are relatively prime positive integers and [XYZ][XYZ] denotes the area of △XYZ\vartriangle XYZ. Compute p+q+rp + q + r.
geometry
[AEFB] = [DEFC] = 2[AED] = 2[BFC] unit square 2020 BMT Team 8

Source:

1/7/2022
Let ABCDABCD be a unit square and let EE and FF be points inside ABCDABCD such that the line containing EF‾\overline{EF} is parallel to AB‾\overline{AB}. Point EE is closer to AD‾\overline{AD} than point FF is to AD‾\overline{AD}. The line containing EF‾\overline{EF} also bisects the square into two rectangles of equal area. Suppose [AEFB]=[DEFC]=2[AED]=2[BFC][AEF B] = [DEFC] = 2[AED] = 2[BFC]. The length of segment EF‾\overline{EF} can be expressed as m/nm/n , where m and nn are relatively prime positive integers. Compute m+nm + n.
geometryareassquare
2020 BMT Individual 8

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1/9/2022
By default, iPhone passcodes consist of four base-1010 digits. However, Freya decided to be unconventional and use hexadecimal (base-1616) digits instead of base-1010 digits! (Recall that 1016=161010_{16} = 16_{10}.) She sets her passcode such that exactly two of the hexadecimal digits are prime. How many possible passcodes could she have set?
number theory
2020 BMT Discrete #8

Source:

3/10/2024
Dexter is running a pyramid scheme. In Dexter's scheme, he hires ambassadors for his company, Lie Ultimate. Any ambassador for his company can recruit up to two more ambassadors (who are not already ambassadors), who can in turn recruit up to two more ambassadors each, and so on (Dexter is a special ambassador that can recruit as many ambassadors as he would like). An ambassador's downline consists of the people they recruited directly as well as the downlines of those people. An ambassador earns executive status if they recruit two new people and each of those people has at least 7070 people in their downline (Dexter is not considered an executive). If there are 20202020 ambassadors (including Dexter) at Lie Ultimate, what is the maximum number of ambassadors with executive status?
combinatorics