MathDB

10

Part of 2019 BMT Spring

Problems(5)

Oh, casework bashable inequality, how I've missed you (BMT 2019 Algebra #10)

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5/6/2019
Find the number of ordered integer triplets x,y,z x, y, z with absolute value less than or equal to 100 such that 2x2+3y2+3z2+2xy+2xz−4yz<5 2x^2 + 3y^2 + 3z^2 + 2xy + 2xz - 4yz < 5 .
inequalities
Coaxial Apollonius circles (BMT 2019 Geo #10)

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5/18/2019
A 3-4-5 point of a triangle ABC ABC is a point P P such that the ratio AP:BP:CP AP : BP : CP is equivalent to the ratio 3:4:5 3 : 4 : 5 . If △ABC \triangle ABC is isosceles with base BC=12 BC = 12 and △ABC \triangle ABC has exactly one 3−4−5 3-4-5 point, compute the area of △ABC \triangle ABC .
&quot;I knew this was sum-of-squares bashable!&quot; (BMT 2019 Discrete #10)

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5/26/2019
Let S(n) S(n) be the sum of the squares of the positive integers less than and coprime to n n . For example, S(5)=12+22+32+42 S(5) = 1^2 + 2^2 + 3^2 + 4^2 , but S(4)=12+32 S(4) = 1^2 + 3^2 . Let p=27−1=127 p = 2^7 - 1 = 127 and q=25−1=31 q = 2^5 - 1 = 31 be primes. The quantity S(pq) S(pq) can be written in the form p2q26(a−bc) \frac{p^2q^2}{6}\left(a - \frac{b}{c} \right) where a a , b b , and c c are positive integers, with b b and c c coprime and b<c b < c . Find a a .
2019 BMT Team 10

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1/7/2022
Compute the remainder when the product of all positive integers less than and relatively prime to 20192019 is divided by 20192019.
number theory
area of BMT, MATH square, &lt;MBT = 3.5 &lt; BMT 2019 BMT Individual 10

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1/5/2022
Let MATHMATH be a square with MA=1MA = 1. Point BB lies on ATAT such that ∠MBT=3.5∠BMT\angle MBT = 3.5 \angle BMT. What is the area of △BMT\vartriangle BMT?
geometry