MathDB

8

Part of 2019 BMT Spring

Problems(5)

The most complex coin ever (BMT 2019 Algebra #8)

Source:

5/6/2019
A biased coin has a 6+2312 \dfrac{6 + 2\sqrt{3}}{12} chance of landing heads, and a 6−2312 \dfrac{6 - 2\sqrt{3}}{12} chance of landing tails. What is the probability that the number of times the coin lands heads after being flipped 100 times is a multiple of 4? The answer can be expressed as 14+1+abc⋅de \dfrac{1}{4} + \dfrac{1 + a^b}{c \cdot d^e} where a,b,c,d,e a, b, c, d, e are positive integers. Find the minimal possible value of a+b+c+d+e a + b + c + d + e .
Coordbash? Call me crazy... (BMT 2019 Geo #8)

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5/18/2019
Let △ABC \triangle ABC be a triangle with AB=13 AB = 13 , BC=14 BC = 14 , and CA=15 CA = 15 . Let G G denote the centroid of △ABC \triangle ABC , and let GA G_A denote the image of G G under a reflection across BC‾ \overline{BC} , with GB G_B the image of G G under a reflection across AC‾ \overline{AC} , and GC G_C the image of G G under a reflection across AB‾ \overline{AB} . Let OG O_G be the circumcenter of △GAGBGC \triangle G_AG_BG_C and let X X be the intersection of AOG‾ \overline{AO_G} with BC‾ \overline{BC} . Let Y Y denote the intersection of AG‾ \overline{AG} with BC‾ \overline{BC} . Compute XY XY .
[a phi function butterfly flies by] Is this brute force? (BMT 2019 Discrete #8)

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5/25/2019
For a positive integer n n , define ϕ(n) \phi(n) as the number of positive integers less than or equal to n n that are relatively prime to n n . Find the sum of all positive integers n n such that ϕ(n)=20 \phi(n) = 20 .
function
2019 BMT Team 8

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1/7/2022
Let (ki)(k_i) be a sequence of unique nonzero integers such that x2−5x+kix^2- 5x + k_i has rational solutions. Find the minimum possible value of 15∑i=1∞1ki\frac15 \sum_{i=1}^{\infty} \frac{1}{k_i}
algebranumber theory
2019 BMT Individual 8

Source:

1/9/2022
Let ϕ=12019\phi =\frac{1}{2019}. Define gn={0 if    round(nϕ)=round    ((n−1)ϕ)1 otherwise..g_n =\begin{cases} 0 & \text{ if} \,\,\,\, round (n\phi) = round \,\,\,\, ((n - 1)\phi) \\ 1 & \text{ otherwise} .\end{cases}. where round (x)(x) denotes the round function. Compute the expected value of gng_n if nn is an integer chosen from interval [1,20192][1, 2019^2].
algebra