# michaelcai

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### Determine a constant $k$ such that the polynomial$$P(x, y, z) = x^5 + y^5 + z^5 + k(x^3+y^3+z^3)(x^2+y^2+z^2)$$is divisible by $x+y+z$.

michaelcai  Feb 20, 2018
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### Last logarithm problems

michaelcai  Feb 12, 2018
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### Logarithm problem

michaelcai  Feb 12, 2018
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michaelcai  Feb 12, 2018
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michaelcai  Feb 12, 2018
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### Let $f(x)$ be a polynomial with integer coefficients. Suppose there are four distinct integers $p,q,r,s$ such that$$f(p) = f(q) = f(r) = f(s michaelcai Jan 29, 2018 +1 1132 7 +598 ### Problem: If f is a polynomial of degree 4 such that$$f(0) = f(1) = f(2) = f(3) = 1$$and$$f(4) = 0, then determine $f(5)$.

michaelcai  Jan 29, 2018
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### Suppose $f(x)$ is a polynomial of degree $4$ or greater such that $f(1)=2$, $f(2)=3$, and $f(3)=5$. Find the remainder when $f(x)$ is divid

michaelcai  Jan 26, 2018
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### Evaluate the infinite geometric series:​

michaelcai  Jan 25, 2018
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### Find a polynomial $f(x)$ of degree $5$ such that both of these properties hold: $\bullet$ $f(x)-1$ is divisible by $(x-1)^3$. $\bullet$ $f(x michaelcai Jan 23, 2018 0 1015 1 +598 ### Suppose the polynomial$f(x)$is of degree$3$and satisfies$f(3)=2$,$f(4)=4$,$f(5)=-3$, and$f(6)=8$. Determine the value of$f(0)$. michaelcai Jan 23, 2018 0 849 1 +598 ### Find all integers$n$for which$\frac{n^2+n+1}{n-1}$is an integer. michaelcai Jan 23, 2018 +1 1141 2 +598 ### Suppose$f(x)$is a polynomial of degree$4$or greater such that$f(1)=2$,$f(2)=3$, and$f(3)=5$. Find the remainder when$f(x)$is divide michaelcai Jan 23, 2018 0 992 4 +598 ### Let$f(x)=ax^2+bx+a$, where$a$and$b$are constants and$a\ne 0$. If one of the roots of the equation$f(x)=0$is$x=4$, what is the other michaelcai Dec 20, 2017 +2 773 6 +598 ### Prove that if$w,z$are complex numbers such that$|w|=|z|=1$and$wz\ne -1$, then$\frac{w+z}{1+wz}\$ is a real number.

michaelcai  Dec 14, 2017