# General Practice Test

 1. (10 p.) The equation $$2^{333x-2} + 2^{111x+2} = 2^{222x+1} + 1$$ has three real roots. Assume that their sum is expressed in the form $$\frac mn$$ where $$m$$ and $$n$$ are relatively prime positive integers. Find $$m+n$$.

 2. (10 p.) Let $$0 < a < b < c < d$$ be integers such that $$a$$, $$b$$, $$c$$ is an arithmetic progression, $$b$$, $$c$$, $$d$$ is a geometric progression, and $$d - a = 30$$. Find $$a + b + c + d$$.

 3. (16 p.) The sequence of complex numbers $$z_0,z_1,z_2,\dots$$ is defined by $$z_0=1+i/211$$ and $$z_{n+1}=\frac{z_n+i}{z_n-i}$$. If $$z_{2111}=\frac ab+\frac cdi$$ for positive integers $$a,b,c,d$$ with $$\gcd(a,b)=\gcd(c,d)=1$$, find $$a+b+c+d$$.

 4. (41 p.) Let $$\triangle ABC$$ have $$AB=6$$, $$BC=7$$, and $$CA=8$$, and denote by $$\omega$$ its circumcircle. Let $$N$$ be a point on $$\omega$$ such that $$AN$$ is a diameter of $$\omega$$. Furthermore, let the tangent to $$\omega$$ at $$A$$ intersect $$BC$$ at $$T$$, and let the second intersection point of $$NT$$ with $$\omega$$ be $$X$$. The length of $$\overline{AX}$$ can be written in the form $$\tfrac m{\sqrt n}$$ for positive integers $$m$$ and $$n$$, where $$n$$ is not divisible by the square of any prime. Find $$m+n$$.

 5. (20 p.) Let $$a$$ and $$b$$ be positive real numbers such that $$ab=2$$ and $\dfrac{a}{a+b^2}+\dfrac{b}{b+a^2}=\dfrac78.$ Find $$a^6+b^6$$.

2005-2021 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax