# General Practice Test

 1. (25 p.) There are 27 candidates in elections and $$n$$ citizens that vote for them. If a candidate gets $$m$$ votes, then $$100m/n \leq m-1$$. What is the smallest possible value of $$n$$?

 2. (20 p.) Find the largest possible integer $$n$$ such that $$\sqrt n + \sqrt{n+60} = \sqrt m$$ for some non-square integer $$m$$.

 3. (20 p.) The area of the triangle $$ABC$$ is 70. The coordinates of $$B$$ and $$C$$ are $$(12,19)$$ and $$(23,20)$$, respectively, and the coordinates of $$A$$ are $$(p,q)$$. The line containing the median to side BC has slope -5. Find the largest possible value of p+q.

 4. (6 p.) Let $$\alpha$$ be the angle between vectors $$\vec a$$ and $$\vec b$$ with $$|\vec a|=2$$ and $$|\vec b|=3$$, given that the vectors $$\vec m=2\vec a-\vec b$$ and $$\vec n=\vec a+5\vec b$$ are orthogonal. If $$\cos\alpha=\frac pq$$ with $$q>0$$ and $$\gcd(p,q)=1$$, compute $$p+q$$.

 5. (27 p.) A bug moves around a triangle wire. At each vertex it has 1/2 chance of moving towards each of the other two vertices. The probability that after crawling along 10 edges it reaches its starting point can be expressed as $$p/q$$ for positive relatively prime integers $$p$$ and $$q$$. Determine $$p+q$$.

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