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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 \( 100m/n \leq m1 \). 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 nonsquare 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 \( p+q \).
20052021
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