IMOmath

General Practice Test

1. (5 p.)
Find the product of the real roots of the equation \( x^2+18x+30=2\sqrt{x^2+18x+45} \) (the answer is an integer).

2. (57 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 \( m+n \).

3. (3 p.)
Let \( S \) be the set of vertices of a unit cube. Find the number of triangles whose vertices belong to \( S \).

4. (24 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 \( p+q \).

5. (9 p.)
Given a regular 12-gon D, determine the number of squares that have two or more vertices among the vertices of D.





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