IMOmath

General Practice Test

1. (17 p.)
Let \( ABC \) be a rectangular triangle such that \( \angle C=90^o \) and \( AC = 7 \), \( BC = 24 \). Let \( M \) be the midpoint of \( AB \) and \( D \) point on the same side of \( AB \) as \( C \) such that \( DA = DB = 15 \). The area of the triangle \( CDM \) can be expressed as \( \frac{p\sqrt q}r \) for positive integers \( p \), \( q \), \( r \) such that \( q \) is not divisible by a perfect square and \( (p,r)=1 \). Find area \( p+q+r \).

2. (39 p.)
Suppose \( m \) and \( n \) are positive integers with \( m> 1 \) such that the domain of the function \( f(x) = \text{arcsin}(\log_{m}(nx)) \) is a closed interval of length \( \frac{1}{2013} \). Let \( S \) be the smallest possible value of \( m+n \). Find the remainder when \( S \) is divided by \( 1000 \).

3. (7 p.)
Find the least positive integer \( n \) such that when its leftmost digit is deleted, the resulting integer is equal to \( n/29 \).

4. (17 p.)
A circle of radius 1 is randomly placed inside a \( 15 \times 36 \) rectangle \( ABCD \). The probability that it does not intersect the diagonal \( AC \) can be expressed as \( p/q \) where \( p \) and \( q \) are relatively prime integers. Find \( p+q \).

5. (18 p.)
Let \( ABCD \) be a convex quadrilateral with \( AB = CD = 180 \). Assume further that the perimeter of \( ABCD \) is 640, \( AD \neq BC \), and \( \angle A = \angle C \). Then \( \cos \angle A \) can be represented as \( p/q \) for relatively prime positive integers \( p \) and \( q \). Calculate \( p+q \).





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