IMOmath

General Practice Test

1. (29 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 \( p+q \).

2. (12 p.)
Find the least positive integer \( n \) such that when its leftmost digit is deleted, the resulting integer is equal to \( n/29 \).

3. (26 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 \( p+q+r \).

4. (24 p.)
Find the largest possible integer \( n \) such that \( \sqrt n + \sqrt{n+60} = \sqrt m \) for some non-square integer \( m \).

5. (7 p.)
Let \( a \), \( b \), \( c \) be positive integers forming an increasing geometric sequence such that \( b-a \) is a square. If \( \log_6a + \log_6b + \log_6c = 6 \), find \( a + b + c \).





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