IMOmath

General Practice Test

1. (22 p.)
Assume that all sides of the convex hexagon \( ABCDEF \) are equal and the opposite sides are parallel. Assume further that \( \angle FAB = 120^o \). The \( y \)-coordinates of \( A \) and \( B \) are 0 and 2 respectively, and the \( y \)-coordinates of the other vertices are 4, 6, 8, 10 in some order. The area of \( ABCDEF \) can be written as \( a\sqrt b \) for some integers \( a \) and \( b \) such that \( b \) is not divisible by a perfect square other than 1. Find \( a+b \).

2. (19 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 \( p+q \).

3. (24 p.)
Let \( K \) and \( L \) be the points on the sides \( AB \) and \( BC \) of an equilateral triangle \( ABC \) such that \( AK=5 \) and \( CL=2 \). If \( M \) is the point on \( AC \) such that \( \angle KML=60^o \), and if the area of the triangle \( KML \) is equal to \( 14\sqrt3 \) then the side of the triangle \( ABC \) can assume two values \( \frac{a\pm \sqrt b}c \) for some natural numbers \( a \), \( b \), and \( c \). If \( b \) is not divisible by a perfect square other than 1, find the value of \( b \).

4. (14 p.)
Let \( ABC \) be a triangle with sides 3, 4, 5 and \( DEFG \) a \( 6 \times 7 \) rectangle. A line divides \( \triangle ABC \) into a triangle \( T_1 \) and a trapezoid \( R_1 \). Another line divides the rectangle \( DEFG \) into a triangle \( T_2 \) and a trapezoid \( R_2 \), in such a way \( T_1\sim T_2 \) and \( R_1\sim R_2 \). The smallest possible value for the area of \( T_1 \) can be expressed as \( p/q \) for two relatively prime positive integers \( p \) and \( q \). Evaluate \( p+q \).

5. (19 p.)
Consider the set \( S\subseteq(0,1]^2 \) in the coordinate plane that consists of all points \( (x,y) \) such that both \( [\log_2(1/x)] \) and \( [\log_5(1/y)] \) are even. The area of \( S \) can be written in the form \( p/q \) for two relatively prime integers \( p \) and \( q \). Evaluate \( p+q \).





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