IMOmath

Geometry

1. (17 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 \( p+q \).

2. (13 p.)
Given a rhombus \( ABCD \), the circumradii of the triangles \( ABD \) and \( ACD \) are 12.5 and 25. Find the area of \( ABCD \).

3. (19 p.)
Let \( AXYZB \) be a convex pentagon inscribed in a semicircle with diameter \( AB \). Suppose \( AZ-AX=6 \), \( BX-BZ=9 \), \( AY=12 \), and \( BY=5 \). Find the greatest integer not exceeding the perimeter of quadrilateral \( OXYZ \), where \( O \) is the midpoint of \( AB \).

4. (29 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 \( b \).

5. (19 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.





2005-2017 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax
Home | Olympiads | Book | Training | IMO Results | Forum | Links | About | Contact us