Log In
Register
IMOmath
Olympiads
Book
Training
IMO Results
Forum
IMOmath
Geometry
1.
(12 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.
(14 p.)
Let \( A,B,C \) be points in the plane such that \( AB=25 \), \( AC=29 \), and \( 45^\circ< \angle BAC< 90^\circ \). Semicircles with diameters \( \overline{AB} \) and \( \overline{AC} \) intersect at a point \( P \) with \( AP=20 \). Find the length of line segment \( \overline{BC} \).
3.
(7 p.)
Let \( K \) and \( M \) be the points on the sides \( AB \) and \( AC \), respectively, of an equilateral triangle \( ABC \) such that \( BK=10 \), \( MK=12 \), and \( MC=8 \). Then the side of the triangle \( ABC \) is equal to \( p+\sqrt q \) for some integers \( p \) and \( q \). Evaluate \( p+q \).
4.
(56 p.)
Let \( \triangle ABC \) be a triangle with \( AB=13 \), \( BC=14 \), and \( CA=15 \). Let \( O \) denote its circumcenter and \( H \) its orthocenter. The circumcircle of \( \triangle AOH \) intersects \( AB \) and \( AC \) at \( D \) and \( E \) respectively. Suppose \( \tfrac{AD}{AE}=\tfrac mn \) where \( m \) and \( n \) are positive relatively prime integers. Find \( mn \).
5.
(9 p.)
Given a rhombus \( ABCD \), the circumradii of the triangles \( ABD \) and \( ACD \) are 12.5 and 25. Find the area of \( ABCD \).
20052017
IMOmath.com
 imomath"at"gmail.com  Math rendered by
MathJax
Home

Olympiads

Book

Training

IMO Results

Forum

Links

About

Contact us