2009

Prove $C = \sqrt{AB}$.

2008

Prove $a+b+c = 2(r+R)$.

2007

Prove the red triangles are similar.

2006

Prove the red line bisects the blue angle.

2005

The blue lines are parallel to the sides of the red triangle. Prove the area of the red triangle is $\left( \sqrt{A}+\sqrt{B}+\sqrt{C} \right)^2$.

2004

The green points are the midpoints of the sides of the yellow orthic triangle. Prove the red lines are concurrent.

2003

Prove the sum of the blue line segments equals the red line segment.

2002

The yellow hexagon is centrally symmetric. The blue triangles are equilateral. The red points are the midpoints of the green line segments. Prove the red hexagon is regular.

2001

The blue point is the orthocenter of the blue triangle, the yellow point is the orthocenter of the yellow triangle, and the green point is the orthocenter of the green triangle. Prove the red triangles have the same area.

2000

Prove radii of the red circles are $\dfrac{Rh_a}{a+h_a}$, $\dfrac{Rh_b}{b+h_b}$, and $\dfrac{Rh_c}{c+h_c}$.

1999

Given a regular $30$-gon. Prove the red lines are concurrent, the blue lines are concurrent, and the green lines are concurrent.

1998

Prove $R = a + b + c$.

1997

Prove the product of the radii of the red circles equals the product of the radii of the blue circles.

1996

The yellow triangle is equilateral. Prove the sum of the radii of the blue circles equals the sum of the radii of the red circles.

1995

Given an isosceles trapezoid. Prove $r = \frac{1}{2}\left( s + t + \sqrt{s^2+t^2} \right)$.

1994

Given a square. The red and blue triangles are equilateral. Prove the radius of the green circle is double the radius of the yellow circle.

1993

Prove the red line segments are congruent and perpendicular.

1992

The blue line is the perpendicular bisector of the green line segment. Prove the red and blue lines are concurrent.

1991

The blue circle bisects two sides of the yellow parallelogram. Find the measure of the red angle.

1990

Prove the red lines are parallel.

1989

The circles are congruent. Prove the sum of the red line segments equals the sum of the blue line segments.

1988

Prove the red points are collinear.

1987

The yellow and green shapes are squares. Prove the blue and red triangles have the same area.

1986

Prove the red lines are concurrent.

1985

The blue points are the midpoints of the sides of the triangle. Prove the red line segments are congruent.

1984

Prove the red line segments are congruent.

1983

Given a square. The red and blue triangles are congruent. Find the measures of their angles.

1982

Prove the red angles are congruent.

1981

Prove the red lines are perpendicular.

1980

Congruent circles are centered at the vertices of a regular hexagon. Prove sum of the measures of the colored arcs is six times the measure of the red angle.

1979

Prove the blue lines are parallel and the red line is tangent to the circle.

1978

Prove the red line segments are congruent.

1977

The red, green, and blue line segments are parallel to the sides of the yellow triangle. Prove $\dfrac{p}{a}+\dfrac{q}{b}+\dfrac{r}{c}=2$.

1976

The red line segments are congruent and parallel to the sides of the yellow triangle. Prove their length is $\dfrac{2abc}{ab+bc+ca}$.

1975

Prove the red lines are concurrent.

1974

Prove the red lines are perpendicular.

1973

Prove the red shape is a parallelogram.

1972

The circles are congruent. Prove the red lines are parallel.

1971

Prove $PC = \sqrt{ PA \cdot PB }$.

1970

Prove the red points are concyclic.

1969

A square is inscribed in a circle. Prove the red and blue areas are the same.

1968

The blue shape is an isosceles trapezoid. Prove the red line segments are congruent.

1967

Prove the red line segments are congruent.

1966

Prove the red lines are parallel.

1965

Prove the sum of the green line segments is double the red line segment.

1964

The blue shape is a parallelogram. Prove the red points are concyclic.

1963

Prove the red line bisects the blue diagonal.

1962

Prove the red lines are perpendicular.

1961

The blue vectors are parallel to the altitudes of the triangle. Prove the sum of the blue vectors is the red vector.

1960

Prove the red lines are perpendicular.

1959

Prove the red lines are concurrent.

1958

Prove the red lines are perpendicular.

1957

The yellow shape is a square. Prove the red points are collinear.

1956

Prove the red points are collinear.

1955

The blue points are midpoints of opposite sides of the yellow quadrilateral. Prove the red line segments are congruent.

1954

The blue line segments are congruent. Prove the red triangle is isosceles.