IMO Practice Test — Loci
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
medium
Two fixed points A and B are separated by a distance of 10 cm. The locus of a moving point P is handled such that the area of the triangle PAB always stays exactly equal to 40 square cm. Describe the complete geometric locus of P.
A circle of radius 4 cm centered at the midpoint of AB
A single straight line running parallel to segment AB at a distance of 4 cm
Two separate lines parallel to AB, one on each side, at a distance of 8 cm
A perpendicular bisector of segment AB
Explanation: Area = 0.5 * base * height => 40 = 0.5 * 10 * height => height = 8 cm. The altitude from P to AB must be 8 cm, which yields two parallel lines.
Question 2 of 6
medium
A ladder of fixed length L stands vertically against a high wall. The foot of the ladder begins to slide out horizontally along the floor while the top slides down the wall. What is the path traced out by the exact midpoint of the ladder during this collapse?
A straight slanted line
A parabolic curve
A circular arc of radius L/2 centered at the corner
A quarter-ellipse path
Explanation: In a right triangle, the median to the hypotenuse is half the hypotenuse length. The midpoint stays exactly L/2 away from the corner vertex.
Question 3 of 6
medium
Find the number of distinct points in a two-dimensional plane that are simultaneously at a distance of 3 cm from a fixed point K and equidistant from two other fixed points M and N, given that K lies directly on the perpendicular bisector of MN.
0 points
1 point
2 points
Infinite points
Explanation: The perpendicular bisector passes through the center of the 3 cm circle. Any line passing through the center of a circle cuts it at exactly 2 points.
Question 4 of 6
medium
A point P moves in a plane such that the difference of the squares of its distances from two fixed points A and B is a constant non-zero value k. The locus of P is a...
Circle centered at the midpoint
Straight line perpendicular to AB but not bisecting it
Straight line parallel to AB
Hyperbolic curve
Explanation: Let A=(-c,0), B=(c,0). (x+c)^2+y^2 - ((x-c)^2+y^2) = k => 4cx = k => x = k/(4c), which represents a vertical line perpendicular to the axis.
Question 5 of 6
medium
Three distinct non-collinear points A, B, and C are given in a plane. How many unique points can be found in the entire plane that are completely equidistant from all three points simultaneously?
0 points
1 point
3 points
Infinite points
Explanation: The equidistant points are found by intersecting the perpendicular bisectors. For three non-collinear points, they intersect at exactly 1 point (the circumcenter).
Question 6 of 6
medium
Two lines intersect at an angle of 60 degrees at point O. A point P moves such that the sum of the perpendicular distances from P to both lines is always equal to a fixed length standard s. In any single quadrant region, the path traced by P forms a...
Segment of a circle
Straight line segment
Portion of a parabola
Elliptical arc
Explanation: Let distances be x and y. The rule is x + y = s. In linear coordinate geometry, any first-degree equation of the form x + y = s plots as a straight line segment.