USA Math Olympiad

AMC 10 Paper Folding Geometry

We fold a paper using GeoGebra and explore a problem from American Mathematical Contest (AMC 10)

Problem: A rectangular piece of paper whose length is ( \sqrt 3 ) times the width has area A. The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area B. What is the ratio B: A?

You may also try another paper folding scenario. Here we make the crease a variable!

AIME I Math Olympiad USA Math Olympiad

Geometry Problems in AIME; problems and discussions.

Let’s have a problem discussion of Geometry problems in AIME. (American Invitational Mathematics Competitions). Give them a try.

  1. In \Delta ABC, AB = 3, BC = 4 , and CA = 5. Circle \omega intersects\overline{AB} at E and B, \overline{BC} at B and D, and \overline{AC} at F and G. Given that EF=DF and \displaystyle \dfrac{DG}{EG} = \frac{3}{4} , length \displaystyle DE=\dfrac{a\sqrt{b}}{c} , where a and c are relatively prime positive integers, and b is a positive integer not divisible by the square of any prime. Find a+b+c. (2014 AIME I Problems/Problem 15)
  2. Circle C with radius 2 has diameter \overline{AB}. Circle D is internally tangent to circle C at A. Circle E is internally tangent to circle C, externally tangent to circle D, and tangent to \overline{AB}. The radius of circle D is three times the radius of circle E, and can be written in the form \sqrt{m}-n, where m and n are positive integers. Find m+n. (2014 AIME II Problems/Problem 8)
  3. A rectangle has sides of length a and 36. A hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length 36. The sides of length a can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length a parallel and separated by a distance of 24, the hexagon has the same area as the original rectangle. Find a^2. (2014 AIME II Problems/Problem 3)
  4. In \triangle{ABC}, AB=10, \angle{A}=30^\circ , and \angle{C=45^\circ}. Let H, D, and M be points on the line BC such that AH\perp{BC}, \angle{BAD}=\angle{CAD}, and BM=CM. Point N is the midpoint of the segment HM, and point P is on ray AD such that PN\perp{BC}.  Then AP^2=\dfrac{m}{n}, where m and n are relatively prime positive integers. Find m+n. (2014 AIME II Problems/Problem 14)
  5. In triangle RED, measured \angle DRE=75^{\circ} and measured \angle RED=45^{\circ}. abs{RD}=1. Let M be the midpoint of segment \overline{RD}. Point C lies on side \overline{ED} such that \overline{RC} \perp \overline{EM}. Extend segment \overline{DE} through E to point A such that CA=AR. Then AE=\frac{a-\sqrt{b}}{c}, where a and c are relatively prime positive integers, and b is a positive integer. Find a+b+c. (2014 AIME II Problems/Problem 11)
  6. In triangle ABC, AB= \frac{20}{11} AC. The angle bisector of angle A intersects BC at point D, and point M is the midpoint of AD. Let P be the point of the intersection of AC and BM. The ratio of CP to PA can be expressed in the form \dfrac{m}{n}, where m and n are relatively prime positive integers. Find m+n. (2011 AIME II Problems/Problem 4)
  7. The degree measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle. (2011 AIME II Problems/Problem 3)
  8. On square ABCD, point E lies on side AD and point F lies on side BC, so that BE=EF=FD=30. Find the area of the square ABCD. (2011 AIME II Problems/Problem 2)
  9. Point P lies on the diagonal AC of square ABCD with AP > CP. Let O_{1} and O_{2} be the circumcenters of triangles ABP and CDP respectively. Given that AB = 12 and {\angle O_{1}PO_{2}} = 120^{\circ}, then AP = \sqrt{a} + \sqrt{b}, where a and b are positive integers. Find a + b. (2011 AIME II Problems/Problem 13)
  10. Gary purchased a large beverage, but only drank m/n of it, where m and n are relatively prime positive integers. If he had purchased half as much and drunk twice as much, he would have wasted only 2/9 as much beverage. Find m+n. (2011 AIME II Problems/Problem 1)
  11. Let ABCDEF be a regular hexagon. Let G, H, I, J, K, and L be the midpoints of sides AB, BC, CD, DE, EF, and AF, respectively. The segments \overbar{AH} , \overbar{BI}, \overbar{CJ}, \overbar{DK}, \overbar{EL}, and \overbar{FG} bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of ABCDEF be expressed as a fraction \frac {m}{n} where m and n are relatively prime positive integers. Find m + n. (2010 AIME II Problems/Problem 9)
  12. Triangle ABC with right angle at C, \angle BAC < 45^\circ and AB = 4. Point P on \overbar{AB} is chosen such that \angle APC = 2\angle ACP and CP = 1. The ratio \frac{AP}{BP} can be represented in the form p + q\sqrt{r}, where p, q, r are positive integers and r is not divisible by the square of any prime. Find p+q+r. (2010 AIME II Problems/Problem 14)
  13. Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is 8: 7. Find the minimum possible value of their common perimeter. (2010 AIME II Problems/Problem 12)
  14. In triangle{ABC} with AB = 12, BC = 13, and AC = 15, let M be a point on \overline{AC} such that the incircles of triangle{ABM} and triangle{BCM} have equal radii. Let p and q be positive relatively prime integers such that \frac {AM}{CM} = \frac {p}{q}. Find p + q. (2010 AIME I Problems/Problem 15)
  15. Rectangle ABCD and a semicircle with diameter AB are coplanar and have nonoverlapping interiors. Let \mathcal{R} denote the region enclosed by the semicircle and the rectangle. Line ell meets the semicircle, segment AB, and segment CD at distinct points N, U, and T, respectively. Line ell divides region \mathcal{R} into two regions with areas in the ratio 1: 2. Suppose that AU = 84, AN = 126, and UB = 168. Then DA can be represented as m\sqrt {n}, where m and n are positive integers and n is not divisible by the square of any prime. Find m + n. (2010 AIME I Problems/Problem 13)
  16. Let \mathcal{R} be the region consisting of the set of points in the coordinate plane that satisfy both |8 - x| + y \le 10 and 3y - x \ge 15. When \mathcal{R} is revolved around the line whose equation is 3y – x = 15, the volume of the resulting solid is \frac {m\pi}{n\sqrt {p}}, where m, n, and p are positive integers, m and n are relatively prime, and p is not divisible by the square of any prime. Find m + n + p. (2010 AIME I Problems/Problem 11)