MathDB

13

Part of 2013 AIME Problems

Problems(2)

Repeated Sequence of Triangles

Source: 2013 AIME I Problem 13

3/15/2013
Triangle AB0C0AB_0C_0 has side lengths AB0=12AB_0 = 12, B0C0=17B_0C_0 = 17, and C0A=25C_0A = 25. For each positive integer nn, points BnB_n and CnC_n are located on ABn−1‾\overline{AB_{n-1}} and ACn−1‾\overline{AC_{n-1}}, respectively, creating three similar triangles △ABnCn∼△Bn−1CnCn−1∼△ABn−1Cn−1\triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1} \sim \triangle AB_{n-1}C_{n-1}. The area of the union of all triangles Bn−1CnBnB_{n-1}C_nB_n for n≥1n\geq1 can be expressed as pq\tfrac pq, where pp and qq are relatively prime positive integers. Find qq.
geometrytrigonometrysimilar trianglesnumber theoryrelatively primegeometric sequencetrig identities
Find the area of isosceles ABC

Source: AIME II 2013, Problem 13

4/4/2013
In △ABC\triangle ABC, AC=BCAC = BC, and point DD is on BC‾\overline{BC} so that CD=3⋅BDCD = 3 \cdot BD. Let EE be the midpoint of AD‾\overline{AD}. Given that CE=7CE = \sqrt{7} and BE=3BE = 3, the area of △ABC\triangle ABC can be expressed in the form mnm\sqrt{n}, where mm and nn are positive integers and nn is not divisible by the square of any prime. Find m+nm+n.
geometryratioPythagorean Theoremarea of a triangleHeron's formulaalgebrasystem of equations