MathDB

7

Part of 2015 AIME Problems

Problems(2)

Many Inscribed Squares

Source: 2015 AIME I Problem 7

3/20/2015
In the diagram below, ABCDABCD is a square. Point EE is the midpoint of AD‾\overline{AD}. Points FF and GG lie on CE‾\overline{CE}, and HH and JJ lie on AB‾\overline{AB} and BC‾\overline{BC}, respectively, so that FGHJFGHJ is a square. Points KK and LL lie on GH‾\overline{GH}, and MM and NN lie on AD‾\overline{AD} and AB‾\overline{AB}, respectively, so that KLMNKLMN is a square. The area of KLMNKLMN is 99. Find the area of FGHJFGHJ. [asy] pair A,B,C,D,E,F,G,H,J,K,L,M,N; B=(0,0); real m=7*sqrt(55)/5; J=(m,0); C=(7*m/2,0); A=(0,7*m/2); D=(7*m/2,7*m/2); E=(A+D)/2; H=(0,2m); N=(0,2m+3*sqrt(55)/2); G=foot(H,E,C); F=foot(J,E,C); draw(A--B--C--D--cycle); draw(C--E); draw(G--H--J--F); pair X=foot(N,E,C); M=extension(N,X,A,D); K=foot(N,H,G); L=foot(M,H,G); draw(K--N--M--L); label("AA",A,NW); label("BB",B,SW); label("CC",C,SE); label("DD",D,NE); label("EE",E,dir(90)); label("FF",F,NE); label("GG",G,NE); label("HH",H,W); label("JJ",J,S); label("KK",K,SE); label("LL",L,SE); label("MM",M,dir(90)); label("NN",N,dir(180)); [/asy]
AMCAIMEAIME I2015 AIME I
Area of Rectangle Inscribed in Triangle

Source: 2015 AIME 2 Problem 7

3/26/2015
Triangle ABCABC has side lengths AB=12AB=12, BC=25BC=25, and CA=17CA=17. Rectangle PQRSPQRS has vertex PP on AB‾\overline{AB}, vertex QQ on AC‾\overline{AC}, and vertices RR and SS on BC‾\overline{BC}. In terms of the side length PQ=wPQ=w, the area of PQRSPQRS can be expressed as the quadratic polynomial Area(PQRS)=αw−β⋅w2\text{Area}(PQRS)=\alpha w-\beta\cdot w^2 Then the coefficient β=mn\beta=\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
geometryrectangle