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14

Part of 2014 AIME Problems

Problems(2)

Degree Six Polynomial's Roots

Source: 2014 AIME I Problem 14

3/14/2014
Let mm be the largest real solution to the equation 3x−3+5x−5+17x−17+19x−19=x2−11x−4.\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}= x^2-11x-4. There are positive integers a,b,ca,b,c such that m=a+b+cm = a + \sqrt{b+\sqrt{c}}. Find a+b+ca+b+c.
quadraticsalgebrapolynomialsymmetrySFFTlinear algebraContrived
Find a Length in a Rigid Triangle

Source: 2014 AIME II Problem 14

3/27/2014
In △ABC\triangle ABC, AB=10AB=10, ∠A=30∘\angle A=30^\circ, and ∠C=45∘\angle C=45^\circ. Let H,DH,D, and MM be points on line BC‾\overline{BC} such that AH‾⊥BC‾\overline{AH}\perp\overline{BC}, ∠BAD=∠CAD\angle BAD=\angle CAD, and BM=CMBM=CM. Point NN is the midpoint of segment HM‾\overline{HM}, and point PP is on ray ADAD such that PN‾⊥BC‾\overline{PN}\perp\overline{BC}. Then AP2=mnAP^2=\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
trigonometryAMCAIMEnumber theoryrelatively primegeometrytrig identities