MathDB

2

Part of 1992 Iran MO (2nd round)

Problems(2)

Sequence Inequality

Source:

7/15/2010
In the sequence {an}n=0∞\{a_n\}_{n=0}^{\infty} we have a0=1a_0=1, a1=2a_1=2 and an+1=an+an−11+an−12∀n≥1a_{n+1}=a_n+\dfrac{a_{n-1}}{1+a_{n-1}^2} \qquad \forall n \geq 1 Prove that 52<a1371<6552 < a_{1371} < 65
inequalitiesinductioninequalities unsolved
Prove that MDC is not greater than 45 degrees [Iran 1992]

Source:

11/28/2010
In triangle ABC,ABC, we have ∠A≤90∘\angle A \leq 90^\circ and ∠B=2∠C.\angle B = 2 \angle C. The interior bisector of the angle CC meets the median AMAM in D.D. Prove that ∠MDC≤45∘.\angle MDC \leq 45^\circ. When does equality hold?
[asy] import graph; size(200); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqqqzz = rgb(0,0,0.6); pen ffqqtt = rgb(1,0,0.2); pen qqwuqq = rgb(0,0.39,0); draw((3.14,5.81)--(3.23,0.74),ffqqtt+linewidth(2pt)); draw((3.23,0.74)--(9.73,1.32),ffqqtt+linewidth(2pt)); draw((9.73,1.32)--(3.14,5.81),ffqqtt+linewidth(2pt)); draw((9.73,1.32)--(3.19,2.88)); draw((3.14,5.81)--(6.71,1.05)); pair parametricplot0_cus(real t){ return (0.7*cos(t)+5.8,0.7*sin(t)+2.26); } draw(graph(parametricplot0_cus,-0.9270442638657642,-0.23350086562377703)--(5.8,2.26)--cycle,qqwuqq); dot((3.14,5.81),ds); label("AA", (2.93,6.09),NE*lsf); dot((3.23,0.74),ds); label("BB", (2.88,0.34),NE*lsf); dot((9.73,1.32),ds); label("CC", (10.11,1.04),NE*lsf); dot((3.19,2.88),ds); label("XX", (2.77,2.94),NE*lsf); dot((6.71,1.05),ds); label("MM", (6.75,0.5),NE*lsf); dot((5.8,2.26),ds); label("DD", (5.89,2.4),NE*lsf); label("α\alpha", (6.52,1.65),NE*lsf); clip((-3.26,-11.86)--(-3.26,8.36)--(20.76,8.36)--(20.76,-11.86)--cycle); [/asy]
geometry proposedgeometry