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CGMO-2012 (China Girls Math Olympiad 2012) Problem 8


Find the number of integers k in the set \{0, 1, 2,\cdots, 2012\}   such that  \binom{2012}{k} is a multiple of 2012

CGMO – 2012 ( China Girls Math Olympiad 2012 ) Problem 7


Let  \{a_n\} be a sequence of nondecreasing positive integers such that  \frac{r}{a_{r}}= k+1 for some positive integers k and r. Prove that there exists a positive integer s such that  \frac{s}{a_s} = k

CGMO 2012 ( China Girls Math Olympiad 2012) Problem 5


As shown in the figure below, the in-circle of ABC is tangent to sides AB and AC at D and E respectively, and O is the circumcenter of BCI. Prove that \angle ODB = \angle OEC.
import graph; size(5.55cm); pathpen=linewidth(0.7); pointpen=black; pen fp=fontsize(10); pointfontpen=fp; real xmin=-5.76,xma...

CGMO-2012 (China Girls Math Olympiad 2012) Problem 4


There is a stone at each vertex of a given regular 13-gon, and the color of each stone is black or white. Prove that we may exchange the position of two stones such that the coloring of these stones are symmetric with respect to some symmetric axis of the 13-gon

CGMO 2012 (China Girls Math Olympiad 2012)- Problem 2


Circles Q_1 and Q_2 are tangent to each other externally at T. Points A and E are on Q_1, lines AB and DE are tangent to Q_2 at B and D, respectively, lines AE and BD meet at point P
Prove that

(1) \frac{AB}{AT}=\frac{ED}{ET};
(2) \angle ATP + \angle ETP = 180^{\circ}.
import graph; size(5.97cm); real lsf=0.5; pathpen=linewidth(0.7); pointpen=black; pen fp=fontsize(10); pointfontpen=fp; real ...

CGMO 2012 (China Girls Math Olympiad 2012) Problem 1


Let  a_{1}, a_{2},\ldots, a_{n} be non-negative real numbers. Prove that                                                                          \LARGE \frac{1}{1+a_{1}}+\frac{ a_{1}}{(1+a_{1})(1+a_{2})}+\frac{ a_{1}a_{2}}{(1+a_{1})(1+a_{2})(1+a_{3})}\cdots+\frac{ a_{1}a_{2}\cdots a_{n-1}}{(1+a_{1})(1+a_{2})\cdots (1+a_{n})}\le 1.