MathDB
1999 Chile NMO Juniors XI

Source:

October 18, 2021
algebrageometrynumber theorycombinatoricschilean NMO

Problem Statement

[url=https://artofproblemsolving.com/community/c1068820h2917815p26064153]p1. In a triangle ABCABC the median CMCM is such that ∠ACB\angle ACB is divided in the ratio 1:21: 2. The median CMCM is extended to point DD such that ∠DAC=90o\angle DAC = 90^o. Show that CD=2BCCD = 2BC.
[url=https://artofproblemsolving.com/community/c1068820h2917816p26064158]p2. Given two tangent equal circles with centers OO and O1O_1 respectively . Determine the distance at which a parallel line OO1OO_1 should be drawn so that the hatched areas in the figure are equal. https://cdn.artofproblemsolving.com/attachments/0/3/96182b4e3515734d7c5531b0e1a8e80b70ea16.png
p3. There are 19991999 balls numbered from 11 to 19991999. Find all positive integers kk with the following property: The total of balls can be separated into kk groups, so that the sum of the numbers of the balls in each of the groups is the same.
p4. Three married couples, the Alvarezes, the Barros and the Castros, went to buy Christmas gifts. Each woman bought a quantity pp of gifts in 1000p21000p^2 pesos and each woman (Miriam, Francisca and Gladys) spent 147,000 147,000 pesos more than her respective husband. Miriam is known to have bought 99 fewer gifts that Mr. Barros and that Francisca bought 4747 less gifts than Mr. Castro. Who is married with Gladys?
[url=https://artofproblemsolving.com/community/c1068820h2917817p26064161]p5. In the figure, ADAD and CBCB are two chords of a circle, which intersect at EE, and FGFG is the bisector of ∠AED\angle AED. Show that AF⋅BG=CG⋅FDAF \cdot BG = CG \cdot FD. https://cdn.artofproblemsolving.com/attachments/6/0/0e29eeaae46afea0f8c646ec223a27e1134e6a.png
p6. Does the equation x2+y2=1999xyx^2 + y^2 = 1999xy has positive integer solutions?
[url=https://artofproblemsolving.com/community/c1068820h2946119p26374834]p7. In a circle, 140140 points are located. Show that, among them, you can choose 19991999 pairs so that the arc subtended by the two points of each pair is less than or equal to 120o120^o.
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