MathDB

1997 IberoAmerican

Part of IberoAmerican

Subcontests

(3)
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2

12nd ibmo - mexico 1997/q3.

Let n≥2n \geq2 be an integer number and DnD_n the set of all the points (x,y)(x,y) in the plane such that its coordinates are integer numbers with: −n≤x≤n-n \le x \le n and −n≤y≤n-n \le y \le n.
(a) There are three possible colors in which the points of DnD_n are painted with (each point has a unique color). Show that with any distribution of the colors, there are always two points of DnD_n with the same color such that the line that contains them does not go through any other point of DnD_n.
(b) Find a way to paint the points of DnD_n with 4 colors such that if a line contains exactly two points of DnD_n, then, this points have different colors.

12nd ibmo - mexico 1997/q6.

Let P={P1,P2,...,P1997}P = \{P_1, P_2, ..., P_{1997}\} be a set of 19971997 points in the interior of a circle of radius 1, where P1P_1 is the center of the circle. For each k=1.…,1997k=1.\ldots,1997, let xkx_k be the distance of PkP_k to the point of PP closer to PkP_k, but different from it. Show that (x1)2+(x2)2+...+(x1997)2≤9.(x_1)^2 + (x_2)^2 + ... + (x_{1997})^2 \le 9.
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12nd ibmo - mexico 1997/q2.

In a triangle ABCABC, it is drawn a circumference with center in the incenter II and that meet twice each of the sides of the triangle: the segment BCBC on DD and PP (where DD is nearer two BB); the segment CACA on EE and QQ (where EE is nearer to CC); and the segment ABAB on FF and RR ( where FF is nearer to AA).
Let SS be the point of intersection of the diagonals of the quadrilateral EQFREQFR. Let TT be the point of intersection of the diagonals of the quadrilateral FRDPFRDP. Let UU be the point of intersection of the diagonals of the quadrilateral DPEQDPEQ.
Show that the circumcircle to the triangle △FRT\triangle{FRT}, △DPU\triangle{DPU} and △EQS\triangle{EQS} have a unique point in common.

12nd ibmo - mexico 1997/q5.

In an acute triangle △ABC\triangle{ABC}, let AEAE and BFBF be highs of it, and HH its orthocenter. The symmetric line of AEAE with respect to the angle bisector of ∢A\sphericalangle{A} and the symmetric line of BFBF with respect to the angle bisector of ∢B\sphericalangle{B} intersect each other on the point OO. The lines AEAE and AOAO intersect again the circuncircle to △ABC\triangle{ABC} on the points MM and NN respectively.
Let PP be the intersection of BCBC with HNHN; RR the intersection of BCBC with OMOM; and SS the intersection of HRHR with OPOP. Show that AHSOAHSO is a paralelogram.
1
2

12nd ibmo - mexico 1997/q1.

Let r≥1r\geq1 be areal number that holds with the property that for each pair of positive integer numbers mm and nn, with nn a multiple of mm, it is true that ⌊nr⌋\lfloor{nr}\rfloor is multiple of ⌊mr⌋\lfloor{mr}\rfloor. Show that rr has to be an integer number.
Note: If xx is a real number, ⌊x⌋\lfloor{x}\rfloor is the greatest integer lower than or equal to xx}.

12nd ibmo - mexico 1997/q4.

Let nn be a positive integer. Consider the sum x1y1+x2y2+⋯+xnynx_1y_1 + x_2y_2 +\cdots + x_ny_n, where that values of the variables x1,x2,…,xn,y1,y2,…,ynx_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n are either 0 or 1.
Let I(n)I(n) be the number of 2nn-tuples (x1,x2,…,xn,y1,y2,…,yn)(x_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n) such that the sum of the number is odd, and let P(n)P(n) be the number of 2nn-tuples (x1,x2,…,xn,y1,y2,…,yn)(x_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n) such that the sum is an even number. Show that: P(n)I(n)=2n+12n−1 \frac{P(n)}{I(n)}=\frac{2^n+1}{2^n-1}