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Source: ILL 1970 - Problem 13.

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#1 h May 24, 2011, 3:32 AM • 1 Y

Y by Adventure10

Each side of an arbitrary $\triangle ABC$ is divided into equal parts, and lines parallel to $AB,BC,CA$ are drawn through each of these points, thus cutting $\triangle ABC$ into small triangles. Points are assigned a number in the following manner:
$(1)$ $A,B,C$ are assigned $1,2,3$ respectively
$(2)$ Points on $AB$ are assigned $1$ or $2$
$(3)$ Points on $BC$ are assigned $2$ or $3$
$(4)$ Points on $CA$ are assigned $3$ or $1$
Prove that there must exist a small triangle whose vertices are marked by $1,2,3$ .

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jred

#2 h Apr 28, 2022, 3:06 AM

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Suppose there is no small triangle whose vertices are marked by $1,2,3$ . Now we label all segments, without any cutting point inside, whose ends are marked by $1$ and $2$ . Let $p, q$ be the number of labeled segments on the boundary of $\triangle ABC$ (more specifically on side $AB$ ) and inside $\triangle ABC$ . It is easy to see that $p$ must be an odd number.
On the other hand, let $t$ be the number of small triangles whose vertices are marked by $1, 1, 2$ or $1, 2, 2$ . Since each of such small triangle contains two labeled segments, then there are $2t$ labeled segments in total. Notice that every labeled segment lied inside $\triangle ABC$ has been counted twice, hence $2t=p+2q$ , implying $p$ is even, which is a contradiction.

This post has been edited 1 time. Last edited by jred, Apr 28, 2022, 3:20 AM

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