Function from the plane to the real numbers

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Function from the plane to the real numbers

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Source: Balkan MO Shortlist 2024 G7

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210 posts

#1 h Apr 28, 2025, 6:50 AM • 1 Y

Y by Ahmed_mallek

Let $f:\pi\rightarrow\mathbb{R}$ be a function from the Euclidean plane to the real numbers such that $f(A)+f(B)+f(C)=f(O)+f(G)+f(H)$ for any acute triangle $ABC$ with circumcenter $O$ , centroid $G$ and orthocenter $H$ . Prove that $f$ is constant.

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#3 h Apr 28, 2025, 8:56 AM

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Fix a point $K$ , and let $f(K)=c$ . Let $X$ be any points on the plane
Consider a rectangle $ABCD$ , such that the centroid of $\triangle ABC$ , $\triangle CDA$ are $K,X$ respectively [*]. Suppose $AC$ and $BD$ intersects at $O$
Apply the condition on $\triangle ABC$ and $\triangle CDA$ yeilds
$f(A)+f(B)+f(C)=f(O)+f(K)+f(B)\implies f(K)=f(A)+f(C)-f(O)$ $f(C)+f(D)+f(A)=f(O)+f(X)+f(D)\implies f(X)=f(A)+f(C)-f(O)$ thus $f(X)=c$

@below ,sorry my eyes are bad

This post has been edited 1 time. Last edited by shanelin-sigma, Apr 28, 2025, 3:43 PM

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#4 h Apr 28, 2025, 9:00 AM

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@above You cannot use the condition on right triangles. `any acute triangle $ABC$ '

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#5 h Apr 28, 2025, 10:24 AM

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Why this is G7 lol

Consider $\triangle ABC$ with circumcenter $O$ , centroid $G$ and orthocenter $H$ .
$D,E,F,M_{OG},M_{OH},M_{HG}$ is midpoint of $BC,CA,AB,OG,OH,HG$
Define $O_A,G_A,H_A$ be circumcenter, centroid and orthocenter of $\triangle AEF$ .
and $H_B,O_B,G_B,H_C,O_C,G_C$ analogously
by $\triangle ABC, \triangle AFE , \triangle BFD , \triangle CDE , \triangle DEF$ and some calculation
known that $f(O)+f(G)=f(M_{OG})+f(M_{HG})$
Let $M'$ be reflection of $M_{HG}$ wrt point $M_{OG}$ notice that $f(M_{HG}=f(M')$
so f is constant. done $\square$

This post has been edited 1 time. Last edited by ItzsleepyXD, Apr 28, 2025, 10:25 AM
Reason: rutthee again.... no he didn't do anything wrong but idk what the reason is

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#6 h Apr 28, 2025, 12:04 PM

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I had already posted this problem, kindly delete post

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#7 h Apr 28, 2025, 1:34 PM

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GreekIdiot wrote:

I had already posted this problem, kindly delete post

umm you posted before lol
should I post solution in your post?

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#8 h Apr 28, 2025, 1:50 PM

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yeah go ahead. thanks

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