maximal number of 120^o angles in specific 7-gons

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0 replies

jlacosta

Apr 2, 2025

0 replies

Strange angle condition and concyclic points

lminsl 126

N 22 minutes ago by cj13609517288

Source: IMO 2019 Problem 2

In triangle $ABC$ , point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$ . Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$ , respectively, such that $PQ$ is parallel to $AB$ . Let $P_1$ be a point on line $PB_1$ , such that $B_1$ lies strictly between $P$ and $P_1$ , and $\angle PP_1C=\angle BAC$ . Similarly, let $Q_1$ be the point on line $QA_1$ , such that $A_1$ lies strictly between $Q$ and $Q_1$ , and $\angle CQ_1Q=\angle CBA$ .

Prove that points $P,Q,P_1$ , and $Q_1$ are concyclic.

Proposed by Anton Trygub, Ukraine

126 replies

lminsl

Jul 16, 2019

cj13609517288

22 minutes ago

Geo with unnecessary condition

egxa 7

N 23 minutes ago by ehuseyinyigit

Source: Turkey Olympic Revenge 2024 P4

Let the circumcircle of a triangle $ABC$ be $\Gamma$ . The tangents to $\Gamma$ at $B,C$ meet at point $E$ . For a point $F$ on line $BC$ which is not on the segment $BC$ , let the midpoint of $EF$ be $G$ . Lines $GB,GC$ meet $\Gamma$ again at points $I,H$ respectively. Let $M$ be the midpoint of $BC$ . Prove that the points $F,I,H,M$ lie on a circle.

Proposed by Mehmet Can Baştemir

7 replies

egxa

Aug 6, 2024

ehuseyinyigit

23 minutes ago

Functional equations

hanzo.ei 19

N 24 minutes ago by GreekIdiot

Source: Greekldiot

Find all $f: \mathbb R_+ \rightarrow \mathbb R_+$ such that $f(xf(y)+f(x))=yf(x+yf(x)) \: \forall \: x,y \in \mathbb R_+$

19 replies

hanzo.ei

Mar 29, 2025

GreekIdiot

24 minutes ago

Functional Equation

AnhQuang_67 3

N 28 minutes ago by GreekIdiot

Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $2\cdot f\Big(\dfrac{-xy}{2}+f(x+y)\Big)=xf(y)+y(x), \forall x, y \in \mathbb{R}$

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AnhQuang_67

4 hours ago

GreekIdiot

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No more topics!

maximal number of 120^o angles in specific 7-gons

parmenides51 1

N Oct 24, 2021 by JustinLee2017

Source: Nordic Mathematical Contest 1999 #2

Consider $7$ -gons inscribed in a circle such that all sides of the $7$ -gon are of different length. Determine the maximal number of $120^\circ$ angles in this kind of a $7$ -gon.

1 reply

parmenides51

Oct 3, 2017

JustinLee2017

Oct 24, 2021

maximal number of 120^o angles in specific 7-gons

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Source: Nordic Mathematical Contest 1999 #2

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parmenides51

30629 posts

parmenides51

#1 h Oct 3, 2017, 5:44 PM • 2 Y

Y by Adventure10, Mango247

Consider $7$ -gons inscribed in a circle such that all sides of the $7$ -gon are of different length. Determine the maximal number of $120^\circ$ angles in this kind of a $7$ -gon.

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JustinLee2017

1703 posts

JustinLee2017

#2 h Oct 24, 2021, 8:57 PM

Y by

We claim the answer is $2$ angles.
Consider a cyclic heptagon $ABCDEFG$ . If two adjacent angles are equal, then the sides are symmetric with respect to their perpendicular bisector, contradicting our original condition of different-length sides.
Now we examine the case where $3$ non-adjacent angles are equal to $120$ , say $\angle B = \angle D = \angle F = 120^{\circ}$ . If $O$ is the circumcenter of our circle, then $\angle COA = \angle COE = \angle EOG = 120^{\circ}$ , but this means $\angle GOA = 0$ , contradiction. $\quad \blacksquare$

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