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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
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0 replies
jlacosta
May 1, 2025
0 replies
Trigo relation in a right angled. ISIBS2011P10
Sayan   11
N a few seconds ago by Project_Donkey_into_M4
Show that the triangle whose angles satisfy the equality
\[\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2\]
is right angled.
11 replies
+1 w
Sayan
Mar 31, 2013
Project_Donkey_into_M4
a few seconds ago
Expressing polynomial as product of two polynomials
Sadigly   3
N 5 minutes ago by Sadigly
Source: Azerbaijan Senior NMO 2021
Define $P(x)=((x-a_1)(x-a_2)...(x-a_n))^2 +1$, where $a_1,a_2...,a_n\in\mathbb{Z}$ and $n\in\mathbb{N^+}$. Prove that $P(x)$ couldn't be expressed as product of two non-constant polynomials with integer coefficients.
3 replies
Sadigly
Yesterday at 9:10 PM
Sadigly
5 minutes ago
Help me this problem. Thank you
illybest   0
15 minutes ago
Find f: R->R such that
f( xy + f(z) ) = (( xf(y) + yf(x) )/2) + z
0 replies
illybest
15 minutes ago
0 replies
Product of consecutive terms divisible by a prime number
BR1F1SZ   2
N 22 minutes ago by bin_sherlo
Source: 2025 Francophone MO Seniors P4
Determine all sequences of strictly positive integers $a_1, a_2, a_3, \ldots$ satisfying the following two conditions:
[list]
[*]There exists an integer $M > 0$ such that, for all indices $n \geqslant 1$, $0 < a_n \leqslant M$.
[*]For any prime number $p$ and for any index $n \geqslant 1$, the number
\[
a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p}
\]is a multiple of $p$.
[/list]


2 replies
BR1F1SZ
Yesterday at 12:09 AM
bin_sherlo
22 minutes ago
No more topics!
50 points in plane
pohoatza   13
N May 6, 2025 by cursed_tangent1434
Source: JBMO 2007, Bulgaria, problem 3
Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color.
13 replies
pohoatza
Jun 28, 2007
cursed_tangent1434
May 6, 2025
50 points in plane
G H J
Source: JBMO 2007, Bulgaria, problem 3
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pohoatza
1145 posts
#1 • 13 Y
Y by Adventure10, Mathlover_1, OronSH, aidan0626, Blue_banana4, PikaPika999, and 7 other users
Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color.
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bodan
267 posts
#2 • 24 Y
Y by amirmath1995, Catalanfury, Nguyenhuyhoang, silouan, raven_, DepressedCubic, Adventure10, Mathlover_1, Mango247, aidan0626, Blue_banana4, Funcshun840, PikaPika999, kiyoras_2001, and 10 other users
Lemma. Among $n$ points in a plane positioned generally (no three collinear) we have at least $\frac{n(n-1)(n-8)}{6}$ scalene triangles.
Proof. Suppose that $n$ points are fixed and the number of isosceles triangles is $\alpha$. Set $\beta$ to be the number of bases of these triangles (we count three bases for each equilateral and one for each nonequilateral isosceles triangle). Clearly $\beta\geq \alpha$. Then each segment connecting a pair of points can be a base of at most two triangles, as if this is not the case three points will lie on this pairs' perpendicular bisector. Thus there are at most $2{n\choose 2}$ isosceles triangles, yielding that there are at least ${n\choose 3}-2{n\choose 2}=\frac{n(n-1)(n-8)}{6}$ scalene triangles. $\square$


In the problem there are at least $13$ points of the same color and we apply the lemma.
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reveryu
218 posts
#3 • 3 Y
Y by Adventure10, Mango247, PikaPika999
why the fact "there are at most $2{n\choose 2}$ isosceles triangles"
does not imply "#scalene triangle + #acute triangle(except isosceles and equilateral triangle) is at least ${n\choose 3}-2{n\choose 2}=\frac{n(n-1)(n-8)}{6}$ " ??
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huashiliao2020
1292 posts
#4 • 1 Y
Y by PikaPika999
bodan wrote:
Lemma. Among $n$ points in a plane positioned generally (no three collinear) we have at least $\frac{n(n-1)(n-8)}{6}$ scalene triangles.
Proof. Suppose that $n$ points are fixed and the number of isosceles triangles is $\alpha$. Set $\beta$ to be the number of bases of these triangles (we count three bases for each equilateral and one for each nonequilateral isosceles triangle). Clearly $\beta\geq \alpha$. Then each segment connecting a pair of points can be a base of at most two triangles, as if this is not the case three points will lie on this pairs' perpendicular bisector. Thus there are at most $2{n\choose 2}$ isosceles triangles, yielding that there are at least ${n\choose 3}-2{n\choose 2}=\frac{n(n-1)(n-8)}{6}$ scalene triangles. $\square$


In the problem there are at least $13$ points of the same color and we apply the lemma.


Thanks. I had almost the exact same solution, except one question. Why do you care about “ we count three bases for each equilateral and one for each nonequilateral isosceles triangle”? It seems that already 2 bases of triangles at most x n choose 2 pairs of points to form an isosceles triangle is enough.
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megarnie
5607 posts
#5 • 2 Y
Y by OronSH, PikaPika999
We solve the following problem first:

Given are $13$ points in the plane, no three of them belonging to a same line. Prove that there are at least $130$ scalene triangles with vertices in the plane.

Notice that any edge between two points in the plane can be a base of at most two isosceles triangles (because otherwise we would have $3$ points on the perpendicular bisector). Hence there are at most $\binom{13}{2} \cdot 2 = 256$ isosceles triangles, so at lesat $\binom{13}{3} - 256 = 130$ scalene triangles.



This solves the original problem because there must exist a color with at least $\left\lceil \frac{50}{4} \right\rceil = 13$ points of that color.
This post has been edited 3 times. Last edited by megarnie, Aug 9, 2023, 9:11 PM
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asdf334
7585 posts
#6 • 1 Y
Y by PikaPika999
We prove that given 13 points in standard position, there are 130 scalene triangles. This clearly solves the original question.

Note that every base (i.e. two points) gives at most two isosceles triangles, else we have three points along the perpendicular bisector.

Therefore there are at most $\binom{13}{2}\cdot 2=156$ isosceles triangles. At the same time there are $\binom{13}{3}=286$ triangles in total, making 130 non-isosceles AKA scalene triangles. $\blacksquare$
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dolphinday
1325 posts
#8 • 1 Y
Y by PikaPika999
By pigeonhole, there is a color that has at least $13$ points of the same color. This means there are $\binom{13}{3} = 286$ triangles that can be formed.
Let $I$ be the number of isosceles triangles in the $13$ points.
Then each pair of two points can contribute at most $2$ to $I$(due to the collinear condition), so $\binom{13}{2} \cdot 2 = 156$ is the maximum number of isosceles triangles. Then the minimum number of scalene triangles is $130$.
This post has been edited 1 time. Last edited by dolphinday, Dec 9, 2023, 3:09 PM
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mannshah1211
651 posts
#9 • 1 Y
Y by PikaPika999
By Pigeonhole, some color has at least $13$ points of that color. So, assume there are exactly $13$ points, we'll show that we can find at least $130$ scalene triangles whose vertices all belong to those $13$. First, the total number of triangles is $\binom{13}{3} = 286$. Also, note that for each pair of points $(A, B)$, there are at most two isosceles triangles $ABC$ which have $\overline{AB}$ as a base, since in order for that to happen, $C$ must lie on the perpendicular bisector of $\overline{AB}$, but if three or more $C$ exist, this is a contradiction to the fact that no three points are collinear. So, there are at most $2 \cdot \binom{13}{2} = 156$ isosceles triangles, thus at least $130$ scalene triangles among those $13$ points (basically at least $130$ of that color), so done.
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InterLoop
280 posts
#10 • 1 Y
Y by PikaPika999
smol
solution
This post has been edited 1 time. Last edited by InterLoop, May 1, 2024, 5:00 AM
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RedFireTruck
4223 posts
#11 • 1 Y
Y by PikaPika999
https://cdn.aops.com/images/b/8/2/b824a1d2347b72751984bb68d01bd220d96d5d63.png
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kotmhn
60 posts
#12 • 1 Y
Y by PikaPika999
On each base at most 2 isosceles can lie else the collinearity condition is violated.
By PHP 13 dots of same color exist. then at most $2{13 \choose 2} = 156$ isosceles triangles and total triangles equal ${13 \choose 3}=286$ so at least $286-156=50$ triangles.
done
This post has been edited 1 time. Last edited by kotmhn, Aug 13, 2024, 11:14 AM
Reason: arithmetic skill issue
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ihategeo_1969
235 posts
#13 • 1 Y
Y by PikaPika999
Start with a claim.

Claim: For any $x$ points in the plane (no three collinear), there are atleast \[\binom{x}{3}-2\binom{x}2\]scalene triangles formed by it.
Proof: The maximum number of isosceles triangles by choosing a fixed base is atmost $2$ or else there will be $3$ points on the perpendicular bisector of the base.$\blacksquare$.

And hence choose the colour with atleast $\left \lceil \frac{50}4 \right \rceil=13$ points and the number of scalene triangles it form are atleast \[\binom{13}3-2\binom{13}2=130\]as desired.
This post has been edited 1 time. Last edited by ihategeo_1969, Sep 11, 2024, 8:20 AM
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de-Kirschbaum
199 posts
#14 • 1 Y
Y by PikaPika999
Note that by pigeonhole there exists a color that repeats at least $13$ times. Then, since there are no three points that are colinear, each line segment drawn in these $13$ points can be the base for at most $2$ isoceles triangles, thus there are at least $\binom{13}{3}-\binom{13}{2}2=130$ scalene triangles with same colored vertices.
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cursed_tangent1434
631 posts
#15
Y by
Let $x$ denote the number of points of the color with the highest frequency. There are clearly $\binom{x}{3}$ triangles. For any line segment connecting any two of these points, at most 2 isosceles triangles with this segment as the base may be constructed since we are given that no three points are collinear as any vertex for which the resulting triangle is isosceles must lie on the perpendicular bisector of this segment. Hence we have atleast,
\[S\ge \binom{x}{3} -2\binom{x}{2} = \frac{x(x-1)(x-2)}{6} - x(x-1) = \frac{x(x-1)(x-8)}{6}\]scalene triangles all of whose vertices are of this same color. However since we have 50 points and only 4 colors, there exists some color with atleast, $\lceil \frac{50}{4} \rceil = 13$ points of that color. But then,
\[S \ge  \frac{x(x-1)(x-8)}{6} \ge \frac{13\cdot 12 \cdot 5}{6} = 130\]scalene triangles with all vertices of the same color, as desired.
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