equal angles in equilateral grid (2016-17 Savin Competition 7-9 p18)

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0 replies

jlacosta

Apr 2, 2025

0 replies

School Math Problem

math_cool123 5

N 8 minutes ago by math_cool123

Find all ordered pairs of nonzero integers $(a, b)$ that satisfy $(a^2+b)(a+b^2)=(a-b)^3.$

5 replies

math_cool123

Apr 2, 2025

math_cool123

8 minutes ago

KSEA NMSC Mock Contest Group B (Last Problem)

Shiyul 3

N an hour ago by Giant_PT

Let $a_n$ be a sequence defined by $a_n = a^2 + 1$ . Then the product of four consecutive terms in $a_n$ can be written as the product of two terms in $a_n$ . Find $p + q$ if $(a_(11))(a_(12))(a_(13))(a_(14)) = (a_p)(a_q)$ .

3 replies

Shiyul

Today at 3:09 PM

Giant_PT

an hour ago

Geo Mock #10

Bluesoul 2

N 3 hours ago by Bluesoul

Consider acute $\triangle{ABC}$ with $AB=10$ , $AC<BC$ and area $135$ . The circle $\omega$ with diameter $AB$ meets $BC$ at $E$ . Let the orthocenter of the triangle be $H$ , connect $CH$ and extend to meet $\omega$ at $N$ such that $NC>HC$ and $NE$ is the diameter of $\omega$ . Draw the circumcircle $\Gamma$ of $\triangle{AHB}$ , chord $XY$ of $\Gamma$ is tangent to $\omega$ and it passes through $N$ , compute $XY$ .

2 replies

Bluesoul

Apr 1, 2025

Bluesoul

3 hours ago

Inequalities

sqing 4

N Today at 3:28 PM by sqing

Let $a, b,c\geq 0$ and $2a+3b+ 4c=11.$ Prove that
$a+ab+abc\leq\frac{49}{6}$ Let $a, b,c\geq 0$ and $2a+3b+ 4c=10.$ Prove that
$a+ab+abc\leq\frac{169}{24}$ Let $a, b,c\geq 0$ and $2a+3b+ 4c=14.$ Prove that
$a+ab+abc\leq\frac{63+5\sqrt 5}{6}$ Let $a, b,c\geq 0$ and $2a+3b+ 4c=32.$ Prove that
$a+ab+abc\leq48+\frac{64\sqrt{2}}{3}$

4 replies

sqing

Apr 1, 2025

sqing

Today at 3:28 PM

No more topics!

equal angles in equilateral grid (2016-17 Savin Competition 7-9 p18)

parmenides51 4

N Jun 21, 2021 by sunken rock

Two angles are marked on a grid of equilateral triangles. Prove that they are equal.
IMAGE

4 replies

parmenides51

Jun 19, 2021

sunken rock

Jun 21, 2021

equal angles in equilateral grid (2016-17 Savin Competition 7-9 p18)

G H J

Reveal topic tags

geometry

equal angles

Equilateral

G H BBookmark kLocked kLocked NReply

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parmenides51

30629 posts

parmenides51

#1 h Jun 19, 2021, 7:25 PM

Y by

Two angles are marked on a grid of equilateral triangles. Prove that they are equal.

https://cdn.artofproblemsolving.com/attachments/6/a/23569f1494efeaf7ae471fde42a9eb65beabc4.png

This post has been edited 1 time. Last edited by parmenides51, Jun 19, 2021, 7:25 PM

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parmenides51

30629 posts

parmenides51

#2 h Jun 19, 2021, 7:25 PM

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posted for the image link

Attachments:

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BakedPotato66

747 posts

BakedPotato66

#3 h Jun 20, 2021, 1:24 AM

Y by

We can use Law of Cosines (but that would be a bit bashy).

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vanstraelen

8949 posts

vanstraelen

#4 h Jun 20, 2021, 2:43 PM

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Point left $O(0,0)$ , point right above $(6,\sqrt{3})$ , $\tan \alpha=\frac{\sqrt{3}}{6}$ .

Points $A(2,0)$ and $B(\frac{7}{2},-\frac{\sqrt{3}}{2})$ .
Slope of the line $AB\ :\ m_{AB}=-\frac{\sqrt{3}}{3}$ , slope of the line $OB\ :\ m_{OB}=-\frac{\sqrt{3}}{7}$ , then $\tan \beta=\frac{\sqrt{3}}{6}$ .

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sunken rock

4379 posts

sunken rock

#5 h Jun 21, 2021, 4:13 PM

Y by

My solution at https://www.blogger.com/blog/post/edit/4971068601603455662/3031546455599395755?hl=ro

Best regards,
sunken rock

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