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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 2, 2025
0 replies
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Valuable subsets of segments in [1;n]
NO_SQUARES   1
N 11 minutes ago by NO_SQUARES
Source: Russian May TST to IMO 2023; group of candidates P6; group of non-candidates P8
The integer $n \geqslant 2$ is given. Let $A$ be set of all $n(n-1)/2$ segments of real line of type $[i, j]$, where $i$ and $j$ are integers, $1\leqslant i<j\leqslant n$. A subset $B \subset A$ is said to be valuable if the intersection of any two segments from $B$ is either empty, or is a segment of nonzero length belonging to $B$. Find the number of valuable subsets of set $A$.
1 reply
NO_SQUARES
Thursday at 8:34 PM
NO_SQUARES
11 minutes ago
J
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geometry party
pnf   0
13 minutes ago
https://artofproblemsolving.com/community/c4260013_geometry_party
0 replies
+2 w
pnf
13 minutes ago
0 replies
J
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All heads to tails?
smartvong   0
34 minutes ago
Source: CEMC Euclid Contest 2025
An equilateral triangle is formed using $n$ rows of coins. There is 1 coin in the first row, 2 coins in the second row, 3 coins in the third row, and so on, up to $n$ coins in the $n$th row. Initially, all of the coins show heads (H). Carley plays a game in which, on each turn, she chooses three mutually adjacent coins and flips these three coins over. To win the game, all of the coins must be showing tails (T) after a sequence of turns. An example game with 4 rows of coins after a sequence of two turns is shown.

IMAGE

Below (a), (b) and (c), you will find instructions about how to refer to these turns in your solutions.

(a) If there are 3 rows of coins, give a sequence of 4 turns that results in a win.

(b) Suppose that there are 4 rows of coins. Determine whether or not there is a sequence of turns that results in a win.

(c) Determine all values of $n$ for which it is possible to win the game starting with $n$ rows of coins.

Note: For a triangle with 4 rows of coins, there are 9 possibilities for the set of three coins that Carley can flip on a given turn. These 9 possibilities are shown as shaded triangles below:

IMAGE

IMAGE

[You should use the names for these moves shown inside the 9 shaded triangles when answering (b). You should adapt this naming convention in a suitable way when answering parts (a) and (c).]
0 replies
smartvong
34 minutes ago
0 replies
J
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Inequality with three variables
crazyfehmy   14
N an hour ago by TopGbulliedU
Source: Turkey JBMO Team Selection Test 2013, P4
For all positive real numbers $a, b, c$ satisfying $a+b+c=1$, prove that

\[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]
14 replies
crazyfehmy
May 31, 2013
TopGbulliedU
an hour ago
J
No more topics!
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J
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Proof that BQC is isosceles
Dytto   5
N Oct 5, 2017 by ythomashu
$ABCD$ is a rectangle , $P$ is the midpoint of $AB$ and $Q$ is the $PD$ such that $CQ$ is perpendicular to $PD$ . Prove that triangle $BQC$ is isosceles. Can anybody please provide solution using both complex numbers and by normal method. Thanks in advance.
5 replies
Dytto
Oct 5, 2017
ythomashu
Oct 5, 2017
J
Proof that BQC is isosceles
G H J
Reveal topic tags
geometry
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Dytto
107 posts
Dytto
#1 h  Oct 5, 2017, 12:56 PM • 1 Y
Y by Adventure10
$ABCD$ is a rectangle , $P$ is the midpoint of $AB$ and $Q$ is the $PD$ such that $CQ$ is perpendicular to $PD$ . Prove that triangle $BQC$ is isosceles. Can anybody please provide solution using both complex numbers and by normal method. Thanks in advance.
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ythomashu
6322 posts
ythomashu
#2 h  Oct 5, 2017, 1:23 PM • 1 Y
Y by Adventure10
Let the midpoint of $CD$ be $E$. $ECQ$ is isosceles. $EB$ is on the perpendicular bisector of $CQ$, making $BQC$ isosceles.
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Dytto
107 posts
Dytto
#3 h  Oct 5, 2017, 2:00 PM • 2 Y
Y by Adventure10, Mango247
How is ECQ isosceles ,Can you please explain???
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Dytto
107 posts
Dytto
#4 h  Oct 5, 2017, 2:53 PM • 2 Y
Y by Adventure10, Mango247
Bump…\Can somebody please help?
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ythomashu
6322 posts
ythomashu
#5 h  Oct 5, 2017, 2:53 PM • 2 Y
Y by Adventure10, Mango247
Dytto wrote:
How is ECQ isosceles ,Can you please explain???

QDC is a right triangle. The median from the right angle always forms two isosceles triangles. Also, QED is isosceles.
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ythomashu
6322 posts
ythomashu
#6 h  Oct 5, 2017, 2:56 PM • 2 Y
Y by Adventure10, Mango247
Another approach to this problem is to use similar triangles. It's very similar to the solution I gave.
This post has been edited 1 time. Last edited by ythomashu, Oct 5, 2017, 2:56 PM
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