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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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The special Miquel's point from a familiar problem
danil_e   8
N a minute ago by anantmudgal09
Problem. Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre. The line through $A$ perpendicular to $BC$ intersects circle $(O)$ again at $T$. The tangents at $B$ and $C$ of $(O)$ intersect at $S$. $AS$ intersects $(O)$ at $X \neq A$. $OB$ intersects $AT$ at $P$. Let $N$ be the midpoint of $TC$.
Prove that $T, P, N, X$ are concyclic.
8 replies
danil_e
Jul 23, 2023
anantmudgal09
a minute ago
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\frac{a}{(b-c)^2}+\frac{b}{(a-c)^2}+\frac{c}{(b-a)^2}
Jjesus   1
N 24 minutes ago by AlexCenteno2007
Source: Peru Ibero TST 2018
Let $p, q$ be real numbers. Knowing that there are positive real numbers $a, b, c$, different two by two, such that
$$p=\frac{a^2}{(b-c)^2}+\frac{b^2}{(a-c)^2}+\frac{c^2}{(a-b)^2},$$$$q=\frac{1}{(b-c)^2}+\frac{1}{(a-c)^2}+\frac{1}{(b-a)^2}$$calculate the value of
$$\frac{a}{(b-c)^2}+\frac{b}{(a-c)^2}+\frac{c}{(b-a)^2}$$in terms of $p, q$.
1 reply
Jjesus
Oct 12, 2020
AlexCenteno2007
24 minutes ago
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Inspired by old results
sqing   1
N 35 minutes ago by lbh_qys
Source: Own
Let \( a, b, c \) be real numbers.Prove that
$$ \frac{(a - b + c)^2}{ (a^2+ a+1)(b^2+b+1)(c^2+ c+1)} \leq 4$$$$ \frac{(a + b + c)^2}{ (a^2+ a+1)(b^2 +b+1)(c^2+ c+1)} \leq \frac{2(69 + 11\sqrt{33})}{27}$$
1 reply
1 viewing
sqing
an hour ago
lbh_qys
35 minutes ago
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too many equality cases
Scilyse   17
N 40 minutes ago by Confident-man
Source: 2023 ISL C6
Let $N$ be a positive integer, and consider an $N \times N$ grid. A right-down path is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A right-up path is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence.

Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses $5$ paths.
IMAGE
Proposed by Zixiang Zhou, Canada
17 replies
Scilyse
Jul 17, 2024
Confident-man
40 minutes ago
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Indonesia Regional MO 2019 Part A
parmenides51   23
N 4 hours ago by chinawgp
Indonesia Regional MO
Year 2019 Part A

Time: 90 minutes Rules


p1. In the bag there are $7$ red balls and $8$ white balls. Audi took two balls at once from inside the bag. The chance of taking two balls of the same color is ...


p2. Given a regular hexagon with a side length of $1$ unit. The area of the hexagon is ...


p3. It is known that $r, s$ and $1$ are the roots of the cubic equation $x^3 - 2x + c = 0$. The value of $(r-s)^2$ is ...


p4. The number of pairs of natural numbers $(m, n)$ so that $GCD(n,m) = 2$ and $LCM(m,n) = 1000$ is ...


p5. A data with four real numbers $2n-4$, $2n-6$, $n^2-8$, $3n^2-6$ has an average of $0$ and a median of $9/2$. The largest number of such data is ...


p6. Suppose $a, b, c, d$ are integers greater than $2019$ which are four consecutive quarters of an arithmetic row with $a <b <c <d$. If $a$ and $d$ are squares of two consecutive natural numbers, then the smallest value of $c-b$ is ...


p7. Given a triangle $ABC$, with $AB = 6$, $AC = 8$ and $BC = 10$. The points $D$ and $E$ lies on the line segment $BC$. with $BD = 2$ and $CE = 4$. The measure of the angle $\angle DAE$ is ...


p8. Sequqnce of real numbers $a_1,a_2,a_3,...$ meet $\frac{na_1+(n-1)a_2+...+2a_{n-1}+a_n}{n^2}=1$ for each natural number $n$. The value of $a_1a_2a_3...a_{2019}$ is ....


p9. The number of ways to select four numbers from $\{1,2,3, ..., 15\}$ provided that the difference of any two numbers at least $3$ is ...


p10. Pairs of natural numbers $(m , n)$ which satisfies $$m^2n+mn^2 +m^2+2mn = 2018m + 2019n + 2019$$are as many as ...


p11. Given a triangle $ABC$ with $\angle ABC =135^o$ and $BC> AB$. Point $D$ lies on the side $BC$ so that $AB=CD$. Suppose $F$ is a point on the side extension $AB$ so that $DF$ is perpendicular to $AB$. The point $E$ lies on the ray $DF$ such that $DE> DF$ and $\angle ACE = 45^o$. The large angle $\angle AEC$ is ...


p12. The set of $S$ consists of $n$ integers with the following properties: For every three different members of $S$ there are two of them whose sum is a member of $S$. The largest value of $n$ is ....


p13. The minimum value of $\frac{a^2+2b^2+\sqrt2}{\sqrt{ab}}$ with $a, b$ positive reals is ....


p14. The polynomial P satisfies the equation $P (x^2) = x^{2019} (x+ 1) P (x)$ with $P (1/2)= -1$ is ....


p15. Look at a chessboard measuring $19 \times 19$ square units. Two plots are said to be neighbors if they both have one side in common. Initially, there are a total of $k$ coins on the chessboard where each coin is only loaded exactly on one square and each square can contain coins or blanks. At each turn. You must select exactly one plot that holds the minimum number of coins in the number of neighbors of the plot and then you must give exactly one coin to each neighbor of the selected plot. The game ends if you are no longer able to select squares with the intended conditions. The smallest number of $k$ so that the game never ends for any initial square selection is ....
23 replies
parmenides51
Nov 11, 2021
chinawgp
4 hours ago
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VOLUNTEERING OPPORTUNITIES OPEN TO HIGH/MIDDLE SCHOOLERS
im_space_cadet   13
N 5 hours ago by im_space_cadet
Hi everyone!
Do you specialize in contest math? Do you have a passion for teaching? Do you want to help leverage those college apps? Well, I have something for all of you.

I am im_space_cadet, and during the fall of last year, I opened my non-profit DeltaMathPrep which teaches students preparing for contest math the problem-solving skills they need in order to succeed at these competitions. Currently, we are very much understaffed and would greatly appreciate the help of more tutors on our platform.

Each week on Saturday and Wednesday, we meet once for each competition: Wednesday for AMC 8 and Saturday for AMC 10 and we go over a past year paper for the entire class. On both of these days, we meet at 9PM EST in the night.

This is a great opportunity for anyone who is looking to have a solid activity to add to their college resumes that requires low effort from tutors and is very flexible with regards to time.

This is the link to our non-profit for anyone who would like to view our initiative:
https://www.deltamathprep.org/

If you are interested in this opportunity, please send me a DM on AoPS or respond to this post expressing your interest. I look forward to having you all on the team!

Thanks,
im_space_cadet
13 replies
im_space_cadet
Yesterday at 2:27 PM
im_space_cadet
5 hours ago
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In a school of $800$ students, $224$ students play cricket, $240$ students play
Vulch   1
N 6 hours ago by RollingPanda4616
Hello everyone,
In a school of $800$ students, $224$ students play cricket, $240$ students play hockey and $336$ students play basketball. $64$ students play both basketball and hockey, $80$ students play both cricket and basketball, $40$ students play both cricket and hockey, and $24$ students play all three: basketball, hockey, and cricket. Find the number of students who do not play any game.

Edit:
In the above problem,I just want to know that why the number of students who don't play any game shouldn't be 0, because,if we add 224,240 and 336 it comes out to be 800.I have solution,but I just want to know how to explain it without theoretically.Thank you!
1 reply
Vulch
Yesterday at 11:41 PM
RollingPanda4616
6 hours ago
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100th post
MathJedi108   1
N Yesterday at 11:10 PM by mdk2013
Well I guess this is my 100th post, it would be really funny if it isn't can yall share your favorite experience on AoPS here?
1 reply
MathJedi108
Yesterday at 10:59 PM
mdk2013
Yesterday at 11:10 PM
J
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Find all triples
pedronis   2
N Yesterday at 10:43 PM by Kempu33334
Find all triples of positive integers $(n, r, s)$ such that $n^2 + n + 1$ divides $n^r + n^s + 1$.
2 replies
pedronis
Apr 19, 2025
Kempu33334
Yesterday at 10:43 PM
J
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Median geometry
Sedro   4
N Yesterday at 10:01 PM by Sedro
In triangle $ABC$, points $D$, $E$, and $F$ are the midpoints of sides $BC$, $CA$, and $AB$, respectively. Prove that the area of the triangle with side lengths $AD$, $BE$, and $CF$ has area $\tfrac{3}{4}[ABC]$.
4 replies
Sedro
Yesterday at 6:03 PM
Sedro
Yesterday at 10:01 PM
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geometry
carvaan   1
N Yesterday at 6:38 PM by Lankou
The difference between two angles of a triangle is 24°. All angles are numerically double digits. Find the number of possible values of the third angle.
1 reply
carvaan
Yesterday at 5:46 PM
Lankou
Yesterday at 6:38 PM
J
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weird permutation problem
Sedro   1
N Yesterday at 6:07 PM by Sedro
Let $\sigma$ be a permutation of $1,2,3,4,5,6,7$ such that there are exactly $7$ ordered pairs of integers $(a,b)$ satisfying $1\le a < b \le 7$ and $\sigma(a) < \sigma(b)$. How many possible $\sigma$ exist?
1 reply
Sedro
Yesterday at 2:09 AM
Sedro
Yesterday at 6:07 PM
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Recursion
Sid-darth-vater   6
N Yesterday at 5:59 PM by vanstraelen
Help, I can't characterize ts and I dunno what to do
6 replies
Sid-darth-vater
Yesterday at 3:02 AM
vanstraelen
Yesterday at 5:59 PM
J
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geometry
carvaan   0
Yesterday at 5:48 PM
OABC is a trapezium with OC // AB and ∠AOB = 37°. Furthermore, A, B, C all lie on the circumference of a circle centred at O. The perpendicular bisector of OC meets AC at D. If ∠ABD = x°, find last 2 digit of 100x.
0 replies
carvaan
Yesterday at 5:48 PM
0 replies
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Abusing surjectivity
Sadigly   7
N Apr 13, 2025 by Sadigly
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ and $g:\mathbb{Q}\rightarrow\mathbb{Q}$ such that

$$f(f(x)+yg(x))=(x+1)g(y)+f(y)$$
for any $x;y\in\mathbb{Q}$
7 replies
Sadigly
Apr 13, 2025
Sadigly
Apr 13, 2025
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Abusing surjectivity
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Reveal topic tags
algebra
functional equation
TST
AZE IzHO TST
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G H BBookmark kLocked kLocked NReply
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Sadigly
147 posts
Sadigly
#1 h  Apr 13, 2025, 6:54 PM
Y by
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ and $g:\mathbb{Q}\rightarrow\mathbb{Q}$ such that

$$f(f(x)+yg(x))=(x+1)g(y)+f(y)$$
for any $x;y\in\mathbb{Q}$
This post has been edited 1 time. Last edited by Sadigly, Apr 13, 2025, 7:01 PM
Z K Y
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MuradSafarli
86 posts
MuradSafarli
#2 h  Apr 13, 2025, 8:34 PM
Y by
why you posted secret Izho tst questions?
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Sadigly
147 posts
Sadigly
#3 h  Apr 13, 2025, 8:39 PM
Y by
MuradSafarli wrote:
why you posted secret Izho tst questions?

Secret :rotfl:

Anyways here is a hint:

Prove that $f$ is surjective, and consider $P(a;0)$ where $f(a)=0$
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MuradSafarli
86 posts
MuradSafarli
#4 h  Apr 13, 2025, 8:44 PM
Y by
Let P(x,y) be assertion of the given equation.
P(x,0) : f(f(x)) = (x+1)g(0)+f(0) (1)
If g(0)= 0, from substitutions P(a,0) and P(b,0) we get f(a) = f(b) -> a =
b. Hence, either g(0) = 0, or f is injective.
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alexheinis
10550 posts
alexheinis
#5 h  Apr 13, 2025, 8:50 PM
Y by
@sadigly: You can't prove that $f$ is surjective since $g\equiv 0, f\equiv c$ is a class of solutions.
If $g(y)\not=0$ for some $y$ then $(x+1)g(y)+f(y)$ is surjective hence so is $f$.
So next time be more careful when you state something as a fact.
It's a nice problem though.
This post has been edited 1 time. Last edited by alexheinis, Apr 13, 2025, 8:59 PM
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MuradSafarli
86 posts
MuradSafarli
#6 h  Apr 13, 2025, 8:59 PM
Y by
Solution page #1
Attachments:
This post has been edited 1 time. Last edited by MuradSafarli, Apr 13, 2025, 9:01 PM
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MuradSafarli
86 posts
MuradSafarli
#7 h  Apr 13, 2025, 9:00 PM
Y by
Solution page #2
Attachments:
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Sadigly
147 posts
Sadigly
#8 h  Apr 13, 2025, 9:08 PM
Y by
alexheinis wrote:
@sadigly: You can't prove that $f$ is surjective since $g\equiv 0, f\equiv c$ is a class of solutions.
If $g(y)\not=0$ for some $y$ then $(x+1)g(y)+f(y)$ is surjective hence so is $f$.
So next time be more careful when you state something as a fact.
It's a nice problem though.

Well there was going to be a small case you need to consider when you try to prove surjectivity, but that's it
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