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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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0 replies
jlacosta
May 1, 2025
0 replies
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k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
J
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k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
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orthocenter on sus circle
DVDTSB   1
N 21 minutes ago by Double07
Source: Romania TST 2025 Day 2 P1
Let \( ABC \) be an acute triangle with \( AB < AC \), and let \( O \) be the center of its circumcircle. Let \( A' \) be the reflection of \( A \) with respect to \( BC \). The line through \( O \) parallel to \( BC \) intersects \( AC \) at \( F \), and the tangent at \( F \) to the circle \( \odot(BFC) \) intersects the line through \( A' \) parallel to \( BC \) at point \( M \). Let \( K \) be a point on the ray \( AB \), starting at \( A \), such that \( AK = 4AB \).
Show that the orthocenter of triangle \( ABC \) lies on the circle with diameter \( KM \).

Proposed by Radu Lecoiu

1 reply
DVDTSB
Today at 12:18 PM
Double07
21 minutes ago
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Maximum number of pairs adding to powers of n
MathMystic33   0
33 minutes ago
Source: 2025 Macedonian Team Selection Test P6
Let $n>2$ be an even integer, and let $V$ be an arbitrary set of $8$ distinct integers. Define
\[ E(V,n) \;=\; \bigl\{(u,v)\in V\times V : u < v,\ u+v = n^k\text{ for some }k\in\mathbb{N}\bigr\}. \]For each even $n>2$, determine the maximum possible size of the set $E(V,n)$.

0 replies
MathMystic33
33 minutes ago
0 replies
J
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Incircle triangles inequality
MathMystic33   0
36 minutes ago
Source: 2025 Macedonian Team Selection Test P5
Let $\triangle ABC$ be a triangle with side‐lengths $a,b,c$, incenter $I$, and circumradius $R$. Denote by $P$ the area of $\triangle ABC$, and let $P_1,\;P_2,\;P_3$ be the areas of triangles $\triangle ABI$, $\triangle BCI$, and $\triangle CAI$, respectively. Prove that
\[ \frac{abc}{12R} \;\le\; \frac{P_1^2 + P_2^2 + P_3^2}{P} \;\le\; \frac{3R^3}{4\sqrt[3]{abc}}. \]
0 replies
MathMystic33
36 minutes ago
0 replies
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Functional equation with extra divisibility condition
MathMystic33   0
38 minutes ago
Source: 2025 Macedonian Team Selection Test P4
Find all functions $f:\mathbb{N}_0\to\mathbb{N}$ such that
1) \(f(a)\) divides \(a\) for every \(a\in\mathbb{N}_0\), and
2) for all \(a,b,k\in\mathbb{N}_0\) we have
\[ f\bigl(f(a)+kb\bigr)\;=\;f\bigl(a + k\,f(b)\bigr). \]
0 replies
MathMystic33
38 minutes ago
0 replies
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Cyclic inequality with rational functions
MathMystic33   0
41 minutes ago
Source: 2025 Macedonian Team Selection Test P3
Let \(x_1,x_2,x_3,x_4\) be positive real numbers. Prove the inequality
\[ \frac{x_1 + 3x_2}{x_2 + x_3} \;+\; \frac{x_2 + 3x_3}{x_3 + x_4} \;+\; \frac{x_3 + 3x_4}{x_4 + x_1} \;+\; \frac{x_4 + 3x_1}{x_1 + x_2} \;\ge\;8. \]
0 replies
MathMystic33
41 minutes ago
0 replies
J
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Hexagonal lotus leaves
MathMystic33   0
43 minutes ago
Source: 2025 Macedonian Team Selection Test P2
A lake is in the shape of a regular hexagon of side length \(1\). Initially there is a single lotus leaf somewhere in the lake, sufficiently far from the shore. Each day, from every existing leaf a new leaf may grow at distance \(\sqrt{3}\) (measured between centers), provided it does not overlap any other leaf. If the lake is large enough that edge effects never interfere, what is the least number of days required to have \(2025\) leaves in the lake?
0 replies
+1 w
MathMystic33
43 minutes ago
0 replies
J
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Collinearity of intersection points in a triangle
MathMystic33   0
an hour ago
Source: 2025 Macedonian Team Selection Test P1
On the sides of the triangle \(\triangle ABC\) lie the following points: \(K\) and \(L\) on \(AB\), \(M\) on \(BC\), and \(N\) on \(CA\). Let
\[ P = AM\cap BN,\quad R = KM\cap LN,\quad S = KN\cap LM, \]and let the line \(CS\) meet \(AB\) at \(Q\). Prove that the points \(P\), \(Q\), and \(R\) are collinear.
0 replies
MathMystic33
an hour ago
0 replies
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3 variable FE with divisibility condition
pithon_with_an_i   2
N an hour ago by ATM_
Source: Revenge JOM 2025 Problem 1, Revenge JOMSL 2025 N2, Own
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that $$f(a)+f(b)+f(c) \mid a^2 + af(b) + cf(a)$$for all $a,b,c \in \mathbb{N}$.
2 replies
pithon_with_an_i
3 hours ago
ATM_
an hour ago
J
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Dissection into equal‐area pieces using diagonals
MathMystic33   0
an hour ago
Source: Macedonian Mathematical Olympiad 2025 Problem 5
Let \(n>1\) be a natural number, and let \(K\) be the square of side length \(n\) subdivided into \(n^2\) unit squares. Determine for which values of \(n\) it is possible to dissect \(K\) into \(n\) connected regions of equal area using only the diagonals of those unit squares, subject to the condition that from each unit square at most one of its diagonals is used (some unit squares may have neither diagonal).
0 replies
MathMystic33
an hour ago
0 replies
J
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Brazilian Locus
kraDracsO   15
N an hour ago by Ilikeminecraft
Source: IberoAmerican, Day 2, P4
Let $B$ and $C$ be two fixed points in the plane. For each point $A$ of the plane, outside of the line $BC$, let $G$ be the barycenter of the triangle $ABC$. Determine the locus of points $A$ such that $\angle BAC + \angle BGC = 180^{\circ}$.

Note: The locus is the set of all points of the plane that satisfies the property.
15 replies
kraDracsO
Sep 9, 2023
Ilikeminecraft
an hour ago
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Divisibility with the polynomial ax^{75}+b
MathMystic33   0
an hour ago
Source: Macedonian Mathematical Olympiad 2025 Problem 4
Let $P(x)=a x^{75}+b$ be a polynomial where \(a\) and \(b\) are coprime integers in the set \(\{1,2,\dots,151\}\), and suppose it satisfies the following condition: there exists at most one prime \(p\) such that for every positive integer \(k\), \(p\mid P(k)\). Prove that for every prime \(q \neq p\) there exists a positive integer \(k\) for which $q^2 \mid P(k).$
0 replies
MathMystic33
an hour ago
0 replies
J
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Maximum reach of splitting tokens
MathMystic33   0
an hour ago
Source: Macedonian Mathematical Olympiad 2025 Problem 3
On a horizontally placed number line, a pile of \( t_i > 0 \) tokens is placed on each number \( i \in \{1, 2, \ldots, s\} \). As long as at least one pile contains at least two tokens, we repeat the following procedure: we choose such a pile (say, it consists of \( k \geq 2 \) tokens), and move the top token from the selected pile \( k - 1 \) unit positions to the right along the number line. What is the largest natural number \( N \) on which a token can be placed? (Express \( N \) as a function of \( (t_i;\ i = 1, \ldots, s) \).)
0 replies
MathMystic33
an hour ago
0 replies
J
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Inequality with rational function
MathMystic33   0
an hour ago
Source: Macedonian Mathematical Olympiad 2025 Problem 2
Let \( n > 2 \) be an integer, \( k > 1 \) a real number, and \( x_1, x_2, \ldots, x_n \) be positive real numbers such that \( x_1 \cdot x_2 \cdots x_n = 1 \). Prove that:

\[ \frac{1 + x_1^k}{1 + x_2} + \frac{1 + x_2^k}{1 + x_3} + \cdots + \frac{1 + x_n^k}{1 + x_1} \geq n. \]
When does equality hold?
0 replies
MathMystic33
an hour ago
0 replies
J
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Circumcircle of MUV tangent to two circles at once
MathMystic33   0
an hour ago
Source: Macedonian Mathematical Olympiad 2025 Problem 1
Given is an acute triangle \( \triangle ABC \) with \( AB < AC \). Let \( M \) be the midpoint of side \( BC \), and let \( X \) and \( Y \) be points on segments \( BM \) and \( CM \), respectively, such that \( BX = CY \). Let \( \omega_1 \) be the circumcircle of \( \triangle ABX \), and \( \omega_2 \) the circumcircle of \( \triangle ACY \). The common tangent \( t \) to \( \omega_1 \) and \( \omega_2 \), which lies closer to point \( A \), touches \( \omega_1 \) and \( \omega_2 \) at points \( P \) and \( Q \), respectively. Let the line \( MP \) intersect \( \omega_1 \) again at \( U \), and the line \( MQ \) intersect \( \omega_2 \) again at \( V \). Prove that the circumcircle of triangle \( \triangle MUV \) is tangent to both \( \omega_1 \) and \( \omega_2 \).
0 replies
MathMystic33
an hour ago
0 replies
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weird symmetric equation
giangtruong13   1
N Apr 27, 2025 by pooh123
Solve the equation: $$8x^2-11x+1=(1-x)\sqrt{4x^2-6x+5}$$
1 reply
giangtruong13
Apr 27, 2025
pooh123
Apr 27, 2025
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weird symmetric equation
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giangtruong13
146 posts
giangtruong13
#1 h  Apr 27, 2025, 4:29 AM
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Solve the equation: $$8x^2-11x+1=(1-x)\sqrt{4x^2-6x+5}$$
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pooh123
54 posts
pooh123
#2 h  Apr 27, 2025, 8:25 AM • 1 Y
Y by ehuseyinyigit
giangtruong13 wrote:
Solve the equation: $$8x^2-11x+1=(1-x)\sqrt{4x^2-6x+5}$$

The domain of the equation is all real \( x \).
Let \( y = \sqrt{4x^2 - 6x + 5} \), \( z = 1 - x \), then we obtain the system of equations:
\[ 5y^2 - 2yz - 4y^2 = 19 \quad (1) \]\[ y^2 - 4z^2 = 3 - 2z \quad (2) \]\((2) \times 7 - (1)\) gives:
\[ 7(y^2 - 4z^2) - (5y^2 - 2yz - 4y^2) = 7(3 - 2z) - 19 \]\(\iff 2y^2 + 2yz - 24z^2 = 2 - 14z\)
\(\iff y^2 + yz - 12z^2 + 7z - 1 = 0\)
\(\iff (y - 3z)(y + 4z) + (y + 4z) - (y - 3z) - 1 = 0\)
\(\iff (y - 3z + 1)(y + 4z - 1) = 0\)
Therefore \( y + 4z = 1 \) or \( y - 3z = -1 \). Consider 2 cases:
Case 1: \( y + 4z = 1 \)
\(\iff \sqrt{4x^2 - 6x + 5} + 4(1 - x) = 1\)
\(\iff \sqrt{4x^2 - 6x + 5} = 4x - 3\)
\(\Rightarrow 4x^2 - 6x + 5 = (3 - 4x)^2\)
\(\iff 6x^2 - 9x + 2 = 0\)
\(\iff x = \frac{9 \pm \sqrt{33}}{12}\)
Substituting in the values of \( x \), we see that \( x = \frac{9 + \sqrt{33}}{12} \) is a solution.
Case 2: \( y - 3z = -1 \)
\(\iff \sqrt{4x^2 - 6x + 5} - 3(1 - x) = -1\)
\(\iff \sqrt{4x^2 - 6x + 5} = 2 - 3x\)
\(\Rightarrow 4x^2 - 6x + 5 = (2 - 3x)^2\)
\(\iff 5x^2 - 6x - 1 = 0\)
\(\iff x = \frac{3 \pm \sqrt{14}}{5}\)
Substituting in the values of \( x \), we see that \( x = \frac{3 - \sqrt{14}}{5} \) is a solution.
Hence the equation has 2 roots \( x = \frac{3 - \sqrt{14}}{5} \) and \( x = \frac{9 + \sqrt{33}}{12} \).
This post has been edited 1 time. Last edited by pooh123, Apr 27, 2025, 8:34 AM
Reason: typo
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