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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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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
Cyclic ine
m4thbl3nd3r   2
N a minute ago by m4thbl3nd3r
Let $a,b,c>0$ such that $a^2+b^2+c^2=3$. Prove that $$\sum \frac{a^2}{b}+abc \ge 4$$
2 replies
m4thbl3nd3r
Today at 3:34 PM
m4thbl3nd3r
a minute ago
Sequence inequality
BR1F1SZ   0
12 minutes ago
Source: 2025 Francophone MO Seniors P1
Let $a_1, a_2, a_3, \ldots$ be a sequence of positive integers satisfying the following property: for all positive integers $k < \ell$, for all distinct integers $m_1, m_2, \ldots, m_k$ and for all distinct integers $n_1, n_2, \ldots, n_\ell$,
\[
a_{m_1} + a_{m_2} + \cdots + a_{m_k} \leqslant a_{n_1} + a_{n_2} + \cdots + a_{n_\ell}.
\]Prove that there exist two integers $N$ and $b$ such that $a_n = b$ for all $n \geqslant N$.
0 replies
BR1F1SZ
12 minutes ago
0 replies
GCD and LCM operations
BR1F1SZ   0
20 minutes ago
Source: 2025 Francophone MO Juniors P4
Charlotte writes the integers $1,2,3,\ldots,2025$ on the board. Charlotte has two operations available: the GCD operation and the LCM operation.
[list]
[*]The GCD operation consists of choosing two integers $a$ and $b$ written on the board, erasing them, and writing the integer $\operatorname{gcd}(a, b)$.
[*]The LCM operation consists of choosing two integers $a$ and $b$ written on the board, erasing them, and writing the integer $\operatorname{lcm}(a, b)$.
[/list]
An integer $N$ is called a winning number if there exists a sequence of operations such that, at the end, the only integer left on the board is $N$. Find all winning integers among $\{1,2,3,\ldots,2025\}$ and, for each of them, determine the minimum number of GCD operations Charlotte must use.

Note: The number $\operatorname{gcd}(a, b)$ denotes the greatest common divisor of $a$ and $b$, while the number $\operatorname{lcm}(a, b)$ denotes the least common multiple of $a$ and $b$.
0 replies
BR1F1SZ
20 minutes ago
0 replies
Concurrency of two lines and a circumcircle
BR1F1SZ   0
24 minutes ago
Source: 2025 Francophone MO Juniors P3
Let $\triangle{ABC}$ be a triangle, $\omega$ its circumcircle and $O$ the center of $\omega$. Let $P$ be a point on the segment $BC$. We denote by $Q$ the second intersection point of the circumcircles of triangles $\triangle{AOB}$ and $\triangle{APC}$. Prove that the line $PQ$ and the tangent to $\omega$ at point $A$ intersect on the circumcircle of triangle $\triangle AOB$.
0 replies
BR1F1SZ
24 minutes ago
0 replies
Range if \omega for No Inscribed Right Triangle y = \sin(\omega x)
ThisIsJoe   1
N 6 hours ago by Lankou
For a positive number \omega , determine the range of \omega for which the curve y = \sin(\omega x) has no inscribed right triangle.
Could someone help me figure out how to approach this?
1 reply
ThisIsJoe
May 8, 2025
Lankou
6 hours ago
2012 preRMO p17, roots of equation x^3 + 3x + 5 = 0
parmenides51   11
N Today at 3:29 PM by Pengu14
Let $x_1,x_2,x_3$ be the roots of the equation $x^3 + 3x + 5 = 0$. What is the value of the expression
$\left( x_1+\frac{1}{x_1} \right)\left( x_2+\frac{1}{x_2} \right)\left( x_3+\frac{1}{x_3} \right)$ ?
11 replies
parmenides51
Jun 17, 2019
Pengu14
Today at 3:29 PM
Interesting question from Al-Khwarezmi olympiad 2024 P3, day1
Adventure1000   3
N Today at 2:38 PM by sqing
Find all $x, y, z \in \left (0, \frac{1}{2}\right )$ such that
$$
\begin{cases}
(3 x^{2}+y^{2}) \sqrt{1-4 z^{2}} \geq z; \\
(3 y^{2}+z^{2}) \sqrt{1-4 x^{2}} \geq x; \\
(3 z^{2}+x^{2}) \sqrt{1-4 y^{2}} \geq y.
\end{cases}
$$Proposed by Ngo Van Trang, Vietnam
3 replies
Adventure1000
May 7, 2025
sqing
Today at 2:38 PM
Malaysia MO IDM UiTM 2025
smartvong   1
N Today at 2:20 PM by jasperE3
MO IDM UiTM 2025 (Category C)

Contest Description

Preliminary Round
Section A
1. Given that $2^a + 2^b = 2016$ such that $a, b \in \mathbb{N}$. Find the value of $a$ and $b$.

2. Find the value of $a, b$ and $c$ such that $$\frac{ab}{a + b} = 1, \frac{bc}{b + c} = 2, \frac{ca}{c + a} = 3.$$
3. If the value of $x + \dfrac{1}{x}$ is $\sqrt{3}$, then find the value of
$$x^{1000} + \frac{1}{x^{1000}}$$.

Section B
1. Let $\mathbb{Z}$ be the set of integers. Determine all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that for all integer $a, b$:
$$f(2a) + 2f(b) = f(f(a + b))$$
2. The side lengths $a, b, c$ of a triangle $\triangle ABC$ are positive integers. Let
$$T_n = (a + b + c)^{2n} - (a - b + c)^{2n} - (a + b - c)^{2n} - (a - b - c)^{2n}$$for any positive integer $n$.
If $\dfrac{T_2}{2T_1} = 2023$ and $a > b > c$, determine all possible perimeters of the triangle $\triangle ABC$.

Final Round
Section A
1. Given that the equation $x^2 + (b - 3)x - 2b^2 + 6b - 4 = 0$ has two roots, where one is twice of the other, find all possible values of $b$.

2. Let $$f(y) = \dfrac{y^2}{y^2 + 1}.$$Find the value of $$f\left(\frac{1}{2001}\right) + f\left(\frac{2}{2001}\right) + \cdots + f\left(\frac{2000}{2001}\right) + f\left(\frac{2001}{2001}\right) + f\left(\frac{2001}{2000}\right) + \cdots + f\left(\frac{2001}{2}\right) + f\left(\frac{2001}{1}\right).$$
3. Find the smallest four-digit positive integer $L$ such that $\sqrt{3\sqrt{L}}$ is an integer.

Section B
1. Given that $\tan A : \tan B : \tan C$ is $1 : 2 : 3$ in triangle $\triangle ABC$, find the ratio of the side length $AC$ to the side length $AB$.

2. Prove that $\cos{\frac{2\pi}{5}} + \cos{\frac{4\pi}{5}} = -\dfrac{1}{2}.$
1 reply
smartvong
Today at 1:01 PM
jasperE3
Today at 2:20 PM
Nice problem
gasgous   2
N Today at 1:47 PM by vincentwant
Given that the product of three integers is $60$.What is the least possible positive sum of the three integers?
2 replies
gasgous
Today at 1:30 PM
vincentwant
Today at 1:47 PM
Angle Formed by Points on the Sides of a Triangle
xeroxia   1
N Today at 1:28 PM by vanstraelen

In triangle $ABC$, points $D$, $E$, and $F$ lie on sides $BC$, $CA$, and $AB$, respectively, such that
$BD = 20$, $DC = 15$, $CE = 13$, $EA = 8$, $AF = 6$, $FB = 22$.

What is the measure of $\angle EDF$?


1 reply
xeroxia
Today at 10:28 AM
vanstraelen
Today at 1:28 PM
Inequalities
sqing   1
N Today at 1:08 PM by sqing
Let $ a,b,c\geq 0 , (a+8)(b+c)=9.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq  \frac{38}{23}$$Let $ a,b,c\geq 0 , (a+2)(b+c)=3.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq  \frac{2(2\sqrt{3}+1)}{5}$$
1 reply
sqing
Today at 12:50 PM
sqing
Today at 1:08 PM
Is this true?
Entrepreneur   1
N Today at 12:56 PM by revol_ufiaw
Define the $\text{\textcolor{red}{Pell Sequence}}$ as $$P_0=0,P_1=1,\;P_{n+2}=2P_{n+1}+P_n.$$Prove that $4P_{2k}^2+1$ is prime for all $k\in\mathbb N.$
1 reply
Entrepreneur
Today at 9:32 AM
revol_ufiaw
Today at 12:56 PM
Geometry
AlexCenteno2007   4
N Today at 12:05 PM by Raul_S_Baz
Let ABC be an acute triangle and let D, E and F be the feet of the altitudes from A, B and C respectively. The straight line EF and the circumcircle of ABC intersect at P such that F is between E and P, the straight lines BP and DF intersect at Q. Show that if ED = EP then CQ and DP are parallel.
4 replies
AlexCenteno2007
Apr 28, 2025
Raul_S_Baz
Today at 12:05 PM
Inequalities
sqing   1
N Today at 11:45 AM by sqing
Let $ 0\leq x,y,z\leq 2. $ Prove that
$$-48\leq (x-yz)( 3y-zx)(z-xy)\leq 9$$$$-144\leq (3x-yz)(y-zx)(3z-xy)\leq\frac{81}{64}$$$$-144\leq (3x-yz)(2y-zx)(3z-xy)\leq\frac{81}{16}$$
1 reply
sqing
Yesterday at 8:50 AM
sqing
Today at 11:45 AM
FE solution too simple?
Yiyj1   9
N Apr 23, 2025 by jasperE3
Source: 101 Algebra Problems from the AMSP
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the equality $$f(f(x)+y) = f(x^2-y)+4f(x)y$$holds for all pairs of real numbers $(x,y)$.

My solution

I feel like my solution is too simple. Is there something I did wrong or something I missed?
9 replies
Yiyj1
Apr 9, 2025
jasperE3
Apr 23, 2025
FE solution too simple?
G H J
G H BBookmark kLocked kLocked NReply
Source: 101 Algebra Problems from the AMSP
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Yiyj1
1266 posts
#1
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Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the equality $$f(f(x)+y) = f(x^2-y)+4f(x)y$$holds for all pairs of real numbers $(x,y)$.

My solution

I feel like my solution is too simple. Is there something I did wrong or something I missed?
Z K Y
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InterLoop
280 posts
#2 • 1 Y
Y by Yiyj1
You cannot immediately "cancel" the $f$ without further conclusions.

For example $f(3) = f(2) = 1$ is possible for a function - this does not mean that $3 = 2$.
The property that leads you to $f(a) = f(b) \implies a = b$ or the "cancellation" of $f$ is called injectivity. You have to prove the function is injective first before cancellation.

Another example is simply the fact that you have not "excluded" the solution $f(x) \equiv 0$ from the equation $f(f(x)) =f(x^2)$ in any way - so $f(x) = x^2$ is wrong for that function as well. (thus $f(x) \equiv 0$ is not injective)
This post has been edited 2 times. Last edited by InterLoop, Apr 9, 2025, 3:39 AM
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Yiyj1
1266 posts
#3
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InterLoop wrote:
You cannot immediately "cancel" the $f$ without further conclusions.

For example $f(3) = f(2) = 1$ is possible for a function - this does not mean that $3 = 2$.
The property that leads you to $f(a) = f(b) \implies a = b$ or the "cancellation" of $f$ is called injectivity. You have to prove the function is injective first before cancellation.

ahh ic. I'll try to prove the injectivity. ty!
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AshAuktober
1005 posts
#4
Y by
This is in fact from Iran TST.
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davichu
8 posts
#5
Y by
Clearly, $f(x)\equiv0$ is a trivial solution, from now on, we assume it is not the case
Let $P(x,y)$ denote the assertion $f(f(x)+y) = f(x^2-y)+4f(x)y$
$$P(x,-f(x))\rightarrow f(0)=f(x^2+f(x))-4f(x)^2$$$$P(x,x^2)\rightarrow f(x^2+f(x))=f(0)+4f(x)x^2$$Adding these two together we get:
$4f(x)^2=4f(x)x^2\rightarrow f(x)^2=f(x)x^2$
Since $f(x)\neq0$,we can divide by $f(x)$ on both sides to get $f(x)=x^2$
so the only solutions are $f(x)\equiv0$ and $f(x)=x^2\forall x \in \mathbb{R}$
Z K Y
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Primeniyazidayi
98 posts
#6
Y by
davichu wrote:
Since $f(x)\neq0$,we can divide by $f(x)$ on both sides to get $f(x)=x^2$

You must at first prove that $f(x) =0 \text{ iff } x=0$(or simply avoid pointwise trap).
This post has been edited 1 time. Last edited by Primeniyazidayi, Apr 22, 2025, 11:12 AM
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Primeniyazidayi
98 posts
#7
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The finish for @2above(hopefully correct):We will avoid pointwise trap.We of course have $f(0) =0$.Let $f(t) =0$ for $t \neq 0$.$P(t,y)$ gives $f(y) =f(t^2-y)$.Take some $u$ such that $f(u) =u^2 \neq 0$.Then we have $u^2=t^2(t^2-2u) +u^2$ or $u=\frac{t^2}{2}$.But $P(0, x) $ gives that $f$ is even which means $\frac{t^2}{2}=-\frac{t^2}{2}$ or $t=0$, contradiction. Thus we are done.
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ariopro1387
16 posts
#8
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Let $P(x,y)$ be the assertion of the problem.
$P(x,\frac{x^2-f(x)}{2});$ $\frac{x^2-f(x)}{2}.f(x) = 0$
$\forall x \in \mathbb{R}$
1. $f(x)\equiv0$
2. $f(x)=x^2$
we have to just check that both won't happen:
if $f(x_{1}) = 0:$
$P(x_{1},y);$ $f(y) = f(x_{1}^2-y)$
then by changing $y$ value we get that $x_{1} = 0$ or $f(x)\equiv C$ (Just $C=0$ works).
This post has been edited 1 time. Last edited by ariopro1387, Apr 22, 2025, 4:06 PM
Reason: edit
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lksb
171 posts
#9 • 1 Y
Y by Yiyj1
one-liner
This post has been edited 1 time. Last edited by lksb, Apr 22, 2025, 7:15 PM
Reason: typo
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jasperE3
11320 posts
#10
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lksb wrote:
one-liner

pointwise trap
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