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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
This question just asks if you can factorise 12 factorial or not
Sadigly   1
N 6 minutes ago by COCBSGGCTG3
Source: Azerbaijan Junior MO 2025 P1
A teacher creates a fraction using numbers from $1$ to $12$ (including $12$). He writes some of the numbers on the numerator, and writes $\times$ (multiplication) between each number. Then he writes the rest of the numbers in the denominator and also writes $\times$ between each number. There is at least one number both in numerator and denominator. The teacher ensures that the fraction is equal to the smallest possible integer possible.

What is this positive integer, which is also the value of the fraction?
1 reply
Sadigly
Friday at 7:34 AM
COCBSGGCTG3
6 minutes ago
Prime sums of pairs
Assassino9931   5
N 13 minutes ago by aidan0626
Source: Al-Khwarizmi Junior International Olympiad 2025 P5
Sevara writes in red $8$ distinct positive integers and then writes in blue the $28$ sums of each two red numbers. At most how many of the blue numbers can be prime?

Marin Hristov, Bulgaria
5 replies
Assassino9931
Yesterday at 9:35 AM
aidan0626
13 minutes ago
Inequality, inequality, inequality...
Assassino9931   11
N 20 minutes ago by Assassino9931
Source: Al-Khwarizmi Junior International Olympiad 2025 P6
Let $a,b,c$ be real numbers such that \[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]Find the smallest possible value of $a^2 + b^2 + c^2$.

Binh Luan and Nhan Xet, Vietnam
11 replies
1 viewing
Assassino9931
Yesterday at 9:38 AM
Assassino9931
20 minutes ago
Anything real in this system must be integer
Assassino9931   2
N 27 minutes ago by Assassino9931
Source: Al-Khwarizmi International Junior Olympiad 2025 P1
Determine the largest integer $c$ for which the following statement holds: there exists at least one triple $(x,y,z)$ of integers such that
\begin{align*} x^2 + 4(y + z) = y^2 + 4(z + x) = z^2 + 4(x + y) = c \end{align*}and all triples $(x,y,z)$ of real numbers, satisfying the equations, are such that $x,y,z$ are integers.

Marek Maruin, Slovakia
2 replies
Assassino9931
Friday at 9:26 AM
Assassino9931
27 minutes ago
Calculate the distance AD
MTA_2024   6
N Yesterday at 10:17 PM by WheatNeat
A semi-circle is inscribed in a quadrilateral $ABCD$. The center $O$ of the semi-circle is the midpoint of segment $AD$. We have $AB=9$ and $CD=16$.
Calculate the distance $AD$.
6 replies
MTA_2024
Friday at 3:50 PM
WheatNeat
Yesterday at 10:17 PM
Question from Gazeta matematica
abcdefghijklmop   5
N Yesterday at 9:25 PM by abcdefghijklmop
Determine how many subsets formed by 7 elements which are in geometric progession are in the set
{1,2,....,2025}.
5 replies
abcdefghijklmop
Yesterday at 7:30 PM
abcdefghijklmop
Yesterday at 9:25 PM
Unknown triangle area
smartvong   2
N Yesterday at 8:09 PM by vanstraelen
The diagram shows a convex quadrilateral $ABCD$. The points $E$ and $F$ divide $AB$ into three equal parts while the points $G$ and $H$ divide $CD$ into three equal parts. The line segments $AH$ and $ED$ intersect at $I$. The line segments $CF$ and $BG$ intersect at $J$. Given that the areas of the triangles $AID$, $EHI$ and $FJG$ are $154$, $112$, and $99$ respectively, find the area of the triangle $BJC$.

IMAGE
2 replies
smartvong
May 8, 2025
vanstraelen
Yesterday at 8:09 PM
Range if \omega for No Inscribed Right Triangle y = \sin(\omega x)
ThisIsJoe   1
N Yesterday at 5:45 PM 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
Yesterday at 5:45 PM
2012 preRMO p17, roots of equation x^3 + 3x + 5 = 0
parmenides51   11
N Yesterday 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
Yesterday at 3:29 PM
Interesting question from Al-Khwarezmi olympiad 2024 P3, day1
Adventure1000   3
N Yesterday 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
Yesterday at 2:38 PM
Malaysia MO IDM UiTM 2025
smartvong   1
N Yesterday 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
Yesterday at 1:01 PM
jasperE3
Yesterday at 2:20 PM
Nice problem
gasgous   2
N Yesterday 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
Yesterday at 1:30 PM
vincentwant
Yesterday at 1:47 PM
Angle Formed by Points on the Sides of a Triangle
xeroxia   1
N Yesterday 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
Yesterday at 10:28 AM
vanstraelen
Yesterday at 1:28 PM
Inequalities
sqing   1
N Yesterday 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
Yesterday at 12:50 PM
sqing
Yesterday at 1:08 PM
Another System
worthawholebean   3
N Apr 21, 2025 by P162008
Source: HMMT 2008 Guts Problem 33
Let $ a$, $ b$, $ c$ be nonzero real numbers such that $ a+b+c=0$ and $ a^3+b^3+c^3=a^5+b^5+c^5$. Find the value of
$ a^2+b^2+c^2$.
3 replies
worthawholebean
May 13, 2008
P162008
Apr 21, 2025
Another System
G H J
G H BBookmark kLocked kLocked NReply
Source: HMMT 2008 Guts Problem 33
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worthawholebean
3017 posts
#1 • 2 Y
Y by Adventure10, Mango247
Let $ a$, $ b$, $ c$ be nonzero real numbers such that $ a+b+c=0$ and $ a^3+b^3+c^3=a^5+b^5+c^5$. Find the value of
$ a^2+b^2+c^2$.
Z K Y
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The QuattoMaster 6000
1184 posts
#2 • 2 Y
Y by Adventure10, Mango247
worthawholebean wrote:
Let $ a$, $ b$, $ c$ be nonzero real numbers such that $ a + b + c = 0$ and $ a^3 + b^3 + c^3 = a^5 + b^5 + c^5$. Find the value of
$ a^2 + b^2 + c^2$.
Solution
EDIT: Sorry everyone, arqady is correct. I made a small error in my solution; it has been edited.
This post has been edited 1 time. Last edited by The QuattoMaster 6000, May 14, 2008, 1:49 AM
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arqady
30244 posts
#3 • 3 Y
Y by Adventure10, Mango247, P162008
I have got $ \frac {6}{5}.$ :maybe:
Z K Y
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P162008
187 posts
#4
Y by
Let $a,b$ and $c$ be the $3$ roots of the equation $x^3 + px + q = 0$

By Vieta's Relation, we've

$$\sum_{cyc}{a} = 0, \sum_{cyc}{ab} = p$$and $$\prod_{cyc}{a} = -q$$
Then, $$\sum_{cyc}{a^3} = 3\prod_{cyc}{a} = \boxed{-3q}$$and $$\sum_{cyc}{a^2} = -2\sum_{cyc}{ab} = \boxed{- 2p}$$
Now, define $$ r_n = a^n + b^n + c^n$$such that $$r_3 = -3q$$and $$r_2 = -2p$$then by Newton's Sum we get

$$r_5 + p.r_3 + q.r_2 = 0$$
$$r_5 = \boxed{5pq}$$
Now, $$r_3 = r_5$$
$$p = \boxed{\frac{-3}{5}}$$
Therefore, $$r_2 = \sum_{cyc}{a^2} = \boxed{\frac{6}{5}}$$
This post has been edited 3 times. Last edited by P162008, Apr 21, 2025, 11:46 PM
Reason: Typo
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